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Th e ai m of thi s featu r e is to shar e up -dates, desi gn ti ps an d an sw ers to qu eri es. The Steel Constr uc ti on

In st i tu te pr ovides i tem s wh ich, i t i s hoped, wi l l p r ove useful to the in dustr y.

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Design of Members Subjectedto TorsionWe have recently received a number of questions in relation to structuralsteel members subjected to torsion. From these calls it is evident that there is

some confusion over the difference in behaviour of open and closed cross-sections. In response to these queries, it therefore seems timely to present aqualitative background to the theory of torsion.Torsional loading can arise within members in two ways: an externallyapplied torque; or when the applied load acts eccentrically to the shear centreof the cross-section. In both cases, the member will twist about itslongitudinal axis, which passes through the shear centre of the cross-section.

Categories of cross-sectionsTorsional loading has a significant influence on the initial choice of sectionfor maximum structural efficiency. I-shaped sections are particularly poor inresisting torsion while hollow sections can be very effective. A distinction isnormally made between these two types of sections, by calling I- andChannel sections (which are poor in resisting torsion), Open Sections(fig. 1);while rectangular and circular hollow sections (which are more effective inresisting torsion), are referred to asClosed Sections(fig. 1).

Fig. 1. Open and closed str uctural sections

Location of shear centre and its significance

The position of the shear centre is of particular importance in design, sincewhen a load is applied to a member, a torque will develop if the applied loaddoes not act through the shear centre of the cross-section. In thesecircumstances, the torque is simply equal to the applied load multiplied byits eccentricity from the shear centre. A cross-section having two axes ofsymmetry has its shear centre located at the centre of gravity of the cross-section (fig. 2a). With one axis of symmetry, the shear centre lies on that axis,but will in general not coincide with the centre of gravity (fig. 2b). However,for a section having skew-symmetry, the shear centre and the centre ofgravity do coincide (fig. 2c). For angle sections, the shear centre is located atthe intersection of the two legs (fig. 2(d)).

Fig. 2. Location of shear centre s and centre of gr avit y c

Reference will be made below to warping, which is best described byconsidering the rectangular hollow section shown in Fig. 3. In the initialconditions, the ends of the hollow section are rectangular and plane. Supposethat a slit is made along one side of this section (thereby transforming itfrom a closed section to an open section), and one end is twisted relative tothe other. As can be seen in Fig. 3, in response to this applied torque, theends of the hollow section remain rectangular, but are no longer plane. Thisdistortion of the cross-section is called warping, and is particularlypronounced in I-beams.

Fig. 3 Warping of a split (open) rectangular hollow section

The total resistance of a structural member to torsional loading may beconsidered to be the sum of two components namely, uniform torsionand

warping torsion. In some cases, only uniform torsion occurs. Whereas, whenboth uniform torsion and warping torsion are included in the torsionalresistance, the member is in a state ofnon-uniform torsion. A diagrammaticrepresentation of uniform and non-uniform torsion, on a member composedof an I-section, is shown in Fig. 4.

Fig. 4. Uni form and non-uniform torsion of a member composed of an I-

section (viewed on plan)

Uniform torsionWhen a member is subjected to uniform torsion (sometimes referred to aspure or St Venant tor sion), the rate of change of the angle of twist is constantalong the member, and the longitudinal warping deflexions are also constant

along the member (fig. 4a). In this case, the torque acting at any cross-sectionis resisted by a single set of shear stresses distributed around the cross-section (fig. 5). The ratio of the torque Tto the twist rotation per unit

length, is defined as the torsional rigidity GJof the member; whereGis theshear modulus and Jis the torsion constant (sometimes called the St Venanttorsion constant).

ADVISORY DESK

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Fig. 5. Shear stresses due to un iform torsion of (a) closed sections; and (b)

open sections

Non-uniform torsionWhen a member is subjected to non-uniform torsion, the rate of change ofthe angle of twist varies along the member (fig. 4b and c). In this case, the

warping deflexions vary along the member and, to resist the applied torque,an additional set of shear stresses act in conjunction with those due touniform torsion. The stiffness of the member associated with theseadditional shear stresses is proportional to the warping rigidity EH; whereEis the modulus of elasticity andHis the warping constant.For a member composed of an I-section, the action of warping resistance canbe visualised as follows: the torqueTis resisted by a moment comprising offorces equal to the shear forces in each flange, which are separated by thelever arm, dfequal to the depth between the centroids of the two flanges. Ifeach flange is treated as a beam, the bending moments produced by theseforces lead to the warping normal stresses, as shown in Fig. 6b.For an I- or H-section, this approach provides a reasonable approximation,but will generally over-estimate the warping normal stress whilst under-estimating the warping shear stress (since the approach ignores the shearstresses from uniform torsion). However, it cannot be readily applied tochannel sections; in such circumstances, more rigorous methods of analysis

need to be adopted.

Fig. 6. Warping stresses in an open section member composed of an I-beam

Effect of cross-section on torsional behaviourBefore examining how different types of sections perform in resistingtorsion, it is useful to first introduce the non-dimensional torsion parameterfor the member:

For sections that have a very high torsional rigidity GJcompared to theirwarping rigidity EH , Kbecomes small; in these circumstances the memberwill effectively be in a state of uniform torsion (as indicated in Fig. 7). Closedsections, whose torsional rigidities are very large, behave in this way, as do

sections whose warping rigidities are negligible, such as angle and T-sections.Conversely, for sections whose warping rigidity EHis very high compared totheir torsional rigidity GJ, Kbecomes very large, and the member is in thelimiting state of warping torsion (as indicated in Fig. 7). Very thin-walledopen sections, such as light gauge cold-formed sections, whose torsional

rigidities are very small, behave in this way. Between these extremes, theapplied torque is resisted by a combination of the uniform and warpingtorsion components, and the member is in the general state of non-uniformtorsion. This occurs for intermediate values of the parameter K, as shown inFig. 7, which are appropriate for most open sections such as hot-rolled I- orchannel sections.

Fig. 7. Effect of cross-section on torsional behaviour

Whether a member is in a state of uniform or non-uniform torsion alsodepends on the loading arrangement and the warping restraints. If thetorsion resisted is constant along the member and warping is unrestrained(as shown in fig. 4a), then the member will be in uniform torsion, even if thetorsional rigidity is very small. If, however, the torsion resisted varies alongthe length of the member (fig. 4b), or if the warping displacements arerestrained in any way (fig. 4c), then the rate of change of the angle of twistrotation will vary, and the member will be in a state of non-uniform torsion.As a result of applying a torque to a member, the torsional stresses inducedwithin the section, which should be considered in design, are:

(a) Shear stresses due to uniform torsion.(b) Shear stresses due to warping torsion.(c) Bending stresses due to warping.Each of the above stresses is associated with the angle of twist , or its

derivatives. Hence, if is determined for different positions along the

member length, the corresponding stresses can be evaluated at each position.

Torsion of closed sectionsAs discussed above, the torsional rigidity GJof a closed section is very largecompared with its warping rigidity EH(fig. 7). Therefore, a membercomposed of a closed section may be considered to be subject only touniform torsion. For a rectangular or circular hollow section, the uniformtorsion will result in a uniform shear stress developing within the walls ofthe cross-section (fig. 5a). In these circumstances, the problem is staticallydeterminate, and the shear stress as well as the angle of twist may be

determined from simple statics.

Torsion of open sectionsAs discussed above, for members composed of open sections such as hot-rolled I- or Channel sections, the section may be considered to be subject tothe general state of non-uniform torsion (fig. 7). In these circumstances, theapplied torque is resisted by a combination of uniform torsion and warpingtorsion components.

Uniform torsion

If a torque is applied to the ends of a member, in such a way that the ends arefree to warp, then the member will only develop uniform torsion (fig. 4a). Theresulting shear stresses will vary linearly across the thickness of the element(fig. 5b): they are maximum at the element surfaces, with two equal values,but opposite in direction. These stresses are a function of the rate of changeof the angle of twist, and are greatest in the thickest element of the cross-

section i.e., typically the flanges in an I-beam. (At junctions between the weband the flanges, the local shear stresses may exceed the stresses in thethickest element of the cross-section; for rolled sections, this effect may beneglected by the designer, as allowance for the root fillet radii are made indetermining the torsional constant Jin section property tables).

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Warping torsion

When a uniform torque is applied to a member restrained against warping,the section itself will be subject to non-uniform torsion with the rate ofchange of the angle of twist varying along the length of the member (fig. 4c).The rotation of the section with respect to a restrained end will beaccompanied by bending of the flanges in their own plane (sometimesreferred to as the Bimoment). The warping normal and warping shearstresses developed by this condition are shown in Fig. 6. Warping stresses

are also generated in members of open section when the applied torquevaries along the length, even if the ends are free to warp (fig. 4b).

End conditionsAs discussed above, the end conditions will also greatly influence thetorsional stresses along the member. Note that end conditions for torsioncalculations may be quite different from those for bending e.g., a beam maybe supported at both ends, but torsionally restrained at only one end: thetorsional equivalent of a cantilever. Torsional fixity must be provided by atleast one point along the length of the member, otherwise it will simplyphysically twist when the torque is applied. Also, warping fixity cannot beprovided without also providing torsional fixity. As a result, there are threepossible boundary conditions which may sensibly considered in torsioncalculations:

(i) Torsion fixed, warping fixed

This condition is satisfied when, at the ends of the member, both twistingabout the longitudinal axis and warping of the cross-section areprevented (sometimes referred to as a Fixedtorsional end condition).Effective warping fixity is practically almost impossible to achieve inmost structures. A connexion providing fixity in both directions is notsufficient, it is also necessary to restrain the two flanges either side of theweb. Details such as those shown in Fig. 8a need to be provided toachieve this type of boundary condition. It should be noted, however, thatthe provision of warping fixity does not produce such a large reduction intorsion stresses as is obtained from bending fixity. Therefore, it is morepractical to assume warping free connexions (see (ii) below), even whenfixity is provided in terms of bending.

(ii) Torsion fixed, warping freeThis is satisfied when the cross-section at the ends of the member is

prevented from twisting, but is allowed to warp freely (sometimesreferred to as aPinnedtorsional end condition). Such a condition may bereadily achieved by providing the relatively simple standard connexions,such as shown in Fig. 8b.

(iii)Torsion free, warping freeThis condition is achieved when the end is free to warp and twist(sometimes referred to as a Freetorsional end condition): theunsupported end of a cantilever illustrates this case.

Fig. 8. Practical end condit ions

The above gives a basic overview of the considerations that should be madein the design process for structural steel sections, which are to be subjectedto torsion. Design equations for estimating the stresses due to torsion, incombination with bending, may be found in SCI publication 057 entitled

Design of Members Subject to Combined Bending and Torsion.

Contact: Dr Stephen Hicks: e-mail: s.hicks@steel-sci.com