Cheng Yu, Benjamin W. Schafer The Johns Hopkins University August 2004 D ISTORTIONAL B UCKLING O F C...
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Transcript of Cheng Yu, Benjamin W. Schafer The Johns Hopkins University August 2004 D ISTORTIONAL B UCKLING O F C...
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
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)