Narrow Results

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, we place a certain amount of denser material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In this paper, we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.

This paper is concerned with the elastic buckling problem of vertical plates under body forces/selfweight. The vertical plate is either clamped or simply supported at its bottom edge while its top edge is free. The two sides of the plate may either be free, simply supported or clamped. For plates with simply supported sides, exact critical buckling solutions are derived using the Levy approach. For other boundary conditions, accurate buckling solutions are obtained for very wide plates to very tall plates using the Ritz method.

The paper presents a Rayleigh–Ritz based non-discretization method of analysis for the inelastic local buckling of rectangular steel plates subjected to applied in-plane axial, bending and shear actions with various boundary conditions. Use is made of the pb-2 representation of the displacement function as the product of a domain polynomial and a boundary polynomial. The constitutive model for the plate is an adaptation of the von Mises yield criterion and the associated flow rule presented in an infinitesimal form, and which leads to an incremental and iterative method of solution. The convergence and accuracy of the solutions are demonstrated, and it is shown that the method of analysis is computationally feasible for the solution of inelastic bifurcative plate instability problems. A parametric study is then undertaken for a range of plate boundary conditions under various regimes of applied loading.

A thin circular plate with a free or movable boundary, and weakened along an internal concentric circle, is considered. The frequencies of the circular plate varying with the radius of the weakened circle and the extent of the weakening represented by an elastic rotational restraint are determined. The results for the first four modes are tabulated. For the free plate, the internal weakening only slightly decreases the fundamental frequency of the plate, which is advantageous. For the plate with a movable edge, the internal weakening greatly decreases the fundamental frequency, especially for the purely hinged case.

A procedure is presented for stability analysis of thin plates subject to any combination of in-plane shear, biaxial compression, and bending. Unlike finite element or finite strip methods, where the plate is discretized into a set of elements or strips, the plate here is treated as a single element. An energy-based formulation is used to express the buckling coefficient, K, in terms of general functions that describe the longitudinal and transverse displacement profiles. The longitudinal edges are treated as partially restrained against rotation (PRR) and in-plane translation. Unconstrained optimization technique is then used to determine the minimum combinations of pre-selected geometric plate parameters. Accuracy of the derived expressions is compared with the Galerkin method for the limiting simply supported and clamped boundary conditions. Results are then presented for plates PRR and in-plane translation.

This paper proposes a finite-element (FE) and perfectly matched layer (PML)modeling of three-dimensional (3D) scattering of transient elastic waves in a cracked infinite plate with rectangular cross-section. The FE predictions are validated against 3D semi-analytical literature results. The effects of PML parameters on a root-mean-square error estimate are measured against the reference FE predictions computed using extended meshes. The proposed model is shown, through the numerical examples, to offer huge saving in real run-time at a slight degradation in accuracy. Practical applications indicate its potential in modeling elastic-wave-based nondestructive evaluation of engineering structures.

In this paper chaotic vibrations of flexible plates of infinite length are studied. The Kirchhoff–Love hypotheses are used to derive the nondimensional partial differential equations governing the plate dynamics. The finite difference method (FDM) and finite element method (FEM) are applied to validate the numerical results. The numerical analysis includes both standard (time histories, fast Fourier Transform, phase portraits, Poincaré sections, Lyapunov exponents) as well as wavelet-based approaches. The latter one includes the so called Gauss 1, Gauss 8, Mexican Hat and Morlet wavelets. In particular, various plate dynamical regimes including the periodic, quasi-periodic, sub-harmonic, chaotic vibrations as well as bifurcations of the plate are illustrated and studied. In addition, the convergence of numerical results obtained via different wavelets is analyzed.

Presented herein is a new method for the analysis of plates with clamped edges. The solutions for the natural frequencies of the plates are found using static analysis. The starting are the equations of motion of an isotropic rectangular plate supported on Winkler elastic foundation, with a positive or negative value. In either case, one can solve the displacements of such a plate under a given concentrated load. This deflection will be infinite if the plate losses its stiffness, or in other words, the generalized foundation is causing the plate to be unstable. The solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution for a clamped plate is decomposed as the sum of three cases of plates resting on elastic foundation: simply supported plate with a concentrated load, and two cases of distributed moments along opposite edges. The solution for simply supported plates with elastic foundation is found using Navier's method. For zero force, the vibration frequencies are found up to the desired accuracy by careful calculations at the neighborhood of the roots.

In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: C — clamped, S — Simply supported, F — free, and G — guided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution is composed of three parts, the sum of which satisfies exactly both the field equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the first six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the first time.

In this research, a new analytical solution is used for finding the buckling loads of rectangular plates with vertically and rotationally restrained edges. The solution method in this study is based on the development of a static solution for a plate. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution fits all the combinations of possible boundary conditions, of the deflection, slope, shear force and bending moment along the edges of the plate. In the case of springs, the edge force and moment boundary conditions are modified to include these effects. Any number of edges, from one to four, with both types of stiffening springs can be solved. Using this new method, the exact buckling loads and modes are found. The results are verified with published data, and many new cases are presented for uni-axially and bi-axially loaded isotropic, orthotropic, and composite plates.

The aim of this paper is to study the asymptotic behavior of a structure made of plates of thickness 2δ when δ → 0. This study is carried out within the frame of linear elasticity by using the unfolding method. It is based on several decompositions of displacements of the structure and on the passing to the limit in fixed domains.

We begin by studying the displacements of a plate. We show that any displacement is the sum of an elementary displacement concerning the normal lines on the middle surface of the plate and a warping. An elementary displacement is linear with respect to the variable x₃. It is written where is a displacement of the mid-surface of the plate. We show a priori estimates and convergence results when δ → 0. We characterize the limits of the unfolded displacements of a plate as well as the limits of the unfolded strained tensor.

Then, we extend these results to structures made of plates. We show that any displacement of a structure is the sum of an elementary displacement of each plate and of a residual displacement. The elementary displacements of the structure (e.p.s.d.) coincide with elementary rod displacements in the junctions. Any e.p.s.d. is given by two functions belonging to H¹(S; ℝ³) where S is the skeleton of the structure (the set formed by the mid-surfaces of the plates constituting the surface). One of these functions, , is the skeleton displacement. We show that is the sum of an extensional displacement and of an inextensional one. The first one characterizes the membrane displacements and the second one is a rigid displacement in the direction of the plates and it characterizes the flexion of the plates.

Eventually, we pass to the limit as δ → 0 in the linearized elasticity system. On the one hand, we obtain a variational problem that is satisfied by the limit extensional displacement, and on the other hand, a variational problem satisfied by the limit of inextensional displacements.

This paper is dedicated to the homogenization and dimension reduction for a nonlinear elasticity problem of a textile structure. The structure is represented as a squared piece of cloth and modeled as a woven canvas made of long and thin fibers, crossing each other in a periodic pattern. The squared domain is partially clamped on the left and bottom. The fibers are assumed to be glued, a condition that allows to extend the woven structure to a non-perforate plate. The nonlinear elasticity problem is stated via minimization over the energy functional. The homogenization is made via the periodic unfolding method, with an additional dimension reduction. Since the existence of a minimum of the limit energy functional is not ensured, sufficient conditions for the existence are given by assuming that the material is homogeneous and isotropic.

Reducing the general problem of computing three-dimensional Green's function in a transversely isotropic plate to a finite summation of contributions from a series of planar problems can efficiently yield an accurate solution. Hence, solving the planar scattering problem, of the Pressure-Shear-Vertical (PSV) type or the Shear-Horizontal (SH) type, was performed by three different techniques: The boundary element method; the hybrid method; and the perfectly matched layer method. In the pursuit of these methods, the objective was to highlight their pros and cons in terms of accuracy and efficiency.

This paper introduces a novel approach called the five-unknown trigonometric shear deformation (FTSD) theory for analyzing the flexural and buckling behavior of porous functionally graded (PFG) plates mounted on an elastic foundation. The FTSD theory features a unique trigonometric displacement field distribution that accurately captures the parabolic transverse shear stress distribution and satisfies the zero transverse shear stress conditions on the plate’s top and bottom surfaces while accounting for thickness stretching effects. The elastic foundation is modeled using a three-parameter Kerr-type model, consisting of two spring layers connected by a shear layer. The study considers both isotropic and sandwich PFG plates with various configurations of constituent layers, as well as both regular and irregular porosity types. The validity of the proposed methodology is evaluated by comparing its results with those from previously published research. Additionally, an extensive parametric analysis is conducted to explore the effects of porosity parameters and foundation parameters on the bending and buckling performance of PFG plates.

The influence of auxeticity on the mechanical behavior of isotropic plates is considered herein by evaluating the plate flexural rigidity as the Poisson's ratio changes from 0.5 to -1. Since the change in plate's Poisson's ratio is followed by a change in at least one of the three moduli, any resulting change to the plate flexural rigidity is only meaningful when at least one of the moduli is held constant. This was performed by normalizing the plate flexural rigidity by a single modulus, a square root of two moduli product, or a cube root of three moduli product to give a dimensionless plate flexural rigidity. It was found that the plate flexural rigidity decreases to a minimum as the plate Poisson's ratio decreases from 0.5 to 0 when only the Young's modulus is held constant. Thereafter the plate flexural rigidity increases with the plate auxeticity. Results also reveal that when only the shear modulus or when the bulk modulus is held constant, the plate flexural rigidity decreases or increases, respectively, with the plate auxeticity. Intermediate trend in the plate flexural rigidity is observed when the product of two moduli is held constant. When the product of all three moduli is held constant, the plate flexural rigidity increases with the plate auxeticity, and the change is especially drastic when the plate Poisson's ratio is near to the upper and lower limits of Poisson's ratio for isotropic solids. Results from this work are useful for structural designers to control the flexural rigidity of plate made from auxetic materials.

A 3D free vibration analysis of multilayered structures is proposed. An exact solution is developed for the differential equations of equilibrium written in general orthogonal curvilinear coordinates. The equations consider a geometry for shells without simplifications and allow the analysis of spherical shell panels, cylindrical shell panels, cylindrical closed shells and plates. The method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structures. Results are given for multilayered composite and sandwich plates and shells. A free vibration analysis is proposed for a number of vibration modes, thickness ratios, imposed wave numbers, geometries and multilayer configurations embedding isotropic and orthotropic composite materials. These results can also be used as reference solutions for plate and shell 2D models developed for the analysis of multilayered structures.

In this paper, a displacement-based unified shear deformation theory is developed for the analysis of shear deformable advanced composite beams and plates. The theory is developed with the inclusion of parabolic (PSDT), trigonometric (TSDT), hyperbolic (HSDT) and exponential (ESDT) shape functions in terms of thickness coordinate to account for the effect of transverse shear deformation. The in-plane displacements consider the combined effect of bending rotation and shear rotation. The use of parabolic shape function in the present theory leads to the Reddy’s theory, but trigonometric, hyperbolic and exponential functions are first time used in the present displacement field. The present theory is accounted for an accurate distribution of transverse shear stresses through the thickness of plate, therefore, it does not require problem dependent shear correction factor. Governing equations and associated boundary conditions of the theory are derived from the principle of virtual work. Navier type closed-form solutions are obtained for simply supported boundary conditions. To verify the global response of the present theory it is applied for the bending of both one-dimensional (beams) and two-dimensional (plates) functionally graded, laminated composite and sandwich structures. The present results are compared with exact elasticity solution and other higher order shear deformation theories to verify the accuracy and efficiency of the present theory.

This paper is concerned with the elastic buckling of rectangular plates subject to both intermediate and end uniaxial loads. The aforementioned buckling problem is solved by decomposing the plate into two subplates at the boundary where the intermediate load acts. Each sub-plate buckling problem is solved exactly using the Levy approach and the two solutions brought together by the continuity equations at the separated edge. It is worth noting that there are 5 possible solutions for each sub-plate and consequently there are 25 solution combinations to be considered. The final solution combination depends on nature of the ratio of the intermediate load to the end load. The exact stability criterion for the plate is presented both in tabulated and in a graphical forms which should be useful for engineers designing walls or plates that have to support intermediate floors/loads.