Torsion Analysis for Cold-Formed Steel Members Using Flexural Analogies

Proceedings of the Cold-Formed Steel Research Consortium Colloquium 20-22 October 2020 (cfsrc.org) Torsion Analysis for Cold-Formed Steel Members Using Flexural Analogies Robert S. Glauz, P.E.1 Abstract The design of cold-formed steel members must consider the impact of torsional loads due to transverse load eccentricity. Open cross-sections are particularly susceptible to significant twisting and high warping stresses. Design requirements for combined bending and torsion were introduced in the American Iron and Steel Institute Specification in 2007, and more recently in the Australian/New Zealand Standard 4600:2018. These provisions require an understanding of the distribution of internal forces and stresses due to torsional warping, which is not commonly taught in engineering curriculums. Furthermore, most structural analysis programs do not properly consider torsional warping stiffness and response. The purpose of this paper is to educate the structural engineer on torsion analysis using analogies to familiar flexural relationships. Useful formulas are provided for determining torsional properties and stresses. 1. Introduction Current editions of design specifications AISI S100 [1] and AS/NZS 4600 [2] have provisions to account for stresses Cold-formed steel members of open cross-section are often produced by torsional loads. These provisions consider the susceptible to twisting and torsional stresses. The shear effect of combined longitudinal stresses resulting from center for many shapes is outside the envelope of the cross- flexure and torsional warping. Future provisions may section so it can be difficult to apply transverse loads without address combined shear stresses from flexure and torsion, producing torsional effects. Open thin-walled members also and may consider the impact of combined longitudinal and have inherently low torsional stiffness, thus even small shear stresses from all types of loading. magnitudes of torsion can result in a significant amount of twisting. Understanding the distribution of internal torsional forces is essential to performing the required member design Design strategies can be employed to reduce or eliminate calculations. In a flexural analysis, the distribution of torsion. Locating beams in line with transverse loads is often moments and shear forces along a beam are independent the most obvious approach. Remaining load eccentricity of the selected member size. But in a torsion analysis, the from the beam shear center can be effectively reduced by distribution of internal forces can vary with certain properties transferring loads through components that restrain twisting, of the cross-section. This paper provides an overview of thereby causing the loads to act through the shear center. torsion analysis for cold-formed steel members. It is For cases where beams cannot be aligned with transverse intended to educate the engineer in the characteristics of loads, supplemental perpendicular members could be torsional internal forces, how they are distributed, and ways designed to carry the loads in flexure, rather than subjecting to simplify these concepts. the supporting beam to torsion. 2. General Torsion Theory If torsional loads cannot be avoided, the member must be designed with consideration for the resulting twist and The development of the elastic torsional response of general torsional stresses. Large torsional loads are best handled prismatic members can be found in the works of using closed cross-sections with high torsional stiffness and Timoshenko [3] and Vlasov [4], which remain the basis for minimal warping. Applications using open cross-sections torsional design methods as presented in numerous texts, may require additional bracing to restrain twist at spacings including Heins [5], Seaburg and Carter [6], and SCI P385 close enough to reduce the amount of twist and torsional [7]. In the classical theory, the relationship between applied stresses to acceptable levels. torsion and angle of twist is given by Equation 1: 1 RSG Software, Inc., Lee’s Summit, Missouri, USA, [email protected] 퐸퐶 휙′′′′ − 퐺퐽휙′′ = 푚 (푧) (1) 푤 푡 Mt where E is the modulus of elasticity, Cw is the warping torsion constant, G is the shear modulus, J is the St. Venant ϕ torsion constant, ϕ is the shape function for angle of twist, and mt(z) is the applied torsional moment intensity as a M function of the longitudinal z axis of the member. The 푀 퐿 t 푡 L internal torque T is obtained by integrating the applied 휙 = 퐺퐽 torsional moment, as given by Equation 2: Figure 3: Pure torsion – plane sections remain plane ′′′ ′ 푇(푧) = 퐸퐶푤휙 − 퐺퐽휙 = 푇푤 + 푇푠푣 (2) For the loading in Figure 3 where torsional moments Mt are This internal torque is resisted by two types of shear applied to the ends of the member, the internal torque is corresponding to the two terms in Equation 2. The first term constant throughout the length of the member, and the total is the warping torsion component Tw, and the second term angle of twist is MtL/GJ. is the St. Venant (pure) torsion component Tsv. Some cross- sections as shown in Figure 1 are dominated by pure torsion Pure torsion can be evaluated using an elastic membrane 2 where ECw/L is much less than GJ, and others as shown in analogy, which was first introduced by Prandtl [8]. In this Figure 2 are often dominated by warping torsion where analogy a membrane over the cross-sectional area displaces 2 ECw/L is much greater than GJ, particularly for thin material. due to an internal pressure. The volume under the displaced membrane is proportional to the internal torque, and the slope of the membrane is proportional to the shear stress. For common shapes, the torsion constant and shear stresses can be closely approximated with simple formulas. The development of these can be found in several texts such 2 Figure 1: Sections dominated by pure torsion: ECw/L << GJ as Muvdi and McNabb [9]. For closed cross-sections with uniform thickness t, the torsion constant is calculated using Equation 3, where Am is the area enclosed by the midline of the thickness, and Sm is the length of the midline perimeter, as illustrated in Figure 4. 2 4퐴푚푡 퐽 = (closed sections) (3) 푆 푚 2 Figure 2: Sections dominated by warping torsion: ECw/L >> GJ 2 For the special case of a cylindrical tube, 퐴푚 = 휋푟푚 and For cross-sections where torsion is resisted by both pure 푆 = 2휋푟 , giving the torsion constant as: torsion and warping torsion, the solution to Equation 1 is 푚 푚 more complex than beam flexure, requiring the use of 3 퐽 = 2휋푟푚푡 (cylindrical tubes) (4) hyperbolic sine and cosine functions. This solution is discussed in Section 5. It is helpful to first understand the two torsional mechanisms individually. These are presented r in Section 3 for pure torsion and Section 4 for warping m torsion. t 3. Pure Torsion t A member undergoes pure torsion if the plane of each cross- section remains plane as the member twists (see Figure 3). This type of torsion occurs only with closed or solid circular cross-sections. However, pure torsion is predominant for other closed cross-sections, and for cross-sections with straight elements passing through the shear center, where Figure 4: St. Venant torsion constant for closed sections the warping constant Cw is relatively small. These are the cases shown in Figure 1. 2 For open sections, the torsion constant is calculated using The area under the internal torque curve between two points Equation 5. For uniform thickness, this simplifies to 퐽 = is equal to the change in –GJϕ between those two points, ⅓퐴푡2, where A is the cross-sectional area. and the slope of the twist angle curve at any point is equal to –Tsv/GJ. 1 3 3 퐽 = ∫ 푡 푑푠 = ⅓ Σ 푏푡 (open sections) (5) mt 3 The flow of shear stress caused by pure torsion is illustrated in Figure 5. Closed sections have a nearly constant stress across the thickness, and open sections have linearly varying stresses across the thickness. Torsion Intensity mt = –GJϕ′′ Internal Torque Tsv = –GJϕ′ Twist Angle ϕ Figure 5: St. Venant shear stresses (rad) For closed sections with uniform thickness, shear stress is calculated using Equation 6, which simplifies to Equation 7 Figure 6: Pure torsion distribution for cylindrical tubes. For open sections, the maximum shear stress at the material surface is calculated using Equation 8. For a simple beam with uniform torsion, the diagrams for internal torque and twist angle are shown in Figure 7. The 푇푠푣 expressions for maximum torque and twist angle should look 휏푠푣 = (closed sections) (6) 2퐴푚푡 familiar to the structural engineer, as they mirror the flexural diagrams for shear and moment for a simple beam with 푇푠푣푟푚 휏푠푣 = (cylindrical tubes) (7) uniform load. 퐽 mt 푇푠푣푡 휏푠푣 = (open sections) (8) 퐽 Determining the internal torque Tsv at any point along the mtL/2 member requires an understanding of the torsion distribution. For pure torsion, the first term in Equation 1 is Torque omitted, and the second derivative of the displacement function is proportional to the torsion intensity mt. Integration of the torsion intensity gives the internal torque Tsv, which is –mtL/2 proportional to the first derivative of the displacement Twist function. Finally, integration of internal torque divided by the torsional stiffness (Tsv/GJ) and changing sign, gives the displacement twist angle ϕ. –mtL²/8GJ These relationships are illustrated in Figure 6. Structural Figure 7: Pure torsion distribution engineers learn the relationships between load, shear, and moment diagrams for beam flexure, and these concepts can This analogy between beam flexure and pure torsion can be be applied to pure torsion relationships. The area under the extended to multiple span applications and different loading torsion intensity curve between two points along the patterns (see Appendix A for more examples).

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