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
μ
σ
μ
σ
μ
σ
μ
σ
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
M
test
/M
y
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 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.
loading point
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
web slenderness =  web = (f y/f cr_web)
0.5
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
Extended Finite Element Analysis
1.1
1
0.9
M
test
/M
y
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
0.5
4
2
0
3.5
0
1
2
3
4
5
6
7
8
9
Stress diagram of top flange
3
2.5
2
10
1
0.5
0
0
-1
-0.5
-2
1.5
1
-1
-3
-1.5
0.5
0
-4
1
0
1
2
3
4
5
6
7
8
Stress diagram of bottom flange
9
-2
10
0.8
10
0.6
8
-2.5
6
0.4
4
0.2
2
0 0
tension
-3
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

o

 min
X
o
simply supported
b

 max
a

Y
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
mx
ny
Deflection function:
(by Libove 1949)
w   wmn sin
sin
a
b
m 1 n 1
Stress Gradient Effect on Thin Plate –
Analytical model

o

X
o
elastic restrained edge
 min
b
 max
a
free edge
Y
Stress distribution:
Unstiffened element
 (1  r ) x

 x   max 
 r
 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 model
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
kmax vs. plate aspect ratio (β) for ss-ss stiffened element
(recalculation of Libove’s equations 1949 )
Stress Gradient Effect on Thin Plate –
Unstiffened Element Results
kmax vs. plate aspect ratio (β) 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=0
kmax= buckling coefficient at the maximum stress edge
k0= buckling coefficient for plates under uniform compression stress
Stress Gradient Effect on Thin Plate –
Ultimate strength
1.2
1.0
Winter curve
0.8
ABAQUS r=1
ABAQUS r=0
0.6
0.4
0.2
0.0
0.00
0.50
1.00
1.50
2.00
 0.22 


1 
  1
Winter curve -- 

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 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
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
 xy   

2
a



S
w
( term for the elastic restraint if exists)
+     dx
2 0  y  y 0 


b a
2
2
b a


t

w

w
w w 
 
T      x     y    2 xy
 dxdy
2 0 0   x 
x y 
 y 


When buckling happens:   0
i  1, 2 N 
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
   ix 
Sb  y 
y
y
y






1. w   wi  

   a1    a 2    a3     sin 
b
b
 b     a 
i 1
 b 2a3 D  b 
N
2. w 
 ix 
2
3
4
w
c
y

c
y

c
y

c
y
sin



i i1
i2
i3
i4
 a 
i 1
N
N

 




 ix 

 a 
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 
i 1
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.
( w) x 0  0
2.
( w) x a  0
 w
 w
4. D
 y 2   x 2   0

 y b
2
2
( w) y 0  0 (no deflection)
3.
(no moment)
 2w
 w 
2w 


 
D



S
5. 
2
2 
x  y 0
 y  y 0
 y

3w
3w 

6. D  3  (2   ) 2 
0
x y  y b
 y
o
X
o
elastic restrained edge
 min
(no shear force)
b
a
free edge
Y
 max
