Cheng Yu, Benjamin W. Schafer

The Johns Hopkins University

August 2004

DISTORTIONAL BUCKLING OF C AND Z MEMBERS

IN BENDING

Progress Report to AISI

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.

Mtest/ MAISI

Mtest/ MS136

Mtest/ MNAS

Mtest/ MAS/NZS

Mtest/ MEN1993

Mtest/ MDSM

μ 1.01 1.06 1.02 1.01 1.01 1.03 Controlling specimens σ 0.04 0.04 0.05 0.04 0.06 0.06

μ 1.00 1.05 1.01 1.00 1.01 1.03

Local buckling

tests Second specimens σ 0.05 0.06 0.07 0.05 0.06 0.07

μ 0.84 0.92 0.88 1.02 0.96 1.02 Controlling specimens σ 0.08 0.08 0.09 0.07 0.09 0.07

μ 0.85 0.90 0.87 1.00 0.94 1.00

Distortional buckling

tests Second specimens σ 0.08 0.07 0.09 0.07 0.09 0.07

Comparison with design methods

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Mte

st/M

y

(My/Mcr)0.5

DSM local curveDSM distortional curveLocal buckling testsDistortional buckling tests

Test results vs. Direct Strength predictions

Test Summary -Performance of Direct Strength Method

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.lo a d in g p o in t

Finite Element Modeling -Comparison with test results

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

web slenderness = web = (fy/fcr_web)0.5

FE

M-t

o-t

est

ra

tio

25% CDF

75% CDF

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

On average:

Extended Finite Element Analysis

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Mte

st/M

y

(My/M

cr)0.5

DSM Local curveDSM Distortional curveLocal buckling failuresDistortional buckling failures

FEA results vs. Direct Strength predictions

Stress Gradient Effect on Thin Plate -Moment gradient on beams

Stress diagram of top flange

p

Moment diagram

- 3

- 2 . 5

- 2

- 1 . 5

- 1

- 0 . 5

0

0 . 5

1

1 . 5

02

46

81 0

0

0 . 2

0 . 4

0 . 6

0 . 8

1- 4

- 3

- 2

- 1

0

1

2

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Stress diagram of bottom flange

compression

tension

Stress Gradient Effect on Thin Plate -Plate buckling

Buckling of uniformly compressed rectangular plates

Hat section

C section

Stress Gradient Effect on Thin Plate –Analytical model

Stiffened element

M

m

N

nmn b

yn

a

xmww

1 1

sinsinDeflection function: (by Libove 1949)

Stress distribution:

r

a

xrx

)1(max 0y

max

min

r

y

b

a

rxy 2

)1(max

Unstiffened element

Stress Gradient Effect on Thin Plate –Analytical model

Stress distribution:

r

a

xrx

)1(max 0y

b

y

a

rbxy 1

)1(max

Finite element analysis by ABAQUS is used to verify these 3 deflection functions.

Stress Gradient Effect on Thin Plate –Analytical model

Bucking shape by FEA

Average analytical result-to-FEA ratios are

Bucking shape by analytical model

Deflection function 1: 102.4%

Deflection function 2: 99.7%

Deflection function 3: 99.6%

selected

kmax vs. plate aspect ratio (β) for ss-ss stiffened element

(recalculation of Libove’s equations 1949 )

Stress Gradient Effect on Thin Plate –Stiffened Element Results

ss

ss

ss

ss

ss

ss

max

max

max

max

tb

Dkcr 2

2

maxmax )(

max

kmax vs. plate aspect ratio (β) for ss-free unstiffened element

Stress Gradient Effect on Thin Plate –Unstiffened Element Results

ss

free

ss

free

ss

free

max

max

max

max

tb

Dkcr 2

2

maxmax )(

max

Stress Gradient Effect on Thin Plate –(r=0) Results

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

0

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

122.0

1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

Winter curve

ABAQUS r=1

ABAQUS r=0

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 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.

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

TU

dxdyyx

w

y

w

x

w

y

w

x

wDU

b a

0 0

22

2

2

2

22

2

2

2

2

)1(22

Total potential energy:

b a

xyyx dxdyy

w

x

w

y

w

x

wtT

0 0

22

22

Niwi

2,10 When buckling happens:

Need two assumptions to solve the elastic buckling stress:

• the stress distribution in plate:

• the deflection function:

,, yx

w

a

y

dxy

wS

0

2

02+ ( term for the elastic restraint if exists)

3 deflection functions are considered for the unstiffened element:

Stress Gradient Effect on Thin Plate –Analytical model

1.

a

xi

b

ya

b

ya

b

ya

b

y

Da

Sb

b

yww

N

ii

sin

21

2

3

3

2

4

1

5

3

2.

3.

a

xiycycycycww

N

iiiiii

sin

1

44

33

221

N

iiiiiiiiii a

xiyqyqyqwypypypypww

1

53

42

312

44

33

2211 sin

The coefficients in the assumed deflection function are determined by applying to

the 6 boundary conditions:

Stress Gradient Effect on Thin Plate –Analytical model

1. 2. 3.0)( 0 xw 0)( axw 0)( 0 yw

002

2

2

2

yyy

wS

x

w

y

wD 0

2

2

2

2

byx

w

y

wD

0)2(2

3

3

3

byyx

w

y

wD

4. 5.

6.

(no deflection)

(no moment)

(no shear force)