Numerical solution of nonlinear problems on deformation of elastic shells of revolution at eigenstates

The quasi-static problems on axisymmetric deformation of shells of revolution made from an elastic material with consideration for its geometric nonlinearity are numerically solved. The special attention is given to the determination of eigenstates of shells appropriate to the non-trivial solution of a homogeneous rate problem. The Riks-Wempner approach is used wherein the parameter of the external load intensity is entered as unknown. The basic nonlinear problem in determination of eigenvalues and appropriate eigenvectors reduces to the linearized (generalized) problem in eigenvalues. The developed algorithm of solving problems on deformation of shells in the vicinity of eigenstates is approved for a problem on deformation of the longitudinally compressed simply supported circular cylindrical shell. Numerical solutions are compared with analytical ones. It has been found that the behaviour of solution of the problem in the vicinity of an eigenstate is sensitive to perturbations of geometric parameters of the shell due to the dense spectrum of eigenvalues.