Nonlinear equations of ribbed shells balance taking into account the different properties of material

In previous works isotropic and orthotropic shells of general form, under the influence of static and dynamic mechanical loads in conditions of elastic, nonlinear elastic and viscoelastic properties of the material development were considered.

It was assumed that the envelope contained a certain way along the contour can be supported by ribs, spaced along the coordinate lines (directed along the lines of curvature) from the inside (by the concavity in the case of convex hulls).

In this paper on the basis of variational principle of Hamilton-Ostrogradskii the total energy functional of deformation ribbed shells of general form under dynamic loads (action) is obtained and, under certain assumptions from the stationarity conditions the equations of motion (with appropriate boundary and initial conditions) for the shallow ribbed shells are derived. On the basis of Lagrange variational principle, the total energy functional of general form ribbed shells deformation under static load (the difference between potential energy and the work of external forces) is obtained and the general equation equilibrium of ribbed shells, as well as the natural boundary conditions from the condition of Lagrange functional stationarity are derived.