Postbuckling of shear deformable laminated plates under biaxial compression and lateral pressure and resting on elastic foundations

Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to biaxial compression combined with uniform lateral pressure and resting on an elastic foundation. The lateral pressure is first converted into an initial deflection and the initial geometrical imperfection of the plate is also taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and including the plate-foundation interaction. The analysis uses a perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates under combined loading and resting on Pasternak-type or softening nonlinear elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, the biaxial load ratio and initial lateral pressure are studied.     

The refined sinusoidal theory for FGM plates on elastic foundations

Using the refined sinusoidal shear deformation plate theory and including plate-foundation interaction, a thermoelastic bending analysis is presented for a simply supported, rectangular, functionally graded material plate subjected to a transverse uniform load and a temperature field, and resting on a two-parameter (Pasternak model) elastic foundation. The present shear deformation theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The equilibrium equations of the present plate are given based on various plate theories. A number of examples are solved to illustrate the numerical results concerning bending response of homogeneous and functionally graded rectangular plates resting on two-parameter elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, volume fraction distributions, and elastic foundation parameters are studied.     

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.     

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.     

Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments

Nonlinear bending analysis is presented for a simply supported, functionally graded rectangular plate subjected to a transverse uniform or sinusoidal load and in thermal environments. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded plate are based on Reddy's higher-order shear deformation plate theory that includes thermal effects. Two cases of the in-plane boundary conditions are considered. A mixed Galerkin-perturbation technique is employed to determine the load-deflection and load-bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded rectangular plates with two constituent materials. The influences played by temperature rise, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio and volume fraction distributions are studied.     

Postbuckling analysis of composite laminated plates on two-parameter elastic foundations

A postbuckling analysis is presented for a simply supported, composite laminated rectangular plate under uniaxial in-plane loading and resting on a two-parameter (Pasternak-type) elastic foundation. The analysis uses a perturbation technique to determine buckling loads and postbuckling equilibrium paths. The initial geometrical imperfection of the plate is taken into account. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated rectangular plates. Typical results are presented in dimensionless graphical form.     

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.     

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.     

Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations

Approximate solutions for the bending of moderately thick rectangular plates on two-parameter elastic foundations (Pasternak-type) as described by Mindlin's theory are presented. The plates are subjected to an arbitrary combination of clamped and simply-supported boundary conditions. An efficient computational technique, the differential quadrature (DQ) method, is employed to transform the governing differential equations and boundary conditions into a set of linear algebraic equations for approximate solutions. These resulting algebraic equations are solved numerically. In this study, the accuracy of the DQ method is established by direct comparison with results in the existing literature. The convergence properties of the method are illustrated for different combinations of boundary conditions. The deflections, moments and shear forces at selected locations are tabulated in detail for different elastic foundations. The efficiency and simplicity of the solution method are highlighted.     

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.     

A post-buckling analysis for isotropic prismatic plate assemblies under axial compression

This paper presents a post-buckling analysis for prismatic plate assemblies made of isotropic materials. The structures are assumed to consist of a series of long flat strips rigidly connected together at their edges, subjected to longitudinal in-plane compressive load. The buckling load and corresponding buckling mode of the structure are first obtained as the results of transcendental eigenvalue problems, which arise when exact solutions to the member differential equations are used to form the stiffness matrix of the plate assemblies. The other post-buckling field functions are also obtained analytically as exact solutions to the member differential equations. Results for the load end-shortening and load–deflection relationships for long prismatic plate assembly examples are obtained and compared with results obtained by other authors.     

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.     

Free vibration of a laminated composite shell of revolution: Effects of shear non-linearity

Effects of shear non-linearity on free vibration of a laminated composite shell of revolution are investigated using a semi-analytical method based on the Reissner–Mindlin shell theory. The coupling between symmetric and anti-symmetric vibration modes of the shell is considered in the shear deformable shell element employed in this study. The Hahn–Tsai non-linearly elastic shear stress–shear strain relation is adopted. Numerical examples are given for laminated composite circular cylindrical and conical shells with various boundary conditions. The numerical results indicate that shear non-linearity may reduce significantly the fundamental frequencies of cross-ply composite shells of revolution.     

Flat-topped conical shell under axial compression

Based on experimental observations of a grid-domed textile composite under axial compression, the large deformation mechanisms of a flat-topped conical shell are identified. Accordingly, both elastic model and rigid-plastic model are proposed to describe the collapse process and predict the load–displacement characteristics. In the rigid-plastic analysis, the energies dissipated in bending along plastic hinge lines and in stretching of the thin-wall segments between the plastic hinge lines are taken into account. Analytical expressions describing the load–displacement and energy–displacement relationships during the large deformation process are derived. Illustrated by typical numerical examples, the effects of apical angle of a flat-topped conical shell on its energy absorption capacity are revealed. The respective strain distributions on the conical shell resulted from bending deformation and membrane deformation are presented. A good agreement is shown between the theoretical predictions and experimental results.     

A twelfth-order theory of antisymmetric bending of isotropic plates

In this paper the transverse shear and normal strain and stress effects on antisymmetric bending of isotropic plates are considered. A set of twelfth-order partial differential governing equations as well as a set of fourth-order ordinary differential equations for ƒ(z) and φ(z), which represent the transverse shear and normal effects, are derived from a mixed variational theorem. There exists coupling between the partial differential equations and the ordinary ones. In the homogeneous solutions for the former, besides an interior solution contribution, there exist two types of edge-zone solution contributions. One of them is similar to the edge-zone solution in the Reissner—Mindlin theory. The other one is an edge-zone solution consisting of a pair of conjugate functions. Two sample examples are calculated using the present theory. In the former the present two-dimensional theory obtains the three-dimensional exact solution. The latter gives the stress couple and maximum-stress concentration factors at the free edge of a circular hole in a large bent plate. The numerical results still approximate to exact solutions.     

Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass

The elastodynamic response of a rectangular Mindlin plate subjected to a distributed moving mass is investigated. The set of governing characteristic partial differential equations that include the effects of shear deformation and rotary inertia is expressed in its dimensionless form. A finite difference algorithm is employed to transform the differential equations into a set of linear algebraic equations. Simply supported edge conditions were used as an illustrative example. The analysis is also valid for other edge conditions. It is found that the maximum shearing forces, bending and twisting moments occur almost the same time. Also, the values of the maximum deflections are higher for Mindlin plates than for non-Mindlin plates.     

The influence of elastic shear deformation on the transverse shear failure of a fully clamped beam subjected to idealized blast loading

The influence of elastic shear deformation on the transverse shear response of a fully clamped beam is investigated in the present paper. The beam is made from a rigid, perfectly plastic material and subjected to a uniformly distributed pressure pulse loading. The elastic shear deformation is idealized by an elastic, perfectly plastic spring with a constant spring coefficient. Analytical solutions are obtained for the transverse shear response, which are then used to predict the occurrence of a transverse shear failure. The method presented in the paper may be extended to study the blast-induced shear failure of other structural elements when the elastic shear deformation needs to be considered.     

A composite view on Windenburg's problem: Buckling and minimum stiffness requirements of compressively loaded orthotropic plates with edge reinforcements

This paper discusses the buckling behaviour of orthotropic composite plates under uniform uniaxial compression with one free reinforced unloaded edge. A typical application example for use of such a mechanical model is the web of stiffeners and frames attached to the fuselage skin of an aircraft. The considered plates are rectangular and simply supported at the loaded transverse edges. One of the longitudinal unloaded edges is also simply supported, while the second unloaded edge is not supported at all but is reinforced by a flange of arbitrary cross-section. At first, an exact solution for the elastic buckling problem is derived from the governing differential equation by imposing the underlying boundary conditions. Thereafter, two approximate closed-form solutions for the buckling load are derived, which can be conveniently used for practical application purposes. Generic buckling curves using characteristic non-dimensional quantities are also presented. Finally, the question of the required bending stiffness EI_min of the flange is treated, to ensure that the flange withstands buckling and provides simply supported boundary conditions to the free reinforced plate edge.     

Effect of compressive axial load on forced transverse vibrations of a double-beam system

Based on Bernoulli–Euler beam theory, the forced transverse vibrations of an elastically connected simply supported double-beam system under compressive axial load are investigated. It is assumed that the two beams of the system are continuously joined by a Winkler elastic layer. The dynamic responses of the system caused by arbitrarily distributed continuous loads are obtained. The effects of compressive axial load on the forced vibrations of the double-beam system are discussed for two cases of particular excitation loadings. The properties of the forced transverse vibrations of the system are found to be significantly dependent on the compressive axial load.     

Dynamic stability of a cantilevered Timoshenko beam on partial elastic foundations subjected to a follower force

This paper presents the dynamic stability of a cantilevered Timoshenko beam with a concentrated mass, partially attached to elastic foundations, and subjected to a follower force. Governing equations are derived from the extended Hamilton’s principle, and FEM is applied to solve the discretized equation. The influence of some parameters such as the elastic foundation parameter, the positions of partial elastic foundations, shear deformations, the rotary inertia of the beam, and the mass and the rotary inertia of the concentrated mass on the critical flutter load is investigated. Finally, the optimal attachment ratio of partial elastic foundation that maximizes the critical flutter load is presented.