D ISTORTIONAL B UCKLING O F C A ND Z M EMBERS I N B ENDING

1. DISTORTIONAL BUCKLING OFCAND ZMEMBERSIN BENDING Progress Report to AISI Cheng Yu, Benjamin W. Schafer The Johns Hopkins University August 2004

2. Overview • Test Summary • Finite Element Modeling • Extended FE Analysis • Stress Gradient Effects • Conclusions

3. Local buckling tests

4. Distortional buckling tests

5. Test Summary • 5 more tests were performed since last report in February 2004 • Total 24 distortional buckling tests have been done. All available • geometry of sections in the lab have been tested. Comparison with design methods

6. Test Summary - Performance of Direct Strength Method Test results vs. Direct Strength predictions

7. Finite Element Modeling • Shell element S4R for purlins, panel and tubes, solid element C3D8 • for transfer beam. • Geometric imperfection is introduced by the superposition of local • and distortional buckling mode scaled to 25% or 75% CDF. • Residual stress is not considered. • Stress-strain based on average of 3 tensile tests from the flats of every specimen • Modified Riks method and auto Stabilization method in ABAQUS were considered for the postbuckling analysis. The latter has better results and less convergence problems therefore the auto Stabilization is used. • The FE model was verified by the real tests.

8. Finite Element Modeling - Comparison with test results On average: FEM-to-test ratio= 106% for 25% CDF; 93% for 75% CDF --- local buckling tests FEM-to-test ratio= 109% for 25% CDF; 94% for 75% CDF --- distortional buckling tests

9. Extended Finite Element Analysis FEA results vs. Direct Strength predictions

10. p Moment diagram Stress Gradient Effect on Thin Plate - Moment gradient on beams compression Stress diagram of top flange Stress diagram of bottom flange tension

11. Stress Gradient Effect on Thin Plate - Plate buckling Hat section C section Buckling of uniformly compressed rectangular plates

12. Stress Gradient Effect on Thin Plate – Analytical model Stiffened element Stress distribution: Deflection function: (by Libove 1949)

13. Stress Gradient Effect on Thin Plate – Analytical model Unstiffened element Stress distribution:

14. Stress Gradient Effect on Thin Plate – Analytical model Finite element analysis by ABAQUS is used to verify these 3 deflection functions. Bucking shape by FEA Bucking shape by analytical model Average analytical result-to-FEA ratios are Deflection function 1: 102.4% selected Deflection function 2: 99.7% Deflection function 3: 99.6%

15. Stress Gradient Effect on Thin Plate – Stiffened Element Results kmax vs. plate aspect ratio (β) for ss-ss stiffened element (recalculation of Libove’s equations 1949 )

16. Stress Gradient Effect on Thin Plate – Unstiffened Element Results kmax vs. plate aspect ratio (β) for ss-free unstiffened element

17. Stress Gradient Effect on Thin Plate – (r=0) Results 0 Comparison of stiffened and unstiffened elements subject to stress gradient r=0 kmax= buckling coefficient at the maximum stress edge k0= buckling coefficient for plates under uniform compression stress

18. Stress Gradient Effect on Thin Plate – Ultimate strength Winter curve --- ABAQUS r=1 --- plate under uniform compression stress ABAQUS r=0 --- plate under stress gradient, stress is only applied at one end

19. Conclusions • Tests that separate local and distortional buckling are necessary for understanding bending strength. • Current North American Specifications are adequate only for local buckling limit states. • The Direct Strength expressions work well for strength in local and distortional buckling. • Nonlinear finite element analysis with proper imperfections provides a good simulation. • Extended finite element analysis shows that DSM provides reasonable predictions for strengths in local and distortional buckling.

20. Conclusions - continued • An analytical method for calculating the elastic buckling of thin plate under stress gradient is derived and verified by the finite element analysis. • Plate will buckle at higher stress when stress gradient exists. The stress gradient has more influence on the unstiffened element than stiffened element. • Study on the ultimate strength of plate under stress gradient has been initialized. Up-to-date results show Winter’s curve works well for stiffened element under stress gradient. • More work on restraint and influence of moment gradients will be carried out by the aid of the verified finite element model.

21. Acknowledgments • Sponsors • MBMA and AISI • VP Buildings, Dietrich Design Group andClark Steel • People • Sam Phillips – undergraduate RA • Tim Ruth – undergraduate RA • Jack Spangler – technician • James Kelley – technician • Sandor Adany – visiting scholar

22. Stress Gradient Effect on Thin Plate – Energy method Total potential energy: + ( term for the elastic restraint if exists) When buckling happens: • Need two assumptions to solve the elastic buckling stress: • the stress distribution in plate: • the deflection function:

23. Stress Gradient Effect on Thin Plate – Analytical model 3 deflection functions are considered for the unstiffened element: 1. 2. 3.

24. Stress Gradient Effect on Thin Plate – Analytical model The coefficients in the assumed deflection function are determined by applying to the 6 boundary conditions: 1. 2. 3. (no deflection) (no moment) 4. 5. 6. (no shear force)