Vibration of buckled elastic structures with large rotations by an asymptotic numerical method

A methodological approach based on the finite element and perturbation methods (asymptotic numerical method) has been developed for small vibration analyses of post-buckled shells with large displacements and large rotations. The coupled non-linear static and dynamic problems based on non-linear shells theory are transformed into a sequence of linear problems. Only two linear operators have to be inverted and a large number of terms of the polynomial approximation are numerically computed. The static non-linear response, the load–displacement solution, and the load–frequency dependence are investigated for large amplitudes. The load–frequency curves are obtained for various natural frequencies at any desired load level. A continuation procedure based on Padé approximants is used to get the whole solution. Limit points, bifurcation points and stability zones are analysed. The efficiency of this procedure is tested in some benchmark problems such as rectangular plates, thin and thick cylindrical roofs and deep arches.