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

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**1. **Buckling mode decomposition and identification of open thin-walled members the constrained finite strip method (cFSM) B.W. Schafer Dept. of Civil Engineering Johns Hopkins University Structural Stability
May 1, 2006

**2. **acknowledgments Sandor Ádány Budapest University of Technology and Economics
Cheng Yu University 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 Peköz 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, Höglund 1980, König 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 90º
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 90º
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 Peköz 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, Höglund 1980, König 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?