Keywords: non-linear, buckling, adaptive bisection, arc-length method.

Conventional nonlinear buckling analysis is performed by extrapolation in the vicinity of the buckling point. MSC.Nastran provides such a capability assuming that the tangential stiffness is proportional to the displacement increment[1]. The accuracy of the solution from this extrapolation procedure depends heavily on the data points used for extrapolation. This lack of certainty dictates that the nonlinear buckling analysis be conducted repeatedly along the incremental processes until an instability condition is encountered. MSC.Nastran provides a user-friendly procedure to allow repetitive buckling analysis.

There are two more ways to estimate the critical buckling load. One method is the arc-length method that can provide solutions past the critical buckling load into the post-buckling state. Using the arc-length method, the nodal displacement of a point with the most noticeable motion should be traced to illustrate the peak point. The applied load at this peak resembles the critical buckling load. Another method is to use the Newton's method until the solution cannot be obtained due to divergence, in which case adaptive bisection method is activated in the vicinity of the critical buckling load and stops at the limit load, close to the critical buckling load. If the problem does not involve nonlinear materials, a linear analysis approach may be used for an approximation. To this end, MSC.Nastran allows linear buckling analysis on a preloaded structure.

A thick spherical cap with a double curvature is analyzed by eigenvalue analysis to render 3528.38 psi as the critical buckling load. When the example problem was analyzed using an arc-length method, the solution went through the peak load of 3559.2 psi. When the same problem was analyzed using Newton's method with adaptive bisection procedure, the solution traced up to the maximum load of 3540.63 psi before divergence due to instability. When the model was remeshed with about 100 times more elements, the buckling analysis procedure predicted a critical buckling load at 3545.43 psi. In this refined model, Newton's method with bisection converged up to 3546.88 psi and the arc-length method peaked at 3555.72 psi, respectively.

Results also vary depending on the coarseness of the mesh or the type of element used for the model, although solutions are more accurate for a more refined model. The eigenvalue analysis for buckling predicts a more accurate buckling point as the two solution points for extrapolation approach the actual buckling point, unless the two points are too closely spaced. Comparison of the results confirms that the limit load obtained from Newton's method with adaptive bisection process closely predicts the critical buckling load. The arc-length method tends to predict a slightly higher value for the buckling load.

1

S. H. Lee and D. N. Herting, A note on "A Simple Approach to Bifurcation and Limit Point Calculation", Int. J. Numerical Methods in Engineering, John Wiley & Sons, Vol. 21, pp. 1935-1937, 1985. doi:10.1002/nme.1620211016

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