Discussion of Global Modes
A comparison of different solution methods for global modes highlight some important differences between constrained FSM, conventional FSM, and classically used analytical solutions. In Figure 7a the constrained FSM solution is compared to the classical analytical solution for flexural-torsional buckling (Timoshenko and Gere 1936). In the Figure, “Theory G1 – G3” represent the first three roots of the classical cubic equation that is solved for flexural-torsional buckling, and “Gi – G3” represent the first three eigenvalues of an FSM model constrained to only the deformations consistent with global modes (see Figure 4a and b). Constrained FSM gives higher critical forces. The difference is near 10% for any buckling lengths of practical importance. This difference is a direct consequence of the basic assumptions between beam and plate theory.
Constitutive Relations and Global Modes
In classical analytical solutions for global flexural buckling only the longitudinal normal stresses are considered, while the transverse normal stresses are assumed negligible. The problem is handled by a beam model, and only a one-dimensional constitutive relation, e. g.:
ax = 0
a y = Es y az = 0
Although rarely considered, transverse strains are not zero, even if they are small. FSM calculations are predicated on a plate model, and thus a two-dimensional constitutive relation, e. g.:
It is assumed that the plates that form the cross-section are thin enough (compared to their width and length) that the Poisson effect should be considered. Thus, neither cx nor cy is negligible, while cz is implicitly assumed to be zero. If we calculate the pure global or distortional modes by applying the constraint matrix (R), we generate deformations that must satisfy the conditions that ex and yxy are zero. Thus, for constrained FSM the normal stresses can be expressed as follows:
The difference between the beam model, Eq. (14), and the constrained FSM model, Eq. (16), is conspicuous. For longitudinal stresses the difference is equal to 1/(1-v2). Considering that for steel the Poisson’s ratio is approximately 0.3, the difference in the longitudinal stresses is approximately 10%. For flexural (global) buckling the only non-zero stress component (according to a beam model) is the longitudinal stress, consequently this difference in the longitudinal stresses directly transfers into a difference in the critical loads. The difference between the flexural buckling load calculated by the beam model and the constrained FSM is 1/(1-v2) or 10% for v=0.3. Flexural-torsional global buckling involves shear stresses which are not affected by the Poisson’s ratio, therefore, a smaller difference (i. e., < 10%) occurs between analytical and constrained FSM solutions.