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Understanding and classifying local, distortional and global buckling in open thin-walled members by: B.W. Schafer and S

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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.

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slide1

Understanding and classifying local, distortional and global buckling in open thin-walled membersby: B.W. Schafer and S. Ádány

SSRC Annual Stability Conference

Montreal, Canada

April 6, 2005

slide2
Motivation and challenges
  • Modal definitions based on mechanics
  • Implementation
  • Examples
what are the buckling modes
What are the buckling modes?
  • member or global buckling
  • plate or local buckling
  • other cross-section buckling modes?
    • distortional buckling?
    • stiffener buckling?
buckling solutions by the finite strip method
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

y

typical modes in a thin walled beam

local buckling

distortional buckling

lateral-torsional buckling

Typical modes in a thin-walled beam

Mcr

Lcr

are our definitions workable
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 agreed
  • upon joint AISC/AISI terminology
we can t effectively use fem
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-section

100 nodes along the length

5 DOF elements

15,000 DOF

15,000 buckling modes, oy!

  • Modal identification in FEM is a disaster
generalized beam theory gbt
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
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.
slide16
#1 membrane strains:

gxy = 0, membrane shear strains are zero,

ex = 0, membrane transverse strains are zero, and

v = f(x), long. displacements are linear in x within an element.

#1

#2

#3

slide17
#2 warping:

ey 0,

longitudinal membrane strains/displacements are non-zero along the length.

#1

#2

#3

slide18
#3 transverse flexure:

ky = 0,

no flexure in the transverse direction. (cross-section remains rigid!)

#1

#2

#3

slide19
Distortionalmodes are those deformation patterns that satisfy criteria #1 and #2, but do not satisfy criterion #3 (i.e., transverse flexure occurs).

#1

#2

#3

slide20
Localmodes 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

slide21
Othermodes(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

constrained deformation fields

so

a GBT criterion is

or

therefore

Constrained deformation fields

FSM membrane disp. fields:

impact of constrained deformation field
impact of constrained deformation field

general FSM

constrained FSM

modal decomposition
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
lipped channel in compression
lipped channel in compression
  • “typical” CFS section
  • Buckling modes include
    • local,
    • distortional, and
    • global
  • Distortional mode is indistinct in a classical FSM analysis

50mm

20mm

200mm

P

t=1.5mm

i beam cross section
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?

80mm

tf=10mm

200mm

M

tw=2mm

concluding thoughts
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
acknowledgments
  • Thomas Cholnoky Foundation
  • Hungarian Scientific Research Fund
  • U.S., National Science Foundation
varying lip angle in a lipped channel

?

P

varying lip angle in a lipped channel
  • lip angle from 0 to 90º
  • Where is the local – distortional transition?

120mm

q

10mm

200mm

t=1mm

classical finite strip solution39
classical finite strip solution

q

q = 0º

= 18º

= 36º

= 54º

= 72º

= 90º

Local? Distortional? L=700mm, q=54-90º

Local? Distortional? L=170mm, q=0-36º

slide40

q = 0º

= 18º

= 36º

= 54º

= 72º

= 90º

q=0

q=18º

lipped channel with a web stiffener
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?

50mm

20mm

200mm

P

20mm x 4.5mm

t=1.5mm

fsm solution
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
Classical FSM
  • Capable of providing complete solution for all buckling modes of a thin-walled member
  • Elements follow simple mechanics

membrane

      • 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 workable53
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
finite strip method
  • Capable of providing complete solution for all buckling modes of a thin-walled member
  • Elements follow simple mechanics

bending

      • 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
experiments on cold formed steel columns
Experiments on cold-formed steel columns

267 columns , b = 2.5, f = 0.84