Thermal post-buckling analysis of imperfect shear-deformable plates on two-parameter elastic foundations

A thermal post-buckling analysis is presented for a shear-deformable rectangular plate subjected to uniform or non-uniform tent-like temperature loading and resting on a two-parameter elastic foundation. The initial goemetrical imperfection of the plate is taken into account. The formulations are based on the Reissner-Mindlin plate theory considering the first order shear deformation effect and including thermal effects. The analysis uses a deflection-type perturbation technique to determine the thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, shear-deformable plates resting on Winkler or Pasternak-type elastic foundations.     

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigen mode of the panel.In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths.The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.     

A three-dimensional nonlinear analysis of cross-ply rectangular composite plates

The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropie composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement     

A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges

On the basis of three-dimensional elasticity, this paper presents a free vibration analysis of cross-ply laminated rectangular plates with clamped boundaries. The analysis is based on a recursive solution suitable for three-dimensional vibration analysis of simply supported plates. Clamped boundary conditions are imposed by suppressing the edge displacements of a number of planes which are parallel to the mid-plane. This is achieved by coupling a number of different vibration modes of the corresponding simply supported plate using a Lagrange multiplier method. A satisfactory solution can be obtained by choosing suitably larger numbers of the coupled vibration modes and the fixed planes across the thickness of the plate. Numerical results are presented to show the convergence of the solution. Results are also obtained for either isotropic or cross-ply laminated plates having different combinations of simply supported and clamped boundaries.     

Finite-element analysis of laminated bimodulus composite-material plates

Finite element analysis of the equations governing the small-deflection elastic behavior of thin plates laminated of anisotropic bimodulus materials (which have different elastic stiffnesses depending upon the sign of the fiber-direction strain) is presented. Single-layer and two-layer cross-ply, simply-supported rectangular plates subjected to sinusoidally distributed normal pressure and uniformly distributed normal pressure are analyzed. For the sinusoidally loaded case, the finite element solutions are gratifyingly close to the exact closed-form solutions.     

Stability analysis of anisotropic laminated composite plates by finite strip method

The finite strip method has been applied to the stability analysis of rectangular shear-deformable composite laminates. However, for the plates with two opposite simply supported sides, the existing analysis was restricted to the symmetrical cross-ply laminates under compression loading.In the present study, by selecting proper displacement functions and including the coupling between different series terms, the finite strip method is extended to the stability analysis of any anisotropic laminated plates under arbitrary in-plane loading. Furthermore, a number of numerical results are presented to show the effects of thickness, fibre orientation and stacking sequence on the buckling loads.     

Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation

A modified finite element formulation to study the post-buckling behaviour of elastic circular plates is presented in this paper. A discussion on the derivation of nonlinear stiffness matrix for post-buckling analysis is included and the present results are compared with continuum solutions.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

Analysis of laminated shells with laminated stiffeners using rectangular shell finite elements

A finite element analysis of laminated shells reinforced with laminated stiffeners is described in this paper. A rectangular laminated anisotropic shallow thin shell finite element of 48 d.o.f. is used in conjunction with a laminated anisotropic curved beam and shell stiffening finite element having 16 d.o.f. Compatibility between the shell and the stiffener is maintained all along their junction line. Some problems of symmetrically stiffened isotropic plates and shells have been solved to evaluate the performance of the present method. Behaviour of an eccentrically stiffened laminated cantilever cylindrical shell has been predicted to show the ability of the present program. General shells amenable to rectangular meshes can also be solved in a similar manner.     

Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: the effects of large vibration amplitudes on the fundamental mode shape

The geometrically non-linear free vibrations of thin isotropic and laminated rectangular composite plates with fully clamped edges have been successfully investigated in previous series of works using a theoretical model based on Hamilton’s principle and spectral analysis. The objective of this work is the extension of the above model to the case of clamped clamped simply supported simply supported rectangular plates, denoted by CCSSSSRP, in order to determine their fundamental non-linear mode shape, and associated amplitude-dependent resonant frequencies, and flexural stress distribution. Numerical data are given for both linear and non-linear analysis, for various plate aspect ratios and vibration amplitudes. Good agreement was found with previous published results.     

Free vibrations of rectangular unsymmetrically laminated composite plates with internal line supports

Free vibrations of rectangular unsymmetrically laminated composite plates with internal line supports in one or two directions are studied. The static beam functions, which are the complete solutions of a unit width strip taken from the plate in the direction parallel to the edges of the plate and under a series of transverse static sinusoidal loads, are used as the trial functions of the transverse bending deflection of the plate. Besides, the vibrating bar functions are used as the trial functions of the in-plane stretching deformation of the plate. The Rayleigh–Ritz method is applied to arrive at the governing eigenfrequency equation in which the coupling effect of bending and stretching is contained. Different transverse boundary conditions combining with various in-plane constraints are considered in the analysis. The effects of fiber orientation angles and locations of internal line supports on eigenfrequencies of cross-ply and antisymmetric angle-ply laminated composite plates of two layers are investigated in detail. Convergence and comparison study show the correctness and effectiveness of the proposed method. Sets of valuable results are presented for the first time, which can serve as benchmarks for future reference.     

Post-buckling behaviour of moderately thick circular plates with edges elastically restrained against bending rotation

A finite element formulation is presented to study the post-buckling behaviour of circular plates with the effects of shear deformation and elastic restraints against bending rotation at the edges of the plates considered. A set of numerical results are included which clearly bring out variation of post-buckling characteristics with the varying values of thickness of the plate and the stiffness of the spring supports.     

Finite elements for post-buckling analysis. II—Application to composite plate assemblies

In a companion paper the authors presented a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path and the existence of distinct critical loads. The formulation developed is known as the W-formulation, where the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the present paper a number of applications of finite elements for post-buckling analysis on composite plate assemblies are presented. Thin-walled composite plates, I-beams, angle sections, and a specially designed box-beam with flanges (unicolumn) are studied in post-buckling when axially loaded. The results are in good agreement with previous studies. Moreover, a parametric study involving critical buckling load and geometry is presented for the case of the unicolumn.     

Elastic-plastic dynamic analysis of anisotropic laminated plates

This paper presents the application of a refined finite element model to the elastic and elastic-plastic dynamic analysis of anisotropic laminated plates. Dynamic analysis is based on Newmark's algorithm used in conjunction with the Hughes and Liu predictor-corrector scheme resulting in an ‘effective static problem’ which is solved using a Newton-Raphson-type process. Flow theory is used in the inelastic range and the Huber-Mises yielding surface extended by Hill for anisotropic materials is adopted. Numerical results obtained for two categories of anisotropic structures, namely cross-ply laminated plates and angle-ply laminated plates, are presented and the effects of anisotropy and bending/ stretching coupling on the dynamic elastic and elastic-plastic responses are discussed. The effects of lamina stack sequences and lamina angle sequences on the dynamic responses are also considered.     

Post-buckling analysis of laminated stiffeners

An analytical-numerical procedure is extended to investigate the local buckling and post-buckling behaviour of thin-walled laminated stiffeners under uniaxial compression. Two stiffener sections are considered: I and channel sections. The stiffener section is treated as a plate assembly. The analysis uses a deflection-type perturbation technique to determine the buckling load and the post-buckling equilibrium path, ensuring compatibility at the plate junctions, adding extra in-plane terms in the strain-displacement relations, and including shear stresses along the flange-web junctions. Meanwhile, the effects of geometrical imperfections are taken into consideration.     

Dynamic instability of composite laminated rectangular plates and prismatic plate structures

Periodic dynamic loadings may cause dynamic instability of a structure through parametric resonance. In this paper, a B-spline finite strip method (FSM) is presented for the dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures, based on the use of first-order shear deformation plate theory (SDPT). The equations of motion of a structure are established by using Lagrange's formulation and they are a set of coupled Mathieu equations. The boundary parametric resonance frequencies of the motion are determined by using the method suggested by Bolotin through a novel development which incorporates the Sturm sequence method and the multi-level substructuring technique to achieve reliability, efficiency and accuracy. Various loading patterns, arbitrary lamination and general boundary conditions are accommodated. A variety of numerical applications is presented to test the developed method and to study the dynamic instability behaviour of single plates and of complicated plate structures under various types of dynamic loading. A dynamic instability index (DII) is devised to measure the degree of instability against certain parameters which include the thickness-to-length ratio, the degree of orthotropy, the fibre orientation, the loading pattern and the boundary conditions.     

Thermal post-buckling characteristics of clamped skew plates

Thermal post-buckling characteristics of clamped skew plates with restrained edges subjected to planar temperature distributions are studied. The problem is formulated in terms of Von Kármán equations expressed in terms of displacements generalised to include thermal effects. A perturbation method is employed to obtain a linear set of partial differential equations. The solution to this linear set is then obtained by using the Galerkin method. Numerical results are presented for different skew plate configurations for both uniform and two-dimensional temperature distributions. Plots of central deflection, membrane and bending stress distributions are presented and the post-buckling characteristics are discussed in detail.     

Effect of geometric nonlinearity on the free flexural vibrations of moderately thick rectangular plates

The effect of geometric nonlinearity on the free flexural vibrations of moderately thick rectangular plates is studied in this paper. Finite element formulation is employed to obtain the non-linear to linear period ratios for some rectangular plates. A conforming finite element of rectangular shape wherein the effects of shear deformation and rotatory inertia are included, is developed and used for the analysis. Results are presented for both simply supported and clamped boundary conditions.