Member Stability Solutions by the Finite Strip Method
Cross-section stability of open thin-walled members may be readily examined using the finite strip method. In a conventional stability solution the member is modeled as an inter-connected series of
M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 411-422.
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strips, as highlighted in Figure 1. The nodal lines of the strips have four degrees of freedom (DOF) each. In the local coordinates of the strip, the membrane DOF (ui, vi) follow plane stress assumptions, while the bending DOF (wi,0i) follow thin plate bending assumptions. The membrane DOF are allowed to vary linearly in the transverse direction, while the bending DOF vary as a cubic (i. e., the typical beam shape function). Longitudinal deformation typically employs a trigonometric function, in its simplest form a single half-sine wave is enforced. The strips are transformed to global coordinates and assembled in a conventional manner to form the global stiffness matrices. A complete discussion of the finite strip method is available in Cheung and Tham (1998) and explicit details for construction of the elastic and geometric stiffness matrices as used in the open source program CUFSM are available in Schafer (1997).
The desired stability solution takes the form of an eigenvalue problem:
Kd =^Kgd (1)
where K is the global elastic stiffness matrix, Kg is the global geometric stiffness matrix, X is the buckling load multipliers, and d is the buckling mode shapes. K is dependent on cross-section geometry and material, while Kg is dependent on geometry and the applied longitudinal stress. The size of the eigenvalue problem is equal to 4n, where n is the number of nodal lines. Since both K and Kg vary as a function of length, the conventional approach is to sweep through all lengths of practical interest and construct the FSM buckling load multiplier vs. half-wavelength curve. Alternatively, one could fix the length and instead sweep through all sine wave “frequencies” of interest; this is in essence the approach in a conventional finite element method (FEM) stability analysis.
As an example of the information gained from a typical FSM analysis, consider the C-section of example (a) as shown in Figure 2. The developed FSM model has numerous internal nodal lines to ensure the accuracy of the local buckling solution. The model has a total of 21 nodal lines, and thus 84 independent DOF. The eigenvalue problem size is thus 84×84. Further, the eigen solution is performed at 59 different lengths to generate the results of Figure 2b. The analysis results demonstrate FSM’s ability to capture all cross-section stability modes of interest, from local plate instabilities to global member instability, and FSM provides a means to perform an initial classification. Based on the halfwavelength, the presence of minima, and the observed cross-section deformations, out of 84 possible buckling modes examined at 59 different lengths, three classes of buckling modes are approximately defined: local, distortional, and global.
FSM greatly reduces the problem size below a conventional FEM (shell/plate element) stability analysis. However, the further conceptual reduction of the solution (e. g., our 84×84 eigen problem solved 59 times) down to three buckling classes: local, distortional, and global, is necessary for design. Without this reduction the possibilities remain too numerous, and it is impractical to provide engineers with reasonable guidelines on the post-buckling and collapse response of all the modes. As detailed in
Schafer and Adany (2005) exceptions exist where the FSM analysis is not sufficient to definitively identify the classes. If the buckling classes are defined properly from the start it should be possible to perform the solution directly for classes of modes; instead of indirectly as is done in a conventional FSM analysis. In this way, the conceptual reduction performed at the conclusion of an FSM analysis could become a mechanical reduction performed at the beginning of an analysis.