Free vibration of cantilevered arbitrary triangular Mindlin plates

This paper presents a free vibration analysis of thick cantilevered arbitrary triangular plates based on the Mindlin shear deformation theory. The solutions are computed using the recently developed pb-2 Rayleigh—Ritz method. The actual triangular plate is first mapped onto a basic square plate, and the deflections and rotations of the plate are approximated by Ritz functions defined as products of two-dimensional polynomials in the basic square plate domain and a basic function. The basic function satisfies the geometric boundary conditions at the outset and is chosen as the boundary expression of the cantilevered edge. Stiffness and mass matrices are integrated numerically over the domain of the basic square plate using Gaussian quadrature. Wherever possible, the present results are verified by comparison with existing analytical and experimental values from the open literature. To the authors' knowledge, first known results of natural frequencies for cantilevered arbitrary triangular Mindlin plates are presented for a wide range of geometries and thicknesses. These results are valuable to design engineers for checking their natural frequency calculations and may also serve as benchmark values for future numerical techniques and software packages for thick plate analysis. The influence of shear deformation and rotary inertia on the natural frequency parameters are examined.     

DSC-element method for free vibration analysis of rectangular Mindlin plates

A novel DSC-element method is proposed to investigate the free vibration of moderately thick plates based on the well-known Mindlin first-order shear deformation plate theory. The development of the present approach not only employs the concept of finite element method, but also implements the discrete singular convolution (DSC) delta type wavelet kernel for the transverse vibration analysis. This numerical algorithm is allowed dividing the domain of Mindlin plates into a number of small discrete rectangular elements. As compared with the global numerical techniques i.e. the DSC-Ritz method, the flexibility is increased to treat complex boundary constraints. For validation, a series of numerical experiments for different meshes of Mindlin plates with assorted combinations of edge supports, plate thickness and aspect ratios is carried out. The established natural frequencies are directly compared and discussed with those reported by using the finite element and other numerical and analytical methods from the open literature.     

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.     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

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.     

Nonlinear bending of Reissner–Mindlin plates with free edges under transverse and in-plane loads and resting on elastic foundations

A nonlinear bending analysis is presented for a rectangular Reissner–Mindlin plate with free edges subjected to combined transverse partially distributed load and compressive edge loading and resting on a two-parameter (Pasternak-type) elastic foundation. The formulations are based on the Reissner–Mindlin plate theory considering the first-order shear deformation effect, and including the plate-foundation interaction. The analysis uses a mixed Galerkin-perturbation technique to determine the load–deflection curves and load–bending moment curves. Numerical examples are presented that relate to the performances of moderately thick rectangular plates with free edges subjected to combined loading and resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, loaded area, the plate aspect ratio and initial compressive load are studied. Typical results are presented in dimensionless graphical form.     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

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.     

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.     

Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations

Closed-form solutions for the vibration problem of initially stressed thick rectangular plates as described by Mindlin theory are presented. The plates are simply supported and resting on Pasternak foundations. The subset problem of buckling of Mindlin plates on Pasternak foundations is automatically solved by setting the frequency parameter to be equal to zero. The solution also applies to Winkler foundations where the shear modulus of the Pasternak model is taken to be zero. The closed-form solutions should be useful for checking the accuracy of numerical solutions.     

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.     

Analysis of annular Reissner/Mindlin plates using differential quadrature method

The axisymmetric flexure responses of moderately thick annular plates under static loading are investigated. The shear deformation is considered using the first-order Reissner/Mindlin plate theory and the solutions are obtained using the differential quadrature (DQ) method. In the solution process, the governing differential equations and boundary conditions for the problem are initially discretized by the DQ algorithm into a set of linear algebraic equations. The solutions of the problem are then determined by solving the set of algebraic equations. This study considers the plate subjected to various combinations of clamped, simply-supported, free and guided boundary conditions and different loading manners. The accuracy of the method is demonstrated through direct comparison of the present results with the corresponding exact solutions available in the literature.     

Free vibration of cantilevered symmetrically laminated thick trapezoidal plates

To account for the effect of transverse shear deformation, the p-Ritz method incorporating Reddy’s third-order shear deformation theory has been developed for the vibration analysis of cantilevered, thick, laminated, trapezoidal plates. In the p-Ritz method, a set of uniquely defined polynomial functions, consisting of the product of a two-dimensional function and a basic function, are used as the admissible trial displacement and rotation functions in the Ritz minimization procedure. The energy integral is formulated based on Reddy’s third-order shear deformation theory. From the p-Ritz method, the governing eigenvalue equation is derived which is used to compute the vibration frequency parameters and mode shapes of the laminated plate. Convergence and comparison studies have been presented to demonstrate and verify the accuracy of the results.     

Vibration analysis of piezoelectric FGM sensors using an accurate method

In this study, free vibration analysis of moderately thick smart FG annular/circular plates with different boundary conditions is presented on the basis of the Mindlin plate theory. This structure comprised a host FG plate and two bonded piezoelectric layers. Piezoelectric layers are open circuit therefore this plate can be used as a sensor. According to power-law distribution of the volume fraction of the constituents, material properties vary continuously through the thickness of host plate while Poisson's ratio is set to be constant. Using Hamilton's principle and Maxwell electrostatic equation yields six complex coupled equations which are solved via an exact closed-form method. The accuracy of the frequencies is verified by the available literature, finite element method (FEM) and the Kirchhoff theory. The effects of plate parameters like boundary condition and gradient index are investigated and significance of coupling between in-plane and transverse displacements on the resonant frequency is proved.     

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.     

Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

Reflection characteristics of longitudinal waves in a semi-infinite cylindrical rod connected to an elastic infinite plate

The reflection characteristics of longitudinal strain waves in a semi-infinite cylindrical rod connected to an elastic infinite plate are investigated theoretically and experimentally. In the analysis, both the Mindlin plate theory which includes the effects of plate rotary inertia and shear deformation and the classical plate theory are employed. Taking account of the experimental results of the incident strain pulses produced by the impact of spherical balls, the shapes of the incident pulses for the numerical calculations are assumed to be square of a half sinusoid. Applying Laplace transformations with respect to time and numerical inverse Laplace transformations, the time histories of the longitudinal strain at the arbitrary points of the rod are presented. The differences between the reflected waves obtained from the Mindlin plate theory and those from the classical plate theory are large when the plate thickness is large and/or the plate is soft compared with the rod. Thus, the effects of plate rotary inertia and shear deformation are large in the case of thick and/or soft plate. Experimental results are closer to the numerical results obtained from the Mindlin plate theory than those obtained from the classical plate theory.     

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.     

Stability and free vibration analysis of thick piezoelectric composite plates using spline finite strip method

In the present study, a spline finite strip with higher-order shear deformation is formulated for stability and free vibration analysis of piezoelectric composite plates. At each knot, the electric potentials on the surfaces and middle plane of each piezoelectric layer are taken as nodal degrees of freedom. However, if a continuous electrode is installed on the surface of the layer, the electric potential on the electrode is changed to structural degree of freedom, so that the equipotential condition on the electrode is automatically satisfied. The analysis can be conducted based on Reddy's third-order shear deformation theory, Touratier's “Sine” model, Afaq's exponential model or Cho's higher-order zigzag laminate theory. Consequently, the shear correction coefficients are not required in the analysis, and an improved accuracy for thick plates over the first-order shear deformation theory is achieved at only little extra computational cost.The numerical results obtained based on different shear deformation theories are presented in comparison with the three-dimensional solutions. The effects of length-to-thickness ratio, fiber orientation, boundary conditions and electrical conditions on the natural frequency and critical buckling load of piezoelectric composite plates are investigated through numerical examples.     

Optimum design of composite laminated plates via a multi-objective function

This paper deals with the optimum design of composite laminated plates under stiffness and gauge constraints. A multi-objective function which combines the plate weight and the strain energy stored in the plate by weighting parameters is introduced. This objective function is minimized while satisfying constraints such as the structural deformation and the limits on design variables. Both ply orientation angles and ply thicknesses of the composite plate are used as the design variables. The stiffness analysis is performed by the finite element method in which a triangular element is used that is suitable for the analysis of thin to thick plates and includes the transverse shear effects. Analyses of the derivatives of the objective function and the constraint functions with respect to the design variables is performed analytically. The mathematical programming method called the constrained variable metric is used to solve this optimum problem. An example is provided for the optimal design of a rectangular laminated plate.