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

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