Plate buckling

Eigenvalue methods for computing buckling loads

Plate and shell buckling loads are classically computed by solving the eigenvalue problem $\mathbf {K} -\lambda \mathbf {G} (\mathbf {N} )$ where $\mathbf {K}$ is the shell's/plate's stiffness and $\mathbf {G} (\mathbf {N} )$ is its geometric stiffness, dependent on the in-plane (membrane) "stresses", and $\lambda$ is the proportional load factor.

To compute the buckling load in FEAP, one first solves for the membrane stresses and shell stiffness, then one forms the geometric stiffness, and finally one uses an eigensolver to determine the buckling load(s).

The basic MARCO commands are as follows:

TANGent,,1
GEOMetric
SUBSpace,,5

Optionally one can use ARPAck,,5 (if optionally built).

As an example consider a cantilevered plate that is subjected to an in-plane compression, the buckling factor can be computed from the following input file. With this course mesh the minimum eigenvalue is found to be 1.30888e+4. Thus the buckling force is found to be 1e-3*0.1375*1.30888e+4 = 1.8 in the units of the input file. The validity of the result can be checked using the Euler load $P_{\mathrm {euler} }={\frac {1}{4}}{\frac {\pi ^{2}EI}{12(1-\nu ^{2})L^{2}}}$.