Buckling mode decomposition and identification of open thin-walled members the constrained finite strip method cFSM B.W

1. Buckling mode decomposition and identification of open thin-walled membersthe constrained finite strip method (cFSM)B.W. SchaferDept. of Civil EngineeringJohns Hopkins University Structural Stability May 1, 2006

2. acknowledgments Sandor �d�nyBudapest University of Technology and Economics Cheng YuUniversity of North Texas National Science Foundation American Iron and Steel Institute Metal Building Manufacturers Association Thomas Cholnoky Foundation Hungarian Scientific Research Fund

3. Introduction to thin-walled members Motivation and challenges Mechanics-based modal definitions Modal decomposition and identification Implementation and cFSM Examples

4. thin-walled members and applications cold-formed steel framing

5. thin-walled members and applications cold-formed steel trusses and decks

6. thin-walled members and applications Hot-rolled steel frames, e.g., metal buildings

7. thin-walled members and applications aluminum members, plastic members

8. thin-walled members and applications one answer to costly materials is the creation of thin-walled members and systems. thin-walled members suffer from cross-section instability, and that makes their behavior and design far more complex (interesting!) than typical �compact� sections used in civil/structural engineering.

9. what are these instabilities/modes? member or global buckling plate or local buckling other cross-section buckling modes? distortional buckling stiffener buckling

10. buckling solutions by the finite strip method

11. typical modes in a thin-walled beam

12. Why are elastic buckling modes important?

13. tests on C- and Z-section CFS beams (Yu and Schafer 2004, 2005)

15. why bother? modes ? strength

16. Direct Strength Development 267 Columns Kwon and Hancock 1992, Lau and Hancock 1987, Loughlan 1979,Miller and Pek�z 1994 Mulligan 1983, Polyzois et al. 1993, Thomasson 1978 569 Beams (C & Z) Cohen 1987, Ellifritt et al. 1997, LaBoube and Yu 1978, Moreyara 1993,Phung and Yu 1978, Rogers 1995, Schardt and Schrade 1982, Schuster 1992,Shan 1994, Willis and Wallace 1990 (Hats & Deck) Acharya 1997, Bernard 1993, Desmond 1977, H�glund 1980,K�nig 1978, Papazian et al. 1994

17. Columns

18. Beams

19. Reliability

20. What�s wrong with what we do now?

21. 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. Lateral-torsional buckling. A mode of buckling in which flexural members can bend and twist simultaneously without change of cross-sectional shape.

22. 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�

23. so, what mode is it?

24. 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 A US Supreme Court Judge � Justice Potter Stewart � once said about pornography that �you know it when you see it�. A US Supreme Court Judge � Justice Potter Stewart � once said about pornography that �you know it when you see it�.

25. special purpose finite strip can fail too

26. we need an efficient means to identify thin-walled member buckling modes: modal identification it would be advantageous if we could use such definitions to focus our analysis on a pre-selected type of behavior/mode: modal decomposition

27. 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, ...)

28. 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 �modal� solutions.

29. mechanics-based modal buckling definitions

30. Global modes are those deformation patterns that satisfy all three criteria.

31. #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.

32. #2 warping: ey�?�0, longitudinal membrane strains/displacements are non-zero along the length.

33. #3 transverse flexure: ky�=�0, no flexure in the transverse direction. (cross-section remains rigid!)

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

35. 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.

36. 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.

38. an example(to whet the appetite before the derivation)

39. lipped channel column example FSM DOF: 4 per node, total of 24

40. G and D deformation modes

41. L deformation modes

42. O deformation modes

43. 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

44. modal decomposition

45. modal identification

46. Note on L deformation modes

47. implementation into FSM

48. FSM implementation details...

49. 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) V is all main nodal line V for single branched, but less (by branches) for multi-branched, V not zero is criterion 2V is all main nodal line V for single branched, but less (by branches) for multi-branched, V not zero is criterion 2

50. u(i)-v1,2 relation

51. impact of membrane restriction

52. u(i-1),(i)-Vi-1,i,i+1 relation

53. u(i-1),(i)-UiWi relation

54. UiWi-Vi-1,i,i+1 relation

55. 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) V is all main nodal line V for single branched, but less (by branches) for multi-branchedV is all main nodal line V for single branched, but less (by branches) for multi-branched

56. further examples

57. lipped channel in compression �typical� CFS section Buckling modes include local, distortional, and global Distortional mode is indistinct in a classical FSM analysis

58. classical finite strip solution

59. modal decomposition

60. modal identification

61. 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?

62. classical finite strip solution

63. modal decomposition

64. modal identification

65. varying lip angle in a lipped channel lip angle from 0 to 90� Where is the local � distortional transition?

66. classical finite strip solution

68. 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?

69. classical finite strip solution

70. modal decomposition

71. modal identification

72. 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

75. varying lip angle in a lipped channel lip angle from 0 to 90� Where is the local � distortional transition?

76. classical finite strip solution

78. What mode is it?

79. 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?

80. classical finite strip solution

81. modal decomposition

82. modal identification

83. Coordinate System

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

85. FSM Solution Ke Kg Eigen solution FSM has all the cross-section modes in there with just a simple plate bending and membrane strip

87. 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

88. 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.

89. 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

90. Special purpose FSM can fail too

91. Experiments on cold-formed steel columns

92. Direct Strength Development 267 Columns Kwon and Hancock 1992, Lau and Hancock 1987, Loughlan 1979,Miller and Pek�z 1994 Mulligan 1983, Polyzois et al. 1993, Thomasson 1978 569 Beams (C & Z) Cohen 1987, Ellifritt et al. 1997, LaBoube and Yu 1978, Moreyara 1993,Phung and Yu 1978, Rogers 1995, Schardt and Schrade 1982, Schuster 1992,Shan 1994, Willis and Wallace 1990 (Hats & Deck) Acharya 1997, Bernard 1993, Desmond 1977, H�glund 1980,K�nig 1978, Papazian et al. 1994

93. Columns

94. Beams

95. Reliability

97. brief example...

98. decomposition and identification of an I-beam

102. Constrained deformation fields

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

104. FSM Ke = Kem + Keb Thin plate bending

105. what mode is it?