Consistent Equations of Nonlinear Multilayer Shells Theory in the Quadratic Approximation

For the laminated shells on the basis of the Timoshenko model, taking into account the transversal compression used for each layer, two versions of two-dimensional equations describing geometrically nonlinear deformation for arbitrary displacements and small strains are constructed. They are based on previously proposed consistent relationships of the non-linear elasticity theory, usage of which does not lead to the appearance of false bifurcation solutions. The first version corresponds to the contact problem statement, in accordance with which the contact stresses into the contact points of the layers as unknowns are introduced, and the second version corresponds to the preliminary satisfaction of the kinematic coupling relations for the layers along the displacements. An example is given of the application of the derived equations for solving the linear problem of a plane stress-strain state of a straight beam under the action of normal surface forces applied to the front boundary surfaces is given.