2. 990731_262423_380v3.i NATIONAL RESEARCH LABORATORY Jaehong Lee Dept. of Architectural Engineering Sejong University October 12, 2000 Energy-Based Approach for Buckling Problems in Steel Structures

3. 990731_262423_380v3.i OBJECTIVES To Present Energy Method in Buckling Analysis State-of-the Art Review of the Analysis of Thin-walled Structures Structural Behavior of Cold-formed Channel Section Beams

4. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Flexural-Torsional Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel Next Steps

5. 990731_262423_380v3.i Cold-formed steel represents over 45 percent of the steel construction market in U.S. Sophisticated structures such as schools, churches and complex manufacturing facilities. COLD-FORMED STEEL OFFERS VERSATILITY IN BUILDINGS Ease of Prefabrication and Mass Production Light Weight Uniform Quality Economy in Transformation and Handling Quick and simple erection

6. 990731_262423_380v3.i ANALYSIS & DESIGN OF COLD-FORMED CHANNEL-SECTION BEAMS ARE NOT EASY How to Take Care of These Complecated Behavior Finite Element Analysis & AISI Aisi Code Effective Width Linear Method & Iterative Method Bending + Torsion Things to Consider in Analysis and Design of Beams Elastic Lateral Buckling Inelastic Lateral Buckling Local Buckling Sectional Properties Center of Gravity  Shear Center

7. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Flexural-Torsional Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel Next Steps

8. 990731_262423_380v3.i LATERAL BUCKLING MAY OCCUR WELL BELOW THE YIELD STRENGTH LEVEL  u v Original position Final position for inplane bending FyFy Elastic Lateral Buckling Strength

9. 990731_262423_380v3.i FINITE ELEMENT MODEL IS THE BEST Kinematics Constitutive Relations Variational Formulation Lateral Buckling Equations Finite Element Model Build the appropriate displacement fields Derive the strain tensor Strain energy Potential of transverse load at shear center Stress resultants vs. strains Can be derived by integrating by parts Coupled differential equations Setup the eigenvalue problem Buckling loads and mode shapes Kinematics Variational Formulation Constitutive Relations Lateral Buckling Equations Finite Element Model

10. 990731_262423_380v3.i KINEMATICS OF THIN-WALLED SECTION x y P z  O s n q r Contour Coordinate Displacement Field Plate Action Beam Action Kirchhoff-Love assumption Shear strain at midsurface is zero. Basic Assumptions

11. 990731_262423_380v3.i VARIATIONAL FORMULATION IS USED TO FORMULATE THE GOVERNING EQUATIONS WEAK FORM CONSTITUTIVE MODEL Load Type s.c a

12. 990731_262423_380v3.i GOVERNING LATERAL BUCKLING EQUATIONS CAN BE DERIVED BY INTEGRATION BY PARTS THE VARIED QUATITIES Lateral Buckling Equations

13. 990731_262423_380v3.i FINITE ELEMENT MODEL IS DERIVED FROM THE WEAK FORM Finite Element Model (Standard Eigenvalue Problem) : eigenvalue (buckling parameter)  : eigenfunction (buckling mode shape)

14. 990731_262423_380v3.i CLOSED-FORM SOLUTION FOR ELASTIC LATERAL BUCKLING IS LIMITED M M Simply-supported Beam Under Pure Bending Buckling moment Buckling mode shape (For H-section) Buckling stress

15. 990731_262423_380v3.i UNEQUAL END MOMENTS AND VARIOUS BOUNDARY CONDITIONS SHOULD BE CONSIDERED Bending coefficient (moment gradient factor) C b ()() (+) M 2 >M 1 M >M 2 C b =1 M1M1 M1M1 M2M2 M2M2 AISI Specification 1968-1980 edition St. Venant torsion neglected 1989 edition Pekoz & Winter For singly-symmetric section: torsional-flexural buckling considered

16. 990731_262423_380v3.i BEAM UNDER UNEQUAL END MOMENTS M MM

17. 990731_262423_380v3.i BUCKLING MODES OF A BEAM UNDER UNEQUAL END MOMENTS M MM

18. 990731_262423_380v3.i EFFECT OF LOADING POINT ON A CANTILEVER BEAM UNDER POINT LOAD AT FREE END P

19. 990731_262423_380v3.i EFFECT OF LOADING POINT ON A SIMPLY-SUPPORTED BEAM UNDER UNIFORMLY-DISTRIBUTED LOAD w

20. 990731_262423_380v3.i LATERALLY-SUPPORTED BEAM UNDER UNEQUAL END MOMENTS M MM x

21. 990731_262423_380v3.i LATERALLY-SUPPORTED BEAM UNDER UNEQUAL END MOMENTS M MM x

22. 990731_262423_380v3.i INELASTIC LATERAL BUCKLING SHOULD BE CONSIDERED FOR REAL PROBLEMS FyFy Elastic Lateral Buckling Inelastic Lateral Buckling When buckling stress exceeds the proportional limit The beam behavior is governed by inelastic buckling  pr For accurate solution, rigorous iterative method is required

23. 990731_262423_380v3.i AISI CODE PROVIDES CONSERVATIVE INELASTIC BUCKLING MOMENT M cr /M y M y /M e 0.5 1.0 2 13

24. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Flexural-Torsional Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel Next Steps

25. 990731_262423_380v3.i SAME PROCEDURE EXCEPT THE WORK DONE BY FORCES Kinematics Constitutive Relations Variational Formulation Lateral Buckling Equations Finite Element Model Build the appropriate displacement fields Derive the strain tensor Strain energy Potential of external forces Stress resultants vs. strains Can be derived by integrating by parts Coupled differential equations Setup the eigenvalue problem Buckling loads and mode shapes Kinematics Variational Formulation Constitutive Relations Lateral Buckling Equations Finite Element Model

26. 990731_262423_380v3.i VARIATIONAL FORMULATION IS USED TO FORMULATE THE GOVERNING EQUATIONS WEAK FORM CONSTITUTIVE MODEL s.c c.g

27. 990731_262423_380v3.i GOVERNING FLEXURAL-TORSIONAL BUCKLING EQUATIONS CAN BE DERIVED BY INTEGRATION BY PARTS THE VARIED QUATITIES

28. 990731_262423_380v3.i FINITE ELEMENT MODEL IS DERIVED FROM THE WEAK FORM Finite Element Model (Standard Eigenvalue Problem) : eigenvalue (buckling parameter)  : eigenfunction (buckling mode shape)

29. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Flexural-Torsional Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel Next Steps

30. 990731_262423_380v3.i WHEN THE TRANSVERSE LOADS DO NOT PASS THROUGH THE SHEAR CENTER, THE MEMBER WILL BE SUBJECTED TO BOTH BENDING AND TORSION  v c.g s.c v Bending Bending + Torsion Loads applied at shear center Loads applied at center of gravity Lateral Buckling

31. 990731_262423_380v3.i VARIOUS NORMAL AND SHEAR STRESSES CAN BE GENERATED BENDING TORSION 1.Longitudinal bending stress 2.Shear stress 1.Longitudinal bending stress 2.Shear stress 1.Warping longitudinal stress 2.Pure torsional shear stress 3.Warping shear stress 1.Warping longitudinal stress 2.Pure torsional shear stress 3.Warping shear stress

32. 990731_262423_380v3.i WARPING CHARACTERISTICS OF CHANNEL SECTION Shear Center Location Normalized Unit Warping  n1  n2  n3  n4 x y b d’ c.g s.c xpxp t Definition : Channel Section: Definition : Channel Section:

33. 990731_262423_380v3.i WARPING CHARACTERISTICS OF CHANNEL SECTION Warping Moment of Inertia b3b3 Warping Static Moment S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 Definition : Channel Section: Definition : Channel Section:

34. 990731_262423_380v3.i STRESS ANALYSIS OF CHANNEL SECTION BEAM – EXAMPLE PROBLEM Load applied at shear center Load applied at center of gravity Shear stress + + + - - + + + + + - - + + + - - Normal stress 0.3k/ft 10’ 1.5’’ 7’’ 0.135’’ bb ww bb vv ww vv tt

35. 990731_262423_380v3.i RESULTS OF STRESS ANALYSIS – EXAMPLE PROBLEM point bb ww b+ wb+ w 120.10-14.645.46 220.105.6425.74 3-20.10-5.64-25.74 4-20.1014.64-5.46 point vv ww tt  v +  w +  t 60.680.3699.249.551 20.9410.3149.249.867 52.0700.1589.2411.468 1 2 3 4 5 6 A member exhibiting bending-torsion coupling shows significantly different stress distribution

36. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel Next Steps

37. 990731_262423_380v3.i LOCAL BUCKLING CAN OCCUR BEFORE GLOBAL BUCKLING Reduce the ultimate load-carrying capacity significantly

38. 990731_262423_380v3.i BEHAVIOR OF STIFFENED AND UNSTIFFENED COMPRESSION ELEMENTS ARE NOT IDENTICAL A flat compression elements stiffened by other components (web, flange, lip, stiffener) along both longitudinal edges Stiffened compression elements (s.c.e) Unstiffened compression elements (u.c.e) A flat compression element stiffened only along one of the two longitudinal edges u.c.e s.c.e

39. 990731_262423_380v3.i PLATES DO NOT COLLAPSE WHEN BUCKLING OCCURS, BUT CAN STILL CARRY LOAD AFTER BUCKLING - POSTBUCKLING STRENGTH p  p cr Plate Buckling Equation Plate Buckling Stress Rigorous solution of postbuckling is difficult (Nonlinear numerical Analysis needed) Can define EFFECTIVE width

40. 990731_262423_380v3.i EFFECTIVE DESIGN WIDTH “b” CONCEPT IS WIDELY USED IN DESIGN PROCEDURE DUE TO THEIR SIMPLICITY effective width, b, represents a width of the plate which just buckles when  =  y w 11  1   cr w 22  cr   2   y w 33  3 =  y w w b/2 First introduced by von Karman (1932)  max The initially uniform compressive stresses become redistributed Relation of b and w

41. 990731_262423_380v3.i AISI SPECIFICATION FOR EFFECTIVE WIDTH HAS BEEN DEVELOPED Winter (1946) presented the formula for effective width AISI design provision (1946-1968) Winter (1970) presented more realistic equation AISI design provision (1970- )

42. 990731_262423_380v3.i AISI DESIGN PROVISION FOR EFFECTIVE WIDTH 0.673 1  Effective Design Width Equation Individual plates subjected to different boundary conditions Need to calculate k

43. 990731_262423_380v3.i BUCKLING STRESSES CAN BE DETERMINED VIA COEFFICIENT K Boundary condition Types of stress k Comp.4.0 Comp.6.97 Comp.0.425 Comp.1.277 Comp.5.42 s.s. free fixed free Boundary condition Types of stress k Shear5.34 Shear8.98 Bending23.9 Bending41.8 s.s. fixed s.s. fixed

44. 990731_262423_380v3.i CALCULATION OF EFFECTIVE WIDTH OF COMPRESSION FLANGE FOR CHANNEL IS STRAIGHTFORWARD k=0.425 Check if Buckling coefficient for ss-ss-ss-free Check the width-to-thickness ratio Calculation of slenderness ratio Determine the effective width w b Calculation of efffective width parameter

45. 990731_262423_380v3.i EFFECTIVE WIDTH OF WEB SHOULD BE CALCULATED BY ITERATION PROCESS (NOT SIMPLE) Assume fully effective w b f1f1 f2f2 b1b1 b2b2 Check if hchc Recalculate the neutral axis Web is fully effective! b 1 & b 2 calculated no yes n=1 n>1

46. 990731_262423_380v3.i CALCULATION OF EFFECTIVE WIDTH OF COMPRESSION FLANGE FOR LIPPED CHANNEL IS DEPENDENT TO THE RIGIDITY OF THE LIP Check if w D d and No edge stiffnener needed for for lip stiffener dsds ds’ds’ for for lip stiffener For edge stiffener k=0.425

47. 990731_262423_380v3.i ANALYSIS AND DESIGN OF COLD-FORMED STEELS ARE INTEGRATED PROCEDURE AISI code Design FEM Analysis Stresses Ideas for Inelastic buckling Ideas for local buckling Ideas for stress analysis Ideas for lateral buckling

48. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel Next Steps

49. 990731_262423_380v3.i DESIGN STRENGTH CAN BE CALCULATED VIA COMPLICATED PROCEDURE Sectional Properties Calculate the sectional properties (A, x, y, S, J, I x,I y,I w ) of full section by linear method Elastic Lateral Buckling Moment Inelastic Lateral Buckling Moment Effective Width of Flange and Lip Assume fully effective web and check the effectiveness by iteration Effective Width of Web Effective Sectional Modulus Recalculate the neutral axis until the effective web width is determined Nominal and Design Strength Determine buckling moment and mode using accurate finite element analysis or AISI code Determine inelastic buckling moment using AISI code Determine the effective width of compression flange and edge stiffener The interaction of the local and overall lateral buckling results in a reduction of the lateral strength

50. 990731_262423_380v3.i DESIGN STRENGTH OF CHANNEL BEAM - EXAMPLE PROBLEM 2.5’ Sectional Properties Elastic Critical Moment is calculated from AISI code or FEM Inelastic Critical Moment is calculated from AISI code Nominal Moment is based on the effective sectional modulus By linear method P (4% reduction) (31% reduction)

51. 990731_262423_380v3.i RIGOROUSLY ANALYSE THE TECHNOLOGY TREE OF COLD-FORMED STEEL MEMBER Shear Diaphragms Beam Webs Lateral Buckling Bending + Torsion Flexural Members Cylindrical tubular members Flexural Buckling Torsional Buckling Effective Length Compressive Members Bending Bending Strength Stress Analysis Deflection COLD- FORMED STEEL MEMBER Composite Design Corrugated Sheets Local Buckling Warping Pure Torsion Purlins Wall Studs Compressive Strength Cold Work Local Buckling Effective Sectional Prop. other cross-sections: Distortional Buckling

52. 990731_262423_380v3.i CONTENTS Introduction –Impact of cold-formed steel –Structural Consideration of Channel section Lateral Buckling Stress Analysis Local Buckling & Effective Width Analysis & Design of Cold-formed Channel How do we take care of the combined effects?

53. 990731_262423_380v3.i VARIOUS TYPES OF BUCKLING CAN OCCUR Global Buckling: profile of cross section does not change Distortional Buckling: Lateral deflection of the unsupported flange Local Buckling: Each plate element can buckle

54. 990731_262423_380v3.i DIFFERENT AVAILABLE NUMERICAL METHODS Plate Finite Elements Finite Strip Method Beam Models –Effective Width Concept –Special Constitutive Law –Enriched displacement field –Plate FE with static condensation of d.o.f.’s

55. 990731_262423_380v3.i PLATE FINITE ELEMENTS –Can model local effects –Requires a fine mesh –Practical Difficulties

56. 990731_262423_380v3.i FINITE STRIP METHOD –D.O.F.’s can be reduced –Limited to prismatic simply-supported members with constant forces

57. 990731_262423_380v3.i BEAM MODELS –Good for overall stability –Nondeformability of the profile cross section –Cannot account for local effects  Effective Width Concept:  limited to local buckling  Try to represent the effect rather than the phenomenon itself  Enriched displacement field:  Local deformation of the cross section is superimposed in a displacement field  Assumed that the shape of the local field is unchanged during the process  Plate FE with static condensation of d.o.f.’s:  Modeled as plate finite elements with restrained d.o.f.  Classical beam d.o.f. + magnitude of the local deformation  Timoshenko beam model  Transverse shear deformation

58. 990731_262423_380v3.i CONCLUDING REMARKS  The geometric coupling depends on the shape of the cross section.  Needs fully geometrically nonlinear model to predict the structural behavior accurately. (tremendous efforts)  Beam model seems reasonable, but can be improved by considering local effects or shear deformation.  Consideration of material nonlinearity including inelastic buckling can be achieved by stress analysis or global assumption of plastic process.  Structural members with anisotropic materials (pultruded composites) awaits future attention.

59. 990731_262423_380v3.i