THE BEHAVIOUR OF SPACE TRUSSES INCORPORATING

NOVEL COMPRESSION MEMBERS

by

Gerard Andrew Roger Parke

A thesis submitted in accordance with the requirements of the University of Surrey for the degree of Doctor of Philosophy

1>)ýý 10 -5 ý

Department of Civil Engineering University of Surrey Guildford, Surrey May 1988

SUMMARY

Double-layer space trusses are structural systems which may

possess significant reserves of strength beyond their elastic limit

load. However, their behaviour and ultimate load-carrying capacity depend on the topology of the structure, the support positions, and the load-displacement response of the individual members forming the

space truss. If plastic buckling of the compression members is

probable then the collapse of these structures may be sudden,

exhibiting a brittle load-displacement response.

The present investigation has considered ways of improving the

load-displacement response of square-on-square double-layer space trusses. To undertake this investigation an analysis program has

been written which is c. apable of tracing the full non-linear load-displacement behaviour of these structures. This program has

been used to study the response of space trusses incorporating

special ductile soft compression members and tension members which

were permitted to yield and deform plastically.

An extensive test program has been undertaken to develop the

novel soft compression members and refine their load-displacement

response. Also four model square-on-square double-layer space truss

structures have been fabricated and tested to collapse. The

load-displacement response obtained from the first two model

structures tested, in which tensile yielding of the lower chord

members was permitteb, showed that it is possible to create

extensive ductility in the post-elastic behaviour of these

structures.

The experimental results presented in this study indicate that

the assumptions used in the analysis program to model both the

unusual load-displacement response of the soft members, and to

predict the collapse behaviour of double-layer space trusses, are

valid. This approach may be used with confidence to investigate the

collapse behaviour of other double-layer space truss types either

with or without the novel soft compression members.

ACKNOWLEDGMENTS

I wish to express my sincere gratitude to Professor Z. S. Makowski for all his continuous help, advice and compelling enthusiasm which sustained my research. I

I ali also greatly indebted to Mr D. I. Retief for his unfailing

assistance during the experimental work.

My thanks are also due to Mr A. Smith who carefully fabricated the

test structures, and also to Mr P. Disney for his valued advice. on the experimental investigations.

I would also I ike to express my thanks to both Or H. Nooshin and Or P. J. Wicks for their adv Ice and encouragement g iven freely throughout this work.

The Thesis was typed by Mrs E. Ryan, whom I would like to thank for her patience and care in preparing the manuscript.

The work was supported financially by the Department of Civil Engineering, University of Surrey; Constrado (the Steel Construction Institute); and the Science and Engineering Research Council, to

whom I am most grateful.

CONTENTS

Page

Summary ...................................................... 1

Acknowledgements ............................................. ii

Contents ..................................................... m

INTRODUCTION ................................................. 1

CHAPTER 1 Material, Member And Space Truss Behaviour 4

CHAPTER 2 Non-Linear Collapse Analysis Of Space Trusses ..... 70

CHAPTER 3 Methods Of Improving Space Truss Behaviour ........ 97

CHAPTER 4 Novel Force-Limiting Device ...................... 142

CHAPTER 5 The Experimental Behaviour Of Double-Layer Space Trusses .................................... 211

CHAPTER 6 General discussion ............................... 326

References .................................................. 344

INTRODUCTION

Double-layer space trusses are reticulated three-dimensional

frameworks, constructed from an assembly of discrete elements,

possessing high strength to weight ratios, which are frequently used to cover large, column-free areas (Makowski, 1987). Their use is

not restricted to roof structures, for in the recently completed Javits Convention Centre in New York City, double-layer space trusses have been used to form intermediate floors, and both

vertical and sloping walls.

At present, double-layer space trusses are normally designed

elastically to comply with either 'working stress' or 'limit state' design requirements (BS449,1969; BS5950,1985). In a 'working

stress' design the maximum probable elastic load carried by a

member in the structure is compared to an allowable stress which is

a fixed proportion of the critical stress causing yielding, rupture

or member buckling. This results in the same factor of safety being

applied to both dead and imposed loads acting on the structure. As

variations in the imposed load are normally much greater than

variations in the dead load, this design approach is undesirable

especially for double-layer space truss roof structures, where the

imposed load forms the major portion of the total load effect. In

'limit state' design, separate partial safety factors are applied to

both the dead and imposed loads. A series of checks are implemented

to ensure that the truss performs satisfactorily under

serviceability loading and also that the ultimate strength of the

structure is sufficient to prevent collapse occurring.

The strength, or load-carrying capacity of a double-layer space truss Is normally assessed by considering the elastic load-displacement response of the structure up to the point where the first compression member falls. Normally, no account is taken

of any post-buckling reserves of strength which the double-layer

space truss may, possess. The inelastic behaviour of the structure is usually ignored in the design process because it is considered that, due to their high degree of statical indeterminacy, double-layer space trusses, fabricated from ductile materials, will possess reserves of strength in excess of their elastic capacity. However, this is not necessarily true. The complete

1

load-displacement behaviour of a space truss depends on the. topology

of the structure, the support conditions and the load-displacement

response of the individual members forming the structure. Post-elastic reserves of strength will only exist in a double-layer

space truss if after the failure of the first compression member the

adjacent members have sufficient strength to carry the additional load transferred into the structure from the buckled member.

If the compression members in a space truss are stocky then their post-buckling, load-displacement response will be ductile,

exhibiting a steady load pI ateau as the member squashes. Alternatively, if the compression members are very slender, elastic buckling of the struts occurs exhibiting a gradual reduction in the load carrying capacity of the elements. However, practical considerations usually dictate that the compression members used in

a double-layer space truss have intermediate slenderness and consequently buckle inelastically. The inelastic or plastic buckling of a compresion member is a dynamic process resulting in a sudden loss of both member stability and load-carrying capacity. If the double-layer space truss encompassing the failed member can absorb this release of energy then the structure may also be able to

support an increase in the imposed loading. However, if the force

redistribution resulting from the failed member causes further

members to fail, then it is likely that progressive inelastic buckling of the compression members will occur, resulting in a complete collapse of the structure.

The sudden collapse in January 1978 of the steel double-layer

space truss forming the roof of the Hartford Coliseum in Hartford, Connecticut, USA has necessitated the accurate determination of both

the complete load-displacement response and the ultimate load-carrying capacity of double-layer space trusses.

The collapse analysis of double-layer space trusses has been

studied by several investigators and their work has indicated that certain double-layer trusses are prone to progressive collapse precipitated by the inelastic buckling of the compression members (Schmidt, et a], 1979; Collins, 1981; Smith, 1984b; Hanaor 1979). Their investigations have also shown that space trusses prone to

2

progressive collapse, like the Hartford Coliseum roof structure, fall suddenly without warning, exhibiting a brittle load-displacement response.

The major objective of the present study was to investigate

means of inducing ductility into the load-displacement response of double-layer space truss, and to quantify, for design purposes, any post-elastic reserves of strength which may exist in these

structures. To facilitate this work it was first necessary to write a computer program capable of tracing the full load-displacement

response of a double-layer space truss through the elastic range of behaviour up to the collapse of the structure. Using this program the collapse behaviour of space trusses incorporating members with pre-defined characteristics were studied.

Two methods to improve the load-displacement response of double-layer space trusses have been investigated. The first method considered the effects of permitting extensive tensile yield to

occur in the bottom chord members of a double-layer truss before instability occurred in any of the top chord compression members. The second method was more radical, and involved the development of a novel soft member which, when loaded in compression, exhibits an elastic-plastic load-displacement response. ' This second investigation has considered the effects of replacing the most heavily stressed compression members in a double-layer space truss

with the novel soft members. The design and development of the soft member has required extensive testing to determine and improve the behaviour of both full size and model soft members.

In order to check the assumptions used in the analytical program and to determine the load-displacement response of double-layer space trusses incorporating soft members, and permitting tensile yield, four carefully fabricated steel model structures have been tested to collapse. The behaviour of the four

separate test structures has been extensively monitored and the load-displacement response of each space truss recorded on video tape for information for future investigators.

3

CHAPTER 1

MATERIAL, MEMBER AND SPACE TRUSS BEHAVIOUR

INTRODUCTION

In order to determine the collapse load and assess the inelastic response of double-layer grids, careful consideration must be given to the strength, stiffness and stab iI ity of the individual

members forming the complete structure. In double-layer grids the imposed external loads are transmitted throughout the structure primarily as axial forces and, consequently, the collapse of these

structures can involve both the yield and plastic flow of tension

members and, more importantly, the instability and buckling of

axially-loaded compression members. Usually the collapse behaviour

of double-layer grids is dominated by the instability

characteristics of the compression members and this behaviour, in

addition to the grid configuration and support conditions, can determine whether the complete structural response after first

yield, will be ductile or brittle.

Typical compression members used in the fabrication of double-layer grids are neither very stocky nor very slender, and the

buckling failure of these members is likely to take place inelastically, associated with a sudden and rapid loss of member

strength. The critical buckling load of these columns is

particularly sensitive to imperfections, including initial column

curvatures and residual stresses formed during manufacturing.

Consequently, to evaluate both the collapse load and

post-collapse behaviour of double-layer grids it is necessary to

trace, step by step, the behaviour of the entire structure under the

action of an increasing app] led load. This will require accurate

numerical modelling of both the post-buckled response of the

compression members and the post-yield characteristics of the

tension members.

Before this numerical modelling can be undertaken it is

necessary to consider the material properties of certain steels in

addition to the post-yield behaviour of typical tension and compression members used in the fabrication of double-layer space trusses.

4

STEEL MATERIAL PROPERTIES

The most important material properties which influence the behaviour of steel tension and compression members are the modulus of elasticity, E, and the yield stress, ay. Values for both the modulus of elasticity and the yield stress may be obtained from tensile tests on coupon specimens or from compression tests on stub columns.

Figures 1.1 and 1.2 show typical stress-strain relationships for steel tested in axial tension and compression. The curve shown in Figure 1.1, obtained from a tensile specimen, exhibits a linear

elastic relationship until the specimen yields and deforms plastically at a constant load. After plastic deformation has

occurred the material strain hardens and is capable of supporting an increase in load before final rupture. The value of stress at yield is termed the upper yield stress, ayu, and the corresponding strain at yield for a structural steel specimen is in the order of 0.12%. As straining progresses, the mean stress in the specimen reduces to the lower yield stress value ayl and then increases

slightly to remain substantially constant at a value termed the dynamic yield stress ayd- The value of the dynamic yield stress is influenced by the rate of strain applied to the specimen and as the strain rate reduces to zero the yield stress reduces to a limiting value, termed the static yield stress, ays. It has been found that for an annealed cold-drawn steel tube the dynamic yield stress could be as much as 15% greater than the static yield stress (Smith et a], 1979).

As the static yield stress increases for different grades of structural steel the length of the plastic plateau, and overall elongation before rupture both decrease. For a typical grade 50c

steel coupon, cut from a hot finished seamed steel tube, the amount of plastic strain occurring after yield is in the order of ten times the yield strain and the percentage overall elongation at failure

would be greater than 20% (BS4360,1986).

The stress-strain curve shown in Figure 1.2, obtained from a tubular stub column compression test also exhibits dynamic and static yield stress values, but does not show a well defined yield point. Consequently the static yield stress is usually defined by the 0.2% proof stress which is the stress value which leaves the specimen with a permanent set of 0.2% when the specimen is unloaded.

5

b In Ifl L 1.3 (J

Strain 6

Figure I. I. Typical Stress-Strain Relationship For A Carbon Structural Steel Coupon Tested In Tension. The Figure shows a typical stress- strain relationship for a carbon structural steel coupon tested in tension. The elastic modulus E may be taken as 2.06 x 10s N/MM2 for all structural steels while the strain hardening modulus EST is variable depending on prior strain history.

r0yu is the upper yield stress. 6yi is the lower yield stress. 6yd is the dynamic yield stress. 6ys is the static yield stress.

b

tn ul a) U,

0-2% 0.5%

Strain

Figure 1.2. Typical Stress-Strain Relationship For A Carbon Structural Steel Stub Column Tested In Compression... The Figure shows a typical stress-strain relationship for a carbon structural steel stub column tested in compression. The static yield stress 6ys is usually defined by 0.2% proof stress which is the stress value which leaves the specimen with a permanent set of 0.2%. Alternatively of 0.5% proof stress, corresponding to the value of stress at a total strain of 0.5% may be used (ECCS, 1984).

E is the modulus of elasticity. 6yd is the dynamic yield stress. 6yz is the static yield stress. 61P is the limit of proportionality stress.

6

Alternatively a 0.5% proof stress, corresponding to the value of stress at a total strain of 0.5%, may be used (ECCS, 1984). The lack of a clearly defined yield point is attributed to the presence of residual stresses occurring in the tubular stub column.

To predict the strength of compression members it is preferable to determine the material yield stress from stub column compression tests. These tests tend to be vulnerable to locallsed bulging at the ends of the specimens due to the end restraint restricting the Poisson expansion of the tube in the radial direction. As a result of the 6ulges the compressive yield stress of the material appears to be 5-10% lower than the tensile yield stress of the material. This apparent discrepancy between tensile and compressive yield stress has been invesiigated by several researchers. Korol (1979),

has indirectly undertaken a comparison by providing test data for an experimental investigation into the inelastic bending of thirteen

small-diameter, circular, hollow, tubular beams. Figure 1.3 shows the mater 1 a] stress-strain relationship in both tension and compression for specimens cut from the same cold-formed steel tube. It can be seen from the Figure that the tensile and compressive stress are almost identical for the 0.2% stress indicating a very small discrepancy between the tensile and compressive yield stresses. Wolford, et al, (1958), have also undertaken a comparison, forming part of an extensive investigation into the behaviour of small diameter welded and seamless steel tubes. Figure 1.4 shows a comparison between the compressive and tensile yield stresses obtained by Wolford. For the high strength structural steels the tensile yield stress was found to be generally higher than the compressive yield stress. This disparity tended to decrease for the lower strength steels. Similar results shown in Figure 1.5, were obtained by Yeomans (1976), who found that for 114

mm diameter steel tubes, the mean compressive yield stress was approximately 5% less than the mean tensile yield stress. Conversely, from a limited series of tests on samples cut from one 8

mm thick steel plate it was found that the mean compressive static yield stress was on average 5% greater than the mean tensile yield stress (D. O. E., 1977).

Variations in Static Yield Stress

Significant variations in yield stress can occur in nominally identical steel members. These discrepancies may be due to small,

7

04

E ý31 2 b L4

ul

Btrain 6[%] Figure 1.3.

' Compression And Tension Stress-Strain Relationships For

Specimens Taken From The Same Cold Formed Seamed Steel Tube. The Figure shows a comparison between the compression and tension stress-strain relationships obtained from one coupon and one stub column test specimen cut from the same steel tube. The tube was cold formed and not stress relieved before the samples were taken. Stub column or oss-section dimensions: Diameter = 114.3mm, Thickness = 3.96mm, (from, Korol, 1979).

E JE 2 tr

U)

CL

E a u

Tensile Yield Btress cry [N/mmal

Figure 1.4. Comparison Between Compressive And Tensile Yield Stress For Steel Tube Specimens. The Figure shows a comparison between the compressive and tensile yield stress obtained for test specimens taken from galvanised steel tubes. The tubes ranged in diameter from 9-5mm to 75. Omm and were made by cold forming and welding 'Armco, Zincgrip' flat rolled steel sheet. In addition standard galvanised steel pipe was also tested for comparison (Wolford, et al, 1958).

Key: A Low carbon (0.02 to 0.06%) welded galvanised steel tubes - 0.2% proof stress.

* Medium carbon (0.21 to 0.29%) welded galvanised steel tubes - 0.2% proof stress.

* Standard galvanised steel pipe - yield point. (Figure from, Ellinas, 1985).

8

0-2 0-5 1-0 1-5 2-0

local variations in the steel composition in addition to changes in

heat treatment, cold working, and to residual stresses occurring throughout the member length. Several investigators have assessed the magnitude of the variations in tensile and compressive yield

stress and the work of Yeomans (1976) is of particular value. Yeomans obtained tensile and compressive yield stresses for a series

of specimens cut from 114 mm diameter steel tubes. Three different

groups of tubes were used in the investigation, each group of tubes

manufactured by a different process. Figure 1.5 summarises the

results and indicates that variations in yield stress of up to t 13.5% from the mean value were obtained for the welded tubes and t 16.3% from the mean yield value for the seamless tubes.

Small variations in the tensile yield stress were obtained from

tests on 3 and 8 mm thick steel plates (D. O. E., 1977). This investigation showed that local variations in'yIeld stress, although small, were slightly higher in the thinner plate than in the thick

plate. This was considered to be the combined result of differential rates of cooling and local differences in chemical composition.

It is apparent that significant variations in both tensile and

compressive yield stress do occur in steels of nominally identical

strength. In addition, for manufactured tubular members, often used in double-layer grids, the compressive yield stress is normally less

than the tensile yield stress. Consequently the generally accepted

practice of assessing compression member strength using the tensile

yield stress must be used with caution.

Strain Aging

Strain aging is an important physical property of steel which can significantly affect the behaviour of steel tension and compression members. When an annealed or normalized mild steel is

strained beyond the yield strain, unloaded and then immediately

reloaded the stress-strain relationship of the material exhibits a form very similar to the relationship obtained for a continuously strained material. However, if after the material has been strained into the strain hardening region, unloaded and then allowed to remain unstressed at room temperature for a period of approximately three months before reloading, the stress-strain relationship

9

4 0 0 2

ul

L CL E

Tensile Yield Btress cry [N/mm']

Figure 1.5. Comparison Between Compressive And Tensile Yield Stress For Steel Tube Specimens. The Figure shows a comparison between the compressive and tensile yield stress obtained for test specimens taken from continuously welded, electric welded and seamless steel tubes. The tests were carried out on tubes of the same size with a diameter of 114.3mm and a wall thickness of 3.6mm (Yeomans, 1976).

Key: a Continuously welded tubes - 0.2% proof stress. A Electrically welded tubes - 0.2% proof stress. 0 Seamless tubes - 0.2% proof stress.

(Figure from, Ellinas, 1984)

10

Btrain

Figure 1.6. Stress-Strain Relationship Showing The Effects Of Strain Aging. The Figure shows the stress-strain curve of an annealed or normalized low carbon steel strained to point Al unloaded and then restrained immediately (curve a) and after aging (curve b). AY is the change in yield stress due to strain aging. AU is the change in the ultimate tensile strength due to strain aging. , &E is the change in elongation due to strain aging. (from, Baird, 1963a and b).

10

275 300 350 400 450

exhibits the behaviour, shown in Figure 1.6. Upon reloading, the discontinuous yielding behaviour returns and the value of the yield stress increases. In addition there may also be an increase in

ultimate tensile strength and a decrease in the overall elongation to fracture. Baird (1963a and b) in an extensive review on strain aging, has shown that other properties are also affected, notably the ductile-brittle transition temperature, high-temperature

strength, fatigue strength, and electrical and magnetic properties.

The phenomenon of strain aging was not adequately explained

until publication of the dislocation theory (Cottrell and Bilby, 1949). It is now generally accepted that the effects of strain aging are caused by carbon and nitrogen atoms migrating to lock and f ix dislocations. This theory has been reinforced by the earl jer

work of Low and Gensamer (1944), who showed that if both carbon and

nitrogen are removed from steel both the initial discontinuous

yielding behaviour and the strain aging phenomenon are eliminated.

Strain aging is most pronounced in steel specimens which have

been pre-strained in tension and then reloaded again in tension

after aging. However if the pre-straining after aging is carried

out in compression then the return to a well defined tensile yield

point is suppressed. Other characteristics of strain aging are not

affected by the mode of pre-strain (Tardif and Ball, 1956). In

addition the amount of tensile pre-strain has only a small effect on the change in yield stress produced by subsequent aging. However, a repeated straining and aging cycle produces a greater increase in both yield and ultimate tensile strength than that obtained If aging is carried out only after the completion of straining, (Low and Gensamer, 1944).

BauschingerEffect

When a metal has been plastically strained, the stress-strain behaviour during any subsequent straining is affected by both the

magnitude and relative direction of the initial prestrain. If a steel specimen is strained plastically in tension and then strained in compression, a decrease in compressive yield stress is exhibited In comparison with the compressive yield stress obtained from a virgin specimen. This drop in yield stress, which also occurs if a specimen is prestrained in compression and subsequently strained in tension, was first observed by Bauschinger (1886), in wrought iron,

and is referred to as the Bauschinger effect.

11

Several specific mechanisms have been proposed to explain the Bauschinger effect and it is now accepted that this phenomenon may be due to the pile-up of dislocations moving against randomly positioned obstacles (Orowan, 1966). These obstacles may be foreign

atoms such as nitrogen, oxygen or carbon but are more likely to be

grain boundaries existing between adjacent crystals.

The Bauschinger effect is most pronounced when the secondary straining direction is directly opposed to the prestrain direction. When subsequent prestrain is in a direction perpendicular to the

original prestrain, the effect can be significantly reduced. Rolfe,

et a], (1968), investigated the effects which the state-of-stress and yield criteria have on the Bauschinger effect. In their investigation test specimens were cut in the longitudinal and transverse directions from each of three large steel plates cold-formed to small radii. Both longitudinal and transverse

specimens were tested in tension and compression to assess the

magnitude of the Bauschinger effect. As a result of extensive testing no Bauschinger effect in the transverse straining direction

was observed only work hardening (Rolfe, et a], 1968). However, the Bauschinger effect was observed during transverse straining of simple tension specimens (Chajes, et al, 1963). The discrepancy was considered to be due to the differences in prestraining conditions (Rolfe, et al, 1968). Rolfe, et al, (1968), strained their material by the curvature of wide plates so that a plane-strain condition applied, whereas Chajes, et a], (1963), strained their thin specimens in tension creating transverse compressive strains. These transverse compressive strains gave rise to a marked reverse Bauschinger effect resulting in the compressive yield strength being

greater than the tensile yield strength for the transverse

specimens.

The effect which the prestrain direction has on the Bauschinger

effect has been further investigated by Pascoe (1971). In this investigation a 915 mm x 150 mm x 12 m-n thick strip was cut from a larger plate of high-yield low-alloy steel. The strip was strained in tension and then specimens of square cross-section were cut from the plate in directions spaced by 150 Intervals from the strain direction as shown in Figure 1.7. Compression specimens 50 mm long and tension specimens 150 mm long were subsequently tested. The results for both tension and compression tests are shown in Figure 1.8. From these results it is evident that the effect of the

12

...................

c 00

TOO

Direction of tensile pre-strain.

Figure 1.7. Cutting Plan For Steel Specimens. The Figure shows the cutting plan for test samples cut from a steel plate 300mm long by 150MM

wide by 11.4mm thick. The angle of each specimen to the pre-straining direction is also given. The plate was made from a high-yield low-alloy

steel and was strained to 1.3% before the individual specimens were cut. T= Tensile specimen. C= Compression specimen.

(From, Pascoe, 1971).

go* 45*

30' 500

z

0. 400

300 Values on curves are angles of test specimens to pre-strain direction. M 200 --------- Compression of unstrained material.

Compression Tests After 1.3% Pre-tension

IC, O

0 0-2 0-4 0 E3 0.8 1-0 1-2

Strain SM

E 45. Go* 75* go* E z 'ý400

300 11 Values on curves are angles of test specimens to pre-strain direction. ---------- Tension of unstrained material.

200 Tension Tests After 1.3% Pre-tension

0 0.2 0-4 0-6 0.8

Strain 61%)

FijZure 1.8. Compression And Tension Tests On Specimens After 1.3% Pre-Tension. The Figure shows how pre-straining a steel specimen in tension influences the subsequent tensile and compressive yield stress of the material. If a pre-strained specimen is re-strained in compression then reductions in compressive yield stress are obtained. The magnitude of the reduction depends on the orientation of the specimen to the original pre-strain direction (From, Pascoe, 1971).

13

tensile prestraining produced a marked Bauschinger effect for

subsequent compressive loading in the same direction. Also, as the direction of loading changed from being parallel to the prestrain direction, to being perpendicular to the prestrain direction, the Bauschinger effect decreased.

It is apparent from the work of Rolfe et a], (1968), Chajes et a], (1963) and Pascoe (1971) that in addition to both the type and direction of prestrain, the chemical composition of the steel has a significant effect on its behaviour. Killed, semi-killed and rimmed carbon steels all with approximately the same yield strength, ultimate strength and ductility, behave in markedly different ways for the same amount of pre-strain. The Bauschinger effect is

apparent in varying degrees in all steels, whereas strain aging may not be present in cold-finished killed steels.

All material characteristics mentioned will occur to some extent in structural steel members and consequently the mode of manufacture of both the steel, and later the steel members, plays a significant role in determining overall member behaviour.

MEMBER MANUFACTURE

In the United Kingdom steel double-layer grids are usually fabricated from grade 43c or grade 50c structural steels (BS4360,

1986). Several cross sectional shapes are used but due to the

advantages offered by hollow sections, the use of these

predominates. Round, square and rectangular hollow sections are

available and their mode of manufacture depends on their

cross-sectional size and desired use. Small diameter seamless tubes

are produced by hot working methods which involve piercing a steel billet with a mandrel and rolling it externally to obtain the

required cross-section. Alternatively, both small and medium size tubes ranging from 40 mm to 600 mm in diameter are fabricated from

coiled sheet steel. The sheet steel is first decolled and cold formed into a circular tube shape. The tube's longitudinal edges

are then heated and joined by electrical resistance welding. The

complete tube is heated by passing through furnaces and is then hot

rolled to produce circular, square or rectangular hollow sections. Larger diameter tubes are fabricated from single plates which are cold formed into a tubular shape and joined using submerged arc

we I ding.

14

The method of manufacture has a significant influence on the overall behaviour of the member. Large diameter cold formed tubes possess complex longitudinal and circumferential residual stresses which must be taken into consideration when assessing column strength (Ross, et a], 1976). Smaller diameter cold formed tubes also possess residual stresses which have produced reductions of up to 40% in column strength (Sherman, 1971). Fortunately, double-layer

space trusses are usually fabricated from cold-formed hot-finished,

seamed tubes which have very small residual stress levels, resulting in negligible reductions in column strength (Dwyer, et al, 1965).

TENSION MEMBER BEHAVIOUR

Tension members are efficient structural elements which are normally in a state of uniform axial stress. The behaviour of axially loaded steel tension members is very similar to the behaviour exhibited by a tensile test coupon cut from the member. Differences between coupon and member behaviour may be due to differences in the end fixing methods, but it is generally the

result of residual stresses present in the manufactured or fabricated steel member. Residual stresses present in steel tension

members do not affect the yield or ultimate strength of the member. However, they do induce early local yielding within the member which causes a loss of stiffness as the yield load is approached. The

premature local yielding causes early local strain hardening, and consequently the plastic range of deformation is shortened. A

similar effect is caused by an initial crookedness occurring within the member. The axial load applied to the member induces bending

stresses which again cause early yielding followed by local strain hardening.

In order to assess the difference between member and test

coupon behaviour for typical steel members used in double-layer

grids,. the author tested in tension twenty one steel hollow sections

and forty two coupon specimens. Two coupons were cut from each of the hollow sections before they were tested in tension. Both square

and circular hollow sections manufactured from grade 43c and 50c (BS4360,1986) structural steels were tested. The hot finished tubes were manufactured from steel strip with the tube seam formed by electrical resistance welding. Figure 1.9 shows the relationship between the axial stress and strain for one of the 2.0 metre long,

15

500 Z

400

soo

(1

ri 200 ti-

lzo

Figure 1.9. Tensile Stress Vs. Strain RelationshiD For A Grade 50C Steel Square Hollow Section. 7be experimental relationship was obtained for a 2. Om long 76.1mm by 76.1mm grade 50C square hollow tube with a wall thickness of 3. Omm tested in tension at a straining rate of 9.25 x 10- 7 IS. Cy = 451 N/mm2 ;E=2.06 x 105 N/mm2 .

Soo

400

300

200

loo

TEST SPECIMEN LONGITUDINAL STRIP 9A

Figure 1.10. Tensile Stress-V-Strain RelationshiD For A Grade 50C Steel Coupon. The steel coupon was cut from a 2. Om long 76.1mm by 76.1mm grade 50C square hollow tube with a wall thicImess of 3. Omm. The coupon was tested in tension at a straining rate of 2.22 x 10-5/S. cy = 464 N/mrn2 ;E=2.06 x 105 N/mm2 .

16

. 05 1 . 1-5

STRAIN

. lu.;, ) .1 . 1.5 .0 . 25 .3 . 3s

STRAIN

76.1 mm x 76.1 mm grade 50c steel square hollow tubes tested in tension. Figure 1.10 shows the axial stress versus strain relationship for one-of-the-coupons--cut-from-the-corresponding -steel- member. From the test results it is evident that the behaviour of the tube and the corresponding coupons is very similar although the tube members did experience a reduction in overall elongation. In

addition, for both grades of steel, the rectangular hollow sections exhibited greater plastic deformation and ductility than the

circular tube members.

COMPRESSION MEMBER BEHAVIOUR

Column Strength

Numerous theoretical and experimental studies have been

undertaken to determine and explain the behaviour of axially loaded

compression members. The first empirical relationship to be

published was derived from experimental work on timber columns and

showed column strength to be inversely proportional to the square of the column length (van Musschenbroek, 1729). This early work was developed by Euler (1757) who produced a quantitative expression for

the experimental relationship derived by van Musschenbroek.

Euler's expression is given as:

P2 Ek2 L7

Euler named the term Ek2 the "moment du ressort", where E is the

qual ity factor and the term k2 represents a dimensional factor.

Euler realised that column strength is primarily a problem of

stability and his fundamental work was confirmed and extended with the use of an improved theoretical analysis (Lagrange, 1770).

Experimental work undertaken to verify Euler's expression showed that column critical loads were over estimated for both short and

medium length columns. In his calculations Euler had neglected the

effect of both direct axial compression and shearing forces. At first these two omissions were considered to be the cause of the discrepancy between experimental and theoretical results. However, it was not until 1845 that it was observed that the major reason for this discrepancy was that the elastic limit of the column material was exceeded and that column failure by crushing, followed by the

possibility of inelastic buckling, occurred before failure due to

elastic buckling (Lamarle, 1846).

17

Due to the limitations imposed by Euler's equation several attempts have been made to modify his expression to determine the theoretical strength of perfect columns which buckle inelastically

or plastically. A tangent-modulus theory was proposed in which it

was assumed that the perfect column would remain straight until failure and that the modulus remained constant across the

cross-section even beyond the elastic limit. When the elastic limit

of the material is exceeded the material modulus decreases, and the

elastic modulus E, used in the Euler equation was replaced by the tangent modulus ET (Engesser, 1889). This approach was rejected by Consid6re who in 1889 suggested that as an ideal column stressed beyond the elastic limit begins to bend, the stress on the concave side increases according to the material stress-strain relationship, and the stress on the convex side decreases elastically. Consequently the strength of a perfect ' column may be determined by using an average modulus E in the Euler equation. This is based on the assumption that the value of E lies between the

modulus of elasticity E and the tangent modulus ET-

Engesser acknowledged the error in his original theory and subsequently developed an improved average modulus theory called the

reduced modulus theory, or double-modulus theory, based on the

elastic modulus, tangent modulus and the column cross-section (Engesser, 1893). This analytical work was verified by von Karman

who in addition checked his results by undertaking a series of tests

on rectangular columns. Several expressions for the reduced modulus ER, each depending on the column cross section, were evaluated and the column critical load obtained by substituting ERs, instead of the elastic modulus E, in the Euler equilibrium equation.

The reduced-modulus theory became the accepted theory

describing the behaviour of perfect columns in the inelastic range. However, the results from several careful experimental investigations indicated that the reduced-modulus theory

consistently over-estimated Column strength, with the actual column buckling loads being closer to the tangent modulus values than to

the reduced modulus values. Both this discrepancy and the

difficulty in use of the reduced modulus theory, led engineers to

adopt the tangent-modulus theory to calculate the plastic buckling loads of perfect columns.

The difference between the tangent modulus and reduced modulus

critical loads was not understood until the basic assumptions used 18

in the derivation of the reduced modulus theory were re-examined in

an important paper on inelastic column theory (Shanley, 1947). In the derivation of the reduced modulus theory it was implied that the

column remains straight while the axial load is increased to the

calculated critical value, after which the column begins to bend. To obtain this predicted critical load, which is greater than the tangent modulus load, some strain reversal is needed in order to

provide the additional increase in stiffness required beyond the

tangent modulus load. The strain reversal required for the column to obtain the reduced modulus load cannot occur in a straight

column. Shanley reallsed that there is nothing preventing a column from bending simultaneously with increasing axial load, and reasoned that column bending will begin as soon as the tangent modulus load is exceeded. Consequently the maximum column load in the inelastic

range will be greater than the tangent modulus load but less than

the reduced modulus load.

Perfect Compression Member Behaviour

Figure 1.11 shows the theoretical relationship Petween the

axial stress and end shortening for a series of steel columns Of different slenderness, loaded in compression. Curve A describes the

typical characteristics for a very short steel column with a low

slenderness ratio in the order of L -c 40. These stubby columns r

will deform elastically until the material yield stress has been

reached throughout the column cross section initiating plastic deformation without buckling. Columns which have a high

slenderness ratio in the order of L> 100 are long and slender r

and buckle elastically before the material compression yield stress is reached in theýmember cross section. The typical behaviour of these slender columns is shown by curve B in Figure 1.11. The

post-buckling path is characterised by a load plateau range followed

by a soft and ductile unloading path. For steel columns with an intermediate slenderness ratio (40 -c L -c 100) plastic buckling

r occurs and the post-buckling behaviour shown by curve C in figure

1.11 is characterised by a sharp unloading path. This behaviour is

typical for compression members commonly used in double-layer grids, and the term "brittle-type" is used to describe the rapid load-shedding, post-buckling path characteristic of these member types.

19

cry

c) L 43 ul

*i x 4 CrF

End Shortening

Figure 1.11. Axial Stress-End Shortening Relationshii) For A Rectangular Bar In Compression. The Figure shows the axial stress-end shortening relationship for a rectangular bar with three different slenderness ratios, loaded in compression. Columns with a low slenderness ratio (L/R 4 40; curve A), deform elastically until the material yield stress is reached initiating plastic deformation without buckling. Columns with a high slenderness ratio (L/R ý 100; curve B), buckle elastically before the material yield stress is reached. Columns with an intermediate slenderness ratio (40 \< L/R ý 100; curve C), buckle plastically and exhibit a rapid load shedding post-buckling path. (from, Wolf 1973).

20

To explain further the behaviour of perfect compression members it is beneficial to consider the theoretical stress distribution in

perfect struts exhibiting perfect elastic-plastic behaviour. Figure 1.12 shows the stress distribution at the middle cross-section for a column which buckles elastically at a uniform stress equal to the Euler stress OE, which is less than the

material compressive yield stress ay. The buckling of the column causes a lateral deflection to occur in the column, which in turn

creates a bending moment on the section, which changes the stress distribution as shown in Figure 1.12. Equilibrium can be maintained throughout the early stages of this process while the stress increases on the concave side of the buckled column. When the

stress in this region reaches the material compressive yield stress the material behaviour becomes ductile and, in order to maintain equilibrium between the external moment and the internal moment of resistance of the column the external axial load must decrease as the column deformation increases.

The changes in stress distribution which occur during inelastic buckling are shown sequentially in Figures 1.13, a, b and c. Prior to buckling the entire column cross-section is at yield as indicated in Figure 1.13a. Any slight disturbance causes the member to displace laterally and again causes a bending moment to act on the

column cross-section. Similar to the elastic buckling procedure, the formation of the bending moment will try to increase the stress on the concave side of the buckled column, but as the stress throughout the cross-section is already at the yield value and cannot be exceeded, the stress distribution must change rapidly to

maintain equilibrium to the level shown in Figure 1.13c. Associated

with this dynamic jump from the pre-buckling equilibrium to the

post-buckling equilibrium is a sudden decrease in the column load-

carrying capacity. This can be seen by considering the stress distribution shown in Figure 1.13c where the compressive yield stress is no longer attained over the entire column middle cross-section. These characteristic differences between elastic and plastic buckling, which depend on the column slenderness ratio, contribute towards the large differences experienced in the post-buckled behaviour of steel columns.

The critical slenderness ratio which determines whether plastic

squashing or elastic buckling is initiated is obtained by equating

21

compression 7

ýE 111,

tension

Figure 1.12. Sequential Changes In Stress Distribution During Elastic Buckling Of Columns - The diagrams show the sequential changes ;., -hich occur in the theoretical stress distribution in a perfect column cross- section at mid-height during elastic buckling. When the column slenderness A is greater than the transition slenderness ratio At the perfect column buckles at the Euler stress CE- Lateral deflection of the column increases the compressive stress on the concave side of the column and reduces the stress on the convex side of the column. This change in stress distribution is associated with a constant axial load.

compression

tension Xý

[a]

Figure 1.13. Sequential Changes In Stress Distribution During Plastic Buckling-Of Columns. The diagrams show the sequential changes which occur in the theoretical stress distribution in a perfect column cross- section at mid-height during plastic buckling. If the column slenderness X is less than the transition slenderness ratio It plastic buckling occurs with an associated reduction in axial load.

22

the Euler buckling stress OE to the material compressive yield stress ay.

For a perfect pin-ended column fabricated from a material

exhibiting perfect elastic-plastic behaviour the Euler buckling

stress GE can be shown to be:

T12 E (

-V)

where L is the column length

r is the radius of gyration E is the material elastic modulus.

Denoting the slenderness ratio. L by x gives: r

112E x Ir-

equating the Euler stress to the compressive yield stress, gives:

a n2E (3) y x=

so that the transition slenderness ratio of xt is given by:

xt IJIL . ................ ayy

Consequently for perfect pin-ended struts having a slenderness ratio of X<xt failure is initiated by plastic squashing while for

struts with Wt failure is by elastic buckling. The theoretical

column strength curve is shown in Figure 1.14. For a grade 43C

steel having a minimum yield stress ay of 275N/mm2 the transition

slenderness ratio Xt is of the order of 86.

Imperfect Compression Member Behaviour

The preceeding section considered the behaviour of perfect

columns which were assumed to be perfectly straight, axially loaded

through the centrold and fabricated from a perfectly elastic-plastic

material. It is beneficial to look at some of the imperfections

present In real compression members and to consider how these affect the column strength. Real struts are never perfectly straight and

23

cry

b

C, C.. 43 U,

Plastic squashing

Elastic buckling EULER curve

0

Slenderness Ratio L/r

Figure 1.14. Theoretical Strenath Curve For Perfect Pin-Ended ColumncL.

The Figure shows the theoretical strength curve for a perfect pin-ended

column fabricated from perfect elastic-plastic material. When the column

slenderness ratio X is less than 'At the material yield stress is reached throughout the column mid-height cross-section and plastic squashing is

initiated followed by the possibility of plastic buckling.

cry

b

U,

43 (I,

Slenderness Ratio A= L/r

Figure 1.15. Reduced Strength Curve For A Pin-Ended Column With Imperfections - The Figure shows the reduced strength curve due to imperfections for a pin-ended column showing the combined effect of initial column out of straightness, eccentricity of loading, material strain hardening and residual stresses. The greatest reduction in strength occurs when the column slenderness ratio A is equal to the transition slenderness ratio 7ýt.

24

At

they are always eccentrically loaded. In addition the steel from which they are fabricated will have physical properties deviating from the ideal elastic-plastic behaviour. The steel may not show a definite yield point and at large strains the material will strain harden. Also the physical properties exhibited may not be consistent throughout the column length and cross-section. Residual stresses will be present, their magnitude depending to a large

extent, on the column cross section and fabrication process.

Young (1807), was the f irst to investigate the effect which imperfections have on the strength of columns. He considered the behaviour of eccentrically loaded columns of rectangular cross section and produced the first theoretical treatise which considered the reduction in column strength due to eccentric loading and initial column curvature. Young concluded from his investigation that both eccentric loading and initial column curvature decrease

Column strength. This reduction is most pronounced at intermediate

values of column slenderness, with the greatest loss of strength occurring at the transition slenderness ratio, when X=Xt. The

presence of residual stresses also decreases column strength except for very stocky or very slender struts. Conversely, if the column material experiences strain hardening after initial yield this will enhance column strength especially at very low slenderness values where plastic deformation predominates. The combined effect of these imperfections is to decrease the column strength for most of the slenderness range as shown in Figure 1.15. The greatest overall reduction in column strength occurring at the transition slenderness ratio.

Post-buckling Behaviour Of Compression Members

In order to predict the post-elastic response of double-layer

grids it is necessary to consider in detail the post-buckling behaviour of the compression members used in the structure. In indeterminate structures failure of an individual compression

member, causes a transfer of forces from the failed member to

adjacent compression members. If the adjacent members are capable of both carrying the increased load and absorbing any release of energy from the failed member, then the structure will remain in

equilibrium with the external loading. Supple and Collins (1981), have shown that the characteristics of the post-buckling curve of

25

the compression member has a commanding effect on the overall behaviour of the structure.

Paris (1954), proposed a means of following the post-buckl ing behaviour of rectangular compression members, and this successful approach has been extended by Supple and Collins (1980), to predict accurately the post-buckling behaviour of both thin- and thick-walled tubes, typical of members used in double-layer grid structures.

In this method the axial deformation of a column during plastic buck] Ing can be considered as the sum of the deformat Ions due to direct stress and column flexure.

Using the nomenclature of Figure 1.16:

LL

PL + (ds - dx) ............ (5) Tr 17 0

Using ds = dx + dy 2 the integral in equation 5 may be expressed dxi)

as: LL L

(cls - dx) = ! /2 dV\2 dx 2 NX-) L

0

011( on substituting into equation 5 this gives:

PL dy x

LL2

dx ............ (6) ýT +2 EFX

0

In order to evaluate the integral it is necessary to assume a deflected shape for the column. It has been shown that a good approximation is achieved by assuming the column deflection to be

sinusoldal (Lin, 1950). Then:

y=a sin TIX (7) (7) ..............

differentiating equation (7) with respect to x gives:

dy a ]I IIIXN d-x L cosýr-)

26

ýds

Figure 1.16. Column Parameters. Column parameters and dimensions used in the derivation of the post-buckling load-displacement relationship, (from, Paris, 1954).

h

tension

Figure 1.17. Assumed Plastic Stress Distribution At The Central Cross- Section Of Column. The assumed rectangular plastic stress distribution for a perfect pin-ended column at the mid height cross-section, (Van den Broek, 1948). This simplified stress distribution has been used by Paris (1954) in the derivation of the post-buckled load-displacement relationships for columns with rectangular cross-sections.

27

(h-g)

Substituting equation 8 back into equation (6) and integrating

gives:

PL +

(an )2 7T -4 -E (9)

In order to el iminate the unknown displacement 'a' it is

necessary to consider the strut equilibrium at mid height. Then:

Pa .................... (10)

The internal moment of resistance of the column at mid height IMI can be evaluated by considering an assumed plastic stress distribution at the central cross section of the column as shown in

Figure 1.17, (Paris, 1954).

From equilibrium requirements:

P Al

ay A ............. (11)

and

MCZ dA ............. (12)

Af ry

Substituting equation (10) into equation (9) gives:

6 PL + 11 2 (m 2

WE -T 7)............ (13)

For a rectangular column cross section of area A, depth h and width b where b>h:

P= ayb(h - 2g) .............. (14)

and M= cyybg(h - g) ..............

(15)

Eliminating the unknown g from equations (14) and (15) and substituting into equation (13) gives the required P-S relationship

112 [h

A

p

LL +p 16 AE 4L p CTYA

28

Using the approach outlined above, Supple and Collins (1981)

have extended this work to obtain the post-buckling relationship between the axial load and end displacement for thin- and thick-walled tubes with both pinned and fixed ends. The

relationship between the axial load and end displacement for a thin

walled fixed-end tubular strut is given as:

-2 PL Aý4 .R. cos p

L[p +I- (11 w)....... We L ay

Providing the tubes are not prone to local buckling Equation (17) is

valid for tubular members with 1> 25, where R is the radius of t

the tube and t is the tube wall thickness.

From experimental investigations Paris (1954), and Collins (1981) have shown that there is good agreement between experimental results and theoretical predictions especially in the advanced

post-buckling regions. Figure 1.18 shows the early portion of both

the experimental and theoretical post-buckled load-versus-end

shortening curves for a 12.7mm square annealed, pin-ended column (Paris, 1954). The discrepancy between the theoretical and experimental values shown in the immediate post-buckled region is

primarily due to the Hnear response reaction from the structure or test machine loading the column.

If the column is tested in a displacement controlled test

machine the experimental values will follow the line AD, the slope

of which relates to the stiffness of the machine. If the test

machine was Infinitely rigid then the vertical line AB instead of line AD would be followed. For a column forming part of a larger

redundant structure the slope of the line AD would be proportional to the stiffness of the structure without the column present. Paris, has shown that the area represented by ABCD in Figure 1.18 is

representative of the magnitude of energy dissipated during 'plastic

dynamic buckling' as the column undergoes a rapid change in

equilibrium states. Furthermore, he has shown that 'plastic dynamic

buckling' is most pronounced for columns with slenderness equal to

the transition slenderness ratio.

29

40

30

r--l

m 20 to

10

0

A

X=76-3

Dynamic Instability

r. 3 , --Test Data %eZ

c *-% -% -. \: -*"Theory

Equation 16]

0-25 0-50 0-75 End Shortening [mm]

Figure 1.18. Theoretical And ExDerimental Load Vs. Deformation Relationship For A Pin-Ended Square Column. This Figure shows the phenomenon of plastic dynamic buckling. The experimental valves were obtained during a series of column tests undertaken by Paris (1954). The column material was an annealed hot-rolled steel with an average yield stress of Cy = 239 WmM2. The axial deflection was measured in the direction and at the point of application of the load.

30

NON-LINEAR BEHAVIOUR OF SPACE TRUSSES

The complete range of behaviour of a space truss can be

controlled to a large extent by the designer. With a suitable choice of member type and size, the tensile chords, compressive chords or web members may be allowed to yield or buckle

. as appropriate, as the imposed load acting upon the structures is increased. Space trusses are usually designed to remain elastic when supporting an imposed load equal to the design load multiplied by a suitable minimum safety factor or load factor. It may then be

assumed that, because these structures are usually highly indeterminant and will contain several under-stressed members in

addition to the fully stressed members, even further increases in

loading can be supported before collapse is imminent. However this

may not always be true. Several investigators have reported the

progressive collapse of model square-on-square space trusses

occurring after the failure of the first compression member (Schmidt, et a], 1976; Collins, 1981). In addition it has also been

suggested that square-on-square space trusses even may collapse before reaching their designed maximum capacity (Hanaor, 1979;

Howles, 1985). This reduction in safety factor was thought to be

due to the common practice of using the concept of effective length

to design the truss compression members. This approach is only

valid if the struts are perfectly straight, axially loaded and buckle by bending only. Lateral torsional buckling is excluded and the concept is completely void if the compressive stress in the

strut varies along the length and exceeds the yield stress at any

position. This situation may arise in rigidly-jointed sway frames

but it has been shown that in double-layer space trusses, in which axial forces predominate, the concept of effective length used in

the design of the -compression members is acceptable (Massonnet, et

a], 1986).

Tensile Chord Yield

By allowing yield to occur in the lower chord tensile members of a space truss it is possible to introduce a large amount of post-yield ductility into the structure, provided the compression members can remain stable throughout the yielding process. Several

analytical and experimental studies into the behaviour of space trusses in which tensile yield predominates have been reported.

31

Two model square-on-square double-layer space trusses have been fabricated and tested to collapse (Schmidt, et al, 1978). The

nominally identical structures shown in Figure 1.19 were 1.83 metres square, 0.22 metres deep and were simply supported at each of the lower chord perimeter nodes. The upper chord, perimeter lower

chord, and the web diagonal members were manufactured from 12.7 rWn outside diameter aluminium tube which had the ends coined to fit

aluminium TrIodetic nodes. All of the tensile chord members, apart from the perimeter members, were formed from 1.6 mm diameter soft iron wire. Figure 1.20 shows both the theoretical and experimental load-deflection relationships obtained for the structures. It can be seen from the figure that a large reserve of strength exists beyond the-elastic limit. This ductility exhibited by both of the

structures was terminated by buckling of the top chord compression members which led to a sharp drop in load carrying capacity. Figure 1.21 shows the theoretical and experimental compressive chord forces for one half of one truss occurring at the elastic limit and also just prior to collapse. The Figure shows that as tensile yield occurs in the bottom chord members a significant redistribution of force occurs throughout the truss. During tensile yield a compressive membrane force is established which increases the

compressive chord forces in a non-linear manner as yielding progresses. A four-fold increase in the external load resulted in a nine-fold increase in force in the most heavily stressed compression member. In addition the peripheral compressive chord members underwent a force reversal and carried a substantial tensile force just prior to collapse of the structure. The non-linear development

of the compressive membrane force associated with yield in the tension chord prevents the use of the yield line method of analysis to determine the collapse load of the structure. The yield I ine

approach requires that the limit moment be maintained at any point within the structure and this cannot be obtained with the marked non-linear behaviour exhibited by the compression members. The behaviour of the two square-on-square double-layer structures shows clearly that it is possible to create a large amount of post-yield ductility and strength reserve within these structures. However, for both of these structures the elastic limit load was significantly less than the collapse load. This difference could be

reduced to provide a suitable safety factor by increasing the area of the tensile members in addition to using a material which In tension exhibits a definite yield plateau and a smaller yield to

ultimate load ratio. 32

aT nil

Figure 1.19. Square-On-Square Double-Layer Space Truss. The double-layer space truss shown in the Figure was designed to favour tensile chord yield with a ratio of tensile chord cross-sectional area to compressive chord cross-sectional area of 1: 28. The structure was simply supported at each lower-chord perimeter joint. Upper-chord, perimeter lower chord and web members were 12.7mm outside diameter aluminium alloy tube and all tensile chord members, except the perimeter members, were 1.60mm diameter solid soft iron wire (From, Schmidt, 1978).

7

Lli

_AMT Al

Figure 1.20. Theoretical And Experimental Load-Deflection Behaviour. The Figure shows the theoretical and experimental load-def lection behaviour for two nominally identical space trusses with the configuration shown in Figure 1.19. Pi is the test truss load at which the first tensile member yielded and Ai is the corresponding truss deflection. The diagram shows a substantial reserve of strength after first yield (From, Schmidt, et al, 1978).

33

10 is 20 25

Z

L

tL

E

.0

9

0' .. I. Experimental iTheoretical

8-

6

4-

2.

0-

[A

2

I2 56

Member Number

123456

Fb-, ure 1.21. Experimental And Theoretical Compressive Chord Forces For Pin-Jointed Space Truss. The experimental and theoretical chord forces

are shown for one half of the structure only. Represent the member forces at the elastic limit of the truss. Represent the member forces at imminent collapse of the truss.

(From, Schmidt, et al, 1978).

1

970 mm

i 9600 mm t

Upper chord and Lower chord Web members members

Figure 1.22. Square-On-Diagonal Truss --

Layout. The structure was fabricated from mild steel tubular members and simply supported at all upper chord perimeter nodes. The structure was loaded equally at each of the lower chord nodes (Fromy Schmidt, 2_t al, 1980a).

34

Another double-layer space truss designed to have substantial tensile chord yield before collapse is shown in Figure 1.22 (Schmidt, et a], 1980a). This square-on-diagonal space truss was fabricated from mild steel tubular members and was simply supported

at all of the upper chord perimeter nodes. The structure was loaded

equally at each of the in ner lower chord nodes. Figure 1.23 shows a typical load-deformation relationship obtained for both tensile and

compressive members. Figure 1.24 shows both the theoretical and

experimental load-deflection behaviour obtained for the structure. As the imposed load acting on the structure increased, several lower

chord members yielded. However before any substantial ductility

occurred in the structure, buckling of some upper chord members led

to a further reduction in the stiffness of the structure and a loss

of load carrying capacity. This behaviour was not as predicted by

the theoretical analysis undertaken by Schmidt et al (1980a). They

observed that although the first compression member to buckle

collapsed at a load close to the average maximum load obtained from

individual element compression tests, subsequent buckled compression

members failed at loads well below 70% of the control test results. This reduction in member capacity was due to a large amount of

rotation occurring at the end nodes caused by the collapse of

adjacent compression members. Schmidt et a] (1980a) also reported that although both joints and members were fabricated to normal tolerances, increasing difficulty was encountered in inserting

members into the structure as construction progressed. Initial

member force measurements were made for the structure. The maximum initial lower chord tensile force was 9% of the yield force and the

maximum initial compressive chord force was 8% of the member buck] ing load. These initial forces were of the same sign as the forces caused by the imposed loading and were taken into

consideration in the theoretical analysis. This slightly improved

the poor correlation between the theoretical analysis and

experimental results and suggested that other factors may have a

predominant effect on the truss behaviour.

The Bamford jointing system used in the construction of this

structure, will allow a certain amount of joint slip, and this together with excessive joint rotation may account for the large discrepancy between the theoretical and experimental truss behaviour. Nevertheless the experimental results do show that the

structure possesses some post-yield ductility which highlights the benefits of allowing tensile yield to dominate the truss behaviour.

35

r. -"

.D 0 -I

z m co

Disp lacement [nnm]

TENSION COMPRESSION

Figure 1.23. Load-Displacement Relationships For Tension And Compression Members. The Figure shows typical load-displacement behaviour for mild steel tubular tensile and compressive members usedin the square -on-diagonal grid shown in Figure 1.22. The compression members were tested with end nodes and had a slenderness ratio of 87 (From, Schmidt, et al, 1980a).

04

r-

0

Figure 1.24. Theoretical And Experimental Load-Displacement Relationships For Square-On-Diagonal Space Truss. Curves A, B and C represent three different theoretical analyses of the square-on-diagonal space truss shown in Figure 1.22. Curve A was derived using the full member behaviour as shown in Figure 1.23. Curve B represents the theoretical load-displacement relationship for the structure allowing for initial member forces due to lack of fit. Curve C also represents the theoretical structure behaviour allowing for lack of fit forces but with an assumed 20% reduction in compression member capacity. Curves A, B and C were terminated when the first compressive chord member was about to buckle (From, Schmidt$ et al, 1980a).

36

Displacement Lmm]

misplacoment [mm]

One of the leading, commercially available systems, Space Deck, fabricated by Space Decks Ltd. of Chard, Somerset, is also designed to allow a limited amount of tensile yield to occur within the bottom chord members before the structure becomes unstable. A large square-on-square double-layer space truss constructed from Space Decks 900 zone roof units, was tested to assess the amount of load redistribution that takes place within the structure before

complete collapse (Space Decks Ltd, 1973). Figure 1.25 shows details of a typical pyramid unit and a plan of the complete structure. The structure was erected in three sections and when complete, a predetermined camber was built into the space truss by

successively tightening the tie bars forming the lower chord members. The structure was symmetrically loaded at the top nodes by

using a system of spreader beams and cables pulling from beneath the

structure. As the imposed load was increased several bottom chord members in the central region of the structure f ailed in tension. In addition some bottom chord members adjacent to columns and

several diagonal web members buckled. Figure 1.26 shows the

load-displacement relationship obtained for the structure. It is

evident that for this particular load distribution and support condition the structure exhibits both a large reserve of strength and ductility beyond the elastic limit load. This will ensure an acceptable collapse mode exhibiting large visible deflections if the

structure is grossly overloaded.

CompressionChord Collapse - Theoretical Studies

The post-yield response of a space truss depends, to a considerable extent on the behaviour of the compression members within the structure. Most space trusses will have compression members with slenderness ratios, L/r, within the range 40 c L/r

4140 and consequently these members will exhibit a "brittle type"

post-buckling response when axially loaded.

Several analytical investigations have considered the effect

which the individual compression member post-buckled characteristics has on the collapse behaviour of the complete structure. The two dimensional "compression fan" shown in Figure 1.27 was used to study the effect of changing the post-critical slope of the Individual

compression members forming the fan structure (Supple, et al, 1981).

A vertical load was applied to node 1 and incrementally increased to

produce a series of failures in the individual members. The

post-critical slope a, of the compression members shown in Figure

1.28 was varied from 101 to 801 in 100 increments and the resulting 37

18-000 Top Chord Members All Members

1200 m sqýale PERIMETER

ANGLE TOP CHORD 50 38 1.6 35 L, Light Llrýd (Standard 50x 38.1 x38 1

50 50 x6 35 L, Shear Und (Q 01, r-) v

E E

V

BOTTOM CHORD MEMBERS ALS 20 57mm DIA GON 9 so lid bar

, 0_ 0 OD D ýb be 27-0- OD I ubt L, g ýh Lg. j ht

ý

28-5- o ýohd bar.. Shear Und

Bottom Chord Members

Figure 1.25. Plan Of Square-On-Square Double-Layer Space Truss. The square - on- square double-layer space truss was supported at twelve boundary nodes and was symmetrically loaded at each top node. The load-displacement behaviour for the structure is shown in Figure 1.26 (Space Decks, 1973).

3-0

T2° 1.0

0

Displacement [mrn I

Figure 1.26. Load-Displacement Behaviour For Square-On-Square Double- Layer SDace Truss.

38

0 racinu Iv, = mLrnr

100 200 300 400

Fi

ion

F:

48 12 6 20 24 - verticEj defln. r, o.:! e 1 trnm)

Figure 1.29. Collapse Curves For Compression Fan For Various Values Of(x. The Figure shows the theoretical load-displacement relationships for the compression fan structure shown in Figure 1.27. The curves and numbers correspond to different values of the compression member post- buckling slope, m, shown in Figure 1.28 (From, Supple, et al, 1981).

39

theoretical load-deflection relationships are shown in Figure 1.29. From their analysis Supple, et a] (1981), showed that for low values of a where the strut failure characteristic is not abrupt, failure of the individual members did not produce a sudden collapse of the structure. However, when the strut failure characteristics were sudden the structure exhibited several unstable modes.

Earl ier work considered in detail the variation in truss response due to changes in the post-buckled characteristics of the compression members (Schmidt, et a], 1980b). The double-layer square-on-square grid shown in Figure 1.30 was analysed assuming six different post-buckling paths for the compression members. Figures 1.31 and 1.32 show the assumed post-buckled column behaviour and the corresponding response of the double-layer structure. As the post-buckling response of the compression members becomes more abrupt the post-yield behaviour of the structure becomes critical. However the results given in Figure 1.32 show that for this

structure individual compression member capacities can be reduced by

almost 30% below their initial ultimate load before the complete truss post-yield capacity falls below its initial yield capacity.

A more general theoretical model of column behaviour, -shown in Figure 1.33 has been used to assess the effect which changes in both

the plateau length a and in the magnitude of the residual force

ratio 0, will have on the post-yield response of space trusses (Schmidt et a], 1976). The double-layer square-on-square grid shown in Figure 1.30 was studied in this analytical investigation. Figure 1.34 shows the response of the truss due to changes in a and 0,

assuming that the structure behaves symmetrically throughout the loading sequence and finally collapses by the formation of a symmetrical pattern of buckled members. From consideration of the theoretical load-deflection curves given in Figure 1.34, it is

evident that the larger the ductile region assumed in the theoretical column behaviour, the greater the ductility exhibited in the post-yiel d response of the truss. For values of a less than 0.5

and 0 less than 0.6, the initial Yield load for the truss represents the maximum load-carrying capacity for the structure. When the

plastic plateau length a is greater than 0.5 and 0 is greater than 0.44 the maximum load-carrying capacity of the structure is greater than the yield load capacity and a reserve of strength is obtained.

40

121S

-.. -_. S"-S---.

I ""

".

" I "..

".

"

I "".

".

" I "..

"""

""""""

I """_!

S"S I

I. ." ""

"" ; "" "" ""

". ""

"

" -: - " "

-: -- -

I

"" " -: -

..

-.; - -

.

I ::

..

" I "..

:

".

" "".

:i

".

" I "".

I ..

""" I "".

: .

".

" I "..

S..

""S

-

".. I "..

-: E -

I

""�

j"". b

- --: -

I

"".

I . ""

-- -

I

"".

".

"

-

I" i " ---

""S" ISS...

-- -

..

"" I "..

"

--

. """ I .�

--

.I.

".

-

.I

"".. I ."

- -.. ---

p.

.I.

--- -

I

"" I. - --: --

I

.I.

-: -- -

I

.I.

-

I

"". I -.

"1 I

P.

S.. I "".. I

-.

"1 :I

-. .J !

""I I .. I

.

I.. . "". I

""" III """.

-. -"-"

. _i .

___ _. _. -S

"__ "

__

1B30 mm

Top members

Bracing members

Bottom members

Loading points

E E 0 m co rl

Figure 1.30. Plan And Elevation Of S uare-On-Square Double-Layer Spac Truss. The structure was simply supported at all boundary nodes and was symmetrically loaded at four lower chords nodes. Member properties: upper chord, AE = 2.28MN; web members, AE = 2.52MN; lower chord, AE = 2.76MN (From, Schmidt, et al, 1980b; Schmidt, et al, 1976).

41

1.0

0.5

8 8y

Figure 1.31. Theoretical ComDression Member Load-Dis-Dlacement Relationships. The idealised compression member load-displacement relationships are shown in non-dimensionalised form, where by is the member end displacement at buckling and Py is the maximum member load. These six relationships have been used to investigate the theoretical response of the double-layer grid shown in Figure 11.30 (From, Schmidt, et al, 1980b).

15

1

Q QY

05

A Ay

Figure 1.32. Theoretical Load-Displacement Relationships For A Square-On-Square Double Layer Grid. The Figure shows the theoretical load-displacement relationships for the structure shown in Figure 1.30. The lines marked A to E correspond to the assumed theoretical compression member behaviour shown in Figure 1.31. The truss load has been non-dimensionalised by dividing by the theoretical truss peak load, Qy, derived from compression member behaviour D of Figure 1.31. The central vertical deflection A has also been non-dimensionalised by dividing by Ay, which is the displacement of compression member behaviour D at buckling (From, Schmidt, et al, 1980b).

42

0123456

1.0 p

py

0.5

01111 0246 10 %Y

Figure 1.33. General Theoretical Model Of Compression Member Behaviour.

The Figure shows the general theoretical model of the compression member behaviour used to study the response of the square-on-square double-layer grid shown in Figure 1.30. The effect of changes in load

plateau length, (5 and residual load level 0 have been assessed. Py and 8y are the maximum column load and column displacement at buckling,

respectively (From, Schmidt, et al, 1976).

1.5 Q O. y

1.0

0.5

0 'A

CLY

1.0

0.5

0

'cý

load-deflection curves 1.5 ýI

0 QY

I. 0

load-deflection curves Q3 = 0.5) 0-5

0

'A

load-def lection curves (P = 2.0)

Figure 1.34. Space Truss Load-Displacement Relationship. The Figure

shows the theoretical load-displacement behaviour for the space truss shown in Figure 1.30 for three valves of 0 (0,0.44,0.88) each associated with three separate valves of (, 3 (0,0.5,2.0). Point A represents the initial yield load of the structure and point B represents the development of lines of collapsed members occurring across the structure (From, Schmidt, et al, 1976).

43

1

1

1

Additional work investigating the effect which changes' in the length of the plastic plateau has on the behaviour of trusses has been reported by Stevens (1968). The investigation was limited to the study of plane trusses and concluded that in order to obtain full potential load capacity of the structure, the length of the

plastic plateau required in the compression member behaviour must be

at least several times the yield strain. Figure 1.35 shows details

of the plane truss used by Stevens (1968) in his analytical investigation. For the collapse mechanism shown in the Figure the

member DE would require a plastic load plateau length of 5.8 times

the yield strain to permit sufficient redistribution to occur, to

obtain the maximum truss capacity. For plane trusses, in which only- in-plane behaviour is considered it is important to consider lateral

buckling of the compression members which may, in practice, be the

most critical design requirement.

The investigations related to the post-yield behaviour of space trusses have all assumed the formation of a symmetrical collapse

pattern of buckled compression members. However, it is most

unlikely for a completely symmetrical collapse pattern to occur in

practice. One member in a group of nominally identical,

symmetrically loaded mqMbers, will always buckle before the others due to larger imperfections. This may r'esult in a completely different collapse pattern of buckled members compared with that

which would be obtained if full symmetry was imposed on the

structure throughout the collapse sequence. The theoretical investigations undertaken into the post-yield behaviour of space trusses would be erThanced by considering how sensitive the structure is to the premature buckling of one compression member. In

addition, it would be beneficial to assess the level of imperfections required to alter the collapse mechanism of the

structure, assuming symmetry is not enforced on the collapse

sequence.

Compression Chord Collapse - Experimental Studies

One of the first double-layer grids to be tested to collapse

consisted of two sets of interconnected orthogonal plane trusses (Canon, 1969). The complete structure was 3.8 metres square and fabricated from rectangular, solid, mild steel sections, joined

together by bolts. The test was undertaken using a load control

system where the structure was uniformly loaded by the use of a

water filled plastic membrane. The load-displacement relationship

obtained for the simply supported structure is shown in Figure

44

Figure 1.35. Collapse Mechanism Of Plane Truss. To develop the full

potential load capacity of the plane truss, the compression member DE

would be required to deform at the squash load and exhibit a total

strain of 5.8 times the yield strain (From, Stevens, 1968).

6

LJ

-0 Co 0

-J

E L2 0

C

Centre Deflection Fmm

Be

Figure 1.36. Experimental Load-Displacement Relationship For A Square Lattice-Grid. The square lattice-grid test structure was simply supported at all boundary nodes and was fabricated from rectangular, solid, mild steel section.

Top chords 4.76mm by 19. Omm Bottom chords 1.6mm by 25.4mm

(From, Cannon, 1969).

45

IIII

PC PC p PC

c

10 20 - 30 40 50 60

1.36. A theoretical analysis was undertaken by Canon (1969) who presented a 'yield line' procedure which assumed perfect elastic-plastic member behaviour. Extensive moment redistribution was assumed to occur throughout the grid resulting in a failure mode similar to that expected from an orthotropically reinforced concrete slab. The actual collapse mode involved the plastic buckling of the

compression members and did not result in the formation of yield or collapse lines as assumed in the theoretical analysis. However, the

proposed theoretical collapse load was within 5% of the actual

collapse load as shown in Figure 1.36. Unfortunately, due to the

method of load application the full collapse load-displacement

relationship for the structure was unobtainable.

A large model tubular steel square-on-square double-layer space truss was fabricated and tested to study both its static and dynamic

behaviour (Van Laethem, et a], 1975). The structure was a scaled

model of the double-layer space truss used for the symbol zone of

Expo 117011 in Osaka, Japan (Tsuboi, 1971). The space truss was 0.2

metres deep, and was simply supported at four corners, creating a

clear span of 4.0 metres span along each side. The grid was tested

under load control with a uniformly distributed load applied to the

structure by steel plates attached to the top nodes. The resulting load-displacement relationship is shown in Figure 1.37. The

structure collapsed by the formation of only one line of buckled

compression members bisecting the structure through the centre. Van

Laethem, . 2t a], (1975) undertook only a linear analysis of the

structure. However, they obtained a realistic estimate of the

collapse load by calculating a collapse bending moment equal to the

product of the critical buck] ing load for one pin-ended member

multiplied by the number of buckled bars and the depth of the

structure. This simple procedure predicted a collapse load 15%

greater than the actual collapse load of the structure.

The elastic-plastic behaviour of another steel square-on-

square double-layer space truss has been reported (Mezzina, et a], 1975). The structure shown in Figure 1.38 was simply supported at the four corners and had a clear span of 6.0 metres. All the

members were fabricated from cold-formed welded steel tubes with an outs 1 de di ameter of 60.3 mm and a wal 1 th 1 ckness of 2.9 mm. Each

member end was slotted to accept welded end plates which were in turn welded together to form the nodes. To determine the collapse behaviour of the structure a single concentrated load was applied

46

r--l cl,

_E Z

m tu

Central Displacement IMM ]

Fioure 1.37. Load-Displacement Behaviour Of A Double-Layer Square-On-Square Space Tnisst The Figure shows both the theoretical and experimental load-displacement relationship for a square-on-square space truss. The steel structure Was simply supported at the four comers and loaded by attaching steel plates to the top nodes (From, Van Laethem, et al, 1975).

Key: ------ theoretical load-displacement response.

47

10

In

161

-4 -1.1 -A

750mm

T 6000mm 11500

Figure 1.38. Square-On-Square Steel Space Truss. 7he steel structure

was simply supported at the four comers and was subjected to a point load applied at the top central node. The test was undertaken using a load control system. All members were cold-formed welded steel tubes

with an outside diameter of 60.3mm. and a wall thickness of 2.9mm.

(From, Mezzina, et al, 1975).

300

200

13

100

Displacement IMMI

Figure 1.39. Load-Displacement Relationship For A Steel Square-On- Square Double-L; ýY-er Space Truss* The Figure -shows the load-displacement relationship for the space truss shown in Figure 1.38. The structure collapsed under a central point load of 300XN with the simultaneous yield of the lower chord tensile members (16-16; Figure 1.38) the buckling of both the upper chord compression member (9-10; Figure 1.38) and web member (10-16), (From, Mezzina, Et 1q, 1975).

48

10 15 20 25

to the top central, node by a series of interconnected jacks. Under increasing load the top central node was effectively pulled through the structure resulting in a locallsed failure. Yield occurred in

adjacent lower chord members shortly followed by buckling of the four upper chord and the four web members joined together at the loaded node. The load- displacement relationship obtained for the

structure is shown in Figure 1.39. Unfortunately, because the

experiment was performed under load control, the full collapse relationship showing the post-buckling unloading behaviour of the

structure was unobtainable. To correlate the experimental and theoretical behaviour of the space truss an extensive series of complementary tests were undertaken on -individual members. The- tensile and compressive static yield stresses were obtained in

addition to the load-deflection relationship of the full size members, tested in tension and compression.

Mezzina et a] (1975) presented two methods of analysis used to

predict the theoretical behaviour of the space truss. In the first

method a step-by-step approach was used assuming perfect elastic-

plastic behaviour for the tension members and an elastic response up to the critical buckling load followed by a zero residual load for

the compression members. The second method used a more approximate linear programming yield-line approach. Both theoretical analyses produced collapse loads in very close agreement with each other, but

they were both approximately 13% less than the experimental collapse value.

Further experimental work undertaken to assess in particular the post-collapse behaviour of square-on-square space trusses has

been reported (Schmidt, et a], 1976). Three nominally identical,

six by six bay, model structures were designed so that compressive

chord collapse would occur before any tensile chord yield. The

models were fabricated from aluminium tubes which had an outside diameter of 12.0 on and a wall thickness of 1.5 mm. The ends of the

tubes were coined to fit into Triodetic aluminium nodes. Figure

1.30 shows a plan and elevation of one of the model structure giving the principal dimensions. The structures were simply supported at

each'of the lower chord perimeter nodes and four equal point loads

were applied to each structure through a four-point load sharing

system. Controlled deflections were then imposed on the structure by the use of a turnbuckle. The load-displacement relationships

obtained for each structure are shown in Figure 1.40. The three trusses all behaved similarly, however, the test on the first model

was halted prematurely due to the collapse of one link support. The

49

40

r---l

30

20

10

40

r--l z 30

t !e1

20

10

40

r--n z 30

20

10

r -1

0 10 20 30

C31placement [mm]

Figure 1.40. Experimental Load-Displacement Behaviour For Three Nominally Identical Square-On-Square Double-Layer Space Trusses. The Figure shows the load-displacement behaviour obtained from displacement controlled tests on three nominally identical space trusses. The layout and dimensions of a typical tests grid is shown in Figure 1.30. The structures were simply supported at each of the lower chord perimeter nodes and each structure was loaded by applying equal point loads to four lower chord nodes forming a 600mm square, synnetrically about the centre of the truss (From, Schmidt, et al, 1976).

50

0 10 20 30

0 10 20 30

Diplacement [MM]

structure was unloaded, the support reinstated and the test

resumed. As the load was increased several top chord compression members buckled which resulted in a drop in load carrying capacity. As the test progressed, moderate increases in capacity were obtained as shown in Figure 1.40. The tests were stopped when two complete lines of collapsed compression members had formed perpendicular to

each other in the top chord.

Schmidt, et al, (1976) undertook a theoretical analysis of the

structure using a step-by-step procedure which used ]! near approximations to represent the real compression members behaviour

as shown in Figure 1.41. The analysis predicted a theoretical

collapse load 27% greater than the actual collapse load. This discrepancy between the theoretical and actual collapse load was explained by Schmidt, et al, (1976) to be due to several factoýs.

They noted that during assembly of the models there was a variation in the ease in which the members fitted into the nodes. This

introduced slip into some joints which resulted in a non-

symmetrical strain distribution within the structures even during

the elastic range. Figure 1.42 shows both the theoretical

and experimental member forces within the elastic range in a section taken through the top chord members across a centre I ine of the

grid. Schmidt et a] (1976) also suggested that the difference

between experimental and theoretical collapse loads may have

resulted from imperfections inherent in the compression members. The compression members used in the trusses had a slenderness ratio of 67 which is close to the transition length of the strut and consequently they were particularly sensitive to imperfections.

Apart from the difference in the collapse loads, the experimental

and theoretical post-yield behaviour of the space trusses is very similar. A large amount of ductility was exhibited by the trusses

even though the individual compression members exhibited brittle

type load-shedding characteristics. However there was no reserve of

strength in the structure after the initial collapse of the first

compression members.

A series of tests undertaken to assess the load-carrying

capacity of square-on-diagonal space trusses has recently been

reported (Saka, et a], 1986). Three different square-on-diagonal space trusses were investigated and three nominally identical models were fabricated for each type. The nine models were constructed from brass tubes and brass balls and were simply supported at each of the boundary nodes in the bottom layer. The models were

51

10 , Experimental

r-n 8 ------ Theoretical approximation 2

J 6

C) .J4

L ---------------------- ------------ 7 2

2468 10 12 14

End Displacement [mm]

Fizure 1.41. Experimental And Theoretical Compression Member Behaviour. Both the experimental and the assumed linear approximation to the compression member behaviour are shown in the Figure. The aluminium compression members had an outside diameter of 12.0mmv with a wall thickneSB Of 1.5mm. and had a slenderness ratio, L/r of 67 (From$ Schmidt, et al, 1976).

Experimental

Theoretical

81

0)

LL

(0 21

ol IIIII 12

Member Number

Figure 1.42. Theoretical And ExT)erimental Top Chord Member Forces. The Figure shows both the theoretical and experimental elastic forces in the top chord members across a centre line of the double-layer space truss shown in Figure 1.30. The lack of symmetry in the experimental results was considered to be due to slip occurring between the members and nodes in addition to the presence of an initial force distribution caused by the lack of fit of members (From, Schmidt, et al, 1976).

52

subjected to a uniformly distributed load, applied under displacement control, to the nodes in the top layer. Figure 1.43

shows a plan and elevation of the three model types tested. The

experimental load-deflection relationships for two model types are given in Figure 1.44. Both the elastic limit load and collapse load for the trusses have been estimated using the continuum analogy method (Saka, et al, 1968). Their theoretical values for the four-by-four bay model were within 15% of the experimental values for the elastic limit load and within 23% of the experimental values for the collapse load. However these theoretical values were improved for the six-by-six bay models which had a denser mesh of members in comparison with the other model types. The experimental load-displacement relationships obtained for each of the model trusses shows a large amount of ductile behaviour with a load

plateau followed by a -gradual load-shedding phase (Figure 1.44). This ductile post-yield behaviour of the model trusses is a direct

result of the load-displacement relationships exhibited by the brass

compression members used in the structures. Typical load-displacement

relationships for the two lengths of compression members are shown in Figure 1.45, (Saka, et al

', 1984). From Figure 1.45 it can be

seen that the 330 mm long members show an extensive load plateau

which has in turn led to the load plateau exhibited in the

experimental model truss behaviour. The load-deflection behaviour

shown in Figure 1.44 is uncharacteristic of the post-yield behaviour

exhibited by steel space trusses.

In the brass models of Saka, et al, (1986) elastic buckl ing

occurred in the compression members, whereas in steel space trusses

which are compression chord critical, plastic buckling of the

compression members occurs, usually resulting in an immediate loss

of load-carrying capacity of the structure.. Nevertheless, it is

possible to obtain a ductile post-yield response from a compression

chord critical space truss. This has been achieved to a large

extent, by introducing into double-layer space trusses continuous

chord members which are eccentrically connected at the joints

(Schmidt, et a], 1977). In this investigation a seven-by-seven bay

square-on-square steel double-layer space truss was tested to

collapse. Figure 1.46 shows a plan and elevation of the structure

which was fabricated with square hollow section chords and circular hollow section web members. Continuity of the chord members was achieved resulting in joint eccentrities. Figure 1.47 shows a plan

and elevation of the joint detail used in the structure. The space truss was 8.5 metres square by 0.86 metres deep and was supported at

53

Lr) %D

Cl)

c: b

-. 0 -1

-1320

Type B

Lr) to

fl) (11

(D

CDI m CI)

%0

CD,

CD cli cli

CD cli C%j

C) C\i cli

ID cli Cýj

CD cli cli CD

1320

Type A

"I A, . 01 . 1-1 A,

01 > >

> > > Nor 1w, I.,

-1320

Type

Fioure 1.43. Plan And Elevation Of Three Model Diagonal-On-Square Space Trusses. The double-layer space trusses were simply supported at the bottom layer boundary nodes and were subjected to a uniformly distributed load at the top nodes. The structures were constructed in brass with tubular members with an outside diameter of 5. Omm and a wall thickness of 0.8mm bolted to brass spherical nodes of diameter 15.9mm (From, Saka et al, 1986).

54

13

J

0

Load-Displacement Relationship For The Central Wes A and A' Of Space Truss Type B.

2 75

60

45

to 30

15

Figure 1.44. Load-Dis-Placement Relationships For The Central Nodes A And A' Of Space Truss Type C. The Figure shows typical load-displacement relationships for the two diagonal-on-square space trusses Types B and C (Figure 1.43). The average values were obtained from three nominally identical models tested for each configuration (From, Sakai et al, 1986).

55

2468 10 12 14 16

Displacement at. node A [MMI

2468 10 12 14 16 18 20

Displar-E3ment at nacie A [MM 1

r--n

2 30

Je 25

to 20 0 A

oi 15

X lo 4

5

Enci Misplaaernent [mm]

Figure 1.45. Average Load-Displacement Relationships For Brass ession Members. The compression load-displacement relationships

were obtained from an average of seven test members. Two tube lengths were tested for the pin-ended brass tubular members. Curve A represents the average behaviour of a 220mm long member while Curve B corresponds to a member length of 330mm. The brass tubular members had an outside diameter of 5mm and a wall thickness of 0.8mm (From, Saka, et 2.1,1984).

56

0 1-0 2-0 3-0 4-0 5-0

T-8 60mm

T '

T T 1

: r. 4 -

- - - -I- _ _

8540mm

Loaded Nodes 0

E E

Co

Figure 1.46. Plan And Elevation Of Square -Orn-Square Double-Layer Space Truss. The square-on-square steel double-layer space truss was simply supported at all of the lower chord boundary nodes. The top and bottom chord members were 25mm by 25mm by 3.2mm steel rectangular hollow sections with a cross-sectional area of 277mM2. The web members were 26.9mm outside diameter circular hollow sections with a wall thickness of 3.2mm and a cross-sectional area of 242mM2 (From, Schmidt, et al, 1977).

hi S in terýlle bolt.

Black bolt

SZC'rIO-4 A-A

A A

Figure 1.47. Plan And Elevations Of Space Truss Joint. The Figure shows the joint detail for the square-on- square space truss shown in Figure 1.46. The chord members in the structure are continuous through the panel points and pass over each other. The web members do not intersect at the panel points and consequently large joint eccentricities exist (From, Schmidt, et al, 1977).

57

a] I lower chord boundary nodes. The structure was loaded, under displacement control, at twelve lower chord nodes as shown in Figure

1.46. To assess the importance of member continuity, joint details,

and the adjoining members on the strut behaviour, three different

sub-structures were tested to collapse. A typical sub-structure and the resulting load-deflection relationship are shown in Figure

1.48. It can be seen from Figure 1.48 that member continuity, joint

details and adjoining members dramatically affect the overall behaviour. Figure 1.49 shows both the theoretical and experimental load-deflection relationship obtained from the full size truss. The

theoretical relationships were obtained from a simplified non-Hnear analysis assuming different Hnear idealisations representing the

strut load-deflection relationship as shown in Figure 1.50. Curve A

shown in Figure 1.49 represents the upper bound truss capacity assuming a concentrical ly-loaded, fixed-ended strut. Curve P,

represents the theoretical behaviour assuming a concentr ical ly- loaded, pin-ended strut. Curve S shown in Figure

1.49 represents the theoretical behaviour assuming the

strut capacity and behaviour obtained from the sub-assembly tests. For this last analysis two different sets of strut parameters were used to represent the orthotropic strength of the truss resulting from the joint characteristics.

From Figure 1.49 1t is ev Went that there is aI arge discrepancy between theoretical and experimental results. The

closest theoretical analysis obtained using the strut behaviour derived from the sub-assembly tests, considerably underestimates the,

maximum capacity and post-yield ductility of the truss. To improve

the theoretical estimate a rigorous non-linear analysis is required which will accurately reflect the element behaviour in the actual truss by modelling the complex interaction between member continuity and joint eccentricity. The experimental behaviour exhibited by the truss does, however, highlight the advantages of using continuous members with eccentric jointing system. A large amount of post-yield ductility is created and a substantial increase in

strength exists beyond the ultimate load predicted by a non-linear pin-jointed analysis.

Further work undertaken to investigate the collapse behaviour

of space trusses has been reported (Collins, 1981). This fundamental work is of value because four models were accurately manufactured to minimise Imperfections and carefully tested under full displacement control. Figure 1.51 shows the model geometry,

58

Critical strut 1E

/ to /

Id point

-- - Tests 2 and 3 only

Sub-Structure Layout And Dimensions.

Test No. 1

50

M 40

30

20

10

0 "'

Central Def lect ion Im M]

Fij7. ure 1.48. Total Ix)ad Vs. Central Deflection Plots For Sub-Structure. Total load vs. central deflection relationships are shown for the top compression member in the sub-structure tested with three different end conditions. The Figure shows the influence of member continuity, joint details and the presence of adjacent members (From, Schmidt,

-et al, 1977).

59

0 10 20 30 40 50 60

400

X

300

to

200

to

100

Central Oeflection IM mI

ental

Figure 1.49. Theoretical And Experimental Truss Behaviour. The theoretical and experimental truss behaviour is shown for the space truss shown in Figure 1.46. Curve A represents the theoretical upper bound truss capacity assuming a concentr ical ly- loaded, fixed-ended strut. Curve P represents the theoretical behaviour assuming a concentrical ly- loaded, pin-ended strut. Curve S represents the theoretical behaviour assuming the strut capacity and behaviour obtained from the sub-assembly test. Curve E represents the theoretical behaviour obtained from excentr ical ly- loaded continuous chord members (From, Schmidt, et al, 1979).

70

r---l 60

2 50

13

40 J

30

20

10

Axial Deformation IM M]

Figure 1.50. Compression Member Load-Deformation Behaviour. Both experimental behaviour and assumed member idealisation are shown in the Figure. Curve F shows the experimental behaviour for a fixed-ended member and lines FT represent the assumed linearisation of the member behaviour. Curve P and lines PT correspond to the pin-ended member and curve E and ET correspond to excentrically loaded continuous members (From, Schmidt, et al, 1977).

60

0 10 20 30 40 50 60

10 20 30

1 1800 mm bý

1

i 360 mmj

6 17 28 39 50 61

ý'jl 1 - "2 2 -/33 55 ,

/* 5 I 6 ý7

-9

/* -2 1 /*

54 t

`15

J

-ý7 /*

'19 /* /*

* * /*

14 1 1 ýs

-- ./5 ý\

-36 47

-, l 9 \ .:. /30 ýl

/* . 152

- -24 6

-18 9 51

-12 34

-------------- ------

56 57 58 59 60

LO 0 LO 0 tr) C14 cli

51 52 53 54 55

C)

46 47 48 49 50

Cl) co Cl) co Cl) cli

41 42 43 44 45

r- C14 cli

36 37 38 39 40

CN

31 32 jj 154 15-ID

60

59

1800 mm

53

57

56

1 254.56 mm

"t CO M (0 (D CD

73- -9 11111 iM ir, , V) i cý

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L

81 82 83 84

Figure 1.51. Geometry, Node And Member Numbers For Model Sauare-On- Square Double-Layer Space Trusses. The diagram shows the geometry, node and member niutýoers for the model designed and tested by Collins (1981). This model was fabricated from bright seamless mild steel tubes with typical member properties of: A= 22mm2; r=3.09mm; E= 203000 N/MM2; cTy = 285 N/MM2

61

node and member numbers. The first test is of particular interest

due to the sudden brittle collapse sequence exhibited. The model

was fabricated from bright, seamless, mild steel tube of nominal diameter 9.52 mm and with a wall thickness 0.81 mm. The steel

members were welded into steel nodes and the complete model

structure was loaded by two matched displacement-controlled

hydraulic actuators located at nodes 20 and 42. The theoretical

analysis for this grid, undertaken by Collins, indicated that the

f1 rst members to fa 11 wou Id be members 62,63,78 and 79. The

actual displacement behaviour of the test model is shown in

Figurel. 52. As increments of displacement were applied to the

structure the model behav ! our was both syrrnetr ical and I inear unt iI

a load of 7800 Newtons was obtained at each of the two loaded

nodes. Beyond this load successive increments of equal displacement

induced larger increments of load on node 20 than on node 42. As

the loads increased, member 63 buckled at a load of 8864 Newtons at

node 20 and 7956 Newtons at node 42. This initial buckling of

member 63 led to a rapid buckling of members across the grid as

members 67,71 and 79 failed in quick succession. Due to these

failures the structure suffered a sudden loss of load-carrying

capacity as it developed into a mechanism.

Such rapid losses In load-carrying capacity were not

experienced in the remaining three model tests. Model four was similar to model one except that the tubes used in model four were annealed and had a higher yield stress than those used in model one. The model was loaded at nodes 9 and 29 and under test produced the load-displacement behaviour shown in Figure 1.53. The

experimental results given in Figure 1.53 show that the structure has a reserve of strength beyond the elastic limit and that the

ultimate load was attained after members In the grid had buckled. The theoretical analysis undertaken by Collins (1981) also predicted a reserve of strength for this particular loading and model structure.

The diverse behaviour of these models tested by Collins (1981)

shows the importance which compression member behaviour and loading

conditions have on the overall behaviour of the structure.

Hartford coliseum roof collapse

The sudden collapse in January 1978, of the steel double-layer

space truss forming the roof of the Hartford Coliseum, Hartford,

62

EI

T6 C 0

5 C C 0

"0 S 0

-I 2

Vertidel Deflection trnrn)'

FiMare 1.52. Load Vs. Displacement Curves For 7he LceAed Nodes Of The Square-On-Square Double-Layer Space Truss. The Figure shows both the experimental and theoretical relationship between the load and displacement for the loaded nodes of the space truss shown in Figure 1.51. The test structure was symmetrically loaded at nodes 20 and 42 using two displacement controlled hydraulic actuators (From, Collins, 1981).

. .. . t .......

A

C 1. V .....................

3 0 C 2

lyý 2 -

.......... Actuator at Node No 9 0 1 Actuator at Node No 29

- Theoretical Result

12 Iß 24

Vereical Cleflecticm [rnm)

Figure 1.53. Load Vs. Displacement Curves For 7be Loaded Nodes Of 7he Model Square-Qý---Square Double-Layer -Space Truss. The Figure shows the experimental and theoretical relationships between the applied external load and the deflections at the loaded nodes for the model square-on- square space truss. The structure was symmetrically loaded at nodes 9 and 29 using two displacement controlled hydraulic actuators. The members used in this model were annealed cold-drawn seamless steel tube with typical properties of: A= 22MM2 r=3.09mm; E= 203000 Wmm2; cy = 375 N/mm2 (Fran, Collins, 1981).

.......... Actuator at Nods No 20

-----Actuator at Nods No 42

- Theoretical Result

63

10 20

Connecticut, USA, led to the speculation that all space trusses are prone to failure by progressive collapse initiated by the failure of one member. Several investigations into the collapse of the

structure have been published and their findings show how the interaction between failed compression members and compression member bracing influence load shedding during the collapse of the

structure.

The primary structure of the roof was -a square-on-square double-layer grid with plan dimensions of 92.4 by 109.7 metres and a depth of 6.46 metres. The structure was supported at four interior lower chord nodes which provided a clear column-free rectangular area of 64.0 by 82.3 metres and an overhang on each side of 13.7

metres past the column supports. The chord members which were 9.2

metres long were formed from four steel equal-leg angles bolted together to form a cruciform section. The top chord members were braced in their centre by two steel angles forming part of an additional bracing system as shown in Figure 1.54. The

primary aim of the additional bracing was to halve the effective length of the compression chord members and consequently they were designed with an assurned effective length of 4.6 metres (Thornton, et al, 1984).

The roof structure collapsed co-. -npletely in the early morning during a freezing rain and snow storm (Figure 1.55). Two major orthogonal fold lines formed across the structure resulting from the

progressive buckling of upper chord compression members. It has been estimated that the snow and rain supported by the roof just

prior to collapse was equivalent to an imposed load of 623N/M2 9 approximately half the imposed load which should have initiated

yield in the structure, (Lev Zetlin Associates, Inc., 1978).

All of the published investigations into the collapse of the

roof structure have concentrated on the behaviour and design of the

unusual cruciform compression chord members. It has been suggested that these members had been designed to resist flexural buckling instead of torsional buckling, which was considered to be

significantly more critical (Loomis, et al, 1980). However, a detailed theoretical investigation into both torsional. and flexural buckling of cruciform members has highlighted several discrepancies In the work of Loomis et a] (Smith, 1983).

64

-uciform chord

members

memoera

Figure 1.54. Additional Bracing System To The Top Chord Compression Member. The top chord compression members were braced at mid-span by two steel angles. The single angles (125 by 125 by 7.9mm) were attached to the chord members with bent plates resulting in large eccentricities between the angle bracing and cruciform chord members (From, Thornton, et al, 1984; Smith, et al, 1980).

65

in. ei lý-ý:::, -

1 "4_ø" J? -

1

4p'.

, 00: d.

FiclurQ 1 5ý QOiiaj25Q Qt ThQ Hartford CQ115QUITI RQQLHartford.

Connegliclil U, S A The coliseum root was a square on square

double layer space truss with a plan dimension of 92.4 by 109 7

metres and a depth of 6.46 metres. The structure was supported at

tour interior lower chord nodes providing a column tree area of

64 0 by 82 3 metres and an overhang on each side of 13 7 metres

past the column supports (Smith pL,, 11,1980. Thornton ! ýJ_al. 1984)

66

Smith (1983) noted that for a given boundary condition, the

effective length used to determine the critical load for flexural buckling would not be the same as the effective length used to determine the critical load in torsional buckling. He extended existing work on elastic buckling of cruciform sections (Blelch,

1952) and presented relationships between critical buck] ing stress and column slenderness ratios covering the Inelastic buckling of cruciform members with and without residual stresses.

In order to assess the ultimate load capacity and collapse mode

of the Hartford structure several independent collapse analyses of the space truss were undertaken. Loomis et a], (1980) undertook an

elastic analysis of the structure supporting only dead loads and

estimated that under this loading, seventy four compression members

should have buckled. The analysis was continued by effectively

removing the failed members from the structure by applying the

critical buckling loads to the nodes at each end of the buckled

members. The structure was then reanalysed with the seventy four

members removed and it appeared that an additional sixteen members

should have also buckled without any increase in external load.

This indicates a breakdown in both compatibility and equilibrium

requirements, which should have been apparent If a check was incorporated into the analysis to correlate node deflections with the assumed compression member shortening behaviour. Loomis et al (1980) assumed that compression member failure would be by torsional buckling and not by flexural buckling. By coincidence, their

calculation of the torsional buckling capacity of the members

approximated to the flexural buckling capacity of the members, including the effects of the secondary bracing system, and

consequently they. obtained an estimate of the collapse load close to

the. actual collapse load.

An improved analysis of the Hartford roof was presented by

Smith and Epstein (1980). Their non-linear analysis assumed flexibly-braced pin-ended members with the load-shortening behaviour

shown In Figure 1.56. This load-shortening relationship was obtained by analysing a series of sub-structures which were used to

assess the influence of the angle bracing restraining the middle of the cruciform compression members. The non-linear iterative

analysis followed the behaviour of the structure up to collapse by

replacing buckled compression members by pairs of equivalent nodal forces. Equilibrium and compatibility requirements were satisfied

67

to

x 4

Chord Shortening

buckling oading

Figure 1.56. Axial Load Vs. Chord Shortening Relationship Fo Compression Members. Curve A is generated by assuming that the pin-jointed compression member remains elastic while for curve B it is assumed that all fibres have yielded at the central cross-section. Point Y on curve A indicates approximately where extreme fibre yielding begins and the intersection of curves A and B at point M indicates the maximum load attainable (Smith, et al, 1980).

68

before each new load increment was app] ied to the structure. The

results obtained by Smith and Epstein (1980) predicted a collapse load close to the actual collapse load and in addition, produced

collapsed lines of failed compression members in the same bays as in

the collapsed truss. The collapse investigation showed the importance of the post-buckled behaviour of the compression members

and the possible disastrous effects resulting from rapid load

shedding when a compression member buckles.

The premature collapse of the Hartford Coliseum space truss has

been attributed to the underdesign of the principal compression

members resulting from an incorrect assessment of their effective length. The structure did not collapse. from an overload, but from

an under capacity. However, it is not unreasonable to assume that

if the compression members had been correctly designed to comply

with the present codes of practice, and the structure had been

subjected to a gross overload, rapid progressive collapse of the

structure would have occured without any warning, in a manner

similar to the actual collapse of the roof truss. This highlights

the importance of assessing both the ultimate capacity and collapse

mode of space trusses and also the need to reapralse the integrity

of the structure after the removal of one, or a small group of the

key members.

69

CHAPTER 2

NON-LINEAR COLLAPSE ANALYSIS OF SPACE TRUSSES

INTRODUCTION

In order to trace both the I ! near and non-] Inear behav ! our of double-layer space trusses up to collapse, it is necessary to adopt a versatile method of analysis which is capable of dealing with plastic yielding and strain hardening as well as the highly

non-Hnear unloading characteristics associated with the

post-buckling of compression memb er s These and other prerequisites can be met by using the standard stiffness method of analysis. To deal with the non-Hnear behaviour the usual approach is to represent the behaviour of the structure by a series of linear approximations which can be solved in the normal way. The

external loads acting on the structure are applied in small increments and within each of these increments of loading, the

equilibrium, compatibility and yield requirements are satisfied by

a process of iteration.

There are two main formulations associated with the iteration

process both of which calculate and eventually minimise Out-Of- balance residual nodal forces. The first formulation is a

strain-to-stress approach where imposed strain Phanges uniquely determine stress changes. In the second type of formulation,

stress-to-strain, imposed stress changes uniquely determine

strain changes. In problems associated with ideal plasticity, it

is necessary to adopt the first formulation because Increments of

plastic strain uniquely determine the stress system, whereas Increments of stress cannot uniquely determine Increments of

plastic strain.

METHODS OF COLLAPSE ANALYSIS

Initial Stress Method

A procedure for handling elasto-plastic problems was proposed by Zienkiewicz, et a]., (1968). This non-] inear method of analysis has been extended. to include the post-bucki ing behaviour of the members (Wolf, 1973). Wolf describes the analysis of a large space truss aircraft hangar in which the non-Hnear stress-strain equations, resulting from the non-] Inear post-bucki ing behaviour

70

of the compression members are solved, using the 'initial stress

method'. The 'Initial stress methodl,, Iater renamed the 'residual

force method' is best illustrated in terms of a one degree of freedom problem as shown in Figure 2.1.

The assumed Hnear response is represented by the straight line

OA wh iIe the true non- 11 near response 1s represented by the curve OB. Using the stiffness method of analysis the displacement is

calculated by solving:

JA6 'I = [K]-IIPII ................. 2.1

where [K] is the initial stiffness matrix, 1P. 11 is the load vector and JA6 01 is the nodal displacement vector.

Using the strain to stress approach, the member strains are

calculated from:

jAe 'I = [S]JAS 'I .................. 2.2

where [B] is the strain-displacement matrix.

In turn the member stresses can be obtained from:

fal = [D]JAc 'I .................... 2.3

where [D) is an el ast ic itY matr ix def In Ing the member const Itut Ive

relationship for the linear range of the material.

Another set of member stresses can be calculated using the true

member constitutive relationships, representing the non-]! near behaviour of the material. Hence:

fal = [Dep]IAE 01 .................. 2.4

Referring to Figure 2.1, Equation 2.3 evaluates the stress at point A and Equation 2.4 determines the stress at point C. The difference between these two stress values, RI is a residual

stress, which can be treated as a residual force and visualized as additional fictitious nodal forces, required to fulfil nodal

equilibrium and compatibility requirements.

The iteration procedure is implemented by treating the

residuals as external nodal forces, and solving again for displacements using equation 2.1, ie:

jall = [K]-IIRII ................... 2.5

The iteration Is continued with the Increments of strain added, until the calculated residual Is considered to be negligible. A new

71

U) to ei

to

cr

0

rigure 2.1- The Tnitial Stress Method. The diagram illustrates the initial stress method or residual force method in terms of a one degree of freedom problem (from, Zienkiewiczrrt a. 1,1968). The line

OA represents the assumed linear response while the true

non-linear response is represented by the curve OB. The difference between the true stressrgiven by the ordinate of point C and the assumed linear stress, given by the ordinate of point A, is equivalent to a residual stress Ri. This residual stress may be treated as a residual force and visualized as an additional fictitious nodal force required to fulfil nodal equilibrium and compatibility requirements.

72

Strain

increment of external load is applied and the iteration process repeated. The major advantage of this method is that the original elastic stiffness matrix is used throughout the analysis, and consequently it is not necessary to reform the stiffness matrix, or solve a new set of linear equations at each stage of the iteration,

or at each new load increment.

Unfortunately for highly non-] ! near problems the procedure has

a slow convergence rate. Consequently, difficulties can be expected when analysing space trusses in which the majority of members have

either yielded in tension or buckled in compression.

Dual Load Method

To overcome this difficulty Schmidt and Gregg (1980b) have

proposed a method which is capable of following the highly

non-linear post buckling behaviour of typical compression members found in double-layer space trusses.

In this method called the 'Dual Load Method' the known

non-linear member behaviour Is modelled using a series of linear

approximations as shown in Figures 2.2 A and B. Using a plecewise linearization technique, each linear . approximation is only valid within a specified range df member end displacements.

The elastic limit of the pin-jointed space structure Is first determined using the direct stiffness method of analysis. Assuming that a group of tensile members are the first members to become

non-linear and that all remaining members are elastic, the yielded members which are following the load-displacement characteristics of Figure 2.2A may be effectively removed from the structure, and replaced by two nodal forces of value R,, as shown In Figure 2.3. The external imposed load may be Increased by Incrementing the load factor X until the extension of the yielded members, undergoing pure plastic flow, have reached the value 62 (Figure 2.2A). To

continue the analysis the yielded members must now be given a stiffness represented by the slope of the line BC and the value of the nodal residual forces must change from R, to R2 (Figure

2.2A).

A similar approach is adopted for modelling the behaviour of the compression members. However, It should be noted that the post- buckled stiffness of the pin-ended members is negative, and

73

E R4-

r-13- B Rj- I R2

Ic I POSSIBLE 1--ýELASTIC M RECOVERY

.j

End Displacement

rTr. uRF, 2.2A Tension Member

RI

13 0 0 J

r-12

143

r-I

Compression Member

Figure 2.2A & B. Plecewlse Linearization Of Tension And Compression Members usea in ine uual Loaa metnoa. ine memuer benaviour ot botn tension and compression members is Real ised by a series of I inear approximations. Each I inear approximation is only valid over a specific range of end displacements. During each linear range the non-linear member behaviour may be modelled by applying a fictitious nodal force R, at each of the member end nodes in addition to using a member stiffness equal to the slope of the relevant line approximating the member behaviour throughout the range.

74

End Displacement

Figure 2.3. Pin-Jointed Double-Layer Space Truss With, Yielded -Tension Members. WItR the dual load method ot analysis yielded tension members are initially removed from the structure and replaced by two equal and opposite nodal forces (RI) equivalent to the yield load of the members removed.

75

represented by the slope of the I ines EF, FG, GH and HJ, where each

value of the stiffness is only valid for the specified range of end displacements 61-62,62-63P 63-64 and 64-65 respectively (Figure

2.213). Using this approach, the actual value of the force in a

member is given by the addition of the nodal residual force acting

at the member ends, and the value of the member force determined from the analysis of the structure loaded simultaneously by both the

external imposed loads and the nodal residual forces.

After each analysis and increment of the load factor, every member in-the structure must be checked to make sure that it is

satisfying equilibrium and compatibility requirements dictated by both the structure and the linearization approximation of the member behaviour. This may be undertaken by checking that the member displacements are within their current limits, and also that the

member forces correlate with member end displacements. If a discrepancy has arisen, one or a group of members will-require a

phase change from one linear range to the next. This required phase change may entail a straightforward shift from one region to the

next, for example from range FG to range GH in Figure 2.2B, or the

phase change may involve an elastic unloading sequence or elastic recovery sequence as shown by the dashed line in the Figure. During the elastic unloading phase the member stiffness reverts or recovers back to its initial linear pre-buckling. value.

. Whichever of the phase changes are occurring, changes in both

the member stiffness and nodal residual forces are required to

maintain equilibrium. Successive linear analyses are then

undertaken, until either the deflection of the structure has

exceeded specified limits, or the structure has degenerated into a

mechanism. The maximum load carried by the structure is determined

by the maximum value of the load factor X.

Supple and Collins Algorithm

A modification to simplify the 'Dual Load Method' has been made by Supple and Collins (1981). In this method the non-linear range of the member behaviour is still represented by a series of linear

approximations, but the need for the nodal residual forces required in the 'Dual Load Method' to maintain equilibrium and compatibility between the member and structure, is removed. This is achieved by

adding and updating displacement and force increments at each

76

incremental change in the load factor. This analytical method has been used to accurately predict the initial collapse behaviour of four steel double-layer space trusses (Collins, 1981). In his

non-linear analysis, Collins used the mathematical model shown in Figure 2.4, which consisted of nine linear phases to represent the

entire tension and compression member loading and unloading paths.

Smith's Algorithm

Another non-Hnear method of analysis which, like the 'Residual Force Method', does not require the repeated formation and inversion of the structure stiffness matrix has been published by Smith (1984a). A residual force system of fictitious nodal forces

is created, which, when applied to the original structure along with the real external loads, produces a set of internal forces and displacements that correspond to the required non-linear behaviour

of the members. In this method the member non-linear

characteristics are modelled using the stepwise linearization

technique shown in Figure 2.5.

The full member response shown in the Figure is represented by four distinct phases which cover the initial linear elastic range A

to B, a constant force plateaux B to C, a reversing linear elastic

phase D to E and a step down sequence F to J.

The analysis is implemented by first increasing the external load, until the first member, or group of members, reaches the end of the initial linear elastic range. The external load is then Increased by incrementing the load factor X by a pre-determined amount just sufficient to cause any of the members within the

structure to change regime. The required increase in X, represented by &X, is determined by a series of iterations.

Using a procedure similar to that adopted for the 'Dual Load

Method', the internal force in each of the members in the structure can be represented by the expression:

{P) = {Poj + {b); k + [A]{Rl - {Rj ................... 2.6

where (P) is a vector of internal member forces.

{Poj is a vector of initial member forces.

is an influence coefficient vector of member forces.

[Aj is an influence coefficient matrix containing values of

77

Figure 2.4. Member Idealisation. The Figure shows the nine linear approximations used to repF-e-sent member load-displacement behaviour (from, Collins 1981).

78

; Ijurre 2 5. Stepwise Linearization Used To Model Tension And r r ompress; on Member Behaviour. Smith modelled tfie full member

response by four distinct linear phases. A to B represents the initial linear elastic range, B to Ca constant force plateaux with no strain hardening, D to Ea reversing linear elastic or elastic recovery phase and F to Ja step down sequence (from, Smith, 1984). Physical model (- ); numerical model ( ); step down

79

the member forces due to f ict it ious un it res ! dual forces appl ied at the ends and directed along the axis of each of the members in turn.

{R) is a vector of fictitious residual forces.

Equation 2.6 may be rewritten as:

{Pj = {Poj + {b), X + [A*]{Rl .................... 2.7

where [A*] =

Change in the member forces {AP) are given by the expression:

{AP} = {b){A, \} + [A*J{ARI ............... 0* ... e. 2.8

In order to determine the value of AX required to change phase,

the matrices in equation 2.8 are partitioned into three sets, each

set containing all the members which have an identical behaviour.

The three groups considered are: group 1, the initial linear

elastic members; group 2, the constant force plateaux members; and

group 3, the reversing linear elastic members.

Hence, partitioning equation 2.8 gives:

- it Pj) b Ail A12 AO &R i

AAA 23 &R2 ******* 2.9 P2 b2 &x + A21 22 A

bAAA AR &P3 3) 31 32 33 3

For the I ! near elastic members in groups 1 and 3, an increase

in AX will be associated with an increase in the internal member forces JAP11 and II&P31, but no increases are required in the

fictitious nodal residual forces, hence J, &R. 1 = 101 and 1, &R31 0 01

For the non-] ! near members in group 2 an increase in AX must be

associated with a change in the fictitious nodal residual forces JAR21 but must not produce an increase in the internal forces in

these members, hence 1AP21 '4 1011

Substituting these requirements into equation 2.9 yields:

b! A12 'j-1

. 6p bA --- A22 J-b2l

31 3 32 -pi

which may in turn be written as:

Ax 2.10

80

P,

.................. 2.11 AP3 C3

where C, bi A12 + ---

[A 1-b2l C3 b3 A32

Using equation 2.11 a value for Ax can be chosen, so that a

member from either group 1 or 3 will reach the limit of their

particular region. In addition, the constant force members in group 2 must be checked to determine if a strain reversal occurs, and if

the members are elastically unloading during a positive increment in

the load factor. The change in strain can be assessed by using the

apparent force in the member, 7 given by:

«F = K6 ....................... 2.12

where K is the initial axial stiffness of the member and 6 is the

member axial displacement.

The internal forces for the members in group 2 are given by:

IP21 0 f721 -

JR21 ............. 2.13

and hence:

I AXIIT 21 ý 1AP21 + fAR21 7' 1, &R21 ....... 2.14

substituting for:

JAR21 = [A *J-'If-b2l Aý' 22

gives: JEP-21 = [A211-1 1-b21 AX

........... 2.15

Any stress reversal occurring in any of the members In group 2, for a positive increment AX, can be determined by comparing the

signs of 7 and TP- for each member in turn. If a stress reversal Is

sensed in a particular member or group of members, then they are removed from set 2 and added to set 3, so that the actual increment

AX can be calculated.

In order to evaluate the required increment in the load factor

only the coefficients of {b), [A121s, [A21J and [A32J are required. These coefficients are determined and stored when a member leaves

set 1 and enters set 2. When a member leaves set 2 and enters set

81

3, the member coefficients are retained and matrix [A*] is

repartitioned. At the end of every load increment the load vector {Poj is updated providing the initial forces for the start of the

next load increment.

Members in group 2 may also shed load by plastic buck] ing, and in the stepwise linearization approximation such behaviour would cause the member to 'step down' to the next lower load plateau. Throughout the 'step down' sequence the external load is held

constant i. e. AX = 0. The force in the particular member is reduced by changing all JR21 values only. Hence substituting for AX = 0,

j, dRjj = 101 and fAR31 = 101 in equation 2.9 gives: 1AP21 [A'*] JAR21

................. 2.16 22

and

API A, 12 JAR21 ..............

2.17 AP 31 A32

and from equation 2.11

JAP*21 = [A221-1 1'ýP21

............. 2.18

From equation 2.17 it is evident that the incremental change 1, &R21 may cause other members to change state which will in turn

alter the number of members in set 2 and change [A* J. The change 22 JAR21 is appl led in Increments and set 2 is modified as required. Other members may also reach the 'step down' point and they may be

effectively removed from the group until their 'step down' is

initiated by setting &P21 ý 'ýP21 =0 for these particular members.

After the completion of a 'step down' all the members are

checked to determine if any members are at the end of a load plateau

or at the top of a step. If all 'step downs' have been completed,

and all member behaviour is compatible with the assumed linear

approximation, then the external load may be incremented and the

iteration procedure repeated for the new load increment.

Smith (1984a) has successfully developed an algorithm implementing the analysis procedure previously described and has

used this method to accurately model the test results reported by Schmidt, et a], (1976), and Mezzina, et a], (1975). The main advantage of Smith's algorithm Is that the procedure is based on the initial stiffness of the structure, and as such does not require any

82

modification or updating to the structure stiffness matrix. This

eliminates the need for the repeated solution of large sets of linear equilibrium equations, and will undoubtedly provide valuable savings in processor time when undertaking the collapse analysis of large space trusses.

MATERIAL AND GEOMETRIC NON-LINEAR ANALYSIS

All the methods of analysis, previously outlined, are capable of modelling the non-linear behaviour exhibited by a double-layer

grid loaded beyond its elastic limit. Inherent in a] I of these

methods, is the assumption that node displacements are small in

comparison with member lengths, and that after displacements have

occurred, members remain parallel to their original directions.

These assumptions are fully justified for the linear elastic analysis of space trusses, and also for the collapse analysis of space trusses, provided the analysis is stopped before the displacements become excessive. However, for certain space trusses it is beneficial, to continue the analysis, past the small displacement limit, by taking into consideration changes in

geometry occurring with the structure. 141th this modification implemented it is possible to trace the full behaviour of double layer space trusses, which exhibit only tensile yield throughout the collapse sequence. In addition, it is also possible to assess the load carrying capabilities of a grossly deformed space truss

which may have deteriorated into behaving as a catenary.

Changes in geometry can be taken into consideration by

modifying the- iteration procedure adopted to model the member non-linear behaviour characteristics. For the 'Dual Load Method', this can be achieved by adding either the Newton-Raphson or Modified Newton-Raphson iteration to the existing iterative

procedure, and this modified sequence is then undertaken during

each increment of the external load. Figure 2.6 shows the Newton-Raphson method applied to a single degree of freedom

softening system.

The load increment 101 is applied to the structure and the node displacements 16-LI are calculated using equation 2.19

f6il = [K]-l JAPI ................ 2.19

83

0 -j

AP

Figure 2.6. Newton-Raphson Method. The figure illustrates the Newton-Raphson method applied To-a single degree of freedom softening system (From, Zienklewicz et a]. 1968). The true non-linear response is represented by _t7e _Furve OE. For a load increment of AP the initial estimate of displacement f6jj is calculated using a stiffness [K] equal to the slope of the line OD. The displacement fSj is then used to update the stiffness to [K, j which when post-multiplied by the displacement 16ý1 yields a new nodal force equal to CB. The residual force R, is found by subtracting the ordinate of point C from AP and Is then used with the updated stiffness [K, j to calculate the next increment of displacement 1621- The displacements are updated by summing the displacement increments and the iteration is repeated until acceptably small residuals are obtained.

84

Displacement

where [K] is the structure stiffness matrix. The displacements f611

are used to update the node positions, the member stiffness submatrices and finally the structure stiffness matrix. The

updated structure stiffness' matrix [K, ] is post multiplied by the displacements 1611 to obtain a new force vector fFjj. For the

single degree of freedom system shown in Figure 2.6, the force

vector JFIJ is represented by the distance BC. The out of balance

nodal residual forces are then calculated from equation 2.20.

IR, 1 = JAPI - IF, 1 ........... 2-. 20

These residual forces are then treated as the new external load,

and the corresponding displacements 1621 are obtained from equation 2.21:

1621 = [K 11-1

IRIJ ............. 2.21

The node displacements are updated by add! ng 1621 to f6j, and the

procedure is repeated by again updating the member stiffness

submatrices and hence, the structure stiffness matrix. The iteration may be terminated when the vector of residual forces IRI

has become acceptably small.

The Newton-Raphson method can be modified, by using the inverse of the initial structure stiffness matrix [K]-l in equation 2.21 instead of the inverse of the updated stiffness matrix [K

II". This important difference between the Newton-Raphson and

Modified Newton-Raphson methods will increase the number of iterations required for convergence but, because only one inverse

of the stiffness matrix is required, it will significantly reduce the overall computation process and hence the cost.

PRESENT ALGORITHM FOR NON-LINEAR COLLAPSE ANALYSIS

Introduction

one of the principal alms of the present study, was to

implement a non-linear computer program that could determine the Inelastic structural behaviour of a pin-jointed space truss. This

program has been written by the author in Fortran 77, and has been

implemented to run on Prime computers. The algorithm is formulated

using the direct stiffness approach, and is based on the Dual Load Method with an optional Newton Raphson iteration available between load steps.

85

Any computer program is inevitably a compromise between the degree of accuracy of the mathematical model, and the amount of numerical computation required. The validity of the mathematical idealisation, together with the subsequent analysis, can only be

checked by physical measurements on actual structures. This final

check involving four model space structures is described in Chapter five. Other fundamental checks have been undertaken to ensure that the results of the analysis satisfy the conditions of equilibrium. The reactions acting at the points of constraint have been checked, to ensure that they are in equilibrium with the externally applied loads. Also, the internal member force distribution has been

checked, by analysing a structure in which the internal forces were deduced from the conditions of static equilibrium. In addition, the finite element stress analysis prograrn LUSAS (1984), has been

used to provide check results for the linear analysis of several space trusses.

Member Idealisation

Several types of structures may be accurately modelled as

pin-jointed structural systems. These include plane trusses,

transmission towers, and double and triple-layer grids (Makowski,

1965). The initial response of these structures is almost independent of the flexural and torsional member properties, however, these parameters, in addition to joint rigidities, may have an influence on the post-collapse behaviour of the

structures. Pin-jointed space trusses may be considered to be

constructed from individual structural elements, consisting of

single members connected by perfect hinges to end nodes. The

individual member behaviour is assumed to be completely independent

of adjacent members, and progressive collapse of the structure is

considered to result from the sequential failure of individual

members.

The non-linear behaviour of both tension and compression members has been idealised, by using a sequence of linear

approximation. Figure 2.7A shows the relationship between load and strain, obtained for an annealed steel small diameter tube tested in tension. The behaviour shown, is typical of that exhibited by the tension members used in the first physical test model, described in Chapter five. The non-linear response has been

represented by six linear phases shown In Figure 2.7B, which cover

86

12.5

10

7.5 'ýt: Lij

is

PERCENTAGE STRAIN

SMA LL HOL L0W TUBE (BL UE)

5

2.5

Figure 2.7B Linear Ideal Isation Of The Load-Strain Behaviour For Tffe-Annealed Steej Tensile Member. The Figure shows both the expFr-imental loatT-strain relaElon p and the six linear phases, T1 to T6, used to model the member behaviour. The phase T6 allows for elastic unloading or recovery of the member from any point on the load-strain curve.

87

. 12 1.75 5.0 B. 5 11.6

the elastic, load plateau, strain hardening, and unloading member behaviour. Elastic unloading is possible from any point on the loading path, and this implies that the member modulus will revert to the original value upon reversal of strain. Table 2.1 gives the

values of the elastic modulus, nodal residual forces and their

corresponding range of validity for each of the six linear phases.

Figure 2.8A shows the load-strain relationship obtained from a compression member, typical of the members used in the test

models. The non- I ! near behaviour of the compression member has been represented by seven I ! near phases shown in Figure 2.8B, six of which cover the post-buck] ing response. Table 2.2 gives the

values of the nodal residual forces, elastic modulus, range and phase number for each of the seven regions representing the full

compression member behaviour.

Figure 2.9A shows the load-displacement relationship obtained for a typical model soft member loaded in compression. This

behaviour has been represented by ten linear ranges as shown in Figure 2.9B. Two different values of the elastic modulus have been

given to represent the elastic loading and unloading of the member occurring before or after the end of the load plateau. Table 2.3

gives the relevant values of elastic modulus, nodal residual forces, strain range and phase numbers for a typical model soft member.

The numerical models, representing the post-buckling

compression member behaviour, do not allow for perfect rigid strut

post-buckling behaviour, as this would require both an infinite

negative stiffness, and an infinite positive nodal residual force.

However, the tension member model does allow for plastic behaviour

which requires zero stiffness, and a constant nodal residual force

equal to the yield load of the member.

Algorithm For Non-Linear Analysis

Figure 2.10 shows a simplified flow chart, outlining the key

steps in the present program. The majority of the data required for the analysis of the model square-on-square double-layer space structures was generated using the principles of formex algebra (Nooshin, 1984). The data consisted of member topology, member type and node co-ordinates. In addition, member properties, member model parameters, joint constraints and the imposed unit loads are read into the program from data files.

88

U 4- '4"3 -0 S- 0 0 Z. 2

4- 01- f- (0

>

t.. -0 r t, ro U.

ý -1. -1 M (A ro c EE c2. S..

-0 r= c) :3 0 (L, f- u CZ U 4- V) u

. du 19 0

LA c2. -ýJ

= 10 S- 0E m

r= 00

L)

(A m .- 00 *0 - 4-

G) 0) 00

4- Li - #A CM

0 re c tA. C = :3 ro -

X. - (D0 0 11

u

1- CA S- > C- fo 0

OL 4- (1) L)

04- ro 4-3

(V m >.. - - >

s- c:

CL c -0

F- 0MME

4- cý 0 0 4-

c: Zxo (L) . - 10

in 3- Z LA ý rcs . - ý>

m tj (A ý ci r 00 eKj

c) >> Zm cu

c: 0 0

c: cm . 0) G) c 5- E- c: F `o c) e) 0 S- 0u r= ro o

.C (D -0 ýA - r= CL a

. 10 vi C)

du a (L) 0c 1-- ozj . 6. J - rd

c c:

tA CD.

C) CD cM

Ln cý M t. 0 8.0 LO cli la ro ll C%i c7)

C\i

Cif

eri IV 4J Z fli

0

C: ) C: )

4 X x

cm CD 0 CD CD CD CD C) CD Z cý cý u; (lý cý L) C: ) --4 CD

LU

Ln CD CD rlý CD LO to

c9 cm r_ (Ki

Ln CD C: ) CD i pl C: lx C (0 - : ! ý ! s- C: ) (D P-4 Ln CD CD 4-3 vi

.0 0

- E m

Ln (M u

CL

11 1 -1 -1

1 1 w1

89

10.0

: 02 7.5

5.0

2.5

PERcENTAGE sTRAIN

LARGE HOLL OW YELL OW iental Load-strain Relationshi

-Compression Member.

10.0

7.5

c

2.5

A For A Typical

Figure 2.8B Linear Idealisation Of The Load-Strain Behaviour For A I Compres-sion Member. The Figure shows both the experim

load-strain relationships and the seven I inear phases, CI to C7, used to model the member behaviour. The phase C7 allows for elastic unloading or recovery of the member from any point on the load-strain curve.

90

15.0 30.0

15.0 30.0 PERCENTAGE STRAIN

ý 4-) 4-) "I C: a r': U, C: = 0) 11) C-) S- (1)

10 E Ln 0 E-= .- (1) Q) ci 4- (D (A u S.. IA (Ij cn. "I (U S- (U -a C, &- CL

ko (1) 4-) -0 'D U

-a 0 01 1-- a) 4- C'o 0 S. -

0 Al 4-)

.0 CY) CA ý C\1 CIJ r

>

wo 0 (1) oil

E -a C:

AI ZE (D .- 4- '- 'A .- -0 0 AJ , ts x (A

0

CDL =3 Zu

4- CL O'D ý

(: ) fo r- 0 oc$ C: a) E>

Ln 0 10 0U E

r 4_3 , F- %I_ ra 00 (1)

cl) 0 >

'm 4) (D

W

FF

ý: X: .EC:

P CL C:

Q) CD

x S- r

to

L- 4- C: (2) C: (1) 0 >> Q) 0 Oý V) 'o 4- u (U

(1) CD > C: tA -0 S- CU

W=C:

(L) 4-3 4-) 0

;z0E -0 WE to E=- (1) 0

u C: (A 0-4 eo 0 (1) S- 'a

.- -C= 0 C=

(2) tn al CD-

-0 b (V 4- r (1)

E S- 0. --o 4- CD.

E CM'o cu m CU 00 "J a ol U. 9 ý C) C:

ý "0 C: -a CL 410 0

4A uý a U. -

ca. :3 (2.

>-) -I-) 4-J ;; 4-) -0 tA E (1) 10 = o eO ý 4-)

he (ij -- 4- u 3: 0= S-

Cý a X) 0 (D 4-

4_3 0 tA Al *

0 -- 0 (A .- fA (U > CL ý1: u s- U fil 00 =3

.0L 0) V) U

ý- t; -- L 4-1 '. 0 ,1o

CZ 00

ci

u t- 0

U-

C: ) CD C: ) CD CD de du 0. m CD CD CD C: ) C) CD CD CD LO C) -4 .. C LO (n C) m cu Z m Ln CM -4 --4 -0 4-3

a2 fli fli

de fli Z

CD CD

x x

CD Co A: t C> C: )

cý tý rl: c; -4 uý cz + %2» 03 m Ln Pli + 4.0 Lr) (2, b gt cn 9-4 CM 9 1

LU

Lr) rý. -4 r--i cm -e CD C: ) c; c; c; c;

le le

tn rý. M cý "i C*,! Kt Lr) C: ) C: ) CD CD c;

vi

1

c7)

t . (D ro

. c2 0 F

Z=

LLJ

91

Ltj

6 -j

1

-j

PERCENTAGE STRAIN

Flaure 2.9A. Experimental Load-Strain Relationship For A Soft R -emB er

I PERCENTAGE STRAIN

Figure 2.9B. L inear Ideal isat ! on Of The Load-Strain Rel at lonsh lp uf--A---soft member. ine tigure Shows both the experimentaff To-aT--Tfrajn relatio ships and the ten I ! near phases, S1 to S10, used to model the soft member behaviour. The phase S9 allows for elastic unloading to occur during phases S1 to S3 and phase S10 allows for elastic unloading to occur during phases S4 to S8.

92

5.0 10.0 15.0

5.0 10.0 15.0

0

(D 0> tn

u ro 0 -ll ni c

qe ci. -4--3 c)- 4- vi (L)

(MM >o

z3 j- 4-3 s- (n > -0 c: m (1) (A (1) m -- ni fli -- -0

OKJ c2 -C-

4j 0

, cs E .- -c m0- s--

c: r

c3 C C- 00 , 4. j -0 X-- 4-

, m-a - S- C) 0 -a M0 4-3 » 00 .-0U

(A 4-

0 m IA ,

.2 ci, -M (L) m"Z wX 4-3 - t-

(L) c: (U

tn 4J CJ. C) CD tn (A (1) (A G) _. j M v)

0 0-

4- E- (n O_ ni «a 0o V) 0m cz (1) S- (A c: = 0- fgö Z Z$

fZ5 tu tu 4-) > -4.1 L) (D (L)

0 4- (U

(D (A V) 0

_c 41 0(1) Z 4-

tn (1) M00 -0 (1) (A 0 4- fe >

u;; > cn

eij «0 cz

. 4-J c: CM c0 _C vi m0 10 w'

r- m 41 LA S- -U cn ý

-0 _C c:

. (A (U

m r- c: aj c). ,0

Jol 4.3

r=

M 00 G) M CD l) -- - -&J

(1) 4-3 . - r= ý> M f-. c: V)

aj

CD-. 1.. ) -£Z

o S- . - 2X ni M b-) C 0 CZ r= C: n L)

X0

tA M S- f- 0 ei

4-b tn

#0 4-3 12. ., c: -a

c: c: .- r= c7) t (1) c: =; r=

4- CO .- cz 0 -a v) , -0 -0 G) (L) u (1) eij m

C)

Jz U tö tA U CZOO c: E- eu _C eij M :3 (n . - (1) (L) e- c2. m j2 2: w10C 41 t-

4-b. J . 6-3 . -

4- 04- -00 E-= 0. V) 4.. )

V) E= V)

CM 4- (V4- c:

C: 4- C]. (Lj cm (1) >

ý- (0 , -j (1) s- 0 . - m s- CL . 1. J 0- G)

4- 4- (1) 0 -1-1 0 r: C s- fo -ci 41

rd nj &- S-

4-3 4.3 . 4-3 4-3

c0 c0 0m 0 D.

LL-

4^ (M Ln En C) CD co m Cl%j to LO 10 a MO C: ' : c) c; C; C6 -4 0ý C; r,: ( L)

Zl) G C) to C\j C%j co LO CL CO Ca. q15 r-. wil 1-4 (n tn LO Wo cu 0

fA CY) týo C\j to en C\j r-4 L-3 I (3) + r-4 + + + + C: C: Of + .. :3 .. :3

W (U r- 4-J r- 4-3

-0 to in did

0 S- S- t" ro

C14 Ln E CD Ln C)

X x LO U') LA r%. C> CD C) co C) pl, LO r.. C)

C\i 1.4 Cý Cý rý C8 C4 Lý Cý clý Lf) ILO U-) C\j C\i m LO 4m to C) m M r-4 CYN

0 to C\j cn C\i I I t. 0

to I

C\i ko ko (71 0) -4 m C) *10 C) (Y) U-) týo M C"i CD C7% CD Lf) C)

Cý C; Cý cý c; C: ro

Oj ILO ko M cn m

ly! Lý llý (71 Cý Cý Cý 9 Lý f13 . c) C: ) a o cj --d- (::, c) CD

Q) 0 0

C: C: 2= C)

--4 C\i m W. P LO to r-. co CY) u -I U LIO V) V) V) VI V) V) . -

41 V) 4-3

CL r I LLJ LU

93

I Read Data I

Form external load vector JXP) Calculate member stiffness sub-matrices

Form primary stiffness matrix [X) Constrain [IC) and form structure stiffness matrix [K]

Solve for elastic displacements

JAI a [K]-1 fXP)

Calculate member forces

Increase ). to determine first or next i critical member

I

Modify critical member stiffness submatrices to represent required change in member stiffness

Assemble modified structure stiffness matrix I [Knl

I

Modify external load vector to include new nodal residual forces JXP + R)

Yes # structure is is [KnI singular

STO a mechanism I

No

w total displacem6nt

[KnI-I fXP + R)

No # Are deflections within predefined

limits?

I Yes

No Newton Raphson iteration ---!

LF Ac ceptable Residuals?

-TY es

Calculate current member strains and forces

I

Ves

Check if all critical member beha Reduce X to is consistent witft assumed linear previous value

member behaviour elastically

Yes unloading

Fig 2.10

Flqure 2.10. Flow Chart For Non-Linear Analysis Algorithm. The 'F`ig-ureshows the main Reps in the present non-linear analysis programme. The algorithm is based on the 'Dual Load Method' with an optional Newton Raphson iteration available between load steps.

A critical member, mentioned in the flow chart, is a member with a displacement and force corresponding to the intersection of any two of the linear ranges assumed in the representation of the member behaviour.

94

This data is used to construct the member stiffness sub-matrices, which are used in turn to form the primary stiffness matrix. The boundary constraints are app] ied to the primary matrix, to form the structure stiffness matrix which is stored in a banded form, to minimise storage requirements and processor time. The stiffness matrix and load vector form a linear systern of equilibrium equations, which are solved to obtain the components of joint displacements. The joint displacements are then used to

determine the member forces and the minimum load factor required to

cause the first members to become critical is found.

The critical members then undergo their first phase change,

which requires the modification of the member stiffness, and the formation of residual forces applied at the nodes of the critical

members. These nodal residual forces are stored in a vector separate from the external load vector, because only the latter

requires multiplication by the load factor. Once the stiffness of the failed member has been modified, the structure stiffness matrix is updated, and the equilibrium equations solved to yield the new total displacements. The Newton Raphson iteration is applied to

update changes in geometry, and when acceptably small out-of- balance residual forces have been obtained, the current member forces and strains are calculated.

The checks in the next section of the program, ensure that all critical member behaviour is consistent with the assumed linearised

member behaviour. Any failed member within the structure has a choice of two options at any time. The member may either follow

the pre-set sequence of Hnear phases or unload elastically. This

means that a yielded tension member must be extending, and a buckled compression member must be shortening unless they are unloading elastically. If there are N failed members then there

are a total of 2N-1 possible alternative paths only one of which

will fulfil the requirement of equilibrium and compatibility. Each

possible choice of member. path requires modification to the

stiffness matrix and the nodal residual force vector, before

solution of the updated equilibrium equations, to assess if the

correct combination of member loading paths has been chosen. Consequently, this procedure can become time consuming, resulting in numerous Iterations before compatibility is achieved. To reduce the amount of computation required, this section of the program was made interactive to enable the analyst to pre-select possible

95

member loading paths, and to take into consideration any identical

member behaviour resulting from symmetry.

Once all of the failed members are conforming with their linearised behaviour, then the load factor can be incremented to determine the next member, or members, to fail. These members are then set to follow their post-critical paths. The analysis and checking cycle is repeated, until the structure degenerates into a mechanism, resulting in singularity of the stiffness matrix or, until nodal displacements exceed a predetermined limit.

At each member phase change occurring throughout the collapse

analysis, the current value of the load factor, joint

displacements, member stresses, member critical stress ratios, and failed member phase numbers are read into an output file. The

critical stress ratio is a ratio of member stress to yield stress for tension members, and member stress to buckling stress for

compression members, and provides an indication of how close a

member is to failure. Detailed examination of the results stored in the output file was simplified by the use of a plotting program

which provided a graphical representation of the data.

96

]is.

CHAPTER 3

METHODS OF IMPROVING SPACE TRUSS BEHAVIOUR

INTRODUCTION

A review of the theoretical and experimental investigations,

undertaken to determine the post-yield behaviour of double-layer

grids, has shown their behaviour to be highly dependent on the individual characteristics of the members forming the structure. The results of physical tests on several different space trusses,

all of which have failed by compression chord buckling, have shown that theoretical ultimate load capacities have overestimated the

actual truss capacities by some 20 to 25%. This has been apparent even when the theoretical ultimate load capacities have been determined, using precise strut behaviour obtained from physical tests on individual compression members. This discrepancy between theoretical and actual truss capacity, is due to both the high imperfection sensitivity of the transition length compression members, and to an initial force distribution occurring within the

structure resulting from any initial lack of fit of members.

Both the uncertainty in assessing the ultimate load capacity

of compress ! on-chord-crit ical space trusses in addition to their

brittle post-yield behaviour, has led engineers to investigate

means of improving space truss behaviour. Post-yield ductility can be created by allowing tensile yield to occur in bottom chord members or, to a lesser extent, by using eccentrically connected top chord compression members. Alternative methods, undertaken

with the aim of improving space truss behaviour, have considered the removal of selected web members, the introduction of an internal force system obtained by pre-stressing certain members and the incorporation of force-limiting devices.

MEMBER REMOVAL IN DOUBLE-LAYER GRIDS

At present, only a small amount of work has been undertaken to assess possible improvements in the force distribution In space trusses, due to the removal of selected web members. The theoretical elastic behaviour of a double-layer square-on-square space truss, supporting a uniformly distributed load has been studied assuming three different support conditions (Marsh, 1986). The first structure is considered to be supported only at the

centre of the grid and has thirty-six web members removed in an 97

attempt to even out the chord member forces. Figure 3.1 shows a diagram of the structure and the distribution of chord forces along the centre line. It can be seen from the Figure that removal of the thirty-six web members improves the force distribution within the chords, and as a result, the structure is capable of supporting

an increase in load of 24% before the most highly stressed member

reaches its full capacity. Two additional trusses also considered are a square-on-square truss supported at the corners, and a

square-on-square truss spanning continuously over several columns. Figure 3.2 shows a diagram of the corner supported truss and the force distribution in the chord members taken along a centre line,

when f if ty-s ix web members have been removed from the pr imary

structure. The corresponding diagrams for the continuous

structure, which has forty web members removed from each bay, is

shown in Figure 3.3. Marsh (1986) estimates that due to the

removal of these web members, both of the structures are capable of

supporting an increase in load of 20% before failure of the most heavily stressed chord member occurs.

The internal web members omitted from the structures are

removed to prevent the transfer of shear forces which, in turn,

prevents an increase in tensile and compressive forces in adjacent

chord members. The object of the study was to determine the

optimum pattern of retnoved members, but no recommendat ions or

guidance has been presented by Marsh (1986). However, Marsh has

suggested that this procedure will increase the ultimate capacity

of space trusses with uniform chords and in addition, that it is

only necessary to consider the linear behaviour of the trusses.

Unfortunately, the post-critical response of the structures both

with and without the optional web members has not been

investigated, so it is not possible to determine If the ultimate

capacity of each structure is reached with the failure of the first

member.

PRESTRESSING IN DOUBLE-LAYER GRIDS

Most attempts at improving the load carrying capacity, and post-buckling behaviour of compress ! on-chord-critical double-layer

space trusses have concentrated on methods which introduce a small amount of ductility into compression member behaviour. However, if a favourable initial force distribution can be created in these

structures which will pre-stress the critical top chord

98

17ýý

36 Diagonals Diagonals Present Removed

Load 1.24 1.52

-Top Chord

Bottom Chord Diagonal Web Members

Removed Members

Chord Forces Along Centre Line

[Uniform Load, Uniform Chords

Figure 3.1 Chord Forces In A Square-On-Square Space Truss Supported 7ýF-Th-e-Centre. 1he Figure snows the theoretical top and bottom chord forces along a centre line of the square-on-square double-layer space truss carrying a uniformly distributed load. The chord forces are shown when all bracing members are present and when thirty six bracing members have been removed from the complete structure. Removal of the bracing members has improved the theoretical force distribution within the chords and as a result the structure is capable of supporting an increase in load of 24% before the most highly stressed member reaches its full capacity (from, Marsh, 1986).

99

56 Oiagonals Diagonals Present Removed

-Top Chord

Bottom Chord

Oiagonal Web Members

--- Removed Members

Load 1.0

Chord Forces Along Centre Line

Uniform Load, Uniform Chords

Figure 3.2. Chord__Forces In A Square-On-Square Space Truss Supported At The Corners. The diagr&fl shows the theoretical top and bottom chord forces aTong a centre I ine of a square-on-square double-layer space truss supported at the corners. The structure is supporting a uniformly distributed load and the chord forces are shown before and after the removal of fifty six web bracing members. The removal of these memebers has improved the force distribution within the chords and the structure is capable of supporting a 20% increase in load before failure of the most heavily stressed chord member (from, Marsh, 1986).

100

40 Diagonals Diagonals Present Removed

-Top Chord

- Bottom Chord

- Diagonal Web Members

--- Removed Members

Load 1.0

Chord Forces Along Centre Line

Uniform Load, Uniform Chords

Fi Qure 3.3 Chord Forces inA Continuous Square-on-Square -D-ou-ble-Layer Space Iruss. ine diagram shows the theoretical top and bottom chord torces along a centre line of a continuous square-on-square double-layer space truss. The structure is supporting a uniformly distributed load and the chord forces are shown before and after the removal of forty web bracing lilembers per bay. By the removal of these membersthe structure is capable of supporting a 200/10 increase in load before failure of the most heavily stressed chord member (from, Marsh, 1986).

101

compression members in tension, then the possibility also exists to

enhance the load carrying capacity of the structure. The favourable initial force distribution can be created by modifying member lengths and has

' been used as a means of improving truss

design in a number of studies (Holnicki-Szulc, 1979; Spillers, et a], 1984). An initial state of pre-stress can be created in any statically indeterminate structure, and optimising the state of pre-stress for a particular structure and loading condition has been achieved using linear programming techniques (Hanaor, et, a], 1985).

AI im 1 ted amount of experimental work has been undertaken to

val ! date theoretical studies undertaken on pre-stressed double-layer space trusses. However, two square-on-square double-layer space trusses, each with the configuration shown in Figure 3.4, have been tested to compare the effects of providing an initial pre-stress (Hanaor, et a], 1986). Both of the truss

structures were constructed using compression members which exhibited a small load plateau together with tensile members which exhibited a long plastic plateau. One structure was tested without any prescribed pre-stress but with an unknown internal force distribution due to the random lack of fit of members. The other structure was tested after an initial pre-stress was imposed by

shortening twe. lve members. Figure 3.5 shows the experimental load-displacement relationships obtained for both trusses. From the Figure it is evident that pre-stressing the structure has improved its load carrying capacity but reduced the post-buckling ductility. The physical testing of these double-layer grids highlights the benefits which can be achieved by pre-stressing, but it is apparent that to achieve similar increases in load carrying capacity in full size structures, the pre-stressing must be

undertaken with care to ensure that actual initial member forces

are in close agreement with their initial theoretical values.

FORCE-LIMITING DEVICES

In order to improve the load carrying capacity and post-buckl ing behaviour of compress ion-chord-cr it ical double-layer

grids, several engineers have I nvest ig ated the poss 1biIi ty of introducing artificial ductility into compression member behaviour (Schmidt, et al, 1979; Hanaor, et a],, 1980). This ductility will,

102

4116 mm

Eý E

qq

p

Figure 3.4(A)

Mwe

' \ gýý \ Z .

x Figure 3.4(B)

Fiqure 3.4. Experimental Space Truss Dimensions. Figure 3.4(A) sKoWs a plan and elevation of a typical space truss used to assess the effects of a known imposed lack of fit of members. The structures were supported at all perimeter nodes and were constructed with the MERO jointing system using uniform tubular members with an outside diameter of 41.2 mm and a wall thickness of 1.6 mm. The first structure was tested without any prescribed prestress, but with an unknown internal force distribution due to random lack of fit of members. The second structure was prestressed by shortening twelve members as shown in Figure 3.4(B) (from, Hanaor, et al, 1986).

103

RR

r. -. % 2

, 100

13 (9

'i 50

(U 0

Central Deflectian Imm]

Figure 3.5. Experimental Load-Displacement Relationships For A Prestressea Ana Non-prestressea

_ Uoub I e-Layer Space Truss. The

F-igure shows the experimental load-displacement relationships for both the prestressed and non-prestressed double-layer space trusses shown in Figure 3.4(A). It can be seen from the relationships shown in Figure 3.5 that prestressing the structure has improved the load carrying capacity of the structure but reduced the post-buckling ductility (from, Hanaor, et a], 1986).

104

10 20 30

in turn, introduce ductility into the structure and decrease the

possibility of progressive collapse occurring after failure of the

most heavily stressed compression member. Artificial ductility can be created in compression member behaviour, by incorporating into the member a force-limiting device. This device would, ideally, limit the compression force in the member to a pre-determined level

which would remain constant under increasing deflection. Consequently a compression member, protected by a force-limiting device, would exhibit the elastic-plastic load deflection

characteristics, shown in Figure 3.6, instead of the highly

unstable, brittle, post-buckling characteristics typical of transition length compression members. The value of the load

plateau set by the force-] imiting-dev ice must be lower than the

average compression member buckling load to ensure that the device becomes operative before the member buckles. However, it should be

appreciated that the introduction of the force-limiting-device itself, may alter the buckling load of the protected member.

The characteristics required from the force-limiting devices

in addition to the ideal elastic-plastic behaviour are reliability

and repeatibility. The device must be capable of providing a

constant limit force with a load plateau of sufficient length to

allow redistribution of member forces to occur within the

encompassing space structure. In addition, the device should function with the m1n imum of maintenance and the behaviour

characteristics should be independent of loading sequence and time. To a large extent, these operating requirements are shared by energy absorbing devices, and the design concepts used in energy

absorbers can be modified where necessary for use in

force-limiting devices. Energy absorbers are rated on both their

energy absorptions per unit weight, and their ratio of stroke length to device length. These parameters are generally not

critical in assessing force-limiting devices in which acceptable load-displacement requirements predominate. Several different

types of energy absorbers are available, all of which may be

considered for use as force-I imit ing-dev Ices. To assist In their

comparison, they have been classified according to their mode of

operation which may involve material deformation, extrusion or friction (Ezra, et a], 1972).

105

Ideal Behaviour of Limiting Device

Typical Behaviour of Transition Length Member

figure 3.6. Ideal Behaviour Of A Force Llmltlný Device. The Figure shows the ideal behaviour of a force limiting device when loaded In tension and compression. The Ideal characteristics are a perfectly elastic-plastic load displacement relationship.

106

Material Deformation

A wide range of energy absorbers, offering potential as force-limiting devices, rely on the deformation of material for the

absorption of energy. Energy is absorbed by rods, wires, cables or tubes extended plastically or deformed by buckling, bending or shearing. Thin metal tubes are particularly efficient in absorbing energy. Tubes can be flattened, made to turn inside out, made to

expand or contract, or made to fracture as shown in Figure 3.7 (Ezra, et a], 1972). Axially loaded tubes which absorb energy by buckling or fracture, exhibit a fluctuating load-displacement behaviour, which has a mean value of load smaller than the initial

peak load. Alternatively, laterally loaded tubes provide a smooth load-displacement response which is not affected by the direction

of the applied load. The lateral compression of a single tube between rigid plates has been studied by several investigators (De

Runtz, et al, 1963; Reddy, et al, 1979).

De Runtz, et aI, (1963) assumed the collapse mode of deformation of a laterally loaded unrestrained tube to consist of four quadrants, separated by concentrated plastic hinges as shown in Figure 3.8. Their analysis of the deforming tube accounted for large changes in geometry and assumed rigid, perfectly plastic material behaviour. However, these assumptions led to an underestimate of the stiffness of the system when compared with experimental results. The theoretical model of De Runtz, et al, (1963) was later improved by allowing for the effects of strain hardening in the hinge regions (Redwood, 1964). Even further improvements between theoretical and experimental results were obtained by using a more complex model which assumed regions of plasticity instead of localised plastic hinges (Reid, 1978; Reddy,

et a], 1980). Figure 3.9 shows the non-dimensional load-deflection

relationship obtained by crushing laterally a welded mild steel tube between two rigid flat plates. Figure 3.10 shows the load-displacement characteristics obtained by laterally compressing a constrained tube. By constraining the tube and preventing the horizontal diameter from increasing, twice as many plastic hinges

are required to form a collapse mechanism In comparison with the

number required in the collapse of an unrestrained tube. Crossed-layers of tubes compressed laterally can also provide suitable load-displacement behaviour characteristics (Reid, 1983).

107

rgl

///////1// //

Flattening Tube

t

Inverted Tube Expanding Tube Contracting Tube

Tube and Mandrel

Figure 3.7. Metal Tubes Used As Energy Absorbers. The plastic d-eformation of thin metal t7es is an efficient means of absorbing energy. Metal tubes can be flattened, made to turn inside out, made to expand or contract or made to fracture as shown in the Figure (from, Ezra, et a], 1972).

108

P< Po

t

P/2 P/2

-L 5/2 T

H

< Po o Plastic hinge

Collapse Mode Of A Tube Compressed Between Rigid a es. e collapse mode of deformation of the tube is assume

consist of four quadrants separated by concentrated plastic hinges. The analysis of the deforming tube accounted for large changes in geometry and assumed rigid, perfectly plastic material behaviour (from, De Runtz, et a], 1963).

109

P/Z P/2

4

J

CL

6/0

Fiqure 3.9 Experimental Load-Deflection Relationship Obtained From Crushing A Mild Steel Tube. The Figure shows a typical experimenta load-deflection rela: F'onship obtained by crushing laterally a welded mild steel tube between two rigid flat plates. P is the app] ied load, 6 is the lateral def lectionL and D are the length and mean diameter of the tube respectively, Po is the initial collapse load per unit length of tube (from, Reddy, et a], 1979).

110

0.1 0-2 0-3 0-4 0.5 0-6 0-7 0-8

120

100

80

J 11\ CL

60

40

20

31 r. 3

Figure 3.10. Experimental Load-Displacement Relationship Obtained 7-rom-77u-shing Laterally A Constrained Aluminjum Tube. The Fig-ure shows tFe- experimental load-displacement relatiTnF's-hip obtained by crushing a constrained aluminium alloy tube with an outside diameter D of 25 mm and wall thickness t of 0.9 mm,. P is the applied load, 6 is the lateral deflection and L is the length of the tube. By constr aini ng the tube and preventing the horizontal diameter increasing, twice as many plastic hinges are produced in the co II apse mechan i sm comp ared wi th the co II apse of af ree tube (f rom, Reid, 1983).

ill

0.1 0-2 0-3 0-4

Figure 3.11 shows the experimental load-displacement relationships obtained from both 'open' and 'closed' crossed-layer tube systems. In an open crossed-layer tube system, the lateral separation between adjacent tubes is sufficiently large so that no contact occurs between them as the system deforms. In this system each tube deforms in a similar way to a single tube crushed between flat

plates. In a closed crossed-layer tube system, adjacent tubes are kept in contact with each other, before and during deformation of the system (Reddy, et_al, 1979).

From Figure 3.11 it is evident that the load-displacement

relationships obtained from a closed crossed-tube systems exhibits large fluctuations in load, and would be difficult to use in a load limiting capacity. However, the load-displacement behaviour

obtained for a single unrestrained tube shown In Figure 3.9, is

particularly suited to the compression characteristics required in

a force-limiting device.

Tube Inversion

Tubes which are made to turn inside out and absorb energy by

plastic bending and stretching, are also capable for use as force-limiting devices. This process consists of pushing a thin tube on to a radiused die, which will cause either internal or external inversion of the tube (M-Hassani, et a], 1972). Figures 3.12A and B show the process and the resulting load-displacement

relationship. Figure 3.13 shows how this mechanism could be

adapted to provide a force- I imit ing-dev ice to limit the load In a tubular compression member (Ezra, et a], 1972). Inversion tubes

must be loaded axially to ensure stability of the inversion

process, and consequently this could be a major disadvantage

prohibiting their use in double-layer grids.

Plastic Hinge Systems

A versatile device which may be carefully proportioned to

obtain the required load-deflection characteristics is shown in Figure 3.14 (Johnson, 1972). The W-frame shown in the Figure

resists axial compressive forces by acting in flexure resulting in

plastic hinges forming at the elbows. Figure 3.15 shows the load-displacement relationship obtained for the W-frame. The

112

150 Closed Crossed Tube System

E 100- P

J Open Crossed Tube System

CL 50-

0.1 0-2 0-3 0-4 0.5 0-6 07 0.8

B/D

Figure 3.11. Load-Deflection Characteristics of Closed And Open 7r-ossec-F-Tube Systems. The experime a oa -e ec ion FeT-ationships were obtained from crushing both closed and open crossed-tube systems. The aluminium alloy tubes had an outside diameter D of 25 mm and a wall thickness t of 0.9 w. P is the applied load, 6 is the deflection and L is the length of the tube. In an open crossed-layer tube system the lateral separation between adjacent tubes is sufficiently large so that no contact occurs between the tubes as the system deforms. In a closed crossed-layer tube system adjacent tubes are held in contact with each other as the system deforms (from, Reddy, et a], 1979).

113

m to 0 J

m tu

I" (D

" >

\ ý, C

\ m (D - CL

P C L 0) 41 C

m

114

13 M 0 J

CL

.D 4. )

C

C L

x w

(D V) > G) c) CD ro c c2-

-0 ý- =

C)

> 73 CU

ý 'Cl _

4- '-

4-) l<

4-J Q) cu C)

4-1

ci -C3 (2) 1- 10 ý0 >Z (1) CO C) Z c7) _c -

'4- CD

-0 4-> CD >

Q) 7f5

CD vý

<Z tn (3) c: vl - cm _Z c (M

CY) rn ---1

(A '0 4-J cu

-0 u

-0 CD :3

.- :3 ý- c: Z 4--, , LL- 4-3 C-l -- C) ý

20-0 r--n

2 .X 13 co 0 J 10.0

Displacement [mm]

F igUre 3 . 12B Load-Displacement Relationship Obtained From The r`xtýTr-naT-_T_nvers ion Of An Aluminium Tube. The Io aýd --d i nt relationship shown i 11 the Figure was obtained from the external inversion of an alu, -ninium alloy cold drawn seamless tube. The tube had an outside diameter of 50 mm, a length of 76.2 rnm and a wall thickness of 1.625 mm. The tube was supplied in the 'half-hard' annealed condition. The external inversion was undertaken under quasi-static conditions and the tube and mild steel die were lubricated throughout the test with polytetrafluorethylene (from, M-Hassani, et a], 1972).

Figure 3.13. Internal Invertube Force Limiting Device. The Figure shows a cross-sect ion throujF-a--J`evjcý- which couTT-'Te used as a load- I imiter' for tubular members. However, to ensure stability during operation, the device must be loaded axially (from, Ezra, et al, 1972).

115

0 10.0 20-0 30-0 40-0 50-0

C

p

Figure 3.14. Typical -CETEFr-ess i ve Loads .

750

500

(a

250

Geometry Of A W-Frame Under 0

50 100 150 200 Oeflection [mm]

Figure 3.15 Load-Displacement Relationship For The W-Frame Loaded In Compression . The load-displacement relationship shown in Figure 3.15 was obtained from the quasi-static testing of a W-frame fabricated from 5 m, Ti square mild steel. When loaded in compression the spechilen deforms elastically at first until a plastic hinge forms at C. As the joint deforms plastically and closes, the ]ever arrn increases and the load correspondingly decreases. 'When the joint at C has fully closed the load increases rapidly until plastic hinges form at joints B and D (from, Johnson, 1972; Rawlings, 1967).

116

p

osed

I

s E3

complete load-displacement behaviour is well suited to the

requirement of a force-limiting device if the gradual drop in load

carrying capacity occurring after yield can be eliminated. This

should be possible to achieve by fabricating the W frame from a

material which exhibits moderate strain hardening.

Cyclic Bending Device

An unusual device which uses cycl ic bending to absorb energy has a high potential for use as a force-] imiting device. Figure 3.16 shows a cross-section thýough the, Idad-1 imiting device which has been termed the "rol I ing torus load-] ! miter" (U. S. Army, 1971; Johnson 1973). The device uses a continuous helix of wire gripped in the gap between two axial telescoping cylinders. An interference fit between the wire and tubes ensures that the wire is rotated when the two cylinders telescope under the action of an applied axial load. It has been shown by Johnson (1973) that the

mode of deformation of the wire is essentially bending and an analysis, assuming rigid-perfectly plastic material behaviour,

gives an operating force per ring of:

P=I Tiaot (Johnson, 1973) 3

where a is the radius of the wire and at is the tensile yield stress of the wire.

An analysis of the rolling torus assuming elastic-perfectly

plastic material behaviour has also been undertaken (Ezra, 1968).

However, experimental testing of a wide range of torus devices has

shown that the estimate of Johnson (1973) and Ezra (1968)

considerably underestimate the limit load of the devices (Johnson,

et al, 1975). Johnson, et a], (1975) found that the 'rolling'

friction force produced between the wire and tubes had a

considerable influence on the operational limiting load of the

devices.

Figure 3.17 shows both the shortening and lengthening

load-displacement relationships for a torus constructed from copper

wire gripped between mild steel tubes. From the Figure it is

evident that although large fluctuations of force are present in

the first shortening cycle, they are greatly reduced in the second and subsequent cycles, resulting in an almost constant limit load

117

tu

in

4

10

Int 32 Z \, F- L ID 4.3 Z

0

a

m

-13 5 4) CL - CL

ul

> in

= 93

12 12

4

to 41 ID 13

9A 4- C CZ 4- ü) ci cA 0 Q) 00 1- m0c

(1) .- :3 4-3 .-0 3: 4J (1) (n 4-3 V) 4-3 u a) r- 00 r- (Ufo f_U) CEM

CD

m=- 1-- ,u0- lu

-4-3 0 «c3 .-m ro (L) (1) c .-. 0) %-

=O 0) 4-3 U ý- L) (1) c)- - f- >

-j %A CD -cy (1) S- 0- 0.

vý S- to G) (A (1)

4-3 fo (L) 4-3 c: tA �,

(A 4-3 (L) e0

s- -ý; v 4-3 41 *ý «CC3:

-i-i

(A #0 0M c7 0 4. -3

ý, ' -T

4- Ei c: m0 4-3 -- IV w VIO

z3 tn L) 3: u L) (A

0 4- 4-

= (A .-0 (L) Ni (L) 4-1 m c fo (L) 4-3 fe 4.1 r, -ý

10 4-3 M 4- r- M r- 4- (n u0 -4.. ) 0 -- 41 -- 0 "f

118

0 6 co

0

to

0

0 I0

r--l

E (? L_E

J

c C) 43

CI)

0 0

(A >ý, S- WC CM Ln (L) 0 4- =m (1). 0 4-1 r_ (A m 41 0 1- 0- Z .-0 (1) 0 T3 CL -4-3 3: -c kv0 -ý, ' -0 mu U L- -EUcZ 45 vIE- ý2 -V rr. - od 41 (L) CL

. 4-3 c: CZ CD w to CD

-G 0 (A (1) CD U)

LY vi .0

ý- -f 3: = to 0 s-

ta (1) c 4- (D- 0 c2.

LL- tn 4.., Lr) c3t: 10 3:

of m (A to 3:: 0)

CL CD M t- . _C c;

(1) (1) E 4-3

:2 4-) JL2 t= a) :3

4- r= -1-3 c CD zi G) 0 ro e 0)

cn (L)

tu 6M «a c) - 4-3 s-

. 4-3 1- -, vu (L) , (1) 4- S- (L) :j to 0

Q) 4-

tu -4 (1)4- (V '72. -0 u -c

CM c VI s-

4j mw cý rZ CZ

mE0 10 (L) 4-3 M

ro 0 r- r- .A

r=

rý s: 0 0) .2GE tn p--4 4.. ) (L) ' r= (2) 4- - tu m CD M 0%$ r= h-

c; 4-J CO

Lt) - du --4 r1- E 9-4 ý; m "ý ý; LO m «a 4--1 (L) cm xt

V Lr) :0

_) ro C 4- 4-J

-i , 3: ýc 0 to 0 (A (ý cý 0

0 I-

119

0 6 [NX] Gojo=j

response. Johnson, et a], (1975) report considerable difficulty in

constructing a working assembly of the torus. Several materials were used over a range of interference depths. For a smal I interference sliding occurred between the wire and tubes, while for large interferences the outer tube grooved and material was sheared off the contact faces of the tubes.

The load-displacement relationships shown in Figure 3.17, show that the rolling torus is capable of producing a constant limit force in both tension and compression. However, it is not clear if the large fluctuation experienced in the first shortening of the torus, but absent from subsequent cycles, would return after a period of recovery. Both the possibility of time dependent load fluctuations and the accurate fabrication required to ensure the

required displacement characteristics severely limit the use of the

rolling torus as a force-limiting device in double-layer space trusses.

Shearing and Machining

The shearing and machining of metal requires energy, and this

process has been used in both energy absorbing mechanisms and in

the construction of a force-limiting device (Kirk, 1977; Hanaor, 1979). The force-limiting device consisted of a metal rod pushed through the centre of a four bladed cutting too]. A cross-section through the device, which has been incorporated into several space trusses is shown in Figure 3.18 (Hanaor 1979). In order to obtain the required load-displacement characteristics, three different

metals were tested in the device. Brass, aluminium and mild steel

rods were used over a range of different cut depths. The brass rod

produced discontinuous chips and an undulating load-displacement

characteristics at all cutting depths. Both aluminium and mild

steel proved more ductile, producing continuous chips at cut depths

of less than 0.32 m. m. Figure 3.19 shows typical experimental load-displacement relationships obtained by Hanaor (1979). His

test results show that it is possible to obtain almost ideal

characteristics from this device, provided continuous chips are formed during the cutting mechanism. However, from Figure 3.18 it is apparent that the successful operation of this device is dependent on the level of resistance provided to the cutting tools by the restraining bolts. Consequently these bolts should be set at a predetermined torque and in addition the complete assembly must be protected from corrosion.

120

p

I

Figure 3.18. Metal Cutting Force-Limiting Device. The device Tp-er, it-es by pushing-or pulling a metal Foý__t'Frou`J-FF_the centre of a four bladed cutting too]. To obtain the required load-displacement characteristics the depth of cut must be accurately set by adjusting the cutting too] restraining bolts (from, Hanaor, 1979).

121

m tu

Brass Rod

("0 0 -J

AJUMiniUM & Mild StE3el ROCI

Fiqure 3.19. Load-Displacement Relationships Obtained From The Metal cutt i ng i-orce-L Irn ILi ng yev ice. I ne io aa-a i sp i acement relationships are shown for three different materials. Brass, aI um ini um and miId stee I rods were each used in the dev i ce Ai ch was tested over a range of cutting depths. The brass rod produced discontinuous chips and undulating characteristics at all cut depths. Both the aluminjurn and steel rods produced a duct iIe reponse for all cut depths below 2.0 an (from, Hanaor, 1979).

122

Oisplacement

Displacement

Crushinq

A honeycomb core is a useful structure which can be used in a force-limiting device to provide a constant limit force over a large strain. A honeycomb structure is essentially a group of hexagonal cells, which under axial load has a collapse behaviour

similar to that of a cylindrical shell. Figure 3.20 shows a typical load-displacement behaviour obtained from crushing an aluminium honeycomb structure (Coppa, 1968). As the compression load increases, the response is Hnear elastic until the honeycomb

structure buckles, producing six circumferential waves in each hexagonal cell. At this point the load rapidly decreases to a lower level, which is maintained at an almost constant value for

the remainder of the displacement.

Phenol ic-g I ass re inforced honeycomb structures a] so exh ib It

acceptable load-displacement characteristics suitable for use in

force-limiting devices. Figure 3.21 shows the stress-strain

relationship for a phenolic-glass honeycomb structure, and gives an indication of how ductile a honeycomb structure can be even when constructed from a brittle material (Coppa, 1968). Under a compressive load the phenolic honeycomb fails by progressive brittle fracture, forming small particles adjacent to the loaded

ends. The process continues under an almost constant external load, until all the material is crushed. Although the load-displacement characteristics for the phenolic-glass reinforced honeycornb structure is more acceptable than the al um In1 um honeycomb, the I atter structure Is more adaptable for use as a force-I imit Ing dev ice in space trusses. An aluminium honeycomb

would be capable of resisting tensile forces, in addition to

providing a limit load in compression, although the device would have to be deployed in a post-buckled condition to avoid the initial load peak which is a typical characteristic of the load-displacement behaviour.

Extrusion

A large number of devices have been constructed which use the

extrusion of a material as an efficient means of absorbing energy. An extrusion damper designed to be incorporated into a structure to

suppress earthquake oscillations exhibits load-displacement

character ist ics ideally suited to the requirements of a force-] Imiting device (Robinson, 1977). A cross-section drawn

123

80-0

60-0

2

13 (0 40-0

2 0'0

0

Displace me nt [rnrn]

Figure 3.20. T ýIcal Load-Displacement Relationship For An Aiuminjum Honeycomb Structure. The Migure sho -ws---ETFe Joad-displacement relationship o5tained from crushing a low density (128.1 kg/M3) aluminium alloy honeycomb structure. Initially a Hnear response is obtained with the load, increasing until the structure buckles producing six circumferential waves in each hexagonal -cell. After this initial buckling the load rapidly decreases to a lower level which is almost constant for the remainder of the displacement (from, Coppa, 1968).

124

50-0 100.0

r--l CV

E ýE

15-0

z

U, U, C) C- 4.3 U)

10.0

L 12. E 0 u 5-0

a

Fiqure 3.21. Stress-Strain Relationship for A Phenolic-Glass Honeycomb ýtructure. Under a compressive load the phenolic glass honeycomb structure behaves plastically, although the actual behaviour of the phenol ic glass material is brittle. The honeycomb structure fails by progressive brittle fracture occurring locally at the loaded ends. This failure process is continuous, resulting in an almost steady load plateau until the majority of the honeycomb structure is crushed (from, Coppa, 1968).

125

0 25-0 50-0 75-0 100.0 Compressive Strain[9/-]

through the extrusion damper is shown in Figure 3.22. The device

consists of a thick walled tube encasing a piston surrounded by lead. Tensile or compression loading on the system forces

extrusion of the lead through a restricted orifice, producing the load-displacement characteristics shown in Figure 3.23. Figure

3.24 shows the load-displacement relationship for a similar device, in which a bulge on a shaft is moved through the lead. During the

extrusion process, changes occur in the microstructure of the

lead. Elongation of the grains occurs adjacent to the orifice, and

after the material has been extruded recovery occurs with the

recrystallisation and grain growth of the material. The rate of

recovery of a deformed material depends on the type of material, temperature, time and degree of deformation (Rollason, 1982).

Deformed lead will completely recover its properties at room temperature in under ten seconds, whereas copper under the same

conditions takes about one hundred years to completely

recrystallise.

The load-displacement characteristics shown in Figures 3.23

and 3.24 show that the extrusion device is particularly suited for

use in a force-limiting capacity. The identical behaviour in

tension and compression, is a valuable characteristic of the

device, which when used in space trusses, would permit

redistribution of forces to occur under several different loading

conditions.

Friction

A var 1 ety of dev 1 ces have been conce I ved AI ch mob I 11 se friction forces to absorb energy. The load control device shown in

Figure 3.25, has been used in the foundations of a large power

station, to minimise the effects of differential settlement on the

rigid jointed framed structure (Clark, et al, 1973). The device

was designed so that a constant friction force would be developed

on each of two faying surfaces, prestressed together by high

strength friction grip bolts. Considerable difficulty was encountered in obtaining the desired load-displacement

characteristics from the load control device. Several different

faying surfaces were tested and the best characteristics were obtained with sliding surfaces of firm unrusted mill-scale. Figure 3.26 shows a typical load-displacement relationship obtained from

the load control device shown in Figure 3.25. Under a compression load a peak value was reached initiating Intermittent sliding,

126

)R IGI NAL : RAINS

SEALS

c AD

ELONGATED GRAINS

Ex -r; tv s CRIFICi

RECRYSTAL-

IMATION

GRAIN G m. ow-r H

Figure 3.22. Longitudinal Section Through An Extrusion Energy Wsorber. The diagram Shows a longitudinal section tHrough an Fxtrusion energy aborber and the changes in m1crostructure of the working material. During operation of the device the working material is extruded through a restricted orifice and elongation of the material grains occurs. After extrusion the material recovers by recrystallisation and grain growth (from, Robinson, 1977).

127

20-0

15.0

10.0

L1 5-0

0-0 0 IL

5-C

10-C

15-(

20-(

Figure 3.23. Load-Displacement Characteristics Of A Constricted T-ube-Energy Absorber ..

40-C

30-(

r--l Z 20-C

L--ýj 0 10.1 u L 0 0.1 LL

10.1

20-

30,

40-

Fiqure 3.24. Load-Displacement Characteristics Of A Bulged Shaft -Energy Absorber. The diagrams show the load-displacement relationships for two lead filled extrusion energy absorbers. Figure 3.23 relates to the device shown in Figure 3.22 while Figure 3.24 is for a device in which a bulge on a shaft is moved through the lead. The devices were tested at a rate of 10 mrn of movement per minute (from, Robinson, 1977).

128

E E

co (D co

r

lu Co

k- ýCq

0-

CD- CD

k-

C) 4-J ro

cl

4--) 4-) 0

cn

>1 to

LL-

cli -a 1

Z,, 1)

CE: D

Ca- ro CD

-0

cr

-ci ei

co M

m LLJ

C) z LU 0-1

LLJ

CL

CD r",

Li

C)

;m CD

h- im :: Z

LIJ

-A __j L'i

CD

LLJ V)

0 _0 -C (3) 00

(A Wo (1) U 4- S- (n

>C Ln CT)

4-) c-

4- M

uc

(1) Lo M C: ) c: c

01-

S- 110 4- 0

0 4-) 0-

4. ) Ln

4-J 10 C) -

-1 0

-) -0 0

C: 4- C-ý 0 _0 0-

C) c Ln Ts C)

LL- (3) 4-, )

. L)

LO 0

S- ('ý

10 -4-

0) C) cn S- S- - C= = cn (1) (L)

U- _0 LA

129

WWOBS I

80

Load 60,

r---l

40

20

Displacement Imm]

Fiqure 3.26. Typical Load-Displacement Relationship Obtained From 7Fe- Friction Load-Control Device. The load-displacement relationship shown in the diagram was obtained by loading the friction load-control device shown in Figure 3.25 at a rate of 3.3 kN per second. The device behaves elastically until the first slip occurs after which the load decreases to a residual value about which subsequent slipping takes place. Very large fluctuations in behaviour have been reported depending on the faying surfaces, the test loading rate and the test machine characteristics (from, Clark, et al, 1973).

130

0 20-0 40-0 60-0

producing a fluctuating residual load. Clark, et al, (1973) also investigated the short-term behaviour of the device. They found that if the device was re-tested after a period of several hours had elapsed then the first slip in the re-test occurred at a peak load approximately 14% higher than the original residual load. No

explanation has been given by Clark, et al, for this time dependent discrepancy occurring between the residual and peak loads. However, they have shown that the magnitude of the long term slip or residual load is unaffected by time.

A similar load control device to that tested by Clark, et a], (1973) has been used in the construction of a long span composite steel and concrete bridge to absorb horizontal forces resulting from earthquakes. Figure 3.27 shows details of the device which relies on the friction resistance of polished stainless steel plates rubbing against ferrobestos pads to absorb energy (Loo, et a], 1977). Loo, et a], tested several different materials for the faying surfaces over a small range of bolt tensions in order to

optimise the load-displacement characteristics.

Loo, et a] , (1977) and Clark, et al, (1973) found these devices to be extremely sensitive to both the type of material used for the faying surfaces and the magnitude of tensile force in the

prestressing bolts. Similar sensitivity was encountered in load limiting friction devices incorporated into reinforced concrete framed structures (Baktash, et a], 1983). The sensitivity of the friction devices to changes in faying surfaces and bolt tension is

a major drawback to their use as load limiting devices in space trusses. Although acceptable load-displacement characteristics can be obtained, the repeatability of the behaviour cannot be

guaranteed.

SPACE TRUSSES WITH FORCE-LIMITING DEVICES

Although the concept of improving space truss behaviour by incorporating force-limiting devices has been the subject of theoretical studies, only a very limit

' ed amount of. experimental work has been undertaken to assess the physical performance of these modified structures. Three different sets of space trusses have been tested In order to compare the behaviour of similar structures, both with and without, the force-limiting devices (Hanaor, et a], 1980).

131

I

I

: errobestos Pads

3ix High Tensile

Bolts

Figure 3.27. Friction Load-Limiting Device. The diagram shows sections through a f_Fýiction load- I imit i-n-g-revice used to protect a composite bridge f rom horizontal displacement resulting from earthquakes. The faying surfaces are provided by polished stainless steel plates rubbing against ferrobestos pads. During the experimental testing of the device it was found that proper alignment of the two sliding components was vital in preventing both premature damage to the ferrobestos surfaces and ensuring a steady load response during operation (from, Loo, et a], 1977).

Slots

132

The first group of space trusses tested were all statically determinate structures, chosen to eliminate the effect of an initial force-distribution resulting from the lack of fit of

members. Figure 3.28 shows the general layout of the trusses,

which were constructed using aluminium alloy tubes and nodes. Three statically determinate structures were tested, one with force-limiting devices and two without the devices. The tests were

carried out under displacement control with the structures simply

supported at the boundary nodes and loaded at the centre bottom

node. Figure 3.29 shows both the experimental and theoretical

relationship between the external load and central deflection for

the trusses without force-limiting devices. Figure 3.30 shows the

relationship between external load and central deflection for the

structure which has four of the top chord compression members

equipped with force-limiting devices. It can be seen from Figure

3.29 that-the two structures without force-limiting devices show a

brittle post-buckling behaviour, whereas the structure incorporating four force-limiting devices exhibits a ductile

behaviour, and is capable of supporting an additional load in

excess of the elastic limit load.

The undulating response of the load-displacement relationship

shown in Figure 3.30, is a direct result of the poor behaviour

characteristics of the force-limiting devices used by Hanaor

(1980). The devices Used a hydraulic cylinder and ram, fitted with

a relief valve which could be set to open at a predetermined

pressure. A typical load-displacement relationship obtained for

the force-limiting device is shown In Figure 3.31. Hanaor, et a],

reported that the oscillating nature of the response, resulting in

large fluctations in load carrying capacity, was primarily due to

the difference between the effective relief valve area In the open

and closed positions.

The second group of space trusses tested by Hanaor (1980) were

small square-on-square double-layer grids. The steel structures

were again simply supported at the boundary nodes, and loaded under displacement control from the central lower chord node. Only one

of the three structures incorporated force-limiting devices In the

compression chord. For this structure the hydraulic force-limiting devices were not used. Instead, devices operating on the metal cutting principle were used (Hanaor, 1979). Figure 3.18 shows a

133

RR

R

R

R

R

t Plan t

R S13.5 mm

T 2-15-3 mm

p

Elevation

Fiqure 3.28. Plan And Elevation Of Statically Determinate Space 7-r US -S. -The trusses were Simply supported at the boundary nodes marzlE by R on the drawing, and loaded at the centre bottom node. All members in the structure were nominally identical and fabricated from aluminium alloy tubes with an outside diameter of 12.7 mm and a cross-sectional area of 55.4 mm2 . All of the members were 305 mm long and had an equivalent elastic modulus of 41.1 GN/m 2 in compression and 49.8 GN/m 2 in tension (from, Hanaor, et a], 1980).

134

RR

Theoretical 2 Test with stiffened

20 joints 13 .......... Test with unstiffened 0 joints

L 10

x ------------- ........................

10 20 30

Central Deflection [Mml

Fiqure 3.29. Load-Deflection Relationship For The Statically Te-termina-te Space Trusses tho t orce-Limiting Devices. he space truss tested had members which were effec ive y pin-jointed in the horizontal planes and partially fixed in the vertical plane. In an attempt to assess the influence of joint instability, the nodes of the four central top chord compression members were stiffened by applying epoxy cement before the structure was re-tested (from, Hanaor, et al, 1980).

135

2

20

L1 0 41 x ul

Theoretici

Top chord

......................................

Test without F. L. Ds. [stiffened joints]

10 20 30 Central Def lection [ mm

Fiqure 3.30. Load-Deflection Relationship For The Statical] -D-eter-m in AeSp ace Truss With Force-Limiting Devices. Ihe rigure shows how tMe incorporation of force-lim-iting devices has introduced post-yield ductility into the structure. Without the devices the structure behaves in a brittle manner as shown by the dotted curve in the Figure (from, Hanaor, et a], 1930).

Q u L 0 LL

Oef lection

Figure 3.31. Force-Deflection Relationship For The Hydraulic Force-Limiting Device. The device consists of a hydraul ic cyl in-Ue-r and ram f itted with a rel ief valve which can be set to open at a predetermined pressure. During operation the device exhibited large fluctuations in load which was reported to be due to the difference between the effective rel ief valve area in the open and closed positions (from, Hanaor, et al, 1930).

136

cross-section through the device and a typical force-displacement

relationship is shown in Figure 3.19. The gently undulating behaviour exhibited by the metal cutting device is an improvement

on the behaviour characteristics of the hydraulic device. However,

this improved behaviour was obtained at the expense of some loss in

flexibility in setting the required operating limit load. Figure

3.32 shows the experimental load displacement relationship obtained for the three trusses, in addition to the theoretical behaviour for

the truss with force-limiting devices. Hanaor, et al, reported that the large differences between theoretical and experimental behaviour shown in Figure 3.32, was due to an initial force

distribution occurring throughout the structures resulting from a lack of fit of members. Nevertheless, the response of the space truss with force limiting devices is ductile, and the structure

exhibits a load carrying capacity well in excess of its elastic limit load.

In order to observe the effect of incorporating force-limiting

devices in full size trusses, Hanaor, et al, (1980), undertook four

separate tests on one square-on-diagonal steel space truss. A plan

and elevation of the space truss are shown in Figure 3.33. The

structure was originally designed and tested for a tensile yield failure, but in order to achieve a compression chord critical

structure, the loading system was altered and the four central bottom chord members were stiffened. In addition, several top

chord compression members were welded to their end nodes as shown in Figure 3.33. The structure was simply supported at the edge

nodes and loaded, under displacement control, at the four central bottom chord nodes. For the first test on the structure, the

bottom tensile chord members were unstiffened and eight force-limiting devices were attached to the critical top chord

compression members. The hydraulic devices were used In this

investigation and for the first test, the limit force was set at

such a level as to make the structure compression chord critical. In the second test, the limit force of the devices were raised to

enable tensile yield to occur in the lower chord members before any top chord compression members became critical. For the third test,

the four central bottom chord members were stiffened and consequently, the structure reverted to being compression chord critical. In the fourth test, the force-limiting devices were removed and the compression members allowed to buckle. Figure 3.34

137

200-0

180.0

160-0

140-0

m tu 120-0

100-0

c L

80-0

w

40-0

20-0

No. 3

Central Deflection Imm]

1&2 Theoretical

Figure 3.32. Load-Displacement Relationship For Nominally Identical Space Trusses Both With Ana Without Force Devices. The 3Ta-gram77h, To_wsboth the experimental load-3isplacement relationship obtained from a steel space truss test. Theoretical behaviour

Experimental behaviour (-). The structure was fabricated from steel tubular members which had an outside diameter of 33.7 qrn and wa II th 1c kness of 3.2 rnii. The members were 1372 mm

15 0 U^! q p long and had an equivalent elastic modulus of n2 both in tension and compression. For test number 1 the structure was without any f orce- I im iti ng devices and Hanaor, et al, reported significant lack of member fit occurring during asser-nbly of the mode I. This lack of fit of members was reduced for the second test on the structure, which also had no force-limiting devices present. For the third test force- I imit ing devices were fitted to the four central top chord compression members (from, Hanaor, et a], 1980).

Theoretical

138

4-0 8-0 12-0 16 ýO 20-0 24-0 28-0

9601 mm

Eý E

I

XIK

.. 4-

/\

X

I

0

' x

1 X , II IV F )< X ,ý \\Q /N, ý \\ T \\ / X

I

. \, a ýA \-- Z

Plan

r, C0 VN Cj)E AV V yy

-y-

Elevation

OStiffened joints

Figure 3.33. -Plan

And Elevation Showing Details Of A 7qTu -are---Un--Z i -aqon-a-I 5'p-ace-T-rus s. The square-on-diagonal spac--e-l-r-uss was simply supported at al I edge nodes and loaded at four central lower chord nodes. The structure was fabricated from steel tube with the top chord compression members having an outside diameter of 48.3 mm, wall thickness of 4.0 ; nrn and an equivalent elastic modulus of 120 GN/m 2. The tubular tensi le me-mbers had an outside diameter of 33.7 mm, a wall thickness of 3.2 ffn and an equivalent elastic modulus of 14-8 GN/rn 2- The tubular web members had an outside diameter of 33.7 mm, a wall thickness of 3.2 mm and an equivalent elastic modulus of 150 GN/M2 (from, Hanaor, et a], 1980).

139

200-0

r--l 2

M150-0 (a 0 J

c L G)IOO-o 4. ) x

ui

0

Theoretical No. 3

r

Experimental No. 3

Experimental No. 4

Theoretical No. 4

20-0 40-0 60-0 80-0

Central Deflection [mm]

Fiqure 3.34. Load-Deflection Relationships For Square-On-Diagonal T_ruýs -Wiitfý_And Without Force-Limiting Devices. Tfie di ag r am

Th-ows the load-defle ion relationship' for the steel space truss both with and without force-] imiting dev ices . Eight hydraulic devices were fitted to top chord compression members for test number three. These devices were removed for test number four and the compression members were allowed to buckle (from, Hanaor, et a], 1980).

140

shows the experimental load displacement relationship oýbtalned for tests three and four. These relationships shown in Figure 3.34,

also give an indication of the influence of including the force-limiting devices in the structure. It is apparent that the inclusion of the force-limiting devices has had a slight detrimental effect on the behaviour of the structure. With the devices present, the experimental load displacement behaviour has become erratic and deviates widely from the theoretical behaviour.

The investigation of Hanaor, et a], (1980) have highlighted the importance of obtaining a steady and consistent response from the force-limiting devices. The hydraulic system used in the first

and last group of tests, exhibited large fluctuations in load

capacity which dominated and adversely affected the experimental tests. In their investigations Hanaor, et al, used three different

space trusses for a total of ten non-]! near tests. It is assumed that a] I the tensile members which have yielded, and compression members which have buckled are replaced after each test. However, this can only be achieved for all the failed members if the forces

are monitored in the majority of elements in the structure. The large discrepancy between theoretical and experimental behaviour

obtained in the investigation, may be a direct result of using one structure for several tests, where each non-linear investigation involves a completely different collapse sequence.

141

CHAPTER 4

NOVEL FORCE-LIMITING DEVICE

INTRODUCTION

One of the recurring problems arising from the use of force-limiting devices, is the general unsuitability and lack of control of their load-displacement characteristics. Several of the devices considered in the preceeding chapter, showed either large fluctuations in their working limit loads, or a high initial load

occurring before the load decreased and stabilised at the working I imit load. In an attempt to overcome these problems, a novel compression member has been designed to act both as a 'soft'

member, with reduced axial stiffness, and as a load-limiting device. The initial concept of a 'soft' member, with reduced axial stiffness was proposed by Constrado (The Steel Construction Institute). The aim of the present work was to ascertain if improvement could be made in the force distribution occurring within a double-layer space truss, by replacing selected compression members with a 'soft' member. These initial concepts, have been investigated and the role of the 'soft' member extended to incorporate a load-limiting potential.

Mode Of Operation Of Soft Members

Details of a full size soft member are shown in Figure 4.1. The compression member consists of two square hollow section tubes

and four rectangular strips. The two hollow section tubes

are proportioned so that the smaller tube plus strips just fit inside the large tube.

Figure 4.2 shows the load-strain relationship obtained for the

soft member loaded in compression. Under a compression load the

soft member initially behaves elastically, but with a reduced elastic stiffness (A to B, Figure 4.2). At a pre-determined compression load acting on the soft member, the four middle strips yield in tension and provide a load plateau for the complete soft member (B to C, Figure 4.2). At this stage, the member is only capable of supporting a constant compression force. Under this

steady compression, yielding of the middle strip will continue until the outer and inner tubes mate simultaneously at both ends of the soft member, represented by point C in Figure 4.2. When this

142

40

A

0

N-

0i

L -1

Lv 0I '-I

INNER TUBE

PLUG WELDED THROUGH OUTER TUBE TO MIDDLE STRIP

OUTER TUBE

MIDDLE STRIPS

INNER TUBE

5mm FILLET WELD

MIDDLE STRIP

BOTTOM DETAIL

60 POSITION OF TOP OF LOWER END BLOCK SEE FIGURE 4.9

6Ox6Ox4 S. H. S. Notes 25x5 STRIP 1/ ALL DIMENSIONS IN MILLIMETRES.

40X40X4 S. H. S. 2/ GRADE 43c STRUCTURAL STEEL.

SECTION A-A

SOFT MEMBER SM5

Figure 4.1. Soft Member SM5. The Figure shows fabrication details of the full size soft member SM5. End blocks shoun in Figure 4.9 were f itted into the member at both ends and allowed a total vertical movement between the inner and outer tubes of 40. Omm. The pin-ended soft member was tested in compression under displacement control at an initial strain rate of 0.006% per minute. This rate was continued until a total strain of 4% had occurred in the middle strips whereupon the strain rate was increased to 0.06% per minute.

LO 04 r-

0 04

0 Cl)

0 Cl)

0 C11

TOP OETAIL A

143

25

i 40

500

400 0 z

Ld 300 z

200 PI

ED -i

100

PERCENTAGE STRAIN

SOFT MEMBER No, SM5

Figure 4.2. Compr ssive Load-Strain Relationship For Soft Member SM5. The Figure shows the experimental load-strain relationship exhibited by

soft member SM5 tested in compression. The member initially beha,,, -es elastically but with a reduced elastic stiffness (points A to B). At a pre-set compression load acting on the member, the four middle strips yield in tension and provide a load plateau for the complete soft member (points B to C). Yielding of the middle strips continues until the

outer and inner tubes mate simultaneously at both ends of the soft member (point C). When this has occurred, the stiffness of the soft member increases and the compression member acts as a pre-stressed column which is capable of supporting a further increase in load (points C to D). At point D the column buckles exhibiting a 'brittle type' strut buckling path.

144

10

has occurred, the stiffness of the soft member increases and the

compression member acts as a pre-stressed column which is capable of supporting a further increase in compression loading (C to 0, Figure 4.2). At point D, the column buckles exhibiting a 'brittle type' strut buckling path.

The characteristics of the soft member shown in Figure 4.2, can be significantly altered by varying both the length and cross-sectional area of the four steel strips. The initial

stiffness of the systei can be increased by shortening the length

of the four strips, and the magnitude of the load plateau can be

enhanced by increasing the cross-sectional area of the strips. In

addition, the length of the load plateau can be controlled by

varying the end distance occurring between the inner and outer square tubes. Figure 4.3 shows the tensile load-strain behaviour

obtained from the soft member shown in Figure 4.6. If the soft

member is loaded in tension, then the four steel strips are

subjected to compression forces, and both the inner and outer square tubes are stressed in tension. If the gap between the

strips and tubes is very small, then the steel strips will yield and squash in compression. If this is not the case, the strips

will buckle and deflect until they are restrained by the inner and

outer tubes, whereupon the member will carry additonal load, until the strips yield in compression and deform plastically. When the

strips have yielded, the soft member will carry an almost constant tensile load until the strip material strain hardens, and the

member supports an additional tensile load. The tensile load

carried by the member will increase, until either the end welds fall, or the inner or outer square tubes yield, and finally rupture in tension.

Several improvements to the type of soft member shown in

Figure 4.1 have been made. The relative maximum load capacity of the device was enhanced, by changing the two square tubes to round tubes, and more significantly, by replacing the four separate steel strips also by a round steel tube. The three tubes were fitted

closely one inside the other, and the assembly was fabricated using a similar procedure to that adopted for the square soft members. These triple-tube soft members have a simple mode of operation, but the ultimate capacity of the novel triple-tube compression member is difficult to assess, due to the inderdependence of the three tubes.

145

500

400

300

z 1--f 200

-J 100

5

PERCENTAGE STRAIN

SOFT MEMBER No, SM3 Figure 4.3. Tensile Load-Strain Relationship For Soft Member SM3. The Figure shows the experimental load-strain relationship exhibited by soft member SM3 tested in tension. When the soft member is loaded in tension the four middle strips are strained in compression. The middle strips are restrained from buckling about their weak axis by both the outer and inner tubes and an increasing tensile load applied to the soft member causes the strips to yield in compression and deform plastically. This behaviour results in the short load plateau shown in the figure. The load plateau was curtailed when the four strips buckled about their strong axis with a small displacement parallel to the tube walls. This caused additional compressive strain in the concave edges of the SUPS and the onset of strain hardening in the material. The soft member supported a total tensile load of 462KN before the test was halted to prevent damage occurring to the test machine.

146

Theoretical Behaviour And Ultimate Capacity Of Triple-Tube Soft Members

The theoretical response of a perfect soft member supporting a compressive load, can be investigated by considering the stiffness of the individual tubes and their joint interaction. Figures 4.4 A to F show the changes in stress occurring within the three tubes

of the soft member as the external compressive load is increased.

Before the soft member has closed, each of the three tubes will

carry a force equal in magnitude to the external load. This will result in a. compressive stress in the inner and outer tubes, and a tensile stress in the middle tube (Figure MA). As the external load is increased, the force carried by each of the three tubes

will increase by the same amount. Provided that the compression force in the inner and outer tubes is always insufficient to cause

premature elastic buckling of both of these members, the tensile

force in the middle tube will continue to increase ]! nearly. When

the tensile force acting on the middle tube is sufficient to cause this tube to yield, plastic flow occurs within this member and the

soft member will not support any increase in external load (Figure

4.4B). The middle tube continues to yield until the three tubes

close up and form a pre-stressed triple tube compression member. As the external load is increased, the force within each of the three tubes now alters. The middle tube unloads elastically, while the compressive stress in the inner and outer tubes increases (Figure 4.40. A further increase in the external loading will cause a corresponding increase in the compressive stress in the inner and outer tubes, and consequently increase their tendency to buckle. However, because the three tubes are in very close

proximity to each other, premature elastic buckling of either the inner or outer tubes will not occur due- to the restraint provided by the middle tube still in tension. As the load is further

increased, the compressive stress in the inner and outer tubes will

approach yield and the middle tube will continue to unload

elastically. If perfect elastic-plastic material behaviour is

assumed, then when the inner and outer tubes yield in compression they are unable to support any further increase in external load (Figure 4.4D). In addition, when the inner and outer tubes have

yielded, the bending stiffness of the triple column will have decreased to a value equal to the bending stiffness of the middle tube only. Any further increase in the external load must now be

147

Outer tube riddle tube lInner tube

(A)

Tension

Compression

(B)

Tension

Compression

(C)

Tension

compression

(B)

Tension

Compression

(E)

INCREASING LCAD 13N SCFT MEMBER

Elastic with

-city reduced stiffness

C13NSTANT LOAD ON SOFT MEMBER

Yield of middle tube Zrro stiffness

INCREASlNG L13AD

Soft member- closed Middle tube unloading Normal stiffness.

I-edy

INCREASING L13AD I 13N S13FT MEMBER

CFV

II

(ýf

Inner and outer 'tubes at yletcL

imuta TUE£

MIDDLB TUBC

Oma ""Z

railure of the soft Comprevislon member with the d'y elastic buckling

of the mldcftt tube IDEAUSED SOFT MEMBER (CLOSED)

(F)

Fagurt of the soft member with the CompresgillOn CIV ptastic buckling

1

1, of the midero tube

jOuter tube riddle tube Inner tub

Stress Diagrams Showing The Theoretical Response Of-A Perfect Soft Member Supporting An Axial Compression Load. 'I'he diagram shows the changes in stress occurring within the three tubes of the soft member as the external compression load is increased. Initially the member behaviour is elastic but with a reduced elastic stiffness (Figure 4.4A). An increase in compression load on the soft member causes the middle tube to yield in tension. Ubile the middle tube Is yielding the soft member cannot support any further increase in compression loading (Figure 4.4n). Yielding of the middle tube stops when the soft member closes up to form a triple tube compression member. A further increase in the compression load on the soft member increase the compressive stress in both the inner and outer tubes and causes the middle tube to unload elastically (Figure 4.4C). Additional load causes both the inner and outer tubes to yield in compression and a further decrease in the tensile stress in the middle tube (Figure 4.4D). If the slenderness ratio of the middle tube is sufficiently large to allow this member to fail by elastic buckling, collapse of the complete soft member will occur when the stress in the middle tube reaches the elastic critical flexural buckling value for this component member (Figure 4.4E). If the slenderness ratio of the middle tube Is less than its transition slenderness ratio# the middle tube does not buckle elastically and collapse of the soft member occurs when the stress in the middle tube reaches the yield stress of the tube material (Figure 4.4F).

I

INNER Tust

MIDDLE TUB&

. OmR TUB&

IDEAUSED SOFr MEMBER (OPEM

LOAD

LOAD

148

supported only by the middle tube. Consequently, the stress in the

middle tube will change from tensile to compressive, and will increase until the tube becomes unstable, and the middle portion of the tube begins to displace horizontally. This horizontal displacement will occur with buckling at the Euler buckling load, if the middle tube falls by elastic buckling, or will occur at the tangent-modulus load if the tube is prone to failure by

plastic buckling. If elastic buckling of the middle tube occurs, the bending stiffness of the member has been decreased to zero, and

all lateral restraint provided to both the inner and outer tubes by the middle tube vanishes, causing the complete soft member to buckle (Figure ME). If the middle member falls by plastic buckling it will support a load greater than the tangent modulus load but less than the double-modulus load. However, buckling of the middle tube will still cause buckling of the complete soft

member (Figure 4AF). Consequently, the buckling load of the

perfect soft member will be controlled by the buckling load of the

middle tube. If the middle tube fails by elastic buckling, a

perfect soft member will fall at a theoretical load, equal to the

s um of the squash loads of the inner and outer tubes, plus the Euler critical buckling load of the middle member. However, if the

middle tube falls by plastic buckling a perfect soft member will fall at a theoretical load equal to the sum of the plastic buckling load of all three tubes.

If perfect elastic-plastic material behaviour is not assumed, then material strain hardening will allow the inner and outer tubes to sustain a compressive stress greater than the yield stress,

provided the stress in the middle tube remains below Its critical buck] Ing value. In addition, because the middle tube has been

extensively strained in tension before being stressed in

compression, both the compressive yield stress and the member

stiffness in compression will be decreased as a direct result of the Bauschinger effect. These small decreases in compression stiffness and yield stress, will decrease both the elastic and

plastic buckling loads of the middle tube, which will in turn

reduce the collapse load of the complete soft member.

The presence of imperfections will also decrease the collapse load of the soft member. If both the inner and outer tubes yield in compression before the middle member fails, It will be imperfections occurring within the middle member, which will have

149

the dominating influence on the collapse load of the entire soft member.

EXPERIMENTAL

An experimental investigation has been undertaken to assess the behaviour of soft members loaded in both tension and

compression. Accurate and reliable values are required for the

initial stiffness, yield load, collapse load and the post-buckled

response of the soft member. This data was used to compare the

theoretical and experimental behaviour of the soft member in

addition to obtaining the theoretical response of space trusses incorporating soft members.

Several groups of soft members have been tested. The

load-displacement behaviour has been obtained for. both model soft

members and a limited number of full size soft members. In

addition, several small diameter tubes have been tested along with test coupons cut from each steel tube and strip.

Full Size Soft Members

Six ful I size soft members have been fabricated and tested.

Five have been loaded in compression and only one member tested in

tension. During this investigation modifications were made to improve the behaviour of the soft member. Changes were made to both the connection details and also to the lengths of the individual components.

Fabrication of members

All six of the soft members were fabricated from grade 43C

structural steel (BS4360,1986). This grade of steel was chosen because of its high ductility and good weldability. The f irst

three members SM1, SM2 and SM3 were fabricated from one six metre length of 100 x 100 x4 mm thick square hollow sections, one six

metre length of 80 x 80 x5 qrn thick square hollow section and four, eight metre lengths of 50 x5 mm thick flat steel strips. The-second group of soft members SM4, SM5, and SM6 were fabricated

from one six metre length of 60 x 60 x4 mm thick square hollow

section, one six metre length of 40 x 40 x4 mm thick square hollow

section and four, eight metre lengths of 25 x5 an thick flat steel strips. All of the square hollow sections were hot finished seamed

150

tubes supplied to meet the dimensional tolerances given in BS4848 Part 2, (1975).

Figure 4.5 shows the fabrication details of soft member SM1. Fabrication details of members SM2 and SM5 are shown in Figures 4.6

and 4.1 respectively. The fabrication of each of the soft members was undertaken by first cutting and then milling to the correct length the required steel tube and strip components. The four

strips were then mig (metal Inert gas) welded, -

one each side, to

the bottom of the smaller square hollow section. This assembly was carefully slid inside the larger square hollow section and the four

strips were then attached, one on each of the four sides, to the

top of the outside tube. For the first soft member, SM1, the top

connection, made between the tube and strips, was formed using a combination of high strength bolts and plug welds (Figure 4.7).

This joint proved to be unnecessarily complex so the top joints on the remaining members were formed using two plug weýlds for each

strip.

During fabrication of the soft members the width and thickness

of each of the strips was carefully measured at three different

cross-sections. In addition these measurements were taken at both

ends of the tubes but only the outside dimensions and not the wall thickness could be measured along the tube length. Also the

curvature and twist of the completed soft members were carefully measured using a straight edge and depth gauge. The straight edge was placed adjacent to the soft member and raised from the tube

side by two gauge blocks positioned at each end of the member. Measurements were made along the tube from the top of the straight edge to the tube side using a vernier depth gauge.

Two test coupons were taken from each of the four square hollow sections and the eight flat strips. These test coupons were

used to determine the material tensile properties and were cut and

machined to comply with the requirements given in BS18, Parts 2 and 4, (1971).

Equipment

Loading machine: A SATEC screw-type universal testing machine controlled by a SATEC Control Unit was used for each of the six tests - The machine has a capacity of 500 kN and will allow the

cross-head to be moved at a constant rate of displacement. Before

any tests were undertaken the accuracy of the load cell was 151

!4 80 vi

0 0

F A

cli

FILLET WELD-

ER TUBE II I

-111 qo. ' 11mm DIA iji

LES EACH FACE, -1 I 10mm DIA C/S JS GRADE 8.8--

SEE DETAIL 1 -41

rER TUBE

EE DETAIL 2

45

50

OETAIL I

50 x5 mm

91

Ii '1 Ii

[I

r1

ILI

0

100

1

1/1 1111 77J

, oo(- 100 x 100 x4 mm S. H. S. 80 "i

4 No. 50x5mm STRIPS [1-

80 x 80 x 5mm S. H. S. DETAIL 2

SECTION A-A

PLUG WELD THROUGH OUTER TUBE TO MIDDLE STRIP

INNER TUBE

5mm FILLET WELD

MIDDLE STRIP

Figure 4.5. Soft Member SM1. The Figure shows the fabrication details of soft member SM1. This member was the first soft member to be tested and was not designed to support additional compression loading after the middle strips had yielded in tension. The pin-ended member was tested under displacement control at a cross-head movement of O. 1mm per minute. This rate was increased to 1.0mrn per minute after a total of 4% strain had occurred in the middle strips.

L 15 j- 20 1 151 r- -- -T--r---l

0

I I ii I I ii I, I'I 1I I

j. J' ill

\

152

ý 80

80

1 A

C0

1: jI

--I II

II II

0 0

cli 0 '4

i TOP OETAIL

A

(0

T

0 Lf)

INNER TUBE

PLUG WELDED THROUGH OUTER TUBE TO MIDDLE STR; P

OUTER TUBE

MIDDLE STRIPS

INNER TUBE

5mm FILLET WELD

MIDDLE STRIP

BOTTOM DETAIL

POSITION OF TOP OF LOWER 00 END BLOCK SEE FIGURE 4.9

Z- 100x100x4 S H. S. Notes

, 50X5 STRIPS 1/ ALL DIMENSIONS IN MILLIMETRES.

80x8Ox5 S. H. S. 2/ GRADE 43c STRUCTURAL STEEL.

SECTION A-A

SOFT MEMBER SNA2

Figure 4.6. Soft Member SM2. The Figure shows the fabrication details

of the full size soft member SM2. The member was proportioned to allow 150.0mm of relative vertical movement to occur between the inner and outer tubes. The pin-ended soft member was tested in compression under displacement control at an initial strain rate of 0.006% per minute. This rate wa-s continued until a total strain of 4% had occurred in the middle strips whereupon the strain rate was increased to 0.06% per minute.

153

50

i 80 "1

Figure 4.7. Bolted Connection Of Soft Member SMI,

154

certified by R. D. P. - Electronics Ltd., to comply with the British Standards Institution (B. S. I) Grade 1.0 requirements given in BS 1610, (1985).

Displacement measurement: To obtain axial deformation of the soft

members six I inear variable differential transformer tranducers

were used. To determine the strain in the outside tube three

R. D. P. D5/2000 transducers were attached to three different sides

of the member. Each of these transducers had a working range of ± 50 mm and were cert if1 ed by R. 0. P. -E I ectron 1 cs Ltd. , to h ave a linearity better than ± 0.15% of their working range. The

remaining three transducers were R. D. P. D5/IOOOOC transducers which had a working range of ± 250 mm. These transducers were certified by R. D. P. -Electronics Ltd., to have a linearity better than ± 0.25%

of their working range. Two of the D5/10000C transducers were

carefully arranged to measure the displacement of the middle tube

and strips. Steel wires were threaded through opposite sides of the soft members adjacent to the steel strips and attached to the

bottom of the middle tube. Because the two steel wires passing through the soft member rested on the top of the outside tube (Figure 4.8) the readings obtained from these two transducers were

added to the displacement of the outside tube to obtain the correct

values of displacements for the middle tube and strips. The third

D5/10000C transducer was used to measure the overall displacement

of the soft member. This was achieved by attaching the transducer

body to a collar at the bottom of the member and the associated ferromagnetic plunger to a collar at the top of the member. Figure

ý4.8 shows the transducers mounted on the second soft member -tested. The same transducers were used for each of the soft member test.

At the start of each test each transducer was carefully

cal ibrated over its full working range. The transducers were calibrated with their respective amplifiers using accurately

measured slip gauges. The complete procedure was monitored using a simple computer program. The transducer displacements were scanned

at each pre-set Interval and the best straight line was fitted

through the data points using the method of least squares to obtain the millimetre - voltage calibration coefficient.

Temperature measurement: The temperature was measured continuously throughout each test using a Digitron thermometer. Al I of the

155

i,.

M,

iq ur e_4_. 8 F(i Iiz, 2 So fL ýle: nber SM2

156

experimental work, reported in this investigation including load

cell and transducer calibrations were undertaken at an ambient temperature of 20 0±3 'C. Before any experimentation was undertaken, all of the test equipment was kept within this temperature range for at least one hour before the test commenced. This minimum period allowed for near stable conditions to be

achieved and mininised temperature induced drift of the amplifiers and power supply.

Computer and data-logger: A Spectra-xb data-logger made by Interco] Measurements and Control Systems was used to monitor and record data obtained from all of the experimental work reported in

this investigation. The load cell and transducers fed continuous signals to the data logger via their amplifiers and these signals were scanned at pre-set time intervals under the control of a 16-bit microcomputer. The microcomputer is configured around the Digital Equipment Corporation LSI-11/2 micro-processor. Software

was written in the BASIC language which enabled the operator to

select the number and frequency with which the measuring devices

were read. This data was stored on floppy discs and a subroutine was incorporated into the software which enabled data files to be

opened and closed and the discs changed during a test. This was necessary to prevent the possible loss of test data resulting from

a power failure or more frequently a power surge. Interruption to the power source erased all data stored in an open data file. The data stored on discs was re-read into a plotting program which converted voltages from the load cell and transducers into load and displacement values respectively. These results were then plotted on a Hewlett Packard T221A Graphics Plotter driven by the Digital

microcomputer.

Test procedure

An endeavour was made to test f1 ve of the fuIIs1 ze sof t

members as pinned-ended compression members. A perfect pinned-end condition should not give any restraint to both column-end rotation and warping. Any restraint which is present will influence the

effective slenderness ratio and ultimate load of the column. Figure 4.9 shows details of the end fixings used to provide a pinned-end for the soft members. The end blocks shown in Figure 4.9 fitted tightly into the tube ends and the tapered holo in the

end blocks fitted over a rounded plinth projecting from each of the

157

, iare Hallow =tIon

E E LO VM!

E E 0 (D

E E

c4

Z CY 4 U)

1-52 (32) mm. -4

E 0

Figure 4.9. Soft Member End Fixing Details. 7he diagram gives details of the end fixings used to provide pinned-end conditions for the full size soft member compression tests. The end blocks were H=hined out of toughened steel and were placed in the test machine on top of rounded plinths projecting from each of the cross-heads of the machine. The dimensions shown on the figure in brackets correspond to the end block which fits into the end of the inner tube of the soft member.

158

loomml

cross-heads of the testing machine.

The soft member test specimens were positioned and a] igned in the test machine with great care. The centre line of the outside tube was accurately al igned with the centres of the top and bottom

crossheads. The displacement transducers were fixed to the members by spec fal hangers and were al Igned using a spirit level. The

angle of inclination of the sloping transducers, used to measure the displacement of the inner tube, were also measured. After the

soft members were positioned in the test machine they were loaded

up to approximately 10 kN to ensure bedding In of the end fixings

and correct operation of the data logging and test equipment. Before the test was undertaken the load was removed from the

specimen and the initial readings of the measuring devices taken.

The specimens were tested under displacement control at a

crosshead movement of 0.1 mm per minute until 4% strain had

occurred in the four middle strips. At this stage the cross-head displacement was increased to 1 mn per minute for the remainder of the test. These cross-head displacements correspond to a constant strain rate in the specimen of approximately 0.006% per minute and 0.06% per minute respectively. This very low strain rate was

chosen so as to approach quasi-static test conditions which would form a basis for comparison of all of the experimental work reported in this investigation.

Model Soft Members

Twenty-four model soft members have been fabricated and tested. Sixteen members have been tested in compression and eight members tested in tension. To determine both the tensile yield stress and the critical buckling stress of the individual component

parts of the soft members, twenty-six tubes have been tested in

tension and twelve in compression. When the model soft member Is loaded in compression, the middle component, which is sandwiched between the outer and inner tubes,, is stressed in tension. After

the soft member has closed, the middle component unloads

elastically and finally becomes stressed in compression. To assess the influence of initially straining the middle components in tension before loading the component in compression, tests on twenty-seven individual tubular members have been undertaken.

159

Additional tests have also been undertaken, to assess the influence

of strain aging on the soft member. In this investigation several tubes have been initially strained in tension, aged and then restrained in tension.

Fabrication of members

Each of the model soft members were fabricated from three

small diameter, cold-drawn, seamless, annealed steel tubes BS6323, Part 1, (1982). The three tubes had different diameters and wall thicknesses and were carefully proportioned so that they would just fit inside each other as shown in Figure 4.10. The members were fabricated by first cutting and accurately machining the three tubes to the required length. Four small slots were then milled into the middle and outer tubes close to one end. These slots enable the three tubes to be plug welded together using a sequence similar to that previously described in the fabrication of the large soft members.

The design details of the model soft member have slowly evolved in an attempt to improve the behaviour of the device. Figure 4.11 shows the three principal stages in the development of the model soft member. The members shown as type 1 in Figure 4.11 behaved satisfactorily in the compression test machine but

premature rotation at the top of the Inner tube occurred in some of the soft members incorporated into the test model double-layer

space truss structures described in the next chapter. This failure

characteristic was removed by incorporating a small collar at the top of the soft member as shown in the type 2 soft member (Figure 4.11). In some preliminary tests undertaken on both the type 1 and type 2 members the weld at the bottom of the member, connecting the member to the node, constricted the movement of the inner and middle tubes. This difficulty was overcome by modifying the tube lengths and increasing the height of the bottom plug as shown in the type 3 members (Figure 4.11).

During the fabrication of the soft members both the external diameter and the wall thicknesses of all of the component parts were carefully measured. In addition the initial bow of the fabricated soft members was measured using a machined steel surface plate and a vernier depth gauge.

160

Outer Middle

Tube Tube

Figure 4.10

Inner Section Through

Tube Soft Member

Model Soft Member

Fabricated Model Soft Member

161

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Experimental equipment

Both the data logging equipment and microcomputer used to monitor all of the full size soft members tests were also used in the same capacity to monitor the tests undertaken on the model soft members and their component tubes. However, a different test

machine and displacement measurement system was used for the model soft member and component tests.

Loading_machine: A Howden EU500 twin screw drNen testing machine was used for all of the tests on the model soft members and component tubes. The test machine has a maximum capacity of 50.0 kN and was controlled by a Howden E179A Control Unit. Both before

and after the tests were undertaken the linearity of the load cell was certified by R. D. P. -Electronics Ltd. to comply with the British Standards Institution Grade I requirements given in BS1610 (1985).

Displacement measurement: One R. D. P. D5/2000 linear variable differential transformer transducer was used to obtain the total

axial deformation of the test samples. The same transducer, which was used for all of the tests, had a working range of t 50 mm and was certified by R. D. P. -Electronics Ltd., to have a linearity better than t 0.15% of its working range. To determine accurately the strain in the test specimens a R. D. P. DHE 25/50 dual

extensometer was used. The extensometer had a gauge length of 50 mm and incorporated two D5/4OG8 tranducers each with a working range of t1 mm. Before the test program commenced both of the transducers in the extensometer were calibrated by R. D. P. Howden Ltd. and compiled with the Grade C requirements given In BS3846, (1970). Throughout the test program these transducers were regularly calibrated over their full working range of ±lm., n using the data logger and a specially modified micrometer.

Test procedure

Soft member and component tubes: Four separate sets of steel tubes have been used in the fabrication of the soft members. Each of these groups termed white, yellow, red and green consisted of three different diameter six metre long tubes. One length of tube from

each group was used for each of the outer, middle and Inner components of the soft member. Four, type 3 model soft members, (Figure 4.11), were fabricated frorn each of the white and yellow

163

tube groups. All of these eight members were tested to failure in

compression. In addition, four samples were taken from each of the

six metre lengths of tube and two of each of these four samples

were tested to failure in tension and two were tested to failure in

compression. All of the four samples taken from each of the six tubes were cut to the saine length as the corresponding component tubes used in fabrication of the soft members.

A further eight type 3 model soft members also were fabricated

from each of the red and green tube groups. Two tensile specimens

were taken from each of the three tubes used in both groups. The

eight soft members fabricated from the red tube group were tested

to failure in compression while the eight members fabricated from

the green tube group were tested to failure in tension.

All of the model soft members and component tubes tested in

this investigation were first carefully mig welded into recessed

end blocks using four symmetrically positioned 5 mrn fillet welds. Every specimen was accurately aligned in the test machine using a

specially made jig which moved up and down the main frame of the

machine. The extensometer was lightly clamped to the outside of the specimens at mid height and the body of the larger tranducer

was fixed vertically to the top cross-head of the test machine. All of the specimens were tested under displacement control at an initial cross-head movement of 0.2 mil per minute. This displacement was continued until 4% of tensile or compressive strain had occurred in the specimen whereupon the cross-head displacement was increased to 2.0 mm per minute. These cross-head displacements correspond to a strain rate in the specimens of 0.063% and 0.63% per minute respectively.

Pre-strained middle tubes: In order to determine how the critical buckling load of the middle tube of the soft member is affected by

varying amounts of initial tensile pre-strain, test specimens were

prepared from two, nominally identical, six metre lengths of the

middle tube. Twelve tube samples, each 313 mm long, were cut from

the first six metre length of tube. One sample was failed in

tension and three samples were failed in compression. The

remaining eight samples were pre-strained in tension before being

tested to failure in compression. Six of these eight samples were yielded in tension and extended 2.0 mm before the loading was

reversed and the samples failed in compression. This test sequence

164

models the behaviour of the middle member in the type 3 model soft members (Figure 4.11). The two remaining samples cut from the first tube, were extended in tension by 4 an dnd 6 mm respectively before they were failed in compression.

A similar procedure was adopted for the fifteen samples cut from the second six metre length of tube. Three of the fifteen

samples were tested to failure in tension and another three samples were tested to failure in compression. Four samples were failed in

compression after an initial tensile pre-strain of 0.639% resulting from a 2.0 mm extension in tension. The remaining five samples cut from the second tube were given initial tensile pre-strains resulting from extensions of 4,6,8,10 and 12 mm, before the loading was reversed and the samples failed in compression.

All of these samples were welded onto end blocks and tested

using an identical procedure to that outlined above for the model soft member and component tests.

Strain aging: Additional tests on samples cut from another six metre length of middle tube have been undertaken to assess changes in yield stress and ductility due to strain aging. Eleven samples each 313 mm long were cut from the same tube and two of the samples were tested to failure in tension. The remaining nine samples were initially strained in tension, each to a different predetermined level, removed from the test machine and iramersed in boiling water for one hour. After each sample was boiled it was allowed to cool at room temperature for ten minutes before being returned to the test machine and failed in tension. Each of these samples were also welded onto end blocks before they were tested In tension. However, these samples were tested at the lower strain rate of 0.063% per minute until the strain in the specimens had exceeded 13% whereupon the strain rate was increased ten fold to 0.63% per minute.

RESULTS

Full Size Soft Member Tests

Soft member and component measurements: Table 4.1 gives the

measurements of the individual component parts of the soft member SM5 and Table 4.2 gives the initial out-of-straightness and twist

165

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of the completed soft members SM5 and SM6. The strips used in the fabrication of the members were all wider and thicker than the

specified nominal dimensions whereas the tubes were all slightly

smaller and thinner than specified. These measurements were-made for all of the full size soft members and have been used to compare their theoretical and experimental behaviour.

Material coupon tests: Table 4.3 gives the experimental values

obtained for each of the two tensile test coupons cut from every tube and strip used in the manufacture of the full size soft

members. All of the material tested exhibited a relatively long

plastic plateau and a large amount of ductility before failure.

Large differences In upper yield stress were obtained from some of the pairs of coupons cut from the sarne square hollow tubes, however

these differences did not occur in either the lower yield stress or

ultimate stress values. Both the upper and lower yield stresses

obtained from the tube sections are significantly higher than the

minimum value of 275 N/mm 2 specified for grade 43C structural steel (BS4360,1986). The upper yield stress values obtained from the

tube samples had a mean value of 376.7 N/n, 112 and a standard deviation of 34 N/. mm2. The mean value minus three times tKe

standard deviation gives the minimum specified value for the yield

stress of 275 N/mm 25 indicating that there is a three In one thousand chance of a result falling below the minimum specified

value. The test values for upper yield, lower yield and ultimate stress obtained from the strips, were generally lower than the

results obtained from the square tubes. The mean value for the

upper yield stress was 309 N/MM2 with. a standard deviation of 34

N/mm 2 indicating that only sixty eight percent of the test results

can be expected to be above the minimum specified yield stress of 275 N /Mm 2.

Soft member SM1: The load-strain relationship obtained from the f irst soft member SM1 is shown In Figure 4.12. This member which was tested in compression was designed only for the four

middle strips to yield in tension. The member yielded at a load of 305 kN and 156 mm of movement of the Inner tube, relative to the

outer tube, occurred before the test was halted. The initial

elastic stiffness of the soft member, obtained from the Hnear

portion of the experimental load-displacement relationship, was 69.4 kNImm. This experiment value was less than the theoretical

value of 77.4 kN/mm, which was calculated by estimating an

168

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PERCENTAGE STRAIN

SOFT MEMBER No, SM1

ft 4.12. Load-Strain Relationshi EigLw Soft Member SH1. The Figure shows the load-strain relationship obtained from soft member SM1 tested in compression. The member was tested under displacement control at a cross-head movement of O. 1mm per minute until 4% of strain had occurred in the four middle strips. At this stage the cross-head movement was increased to 1.0mm per minute. The member yielded under a compression force of 305KN and 156mm of movement of the inner tuberelative to the outer tube, occurred before the test was halted.

170

10

effective length for each individual component of the soft member and did not take into consideration the flexibility of the long

bolted connection.

Soft member SM2: The second soft member SM2 was designed to

close-up to form a compound compression member after 150 mm of

relative movement had occurred between the inner and outer tubes.

The experimental load-strain relationship obtained from testing the

soft member in compression is shown in Figure 4.13. The soft

member yielded at a load of 308.4 kN and exhibited an initial

elastic stiffness of 59.4 kN/mm compared with a theoretical

stiffness of 63.4 kN/mm. After a total of 150 an of extension had

occurred in the middle strips the soft member closed to form a

compound column. This was accompanied by both an increase in

stiffness and load carrying capacity of the member. The test was halted prematurely before the member failed at a compression load

just below the maximurn capacity of the test machine.

Soft member SM3: The third soft member SM3 was nominally identical to the second soft member SM2. Figure 4.3 shows the

load-percentage strain relationship obtained from testing member SM3 in tension. The soft member had a physical elastic stiffness

of 63.0 kN/mm compared with a theoretical stiffness of 63.4 kN/mm.

When the soft member is loaded in tension the four steel strips inside the soft member are in compression. Due to the closefit of these strips sandwiched between the two tubes, the strips should

yield in compression and deform plastically when an increasing

tensile load is applied to the soft member. This behaviour was

exhibited during the early part of the test and a small plastic load plateau is evident in the load-strain relationship shown in

Figure 4.3. The load plateau was curtailed when the four strips

buckled laterally with a small displacement parallel to the tube

walls. This caused additional compressive strain in the concave

edges of the strips and the onset of strain hardening in the

material. The soft member supported a total tensile load of 462 kN

before the test was stopped in order to prevent damage occurring to

the test mach'ine.

Soft member SM4: The last three soft members tested were designed to be more slender than the first three soft members. This was undertaken so that an assessment could be made of the

critical buckling load of the soft members. The fourth soft member to be tested SM4, had an overall length of 2.0 metres and provision

171

500

400

0 z C3

300

z C3 -j

z 200 S-4 p . CE

100

PERCENTAGE STRAIN

SOFT MEMBER No, SM2

Figure 4.13. Load-Strain Relationship For Soft Member SM2. The figure shows the load-strain relationship obtained from testing soft member SM2 in compression. The member was tested under displacement control at the same rate as soft member SM1. The member SM2 yielded at a load of 308AHN and exhibited an initial elastic stiffness of 59.4KN/mm. The soft member closed up and supported an additional compression load after a total of 150mm of relative movement had occurred between the inner and outer tubes.

172

5 10

was made for 150 mm of movement to occur before the soft member

closed. The experimental load-strain relationship obtained from

testing the member in compression is shown in Figure 4.14. The

member failed by premature buckling of the top of the inner tube

after only 50.8 mm of relative movement had occurred between the

inner and outer tubes. The soft member failed under a compression load of 189.5 kN which caused an average compressive stress in the

inner tube of 336 N/mm2-. This short portion of the inner tube

projecting 98.2 mm beyond the outer tube should have been capable

of supporting a compression load of 220 kN which would just have

caused the tube to yield in compression. Due to the fabricating

tolerances in this soft member it is possible for the inner tube to

move 1.5 mm out of line with the outer tube, which is equivalent to

a large imperfection occurring in the top portion of the inner

tube. A small lateral displacement of the inner tube can occur if

one of the four inner strips has a higher yield stress than the

other three strips which under load will cause an unsymmetrical force to act on one side of the inner tube. In soft member SM4 the

average yield stress of one of the four inner strips was 27.5%

greater than the average yield stress of the remaining three strips

and this may have resulted in the premature buckling of the top of

the inner tube and overall failure of the soft member.

Soft member SM5 and SM6: The soft members SM5 and SM6 were nominally identical and were proportioned to allow 40 mm of movement to occur before closing up to form a compound compression

member. Figures 4.15 and 4.16 show the respective load-strain

relationship obtained from testing the members SM5 and SM6 in

compression. Member SM5 yielded initially at a load of 152.9 kN

and finally buckled at a load of 331.6 kN. The member had an initial physical elastic stiffness of 27.3 kN/mm and a theoretical initial elastic stiffness of 28.0 kN/mm. Member SM6 yielded at a load of 147.4 kN and buckled at an ultimate load of 298.9 kN. The

experimental initial elastic stiffness of the member was 27.5 kN/Mm, almost identical to the experimental initial elastic

stiffness of member SM5.

Theoretical collapse load of soft members SM5 and SM6: The theoretical collapse load of members SM5 and SM6 can be obtained by

considering both the behaviour of the Individual components of the

soft member and their joint interaction with each other. An

173

500

400 0 z C3 F-

L'i 300 z C3 -j

z 1--f 200

C3 -i

100

5

PERCENTAGE STRAIN

SOFT MEMBER No, SM4

Figure 4.14. Load-Strain Relationship For Soft Member SM4. The Figure shows the load-strain relationship obtained from testing soft member SM in compression. 7be member was tested under displacement control at the same rate as soft member SM1. Soft member SM4 was more slender than the first three soft members tested and was designed to assess the flexural critical buckling load of the member. The member failed prematurely by buckling of the top of the inner tube after only 50.8mm of relative movement had occurred between the inner and outer tubes.

174

0 z 13

Soo Ld z D

z 200 I. -I

D -j

100

PERCENTAGE STRAIN

SOFT MEMBER No, SM5 Fimwe 4.15. Load-Strain Relationship-For Soft Member SM5.

0 z C3

300 Lij z C3 -i

.4 200

C3 -j

100

PERCENTAGE STRAIN

SOFT MEMBER No, SM6 FiLfUre 4.16. Inad-Strain Relationship For Soft member SM6. The Figures show the load-strain relationship obtained from testing soft members SM5 and SM6 in compression. These two members were nominally identical and were proportioned to allow 40mm of movement to occur before closing up. The members were tested under displacement control at a cross-head movement of O. 1mm per minute until 4% of strain had occurred in the four middle strips. At this stage the cross-head movement was increased to I. Omm per minute. Member SM5 yielded at a load of 152.91N and buckled at a load of 331.6KN. The member had an initial elastic stiffness of 27.3HN/mm. Soft Member SM6 yielded at a load of 147AHN and buckled at an ultimate load of 298.8KN. The member exhibited an initial elastic stiffness of 27.5HN/mm.

175

10

5 10

estimate . of the individual buckl ing load of imperfect tubular

members, may be obtained by representing the initial bow occurring in the unloaded pin-ended member by the relationship:

x Wo = aj sin-IL .... (4.1), (Ayrton, et a], 1886),

where Wo represents the transverse displacement of the unloaded

member at po int X and a, represents the maximum central displacement of the unloaded member (Figure 4.17).

p

L

wo 1

p

Initial Untoaded Position

Loaded Position

Figure 4.17. Pin-Ended Colurrin With Initial Deformation. 7be f igure Aows the parameters used in the derivation of the Perry-Robertson formula. 7he dashed line indicates the position of the unloaded column and WO represents the transverse displacement of the unloaded member at point X- The maximum central displacement of the unloaded member is given by ai.

.L (1980), the Following the procedure outlined by Allen, I,

transverse displacement of the loaded member is given by:

lix (4.2)9 31

where aL

p .......... 'Fe

The maximum stress in the loaded compression member will be at the

centre of the strut and is the sum of both the axial and bending

stresses:

"imax =P+ PNI Z 7 -1

176

where A is the cross-sectional area of the compression member, I is the second moment of area of the compression member and Z is the distance of the centrold of the section to the extreme fibre on the concave side.

Substituting I= Ar2 into equation 4.4 gives:

"max = cr 1+ (4.5),

Oel

where a, Z

.... (4.6)9 r2

r= radius of gyration of the member, a is the mean stress and 17e is the Euler buckling stress of the member.

As the external load acting on the column is increased the

mean stress in the column also increases. However, an Increase In the mean stress is accompanied by a larger increase in the maximum stress given by equation'4.5. When the maximum stress in the

column reaches the yield stress of the column material a zone of plasticity occurs on the concave side at the mid-height cross-section of the column and the member loses stiffness. This decrease in column stiffness indicates the onset of collapse and consequently the value of load P at which the maximum stress equals the yield stress may be taken as a lower bound to the collapse load PC.

Substituting amax '2 ay in equation 4.5 yields a quadratic equation in terms of a,

(ae-a)(ay-a) = naea ..........

The smaller root of equation 4.7 is:

ae cly 2*...

(4.8) a+ (a -aeay)"

where a= ay ' (n + 1) ce

.»0. (4.9) 2

177

Equations 4.1 to 4.9 can be used in conjunction with the initial measurements recording the out-of-straightness of the soft members, to estimate the critical buck] ing loads of both the

components of the member and the complete soft member itself.

Estimate of the critical flexural buckling load of members SM5 and SM6.

Average values measured for the width and thickness of the

component tubes have been used to calculate the section properties.

For the outer tube of soft member SM5 the section properties are as follows:

average tube width = 59.67 mm

average tube thickness 3.94 mm

second moment of area 1 457287.41 rrrn4

cross-sectional area A 879.13 MM2

radius of gyration r 22.81 mrn

slenderness ratio x L/r = 76.29

average yield stress (from test coupons) = 370 N/MM2

average elastic modulus (from test coupons) = 205 kN/mm 2

From table 4.2 the maximum measured value for the out-of- straightness of soft member SM5 is 1.21 mm occurring 1044 rn-n up from the bottom of the member. Representing the Initial lateral deflection measurements for the soft members by the sin curve given in equation 4.1, gives a value for a,, of 1.27 mm.

From equation 4.6

1.27 x 59.67 = 0.0729

2x 22.812

The Euler buckling stress ae is given by:

7r 2E_ 7r2 x 2.05 X 105 cro == 347.6N/MM2

79.2.79

178

From equation 4.9

370 + (1 + 0.0729) 347.6 -, 1.48N/MM2

2

From equation 4.8

a= 347.6 x 370

371.48 + (371.482 - 347.6 x 370ý; '

274.60 N/MM2

Repeating the steps outlined above, the critical buckling

stress for the inner tube of member SM5 was calculated to be 143.37 2 N/mm . Consequently if the inner tube of the soft member was

unrestrained by both the strips and outer tube the tube would buckle under an applied compressive force of 80.75 kN. However, the middle tube is restrained against buckling by the other components of the soft member and consequently the buckling of the

complete soft member will occur when the outer tube becomes

critical.

Figure 4.18 A to D shows how the stresses In each of the

component parts of soft member SM5 change under an increasing

external compressive force. Before the soft member has closed a constant equal force is acting on the outer tube, Inner tube and the middle strips. The force In each of these components is equal to the externally applied force (Figure 4.18A). When the strips have just yielded the average stress In each component Is shown in Figure 4.18B. These stress levels are maintained through the

yielding process until the soft member closes. When the member has

closed each component Is subjected to the same Imposed displacement

causing equal changes in stress to occur in the inner tube, outer tube and strips. Figure 4.18C shows the stresses In the component parts of the soft member after the closed-up member has been

subjected to a displacement of 0.5 mm. At this stage the soft member can carry an additional load until the stress In the outer tube reaches its critical buckling value of 274.0 N/MM2. When this level of stress has been attained in the outer tube, this tube

179

Puter tube_ Middle strip i1nner tube i

Area ITotal Areal Area I 879-13mm2 534-66MM2 563-26MM2

EXTERNAL FORCE ACTING 13N SOFT MEMBER

Tension IE37-03N/mm2

All components Compression Compression elastic

I

113-75N/nim2 177-64N/mm2

Tension 285-96N/MM2

Sof t member just closed. Inner and outer tubes

Cornpressior I Compressior

e ielded St i h 2 173. SIN mM2 P 71-43N av y . elastic. r ps /,, 1

FIGURE

- /

4-185

Tension

227-C35N)

Sof t member closed. A 0.5mm displacement imposed on alt component parts causing a stress change of 50.91N/mm'.

Tension

IBB-27N,

Soft member at ultimate load

Cornpression 1274-SON/mM2

;; ompres 330-33

=ompressior

372*11N/mm2

100-OkN

152-89kN

269-35kN

CRITICAL BUCKLING L13AD

351-95kN

Fioure 4.18. Stress Diagrams For Comment Parts Of Soft Member SM5. The Figure shows the changes in stress which occur in the component parts of soft member SM5 due to an increasing compression load. When the member is supporting a compression load of 100KN all the component parts of the soft member are behaving elastically (Figure 4.18A). The four middle strips and the soft member yield under a load of 152.89XN. This load is maintained at a constant level until the soft member closes up (Figure 4.18B). Once the soft member has closed the device can support additional compression loading (Figure 4.18C). 7he soft member fails at a load of 351.98KN when the outside tube looses stiffness and buckles (Figure 4.18D).

180

loses stiffness and no longer provides any restraint to the strips and inner tube and consequently the complete soft member buckles. The force in each component part of the soft member can be obtained by multiplying the stresses given in Figure 4.18D by the relevant areas of the tubes and strips. The critical buckling load of the

soft member is then obtained by summing up the forces in each of the component parts of the member. From Figure 4.18D the theoretical critical buckling load of the soft member SM5 is 351.95 kN which compares favourablY with a buckling load of 331.58 kN

obtained from the experimental investigation.

Using the s ame procedure as previously outlined, the theoretical critical flexural buckling load for the soft member SM6

was calculated to be 328.17 kN. This value is less than the

estimated theoretical buckling load of mernber SM5 due to the

slightly larger initial bow of member SM6 compared with the initial

bow measured in member SM5 (Table 4.2). The flexural buckling load

of soft member SM6 obtained from the experimental investigation was 298.89 kN.

The critical flexural buckling load of both of the soft members can also be calculated using the column curves given in the British Standard Code of Practice for the structural use of steelwork in buildings BS5950, Part 1, (1985). These c6lum6 curves are also derived using the Perry-Robertson formula and are based on the assumption that the column has an Initial bow of L/1000, and and a parabolic distribution of residual stresses (Young, 1971). Four column curves are presented, each curve depending on the type

of column cross-section and the axis about which buckling can occur. To estimate the critical flexural buckling load for square hollow sections it is permitted to use the most favourable of the four column strength curves. This curve is based on a Perry factor

n given by the expression:

n=0.00la(x-xo) where a is the Robertson constant equal to 2.0

x is the column slenderness ratio

and xo is the limiting slenderness which should be taken as:

]12 2 Xo = 0.2

(ay E (BS5950), Part 1, (1985)

181

Using the tabulated values of the column strength curve given in BS5950 (19853, the critical flexural buckling stress for the

outer tube of the soft member is given as 253.50 N/mm2. This will result in a theoretical tensile stress of 206.37 N/mm2 in each of the four strips and a theoretical compressive stress of 351.01 N/mm 2 in the inner tube at failure of the soft member. This stress distribution leads to a theoretical critical flexural buckling load

of 310.23 kN for both of the soft members SM5 and SM6.

Comparison between theoretical and experimental buckling loads for

soft members SM5 and SM6

Table 4.4 gives a summary of both the theoretical and experimental results obtained for the critical flexural buckling load of the soft members SM5 and SM6. The differences occurring between the theoretical and experimental flexural buckling loads

obtained for the members SM5 and SM6 can be due to several factors. The estimate of the flexural buckling load, allowed for

the initial bow of the column but did not take into consideration the twists of the columns or any residual stress distribution

occurring within the soft member. The method of fabrication of the

soft members requires a relatively large amount of welding to form the internal connections and this may cause a significant residual stress distribution to occur within the member. If the estimate of the flexural buckling load of the soft member Is to be Improved an assessment must be made and account taken of these residual stresses.

The experimental data could also be enhanced by measuring, during the loading process, the lateral deflections occurring at the centre of the members. These displacements can also be used to

estimate the critical buckling load of the soft members (Southwell,

1932). In addition to these extra measurements an Increase in the

scanning rate of the data logger as the member approached the

critical load would have improved the accuracy of the measurement of the maximum load attained by the members.

Model Soft Member Tests

Soft member component tubes in tension: A total of twenty-five component tubes have been tested to failure in tension. Two

samples have been taken from each of the twelve tubes used in the

182

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4-

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ta

CS (1) 0 P

(1) DI

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183

fabrication of the soft members and one additional sample has been

prepared from extra material remaining in the red tube set. Table 4.5 gives the experimental values of upper yield stress, ultimate stress, elastic modulus, plastic and overall percentage elongation, obtained from the tensile tests. Figure 4.19 shows typical

experimental load-end displacement relationships obtained from an inner, middle and outer tube. Both the yield stress and ultimate stress values obtained from all the samples exceeded the minimum requirements of 170N/mm2 and 340N/mm2 respectiv ely, specified for

this particular type, grade and finish of steel tube, CFS3GBK, BS6323, Part 1 (1982). Table 4.5 shows close agreement between the two values of yield stress and ultimate stress obtained from each

pair of samples taken from the same tube. However, there is a large difference in each of these values obtained from different

but nominally identical tubes. All of the tube samples exhibited a large amount of ductility and load plateaux were obtained from all

of the middle and inner tube specimens.

Soft member component tubes in compression: Two test samples have

been taken from each of the inner, middle and outer tubes of the

yellow and white tube groups. Unfortunately, one sample taken from

the middle tube in the white group was damaged during welding and insufficient material was available to replace the specimen. The

experimental values of the critical buckling stress and elastic modulus obtained from the compression tests are given in Table 4.6. The results presented in Table 4.6 show differences varying from 1.4% to 9.4% in the critical buckling stress obtained from

each pair of nominally identical test specimens. Table 4.7

compares the experimental critical buckling stress obtained from

each compression sample with the theoretical values calculated using the measured out-of-straightness of the specimen and the

average tensile yield stress of the material. In addition, the theoretical critical flexural buckling stress has been calculated for the test specimens using two different values of the Robertson

constant of 2.0 and 3.5 assuming an initial member bow of 1/1000 (BS 5950, Part 1,1985).

Soft members in tension

A total of eight nominally identical model soft members were tested in tension. All of the members were type 3 (Figure 4.11)

and were fabricated from the three steel tubes allocated to the

184

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END DISPLACEMENT (mm)

e% z 6-

4,1 C3 I

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END DISPLACEMENT (mm)

SAMPLE No. GTIlGA Figure 4.19. Load-Displacement Behaviour Of Soft Member Component Tubes . The Figure shows typical tensile load-displacement behaviour obtained from the outer, middle and inner tubes used in the fabrication of the model soft members. All three specimens were tested under displacement control at an initial strain rate of 0.063% per minute. This rate was increased to 0.63% per minute after a total of 4% strain had occurred in the members. The relationships shown in the Figure are for the samples cut from the three tubes allocated to the green group.

186

SAMPLE No. GT02G

SAMPLE No. GTMlG

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:c X :x 2: -4 " -4 ri -4 cli cli 9

CD C: ) CD (D - - - - >- (-) (-) ti (.. > C-) u ci u L> #. -) f-. ) c) (D CO (D (D CD LD ýD (D f. M CM C-13 (D

La

0

w 4-)

'S,

llo, s

sl

4-4

C3

rL)4 4 C44 o 0 ;. 4 0

cc 0) C) to 0

LO P0 4-3 -H 4-3

C13

.0 f+4 -4

::,

.0

4-)

ý74 a) E(I -4

0

44 0

0 E 4J Cý. 4 rj)

.H04. ) P 44 ;4

0 ý: 9ý 0

D)

4.4 a) 0

88 AN Oo .HMM 4. )

ý4 00

-P4 w 4-)

CI-4 0

9t Q) w U) C)

P-4 LO :s0 ': 2 "I

4L

4-) 4-4 o

P0 44 0) P-. 4

.0ý0ýý: 9z vs 2

188

green group. Figure 4.20 shows the experimental load-displacement

relationship obtained from the fifth test specimen. All of the

remaining test samples, apart from the sixth sample which failed

prematurely due to a poor weld, exhibited very similar load-displacement relationships to that shown in Figure 4.20. Table 4.8 gives the experimental values of the yield load, ultimate load, percentage plastic extension, and percentage overall elongation at failure for each of the soft member tensile tests.

The values obtained for the member yield and ultimate loads have

been used to calculate the corresponding stress levels in the inner, middle and outer tubes. These member stresses are given in

Table 4.8 in addition to both the experimental and theoretical

member stiffnesses.

When the triple-tube soft members are strained in tension the

middle tube is strained in compression and the outer and inner

tubes are correspondingly strained in tension. The cross-sectional

areas of the three tubes are proportioned so that the middle tube

will yield and deform plastically in compression before the outer

and inner tubes yield in tension. The individual tube stresses

given in Table 4.8 show that when the soft member yields in tension

the average compressive stress in the middle tube is 30% greater than the average tensile yield stress of the tube material. This indicates that when the middle tube reaches yield in compression, plastic squashing and lateral expansion of this tube is prevented by the outer tube. As the middle tube is restrained from deforming

plastically, an increase in the tensile load applied to the soft member will cause the compressive stress in the middle tube to

exceed the yield stress of the tube material.

For the soft members under test, the tensile load required to

yield the inner tube is 12% less than the tensile load required to

yield the outer tube. Consequently the tensile load acting on the

soft member can increase until the tensile stress in the Inner tube

reaches yield. At this point the inner tube deforms plastically In tension, and effectifly controls the yield and post-yield behaviour

of the soft member.

Soft members in compression

A total of sixteen type 3 (Figure 4.11) model soft members have been tested to failure in compression. Figure 4.21 shows the compressive load-displacement relationship obtained from the third

189

z "4

C3 I

Ld 1

z Ld f--

7-0

6-0

5-0

4-0

3-0

2-0

1.0

END DISPLACEMENT (mm)

SAMPLE No. SMGT5 Figure 4.20. Tensile Load-Displacement Behaviour Of Model Soft Member. The Figure shows a typical load-end displacement relationship obtained from testing a model soft member in tension. The member yielded under a force of 440ON and supported a maximum tensile force of 7388N. In addition the initial stiffness of the member was 2745 Nlmm. The soft member yielded when the stress in the middle tube reached a value 3W. greater than the average tensile yield stress of the tube material. The yield and post-yield behaviour of the soft member is governed by the tensile load-displacement behaviour of the inner tube. The soft member failed when the inner tube necked and ruptured in tension.

dol% Z

. Y. 7-0

6-0

5.0

13 >-4 4-0

3.0

2-0

1-o

END DISPLACEMENT (mm)

SOFT MEMBER SAMPLE No. GS3Y Figure 4.21. Load-Displacement Relationship For AModel Soft Member In Compression. 7he Figure shows the experimented load-end displacement relationship obtained from testing model soft member GS3Y in compression. The member yields under a force of 3916N and buckles under a force of 7865N. The member has an initial elastic stiffness of 3258.8 N/mm.

190

, *I 0,

w C-i v 'o

--r Ln Pý cn Ci "i Ln w

W- C4- Q. w o

w t"

3 Z Zu co

I N

.uc al a. Lj c -j

41 s

4! 41 zS, co 47, f, .2

Ln fn

- en m en m m m

+ 0-= o cm 41) -0 (1)

.; 71 Www-a

4: c -9 - .0 'n Z , 0- ý- 10 P. In ý cli 'a 1% N

CL wm 10 In

ýo wi

10 ul

0-2 0

14

E 0 LI) r ý41 Cc

co 4m -

t, %a

rý V) en m m m m

t, r Ci 1ý 2: C4 Cý ýg 14 Cý - 01 e co 00 00

O; m Vb=)-J-

w. 9 : 7' ul " en m co V! 't ; 4

QP V f cv CD r C. CIA . . . + +

c w

at r, r CO (n at Pý M m " rn m m m

C%- w m I m I en (n en M LJ 31.

'A

'9 L!

C5 r- w ri C>

+ +

c 4 0c

ý? 'I "ý '! 9 In .0 9 LI!

- !! ZL ý ol ON co 8 Z U-0 C'i

. en %0 (D " ý m

V)

c0 4,0 a 4 ý! 0 1 CO

- . - S. ý 4-

-4 40 co rn o

E ri r

A 4) ý. a 2

; r

6 Zo "; in -; (),

a -W C,; Ln

z co

X. -= co 03 Go %D r, (L r, C- to cv ev i cm

C9 C9 9 Ci eq Ci eq Ci a) c ýý .1 W$ Vý ul Ul Ul %n Ln 1 0 t:

J- 41 Ln 19) Ln In W) m rn cn 4" fn M en gn

w U

m to R

io 2 2 ;F ;2 v) 'n 'n vb vi

4-4 4-4 00

4-)

4-1

4-) rn

4, )

C, .5 4-)

pq 11 r8 44 0) 0

4-) -rj 4-)

r, nL) p

4-) V2

C14 R .0ý0 0 ca a) 4-ý Ea :3

4-ý (1) rA 19 9

a) CH >

4-) po ý !ý

a) -tf)

bO W 4-)

P 4j ba

w

4-) 4-3

0

DI 4 4-)

ý "',

L) 0. ;4

'8 a)

0 Cd 0 . r-I a) w () . r.. C) 8, 0 0) 4J H

4J 4-)

g CO

A ; -4

* ro Z0

cs r.

ý4

4-) 0 a) m .00': 0 4-) -rq

I

4-) 4 1*

$4 4) 44 4-)

Id 4)

co

(L) t§o 4, R

:ý

0 ;4 a)

44 4-) -t: r 4J

191

test member which was fabricated from steel tube allocated to the yellow group. The soft member exhibited an initial sitffness of 3259 N/mm, yielded under a compressive load of 3916 N and buckled

at a load of 7864 N. The soft member characteristics shown in Figure 4.21 are typical of the load-displacement behaviour obtained from the other fifteen test members. The experimental values of yield load, buckling load and member initial stiffness obtained from each test member are given in Table 4.9. In addition Table 4.9 gives the average stress levels in the three'component tubes at the yield load of the soft members and the average stress in the

outer tube at the buckling load of the soft members. The stresses in the outer tube have been obtained from strain measurement made by the dual extensometer while the stress in the middle and inner tubes have been calculated from the member loading and the tube

cross-sectional areas. Table 4.9 also gives the theoretical initial stiffness of each of the test soft members. These member stiffnesses have been calculated using the measured tube

cross-sectional areas, the average elastic modulus obtained from the tensile tube tests described previously and the lengths of the

component tubes. The tube lengths were taken as the distance between the ends of the members to the centre of the plug welds for both the outer and inner tubes and the distance between the two end plug welds for the middle tube.

Theoretical critical flexural buckling loads of model soft members

Estimates of the flexural buckling load of the model soft members can be calculated, using the physical dimensions of each component tube together with the measured bow of the complete soft member. Two different values of the soft member buckling loads have been calculated assuming two different stress conditions to be

valid when the soft members buckle. The first estimate of the buckling load is calculated assuming that the soft member loses

stability when the stiffness of the outside tube becomes zero. The

second estimate is calculated assuming that the soft member buckles

when the stiffness of the middle tube becomes zero. Both these values depend on assessing the flexural buckling stress of one of the component tubes of the soft members and this has been undertaken using equations 4.1 to 4.9 described previously. The flexural buckling loads of the complete soft members have been calculated using a procedure similar to that ut! I ised to estimate

192

A A

4 A

14 14

m A Pý ;4 A

R C! C! I'! R li ... T I

-9 V

a 4 A 4

ol

b-

A 0- A .1 s: X R W, Iz c4

lo

I all 14 1181 1 is Jý bT j3

bt u. IS

b 1. IS

1.2 13

H j b 15

ýg Rig 9

Fl 1 - ig 0

-0 05

-0 0--

x x y x x x x

X

4--l Cd W t4 Cd 4J

S *4 5 r-4 5 (D p Cd 0) 4--) 4-) e-I

44 M :3F! ý4 ý Ld C) a) P4 r-I 0 la) 0 rq 44 [a +) k 4-) 0

:5 44

10. $4 M Id

A [a 0 C)

0 ce) 41

co . ý4 ý, ý4 .

M a) P, 0 (L)

10 Cd Z 0) -P 0 ;4 0 $4

,8 . "-,

4 -s ', q +)

4-) :ýs 0) D) EO 0

-QO) 5

(L) -r-, 4-) '0,4-1 a) a) 4-)

ý4 4-ý JZ (L) rMc . 4J ýr

ý -H

.2wF. 4-) to 0

0 rI rj a) $. 4 P, 4 C)

4-)

4-) 0) 00 ý4 4. )

44

ý4 0

p r-4 Q) P

; to B Cd :3 Q)

En t 0) 0 f--4 "0

C') R., 04 ý9 to . 49 P C) 0

41 8

10C)a 41 r. 0 Cd D)

44 4-3 1 ý4 0 -8 x" ") 49

U) > 0 Ld

EQ ý4 r

. e4 . rz.

.)m00 -ýj ,

44 r-ý 0 C3 0 001 ý4 DI 0 V) -H 44 Q)

4. ) ? It ; -ý W 44

f... 4 ., A qV ý r-q P-4 0, r. T4-4

0 0

4-) 9 Cd

4-) Ea > (a ": r .I La td cd 0) La

4 ;: 4S

00 a) rj) 4-) ,8

9U 4-) ;4 ý$ U) F-

193

the buckling loads of the full size soft members SM5 and SM6. The

procedure described has been used to calculate both estimates of the flexural buckling load of the soft members, each value depending on the stress conditions assumed at failure. The

calculations presented have been based on the average physical properties obtained from the four soft members fabricated from the

steel tubes allocated to the white group.

Estimate of the flexural buckling load of the model soft members when buckling is controlled by the stability of the

outside tube.

The parameters given in Table 4.10 have been used to calculate the critical flexural buckling stress for the outside tube. Assuming the outside tube to be restrained in both position and direction at one end and restrained in position only at the other end, then the effective length Le Of the tubular strut may be

taken as 0.7 times the actual length of the tube.

The tube slenderness ratio x is given by X Le

r

0.7 x 318 = 71.23

3.125 -

Representing the average initial lateral deflection measurements for the four soft members by the sin curve given in equation 4.1,

gives a value for a, equal to 0.14 mm. From equation 4.6;

TI _ a,

2Z0.14 x 9.52

2=0.06824 r2x3.125

The Euler buckling stress ae is given by;

ae _n2E_, n 2x2.09 x 105

= 406.56 N/MM2 X7 71.232

194

LM C) 1.4 =3 x

cy) tD q: r C% 9- m 4.0 LO 4D km (1) CV) co CD CY)

co all co C: ILD 1-4 %0 1-4 cli

.0 C) =3

W Cý co rl_ LO (71 ci -4 -4

,a %0 C) a co Csi co

co --I Pý% Cl%i cli C%i

ON X s- co LO Ln Ul) W cli CY)

41 LC! C! en :3 CI C:

4 c::,

C) 1-4 1-4 C73 M (1)

C4 -1 I E =

LLJ fo S- in c u

0 4- o 41 Ln >1

413 4ý to 013 t) a cl) - 0 E W %A

4- (1) 0 u 4- 0 S-

aj 0 4-A %A cu 0 V)

0 %A 0 u 0 . 6-1 "a (L) 1w S- #10 0

V) ct: >-

EQ 0)

p 0 CH

w

En

0

be CH ., I

(a 4-4

4-)

4-4

94 a) Q)

e. I

Q) 4J ;4

$4 0

4J

.8 ý4 0

4-4

w 4)

r-q

4--) c+4 0

0 ý4

44

195

From equation 4.9;

303.53 + (1 + 0.06824) 406.56 2

368.91 N/mm

Substituting into equation 4.8 gives the critical buckling stress Ocrit for the outside tube as;

Ocrit = 406.56 x 303.53

368.91 + (368.91ý' - 406.56 x 303.53ý/'

. n2 "crit = 256.25 N/M,

Considering the behaviour of the complete soft member, the average

yield load for the four soft members fabricated from the steel

allocated to the white group is given by;

Average yield load = 286.02 x 19.98 = 3712.47 N

where:

the average yield stress of the middle tube ay = 286.02 N/MM2 the average cross-sectional area of the middle tube A= 19.98 MM2

When the soft member yields in compression the magnitude of the force acting on each of the three tubes is identical and the

stress di agram at yield wi II be as shown 1n Figure 4.22

286-02N/m

Tension

IBB-S3N/mM'

: ompression

iaa. a3 N/m

Compressic

Yield Load 3712-47N

Outer tube Middle tub?, Inner tube

Figure 4.22. Stress Diagram For Model Soft Member At Yield.

196

The stress diagram shown in Figure 4.22 will remain unchanged during the yielding sequence of the soft member. After the

yielding phase is completed, the soft member closes up and all three component tubes are subjected to the same end displacements. Any small end displacement of the soft member will now cause equal strain and stress changes to occur in each of the three tubes.

Consider the soft member to be subjected to a small end displacement which will raise the stress in the outside tube to its

critical buckling value. This will cause a stress change in all three tubes of:

256.25 - 188.93 = 67.32 N/MM2

It is proposed that when the outside tube of the soft member becomes unstable the complete soft member buckles. Consequently, the stress diagram at failure is given adding 66.23 N/mm 2 to the

compressive stresses in both the outer and inner tubes and subtracting 66.23 N/MM2 from the tensile stress in the middle tube. The stress diagrarn at failure of the soft member is given in Figure 4.23.

Failure Load 7232-5N

2IE317ON/md Tension

256-25N/mM2 256-15N/mm Compressio . -ompression 11

Outer tube Middle tube Inner tube

FigUre 4.23. Stress Diagram For Model Soft Member At Failure, TAliere Collapse Is Govermed By Buckling Of Outside Tube.

Multiplying the component tube stresses shown in Figure 4.23 by the corresponding tube areas gives an average theoretical failure load for the four soft members of 7232.5 N.

(11) Estimate of the flexural buckling load of the model soft members when buckling is controlled by the stability of the middle tube.

197

The parameters given in Table 4.10 for the middle tube have been used to calculate the critical flexural buckling stress for this component. Before the middle tube becomes stressed in

compression both the outer and inner tubes will have yielded. in

compression. This will reduce the restraint offered to the middle tube by both the outer and inner tubes and as a result the middle tube may be considered to be heldin position but not restrained in direction at both ends.

The tube slenderness ratio x-1.0 x 318 = 121.53

Using an average member bow of 0.14 mm in equation 4.6, gives:

2ýj= 0.14 x 7.94 = 0.08115

2x2.6177

The Euler buckling stress ae is given by;

2E- ]12 x 2.01 X 105

ý 134.32 N/MM2 x2 121.53 2

From equation 4.9

cc = 286.52 + (1 + 0.08115) 134.32

= 215.87 N/rnm2 2

Substituting into equation 4.8 gives the critical buckling stress acrit for the middle tube as;

Ocrit = 134.32 x 286.52

215.87 + (215.872 - 134.32 x 286.52)V2

acrit = 125.79 N/inn2.

At failure of the soft member both outside and inner tubes have yielded in compression and the stress in the middle tube will

2 be 125.79 Nlmm The stress diagram at failure is shown In Figure 4.24.

198

303-53 Nlm

Compression Compressio

Outer tube Middle tube Inner tube

13161- 5N

Figure 4.24. Stress Diagram For Model Soft Member At Failure, Where Collapse Is Governed By Buckling Of The Middle Tube.

Multiplying the tube stresses given in Figure 4.24 by the

corresponding tube areas gives an average theoretical failure load for the four soft members of 13161.5 N.

Similar calculations have also been undertaken to obtain the two estimates of the flexural buckling load for the remaining two groups of model soft members. In addition the flexural buckling loads have been calculated using the values of critical buckling stress for both the outer and middle tubes obtained from BS 5950 Part 1, Table 27A, (1985). Table 4.11 gives a summary of the estimated flexural buckling loads for the groups of soft members together with the average buckling loads obtained from the experimental investigation. From the results presented In Table 4.11 it is evident that the model soft members under Investigation buckle when the stiffness of the outside tube decreases to zero.

Pre-strained middle tubes

To determine the influence of pre-straining the middle tube in tension, twenty seven nominally identical tubular samples have been

prepared from two stock lengths of the middle tube. Four samples have been used to determine the yield stress and tensile behaviour of the material, six samples have been used to obtain an average value of the flexural buckling load of the specimens and seventeen

125 - 7S N/MM2 I 2E33-03N/m

199

c7) (L) c cm

-a .-a :jCj. 8- -ý %A 41 0 CD

. le E: - - ti) uZ -- 0) «c3 Cn Z V) du ý (1) Ln M (n rý

Co 0 4- -0 vý (A 4- ýIr _A to < -0 00 Co 0 CD

1- -- Co r= - 3: cý cý cý '; 0 V-t ". 1 1-1 (L) . r,

be 0) m vi C%i FE- G

C: ) «a 4j r- >e x: C)

CU #U 4- (V U

. r. ei G)

,0C 4- M- :i C> 413. - -0 Ln c 4A r--

ý0 C- $- -- 0) #0 M" _A --4 m

du _i

:3 0) EE ý ei C%i to M 4.0.

u2 --4 a r=

CY) 4--0 r-

4. -0 jo '0 CM 4- r-f r-A W-4

(V

4- G) JD 113 0) 3: m zi _U, 0 M- Z ra >0 1- CO - VI 3t 4ý - 40 0

4- 0 c: 0.. vi Q) cu CD

m -0 m Lr)

cm X :3 (U et c: 4. -b VI LO 4-

-ý 0 415 tn 0 ý02 r- 0) OD Co ý-1 to rý M _j :3 Co rý -4

vi OKJ .- #0 0 C7) tA 4- (A -

41 ;

c: < c"i de

(V CD 0 -, Z -JZ -! ý. e -M C> 0) u= P- (U U

. r: Z je ei J-- :3 ý- 02 ---' E F- 3:. -4

(V 4- cn ai vi 0

J-- du (U 4-b 0 02 17. ýr

ý- 0 s- j2 du 0 fe

_i (U l= :3 S- im

0) 4J w u V) .

alt 9 .ýM vi m. > *C) . M CD C%i

(U X cu tA 00 0 --- CM cli cu EE m ai 1- CD

tn .- ý- C= rý 0 je ý 41 r- tA he du V) " G) U 4- -ý 4-3 U du C%i

m :30 tu :3MC u) ý- F-Coýv)4- 0. a 0

r- #0 dqj 0

to C% m V) CD Lt) Lr) V-1 LO

EE ý z c21. - c r

0) c2. u= >x :1 -2 LU CO

4- F-

VI t41-- 4-

C) c) du -

m iv (1) ýgiE: ýn u :x >- eie

.00 . 43

CL) 4.4 0 0 to Lo 4.4 -, q

rA a) 4) > ýq a) (L) 40

.H :q:: s 0 pw co "j 4-) ý4 Cý

0 2 *r-4 4.4

4-) 0ý 44

14 44 Cý-4 g 0 J8

0 sz 41 0

La 4.3 4J -00 4J P. 44 9 0o

4J ý: w Icno a) Ul 4. ) ;4 0)

0

4-1 41 0 4J 000

0 45 0sC.

! -ý' A 4) 0 a)

9sý:

"ow r-A 12,8'0 m 44 4-) CH ý4

4Q')

Cc) 4-) 0 C)

0M 4-4 4-) 4-4 Q) C61-4 0

0 U) r

, -ý 40) ý, 8"o,, I

>0 ý4

a) 0 4-) D) r-q 0 00

to

A cd

bo 0: 3

, rp

w

8 0 4-4 403

rs 0 91 E . 4. ) EQ EA

Cd 902 (D

4.4 41 $4

0 rA

Q) ra o4

4-) > -4 M cd 0) (a

cc 4-) a) r-I A

'. 4 Q) a) ., >4

rQ-),

.> ;4

ýq om4.4

#--I r-ý r-4 -H 4. )

.0 ID 0 a) a) 4-) rn U)

U) Q)

I

Q) 9 E-'

9 !ýIý 0) kw 4-)

200

samples have been pre-strained in tension by varying amounts before being stressed to failure in compression. Tables 4.12 and 4.13

give the experimental values of yield load and ultimate load

obtained from the samples tested from each of the two parent tubes. Figures 4.25 A, B and C show the load displacement

relationships obtained from three of the samples pre-strained in

tension. From the results given in Tables 4.12 and 4.13 it is

evident that pre-straining the tubular samples in tension reduces their flexural buckling loads. The specimens taken from the first

parent tube, tube L, show a 10% decrease in the average flexural buckling load due to a tensile pre-extension in the specimens of 2.0 mm. A smaller decrease of 4% has been obtained for the

corresponding samples taken from the second parent tube, tube K, (Table 4.13). In addition, both test groups show larger decreases in flexural buckH-ng loads corresponding directly with increasing

amounts of pre-strain applied to the samples. A tensile

pre-extension of 6 rnm resulted in decreases in the flexural

buckling load of 25% and 30% for the two corresponding samples, taken from tubes L and K. Figure 4.25C shows that for

pre-extensions of 12 mrn or greater the material strain hardens. This in turn raises the yield stress of the material and prevents further decreases in the flexural buckling load.

Strain aging

A total of eleven nominally identical test samples have been

prepared from one stock length of middle tube. These samples have

been used to determine changes in yield stress and ductility due to

strain aging. Two samples have been used as control specimens and have been tested to failure In tension without any aging, while the

remaining nine samples have been pre-strained in tension, aged and then failed in tension. Figures 4.26 A, B and C show the

load-displacement relationships obtained from one of the control

samples and two of the aged specimens. Table 4.14 gives the

experimental values of first and second yield loads together with the ultimate loads obtained from the test specimens. In addition, the lengths of the load plateaux and the percentage elongation at failure are also given.

A comparison of the load-displacement relationships given In Figures 4.26 A, B and C shows that a prominent yield point and load plateau return to the tensile load-displacement relationship

201

cý cý %9 19 9 li cý N Ui 19 N Lr! to r- Co P- C: ) LA (D cli tn le

.2 m m cm cm C) (D -4 C% P- «: r m Ln cli cli cli cli tli cm 14 -4

CL- 4,

2

-0

Co --0 C> M Ul) Ul) %D el cli

fl; ri ý lý 14 tý lý cý vý cý Z 4, tA t 1 1 r, Co %0 Ln CD %D f-) CY, rý r- NO tm M (L) 0 CD CA CYN LO %0 r- «& cli clo f- (:

s- ei m " C\i CM cli cm -4 -4 -4

c2

ß- Z.

Ulb 0 cli 41 tn c ei fý tý 1 1 Co m CD r, -t Co -4

-: r -e m -e -t CD m m -w r4 -: r

.0 Z , , 0 ci m cli -4 to r_ U*) CY% 0 rý d" 14

-i c 0 cý A; .4 1 1 cý c; ei cý 4 Z c; tý (4 e tn to %0 %0 -4 w ýr (n m Co Ln

-4 C> CYN -4 A %0 CD C> . -4 Co ei m M cli m M cli m m M cu m

C 2 =O Co r- 9 W 9 C 2 0 . =D z2 =:. . - => - =D - : 2. =I, U ý ;

0) c; gj c (L) c w c cu 0 (U ej

m 0 c21 c2. a o CL ýn c2. E E r

(L) 0 c0 c0 c0 c0 c0 r- 0 gi 0 cu 0 a) u 41

4) u 41

0, ti 0) fi 4ý

0, u 4,

vi u (L) u 41

Zu 4N u

C) w r .5 C -ýE

e C-

c c cc

CL) 0) ci w tu ci (i) c 4) c 4)

4, AZ %A -2 %A -:; A- %A -:; 0 -Z 0 -Z c . - c -:;

, , , c c c cm c c c G) m cu (U 4- C CL) (U 4- ai4- 0.. tu 0)4- 0)4- 41 44. - *, 4-

(L) CL) tu 41 4ý b. - x x x C) wZ

x0 w 4,

x0 (11 2

x0 iv 2

x0 w 4,

x0 cu

x0 cu Z

w0 w0

41 41 41 %A tn &A W IM 4,

w 4, Z

(L) «o , *0 , 10 4, «0

4,0 0 «0

4,4) 010

i tn ýA om 0-0 om 0 -0 0m 0-0 0,0 2M 4ý -" (U cu cu ý0 ý 2 2 4, 2 4,0 42 2 4,

- t- c2

1- CL

- 1 . - 3

r %A

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Figure 4.25 AB and C. Load-Displacement Behaviour Of Pre-Sfrained Tubes. The figure shows the load-displacement relationships obtained from three test samples which were pre-strained in tension by varying amounts before loaded to failure in compression. Sample number GTC4L was pre-strained in tension by 2-Omm and failed in compression under a force of 2736AN (Figure 4.25A). Sample number GTC7L was pre-strained in tension by 8. Omm and failed in compression under a force of 1878.6N (Figure 4.25B). Sample number GTC9L was pre-strained in tension by 12. Omm and failed in compression under a force of 1967.2N (Figure 4.25C). The average flexural buckling load of the samples which were not pre-strained in tension was 3041.7N (Table 4.12).

204

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205

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exhibited by an aged specimen. Also the experimental values of the

second yield stress given in Table 4.14, indicate that this stress increases with a corresponding increase in the amount of initial

tensile pre-strain given to the specimens before aging. To a lesser extent, there is also an increase in the ultimate tensile

strength of the specimens corresponding to the increase in

pre-strain given to the test samples. Both the load-displacement

relationships shown in Figures 4.26 B and C and the experimental

values given in Table 4.14 show that these increases in both the

second yield stress and ultimate tensile strength are accompanied by a decrease in the ductility exhibited by the aged specimens.

DISCUSSION

The investigations outlined in this chapter have been

undertaken to determine the behaviour characteristics of the novel force-limiting device. The results obtained show that a

compression member has been constructed capable of exhibiting a

steady load plateau followed by a reserve of strength before

failure.

All of the tests reported on both the soft members and

component tubes have been undertaken with the upmost care, however

some results give cause for further discussion. Relatively large

differences, in the region of ± 5%, have been reported in the

values of elastic modulus calculated for the component tubes.

These variations are most likely to be due to the difficulty

encountered in -measuring the tube wall thicknesses which in turn

cause inacuracies in calculating the cross-sectional areas of the

members. The inacuracles associated with determining the member

cross-sectional areas also present themselves in the calculation of the member stresses. However, the recorded readings of both strain

and load are more precisely determined using a combination of

strain gauges and the accuracy of these results are within ± 1.0%

of the true readings.

A major part of this investigation has been related to

assessing the buckling behaviour and critical buckling loads for

both the full size and model soft members. A proposal describing

the theoretical buckling behaviour and ultimate capacity of the

triple tube soft members has been included in the introduction to

the chapter. This proposal postulates that both the inner and

outer tubes should yield in compression before the stiffness of the

207

mi dd Ie tube dec re ases to zero c aus i ng the So ft member to b uc kIe.

However, a comparison of the theoretical results of ultimate load

based on these assumptions with the experimental results, both

given in Table 4.11 shows these proposals to be unsound. It is

apparent that the model soft members buckled when the stiffness of the outside tube has decreased to zero. For the stability of the

soft member to be governed by the stability of the middle tube, it

is necessary for the middle tube to provide sufficient restraint to

both the inner and outer tubes to prevent them buckling prematurely

before squashing in compression. The model soft members which have

been fabricated from three different diameter tubes, have clearance

gaps of 0.08 m,,, n between the outer and middle tubes and 0.235 m. -n

existing between the middle and inner tubes, as shown in Figure

4.27.1 OUTER TUB DIA. 9-52

MIDDLE TUBE DIA. 7,94

INNER TUBE DIA. 6-35

X

i O-OEý Q-235, pii%ý4 P-724

Figure 4.27. Tube Clearances In The Model Soft Member. The Figure shows the clearances in millimetres which exist between the three tubes used in the fabrication of the soft member. The tube diameters given at the top of the diagram are average valves obtained from thirty six measurements for both the outer and inner tubes and from one hundred and fifty three measurements for the middle tube.

As the compression load on the soft member is increased the

diameter of the middle tube decreases whi le the diameter of the

outer and inner tubes increase. After the middle tube has yielded

in tension the diameter of the tube wi II decrease even further,

increasing the clearance gap existing between the inner and outer

tubes. It is possible for this clearance gap to increase to 0.315

rn: -n each side, a] lowing a total lateral movement of 0.630 mm to

Occur between the two tubes. This indicates that it is no longer

possible for the middle tube to offer I ater aI restraint to the

outer tube and consequently the outside tube wi II fail at its

critical buckling load. Once the outside tube has buckled then the

stiffness of this member is negative and it must shed load

resulting in the failure of the complete soft Tiember.

208

Reviewing the fabrication details of the type 3 soft member

shown in Figure 4.11 it is questionable if the middle tube provides

any restraint to the outside tube even when the three component tubes are behaving elastically. The top of the middle tube Is plug

welded to the outside tube and when the outside tube becomes

unstable, rotation of both tubes will occur at this connection. However, the middle tube does offer restraint to the inner tube.

At failure of the soft member, the strain in the inner tube

indicates the stress in this member to be double the theoretical

flexural buckling stress of the tube. This buckling stress for the inner tube has been calculated assuming the effective length to be

0.7 times the actual- length and the tube to be laterally

unrestrained at its centre.

In the full size soft members, the middle tube is replaced by

four steel strips. These strips have negligible flexural stiffness

even when they are stressed in tension but they will prevent the

middle tube from buckling at its critical buckling load provided they in turn, are laterally restrained by the outside tube.

Consequently, both the full size soft members and the model soft

members fall when the outside tube buckles.

The investigation undertaken to assess decreases in the

flexural buckling load of the middle tubes due to tensile

pre-strain have shown only relatively small decreases of about 7%

resulting from pre-extensions of 2 mm. Because both the full size

and model soft member have failed when the outside tube buckles,

the reduction in the buckling load of the middle tube due to

pre-straining will have no effect on the collapse loads of the soft

members. Figure 4.23 shows the stress diagram for the model soft

members at failure. It can be seen from this Figure that the

stress In the middle tube at failure of the soft member Is In fact

tensile, and consequently any reductions in the buckling load of the middle tube cannot influence the ultimate capacity of the soft

members.

The influence of strain aging on the middle tube may however

have an effect on the long term behaviour of the soft members. The

limited experimental investigation undertaken has shown Increases

to occur in both the yield and ultimate stress when the member is

re-stralned in tension after a period of aging. It is envisaged

that soft members incorporated into space truss structures will

209

remain fully elastic under working loads and any significant overload will cause the soft member to yield and behave like a load-limiting device. Once the middle tube of the member has

yielded in tension then the soft member is prone to the effects of strain aging. Subsequent overloading of the structure and soft member will raise the value of the yield stress in the middle tube

of the soft member and alter the characteristics of its load

plateau. The load displacement characteristics given in Figure

4.26B indicates that provided the initial pre-strain of the middle member is insufficient to cause the material to strain harden,

re-straining after aging will cause a sharp rise in t he limit load followed by a return to the normal load-displacement behaviour. However, if sufficient pre-strain has occurred to strain harden the

middle tube material, re-straining after aging will increase the

magnitude of the limit load of the soft member. Any significant increase in the predicted limit load offered by the soft member

will in turn alter the stress distribution within the encompassing space truss and may change the collapse mode of the structure.

210

CHAPTER 5

THE EXPERIMENTAL BEHAVIOUR OF DOUBLE-LAYER SPACE TRUSSES

INTRODUCTION

To investigate the viability of improving space truss behaviour, four model double-layer space truss structures have been fabricated and tested to collapse. The first two model structures tested have been designed to exhibit ductile load-displacement

characteristics by allowing extensive tensile yield to occur in the bottom chord members. The last two structures tested have also been designed to permit yield in the bottom chord members, but in

addition, several of the novel force-limiting devices described in

the preceding chapter have also been incorporated into these

structures.

The experimental testing of model structures cannot by itself

constitute a basis for the formulation of new or updated design

methods, but when experimentation is used In conjunction with a

verified mathematical analysis the concepts can be extended, with due caution, to cover the design of a wide range of related structures. The primary objectives of this experimental investigation were to obtain reliable experimental data of the load-displacement response of these structures and to use these data to assess the validity of the assumptions made In the idealised

theoretical collapse analysis program outlined in Chapter two.

EXPERIMENTAL

Fabrication of Models

The four model structures tested were al I square-on-square double-layer space structures with a mansard edge detail. The

models were 1.80 metres square on plan with a bottom grid of five

square bays in each of the two principal directions and a top grid of four bays In each direction. Each structure was 254.56 mm deep

making all of the members in each structure the same length and fixing the angle of inclination of the members, used to join the top

and bottom chords, at 451 to the horizontal plane. Figure 5.1 shows

211

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Fiqure 5.1. Node And Member Numbers For Models 1 To 4. The Figure

shows a plan and elevation of the top Chord members, web' members and bottom chord members with their corresponding node and member numbers. Each member in each structure had a length of 339.5mm.

212

6 56 17 57 '2 8 58 -9 ! 59 50 so 61

a plan and elevation of one of the model structures. Also shown on the Figure are the node and member numbers which apply to each of the four structures.

To overcome the difficulty of constructing perfectly pinned joints which allow equal freedom of rotation about the three

principal axes, the model structures were fabricated with rigid joints. Although this violates one of the basic assumptions used in

the collapse analysis program, it was considered beneficial to have

a realistic assessment of the joint rigidity instead of an indeterminate joint rigidity failing between the two extremes of the

perfectly-pinned and fixed-joint idealizations.

Member Manufacture:

The four model structures were fabricated from six different

member types. Five of the member types were annealed, cold-drawn,

seamless, mild steel tubes ordered from the supplier to comply with the requirement given in the British Standard Specification BS 6323 (1982). The sixth member type was a solid, bright steel bar ordered to comply with the requirements given in BS 970, Part 1, (1983).

Table 5.1 summarises the member types and gives the nominal outside diameters, wall thickness and member areas. Member types classified in Table 5.1 as T2, T3 and T4 have been used in the fabrication of the model soft members incorporated into the third and fourth double-layer grid models.

A suff ic lent quantity of each tube and bar was purchased in an

attempt to ensure that each member type came from only one batch of

material. Each separate tube and bar used in the fabrication of the

model structures was colour coded and samples were taken from each

stock length to determine the material properties. The annealed

steel tube was delivered in random lengths approximately 6.0 to 7.0

metres long while the solid bar was supplied in stock lengths of 2.440 metres. All the members were prepared from straight undamaged tubes and bars. Each of the long annealed tubes were f irst cut In

half and then the individual members were cut and accurately

machined in a lathe to a length of 339.5 mm. Two of the members cut from each stock length of tubes T1 and T2 (Table 5.1) were tested to failure in tension. One of the members cut from each stbck length

of tube T5 and bar T6 (Table 5.1) were also tested in tension but,

in addition, two members from each of these tubes and bars were also

213

Member type Outside Wall thickness Cross-sectional Use Diameter (mm) area (MM2)

(mm)

T1 4.76 0.91 11.00 Tension members Models 1,2 and 3.

T2 6.35 1.22 19.66 Tension members models 4. Soft member inner tube.

T3 7.94 0.56 12.98 Soft Member middle tube.

T4 9.52 0.71 19.65 Soft Member outer tube.

Web members models 1.2,3 and 4. Top

TS 9.52 0.91 24.61 chord members models 3 and 4.

T6 10.00 Solid 78.54 Top chord members models 1 and 2.

Table 5.1. Member Types. The Table lists the six different member types usFd in the construction of the four double-layer grid structures. Member types TI to T5 were annealed, cold-drawn seamless, . mild steel tubes complying with the British Standard Specification BS6323 (1982). Member type T6 was a solid bright mild steel bar specified to comply with the requirements given in British Standard Specification BS970, Part 1 (1983). The properties given in the Table are the mean values obtained from the test samples.

Linear variable differential Mechanical dial qauges transformer transducers positioned at node numbers: - positioned at node numbers: -

Models 31,37,26,25,27,15,38, Models 7,18,8,19,42,53,29,30, 1,2, 14.28,4,39,3,17,5.32, 1 and 2 36,24,23,12,13,35,34,

3 and 4 20,33,9,22,10,21,16,11. only. 2,46,57,47,

Table 5.2. Positions Of L. V. D. T. Transducers And Mechanical Dial Gauges U In Models 1,2,3 And 4. The Table gives the--no-cre- numbers at which the vertical displicements were measured using linear variable differential transformer transducers and mechanical dial gauges. The dial gauges were used as a precautionary measure in case voltage readingsfrom the L. V. D. T. transducers were lost due to interuptions to the power supply. The mechanical gauges were only used to measure certain node displacements in models 1 and 2 but these readings were not used in the-experimental investigation.

214

tested to failure In compression. These ninety-five individual

member tests were undertaken using an identical procedure to that

adopted for the component tubes of the model soft member, outlined in the preceding chapter. After the members were prepared for each model, the diameter and wall thickness of each component was carefully measured using a vernier calliper.

Joint manufacture: The joints used in each of the four model space structures were very similar to those used and described by Collins (1981). Figure 5.2 shows details of a typical joint, all of which were carefully machined from 28.58 mm square mild steel bars. The joints were rigidly held in a jig while each face was machined flat

and each member-locating hole drilled using a milling machine. The

member-locating holes were accurately drilled to a depth of 8.0 mm giving the members the possibility of 8.0 mm of longitudinal

movement when they are positioned between these two end nodes, prior to welding.

Model assembly: Each of the four models were accurately assembled

on a special steel jig manufactured for the experimental work reported by Collins (1981). The jig was fabricated from rectangular hollow steel sections and was pivoted about the axis passing through its centre of gravity, so that the whole jig plus model could be

inverted if necessary. The nodes were fixed to the Jig using special supports which were adjusted at their base to position the

node accurately in the correct location. This proved to be a very time-consuming procedure because small alterations made to the height of the node also changed its position in the horizontal

plane. The four bottom layer corner nodes were first accurately

positioned and levelled relative to each other with check measurements made between diagonal ly-opposite nodes. The remaining bottom chord nodes were then positioned and levelled, working from

the four corner nodes using a 2.0 metre long steel straight edge. The top central node and top corner nodes were positioned using a Jig which bolted onto the top of four bottom nodes and provided the

correct grid depth. After the five key top nodes had been

accurately positioned and levelled, the remaining top nodes were positioned and levelled again using the steel straight edge. The

position and level of all of the nodes were rechecked and small inaccuracies corrected. The nodes for the first model took almost three days to position before the overall geometry of the model

215

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rigure 5.2. Joint Details- Each joint was manufactured from a 28.58mm square mild steel bar using a similar procedure to that outlined by Collins (1981). The joints were held rigidly in a jig and each sloping face and hole was prepared using a milling machine.

216

structure was within acceptable limits, with the location of each node within ± 0.25 mil of its true grid position.

After the nodes had been accurately located on the jig the

members were inserted into their correct positions. Each member was checked to ensure that it was free to move along its axis before

welding commenced. The members were welded to the nodes using four

symmetrical ly-placed spot welds, made using a versatile mig welder. A strict operational procedure was adopted for the welding of each of the model structures. The f irst members to be welded were the

eight members surrounding the top central joint, node 31 in Figure 5.1. Welding then progressed in a clockwise sequence with the

welding of the members at the bottom nodes 26, . 37,36 and 25

followed by the members at the top nodes 20,21,32 etc. (Figure 5.1). After all the members were welded to the joints, each spot weld was visually checked to ensure that a suitable degree of penetration of the joint had been achieved and that the tube walls of the members had not been burnt through. Any tubes which were damaged during the welding sequence were cut out and replaced by new members. Three members were replaced during the fabrication of the four structures which required a total of three thousand two hundred

separate spot welds.

After the model structures were f inally checked, they were carefully removed from the jig by first releasing the perimeter nodes followed by the remaining nodes. Once the structures were lifted free from the jig they were placed on a flat table top and rechecked for alignment and level using the steel straight edge. No

apparent change in joint position had occurred in any of the models, however one of the thin bottom edge members in model 1 (number 33, Figure 5.1) had a discernible bow, indicating some joint movement and the presence of residual stresses. This member was subsequently

cut out and replaced by a new tubular member.

Although care was taken in the fabrication of the steel

structures to prevent member stresses due to initial lack of member fit, it is apparent that some residual stress distribution was

present in each of the structures, resulting from the heat generated during welding of the members. These residual stress distributions

can only be decreased by stress-relieving the complete structure and unfortunately, it was not possible to undertake this within the existing budget.

217

Each model structure was constructed from a different

combination of the members given in Table 5.1. Figures 5.3,5.4,

5.5 and 5.6, show the member types and colour codes used for models 1,2,3 and 4 respectively. The soft members used in model 3 were type 1 soft members (Figure 4.11) while the soft members used in

model 4 were the type 2 soft members also shown in Figure 4.11.

Equipment

Loading system: _

The system used to load each of the four space

structure models consisted of a 50 kN hydraulic actuator, a hydraulic power pack and a control unit all manufactured by R. D. P.

Howden Ltd. The hydraulic actuator contained its own load cell and displacement transducer, which were both incorporated into a closed loop control system monitored by the control unit. The loading

system was arranged so that the actuator would impose pre-set displacements on the structures, providing the opportunity to

investigate the post-ultimate strength behaviour of the models. Both the rate and magnitude of the actuator displacements were

controlled by the D. E. C. LSI-11/2 micro-processor. Using the

analogue to digital converter incorporated into the 16-bit

microcomputer, displacements -as small as ± 0.05 mm could be imposed '

by the actuator over its ful I working range of ± 100.0 mm. Before

any tests were undertaken, the complete system was calibrated by R. D. P. -Howden Ltd. The actuator load cell used in conjunction with the control unit was certified to comply with the British Standards

Institution Grade 1 requirements given in BS 1610 (1985).

To load the test models the. actuator was bolted to a sliding joint which was in turn bolted onto the loading frame placed beneath

the structure. The sliding joint consisted of a series of parallel horizontal plates separated by hardened steel balls which permitted

a free horizontal movement of 11.0 mm in any direction. Test models 1 and 4 were loaded through the actuator at the top central joint

(node 31, Figures 5.1). This was achieved by bridging over the top

of the joint using a steel plate fixed above and attached to the

actuator by two 500 mm long threaded steel rods. The steel plate

rested on top of a 10 mn diameter hardened steel ball which was

seated in a small cup positioned in the top of the node. Figure 5.7

shows the device in operation during the testing of model 1.

218

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Tube types: T1 small hollow tube; 4.76mm outside diameter; 0.91mm wall thickness.

T5 large hollow tube; 9.52mrn outside diameter; 0.91mm wall thickness.

T6 solid bar; 10. Omm diameter.

Member Colour Code: G= Green; Y= Yellow; W= White; B= Blue; R= Red; WY = White Yellow; YR = Yellow Red; BY = Blue Yellow; WR = White Red; YG = Yellow Green; W3 = White Blue; WG = White Green.

219

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BGW evi ew , BC, ýv TJ T r. 76 G

ID 0 ý9 -3 .p

Co 4 ID ý 0 ý e ý ýz

P- Q EG EG BR TG TG TG 76

e cc

e ý

-2 5 ,

, 2c

ID

I ý. 1 1.4 $o

TOP CHORD MEMBERS

WEB MEMBERS

,Z ID 2 ýo , m clý ý c

r6 Tro rG TG

Ic ID e (ec

l

,rý n , tc . tc ý Z e ,

co ID 7 3: 1 .v

- - - - :n ý- c ý cc in

I ý

I "'

p -,

2 BG E5G Tr. TG TG J

to 'D uo . co CD -

C, , ýI., rý li F vi PIW

rI ri TI TI TI

:0 ý- cc ý cc - cr CC

C, ED ý

C, m ý

aw S4 a" 81-J TI rI TI rI

co ý CD ý CD cc t- C ý- CD a r. 5G BG BG 6G TI TI ri rI ri

BOTTOM CHORD cc cc CC, MEMBERS

T1 71 TI TI

ED i- cr C

cf IL

cc C: )

TI TI TI TI TI

rI r1 71 TI TI

a - cc CC ED m

81-J TI rI TI rI rI

0ý -

SG 5G EG 51 6G TI TI T rI

cr CD cc CC) CC)

eG sa EG BG BG rl Ti 71

> - cr ED i- C

BG BG BG eG

IIIIIIIIII

Figure 5.4. Member Types With Colour Code Identification Used For T-hý-To-nst! Fu-ction Of Model 2. The Figure shows the member types and tLFe-coTrrespond1nFg colFour (codes used to identify each member used in the fabrication of model 2.

Tube types: T1 sma II hol low tube; 4.76mm outside diameter; 0.91mm wall thickness.

TS large hol low tube; 9.52mm outside diameter; 0.91m, n wall thickness.

T6 solid bar; 10.0ran diameter.

Member Colour Code: BG Blue Green; BW = Blue White; SR = Blue Red; BY = Blue Yellow; BYW Blue Yellow White; B8 = Blue Blue; 3GW Blue Green White; BBW Blue Blue White.

220

0(- p0Q (-,

T5 SM2 SM2 T5

PG RG PG Pci T5 T5 T5 T5

r co In co T) co Vý. - a CY

PG RG PG RG T5 T5 T5 T5

co n cc n (X V)

RG PG PG G T5 l 75 T5 T5

in co V) CD Lo m Lo co cc CC ý- Cý L.

RG

I

R PG T5 SM2 5 M2 T. *

TOP CHORD MEMBERS

WEB MEMBERS

0 cc

ko ct

(2 0.1

(D 1. ) cc V') (n vý CL u3 CD n

CD m pG RG PG Pci r5 75 TÖ Tö

Co vý CD (D cz vý. a CY

PG RG PG T5 T5 T5 T5

m n CD n cü n cc

RG PG PG PG T5 75 75 T5

o ', Co %n cz Lr) (Z 41 e n

D9Q2a ca 0A0A

ri rI rI TI TI

rr a cl, Re Re Re Re PB TI T T TI TI

cjý Re RB RB P13 pa TI ri F TI TI

py py py RY py Ti rl 71 T1 71

CC Cf cr

RY Py py Ry T ri ri TI TI

-? cr cc

ply RY py P LIj RY

BOTTOM CHORD MEMBERS

IIFIIIIITI

r -igure 5.5. Member Types With Colour Code Identification Used For The Construction7f- Model 3. The Figure shows the member fy-p-e-s-ani-ff the corresponding Eý-Iourcodes used to identify each member used in the fabrication of model 3.

Tube types: T1 small hollow tube; 4.76 outside diameter; 0.91: nm wal I thickness

T5 I at, ge ho II ow tube; 9.52 ou ts i de di a-me ter; 0.9 1mm wall thickness

T6 solid bar; 10.0m-n diameter Member Colour Code: R= Red; RG = Red Green; RB = Red Blue; RBG Red Blue Green; RBB = Red Black Blue; RY = Red Yellow; RBW = Red Blue White; RW = Red White. SM2--Soft Member Typel

221

vr. Fk vI . -I

T5 61-15 5M3 T5

, :, 'n co

YG5 YG8 Y6 5 T5 r5 T5 T5

YGB YCB YGB YGB T5 T5 TS 75

u) > L,

YGB YGB YGO YCB T5 T5 TS T5

YG 13 y y 13G Y F5 SM3 SM3 T5

TOP CHORD MEMBERS

WEB MEMBERS

u' - II- >-

YGB YCB YGB YGB T5 T5 TS 75

n n to ý, - ) ý- L, )- o> ' Z

YGB YGB YGO YCB T5 T5 TS T5

co

Y G. Q YGQ YGIZ YGR YGQ

o D m CD c C I I

YWR Yý-Jp YWR YWR Y114P

r2 T2 T2 T2 72

co C4 M C\l , CD

'JI- Cc cli

> - cr cr cr YWR YLVQ YWA YWR YP, 'Q T2 T2 T2 T2 T2

to Q j T

')- M N CD co ý to

! GQY GRY GRY GRY T? 72 T2 T2 T2

co r's ý, - CD "I -

'ý, co N CD N CO CD

a CY cc cr

G; z y Gay GRY GRY T2 T2 TZ T2 72

CO co Cj >

cr, CY CD C4

i

co

d

cli ce cr cr cr I

T2 T2 T2 T2 T2

D CD I

YWR Yývp YWR YWR Y1,4P r2 T2 T2 T2 72

co CD Cc cr cr cr

YWR YLVQ YWA YWR YK'Q T2 T2 T2 T2 T2

N T m CD - co ý

GQY GRY GPY GRY GRY T? 72 T2 T2 T2

r's CD "1 co CD N '3- CO cy CD

a CY cc cr GRY GQY Gay GRV Gy T2 T2 TZ T2 72

co C\j ý

> cr, CY

ý > CD CD

ce cr cr I Q

YGA VGR y6p YGQ YGQ .C 14 14 i, _, 14

Figure 5.6. Member Types With Colour Code Identification Used For The ConsUr--uction Of Model 4. The Figure shoqs the me-mE-e-r types and the corresponding colour codes. used to identify each member used in the fabrication of model 4. Tube types: T2 = hol low tube; 1. ). 35mm outside diameter; 1.22min

wall thickness. T5 large hol low tube; 9.52mrn outside diameter;

0.91-nm wal I thickness. T6 solid bar; 10. Omm diameter.

Member Co I our Code: BBI ue; Y= Ye II ow; YG3 = Ye II ow Green 13 1 ack; GY Green Yellow; BRY Black Red Yellow; WBY = White Black Yellow; YR Yel low Red; YW = Yel low White; Y3 Yel low Blue; YG = Yel low Green; BGY =BI ue Green Ye II ow; BBY 31 ue BI ack Ye II ow; YGR = Ye II ow Gr ey Red; YWR = Ye II ow Wh i te Red; 6RY = Gr ey Red Ye II ow; YBG = Yellow Blue Green; RBY = Red Blue Yellow; GBY = Green Blue Yellow. SM3 = Soft Member Type 2

222

rJ

C\j

I-

BOTTOM CHORD MEMBERS

Figure 5.7. Loadiný Attachment Used For -

Mode 171. The photograph sho4s the deviice used to transmit displacements rom the actuator to the top central node of ! nodel 1. The 15 wi thick steel plate rested on top of a 10 mm diameter hardened steel b3ll which was seated in a small cuQ positioned in the top of the node.

Figure b. 8- Loadin, ý Rf-3ifl SY, ý. tem Used Fotý N1011els 2 atil 3. The photoqY'aF77. ýho4_77e heam sy,; t. f-, q jcej tý) 1j)II eqýjajjv folp- (19

, ?I, 43 and 41 ) : )f no]e Isý and 3. Facýi of the fr)tjy- I oaded at the top viaah at, lened s tee I ba IId nd two Pdr- aIIe b r- idgingpIates connected t0qet 11 e r- one ib, ) vein, l Uie other I)e Ioý, q each node. Each )f these four, devices qor- atLaOied ser)at-, jtejy to the f out- ends of tkqo pat' aI le I be a! n-, qh -'I Ci wer e 11 tur, rl 10 3deýj at their, centre by one horzontal c r, o S,; beam. The ; ectio(I cr-oss be arn was Io aded at i ts cen ttý P- thr- otj qh )a II 3nd 1) 1 ate ý nd ir--c tIv 3tt-ichod tn the +-, ý, I it w-

223

Test models 2 and 3 were symmetrically loaded at the four top

joints 19,21,43 and 41 (Figure 5.1). Each of the four joints were loaded at the top via a hardened steel ball and two parallel bridging plates connected together one above and the other below

each node. Each of these four devices were attached separately to

the four ends of two parallel beams, which were in turn loaded at their centre by one horizontal cross beam. The 'I' section cross beam was loaded at its centre through a small ball and plate indirectly attached to the actuator. All of the three beams in the

loading system were arranged to be simply supported so that equal displacements were applied to each of the four top nodes. The

complete loading system, used in the test on model 2 is shown in

Figure 5.8.

Test frame: The test frame used in the experimental investigation

was a modified form of the test frame used by both Butterworth

(1975) and Collins (1981). The frame consisted of eight universal beams (1611 x 51/2' x 26 lbs/ft run) welded and bolted together to

form a square grillage. The models were supported at their four

corners above the test frame on four steel square hollow section

columns, (100 x 100 x5 mm).

Each of the four columns had 15 an thick end plates butt-welded

to them. The columns were then bolted, in the correct location, to

the test frame using four M20 bolts for each connection.

Support conditions: As mentioned previously each of the four models

was supported at its four bottom corner joints, numbers 1,6,61 and 56 (Figure 5.1). All of the four supports were designed to

provide vertical constraint and free rotation about the three

principal axes. However, the support at joint 1 was constrained to

prevent any translation in the horizontal plane, while the supports

at joints 6 and 56 were designed to allow horizontal translation

along one axis only. The support at node 61 was not constrained

against any horizontal translation and was only constrained to

prevent vertical displacement. Figure 5.9 shows the details of the

support at node 6. This support consisted of two flat plates

surfaced-hardened and separated by a bearing race containing hardened steel balls. Two bearing races were also fixed parallel to

each other in the side walls of the device and these constrained all horizontal movement to one direction only. The bottom plate of the

sl Wing support was positioned on four adjustable 12 mm diameter

224

Q) c

M5

77 C)

Q) c

0. -) c: ýýj =F0

N (1)

(L) 7C

a-

7)

> c

Ln Ln c0

-73

C3

4ý

ul)

Ln

0 (71 0- 1-

V)

i Ol -3 Qo

LI; j) eLi

CT -- -C7

(1) Li- r, ý -, >

225

bolts fixed to the top plate of the support column. The end node of the model was supported off the top plate of the sliding support by

means of a spherical end piece which fitted into the node. This end piece was machined, so that the centre of 'rotat-lon of the spherical surface coincided with both the centre of the joint and the point of intersection of the three members meeting at the node. These end pieces were identical to the spherical pieces used by Collins (1981)

and were designed to prevent the vertical support reaction inducing

bending moments on the model structures.

Displacement measurements

The vertical displacements of fourty-three nodes were measured for both model structures 1 and 2. This was achieved by using twenty-three linear variable differential transformer (L. V. D. T. )

transducers and twenty mechanical dial gauges. The mechanical

gauges were used as a precautionary measure in case voltage readings from the transducers were lost due to interrupt. ions to the power

supply. However, no transducer readings were lost during the first

two tests so the mechanical gauges were not used duri'ng the third

and fourth model tests. For these grids the twenty-three L. V. D. T.

transducers were the sole means of measuring node displacements.

Ten D5/2000 and twelve D5/4000 transducers, with working ranges of 100 mm and 200 mm respectively were used to measure the vertical displacements of the unloaded nodes. The vertical displacements of the loaded node was measured using an L. V. D. T. transducer incorporated into the actuator. The twenty-two D5 transducers were held in position beneath the nodes using the adjustable holders developed by Dianat (1979). These transducer holders supported the

main body of the transducers and bolted onto a 10.0 mm thick mild steel plate supported just above the test frame. The moving armatures of the transducers were suspended from the monitored nodes using thin steel wires.

All of the transducers used in this investigation were certified by R. D. P. -Electronics Ltd. to have a linearity better than

± 0.20% over their working range. Before each of the four model tests commenced, every D5 transducer was calibrated separately. The

ten D5/2000 transducers were all switched in turn through one amp I if 1 er. They were calibrated in conjunction with this amplifier over their working range of 100 mm in steps of 5 mm. The twelve

226

D5/4000 transducers were switched in turn through another amplifier and they were calibrated from 0 to 200 = in increments of 10 mm. Several voltage readings were taken at each displacement increment

and the calibration factor for each transducer was calculated using the method of least squares to obtain the best straight line through the data points.

Table 5.2 gives the position of both the L. V. D. T. transducers

and dial gauges used to measure node deflections In each of the four

test models.

Strain measurement

Precision electrical resistance strain gauges have been used to

measure member strains in each of the four model structures. All of the gauges used had a resistance of 350 Ohms and were manufactured by Micro-Measurements Incorporated. Type EA/06/125BZ-350 gauges

were used for the first three test structures and type

CEA/06/125UN-350 gauges were used for the fourth structure. Both

types of gauges are manufactur. ed using a constantan alloy foil in a

se If -temper at ure-compen sated form with the CEA gauges containing integral copper coated terminals making thein slightly larger and

easier to solder than the EA gauges. Table 5.3 gives the strain

gauge sizes, gauge lengths and resistances for both gauge types.

In order to determine the magnitude of the member axial force

and bending moments, two strain gauges were fixed onto each of the

monitored members in both models 1 and 2. Two gauges were also used to measure the strains in selected tension members in models 3 and 4, however the compression members to be monitored in these

structures had three gauges fixed to them instead of two, so that a

more accurate assessment of both the axial forces and bending

moments existing in these members could be made. All of the gauges

were carefully fixed to the members after the models were fabricated. The gauges were cemented to the members using M-Bond

200 and every gauge was protected after installation using M-Coat

D. Both of these products were recommended and manufactured by

Micro-Measurements Incorporated. All of the strain gauges were fixed at the mid-length of the member and where two gauges were used for each member, they were placed diametrically opposite each other

on the top and bottom of the member using a special jig to mark the

location. Where three strain gauges were fixed to one member, they

227

,a s- -0 C) ci (D 4A 'a 14ýpt

4-ý ý0 r- (Y) ro = D. 0 +1

0.010 110 lb-Z (L) E Ol c S- Lfl) 4-j 0= 00 0) I'd u to (1) m C) E

VI -0 4-J r. LO 0) .ýC C; +1 X CY) S- r- S- -ý LO 0

=-ý M +1 c: ) ON -t: r Lf) 4.0 S- 4- o -0 C) r-4 al LCI C) Cý C3. M 4- '0 Q) Ln 0.

Ln S- (1) 4-4 CY) ce) Cý Cý Cý to cli (1) r_ 4-)

ca. to ed E 4--) - L) Qj C: :3

C) 4--) tio Ln S-

CL C: u C: )- .0

LU Woco eo u V) u a) U 4-1

CD C: ux m a) . =3 W +1 CD. (L) S- ý SM 4ý 4- J: ) 0 :3 %n u (0 a la-k (D m0 Irt Lil) 4-3

U Ln 93 C) E S- C) E-= Ln =3 (Y) 4--l 0 4- C; +1

I rO 4- LO 0 r14 S- +1 co Olt r*_ r*ý ILO S- co (1) a -a C) r-4 Lr) Ln LO CD . CL LO C. eo oj C) (u LO

- cli E . 1-i L) CY) m Lr; 14 14 Cý f2 1-4 cu C: fKj 4-J fO 4- E

4-) 1 4- 0a

< CL) 0 CD. r_ 0 LLJ V) u0 to Ca.

0 0 4)

4-ý u

to C)

u 4J Ln 4-) ci a CLS E

L) a

fo 9 Ln (1) to #0 u r. CLI M S- a S- #0 a %A :3 (1) (U 0. W CU 610 > >

C-0 of CD Cl (D C) V)

0) W

4j 4-) 41

(U

45 C; 0

(A ., E to (1)4- 4--) W

'0 0 ed CE M., -

4- (A 0 Q) (1) (A (3) >

(I) a 4A (0 cl. r- 0 :3 0

(A s- - a) o) >

r 0

. 6- CA Q)

C: CX 0 =3 0x r. W CA a (u C: o

ý- ai = to

*0 a kn =p

1ý r. :3 =D (31 - Cl 10 .0 a) Ln L)

LA

C3.

LLJ

.ý 4-S IM C: ) LC)

4-) 4-J CY) 4--l (L) V) Ia

S- 1ý4 *C3 0 c: 4j ca C: r= 0 &A (0

Ln a 4- C%J

Lr) -4 T3 aj 'a A tO "C3 -ý

- (0 (1) (=ý c fo .00.1-1 S-

UJ

228

were accurately positioned equidistant around the circumference of the member, also at mid-length.

Each of the strain gauges was connected with its own high

precision dummy resistor to form a quarter bridge system. Both the

strain gauge and dummy resistor were connected to the measurement system incorporated in the data logger using the five wire system shown in Figure 5.10. The measuring system provides twin, constant- current energising for each of the strain gauges and this was only switched through each gauge during the actual measurement period. This minimised the heating of the gauge and also decreased lead wire errors to a minimum.

All of the output voltage signals from each of the strain

gauged were stored on floppy discs. Ncomputer program was written to convert these voltages into strains using a constant value for

the energising current and the gauge factor corresponding to the

particular strain gauge under consideration. The values of strain were then used to obtain the member axial forces and bending moments which were plotted, to ease interpretation, using the Hewlett Packard T221A Graphics Plotter.

Table 5.4 gives a list of the members which have been strain-

gauged with their corresponding strain gauge numbers for each of the four test models.

Test procedure

Before each of the model structures was tested to fai lure the four adjustable supports located at the top of the four tubular

support columns were accurately levelled using a steel straight-edge and a precision level. When this was completed the model structure was carefully positioned on top of the four supports and the steel wires supporting the transducer armatures were attached to the

monitored nodes. The transducer bodies held in the transducer

supports were then plumbed and the strain gauges wired into the

switching boxes attached to the data logger. Each transducer was independently calibrated and set up to operate over the middle of its working range. The actuator and loading beam system were connected to the correct nodes and the structure given a small

, vertical displacement to check the operation of the test equipment. In addition the voltage readings from each strain gauge were checked

229

S+

B2

s-

Bi

DL RE

G

Figure 5.10. used to obtain was connected energised by mi IIi amps.

S+ - SIGNAL LEAD (+Ve)

B2 = CONSTANT - CURRENT ENERGISING SINK (-Ve)

S- - SIGNAL LEAD (-Ve)

Bl = CONSTANT - CURRENT ENERGISING SINK (-Ve)

G- GUARD AND ENERGISING SOURCE (+Ve)

Strain Gauge Circuit. The diagram shows the circuit a quarter bl'idge operating system. Each strain gauge with its own high precision dummy resistor and was a switched twin -constant current supply set at 5

230

Model I Model 2 Model 3 Model 4

Member Strain Gauge Member Strain Gauge Member Strain Gauge Member Strain Gauge Numbers Numbers Numbers Numbers Numbers Numbers Numbers Numbers

58 39,40 63 9,10 70 20,18,17 70 20,18,17

53 37,38 62 35,36 69 33,34,23 69 33,34,23

48 35,36 61 25,26 66 21.11,12 66 21,11,12

43 33,34 82 1,2 65 26,25,24 65 26,25,24

38 31,32 101 29,30 62 7,40,8 62 7,40,8

33 29,30 184 23,24 61 36,35,22 61 36,35,22

3 27,28 1 31,32 101 29,3,30 101 29,3.30

8 25,26 2 17,13 199 5,6,19 199 5,6 19

13 23,24 3 7,8 82 4,1,2 82 4,1,2

18 21,22 7 21,22 13 37.38 13 37,38

23 19,20 8 37,38 8 13,14 8 13,14

28 17,18 9 39,40 7 39,9 7 39,9

118 15,16 12 19,20 3 15,16 3 15,16

134 13,14 13 11,12 2 27.28 2 27,28

150 11,12 37 S. 6 1 31,32 1 31,32

98 9,10 38 33,34

94 7,8 32 3,4

90 5,6 33 27,28

86 3,4

82 1,2

Table 5.4

Table 5.4. Member Numbers And Associated Strain Gauge Numbers For '19-0-6e] StrRtures-1,2, -3 and 4. Te Ta -e correlates member numbers with- strain gauge numbers for each of the four test model structures. Two strain gauges were used to measure the strain in selected members of models 1 and 2 and three gauges were used to measure the strain in selected compression members of models 3 and 4.

231

and any faulty gauges removed and replaced with new gauges. The

reading from the strain gauges were also checked to ensure that the

model was deforming symmetrically. For the last structure tested, load cells were placed under each of the four supports during the

pre-test load ing sequence. Each of these load cells was independently monitored to check that equal reactive forces were obtained for each support.

Each of the four structures was tested . under displacement

control with an initial displacement of the loaded node of 0.1 mm per minute. This rate of displacement was imposed on the loaded

node by the actuator while the load-displacement response of the

structure was linear. When non-linear behaviour was apparent the

rate of displacement of the actuator was increased to 1.0 mm per

minute. After a total vertical displacement of the actuator of

approximately 100 mm the displacement rate was increased further to

2.0 mm per minute.

The voltage signals from the actuator, transducers and strain

gauges were read and recorded on floppy disc every thirty seconds throughout the test. During each test the load-displacement

behaviour of the structure was continuously plotted on a Bryant x-y

plotter by monitoring both the load and displacement signals

emanating from the actuator.

RESULTS

Component members

A total of ninety-f ive members have been tested to determine

both the physical properties of the different component materials

and the load-displacement behaviour of the individual tension and

compression members. The majority of these tests have been

undertaken on tube types T1, T5 and T6 (Table 5.1) and complement the tests undertaken on tube types T2, T3 and T4 used in the

fabrication of the model soft members, reported in Chapter 4. To

assist in assessing the physical properties of the different

component members, a small sample from e, ach of the six member types

was analysed to determine its chemical constitutents. Also an

additional sample was taken from each of the six member types and used to determine the microstructure of the different steels. Table 5.5 compares the chemical composition of Samples taken from each of

232

c; o 00 Ln CD _A cm

Cý cý c CD Q CD C: )

. 3d Cm u L2 cý - -

," m Cy% CD 3: w

cý Co nr CNJ

c; cli

k- c:, CD CD C-- 0 CU C% CD 3:

cý

c; Ln CNJ m lu CD C: ) CD E r_ . be CD c; c; cý

-0 (7% Ln rz cý 3: 41

. -4 cv c; r- Co C% ci

cý

F- CD CD C) C: ) CD

0 Ln !2 -0 M CU ý.

c;

r- Ln r- m -4 . be CD CD (D

to r, 0ý cz; c; cý c; :

CD 3:

4A 0 w

%A .- CL) E-

.- `0 U, " z (1) ccco 0 3: Co $- M CD-4 CD CD

$.. m (D Ln (D Uli tA «0 41 CA (D C>

vi 1- LL. le (D Q CD

4- 0

(U 1- 0" 0- 0 :3m

V) 4- 4ý NO

cu 41 41 4J c r_ c 0 (L)

4.1 0

- 1; M IA ZU

c 0 u

u 0) In %-

41

M r= -9

c 0) 0 u tn =3 c 0 c m Z

- 0 a) E 3c 0 u i CL m 2 (3. -- 2 ;Z cu r= m0 10 m L. ) c-) - 9-3 V) Ci- tA

-0-A (A 0) ro c (0 C u03:

CA (1) E -4. -j 0) ýc r- eo r-

J_- = r- . 15 4-3 a

'0 0a

to (D ý C)_-E 4-

0

U a) 4- a Ln M 0. ý (A c

tA (0 (L) C:

:; ýr coo -w

C. 0

10c CL) (L) S. - -se CL - ,a

C, > E- "0 4- u

(A u

0 4--) r- 0) cl 10 >1

E C: - 0 C) C:

S- 0 Cl. tA (A Qj = (1)

= 'a -

0 1,5 U0 4ý- z

-ý Ln (L) 4j C--j 0 fo

+j -0 (1) CO c L., ) 0 cn CL E

V-4 0

.0 4- L9 - CU

4-ý 4-) a kA 'a 0 LO - -ý r-4 (D (1) 4-)

tA 3: C: a ro Wo to .ý S-

cl "m ro o

-0 F=

-C 4-) wo to 0 Lc;

S to #10 wo (0

U ý- CL Eo-

233

the five steel tubes, T1 to T5 (Table 5.1) with the specified chemical composition given in Appendix A of BS 6323, Part 1 (1982).

Table 5.6 allows a similar comparison to be made between the

chemical composition of a sample taken from the solid bar, type T6 (Tab Ie5.1), and the specified chemical composition for this type of steel given in BS 970, Part 1, (1983). From the results of the

chemical analysis presented in Tables 5.5 and 5.6 it is evident that

all of the six materials comply with the composition requirements

given in the relevant British Standard Specification.

Figure 5.11 shows photographs of the microstructure obtained from each of the six samples. These samples were first polished and

etched and then viewed under an optical microscope. The photographs

show the grain size of the material at a magnification of two hundred times the actual grain size. The black crystals present in

varying amounts in all the photographs, are crystals of pearlite and their prominence gives a direct indication of the carbon content of the steel specimens. A high density of pearlite crystals shown in

the photographs of the samples taken from both member types T5 and T6 correspond with the carbon content of 0.08% by weight (Tables 5.5

and 5.6) obtained from the chemical analysis of these two samples. The photographs of the specimens taken from member types T2 and T3

both show a decrease in the density of the pearlite crystals when compared with the photographs from member types T5 and T6. Even larger decreases in the occurrence of pearlite crystals are shown in

the photographs of the samples taken from member types TI and T4.

These visual estimates of the concentration of pearlite crystals and hence the carbon content are also supported by the values obtained from the chemical analysis given in Tables 5.5 and 5.6.

The six photographs shown in Figure 5.11 give a good indication

of the physical grain size of the different material. This In turn

gives an indication of the yield strength and ductility of the different materials which tend to improve as the grain size of a particular material decreases. Table 5.7 gives the grain size of

each of the six samples measured in accordance with ASTMS (1948).

The samples taken from the member types T5 and T6 (Table 5.1) have

the smallest grain size of the six samples. In addition they also have the highest carbon content and should therefore exhibit higher

yield and higher ultimate stress values than the corresponding stress values given by the four remaining samples.

234

Chemical Composition ladle analysis % allowable.

Specified properties for 220MO7 bar BS 970, Part 1 (1983).

Tube T6 10 mm outside diameter solid bar.

Carbon content 0.15 max 0.083

Silicon content 0.40 max 0.03

Manganese content 0.90 to 1.3 1.12

Phosphorus content 0.070 max 0.052

Sulphur content 0.20 to 0.30 0.205

Table 5.6. Steel Bar Composition. The Table compares the chemical composi't'l'on 75T'Mined trom bar t pe T6, with the corresponding properties specified in BS970, Part 1 (1983). The chemical composition was obtained from a ladle analysis undertaken by an independent testing laboratory.

Tube Type

(Table 5 1)

Average ASTM Number

Average Number of Grains per mm 2

T1 7.5 1536

T2 5.5 384

T3 5.5 384

T4 7.5 1536

T5 8.5 3072

T6 8.5 3072

Table 5.7. Grain Size Of Tube Test Samples. The Table gives an average valu4 for the grain size of each of the six samples. The grain size is specified using the ASTM index number-which is based on the formula:

Number of grains per square inch = 2N-1 (at a magnification of 100)

where N is the ASTM index number, (ASTM, 1948).

235

% Aý

TUBE TI

a

. '.

'T ',,

TUBE T2

TUBE T3

l"Oure 5 11 Steel Microstructure The photograph shows the steel microstructure of each of the six tube types (Table 5 1). From the individual photographs an estimation of the material grain size can be obtained. The black crystals present in varying amounts in all the pholographs, are crystals of pearlite and their prominence gives a direct indication of the carbon content of the steel specimens. The black strip shown at the top of the photographs of the specimens taken from tubes T4 and T5 is bakelite material used to back the samples. The photographs were taken at a magnification of two hundred.

TubeT1 4 76 mm outside diameter 0.91 mm wall thickness. TubeT2.6 35 mm outside diameter. 1,22 mm wall thickness. TubeT3 7 94 mm outside diameter 0 91 mm wall thickness TubeT4 9 52 mrn outside diameter. 0.71 mm wall thickness. TubeT5 9 52 mm outside diameter. 0 91 mm wall thickness. BarT6 10,0 mm outside diameter Solid bar.

TUBE T4

07.

.,, ". I. 1,:, 0

236

TUBE T5

:-+ li: y

- __"__i --

lzm:

BAR TS

Tension behaviour: Figures 5.12 A and B show typical tensile load-strain relationships obtained from tube types TI and T2 (Table 5.1). The small diameter tube, (4.76 mm) type T1, was used for the lower chord tensile members in model structures 1,2 and 3 while the

slightly larger diameter tube, (6.35 mm) type T2, was used for the bottom chord tensile members in model 4 only. Both of the load-strain relationships given in Figure 5.12 show a definite yield point followed by a constant load plateau. However, the

relationship exhibited by the type T2 members gives a higher value for the ultimate-stress to yield-stress ratio than the type T1

members.

Figures 5.13 A and B show the tensile load-strain behaviour

obtained for member types T5 and T6 (Table 5.1). The tubular T5

members have been used to carry both tensile and compressive forces in all of the four model structures. In particular all of the top

chord members in models 3 and 4, apart from the eight soft members in each structure, have been fabricated from this tube size. The

solid bar members type T6 have been used for the top chord compression members in both models 1 and'2. The typical load-strain behaviour obtained from the tubular T5 members shows both an upper and lower yield stress followed by a load plateau and an increase in load carrying capacity due to strain hardening (Figure 5.13A). However, the tensile characteristics of the 10.0 mm diameter solid bar, member type T6, d id not exh lb it a def in Ite yield po Int and failure of the test samples occurred after a short load plateau, at an average total strain of 4.7% (Figure 5.13B).

Table 5.8 gives the values of the elastic modulus, upper yield stress, ultimate stress and the percentage elongation of the gauge length after fracture, obtained from the tensile testing of member types T1, T2, T5 and T6. Table 5.9 summarlses these results and compares the mean of the experimental values of yield stress, ultimate stress and percentage elongation with the minimum values. given in the relevant British Standard Specifications BS 6323, Part 1 (1982); BS 970, Part 1 (1983). It can be seen from the values given in Table 5.9 that the yield stress of each 'of the tubular

members plus the solid bar is significantly greater than the minimum specified yield stresses of 170 N/mm2 and 400 N/mm2 respectively. However, the average percentage elongation at failure of each of the different member types is less than the minimum specified values. The apparent discrepancy is due to the difference in gauge length

237

lo

7.5

2.5

PERCENTAGE STRAIN

rTr, TjRE -S--122 SMALL HOLLOW TUBE (WHITE)

8

--i

rTr, uRF, -, i, 1Za MODEL 4 RWY

Fi ure 5.12A and B. Tensi le Load-Displacement Relationships I- From Abit ua

Min e UFF i (1 13

27 rom 7u e 7_ýypes 71 And T 2. The Figure shows typical tensile loa2-displacement behaviour Wtained from tube types T1 and T2. The small diameter tube (4.76mn) type T1 was used for the lower chord tensile members in model structures 1,2 and 3 while the slightly larger diameter tube (6.35mm) type T2, was used for the bottom chord tensile members in model 4 only. All of the tensile specimens were tested under displacement control at an initial strain rate of 0.063% per minute. This rate was increased to 0.63% per minute after approximately 4% strain had occurred in the specimens.

238

5 10

2468 le .

12 H Is 18 28 22

Percentage Strain

40

: k:

30 6-4

1-4 20 0

10

FIGURE 5.13A PERCENTAGE STRAIN

LARGE HOLLOW WHITE A (TENSION)

40

30

Q 20

10

5 10

FTGURE-5-. -U3. PERCENTAGE STRAIN

SOLID RED WHITE (TENSION)

B. Tensi e- Loa Disp acement Re at onships Obtained From lube Types lb -AnT767. The Figure sh i typical

ensi e oa - isp acement ehaviour obtained from tube type T5 and bar T6. The tubular members T5 have been used in the top chord of models 3 and 4 and also as web members in all four models. The solid bar members T6 have been used in the top chord of models 1 and 2. All of the tensile specimens were tested under displacement control at an initial strain rate of 0.063% per minute. This rate was increased to 0.63% per minute after approximately 4% strain had occurred in the specimens.

239

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241

used 1n the material specifications and in the exper ! mental testing. All of the experimental values were based on a gauge length of 331 mm while the minimum specified values are calculated

using a gauge length of 5.65 ýSo, where So is the original cross

sectional area of the gauge length.

The mean values for the yield stress and ultimate stress

presented in Table 5.9 also agree with the relative member strength

predicted from both the chemical analysis- and grain size

comparison. Member types T5 and T6 which have the smallest grain

size and highest carbon content of the six samples also exhibit the

highest average yield stress, 0.5% proof stress and ultimate stress.

Compression behaviour: A total of twenty tubular type T5 members (Table 5.1) and tWenty-seven solid type T6 members have been tested

to failure in compression. Figures 5.14 A and B show typical

compressive load-strain relationships obtained from type T5 and T6

members respectively. Both member types exhibited "brittle type"

strut buckling behaviour characterised by rapid load shedding after buckling. Although the ultimate load of member type T6 is

approximately three times greater than the ultimate load of member type T5, the post-buckling behaviour of both member types Is very

similar. This can be seen by comparing the average residual loads

of both member types, measured at a total strain of 15%, which are both almost one tenth of the corresponding ultimate load of the

members.

Table 5.10 gives the value of elastic modulus, ultimate load

and critical flexural buckling stress for the fourty-seven members tested In compression. Member types T5 have a slenderness ratio of 52.50and a transition slenderness ratio of 77.16 calculated using an

average tensile yield stress of 331.70 N/MM2 (Table 5.9). Because

the actual slenderness ratio is less than the transition slenderness

ratio, these members should fail by plastic buckling at a stress

close to the compressive yield stress of the material. The mean

critical flexural buckling stress for the members, obtained from the

results presented in Table 5.10, is 319.24 N/MM2 with a standard deviation of 27.91 N/mm2. This compares favourably with an

estimated critical flexural buckling stress for these members of 290.72 N/MM2, calculated using the "Perry" formula represented by

equations 4.1 to 4.9, (Chapter 4) with an assumed initial member bow of L/1000 and a yield stress of 331.70 N/MM2.

242

12.5

PERCENTAGE STRAIN

Finum . 14A LARGE HOLLOW RED A -,

5

PERCENTAGE STRAIN

rTGURE 5-14B SOLID RED A

10

7.5

2.5

25

20

it:

15

10

5

Figures 5.14A and B. Compressive Load-Displacement Relationships For lube Types T5 And T6. The Figure shows typical compressive load displacement relationshi s obtained from tube type T5 and bar T6. Both member types exhibited 'brittle type' strut-buckling behaviour characterised by rapid load shedding after buckling. Al 1 of the specimens were tested under displacement control at an initial strain rate of 0.063% per minute. This rate was increased to 0.63% per minute after approximately 4% strain had occurred in the specimens.

243

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244

The 10.0 mm diameter so] id bar members, type T6, have a

slenderness ratio of 64.28 and a transition slenderness ratio of 61.90 calculated using the average 0.5% proof stress of 515.11

N/mm 2- With the actual slenderness ratio of this member so close to its transition slenderness ratio, the critical flexural buckling

stress will be significantly less than the compressive yield stress

of the material. The mean critical flexural buckling stress for the

members, obtained from the, experimental investigation is 312.68

N/mm 2 with a standard deviation of 25.73 N/mm 2( Table 5.10).

Because the tensile load-strain relationships of member type T6 (Figure 5.13B) shows no definite yield point, the theoretical

critical flexural buckling load for these members is calculated

using the tangent modulus, Et, corresponding to the stress in the

member when it is supporting its critical load. The tangent modulus

stress at is then given by:

=2 Et ............. (5.1) (Shanley, 1947).

L2

r)

As the theoretical critical load is unknown, this equation can

only be solved by' an iterative process using the tensile

stress-strain relationship for the material (Figure 5.13B).

Substituting for the known slenderness ratio of 64.28 equation 5.1 becomes:

at = 0.00238 Et ............. (5.2)

To solve equation 5.2 iteratively a value of the stress at Is

assumed and the tangent modulus corresponding to this chosen stress level is calculated from the experimental data, by averaging the

strain readings obtained from the dual extenometer. The values of

at and the corresponding values of Et are then substituted into

equation 5.2 until the equality is satisfied. Table 5.11 shows the

iterative sequence using the test data obtained from the solid blue-white-B specimen.

245

Chosen Et from at from at(N/mm 2) test data equation 5.2

(N/mm? -) (N/mm2)

375.80 1.8526xlOs 440.94

381.97 1.7544105 417.56

399.87 1.6728xjO5 398.12

Table 5.11. Iterative Procedure Used to Obtain The Theoretical Critical Flexural Buckling Stress Of MemBer Types-767. -

To solve equation 5.2 (at = 0.00238Et) iteratively values of at are assumed and corresponding values of the tangent modulus Et are obtained from the tensile stress-strain relationship exhibited by the material. The values of at and Et are then substituted into equation 5.2 until the equality is satisfied.

From Table 5.11 it can be seen that the tangent modulus stress

at lies between the two values of 399.87 and 398.12 N/mm2. Us ing

a mean value for the tangent modulus stress of 399.0 N/MM2 gives a

mean value for the flexural buckling load of the member of 31337.0

N. The tangent modulus buckling formula gives the axial load at

which an initially perfect column begins to deflect laterally and for design purposes this value must be decreased to allow for

imperfections. This can be achieved by using the "Perry" formula

and a modified value for the yield stress ay (Calladine, 1973).

Calladine proposed that the modified value of yield stress, a* which

corresponds to a tangent modulus Et = ME is located from the

material stress-strain curve. An estimate of the member critical

buckling load accounting for an initial column curvature can then be

obtained by substituting a* for cy in the "Perry" formula.

Using the experimental data obtained from the member tests, the

mean stress value of a* corresponding to a tangent modulus Et = ME was found to be 477.63 N/mm2. Using this value to represent the yi eld stress In equations - 4.1 to 4.9, the average member buckling stress, assuming an initial member bow of L/1000 was calculated to be 277.75 Nlmm 2 This yields a buck] Ing load of 21814.43 N which will be a lower bound to the mean value of 24557.59 N obtained from the experimental investigation.

246

Model 1

The first double-layer space truss model tested to collapse was designed to exhibit a ductile load-displacement behaviour by permitting extensive tensile yield to occur in the lower chord members. All of the bottom chord members in this structure were type T1 members (Table 5.1) and all of the top chord members were type T6 members. The web members, connecting together the top and bottom chords were all type T5 members (Table 5.1) except for the four diagonal web m' embers in the centre of the structure and the four corner web members, al I of which were the so] id bar, type T6

members. The structure was loaded only at the top central node by the 50 kN actuator working under displacement control. The vertical displacements were measured throughout the tests at 43 nodes and in

addition the strain was measured in twenty members. Table 5.2 lists the nodes in the structure at which the displacements were recorded and Table 5.4 lists the members which have been strain gauged. Figure 5.15 shows the model structure under test.

Linear Behaviour: As the imposed central point load acting on the

model space truss was gradually incremented from zero, the structure behaved elastically exhibiting almost identical deflections at

symmetrically positioned nodes. Table 5.12 compares the measured deflections of several nodes, symmetrically positioned in the space truss, with their theoretical values when the structure 'was

supporting, an imposed load of 7458 N corresponding to the theoretical limit of its elastic behaviour. Table 5.13 gives both the theoretical and experimental values of deflection for all of the

nodes monitored by transducers, for two values of the imposed load. The lower load of 4052 N is approximately In the middle of the

elastic range, while the higher load of 7458 N corresponds to the

end of the linear elastic range. Both Tables 5.12 and 5.13 show all of the deflections to be within 11% of their theoretical values, but

with the measured deflections of the boundary nodes generally greater than their theoretical values and the measured deflections

of the central nodes slightly less than their theoretical values. However, the measured deflections at the loaded central node, number 31, go against this general trend, with the experimental

measurements apDroximately 2% greater than the corresponding theoretical values. This apparent discrepancy is probably due to

smal I add it ional d ispl acements ar is 1 ng f rom bear 1 ng stresses occurring between the threaded bars and nuts and also the node and

247

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to 41 uu m

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248

Total Imposed Load 4052 N

Total Imposed Load 7458 N

Theoretical Experimental Theoretical Experimental Node Vertical Vertical Node Vertical Vertical

Numbers Displacements(mm) Displacements(mm) Numbers Displacements(mm) Displacements(mm)

31 1.960 2.007 31 3.607 3.668 37 1.746 l. e64 37 3.224 3.145 26 1.746 1.572 26 3.214 2.994 25 1.746 1.598 25 3.214 3.001

27 1.430 1.335 27 2.631 2.561

15 1.430 1.345 15 2.631 2.527

38 1.430 1.300 38 2.631 2.509

14 1.430 1.313 14 2.631 2.450 28 1.111 1.144 28 2.044 2.225 4 1.111 1.107 4 2.044 2.070

39 1.111 1.113 39 2.044 2.170 3 1.111 1.119 3 2.044 2.062

17 0.748 0.791 17 1.376 2.516 5 0.748 0.798 5 1.376 1.474

32 1.686 1.506 32 3.104 2.938

20 1.686 1.588 20 3.104 2.969 33 1.336 1.274 33 2.459 2.471 9 1.336 1.291 9 2.459 2.412

22 1.161 1.163 22 2.137 2.234 10 1.161 1.152 10 2.137 2.159 21 1.494 1.325 21 2.750 2.559 26 1.145 1.113 16 2.207 2.099 11 0.679 0.731 11 1.249 1.349

Table 5.13. Comparison Of Theoretical And Experimental Deflections I nitored Rot"- A 11 140-1 Node-s-7 Model I. Ihe lable gives both

5ýeoretical ind- experimentally measured vertical displacements for aII of the nodes monitored by transducers for two values of the imposed load. The lower load value of 4052N causes the structure to be approximately in the middle of its elastic range of behaviour while the higher load of 7458N corresponds to the theoretical end of the linear elastic range.

The Table shows all of the experimentally measured deflections to be within 11% of their theoretical values but with the measured deflections of the boundary nodes genernally greater than their theoretical values and he measured deflections of the central nodes slightly less than their theoretical values.

249

I! 1)

F-

ci,

0

If)

I'-)

u

,x0

250

hardened steel ball, all of which were used in the loading system positioned between the deflecting node and the recording transducer.

Figure 5.16 gives the member numbers and also values of the

member stress ratios for the model structure at the end of the Hnear elastic range. The stress ratio is a ratio of member stress to yield stress for tension members and a ratio of member stress to

critical flexural buckling stress for compression members. From the figure it is evident that members 62,63,78-etc. are the most highly stressed compression members with a stress ratio of 0.13 and members 13,18,43 and 48 are the most highly stressed tension

members, with a stress ratio of 1.00.

As the magnitude of the central point load acting on the

structure was increased, the actual load-displacement behaviour

exhibited by the model space truss followed very closely the theoretical behaviour predicted by the analysis procedure. Figure 5.17 shows the relationship between the applied load and both the

theoretical and experimental vertical displacements of the top

central node of the test model. The drop in appi led load and subsequent elastic recovery occurring in the experimental results, just beyond the elastic I imit of the structure, was due to a power failure occurring at the hydraulic pump supplying oil to the

actuator.

Theoretical Non-Linear Behaviour: The theoretical analysis of the

structure predicted that yield in the tension members would spread from the centre of the grid outwards towards the sides until two

complete yield lines had formed symmetrically across the bottom

chord of the truss structure. The f irst set of members to yield

were members 13,18,43 and 48 (Figure 5.17) under an Imposed load

of 7458 N acting on the structure. As the Imposed load was increased, members 8,23,38 and 53 yielded followed by members 3,

28,33 and 58 at the corresponding loads of 8701 N and 9447 N

respectively. At this stage, If strain hardening Is ignored in the

tension members, and they are assumed to remain continuously on a

constant load plateau, with zero elastic stiffness, the theoretical

structure becomes a mechanism and the numerical analysis cannot

continue. However, if strain hardening of the tension members is

taken into consideration, the members possess a relatively small elastic stiffness and the analysis can proceed. The next members to

251

0.10 0.13 0.13 0.10

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C,

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0.01 0.09 0.09 0.01

89 so 91. 92 a) r, M

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0.02 0.07 0.07 0.02 85 66 87 68

(14 CD FD CD ý ; CD LO C (ý Q C

0.10 0.13 0.13 0.10

D

81 82 83

0-91 0-73 A -71-; -C; l

56 57 58 59 60

A CD ý Z

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CD cý Z

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q1

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0.00 0.52 0.77 0.52 0.00 51 52 53 -5 di -ý b

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Co (ý m uý

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0.00 0.28 1.00 0.28 0.00 116 A7 A8 ti u

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OD rý N

N LO cy

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0.51 0.73 0.79 1 0.73 1 0.51

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BOTTOM GRIT) 31 32 33 31,

ical Stress Ratios And Member Numbers For Model

st Yield. The diagram shows the member numbers and

critical stress ratios for both the top and bottom chord members of model 1. The model was loaded at the top central node and supported at the four corner nodes of the bottom chord.

For tension members (bottom chord): member axial stress

Critical Stress Ratio = yield stress

For compression members (top chord): member axial stress

Critical Stress Ratio = flexural buckling stress

252

20

15

CL

-. a--

cs

-6-> C)

F--

5

50 100 150 200

Vertico. L Disptacement in mm.

Figure 5.17. Total Applied Load Vs Central Node Displacement For Mode II. Ie diagram shows both the theoretical and experimental verticaT- displacements of the loaded top central node of model 1. The actuator imposed a vertical displacement on node 31 of 'J. 1m, n per minute for the first 26min of displacement after which the deflection rate was increased ten fold to 1.0mm per minute. 'When the total vertical displacement of the actuator was approximately 100m, n the displacement rate was increased further to 2. Omm per minute. The drop in applied load and subsequent elastic recovery occurring in the experimental results, just beyond the elastic limit of the structure, was due to a power failure occurring at the hydraulic pump supplying oil to the actuator. The model structure was analysed as a gin-jointed soace truss and the theoretical displacements were obtained using a modified form of the Dual Load method accounting for changes in model geometry. Points on the theoretical load-displacement curve.

A= Mid-point in the-elastic response. B= Elastic limit. Yield of members 13,18,43,43. C= Yield of members 8,23,38,53. D= Yield of members 3,28,33,58. E= Yield of members 2,4,57,59, etc. F= Yield of members 7,9,52,54, etc. G= Change in stiffness for members 7,9,52,54, etc. H= Buckling of compression members 70,71,90,91.

253

yield are members 2,4,57,59 etc. and then members 7,9,52,54 etc. at loads of 11435.6 N and 12430 N respectively. Yielding continues in the bottom chord members until the imposed load reaches a value of 13325 N when the compression members 70,71,90 and 91 should theoretically become unstable and buckle. This last value of load corresponding to the buckling of the first compression members was very dependent on the value of post-yield stiffness given to the last group of tension members to yield. At this point in the

analysis, the stiffness of the entire structure was very small in

comparison with its original stiffness and the force residuals generated in the non-linear analysis computer program at first tended to decrease and then converged towards a minimum, but later increased and diverged if the number of analysis cycles was increased. Due to this numerical instability in the analysis program, resulting from the small stiffness of the structure, this last theoretical value, taken when the force residuals were at a minimum, should be accepted with caution.

Experimental Non-Linear Behaviour: As the imposed load acting on the structure was increased, yield spread through the bottom chord members as predicted, but the structure did not begin to fall by the

buckling of one of the compression members as indicated by the

theoretical analysis. Instead, the first ýember to fall was the bottom chord tension member 43 (Point A, Figure 5.18). This member ruptured when the model structure was supporting an imposed load of 12699.13 N. Failure of member 43 was followed by the tensile failure of member 38 and then the failure of tension member 33 (Points B and C respectively, Figure 5.18). As the bottom chord of the structure began to open up due to the three failed tension

members, the stress in member 58 reversed and this member failed in

compression (Point D, Figure 5.18). At this point, the structure

was capable of supporting an imposed load of 3410.73 N,

approximately 27% of the maximum load carried by the structure. A

short time after member 58 had buckled, member 48 failed in tension (Point E, Figure 5.18) and the imposed load supported by the

structure immediately decreased to 2156.35 N (Point F, figure 5.18). After member 48 had failed, the imposed load supported by the structure slowly began to recover (Points F to G, Figure 5.18). The top chord of the model structure had deflected significantly and the structure was supporting the additional load as a catenary. The testing of the structure was halted when the displacement capacity

254

TOTAL APPLIED LOAD vs NODE DISPLACEMENT CURVES OBTAINED FROM MODEL 1

28

SULT

15

:Z

C CD 1--

Verticat DispLacement in mm.

Fiqure 5.18. Experimental Load-Displacement Behaviour Obtained From Moae i I-TK-e- Figure shows the experimental load-displacemenE behaviour obtained from Model 1. The structure was tested under displacement control with an initial imposed vertical displacement on node 31 of 0.1mm, per minute. The points (A to E) shown on the Figure, correspond with the failure of individual members leading to the complete collapse of the structure.

Point A Tensile failure of member 43 B Tensile failure of member 38 C Tensile failure of member 33 D Compression failure of member 58 E Tension failure of member 48

Points F to G show a gradual recovery in the stiffness of the structure as excessive vertical displacements allow the structure to behave as a catenary.

255

so lea 158 283

of the actuator was at its limit and the total vertical deflection

of the loaded node was 200 mm. At this stage the structure

supported an imposed load of 3697.44 N.

Figures 5.19 A, B and C show how the theoretical and

experimental values for the strain in the tension members 18,23 and 3 varied with an increase in the external load. The Figures show the recorded strain from both of the strain gauges which were

positioned centrally and diametrically opposite. each other on each

member. Examination of the numerical results of the recorded strain indicate that yield in these members followed the proposed theoretical sequence with member 18 yielding first, followed by

members 23 and 3. From the plots shown in Figure 5.19 it is

apparent that bending effects are negligible in these members when they have yielded in tension. Unfortunately, the members in the

fourth and fifth groups to yield were not strain-gauged so it was

not possible to monitor their behaviour or determine if they yielded in the predicted theoretical sequence.

Figures 5.20 A, B and C show how both the theoretical and

experimental values of axial stress in compression members 98,94

and go change with increasing external applied load. It Is

interesting to note that as the external imposed load increased and the bottom chord members yielded and exhibited plastic flow, the

force in the most heavily stressed compression members Increased at

a rate much greater than the rate of increase In the external

applied load. Figure 5.20 A shows that for a 5% increase in

the external imposed load there was a 14% theoretical and 12%

experimental increase in the axial force carried by compression

member 98. Figure 5.20 C shows a similar behaviour for member 90,

where a 5% increase in the external load after first yield caused a 16% theoretical and 15% experimental increase in the axial

compression force carried by the member. However, these sharp increases in axial force, corresponding to small increments In the

imposed load exhibited by members 98 and 90, were not shown In the

behaviour of compression member number 94. ý Figure 5.20 B shows that

the axial force in member 94 decreases significantly, with increases

in the imposed load after the first three sets of tension members have yielded, forming the two complete yield lines across the bottom

chord of the structure.

256

TOTAL APPLIED LOAD vs MEMBER STRAIN CWVES-' OBTAINED FROM MODEL 1

(Positive Strains are TensiLe) 15

..............

MEMBER NO. 18 Cl_

EXPERIMENTAL RESULTS Strain Gauge No. 21

-------- Strain Gauge No. 22 ANALYTICAL RESULT

F'TQURE 5-19A

C: 9

23 Percentage Strain

-

MEMBER NO. 23 EXPERIMENTAL RESULTS Strain Gauge No. 19 Strain Gauge No. 20 ANALYTICAL RESULT

FIGURE 5.19B

k21 ,n

23

Percentage Strain

v-. ... 0

MEMBER NO. 3 EXPERIMENTAL RESULTS Strain Gauge No. 27 Strain Gauge No. 28 ANALTTICAL RESULT

3

FIGURE 5.19C 12343 Percentage Strain

Fiqure 5.19A, 'B and C. Experimental And Theoretical Applied Load Vs Member strain Curves Ubtalned For Model 1. lhe I-igures show how-77-Fe theoretical and experimental values of the strain in the tension members 3,18, and 23 vary with an increase in the external applied load. The experimental values were obtained from each of two strain gauges placed diametrically opposite each other at the centre of each member.

257

28

15

8

«o le

-10 -28 -36 -10 -58 -08 -78 -60 ýA -lK Axiat Stress (N/sq. mm)

FIGURE 5.20A

(Positive Stresses are TensiLe) 201 MEMBER NO. 91

- EXPERIMENTAL RESULT --- 0--- ANALYTICAL RESULT

15

rý

-0

a

58 is 30 20 is -10 -20 -30 -40 -58

FIGURE 5-20B Axial Stress (N/sq. mm)

20

15

-c -ti la

. e:

(Positive Stresses are Tensite)

-10 . 20 -3e -io -58 -0 -78 -0 -106

Axidt Stress (N/sq. ra) FTGURE 5.20C

d C. Experimental And Theoretical Applied Load Vs

Member Axial Stress rurves For Model 1. T igures show how botF the theoretical and experimental va s of stress in compression members 98,94 and 90 change with increasing external applied load. The axial stress was obtained for each member by averaging the

strain readings obtained from two strain gauges placed diametrically

opposite each other at the centre of each member.

258

(Positive Stresses are Tensite)

Figures 5.21 A, B and C give the experimental load-strain

relationships obtained from each of the two strain gauges fixed at the centre, diametrically opposite each other, on the three top

chord compression members 98,94 and 90. It can be seen from these figures that yielding in the structure was accompanied by bending in

these compression members. The maximum strain recorded frorn strain

gauge number 5 positioned on member 90 was almost 0.186%

corresponding to a stress of 379 N/MM2. This st ' ress level is below

the mean value for the 0.5% proof stress of the material (515.11

N/mm2 Table 5.10), indicating that these members were capable of

supporting a small additional axial load and bending moment before

failure.

Model 2

The second double-layer space truss model tested to collapse

was also designed to exhibit a ductile post-elastic load-displacement response. The second model structure was almost identical to the first model except that the four diagonal bracing

members in the centre of the structure were type T5 members (Table

5.1) instead of the solid type T6 members used in the first model

structure. The second model space truss was loaded simultaneously

at four top chord nodes, numbers 19,21,41 and 43. The four point imposed load was applied through the simply supported beam system,

shown in Figure 5.8, using the 50 kN actuator working under displacement control. The vertical displacements were measured at 43 nodes (Table 5.2) and the strain was measured in eighteen members (Table 5.4). Figure 5.22 shows the model structure under test.

Linear Behaviour: In order to determine if the structure behaved

symmetrically under load, the vertical displacements of several

nodes symmetrically positioned in the model structure were

measured. Table 5.14 gives both the theoretical displacements and the measured displacements for groups of nodes which should theoretically exhibit identical vertical displacements when the

structure is loaded by four symmetrically positioned point loads

each of 2000 N. The measured displacements given in Table 5.14 show

relatively small differences occurring In the experimental values

obtained from nodes in the same group. The largest difference of 13.5% occurring between the measured displacement of nodes 17 and 5. Smaller differences obtained from similar nodes in other groups indicate that, in general, a symmetrical respon. se, was obtained from

259

20 WEER NO. 98 28

EXPERIMENTAL RESLLTS Strain Gouge No. 9

-------- Strain Gouge No. 10

2

0 -i la . -0 -2: --Z

-3 -2

FIGURE 5.21A

1_f__f II

.11

Percentage Strain (ICE-01)

" is

"5

23

20 mom NO. 91 28

EXPERIML RESLLTS Strain Gauge No. 7 Strain Gauge No. 8

le

, 3. -2 .1123

FTGURE 5.21B Percentage Strain ME-el)

TOTAL APPLIED LOAD vs MEMBER STRAIN CURVES OBTAINED FROn MODEL I

20 MEMBER NO. 90 29 EXPERIMENTAL RESLUS Strain Gauge No. 5

..... .. Strain Gauge No. 6 15 Is

RI

-0 Iq

-3 -2 -t 3

PT(; uRE 5.21C Pwcentage Stmin OKAD Figure 5.21A, B And C. Experimental Applied Load VS Member Strain Curves Obtained Rro-m-7odel 1. The Figure Shows how the strain 7 compression members 98, Tr and 90 varies with an increasing externally applied load. The experimental values were obtained from each of the two strain gauges placed diametrically opposite each other at the centre of each member. The Figure shows that yielding in the strqctur. e was accompanied by bending- in these compression members .

Tensile strains are shown as Positive. 260

Figure 5.22 Model 2 Under Te-, t.

261

mstý -71 -

IF E

c LO 0 -cr LO

Lr! M 4ý uu cli

.- (U AJ - S- 4- cu (U

cn C) Ci

uu cli cli .-W 41 - S.. 4-

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0 =3

LLJ

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C%j m CYN cn C) -4

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(1) Ln %0 tn 4D CN C5 d)

-R ý4 -tt cli cli -4 -0 0 :3

E A

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; ., o 41 cl; Cý Cý Cý cl Cý

S- 4- 4)

r- co r- cli 4" cli 0)

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m

LU

0

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LC) tD C> C)ý Ln cli (n

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cli

u 4ý

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>> 4- -o (L) 0) 0. CL

Ln CL

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V, to .0 .0 -i-. ) to - LA C ý- ro to (: ) (31 u E

aj (L) S- cu 0) =3 CL > (1) 4-J '&j X. -0 UU LLJ C%J

-a o :3= QJ E-= S. -

&- s- 1.., 4-3

tA C

urE

-4--) >, -C3 C0 'A 0 C: .ý I-

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-0 UW ý- aEr (L) 4- (L) ý04- 8a s- S. - C) o

o C) C)

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E=- 4-- to 010 0

Cl. a Q) CL 10 u Ez to ., - 00

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u U., A-0- (1)

LA

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Lr; E0 (L) E ., 4-1 A (0 4-

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F. L. -IJ 4- '0 OJ (u

262

the model, provided that the imposed load acting on the structure

maintained the node deflections within the elastic range. Table

5.15 gives both the theoretical and experimental values of deflection for all of the nodes monitored by transducers. The node displacements given in Table 5.15 correspond to two separate values

of the total imposed load. The first load value of 8000 N

corresponds with theoretical displacements which are still within the elastic response of the structure, while the second load value

of 10240 N corresponds to the theoretical end of the linear-elastic

behaviour of the structure. A comparison of the theoretical and

measured displacements given in Table 5.15 show all of the

experimental values, obtained when the structure was supporting a total load of 8000 N, to be within 18% of their theoretical values. However, this large discrepancy occurring between the theoretical

and experimental displacements was only apparent in three of the

lower chord edge nodes, (3,4 and 5), which were adjacent to each

other on the south side of the model structure. The measured displacements of the remaining nodes were within 9% of their

theoretical values, with the measured deflections of the boundary

nodes approximately 4% greater than their theoretical values, and

the measured deflections of the central nodes approximately 5% less

than their theoretical values.

The node displacements, measured when the model structure was

supporting a total applied load of 10240 N corresponding to the end

of the I ! near elastic range, were aII greater than their

corresponding theoretical values. The measured displacements of the

three lower chord edge nodes (3,4 and 5) were approximately 25%

greater than their theoretical values, while the measured deflections of the remaining nodes were, on average, 16% greater

than their theoretical values. As all of the measured displacements

were greater than the theoretical values it is evident that the

elastic limit of the structure was exceeded when it was supporting

the total imposed load of 10240 N.

Figure 5.23 gives both the member numbers and the values of the

member stress ratios for the model structure at the end of the linear elastic range. It can be seen from the Figure that members 3,28,33 and 58 are the most heavily stressed tension members with a stress ratio of 1.02, indicating that the force in the member Is just sufficient to cause the member to yield. However, the

compression members in the. model .

structure are understressed In

263

Total Imposed Load 10240 N

Total Imposed Load 8000 N

Theoretical Experimental -Theoretical Experimental Node Vertical Vertical Node Vertical Vertical

Numbers Displacements(mm) Displacements(mm) Mrnbers Displacements(mm) Displacements(mm)

S 31 3.780 9.803 31 2.949 7.674 37 3.660 4.441 37 2.855 2.824 26 3.660 4.265 26 2.855 2.732 25 3.660 4.342 25 2.855 2.739 27 3.333 3.730 27 2.601 2.408 15 3.333 3.875 15 2.601 2.410 38 3.333 3.795 38 2.601 2.394 14 3.333 4.048 14 2.601 2.545 28 2.860 3.613 28 2.232 2.478

4 2.860 3.836 4 2.232 2.623 39 2.860 3.701 39 2.232 2.404

3 2.860 3.956 3 2.232 2.651 17 1.893 2.291 17 1.477 1.575 5 1.893 2.531 5 1.477 1.797

32 3.646 3.839 ss 32 2.844 2.410 20 3.646 4.421 20 2.844 2.837 33 3.231 3.879 33 2.521 2.476

9 3.231 4.177 9 2.521 2.703 22 2.848 3.514 22 2.222 2.377 10 2.848 3.501 10 2.222 2.311 21 3.518 7.140 21 2.745 5.252 16 2.777 3.162 16 2.167 2.032 11 1.689 2.086 11 1.317 1.411

Key: Displacement Measured By An LVDT Transdu&er Incorporated Into The Actuator. Displacement Readings Not Accurate Due To The Interference Of The Cross-Beam (Transducers Not Attached To The Nodes But Attached To The Cross-Beam).

Table 5.15. Comparison Of Theoretical And Experimental Deflections 'For---KTT- Monitored Nodes in Model 2. The Table gives both theoretical and experimentally measuFed vertical displacements for

all of the nodes monitored by transducers for two values of the imposed load. The first load value of 800ON corresponds with displacements which are still within the elastic response of the structure while the second load value of 10240N corresponds to the theoretical limit of the elastic behaviour of the structure.

The Table shows all of the experimental values obtained when the structure was supporting a total load of 800ON to be within 18% of their theoretical values. In addition the measured deflections of the boundary nodes were generally greater than their theoretical values and the measured deflections of the central nodes were slightly less than their theoretical values.

264

cs

a

(S

0

a

Cs

cs

S112 0-1 rl A-1 rz a V2

97 98 99 100 rl'

0 CD 0 CD r-, (Z N CD

0.02 4f 0.05 0.05 -9 0.02 93 85 96

CD C; D CD

0.02 0.02 0.01 63 90 91 92

r9 cý Cý

im (D (o CD N CD N CD

0.02 0.03 o. es 0.02 85 4 86 G7 -e 68

rIli - r%, (N lý tr) . C) :ý ,1 70 CD Lo V, tD CZ f% cz

L- 0.12 0. IS 0.16 0.12

0

7)

:0 N

N.

81 u e- U15

91-70 A- ý17 1 -n, 2 A- q7 56 57 58 59 60

Cs t,

0.00 0.70 0.69 0.70 0.03 51 52 53 5 55

C: Cý

0.00 0. "11 0.38 0.411 0.00 ý6 "17 lei 6 418 00

0ý m M 0) N

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0.00 0.41 0.38 0.41 O. ea 111 412 413 144 A15 --

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r- C)

CD "y

0.00 0.70 2.69 -

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(S) Cý ýD 9

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BOTTOM GRTD

Fiqure 5.23. Critical Stress Ratios And Member Numbers For Model 2 At Firs t Yield. The diagrams show the member numbers and critical stress ratios for both the top and bottom chord members of model 2. The model was loaded symmetrically at four top chord nodes and was supported at the four corner nodes of the bottom chord. Loaded nodes shown by *

For tension members (bottom chord):

Critical Stress Ratio -- member axial stress

yield stress

For compression members (top chord): Critical Stress Ratio member axial stress

flexural buckling stress

265

comparison with the tension members, with the members 62,63 etc. being the most heavily stressed compression members with a stress ratio of 0.16. Other compression members are also stressed at a similar level with the four corner diagonal bracing members having a stress ratio of 0.15.

Theoretical Non-Linear Behaviour: The non-linear pin-jointed analysis of the structure, indicated that under an increasing

symmetrical four point load, yield in the tension members would spread from the outside bottom chord edge members towards the centre

of the structure. The first set of members to yield were members 3, 28,33 and 58 (Figure 5.23) which yielded when the structure was supporting a total imposed load of 10240 N. As the imposed load was increased, members 2,4,32,34 etc. became critical, yielding when the structure supported a total load of 10800 N. The next group of

members to yield were members 8,38,23 and 53, which yielded when the structure was supporting a total load of 12240 N, followed by

members 7,9,37,39 etc. which became critical when the imposed

load was increased to 12480 N. When the imposed load reached 14648

N, tension members 1,5,31,35 etc. yielded, followed by the four

central bottom chord tension members, 13,18,43 and 48 which

yielded when the imposed load was increased by a further 56 N.

Yield of the four central bottom chord tension members was almost Immediately followed by yielding of the adjacent members 12,14,47

and 37 etc. When the imposed load had reached a value of 1504 N the four corner diagonal bracing members 103,120,197 and 182 had also

yielded in tension. At this stage in the analysis no other tension

members became critical. However, the analysis continued with the

behaviour of each group of members conforming to their Ideallsed

load-displacement relationship. When the Imposed load acting on the

structure reached 17596 N, the post strain-hardening axial stiffness

of the lower chord edge tension members was insufficient to maintain

stability of the numerical procedure, and the theoretical analysis

of the model structure was unable to continue.

Experimental Non-Linear Behaviour: Figure 5.24 shows both the

experimental load-displacement behaviour, resulting from testing to

collapse the second model space truss, together with the theoretical load-displacement relationship obtained from the analysis program. it can be seen from the Figure that there is close agreement between

the theoretical and experimental displacements over the linear range

of behaviour (A to B, Figure 5.24) and even over the early portion

266

20

15

0 0

-J -a

-J

-I-) 0 I-

5

A

NODE NO. 25 EXPERIMENTAL RESULT

--*--ANALYTICAL RESULT

50 103 150

VerticaL DispLacement in mm. Figure 5.24. Experimental And Theoretical Load-Displacement Behaviour Of Model 2. The Figure shows both tHe experimentaT-anU -i-heoretical- load-displacement behaviour of model 2. The structure was loaded simultaneously at four top chord nodes (19,21,24 and 43) by imposing a vertical displacement of O. 1mm per minute on each of the four nodes. This displacement rate was continued until approximately 25mm of vertical displacement has occurred, whereupon the displacement rate was increased ten fold to 1.0mm per minute. After a total vertical displacement of the actuator of approximately 100mm the displacement rate was increased further to 2. Omm per minute.

Points shown on the theoretical load-displacement curve. A-B = Elastic response.

B= Yield of members 3,28,33,58. C= Yield of members 2,4,32,34, etc. 0= Yield of members 8,38,23,53. E= Yield of members 7,9,37,39, etc. F= Yield of members 1,5,31,35, etc. G= Yield of members 13,18,43,48,12,14,47,37, etc. H= Yield of members 103,120,197,182.

I-L = Change in axial stiffness of yielding members. M= Loss of numerical stability. Points shown on the experimental load-displacement curve. B= Elastic limit. G= Yield of members 12,14,47,37, etc. N= Buckling of corner compression member 118. P After failure of member 118, bottom chord members 5 and 56

come into contact with the support. P-Q Increase in stiffness due to members 5 and 56 supporting the

corner reaction force in bending. Q Members 5 and 56 fail in bending. R The "knee" of the buckled corner compression member 118 came

into contact with the corner support. S Due to excessive deformation in the model, the structure

slipped off the corner Support and the test was halted.

LM

200

267

of the non-I ! near range (B to G, Figure 5.24). Point G (Figure 5.24) is the theoretical point at which the penultimate group of members (12,14,47,37 etc. ) yield, and corresponds with the onset of extensive plastic flow occuring in the model structure. The

non-linear range of behaviour G to N (Figure 5.24) is associated with the strain hardening of the yielded tension members. Point N

signifies the failure of the model structure which wai caused by buckling of the corner diagonal compression member (number 118, Figure 5.1) when the model structure was supporting a total imposed load of 18598 N.

The failure of the corner compression member led to the rapid loss of load-carrying capacity of the model structure with the imposed load decreasing to 3739 N (Point P, Figure 5.24). Af ter the failed member had deformed the two corner bottom chord tension

members (numbers 5 and 56, Figure 5.23) came into contact with the base of the corner support of the model and the structure temporarily regained stiffness (P to Q, Figure 5.24). When the imposed load had increased to 5179 N, bending of the two corner tension members occurred and the imposed load decreased to 3635 N (Q

to R, Figure 5.24). Further deformation of the failed member resulted in the 'kneel of the member sitting on the support. The

stiffness of the structure immediately increased and the model finally supported a total imposed load of 19171 N (R to S, Figure 5.24). Excessive deformation of the structure resulted In one corner of the model slipping off its support, effectively terminating the experimental investigation (Point S, Figure 5.24).

Figures 5.25 A to G show how both the theoretical and ex

' perimental values of strain vary In the tension members, as the

imposed load acting on the model structure was increased. The

recorded strains plotted in the Figures 5.25 A to G were obtained from both of the strain gauges fixed to each member, apart from the

strain recorded for member 3 (Figure 5.25A), where one of the gauges separated from the member during the early part of the test. Examination of the Figures 5.25 A to G, and the numerical results of the recorded strain, indicate that the tension members yielded in

almost the exact sequence predicted by the theoretical analysis. However, a comparison of Figures 5.25 A and 5.25 B which give the load-percentage strain relationships for members 3 and 2,

respectively, show that member 2 yielded before member 3 and not vice versa as indicated by the theoretical analysis. Figure 5.23

268

IIEMBR NO. 3 25 DMIMENTAL RESULTS

Strain Gauge No. 8 Strain Gauge No. 8

2B .... 0 .... ANALTTICAL RESULT

.....................

. .............................

15 p ...

FIGURE 5.2-5A

234 Percentage Strain

HEW NO. 2 25 DMIrIENTAL RESULTS

Strain Gauge No. 17 Strain Gauge No. 18

28 .... 0.... MALTTICAL RESULT

R................ ...... W............. 0 15

..............

-0 iI r FIGURE 5.25R

2345 Percentage Strain

OBER NO. 8 25 EXPERIMENTAL RESULTS

Strain Gauge No. 37 Strain Gauge No. 38

20 .... .... ANALYTICAL RESULT

...... . ....

T ........ ......

....................

15 ......

FIGURE 5-9-5jr-

2345

Percentage Strain

25 HM NO. 9 DMIWAL RESUL7S

- Strain Gouge No. 39 - Strain Gauge No. M

28 .... 0... ANUTICAL RESLLT A

R. . *. * ............

.................. ........... -0 Is ........... ..........

FIGURE 5-25T)

234 Percentage Strain

269

(Positive Strains are TensiLe)

mm NO. 1 25 EVERIWAL RESULTS

Strain Gauge No. 31 Strain Gauge No. 32

20 .... 0.... MALYTICAL RESULT

R $

-J

Oj

13

le

2,34

Percentage Strain (Positive Strains are TensiLe)

R -0

MW NO. 13 25 E)MML RESLETS

Strain Gauge No. 11 Strain Gauge No. 12

28 .... .... ANLYTICAL RESLLT

....................

FIGURF 5-25F

234 Percentage Strain

(Positive Strains are Tensile)

R 9

-0

. 2:

MEHK-R NO. 12 23 EXPERIWAL RESLLTS

Strain Gouge No. 19 Strain Gauge No. 28

20 .... 0 .... ANALTTICAL RESLLT

..........

.............. 13 .................

23 Percentage Strain

3

Fiqure 5.25A to G. Experimental and Theoretical Applied Load Vs Member Strain Ci es For Model 2. Ine ýIgures show how -TT7e theoretical and experimental values of the strain in tension members 3,2,8,9,1,13 and 12, vary with an increase in the external applied load. The experimental values were obtained from each of the two strain gauges placed diametrically opposite each other at the centre of. each member, apart from the strain recorded for member 3 (Figure 5.25A) where one of the gauges separated from the member during the early part of the test.

270

gives the member stress ratios for members 2 and 3 at the end of the I Mear elastic range, as 0.97 and 1.02 respectively. The small difference between these two values indicates that either member Is likely to be the first to yield. From a comparison of the load-percentage strain relationships given in Figures 5.25 C to G, it can be seen that the initial yielding of members 2 and 3 was followed by the yielding of members 8,9,1,13 and 12,

corresponding with the sequence outlined by the theoretical analysis

of the model structure.

Figures 5.25 F and G also show that a small amount of bending

was induced in both members 13 and 12 before they yielded In tension. As yielding of these two members progressed, the influence

of bending moments acting on the members decreased.

Figures 5.26 A, B and C indicate how both the theoretical and experimental values of axial stress in compression members 61,62

and 101 change with increasing external applied load. The Figures 5.26 A and B show that as soon as tensile yield has occurred in the

model structure, a fixed increase in the external imposed load

acting on the structure resulted in a larger increase occurring In the cornpression force carried by both members 61 and 62. A 47% increase in the external load carried by the model structure resulted in a 82% theoretical and a 68% experimental Increase In the

axial force carried by compression member 61. Th is non- I ! near response was even more pronounced in the behaviour exhibited by

member 62, where a 47% increase in the external load, after first

yield, resulted in a 99% theoretical and a 96% experimental Increase in the axial compression force carried by the member.

Figure 5.26 C shows that the theoretical axial stress levels

occurring in the corner diagonal compression member are In close agreement with the experimentally measured values. The axial stress readings were obtained from averaging the measured strains recorded from both of the strain gauges fixed to the member. Figures 5.27 A, B and C give the experimental load-strain relationships obtained from each of the two strain gauges fixed at the centre, diametrically opposite each other, on the three compression members 61,62 and 101. It can be seen from these Figures that a relatively small amount of bending was present in these members while the model structure was responding elastically. As yielding occurred in the bottom chord tension members the bending strains recorded In the

271

(Positive Stresses are Tensile) 25 1

3 13 1 «'s,

. e: , EL Q

to I 'a I-

5

OBER NO. 61 DMIENTAL RESULT ANALYTICAL RESULT

-206 -ice -128 -00 -40 Is Be 120 Ice no

FIGURE 5.26A Axid Stmss (N/sq. m)

(Positive Stresses are Tensile) 201 MMER NO. 62

- EXPERIMWAL RESLLT

--- 0--- AMYTICAL RESLLT

9 15

-0

"1 -4 8: 0

I-

5

-280 -168 -120 -ft -is Is

FIGURE 5.26B Axial SUvss (N/sq. mm)

(Positive Stresses are Tensile) 20,

is - .0

5

as 128 IN 200

mm NO. 101 EXPERRENTAL RESILT ANALYTICAL RESLLT

-208 -168 -120 -06 -40 48 128 ice 200

FIGURE 5.26C Axid Stress (N/sq. m)

EigLUyC_5, Z6A, 13 And C. Experimental And Theoretical Applied Load

- Axial Stress Curves For-Model 2. The Figure shows how

both the theoretical and experimental values of stress in compression members 61,, 62 and 101 change with increasing

external applied load. The axial stress was obtained for each member by averaging the strain readings obtained from two strain gauges placed diametrically opposite each other at the centre of each member.

272

(Positive Strains are TensiLe)

23 tMR NO. 61 23 WRIMNTAL RESLETS Strain Gauge No. 25 Strain Gauge No. 28

20 26

9 ,a 13 . .. '. ,. 15

-3 -2 .1122

Percentage Strain (1 "D

FIGURE 5.27A

(Positive Strains ore Tensile)

23 . HM NO. G2

25 EMIWAL RESLLTS - Strain Gauge No. 35

- Strain Gauge No. 3G 20 . 21

15

4:

is

-1 123

Percentage Strain (I "D

FIGURE 5.27B

(Positive Strains are Tensite)

25 rm NO. 101

. 25 EVERUMAL RESIVS Strain Gauge No. 29 Strain Gauge No. 38

20 . 21

-3 -2 .1123 Percentaqe Strain ( ICE-01)

FIGURE 5.27C Experimental Applied Load Vs Member Strain

Lurves Obtained Fro de-T Z. The Figure shows how the strain 7n

compression members 61, and 101 varies with an increasing externally applied load. The experimental values were obtained from each of the two strain gauges placed diametrically opposite each other at the centre of each member. From the Figure it is evident that tensile yielding in the bottom chord of the structure was accompanied by bending in these compression members.

273

three compression members significantly increased. The strain recorded from gauge 30 positioned on the corner compression member 101 indicated that when the adjacent corner compression member (118) buckled, causing collapse of the entire model structue, yield had

not occurred in member 101. The recorded strain of 0.169% measured in member 101 corresponds to a stress of 347 N/MM2 , indicating that the member was capable of supporting additional axial load and bending moment before failure.

Model 3

The third double-layer space truss model tested to failue, incorporated both tension members that were permitted to deform

plastically and eight soft members, which were also allowed to deform plastically when stressed in compression. All of the bottom

chord and web bracing members in model 3 were nominally identical to the members used in the second model structure. However, the

majority of top chord compression members were the tubular type T5 (Table 5.1) instead of the solid bar members used in both models 1

and 2. The eight soft members used in the third structure were all type 1 soft members (Figure 4.11). The soft members were placed symmetrically in the top chord of the structure in pairs at the

centre of each of the four sides (Figure 5.5). This position was chosen so that the soft members would replace the most heavily

stressed compression members in the structure and theoretically

permit a favourable redistribution of forces to occur within the top

chord compression members.

The third model structure was simultaneously loaded at four top

chord nodes (19,21,41 and 43) in an identical procedure to that

adopted for model 2. The vertical displacements were measured at twenty-three nodes (Table 5.2) using only the L. V. D. T. transducers.

In addition, the strain was measured in fifteen members (Table 5.4)

using two strain gauges for each of the six bottom chord tension

members, and three gauges for each of the nine compression members. Figure 5.28 shows the model structure under test.

Linear Behaviour: An assessment of the Hnear behaviour of the

structure supporting an increasing imposed load, can be made by

comparing both the theoretical and experimentally measured vertical deflections of several nodes, symmetrically positioned in the model structure. Table 5.16 gives both the theoretical and experimental displacements of groups of nodes symmetrically positioned in the

274

àa 4

275

E c LM ý? C%i -

0!

1 1 1 1 1

00 4,

S- 4- w Q)

- 1 1 1 1 1

-0 9

4 0 :3

Z,

c 00 Ln Ln ? 4 0% m

- -ý C9 Cý Cý M 4,

-4- V) 41) (V ui

-j 0) 42 co C7% Lr) I I I I

-C a (1) m cli

cc LO E 0-

LAJ

2 1. cli -It ()% C7%

CU -4 C%i C-i C%j C%j s- 4-

CD

W Lr) C) i

cli cli -4

0

Rl

E

-

Cý Ln P, CD to Ln (7% cli r- cli

llý 9 llý Cý 4j -4 cli -4 (%j cli I -, W

(U r- co r- en C%i cli Cl) C%j

Lei

ON LO m to " m 2 cli LC) rý qr U ) qcr

o Cý V! Cl! vi 1ý Cý 4-A C-i eq CQ -4

S. - 4- Q)

C: ) ui

Q) cl; ý Ln C cli m

Lr; ks Cl Co -. T cli Ln 47% ; z z ; ;

0 0 r r cl Cý = I cv " IM -4 1 en V)

z

.0 4ý d2

r_

(A = t- 4- C: (L) c +J 000m 0 04-

.Im %A 0 4. J (L)

> E- mu 4- - (1) > fo :3 :3 (1) cm u 4j 0

cz ex3 -ci =U -r- (L) W CD Z>

-, ý; CL 4.1 CD s- m m (A m 25 . 4. J m 4.. ) (0 -ý - it

c: h- «0 m ei JD g4- w r= ai fo =4-

c2» -4--, M lu 4A cm

LU (V to fo

'11 M c: 4- ý cu 0u

<0 tn 4.. ) -ý 00 41

- 2: tA c2. (A fe 0)

«a r= fzj

%A > (U 4-

,0 tj. -

%A 4-

0 CD -Z3

0,4j c2. W00 ." xCC- E-

0 CL 0

u

to w (D 4.. ) 41 (D u r= um

276

model. The measured displacements given 1'n Table 5.16 show relatively small differences between the measured vertical displacements of symmetrically positioned nodes, but relatively large differences between the theoretical and experimental values of displacements. In general, the theoretical displacements were approximately 15% greater than the corresponding exper imental d1 sp I acements. Table 5.17 gives both the theoretical and experimental values of deflection for all of the nodes monitored by transducers, for two different values of the -imposed load. The f irst load value of 4000 N is approximately in the middle of the

elastic range, while the second load value of 7934 N corresponds with the theoretical end of the linear elastic range. When the structure was supporting a total imposed load of 7984 N, the majority of the theoretical displacements were approximately 10% less than the

experimentally measured values, indicating that the elastic limit of the structures may have been exceeded supporting this load.

Figure 5.29 gives both the member numbers and the values of the

member stress ratios for the model structure supporting a total imposed load of 7984 N. It can be seen from the Figure that the

most heavily stressed tension members are the lower chord edge member numbers 3,8,33 and 58, which have a ratio of actual stress to yield stress of 1.05. The adjacent members 2,4,57,59 etc., have a stress ratio of 0.98 indicating that they are also close to

yielding. The stress ratio of 0.28 given In Figure 5.29 for the

soft member numbers 62,63 etc., indicates that these members are theoretically capable of supporting almost a four fold increase In

compression loading before buckling.

Theoretical Non-Linear Behaviour: The theoretical non-linear behaviour of model 3 is similar to the theoretical behaviour

described for model 2. Using the non-linear pin-jointed analysis program it was

, predicted that members 3,33,58 and 28 (Figure 5.29)

would be the first members to yield when the structure was supporting a total symmetrical four point load of 7984 N. When the total imposed load was increased to 8640 N, members 2,4,57,59

etc. (Figure 5.29) became critical and yielded in tension. The

yield of these edge members was followed by the almost simultaneous yield of members 7,8,9,52,53,54 etc. (Figure 5.29) when the

structure was supporting a total imposed load of 10320 N. When the load acting on the structure had increased to 10880 N, the theoretical force in the eight soft members had increased to 3700 N indicating that these members would no longer remain elastic but

277

Total Imposed Load 4000 N

Total Imposed Load 7S84 N

Theoretical Experimental Theoretical Experimental Node Vertical Vertical Node Vertical -Vertical

ýXnbers Displacements(mm) Displacements(mm) Nmbers Displacements(mm) Displacemcits(mm)

31 2.370 4.658 SO: 31 4.731 8.613

37 2.359 1.820 37 4.709 4.411

26 2.359 2.027 26 4.709 4.802

25 2.359 1.935 25 4.709 4.752

32 2.343 2.037 32 4.677 4.806

20 2.343 2.048 20 4.677 4.929

21 2.257 4.042 21 4.504 7.907

16 1.969 1.747 16 3.731 3.915

38 2.267 1.818 38 4.524 4.361

27 2.267 1.856 27 4.524 4.331

15 2.267 1.923 15 4.524 4.496

14 2.267 1.923 14 4.524 4.722

33 2.256 1.921 33 4.502 4.559

9 2.256 2.196 9 4.502 5.071

10 1.943 2.097 10 3.877 4.534

22 1.943 2.066 22 3.877 4.543

21 1.121 1.173 11 2.238 2.629

39 2.126 1.895 39 4.243 4.245

28 2.126 2.194 28 4.243 4.722

4 2.126 4 4.243

3 2.126 3 4.2243

17 1.275 1.575 17 2.545 3.092

5 1.275 I

5 2.545 I

Key: Transducer Armature Fixed To The Cross Beam. Displacement Measured By An LNMT Transducer Incorporated Into 7be Actuator. Random Readings Obtained Due To An Amplifier Fault.

Table 5.17. Comparison Of Theoretical And Experimental Deflections For Al I go-inito-r-e! Nodes I Mode 1 3. Ihe ]able gives both the theoretical and experimental values -of vertical displacements for all of the nodes monitored by transducers, for two different values of the imposed load. The first load value of 400ON is approximately in the middle of the elastic range while the second load value of 7984N corresponds with the theoretical end of the linear elastic range. When the structure was supporting a total imposed load of 7984N the majority of the theoretical displacement were less than the experimentally measured values indicating that the elastic limit of the structure may have been exceeded supporting this load.

278

cs

0 C,

A 121 M-ý)R S-PA )I

37 sa 99 100 L" n CN CD F,

.r co o

Cý %D Cý t .

o. 12 0.26 0.26 A 0.12 93 91, 85 S5

to ED CD Cý 9 (ý

, - , LD C, to CD N CD N C

0.05 o. eG 0.06 0.05 90 91 92

Lo (D LD a: IR Cý

cm 10 0 r ko CD to C N cr

0.12 0.26 0.26 -0.12 85 4K 66 87 x 68

Lo CS)

M LO CP tZ, Cý c

0.21 0.28 0.28 0.21 81 82 83 Bý

Ir

cs

(X cr

cs

Lr

-9

. 51 jw a 13 41 33

)

D

BOTTOM GRTD

Fiqure 5.29. Critical Stress Ratios And Member Numbers For Model 3 -At FirsU-Ti-eld-- The diagrams show the member numbers and critical Tt-ress ratios for both the top and bottom chord members of model 3. The model was loaded symmetrically at four top chord nodes and was supported at the four corner nodes of the bottom chord. The structure contains eight type I soft members (Figure 4.11), member numbers 62,63,82,83,78,799 98 and 99. Loaded nodes shown by

For tension members (bottom chord): Critical Stress Ratio = member axial stress

yield stress

For compression members (top chord, including soft members): Critical Stress Ratio member axial stress

flexural buckling stress

279

N.

AR4m qQ I- AN A- qR A- Wý4

56 57 58 59 60 CD CD CD

in CD in

CD t)

CD (ý . -y cz

0.00 0.57 0.53 0.57 0. Ple 51 52 53 511 55

r- CD CD r- Co

ýr lý m 3ý T 'lý 3) vý

ýr CD m - a - a -4 cr,

0.00 * 0. Ple 0. OS 0.09 0.00 £16 A7 ji 6 dis z5 0

Co to to p2 vý Co ý Z 9

CD CD 14

O. eß

di 1 A2 A, 4 di 5 r- Co Co r-

.. CM vý N 9 rli cý ý m 9 CD CD CD %j cr,

0.00 0.57 0.53 0.57 0.00 36 37 38 39 410

CD CD CD CD

0.54 0.98 1. a5 0.98 0.54

would yield in compression, exhibiting a constant load plateau in

their load-displacement response. At this stage in the analysis the

structure carried an increase in load without the yield of any additional members, but with the plastic deformation of the soft

members and the gradual strain-hardening of the yielded tension

members. When the imposed load acting on the structure increased to

12000 N the corner, bottom chord tension members 1,5,31,35 etc. (Figure 5.29) became critical and also yielded. This immediately

led to numerical instability in the analysis program resulting from

the loss of stiffness in all of the bottom chord edge tension

members. When the theoretical analysis was terminated, the middle tube of the soft members had yielded a total of 9.78 mm, just short

of the 10.0 mm of movement allowed in the Type 1 soft members (Figure 4.11) before they close to form the triple tube compression

members.

Experimental Non-Linear Behaviour: Figure 5.30 shows the

experimental load-displacement relationships obtained from testing

model 3 to col. lapse, in addition to the theoretical

load-displacement relationship obtained from the analysis program. It can be seen from the Figure that there is only close agreement between the theoretical and experimental values for the Initial

portion of the load-displacement relationship. The experimental load-displacement behaviour plotted in Figure 5.30, shows that after the total applied load acting on the structure exceeded

approximatley 8000 N, the stiffness of the structure decreased as the bottom chord tension members yielded. This Initial ductility

exhibited by the model structure was interrupted by the premature buckling of the soft member, number 62 (Figure 5.29), when the

structure was supporting a total imposed load of 13816 N (Point A,

Figure 5.30). The buckling of this soft member was Immediately

followed by a reduction in load-carrying capacity of the structure

and the rapid transfer of force from the failed member Into adjacent

members. This load shedding from the buckled soft member resulted In the failure of the adjacent top chord compression member, number 66 (Figure 5.29), when the structure was supporting a total load of 11328 N (Point B, Figure 5.30). Failure of member 66 resulted In a further decrease in the load carrying capacity of the structure and

a reversal of stress occurring In three of the bottom chord tension

member numbers 42,43 and 48 (Figure 5.29). These three members buckled in quick succession (Points C, D and E, Figure 5.30),

causing the imposed load acting on the structure to decrease to 7669

N. The buckling of these three lower chord members was followed by

280

15

10

--p

NODE NO. 25 EXPERIMENTAL RESULT ANALYTICAL RESULT

50 100 150

VerticaL DispLacement in mm.

Fiqure 5.30. Experimental And Theoretical Load-Displacemen Behaviour Of Model 3. lhe Figure shows both the Eheoretical and experimental vertical displacement of node 25 in. model 3. The structure was loaded simultaneously at four top chord nodes (19,21, 24 and 43) by imposing a vertical displacement of O. 1mm per minute on each of the four nodes. This displacement rate was continued until approximately 25mm of vert

* ical displacement had occurred,

whereupon the displacement rate was increased ten fold to 1.0min per minute. After a total ' vertical displacement of the actuator. of approximately 100mm the displacement rate was increased further to 2. Omm per minute.

Points shown on the theoretical load-displacement curve. Pi = Mid-point in the elastic response. P2 = Yield of members 3,33,58,28. P3 = Yield of members 2,4,57,59, etc. P4 = Change in stiffness of yielded members P5 = Yield of members 7,8,9,52,53,54, etc. P6 = Change in stiffness of yielded members P7 = Yield of soft members. P8 = Yield of members 1,5,31,35, etc. with ensuing numerical

instabi lity. Points shown on the experimental load-displacement curve. A= Premature buckling of sof mber 62. B= Buckling of member 66. C= Buckling of lower chord member 42. D= Buckling of lower chord member 43. E= Buckling of lower chord member 48. F= Buckling of lower chord member 33.

F-G = Decrease in load carrying capacity due to failure of member 33.

H= Buckling of diagonal web member 127. I= Model slipped off support and test halted.

281

an increase in the stiffness of the structure. The imposed load

applied to the structure slowly increased until the outside lower

chord member, number 33, ruptured in tension (Point F, Figure 5.30)

when the structure was supporting a total load of 12191 N. The tensile failure of member 33 resulted first in a rapid decrease in

the load-carrying capacity of the structure, with the total imposed load dropping to 5116 N (Point G, Figure 5.30), followed by a return in stiffness of the structure. When the imposed load had increased

to 8673 N there was a small lateral deflection of the solid diagonal

web member 127 positioned under the south east point load (Point H,

Figure 5.30). No additional load was carried by the structure and the test was halted when the model slipped off the north west corner

support (Point I, Figure 5.30).

Figures 5.31 A to F show how both the theoretical and

experimentally measured strains vary in the lower chord tension

members as the imposed load acting on the structure is increased.

The experimental values recorded from both of the strain gauges

attached to each member have been plotted, except for member 7 (Figure 5.31C), where one of the strain gauges developed a fault

during the early part of the experimental investigation.

A comparison of the strains recorded for members 3 and 2 given in Figures 5.31 A and B respectively, show that member 2 yielded before member 3 and not vice versa as predicted by the theoretical

analysis. However, a comparison of Figures 5.31 C, D and E shows that the next member to yield was member 7 followed by the yield of

member 8 and finally the yield of member 1. This yielding sequence did correspond with the predicted sequence obtained from the

theoretical analysis. The strains recorded for member 13 given In

Figure 5.31F show the experimental values, to be considerably

greater than the theoretical values especially when the member behaviour is elastic. Member 13 remained elastic throughout the

early part of the test but finally yielded, contrary to the

theoretical analysis, when the total imposed load acting on the

structure was 13323 N.

A comparison of the measured strains recorded from each pair of

strain gauges attached to members 1,2,3,8 and 13 (Figures 5.31 E, B, A, D and F) indicates that bending strains In these members are negligible, when the members are behaving elastically. When the

members have yielded, the bending strains become more significant

especially in members I and 8 (Figures 5.31 E and D).

282

(Positive Strains are TensiLe) 28

79 15

KLO

CZ. la

5

FIGURE 5.31A

20 .

-cl . 15

-0

c21. le

5

20 1.0

79

13 15 cu 0. . 2-

18

3

I

I

MEMBER NO. 2 EXPERIMENT& RESULTS Strain Gauge No. 27 Strain Gauge No. 28 ANALYTIC& RESULT

23

Percentage Strain

MEMBER NO. 7 EXPERIMENTAL RESULTS Strain Gauge No. 39 ANALYTICAL RESULT

123

Percentage Strain ( IOE-01) FIGURE 5.31C

hEMBER NO. 3 EVERIMENTAL RESULTS Strain Gauge No. 15 Strain Gauge No. 1G WLYTICAL RESULT

23

Percentage Strain

283

(Positive Strains are TensiLe)

28

_0

_0 15

FIG URE 5 . 31D

20

: 2ý

-0 15

10

0

FIG URE 5 . 31

v

E

20

15

- 2; Iß

5

FIC, uRE 5.31F Percentage Stra in(I OE-0 1)

MEMBER NO. 8 EXPERIMENTAL RESULTS Strain Gauge No. 13 Strain Gauge No. H ANALTTICAL RESULT

MEMBER NO. 13 EXPERIMENTAL RESULTS Strain Gauge No. 38 Strain Gauge No. 37 ANALYTICAL RESULT

Fiqure 5.31A To F. Experimental And Theoretical Applied Load Vs R-e-ýTeF77TF-ain 7u-rves For-7, -o-del 3. The rigures show how -t-Týo-r-Mi-caT

and experimental values of the strain in tension members 3,2,7,8, 1 and 13 vary with an increase in the external applied load. The experimental values were obtained from each of two strain gauges placed diametrically opposite each other at the centre of each member, apart from the strain recorded for member 7 (Figure 5.31C) where one gauge was not functioning correctly.

284

12 Percentage Strain ( IOE-01)

MEMBER NO. I EXPERIMENTAL RESULTS Strain Gauge No. 31 Strain Gauge No. 32 ANALYTICAL RESULT

1

235

Percentage Strain ( IOE-01)

Figures 5.32 A to H indicate how both the theoretical and experimental values of axial stress vary in compression members 61, 62,65,66,69 and 70 (Figure 5.29) under an increasing imposed load

acting on the model structure. It can be seen from the Figures that

although there are large discrepancies occurring between the theoretical and experimental results, the theoretical results do

predict the general trend in member behaviour.

Figures 5.32 A and B show the changes in axial stress occurring in the outside tube of soft member 62 and in th'e preceeding member number 61 (Figure 5.29). Both of the theoretical and experimental load-stress behaviours plotted in Figures 5.32A and B show that when yielding occurs in the bottom chord tension members the stress in

compression members 61 and 62 increases at a rate greater than the

rate of increase in the imposed load. A 50% increase in the

external load carried by the structure, after yield had occurred in

tension member 2, resulted in a 88% theoretical and a 78%

experimental, increase in the axial force carried by compression

member 61. Similar behaviour was exhibited by the soft member 62,

where a 50% increase in the imposed load acting on the model structure after first yield, caused a 90% theoretical and 240%

experimental increase in the force carried by the soft member. This

large discrepancy occurring between the theoretical and experimental

values of axial stress carried by the soft member (Figure 5.32A) indicates that either the initial stiffness of the soft member was underestimated in the theoretical analysis, or this particular soft

member was not permitting relative movement to occur between the

outside and middle tubes of the assembly.

Figures 5.32 C and D show the changes in axial stress which occur in members 65 and 66 as the imposed load acting on the

structure Is increased. Both the experimental and theoretical

relationships given in Figures 5.32 C and D show that as yielding of the bottom chord tension members progressed, the axial compressive

stress present in members 65 and 66 decreases. However, as the

tension members begin to strain harden and carry additional load,

this behaviour reversed and the compressive stress in members 65 and 66 increased significantly, with small increases In the external load acting on the structure.

Figures 5.32 E and F show, both the experimental and theoretical

load-stress response for top chord compression members 69 and 70.

285

(Positive Stresses are TensiLe)

23

28

15 -J

10

3

MEMBER NO. 62

-20 -is -so -88 -100 -120 -lie -1 Go -198 -280

FIGURE-5.32A Axid Stress (N/sq. mm)

25

28

12 c cl, 15 -i . c,

0.

3

KEMBER NO. 61

-28 -48 --w -M -100 -128 -14a -Im -198 -2W

FIGURE 5.32B Axiat Stress (N/sq. mm)

25 OVER NO. 65

i 20

15

-< la

5

-18 -28 -38 -48 -53 -Ga -70 -80 -se -100 FIGURE 5.32C Axid Stress (N/sq. mm)

286

(Positive Stresses are TensiLe)

25

23

15

-< 10

5

-20 -io

FIGURE 5.32D

MEMBER NO. 6G ----i nrnill T

-60 -80 -100 -128 -140 -160 -180 -280

Axiat Stress (N/sq. mm)

25

20

15

10

100 88

FIGJJRE 5-32E

-80 -100

108 80 GO io 20 -20 -ia -Go -88 -100

FIGUBE 5.32F AxiaL Stress (N/sq. mm)

287

MEMBER NO. 70

io 20 -28 -18 -IGO

AxiaL Stress (N/sq. min)

(Positive Stresses are TensiLe)

C

-t C

C C

C

25

MEMBER NO. 199 .... . ....... ..

MEMBER NO. 101 -- -- ------- --

28

-0

-< Iß

5

-10 -1210 -30 --io -50 -60 -78 -80 -90 -100

FIGURE 5.32H Axid Stress (N/sq. mm)

Figure 5.32A to H. Experimental and Theoretical Applied Load Vs -Member Axial Stress Curves For Model 3. The Figures show how F'ot-F the theoretical and experimental v lues of axial stress in compression members 62,61,66,65,70,69,199 and 101 change with increasing external applied load. The axial stress was obtained for each member by averaging the strain readings obtained from three strain gauges placed symme tr icaIIV around the circumference at mid-length of each member.

288

-10 -20 -30 -10 -58 -60 -70 -80 -90 -100

Axial Stress (N/sq. mm) FIGURE 5.32G

The compressive stress acting in both of these members also decreased as yield occurred in the bottom chord of the model

structure. As yielding of the tension members progressed, the axial

stress in both members 69 and 70 became tensile. This behaviour was

reversed with the premature buckling of soft member 62, which caused

a rapid transfer of load from the failed member onto the adjacent

compression members.

The experimental and theoretical load-stress behaviour obtained from the corner web members 199 and 101 are shown in Figures 5.32G

and H respectively. Both of these Figures show good agreement between the theoretical and experimental results and indicate that

the members remained elastic until the first compression member buckled.

Figures 5.33 A to D show the load-strain relationships

obtained from each of the three strain gauges attached to

compression members 62,65,66 and 70 respectively. A comparison of the three sets of strain readings recorded from the outside tube of

soft member 62 (Figure 5.33A) indicates that a bending moment of

approximately 5.4 kN mm was present at mid-length of the tube when

premature buckling of the member occurred.

Figure 5.33 C shows a similar behaviour for compression member 66, which was the second member to buckle during the test on the

model structure. The measured strains recorded for both members 65

and 70 (Figures 5.33 B and D) show that bending strains in these

members were negligible until large vertical deflections occurred In

the test model under load.

Model 4

The fourth model square-on-square double-layer space truss

tested to failure was also designed to exhibit a ductile

post-elastic load-displacement response. The structure Incorporated

eight soft members located in the top chord, with two soft members

positioned symmetrically, about the centre of each side (Figure

5.6). The soft members used in model 4 were all type 2 soft members (Figure 4.11) which were an improved version of the Type I soft

members used in model 3. The fifty-two remaining top chord members

were the tubular type T5 members (Table 5.1) identical to the

members used in the top chord of model 3. The web bracing members

289

(Positive Strains are TensiLe)

25 - OBER NO. 62

25 EXPERIMENTAL RESULTS - Strain Gauge No. 7 Strain Gauge No. 40

20 - Strain Gauge No. 8- 28

,a 15 .- 15

01

%. e --BUCKLING LOAD a 13-816 kN

--0 c2. la - .<-

la

5- )AI .5

-3 -2 -1 1

Percentage Strain ( 10E-01)

vTGuRE 5.33A

MEMBER NO. 65 25 25 - EXPERIMENTAL RESULTS ' Strain Gauge No. 2G Strain Gauge No. 25

28 . Strain Gauge No. 21

- 20

,a 13 - cl . 13

-0

-3 -2 -1 12

Percentage Strain ( IOE-01)

290

(Positive Strains are TensiLe)

25 . MEMBER NO. 66

25 EXPERIMENT& RESLLTS - Strain Gauge No. 21 Strain Gauge No. 11

28 . Strain Gauge No. 12

- 20

BUCKLING LOAD z 11-328 M

ca. le -.

Is

C%

-3 -2 -1 23

FIGURE 5-33C Percentage Strain IOE-01)

25 -

20 -

ca. le -

-3

. F_TGURF 5.33D

MEMBER NO. 70 25 EXPERIMENTAL RESULTS

Strain Gauge No. 20 Strain Gauge N9.18 Strain Gauge No*. 17 28

-1 1 Percentage Strain ( IOE-01)

. 15

. la

Figure _5.33A

To 0. Experimental Applied Load Vs Member Strain Curves Obtained FFOm Model 3. The Vigure shows how the strain in compression members 62,65,67 and 70 varies with an increasing externally applied load. The experimental values were obtained from each of the three gauges positioned symmetrical ly around the circumference at mid-length of each member. From the Figure it is evident that the soft member number 62 buckled when the structure was supporting an imposed load of 13.8kN and that also member 66 buckled under an imposed load of 11.3kN.

Tensile strains are shown as positive.

291

used in model 4 were also identical to the web members used in both

models 2 and 3, however the bottom chord members of model 4 were type T2 members (Table 5.1) which had both a larger external diameter and cross-sectional area than the bottom chord members used in each of the three preceding models.

The fourth model structure was loaded only at the top central

node (number 31) using an identical procedure to that adopted for

model 1. The vertical displacements were measu'red at twenty-three

nodes using L. V. D. T. transducers (Table 5.2), and the strain was

measured in fifteen members. Three strain gauges were used to

measure the strain in each of the nine compression members, while two gauges were used for each of the six lower chord tension members (Table 5.4). All of these measurements were recorded at thirty

second intervals throughout the experimental investigation. Figure

5.34 shows the model structure under test.

Linear Behaviour: Before the fourth model was tested to collapse, a small number of tests were carried out on the structure to determine if a symmetrical response was exhibited by the model when loaded

within the elastic range. In the first test, a load cell was positioned under each of the four corner nodes at the supports. The

central point load was first increased from zero to approximately 5000 N and then decreased to zero again. Readings from each load

cell were taken at both load values. Table 5.18 gives the average loads recorded in each of the four load cells when the structure was supporting a central point load of 4985 N. It can be seen from Table 5.18 that there is close agreement between the loads recorded at supports 1,6 and 61. However, the load recorded by the load

cell at support 56 was approximately 2.4% less than the load

recorded at both nodes 6 and 61.

For the second preliminary test, one strain gauge was attached to each of the eight soft members, and also to each of the four

corner web members. The strain in each of these members was

recorded as the load was slowly increased from zero to approximately 5000 N. Table 5.19 gives the strain recorded in each of the eight

soft members and the four corner web members, when the model structure was supporting a central point load of 4859.4 Newtons. The strains recorded in web members 118 and 119 were within 2% of their theoretical values. However, the strain recorded In web members. 101 and 184 were approximately 18% greater and 26% less

292

L tý Im, - ý,, 0ý

LIM- , ýI, qm

Ile ,i.:

IBM

F.

$ 1)

(1) -fl

-n

ý C)

(1

4-A I IL I' I

293

u

Model 4

x

Note All four boundary nodes restrained in the vertical Y direction.

Total Imposed

Load Applied At

Load Cell Readings In Newtons

Node 31 (Newtons) Node Nuaibers

1 6 61 56

4985.0 1246.0 1256.0 1256.0 1226.0

Table 5.18. Load Cell Readings For Model 4. The Table gives the average loads recorded from each load -c-e-FT- Positioned under the four supports nodes of model 4, when a total imposed load of 4985N was applied to the structure at the top central node. The individual values of load recorded from the load cells Positioned under nodes 1,6 and 61 are in close agreement, however the load recorded under node 56 was approximately 2.4% less than the loads recorded at nades 6 and 61.

294

'Ný19 79 78 Ný/*

Ce) COI

CD CY oý

1

Co 1 63_ 62

Member Numbers

Member Numbers

Member Strain Under Acting On

A Point Load Of The Structure

4859.4N

Theoretical Measured

101 1.05079 x 10-4 1.2375 x 10-4

118 1.05079 x 10-4 1.0966 x 10-4

119 1.05079 x 10-4 1.0611 x 10-4

184 1.05079 x 10-4 7.7837 x 10-5

62 2.2349 X 10-4 5.8388 x 10-5

63 2.2349 X 10-4 4.4637 x 10-5

98 2.2349 X 10-4 1.3316 x 10-4

99 2.2349 X 10-4 8.1735 x 10-5

79 2.2349 - 10-4 4.9514 x 10-5

18 2.2349 x 10-4 2.1431 x 10-5

83 2.2349 X 10-4 1.3745 x 10-5

82 2.2349 X 10-4 1.2567 x 10-5

Table 5.19. Strain Readings From Model -

4. The Table gives the strains recorded in each of the eight sof-t members and the four corner web members when the model structure was supporting a central point load of 4859.4 Newtons. The strains recorded in web members 118 and 119 are within 2% of their theoretical values. However the strains recorded in web members 101 and 184 are approximately ls% greater and 26% less respectively than the corresponding theoretical values. The strain readings recorded from gauges attached to the eight soft members showed large variations in the measured strain occurring in these members. The strains recorded in the outside tube soft members 78,82 and 83 were sufficiently small to indicate that these soft members were not functioning correctly.

295

respectively, than the corresponding theoretical values. These I arge differences occurring between the theoretical and

experimentally measured strains, in members 101 and 184, are most likely to be due to the presence of bending strains in these

members, the magnitude of which could not be determined because only one strain gauge was used for each of these bars. The strains

recorded in the eight soft members also given in Table 5.19 show large variations In the measured strain occurring between these

members. The measured strain in the outside tube of each of the

eight members was significantly less than the theoretical value,

with the strain in the outside tube of members 78,82 and 83

sufficiently small to indicate that these soft members were not functioning correctly. The malfunction of these soft members may have been due to the inner and middle tubes binding on the outside tube, due to Insufficient tolerance existing between the three

tubes, or movement of the middle tube may have been constrained by

the bottom welds used to attach the soft member to its end node.

Table 5.20 gives both the theoretical and experimental displacements of groups of nodes symmetrically positioned in the

model corresponding to a central point load of 15844.4 N acting on the structure. The Table shows that there are small differences

occurring in the vertical displacements measured for symmetrically positioned nodes, but also that there are relatively large

variations occurring between the measured values of displacements

and the corresponding theoretical values. In general, the measured displacements were found to be approximately 8% less than the

corresponding theoretical vertical displacements.

Table 5.21 gives both the theoretical and experimental values

of the vertical displacements for all of the monitored nodes corresponding to two different values of Imposed load acting on the

structure. The first load value of 8015 N is approximately in the

middle of the elastic range, while the second load value of 15844 N

corresponds with the theoretical limit in the elastic behaviour of the structure. The displacements measured when the structure was

supporting a total imposed central load of 15844 N, were all within 16% of their theoretical values. The majority of the measured vertical displacements were less than the corresponding theoretical

values, indicating that the actual stiffness of the model structure was greater than the stiffness predicted from the theoretical

analysis.

296

E

90 41

c: i

S-

c Uli -4 ýr c71 Cm m Lei cý 1: ý

1 1 1 1

uu %D N: r OD

1-4- vi cu cu

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LU

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to 1-4 -. LI -cr U ) r-

m 41 112: Cý u

to Lt) CD 0% ci

r- tm m C: ) Co LO r- CYN ýr C" ýr

cý N r, ý «Zý Cý! 4.3 %0 to Ulb

2

w f- Co r- r- M w m -4

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:: » Icr 0% ýr Ln Co r, - -? rý fi fi t'ý cý C! cý M 4, u0 r,

zW

S- 4-

c(;

Ln %0 C: ) c> j ti, ýo

.: C.; .: -; -: cli cli

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CL

E KZ

_C 4- - tA -C (D -4- 0 to 3: rö 0 -c 001. - 0 L) ." 4-)

-r- M0M -ý -4-3 CDL Ln -i-» u tA :3 G) t- -CD WWoU (U C) «M

> s.. -> (0 3: 4- -M ro 0 -6.1 a) tvi eö "a r- C) cm 4- 0 F- (1) .-0

rö M0 c7, MV) 4-ä c c:. -

c: o

L) 'm f- -ýJ c3 ci (1) L3

x %- LU LA s- .- CD Mc

1C3 wVGý; cm -C3 0

cC (1) -- < +. ) 0 L)

> Q) 00 > CL

CM ei 0 ý- U)

m= 4-3 CM L) - po G)

c:

m0 (L) Ez r- -4-3 ro c: - CD ,>

CD 'U 4- %_ - 4- +1 .ý C) >

0

% ci- -0 -t --- cu c2» >ý

c: >-, CO 115 cu Z Ln 2 S- %-

4- 0 00

4- uu1. - C) s- tA (0

cm 0 s- r= m s- r= «- CD >, 0 (2) 0)

(1) a) tA -mu

'm -J -.,

cu >. > m4 ro . 1. -

c -ýJ CL -4--3 o

.0 .- cu -0 -ý (0 n5 (0 (0 (D o0m -c a)

ý-- F- c: +i E

297

Total Imposed Load $015 N

- Total Imposed Load 15844 N

Theoretical Experimental Theoretical Experimental Node Vertical Vertical Node Vertical Vertical

Nu; nbers Displacements(mm) Displacements(mm) Numbers Displacements(mm) Displacements(mm)

31 4.651 7.326 31 9.192 12.594 37 4.222 3.999 37 8.344 7.796

26 4.222 4.219 26 8.344 7.866

25 4.222 4.273 25 8.344 8.034

32 4.039 3.951 32 7.983 7.743

20 4.039 4.240 20 7.983 7.849

21 3.658 3.763 21 7.230 7.062

38 3.715 3.202 38 7.343 6.595

27 3.715 3.455 27 7.343 6.852

14 3.715 3.950 14 7.343 7.202

15 3.715 4.214 15 7.343 7.513

16 2.959 3.244 26 5.849 6.035

33 3.553 3.039 33 7.022 6.430

9 3.553 3.600 9 7.022 6.459

22 3.021 2.471 22 5.970 5.248

10 3.021 3.235 10 5.970 5.876 11 1.689 11 3.339 39 3.233 2.453 39 6.392 5.491 28 3.233 2.377 28 6.392 5.372 3 3.233 3 6.392

4 3.233 4 6.392

5 1.910 5 3.715

17 1.910 1.432 17 3.775 3.376

Key: * Random Readings Obtained Due To An Amplifier Fault.

Table 5.21. Comparison Of Theoretical And Experimental Deflections For All itored Nodes In Model 4. The Table gives both the theoretical and experimental values of vertical displacements for all of the nodes monitored by transducers, for two different values of the imposed load. The first load value of 8015N is approximately in the middle of the elastic range while the second load value of 15844N corresponds with the theoretical limit in the elastic range of behaviour of the structure.

The measured displacements corresponding to an imposed load of 15844N are all within 16% of their theoretical values. The majority of the measured vertical displacements are all less than the corresponding theoretical values, indicating that the actual stiffness of the model structure was greater than the stiffness predicted from the theoretical analysis.

298

Figure 5.35 gives both the member numbers and the theoretical

values of the ratios of axial stress to yield stress for tension

members and axial stress to buckling stress for the compression members, when the structure was supporting a total imposed load of 15844 N. The member stress ratios given in the Figure for the soft members numbers 62,63,98,99 etc., give the ratios of member axial stress to the yield stress of the middle tube of the soft member. It can be seen from Figure 5.35 that the most heavily stressed tension members are the lower chord edge members 3,8,33 and 58

which have a stress ratio of 1.0. The adjacent members, numbers 2, 4,57,59 etc. are the next most heavily stressed tension members with a stress ratio of 0.92. The stress ratio of 0.92 shown in Figure 5.35 for the soft members indicate that when the structure is

supporting a total load of 15844 N the soft members are also close to yielding. The most heavily stressed compression members are the top chord members 66,67,94,95 etc., which have a stress ratio of 0.72, indicating that these should be the first members to buckle.

Theoretical Non-Linear Behaviour: The theoretical non-]! near behaviour of model 4 was investigated using the non-Hnear analysis

program outlined in Chapter two. Figure 5.35 shows the stress

ratios for each member in the model at the end of the elastic

response of the structure, and indicates that members 3,33,58 and 28 would be the first members to yield when the structure was supporting a central point load of 15844 N. When the total imposed load was increased to 16800 N the soft members 62,63,98,99 etc. (Figure 5.35) became critical with the middle tube of the soft member yielding in tension when the soft members were supporting a compression force of 3700 N. Yielding of the soft members was followed by the yielding of tension members 8,53,23 and 38 when the structure was supporting a total load of 17840 N.

As the imposed load was increased to 18000 N, the lower chord

edge members 2,4,57,59 etc. (Figure 5.35) yielded in tension.

Yielding of this group of members and the preceding groups of

members continued until the imposed load reached a value of 18320 N

causing the compression members 66,67,94,95, etc., to buckle.

Failure of the top chord compression members 66,67 etc. was immediately followed by a reduction in the load-carrying capacity of the structure and the elastic unloading of the bottom chord tension

members 3,33,58,28,8,53,23 and 28 (Figure 5.35). As the

buckled compression members 66,67,94,95 etc. shed load, the

299

cs

c

is

ts

cs

56 57 38 59 60

60

C - I - C -4 a: . 'Y (S

0.00 0.53 0.72 0.53 a. el3, 51 52 53

! ý

0.00 0.04 0.76 0.0 0.00 d, 6 7 418 8 u

(D tD N CD tý co tý m tý -) 1ý

O. ea 0.01 0.76 0.01 0.00 142 113 115

Vv cli Vý 1ý r" IR Vý '1 9

O. eo 0.53 0.72 0.53 0.00 36 37 38

GO CD o CD 0 - a) - (S) (S)

l

0.149 0.92 1.00 0.92 O. -is

BOTTOM CRTD

li

at Is 49 . 1.3 ",. "1 aQ

Figure 5.35. Critical Stress Ratios And Member Numbers For Model 4 At First Tield. The diagrams show the member numbers and critical stress ratios for both the top and botton chord members of model 4. The model was loaded at the top central node and supported at the four corner bottom chord nodes. The structure contains eight type 2 soft members (Figure 4.11), mem6el- numbers 62,63,82,83,78,79, 98 and 99.

For tension members (bottom chord): Critical Stress Ratio = member axial stress

yield stress

For compression members (top chord, excluding soft members): Critical Stress Ratio = inember axial stress

flexural buckling stress

For soft members: Critical Stress Ratio = member axial stress

yield stress of middle tube

300

A 4A A-Q? Aý qIP A 40k

97 98 99 100 V,

N Cý CD 11 CD

0.21 0.72 0.72 0.21 93 85

cli (s) CD

0.12 0.51 0.51 0.12 89 so 91. 92

rq

CD to CD CD CD

0.21 0.72 0.72 0.21 86 87 88

CD Cý

Q CD to CD

O. Aa 13.92 0.92 1 O. ia

0 CD

cr) N

co

TOP GRID

tse U-3 b4l

0.49 0.92 1 . 00 0.92 O. is 56 57 38 60

adjacent soft members 'closed-up' to form the triple tube

compression members, and consequently carried additional load.

At this stage in the theoretical analysis, the stiffness of the

structure increased slightly and the space truss was capable of supporting additional imposed load. As the point load acting on the

structure was increased, the buckled compression members continued to shed load and the stress in the tension members, which were previously unloading elastically, returned to the yield value. When the imposed load acting on the structure had reached 20776 N the

compression force carried by the soft members was sufficient to

cause them to buckle. After the soft members had buckled, the

stiffness of the structure decreased and load was transferred from

the buckled members onto the adjacent top chord compression members 70,71,90 and 91 (Figure 5.35). These members inturn buckled,

resulting in the complete loss of stiffness and the collapse of the

structure.

Experimental Non-Linear Behaviour: Figure 5.36 shows both the

theoretical and experimental load-displacement behaviour of model 4. It can be seen from the Figure that there is only close

agreement between the experimental and theoretical results for the

elastic part of the load-displacement behaviour. Under test, the

fourth model structure failed locally, with the premature buckling

of one of the four central web members. The failure of this web member number 149 (Figure 5.1) occurred when the structure was

supporting a total load of 15917 N (Point A, Figure 5.36), and was immediately followed by the buckling of the three adjacent members 150,151 and 152. The buckling of the four central web members led

to the vertical displacement of the top central node (node 31,

Figure 5.35), which inturn caused the premature buckling of the four

central top chord compression members 70,71,90 and 91 (Figure

5.35). - The almost simultaneous buckling' of the four central web

members and the four central top chord members, caused a rapid loss

In load-carrying capacity of the structure, with the central imposed

load decreasing to 4973 N. The load shedding associated with the failure of the four central web members was terminated when the

vertical deflection of the top central node was sufficient to'allow the four buckled central top chord members 70,71,90 and 91 to act

as a catenary and carry load in tension. The imposed load carried by the structure slowly increased to 6021 N, causing the compression

301

P7 28

P5 Is P4 Is

P3A 41

P2f it, %% I", "***

% a P6 %%

A-o 4#Pl P8

cl I Or

10 1 Qj r

CL

C, D B IE

-50

NODE NO. 31 EXPERIMENTAL RESULT ANALYTICAL RESULT

G F I �-

H

lea

VerticaL DispLacement in mm. 150

Fiqure 5.36. Experimental and Theoretical Load-Displacement Behaviour of Model 4. The Figure shows both the theoretical and experimental vert displacement of node 31 in model 4. The structure was loaded at node 31 by imposing a vertical displacement of O. 1mm per minute for the first 27mm of displacement after which the displacement rate was increased ten fold to 1.0mm per minute. After a total vertical displacement of the actuator of approximately 100mm the displacement rate was increased further to 2. Omm per minute.

Points on the theoretical load-displacement curve. P1 Yield of membeFs- 3,33,58,28. P2 Yield of soft members 62,63,98,99. P3 Yield of members 8,53,23,38. P4 Yield of members 2,4,57,59, etc. P5 Buckling of members 66,67,94,95, etc. P6 = Soft members 'closed up'. P7 = Soft members buckle. P8 = Buckling of members 70,71,90,91.

Points shown on the experimental load-displacement curve. A= Premature buckling oTmember 149, followed by web members

150,151,152 and top chord members 70,71,90,91. B= Buckling of member 95. C= Buckling of members 74. D= Buckling of members 66. E= Tensile rupture of top chord member 90. F= Buckling of bottom chord member 33. G= Tensile rupture of top chord member 71. G-H = Loss of stiffness due to rupture of memb'er 71.

302

200

member 95 to buckle (Point B, Figure 5.36), followed by members 74

and 66 (Point C and 0 respectively, Figure 5.36). The buckling of the top chord compression member 66 was shortly followed by the tensile rupture of the top chord member 90 (Point E, Figure 5.36)

causing a sudden decrease In load-carrying capacity of the

structure. With the rupture of member 90, the total load carried by

the structure decreased to 4222 N, and the force carried by this

member was transferred into the adjacent top chord members 70,71

and 91.

As the vertical displacement of the top central node increased,

the load-carrying capacity of the structure slowly began to

recover. The bottom chord member 33 buckled when the imposed load

carried by the structure had increased to 6526 N (Point F, Figure

5.36) and the top chord central member, number 71, failed in tension

when the Imposed load had increased further to 7114 N (Point G,

Figure 5.36). The rupture of member 71 caused a rapid loss in

stiffness of the structure, with the imposed load decreasing to 1996

N (Point H, Figure 5.36).

At this stage, the load-carrying capacity of the structure very slowly increased as the top central node (number 31), and the

remaining members attached to it, were pulled down into the

structure. The experimental investigation was terminated when the

vertical displacement of node 31 had reached 200 mm which was the displacement limit of the actuator used to load the structure.

Figures 5.37 A to F show how both the theoretical and experimental strains vary in six of the bottom chord tension

members. A comparison of the Figures indicates that none of these

members had yielded when the structure failed prematurely and, as predicted by the theoretical analysis, member 3 was the most heavily

stressed member, followed by members 2,8,13,7 and 1

respectively. Figure 5.37 A shows a very close agreement between the theoretical and experimental axial strain values for member 1. In addition, it can be seen from this Figure, by comparing the

strains recorded from gauges 31 and 32 attached to the top and bottom of the member, that only small bending strains were present in this member at mid-length. Figures 5.37 B and C show that in both members 2 and 3, the theoretical values of axial strain calculated for any particular load value, were larger than the corresponding experimentally measured values. These Figures also show that bending strains at mid-length in members 2 and 3 are still

303

M[ýBEER NO. I

-v

25

2e

0 a 0

-J '5

0-

a - l0

EXPERIMENIAL RESULTS Strain Gauge No. 31 Strain Gauge No. 32 ANALYTICAL RESULT

ia 12

Percentage Strain ( 10E-02)

MEMER NO. 2 EXPERIMENT& RESULTS Strain GGUge No. 27 Strain 6auge No. 28

------- ANALYTICAL RESULT

FIGURE 5.37A

FIGURE 5.37B 5

25

ýý 28

15

18 12

Percentage Strain ( IOE-02)

MEMBER NO. 3 EVERIMENTAL RESULTS Strain Gouge No. 15 Strain Gcvge No. IG

------- WLYTICAL RESULT

16 ^10

FIGURE 5.37C

-4

25

2$

a 0

r, -

8 12 16

PerCentOge Strain ( 10E-02)

MEMBER NO. 7 EXPERIMENTAL RESULTS Strain Gauge No. 39 Strain Gouge No. 9

ANALYTICAL RESULT

70 24

-i Ia 12

Percentage Strain ( IOE-02) 304

(Positive Strains are TensiLe)

25

20 zt. -ýZ

-0

15

5

MEMBER NO. 8 EXPERIMENTAL RESULTS Strain Gauge No. 13 Strain Gauge No. H ANALYTICAL RESULT

ý1ý

.0 ------------------ 0 ---------------- 4w

FIGURE 5.37E

8 12 1 ro

Percentage Strain ( IOE-02)

(Positive Strains are Tensite)

MEMBER NO. 13 25 EXPERIMENTAL RESULTS

Strain Gauge No. 38 Strain Gauge 11o. 37

20 --- 41--- ANALYTICAL RESULT

15

ra- Iýf

-8 -4 18

Percentage Strain ( IOE-02)

Ar

12 16

21)

Experimental And ýheoretical Applied Load Vs Member Strain Curves F or Mode 1 4.1 he FI gur es show 7i_o_w___tFe_ theoretical and experimental values f strain in tension members 1, 2,3,7 ,8 and 13 vary with an increase in the external applied load. The experimental values were obtained from each of two strain gauges placed diametrical ly opposite each other, mid-length of each member.

305

small, but nevertheless increasing in magnitude the closer the

member is to the mid-side location in the structure. Figures 5.37 D

and E show close agreement between the theoretical and experimental load-axial strain relationships for both members 7 and 8. However, the two load-strain relationships obtained from strain gauge 37 and 38 attached to member 13, plotted in Figure 5.37 F, Indicate that

although there is close agreement between the theoretical and experimental values of axial strain in the member, bending strains

were present in this member even when the structure was supporting a relatively small total imposed load.

Figure 5.38 A to E indicates how both the theoretical and

experimental values of axial stress vary in compression members 101,

199,61,65 and 69 under the action of an increasing displacement imposed on the top central node of the structure. Figures 5.38 A

and B give the load-axial stress relationship for the two corner

members 101 and 199 respectively. Both of these Figures show close

agreement between the theoretical and experimental results with the theoretical values of axial stress slightly greater than the

experimental values obtained for the same value of total load acting on the structure. However, there is very little agreement between

the theoretical and experimental values of axial stress obtained from members 61,65 and 69 shown in Figures 5.38 C, D and E. These three members were in the top chord of the structure and, after the

premature buckling of member 149 (Figure 5.1), exhibit a behaviour

markedly different from their expected behaviour.

Figures 5.39 A to C show both the theoretical and experimental

relationship between the total applied load acting on the structure and the axial strain in compression members 62,66 and 70

respectively. Figure 5.39 A relates the applied load acting on the

structure to the axial strain in the outside tube of the soft member, and it can be seen from the large discrepancy existing between theoretical and experimental results that this soft member was malfunctioning from the start of the experimental testing of model 4. The experimental load-axial strain relationship obtained from member 66 (Figure 5.39 B) shows that this member was Influenced

by the pre-loading of the structure to 5000 N, and also that it

experienced elastic unloading when member 149 buckled prematLýrely. The experimental load-axial strain behaviour exhibited by member 70 (Figure 5.39 C) indicates that the strain In this member was decreasing just before member 149 buckled and, also that when member

306

(positive Stresses are TensiLe)

25

28

-0 0 15

--i c3. CZ.

-< la

C: 9

5

-18 -20 -30 -io -58 -M -70 -88 -90 -188

FIGURE 5.38A Axid Stress (N/sq. mm)

20

0

--e- le

5

-10 -28 -30 -40 -50 -ca -70 . 88 . 90 -100

FIGURE 5.38B AxiaL Stress (N/sq. mm)

2

,a -0

5

. $sow-*

0 -20 --ie -M -88 -108 -120 -lie -ice -180 -200

FIGURE 5.38C Axid Stress (N/sq. mm)

307

(Positive Stresses are TensiLe)

25

28

--p co

5

ý 20 3i

"0

-0 13 ICU

06

--------------- V

50 is 38 20 Is -10 -20 -30 -is -58

Axid Stress (N/sq. mm) FIGURE 5.38E

Figure 5.38A to E. Experimental And Theoretical Applied Load Vs Member Axial StrFs Curves For Model 4. The Figure shows how-7567F the theoretical and experimental vs of stress in compression members 101,199,61,65 and 69 change with increasing external applied load. The axial stress was obtained for each member by averaging the strain reading obtained from the three strain gauges placed symmetrically around the circumference at mid-length of each member.

MEMBER NO. 65 C: YDr-PTMPJTAI PPqlg T

-18 -20 -38 -io -50 -68 -78 -88 -90 -108

FIGURE 5.38D Axid Stress (N/sq. mm)

MEMBER NO. 69 251 EXPERIMENTAL RESULT

------- ANALYTICAL RESULT

308

149 had failed, member 70 unloaded elastically and eventually became

strained in tension.

Figures 5.40 A to H show the load-strain relationships obtained from each of the three strain gauges attached to compression members 101,199,61,65,69,62,66 and 70 respectively. A comparison Of the three load-strain relationships obtained for each member gives an indication of the amount of bending occurring within the member. Figures 5.40 A and B show that bending of t6e two corner web members, 101 and 199 respectively, occurred from the start of the test and that bending strains had reached significant values when failure of the model occurred with the premature buckling of member 149. Even larger bending strains were experienced by members 61,66

and 70 (Figures 5.40 C, G and H). Figure 5.40 F gives the load-strain behaviour for each of the three gauges attached to the

outside tube of soft member number 62. The Figure shows that a small bending moment of approximately 850 Nmm was present at

mid-length of the tube when member 149 buckled. This bending moment

would have hindered the relative movement of the three tubes forming

the soft member and, in addition, would have had a detrimental influence on the load-displacement behaviour exhibited by this

member. However, it is not envisaged that the presence of this bending moment caused this particular member to malfunction, as several of the other seven soft members in the structure yielded and 'closed-up' during the experimental testing of the structure.

DISCUSSION

Component Members

Both the tensi le and compressive behaviour of the individual

component members used in the fabrication of the test models had a significant. influence on the entire load-displacement response of each of the four test structures. The tests undertaken on the

component members reported in this Chapter were used to determine,

as accurately as possible, the various parameters required to describe numerically the complete load-displacement response of the individual members. A consideration of the mean values with the

corresponding standard deviations given in Table 5.9, which were calculated from the experimental results given in Tables 5.8 indicate that the spread in the values, of both yield and ultimate stress obtained from the tensile tests, was significantly less than

309 1--

(Positive Strains are Tensile)

23 IIEWER NO. G2

25 EXPERMITAL RESULTS AXIAL STRAIN

--- MYTICAL RESULT 26 . 'A 28

........... v 11

. ., -, %\\

-W

-�I Is Is - I'

'I 5. :

-12 -4 48 12

FTC, USE -5-12A Percentage Strain 18HO MMER NO. GG

25 EXPERIWAL RESULTS AXIAL STRAIN ANALYTICAL RESLLT

20

15

-12 -0 0 12

rT MJRF -9

3 9B Percentage Strain IBE-02)

tMER NO. 70 23 EXPERIMENTAL RESULTS

AXIAL STRAIN

20 --- 0--- ANUTICAL RESLLT

-0

0 0

5

-12 -8 -4 48 12

VTr, URE 5-3c)r Percentage Strain ( ICE-02)

Figures 5.39A to C. Experimental And Theoretical Applied Load Vs Member Axial Strain Curves For Model 4. The Figures show how the theoretical and experimental values of axial strain in compression inembers 62,66 and 70 change with incr. easing external applied load. The axial strain was obtained for each member by averaging the strain readings obtained from three strain gauges placed symmetrically around the circumference at mid-length of each member. Figure 5.39A relates the applied load acting on the structure to the axial strain in the outside tube of the soft member. It can be seen from the large discrepancy existing between theoretical and experimental results shown in this Figure that this soft member was malfunctioning from the start of the experimental testing of model 4.310

FIGURE 5.40A MEMBER NO. 101

0 EXPERIMENTAL RESULTS Strain Gauge No. 29 Strain Gauge No. 3

15 Strain Gauge No. 30

10-

-12 -8 -1 8

Percentage Strain 10E-02) (Positive Strains a re Tensite)

2 1: 5 MEMBER NO. 199

FIGURE 5.40B 20 0 C5

EXPERIMENTAL RESULTS 0 Strain Gauge No. 5

Strain Gauge No. 6 15 - Cl-

Strain Gauge No. 19

10

5

-12 -8 -1 18

Percentage Strain ( 10E-02)

(Positive Strains are Tensiýe)

FIGURE 5.40C 20 2 MEMBER NO. 61 EXPERIMENTAL RESULTS

0 Strain Gauge No. 36 Strain Gauge No. 35

15- Strain Gauge No. 22 0-

10- ý2-

5

-12 -8 -1 18

Percentage Strain ( 10E-02) (Positive Strains are TensiLe)

311

12

12

12

: 2-'

MEMBER NO. 65 5 . 40D 20 FIGURE 0 EXPERIMENTAL RESULTS

Strain Gauge No. 26 Strain Gauge No. 25

15 CL 01- Strain Gauge No. 24 cl (Positive Strains are TensiLe)

le

-12 -8 -4 18 12

Percentage Strain 10E-02)

MEMBER NO. 69 25 EXPERIMENTAL RESULTS

0 Strain Gauge No. 33

FTGURE 5.40E Strain Gauge No. 31 Strain Gauge No. 23

20 1 c, (Positive Strains are TensiLe)

-12 -8 -1 18 12

Percentage Strain ( 10E-02)

MEMBER NO. 62 EXPERIMENTAL RESULTS

FTGURE 5.40F - 20 Strain Gauge No. 7 -0 Strain Gauge No. Q

CL Strain Gauge No. 8

15 (Positive Strains are TensiLe)

0

-12 -1 18 12

Percentage Strain ( IOE-02) 312

25

20

-0

is

0 le

5

MFJIBER NO. 66 EXPERIMENTAL RESULTS Strain Gauge No. 21 Strain Gauge No. 11 Strain Gauge No. 12

-12 -8 -1 48

Percentage Strain ( IOE-02)

MEMBER NO. 70 25- EXPERIMENTAL RESULTS

Strain Gauge No. 20 Strain Gauge No. 18

cl Strain Gauge No. 17 20

15

10

-12 -8 -1 1

Percentage Strain ( IOE-02)

(Positive Strains are TensiLe)

8

Figure 5.40A To H. Experimental Aoplied Load Vs Member Strain Curves Obtained Mode 1 4. The Figure shows how the strain in compression members 101,179-, 61,65,69,62,66 and 70 varies with an increasing externally applied load. The experimental values were obtained from each of the three gauges positioned symmetrically around the circumference at mid-length of each member.

12

12

313

the spread in the values of the critical flexural buck] ing stress obtained from the compression tests. However, the mean experimental values of the flexural buckling stress, obtained from both types of compression members T5 and T6 (Table 5.1), showed good agreement with the corresponding theoretical values, calculated assuming an initial sinosodial member bow of 1/1000. For both of these member types the theoretical values of the buckling stress were less than the mean of the experimental values, indicating that the assumed initial member bow of L/1000 was probably greater than the average bow of the test samples. An initial member bow of L/1000 was used in the theoretical calculations because of the difficulty

experienced in measuring accurately both the deviation of the test

samples from a 'true' line, and also the wall thickness of the

various tubes. These are both extremely important parameters, which require precise measurement if experimental results are to be used to assess the validity of theoretical assumptions.

Behaviour of Models 1 and 2

Test models 1 and 2 were almost identical structures, designed

so that under the action of an increasing imposed load, extensive tensile yield would occur in the bottom chord members before any of the top chord members buckled. To fulfil these design requirements, the top chord compression members were significantly stronger than the bottom chord tension members, with the flexural buckling load of the compression members over eight times the yield load, and six times the ultimate load of the tension members. This unusual ratio of member strengths led to the required behaviour, with the load-displacement response of both of these structures showing considerable ductility after first yield.

Elastic behaviour: The experimentally observed elastic behaviour of each of the double-layer grids was both linear and fully symmetric. The measured vertical displacements of symmetrically positioned nodes were in close agreement with each other and also In reasonably good agreement with their corresponding theoretical values. In both

model structures the measured vertical deflections of the boundary nodes were approximately 5% greater than their theoretical values, and the measured deflections of the central nodes were approximately 5% less than their theoretical values. This pattern occurring In the difference between the experimental and theoretical deflections was unexpected. It was envisaged that, because the theoretical

314

analysis assumed the structure to be pin-jointed instead of rigidly jointed, the theoretical vertical deflections of all of the monitor

nodes would marginally exceed the experimentally measured values. This apparent discrepancy between the experimental and theoretical

values of displacement may be due to differences in rigidity between

the boundary and middle regions of the structure. The boundary

nodes are held in position by five members framing into one half of the joint whereas the remaining nodes in the structure are held in

position by eight members symmetrically positioned around the node.

Close agreement was obtained between the experimentally

measured and theoretical values of axial strain. In general the

theoretical values of axial strain were slightly greater than the

corresponding measured values. The difference may be due to the

small bending strains present in both structures, which were not

considered in the theoretical analysis.

Inelastic behaviour: A comparison of the theoretical and

experimental load-displacement behaviour obtained from both models I

and 2, given in Figures 5.17 and 5.24 respectively, shows that the

analytical program provides an accurate prediction of the actual

structural behaviour. The non-linear behaviour of both structures

was, in general, symmetric, resulting in a moderately good agreement between the theoretical and measured vertical displacements of the

nodes.

In addition, there was also very good agreement between the

theoretical and measured values of axial strain recorded in the

majority of the tension members that were monitored. However, the

measured strains in members 2,3 and 8 of model 2, (Figure 5.25)

deviated from the theoretical path just after first yield occurred in the model. This discrepancy in the theoretical and experimental

values of axial strain may have resulted from the yield

characteristics of these particular members. One quarter of the

bottom chord members of model 2 were cut from one tube, type

brown-green, (Figure 5.4) which exhibited an extremely small

post-yield stiffness before strain hardening. The actual post-yield stiffness of this member, measured from tensile specimen tests, was

significantly less than the mean value used in the analytical

program, which was calculated from all of the tensile specimen tests

undertaken on tube type T1 (Table 5.1). The use of a larger

post-yield stiffness to model the displacement behaviour of these

315

tension members in the analysis program would lead to the

underestimation of the strain occurring in these members, which is

shown in the early portion of the post-yield load-displacement behaviour given in Figures 5.25.

A comparison of the theoretical and experimental values of axial stress in the top chord compression members of both models 1

and 2, indicates that in general there was a reasonable agreement between these two sets of values throughout the. non-linear range of behaviour. Examination of the test data shows that in both of these

structures the experimental values of axial strain were greater than the theoretical values for the top chord edge compression members, and less than the theoretical values for the middle top chord compression members. These findings are compatible with the

response which the model structures exhibited during the elastic portion of their load-displacement behaviour, where the measured vertical displacements of the nodes were slightly greater than the

corresponding theoretical values for the boundary nodes, and slightly less than the theoretical values for the internal nodes.

Figures 5.20 (A, B and C) and 5.26 ( A, B and C) show how the

axial stress in some of the top chord compression members of models 1 and 2 alter non-linearly as tensile yield progresses throughout the bottom chord tension members. It is interesting to consider further the Interaction that exists between the top and bottom chord axial forces and the changes which occur in the force distribution

as the imposed load acting on the structure is increased. Figure 5.41 gives the theoretical values of axial force occurring in the members at four cross-sections taken through model 1. for each of two different values of the imposed load. A comparison of these diagrams given in Figure 5.41 shows that as yield progresses In the bottom chord tension members, from the centre members outwards towards the edge members, the axial force in the top* chord compression members, positioned above the yielded tension members, increases by a larger percentage than. the increases occurring in the other top chord compression members.

This behaviour was not expected. Instead it was envisaged that

as bottom chord tension members yielded and carried a constant force, the force in the compression chord members above the yielded tension members would also rema 1n almost constant. It was considered that increases in the imposed load would be carried by the adjacent elastic top and bottom chord members. However, both

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the theoretical and experimental results are in close agreement in describing this behaviour, and show that although the tension

members, which are both adjacent and parallel to the yielded tension

members, do carry a significant proportion of the additional tensile load, the top chord compression members, above the yielded tension

members, carry a significantly larger proportion of the additional compression load, compared with that carried by the adjacent parallel top chord compression members.

Figure 5.42 gives the theoretical values of axial forces

occurring In the members at four cross-sections taken through model 2 for each of two different values of the imposed load. It can be

seen, by comparing the member forces obtained for each of the two load cases, that as tensile yielding occurs in the bottom chord edge meý. -nbers and progresses towards the centre of the bottom chord, the

compression members above the yielded tension members carry all of the additional compression force. In addition, the compression force in the adjacent parallel top chord members actually decreases.

This behaviour is similar to that exhibited by model I shown in Figure 5.41. However, in model 1, which is loaded by a central point load, tensile yield results in the initially understressed centrally positioned top chord members (70,71,90,91 Figure 5.41)

carrying the majority of the additional compression chord force. In

model 2, the large additional compression force, resulting from both

an increase in the imposed load, and tensile yielding, was carried by the top chord edge members (62,63,82,83 etc. Figure 5.42)

which were also initially the most heavily stressed compression members.

Behaviour of Models 3 and 4

Both of the test model structures 3 and 4 were designed to

exhibit a duct I le post-yield load-displacement response, by

permitting tensile yield to occur In the bottom chord members In

conjunction with the yielding of eight model soft members, symmetrically positioned in the top compression chord of each structure. ý Test model 3 was similar to models 1 and 2, In that the strength of the compression members was greater than the strength of the tension members. The compression members used in model 3 (type T5, Table 5.1) exhibited a critical flexural buckling load approximately two and a half times the yield load, and almost twice the ultimate load of the tension members. However, larger tension

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members (type T2, Table 5.1) were used in the fabrication of model 4, providing a yield load approximately 70% of the buckling load

exhibited by the T5 (Table 5.1) compression members, also used in

model 4.

Elastic 6ehavlour

Figures 5.30 and 5.36 give both the theoretical and experimental load-displacement response obtained-from models 3 and 4

respectively. It is evident from these Figures that close agreement

exists between the experimental and theoretical results only for the

elastic, and early post-elastic, parts of the load-displacement

behaviour.

In general throughout the elastic range of behaviour, the

theoretical vertical displacements of the nodes for both structures

were greater than the corresponding measured displacements. This

may be partly due to the assumption, used in the theoretical

analysis, that the members were pin-jointed to the nodes of the

structure, but may also be due to possible variations occurring in

the axial stiffness of the soft members incorporated into , both

structures. The experimental values of axial stiffness obtained from the compression tests undertaken on the model soft members,

reported in Chapter 4 (Table 4.9), show significant variations,

with the highest measured value approximately 12% greater than the

smallest value obtained from nominally identical members. Consequently, similar variations in the axial stiffness occurring in

any of the e. ight nominally identical soft members, used in either models, would influence the measured values of vertical displacements of the nodes in these structures.

The preliminary tests undertaken on Model 4, to assess the

early elastic response of the structure, indicated that in spite of almost equal values of reactive forces, recorded from each load cell positioned under each corner support of the structure, large

variations in strain were measured in the outside tubes of some of the eight soft members. This Indicated that several of these soft members were not functioning as designed. With hindsight it would have been sensible to remove and replace these members at this

stage, but it was considered that their poor performance was due to Interference arising between the three close fitting tubes, which would gradually reduce and disappear as the load acting on the soft

320

members Increased. To improve both the model behaviour and the

theoretical analysis, it would be beneficial to test each soft

member separately, but in conjunction with its end nodes, to assess if each member was functioning correctly prior to yielding, and also to determine Independently the axial stiffness of each device.

Throughout the elastic range of behaviour, the measured values of axial strain recorded In both the tension and compression members of models 3 and 4 was, in general, slightly less than the theoretical values of axial strain obtained from the analysis program. Very small bending strains were recorded in the majority of members and because these were not considered in the analysis program, this may have led to the small differences arising between the measured and theoretical values of axial strain.

Inelastic behaviour

The inelastic behaviour of both models 3 and 4 was, in general,

not as predicted by the theoretical analysis. Yield did occur in

three separate groups of tension members In model 3, as indicated by the analysis program, but the desired non-]! near behaviour of the

structure was interrupted by the premature buckling of one of the

soft members. Soft member, number 62, buckled when the structure was supporting a total imposed load of 13816 N. With this value of imposed load acting on the structure, the force applied to the soft member would have been sufficient to cause the middle tube of the

member to yield in tension, causing the vertical gap between the inner and outer tubes to decrease. However, premature buckling of this soft member occurred before sufficient plastic flow had

accumulated in the middle tube to allow the three tubes of the soft member to close. Close examination of the soft member during the

exper ! mental testing of model 3 revealed that small rotations occurred to the end nodes of the soft member prior to buckling. The

rotation of the end nodes imposed bending moments on the soft member, and led to buckling of the member precipitated by the local failure and rotation of the extended portion of the inner tube,

close to the top node (Figure 4.11). This inherent weakness In bending stiffness occurring in the type 2 (Figure 4.11) soft members, was minimised by increasing the section modulus of the device close to the top node, by incorporating a collar located over the top of the Inner tube (Type 3, soft member, Figure 4.11). As an alternative to changing the inember design details, it would have

321

been possible to isolate the soft member from bending moments by

incorporating pinned-joints at each end of the member. However, it

is particularly difficult to design a 'pinned' joint which allows

equal rotation of the member about any axis, and has similar

characteristics when loaded in tension or compression.

The non-linear behaviour of model 4 was also interrupted by the

premature buck] ing of one of the compression members. The central web compression member (Number 149, Figure 5.1) buckled when the

. structure was supporting a total imposed point load of 15917 N

resulting in a theoretical value of axial stress in the member at failure of 227.37 N/mm 2. This value is approximately 29% less than the mean value of the flexural buckling stress obtained from the

twenty T5 samples tested prior to the model test (Table 5.10). In

addition, the buckling stress of member 149 (227.37 N/MM2) is equal to the mean value of the buckling stress of the twenty samples (319.24 N/mm 2), minus 3.29 times the standard deviation (27.91

N/mm 2) of the test group, indicating that there was a probability of less than one in one thousand of member 149 failing at this low

value of axial stress. However, the theoretical value of stress in

member 149 was obtained by ignoring the possibility that bending

strains can develop in this member which will, in turn, have a detrimental effect on the buckling load of the member. Nevertheless, the large difference arising between the theoretical

critical flexural buckling stress of member 149, and the mean value obtained from the twenty test samples suggests that it is most likely that this member was damaged during fabrication of the model, probably during the welding of the member to the end nodes.

To model mathematically the influence which the premature buckling of member 149 had on the theoretical load-displacement

behaviour of the structure, the model was re-analysed, assumming a 29% reduction in the critical flexural buckling load of member 149. Figure 5.43 compares the theoretical -load displacement behaviour,

obtained assuming this reduction in the buckling load of member 149,

with the experimental load-displkement behaviour obtained from the test on model 4.

It Is evident, by comparing the theoretical load-displac. ement behaviour of the model structure assuming no imperfections in member 149, (given in Figure 5.36), with the theoretical behaviour of the

model assuming a reduction in the flexural buckling load of member 149 (Figure 5.43), that imperfections occurring in the compression

322

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It is evident by comparing the theoretical load-displacement behaviour of the model structure assuming no reductions in the flexural buckling load of member 149 given in Figure 5.36, with the theoretical behaviour of the model assuming a 29% reduction in the flexural buckling load of member 149, that imperfections occurring in certain compression members in these structures will have a significant influence on the overall behaviour of the structure.

200

323

members of these structures will have a significant influence on the

overall behaviour of the structure. This is an important

consideration which should be taken into account when assessing both the ultimate load and collapse behaviour of 'compression chord critical' double-layer grids.

General Comments

Table 5.22 gives both the theoretical and experimental values of the ultimate loads supported by each of the four test models in

addition to a brief description of the collapse mode of each structure. It is evident from the results given in Table 5.22, and from the experimental work reported in this Chapter, that attempts to improve the ductility of square-on-square double-layer grids, by

permitting tensile yield in the lower chord members, has been

completely successful. However, the attempts to improve both the

elastic force distribution within square-on-square double-layer

grids, and the post-elastic ductility of the structure, by incorporating soft members into the compression chords, has not been

so rewarding. The disappointing results emanating from the two tests on double-layer grid structures incorporating the soft members have been due to both the poor performance of the type 2 and type 3 soft members (Figure 4.11) when subjected to both axial forces and bending moments, and also to imperfections introduced into the structure during fabrication.

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CHAPTER 6

GENERAL DISCUSSION

INTRODUCTION

Several Investigators studying the behaviour of square-on-square double-layer space trusses have found that although these structures may be highly statically indeterminate, complete collapse of certain types of structure may occur, under an Increasing Imposed load, with the failure of one or a small number of compression members (Schmidt, et a], 1976; Smith, 1984b). The square-on-square double-layer space truss studied in this investigation shown in Figure 5.1 is statically indeterminate with twenty-five redundancies (Affan, et al, 1986). However, if the

compression members in the structure are designed to have equal strength, failure of one of the four corner web members or one of the top chord, mid-side, edge compression members would be

sufficient to cause a complete collapse of the structure. A typical

compression member used in a double-layer space trusss has a slenderness ratio close to the transition slenderness and consequently these members when supporting an increasing compression load buckle plastically with a sudden release of energy. If a double-layer truss structure is to remain stable after buckling of the most heavily stressed compression member, the remaining members in the structure must be capable of absorbing the total energy released from the failed member. If the members surrounding the failed members are unable to support this sudden increase in load,

progressive buckling and collapse of the structure will ensue. The

collapse mode of the structure will be similar to the failure mode exhibited by the individual compression members, where the collapse is sudden, showing no ductility in the load-displacement response.

Supple et al, (1981) along with other investigators have shown the effect which the post-buckling response of the individual

compression members has on the collapse behaviour of square-on-square space trusses. If compression members which exhibit a gradual load shedding after buckling are incorporated into

a space truss, ductility is introduced into the load-displacement

response of the structure (Saka, et a], 1984).

326

Investigations reported by Schmidt, et al, (1979) have shown that some model double-layer space trusses have failed at loads below their theoretical values even when the theoretical analysis has used the actual critical buckling loads of the component members obtained from compression tests undertaken on the individual

members. Schmidt et a], have suggested that this discrepancy

arising between the experimental and theoretical ultimate loads of certain model space trusses is a direct result of an initial force distribution existing within the structure due to the lack of fit of the members. The real possibility of a significant reduction in the

ultimate load of square-on-square trusses due to the initial lack of fit of members calls into question the magnitude of the true factor

of safety against collaps e existing on certain truss structures designed to comply with the present codes of practice.

The problems associated with the initial lack of fit of members

are I lkely to be present in varying amounts in all statically indeterminant structures. However, tne magnitude of the detrimental

effect which an initial force distribution has on a space truss is

directly related to the configuration of the structure, the position of the support points, and tne type of imposed load acting on the

structure. Square-on-square double-layer space trusses supported at a] I boundary nodes carrying a uniformly distributed load, possess reserves of strengtn aoove tne load level sufficient to cause the first compression member to buckle. Consequently, progressive collapse of these structures is most unlikely and adverse initial force distributions occurring from member lack of fit have less

effect on the structural reserve strength index of the structure. This index can be used as a measure of the reserve of strength existing in the structure, and may be defined as the ratio of the

ultimate strength of the structure to the load applied to the

structure (Feng, 1988).

Analytical Methods

The ab 111 ty to be ab Ie to assess the ul t ! mate strength and

collapse behaviour of a space truss is a prerequisite for efficient

structural design. The algorithm which has been written to determine both the collapse load and collapse behaviour of space trusses, outlined in Chapter 2, is a modified form of the dual load

327

method of analysis first proposed by Schmidt, et al, (1980b). The

method has been extended to deal with the elastic-plastic behaviour

exhibited by tension members, the 'brittle type' post-buckling response exhibited by compression members, and changes in geometry which occur as the structure deflects. The non-linear member behaviour has been modelled using several linear approximations to

represent the load-displacement response of the individual members. This technique is similar to the approach adopted by other investigators (Collins, 1981; Howells, 1985), however the present algorithm uses a greater number of linear approximations to model the member behaviour than have been used in previous work.

This in turn has increased the complexity of the checking

procedure undertaken to ensure that the deformation of each member in the structure conforms with the equilibrium and compatibility

requirements of both the structure and the I ! near ization

approximation of the member behaviour. In the present algorith; n, these checks are performed interactively so that the analyst may

pre-select possible equilibrium paths and reduce to a minimull the large amount of computational work required. This has proved to be time consuming for the analyst who may have to try several combinations of alternative load paths for the failed members before

the unique solution is found. To fully automate this checking

procedure it would be necessary to develop a rationale which would give priority to certain combinations of members and load-paths

which were considered to give the highest possibility of fulfilling

the equilibrium and compatability requirements. However, it is

likely that the priorities used to minimise the numerical

computations required to find the correct solution would change after the failure of each new member, or when each failed member changed from one region to the next on the linearized

load-displacement path. At present, Smith (1984a) Is the only investigator who has reported an algorithm capable of undertaking this complex checking procedure automatically. Smith's algorithm, outlined in Chapter 2, partitions the members into three distinct

sets depending on their behaviour, and checks each set in turn to

ensure that equilibrium and compatability requirements are met.

It is possible that the numerical checking procedures mentioned above and described in detail in Chapter 2, could be simplified by

328

describing the post-buck] ing behaviour of the compression members, and the post-yield behaviour of the tension members, by continuous algebraic functions. A modified form of the Ramberg-Osgood (1943)

equation would be suitable for the description of the tensile behaviour and the expression initially derived by Paris (1954) and extended by Supple and Collins (1981) adequately describes the flexural post-buckling behaviour of tubular compression members. These non-Hnear expressions describing individual member behaviour

can then be solved iteratively in conjunction with the solution of the main set of linear equilibrium equations.

Methods of Improving Space Truss Behaviour

At present, relatively few published papers on space truss

behaviour relate to investigations undertaken to study ways of improving the load-displacement response of double-layer grids. The

work which has been carried out to date has concentrated on five

different areas.

Schmidt et a], (1978) have obtained a ductile load-displacement

response by constructing model space trusses with weak tensile

chords. They found that even with a ratio of compressive to tensile

chord areas of 28 to 1 compressive chord collapse limited the

ductile range of behaviour of the truss. This has led to the

misconception (Hanaor, 1979) that every double-layer space truss

must finally collapse by buckling of the compression members. The first double-layer space truss tested during the present investigation had a ratio of compressive to tensile chord area of

approximately 8 to 1, and collapsed with the progressive rupture of

several of the lower chord tension members.

An alternative approach undertaken to introduce ductility into

the load-displacement response of double-layer space trusses

encompassed the removal of selected web members from the structure (Marsh, 1986). The theoretical investigation reported by Marsh

shows that removal of certain web members can improve the force distribution within the chords and increase the load-carrying

capacity of the structure. The diagrams given in Figure 5.41 and 5.42 show how the axial forces change in both the web and chord members as tensile yielding occurs in the lower chord members of test models 1 and 2 respectively. It can be seen from the Figures that large force changes occur in the web members and this gives an indication of the important role that web members undertake

329

In transferring forces throughout the structure. It would be beneficial to extend the preliminary work undertaken by Marsh and investigate both theoretically and experimentally the load-displacement response of double-layer structures which have both a yielding tensile bottom chord and selected web members removed.

Other researchers have investigated the possibility of improving both the load-displacement response an d the ultimate load

carrying capacity of double-layer space trusses by introducing into the structure an initial state of pre-stress (HoInIckI-Szulc, 1979; Spillers, et a], 1984; Hanaor, et a], 1986). For this concept to be

effective it is essential that the correct combination of tensile

and compressive pre-stress is imparted into the statically indeterminate truss structure. This operation may be complicated by

an Initial force distribution of unknown magnitude existing within the structure due to various imperfections such as the lack of fit

of members and small var. lations in support positions. Kowa] et al, (1975) have measured random force distributions occurring within carefully made models where theoretically 'zerb stressed web members were supporting forces approximately equal to 8% of the forces

carried by the most heavily strained web members. Even larger initial force variations were measured in the chord members. Hence,

any initial pre-stress introduced into the structure must negate any detrimental effects arising from an existing random force distribution resulting from initial imperfections within the

structure.

Other attempts to improve the load-displacement response of double-layer space trusses have concentrated on modifying the brittle post-buckling behaviour exhibited by typical compression members used in these structures. Schmidt et al, (1977) and Marsh

and Fard (1984) have used excentr ical ly- loaded compression members to obtain a small amount of ductility in the load-displacement

response of double-layer space trusses but this has always been

accompanied by a reduction in the load-carrying capacity of the

structure.

However, artificial ductility has successfully been introduced into compression member behaviour by incorporating into the member

330

a force-limiting device (Schmidt, et a], 1979; Hanaor, et a], 1980). A wide range of structural devices have been reviewed in Chapter 3 with the aim of assessing their potential use as force-limiting devices. The ideal pre-requisite which a structural device must possess if it is to be of use as a load I Wter, is a perfect elastic-plastic load-displacement response independent of both time and loading sequence.

Several of the structural devices considered in Chapter 3 are capable of producing the required elastic-plastic load-displacement

response when tested in compression. Laterally loaded tubes provide a smooth post-yield load-displacement response exhibiting a gradual increase in stiffness as the tube crushes (De Runtz, et a], 1963; Reddy et al, 1979). This mechanism could be adopted to form a load-limiting device by positioning a short portion of tube,

possibly constrained in a box, at the end of the member to be

protected. The longitudinal axis of the short tube would be

perpendicular to the long itud inal axis of the member, and the

operational load and length of the ensuing load plateau could be tailored to suit the design requirements by a suitable choice of tube diameter and wall thickness.

An acceptable compressive load-displacement behaviour was also obtained from progressive plastic deformation of a W-frame (Johnson,

1972). The load-displacement response exhibited by the W-frame (Figure 3.15) shows a long, almost constant, load plateau after the first plastic hinge has formed followed by a rapid return in the

stiffness of the frame as the two long legs of the W-frame meet. This increase in stiffness occurring after the load plateau would provide a useful reserve of strength In a force-limiting device

constructed using the W-frame mechanism. The load-limiting device

could be fabricated in the form of a circular bellow which would be

positioned along the axis of the member, between the member end and the node. The actual fabrication process would require careful consideration to ensure that the formation and rotation of the

plastic hinges would not be affected by welds positioned at the

seams, or by changes in material properties due to the fabrication technique adopted.

Both the lateral ly-loaded tube and the W-frame would produce force-limiting devices which exhibited only one full operational cycle. However other devices described in Chapter 3 are capable of providing an acceptable load-displacement response over many cycles

331

of operation. The "rolling truss load-limiter" initially exhibits a fluctuating load-displacement response (Figure 3.17) but settles down to produce a slightly undulating load plateau after two or three cycles of operation (U. S. Army, 1971). Although it would be

possible to use a slightly modified form of the torus as a force-limiting device in space trusses, the h. lgh initial cost of the

device and the difficulties experienced during fabrication (Johnson,

et a], 1975) severely limit the possibility.

The extrusion damper designed by Robinson (1977) exhibits

cyclic load-displacement characteristics ideally suited to the

requirements of a force-limiting device. The damper, which operates by pushing or pul I ing a bulged shaft through the centre of a

constricted tube lined internally with lead, could, with little

modification, be incorporated into a space truss to provide a load-limiting potential protecting selected compression members. Unfortunately the damper was only tested under a continuous cycle of tension and compression, and no information was presented on the

dampers ability to support a continuously applied constant axial load smaller than the operational load plateau value. Most alloys

of lead suffer extensively from creep, (Polakowski, et al, 1966), so it is most likely that the bulged shaft would move slowly through

the lead surround when the damper was supporting a load well below

the limit load. Both the creep behaviour of the damper and its load-displacement response at elevated temperatures would require further investigation if these devices were to be be used as load-limiters in space truss structures.

Novel Force-Limiting Device

A large part of the present investigation has been devoted to

the design, testing and improvement of a novel force-limiting

device. The gradual development of both the full size and model soft members has been outlined in Chapter 4. It can be seen from

the compressive load-strain relationship obtained from a full size

soft member (Figure 4.2) that the overall response of the device is

acceptable. It is possible to significantly alter the load-displacement behaviour of the soft member by changing the length and cross-sectional areas of the component parts. The initial reduced axial stiffness exhibited by the device is directly

332

related to the lengths of the individual tubes used in the soft member, and the magnitude of the load plateau is controlled by the

' yield stress and cross-sectional area of the middle tube in the the model soft members or the four middle strips in the full size soft members.

The tensile testing of the individual tubes and strips also reported in Chapter 4 shows that although there is good agreement between the values of yield stress obtained from nominally identical

samples, the mean value of yield stress are significantly greater than the min1 mu.,, ii values given in each of the relevant specifications. This is to be expected as it is necessary for the

steel manufacturers to ensure that there is only a very small probability of the yield stress falling below the minimum specified value. However, any increase or variations occurring in the yield stress of the middle tube or strips will have a commanding effect on the operational load of the soft member. and as a result change the force distribution and possibly the collapse behaviour of the space truss incorporating the device. To reduce this possibility it would be necessary to calculate the yield load of the soft member based on the 'target strength' of the steel material. The 'target strength' is usually calculated by the manufacturers by adding to the minimum specified yield value three times the standard deviation of test

results obtained from tensile coupon tests undertaken to ensure adequate quality control of the material.

The load-displacement characteristics of the soft member may also be adversely affected by strain aging of the middle tube or middle strips. If the soft member is subjected to cyles of loading

and unloading with the compression cycle of sufficient magnitude to

cause the middle member to yield in tension, ' then Increases in the

yield stress of the middle member will occur if there is sufficient time in between the load cycles for the middle member to strain age. This will in turn increase the limit-load of the soft member and may alter the force distribution within the encompassing structure. If the soft members are designed so that the load

plateau is not reached when the structure is supporting working loads then normal fluctuations in the imposed load acting on the

structure will not have any detrimental effect on the load-displacement behaviour of the soft member.

333

An estimate of the ultimate load of both the ful I size and model soft members has been -obtained from several carefully controlled compression tests reported in Chapter 4. A theoretical

step-by-step analysis of the model triple tube soft members indicates that these members should buckle flexurally when the

compression load acting on the 'closed up' member is sufficient to

cause instability in the middle tube. However, the experimental results Indicate that the assumptions made in the theoretical

buckl Ing analysis may not be entirely correct. - The ulti. mate loads

obtained from the compression tests indicate that the soft members buckled when the outside tube, and not the middle tube, becomes

unstable. It was assumed in the theoretical analysis that the

outside tube would be continuously restrained by the middle tube but

the end details of the soft member and the small gap existing between the two tubes may have caused this assumption to be invalid.

The theoretical buckling loads of the soft members have been

calculated using mean values of the yield stress and the measured Initial bow of the members. However, no allowance has been made for

variations occurring in tube wall thickness or residual stresses resulting from the fabrication procedure. The tube wall thickness

proved difficult to measure, and any variations occurring along the length of the tube could not be quantified. Variations In the tube

wall thickness do have a significant influence on the buckling load

of the soft members. A theoretical decrease of 5% in the wall thickness of the outside tube of the model soft member, equivalent to a reduction in thickness of 0.035 mm, would result in a 4.3%

decrease in the flexural buckling load of the soft member.

The fabrication of both the full size and model soft members involved connecting the tubes together using plug welds. This would induce residual stresses in the members which are likely to be

greater in the model soft members due to the larger ratio of weld area to tube size. The model soft members are sufficiently small to have been annealed individually and it would prove useful to

supplement the present investigation by studying the values of flexural buckling loads obtained from annealed model soft members.

To fully understand the failure mechanism of the novel soft member further experimental investigation is necessary. It would be

334

most beneficial to accurately measure the strain changes in all three tubes so that the validity of the assumptions outlined in Chapter 4 could be investigated. It is relatively simple to measure the strain in the outside tube but recording the strain changes in the middle and inner tubes may prove more arduous. It would be

possible to measure the strain in the inner tube of the full size soft member by attaching strain gauges to the inside of the tube but it would be difficult to measure strain changes in the middle tube

without influencing the load-displacement response of the member.

Space Truss Structures With Tensile Chord Yield,

Relatively little published work exists reporting the behaviour

of double-layer space truss in which tensile yielding in the lower

chord members is permitted. Schmidt et a], (1978; 1980a) have tested to collapse three space trusses with weak tensile chords, two

of which exhibited a ductile load-displacement response before

collapse. In the present investigation two model square-on-square space trusses have been carefully designed to allow extensive tensile yielding of certain bottom chord members. The load tests

undertaken on both structures were completely successful with each structure exhibiting extensive ductility before collapse. In

addition the experimental load-displacement response of each structure was accurately modelled by the non-Hnear collapse analysis program.

It is evident from the present experimental investigation that it is possible to design ductility into the load-displacement

response of double-layer space trusses, and consequently avoid the 'brittle' collapse behaviour exhibited by some space trusses. if double-layer space trusses are to be designed to allow tensile yield in lower chord members., certain points require further

consideration. In order to ascertain correctly the magnitude of the imposed load acting on the structure, which is just sufficient to

cause yield to occur in some of the tension members, it Is most important that the yield stress of the material Is accurately known. Experimental values of yield stress obtained from forty-two test coupons cut from both square and round steel tubes of grades 43C and 50C were significantly higher than the minimu. n values of yield stress given In BS4360 (1986). Consequently, if it is not

335

possible or economic to obtain the material yield stress from tensile coupon tests , the t heoretical collapse analysis and subsequent design of the structure must be based on a value of yield stress substantially greater than the minimum value given in the

relevant specification.

The type of member may also influence the amount of tensile

yield capacity available in a space truss structure. To determine the ductility of different member types, tensile tests were undertaken on twenty-one full size square and circular steel hollow

sections of grade 43C and 50C material (Parke et a], 1985). The load-displacement relationships obtained from the tests indicated that although the rectangular hollow sections of both grades of material exhibited greater plastic deformation and ductility than the circular members, sufficient ductility was available in all of the tubes tested to enable them to be used in a space truss

exhibiting tensile yield prior to collapse.

If the collapse behaviour of a double-layer space truss can be

restricted to tensile yield only then the analysis procedure adopted to determine the ultimate capacity of* the space truss can be

simplified. The non-Hnear elastic-plastic behaviour of the tension

members can be modelled using only two or three different linear

ranges depending on the accuracy required. If strain hardening of the tension members is ignored then only two separate linear ranges are necessary to cover the elastic and perfectly plastic response of the member. The analysis

' of the structure may proceed by first

undertaking a linear analysis to identify the most heavily stressed tension members. The imposed load is then increased by an amount sufficient to cause the first group of tension members to yield. The behaviour of these members may then be effectively modellea by both reducing their axial stiffness to zero and in addition by

applying, at both of the end nodes of the yielded members, equal and

opposite forces equivalent in magnitude to the yield load of the

members. All of the members in the structure must now be checked to

ensure that they are not overstressed and in particular that the

yielded members are conforming with their prescribed load-displacement response. Because the yielded members in the

structure have been approximated by only two linear phases the following procedure in the analysis program has been considerably simplified. The imposed load is incremented again until the next

336

group of tension members yielded and require a modification. The

analysis can continue with further load increments applied to the

structure until either a compression member becomes overstressed. or sufficient tension members have yielded to reduce the structure to a mechanism.

This approach has been adopted for a series of preliminary des ign invest ! gat ions undertaken to compare the I imit state des ign

of double-layer square-on-square space trusses with the design of these structures in which tensile yield of the lower chord members has been permitted (Parke et a], 1984). A specimen structure was first designed in the NODUS system (NODUS, 1981) to comply with the design rules given in BS 5950 (1985). This structure was then

modifled by reducing the size of the bottom chord members and joints

sb that the smallest node size available in the Nodus system could be used throughout the bottom chord of the space truss. The

structure was analysed as described previously but the analysis was

not terminated when the first compression members b ec &me overstressed. Instead, the strength of these compression members was enhanced by increasing the wall thickness but not the diameter

of the individual tubes. For several compression members the tube

wall thickness was increased to the maximum value that could be

accommodated without increasing the strength of the node. This

design constraint was imposed because the cost of the NODUS nodes in a double-layer space structure forms a substantial portion of the total cost of the structural framework. The analysis was then terminated when either an enhanced compression member became

overstressed or singularity of the stiffness matrix of the structure had occurred. The structure was then re-analysed to assess both the deflections of the structure and to determine if any of the bottom

chord members had yielded when the structure was supporting serviceability loading.

A comparison of the two design approaches indicated that a

weight reduction in the order of 10% could be achieved by allowing tensile yield to occur in the lower chord members of the structure

under ultimate limit state loading conditions. The design of the

structure in which tensile yield was permitted was normally governed by the serviceability limit state requirements and not by the

ultimate capacity of the structure. However, even greater reductions in weight could be achieved by introducing a slight camber into the structure and by allowing a small percentage of

337

the tension members to yield under the serviceability loading

condition. At present, the plastic deformation of any member in a structure supporting serviceability loading is not permitted in the

relevant design codes BS5950, (1985) and Draft Eurocode, EC3, (1984), but detailed collapse analysis of square-on-square double-layer space trusses in which tensile yield is permitted shows this restriction to be unduly conservative. Members which are allowed to yield and deform plastically when

' the structure is

supporting serviceability loading will permit a moderate amount of force re-distribution to occur within the structure before the

members eventually strain harden. Because these members have

yielded at the serviceability limit state they will be prone to the

effects of strain aging when the structure supports increases In the

serviceability loading. This will increase both the yield stress

and ultimate tensile strength of the member but should not decrease

the ultimate load capacity of the complete structure.

Space Truss Structures Incorporating Soft Members

The post-elastic collapse behaviour of both the model

square-on-square double-layer space trusses which incorporated soft

members described in Chapter 5 was, in general, not as predicted by

the theoretical collapse analysis program. Both structures failed

with the premature buckling of one of the compression members. In

test model 3 it was a soft member which failed while supporting an

axial force approximately 56% of the average failure load of these

members, determined from compression tests undertaken in ,a displacement controlled test machine. In model 4, the central web

compression member also buckled prematurely while supporting a

compression load approximately 29% less than the mean value obtained from testing twenty nominally identical specimens. The behaviour of both models 3 and 4 shows that in general the ultimate load capacity

of space trusses which are "compression chord critical" is sensitive to both imperfections in the structure and any initial force

distribution arising from the fabrication process.

The load-displacement response of the soft members can be

modified, by adjusting both the individual tube lengths and areas, to suit the requirements of the encompassing space truss. However, a considerable amount of additional work is required to ascertain

338

optimum values for the initial stiffness of the soft members, the magnitude and length of the load plateau, and the ratio of the yield to ultimate load of the member. For the soft member to act as a load-limiting device the magnitude of the load plateau should be less than the flexural buckling load of an adjacent equally stressed compression member in the space truss. To ensure a reasonable probability of the soft member yielding before an equally stressed compression member buckles it would be necessary to set the yield load of the soft member significantly below the buckling load of the

remaining members. From a consideration of the individual buckling loads of the type T5 compression members given in Table 5.10 a yield load for the soft member of approximately 75% of the mean value of the buckling load of the T5 members would be sufficient to ensure that only one member in one thousand would fall before the soft member yielded.

The soft members used in both models 3 and 4 had an average yield value of 3690 N which is only 47% of the mean buck] ing load of the other compression members used in the structure., If the value of yield load of the model soft members could have been increased to 5890 N it is likely that a more favourable load-displacement

response would have been obtained from both test models.

For the soft member to be able to fulfil their primary function

of creating duct iI ity in the I oad-disp I acement response of a space truss, a sufficient length of load plateau in the load-displacement

behaviour of the soft member must be available. Schmidt et a], (1980b) have investigated how the theoretical collapse behaviour of

a squ are-on- square space truss, supported at all boundary nodes,

changes with variations in the plateau length of a compression

member which exhibits an elastic-plastic load-displacement

response. Schmidt et al, have reported that even a small plateau length, equal in magnitude to half the total displacement of the

member at yield, would be sufficient to introduce both a limited

amount of ductility into the post-elastic response of the structure

and increase the theoretical collapse load of the truss. An earlier investigation, limited to the study of a plane truss, indicated that the load-displacement behaviour of a compression member In the

structure would require a plastic load plateau length approximately equal to five times the displacement at yield of the member if the full capacity of the truss was to be developed (Stevens, 1968). It

339

Is not unrealistic to assume that the length of the load plateau required from the soft member to create ductility in the

post-elastic behaviour of a double-layer space truss structure will depend extensively on the configuration of the structure, the

support positions and the principal loading condition. Consequently, to be able to quantify the minimum length of load

plateau required from a soft member positioned in a particular structure, a detailed investigation of the collapse behaviour of the

space truss would be required.

The soft members used in this investigation, described in Chapter 4, are capable of exhibiting a load plateau in their load-displacement response which has a total length of eight times the displacement of the soft member at yield. However, the low initial axial stiffness of the soft member also contributes to the total ductility exhibited by the soft member, resulting in an equivalent load plateau length of appproximately eighteen times the total displacement exhibited by the T5 compression members at failure.

The low initial elastic stiffness exhibited by the soft members influences the load-displacement response of the structure throughout the elastic range of behaviour. If the soft members are positioned in the stýucture to replace the most heavily stressed compression members, the relatively low axial stiffness of the soft members theoretically allows a redistribution of axial forces to occur away from the soft member towards the understressed regions In the space truss. The analytical investigations undertaken on both

model structures 3 and 4 indicate that this redistribution of forces

away from the soft member does occur, but that the adjacent parallel compression members rapidly become overstressed as they carry the load transferred from the soft members. To transfer axial forces from the highly stressed regions to the understressed areas in the

structure, several soft members will be required In the space truss to 'guide' the redistribution of forces throughout the structure.

To allow the soft member to have an adequate margin of safety against premature buckling, the flexural buckling load of the 'closed up' soft member should be substantially greater than the yield load of the soft member. If the soft members are designed to yield when the structure is supporting an approximate increase of 25% in the serviceability loading, then a ratio of yield load to buckling load of 1.7 should be sufficient to ensure that premature buckling of the soft members does not occur.

340

If a limited number of soft members are incorporated into a space truss, then to assess the load-displacement behaviour of the

structure up to collapse, a full non-linear analysis of the

structure is required. However, if soft members are positioned right across one bay of the top chord of a structure in which tensile yield of the bottom chord members is also permitted, then

provided the yield load of the soft members Is approximately equal to the yield load of the lower chord tension members, an ultimate

moment of resistance for the structure can readily be calculated. The soft members could be positioned to form 'yield lines' in the

structure. An assessment of the load carrying capacity of the truss

could be obtained by equating the applied moment acting along the

yield line to the ultimate moment of resistance of the structure, calculated by multiplying the total yield load of all of the soft

members positioned along the yield line by the effective depth of the structure.

General Comments

In the preceding discussion it has been implied that the dominant imposed loading controlling the design of a space truss was

either a vertical uniformly-distributed load or a syr. imetrically positioned point load. Under these load conditions it is possible to obtain both an efficient design and a ductile load-displacement

response from the structure by allowing tensile yield in the lower

chord members and by incorporating soft meTbers in the top chord of the space truss. However, if a non-symmetric load case governs the design of the structure then a weak tensile chord, permitting tensile yielding, may not prove to be so efficient. During the last decade, space truss structures have been used extensively to roof major buildings in Hong Kong (Bell et al, 1984). Roof structures constructed in this region are exposed to extensive wind suction, creating average uplift forces of 2.5 kN/M2. This value of imposed load is significantly greater than the dead weight of the structure and consequently bottorn chord members in the structure will be

subject to frequent stress reversals. This member loading will limit the possible advantages to be gained from tensile yield of the lower chord members, but does not preclude the possibility of obtaining a ductile load-displacement response from the structure by incorporating soft members strategically placed in the lower chord of the space truss. .

341

Both symmetric and non-symmetric imposed loading acting on a double-layer truss are likely to cause a non-symmetric collapse of the structure, in case of over-loading. This is especially true for

compression chord critical space trusses where imperfections in the

compression members will always result in one member, in a group of nominally identical members, being weaker than the others. Consequently, a collapse analysis of these structures should include

a series of separate analyses undertaken to determine how sensitive both the ultimate load and collapse behaviour of-the structure is to the premature buckling of one of a group of nominally identical

compression members.

The effect of compression member imperfections is not so important in the collapse analysis of space trusses in which extensive yielding of the lower chord tension members is allowed before the first compression member becomes overstressed. Both of the test model structures 1 and 2 which allowed tensile yield,

exhibited symmetric deflections over a substantial proportion of their post-elastic load-displacement response. This indicates that

possibly only one non-linear analysis of the structure, corresponding to, each dominant load case, would be sufficient to determine the ultimate load capacity of the space truss provided the load-displacement response of the structure was traced throughout the region of tensile yield up to the point where the first

c ompression member buckled.

One of the advantages of the soft member described in this

investigation is the relatively low, initial elastic axial stiffness

of the device. If several of the soft members were Incorporated

into the bottom chord of a continuous multi-bay space truss

structure adjacent to the internal columns, then high bending

moments occurring in the vicinity of the internal supports would be

decreased by the soft members, permitting a redistribution of bending moments and consequently axial forces towards the centre of the span. At present, this is achieved by reducing the height of the internal columns so that the continuous structure is allowed to

deflect before the support becomes effective. A similar result could also be obtained by using soft members for the internal column

supports. Not only would this result in the required redistribution of forces but would also limit the maximum force occurring within the internal soft member supports.

342

The present investigation has involved both the development of the soft member and a study of the behaviour of square-on-square space trusses incorporating soft members. This work has now been

extended to investigate the behaviour of square-on-diagonal space trusses incorporating soft members. Two model structures have

already been successfully tested and the construction and testing of a large portion of a full size space truss incorporating soft members is at present in progress.

343

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