The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green's functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author's method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

Construction of calculation model and development of algorithm of free vibrations investigation in three-layered cylindrical shell with lightweight aggregate, supported by transverse ribsб are considered in paper. Variation equation of transverse vibrations of three-layered shell of symmetric structure, supported by ribs in two perpendicular directions, taking into account the action of longitudinal forces in meddle surfaces of external layers and ribs is achieved. For external bearing layers of shell there are accepted hypotheses of Kirchgoff-Love and for aggregate there is accepted the linear law of tangential displacements change by thickness. Aggregate transverse deformations was not taken into account. The hypotheses of Bernoulli were accepted for ribs. There was taken into account only bending of ribs in vertical surface. Using passage to the limit there was achieved conditions on ribs lines without taking into account shear deformations in ribs. There are given values of parameter of free vibrations first frequency of three-layered sloping cylindrical shell with lightweight aggregate, supported by one and three transverse ribs.

An axisymmetric contact problem concerning the torsion of a circular shaft of an orthotropic-nonhomogeneous half-space is considered. By means of the technique of integral transformations of Laplace and Hankel, with the subsequent application of the orthogonal polynomial method, an approximate solution in the transformant space is constructed. Also was performed reverse transformation. Calculated formulas for the angle of rotation of the shaft and the tangential stress acting on the contact area are obtained. Numerical calculations for certain types of heterogeneity have been performed. Comparison of the obtained results with the previously known results is made.