1、Torsional Analysis ofStructural Steel MembersSteel Design Guide SeriesSteel Design Guide SeriesTorsional Analysisof StructuralSteel MembersPaul A. Seaburg, PhD, PEHead, Department of Architectural EngineeringPennsylvania State UniversityUniversity Park, PACharles J. Carter, PEAmerican Institute of S
2、teel ConstructionChicago, ILAMERICAN INSTITUTE OF STEEL CONSTRUCTIONCopyright 1997byAmerican Institute of Steel Construction, Inc.All rights reserved. This book or any part thereofmust not be reproduced in any form without thewritten permission of the publisher.The information presented in this publ
3、ication has been prepared in accordance with rec-ognized engineering principles and is for general information only. While it is believedto be accurate, this information should not be used or relied upon for any specific appli-cation without competent professional examination and verification of its
4、 accuracy,suitablility, and applicability by a licensed professional engineer, designer, or architect.The publication of the material contained herein is not intended as a representationor warranty on the part of the American Institute of Steel Construction or of any otherperson named herein, that t
5、his information is suitable for any general or particular useor of freedom from infringement of any patent or patents. Anyone making use of thisinformation assumes all liability arising from such use.Caution must be exercised when relying upon other specifications and codes developedby other bodies
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7、 this edition.Printed in the United States of AmericaSecond Printing: October 2003TABLE OF CONTENTS1. Introduction . 12. Torsion Fundamentals. 32.1 Shear Center 32.2 Resistance of a Cross-Section toa Torsional Moment. 32.3 Avoiding and Minimizing Torsion. 42.4 Selection of Shapes for Torsional Loadi
8、ng 53. General Torsional Theory 73.1 Torsional Response. 73.2 Torsional Properties 73.2.1 Torsional Constant J . 73.2.2 Other Torsional Properties for OpenCross-Sections. 73.3 Torsional Functions . 94. Analysis for Torsion114.1 Torsional Stresses on I-, C-, and Z-ShapedOpen Cross-Sections . 114.1.1
9、Pure Torsional Shear Stresses . 114.1.2 Shear Stresses Due to Warping 114.1.3 Normal Stresses Due to Warping 124.1.4 Approximate Shear and NormalStresses Due to Warping on I-Shapes 124.2 Torsional Stress on Single Angles 124.3 Torsional Stress on Structural Tees . 124.4 Torsional Stress on Closed an
10、dSolid Cross-Sections . 124.5 Elastic Stresses Due to Bending andAxial Load . 134.6 Combining Torsional Stresses WithOther Stresses. 144.6.1 Open Cross-Sections 144.6.2 Closed Cross-Sections. 154.7 Specification Provisions 154.7.1 Load and Resistance Factor Design 154.7.2 Allowable Stress Design . 1
11、64.7.3 Effect of Lateral Restraint atLoad Point. 174.8 Torsional Serviceability Criteria 185. Design Examples 19Appendix A. Torsional Properties 33Appendix B. Case Graphs of Torsional Functions. 54Appendix C. Supporting Information 107C.1 General Equations for 6 and its Derivatives . 107C.1.1 Consta
12、nt Torsional Moment . 107C.1.2 Uniformly Distributed TorsionalMoment 107C.1.3 Linearly Varying Torsional Moment. 107C.2 Boundary Conditions . 107C.3 Evaluation of Torsional Properties. 108C.3.1 General Solution 108C.3.2 Torsional Constant J for OpenCross-Sections. 108C.4 Solutions to Differential Eq
13、uations forCases in Appendix B 110References . 113Nomenclature 115Chapter 1INTRODUCTIONThis design guide is an update to the AISC publication Tor-sional Analysis of Steel Members and advances further thework upon which that publication was based: BethlehemSteel Companys Torsion Analysis of Rolled St
14、eel Sections(Heins and Seaburg, 1963). Coverage of shapes has beenexpanded and includes W-, M-, S-, and HP-Shapes, channels(C and MC), structural tees (WT, MT, and ST), angles (L),Z-shapes, square, rectangular and round hollow structuralsections (HSS), and steel pipe (P). Torsional formulas forthese
15、 and other non-standard cross sections can also be foundin Chapter 9 of Young (1989).Chapters 2 and 3 provide an overview of the fundamentalsand basic theory of torsional loading for structural steelmembers. Chapter 4 covers the determination of torsionalstresses, their combination with other stress
16、es, Specificationprovisions relating to torsion, and serviceability issues. Thedesign examples in Chapter 5 illustrate the design process aswell as the use of the design aids for torsional properties andfunctions found in Appendices A and B, respectively. Finally,Appendix C provides supporting infor
17、mation that illustratesthe background of much of the information in this designguide.The design examples are generally based upon the provi-sions of the 1993 AISC LRFD Specification for StructuralSteel Buildings (referred to herein as the LRFD Specifica-tion). Accordingly, forces and moments are ind
18、icated with thesubscript u to denote factored loads. Nonetheless, the infor-mation contained in this guide can be used for design accord-ing to the 1989 AISC ASD Specification for Structural SteelBuildings (referred to herein as the ASD Specification) ifservice loads are used in place of factored lo
19、ads. Where thisis not the case, it has been so noted in the text. For single-anglemembers, the provisions of the AISC Specification for LRFDof Single-Angle Members and Specification for ASD of Sin-gle-Angle Members are appropriate. The design of curvedmembers is beyond the scope of this publication;
20、 refer toAISC (1986), Liew et al. (1995), Nakai and Heins (1977),Tung and Fountain (1970), Chapter 8 of Young (1989),Galambos (1988), AASHTO (1993), and Nakai and Yoo(1988).The authors thank Theodore V. Galambos, Louis F. Gesch-windner, Nestor R. Iwankiw, LeRoy A. Lutz, and Donald R.Sherman for thei
21、r helpful review comments and suggestions.1Chapter 2TORSION FUNDAMENTALS2.1 Shear CenterThe shear center is the point through which the applied loadsmust pass to produce bending without twisting. If a shape hasa line of symmetry, the shear center will always lie on thatline; for cross-sections with
22、two lines of symmetry, the shearcenter is at the intersection of those lines (as is the centroid).Thus, as shown in Figure 2.la, the centroid and shear centercoincide for doubly symmetric cross-sections such as W-, M-,S-, and HP-shapes, square, rectangular and round hollowstructural sections (HSS),
23、and steel pipe (P).Singly symmetric cross-sections such as channels (C andMC) and tees (WT, MT, and ST) have their shear centers onthe axis of symmetry, but not necessarily at the centroid. Asillustrated in Figure 2. lb, the shear center for channels is at adistance eo from the face of the channel;
24、the location of theshear center for channels is tabulated in Appendix A as wellas Part 1 of AISC (1994) and may be calculated as shown inAppendix C. The shear center for a tee is at the intersectionof the centerlines of the flange and stem. The shear centerlocation for unsymmetric cross-sections suc
25、h as angles (L)and Z-shapes is illustrated in Figure 2.1c.2.2 Resistance of a Cross-section to a TorsionalMomentAt any point along the length of a member subjected to atorsional moment, the cross-section will rotate through anangle as shown in Figure 2.2. For non-circular cross-sec-tions this rotati
26、on is accompanied by warping; that is, trans-verse sections do not remain plane. If this warping is com-pletely unrestrained, the torsional moment resisted by thecross-section is:bending is accompanied by shear stresses in the plane of thecross-section that resist the externally applied torsional mo
27、-ment according to the following relationship:resisting moment due to restrained warping of thecross-section, kip-in,modulus of elasticity of steel, 29,000 ksiwarping constant for the cross-section, in.4third derivative of 6 with respect to zThe total torsional moment resisted by the cross-section i
28、s thesum of T, and Tw. The first of these is always present; thesecond depends upon the resistance to warping. Denoting thetotal torsional resisting moment by T, the following expres-sion is obtained:Rearranging, this may also be written as:whereresisting moment of unrestrained cross-section, kip-in
29、.shear modulus of elasticity of steel, 11,200 ksitorsional constant for the cross-section, in.4angle of rotation per unit length, first derivative of 0with respect to z measured along the length of themember from the left supportWhen the tendency for a cross-section to warp freely isprevented or res
30、trained, longitudinal bending results. ThisAn exception to this occurs in cross-sections composed of plate elements having centerlines that intersect at a common point such as a structural tee. For such cross-sections,3(2.1)(2.3)(2.4)Figure 2.1.where2.3 Avoiding and Minimizing TorsionThe commonly us
31、ed structural shapes offer relatively poorresistance to torsion. Hence, it is best to avoid torsion bydetailing the loads and reactions to act through the shearcenter of the member. However, in some instances, this maynot always be possible. AISC (1994) offers several sugges-tions for eliminating to
32、rsion; see pages 2-40 through 2-42. Forexample, rigid facade elements spanning between floors (theweight of which would otherwise induce torsional loading ofthe spandrel girder) may be designed to transfer lateral forcesinto the floor diaphragms and resist the eccentric effect asillustrated in Figur
33、e 2.3. Note that many systems may be tooflexible for this assumption. Partial facade panels that do notextend from floor diaphragm to floor diaphragm may bedesigned with diagonal steel “kickers,“ as shown in Figure2.4, to provide the lateral forces. In either case, this eliminatestorsional loading o
34、f the spandrel beam or girder. Also, tor-sional bracing may be provided at eccentric load points toreduce or eliminate the torsional effect; refer to Salmon andJohnson (1990).When torsion must be resisted by the member directly, itseffect may be reduced through consideration of intermediatetorsional
35、 support provided by secondary framing. For exam-ple, the rotation of the spandrel girder cannot exceed the totalend rotation of the beam and connection being supported.Therefore, a reduced torque may be calculated by evaluatingthe torsional stiffness of the member subjected to torsionrelative to th
36、e rotational stiffness of the loading system. Thebending stiffness of the restraining member depends upon itsend conditions; the torsional stiffness k of the member underconsideration (illustrated in Figure 2.5) is:= torque= the angle of rotation, measured in radians.A fully restrained (FR) moment c
37、onnection between theframing beam and spandrel girder maximizes the torsionalrestraint. Alternatively, additional intermediate torsional sup-ports may be provided to reduce the span over which thetorsion acts and thereby reduce the torsional effect.As another example, consider the beam supporting a
38、walland slab illustrated in Figure 2.6; calculations for a similarcase may be found in Johnston (1982). Assume that the beamFigure 2.2.Figure 2.3.Figure 2.4.4where(2.5) where(2.6)Rev.3/1/03Rev.3/1/035alone resists the torsional moment and the maximum rotationof the beam due to the weight of the wall
39、 is 0.01 radians.Without temporary shoring, the top of the wall would deflectlaterally by nearly 3/4-in. (72 in. x 0.01 rad.). The additionalload due to the slab would significantly increase this lateraldeflection. One solution to this problem is to make the beamand wall integral with reinforcing st
40、eel welded to the topflange of the beam. In addition to appreciably increasing thetorsional rigidity of the system, the wall, because of itsbending stiffness, would absorb nearly all of the torsionalload. To prevent twist during construction, the steel beamwould have to be shored until the floor sla
41、b is in place.2.4 Selection of Shapes for Torsional LoadingIn general, the torsional performance of closed cross-sectionsis superior to that for open cross-sections. Circular closedshapes, such as round HSS and steel pipe, are most efficientfor resisting torsional loading. Other closed shapes, such
42、assquare and rectangular HSS, also provide considerably betterresistance to torsion than open shapes, such as W-shapes andchannels. When open shapes must be used, their torsionalresistance may be increased by creating a box shape, e.g., bywelding one or two side plates between the flanges of aW-shap
43、e for a portion of its length.Figure 2.5. Figure 2.6.Chapter 3GENERAL TORSIONAL THEORYA complete discussion of torsional theory is beyond the scopeof this publication. The brief discussion that follows is in-tended primarily to define the method of analysis used in thisbook. More detailed coverage o
44、f torsional theory and othertopics is available in the references given.3.1 Torsional ResponseFrom Section 2.2, the total torsional resistance provided by astructural shape is the sum of that due to pure torsion and thatdue to restrained warping. Thus, for a constant torque T alongthe length of the
45、member:C and Heins (1975). Values for and which are used tocompute plane bending shear stresses in the flange and edgeof the web, are also included in the tables for all relevantshapes except Z-shapes.The terms J, a, and are properties of the entire cross-section. The terms and vary at different poi
46、nts on thecross-section as illustrated in Appendix A. The tables give allvalues of these terms necessary to determine the maximumvalues of the combined stress.3.2.1 Torsional Constant JThe torsional constant J for solid round and flat bars, square,rectangular and round HSS, and steel pipe is summari
47、zed inTable 3.1. For open cross-sections, the following equationmay be used (more accurate equations are given for selectedshapes in Appendix C.3):wherewherewhereIn the above equations, and are the first,second, third, and fourth derivatives of 9 with respect to z andis the total angle of rotation a
48、bout the Z-axis (longitudinalaxis of member). For the derivation of these equations, seeAppendix C.1.3.2 Torsional PropertiesTorsional properties J, a, and are necessary for thesolution of the above equations and the equations for torsionalstress presented in Chapter 4. Since these values are depend
49、-ent only upon the geometry of the cross-section, they havebeen tabulated for common structural shapes in Appendix Aas well as Part 1 of AISC (1994). For the derivation oftorsional properties for various cross-sections, see Appendix whereFor rolled and built-up I-shapes, the following equations maybe used (fillets are generally neglected):maximum applied torque at right support, kip-in./ftdistance from left support, in.span length, in.For a linearly varying torque(3.3)(3.2)For a uniformly distributed torque t:shear modulus of elasticity of steel, 11,200 ksitorsional constant of cross-sectio
