1. SSRC 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. Ádány
SSRC Annual Stability Conference
Montreal, Canada
April 6, 2005

2. Motivation and challenges
Modal definitions based on mechanics
Implementation
Examples

3. Thin-walled members

4. What are the buckling modes?
member or global buckling
plate or local buckling
other cross-section buckling modes?
distortional buckling?
stiffener buckling?

5. Buckling solutions by the finite strip method
Discretize any thin-walled cross-section that is regular along its length
The cross-section “strips” are governed by simple mechanics
membrane: plane stress
bending: thin plate theory
Development similar to FE
“All” modes are captured y

6. Typical modes in a thin-walled beam
local buckling
distortional buckling
lateral-torsional buckling
Mcr
Lcr

7. Why bother? modes  strength

8. What’s wrong with what we do now?

9. What mode is it? ?
Local
LTB

10. 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”
definition from the Australian/New Zealand CFS standard,
the North American CFS Spec., and the recently agreed
upon joint AISC/AISI terminology

11. 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
30 nodes in a cross-section
100 nodes along the length
5 DOF elements
15,000 DOF
15,000 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

12. 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, ...)

13. 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 GBT-like “modal” solutions.

14. GBT inspired modal definitions

15. #1 #2 #3
Global 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, and
v = 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 #3
Distortional 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 #3
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.

21. #1 #2 #3
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.

22. example of implementation into FSM

23. Constrained deformation fields
FSM membrane disp. fields:
a GBT criterion is so
therefore or

24. impact of constrained deformation field
general FSM
constrained FSM

25. 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

26. examples

27. lipped channel in compression
“typical” CFS section
Buckling modes include
local,
distortional, and
global
Distortional mode is indistinct in a classical FSM analysis
50mm
20mm
200mm P
t=1.5mm

28. classical finite strip solution

29. modal decomposition

30. modal identification

31. 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?
80mm
tf=10mm
200mm M
tw=2mm

32. classical finite strip solution

33. modal decomposition

34. modal identification

35. 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

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

38. varying lip angle in a lipped channel
lip angle from 0 to 90º
Where is the local – distortional transition?
120mm q
10mm P ?
200mm
t=1mm

39. classical finite strip solutionq
q = 0º
= 18º
= 36º
= 54º
= 72º
= 90º
Local? Distortional? L=700mm, q=54-90º
Local? Distortional? L=170mm, q=0-36º

40. q = 0º
= 18º
= 36º
= 54º
= 72º
= 90º
q=0
q=18º

41. What mode is it? ?

42. 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?
50mm
20mm
200mm P
20mm x 4.5mm
t=1.5mm

43. classical finite strip solution

44. modal decomposition

45. modal identification

46. Coordinate System

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

48. FSM Ke = Kem + Keb
Thin plate bending

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

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

52. 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

53. 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.
* definitions from the Australian/New Zealand CFS standard

54. 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

55. Special purpose FSM can fail too

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