understanding and classifying local, distortional and global buckling in open thin-walled members...
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Understanding and classifying local, distortional and global buckling in open thin-walled members
by: B.W. Schafer and S. Ádány
SSRC Annual Stability Conference
Montreal, Canada
April 6, 2005
• Motivation and challenges
• Modal definitions based on mechanics
• Implementation
• Examples
Thin-walled members
What are the buckling modes?
• member or global buckling
• plate or local buckling
• other cross-section buckling modes?– distortional buckling?– stiffener buckling?
2
2
2
112
b
tEkfcr
22
KL
EIPcr
Buckling solutions by the finite strip method
• Discretize any thin-walled cross-section that is regular along its length
• The cross-section “strips” are governed by simple mechanics– membrane: plane stress– bending: thin plate theory
• Development similar to FE• “All” modes are captured
finite element finite strip
Y
m y
am
sin
y
0 1 2 30
50
100
150
200
250
300
350
400
Lcr
Mcr
local buckling
100
101
102
103
0
100
200
300
400
500
half-wavelength
load
fac
tor
BUCKLING CURVE
5.0,172.7620.0,133.65
distortional buckling
100
101
102
103
0
100
200
300
400
500
half-wavelength
load
fac
tor
BUCKLING CURVE
5.0,172.7620.0,133.65
lateral-torsional buckling
100
101
102
103
0
100
200
300
400
500
half-wavelength
load
fac
tor
BUCKLING CURVE
5.0,172.7620.0,133.65
Typical modes in a thin-walled beam
Why bother? modes strength
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
slenderness
strength
Elastic buckling
Yield
Global
Local
Distortional
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
slenderness
strength
Elastic buckling
Yield
Global
Local
Distortional
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
slenderness
strength
Elastic buckling
Yield
Global
Local
Distortional
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
slenderness
strength
Elastic buckling
Yield
Global
Local
Distortional
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
slenderness
strength
Elastic buckling
Yield
Global
Local
Distortional
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
slenderness
strength
Elastic buckling
Yield
Global
Local
Distortional
What’s wrong with what we do now?
What mode is it?
Local LTB
?
Are our definitions workable?
• Distortional buckling. A mode of buckling involving change in cross-sectional shape, excluding local buckling
• Not much better than “you know it when you see it”
•definition from the Australian/New Zealand CFS standard,the North American CFS Spec., and the recently agreedupon joint AISC/AISI terminology
We can’t effectively use FEM
• We “need” FEM methods to solve the type of general stability problems people want to solve today– tool of first choice– general boundary conditions– handles changes along the length, e.g., holes in the section
30 nodes in a cross-section100 nodes along the length5 DOF elements15,000 DOF15,000 buckling modes, oy!
• Modal identification in FEM is a disaster
Generalized Beam Theory (GBT)
• GBT is an enriched beam element that performs its solution in a modal basis instead of the usual nodal DOF basis, i.e., the modes are the DOF
• GBT begins with a traditional beam element and then adds “modes” to the deformation field, first Vlasov warping, then modes with more general warping distributions, and finally plate like modes within flat portions of the section
• GBT was first developed by Schardt (1989) then extended by Davies et al. (1994), and more recently by Camotim and Silvestre (2002, ...)
Generalized Beam Theory
• Advantages– modes look “right”– can focus on individual modes or subsets of modes– can identify modes within a more general GBT analysis
• Disadvantages– development is unconventional/non-trivial,
results in the mechanics being partially obscured – not widely available for use in programs– Extension to general purpose FE awkward
• We seek to identify the key mechanical assumptions of GBT and then implement in, FSM, FEM, to enable these methods to perform GBT-like “modal” solutions.
GBT inspired modal definitions
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
Global modes are those deformation patterns that satisfy all three criteria.
#1
#2
#3
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
#1 membrane strains: xy = 0, membrane shear
strains are zero, x = 0, membrane transverse
strains are zero, and v = f(x), long. displacements
are linear in x within an element.
#1
#2
#3
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
#2 warping: y 0,
longitudinal membrane strains/displacements are non-zero along the length.
#1
#2
#3
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
#3 transverse flexure: y = 0,
no flexure in the transverse direction. (cross-section remains rigid!)
#1
#2
#3
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
Distortional modes are those deformation patterns that satisfy criteria #1 and #2, but do not satisfy criterion #3 (i.e., transverse flexure occurs).
#1
#2
#3
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
Local modes are those deformation patterns that satisfy criterion #1, but do not satisfy criterion #2 (i.e., no longitudinal warping occurs) while criterion #3 is irrelevant.
#1
#2
#3
G modes
D modes
L modes
O modes
xy = 0, x = 0, v is linear Yes Yes Yes No
y 0 Yes Yes No -
y = 0 Yes No - -
Other modes (membrane modes ) do not satisfy criterion #1. Note, other modes typically do not exist in GBT, but must exist in FSM or FEM due to the inclusion of DOF for the membrane.
#1
#2
#3
example ofimplementation into FSM
Constrained deformation fields
a
ymsin
u
u
b
x
b
x1)y,x(u
2
1
a
ymcos
v
v
b
x
b
x1)y,x(v
2
1
FSM membrane disp. fields:
0x
ux
0
a
ymsin
b
uu
x
u 21x
a GBT criterion is so
2
1
2
2
1
1
v
v
u
100
001
010
001
v
u
v
u
rRdd therefore or
general FSM
constrained FSM
impact of constrained deformation field
Modal decomposition• Begin with our standard stability (eigen) problem
• Now introduce a set of constraints consistent with a desired modal definition, this is embodied in R
• Pre-multiply by RT and we create a new, reduced stability problem that is in a space with restricted degree of freedom, if we choose R appropriately we can reduce down to as little as one “modal” DOF
rgre RdλKRdK
rgrrer
rgT
reT
dλKdK
RdKλRRdKR
dλKdK ge
examples
lipped channel in compression
• “typical” CFS section
• Buckling modes include – local,
– distortional, and
– global
• Distortional mode is indistinct in a classical FSM analysis
200m
m
50mm20mm
P
t=1.5mm
classical finite strip solution
0
0,2
0,4
0,6
0,8
1
1,2
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
dist.
local
modal decomposition
0
0,2
0,4
0,6
0,8
1
1,2
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
dist.
local
modal identification
0
0,2
0,4
0,6
0,8
1
1,2
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
dist.
local
0
20
40
60
80
100
10 100 1000 10000
mod
es (
%) global
dist
local
other
I-beam cross-section
• textbook I-beam
• Buckling modes include – local (FLB, WLB),
– distortional?, and
– global (LTB)
• If the flange/web juncture translates is it distortional?
200m
m80mm
tw=2mm
tf=10mm
M
classical finite strip solution
0
5
10
15
20
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
local
modal decomposition
0
5
10
15
20
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
local
modal identification
0
5
10
15
20
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
local
0
2
4
6
8
10
10 100 1000 10000buckling length (mm)
0
2040
6080
100
10 100 1000 10000
mod
es (
%) global
distlocalother
concluding thoughts
• Cross-section buckling modes are integral to understanding thin-walled members
• Current methods fail to provide adequate solutions• Inspired by GBT,
mechanics-based definitions of the modes are possible• Formal modal definitions enable
– Modal decomposition (focus on a given mode)– Modal identification (figure out what you have)
within conventional numerical methods, FSM, FEM..• The ability to “turn on” or “turn off” certain mechanical
behavior within an analysis can provide unique insights
• Much work remains, and definitions are not perfect
acknowledgments
• Thomas Cholnoky Foundation
• Hungarian Scientific Research Fund
• U.S., National Science Foundation
varying lip angle in a lipped channel
• lip angle from 0 to 90º• Where is the local –
distortional transition?
200m
m
120mm
10mm
P
t=1mm
?
classical finite strip solution
0
0,05
0,1
0,15
0,2
0,25
10 100 1000 10000buckling length (mm)
Pcr
/Py
(a)
(b)
(c)
(d)
(e)
(f)
= 0º = 18º = 36º = 54º = 72º = 90º
Local? Distortional? L=170mm, =0-36ºLocal? Distortional? L=700mm, =54-90º
0
0,05
0,1
0,15
0,2
0,25
10 100 1000 10000buckling length (mm)
Pcr
/Py
(a)
(b)
(c)
(d)
(e)
(f)
= 0º = 18º = 36º = 54º = 72º = 90º
0
0,1
0,2
0,3
0,4
10 100 1000 10000
(a)
0
20
40
60
80
100
10 100 1000 10000
mod
es (
%)
globaldistlocalother
(b)
0
20
40
60
80
100
10 100 1000 10000buckl. length (mm)
mod
es (
%)
globaldistlocalother
º
What mode is it?
?
lipped channel with a web stiffener
• modified CFS section
• Buckling modes include – local,
– “2” distortional, and
– global
• Distortional mode for the web stiffener and edge stiffener?
200m
m
50mm20mm
P
t=1.5mm
20mm x 4.5mm
classical finite strip solution
0
0,2
0,4
0,6
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
dist.
local
modal decomposition
0
0,2
0,4
0,6
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
dist.
local
modal identification
0
0,2
0,4
0,6
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
all-mode
global
dist.
local
0
0,1
0,2
0,3
0,4
10 100 1000 10000buckling length (mm)(e)
0
20
40
60
80
100
10 100 1000 10000
mod
es (
%)
globaldistlocalother
Coordinate System
FSM Ke = Kem + Keb
• Membrane (plane stress) u
x
b
x
b
u
uYm
1 1
2
v
x
b
x
b
v
v
a
mYm
1 1
2 '
Y
m y
am
sin
x
y
xy
u x
v y
u y v x
B d N d
'
K t B D B dAT
FSM Ke = Kem + Keb
• Thin plate bending
Y
m y
am
sin
K t B D B dAT
w Yx
b
x
bx
x
b
x
b
x
b
x
bx
x
b
x
b
w
wm
13 2
12 3 22
2
3
3
2
2
2
2
3
3
2
2
1
1
2
2
dB
yx
wy
wx
w
xy
y
x
2
2
2
2
2
FSM Ke = Kem + Keb
• Membrane (plane stress) u
x
b
x
b
u
uYm
1 1
2
v
x
b
x
b
v
v
a
mYm
1 1
2 '
Y
m y
am
sin
K t B D B dAT
w Yx
b
x
bx
x
b
x
b
x
b
x
bx
x
b
x
b
w
wm
13 2
12 3 22
2
3
3
2
2
2
2
3
3
2
2
1
1
2
2
FSM Solution
• Ke
• Kg
• Eigen solution
• FSM has all the cross-section modes in there with just a simple plate bending and membrane strip
Classical FSM
• Capable of providing complete solution for all buckling modes of a thin-walled member
• Elements follow simple mechanicsmembrane
• u,v, linear shape functions
• plane stress conditions
bending• w, cubic “beam” shape function
• thin plate theory
• Drawbacks: special boundary conditions, no variation along the length, cannot decompose, nor help identify “mechanics-based” buckling modes
Are our definitions workable?
• Local buckling. A mode of buckling involving plate flexure alone without transverse deformation of the line or lines of intersection of adjoining plates.
• Distortional buckling. A mode of buckling involving change in cross-sectional shape, excluding local buckling
• Flexural-torsional buckling. A mode of buckling in which compression members can bend and twist simultaneously without change of cross-sectional shape.
* definitions from the Australian/New Zealand CFS standard
finite strip method
• Capable of providing complete solution for all buckling modes of a thin-walled member
• Elements follow simple mechanicsbending
• w, cubic “beam” shape function
• thin plate theory
membrane • u,v, linear shape functions
• plane stress conditions
• Drawbacks: special boundary conditions, no variation along the length, cannot decompose, nor help identify “mechanics-based” buckling modes
Special purpose FSM can fail too
0
0,2
0,4
0,6
0,8
1
10 100 1000 10000buckling length (mm)
(Pcr
/Py)
Experiments on cold-formed steel columns
267 columns , = 2.5, = 0.84