A revisit to the bending problem of a thin elliptic aelotropic plate with simply supported edge and uniform lateral load

The complex solution method of Okubo for the deflection of a thin circular aelotropic plate with simply supported edge and uniform lateral load was extended to an elliptic plate by Ohasi. In his work however several inconsistencies appear, of which at least one disqualifies a central part. From a revisit to the works of Okubo and Ohasi a new solution for the deflection of a thin elliptic aelotropic plate with simply supported edge and uniform lateral load emerged. The solution is a generalisation of Okubo’s solution and is valid for any angle between material and geometric principal axes. Previously known solutions, including those for circular plates, are reproduced as special cases of the new solution and results of numerical calculations in new situations appear reasonable.     

The inverse bending problem for a thin plate with an elastic imperfection

We consider the inverse problem consisting of determining the unknown shape of an elastic imperfection contained in a thin plate from the condition of equal strength in the stressed state along the phase interface surface. It is shown that such a state is attained in the case of an elliptic imperfection whose shape depends on the values of the applied moments and the mechanical properties of the component phases. It is established that for the geometry found for the imperfection the sum of the moments is constant and the second invariant of the deviator of the stress tensor is superharmonic over the entire plate. Numerical computations are carried out. In special cases the results obtained coincide with known data. One figure. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 34–40, 1991.     

Bending of a thin cantilevered orthotropic plate with a concentrated load at the free edge

The problem of a thin rectangular cantilevered plate of constant thickness with a concentrated load in the center of the free edge is considered. The plate is assumed to be orthotropic, the fixed edge coinciding with the principal direction of elasticity. An equation is obtained for the normal stresses at an arbitrary point on the plate. The theoretical results are compared with experiment.A. A. Zhdanov Gor'kii Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 739–741, July–August, 1968.     

Solution of the bending problem of a circular plate with a free edge by using paired equations

Used in the investigation of the bending of a plate with a free edge is a method including the solution of non-standard paired equations that are trigonometric series, which results in a quasi-completely-regular system of linear algebraic equations. The nature of the plate state of stress and strain is investigated. One of the methods reducing to the solution of paired summation equations is elucidated in /1/ in application to the problem under consideration. A combination of support and clamping is examined in the majority of papers while the case of the free edge is inadequately studied.     

The uniform heating and bending of a two-layer plate

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic layer considered (a soft plate) is in ideal contact with a rigid isotropic thin elastically deformed layer. The ends of the plate are load-free. A boundary layer of the soft plate (a thin contact layer) is introduced, which enables the boundary conditions on the ends of the plate to be formulated in such a way that the problem has a bounded smooth solution [1]. The two-layer plate, generally speaking, is bounded along the axis perpendicular to the axes directed along the length and thickness of the plate. The resultant force and the resultant moment, applied to the end transverse sections, are equal to zero. The exact solution of the temperature problem is sought using the equations of the theory of elasticity. The plane problem of the bending of a two-layer plate acted upon by a uniformly distributed pressure applied to the side surface of an anisotropic layer is solved by a similar method. The ends of the rigid isotropic layer are clamped.     

The scattering of a plane flexural wave by a sector of a thin elastic plate with a supported edge

A solution is constructed of the problem of the diffraction of a plane flexural wave at the vertex of a thin elastic plate cut out in the form of a sector, on the edge of which the condition of a supported edge (hinged support) is specified.     

Chaotic and bifurcation dynamic behavior of a simply supported rectangular orthotropic plate with thermo-mechanical coupling

This paper presents a novel analytical approach utilizing fractal dimension criteria and the maximum Lyapunov exponent to characterize the conditions which can potentially lead to the chaotic motion of a simply supported thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections. The study commences by deriving the governing partial differential equations of the rectangular plate, and then applies the Galerkin method to simplify these equations to a set of three ordinary differential equations. The associated power spectra, phase plots, Poincaré map, maximum Lyapunov exponents, and fractal and bifurcation diagrams are computed numerically. These features are used to characterize the dynamic behavior of the orthotropic rectangular plate under various excitation conditions. The maximum Lyapunov exponents and the correlation dimensions method indicate that chaotic motion of the orthotropic plate occurs at η₁ = 1.0, , and for an external force of . The application of an external in-plane force of magnitude causes the orthotropic plate to perform bifurcation motion. Furthermore, when , aperiodic motion of the plate is observed. Hence, the dynamic motion of a thermo-mechanically coupled orthotropic rectangular plate undergoing large deflections can be controlled and manipulated to achieve periodic motion through an appropriate specification of the system parameters and loads.     

Effect of shear compliance in bending and torsion on the stress concentration in a transversely isotropic plate with a hole whose edge is supported

The stress concentration in shear-compliant, transversely isotropic plates with a hole whose edge is supported has been investigated in bending and torsion. Two variants of the boundary conditions are examined. It is shown that the use of the classical Kirchhoff model may lead to a number of qualitative discrepancies.Mathematical Physics Branch, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR. Translated from Mekhanika Polimerov, No. 3, pp. 458–463, May–June, 1975.     

Approximation of a bending plate problem with a boundary unilateral constraint

Summary We study a variational formulation of the unilaterally supported bent plate problem and we analyze the approximation of the problem by a mixed finite element method. We proveO(h) andO(h|lnh|^1/2) error bounds respectively for the moments and the displacement.Work partially supported by M.P.I., by G.N.I.M. of C.N.R. and by I.A.N. of C.N.R. of Pavia     

A counterexample to Hencky plasticity in the case of a thin plate under vertical load

It is shown that the variational problem describing the bend of a perfect plastic plate under vertical load, in general, has no solution. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 192–198.     

Asymptotics of the solution to the Neumann problem in a thin domain with sharp edge

An asymptotic expansion of the solution to the Neumann problem for a second-order equation in a thin domain with peak-like edge is constructed and justified. Owing to the sharpness of the edge, the procedure of dimension reduction leads to a degenerate limit equation on the longitudinal cross-section of the domain and a solution has irregular behavior near the boundary. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 193–219.     

A contact problem for an elastic plate with a thin rigid inclusion

Under study is an equilibrium problem for a plate under the influence of external forces. The plate is assumed to have a thin rigid inclusion that reaches the boundary at the zero angle and partially contacts a rigid body. On the inclusion face, there is a delamination. We consider the complete Kirchhoff–Love model, where the unknown functions are the vertical and horizontal displacements of the middle surface points of the plate. We present differential and variational formulations of the problem and prove the existence and uniqueness of a solution.     

Pointwise estimate of the solution to a quasilinear elliptic problem in a domain with a thin cavity

We derive a pointwise estimate of the solution to Dirichlet's problem for a quasilinear second-order elliptic equation in a domain whose bounded compliment is contained in a small neighborhood of a set {x Rⁿ: ¦x¦ 1/2, x₁=...=x_s=0}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1417–1432, October, 1992.     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

Thermoacoustic response of a simply supported isotropic rectangular plate in graded thermal environments

A theoretical model is developed to investigate the thermoacoustic response of a simply supported plate subjected to combined thermal and acoustic excitations, with two typical graded thermal environments considered. The thermoacoustic governing equation derived by incorporating the thermal moments, membrane forces and acoustic loads into the plate vibration equation is solved using the modal decomposition approach. In combination with the thermal boundary conditions, the Fourier heat conduction equation is solved for the graded temperature distribution in the plate. Fluid-structure coupling between acoustic excitation and the plate is ensured by adopting the velocity continuity condition at the fluid–plate interface. With focus placed on the effect of graded thermal environments on plate vibroacoustic response, numerical investigations reveal the necessity for considering thermal moments in theoretical modeling, particularly when graded thermal environments are of common concern for aircraft structures.     

A problem of the bending of a plate for a doubly connected domain with a partially unknown boundary

The problem of the bending of an isotropic elastic plate, bounded by two rectangles with vertices lying on the same half-line, drawn from the common centre, is considered. The vertices of the inner rectangle are cut by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary). It is assumed that normal bending moments act on each rectilinear section of the boundary contours in such a way that the angle of rotation of the midsurface of the plate is a piecewise-constant function. The unknown part of the boundary is free from external forces. The problem consists of determining the bending of the midsurface of the plate and the analytic form of the unknown part of the boundary when the tangential normal moment acting on it takes a constant value, while the shearing force and the normal bending moments and torques are equal to zero. The problem is solved by the methods of the theory of boundary-value problems of analytical functions.     

A problem of the bending of a plate for a doubly-connected domain bounded by polygons

The problem of the bending of an isotropic elastic plate, bounded by two convex polygons is considered. It is assumed that the internal boundary of the plate is simply supported and normal bending moments act on each section of the external contour in such a way that the angle of rotation of the middle surface of the plate is a piecewise-constant function. With respect to the complex potentials, which express the bendings of the middle surface (Goursat's formula), the problem is reduced to a Riemann-Hilbert boundary-value problem for a circular ring, the solution of which is constructed in analytic form. Estimates are given of the behaviour of these potentials in the neighbourhood of the corner points.     

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.     

The three-dimensional problem of stress concentrations in a plate with stiff elliptic inclusions

We study the three-dimensional problem of steady vibrations of a plate with elliptic holes rigidly clamped along a cylindrical surface. The problem is studied in detail for blocking frequencies, when a biharmonic particular solution appears along with the vortex and potential solutions. We give expressions for the components of the stressed state. For the static problem we give the results of numerical studies based on the asymptotic integration of the Helmholtz equation and application of the machinery of analytic function theory. We study the influence of the thickness of the plate, curvature of the elliptic boundary, and distance between boundary curves on the error of the applied theory. One table. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 3–11.     

Large deflection of a semi-circular plate on elastic foundation under a uniform load

Large deflection of a simply supported semi-circular plate placed on elastic foundation and subjected to a uniform load has been investigated following Berger’s approximate method. Expressions for the deflections and bending moments are obtained and the theoretical results have been presented in the form of graphs.