D.J. Ridley-Ellis, J.S. Owen & G. Davies
1
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
D.J. Ridley-Ellis, J.S. Owen & G. Davies
2
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.
D.J. Ridley-Ellis, J.S. Owen & G. Davies
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It 
t 3hc 4tAh2

3
hc
(1)
Ct 
hc  I t
thc  2 Ah
(2)
hc  2b  t   h  t   4   ro  ri 
(3)
2

4   ro  ri 
Ah  b  t h  t  
(4)
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.
Tel   y Ct
 el 
L y Ct
GIt
(5)

L y
G

hc
2 Ah  thc
(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.

1 Ah  y L


2 thc 2Gt
where
0   1
(7)
D.J. Ridley-Ellis, J.S. Owen & G. Davies
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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).
T   y Ct , plastic  Ct , elastic 
(8)
where
4   2ro   ro  ri  

Ct , plastic  2 t  b  t h  t  

4


(9)
and
 4tAh2, elastic 1   2 t 3hc , elastic 

1   hc,elastic


h
3
c , elastic


Ct , elastic 
1   thc,elastic  2 Ah,elastic
(10)
for which
hc , elastic  2b  h  2t  2 t   4   ro  ri   ro  ri 
(11)
and
Ah , elastic  b  t  t h  t  t  
4   ro  ri   ro  ri 2
4
(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.
T pl  2tAh y
(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).
D.J. Ridley-Ellis, J.S. Owen & G. Davies
 pl 
L y
G

hc
2 Ah  thc
5
(14)
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
D.J. Ridley-Ellis, J.S. Owen & G. Davies
6
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.
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
D.J. Ridley-Ellis, J.S. Owen & G. Davies
7
results with four-noded shells as with eight-noded shells, provided a sufficient number of the former were
used. The models also included:

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).
D.J. Ridley-Ellis, J.S. Owen & G. Davies
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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
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.
D.J. Ridley-Ellis, J.S. Owen & G. Davies
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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.
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.
D.J. Ridley-Ellis, J.S. Owen & G. Davies
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Warping restraint is unlikely to be the cause of the anomalous behaviour in the laboratory for the
following reasons:
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:
D.J. Ridley-Ellis, J.S. Owen & G. Davies
11
1) Although Young’s modulus is difficult to measure accurately with tensile tests, it does not influence
capacity, unless there is instability.
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].
D.J. Ridley-Ellis, J.S. Owen & G. Davies
12
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
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
D.J. Ridley-Ellis, J.S. Owen & G. Davies
13
(17 and 21%) than the von Mises value and one coupon (representing 9%) indicated a shear yield stress some
6% lower.
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
D.J. Ridley-Ellis, J.S. Owen & G. Davies
14
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,
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
D.J. Ridley-Ellis, J.S. Owen & G. Davies
of the torsional capacities are listed in Table 4, and compared with the theoretical predictions.
15
The
calculations indicate that, although the RHS behaved as predicted in the elastic range, the elastic and plastic
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.
D.J. Ridley-Ellis, J.S. Owen & G. Davies
16
Width, b
z
y
y
Height, h
Internal corner radius, ri
Thickness, t
External corner radius, ro
z
Fig 1. A simplified cross-section and corner detail
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Twisting end
17
Jack point (down)
Fixed end
700 mm
Guide columns
Bearing house
700 mm
Calibration shaft
Reaction beam
Specimen
Connecting plates and bolts
Jack point (up)
250 500 mm
2000 mm
mm
Fig 2. The torsion testing apparatus (schematic)
350
250
mm
mm
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Fig 3. The torsion testing apparatus
18
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Torque / Theoretical torque at yield
1.2
1.0
19
E = 195 kN/mm2
G = 75 kN/mm2
y = 199 N/mm2
0.8
0.6
0.4
Thick wall theory
Finite element
Experimental
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Twist / Theoretical twist at yield
Fig 4. Torque-rotation relationship for RHS 200x100x8 grade S275J2H
4.0
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Torque / Theoretical torque at yield
1.2
1.0
20
E = 195 kN/mm2
G = 75 kN/mm2
y = 190 N/mm2
0.8
0.6
0.4
Thick wall theory
Finite element
Experimental
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Twist / Theoretical twist at yield
Fig 5. Torque-rotation relationship for RHS 150x150x6.3 grade S275J2H
4.0
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Torque / Theoretical torque at yield
1.2
1.0
21
E = 203 kN/mm2
G = 78 kN/mm2
y = 220 N/mm2
0.8
0.6
0.4
Thick wall theory
Finite element
Experimental (2 tests)
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Twist / Theoretical twist at yield
Fig 6. Torque-rotation relationship for RHS 200x100x8 grade S355J2H
4.0
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Torque / Theoretical torque at yield
1.2
1.0
22
E = 200 kN/mm2
G = 77 kN/mm2
y = 234 N/mm2
0.8
0.6
0.4
Thick wall theory
Finite element
Experimental
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Twist / Theoretical twist at yield
Fig 7. Torque-rotation relationship for RHS 150x150x6.3 grade S355J2H
4.0
D.J. Ridley-Ellis, J.S. Owen & G. Davies
Fig 8. Typical pattern of Lüders lines
23
D.J. Ridley-Ellis, J.S. Owen & G. Davies
24
Yield stress / average yield stress in
longitudinal direction
105%
100%
95%
90%
85%
Nottingham University
Swinden Technology Centre
80%
0.0
22.5
45.0
67.5
90.0
Angle (degrees from longitudinal axis)
Fig 9. Lack of anisotropy of uniaxial yield stress for RHS 150x150x6.3 grade S275J2H
D.J. Ridley-Ellis, J.S. Owen & G. Davies
25
180
Shear zone
3 element strain
gauge rosette
60
Milled void
5
10
40
5 Dia
10
120
30
10 Dia
Tension leg
5
Compression leg
60
Dims in mm
Fig 10. The coupon for shear yield measurement
D.J. Ridley-Ellis, J.S. Owen & G. Davies
26
-0.40
1.5
Tresca
1.0
Minimum principal stress / Yield stress
Minimum principal stress / Yield stress
Von Mises
Tresca
0.5
0.0
-0.5
-1.0
-1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
Maximum principal stress / Yield stress
1.5
Von Mises
-0.45
-5%
-2%
-0.50
+2%
+5%
-0.55
-0.60
+20%
-0.65
-0.70
-0.75
-0.80
0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80
Maximum principal stress / Yield stress
Fig 11. Measurements of the shear yield stress
D.J. Ridley-Ellis, J.S. Owen & G. Davies
27
Torque / Theoretical torque at yield
1.2
1.0
0.8
0.6
0.4
Thick wall theory
Finite element with uniform material
Finite element with random variation
Experimental
0.2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Twist / Theoretical twist at yield
Fig 12. The effect of random variation in shear yield stress (RHS 150x150x6.3 grade S355)
D.J. Ridley-Ellis, J.S. Owen & G. Davies
28
1.2
Torque (kNm)
1.0
0.8
Test 1
0.6
0.4
0.2
Test 2
Thick wall theory
Experimental
0.0
0
10
20
30
40
50
60
70
Twist (degrees per metre)
Fig 13. Torque-rotation relationships obtained by Moore and Paul [29]
80
D.J. Ridley-Ellis, J.S. Owen & G. Davies
29
Torque / Theoretical torque at yield
1.2
1.0
0.8
0.6
h = 38.0 mm
b = 38.0 mm
t = 1.43 mm
r i = 0.3 mm
r o = 1.7 mm
0.4
0.2
G = 77 kN/mm2
 y = 210 N/mm2 (based on 0.2% proof stress)
Thick wall theory
Finite element
Experimental (6 tests)
0.0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17
Twist / Theoretical twist at yield
Fig 14. Torque-rotation relationships for small-scale torsion tests
D.J. Ridley-Ellis, J.S. Owen & G. Davies
30
Theoretical plastic torsional capacity based on von Mises (kNm)
Difference in plastic torsional capacity
measured - theoretical (% of theoretical)
5%
0%
0
10
20
30
40
50
-5%
-10%
Theoretical plastic torsional
capacity based on Tresca
-15%
-20%
-25%
-30%
This project (grade S275)
This project (grade S355)
This project (small scale tests)
Marshall's tests [1]
Fig 15. Plastic torsional capacity - Measurements vs. theory
60
70
D.J. Ridley-Ellis, J.S. Owen & G. Davies
31
Table 1. Specimen dimensions and thick wall theory predictions
Section
Grade
Measured dimensions
Thick wall theory predictions
hbt (ro/ri) {web}
el
pl
Tel
Tpl
(mm)
(o/m)
(o/m)
(kNm)
(kNm)
RHS 200x100x8
S275J2H
197.599.57.79 (12.5/5.0) {22}
2.14
2.73
48.1
53.7
RHS 150x150x6.3
S275J2H
149.0149.06.09 (11.0/5.0) {21}
1.83
2.17
43.6
47.2
RHS 200x100x8
S355J2H
199.0100.07.93 (12.5/5.0) {22}
2.26
2.90
54.6
61.1
RHS 200x100x8
S355J2H
199.0100.07.93 (12.5/5.0) {22}
2.26
2.90
54.6
61.1
RHS 150x150x6.3
S355J2H
149.5149.55.79 (11.0/5.0) {23}
2.20
2.57
51.8
55.7
D.J. Ridley-Ellis, J.S. Owen & G. Davies
32
Table 2. A summary of the experimental results
Section
Grade
Stiffness
Maximum torque
Measured
% of
% of
Measured
% of
% of
(kNm/o)
theory
FE
(kNm)
theory
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%
D.J. Ridley-Ellis, J.S. Owen & G. Davies
33
Table 3. Experimental measurements of shear strain in the elastic range
Section
Grade
Torsional modulus constant, Ct
Measured (cm3)
% of theory
Shear strain / twist
% 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%
D.J. Ridley-Ellis, J.S. Owen & G. Davies
34
Table 4. Elastic and plastic torsional capacities for Marshall’s tests [16]
Test
Measured dimensions
Grade
Elastic torsional capacity
Plastic torsional capacity
hbt
BS4360
Measured
% of
Measured
% of
(mm)
[31]
(kNm)
theory
(kNm)
theory
A
50.850.84.9
43C
2.78
91.7%
3.67
101.7%
B
63.563.54.9
43C
4.56
88.7%
5.31
89.9%
C
76.276.24.9
50
9.36
85.4%
10.2
82.9%
D
101.6101.66.4
43C
15.9
87.6%
17.7
86.6%
E
101.6101.64.1
43C
11.6
91.8%
12.0
87.8%
F
127.050.86.4
43C
8.35
82.1%
10.4
86.7%
G
76.238.14.1
43C
2.78
92.0%
3.29
93.9%
H
76.250.83.3
43C
3.54
98.3%
3.75
94.0%
I
101.650.83.3
43C
4.56
92.8%
4.81
89.4%
D.J. Ridley-Ellis, J.S. Owen & G. Davies
35
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 12 th 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 19931-1:1992, 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.
D.J. Ridley-Ellis, J.S. Owen & G. Davies
36
[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 Ibeams 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.
D.J. Ridley-Ellis, J.S. Owen & G. Davies
37
[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 360371, 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 8th International Symposium on Tubular Structures, pp 197-175, Singapore,
Balkema, 1998.
[31] BS 4360, “Specification for weldable structural steels”, British Standards Institution, 1972 (superseded).