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- 1. Steel Design Guide SeriesTorsional Analysis ofStructural Steel Members
- 2. Steel Design Guide SeriesTorsional Analysisof StructuralSteel MembersPaul A. Seaburg, PhD, PEHead, Department of Architectural EngineeringPennsylvania State UniversityUniversity Park, PACharles J. Carter, PEAmerican Institute of Steel ConstructionChicago, ILA M E R I C A N I N S T I T U T E O F S T E E L C O N S T R U C T I O N© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 3. Copyright  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 publication 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 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 this 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 and incorporated by reference herein since such material may be mod-ified or amended from time to time subsequent to the printing of this edition. TheInstitute bears no responsibility for such material other than to refer to it and incorporateit by reference at the time of the initial publication of this edition.Printed in the United States of AmericaSecond Printing: October 2003© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 4. TABLE 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 Loading ...... 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 T o r s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 1 14.1 Torsional Stresses on I-, C-, and Z-ShapedOpen Cross-Sections . . . . . . . . . . . . . . . . . . . . . 114.1.1 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 andSolid Cross-Sections . . . . . . . . . . . . . . . . . . . . . 124.5 Elastic Stresses Due to Bending andAxial Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Combining Torsional Stresses WithOther Stresses........................... 144.6.1 OpenCross-Sections ................ 144.6.2 Closed Cross-Sections............... 154.7 Specification Provisions.................... 154.7.1 Load and Resistance Factor Design .... 154.7.2 Allowable StressDesign ............. 164.7.3 Effect of Lateral Restraint atLoadP o i n t . . . . . . . . . . . . . . . . . . . . . . . 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 Constant Torsional Moment ......... 107C.1.2 Uniformly Distributed TorsionalM o m e n t . . . . . . . . . . . . . . . . . . . . . . . . 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 Equations forCases in Appendix B . . . . . . . . . . . . . . . . . . . . 110References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 5. Chapter1INTRODUCTIONThis 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 Steel 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 and other non-standard cross sections can also be foundin Chapter 9 of Young (1989).Chapters 2 and 3 provide an overview ofthe fundamentalsand basic theory of torsional loading for structural steelmembers. Chapter 4 covers the determination of torsionalstresses, their combination with other stresses, 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 information 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 indicated 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 Specificationfor Structural SteelBuildings (referred to herein as the ASD Specification) ifservice loads are used in place of factored loads. Where thisis notthe case, ithas been so notedin the text. For single-anglemembers, the provisions of the AISC Specificationfor 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; 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 thankTheodore V. Galambos, Louis F. Gesch-windner, Nestor R. Iwankiw, LeRoy A. Lutz, and Donald R.Sherman for their helpful review comments and suggestions.1© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 6. Chapter2TORSION FUNDAMENTALS2.1 Shear CenterThe shear center is the point through which the applied loadsmust pass to produce bending without twisting. Ifa shape hasa line of symmetry, the shear center will always lie on thatline; for cross-sections with 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 fordoubly symmetric cross-sections such as W-, M-,S-, and HP-shapes, square, rectangular and round hollowstructural sections (HSS), 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 eofrom the face of the channel; 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 such 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 rotation 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-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 momentresistedby the cross-sectionis 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.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 restrained, 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.where© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 7. 2.3 Avoiding and Minimizing TorsionThe commonly used 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 torsion; 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 Figure 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 of 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 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 the 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 connection 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 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.3/1/03© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.HH
- 8. 5alone resists the torsional moment and the maximum rotationof the beam due to the weight of the wall is 0.01 radians.Without temporary shoring, the top of the wall would deflectlaterally by nearly3/4-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 steel 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 slab is in place.2.4 Selection of Shapes for Torsional LoadingIn general, the torsional performance ofclosed 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 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-shape for a portion of its length.Figure 2.5. Figure 2.6.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 9. Chapter3GENERALTORSIONALTHEORYAcomplete discussion oftorsional 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 of 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 ofthat due to pure torsion and thatdue to restrained warping. Thus, for a constant torque Talongthe length of the 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 points 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 summarized 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 about 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 forthesolution ofthe above equations andthe equations fortorsionalstress presented in Chapter 4. Since these values are depend-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-section, in.4modulus of elasticity of steel, 29,000 ksiwarping constant of cross-section, in.6(3.1)length of each cross-sectional element, in.thickness of each cross-sectional element, in.3.2.2 Other Torsional Properties for Open Cross-Sections2(3.4)2 For shapes with sloping-sided flanges, sloping flange elements are simplified into rectangular elements of thickness equal to the average thickness of the flange.(3.5)(3.6)(3.7)(3.8)(3.9)(3.10)7© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 10. For channels, the following equations may be used:(3.11)(3.12)(3.13)(3.14)(3.15)(3.16)(3.17)(3.18)(3.19)(3.20)(3.21)(3.22)(3.23)(3.24)(3.25)(3.26)(3.27)Figure 3.1.Table 3.1Torsional Constants JSolid Cross-SectionsClosedCross-SectionsNote: tabulated values for HSS in Appendix A differ slightly because theeffect of corner radii has been considered.For Z-shapes:where, as illustrated in Figure 3.1:8© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 11. 3.3 Torsional FunctionsIn addition to the torsional properties given in Section 3.2above, the torsional rotation 0 and its derivatives are neces-sary for the solution of equations 3.1, 3.2, and 3.3. In Appen-dix B, these equations have been evaluated for twelve com-mon combinations of end condition (fixed, pinned, and free)and load type. Members are assumed to be prismatic. Theidealized fixed, pinned, and free torsional end conditions, forwhich practical examples are illustrated in Figure 3.3, aredefined in Appendix C.2.The solutions give therotational response andderivativesalong the span corresponding to different values of theratio of the member span length l to the torsional property aofits cross-section. The functions given are non-dimensional,that is, each term is multiplied by a factor that is dependentupon the torsional properties of the member and the magni-tude of the applied torsional moment.For each case, there are four graphs providing values of ,and Each graph shows the value of the torsionalfunctions (vertical scale) plotted against the fraction of thespan length (horizontal scale) from the left support. Some ofthe curves have been plotted as a dotted line for ease ofreading. The resulting equations for each of these cases aregiven in Appendix C.4.(3.36)(3.28)(3.29)(3.30)(3.31)(3.32)where, as illustrated in Figure 3.2:For single-angles and structural tees, J may be calculatedusing Equation 3.4, excluding fillets. Formore accurate equa-tions including fillets, see El Darwish and Johnston (1965).Since pure torsional shear stresses will generally dominateover warping stresses, stresses due to warping are usuallyneglected in single angles (see Section 4.2) and structural tees(see Section 4.3); equations for other torsional propertieshave not been included. Since the centerlines ofeach elementof the cross-section intersect at the shear center, the generalsolution of Appendix C3.1 would yield0. A value of a (and therefore is required, however, todetermine the angle of rotation using the charts of Appen-dix B.(3.33)For single angles, the following formulas (Bleich, 1952) maybe used to determine Cw:where and are the centerline leg dimensions (overall legdimension minus half the angle thickness t for each leg). Forstructural tees:(3.34)where(3.35)Figure 3.2,9Figure 3.3.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 12. Chapter4ANALYSIS FORTORSIONIn this chapter, the determination of torsional stresses andtheir combination with stresses due to bending and axial loadis covered for both open and closed cross-sections. The AISCSpecification provisions for the design ofmembers subjectedto torsion and serviceability considerations for torsional rota-tion are discussed.4.1 Torsional Stresses on I-, C-, and Z-shaped OpenCross-SectionsShapes of open cross-section tend to warp under torsionalloading. If this warping is unrestrained, only pure torsionalstresses are present. However, when warping is restrained,additional direct shear stresses as well as longitudinal stressesdue to warping must also be considered. Pure torsional shearstresses, shear stresses due to warping, and normal stressesdue to warping are each related to the derivatives of therotational function Thus, when the derivatives of aredetermined along the girder length, the corresponding stressconditions can be evaluated. The calculation ofthese stressesis described in the following sections.4.1.1 Pure Torsional Shear StressesThese shear stresses are always present on the cross-sectionof a member subjected to a torsional moment and provide theresisting moment as described in Section 2.2. These arein-plane shear stresses that vary linearly across the thicknessof an element of the cross-section and act in a directionparallel to the edge of the element. They are maximum andequal, but of opposite direction, at the two edges. The maxi-mum stress is determined by the equation:The pure torsional shear stresses will be largest in the thickestelements of the cross-section. These stress states are illus-trated in Figures 4. 1b, 4.2b, and 4.3b for I-shapes, channels,and Z-shapes.114.1.2 Shear Stresses Due to WarpingWhen a member is allowed to warp freely, these shear stresseswill not develop. When warping is restrained, these are in-plane shear stresses that are constant across the thickness ofan element of the cross-section, but vary in magnitude alongthe length of the element. They act in a direction parallel tothe edge of the element. The magnitude of these stresses isdetermined by the equation:GENERAL ORIENTATION FIGUREpure torsional shear stress at element edge, ksishear modulus of elasticity of steel, 11,200 ksithickness of element, in.rate of change of angle of rotation first derivativeof with respect to z (measured along longitudinalaxis of member)where(4.1)Figure 4.1.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 13. where(4-2a)whereshear stress at point s due to warping, ksimodulus of elasticity of steel, 29,000 ksiwarping staticalmomentatpoints(seeAppendixA),in.4thickness of element, in.third derivative of with respect to zThese stress states are illustrated in Figures 4.1c, 4.2c, and4.3c for I-shapes, channels, and Z-shapes. Numerical sub-scripts are added to represent points of the cross-section asillustrated.4.1.3 Normal Stresses Due to WarpingWhen a member is allowed to warp freely, these normalstresses will not develop. When warping is restrained, theseare direct stresses (tensile and compressive) resulting frombending ofthe element due to torsion. They act perpendicularto the surface of the cross-section and are constant across thethickness of an element of the cross-section but vary inmagnitude along the length ofthe element. The magnitude ofthese stresses is determined by the equation:wherenormal stress at point s due to warping, ksimodulus of elasticity of steel, 29,000 ksinormalized warping function at point s (see Appen-dixA),in.2second derivative of with respect to zThese stress states are illustrated in Figures 4.1d, 4.2d, and4.3d for I-shapes, channels, and Z-shapes. Numerical sub-scripts are added to represent points of the cross-section asillustrated.4.1.4 Approximate Shear and Normal Stresses Due toWarping on I-ShapesThe shearandnormal stresses due to warping maybeapproxi-mated for short-span I-shapes by resolving the torsional mo-ment T into an equivalent force couple acting at the flangesas illustrated in Figure 4.4. Each flange is then analyzed as abeam subjected to this force. The shear stress at the center ofthe flange is approximated as:where is the value of the shear in the flange at any pointalong the length. The normal stress at the tips ofthe flange isapproximated as:12bending moment on the flange at any point along thelength.4.2 Torsional Stress on Single-AnglesSingle-angles tend to warp under torsional loading. If thiswarping is unrestrained, only pure torsional shear stressesdevelop. However, when warping is restrained, additionaldirect shear stresses as well as longitudinal stress due towarping are present.Pure torsional shear stress may be calculated using Equa-tion 4.1. Gjelsvik (1981) identified that the shear stresses dueto warping are of two kinds: in-plane shear stresses, whichvary from zero at the toe to a maximum at the heel of theangle; and secondary shear stresses, which vary from zero atthe heel to a maximum at the toe of the angle. These stressesare illustrated in Figure 4.5.Warping strengths of single-angles are, in general, rela-tively small. Using typical angle dimensions, it can be shownthat the two shear stresses due to warping are of approxi-mately the same order of magnitude, but represent less than20 percent of the pure torsional shear stress (AISC, 1993b).When all the shear stresses are added, the resultis amaximumsurface shear stress near mid-length of the angle leg. Sincethis is a local maximum that does not extend through thethickness of the angle, it is sufficient to ignore the shearstresses due to warping. Similarly, normal stresses due towarping are considered to be negligible.For the design of shelf angles, refer to Tide and Krogstad(1993).4.3 Torsional Stress on Structural TeesStructural tees tend to warp under torsional loading. If thiswarping is unrestrained, only pure torsional shear stressesdevelop. However, when warping is restrained, additionaldirect shear stresses as well as longitudinal or normal stressdue to warping are present. Pure torsional shear stress may becalculated using Equation 4.1. Warping stresses of structuraltees are, in general, relatively small. Using typical tee dimen-sions, it can be shown that the shear and normal stresses dueto warping are negligible.4.4 Torsional Stress on Closed and SolidCross-SectionsTorsion on a circular shape (hollow or solid) is resisted byshear stresses in the cross-section that vary directly withdistance from the centroid. The cross-section remains planeas it twists (without warping) and torsional loading developspure torsional stresses only. While non-circular closed cross-(4.3a)(4.3b)(4.2b)© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 14. wheresections tend to warp under torsional loading, this warping isminimized since longitudinal shear prevents relative dis-placement of adjacent plate elements as illustrated in Fig-ure 4.6.The analysis and design of thin-walled closedcross-sections for torsion is simplified with the assumptionthat the torque is absorbed by shear forces that are uniformlydistributed overthe thickness ofthe element (Siev, 1966). Thegeneral torsional response can be determined from Equation3.1 with the warping term neglected. For a constant torsionalmoment T the shear stress may be calculated as:13The value of computed using from Appendix A is thetheoretical value at the center of the flange. It is within theaccuracy of the method presented herein to combine thistheoretical value with the torsional shearing stress calculatedfor the point at the intersection of the web and flange center-lines.Figure 4.8 illustrates the distribution of these stresses,shown for the case of a moment causing bending about themajor axis of the cross-section and shear acting along theminor axis of the cross-section. The stress distribution in theZ-shape is somewhat complicated because the major axis isnot parallel to the flanges.Axial stress may also be present due to an axial load P.(4.5)(4.6)where(4.4)area enclosed by shape, measured to centerline ofthickness ofbounding elements as illustrated in Fig-ure 4.7, in.2thickness ofbounding element, in.For solid round and flat bars, square, rectangular and roundHSS and steel pipe, the torsional shear stress may be calcu-lated using the equations given in Table 4.1. Note that theequation for the hollow circular cross-section in Table 4.1 isnot in a form based upon Equation 4.4 and is valid for anywall thickness.4.5 Elastic Stresses Due to Bending and Axial LoadIn addition to the torsional stresses, bending and shear stressesand respectively) due to plane bending are normallypresent in the structural member. These stresses are deter-mined by the following equations:normal stress due to bending about either the x or yaxis, ksibending moment about either the x or y axis, kip-in.elastic section modulus, in.3shear stress due to applied shear in either x or ydirection, ksishear in either x or y direction, kipsfor the maximum shear stress in the flangefor the maximum shear stress in the web.moment of inertia or in.4thickness of element, in.Table 4.1Shear Stress Due to St. Venants TorsionSolid Cross-SectionsClosed Cross-Sections© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 15. In the foregoing, it is imperative that the direction of thestresses be carefully observed. The positive direction of thetorsional stresses as used in the sign convention presentedhereinisindicatedinFigures 4.1, 4.2, and4.3. Inthesketchesaccompanying each figure, the stresses are shown acting ona cross-section of the member located at distance z from theleft support and viewed in the direction indicated in Figure4.1. In all of the sketches, the applied torsional moment actsat some arbitrary point along the member in the directionindicated. In the sketches ofFigure 4.8, the moment acts aboutthe major axis of the cross-section and causes compression inthe top flange. The applied shear is assumed to act verticallydownward along the minor axis of the cross-section.For I-shapes, and are both at their maximum valuesat the edges of the flanges as shown in Figures 4.1 and 4.8.Likewise, there are always two flange tips where thesestresses add regardless of the directions of the applied tor-sional moment and bending moment. Also for I-shapes, themaximum values of and in the flanges will alwaysadd at some point regardless of the directions of the appliedtorsional moment and vertical shear to give the maximum1. members for which warping is unrestrained2. single-angle members3. structural tee membersThis stress may be tensile or compressive and is determinedby the following equation:wherenormal stress due to axial load, ksiaxial load, kipsarea, in.24.6 Combining Torsional Stresses With Other Stresses4.6.1 Open Cross-SectionsTo determine the total stress condition, the stresses due totorsion are combined algebraically with all other stressesusing the principles of superposition. The total normal stressis:As previously mentioned, the terms and may be takenas zero in the following cases:and the total shear stress fv, is:(4.7)(4.8a)(4.9a)Figure 4.2. Figure 4.3.14© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 16. shear stress in the flange. For the web, the maximum value ofadds to the value of in the web, regardless ofthe directionofloading, to give the maximum shear stress in the web. Thus,for I-shapes, Equations 4.8a and 4.9a may be simplified asfollows:(4.8b)(4.9b)For channels and Z-shapes, generalized rules cannot be givenfor the determination of the magnitude of the maximumcombined stress. For shapes such as these, it is necessary toconsiderthe directions ofthe applied loading and to checkthecombined stresses at several locations in both the flange andthe web.Determining the maximum values ofthe combined stressesfor all types of shapes is somewhat cumbersome because thestresses and are not all at their maximumvalues at the same transverse cross-section along the lengthof the member. Therefore, in many cases, the stresses shouldbe checked at several locations along the member.4.6.2 Closed Cross-SectionsFor closed cross-sections, stresses due to warping are eithernot induced3or negligible. Torsional loading does, however,cause shear stress, and the total shear stress is:Av= total web area for square and rectangular HSS andhalf the cross-sectional area for round HSS and steelpipe.4.7 Specification Provisions4.7.1 Load and Resistance Factor Design (LRFD)In the following, the subscript u denotes factored loads.LRFD Specification Section H2 provides general criteria formembers subjected to torsion and torsion combined withother forces. Second-order amplification (P-delta) effects, ifany, are presumed to already be included in the elastic analy-sis from which the calculated stressesand were determined.For the limit state of yielding under normal stress:(4.12)For the limit state of yielding under shear stress:(4.13)For the limit state of buckling:(4.14)or(4.15)as appropriate. In the above equations,= yield strength of steel, ksi= critical buckling stress in either compression (LRFD(a) shear stresses due topure torsion(b) in-plane shearstresses due towarping(c) secondary shearstresses due towarpingFigure 4.4. Figure 4.5.3For a circular shape or for a non-circular shape for which warping is unrestrained, warping does not occur, i.e., and are equal to zero.15In the above equation,(4.10)(4.11)where© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 17. Specification Chapter E) or shear (LRFD Specifica-tion Section F2), ksi0.900.85When it is unclear whether the dominant limit state is yield-ing, buckling, or stability, in a member subjected to combinedforces, the aboveprovisions may betoo simplistic. Therefore,the following interaction equations may be useful to conser-vatively combine the above checks of normal stress for thelimit states of yielding (Equation 4.12) and buckling (Equa-tion 4.14). When second order effects, if any, are consideredin the determination of the normal stresses:(4.16a)If second order effects occur but are not considered in deter-mining the normal stresses, the following equation must beused:(4.16b)Figure 4.6.16Figure 4.7.(4.17)(4.18)For the limit state of yielding under shear stress:4.7.2 Allowable Stress Design (ASD)Although not explicitly coveredin the ASD Specification, thedesign for the combination of torsional and other stresses inASD can proceed essentially similarly to that in LRFD,except that service loads are used in place of factored loads.In the absence ofallowable stress provisions forthe design ofmembers subjected to torsion and torsion combined withother forces, the following provisions, which parallel theLRFD Specification provisions above, are recommended.Second-order amplification (P-delta) effects, if any, are pre-sumed to already be included in the elastic analysis fromwhich the calculated stresseswere determined.For the limit state of yielding under normal stress:compressive critical stress forflexural or flexural-tor-sional member buckling from LRFD SpecificationChapter E term), ksi; critical flexural stress con-trolled by yielding, lateral-torsional buckling (LTB),web local buckling (WLB), or flange local buckling(FLB) fromLRFD Specification ChapterF term)factored axial force in the member (kips)elastic (Euler) buckling load.In the above equations,Shear stresses due to combined torsion and flexure may bechecked for the limit state of yielding as in Equation 4.13.Note that a shear buckling limit state for torsion (Equation4.15) has not yet been defined.For single-angle members, see AISC (1993b). A moreadvanced analysis and/or special design precautions are sug-gested for slender open cross-sections subjected to torsion.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 18. For the limit state of buckling:oras appropriate. In the above equations,(4.19)(4.20)yield strength of steel, ksiallowable buckling stress in compression (ASDSpecification Chapter E), ksiallowable bending stress (ASD Specification Chap-ter F), ksiallowable buckling stress in shear (ASD Specifica-tion Section F4), ksiWhen it is unclear whether the dominant limit state is yield-ing, buckling, or stability, in a member subjected to combinedforces, the above provisions may be too simplistic. Therefore,the following interaction equations may be useful to conser-vatively combine the above checks of normal stress for thelimit states of yielding (Equation 4.17) and buckling (Equa-tion 4.19). When second order effects, if any, are consideredin determining the normal stresses:(4.2la)If second order effects occur but are not considered in deter-mining the normal stresses, the following equation must beused:In the above equations,(4.21b)allowable axial stress (ASD Specification ChapterE),ksiallowable bending stress controlled by yielding,lateral-torsionalbuckling (LTB), web localbuck-ling (WLB), or flange local buckling (FLB) fromASD Specification Chapter F, ksiaxial stress in the member, ksielastic (Euler) stress divided by factor of safety (seeASD Specification Section H1).Shear stresses due to combined torsion and flexure may bechecked for the limit state ofyielding as in Equation 4.18. Aswith LRFD Specification provisions, a shear buckling limitstate for torsion has not yet been defined.17Figure 4.8.where is theelasticLTB stress (ksi), whichcanbe derivedfor W-shapes from LRFD Specification Equation Fl-13. Forthe ASD Specification provisions of Section 4.7.2, amplifythe minor-axis bending stress and the warping normalstress by the factorFor single-angle members, see AISC (1989b). A moreadvanced analysis and/or special design precautions are sug-gested for slender open cross-sections subjected to torsion.4.7.3 Effect of Lateral Restraint at Load PointChu and Johnson (1974) showed that for an unbraced beamsubjectedto bothflexure andtorsion, the stress due towarpingis magnified for a W-shape as its lateral-torsional bucklingstrength is approached; this is analogous to beam-columnbehavior. Thus, if lateral displacement or twist is not re-strained at the load point, the secondary effects of lateralbending and warping restraint stresses may become signifi-cant and the following additional requirement is also conser-vatively suggested.For the LRFD Specification provisions of Section 4.7.1,amplify the minor-axis bending stress and the warpingnormal stress by the factor(4.22)© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 19. where is the elastic LTB stress (ksi), given for W-shapes,by the larger of ASD Specification Equations F1-7 and F1-8.4.8 Torsional Serviceability CriteriaIn addition to the strength provisions ofSection 4.7, memberssubjected to torsion must be checked for torsional rotationThe appropriate serviceability limitation varies; the rotationlimit for a member supporting an exterior masonry wall may(4.23)differ from that for a member supporting a curtain-wall sys-tem. Therefore, the rotation limitmustbe selected baseduponthe requirements of the intended application.Whether the design check was determined with factoredloads and LRFD Specification provisions, or service loadsand ASD Specification provisions, the serviceability checkofshould be made at service load levels (i.e., against Unfac-tored torsional moment).The design aids of Appendix B aswell as the general equations in Appendix C are required forthe determination of18© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 20. Chapter 5DESIGN EXAMPLESExample 5.1As illustrated in Figure 5.1a, a W10x49 spans 15 ft (180 in.)and supports a 15-kip factored load (10-kip service load) atmidspan that acts at a 6 in. eccentricity with respect to theshear center. Determine the stresses on the cross-section andthe torsional rotation.Given:The end conditions are assumed to be flexurally and torsion-ally pinned. The eccentric loadcanberesolvedinto atorsionalmoment and a load applied through the shear center as shownin Figure 5.lb. The resulting flexural and torsional loadingsare illustrated in Figure 5.1c. The torsional properties are asfollows:The flexural properties are as follows:Solution:Calculate Bending StressesFor this loading, stresses are constant from the support to theload point.= 12.4 ksi (compression at top; tension at bottom)(4.5)(4.6)(4.6)19Figure 5.1.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 21. Calculate Torsional StressesThe following functions are taken from Appendix B, Case 3,with 0.5:At midspan 0.5)In the above calculations (note that the applied torque isnegative with the sign convention used in this book):The shear stress due to pure torsion is:(4.1)At midspan, since At the support, for the web,and for the flange,20Thus, as illustrated in Figure 5.2, it can be seen that themaximum normal stress occurs at midspan in the flange at theleft side tips of the flanges when viewed toward the leftsupport and the maximum shear stress occurs at the supportin the middle of the flange.Calculate Maximum RotationThe maximum rotation occurs at midspan. The service-loadtorque is:Calculate Combined StressSummarizing stresses due to flexure and torsion:At the support, sinceThe shear stress due to warping is:(4.2a)At midspan,At the support,The normal stress due to warping is:(4.3a)At midspan,© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 22. and the maximum rotation is:Calculate Combined Stress(4.10)Calculate Maximum RotationFrom Example 5.1,Example 5.2Repeat Example 5.1 for a Comparethe magnitudes of the resulting stresses and rotation withthose determined in Example 5.1.Given:Solution:Calculate Bending StressesFrom Example 5.1,(4.5)(4.11)Comparing the magnitudes of the maximum stresses androtation for this HSS with those for the W-shape of Exam-ple 5.1:(4.4)21Figure 5.2.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 23. Solution:In this example, the torsional restraint provided by the rigidconnection joining the beam and column will be utilized.Determine Flexural Stiffness of ColumnThus, the torsional moment on the beam has been reducedfrom 90 kip-in. to 8.1 kip-in. The column must be designedfor an axial load of 15 kips plus an end-moment of81.9kip-in.The beam must be designed for the torsional moment of 8.1kip-in., the 15-kip force from the column axial load, and alateral force Puy due to the horizontal reaction at the bottom ofthe column, whereCalculateBendingStressesFrom Example 5.1,In the weak axis,Figure5.3.22Thus, stresses and rotation are significantly reduced in com-parable closed sections when torsion is a major factor.Example 5.3Repeat Example 5.1 assuming the concentrated force is intro-ducedby aW6x9 column framedrigidly to the W10x49 beamas illustrated in Figure 5.3. Assume the column is 12 ft longwith its top a pinned end and a floor diaphragm provideslateral restraint at the load point. Compare the magnitudes ofthe resulting stresses and rotation with those determined inExamples 5.1 and 5.2.Given:For the W10X49 beam:= 9,910kip-in./rad.Determine Torsional Stiffness ofBeamFrom Example 5.1,For the W6x9 column:Determine Distribution ofMoment© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 24. that used in Example 5.1, the maximum rotation, whichoccurs at midspan, is also reduced to 9 percent of that calcu-lated in Example 5.1 or:CommentsComparing the magnitudes ofthe stresses androtation forthiscase with that of Example 5.1:(4.5)(4.2b)W10x49unrestrained40.4 ksi11.4ksi0.062 rad.Calculate Torsional StressSince the torsional moment has been reduced to 9 percent ofthat used in Example 5.1, the torsional stresses are alsoreducedto9percentofthose calculatedinExample5.1. Thesestresses are summarized below.Calculate Combined StressSummarizing stresses due to flexure and torsion23As before, the maximum normal stress occurs at midspan inthe flange. In this case, however, the maximum shear stressoccurs at the support in the web.Calculate Maximum RotationSince the torsional moment has been reduced to 9 percent of Figure 5.4.W10x49restrained16.3ksi3.00 ksi0.0056 rad.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 25. Thus, consideration of available torsional restraint signifi-cantly reduces the torsional stresses and rotation.Example 5.4The welded plate-girder shownin Figure 5.4a spans 25 ft (300in.) and supports 310-kip and 420-kip factored loads (210-kipand285-kip serviceloads). As illustratedinFigure5.4b,theseconcentrated loads are acting at a 3-in. eccentricity withrespect to the shear center. Determine the stresses on thecross-section and the torsional rotation.Given:The end conditions are assumed to be flexurally and torsion-ally pinned.Solution:Calculate Cross-Sectional Properties(3.4)Calculate Torsional Properties(3.11)24= 26.6 ksi (compression at top; tension at bottom)At pointCalculate Bending StressesBy inspection, points D and E are most critical. At point D:(4.5)(3.5)(3.6)(3.7)(3.8)(3.9)(3.10)© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 26. Between points D and E:For this loading, shear stresses are constant from point D topoint E.Calculate Torsional StressesThe effect ofeach torque at points D and E will be determinedindividually and then combined by superposition.Use Case 3with 0.3 (The effects ofeach load are addedby superposition).The shear stress due to pure torsion is calculated as:and the stresses are as follows:The shear stress due to warping is calculated as:At point D,At point E,The normal stress due to warping is calculated as:(4.2a)(4.3a)25(4.1)(4.6)(4.6)At point E© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 27. At point D,Calculate Combined StressSummarizing stresses due to flexure and torsion:Thus, it can be seen that the maximum normal stress occursat point D in the flange and the maximum shear stress occursat point E (the support) in the web.Calculate Maximum RotationFrom Appendix B, Case 3 with a = 0.3, it is estimated thatthe maximum rotation will occur at approximately 14½ feetExample 5.5The MCl8x42.7 channel illustrated in Figure 5.5a spans 12ft (144 in.) and supports a uniformly distributed factored loadof 3.6 kips/ft (2.4 kips/ft service load) acting through thecentroid of the channel. Determine the stresses on the cross-section and the torsional rotation,Given:The end conditions are assumed to be flexurally and torsion-ally fixed. The eccentric load can be resolved into a torsionalmoment and a load applied through the shear center as shownin Figure 5.5b. The resulting flexural and torsional loadingsare illustrated in Figure 5.5c. The torsional properties are asfollows:from the left end of the beam (point A). At this location,The service-load torques are:The maximum rotation is:Figure 5.5.26© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 28. Solution:From the graphs forCase 7 in Appendix B, the extreme valuesof the torsional functions are located at values of 0, 0.2,0.5, and 1.0. Thus, the stresses at the supports andandmidspan are of interest.CalculateBendingStressesAt the support:(4.5)(4.6)(4.6)(4.5)27(4.6)(4.6)Since is maximum at the support, its value at aswell as the value of is not necessary.Calculate Torsional Stresses= (3.6 kips/ft)(1.85in.)= 6.66 kip-in./ftThe following functions are taken from Appendix B, Case 7:© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 29. At midspanIn the above calculations:The shear stress due to pure torsion is:At the support, and atmidspan, sincefor the web,and for the flange,The shear stress at point s due to warping is:(Refer to Figure 5.5d or Appendix A for locations of criticalpoints s)At midspan, since At the support,At midspan,At the support,The normal stress at point s due to warping is:28© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 30. Calculate Combined StressSummarizing stresses due to flexure and torsion:= 0.012 rad.Example 5.6As illustrated in Figure 5.6a, a 3x3x½ single angle is cantile-vered 2 ft (24 in.) and supports a 2-kip factored load (1.33-kipservice load) at midspan that acts as shown with a 1.5-in.eccentricity with respect to the shear center. Determine thestresses on the cross-section, the torsional rotation, and ifthemember is adequate ifFy= 50 ksi.Given:The end condition is assumed to be flexurally and torsionallyfixed. The eccentric load can be resolved into a torsionalmoment and a load applied through the shear center as shownin Figure 5.6b. The resulting flexural and torsional loadingsare illustrated in Figure 5.6c. The flexural and torsionalproperties are as follows:Solution:Check FlexureSince the stresses due to warping ofsingle-angle members arenegligible, the flexural design strength will be checked ac-cording to the provisions of the AISC Specificationfor LRFDofSingle Angle Members (AISC, 1993b).Thus, it can be seen that the maximum normal stress (tension)occurs at the support at point 2 in the the flange and themaximum shear stress occurs at at point 3 in theweb.Calculate Maximum RotationThe maximum rotation occurs at midspan. The service-loaddistributed torque is:and the maximum rotation is:Figure 5.6.29© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 31. With the tip of the vertical angle leg in compression, localbuckling and lateral torsional buckling must be checked. Thefollowing checks are made for bending about the geometricaxes (Section 5.2.2).For local buckling (Section 5.1.1),Check Shear Due to Flexure and TorsionThe shear stress due to flexure is,Example 5.7The crane girder and loading illustrated in Figure 5.7 is takenfrom Example 18.1 of the AISC Design Guide Industrial30Calculate Maximum RotationThe maximum rotation will occur at the free end of thecantilever. The service-load torque is:From LRFD Single-Angle Specification Section 3,The total shear stress is,© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 32. Buildings: Roofs to Column Anchorage (Fisher, 1993). Usethe approximate approach of Section 4.1.4 to calculate themaximum normal stress on the combined section. Determineif the member is adequate if Fy= 36 ksi.Given:For the strong-axis direction:Note that the subscripts 1 and 2 indicate that the sectionmodulus is calculated relative to the bottom and top, respec-tively, of the combined shape. For the channel/top flangeassembly:Note that the subscript t indicates that the section modulus iscalculated based upon the properties of the channel and topflange area only.Solution:Calculate Normal Stress Due to Strong-Axis BendingCommentsSince it is common practice in crane-girder design to assumethat the lateral loads are resisted only by the top flangeassembly, the approximate solution of Section 4.1.4 is ex-tremely useful for this case.Calculate Normal Stress due to WarpingFrom Section 4.1.4, the normal stress due to warping may beapproximated as:Calculate Total Normal Stress38.9 kip-ft (weak-axis bending moment on topflange assembly)50.3in.3(4.5)(4.5)(4.3b)(4.8a)31© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 33. AppendixATORSIONAL PROPERTIES33W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 34. 34W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 35. 35W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 36. 36W-, M-, S-, and HP-ShapesTorsional Properties Statical MomentsShape© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 37. 37W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 38. 38W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 39. 39W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 40. 40W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 41. 41W-, M-, S-, and HP-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 42. 42C- and MC-ShapesTorsional PropertiesShapeStatical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 43. 43C- and MC-ShapesShapeTorsional Properties Statical Moments© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 44. 44WT-, MT-, and ST-Shapes WT-, MT-, and ST-ShapesShapeTorsional PropertiesShapeTorsional Properties© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 45. 45WT-, MT-, and ST-Shapes WT-, MT-, and ST-ShapesShapeTorsional PropertiesShapeTorsional Properties© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 46. 46WT-, MT-, and ST-Shapes WT-, MT-, and ST-ShapesShapeTorsional PropertiesShapeTorsional Properties© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 47. 47WT-, MT-, and ST-ShapesTorsional PropertiesShape© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 48. 48Single Angles Single AnglesShapeTorsional PropertiesShapeTorsional Properties© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 49. 49Single AnglesShapeTorsional Properties© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 50. 50Square HSS Square HSS Rectangular HSSShape Shape Shape© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 51. 51Rectangular HSS Rectangular HSS Rectangular HSSShape Shape Shape© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 52. 52Rectangular HSSShapeSteel PipeNominalDiameter© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 53. Z-shapes53Torsional Properties Statical MomentsShape© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 54. AppendixBCASE CHARTS OFTORSIONALFUNCTIONSDetermine the case based upon the end conditions and typeof loading. For the given cross-section and span length, com-pute the value of using in inches and a as given inAppendix A. With this value, enter the appropriate chart fromthe loading. At the desired location along the horizontal scale,read vertically upward to the appropriate curve and readfrom the vertical scale the value ofthe torsional function. (Forvalues of between curves, use linear interpolation.) Thisvalue should then be divided by the factor indicated by thegroup of terms shown in the labels on the curves to obtain thevalue of This result may then be used inEquation 4.1, 4.2, or 4.3 to determineSign ConventionIn all cases, the torsional moment is shown acting in acounter-clockwise direction when viewed toward the left endofthe member. This is considered to be a positive moment inthis book. Thus, if the applied torsional moment is in theopposite direction, it should be assigned a negative value forcomputational purposes. A positive stress or rotation com-puted with the equations and torsional constants of this bookacts in the direction shown in the cross-sectional views shownin Figures 4.1, 4.2, and 4.3. A negative value indicates thedirection is opposite to that shown.On some ofthe Case Charts, the applied torsional momentis indicated by a vector notation using a line with two arrow-heads. This notation indicates the "right-hand rule," whereinthe thumb of the right hand is extended and pointed in thedirection of the vector, and the remaining fingers of the righthand curve in the direction of the moment.In Figures 4.2 and 4.3, the positive direction ofthe stressesis shown for channels and Z-shapes oriented such that the topflange extended to the left when viewed toward the left endof the member. For the reverse orientation of these memberswith the applied torque remaining in a counterclockwisedirection, the following applies:the positive direction of the stresses in the flanges isthe same as shown in Figures 4.2b and 4.3b. Forexample, at the top edge of the top flange, a positivestress acts from left to right. A positive stress at theleft edge of the web acts downward; a positive stressat the right edge of the web acts upward.the positive direction ofthe stresses at the correspond-ing points on the reversed section are opposite thoseshown in Figures 4.2d and 4.3d. For example, apositive stress at the tips of the flanges, point o, is atensile stress.54© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 55. 55Fraction of Span LengthFraction of Span LengthCase1Case1© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 56. 56Case 2Case 2Fraction of Span LengthFraction of Span Length© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 57. Case 2Fraction of Span LengthCase 2Fraction of Span Length57© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 58. 58Case 3Case 3 Rev.3/1/03Rev.3/1/03© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.Pinned PinnedConcentrated torque at= 0.1 on member withpinned ends.αPinned PinnedConcentrated torque at= 0.1 on member withpinned ends.α
- 59. 59Case3Case3Rev.3/1/03Rev.3/1/03© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.Pinned PinnedConcentrated torque at= 0.1 on member withpinned ends.αPinned PinnedConcentrated torque at= 0.1 on member withpinned ends.α
- 60. 60Case 3Case3Rev.3/1/03© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.0.0250.050.0750.10.1250.15
- 61. Case3Case 3Fraction of Span LengthFraction of Span Length61© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 62. Case3Case3Fraction of Span LengthFraction of Span Length62© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 63. Case3Case3Fraction of Span LengthFraction of Span Length63© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 64. Case4Case4Fraction of Span LengthFraction of Span Length64© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 65. Case4Case 4Fraction of Span LengthFraction of Span Length65© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 66. Case5Case5Fraction of Span LengthFraction of Span Length66© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 67. Case5Case5Fraction of Span LengthFraction of Span Length© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 68. Case6Case6FractionofSpanLengthFraction of Span Length68© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 69. Case 669Fraction of Span LengthCase 6Fraction of Span Length© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 70. Case 6Case 6Fraction of Span LengthFraction of Span Length70© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 71. Case 6Case6Fraction of Span LengthFraction of Span Length71© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 72. Case 6Case6Fraction of Span LengthFraction of Span Length72© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 73. Case 6Case 6Fraction of Span LengthFraction of Span Length73© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 74. Case7Case7Fraction of Span LengthFraction of Span Length74© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 75. Case7Case7Fraction of Span LengthFraction of Span Length75© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 76. Case8Case8Fraction of Span LengthFraction of Span Length76© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 77. Case8Case8Fraction of Span LengthFraction of Span Length77© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 78. Case 9Case 9Fraction of Span LengthFraction of Span Length78© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 79. Case 9Case9Fraction of Span LengthFraction of Span Length79© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 80. Case 9Case 9Fraction of Span LengthFraction of Span Length80© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 81. Case 9Case 9Fraction of Span LengthFraction of Span Length81© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 82. Case 9Case 9Fraction of Span LengthFraction of Span Length82© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 83. Case 9Case 9Fraction of Span LengthFraction of Span Length83© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 84. Case 9Case 9Fraction of Span LengthFraction of Span Length84© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 85. Case 9Case 9Fraction of Span LengthFraction of Span Length85© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 86. Case 9Case 9Fraction of Span LengthFraction of Span Length86© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 87. Case 9Fraction of Span LengthFraction of Span LengthCase 987© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 88. Case 9Case 9Fraction of Span LengthFraction of Span Length88© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 89. Case 9Case 9Fraction of Span LengthFraction of Span Length89© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 90. Case 10Case 10Fraction of Span LengthFraction of Span Length90© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 91. Case 10Case 10Fraction of Span LengthFractionofSpanLength91© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 92. Case 10Case 10Fraction of Span LengthFraction of Span Length92© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 93. Case 10Case 10Fraction of Span LengthFraction of Span Length93© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 94. Case 10Case 10Fraction of Span LengthFraction of Span Length94© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 95. Case 10Case 10Fraction of Span LengthFraction of Span Length© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 96. Case 10Case 10Fraction of Span LengthFraction of Span Length96© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 97. Case 10Case 10Fraction of Span LengthFraction of Span Length97© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 98. Case 10Case 10Fraction of Span LengthFraction of Span Length98© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 99. Case 10Case 10Fraction of Span LengthFraction of Span Length99© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 100. Case 10Case 10Fraction of Span LengthFraction of Span Length© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 101. Case 10Case 10Fraction of Span LengthFraction of Span Length101© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 102. Case11Case11Fraction of Span LengthFraction of Span Length102© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 103. Case11Case11Fraction of Span LengthFraction of Span Length103© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 104. Case 12Case 12Fraction of Span LengthFraction of Span Length104© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 105. Case 12Case 12Fraction of Span LengthFraction of Span Length105© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 106. Appendix CSUPPORTING INFORMATIONC.1 General Equations for and its DerivativesFollowing are general equations for for a constant torsionalmoment, uniformly distributed torsional moment, and lin-early varying torsional moment. They are developed fromEquation 2.4 in Section 2.2.C.1.1 Constant Torsional MomentFor a constant torsional moment T along a portion of themember as illustrated in Figure C.1a, the following equationmay be developed:wherez = distance along Z-axis from left support, in.A, B, C = constants that are determined from boundaryconditionsC. 1.2 Uniformly Distributed Torsional MomentA member subjected to a uniformly distributed torsionalmoment t is illustrated in Figure C.lb. For this case, examinea small segment of the member of length dz. By summationof torsional moments:C.2. Boundary ConditionsThegeneralequations contain constants thatareevaluatedforspecific cases by imposing the appropriate boundary condi-tions. These conditions specify mathematically the physicalrestraints at the ends of the member. They are summarized asfollows:Differentiating Equation 2.4 and substituting the aboveyields:C.I.3 Linearly Varying Torsional MomentFor a member subjected to a linearly varying torsional mo-ment as illustrated in Figure C.lc, again examine a smallelement of the member length dz. Summing the torsionalmoments:which may be solved as:Differentiating Equation 2.4 and substituting the aboveyields:orFigure C.I.107(C.2)(C.3)(C.4)(C.5)or(C.6)(C.7)(C.8)(C.9)which may be solved as:© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 107. In all solutions inAppendix C, ideal end conditions have beenassumed, i.e., free, fixed, or pinned. Where these conditionsdo not apply, a more advanced analysis may be necessary.A torsionally fixed end (full warping restraint) is moredifficult to achieve than a flexurally fixed end. If the span isone ofseveral for a continuous beam, with each span similarlyloaded, there is inherent fixity for flexure and flange warping.If however, the beam is an isolated span, Ojalvo (1975)demonstrated that a closed box made up of several plates ora channel, as illustrated in Figure C.2, would approximate atorsionally fixed end. Simply welding to an end-plate orcolumn flange may not provide sufficient restraint.C.3. Evaluation of Torsional PropertiesFollowing are the general solutions for the torsional proper-ties given in Chapter 3 and more accurate equations for thetorsional constant J for I-shapes, channels, and Z-shapes.C.3.2 Torsional Constant Jfor Open Cross-SectionsThe following equations for J provide a more accurate valuethan the simple approximation given previously. For I-shapeswith parallel-sided flanges as illustrated in Figure C.4a:C.3.1 General SolutionReferring to the general cross-section and notation in FigureC.3, the torsional properties may be expressed as follows:Additionally, the following conditions must be satisfied at asupport over which the member is continuous or at a point ofapplied torsional moment:wherePhysicalConditionNo rotationCross-section cannotwarpCross-section canwarp freelyTorsional EndConditionFixed or PinnedFixed endPinned or FreeMathematicalCondition(C.10)(C.11)(C.12)(C.13)(C.14)(C.15)(C.16)(C.17)(C.18)Figure C.2. Figure C.3.108© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 108. whereFor I-shapes with sloping-sided flanges as illustrated in Fig-ure C.4b:where for I-shapes with flange slopes of percent only:For flange slopes other than percent, the value of maybe found by linear interpolation between and as givenabove for parallel-sided flanges:For flange slopes otherthan percent, the value of maybe found by linear interpolation between as given aboveand as given below for parallel-sided flanges:(C.19)(C.20)(C.21)(C.22)(C.23)(C.24)For Z-shapes with parallel-sided flanges as illustrated inFigure C.4d:(C.29)(C.30)(C.31)(C.32)(C.33)(C.34)(C.35)Figure C.4.(C.25)(C.26)(C.27)(C.28)where for channels with flange slopes of percent only:For channels as illustrated in Figure C.4c:109© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 109. whereC.4 Solutions to Differential Equations for Cases inAppendix BFollowing are the solutions ofthe differential equations usingthe proper boundary conditions. Take derivatives of to find(C.36)110(C.37)© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 110. where:111where:© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 111. 112where:© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 112. REFERENCESAASHTO, 1993, Guide Specifications for HorizontallyCurved Highway Bridges, AASHTO, Washington, D.C.AISC, 1994, LRFD Manual of Steel Construction, 2nd Ed.,AISC, Chicago, IL.AISC, 1993a, Load and Resistance Factor Design Specifica-tion for Structural Steel Buildings, AISC, Chicago, IL.AISC, 1993b, Specification for Load and Resistance FactorDesign ofSingle-Angle Members, AISC, Chicago, IL.AISC, 1989a, Specification for Structural Steel Buildings,Allowable Stress Design, AISC, Chicago, IL.AISC, 1989b, Specification for Allowable Stress Design ofSingle-Angle Members, AISC, Chicago, IL.AISC, 1986, "Horizontally Curved Girders," Highway Struc-tures Design Handbook, Volume II, Chapter 6, AISC, Chi-cago, IL.Bleich, F, 1952, Buckling Strength of Metal Structures,McGraw-Hill, Inc., New York, NY.Boothby, T. E., 1984, "The Application of Flexural Methodsto Torsional Analysis ofThin-Walled Open Sections," En-gineering Journal, Vol. 21, No. 4, (4th Qtr.), pp. 189-198,AISC, Chicago, IL.Chu, K. H. andJohnson, R. B., 1974, "Torsion in Beams withOpen Sections," Journal of the Structural Division, Vol.100, No. ST7 (July), p. 1397, ASCE, New York, NY.El Darwish, I. A. and Johnston, B. G., 1965, "Torsion ofStructural Shapes," Journal ofthe StructuralDivision, Vol.91, No. ST1 (Feb.), ASCE, New York, NY.Evick, D. R. andHeins, C. P., 1972, "TorsionofNonprismaticBeams of Open Section," Journal of the Structural Divi-sion, Vol. 98, No. ST12 (Dec.), ASCE, New York, NY.Galambos, T. V, 1988, Guide to Stability Design CriteriaforMetal Structures, 4th Ed., John Wiley & Sons, Inc., NewYork, NY.Gjelsvik, A., 1981, The Theory of Thin-Walled Bars, JohnWiley & Sons, New York, NY.Heins, C. R, 1975, Bending and Torsional Design in Struc-tural Members, Lexington Book Co., Lexington, MA.Heins, C. P. and Firmage, D. A., 1979, Design of Modern SteelHighwayBridges, John Wiley Interscience, New York, NY.Heins, C. P. and Kuo, J. T., 1972, "Composite Beams inTorsion," Journal ofthe Structural Division, Vol. 98, No.ST5 (May), ASCE, New York, NY.Heins, C. P. and Potocko, R. A., 1979, "Torsional Stiffness ofI-Girder Webs," Journal of the Structural Division, Vol.105, No. ST8 (Aug.), ASCE, New York, NY.Heins, C. P. and Seaburg, P. A., 1963, Torsion Analysis ofRolledSteelSections, Bethlehem Steel Company, Bethlehem,PA.Hotchkiss, J. G., 1966, "Torsion of Rolled Steel Sections inBuilding Structures," Engineering Journal, Vol. 3, No. 1,(Jan.), pp. 19-45, AISC, Chicago, IL.Johnston, B. G., 1982, "Design of W-Shapes for CombinedBending and Torsion," Engineering Journal, Vol. 19, No.2. (2nd Qtr.), pp. 65-85, AISC, Chicago, IL.Johnston, B. G., Lin, F. J., and Galambos, T. V, 1980, BasicSteel Design, 2nd Ed., Prentice-Hall, Englewood Cliffs,NJ.Liew, J. Y. R., Thevendran, V, Shanmugam, N. E., and Tan,L. O., 1995, "Behaviour and Design of HorizontallyCurved Steel Beams," Journal of Constructional SteelResearch, Vol. 32, No. 1, pp. 37-67, Elsevier AppliedScience, Oxford, United Kingdom.Lin, PH., 1977, "Simplified Design for Torsional Loading ofRolled Steel Members," EngineeringJournal, Vol. 14, No.3. (3rd Qtr.), pp. 98-107, AISC, Chicago, IL.Lyse, I. and Johnston, B. G., 1936, "Structural Beams inTorsion," ASCE Transactions, Vol. 101, ASCE, New York,NY.Nakai, H. and Heins, C. P., 1977, "Analysis Criteria forCurved Bridges," Journal of the Structural Division, Vol.103, No. ST7 (July), ASCE, New York, NY.Nakai, H. and Yoo, C. H., 1988, Analysis and Design ofCurved Steel Bridges, McGraw-Hill, Inc., New York, NY.Ojalvo, M., 1975, "Warping and Distortion at I-SectionJoints"—Discussion, Journal of the Structural Division,Vol. 101, No. ST1 (Jan.), ASCE, New York, NY.Ojalvo, M. and Chamber, R., 1977, "Effect of Warping Re-straints on I-Beam Buckling," Journal of the StructuralDivision,Vol. 103,No.ST12(Dec.),ASCE,NewYork,NY.Salmon, C. G. and Johnson, J. E., 1990, Steel StructuresDesign and Behavior, 3rd Ed., Harper Collins Publishers,New York, NY.Siev, A., 1966, Torsion in Closed Sections," Engineering Jour-nal, Vol. 3, No. 1, (January),pp.46-54, AISC,Chicago, IL.Tide, R. H. R. andKrogstad, N. V, 1993, "EconomicalDesign113© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 113. 114of Shelf Angles," Proceedings of the Symposium on Ma-sonry: Design and Construction, Problems and Repair,STP 1180, American Society for Testing and Materials,Philadelphia,PA.Timoshenko, S., 1945, "Theory of Bending, Torsion andBuckling of Thin-Walled Members of Open Cross-Sec-tion," Journal ofthe Franklin Institute, March/April/May,Philadelphia,PA.Tung, D. H. H. and Fountain, R. S., 1970, "ApproximateTorsional Analysis of Curved Box Girders by the M/RMethod," Engineering Journal, Vol. 7, No. 3, (July), pp.65-74,AISC,Chicago,EL.Vlasov, V. Z., 1961, Thin-Walled Elastic Beams, NationalScience Foundation, Wash., DC.Young, W. C, 1989, Roarks Formulas for Stress & Strain,McGraw-Hill, Inc., New York, NY.© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 114. NOMENCLATUREA area, in.2; also, constant determined from boundaryconditions (see Equations C.1, C.5, C.9, and C.14)B constant determined from boundary conditions (seeEquations C.1, C.5, C.9, and C.14)C constant determined from boundary conditions (seeEquations C.1, C.5, C.9, and C.14)Cwwarping constant of cross-section, in.6D1variableinEquationC.26,in.D2variable in Equation C.29, in.D3variable in Equation C.42, in.D4variable in Equation C.36, in.E modulus of elasticity of steel, 29,000 ksiE0 horizontal distance from Centerline of channel web toshear center, in.F variable in Equation C.30, in.Faallowable axial stress (ASD), ksiFb allowable bending stress (ASD), ksiFe Euler stress divided by factor of safety (see ASDSpecification Section H1), ksiFv allowable shear stress (ASD), ksiFy yield strength of steel, ksiFcr critical buckling stress, ksiG shear modulus of elasticity of steel, 11,200 ksiH variable in Equation 3.37Imomentofinertia,in.4J torsional constant of cross-section, in.4M bending moment, kip-in.P concentrated force, kipsQ statical moment about the neutral axis of the entirecross-section of the cross-sectional area between thefree edges ofthe cross-section and a plane cutting thecross-section across the minimum thickness at thepoint under examination, in.3Qf value of Q for a point in the flange directly above thevertical edge of the web, in.3Qs value of Q for a point at mid-depth of the cross-sec-tion, in.3R fillet radius, in.S elastic section modulus, in.3; also, variable used incalculation of torsional properties (see Equation 3.31or C.38)Swswarping statical moment at point s on cross-section,in.4T applied concentrated torsional moment, kip-in.Tt resisting moment due to pure torsion, kip-in.Tw resisting moment due to warping torsion, kip-in.V shear, kipsVsvariable in Equation C.32 or C.39, in.Wns normalized warping function at point s on cross-sec-tion, in.2a torsional constant as defined in Equation 3.6b width of cross-sectional element, in.b variable in Equation 3.21 or 3.30, in.bfflange width, in.dT incremental torque corresponding to incrementallength dz, kip-in.dz incremental length along Z-axis, in.e eccentricity, in.e0horizontal distance from outside of web of channel toshear center, in.faaxial stress under service load, ksifn totalnormalstressduetotorsionandallothercauses,ksifv total shear stress due to torsion and all other causes,ksih depth center-to-center of flanges for I-, C-, and Z-shaped members, in.depth minus half flange thickness for structural tee,in.leg width minus half leg thickness for single angles,in.k torsional stiffness from Equation 2.1l spanlength,in.m thickness of sloping flange at beam Centerline (seeFigure C.4b), in.s subscript relative to point 0, 1, 2,... on cross-sectiont distributed torque, kip-in, per in.; also, thickness ofcross-sectional element, in.tfflange thickness, in.tw web thickness, in.tl thickness of sloping flange at toe (see Figure C.4b),in.t2 thickness of sloping flange, ignoring fillet, at face ofweb (see Figure C.4b), in.u subscript denoting factored loads (LRFD); also, vari-able in Equation 3.22 or 3.31, in.u variable in Equation 3.32, in.x subscript relating to strong axisy subscript relating to weak axisz distance along Z-axis ofmember from left support, in.angle of rotation, radiansfirst derivative of with respect to zsecond derivative of with respect to zthird derivative of with respect to z115© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 115. 116fourth derivative of with respect to zdistance from support to point of applied torsionalmoment or to end of uniformly distributed torsionalload over a portion of span, divided by span length lvariable in Equation C.19, in.variable in Equation C.22, in.variable in Equation C.35, in.variable in Equation C.28, in.variable in Equation C.30, in.0.90, resistance factor for yielding (LRFD)0.85, resistance factor for buckling (LRFD)perpendicular distance to tangent line from centroid(see Figure C.3), in.perpendicular distance to tangent line from shearcen-ter (see Figure C.3), in.normal stress due to axial load, ksinormal stress due to bending, ksinormal stress at point s due to warping torsion, ksishear stress, ksishear stress at element edge due to pure torsion, ksishear stress at point s due to warping torsion, ksi© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.
- 116. DESIGN GUIDE SERIESAmerican Institute of Steel Construction, Inc.One East Wacker Drive, Suite 3100Chicago, Illinois 60601-2001Pub. No. D809 (5M297)© 2003 by American Institute of Steel Construction, Inc. All rights reserved.This publication or any part thereof must not be reproduced in any form without permission of the publisher.

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