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D.J. Ridley-Ellis, J.S. Owen & G. Davies

Torsional behaviour of Rectangular Hollow Sections

D.J. Ridley-Ellis, J.S. Owen and G. Davies

School of Civil Engineering, The University of Nottingham, Nottingham, NG7 2RD, UK

Abstract

The results of a series of full-scale torsion tests on Rectangular Hollow Sections are presented and

discussed. The observed torque-twist behaviour is compared to that predicted by an extended version of

Marshall’s simplified thick wall torsion theory and by finite element analysis, and significant differences are

highlighted and examined. The behaviour predicted by the finite element models is shown to be identical to

that predicted by Marshall’s thick wall theory, which forms the basis of the British and European design

procedures. However, even though the experimental measurements agree with the finite element and

theoretical predictions in the elastic range, the measurements of torsional capacity are significantly lower

than those calculated, and this has important implications for design that may be wider than just torsion of

RHS. A number of potential causes for this behaviour are examined, but it is yet to be fully explained.

Evidence of similar behaviour in previous large-scale testing is highlighted and discussed.

Keywords: Torsion; Rectangular Hollow Section; Finite element; Experimental; Thick wall theory

1 Introduction

Rectangular Hollow Sections (RHS) are often used to resist twisting action in situations such as

space frames, edge beams, grillages, and gantry stanchions because they are inherently more efficient at

resisting torsional loads than open sections. To provide further design guidance for the use of RHS edge

beams in slim floor construction, the authors have undertaken a project that examined the behaviour of

perforated and unperforated RHS under the actions of torsion and flexure.

The project, which compared the results of full-scale tests with the predictions of established theory

and finite element (FE) analysis, found significant anomalies for the case of torsion of unperforated sections,

despite close agreement for all other load conditions. The results of the full-scale laboratory torsion tests

1


suggest that RHS have lower torsional capacities than predicted by the theory currently used as the basis of

design.

There is little published literature on the torsion testing of large-scale tubular members, but one

important previous study is that of Marshall [1], upon whose equations the British and European design

codes are based. Marshall was concerned primarily with stiffness rather than strength, but close examination

of his data has revealed that his experimental work shows a similar deficiency in torsional resistance. These

anomalies indicate that there is an important aspect of either the structural or the material behaviour that is

not understood, which has important implications for the design of RHS in torsion. The authors are therefore

presenting this paper to stimulate debate and raise awareness of a potentially serious problem.

The paper begins with a synopsis of simplified elastic thick wall torsion theory (Section 2) and goes

on to summarise an extension of the simplified approach to cover the elasto-plastic torque-twist relationship

(Section 3). The results of a series of full-scale torsion tests are compared to the predictions of the extended

simplified thick wall theory and FE analysis (Section 4) and it is shown that while extended simplified thick

wall theory and FE analysis are in agreement, the experimentally measured torque-twist relationships differ

significantly from the theoretical predictions. A number of potential causes for this behaviour are examined

(Sections 5 and 6) and evidence of similar behaviour in previous large-scale testing is highlighted and

discussed (Section 7), but it is yet to be fully explained.

2 Theoretical approximations of the elastic torsional behaviour of RHS

Theoretical approximations of the torsional behaviour of tubular sections fall into two categories

depending on how the variation of shear stress through the thickness is treated, viz ‘thin wall’ and ‘thick

wall’. Thick wall theories (e.g. Abramyan [2], Byrne and Carré [3], and Marshall [1,4,5]) are those which

attempt to account for the variation in shear stress through the thickness and, perhaps, at the corners.

Marshall [4] observed that the variation in Saint Venant shear stress through the thickness is nearly linear for

typical RHS, and proposed simplified equations for the torsional inertia constant (It, Eqn 1) and the torsional

modulus constant (Ct, Eqn 2) that are good approximations to his rigorous analysis. Marshall’s simplified

equations are similar in form to the classic thin wall theory of Bredt [6] in being based on the mean perimeter

(hc, Eqn 3) and the area enclosed by it (Ah, Eqn 4). The dimensions of cross-section are as defined in fig 1.

2


(1)

(2)

(3)

(4)

The elastic torsional capacity of an RHS (Eqn 5), which is proportional to the shear yield stress ( y),

is the torque at which the external surface begins to yield and it marks the limit of linearity in the torque-twist

response. The magnitude of the twist when the elastic torsional capacity is reached (el) can be calculated

using Eqn 6, where L is the length of the member and G is the shear modulus of the material.

(5)

(6)

3 Theoretical approximations of the elasto-plastic torsional behaviour of RHS

Marshall [4] was concerned primarily with elastic torsional behaviour, but his expressions for the

elastic shear stress on the internal and external surfaces can be used to calculate the relationship between the

torque and twist after yielding has begun. This is achieved by considering the boundary of yielding as it

moves inwards through the thickness of the tube and makes use of the fact that the shear strain and the angle

of twist are directly proportional despite the yielding of the outer part of the thickness. The yielded

proportion () of the thickness can be calculated from Eqn 7.

where (7)

The torque (T) that corresponds to this value of can be calculated by adding together the torque

resisted by the yielded and elastic parts of the cross-section (Eqn 8).

3


(8)

where

(9)

and

(10)

for which

(11)

and

(12)

The plastic torsional capacity (Tpl), the torque at which the whole thickness of the tube has achieved

the shear yield stress ( = 1), can be estimated by considering the flow of uniform plastic shear around the

cross-section (Eqn 13). This calculation does not account for strain-hardened shear stresses, which may

permit a higher ultimate torque. The plastic torsional capacity is the same as the torsional capacity as

calculated by thin wall theory.

(13)

The magnitude of the twist (pl) when the plastic torque is reached can be calculated by considering

the twist required to cause shear yield at this location (Eqn 14).

(14)

4


Because of the assumption that the stress is uniform through the thickness, thin wall torsion theory is

not able to describe the transition from fully elastic to fully plastic behaviour, which is a significant feature of

the behaviour of RHS [7]. Thin wall theory incorrectly predicts the range of elastic behaviour and will

under-predict the twist for a torque higher than the elastic torsion capacity. Since the thick wall equations are

no more complicated to apply than the thin wall equations, it is recommended that thick wall theory be used

in design in the place of thin wall theory for all sizes of RHS. This is already the case in the UK and Europe

as the formulae for the torsional constants quoted in BSEN10210 [11] and BSEN10219 [12] contain thick

wall expressions that are the same as Marshall’s approximate formulae.

4 Laboratory torsion testing of RHS at Nottingham University

4.1 Introduction

This section summarises the results of torsion tests on two sizes of RHS conducted at the University

of Nottingham [15]. The tests constituted the control group for an experimental study of the torsional

behaviour of RHS with circular openings in the web, which formed part of a wider investigation into the

fundamental behaviour of perforated RHS under torsion, bending and shear [16]. The project, which was

funded by British Steel Tubes and Pipes (now Corus Tubes) and the University of Nottingham, combined

laboratory testing and FE modelling in order to develop the basis for a set of design rules for perforated RHS

under combined actions [17]. During the initial stages of the project, a number of torsion tests on small-scale

cold-formed square section tube were also conducted. The results of these tests are discussed in Section 7.

4.2 A description of the specimens, apparatus and testing procedure

The specimens tested are listed in Table 1. Four different hot finished bars were used in the

manufacture of the specimens (two section sizes each of two different grades). The grade S275 RHS were

manufactured by British Steel Tubes and Pipes (now Corus Tubes), and the grade S355 RHS were

manufactured by Vallourec and Mannesmann Tubes. Table 1 also includes predictions of the elastic and

plastic torques (Tel and Tpl), and the corresponding rotations (el and pl), as calculated using thick wall theory

and measured values of the dimensions and material properties.

5


The torsion testing rig comprised of a fixed reaction beam and a pivoted rotating beam, between

which a length of RHS could be caused to twist about its longitudinal axis (Figs 2 and 3). The torque was

created by a matched pair of double acting hydraulic cylinders mounted on the guide columns, and

transmitted to the two metre long specimen through bolted connections at both ends.

The torque was measured by monitoring the shear strain on the outside surface of a circular shaft

between the specimen and the rotating beam, which ensured that the measurements were unaffected by

friction and the change in the lever arm with rotation. The shaft was calibrated, prior to the torsion tests, by

comparing the output from the shear strain gauges with applied torques of known magnitude (up to 95 kNm).

The rotation was measured at each end of the specimen using gravity inclinometers mounted on the

connecting plates, and was therefore unaffected by the stiffness of the testing apparatus and bolted

connections. Three of the torsion specimens were instrumented with electrical resistance strain gauges, and

all five specimens were painted with a brittle lime wash to increase the visibility of the Lüders lines.

The loads applied by the hydraulic jacks were measured by load cells so that the vertical forces could

be balanced, and the movement of the end of the rotating beam was measured by a linear potentiometer to

help determine appropriate load increments. This also provided secondary, but less reliable, measurements

of torque and rotation.

4.3 Finite element models

The ABAQUS v5.8-1 analysis program [18] was used to obtain the FE results presented in this

paper. The models of the laboratory tests were based on measured dimensions and material properties. The

latter incorporated in a non-linear material model that allowed for the effects of strain hardening. As this

investigation formed part of a wider study of RHS with circular web openings, for which the correct

modelling of high strains around the openings was essential, four-noded shell elements (S4R) were used.

These were able to model thickness changes, while the equivalent eight-noded shell elements (S8R) could

not. However, preliminary studies indicated that FE models of unperforated RHS in torsion gave identical

results with four-noded shells as with eight-noded shells, provided a sufficient number of the former were

used. The models also included:

6


Geometric non-linearity, as this was required for the correct modelling of the torsional

displacements.

The potential for shear buckling of the tube wall.

A representation of the connecting plates and hence a degree of warping restraint comparable to that

imposed by the testing apparatus.

4.4 A comparison of experimental results, torsion theory and FE predictions

The results of the laboratory tests were compared with the predictions of FE models and thick wall

theory and the results are summarised in Table 2. The full torque rotation relationships are shown in figs 4 to

7, for which the torque and twist are normalised against the thick wall elastic torsional capacities. The

theoretical predictions are based on measured dimensions and measured values of yield stress and Young’s

modulus, but unlike the FE models they do not include an allowance for strain hardening or shear buckling of

the tube wall.

The predictions of the FE models were virtually identical to those of thick wall theory. As is typical

for hot-finished steel, the stress-strain curves for all four bars featured a long yield plateau. This meant that

the shear strains were insufficient for significantly strain-hardened stresses to develop and that the FE

predictions of torsional capacity were effectively limited to the theoretical plastic torsional capacity. None of

the specimens were susceptible to shear buckling of the tube wall prior to development of the plastic

torsional capacity, although the FE model of the grade S355J2H RHS 200x100x8 exhibited slight plastic

buckling behaviour.

In contrast to the agreement between the FE and theoretical predictions, the RHS tested in the

laboratory failed to achieve even their predicted elastic torsional capacities. The maximum torques were

some 20% lower than the predicted plastic capacities in the case of the grade S275 specimens, and some 12%

lower in the case of the S355 specimens. Despite the poor agreement with the theory, the two tests of grade

S355 RHS 200x100x8 (fig 6) gave similar results indicating good experimental repeatability (less than 1%

difference in capacity).

In the elastic range, the measured stiffnesses (Table 2) and shear strains (Table 3) were within an

acceptable tolerance (15%) of those predicted, given the difficulty of obtaining accurate measurements of

7


Young’s modulus with tensile tests. In the three tests in which strains were measured, the shear strains on the

external surface were lower with respect to both the torque and twist.

In all the tests, the visible Lüders lines were almost entirely longitudinal and ran the full length of the

specimen, although a small number of transverse lines were visible in some specimens. Examination of the

specimens after the tests revealed similar patterns on the internal surfaces. The lines, which appeared mainly

on the wall faces rather than the corners, were slightly inclined to the axis of the tube, although there was no

common pattern to the inclination or curve of the lines. These patterns, an example of which is shown in Fig

8, were similar to those observed by Marshall (Section 7).

5 Appraisal of experimental results

The experimental method, the measurements, the theoretical predictions and the FE models were

thoroughly examined in an attempt to provide an explanation for the differences between the measured

behaviour and that predicted by thick wall theory and FE. The arguments supporting the validity of the

experimental results are discussed in detail elsewhere [15,16], but the main points are summarised below.

5.1 Gross measurement error

The possibility that gross measurement error was the cause of the anomalies was thoroughly

investigated by the authors, but was ruled out for the following reasons:

1) The measurements of torque, twist and shear strain in the elastic range are reasonable, and if anything,

these measurements suggest that the measured torque is higher than the true value.

2) The measurements of torque and twist agreed with the secondary measurements obtained from the load

cells and movement of the lever arm.

3) The shear strains measured at all locations on the surface of the circular calibration shaft were equal, and

were consistent with theory for magnitudes of torque up to 95 kNm.

4) The lack of hysteresis in the measured torque-rotation relationships indicates that the measurements were

not affected by frictional losses. Furthermore, friction between the rotating beam and the guide columns

would not have affected the torque measured on the calibration shaft.

8


5) Measured torque-rotation relationships for the RHS with large web openings agreed extremely well with

the FE predictions [7,15,16].

6) The measuring and logging equipment was checked and was found to be functioning within acceptable

tolerances.

5.2 Buckling

The possibility that buckling of the tube walls caused the anomalies was ruled out for the following

reasons:

1) Shear buckling occurs when the shear stress in the wall reaches a critical value. If the torsional capacity

were being limited by buckling, the grade S355 specimens would therefore have had similar capacities to the

S275 specimens. In contrast, the S355 specimens achieved torsional capacities that were closer to the

corresponding theoretical plastic torsional resistances than the S275 specimens.

2) The shapes of torque-rotation curves for sections that buckle are markedly different from those that do not.

Buckling causes a distinct peak in the curve, which is followed by a decrease in capacity with increasing

twist. In contrast, the measured torque rotation curves are the same shape as the thick wall theory prediction.

Although they failed to achieve the theoretical plastic torsional resistance, the experimental specimens were

able to maintain the maximum torque with increasing twist in all cases.

3) The FE models are capable of modelling shear buckling, yet the predicted torque-rotation relationships

show no tendency to buckle prior to development of the theoretical plastic torsional resistance.

5.3 Boundary conditions and warping

The support conditions for an RHS in torsion can, potentially, reduce the maximum torque it can

resist. Warping restraint induces warping stresses, while a restriction of longitudinal deformation (length

change) induces axial stresses. Both sets of stresses act in addition to the Saint Venant shear stress with the

effect of increasing the maximum Von Mises stress in the section.

Warping restraint is unlikely to be the cause of the anomalous behaviour in the laboratory for the

following reasons:

9


1) According to both theory [19-22] and FE [16], square sections with constant wall thickness do not warp,

and are therefore completely unaffected by warping restraint. However, the square sections showed the same

anomalous behaviour as the rectangular sections.

2) According to both theory [19-22] and FE [16], the effect of warping restraint for RHS that do warp is

slight, and warping restraint has little effect on the torque-rotation relationship.

3) A degree of warping restraint, comparable to that imposed by the testing rig, was included in the FE

models of the laboratory tests.

The rig was designed to allow the specimens to lengthen or shorten with twist so as to prevent the

development of axial force with increasing deformation. The FE models of the tests also included this

freedom of axial movement, but FE models of RHS with axial restraint were found to predict identical

capacities to FE models without axial restraint. This is because the changes in length due to deformation are

very small. Furthermore, the strains measured on the surface of the specimens were fully consistent with

pure torsional shear, and the elastic response was linear.

5.4 Specimen dimensions and material properties

If incorrect dimensions or material properties had been used in the calculations, then both the FE and

theoretical predictions would have been in error and yet remained mutually consistent. However, this is an

unlikely explanation for the anomalies for the following reasons:

1) No single dimension reduces the theoretical elastic and plastic torsional capacities without also affecting

the elastic behaviour.

2) The specimens were re-measured after the tests and the dimensions were found to be correct.

3) The RHS with large web openings, tested during the main part of the project, had the same dimensions as

the unperforated sections, yet the measured torque-rotation relationships agreed extremely well with the FE

predictions [7,15,16].

Similarly, there is also evidence that suggests that incorrect measurements of material properties are

not the cause of the anomalous behaviour:

1) Although Young’s modulus is difficult to measure accurately with tensile tests, it does not influence

capacity, unless there is instability.

10


2) Although the value of Poisson’s ratio used in the analysis (0.3) is an assumed rather than a measured

quantity, it does not influence capacity, unless there is instability.

3) The theoretical and FE analyses were based on the assumption that there were no residual stresses.

However, hot-finished steel tubes are normalised after rolling and therefore do not contain residual stresses

of the same magnitude as for cold-formed sections. Furthermore, residual stresses do not normally prevent

the ability of a section to achieve the plastic torsional capacity.

The elastic and plastic torsional capacities of an RHS are governed by the shear yield stress of the

material. In the application of thick wall theory above, and in the FE models, the shear yield stress was

calculated from the uniaxial yield stress using the von Mises failure criterion. The uniaxial yield stress

therefore influences the theoretical torsional capacities without changing the stiffness, but errors in its

measurement are unlikely to be the cause of the anomalous results for the following reasons:

1) At least four tensile tests were conducted on material from each of the four bars, and the measurements of

yield stress did not vary sufficiently to be the sole cause of the anomalous results (7% for grade S275 and

1% for grade S355).

2) Additional tensile tests of coupons inclined to the longitudinal axis of the tube, both at Nottingham and

(independently) Corus Swinden Technology Centre, did not reveal significant anisotropy in the material

behaviour (fig 9). The results do suggest a slight trend of decreasing yield stress towards the transverse

direction, but at less than 5% this is likely to be due to random scatter rather than a genuine feature. Strain

hardening and tensile strength were unaffected by coupon orientation, but the yield plateau was seen to be

around 25% longer in the transverse direction.

3) The RHS with large web openings, tested during the main part of the project, had the same material

properties as the unperforated sections, yet the measured torque-rotation relationships agreed extremely well

with the FE predictions [15,16].

6 Discussion of alternative shear yield criteria

For metals, the main alternative to the von Mises yield criterion, mentioned above, is the Tresca, or

principal shear stress theory, which assumes that the effect of all the shear stresses within the material is

proportional to the maximum shear stress. The Tresca criterion relates the shear yield stress to the uniaxial

11


yield stress by a factor of one half, which results in a shear yield stress that is 13% lower than that calculated

by von Mises. However, experimental work in combined axial load and torsion by investigators such as

Lode [23,24], Taylor and Quinney [25], and Hohenemser and Prager [26] have shown that the von Mises

failure criterion is particularly successful at modelling the behaviour of structural grade steels (for steel, the

experimentally determined ratio of axial to shear yield stress usually lies between 0.55 and 0.60, close to the

von Mises value of 0.58).

6.1 Measurement of the shear yield stress

Since knowledge of the actual relationship between the shear yield stress and the uniaxial yield stress

of the hot-finished RHS was critical for this research, an attempt was made to determine the ratio

experimentally. When direct measurements are required, the shear yield stress of materials is usually

calculated from torsion tests, a method that is clearly inappropriate in this context. Standard alternatives,

such as the Miyauchi shear test [27] and the Marciniak in-plane sheet torsion test [28], require specialist

equipment that was not available at the time of the original research. In order to obtain measurements using

the uniaxial testing equipment available, a method was devised [16] by which the shear yield stress could be

estimated from strain measurements on a specially shaped coupon (fig 10).

The results are presented in fig 11. The shear yield stress was measured in 11 separate tests on

material from the same bar as the grade S355 RHS 150x150x6.3 specimen. In the figure, the measurements

are normalised against the shear yield stress as calculated from the measured uniaxial yield stress, and the

von Mises criterion. A line showing the shear yield stress according to the Tresca criterion is also plotted.

More than half (55%) of the shear coupons produced results within 2% of the von Mises value confirming

the suitability of the von Mises criterion for calculating the shear yield stress from the tensile yield stress. Of

the remaining coupons, two (representing 18%) indicated a shear yield stresses that was significantly higher

(17 and 21%) than the von Mises value and one coupon (representing 9%) indicated a shear yield stress some

6% lower.

12


6.2 The role of variation in the shear yield stress

This apparent variability in the shear yield stress is a possible explanation for the anomalous torsion

test results since yielding will begin where the shear yield stress is lowest. To study the effect of variation in

shear yield stress, the FE model of the grade S355 RHS 150x150x6.3 specimen was repeated and a randomly

selected proportion (approximately 10%) of the elements were given a reduced yield stress (87% of the base

value). Although the torque at onset of non-linearity was reduced by 13%, the section achieved a torque

within 3% of the plastic torsional capacity at a rotation equal to the theoretical value of twist at plastic torque

(fig 12). Despite the fact that this and other FE models indicate that small, isolated, patches of low yield

stress do not have a strong effect on the overall torsional response, the possibility that variation in the shear

yield stress is the cause of the anomalous test results cannot be firmly ruled out. This is because the shear

yield behaviour in the laboratory may be different from the simplified representation in the FE model since

shear yield in one area may promote premature shear yield in adjacent areas.

7 Previous published experimental work

Many of the published experimental studies of the torsional properties of tubes were conducted in

the years leading to and during the Second World War on the types of tubes found in aircraft construction. In

one such series of tests, Moore and Paul [29] measured the torsional behaviour of two tubes of rectangular

section, but found that some aspects of their behaviour were contrary to their expectations. The work,

coming a few years before Abramyan [2], relied on thin wall theory and with the benefit of thick wall theory

the results are easier to explain. The torque-rotation curves for the two sections are shown in fig 13, which

also shows the thick wall theory predictions. The tubes were manufactured from an aluminium alloy that had

a yield stress in compression that was around 20% lower than that in tension. The theoretical curves are

lower bound estimates based on the compressive yield stress.

In the elastic range, the stiffness of both tubes was within 3% of the theoretical value, but the onset

of non-linearity began at around 75% of the theoretical elastic torsional capacity. This was likely to be

caused by stress concentrations at the inside of the tight corner radii, which caused premature yielding but

did not prevent the section eventually reaching its plastic torsional capacity. In fact, due to strain hardening,

13


one of the tubes was able to resist a torque higher than the plastic capacity. Although it is not shown in the

figure, Moore and Paul indicate that buckling limits the resistance of the more slender tube.

The results of Moore and Paul are typical of other published tests of small-scale tubes of rectangular

section in agreeing reasonably well with thick wall theory predictions. Indeed, when the Authors of this

paper tested cold-formed steel tubes, of a size similar to the aluminium tubes above, the torque-rotation

curves were also close to those predicted by thick wall theory and FE models. The results of the tests, which

are discussed in detail elsewhere [16,30], are summarised in fig 14. The figure shows the results of six

separate tests and an FE prediction, for which the torque and twist are normalised against the thick wall

theory elastic torque and twist. The measurements were in agreement with the predictions of thick wall

theory and the FE model, both in terms of elastic stiffness and plastic capacity. As is typical for cold-formed

steel, strain-hardening occurred immediately after yield, and because of this the torque resisted by the

specimens continued to increase with increasing twist until collapse occurred by plastic buckling. The FE

model was able to correctly predict this behaviour, although the buckle formed at a lower twist than in the

laboratory.

Although previous experimental work showed a broad agreement between the measured and

theoretical behaviour of small-scale tubes of rectangular section, there is evidence in previously published

results of torsion tests of full size RHS of behaviour similar to that discussed in Section 4.4. Tests on hot

finished RHS, of sizes similar to those in Section 4, were conducted by Marshall [1], who compared the

results to the predictions made by his thick wall torsion theory. Marshall measured shear stresses on the

internal and external surfaces of the RHS in addition to torque and twist, and concluded that torsional

stiffness and shear stress were adequately predicted by his theoretical expressions. Marshall also measured

the elastic and plastic torsional capacities, but chose to use the Tresca yield criterion rather than the von

Mises criterion in his theoretical calculations. As a result, his theoretical predictions of the torsional

capacities were 13% lower than they would have been if he had used the von Mises criterion, which is widely

regarded as the more appropriate criterion for such calculations. When Marshall’s test data was re-examined,

the results showed evidence of behaviour similar to that described in Section 4.4. Marshall’s measurements

of the torsional capacities are listed in Table 4, and compared with the theoretical predictions. The

calculations indicate that, although the RHS behaved as predicted in the elastic range, the elastic and plastic

14


capacities were up to 18% lower than the theoretical predictions, and typically at least 10% lower for the

larger sections.

8 Concluding remarks

Simplified thick wall theory has been extended to describe the transition between fully elastic and

fully plastic behaviour and has been shown to be consistent with the behaviour predicted by FE models.

However, torsion tests of full size RHS indicated that, although elastic behaviour was consistent with

expectations, the measured elastic and plastic torsional capacities were significantly lower than the

theoretical and FE predictions. Furthermore, a reassessment of previously published research has uncovered

evidence of similar behaviour in the results of full-scale testing of hot-finished RHS by the author associated

with modern simplified thick wall theory (fig 15). This disparity between the theoretical and measured

torsional capacity is sufficiently large to cause concern, as it indicates that current design expressions

overestimate the true capacity by around 15%. Designers need to be aware of these limitations, but since in

practice torsion does not usually govern design the consequences of this shortfall in resistance are unlikely to

be critical.

Although there are a number of possible explanations for this anomalous behaviour, no single

explanation fits all the evidence available. One potential cause that remains plausible is that a variation in the

shear yield behaviour of the steel resulted in premature yielding. However, at present there is no evidence

available to support this conjecture and its plausibility stems only through a lack of contradictory evidence.

The possibility that the von Mises yield criterion, which is used widely in theoretical analysis and FE

modelling is not consistent with the results of the torsion tests has implications that are wider than just the

torsional behaviour of RHS.

15


Thickness, t

Width, b

Hei

ght,

h

z

z

yy

Thickness, t

Width, b

Hei

ght,

h

z

z

yy

Internal corner radius, ri

External corner radius, ro

Fig 1. A simplified cross-section and corner detail

16


Jack point (up)

Bearing house

Jack point (down)

Calibration shaft

Specimen

Connecting plates and bolts

Reaction beam

2000 mm 350500 mm 250250

mm mm mm

700

mm

700

mm

Guide columns

Fixed endTwisting end

Fig 2. The torsion testing apparatus (schematic)

17


Fig 3. The torsion testing apparatus

18


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Twist / Theoretical twist at yield

Torq

ue /

Theo

retic

al to

rque

at y

ield

Thick wall theoryFinite elementExperimental

E = 195 kN/mm2

G = 75 kN/mm2

y = 199 N/mm2

Fig 4. Torque-rotation relationship for RHS 200x100x8 grade S275J2H

19


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


Torq

ue /

Theo

retic

al to

rque

at y

ield


E = 195 kN/mm2

G = 75 kN/mm2

y = 190 N/mm2

Fig 5. Torque-rotation relationship for RHS 150x150x6.3 grade S275J2H

20


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


Torq

ue /

Theo

retic

al to

rque

at y

ield

Thick wall theoryFinite elementExperimental (2 tests)

E = 203 kN/mm2

G = 78 kN/mm2

y = 220 N/mm2

Fig 6. Torque-rotation relationship for RHS 200x100x8 grade S355J2H

21


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


Torq

ue /

Theo

retic

al to

rque

at y

ield


E = 200 kN/mm2

G = 77 kN/mm2

y = 234 N/mm2

Fig 7. Torque-rotation relationship for RHS 150x150x6.3 grade S355J2H

22


Fig 8. Typical pattern of Lüders lines

23


80%

85%

90%

95%

100%

105%

0.0 22.5 45.0 67.5 90.0

Angle (degrees from longitudinal axis)

Yiel

d st

ress

/ av

erag

e yi

eld

stre

ss in

lo

ngitu

dina

l dire

ctio

n

Nottingham University

Swinden Technology Centre

Fig 9. Lack of anisotropy of uniaxial yield stress for RHS 150x150x6.3 grade S275J2H

24


60

30

180

5

10

105

120

10 Dia

60

40

5 Dia

Dims in mm

Tension legCompression leg

Shear zone

Milled void

3 element strain gauge rosette

Fig 10. The coupon for shear yield measurement

25


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5Maximum principal stress / Yield stress

Min

imum

prin

cipa

l stre

ss /

Yiel

d st

ress

Von Mises

Tresca

-0.80

-0.75

-0.70

-0.65

-0.60

-0.55

-0.50

-0.45

-0.40

0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Maximum principal stress / Yield stress

Min

imum

prin

cipa

l stre

ss /

Yiel

d st

ress

-2%

+2%+5%

+20%

-5%

Von MisesTresca

Fig 11. Measurements of the shear yield stress

26


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0


Torq

ue /

Theo

retic

al to

rque

at y

ield

Thick wall theoryFinite element with uniform materialFinite element with random variationExperimental

Fig 12. The effect of random variation in shear yield stress (RHS 150x150x6.3 grade S355)

27


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50 60 70 80

Twist (degrees per metre)

Torq

ue (k

Nm

)

Thick wall theoryExperimental

Test 1

Test 2

Fig 13. Torque-rotation relationships obtained by Moore and Paul [29]

28


0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17


Torq

ue /

Theo

retic

al to

rque

at y

ield

Thick wall theoryFinite elementExperimental (6 tests)

h = 38.0 mmb = 38.0 mmt = 1.43 mmr i = 0.3 mmr o = 1.7 mmG = 77 kN/mm2

y = 210 N/mm2 (based on 0.2% proof stress)

Fig 14. Torque-rotation relationships for small-scale torsion tests

29


-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

0 10 20 30 40 50 60 70

Theoretical plastic torsional capacity based on von Mises (kNm)

Diff

eren

ce in

pla

stic

tors

iona

l cap

acity

mea

sure

d - t

heor

etic

al (%

of t

heor

etic

al)

This project (grade S275)This project (grade S355)This project (small scale tests)Marshall's tests [1]

Theoretical plastic torsionalcapacity based on Tresca

Fig 15. Plastic torsional capacity - Measurements vs. theory

30


Table 1. Specimen dimensions and thick wall theory predictions

Section Grade Measured dimensions Thick wall theory predictions

hbt (ro/ri) {web}

(mm)

el

(o/m)

pl

(o/m)

Tel

(kNm)

Tpl

(kNm)

RHS 200x100x8 S275J2H 197.599.57.79 (12.5/5.0) {22} 2.14 2.73 48.1 53.7

RHS 150x150x6.3 S275J2H 149.0149.06.09 (11.0/5.0)

{21}

1.83 2.17 43.6 47.2

RHS 200x100x8 S355J2H 199.0100.07.93 (12.5/5.0)

{22}

2.26 2.90 54.6 61.1

RHS 200x100x8 S355J2H 199.0100.07.93 (12.5/5.0)

{22}

2.26 2.90 54.6 61.1

RHS 150x150x6.3 S355J2H 149.5149.55.79 (11.0/5.0)

{23}

2.20 2.57 51.8 55.7

31


Table 2. A summary of the experimental results

Section Grade Stiffness Maximum torque

Measured

(kNm/o)

% of

theory

% of

FE

Measured

(kNm)

% of

theory

% of

FE

RHS 200x100x8 S275J2H 12.6 112.0% 111.8% 43.1 80.2% 79.8%

RHS 150x150x6.3 S275J2H 13.4 112.7% 112.4% 38.8 82.2% 81.8%

RHS 200x100x8 S355J2H 12.3 101.6% 101.6% 53.2 87.0% 87.0%

RHS 200x100x8 S355J2H 13.1 108.2% 108.2% 53.6 87.7% 87.7%

RHS 150x150x6.3 S355J2H 11.9 101.1% 99.7% 49.6 89.0% 88.1%

32


Table 3. Experimental measurements of shear strain in the elastic range

Section Grade Torsional modulus constant, Ct Shear strain / twist

Measured (cm3) % of theory % of theory

RHS 200x100x8 S275J2H 270.8 112.1% 99.9%

RHS 150x150x6.3 S275J2H 259.3 113.0% 99.5%

RHS 200x100x8 S355J2H 269.2 108.3% 92.4%

33


Table 4. Elastic and plastic torsional capacities for Marshall’s tests [16]

Test Measured dimensions Grade Elastic torsional capacity Plastic torsional capacity

hbt

(mm)

BS4360

[31]

Measured

(kNm)

% of

theory

Measured

(kNm)

% of

theory

A 50.850.84.9 43C 2.78 91.7% 3.67 101.7%

B 63.563.54.9 43C 4.56 88.7% 5.31 89.9%

C 76.276.24.9 50 9.36 85.4% 10.2 82.9%

D 101.6101.66.4 43C 15.9 87.6% 17.7 86.6%

E 101.6101.64.1 43C 11.6 91.8% 12.0 87.8%

F 127.050.86.4 43C 8.35 82.1% 10.4 86.7%

G 76.238.14.1 43C 2.78 92.0% 3.29 93.9%

H 76.250.83.3 43C 3.54 98.3% 3.75 94.0%

I 101.650.83.3 43C 4.56 92.8% 4.81 89.4%

34


9 References

[1] Marshall, J., “Aspects of torsion of structural rectangular hollow sections”, Ph.D. thesis, University of

Strathclyde, 1972.

[2] Abramyan, B.L., “Torsion and bending of prismatic rods of hollow rectangular section”, National

Advisory Committee for Aeronautics, Technical Memorandum 1319, November 1951.

[3] Byrne, J.G. and Carré, B.A., “Torsional stress concentrations at rounded corners of Rectangular Hollow

Sections”, Journal of Mechanical Engineering Science, Vol. 4, No. 4, pp 334-340, 1962.

[4] Marshall, J., “Derivation of Torsional Formulas for Multiply Connected Thick-Walled Rectangular

Sections”, Journal of Applied Mechanics, Vol. 37, No. 2, pp 399-402, June 1970.

[5] Marshall, J., “Torsional behaviour of structural rectangular hollow sections”, The Structural Engineer,

Vol. 49, No. 8, pp 375-379, August 1971.

[6] Bredt, R., “Kritische Bemerkungen zur Drehungselastizität,” Zeitschrift des Vereines Deutscher

Ingenieure, Vol. 40, pp 785-790 and pp 785-817, 1896.

[7] Ridley-Ellis D. J., Owen J. S. and Davies G., “Theoretical and measured torsional behaviour of

Rectangular Hollow Sections”, Proceedings of the 12th International Offshore and Polar Engineering

Conference (ISOPE), Vol 4, pp 41-47, editors JS Chung, M Sayed, M Kashiwagi, T Setoguchi, SW Hong.

International Society of Offshore and Polar Engineers, May 2002.

[8] BS 5950-1:2000, “Structural use of steelwork in building”, Part 1: “Code of practice for design. Rolled

and welded sections”, British Standards Institution, 2001.

[9] Nethercot, D.A., Salter, P.R. and Malik, A.S., “Design of members subject to combined bending and

torsion”, SCI publication 057, The Steel Construction Institute, 1989.

[10] Eurocode 3, “Design of Steel Structures”, Part 1.1: “General Rules and Rules for buildings”, ENV 1993-

1-1:1992, European Committee for Standardization, UK version, British Standards Institution, 1992.

[11] BS EN 10210-2:1997, “Hot finished structural hollow sections of non-alloy and fine grain structural

steels”, Part 2: “Tolerances, dimensions and sectional properties”, British Standards Institution, 1997.

[12] BS EN 10219-2:1997, “Cold formed welded structural hollow sections of non-alloy and fine grain

steels”, Part 2: “Tolerances, dimensions and sectional properties”, British Standards Institution, 1997.

35


[13] AISC, “AISC Specification for the design of Hollow Structural Sections”, American Institute of Steel

Construction, Chicago, 1997.

[14] AISC, “Load and resistance factor design specification”, American Institute of Steel Construction,

Chicago, 1999.

[15] Ridley-Ellis D. J., Owen J. S. and Davies G., “Torsion-testing of Rectangular Hollow Sections with web

openings”, Report SC2000-010, The University of Nottingham, March 2000.

[16] Ridley-Ellis D. J. “Rectangular Hollow Sections with circular web openings; Fundamental behaviour in

torsion, bending and shear”, Ph.D Thesis, The University of Nottingham, 2000.

[17] Ridley-Ellis D. J., Owen J. S. and Davies G., “Recommendations for the design of hot-finished steel

Rectangular Hollow Sections with circular web openings; Torsion, bending and shear”, Report SC2000-011,

School of Civil Engineering, The University of Nottingham, July 2000.

[18] ABAQUS/Standard version 5.8-1, Hibbitt, Karlsson and Sorensen Inc, Pawtucket, Rhode Island, USA,

2000.

[19] Thomas, F.M., “An analysis of the effects due to restrained warping of thick walled box beams and I-

beams subject to torsion”, Ph.D. thesis, University of Ilinois at Urbana, 1969.

[20] Smith, F.A., Thomas, F.M. and Smith, J.O., “Torsional analysis of heavy box beams in structures”,

American Society of Civil Engineers, Journal of the Structural Division, Vol. 96, No. ST3, pp 613-635, 1970.

[21] Young, W.C., “A practical approach to theoretical stress analysis – or where do you find it in Roark and

how do you use it effectively”, Society for experimental mechanics, 1982.

[22] Megson, T.H.G., “Aircraft structures for engineering students”, Edward Arnold, 2nd Edition, 1990.

[23] Lode, W., Z. Physik, Vol 36, p 913, 1926.

[24] Lode, W., “Der Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle”. Forsch. –Arb. Geb.

Ing. –Wes., VDI-Verlag G.M.b.H, Berlin, Heft 303, 1928.

[25] Taylor, G.I. and Quinney, H., “The plastic distortion of metals”, Phil. Trans. Roy. Soc., Ser. A. Vol.

230, London, pp 323-362, November 1931.

[26] Hohenmser, G.I, and Prager, W., “The mechanics of the plastic deformation of mild steel”, R.T.P.

Translation number 2468, British Ministry of Aircraft Production, 1932.

36


[27] Miyauchi, K., “Stress-strain relationships in simple shear testing on in-plane deformation for various

steel sheets”, 13th Congress of the International deep drawing research group, Melbourne, Australia, pp 360-

371, February 1984.

[28] Marciniak, Z., “Aspects of metal formability”, Metal working research group, McMaster University,

Ontario, Canada, pp 84-91, 1973.

[29] Moore, R.L. and Paul, D.A., “Torsion tests of 24S-T aluminium-alloy non-circular bar and tubing”,

NACA technical note 885, National Advisory Committee for Aeronautics, 1943.

[30] Ridley-Ellis, D.J., Davies G. & Owen J.S., “Fundamental behaviour of Rectangular Hollow Sections

with web openings”, Proc. of the 8 th International Symposium on Tubular Structures, pp 197-175, Singapore,

Balkema, 1998.

[31] BS 4360, “Specification for weldable structural steels”, British Standards Institution, 1972 (superseded).

37

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