Optimal design of stepped circular plates with allowance for the effect of transverse shear deformation

Presented herein is a canonical exact deflection expression for stepped (or piecewise-constant thickness) circular plates under rotationally symmetric transverse loads. The circular plates may be either simply supported or clamped at the edges. As the plates may be very thick or certain portions of the optimal design may become rather thick, the significant effect of transverse shear deformation on the deflections cannot be ignored. This effect was taken into consideration in accordance to the Mindlin plate theory. Based on the analytical deflection expression, necessary conditions are derived for the optimal values of segmental lengths and thicknesses that minimize the maximum deflection of stepped circular plates of a given volume. These optimality conditions are solved using the Newton method for the optimal segmental lengths and thicknesses. Local minima are observed for this nonlinear problem at hand and they may pose some difficulties in getting the solutions. The shear deformation effect increases the plate deflections, but interestingly it affects the thickness variation marginally.     

Elastic small deflection analysis of annular sector Mindlin plates

An analytical method is developed for the bending response of annular sector Mindlin plates with two radial edges simply supported, and exact solutions are presented in the form of Levy-type series. Several different boundary conditions on the two circular edges are considered, viz. simply supported-simply supported, clamped-clamped and free-free. Numerical results for the case of uniform loading are presented to indicate the effect of shear deformation on the deflections and stress resultants at various points in the plate. Twisting stress couple and transverse shear stress resultant distributions along and near the edges of the plate are illustrated graphically, and the principal differences between the results predicted by Mindlin's plate theory and classical thin plate theory are discussed in detail. Results obtained with the present exact analysis may serve as references for approximate solutions and, especially, as a ‘shear locking’ test for thick plate finite element analysis.     

Free vibration of laminated composite plates using two variable refined plate theory

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.     

An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations

In this paper, the analysis of the titled problem is based on classical thin-plate theory, and its numerical solution is carried out by using the small parameter method and Lévy-type approach. The thin rectangular plate considered herein is simply supported on two opposite edges. The boundary conditions at the other two edges may be quite general, and between these two edges the plate may have varying thickness. Closed-form solutions have been developed for the static response of isotropic rectangular plates with non-uniform thickness variation and subjected to arbitrary loading. The accuracy of the present model is demonstrated via problems for which the exact solutions and numerical results are available, and results are also compared with those obtained by using the finite-difference method.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

Buckling of laminated composite plates subjected to mechanical and thermal loads using meshless collocations

Meshless collocations utilizing Gaussian and Multiquadric radial basis functions for the stability analysis of orthotropic and cross ply laminated composite plates subjected to thermal and mechanical loading are presented. The governing differential equations of plate are based on higher order shear deformation theory considering two different transverse shear stress functions. The plate governing differential equations are discretized using radial basis functions to cast a set of simultaneous equations. The convergence of both radial basis functions is studied for different values of shape parameters. Several numerical examples are undertaken to demonstrate the accuracy of present method and the effects of orthotropy ratio of the material, span to thickness ratio of the plate, and fiber orientation on critical load/temperature are also presented.     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

Bending of linearly tapered annular Mindlin plates

This paper presents exact axisymmetric bending solutions for linearly tapered, annular Mindlin plates with various boundary conditions for the inner and outer edges. The Mindlin plate theory has been adopted so as to incorporate the effect of transverse shear deformation which becomes significant in tapered and thick plates. The analytical solutions, hitherto not available, are useful as benchmark solutions for checking the validity, convergence and accuracy of numerical methods and software for tapered plate analysis.     

Natural frequencies of shear deformable rhombic plates with clamped and simply supported edges

The natural vibrations of thick and thin rhombic plates with clamped and simply supported edges are analyzed, using assemblages of nine-node Lagrangian isoparametric quadrilateral C⁰ continuous finite elements based on a higher-order shear deformable thick plate theory. Here, additional nodal displacement degrees of freedom are derived by retaining higher-order powers of the thickness coordinate in the in-plane displacement fields, which in turn allows for the proper representation of the transverse shear strains of thick plates. Essential rotary inertia terms are derived and included in the present analysis. Nondimensional frequencies are calculated for thick and thin rhombic plates having various combinations of clamped and simply supported edge conditions, and skew angles. The efficacy of using higher-order shear deformable plate finite elements for predicting the in-plane vibration modes of rhombic plates is found to increase as the span-to-thickness ratio decreases and the skew angle increases. The present work shows that higher-order shear deformable finite elements essentially eliminate the transverse shear over-correction of thick rhombic plate frequencies that is produced when finite elements based on the widely used first-order Reissner-Mindlin plate theory are utilized.     

Stress singularities at angular corners in first-order shear deformation plate theory

The first-known Williams-type singularities caused by homogeneous boundary conditions in the first-order shear deformation plate theory (FSDPT) are thoroughly examined. An eigenfunction expansion method is used to solve the three equilibrium equations in terms of displacement components. Asymptotic solutions for both moment singularity and shear-force singularity are developed. The characteristic equations for moment singularity and shear-force singularity and the corresponding corner functions due to ten different combinations of boundary conditions are explicated in this study. The validity of the present solution is confirmed by comparing with the singularities in the exact solution for free vibrations of Mindlin sector plates with simply supported radial edges, and with the singularities in the three-dimensional elasticity solution for a completely free wedge. The singularity orders of moments and shear forces caused by various boundary conditions are also thoroughly discussed. The singularity orders of moments and shear forces are compared according to FSDPT and classic plate theory.     

Thick rectangular plates—I: The generalized Navier solution

Navier's classical solution for the problem of a statically loaded, rectangular plate which is simply-supported on all edges is generalized so as to provide a solution for the problem when transverse shear deformations are taken into account. Such solutions are provided for both Mindlin's plate theory and a new plate theory, recently published, which allows for warping of transverse sections of the plate. The results of the analyses are simple expressions for the Fourier coefficients in the expansions representing the transverse deflection and rotations of normals to the plate midsurface. Although both theories predict the same transverse deflection, the rotations and inplane stresses are different for the two theories. Some observations and comments concerning the effects of transverse shears on the deflections of plates and inplane stresses are made.     

Differential quadrature method for Mindlin plates on Winkler foundations

This paper presents an approximate analysis of rectangular plates resting on Winkler foundations based on the Mindlin plate theory. The plates are subject to any combination of free, simply supported and clamped boundary conditions. Solutions to the problem are obtained using the differential quadrature method (DQM) by solving the governing differential equations. Numerical results are compared with existing literature to establish the validity and accuracy of the method. This study shows numerically the effects of shear deformation on the deflections and stress resultants at some selected locations. The distributions of the bending and twisting moments and shear force for several plates are presented graphically by varying the relative thickness ratio h/a to further show the significant effect of shear deformation.     

Deflection and stress-resultants of axisymmetric mindlin plates in terms of corresponding Kirchhoff solutions

The Kirchhoff plate theory, when used for the analysis of bending of plates that are relatively thick, underpredicts the deflections. This is because it does not account for the effect of transverse shear deformation which becomes significant in thick plates. A more refined plate theory proposed by Mindlin allows for this shear deformation effect by relaxing the condition that the normal to the plate midsurface must remain normal to the deformed midsurface. In this paper, new exact relationships are presented between the Kirchhoff and Mindlin solutions for deflection and stress-resultants for axisymmetric plates under general rotationally symmetric loading. These relationships enable engineers and designers to obtain readily the Mindlin solutions, of such loaded axisymmetric plates, from the abundantly available Kirchhoff solutions. Thus, the task of obtaining solutions from complicated shear deformable plate analysis using the Mindlin theory may be avoided.     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Thick rectangular plates—II: The generalized Lévy solution

Lévy's classical solution for the problem of a statically loaded, rectangular plate which is simply supported on two opposite edges and has arbitrary boundary conditions specified on the remaining edges is generalized so as to provide a solution for the problem when transverse shear deformation is taken into account. Such solutions are provided for both Mindlin's plate theory and a new plate theory, recently published, which does not require that normals to the undeformed midsurface of the plate remain straight after deformation of the plate. The results of these analyses are relatively uncomplicated generalizations of the classical results for the transverse displacement, or deflection, of the midsurface of the plate and equally simple expressions for the rotations of normals to the midsurface of the undeformed plate. Several particular examples are presented and numerical results are tabulated for several typical plate geometries in order to compare the predictions of the two theories considered. Pertinent observations and comments concerning the analytical and numerical results are made.     

3-D vibration analysis of skew thick plates using Chebyshev–Ritz method

The free vibration characteristics of skew thick plates with arbitrary boundary conditions have been studied based on the three-dimensional, linear and small strain elasticity theory. The actual skew plate domain is mapped onto a basic cubic domain and the eigenvalue equation is then derived from the energy functional of the plate by using the Ritz method. A set of triplicate Chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometric boundary conditions of the plate is developed as the trial functions of the displacement components. The vibration modes are divided into antisymmetric and symmetric ones in the thickness direction and can be studied individually. The convergence and comparison studies show that rather accurate results can be obtained by using this approach. Parametric investigations on rhombic plates with fully clamped edges and completely free edges are performed in detail, with respect to the thickness-span ratio and skew angle. Some results known for the first time are reported, which may serve as the benchmark values for future numerical technique research.     

Effect of transverse shear and rotatory inertia on the forced axisymmetric response of linearly tapered circular plates

The forced axisymmetric response of linearly tapered circular plates, based on the shear theory is analyzed by the eigenfunction method. Clamped and simply supported plates subjected to constant and half-sine pulse loads, uniformly distributed over a symmetric portion of the plate, are solved as example problems. Numerical results computed for transverse deflection and radial stress of the plate are compared with the corresponding results of classical theory. Results obtained for a plate of constant thickness, as a particular case, are compared with closed form solutions and a very good agreement is found.     

Postbuckling of symmetrically laminated, moderately thick, axisymmetric shallow spherical shells

The paper deals with the buckling and postbuckling behaviour of cylindrically orthotropic, axisymmetric laminated, moderately thick shallow spherical shells under uniformly distributed normal loading. Considering the effects of transverse shear, the governing equations of equilibrium for the shells are derived and expressed in terms of normal deflection , slope qf and stress function gy. An iterative Chebyshev series solution technique is employed for the buckling and postbuckling analyses. Critical loads are estimated and the effects of boundary conditions, material properties, shell parameter, base radius to thickness ratio and number of layers on the postbuckling behaviour are shown.     

Deflections and free vibrations of laminated plates—Levy-type solutions

This study is concerned with the static deflections and natural frequencies of isotropic, orthotropic/laminated composite plates using a Levy-type solution. Mindlin plate theory is applied in conjunction with the state-space concept to find such solutions. A state-space formulation of such plates is composed of variables having physical meanings, such as moments, shear forces, displacements and rotations. The influences of aspect ratio, ratio, fiber orientation angle, laminate-layer arrangement and ratio of moduli have been investigated. Some numerical results from the present analyses are compared with published results and good agreement is found.