The problem of viscoelastic isotropic and orthotropic shells steadiness under the axial dynamic load was examined. The behavior of gently-sloping variable-thickness shells and plates under all these factors is not enough investigated and needs further inquiry.

Using the Bubnov-Galerkin method, the system of nonlinear ordinary integro-differential expressions in partial derivatives was obtained. The numerical method based on the quadrature rules was used for solving it.

The algorithm for the computer solving was obtained. The computer steadiness modeling of the analyzed shells was also held, under varying its physical, mechanical and geometrical parameters.