Ya. M. Grigorenko

Discrete Fourier-series method in problems of bending of variable-thickness rectangular plates

The approach to the solution of the boundary-value problems of bending of elastic rectangular plates of variable thickness is presented. It is proposed to introduce into the resolving system of partial differential equations additional functions which enables the variables to be formally separated and the problem to be reduced to a unidimensional one by representing all the functions as a Fourier series in a single coordinate. In this case the problem can be solved by the stable numerical method of discrete orthogonalization. To calculate the additional functions, Fourier series of discretely assigned functions with allowance for variations in the plate thickness are used. The boundary-value problems for rectangular plates of variable thickness were solved assuming that their weight is unchanged.

The Effect of Inhomogeneity of Elastic Properties on the Stress State in Composite Cylindrical Panels

The effect of inhomogeneity of elastic properties in the circumferential direction on the distribution of stress and displacement fields in orthotropic cylindrical panels is studied. The mechanical properties of the panels and the load acting on them are constant in the axial direction, which makes it possible to neglect the influence of the curvilinear ends. From the initial relations of a three-dimensional problem of the elasticity theory of inhomogeneous anisotropic bodies, a resolving system of partial differential equations is obtained, whose solution is presented in the form of truncated Fourier series, so that the conditions of free support of the rectilinear ends are satisfied. This allows us to separate the variables and to get a system of ordinary high-order differential equations, which is integrated by a stable numerical method. The problem on the stress-strain state of an orthotropic composite panel with a varying relative volume content of reinforcing elements in the circumferential direction is solved. The effect of the change in the reinforcement density on the stresses and displacements of the panel is studied.

Investigation of the Stress State of Pressure Vessels of Inhomogeneous Structure Made of Composite Materials

The problem on the stress-strain state of layered cylindrical shells with bottoms of intricate shape under the action of internal pressure is considered. The elastic system examined is formed by spiral-circular winding. Two variants of the shell bottom structure are investigated. In the first variant, one spiral layer is installed, which leads to great variations in the bottom thickness along the meridian. In the second one, the bottoms are formed according to the zone-winding scheme. The stress state of the shell constructions of the classes considered is determined by solving boundary-value problems for systems of ordinary differential equations. The solution results for cylindrical shells with elliptic bottoms for the two types of winding are given. It is shown that the zone winding leads to smaller deflections and stresses than the conventional ways of reinforcing shell bottoms.

Numerical solution of multipoint problems in statics of shells

We purpose an approach to solving multipoint boundary-value problems for a system of ordinary differential equations in the theory of shells. The technique is based on reduction of the original problem to several two-point boundary-value problems, which are solved by a stable numerical method. Examples of calculation of variable-thickness cylindrical shells are given.

Numerical analysis of stresses and strains in flexible cylindrical shells with variable rigidity in two directions

The paper analyzes the stress-strain state in a cylindrical shell with variable rigidity in two directions. The shell is analyzed in a geometrically nonlinear framework under different loads and boundary conditions. The proposed approach reduces the nonlinear system of partial differential equations to a sequence of linear systems of ordinary differential equations. The latter are solved by discrete orthogonalization.

An approach to numerical solution of multipoint boundary-value problems

An approach is proposed to solving multipoint boundary-value problems for linear differential equation of w-th order, based on reduction to two-point boundary-value problems. The two-point problems are solved by the stable discrete orthogonalization method. Some numerical examples are considered.

SOLUTION OF BOUNDARY-VALUE PROBLEMS OF THE STATICS OF LAYERED ELASTIC BODIES WITH NONRIGID CONTACT BETWEEN THE LAYERS

A technique is proposed for solving three-dimensional problems of the stress-strain state of cylinders, spheres, and shallow elastic bodies with a rectangular projection which are composed of laterally nonhomogeneous anisotropic layers with nonrigid contact between the layers. The solution of the corresponding many-point boundary-value problem is reduced to solving a number of two-point problems by a known numerical apparatus. Solution results are reported for the strain of a three-layer spherical shell with slipping layers.

Calculation and selection of rational parameter values for shell structures made of composite materials

Thin elastic laminate shells made of various composite materials are widely used structural elements. The problem of ensuring their light weight with adequate rigidity and strength is directly related to a selection of rational parameter values under imposed constraints. The optimal design of structural elements poses several complex problems which, because of contradictory requirements and various other indeterminacies, cannot always be formulated mathematically [i]. Various informal approaches can be successfully taken, along with methods in the theory of optimum systems, to the selection of rational parameter values.