Dome is often used by architects for cover of large spans. Only spherical, conical, elliptical, parabolic, and hyperbolic surfaces of revolution among tens of the well-known surfaces of revolution can be taken for middle surfaces of domes. Spherical domes have the most spread in modern building due to simplicity of their form in comparison with other shells of double curvature. But researches of paraboloidal domes do not end. Some new information on strength analysis of parabolic shells, on determination of the frequencies of their natural vibrations and the examples of application of paraboloid of revolution in civil engineering in 1900=2017 are presented in this review paper. The main presented bibliography was published in XXI century.

We give a mathematical description of calculation model for cracking in sheet structural element under a non-uniform stress field. The model of pre-fracture zones in state of plastic flow under constant stresses was accepted. The boundary value problem for interaction of zones of weakened interparticle material bonds in the sheet structural element under influence of the inhomogeneous stress field is reduced to a system of singular integral equations. The integral equations further reduce to a system of nonlinear algebraic equations for solution of which we use the method of successive approximations. Sizes of the prefracture zones and limit value of external loads at which in the sheet structural element the cracking occurs are found.

The article deals with the solution of the problem of bending of a hinged multilayer beam under the normal uniformly distributed load and induced axial forces. The interaction between layers is accomplished by the contact layer in which the substances of adhesive and substrate are mixed. We will consider the contact layer as the transversal anisotropic medium with such parameters that it can be represented as a set of short elastic rods, which are not connected to each other. The solution is obtained in the form of decomposition into Fourier series of sines. There is an example of the calculation of a three-layer beam. The convergence of the obtained solution is analyzed according to the number of accounted members of the decomposition

The article contains a comparative stability analysis of initial equilibrium forms of reticulated one-sheet hyperboloids of revolution. The calculations are performed taking into account the geometric and the dual (geometric and material) nonlinearities. The influence of the shape of generatrix one-sheet hyperboloid of rotation on shells stability in these formulations of the problem is considered. The equilibrium curves of shells with load acting on the upper base are shown.

The mathematical model relations for buckling investigation of structurally-anisotropic panels made of composite materials are presented. The mathematical model of stiffening rib being torsioned under one-side contact with the skin is refined. One takes into account the influence of panel production technology: residual thermal stresses and reinforcing fibers preliminary tension. The resolved eight order equation and natural boundary conditions are obtained with variation Lagrange method. Computer program is developed using operating MATLAB system. The influence of the structure parameters on the level of critical buckling forces for bending form has analyzed.