Materials Science Forum Vol. 968

Abstract: This paper describes some features and analogies of the mathematical models for the elastic elements with movable load and for the elastic elements of changeable length. In these systems two forms of own oscillations - the own component and the accompanying one, displaced in phase to the right angle correspond to every frequency of the system. The accompanying component is caused by the mobile inertia load or by the changeable length and they are not trivial only when this factor exists. As for objects with time-varying length, these problems lie in outside of the scope classical problems of mathematical physics due to that the eigenfrequencies and eigenforms become time-dependent functions. This non-classical section of the mathematical physics is waiting for its development, new researches and generalizations.

Abstract: The construction of a mathematical model and the development of an algorithm of free vibrations investigation in the three-layered circular shell with a light-weight aggregate supported by annular rigidity ribs are considered in paper. The hypotheses of Kirchoff-Lyav are accepted for external bearing layers of shell and for aggregate there is accepted the linear law of tangential displacements change by thickness. The boundary conditions of a shell region closed between the ribs are established. Using the boundary transition, conditions along the lines of the ribs, taking into account and without deformations of displacement in the ribs, but without taking into account the torsional rigidity in the ribs are determined. The equation of motion of supported three-layered shell is obtained. The frequencies of free vibrations were investigated and values of parameter of the first frequency of free vibrations for a shell, supported by one and three rigidity ribs, were calculated. There are given values depending on the physical and mechanical properties of materials and geometric dimensions of the shell, the curvature parameter, and the rigidity parameter of an aggregate.

Abstract: The developing of the effective methodic of elastic orthotropic plates’ calculation and the research on the base of their state under different boundary conditions are of great importance nowadays. The representation of the received results in the form, convenient for practical use, is also important. For practical applications in engineering are important tables for determining deflections and internal forces of structures. Such tables for the isotropic case under various conditions of plate support on the contour are given in many works. As for the anisotropic plates, there are no such tables, with the exception of one Huber table compiled for a freely supported rectangular orthotropic plate, depending on the relationship between the stiffness values. Here is a method of calculating the non-homogeneous anisotropic rectangular plates with arbitrary fixation on the contour is set forth, which is reduced to a boundary value problem. The main idea of a calculated general methodic of linear marginal differential tasks calculation is based on underlying of the main part of a solution. Such approach is proved by means of development and some generalization of common positions of a variational method of marginal tasks of mathematical physics of self-conjugated tasks solution. To solve a system of equations in terms of displacements using finite difference method (FDM) in combination with different variations of analytical solutions. It is advisable to construct a numerical solution of the problem so that in difficult cases the support fixing and uploading solution sought, not directly, but in the form of amendments to the known solution for simple cases of reference to consolidate and uploading at finding the solutions which the analytical methods or the FDM with sparse mesh may be used. Given as examples are the results of calculation for a series of square orthotropic plates with a fixed boundary under the action of uniformly distributed load.

Abstract: The article has deal with investigations of free bending vibrations of uniform cantilever structures with taking into account the dead weight. The investigation uses the exact solution of the partial differential vibration equation with variable coefficients. The formulas for the natural frequencies of a rod structures are obtained in analytical form. An analytic relationship between the frequencies with and without taking into account the dead weight of the structures is established. The nature of the dependence of frequencies on the value of the longitudinal load is revealed. An example is considered, in which the values of the first three nature frequencies obtained by the author's method are given.

Abstract: The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shells is considered. The main problem here is due to the fact that most of the problems of statics, dynamics and stability of shells are reduced to solving an eighth-order differential equation. As a result, all analytical expressions of the NA BEM (fundamental functions, Green functions, external load vectors) turn out to be very cumbersome, and intermediate transformations are associated with eighth-order determinants. It is proposed along with the original differential equation to consider an equivalent system of equations for the unknown state vector of the shell. In this case, calculations of some analytical expressions related to high-order determinants can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point reduces to its calculation at the point , which leads to a significant simplification of all analytical expressions of the numerical-analytical boundary elements method. On the basis of the proposed approach, a solution is obtained of the problem of bending a long cylindrical shell under the action of an arbitrary load, the stress-strain state of which is described by an eighth-order differential equation. The results can be applied to other types of shells.

Abstract: An algorithm for determining the stress state of plates of different shapes with holes due to residual deformations was suggested. The residual stresses in the plates were determined using the calculation and experimental method. The algorithm for determining the stresses the near the holes in the plates due to residual deformations is based on the method of integral equations. The residual stresses and stresses near the holes were investigated. Stresses near circular holes with different distributions of plastic deformations were investigated. Cases were established, where at the boundary of holes, depending on their location, high compression or tensile stresses may arise. Particular, that high compression stresses appear at the point of intersection of the center of the weld with holes or outer boundary, which high compression stresses are approximately the same for all radii of the holes. In case of the radii of the holes that are smaller than the zone of plasticity, high tensile stresses appear, which decrease, when the size of the holes increases. In case of the radii of holes that are larger than the zone of plasticity, the maximum tensile stresses arise at points close to the boundary of the zone of plasticity.

Abstract: An exact solution of the theory of elasticity is presented for the problem of a narrow multilayer bar section transverse bending under the action of a normal uniform load on longitudinal faces. The solution is built using the principle of superposition, by imposing common solutions to the problems of bending a multilayer cantilever with uniform loads on the longitudinal faces and an arbitrary load on the free end, and allows to take into account the orthotropy of the materials of the layers, as well as transverse shear deformation and compression. On the basis of a built-in general solution, a number of particular solutions are obtained for multi-layer beams with various ways of the ends fixing.

Abstract: We proposed the method of the compliance function for the solution of two-dimensional stationary problem of thermoelasticity for a multilayer foundation (stack of elastic layers coupled with a half plane) with a non-ideal thermal contact between its layers. We constructed the recurrence relations for the auxiliary functions and the compliance functions of neighboring layers of the foundation. We analyzed the influence of thermal resistance on the distribution of stresses and temperature at the points of the lower boundary of the upper layer for a two-layer foundation subjected to the action of thermal loads.

Abstract: Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

Abstract: In this article is determined the ratio between effective elastic characteristics of the fibrous transversally isotropic material. Fibrous uniaxial material, which consists of the isotropic elastic matrix and fiber, is in the focus of attention. It is assumed that mechanical properties of components under stretching and compression are different, notably matrix material and fiber material are multi-modular. Transverse stretching and transverse compression of composite cell are considered. Two problems for each type of strain are solved. In the first problem stresses and displacements of matrix and fiber under conditions of their common axisymmetrical deformation are determined. Subsequently similar characteristics for the cell deformation of the homogeneous transversally isotropic material as a composite are determined. Ratio between effective composite’s characteristics is solved from the conditions of equality of the axial displacements of the composite’s cell and radial displacements on its surface. The relation of the calculated ratio from volume fraction of fiber in a composite is analyzed.