LATERAL-TORSIONAL BUCKLING OF STRUCTURES WITH MONOSYMMETRIC CROSS-SECTIONS. Matthew J. Vensko. B.S. Pennsylvania State University, 2003

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1 ATERA-TORSIONA BUCKING OF STRUCTURES WITH MONOSYMMETRIC CROSS-SECTIONS b Matthew J. Vensko B.S. Pennslvania State Universit, Submitted to the Graduate Facult of the Swanson School of Engineering in partial fulfillment of the requirements for the degree of Master of Science Universit of Pittsburgh 8

2 UNIVERSITY OF PITTSBURGH SWANSON SCHOO OF ENGINEERING This thesis was presented b Matthew J. Vensko It was defended on November, 8 and approved b Kent A. Harries, Assistant Professor, Department of Civil and Environmental Engineering Albert To, Assistant Professor, Department of Civil and Environmental Engineering Mortea A. M. Torkamani, Associate Professor, Department of Civil and Environmental Engineering, Thesis Advisor ii

3 ATERA-TORSIONA BUCKING OF STRUCTURES WITH MONOSYMMETRIC CROSS-SECTIONS Matthew J. Vensko, M.S. Universit of Pittsburgh, 8 ateral-torsional buckling is a method of failure that occurs when the in-plane bending capacit of a member eceeds its resistance to out-of-plane lateral buckling and twisting. The lateraltorsional buckling of beam-columns with doubl-smmetric cross-sections is a topic that has been long discussed and well covered. The buckling of members with monosmmetric crosssections is an underdeveloped topic, with its derivations complicated b the fact that the centroid and the shear center of the cross-section do not coincide. In this paper, the total potential energ equation of a beam-column element with a monosmmetric cross-section will be derived to predict the lateral-torsional buckling load. The total potential energ equation is the sum of the strain energ and the potential energ of the eternal loads. The theorem of minimum total potential energ eerts that setting the second variation of this equation equal to ero will represent a transition from a stable to an unstable state. The buckling loads can then be identified when this transition takes place. This thesis will derive energ equations in both dimensional and non-dimensional forms assuming iii

4 that the beam-column is without prebuckling deformations. This dimensional buckling equation will then be epanded to include prebuckling deformations. The abilit of these equations to predict the lateral-torsional buckling loads of a structure is demonstrated for different loading and boundar conditions. The accurac of these predictions is dependent on the abilit to select a suitable shape function to mimic the buckled shape of the beam-column. The results provided b the buckling equations derived in this thesis, using a suitable shape function, are compared to eamples in eisting literature considering the same boundar and loading conditions. The finite element method is then used, along with the energ equations, to derive element elastic and geometric stiffness matrices. These element stiffness matrices can be transformed into global stiffness matrices. Boundar conditions can then be enforced and a generalied eigenvalue problem can then be used to determine the buckling loads. The element elastic and geometric stiffness matrices are presented in this thesis so that future research can appl them to a computer software program to predict lateral-torsional buckling loads of comple sstems containing members with monosmmetric cross-sections. iv

5 TABE OF CONTENTS. INTRODUCTION.... ITERATURE REVIEW...4. EQUIIBRIUM METHOD Closed Form Solutions...7. ENERGY METHOD Uniform Torsion Non-Uniform Torsion..... Strain Energ Solutions Using Buckling Shapes.... ATERA-TORSIONA BUCKING OF BEAM-COUMNS...8. STRAIN ENERGY..... Displacements Strains Stresses and Stress Resultants Section Properties STRAIN ENERGY EQUATION FOR MONOSYMMETRIC BEAMCOUMNS.45. POTENTIA ENERGY OF THE EXTERNA OADS Displacements and Rotations of oad Points...47 v

6 .4 ENERGY EQUATION FOR ATERA TORSIONA BUCKING NON-DIMENSIONA ENERGY EQUATION ATERA-TORSIONA BUCKING OF MONOSYMMETRIC BEAMS CONSIDERING PREBUCKING DEFECTIONS STRAIN ENERGY CONSIDERING PREBUCKING DEFECTIONS Displacements ongitudinal Strain Shear Strain STRAIN ENERGY EQUATION CONSIDERING PREBUCKING DEFECTIONS POTENTIA ENERGY OF THE EXTERNA OADS CONSIDERING PREBUCKING DEFECTIONS Displacements and Rotations of oad Points ENERGY EQUATION CONSIDERING PREBUCKING DEFECTION APPICATIONS SIMPY-SUPPORTED MONOSYMMETRIC BEAM SUBJECTED TO EQUA END MOMENTS, M SIMPY-SUPPORTED MONOSYMMETRIC BEAM SUBJECTED TO CONCENTRATED CENTRA OAD, P SIMPY-SUPPORTED MONOSYMMETRIC BEAM SUBJECTED TO UNIFORMY DISTRIBUTED OAD, q CANTIEVER WITH END POINT OAD, P FINITE EEMENT METHOD EASTIC STIFFNESS MATRIX GEOMETRIC STIFFNESS MATRIX...6 vi

7 6. FINITE EEMENT METHOD CONSIDERING PREBUCKING DEFECTIONS EASTIC STIFFNESS MATRIX CONSIDERING PREBUCKING DEFECTIONS GEOMETRIC STIFFNESS MATRIX CONSIDERING PREBUCKING DEFECTIONS SUMMARY...5 APPENDIX A...8 A. EEMENT EASTIC STIFFNESS MATRIX...8 A. EEMENT GEOMETRIC STIFFNESS MATRIX... A. EEMENT NON-DIMENSIONA STIFFNESS MATRIX...6 A.4 EEMENT NON-DIMENSIONA GEOMETRIC STIFFNESS MATRIX...8 A.5 EEMENT PREBUCKING STIFFNESS MATRIX...4 A.6 EEMENT PREBUCKING GEOMETRIC STIFFNESS MATRIX...6 APPENDIX B...4 B. MATRIX [A] FROM SECTION B. MATRIX [B] FROM SECTON BIBIOGRAPHY...56 WORKS CITED...56 WORKS CONSUTED...58 vii

8 IST OF FIGURES Figure.a Beams of Rectangular Cross-Section...6 Figure.b Beams of Rectangular Cross-Section with Aial Force and End Moments...6 Figure. inearl Tapered Beam Subjected to Equal and Opposite End Moments...9 Figure. Simpl-Supported Beam with Concentrated oad, P, at Midspan... Figure.4a Monosmmetric Beam Subjected to End Moments and Aial oad...4 Figure.4b Cross-Section of Monosmmetric Beam...4 Figure.5 Twisting of Rectangular Beam Free to Warp...9 Figure.6 I-Beam Subjected Fied at Both Ends to End Moments...6 Figure. Coordinate Sstem of Undeformed Monosmmetric Beam...9 Figure. Eternal oads and Member End Actions of Beam Element... Figure. Cross-Section of Monosmmetric I-Beam... Figure.4 Deformed Beam...4 Figure.5 Translation of Point P o to Point P...5 viii

9 Figure 5. Monosmmetric Beam with Subjected to Equal End Moments...65 Figure 5. Monosmmetric Beam with Subjected to Concentrated Central oad...7 Figure 5. Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β 6. )...75 Figure 5.4 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β. )...76 Figure 5.5 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β. )...76 Figure 5.6 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β )...77 Figure 5.7 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β. )...77 Figure 5.8 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β. )...78 Figure 5.9 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Concentrated Central oad ( β 6. )...78 Figure 5. Monosmmetric Beam Subjected to a Uniforml Distributed oad...79 Figure 5. Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β 6. )...84 i

10 Figure 5. Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β. )...84 Figure 5. Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β. )...85 Figure 5.4 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β )...85 Figure 5.5 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β. )...86 Figure 5.6 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β. )...86 Figure 5.7 Buckling oad: Simpl Supported Beam with Monosmmetric Cross-Section Subjected to a Uniforml Distributed oad ( β 6. )...87 Figure 5.8 Monosmmetric Cantilever Beam Subjected to a Concentrated End oad...88 Figure 5.9 Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad( β 6. )...94 Figure 5. Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad ( β. )...95 Figure 5. Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad ( β. )...95

11 Figure 5. Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad ( β )...96 Figure 5. Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad ( β. )...96 Figure 5.4 Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad ( β. )...97 Figure 5.5 Buckling oad: Cantilever Beam with Monosmmetric Cross-Section Subjected to a Concentrated End oad ( β 6. )...97 Figure 6. Element Degrees of Freedom with Nodal Displacements u... Figure 6. Element Degrees of Freedom with Nodal Displacements v... Figure 6. Element Degrees of Freedom with Nodal Displacements φ... i

12 NOMENCATURE Smbol Description A area of member a distributed load height a non-dimensional distributed load height {D} global nodal displacement vector for the structure {D e } global nodal displacement vector for an element {d e } local nodal displacement vector for an element E modulus of elasticit e concentrated load height e non-dimensional concentrated load height F aial load F non-dimensional aial load G shear modulus ii

13 [g e ] element local geometric stiffness matri for initial load set [g e ] P element local geometri stiffness matri for prebuckling h depth of the member I moment of inertial about the ais I moment of inertial about the ais I ω warping moment of inertia J torsional constant K beam parameter [k e ] element local stiffness matri [k e ] P element local stiffness matri for prebuckling k torsional curvature of the deformed element member length M cr classical lateral buckling uniform bending moment M bending moment [N] shape function matri P concentrated load P cr critical concentrated load iii

14 γ P non-dimensional concentrated load q distributed load q cr critical uniforml distributed load γ q non-dimensional distributed load r o polar radius of gration about the shear center r o non-dimensional polar radius of gration [T e ] transformation matri [T r ] rotation transformation matri t p perpendicular distance to P from the mid-thickness surface U strain energ U e strain energ for each finite element u out-of-plane lateral displacement u p out-of-plane lateral displacement of point P o u, u out-of-plane lateral displacements at nodes and u, u 4 out-of-plane rotation at nodes and u out-of-plane rotation u non-dimensional out-of-plane lateral displacement iv

15 v in-plane bending displacement v M displacement through which the applied moment acts v P in-plane bending displacement of point P o v q displacement through which the distributed load acts v, v in-plane displacements at nodes and v, v 4 in-plane rotation at nodes and v in-plane rotation w aial displacement w F longitudinal displacement through which the aial load acts w P longitudinal displacement of point P o o shear center displacement non-dimensional shear center displacement P concentrated load location from left support non-dimensional member distance p non-dimensional distance to concentrated load β monosmmetr parameter β non-dimensional monosmmetr parameter v

16 ε P longitudinal strain of point P o φ out-of-plane twisting rotation φ, φ out-of-plane twisting rotation at nodes and φ, φ 4 out-of-plane torsional curvature at nodes and φ out-of-plane torsional curvature γ P shear strain of point P o λ buckling parameter total potential energ non-dimensional total potential energ ρ degree of monosmmetr σ P longitudinal stress of point P o τ P shear stress of point P o ω warping function Ω potential energ of the loads Ω e potential energ of the loads for each finite element θ rotation of the member cross-section vi

17 . INTRODUCTION The members of a steel structure, commonl known as beam-columns, are usuall designed with a thin-walled cross-section. Thin-walled cross-sections are used as a compromise between structural stabilit and economic efficienc and include angles, channels, bo-beams, I-beams, etc. These members are usuall designed so that the loads are applied in the plane of the weak ais of the cross-section, so that the bending occurs about the strong ais. However, when a beam, usuall slender in nature, has relativel small lateral and torsional stiffnesses compared to its stiffness in the plane of loading, the beam will deflect laterall and twist out of plane when the load reaches a critical limit. This limit is known as the elastic lateral-torsional buckling load. The lateral buckling and twisting of the beam are interdependent in that when a member deflects laterall, the resulting induced moment eerts a component torque about the deflected longitudinal ais which causes the beam to twist (Wang, et al. 5). The lateral-torsional buckling loads for a beam-column are influenced b a number of factors, including crosssectional shape, the unbraced length and support conditions of the beam, the tpe and position of the applied loads along the member ais, and the location of the applied loads with respect to the centroidal ais of the cross section. This paper will focus on the lateral-torsional buckling of steel I-beams with a monosmmetric cross-section. In a beam with a monosmmetric cross-section, the shear center

18 and the centroid of the cross-section do not coincide. The significance of this can be eplained b the Wagner effect (Anderson and Trahair, 97), in which the twisting of the member causes the aial compressive and tension stresses to eert an additional disturbing torque. This torque can reduce the torsional stiffness of a member in compression and increase the torsional stiffness of a member in tension. In I-beams with doubl-smmetric cross-sections, these compressive and tensile stresses balance each other eactl and the change in the torsional stiffness is ero. In I-beams with monosmmetric cross-sections where the smaller flange is further from the shear center, the Wagner effect results in a change in the torsional stiffness. The stresses in the smaller flange have a greater lever arm and predominate in the Wagner effect. The torsional stiffness of the beam will then increase when the smaller flange is in tension and decrease when the smaller flange is in compression. When a structure is simple, such as a beam, an energ method approach ma be used directl to calculate the lateral-torsional buckling load of the structure. Assuming a suitable shape function, the equations derived using the energ method can provide approimate buckling loads for the structure. However, when a structure is comple, this is not possible. In this case, the energ method in conjunction with the finite element method ma be used to calculate the lateral-torsional buckling load of the structure. The finite element method is a versatile numerical and mathematical approach which can encompass complicated loads, boundar conditions, and geometr of a structure. First, element elastic stiffness and geometric stiffness matrices are derived for an element using the energ equations for lateral-torsional buckling. The structure in question must be divided into several elements, and a global coordinate sstem can be selected for that structure. The element elastic stiffness and geometric stiffness matrices are transformed to the global coordinate sstem for

19 each element, resulting in global element elastic and geometric stiffness matrices for the structure. After this assembl process, boundar conditions are enforced to convert the structure from an unrestrained structure to a restrained structure. The derived equilibrium equations are in the form of a generalied eigenvalue problem, where the eigenvalues are the load factors that, when multiplied to a reference load, result in lateral-torsional buckling loads for the structure. The main objective of this thesis is to formulate equations for lateral-torsional buckling of monosmmetric beams using the energ method. Suitable shape functions will be applied to these equations to provide approimate buckling solutions that can be compared to previous data. The finite element method will be used to derive element elastic and geometric stiffness matrices that can be used in future works in conjunction with computer software to determine lateraltorsional buckling loads of more comple structures.

20 . ITERATURE REVIEW This section reviews available literature that eplores lateral-torsional buckling as the primar state of failure for beams used in structures. A beam that has relativel small lateral and torsional stiffnesses compared to its stiffness in the plane of loading tends to deflect laterall and twist out of plane. This failure mode is known as lateral-torsional buckling. Two methods are used to derive the critical load values that result in lateral-torsional buckling beam failure: the method utiliing differential equilibrium equations and the energ method. The differential equilibrium method of stabilit analsis assumes the internal and eternal forces acting on an object to be equal and opposite. The energ method refers to an approach where the total potential energ of a conservative sstem is calculated b summing the internal and eternal energies. The buckling loads for the sstem can then be approimated if a suitable shape function for the particular structure is used, thus reducing the sstem from one having infinite degrees of freedom to one having finite degrees of freedom. This approach is known as the Raleigh-Rit method. This method will provide acceptable results as long as the assumed shape function is accurate. Both the differential equilibrium method and energ methods are eamined in this literature review. 4

21 . EQUIIBRIUM METHOD The closed form solutions for various loading conditions and cross-sections are demonstrated below using the equilibrium method. The beams are assumed to be stationar and therefore the sum of the internal forces of the structure and the eternal forces is assumed to be ero. The equations are rearranged in terms of displacements resulting in a second order differential equation from which the buckling loads can be solved. The beams are assumed in this section to be elastic, initiall perfectl straight, and in-plane deformations are neglected. Rotation of the beam, φ, is assumed small, so for the small angle relationships sinφ φ and cosφ can be used. Consider a simpl supported beam with a uniform rectangular cross section as shown in Figure.a and Figure.b. Note that u, v and w are the displacements in the -, -, and - directions, respectivel. The section rotates out of plane at an angle φ. The differential equilibrium equations of minor ais bending and torsion of a beam with no aial force (F ) are derived from statics as (Chen and ui. 987) EI d u M φ (.) GJ d φ M du M (.) Ehb where EI and GJ Ghb 5

22 EI represents the fleural rigidit of the beam with respect to the -ais and GJ represents the torsional rigidit of the beam with respect to the -ais. M and M are the internal moments of the beam acting about the -ais and the -ais, respectivel. In Eq. (.), the component of M in the -direction is represented b M sinφ which, b wa of the small angle theorem, reduces to M φ. In Eq. (.), the torsional component of M acting in the -direction is represented b M du. b h u Figure.a Beams of Rectangular Cross-Section F M M F Figure.b Beams of Rectangular Cross-Section with Aial Force and End Moments 6

23 .. Closed Form Solutions Case : A beam that is subjected to onl equal end moments M about -ais This loading case is shown in Figure.b, with F. Since there is no torsional component of the moment, let M M and M. The equilibrium equations given in Eq. (.) and (.) reduce to EI d u M φ (.) GJ d φ M du (.4) Solving for u from Eqs. (.) and (.4) ields (respectivel): d u M φ (.5) EI d u GJ M d φ (.6) Eliminating u ields a single differential equation of the form d φ M φ (.7) GJEI Solving the second order differential equation ields the general solution as M φ( ) Asin B cos EI GJ M EI GJ (.8) B appling the boundar condition φ at, B is equal to ero. 7

24 The constant A ma then not be equal to ero because it provides a trivial solution. Therefore at sin M cr EI GJ (.9) Solving for M cr to provide the smallest nonero buckling load ields M cr EI GJ (.) where M cr is the critical value of M that will cause the beam to deflect laterall and twist out of plane. Case : A linearl tapered beam with a rectangular cross section that is subjected to onl equal end moments M about -ais For this case, consider a linearl tapered beam with initial depth h o which increases at a rate δ as shown in Figure. where h () ( δ )h o (.) In Eq. (.), h() is the linear tapered depth of the beam as a function of, h o is the depth of beam at, and ( δ) is the ratio of height of tapered beam at to. 8

25 h o M h () M ( δ ) h o Figure. inear Tapered Beam Subjected to Equal and Opposite End Moments The beam properties of the tapered section can be written as EI EI η (.) o GJ GJ o η (.) where I o hb o, J hb o o, and η δ. (.4) As in Case, there is no torsional component of the moment so that M M and M. Substituting Eqs. (.) and (.) into the equilibrium equations ield d u EI oη Mφ (.5) d GJ oη φ M du (.6) The derivative of η with respect to is dη δ (.7) which enables the following relationships 9

26 dφ dφ dη δ dφ dη dη (.8) and d φ δ d φ (.9) dη Substituting Eqs. (.7) (.9) into the equilibrium Eqs. (.5) and (.6) ield EIoη δ d u Mφ (.) dη d GJoη δ φ dη δ du M (.) dη Differentiating Eq. (.) and combining it with Eq.(.) in order to eliminate u ields φ η η φ d d k φ (.) dη dη where k M EIoGJoδ (.) The general solution of Eq. (.) is given b (ee, 959) φ Asin( kln η) Bcos( kln η) (.4) Appling the boundar condition φ at, B is equal to ero. The constant A ma then not be equal to ero because it provides a trivial solution. Therefore the boundar condition φ at η ( δ) ields sin( k ln( δ )) (.5) Solving for M cr to provide the smallest nonero buckling load ields

27 M cr δ ln( δ) EI GJ o o (.6) where M cr is the critical value of M that will cause the beam to deflect laterall and twist out of plane. It is important to recognie that ifδ, meaning that the beam is not tapered, Eq. (.6) reduces to M cr EI GJ o o (.7) which is the result obtained in Case. Case : A simpl supported beam with a concentrated load, P, at midspan (at /) For this case, consider a non-tapered beam with a concentrated load, P, at midspan as shown in Figure.. P Figure. Simpl Supported Beam with Concentrated oad, P, at Midspan

28 The moments M and M are derived using basic equilibrium concepts as P M (.8) M P ( u u ) (.9) Where u* represents the lateral deflection at the centroid of the middle cross section and u represents the lateral deflection at an cross section. B substituting the relationships for M and M into Eqs. (.) and (.), the differential equations for this case become EI d u P φ (.) GJ d φ P du P ( u u ) (.) Combining the above equations to eliminate the term u ields d φ P φ (.) 4EI GJ For simplicit, the following non-dimensional relationships are used η (.) ζ 4 P 4EI GJ (.4) Eq. (.) can then be reduced to d φ ζηφ (.5) dη

29 The general solution utilies Bessel functions (Arfken, 5) of the first kind of orders /4 and - /4 shown below as φ ζη ζη η AJ4 / BJ 4 / (.6) Appling the boundar conditions φ atη and d φ dη at η gives J 4 / ζ 8 then ζ ielding an epression for the buckling load, P c, as P c EI GJ (.7) where P c is the lateral-torsional buckling load (Wang, et al. 5). Case 4: A simpl supported I beam with a monosmmetric cross-section subjected to equal end moments, M, and an aial load, F, acting through the centroid: In this case, consider a non-tapered monosmmetric I-beam subjected to end moments, M, and an aial force, F, as shown in Figure.4a with the cross-section of the beam shown in Figure.4b.

30 M M F F Figure.4a Monosmmetric Beam subjected to End Moments and Aial Force I T shear center, s centroid, o o h I B Figure.4b Cross-Section of Monosmmetric Beam The minor ais distributed force equilibrium equation and the distributed torque equilibrium equation for the member can be epressed as (Kitipornchai and Wang. 989) EI 4 d u 4 d φ ( M Fo) F d u (.8) ( GJ F r M ) d φ d φ o β EI w Where the warping rigidit is M du (.9) 4

31 EI EI ρ( ρ) h w (.4) and the degree of monosmmetr can be epressed as ρ I T ( I I ) T B I I T (.4) where I T and I B are the second moments of inertia about the -ais of the top and bottom flanges, respectivel, as shown in Figure.4b. Because the beam has a monosmmetric crosssection, the centroid of the beam, o, and the shear center, s, do not coincide. This introduces a term, o, which represents the vertical distance between the centroid and the shear center. The polar radius of gration about the shear center, r o, can be epressed as r I o o A I (.4) The monosmmetric parameter of the beam (Trahair and Nethercot, 984) is da da β I A A o (.4) where and are coordinates with respect to the centroid. β accounts for the Wagner effect, which is the change in effective torsional stiffness due to the components of bending compressive and tensile stresses that produce a torque in the beam as it twists during buckling. Recogniing that, since the beam is simpl supported, the boundar conditions become φ and d φ at,. With the elimination of u and the implementation of the above boundar conditions, Eqs. (.6) and (.9) ield a closed form solution for critical values F and M (Trahair and Nethercot, 984) as 5

32 F F ( M F ) ro FFE F F where F E is the Euler buckling load given b E Mβ r F o (.44) F E EI (.45) and F is given as (Wang, et al. 5) F GJ EI w (.46) r GJ o In order to obtain a non-dimensional elastic buckling moment, use is made of the nondimensional parameters (Kitipornchai and Wang. 989) K EI h 4 GJ (.47) η 4 h I I A (.48) υ h (.49) Λ F F E (.5) β λ υλ ( Λ) K (.5) h γ M EI GJ (.5) Where h is the distance between the centroids of the top and bottom flange and K is the beam parameter. The practical range for values of K is between. and.5, with low values 6

33 corresponding to long beams and/or beams with compact cross-sections, and higher values corresponding to short beams and/or beams with slender cross sections. Using the above nondimensional parameters, Eq. (.44) ma be rewritten as (Wang, et al. 5) { K [ ]} γ λ ± λ υk Λ ( Λ) 4ρ( ρ) ηλ (.5) The non-dimensional buckling equation shown above is the general solution of M for monosmmetric beams. Eq. (.5) is a versatile equation because it also accuratel predicts the lateral-torsional buckling loads for beams of doubl smmetric cross-sections b simplifing the terms in the equation so that the monosmmetric parameter, β, is equal to ero and the degree of monosmmetr, ρ, is reduced to. 7

34 . ENERGY METHOD The second method used for determining lateral-torsional buckling loads in thin-walled structures is the energ method. The energ method serves as a basis for the modern finite element method of computer solution for lateral-torsional buckling problems of comple structures. The energ method is related to the differential equations of equilibrium method in that calculus of variation can be used to obtain the differential equations derived b the first method. The energ method is based on the principle that the strain energ stored in a member during lateral-torsional buckling is equal to the work done b the applied loads. The critical buckling loads can then be obtained b substituting approimate buckled shapes back into the energ equation if the shape function is known. This approach is known as the Raleigh-Rit method. The strain energ stored in a buckled member can be broken down into two categories, the energ from St. Venant torsion and from warping torsion. Pure or uniform torsion eists when a member is free to warp and the applied torque is resisted solel b St. Venant shearing stresses. When a member is restrained from warping freel, both St. Venant shearing stresses and warping torsion resist the applied torque. This is known as non-uniform torsion. 8

35 .. Uniform Torsion When a torque is applied to a member that is free to warp, the torque at an section is resisted b shear stresses whose magnitudes var based on distance from the centroid of the section. These shear stresses are produced as adjacent cross-sections attempt to rotate relative to one another. The St. Venant torsional resistance must directl oppose the applied torque as T sv GJ d φ (.57) whereφ is the angle of twist of the cross-section, G is the shearing modulus of elasticit, J is the torsional constant, and is direction perpendicular to the cross section, as illustrated in Figure.5. T T Figure.5 Twisting of a Rectangular Beam that is Free to Warp 9

36 .. Non-uniform Torsion If the longitudinal displacements in the member are allowed to take place freel and the longitudinal fibers do not change length, no longitudinal stresses are present and warping is permitted to take place. However, certain loading and support conditions ma be present that prevent a member from warping. This warping restraint creates stresses which produce a torsion in the member. Non-uniform torsion occurs when both St. Venant and warping torsion act on the same cross section. The epression for non-uniform torsion can be given as T T T (.58) sv w where T w is the warping torsion, which, for an I section, is T w V h (.59) f where V f is the shear force in each flange and h is equal to the height of the section. Recogniing that the shear in the flange is the derivative of the moment present in the flange, Eq. (.59) becomes T w dm f h (.6) The bending moment in the upper flange, M f, can be written in terms of the displacement in the -directon, u, as M f EI f d u (.6) Recogniing that u h φ (.6)

37 and introducing the cross-sectional propert known as the warping moment of inertia I w I h f (.6) the warping torsion can now be epressed as T w EI w d φ (.64) The differential equation for non-uniform torsion is obtained b substituting Eq. (.57) and Eq. (.64) into Eq. (.58) is T GJ d φ EI w d φ (.65) The first term refers to the resistance of the member to twist and the second term represents the resistance of the member to warp. Together, the terms represent the resistance of the section to an applied torque... Strain Energ The strain energ stored in a twisted member can be broken into two categories, the energ due to St. Venant torsion and the energ due to warping torsion. The strain energ due to St. Venant torsion (Chajes, 99) is du sv Tsv dφ (.66) where it can be seen that the change in strain energ stored in element due to St. Venant torsion is equal to one half the product of the torque and the change in the angle of twist. Solving for dφ from Eq. (.57)

38 Tsv dφ GJ (.67) and substituting it into Eq. (.66) ields du sv Tsv GJ (.68) Substituting Eq. (.57) into Eq. (.68) and integrating results in the epression for strain energ due to St. Venant torsion. U sv GJ d φ (.69) The strain energ due to the resistance to warping torsion of an I-beam, for eample, is equal to the bending energ present in the flanges. The bending energ stored in an element of one of the flanges is equal to the product of one half the moment and the rotation as du w EI f d u (.7) Substituting Eqs. (.6) and (.6) into Eq. (.7) ields du w 4 EI w d φ (.7) Integrating Eq. (.7) over the length of the member,, and multipling b two to account for the energ in both flanges results in the epression for the strain energ in a member caused b resistance to warping. U w EI w d φ (.7) The total strain energ in a member is then represented b the addition of Eqs. (.69) and (.7). U o GJ d φ EI w d φ (.7)

39 ..4 Solutions Using Buckling Shapes Case 5: A simpl supported, doubl smmetric I-beam that is subjected to onl equal end moments M about -ais The loading in this case is identical to Case, but this case consists of a beam with an I crosssection instead of a rectangular cross-section. As in Case, M M and M. The boundar conditions for the case of uniform bending are given below. d u d v u v at, (.74) d φ φ at, (.75) In order to find the critical moment b use of the energ method, it is necessar to find the moment for which the total potential energ has a stationar value. The strain energ stored in the beam consists of two parts; the energ due to the bending of the member about the -ais and the energ due to the member twisting about the -ais. The total strain energ for the section is U EI d u d GJ EI w φ d φ (.76) The strain energ, U, must now be added to the potential energ of the eternal loads, Ω, to determine a stationar value for Π U Ω. For a member subjected to uniform bending, the eternal potential energ is equal to the negative product of the applied moments and the angles through which the act upon the beam.

40 Ω Mψ (.77) where ψ is the angle of rotation about the -ais of the beam and can be epressed as du ψ dφ (.78) Combining Eqs. (.77) and (.78) ields an epression for the potential energ of the eternal loads as Ω M du dφ (.79) and ields the following epression for the total potential energ of the beam. Π EI d u dφ d φ GJ EI M du dφ w (.8) As stated at the beginning of this section, the Raleigh-Rit method for determining critical loads requires the assumption of suitable epressions for buckling modes. The following buckling shapes satisf our boundar conditions: u Asin (.8) φ B sin (.8) Substituting the buckled shapes into Eq. (.8) and identifing that sin cos (.8) the total potential energ of the beam, Π, becomes Π EI A GJB EIwB MAB (.84) 4

41 Setting the derivative of Π with respect to A and B equal to ero, the critical moment can be obtained. d Π EI A MB (.85) da d Π MA GJ EI w B (.86) db If the deformed configuration of the beam is to ield a nontrivial solution, the determinant of the coefficients A and B in Eqs. (.85) and (.86) must vanish leaving EI GJ EI w M (.88) Solving for M in Eq. (.88) ields the critical moment for a simpl supported beam in uniform bending as M EI GJ EI (.89) cr w Case 6: A doubl smmetric I-beam with fied ends that is subjected to onl equal end moments M about -ais For this case, as in Case, M M and M. Consider an I-beam whose ends are free to rotate about a horiontal ais but restrained against displacement in an other direction, as shown in Figure.6. 5

42 M M Figure.6 I-beam Fied at Both Ends Subjected to End Moments The boundar conditions are as follows u du at, (.9) d v v at, (.9) dφ φ at, (.9) The following buckling shapes satisf the geometric boundar conditions u A cos (.9) φ B cos (.94) Substituting the buckled shapes into Eq. (.8) and using the simplification in Eq. (.8), the total potential energ of the beam becomes A Π 4EI GJB 4EI w B MAB (.95) l 6

43 Setting the derivative of the Π equations with respect to A and B equal to ero, ield the following two equations. dπ da 8EI A MB (.96) d Π GJB 8EI w B MA db (.97) Eqs. (.95) and (.96) epressed in matri form is 4EI M M GJ 4EI w A B (.98) If the deformed configuration of the beam is to ield a nontrivial solution, the determinant of the coefficients A and B in Eq. (.98) must vanish leaving 4EI 4 GJ EI w M (.99) Solving for M as the critical moment ields M EI GJ 4EI cr w (.) It is interesting to note that the critical moment for the restrained beam is proportional to that of the simpl supported beam. If the warping stiffness is negligible compared to that of the St. Venant stiffness, the critical moment for the fied beam is twice that of the hinged beam. If the St. Venant stiffness is negligible compared to that of the warping stiffness, the critical moment of the fied beam is four times that of the hinged beam. The reason for this is that lateral bending strength and warping strength depend of the length of the beam, where St. Venant stiffness does not. St. Venant stiffness is therefore unaffected b a change in boundar conditions. 7

44 . ATERA-TORSIONA BUCKING OF BEAM-COUMNS The energ method detailed in Chapter, in conjunction with the Raleigh-Rit method, is useful in determining closed form or approimate solutions, with a high degree of accurac, when a suitable buckling mode can be identified. In more comple structural sstems, identification of the buckling mode is not possible. In this case, a finite element approach is an ideal method that ma be used to calculate the buckling load. In order to formulate element elastic and geometric stiffness matrices that represent different load cases, one approach is to derive total potential energ of a beam-column element with a concentrated force, distributed force, end moments, and an aial force. Therefore, the objective of this chapter is to derive energ equations for a beamcolumn element with the above mentioned loads. ateral-torsional buckling of a beam-column occurs when the loads on an element become large enough to render its in-plane state unstable. When the loads on the member reach these critical values, the section will deflect laterall and twist out of the plane of loading. At critical loading, the compression flange of the member becomes unstable and bends laterall while the rest of the member remains stable restraining the lateral fleure of the compression flange, causing the section to rotate. This is common in slender beam-columns with insufficient lateral bracing that have a much greater in-plane bending stiffness than their lateral and torsional stiffnesses. It is important to know the critical load for lateral-torsional buckling because this 8

45 method of failure is often the primar failure mode for thin-walled structures. The focus of this chapter is to formulate energ equations that can be used to derive element elastic and geometric stiffness matrices of a beam-column with a monosmmetric I cross-section. The basic assumptions used in the following derivations are:. The member has a monosmmetric cross-section.. The beam-column remains elastic. This implies that the member must be long and slender.. The cross-section of the member does not distort in its own plane after buckling and its material properties remain the same. 4. The member is initiall perfectl straight, with no lateral or torsional displacements present before buckling. 5. The member is a compact section. The orientation of the member used to derive the energ equations is depicted in Figure. using the coordinate sstem with the origin being at o. The -ais is the major principle ais and the -ais is the minor principle ais with the -ais being oriented along the length of the member, coinciding with the centroidal ais of the undeformed beam-column. b o ais h u Figure. Coordinate Sstem of Undeformed Monosmmetric Beam 9

46 The displacements in the,, and directions are denoted as u, v, and w, respectivel. If loading occurs in the plane, the member will have an in-plane displacement, v, in the - direction and an in-plane rotation dv. A member loaded along the -ais will have a displacement, w. The result of lateral-torsional buckling is an out-of-plane displacement, u, in the -direction, an out-of-plane rotation, du, an out-of-plane twisting rotation, φ, and an out-of- plane torsional curvature, d φ. It is assumed in this chapter that in-plane deformations w, v, and dv are ver small and therefore can be neglected. In the net chapter, the energ equations derived in this chapter will be epanded to include these displacements, which are known as prebuckling deformations. The applied loads on the beam column include; () a distributed load, q, which acts at a height a above the centroidal ais () a concentrated load, P, which acts at a height e above the centroidal ais () a concentric aial load, F (4) end moments, M and M, as shown below. P q e F F a M M V V Figure. Eternal oads and Member End Actions of Beam Element

47 The energ equations for the lateral-torsional buckling of a member with a monosmmetric cross-section differ from those of a member with a double smmetric crosssection because the centroid and the shear center do not coincide on a monosmmetric beamcolumn, as shown in Figure.. This introduces the term, o, into the derivation representing the distance between the shear center, s, and the centroid, o. I T shear center, s centroid, o o h I B Figure. Cross-Section of Monosmmetric I-beam The change in effective torsional stiffness of the member due to the components of bending compressive and tensile stresses that produce a torque in the beam as it twists during buckling is referred to as the Wagner effect (Anderson and Trahair, 97). In a beam with a doubl-smmetric cross-section, these compressive and tensile stresses balance each other and do not affect the torsional stiffness of the beam. For the case of monosmmetr, these tensile and compressive stresses do not balance each other and the resulting torque causes a change in the effective torsional stiffness of the member from GJ to (GJ M β ) (Wang and Kitipornchai, 986). Because the smaller flange of the beam is farther awa from the shear center than the

48 larger flange, it creates a larger moment arm and predominates in the Wagner effect. This means that when the smaller flange is in tension, the effective torsional stiffness of the beam is increased while the effective torsional stiffness is reduced when the smaller flange is in compression. This inconsistenc adds to the compleit of the energ equation derivations for lateral torsional buckling. The energ equation for an elastic thin-walled member is derived b considering the strain energ stored in the member, U, and the potential energ of the eternal loads, Ω, as Π U Ω (.) where Π represents the total potential energ of the member. The strain energ present in the member is the potential energ of the internal stresses and strains present in the beam-column, while the potential energ of the loads represents the negative of the work done b eternal forces. The total potential energ increment ma be written as ΔΠ δπ δ Π δ Π... (.)!! The theorem of stationar total potential energ states that of all kinematicall admissible deformations, the actual deformations (those which correspond to stresses which satisf equilibrium) are the ones for which the total potential energ assumes a stationar value (Pilke and Wunderlich, 994) or δ Π. As previousl discussed, the structure is unstable at buckling. The theorem of minimum total potential energ states that this stationar value of Π at an equilibrium position is minimum when the position is stable. Therefore the equilibrium position can be considered stable when

49 δ Π> (.) and likewise is unstable when δ Π< (.4) Therefore, the critical condition for buckling would be when the total potential energ is equal to ero, thus representing the transition from a stable to unstable state (Pi, et al., 99). δ Π (.5) Substituting Eq. (.) ields the critical condition for buckling as ( δ δ ) U Ω (.6). STRAIN ENERGY The strain energ portion of the total potential energ ma be epressed as a function of the longitudinal and shear strains as well as stresses. Assume an arbitrar point P o in the cross section of the thin walled member. The strain energ of the member can be epressed as U ( p p p p) εσ γ τ da (.7) A where ε p longitudinal strain of point P o σ p longitudinal stress of point P o γ p shear strain of point P o

50 τ p shear stress of point P o With its second variation being ( p p p p p p p p) da (.8) δ U δε δσ δγ δτ δ ε σ δ γ τ A ε p, σ p, γ p, and τ p and their variations will be epressed in terms of centroidal displacements in the following section in order to derive the energ equation for lateral-torsional buckling... Displacements In order to properl investigate deformations in a beam-column, two sets of coordinate sstems are defined. In the fied global coordinate sstem o, the ais o is fied and coincides with the centroidal ais of the undeformed beam. The aes o and o represent the principle aes of the undeformed beam, as shown in Figure.4. o ô Figure.4 Deformed Beam 4

51 The second set of coordinate sstems is a moving, right hand, local coordinate sstem, o. The origin of this coordinate sstem is at point o located on the centroidal ais of the beam and moves with the beam during displacement, as shown in Figure.4. The ais o coincides with the tangent at o after the centroidal ais has been deformed. The principle aes of the deformed beam are o and o. When the beam column element buckles, point P o on the beam moves to point P. This deformation occurs in two stages. Point P o first translates to point P t b the displacements u, v, and w. The point P t then rotates through an angle θ to the point P about the line on where the line on passes through the points o and o. After the rotation, the moving local coordinate sstem o becomes fied. The transition of point P o to point P can be seen in Figure.5. The directional cosines of the moving aes o, o, and o relative to the fied global aes o, o, and o can be determined b assuming rigid bod rotation of the aes through an angle θ (Pi, et al., 99). o ˆ P o ˆ P P t ô ˆ ˆ n Figure.5 Translation of Point P o to Point P 5

52 The displacements of point P o can be epressed as (Pi, et al., 99) u p v p w p u v TR o o w ωk (.9) Where u p, v p, and w p are total displacements of general point P ( ) are shear center displacements and ( ) o o,,. Note that u, v, and w is the distance between the centroid and the shear center, as seen in Figure.. ω is the section warping function (Vlasov, 96) and ωk is the warping displacement and represents the deformation in the -direction. The first term on the right side of the equation contains the shear center displacement as the point P o translates laterall to point P t. The remaining terms on the right side of the equation represent the rotation of point P t to its final destination at point P. T R is defined as the rotational transformation matri of the angle of rotation, θ. Assuming small angles of rotation T R θ θθ θ θθ θ θ θθ θ θ θ θθ θ θ θθ θ θθ θ θ (.) where θ, θ, and θ are components of the rotation θ in the,, and directions, respectivel (Torkamani, 998). Consider an undeformed element Δ and its deformed counterpart ( ) Δ ε, where ε represents the strain. Δ u Δv,, and ( Δ Δ w) are components of the deformed element length 6

53 ( ) Δ ε on the o, o, and o aes, respectivel. The relationship between the deformed element and its components can be epressed as ( ) ( ) Δ ε N Δui Δv j Δ Δw k (.) where N is a unit vector in the o direction. Δ ( ε) is projected on the and aes as N Δ u Δ ( ε) N i Δ ( ε) l Δ v Δ ( ε) N j Δ ( ε) m (.) (.) where l, m, and n are directional cosines of the o direction with respect to the o fied coordinate sstem. Dividing the previous equations b Δ, and taking the limit as Δ goes to ero gives du dv Δ u lim Δ Δ o o Δ v lim Δ Δ o o Δ ( ε) l lim Δ Δ Δ ( ε) m lim Δ Δ ( ε ) l (.4) ( ε ) m (.5) Where l and m are defined as. (Torkamani, 998) θ θ l θ θ θ m θ (.6) (.7) Therefore du dv θθ θ ( ε) (.8) θθ θ ( ε) (.9) 7

54 B disregarding higher order terms, strain is eliminated and Eqs. (.8) and (.9) become du dv θθ θ θθ θ (.) (.) Solving for θ and θ from the previous equations ields dv du θ θ du dv θ θ (.) (.) Projecting the unit lengths along the o ais onto the o ais and the o ais onto the o ais ield m and l respectivel. (Torkamani, 998) θ θ θ l θ θ θ m (.4) (.5) θθ The projections θ and θ θθ of unit lengths along the o and o aes onto the o and o aes, respectivel, can be used to define the the mean twist rotation, φ, of the o and o aes on the o ais is θθ θθ φ θ θ (.6) Which simplifies to φ θ (.7) 8

55 Substituting the epressions for θ, θ, and θ into the rotational transformation matri, T R, ields T R l l l m m m n n n (.8) Where du l du dv φ φ (.9) du l dv du dv φ φ φ 4 4 (.) du l (.) du m dv dv du φ φ φ 4 4 (.) dv m du dv φ φ (.) dv m (.4) du dv n φ dv du n φ 4 4 du φ dv φ du n dv (.5) (.6) (.7) The torsional curvature is (ove, 944) k dl l dm m dn n (.8) 9

56 Substituting l through n into the previous epression ields a nonlinear epression for torsional curvature as d d u dv k φ d v du (.9) Eliminating second and higher order terms, the epression for torsional curvature ma be simplified to k d φ (.4) Substituting Eqs. (.9) (.7) into Eq. (.9) ield the displacement of an arbitrar point P o in terms of shear center displacements, rotations, and the section warping. The total displacements u p, v p, and w p can be considered the sum of linear and quadratic components of the form u p v p w p u v w pl pl pl u v w pn pn pn (.4) where u v w pl pl pl u ( o ) φ v φ (.4) w du dv d ω φ u v w pn pn pn du du dv du dv du dv du d o φ φ φ ( ) φ φ ω du dv dv du dv du dv dv dφ φ φ ( o ) φ φ ω dv du du dv d du dv φ φ ( ) o φ φ ω φ 4 4 (.4) 4

57 The relationship between the displacement of the shear center, w s, and the displacement of the centroid, w, in the -direction can be epressed as (Pi and Trahair, 99a) dv dv du ws w o φ φ (.44) The derivatives of u p, v p, and w p with respect to are du dv dw p p p du d φ d φ O du dv o,, φ (.45) dv d φ O du dv,, φ (.46) dw d u d v d d dv d v d du d u O du dv ω φ φ φ φ φ,, φ The terms O, O, and O represent terms that are second order or higher and ma be disregarded. (.47).. Strains The longitudinal normal strain ε p of point P can be epressed in terms of the rates of change of the displacements of point P as dwp du p dv p dwp ε p (.48) For small strains, dwp is small compared to du p and dv p and can be disregarded. Therefore 4

58 dwp du p dv p ε p (.49) Substituting in the derivatives of the displacements u p, v p, and w p of point P o ields dw ε ω φ p d u d v d du dv d v φ d u φ du o d φ ( ( o ) ) dφ (.5) The first variation of the longitudinal strain is δε p dδw δ δ ω δφ δ δ δ δ d u d v d d u du d v dv d v φ d v δφ d u φ ( ( ) ) d u d δ u dφ du dδφ dδφdφ δφ o o o (.5) The second variation of the longitudinal strain is ( ( ) ) dδu dδv δ δ δ δφ δφ δ εp d v δφ d u δφ d u d d o o (.5) It is assumed that the second variations of the displacements in Eq. (.5) vanishes. The above equations contain a combination of the strains before and after buckling. In this case, the prebuckling displacements are defined as v and w. During buckling, the displacements are defined as δu and δφ. Therefore, the displacements u, φ, δv, and δ w are equal to ero and ma be eliminated from the above equations. The equations for longitudinal strain and its first and second variation thus become ε p dw d v dv (.5) 4