SSRC Annual Stability Conference Montreal, Canada April 6, 2005 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?

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 Mcr Lcr

Why bother? modes  strength

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 agreed upon 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-section 100 nodes along the length 5 DOF elements 15,000 DOF 15,000 buckling modes, oy! A US Supreme Court Judge – Justice Potter Stewart – once said about pornography that ‘you know it when you see it’. 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

#1 #2 #3 Global modes are those deformation patterns that satisfy all three criteria.

#1 membrane strains: gxy = 0, membrane shear strains are zero, #2 #3 #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 #2 warping: ey  0, longitudinal membrane strains/displacements are non-zero along the length.

#3 transverse flexure: ky = 0, #1 #2 #3 #3 transverse flexure: ky = 0, no flexure in the transverse direction. (cross-section remains rigid!)

#1 #2 #3 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 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 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.

example of implementation into FSM

Constrained deformation fields FSM membrane disp. fields: a GBT criterion is so therefore or

impact of constrained deformation field general FSM constrained FSM

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

examples

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

classical finite strip solution

modal decomposition

modal identification

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

classical finite strip solution

modal decomposition

modal identification

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? 120mm q 10mm P ? 200mm t=1mm

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

q = 0º = 18º = 36º = 54º = 72º = 90º q=0 q=18º

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? 50mm 20mm 200mm P 20mm x 4.5mm t=1.5mm

classical finite strip solution

modal decomposition

modal identification

Coordinate System

FSM Ke = Kem + Keb Membrane (plane stress)

FSM Ke = Kem + Keb Thin plate bending

FSM Ke = Kem + Keb Membrane (plane stress)

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 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 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 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

Special purpose FSM can fail too

Experiments on cold-formed steel columns 267 columns , b = 2.5, f = 0.84