Bracing in Lattice Structures

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MAIN REPORT:

ANGLE BAR BRACINGS IN LATTICE

STRUCTURES

Martin Jespersen s071919

24th January 2011

Report no. 10-052

TECHNICAL U NIVERSITY OF DENMARK


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UNDERGRADUATE S TUDENT:

Martin Jespersen

Student ID: S071919

Technical University of Denmark (DTU)

PROJECT S UPERVISORS:

Peter Noe Poulsen Mogens G. Nielsen

Associate Professor, Senior Cheif Consultant - M.Sc,

Department of Civil Engineering Department of Masts and Towers

Technical University of Denmark (DTU) Ramboll Telecom - Northern Europe


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Preface

This report was written as a bachelor project by which the author is to acquire the title:

Bachelor in Engineering (Civil and Structural Engineering)

The report is the result of a project work spanning from 30th August 2010 to 24th

January 2011 and is rated to 20 ECTS.

The total project consists of three pieces of material: A Main report (this docu-

ment), a DVD with softcopies of all FEM-models and other material (attached to this

report as Appendix F) and a Appendix report(separate document) containing documen-

tation, which is not crucial for understanding the concepts of this report, but serves as

further documentation of the project work. References to the Appendix report are given

as AR.X, X being the actual section in the Appendix report which is referred to.

The project was made in a cooperation between The Technical University of

Denmark(DTU) and Ramboll Telecom - Northern Europe.

The author would like to use this opportunity to thank supervisors and employees

at The Technical University of Denmark as well as Ramboll Telecom, whom have

contributed to the project work.

A special gratitude goes to Mr. Sankara Ganesh and the design team of Ramboll-

IMIsoft Pvt. Ltd. India, whom have provided material for the project.

Lyngby, 24th January

Martin Jespersen

s071919


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Summary

This bachelor project considers the buckling of angle bar bracings in lattice towers.

The ANSI/TIA-222-G:2005 tower design standard (in the following referred to as

TIA-G) specifies various effective slenderness ratio expressions for angle bar bracing

members dependent on the slenderness, eccentricity and end-restraints of the member.

Especially provisions related to angle bar end-restraints are of a very general and

superficial nature, even though the stiffness of a joint is totally dependent on its

design. The main scope of this project was to make a comparison between the

effective slenderness ratios acquired by above mentioned design code expressions and

results obtained by adding rotational stiffness results from detailed FEM-analysis of a

type joint to a overall non-linear FEM-analysis of angle bar members. As a secondary

objective a comparison between the commercial tower analysis program

RAMTOWER and alternative methods such as hand calculations and the FEM was to

be conducted. Both comparisons were based on a sample telecommunications tower.

By comparing the effective slenderness ratios obtained from the FEM-analysis and

TIA-G expressions, it has been observed that the non-linear FEM-analysis tends to ar-

rive at a effective slenderness which is somewhat lower that what is obtained by the

TIA-G standard in the case of weak-axis buckling. However the very limited amount

of experimental data available on joint stiffness, would tend to suggest that the joint

stiffness FEM-models applied in the current study over-predict the stiffness of joints,

hence a effective slenderness ratio which is larger than what has been found from the

current studies may be expected, yielding ratios which are closer to the expressions

given in TIA-G when considering weak-axis buckling. The need of more specific ex-

perimental data on joint rotational stiffness behavior is pointed out and areas in need of

further research are identified. The FEM-models indicate that there is a dependency in

rotational stiffness of angle bar joints by the axis of rotation considered, a phenomena

which is not currently taken into account in the TIA-G effective slenderness ratio ex-

pressions, as it is the case for other tower design standards such as EN1993-3-1. The

effective slenderness ratios obtained by FEM-analysis confirms that there is a differ-

ence between the ratio, which should be applied for parallel and weak axis buckling,

due to the difference in rotational stiffness about each axis considered (the two parallel

axis of the profile). Hence for parallel buckling the FEM-analysis arrives at effective

slenderness ratios which exceeds the expressions given in TIA-G hence indicating the

standard be on the unsafe side in relation to parallel buckling of angle bar members.

Through extensive discussion it has been found that if FEM-models can be cali-

brated (through more extensive experimental data) to fully capture the rotational stiff-

ness behavior of angle bar joints, the application of rotational stiffness models to inves-

tigate buckling failure of tower bracing members can be utilized commercially. Largescale infrastructure projects with great numbers of identical towers or marginally over

utilized towers, where prospects of savings are considerable, has been identified as the

main areas of application.

On the overall scale the comparison between RAMTOWER and other methods,

showed that RAMTOWER performed as per previous experience, yielding no more

than 10% deviation in force distribution compared to equivalent FEM-models. By com-

paring overall tower reactions found from each method, the incorporated wind profile

in RAMTOWER has been found accurate and in accordance with the ANSI/TIA-222-

G:2005 standard.


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Based on these findings RAMTOWER is considered to produce an acceptable dis-

tribution of forces, when comparing to the ease at which a tower model can be definedand analyzed in the program.

Through the sample tower models, which was required in order to perform the

above mentioned comparisons, the consequences of providing towers with non-triangulated

bracings was also experienced. From a detailed study with tower hip-bracings it was

found that the application non-triangulated bracing should not occur in any tower de-

sign, as it is also specified by the TIA-G standard.

Keywords: Buckling, Telecommunication towers, Joint slip, Lattice triangulation,

Non-linear analysis, FEM


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Resumé

Dette diplomafgangsprojekt omhandler udknækning af vinkeljern i gittertårne. Tårn-

design standarden ANSI/TIA-222-G:2005 (i det følgende benævnt TIA-G) specifi-

cerer flere udtryk til bestemmelse af den effektive slankhed for gitterkonstruktion-

selementer afhængigt af deres slankhed, ekscentricitet og rand-betingelser. Specielt

bestemmelserne der vedrører randbetingelserne for vinkeljern er meget generelle og

overfladiske, til trods for at stivheden af samlingerne afhænger af deres udformning.

Det overordnet formål med dette projekt var at lave en sammenligning mellem de

førnævnte udtryk givet i standarden og resultater opnået under anvendelse af rotations

stivheder fundet ved en detaljeret FEM-analyse og siden hen påsat vinkeljern i en mere

overordnet ikke-lineær FEM-analyse. Et sekundært formål var at lave en sammen-

ligning mellem det kommercielle tårndesign program RAMTOWER og andre metoder

der indbefattede håndberegninger og FEM-analyse. Førnævnte sammenligninger blev

begge udført under anvendelse af et telekommunikationstårn. Ved at sammenligne deneffektive slankhed opnået under anvendelse af FEM-analyse og TIA-G standarden, er

det observeret at den ikke-lineære FEM-analyse har en tendens til at komme frem til

effektive slankheder der ligger lidt under det der er specificeret i TIA-G standarden

i tilfælde med svag-akse udknækning. Dog viser det meget begrænsede omfang af

eksperimentelt data der er tilgængeligt for stivhed af samlinger at FEM-modellerne,

der er anvendt i dette projekt, overestimerer samlingens stivhed, og derfor kan en ef-

fektiv slankhed der er større end hvad der er bestemt i dette projekt forventes, og som

dermed også ligger tættere på de værdier der er givet i TIA-G standarden for svag-akse

udknækning. Behovet for mere eksperimentelt data påpeges og områder der kræver

forsat forskning er udpeget. FEM-modellerne indikerer at samlingsstivheden ved ro-

tation afhænger af den betragtede rotationsakse, et fænomen der ikke er inkluderet

ved bestemmelsen af effektive slankheder i den nuværende TIA-G standard, som deter tilfældet i andre standarder såsom EN1993-3-1. FEM-analysen bekræfter at der er

en forskel i de effektive slankheder, som bør anvendes for svag- og parallel-akse ud-

knækning, grundet forskelle i rotationsstivheden omkring de to akser der betragtes for

udknækning af vinkeljern (de to parallelle akser af profilet). FEM-analysen opnår ef-

fektive slankheder der er højere end hvad der er foreskrevet i TIA-G standarden, og

indikerer dermed at udtrykkene givet i standarden er på den usikre side i forbindelse

med parallel-akse udknækning af vinkeljern. Gennem grundig diskussion er det fun-

det at hvis FEM-modellerne kan kalibreres (gennem mere dybdegående forsøg med

stivhed af samlinger) til at kunne skildre rotationsstivheden af vinkeljernssamlinger,

kan rotationsstivhedsmodeller anvendes til at undersøge udknækning af gitterkonstruk-

tionselementer på et kommercielt niveau. Større infrastruktursprojekter med et stort

antal identiske tårne eller marginalt overudnyttede tårne, hvor udsigterne til en større

finansiel besparelse er til stede, er identificeret som det primære anvendelsesområdefor metoden.

Sammenligningen mellem RAMTOWER og andre metoder viste de forventede re-

sultater, hvorved afvigelsen i fordelingen af kræfter i gitteret mellem RAMTOWER og

FEM-analyse ikke var mere end 10 %. Ved at sammenligne de overordnet reaktioner fra

tårnet blev det fundet at det indarbejdede vind profil i RAMTOWER er tilstrækkeligt og

iht. ANSI/TIA-222-G:2005. Baseret på sammenligningens resultater betragtes RAM-

TOWER som et program der giver acceptable resultater, når simpliciteten hvormed at

tårne kan defineres og analyseres tages i betragtning.

Gennem det telekommunikationstårn der blev anvendt til overnævnte sammen-

ligninger, blev konsekvenserne af tårne med ikke-trianguleret gitter tydeliggjort. Fra et


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detaljeret studie af anvendelsen af ikke-trianguleret “hofte-gitter” er det fundet at ikke-

trianguleret gitter ikke bør forekomme i tårnkonstruktioner, som det også er specificereti TIA-G standarden.

Emner: Søjle udknækning, Telekommunikations tårne, Glidning i samlinger, Tri-

angulering af gitter, Ikke-lineære analyser, FEM


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Contents

Preface 3

Summary 5

Resumé 7

Terms and definition 11

Introduction 13

1 Column flexural buckling theory 15

1.1 Effect of boundary conditions on flexural buckling . . . . . . . . . . 16

1.2 Effect of load application on flexural buckling . . . . . . . . . . . . . 16

2 Buckling resistance according to ANSI/TIA-222-G:2005 19

2.1 Effective Yield stress [Section 4.5.4.1] . . . . . . . . . . . . . . . . . 19

2.2 Design axial compression strength [Section 4.5.4.2] . . . . . . . . . . 19

2.3 Effective slenderness ratio [Table 4-3 to 4-7] . . . . . . . . . . . . . . 20

2.4 Lattice web triangulation [figure 4-2] . . . . . . . . . . . . . . . . . . 23

3 Sample tower:

40m Medium duty Tower Design 25

3.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Design loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Hand calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 RAMTOWER Analysis 29

5 Abaqus Joint FEM-analysis 31

5.1 Type joint description . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.4 Steps, incrementation and output requests . . . . . . . . . . . . . . . 33

5.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5.1 Boundary conditions at step: “Initial” . . . . . . . . . . . . . 35

5.5.2 Boundary conditions at step: “Establish bolt tension” . . . . . 35

5.5.3 Boundary conditions at steps: “Load - region 1”,“Load - region

2” and “Load - region 3” . . . . . . . . . . . . . . . . . . . . 36

5.6 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6.1 Bolt load for tensioning of bolt . . . . . . . . . . . . . . . . . 39

5.6.2 Loading from test setup . . . . . . . . . . . . . . . . . . . . 40

5.7 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.8 Joint axial stiffness results . . . . . . . . . . . . . . . . . . . . . . . 42

5.9 Result testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.9.1 Mesh convergence . . . . . . . . . . . . . . . . . . . . . . . 44

5.9.2 Stress discontinuities . . . . . . . . . . . . . . . . . . . . . . 45

5.9.3 Bolt tensioning . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.10 Joint rotational stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.10.1 Modified material parameters . . . . . . . . . . . . . . . . . 47


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5.10.2 Modified boundary conditions . . . . . . . . . . . . . . . . . 48

5.10.3 Modified loads . . . . . . . . . . . . . . . . . . . . . . . . . 485.10.4 Modified steps and incrementation . . . . . . . . . . . . . . . 49

5.10.5 Joint rotational stiffness results . . . . . . . . . . . . . . . . . 49

6 FEM-Analysis 55

6.1 Initial testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.1.1 Simple linear-buckling of angle bar members . . . . . . . . . 55

6.1.2 Linear-buckling load when considering lateral support provided

by incoming members . . . . . . . . . . . . . . . . . . . . . 56

6.1.3 Buckling load for members with eccentric load application . . 59

6.1.4 Non-linear analysis . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3 Test runs of FEM-Models . . . . . . . . . . . . . . . . . . . . . . . 65

6.3.1 Effects of secondary bracings . . . . . . . . . . . . . . . . . 666.3.2 Effects of non-fully triangulated hip bracing . . . . . . . . . . 66

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Comparison 73

7.1 RAMTOWER, hand calculation and FEM-results . . . . . . . . . . . 73

7.2 Buckling of members with joint stiffness results from FEM-analysis. . 76

8 Perspectives 83

9 Conclusion 85

A Literature 89

B Layout drawing: 40m Medium duty sample tower design 91

C Sample tower force distribution 95

D Examples on calculation of effective slenderness ratios based on ANSI/TIA-

222-G:2005 standard and non-linear FEM results 99

E Abaqus type joint. 105

E.1 Layout drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

E.2 Material hardening curves . . . . . . . . . . . . . . . . . . . . . . . 109

E.3 Stress discontinuities in convergence model . . . . . . . . . . . . . . 111

F Digital Documentation 113F.1 Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

F.2 Abaqus FEM-models . . . . . . . . . . . . . . . . . . . . . . . . . . 113

F.3 ROBOT FEM-models . . . . . . . . . . . . . . . . . . . . . . . . . . 113


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Terms and definition

Hip-bracing Secondary bracing fitted inside the tower section (connected between two

perpendicular diagonal members) to reduce the effective buckling length of

diagonal members.

Plan bracing Internal horizontal bracing located at e.g. main member cross-over point,

platforms or tower portions with large horizontal loading

Redundant member Refer to ’Secondary bracing’

Secondary bracing Bracing member in the latticed structure which is not considered

to carry any load, but only meant to reduce the effective buckling length of

primary members(load carrying members)

Square cross section A tower with a square cross section refers to the tower havinga square shape in a section in the tower horizontal plane, e.i. tower has four

legmembers

Staggered bracing Perpendicular bracings are connected to legmember at different

levels as apose to non-staggered where perpendicular bracings are connected

at same level

TIA-G Refers to the structural design standard for antenna supporting structures and

antennas: ANSI/TIA-222-G:2005

Web pattern Pattern formed by the bracing members of a tower


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Introduction

With the rapid increase in the global population and constant development within

telecommunications, the need of electrical transmission and telecommunication towers

is greater than ever before. Especially in 3rd world countries these areas of infrastruc-

ture are in growth. The most common and applicable tower design in these countries is

the angle bar tower, square based self supporting lattice towers with legmembers and

bracings made from hot-rolled angle bar members.

Among the many advantages of the angle bar is its availability at suppliers, and the

ease at which it can be applied to form several types of lattice designs.

Due to the quantity of identical towers required to provide a infrastructure of e.g.

power or telecommunication even small optimizations on the tower design can be jus-

tified as economically sound.

One area of optimization is the effective slenderness ratio considered for buckling

investigation on tower angle bar bracings. The structural standard ANSI/TIA-222-

G:2005 for telecommunication structures, provide designers with effective slenderness

ratio expressions which depend on the slenderness, eccentricity and end-restraints of

the member under investigation. Especially provisions related to the angle bar end-

restraints are of a very general and superficial nature, even though the stiffness of the

joints is totally dependent on their design.

The main objective of this project is to capture the rotational stiffness of a angle

bar joint by application of a detailed FEM model. The joint rotational stiffness model

obtained from this analysis is then to be applied to a more overall non-linear FEM-

analysis of various angle bar members, and the effective slenderness ratio based on the

buckling load of these members may then be compared with the TIA-G standard.It should be stressed that it is not the scope of this project to develop new effective

slenderness ratio expressions for the TIA-G standard. As it will be illustrated in the

report the current expressions on effective slenderness are very general and easy to

apply for design calculations providing a fast and reliable result. The objective is rather

to investigate the gains by determining the effective slenderness of members, applied in

generic designs to be produced in large numbers such as transmission tower designs or

backbone telecommunication infrastructure, by application of this alternative method.

A secondary application is for design checks in relation to code revisions or increases in

tower design load. Rather than being forced to strengthen tower members, this method

could provide a alternative which might declare a design safe if only a marginal extra

capacity of the member is required.

As a secondary objective a comparison of the force distribution obtained by the

commercial toweranalysis program RAMTOWER and alternative methods such as

hand calculations and the FEM is also to be conducted.

The project deals with a sample telecommunication tower, but results may also be

applicable for transmission tower designs.

The project starts off by recapping some of the basic principles related to flexural

buckling of columns.

Next the overall provisions of the TIA-G standard is shortly presented and their

limitations highlighted. From the TIA-G standard RAMTOWER and hand calculations

are performed on the sample telecommunications tower.


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Following is then the detailed analysis of a type joint by use of the FEM-program

Abaqus, from which a joint rotational stiffness model is acquired.Finally a overall non-linear FEM-analysis of the sample tower is performed. On the

basis of buckling loads obtained from this analysis, effective slenderness ratios may be

calculated and compared with equivalent TIA-G provisions.


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1 Column flexural buckling theory

Axially compressed angle bar members are mainly subjected to 3 varieties of buckling

failure:

• Flexural Buckling failure: Member fails by transverse deflection in a direction

normal to itself.

• Local Buckling failure: Member fails by local buckling of angle “leg” (refer to

figure 1).

• Flexural-Torsional Buckling failure: Member fails by simultaneous transverse

deflection normal to itself and twisting around its own axis (shear center of the

section).

Later it will be shown why local buckling failure and flexural-torsional buckling is notrelevant in relation to this project, and only flexural buckling of the bracing members

is to be considered. It should be mentioned that because of this emphasis on flexural

buckling, this type of failure may in the following just be referred to as buckling.

The development of the basic column buckling stability theory applied in today’s

standards, can to great extents be credited L.Euler (1707-1783). He originally solved

the case of the axially loaded the build-in column and published his findings in a book

he titled “Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes”

in 1744. The critical Euler load is determined by solving a differential equation of the

deflection curve for an axially compressed column. The differential equation leads to a

general solution, which contains some integration constants. These constants are then

determined based on the boundary conditions of the column. The general expression

for determining the critical load (Euler’s formula) for an ideal column is given by:

F cr = F E = π 2 · E · I

l2e(1)

In this expression le refers to the effective buckling length of the ideal column,

which is governed by the boundary conditions. Effective column lengths are in general

determined by use of Engineering references, but as it will be shown later this is not

always sufficiently accurate, since the boundary conditions of a column are not ideal in

the real world.

Some also prefer a alternative expression of the Euler’s formula

F cr = F E = (kl)2 · E · I

l2 (2)

where the value of kl is governed by the boundary conditions of the column.


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Figure 1: Principal axis definitions for buckling for angle bar members

1.1 Effect of boundary conditions on flexural buckling

One area of special interest when considering buckling of bracing members is the end

restraints which are provided. From the traditional buckling stability theory the buck-

ling capacity of columns is dependent on the effective column length, as it is incorpo-

rated in the expression for the critical load as shown in expression (1). The effective

column length is as mentioned dependent on the type of restraint, which is provided at

the column ends. For a lattice structure such as a angle bar tower, designers are often

forced to deviate from the classical ideal restraint conditions for which the effectivethe column length is well defined and resort to effective lengths which are for the most

part developed on the basis of experimental data. Lorin and Cuille (1970) were some

of the first to deal with these issues, proving that the stiffness of end gusset plates has

a enormous effect on the buckling capacity of the member, whereas the strength of the

gussets is to some extent irrelevant.

Evaluation of end-restraint stiffness is very difficult to include in structural standards,

since design possibilities are unlimited, thus today’s standards only deal with simple

criteria when including effects from end-restraints. These are described in section 2 of

this project.

1.2 Effect of load application on flexural buckling

Due to the nature and application of the angle bar member in a lattice structure, concen-

tric loading of the member is often not possible, especially not for single angle bracing

members. Connecting the bracing members to other structural components is typically

achieved by bolting or welding the angle bar member by one leg. This type of connec-

tion naturally generates some eccentricity in the load transfer from one member to the

other. When considering slender axially loaded members, the effect of this eccentric-

ity on the critical buckling load varies with slenderness. The effects of eccentric load

application on beam-columns1 has been treated by e.g. Timoshenko in [17]. Results

1It is a necessity to consider the member as a beam-column since it is loaded by moment


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will briefly be presented below, since they are strongly tied to the provisions of today’s

structural standards.Determining the critical buckling stress of an eccentrically loaded beam-column

is based on the Secant formula. Basically we are seeking a critical stress σ c.Y P, forwhich the extreme fibers in the beam-column reaches the yield point stress σ Y P, by theexpression:

σ Y P = σ c.Y P ·

1 +

e

s· sec

l

2r

σ c.Y P

E

(3)

In the Secant formula given by expression (3), e is the eccentricity of the applied

axial compression force, s is the core radius2, l is the geometric length, r is the radius

of gyration and E is the modulus of elasticity. By utilizing the Secant formulation,

curves for the critical stress dependent on the slenderness of the beam-column can be

developed for various eccentricities(quantified as a ratio to s) as it is done in figure 2a.It should be noted that expression (3) only applies for members with same eccentricity

in load application at both ends. Timoshenko also deals with the case of beam-columns

subjected to load application with different eccentricities at the ends, expressing them

by the ratio β = eaeb

, where ea and eb are the eccentricities at the ends. In the case of

varying eccentricities the critical stress σ c.Y P is given by:

σ c.Y P = σ Y P

1 + eas ψ cosec(2u)

(4)

where

2u = kl = lr

σ YP E

and ψ = β 2−2β cos(2u) + 1

For tower bracings this expression is mostly relevant in the case where β = 0 cor-responding to a load application which is concentric at one end and eccentric at the

other. This would be the case for buckling of a member which is continuous at one

end and connected to other structural members by the methods previously described

at the other end. Buckling curves for member with β = 0 is given in figure 2b. Bothfigures are based on and elastic modulus of 210.000MPa and a yield point stress of

σ Y P = 250 MPa. For reference the buckling curve for the corresponding TIA-G caseis included in both figures, refer to section 2 here on. It should be mentioned that the

curves in TIA-G also includes imperfections and thus a complete comparison can not

be made. Also the expression 4 is not defined for β = 0, thus only values very close toβ = 0 can be applied.

2Core radius s = Z A , where Z is the section modulus and A is the cross-sectional area.


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100

150

200

250

300

F c r

[ M p a ]

Buckling curves for eccentrically loaded column, β=1

lr e/s=1

lr e/s=0,5

lr e/s=0,2

lr e/s=0,1

Euler

TIA‐G curve 3

0

50

100

150

200

250

300

0 20 40 60 80 100 120 140 160 180 200 220

F c r

[ M p a ]

Slenderness L/r [-]


lr e/s=1

lr e/s=0,5

lr e/s=0,2

lr e/s=0,1

Euler

TIA‐G curve 3

(a) Buckling curve for β = 1

100

150

200

250

300

F c r

[ M p a ]


lr e/s=1

lr e/s=0,5

lr e/s=0,2

lr e/s=0,1

Euler

TIA‐G curve 2

0

50

100

150

200

250

300

0 20 40 60 80 100 120 140 160 180 200 220

F c r

[ M p a ]

Slenderness L/r [-]


lr e/s=1

lr e/s=0,5

lr e/s=0,2

lr e/s=0,1

Euler

TIA‐G curve 2

(b) Buckling curve for β = 0

Figure 2: Critical load curves for beam-column with various ratios of es compared to relevant

TIA-G buckling curve. Material parameters: f y = 250 MPa and E = 210.000 MPa


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2 Buckling resistance according to ANSI/TIA-222-G:2005

In this section the current practice for determining the design compression strength of

angle bar members in accordance with to the ANSI/TIA-222-G:2005 structural stan-

dard is reviewed (In the following referred to as TIA-G).

The initial part of this section introduces some of the key provisions given in the

TIA-G standard, which may be considered to be specifically directed towards design

of lattice towers and thus outside traditional structural engineering.

References to the TIA-G standard is enclosed by [], throughout this section.

2.1 Effective Yield stress [Section 4.5.4.1]

In order to avoid local buckling of the angle bar leg, TIA-G considers an effective com-

pression yield stress F y , dependent on the width to thickness ratio wt of the member.The characteristic yield stress F y is reduced in order to obtain F y by the following

principle:

w

t ≤ 0.47

E

F yF y = F y

0.47

E

F y<

w

t ≤ 0.85

E

F yF y =

1.677−0.677

wt

0.47

E F y

·F y

0.85 E

F y<

w

t ≤ 25 F y = 0.0332 ·π

2·

E

wt 2

According to the standard the width to thickness ratio should not exceed 25.

2.2 Design axial compression strength [Section 4.5.4.2]

The design axial strength of a member in compression is given by:

P = Pn ·φ c

where

Pn = Ag ·F cr

φ c = 0.9

and for λ c ≤ 1.5

F cr =

0.658λ 2c

·F y

and for λ c > 1.5

F cr =

0.877

λ 2c

·F y


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where

λ c = K · L

r ·π ·

F y

E

Ag = gross area of member [mm2]

K = effective length factor L = laterally unbraced length of member [mm]r = governing radius of gyration about the axis of buckling [mm]

It should be noted that KL is equivalent to the effective buckling length le. The

standard furthermore stipulates that flexural-torsional buckling need not be considered

for single or double angle bar members.

2.3 Effective slenderness ratio [Table 4-3 to 4-7]

TIA-G considers various effective slenderness ratio

KLr

expressions for tower com-

pression members. Expressions for angle bar members are given in table 4-3 and 4-4

of the standard. They are divided into 2 groups: One considering legmembers and

one considering bracings. For legmembers two separate expressions are given for each

type of profile (angle bar or round), dependent on whether or not the bracing pattern is

staggered or symmetrical (non-staggered) . Buckling of legmembers will not be treated

further in this project.

For bracing members the effective slenderness ratio is governed by either the end-

restraint or eccentricity by which the member is loaded. If the bracing is not slender

Lr


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Figure 3: Is the buckling resistance of angle bar members with these end-restraints (connections)

the same? Yes according to the TIA-G standard. 2 bolts (left), 3 bolts (center) and

welding (right)

Curve Slenderness Parameter Effective slenderness

expression

1

Lr


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0

20

40

60

80

100

120

140

160

180

200

220

0 20 40 60 80 100 120 140 160 180 200 220

E f f e c t i v e

s l e n d e r n e s s K L / r [ - ]

Slenderness L/r [-]

Effective slenderness ratio for angle bar bracingsaccording to table 4-4 TIA-222-G:2005

Curve 1/Curve 4

Curve 2/Curve 5

Curve 3/ Curve 6

End‐restraint governs

Eccentricity governs

(a) Effective slenderness ratio to be considered for flexural buckling of bracings

as per TIA-222-G:2005. Curves 1 to 6 refers to the expressions in table 1.Dashed red line indicates the transition from eccentricity to end-restraints be-

ing governing.

100

150

200

250

300

F c r

[ M p a ]

Critical buckling stress Fcr:

TIA-G buckling curve vs. Euler load

TIA‐222‐G

Euler

EN 1993‐1‐1

0

50

100

150

200

250

300

10 30 50 70 90 110 130 150 170 190

F c r

[ M p a ]

Effective slenderness KL/r [-]

Critical buckling stress Fcr:

TIA-G buckling curve vs. Euler load

TIA‐222‐G

Euler

EN 1993‐1‐1

(b) TIA-G buckling curve compared to Euler and EN1993-1-1. Material parame-ters: f y = 250 MPa and E = 210.000 MPa

Figure 4: Graphic representation of provisions in TIA-G in relation to flexural buckling


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2.4 Lattice web triangulation [figure 4-2]

Several tower design standards such as TIA-G (but also EN 1993-3-1) states that thelattice web patterns should be fully triangulated in order to avoid bending considera-

tions. If e.g. secondary bracings in hip or plane web patterns are not fully triangulated

they can not be considered to prevent buckling in their own plane (without bending

considerations). Non-triangulated web patterns are in general not recommended for

lattice tower design, however they do occur either due to negligence or for practical

reasons. Examples of triangulated and non-triangulated patterns are given in figure 5

for hip bracings, and are basic examples from TIA-G.

(a) Typical locations of lattice hip bracing (Sec-

tion A-A)

(b) Triangulated hip bracing (c) Non-triangulated hip bracing

Figure 5: Examples of triangulated and non-triangulated bracings as per TIA-G


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3 Sample tower:

40m Medium duty Tower Design

In order for the project to be as specific as possible a Medium duty Tower Design

was considered. This would not only give an impression of the possible gains by the

methods developed through this project, but also keep the project at a level at which the

methods developed are practically realistic to implement for future design calculations.

Finally the sample tower design could contribute with a realistically proportioned tower

in regards to member sizes, joint details and outer geometry.

In the following the sample tower is shortly described and in the last part of the section

a traditional hand calculation of the sample tower is presented. This will not only

illustrate the application of the TIA-G standard described in section 2, but also the

traditional methods which has been applied before more computational methods were

introduced to the design of lattice towers. Finally the hand calculations were also toserve the comparison of force distribution with results given by RAMTOWER.

3.1 Description

The sample tower is a 40m so-called “Medium duty” tower, medium referring to its

equipment bearing capacity. It consists of 13 sections, with non-staggered X-bracing

patterns. The 4 top sections are parallel in order to accommodate fixture of telecommu-

nication equipment. The 3 bottom sections are fitted with several secondary bracings,

including internal hip-bracing.

If the hip-bracing is studied more closely it is seen to conflict with the provisions

in TIA-G in regards to complete triangulation of the lattice web pattern. Consequences

of this will be illustrated and discussed at a later stage of the project.

A overall layout drawing of the tower is included as Appendix B

3.2 Design loading

The design load on a telecommunication tower is typically dominated by loads related

to wind. Other than wind load from the tower body itself, loads from appurtenances

is also considered. Since the wind load on tower body is usually considered to be

mandatory, the appurtenance loads are often referred to as the design load of the tower.

The effective projected windarea of the appurtenances originally considered for the

design of the sample tower is given in table 2.

Effective projected wind areas are found by rough estimates described in Ramboll

internal note by Mr. Ulrik Støttrup-Andersen. Exact wind load from appurtenances is

dependent on the type and supplier, and should in any case be determined consideringthe actual load-configuration of the tower.

Furthermore it is brought to the attention of the reader that the sample tower is origi-

nally designed according to Indian Standards (IS), and therefore a full utilization of the

design should not be expected, since the considered wind speed in this project is lower

than what was originally considered. The project at hand only deals with the effects of

different approaches to design of towers, hence a full utilization of the tower is not a

requirement, only realistic distribution of loads and tower proportions.


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Load description Level Shielding Effective projected

wind area (EPA)

1 No. 2.4m Dia. MW Dish

Antenna

(Standard Antenna w. Radome)

38.75m 0% 4m2

1 No. 1.8m Dia. MW Dish

Antenna


38.75m 30% 1.6m2

5 Nos. 1.2m Dia. MW Dish

Antenna


31.25m 50% 4m2

3 Nos. CDMA Panel Antenna

(2.62mx0.37m) 33.75m 0% 3m2

9 Nos. GSM Panel Antenna

(1.917mx0.262m) 33.75m 30% 3.78m2

Cable & Access Ladder

(Along tower center line) 0−35m Complete

shielding

from

35-40m

0.3 m2

m

Table 2: Sample tower design load

3.3 Hand calculation

In relation to this project a complete design calculation of the sample tower in ac-

cordance with TIA-G was made “by hand” in the computer software “MathCad”. The

calculation was performed under the assumption that the tower is statically determinate

3D truss. The calculation served two purposes:

• Approximate reference values for check of force distribution in the FEM-Model

and RAMTOWER

• Illustrate the differences in assuming a static determinate 3D structure and a

static indeterminate 3D structure (comparing traditional methods with more ad-

vanced computational models).

The calculation only considers windload from a 0 degree direction (refer to figure 6),

sometimes also referred to as the normal direction. It should however be noted that tow-

ers should be designed for several different wind load directions (and combinations).

In the case of towers with square cross sections a 45 degree wind direction should alsobe considered. Usually the 0 degree wind load case will govern the design of bracings,

whereas the 45 degree case will govern the design of legmembers (and foundations),

however all members should be checked for both cases.

A more thorough study of these calculations is left to the reader, but the results of

the calculation will be applied for comparison with RAMTOWER at a later stage.

The complete calculation is attached this project as Appendix AR.D


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Figure 6: Relevant wind load directions for design of towers with square cross sections.


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4 RAMTOWER® Analysis

RAMTOWER® is a commercial software developed by Ramboll Telecom for the de-

sign and analysis of self-supporting lattice towers. The program features analysis of

towers with triangular or square cross-sections, composed of a wide variety of lattice

and member types.

Other than the force distribution performed by the RAMTOWER analysis, which

was going to be compared with other methods, the analysis was also used to establish

wind areas of the tower body, to be applied in the hand calculation of the sample tower

previously described. Large deviations between the RAMTOWER analysis and hand

calculation is not expected, since both methods assume that the tower is a statically

determinate structure.

The basic assumptions and analysis concept of RAMTOWER is shortly described

in the following:

RAMTOWER is a Visual Basic Application (VBA) based tower analysis and de-

sign software. The program considers the tower as a cantilever beam(free at one end

and fixed at the other) with relevant loads(it be horizontal or vertical from tower body,

appurtenances, ice etc.) applied at relevant levels. For this beam model is then cal-

culated moment, shear and normal force at the top and bottom of each tower section,

upon which axial forces in section members (by equilibrium equations at the center of

each section) is determined. RAMTOWER can consider sections containing multiple

diagonal members (of same profile type), determining member forces only for the bot-

tom member of the section. All this is done while assuming that the tower lattice is

statically determinant, a assumption which is not always correct since a tower some-

times contain horizontal or other members yielding it statically indeterminate. During

the development of RAMTOWER thorough comparisons with FEM-models were per-

formed and these yielded no more than 10% deviation in distribution of section forces.RAMTOWER is programmed with common structural standards within the telecom-

munication tower industry incorporated, defining wind-profiles, buckling curves, ice-

loads, default safety factors and material parameters. On several occasions throughout

its more than 12 years of existence3, RAMTOWER has proved itself as a simple and

fast tool, obtaining results with good accuracy.

The analysis of the sample tower was performed according the TIA-G standard,

when considering buckling curves, safety factors etc. Two different RAMTOWER

analysis were performed: One with a model loaded by the windprofile which is defined

within the program for the TIA-G standard and another model considering point loads,

related to wind on the tower body and appurtenances found in the hand calculation,

defined at the relevant levels in the RAMTOWER model. The differences between

the results obtained from these two models are treated in section 7. For the model

which applied the incorporated wind profile, wind load from secondary bracings had to

be calculated by hand and then included as additional section wind areas, since RAM-

TOWER can not consider bracing patterns containing secondary members. Calculation

of the additional wind load from secondary bracings is given in Appendix AR.C. For

both models the restraint against buckling provided by the secondary bracings had to

be taken into account by effective column length reduction factors in the analysis. A

automatically generated design report from RAMTOWER is given in Appendix AR.A

and AR.B for each of the two models considered.

3RAMTOWER was initially introduced with the name XLMAST


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Figure 7: Illustration of RAMTOWER program concept


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5 Abaqus Joint FEM-analysis

In order to obtain end-restraint stiffness values to be applied in buckling analysis of

angle bar bracing members, a more detailed FEM-analysis of a type joint was per-

formed. The analysis was executed in the FEM-program Abaqus/CAE version 6.10-1.

A soft-copy of each Abaqus FEM-model is given in Appendix F.

5.1 Type joint description

When selecting the layout of the joint, which was to be applied in order to capture

the stiffness behavior of typical angle bar tower bracing connections, there was one

deciding factor. During the literature study a article by N. Ungkurapinan et. al. [12] in

a very thorough manner described the experimental study of joint slip 4 in bolted angle

bar connections under axial load. In relation to this study a idealized stiffness curve

for joints with very specifically described parameters had been developed based on theexperimental results. Using this idealized curve for the axial stiffness behavior of the

joint, the FEM-model could be calibrated to confirm this data, thus increasing overall

reliability of the model. This would also indicate any limitations of a simple FEM-

model w.r.t. the actual psychical behavior of a angle bar connection. When the axial

stiffness of the type joint corresponded to the experimental data, the FEM-model could

be modified to consider the rotational stiffness, which would be of greater interest for

angle bar buckling considerations.

The layout of the Abaqus model which reflects the test setup applied in [12] is

illustrated in figure10. A drawing of the setup with measurements is given in Appendix

E. Note that Abaqus visualizations applies the coordinate system X-Y-Z (axes colored

red, green and blue respectively), however for in- and output in Abaqus this is referred

to as direction 1-2-3. This number coordinate system is applied in the following.The joint consists of two angle bar members overlapping leg to leg, with 2 bolts

transferring angle bar axial loads through shear. Parameters given in table 3, all effect-

ing the joint stiffness according to [12], was considered. All these parameters reflected

the assumptions of the experiments performed in [12]. Further parameters are given in

the subsequent sections.

Parameter Value

Bolt size M16

Hole clearance 1.6mm

Bolt torque 114.27kNmm

Angle bar type L100x100x6

Table 3: Joint parameters effecting stiffness applied in FEM-model

5.2 Material properties

For defining material properties, two literature resources were used. In [12] basic ma-

terial property data from material testing is provided for both angle bars and bolts. It

was considered to be necessary to use this data in order to obtain results which may be

compared with [12]. Several different material models were considered:

4Joint slip is defined as the sudden motion, due to a loss in friction provided by bolt tensioning, made

possible due to bolt in holes with clearance


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• Linear-elastic (In the following referred to as “Elastic”)

• Linear-elastic - perfect plastic (In the following referred to as “Perfect plastic”)

• Linear-elastic - plastic w. hardening (In the following referred to as “Plastic w.

hardening”)

Hardening and other plastic behavior of the material was not described in [12] and was

therefore based on experimental data by Dick-Nielsen and Døssing [7]. Dick-Nielsen

et. al considered several steel material types with certificates retrieving material models

from them by application of reverse engineering:

The test specimens (in [7]) were applied in normal tension testing, and the results

from this consisted of displacements at different force levels exerted on the specimens -

A test specimen work curve. By use of a FEM-model of the test setup material models

were continuously modified until displacements for different force levels matched the

work curve retrieved from the material testing. The results of the material testing byDick-Nielsen et. al. is referenced in Appendix E.2. For the angle bar members material

data on hardening of S355 was applied, which was in good agreement with the overall

material properties of the angle bars described in [12]. For the bolt material experimen-

tal data on hardening of grade 10.9 bolts was used. It should be noted that this grade

has a tensile strength which is somewhat higher than the bolts used in [12], however

this is considered to be of minor importance, since most deformation (from yielding)

is expected from local yielding in angle bar holes (Refer to later discussion in sub-

section 5.8). For the linear-elastic properties of the material a E-modulus of 215GPa

(corresponding to test results in [12]) and a Poisons ratio of 0.3 was considered.

The material model, from the data collected by Dick-Nielsen et. al, was omitted

in tabular data, from which Abaqus can interpolate (linearly) for any given yield stress

state. If plastic strains exceed the tabulated data, Abaqus assumes the yield stress to be

of same magnitude as the last tabulated yield stress for any plastic strain (larger than

the last specified). This last property was used for defining the perfect plastic model,

were reaching yield stress of the material results in “unlimited” plastic strains.

Residual stresses (from rolling of angle bar member, punching of holes etc.) was

not included in the model.

A frictional coefficient of 0.4 was considered for the angle bar and bolt surfaces.

According to [4] frictional coefficients smaller than 0.2 should not be considered in

Abaqus, since serious convergence problems may occur. The friction coefficient of 0.4

corresponds to the provisions of EN1090-2 for metalized surfaces (Class B surface).

5.3 Contact

Modeling the contact between the different model parts is one of the most criticalprocesses. If contact is improperly modeled, results of the analysis will most definitely

not reflect the real life behavior of the joint. The model consist of various surfaces in

contact . These can be categorized as:

• Contact between bolt head, nut and shank to the surface of the two angle bar

members and their holes.

• Contact between the angle bars

The contact surfaces may be viewed in figure 8. A contact pair in Abaqus consist of

2 surfaces, one referred to as a slave and the other a master. The major difference be-

tween these two is that the slave surface may not penetrate the master, but the master


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(a) Bolt head, nut and shank contact surface

(b) Angle bar contact surface for bolt head, nutand shank

(c) Angle bar to angle bar contact surface

Figure 8: Model contact surfaces (colored red)

can penetrate the slave surface (between the nodes of the slave surface), thus it is rec-

ommended5 that the slave surface is the more finely meshed of the two surfaces. In

the case of contact between the bolt and angle bar surfaces, the bolt was defined as the

master surface and the angle bar made slave. In the case of the contact between the twoangle bars, one of the angle bars was of course to be of master type and the other of

slave type.

The master and slave surface is gathered in a interaction6, to which is assigned a

interaction property. In this case two relevant properties were considered: Tangential

and Normal behavior of the contact surface interaction. For tangential behavior was

defined a frictional coefficient of 0.4 and the allowable elastic slip, refer to [4], was set

to a absolute distance of 0.05mm with zero stiffness. Normal behavior was defined as

“hard”. This property assumes that constraints related to contact can only occur, when

the surfaces are touching (no sticking between the contact surfaces).

5.4 Steps, incrementation and output requests

Due to the nature of the joint FEM-model, serious care had to be taken when organizingsteps and increments in order for the model solution to converge. Especially during the

joint slip serious convergence problems may occur. Due to the hole clearance and bolt

tensioning, the joint will experience a slip as it goes from a friction to a bearing type

joint. At this critical stage the analysis tends to abort with errors, since it does not

recognize that the slip has a definite motion governed by the clearance of the joint

holes, but labels it as a infinite motion with zero stiffness to achieve equilibrium (rigid

5In [4].6In this case a total of 5 interactions were defined in the model: 4 containing the bolt contact between the

area in and around each angle bar hole and 1 containing the contact between the angle bars.


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Figure 9: Springs between bolt and hole for convergence during slip. Angle bar material is

shaded and bolt material crossed. Cut through bolt shank(left) and cut through the

entire length of the bolt (right).

body motion). In order for the FEM iterations to converge the following steps (other

than the mandatory “initial step”) were applied:

• “Establish bolt tension” - Bolt tension is established by applying bolt load.

• “Load - region 1” - Load until joint is close to slipping.

• “Load - region 2” - Close to constant load during joint slip.

• “Load - region 3” - Continue loading with bolts in bearing.

This stepwise analysis of the joint ensured that for the critical part of the analysis (at

joint slip), step incrementation was very detailed and for remaining parts of the analy-

sis, were iterations easily converges, incrementation was more coarse. However mod-

ifying the incrementation of the the analysis, was not completely adequate to meet a

converged solution. Convergence problems are almost inevitable at the joint slip, since

Abaqus in this critical phase considers a very small change in stress to cause infinite

displacements (since slope of work curve in this region is zero, refer to figure 14). If

however a small stiffness is included, the analysis does not continue to divide time

increments until they are infinitely small, but obtains a solution. To introduce some

stiffness to the joint slip region, 12 small springs with a stiffness of 30 N /mm wereprovided between each of the bolt shanks and the surface of the holes as illustrated on

figure 9. The springs provide the work curve with a negligible, slope during the joint

slip. It should however be pointed out that non-converged analysis of the model indi-

cates that the slope of the work curve goes towards zero before analysis is interrupted.The loading in each step was determined by methods described later in this section.

In order to retrieve joint slip curves to compare with the experimental data available

(idealized curve from [12]), history output requests were defined for certain nodes in

the model. These locations may be viewed on figure 10.

For the nodes was requested translations in the direction 3 during all increments of

the analysis (Axial direction of the joint - Abaqus variable: U3).


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Figure 10: Nodes for displacement history output requests (marked by red dots)

5.5 Boundary conditions

In this subsection the boundary conditions, that is the displacement degree of freedom

(dof) on the boundary of the model, is described. In the following a restrained dof

refers to the dof having a prescribed displacement of 0, corresponding to a support

in that dof direction. The boundary conditions of the model varies with each of the

previously described analysis steps, and are described for each step in the following:

5.5.1 Boundary conditions at step: “Initial”

In the initial step all parts in the model, had to be restrained in order for the analysis to

run. This meant:

• Bolt center restrained in direction 1

• Bolt head and nut restrained in direction 2 and 3

• Angle bars restrained at edges in direction 1, 2 and 3.

In figure 11 the boundary conditions for the step may be viewed.

5.5.2 Boundary conditions at step: “Establish bolt tension”

In this step the tensioning of the bolts was applied and to avoid disturbances the bound-

ary conditions were eased to:

• Bolt head and nut restrained in direction 2 and 3

• Angle bars restrained at edges in direction 1, 2 and 3.

Hence the boundary conditions for this step is the same as in figure 11, except the

restraint at bolt center is removed.


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5.5.3 Boundary conditions at steps: “Load - region 1”,“Load - region 2” and

“Load - region 3”In this step the tensioning of the bolts can be considered to restrain the bolts and there-

fore further restraints are not required. Furthermore the angle bars are connected to

each other by friction from normal stresses provided by the bolt tension. All the pre-

viously described boundary conditions may be substituted, by boundary conditions

which reflect the actual test setup given in [12].

For the test setup, both ends of the type joint may be considered to be restrained

against displacements out of the joint plane (due to the plates from the compression test

machine). In order for the model to be of type “plane stress”, restraints out of the joint

plane was only provided in the direction of the angle bar leg, as illustrated on figure

12a. In the axial direction of the joint, restraint was applied to the unloaded joint end.

Boundary conditions for the model in steps: “Load - region 1”,“Load - region 2” and

“Load - region 3”, may be viewed in figure 12b.


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(a) BC’s for bolt in step “Initial”

(b) BC’s for angle bar in step “Initial” (Only one angle bar shown)

Figure 11: Boundary conditions(marked orange) for step: “Initial”


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(a) Directional concept of out-of-plane restraint at the sup-ported ends of the type joint (unloaded end shown).

Arrows mark the supported direction.

(b) BC’s on model for steps: “Load - region 1”,“Load - region 2” and “Load - region 3”

Figure 12: Boundary conditions(Marked orange) for steps: “Load - region 1”,“Load - region 2”

and “Load - region 3”


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5.6 Loads

5.6.1 Bolt load for tensioning of bolt

The joint bolts were modeled as a solid bolt model (with head and nut) a method

recommended by Jeong Kim et. al. in [8] to give the best imitation of real bolt behavior

(although larger computational effort is required). The magnitude of the force which

is imposed by the prescribed torque (listed in table 3) was calculated on the basis of

formulas given in [15]:

F M = 2 M A

1.155µ Gd 2 +µ K Dkm + Pπ

(5)

where for a M16 bolt:

M A is bolt installation torque, M A = 114.27kNmm

µ G is the coefficient of friction of bolt thread, µ G = 0.4µ K is the coefficient of friction of bolt (head and nut) surface, µ K = 0.4d 2 is the edge diameter, d 2 = 24mmP is the bolt pitch, P = 2mm

Dkm is the mean bolt diameter which is obtained from (6):

Dkm = d k + D B

2 (6)

where

d k is the inside diameter of the contact surface (diameter of bolt hole) d k = 17.6mm D B is the outside diameter of the contact surface (bolt head outside diameter) D B =

27.7mmFrom (5) a tension force in the bolt of 11kN or 54.7 MPa (for bolt as a solid ø16 rod) isobtained.

The actual tensioning of the bolt was achieved by means of imposing a Abaqus

“bolt load” in a plane at the center of the bolt shank as illustrated on figure 13. This

bolt load will cause the bolt to obtain internal stresses due to contact pressure between

the bolt-head/nut and angle bars.

Figure 13: Abaqus “bolt load” applied on bolt shank center-plane


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Figure 14: Principal force-displacement curve for joint slip (For linear-elastic material, with noplasticity)

5.6.2 Loading from test setup

In order to simulate loading from the test machine, a uniform pressure was applied to

the axially unsupported end of the type joint.

As previously mentioned load application was accomplished in steps and most crit-

ical was the load at which the joint starts to slip. In order to determine this load, a

simple approximation was initially used and then refined once results from initial runs

of the model was completed. The critical force was determined from expression

F cr = nF M µ (7)

where

n is the number of friction planes for one of the adjoined members, n = 4F M is the tension force of the bolt obtained from expression (5), F M = 11kN µ is the coefficient of friction of the adjoined surfaces, µ = 0.4

According to expression (7) slip is initiated when the applied force exceeds F cr =17.6kN corresponding to a uniform pressure of 15.12 MPa on the angle bar cross-section.

A load interval somewhat below and above this approximate slip value was then

applied to the step “Load - region 2” in the initial test runs of the joint model. Load

intervals was however slightly modified by viewing results from some of these initial

test runs. A model which would reflect the real joint slip behavior would have a dis-

placement curve as illustrated in figure 14(when neglecting plasticity). In the initialmodel with the previously stated axial load pressure interval, the transition from the

friction region (region 1) to the slip region(region 2) was more sudden (no rounding

of curve), indicating that the prescribed load in the step “Load - region 2” was not

sufficient to cause slip and slip was therefore initiated in step “Load - region 3” where

the load increases dramatically between each increment. The axial load interval of the

FEM model was shifted in a number of trials until a smooth transition from from “Load

- region 1” to “Load - region 2” step was obtained resembling figure 14.

As a result of this the following final load steps were applied for the model:

• “Load - region 1” - Load interval:0−15.8 MPa


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• “Load - region 2” - Load interval:15.8−16.5 MPa

• “Load - region 3” - Load interval:16.5−100 MPa

5.7 Meshing

For the model was used a combination of 20-node quadratic hex and hex dominated

elements (Abaqus type: C3D20). According to [4] “reduced integration” elements

may cause convergence problems for contact analysis, and hence full integration was

considered (convergence problems was experienced for reduced integration elements

in some of the initial trials). Special attention was paid to the mesh around the bolt

hole, applying a fine symmetric mesh of hex type. The mesh of bolts and angle bars

may be viewed in figure 15

(a) Bolt mesh

(b) Angle bar mesh (Only one angle bar shown - mesh is identical for the two angle bars)

Figure 15: Angle bar joint mesh


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5.8 Joint axial stiffness results

By combining the history output, e.i. the translation and axial load stresses in direc-tion 3 w.r.t. the Abaqus analysis relative time, the solid line work curves in figure 16

were obtained, for the 3 different material models. As previously mentioned the his-

tory data consisted of measurements in 2 points of the joint (refer to figure 10). The

total difference in axial joint displacement in these points was in the order 1/10 of amillimeter, and the displacement of the joint was therefore based on a mean value of

the history displacement data. For comparison and evaluation of the FEM-results a

idealized curve developed in [12], based on experimental results of several identical

testspecimens of the type joint, is added by the dashed line on the figure. As it may be

seen from the figure there are some differences between the results obtained by FEM

analysis and the idealized curve based on test results. Region 1 (refer to figure 14)

shows good agreement, and also the value at which the joint starts to slip is within

7.5% accuracy of the experimental data, which may be considered to be pretty good,since the factors which govern the slip load of the joint are difficult to determine with

high accuracy (bolt tensioning, friction etc.). However larger discrepancy occurs as the

joint deformation approaches the elastic area. It is obvious that the total slip of the joint

(region 2) is not of same magnitude (idealized curve starts to build elastic deformation

after just 0.85mm of slip). This is justified by N. Ungkurapinan et. al., since little orno attention was paid to place the bolts completely centered in the joint holes of the

specimens, as it has been done in the FEM-model. This will also never be psychically

possible, since joint holes will be made with some tolerance. This last psychical factor

is considered to be most likely to cause the deviation. The most concerning discrepancy

is the elastic stiffness of the joint. The idealized curve indicates a relatively large de-

formation with low elastic stiffness, whereas FEM indicates small elastic deformation

with a larger stiffness quickly achieving plastic behavior (for the models containing

plasticity). Some differences between the FEM-model and the experimental test setup

should be pointed out at this stage:

• The FEM-model considers grade 10.9 bolts whereas the experiment applies bolts

with a ultimate strength of some 800 MPa. (Hence experimental bolts starts to

yield at a earlier stage than the ones applied for the FEM-model, however defor-

mation of the bolts is generally considered to be small.)

• The idealized curve is derived from several sets of experimental data and must

also obscure any “noise” on measurements.

However differences between the two methods, due to different bolt grades, should

not appear in the elastic FEM-analysis, and still this analysis indicates same elastic

stiffness behavior as the two models containing plastic properties. Analysis with boltsof perfect plastic material and a yield strength of 640 MPa (yield strength most likely to

correspond to the bolts applied in the tests) shows no changes in stiffness, and it may

therefore be concluded that in this case yielding of the angle bar holes by far gives the

largest contribution to the reduction in joint stiffness. Plots of the plastic strains in the

bolts confirms this observation, since no plastic strains are observed in the shank of the

bolts (which would lead to substantial axial deformation.), plastic strains only occurs

in bolt head and nut, due to contact pressure with the angle bar surface.

It seems reasonable (as indicated by the FEM-model) that if a perfectly circular bolt

shank, goes into bearing with a perfectly circular hole, the area which initially presses

against the hole, will be of infinite size, an thus produce yield stresses in the hole almost


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60

80

100

120

140

160

F [ M p a ]

Force-displacement curve axially loaded joint w.o. bending

FEM model ‐ Elastic

FEM model ‐ Plastic w. hardening

FEM model ‐ Perfectly plastic

Idealized curve ‐ N. Ungkurapinan

0

20

40

60

80

100

120

140

160

0 ,0 0E +0 0 5 ,0 0E‐01 1,00E+00 1,50E+00 2,00E+00 2,50E+00 3,00E+00

F [ M p a ]

Joint deflection [mm]

Force-displacement curve axially loaded joint w.o. bending

FEM model ‐ Elastic


FEM model ‐ Perfectly plastic

Idealized curve ‐ N. Ungkurapinan

Figure 16: Deformation curve for idealized experimental and FEM-model results (Parts of the“Elastic” and “Plastic w. hardening” work curves are obscured by the work curve for

the “Perfectly plastic”.)

Figure 17: Plastic strains in bolts of perfect plastic material with yield strength 640 MPa for joint

under axial load (zero plastic strain colored blue)

instantaneously. Also residual stresses from punching or drilling of bolt holes in the

testspecimens, may produce a difference (This is not captured in the current FEM-

model), since the material around the holes may start to yield earlier than anticipated

by the FEM-model.

All these factors may inflict on the experimental data, yielding a lower stiffness of the test specimen joint, than what can be obtained by a simple FEM-model as described

here.

5.9 Result testing

Since the joint FEM-model showed some discrepancies with respects to the experimen-

tal data (established in figure 16), further testing of the model was performed in order

to validate if other issues, than what has previously been addressed, were inflicting on

the results. Model and result testing was limited to contain: mesh convergence testing,

stress discontinuities and bolt tensioning.


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60

80

100

120

140

160

F [ M p a ]

Convergence: Force-displacement curve axially loaded joint

w.o. bendingFEM model ‐ Elastic

FEM model ‐ Elastic ‐ conv.


FEM model ‐ Plastic w. hardening conv.

0

20

40

60

80

100

120

140

160

0 0,5 1 1,5 2 2,5 3

F [ M p a ]

Joint deflection [mm]

Convergence: Force-displacement curve axially loaded joint

w.o. bendingFEM model ‐ Elastic

FEM model ‐ Elastic ‐ conv.


FEM model ‐ Plastic w. hardening conv.

Figure 18: Result comparison from type joint convergence testing.

5.9.1 Mesh convergence

The FEM is a mathematical approximation to a psychical problem, by application of

approximated field variables. In general the solution given by this approximation con-

verges towards the actual solution by the number of elements which are applied (several

factors such as geometric order of elements etc. governs the convergence rate). When

performing a FEM analysis it is not desirable to apply a large amount of elements in

order to obtain a completely accurate result, since this would require a long time of

computation. The usual aim is have model with a (relative) fast computation and ac-

ceptable deviations from the exact solution. The usual convergence rate in the FEM

is not linear, thus the solution quickly converges towards the exact solution with just a

reasonable amount of elements. In order to determine the state of convergence for the

type joint FEM-model, the model was re-meshed by increasing the amount of seeds

along previously seeded edges by 50%.

Since this project was mostly concerned with the deformation of the joint, compar-

ison of results, between the original and the re-meshed model, will be limited hereto.

In figure 18 the work curve of the re-meshed models is given by dots at outputted in-

crements of the analysis and may be compared with the initially accepted results (solid

line).

The figure illustrates that there is no visible difference between the results obtained

by the re-meshed model and the original.

At the same time it should be mentioned that the re-meshed model has a CPU time

of 10.5 hours and the original only 1.2 hours (for the elastic material model). This

clearly illustrates the importance of doing convergence testing, analysis run time canbe drastically reduced by mesh optimization based on result convergence. If a 3%

difference in results was obtained, the original model may still be accepted in order to

reduce computation time by 90% from the re-meshed model.

The convergence graph also illustrates the critical phases of the joint axial deforma-

tion: at transition from friction to slip and at transition from slip to bolts in bearing. At

these locations the dots from the convergence results are very closely spaced, indicating

that Abaqus is applying a large number of increments at these locations.


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5.9.2 Stress discontinuities

In order to determine the adequacy of most FEM-model meshes, it will be relevant toview the discontinuities of the model field output results. The discontinuity is the dif-

ference between the lowest and highest nodal value common to two or more elements,

and is a good indicator as to where in the model the mesh density is insufficient. In this

project only discontinuities in Von Mise stress was considered. At several locations

discontinuities was however not considered, these were:

• Corners between the two legs of the angle bar

• Locations at which corners of bolt head and nut is pressing against the angle bar

surface

• Bolt shank at locations which is pressing against corners of angle bar holes (Just

under head and nut and at the center of the shank).

These locations are ignored since high values of stress are inflicted at these areas. For a

perfectly meshed sharp corner, such as the corner between the two legs of the angle bar,

stresses would reach infinite levels. Same applies for the contact areas where corners

are pressing against surfaces. If large discontinuity was to be avoided, all corners would

have to be smoothed, which is very demanding, even for a simple detail as considered

in this case. Furthermore effects from discontinuity in these areas is considered to have

little effect on the joint deformation which is required in this project.

Contour plots of the stress discontinuities are given in figure 19 for the type joint

model with the “plastic w. hardening” material model. For the angle bar member the

largest discontinuities are observed in the area around the bolthole (hence only this area

is considered on the figure).

From the contour plots several nodes of interest were selected and their V. Misestress discontinuities were probed and compared with the actual averaged stress. Re-

sults may be viewed in table 4. From the table stress discontinuities are observed to

be high compared to the actual V. Mise stress at the probed node. The discontinuity

should be viewed with respects to the required parameter of the joint. This project

is mainly concerned with the deformation of the joint at a certain load. Further dis-

continuity tests performed in Appendix E.3, with the convergence model described in

the previous subsection, indicates that the discontinuities are reduced by increasing the

mesh of the model, however not even the convergence model provides satisfying dis-

continuities. Considering that there was no effect on joint deformation by increasing

the mesh by 50% along all edges, and this is seen to reduce the discontinuities of the

joint model, it is assumed that the discontinuities given in table 4, does not have a con-

siderable effect on the joint stiffness results. However if stress in the model is requiredthe mesh must be refined and discontinuities rechecked.

5.9.3 Bolt tensioning

Tests of the bolt tensioning was made up of two parts: Initially was modeled a solid

rod resembling the bolt shank and the bolt load was applied to the center plane by same

concept as illustrated on figure 13. Reactions in the axial direction on one side of the

bolt was then summed to confirm that they were in equilibrium with a internal force of

the prescribed bolt tensioning.

Next the entire joint model was considered. A bolt load of 0.11kN (∼ 1% of the

entire bolt load) was applied to the bolt and stresses in the angle bar at the location of


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(a) Discontinuities in angle bar with probed nodes (the neglected areas, primarily corners, are remove for remainingdiscontinuities to be clearly visible).

(b) Discontinuities in bolt with probed nodes

Figure 19: Stress discontinuities in type joint FEM-model for the material model “plastic w.hardening”.


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Node V. Mise stress

[MPa]

V. Mise stress

(Discontinuity)

[MPa]

Discontinuity

percentage of total

stress [%]

2941 397.9 512.8 129%

792 564.0 370.9 66%

2160 310.7 313.7 101%

3090 330.7 211.5 64%

291 313.8 215.6 69%

(a) V. Mise stress discontinuities in angle bar

Node V. Mise stress

[MPa]

V. Mise stress

(Discontinuity)

[MPa]

Discontinuity

percentage of total

stress [%]

1423 457.9 382.2 79.373 556.4 440.2 70.6

(b) V. Mise stress discontinuities in bolt

Table 4: Probed V. Mise stress discontinuities in nodes selected from contour plots compared

with actual stress values.

bolt head and nut was examined. The purpose of this test was to ensure that the contact

formulation between head, nut and angle bars were tight, e.i. contact between the parts

would be established almost instantly. Both test yielded satisfying results.

5.10 Joint rotational stiffness

For the purpose of buckling analysis the previously determined axial joint stiffness

was to some extent, irrelevant and merely a method of confirming that a FEM-model

could generate joint stiffness results with good correspondence to actual joint behav-

ior. From the previous tests and result comparison this is on some level considered to

be confirmed, even though some factors such as residual stresses, imperfections and

tolerances are not captured by the FEM-model leaving some deviations to the actual

stiffness behavior of the joint. On this note the more buckling relevant rotational stiff-

ness behavior of the joint was to be determined. The overall model was the same as

the previous axial stiffness model, leaving only some small modifications in order to

consider rotation of the joint about the two parallel axis of the joint denoted RX and

RY (refer to figure 21a). During the study of the rotational stiffness of the joint, it has

been assumed that there is no difference in stiffness for clock- and counterclockwiserotation about the same axis, this is however not confirmed.

5.10.1 Modified material parameters

Two major changes was implemented on the material parameters of the model:

• The E-modulus was lowered to 200GPa corresponding to the modulus consid-

ered by TIA-G

• Only the material models “Elastic” and “Plastic w. hardening” were considered.


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5.10.2 Modified boundary conditions

The only modification made to the boundary conditions, was the previously consideredout-of-plane restraint at the loaded end of the type joint. This boundary condition

was removed since restraint from the test machine was no longer considered (refer to

subsection 5.5.3) and the joint rotational stiffness was now alone to be provided by the

connection to the other angle bar.

5.10.3 Modified loads

The axial loading of the joint was moved to the other end of the angle bar member

(closer to the bolt holes) as illustrated on figure 20 and only the angle bar leg containing

bolts was loaded. This was all done to prevent unintended rotation, enabled by the

removal of the previously described BC, due to eccentricities in the load application.

The axial load of the joint was important since it would increase the rotational stiffnessof the entire joint, due to the bolts contact with the hole surfaces by bearing. Also

the axial load reflects the actual conditions before buckling of the member, since loads

close to buckling will be present in the member, before considerable rotation of the

joint due to out-of-plane defo

Bracing in Lattice Structures

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