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Mechanics-based modal identification of thin-walled members with multi-branched cross-sections by: S. Ádány and B.W. Schafer McMat Joint ASME/ASCE/SES Conference on Mechanics and Materials Baton Rouge, Louisiana June 2, 2005

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motivation mechanical criteria for modal decomposition / identification framework for FSM implementation brief example details of FSM implementation including multi-branched sections concluding thoughts

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L cr M cr local buckling distortional buckling lateral-torsional buckling stability mode identification in a thin-walled member FEM FSM

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GBT and modal identification 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 (opinion) – not widely available for use in programs – Extension to general purpose FE awkward We identified the key mechanical assumptions of GBT and then implemented them in FSM (FEM) to enable these methods to perform GBT-like “modal” solutions. (as Dinar has described!)

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mechanical assumptions of GBT modes criteria: #1 membrane restriction #2 non-zero warping #3 no transverse bending #1 #2 #3

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FSM modal decomposition (identification) 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 R T 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

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brief example... implemented in open source matlab-based finite strip software

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decomposition and identification of an I-beam

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#1 #2 #3 u,v: membrane plane stress w, : thin plate bending FSM implementation details...

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general displacement vector: d=[U V W ] T constrained to distortional: d=Rd r, d r =[V] u (i) -v 1,2 relation via membrane assumptions (#1) u (i-1,i) -V i-1,i,i+1 relation considering connectivity u (i-1,i),w (i-1,i) -U i,W i by coord. transformation subset of this: u (i-1,i) -U i,W i relation U i,W i -V i-1,i,i+1 through combining above i -U i,W i relation through beam analogy (#3) notations: superscript = elements, subscript = nodes, lowercase = local, uppercase = global

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u (i) -v 1,2 relation FSM shape functions membrane restriction: resulting relation:

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u (i-1),(i) -V i-1,i,i+1 relation element (i-1): element (i): connectivity:

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u (i-1),(i) -U i W i relation local-global transform: element (i-1): element (i):

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U i W i -V i-1,i,i+1 relation local-global transform membrane assumption + connectivity

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multi-branched: u (i) -v 1,2 relation membrane restriction results in:

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multi-branched: u (i.1),(i.m) -V i,..,m relation

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multi-branched: u (i-1),(i) -U i W i relation (cont.) single-branched: multi-branched: The multi-branched case is over-determined (heart of the issue for a multi-branched section):

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multi-branched: U i W i -V i-1,i,i+1 relation

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multi-branch case leads to additional constraints on V....

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concluding thoughts Current general purpose FSM (FEM) methods are uncapable of modal identification / decomposition for thin-walled member stability modes Inspired by GBT, the modes (i.e., G, D, L, O classes of modes) are postulated as mechanical constraints Modal definitions are implemented in an FSM context for singly and multi-branched sections Formal modal definitions enable FSM to perform – Modal decomposition (focus on a given mode) – Modal identification (figure out what you have) Much work remains, and definitions are not perfect

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acknowledgments Thomas Cholnoky Foundation Hungarian Scientific Research Fund U.S., National Science Foundation

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-U,V relation U,W displacements reconciled through and beam analogy U,W are “support displacements”

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Q-U,V relation

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d=Rd r

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Why bother? modes strength

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Thin-walled members

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FSM K e = K em + K eb Membrane (plane stress)

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FSM K e = K em + K eb Thin plate bending

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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

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Special purpose FSM can fail too

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

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