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.     

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.     

Wear resistance of bodies protected by a thin composite coating

The mechanical contact interaction of bodies with a thin composite coating is investigated with account of wear. The thermal effects are not considered. The coating is modeled by a thin plate. Between the body and the coating is an interlayer, which is modeled by a Winkler body with one modulus of subgrade reaction. Under the action of a rigid stamp on the coating, the process of abrasive wear proceeds. The contact interaction of the coating with the base is described by using the model of an intermediate layer. To determine the stress-strain state of the coating, equations of the generalized theory of plates including the shear strains and the compression of normal are utilized. For the contact wear problem formulated, the basic integral equation with a Fredholm-type kernel is derived, and its solution algorithm is proposed. Numerical results are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 319–330, May–June, 2006.     

Experimental investigation of the transverse shear effect in the bending of cantilevered plates with low shear stiffness

The state of stress of cantilevered plates made of different materials (steel and glass-reinforced plastic) has been experimentally investigated. The results obtained are compared with the known theoretical solutions. It is shown that in certain cases it is necessary to take into account the effects associated with the low shear stiffness of the plate.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. 736–738, July–August, 1971.     

On the theory of thermoelastic bending of reinforced shells and plates

The author examines an approach to the construction of a theory of thermoelastic bending of arbitrarily reinforced shells and plates that takes into account the actual structure, the deformation characteristic, and the actual thermomechanical properties of the elements which make up the composite. The final equations are obtained with and without allowance for transverse shear strains. The problem of the thermoelastic bending of a thin, arbitrarily reinforced rectangular plate hinged at the edges, when transverse shear strains can be neglected, is considered as an example.Institute of Hydrodynamics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekhanika Polimerov, No. 5, pp. 861–873, September–October, 1972.     

Dynamic bending of plates with low shear strength

The dynamic problem of the natural vibrations of glass-reinforced plastic plates with low shear strength is examined. The calculations are based on the Timoshenko model (allowance for shear strains and rotational inertia). It is shown that disregarding effects associated with the large shear compliance of glass-reinforced plastic plates leads to appreciable errors in determining the dynamic characteristics.Institute of Polymer Mechanics, Academy of Sciencies of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 698–704, July–August, 1968.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

Temperature stresses in plates made of reinforced laminated material

The heat conduction equations and the basic equations of the quasi-static axisymmetric problem of thermoelasticity are obtained for thin plates with heat transfer and cylindrical anisotropy. The nonsteady temperature stresses are determined for an infinite plate with a circular opening and an annular plate, both made of reinforced laminated material.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 4, pp. 721–726, July–August, 1970.     

A rigid inclusion in the contact problem for elastic plates

A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.     

The effect of transverse shear upon the buckling stability of inelastic orthotropic plates

The problem of buckling of inelastic orthotropic plates is analyzed. The phenomena associated with the transverse strains of the plate are considered in the framework of the Ambartsumyan theory of plates, which excludes the hypothesis of undeformed normals. The problem is solved with the aid of the localized strains theory, and it is shown that in the case of orthotropic materials, i.e., those characterized by a large E/G ratio, the effect of the transverse shear must be taken into account in determining the critical stress for a plate under compressive loads parallel to its surface.Mekhanika Polimerov, Vol. 1, No. 2, pp. 133–136, 1965     

Some problems of the theory and analysis of anisotropic plates and shells with low shear stiffness

Certain problems associated with the theory and analysis of plates and shells made of materials with low shear stiffness are discussed.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Translated from Mekhanika Polimerov, No. 4, pp. 693–714, July–August, 1970.     

Relative stiffness of irregularly perforated plates

Approximation of the average field is used in order to calculate the relative elastic properties of plates irregularly perforated with a large number of different hole. Relative properties are calculated for a plate perforated with a different kind of elliptical hole. It is shown that the Young's modulus and shear modulus of a plate perforated with identical round holes calculated in this approximation have values similar to those calculated accurately for the case of a regularly perforated plate.Translated from Dinamicheskie Sistemy, No. 5, pp. 3–11, 1986.     

Calculation of simply supported circular plates in the statement of the problem of contact interaction in a package of two plates

The possibility of applying the mechanicomathematical model of bending of a package of transversely isotropic plates to approximate calculations of thick plates is investigated. Within the frame work of this model, the problem of axisymmetric bending of a package of two identical plates with simply supported edges is considered. The conditions of rigid contact are given between the plates. Based on the analytical result obtained, the unknown distribution of stresses and displacements across the thickness of the plate is approximated by the distribution of corresponding parameters in the package of two plates. For an isotropic body, the results of numerical calculations are compared with those given by the 3D theory of elasticity. In the case of a transversely isotropic body, a comparison with the results found by the refined bending model of plates (taking into account the transverse compression and shear) and the Timoshenko model is carried out. The accuracy to which the boundary conditions in every model are satisfied is analyzed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 603–616, September–October, 2007.     

Region of applicability of applied theories in problems of the stability of bars and plates with low shear stiffness subjected to uniaxial compression

The question of the region of applicability of applied theories of bending of bars and plates in stability problems is discussed in relation to the case of uniaxial compression. The critical loads given by these theories are compared with the recently obtained solution [4, 5, 7–10] based on the three-dimensional linearized equations of the theory of elasticity [6]. It is established that for the problems in question the Timoshenko and Ambartsumyan theories are accurate enough for engineering purposes over the entire practical region of variation of the geometric and physical parameters of the bar (plate), whereas the use of the Euler-Bernoulli and Kirchhoff-Love theories may lead to unacceptable errors.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev; Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga; Lvov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1124–1126, November–December, 1969.     

Stress state of freely supported multilayered elliptical plates of anisotropic materials

The problem of a stressed state in elliptic plates has been considered in general for a rigid contour fixation. It is much more difficult to obtain a solution for the freely supported plates, even for isotropic materials. In this paper we suggest an approach for defining the stressed state of thin elliptic plates with layered structure under the condition of a freely supported contour. The solution is obtained in a rectangular cartesian coordinate system. The displacements, which are the fundamental unknowns, are given in the form of polynomials with unknown coefficients defined by a system of algebraic equations. The resolving equations and three out of the four boundary conditions are satisfied precisely. One boundary condition, is satisfied by means of collocation method of separate points of the contour. Estimation of the accuracy of the suggested approach is carried out by comparing the obtained results with the known ones. The problem of deformation of a twolayered plate has been discussed, in which the principal direction of elasticity does not coincide with the coordinate directions.S. P. Timoshenko Institute of Mechanics, National Academy of Science of the Ukraine, Kiev. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 496–504, July–August, 1997. Original article submitted March 19     

Creep strength of anisotropic plates taking account of transverse shear strains

Cylindrical creep buckling of an orthotropic glass-reinforced plastic plate is investigated with consideration of shear creep in the plane xz. In this case the rheology of buckling is not described by the Kirchhoff-Love model, therefore the investigation is carried out using the modified theory of plates [3]. The long-time and instantaneous critical forces are obtained.Mekhanika Polimerov, Vol. 1, No. 5, pp. 114–117, 1965     

Analysis of plates of polymeric material subjected to local loads

An approximate method of analyzing plates of polymeric material under local loading is described. The state of stress and strain and the stability of the plate are investigated.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 3, pp. 430–434, May–June, 1976.     

Periodic axial and shear deformation of polymers in the viscous-flow state

Results are given of an experimental investigation of the combined periodic axial and shear deformation of polymers in the viscous-flow state. Specimens in the form of right cylinders were disposed between two parallel plates to which the material adhered completely. The experimental data are compared with those obtained under simple shear. The results indicate that, with combined periodic axial and shear deformation, nonlinear shear deformations affect the axial viscoelastic properties and nonlinear axial deformations affect the shear properties of the material.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 880–885, September–October, 1972.     

Some quasi-static problems for a viscoelastic half-space

Some problems for a viscoelastic half-space are solved in the case of noncommutative operators. A solution of the equilibrium equation analogous to the Boussinesq-Papkovich solution is constructed. The problem of a normal pressure acting on the boundary of a viscoelastic half-space is solved. Two forms of this solution are obtained and both are used in the following problems, the problem of a concentrated load moving over the boundary of a half-space and the problem of a circular rigid stamp. The case of periodic motion of a periodic load is investigated with reference to the example of motion in a circle. At constant Poisson's ratio the solution of the problem of a stamp can be used for determining the creep or relaxation function.Mekhanika Polimerov, Vol. 2, No. 3, pp. 392–402, 1966Presented 12 November 1965 at the Riga Conference on Polymer Mechanics.     

Action of an elliptic stamp moving at a constant speed on an elastic half-space : PMM vol. 42, no. 6, 1978, pp 1074–1079

The action of a rigid stamp moving at a constant speed, on the boundary of an elastic half-space, is investigated. It is assumed that the frictional forces between the stamp and the surface of the half-space are absent. The integral equation obtained in [1] yields formulas for the pressure, for the case when the area of contact between the stamp and the half-space has an elliptic form.