Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

Theoretical and experimental investigation of the state of stress of a cantilevered plate with low shear stiffness

A general solution is obtained for the problem of a cantilevered, transversely isotropic plate in the form of an infinite strip with a clamped edge bent by a concentrated load applied at the free edge. The investigation relies on the equations of the generalized theory of plates [21–23] with allowance for interlaminar shears. In contrast with the Papkovich solution [1], the maximum theoretical bending moment at the clamped edge depends on the relative width of the strip and on the properties of the material. The classical result is asymptotically exact as h 0 or G_Z . The experimental results confirm the principal conclusions based on the theoretical solution.Kamenetsk-Podol'sk General Engineering Department, Khmel'nitskii Technological Institute. L'vov Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR. Translated from Mekhanika Polimerov, No. 3, pp. 476–481, May–June, 1971.     

Bending of a rectangular orthotropic glass-reinforced plastic plate under a transverse load

The Bubnov-Galerkin method is used to solve the problem of the bending of a clamped flexible orthotropic plate. An equation relating the deflection and the load is obtained. An experiment on a plate made of polyester plastic reinforced with satin-weave glass fabric is described. It is shown that within certain limits the deflections at the center of the plate and the combined stresses along the clamped edges are in good agreement with the theoretical values. The limits of applicability of the first-approximation formulas are investigated.All-Union Scientific-Research Institute of Railroad Transport, Ural Division, Sverlovsk. Translated from Mekhanika Polimerov, No. 4, pp. 674–679, July–August, 1969.     

Action of a concentrated load on an elastic ring pressed into a circular hole in an anisotropic plate

A solution is obtained for the elastic equilibrium problem for an anisotropic plate with a circular hole into which is pressed a ring of arbitrary cross section symmetrical about the middle surface of the plate. Before deformation the outside radius of the ring differs from the radius of the hole in the plate by the amount of the permissible elastic displacements. A normal concentrated load is applied to the ring. The ring is analyzed in accordance with the theory of thin curved bars. A numerical example is given.I. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 6, pp. 1054–1059, November–December, 1973.     

On the stress distribution in a rectangular thick plate of a composite material with curved structure under forced vibration

The stress distribution in a rectangular plate of a multilayer composite material with a periodically curved structure under forced vibration is studied. It is assumed that the plate is hinge supported at opposite sides. The investigation is carried out within the exact three-dimensional linear theory of elasticity. The mechanical relationships of the plate material are described by the continuum theory of Akbarov and Guz'. The numerical results obtained by the finite element method show that even in low-frequency dynamic loading of the plate the extreme values of stresses, which appear as a result of the curving in the plate structure, considerably exceed those in the corresponding static loading.Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, Baku, Azerbaijan. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 447–454, July–August, 1999.     

State of stress of flexible viscoelastic plates with an opening

The problem of the state of stress of flexible plates with an opening is investigated with allowance for the viscoelastic properties of the material. The basic equations and the corresponding boundary conditions are formulated. The boundary value problem is solved by numerical integration. The effect of the creep properties of the material on the stress redistribution in the plate is investigated with reference to polymethyl methacrylate. It is shown that the angle of rotation, the ring moment, and the ring forces at the edge of the opening vary with time and as a function of the load.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 4, pp. 708–713, July–August, 1971.     

Optimization of vertical axisymmetric thermal displacements in a round plate using heat sources

We consider the problem of using heat sources for optimal control of the distribution of the vertical axisymmetric thermal displacements of a thin round plate with edge clamped at an angle of revolution. On the basis of the method of the inverse thermoelasticity problem we construct a solution of the control problem. For specific cases of heating the plate we carry out a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Poly a, Issue 36, 1992, pp. 105–110.     

Concentration of electroelastic fields in a composite piezoceramic plate with a hole intersecting the interface of materials

A plane problem of electroelasticity is considered for an infinite compound plate with a hole located in both constituents of the plate. The corresponding boundary value problems is reduced to a system of singular integral equations of second kind, which is solved in numerical quadratures. Calculation results are presented that describe the concentration of electroelastic fields near the hole upon action at infinity of the vectors of mechanical stresses and electric field strength.Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 359–366, May–June, 1999.     

Effect of material properties on the bending of cantilevered plates under a concentrated load

In a continuation of a previous study [4] a solution is obtained for the problem of a transversely isotropic plate in the form of an infinite strip bent by a concentrated load applied at an arbitrary point. The solution obtained is investigated in relation to the geometric and physicomechanical properties of the plate. The results are compared with the corresponding classical Kirchhoff solution [7].Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov; Kamenets-Podol'skii General Engineering Department of Khmel'nitskii Technological Institute. Translated from Mekhanika Polimerov, No. 4, pp. 704–707, July–August, 1971.     

Application of the differential-difference method to the problem of bending for a rectangular orthotropic plate

Under certain conditions on the smoothness of the exact solution of the problem, the author proves the uniform convergence with order h^2p–1 of the 4p+1 -point scheme of the method of lines in the following cases: 1) two opposite edges of the plate are fixed rigidly, while the other two rest freely; 2) all edges of the plate rest freely.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 114–130, 1983.     

Stability of transversely isotropic annular plates with low shear resistance

The stability differential equations of a cylindrically orthotropic circular plate are obtained on the assumption of an axisymmetric buckling mode with allowance for transverse shears. These equations are solved for the case of a transversely isotropic material when the inner and outer edges of the plate are identically loaded by uniformly distributed radial forces. The transcendental equations for the critical load parameter are constructed for various edge conditions. The dependence of this parameter on the boundary conditions and the relative thickness of the plate, Poisson's ratio, and the ratio of the Young's and shear moduli of the material are investigated. Certain conclusions are reached concerning the design of reinforced-plastic plates.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 872–880, September–October, 1969.     

The state of strain in a semi-infinite plate with edge slots

An approximate method is given for determining the stress distribution in a semi-infinite plate weakened by several closely spaced edge slots with complicated shapes. The method is based on representing the displacement vector as the superposition of four types of singular solution in the planar theory of elasticity for a half-plane, with the solutions continuously distributed over rectilinear segments at the plate boundary, which are completely contained within the slots. Numerical results are given for two slots in a half-plate under tension when one of the slots is a half-ellipse and the other is a half-circle and when both slots are of the same shape and are formed by two quarter circles joined by a segment of the common tangent.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 112–116, 1989.     

Extension of a piezoceramic bimorph with a crack crossing the interface

We consider the boundary-value problem of electroelasticity for a composite plate weakened by a crack crossing the line joining the media. The initial boundary-value problem, is reduced to a mixed system of singular integral and algebraic equations. We present the calculation results characterizing the variation in the stress intensity factors as a function of the opening angle of a segmented crack for different types of loading.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 482–488, July–August 1997.     

Investigation of the natural vibrations of plates made of composite materials with the help of the finite-element method

The modes and frequencies of small natural vibrations in a cavity of thin plates are investigated on the basis of the method of finite elements in displacements. The effect of the rigidity characteristics of a material on the natural frequencies and modes of vibrations of flat cantilever vanes whose tapered edge is restrained is studied. An example is given of the use of the finite-element method to determine the natural vibrations of an orthotropic plate in the shape of an airplane wing. Quadrangular and modified triangular bending finite elements are used to simulate a continuous system. A mass-element matrix is constructed on the basis of a four-term ("contracted") polynomial determining the deflection that permits a significant reduction of the order of the solving system.Translated from Mekhanika Polimerov, No. 2, pp. 284–288, March–April, 1976.     

Optimal reinforcement of a glass-reinforced plastic plate

Relations are derived for the optimal arrangement of the reinforcement in a rectangular glass-reinforced plastic plate supported at the edges and stressed in uniaxial compression by a short-term or long-term, uniformly distributed load. (The optimal arrangement is understood to be that which maximizes the critical load.)All-Union Scientific-Research Institute of Railroad Transport, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 741–744, July–August, 1969.     

Optimization of vertical axisymmetric displacements of a thin circular plate under a nonstationary thermal load

We study the problem of optimal control of the distribution of vertical axisymmetric thermal displacements of a thin circular plate fixed along its edge. The displacements are caused by a nonstationary heat load on one of the end surfaces. The thermal action on the other end surface is chosen as the control function. Using the Hankel and Laplace transforms in the space of continuous functions, we construct the solution of the inverse problem of thermoelasticity to which the original control problem has been reduced.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 148–153.     

Free vibrations of a cantilevered plate with a polymer coating

The article considers some of the special characteristics of the free vibrations of a steel cantilevered plate, with the formation of a polymer coating on the plate. It compares the results of an experimental investigation of the effect of internal stresses in the coating on the vibrations and on the static bending of the plate. It demonstrates the possibility of determining the internal stresses in the coating from the change in the frequency of the natural vibrations of a plate with a coating.Institute for the Mechanics of Metal-Polymer Systems, Academy of Sciences of the Belorussian SSR, Gomel'. Translated from Mehanika Polimerov, No. 3, pp. 541–544, May–June, 1970.     

Contact problems for beams and plates with low shear stiffness

The low shear stiffness and strength of unidirectionally reinforced plates and beams predetermines the choice of calculation model in specific problems [4]. The contact problem is considered for a glass-reinforced plastic plate resting on a Winkler foundation; the deformation properties of the plate are described by the equations of an orthotropic material; the investigation is based on generalized applied theories of the Timoshenko and Ambartsumyan types [5–8], which permit the high shear compliance of thin-walled structures to be taken into account.L'vov Polytechnic Institute; Ivan Franko Institute of Petroleum and Gas. Translated from Mekhanika Polimerov, No. 4, pp. 715–720, July–August, 1970.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.