Presentation on theme: "SSRC Annual Stability Conference Montreal, Canada April 6, 2005"— Presentation transcript:
1SSRC 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. ÁdánySSRC Annual Stability ConferenceMontreal, CanadaApril 6, 2005
2Motivation and challenges Modal definitions based on mechanicsImplementationExamples
4What are the buckling modes? member or global bucklingplate or local bucklingother cross-section buckling modes?distortional buckling?stiffener buckling?
5Buckling solutions by the finite strip method Discretize any thin-walled cross-section that is regular along its lengthThe cross-section “strips” are governed by simple mechanicsmembrane: plane stressbending: thin plate theoryDevelopment similar to FE“All” modes are capturedy
6Typical modes in a thin-walled beam local bucklingdistortional bucklinglateral-torsional bucklingMcrLcr
10Are our definitions workable? Distortional buckling. A mode of buckling involving change in cross-sectional shape, excluding local bucklingNot 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
11We can’t effectively use FEM We “need” FEM methods to solve the type of general stability problems people want to solve todaytool of first choicegeneral boundary conditionshandles changes along the length, e.g., holes in the section30 nodes in a cross-section100 nodes along the length5 DOF elements15,000 DOF15,000 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
12Generalized 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 DOFGBT 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 sectionGBT was first developed by Schardt (1989) then extended by Davies et al. (1994), and more recently by Camotim and Silvestre (2002, ...)
13Generalized Beam Theory Advantagesmodes look “right”can focus on individual modes or subsets of modescan identify modes within a more general GBT analysisDisadvantagesdevelopment is unconventional/non-trivial, results in the mechanics being partially obscurednot widely available for use in programsExtension to general purpose FE awkwardWe 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.
15#1#2#3Global modes are those deformation patterns that satisfy all three criteria.
16#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, andv = f(x), long. displacements are linear in x within an element.
17#1#2#3#2 warping:ey 0,longitudinal membrane strains/displacements are non-zero along the length.
18#3 transverse flexure: ky = 0, #1#2#3#3 transverse flexure:ky = 0,no flexure in the transverse direction. (cross-section remains rigid!)
19#1#2#3Distortional modes are those deformation patterns that satisfy criteria #1 and #2, but do not satisfy criterion #3 (i.e., transverse flexure occurs).
20#1#2#3Local 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.
21#1#2#3Other 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.
24impact of constrained deformation field general FSMconstrained FSM
25Modal decomposition Begin with our standard stability (eigen) problem Now introduce a set of constraints consistent with a desired modal definition, this is embodied in RPre-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
35concluding thoughtsCross-section buckling modes are integral to understanding thin-walled membersCurrent methods fail to provide adequate solutionsInspired by GBT, mechanics-based definitions of the modes are possibleFormal modal definitions enableModal 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 insightsMuch work remains, and definitions are not perfect
36acknowledgments Thomas Cholnoky Foundation Hungarian Scientific Research FundU.S., National Science Foundation
42lipped channel with a web stiffener modified CFS sectionBuckling modes includelocal,“2” distortional, andglobalDistortional mode for the web stiffener and edge stiffener?50mm20mm200mmP20mm x 4.5mmt=1.5mm
52Classical FSMCapable of providing complete solution for all buckling modes of a thin-walled memberElements follow simple mechanicsmembraneu,v, linear shape functionsplane stress conditionsbendingw, cubic “beam” shape functionthin plate theoryDrawbacks: special boundary conditions, no variation along the length, cannot decompose, nor help identify “mechanics-based” buckling modes
53Are 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 bucklingFlexural-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
54finite strip methodCapable of providing complete solution for all buckling modes of a thin-walled memberElements follow simple mechanicsbendingw, cubic “beam” shape functionthin plate theorymembraneu,v, linear shape functionsplane stress conditionsDrawbacks: special boundary conditions, no variation along the length, cannot decompose, nor help identify “mechanics-based” buckling modes