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McMat 2005 2005 Joint ASME/ASCE/SES Conference on Mechanics and Materials Baton Rouge, Louisiana

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

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McMat 2005 2005 Joint ASME/ASCE/SES Conference on Mechanics and Materials Baton Rouge, Louisiana

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

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

  3. local buckling distortional buckling lateral-torsional buckling stability mode identification in a thin-walled member FSM Mcr Lcr l FEM

  4. GBT and modal identification (as Dinar has described!) • 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.

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

  6. 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 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

  7. brief example... implemented in open source matlab-based finite strip software www.ce.jhu.edu/bschafer

  8. decomposition and identification of an I-beam

  9. #1 #2 #3 FSM implementation details... u,v: membrane plane stress w,q: thin plate bending

  10. general displacement vector: d=[U V W Q]T constrained to distortional: d=Rdr, dr=[V] • u(i)-v1,2 relation via membrane assumptions (#1) • u(i-1,i)-Vi-1,i,i+1 relation considering connectivity • u(i-1,i),w(i-1,i)-Ui,Wi by coord. transformation subset of this: u(i-1,i)-Ui,Wi relation • Ui,Wi-Vi-1,i,i+1 through combining above • Qi-Ui,Wi relation through beam analogy (#3) notations: superscript= elements, subscript = nodes, lowercase = local, uppercase = global

  11. u(i)-v1,2 relation FSM shape functions membrane restriction: resulting relation:

  12. u(i-1),(i)-Vi-1,i,i+1 relation element (i-1): element (i): connectivity:

  13. u(i-1),(i)-UiWi relation local-global transform: element (i-1): element (i):

  14. UiWi-Vi-1,i,i+1 relation local-global transform membrane assumption + connectivity

  15. multi-branched: u(i)-v1,2 relation membrane restriction results in:

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

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

  18. multi-branched: UiWi-Vi-1,i,i+1 relation

  19. multi-branched: UiWi-Vi-1,i,i+1 relation multi-branch case leads to additional constraints on V....

  20. 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

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

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

  23. Q-U,V relation

  24. d=Rdr

  25. Why bother? modes  strength

  26. Thin-walled members

  27. Membrane (plane stress) FSM Ke = Kem + Keb

  28. Thin plate bending FSM Ke = Kem + Keb

  29. 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

  30. Special purpose FSM can fail too

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

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