DISTORTIONAL BUCKLING OF C AND Z MEMBERS IN BENDING Progress Report to AISI Cheng Yu, Benjamin W. Schafer The Johns Hopkins University August 2004 Overview Test Summary Finite Element Modeling Extended FE Analysis Stress Gradient Effects Conclusions Local buckling tests Distortional buckling tests 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 Controlling specimens Second specimens

Controlling Distortional specimens buckling Second tests specimens Local buckling tests Mtest/ MAISI 1.01 0.04 1.00 0.05 0.84 0.08 0.85 0.08 Mtest/ MS136 1.06 0.04 1.05 0.06 0.92 0.08

0.90 0.07 Mtest/ MNAS 1.02 0.05 1.01 0.07 0.88 0.09 0.87 0.09 Mtest/ Mtest/ MAS/NZS MEN1993 1.01 1.01 0.04 0.06 1.00 1.01 0.05 0.06 1.02 0.96 0.07 0.09 1.00 0.94 0.07 0.09 Mtest/ MDSM 1.03 0.06

1.03 0.07 1.02 0.07 1.00 0.07 Test Summary Performance of Direct Strength Method 1.3 DSM local curve DSM distortional curve Local buckling tests Distortional buckling tests 1.2 1.1 1 Mtest/My 0.9 0.8 0.7 0.6 0.5 0.4 0 0.2

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 (My /Mcr) Test results vs. Direct Strength predictions 2 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 eve 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 use The FE model was verified by the real tests.

lo a d in g p o in t Finite Element Modeling Comparison with test results 1.4 FEM-to-test ratio 1.2 1 0.8 0.6 0.4 25% CDF 0.2 75% CDF 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.5 web slenderness = web = (f y/f cr_web) On average: FEM-to-test ratio= 106% for 25% CDF; 93% for 75% CDF --- local buckling tests Extended Finite Element Analysis 1.1 1 0.9 Mtest/My 0.8 0.7 DSM Local curve DSM Distortional curve Local buckling failures Distortional buckling failures 0.6 0.5 0.4 0 0.2 0.4

0.6 0.8 1 1.2 1.4 1.6 1.8 0.5 (My /Mcr) FEA results vs. Direct Strength predictions 2 Stress Gradient Effect on Thin Plate Moment gradient on beams p 5 4 . 5 4 3 . 5 Moment diagram 3 2

. 5 compression 2 1 . 5 5 1 .5 1 4 . 5 1 0 . 4 5 2 3 . 5 0 0 1 2 3

4 5 6 7 8 Stress diagram of top flange 3 9 1 0 1 0 .5 0 0 2 -1 -0 .5 1 . 5 -2

2 . 5 1 0 . 5 0 0 -1 -3 -1 .5 -4 1 1 2 3 4 5 6 7 8

Stress diagram of bottom flange -2 9 1 0 0 .8 10 0 .6 8 -2 .5 6 0 .4 4 0 .2 2 0 0 -3 tension Stress Gradient Effect on Thin Plate Plate buckling Hat section

C section Buckling of uniformly compressed rectangular plates Stress Gradient Effect on Thin Plate Analytical model Stiffened element min (1 r ) x Stress distribution: x max r y 0 r a max (1 r ) b xy max y a 2 M N mx ny Deflection function: w wmn sin sin (by Libove 1949) a b m 1 n 1 Stress Gradient Effect on Thin Plate Analytical model

Unstiffened element Stress distribution: (1 r ) x r x max a xy max b(1 r ) y 1 a b y 0 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 mode Average analytical result-to-FEA ratios are Deflection function 1: 102.4% Deflection function 2: 99.7%

Deflection function 3: 99.6% selected Stress Gradient Effect on Thin Plate Stiffened Element Results max ss ss max ss ss max max ss ( max ) cr k max 2D b 2t max ss kmax vs. plate aspect ratio () for ss-ss stiffened element (recalculation of Liboves equations 1949 ) Stress Gradient Effect on Thin Plate Unstiffened Element Results max ss

free max ss ( max ) cr k max free max max 2D b 2t max ss free kmax vs. plate aspect ratio () for ss-free unstiffened element Stress Gradient Effect on Thin Plate (r=0) Results 0 Comparison of stiffened and unstiffened elements subject to stress gradient r= kmax= buckling coefficient at the maximum stress edge k0= buckling coefficient for plates under uniform compression stress 1.2 Stress Gradient Effect on Thin Plate Ultimate strength 1.0 Winter curve

0.8 ABAQUS r=1 ABAQUS r=0 0.6 0.4 0.2 0.0 0.00 0.50 Winter curve --- 1.00 1.50 2.00 0.22 1 1 2.50 3.00 3.50 4.00 4.50

ABAQUS r=1 --- plate under uniform compression stress ABAQUS r=0 --- plate under stress gradient, stress is only applied at one end 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. 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 Winters 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. Acknowledgments Sponsors MBMA and AISI

VP Buildings, Dietrich Design Group and Clark Steel People Sam Phillips undergraduate RA Tim Ruth undergraduate RA Jack Spangler technician James Kelley technician Sandor Adany visiting scholar Stress Gradient Effect on Thin Plate Energy method Total potential energy: U T 2 2 2 2 2 2 2 D w w w w w

dxdy U 2 2 2(1 ) 2 2 2 0 0 x y x y xy 2 a S w ( term for the elastic restraint if exists) + 2 y dx y 0 0 b a 2 2 w t w w w 2 xy T x dxdy y 2 0 0 x x y y

When buckling happens: 0 i 1, 2 N b a wi Need two assumptions to solve the elastic buckling stress: the stress distribution in plate: x , y , the deflection function: w Stress Gradient Effect on Thin Plate Analytical model 3 deflection functions are considered for the unstiffened element: 5 4 3 2 y Sb y y y y ix 1. w wi a1 a 2 a3 sin b 2 a D b

b b b a i 1 3 N N ix 2. w wi ci1 y ci 2 y ci 3 y ci 4 y sin a i 1 N 2 3 4 ix 3. w wi1 pi1 y pi 2 y 2 pi 3 y 3 pi 4 y 4 wi 2 qi1 y 3 qi 2 y 4 qi 3 y 5 sin a i 1

Stress Gradient Effect on Thin Plate Analytical model The coefficients in the assumed deflection function are determined by applying the 6 boundary conditions: 1. ( w) x 0 0 2 2. 2 ( w) x a 0 w w 4. D 2 0 2 x y b y 3. ( w) y 0 0 (no deflection) (no moment)

2 w w 2 w D S 5. 2 2 x y 0 y y 0 y 3 w 3 w 6. D 3 ( 2 ) 2 0 x y y b y (no shear force)