Bending analysis of antisymmetric angle-ply laminated plates including transverse shear effects

In this paper an approximate analysis is presented for the bending of rectangular, simply supported, antisymmetric angle-ply laminated plates subjected to sinusoidal transverse loads. A published refined shear deformation theory is used to obtain a closed-form solution. Numerical results are compared with those given by three-dimensional elasticity solutions, first-order shear deformation theory solutions and classical lamination theory solutions. The present theory gives a better estimate of the stresses and deflections.     

Behaviour of laminated plates subjected to conventional blast

Response of simply supported anti-symmetrically laminated angle-ply plates to explosive blast loading is considered. A closed-form solution is obtained for thick plates, it being assumed that the material remains in the elastic range. The effect of transverse shear deformations on the response of thick plates is taken into consideration. The behaviour of thin laminated plates subjected to blast is studied using geometrically non-linear theory. Initial imperfections, which can be important for thin plates, are included in the analysis. The solution for thin elastic laminated plates is obtained numerically using a Runge-Kutta method. The analysis yields the non-dimensional deflection vs time relationship which can be used to determine the stresses and strains in the layers of the plate.     

Three-dimensional exact elasticity solutions for antisymmetric angle-ply laminated composite plates

International Journal of Mechanics and Materials in Design - This paper is concerned with the derivation of the benchmark three-dimensional exact solutions for the static analysis of rectangular...     

Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM

This paper presents a finite element model based on the first order shear deformation theory to investigate the dynamic behavior of laminated composite plates traversed by a moving oscillator. The oscillator model is assumed to be consisting of two nodal masses that are connected by means of a spring-damper unit. The governing equations of motion of two sub-systems are separately integrated by applying the Newmark’s time integration procedure. Then, the obtained equations are coupled and the responses of system components are calculated in each time step. The accuracy of algorithm is verified by comparing the numerical results of static, free vibration and simplified moving force problems analysis with the available exact solutions and other numerical results in the literature. Also, the effects of mass ratio, damping ratio of system components, stiffness of suspension system, velocity and eccentricity of moving oscillator on dynamic responses is parametrically studied. This algorithm can be applied to various boundary conditions, lamination schemes and fiber angels.     

Flexural analysis of cross-ply laminated plates subjected to nonlinear thermal and mechanical loadings

The objective of this paper is to present an equivalent single-layer shear deformation theory for evaluation of displacements and stresses of cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load. A trigonometric shear deformation theory is used. The in-plane displacement field uses a sinusoidal function in terms of the thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of a shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Stresses and displacements for orthotropic, two-layer antisymmetric, and three-layer symmetric square cross-ply laminated plates subjected to uniformly distributed nonlinear thermo-mechanical load are obtained. Numerical results of the present theory for displacement and thermal stresses are compared with those of classical, first-order and higher-order shear deformation plate theories.     

Accurate free vibration analysis of clamped antisymmetric angle-ply laminated rectangular plates by the Superposition-Galerkin method

The Superposition-Galerkin method is exploited for the first time to obtain an accurate analysis of the free vibration of antisymmetric angle-ply laminated plates with combinations of clamped simply supported edge conditions. Effects of transverse shear deformation are taken into consideration. This recently developed method is extremely easy to employ and provides numerous advantages from a computational point of view. Excellent agreement is obtained when computed results are compared with those already available in the literature.     

Nonlinear analysis of bimodular composite plates under compression

The present work deals with the buckling and post-buckling behaviour of bimodular laminated composite plates. The fiber-governed bimodular constitutive model is adopted for the analysis and the geometrical nonlinearities are accounted for by using the consistent small strain and moderate rotation theory for a shear deformable Mindlin plate. Governing equations of the problem are developed and the finite element formulation is given using lagrangian C°-elements. Some numerical examples are developed to analyze the buckling and post-buckling behaviour of laminated bimodular plates.     

Conditions for laminated plates to remain flat under inplane loading

There is considerable confusion in the existing literature as to whether an unsymmetrically laminated, composite plate will remain flat due to the application of inplane compressive or shear loads. If it does not remain flat, then bifurcation buckling would not normally take place, and transverse displacements would occur no matter how small the inplane loads. This paper investigates the conditions under which arbitrarily laminated and arbitrarily loaded plates remain flat, and therefore when buckling can occur. It is demonstrated that for uniform or linearly varying inplane loads no transverse pressure is necessary to keep a plate flat, although edge moments or transverse forces may be required at the boundaries.     

Nonlinear static analysis of composite laminated plates with evolving damage

Considering geometric nonlinearity and damage evolution, the static response characteristics of laminated composite plates subjected to uniformly distributed loading are investigated using finite element approach based on the first-order shear deformation theory. The damage evolution is modeled employing generalized macroscopic continuum theory within the framework of irreversible thermodynamics. The governing nonlinear equations are solved using Newton–Raphson iterative technique. The resulting finite element-based continuum damage model enables to predict the progressive damage and failure load. A detailed parametric study is carried out to investigate the influences of damage evolution, boundary conditions, span-to-thickness ratio, and lamination scheme on the static response of laminated plates undergoing moderately large deformation. It is revealed that the in-plane stretching forces owing to geometric nonlinearity significantly influence the failure load, damage, and stress distribution for immovable thin laminates.     

Scale models for laminated cylindrical shells subjected to axial compression

This study investigates problems associated with the design of scaled down models. Such study is important since it provides the necessary scaling laws, and the factors which affect the accuracy of the scale models. For better understanding the applicability of scaled down models in designing laminated composite structures, an analytical investigation was undertaken to assess the feasibility of their use. Employment of similitude theory to establish similarity among structural systems can save considerable expense and time, provided that the proper scaling laws are found and validated. In this study the limitation and acceptable interval of all parameters and corresponding scale factors are investigated. Particular emphasis is placed on the case of buckling of cross-ply cylindrical shells under uniaxial compressive loads. Both complete and partial similarity are discussed. This analytical study indicates that distorted models with a different number of layers and geometries than those of the prototype can predict the behavior of the prototype with good accuracy. However, it is shown that a scaled down model with different material properties than those of the prototype is incapable of predicting the response of the prototype.     

Nonlinear dynamic and static analysis of laminated anisotropic thick circular plates

Summary A nonlinear shear-deformable theory is presented for dynamic behavior of generally laminated circular plate composed of rectilinearly orthotropic layers. The basic equations derived by use of Hamilton's principle and variational calculus are expressed in terms of the transverse displacement and two in-plane displacements. On the basis of a single-mode analysis a solution is formulated for clamped laminated circular plates with movable and immovable inplane boundary conditions. Two inplane equilibrium equations and all boundary conditions are satisfied exactly. The Galerkin procedure yields a nonlinear ordinary differential equation for the time function which is then solved by using the method of harmonic balance. Numerical results for the static large-deflection behavior and the amplitude-frequency response of laminated angle-ply and cross-ply graphite-epoxy circular plates are presented for various values of the number of layers and the radius-to-thickness ratio.Notation A _ij,B _ij,D _ij Plate stiffnesses defined in Eq. (4) - a radius of circular plate - E _i Young's moduli along thei principal direction - E _ij plate shear rigidities defined in Eq. (4) - f(t) function of time - f _i (x, y) functions of space coordinates defined in Eqs. (24) - G _ij shear moduli - H _mn constants defined in Eqs. (24) - h thickness of plate - I constant defined in Eq. (10) - K _j differential operators defined in Appendix - k _n Fourier coefficients defined in Eq. (27) - L _i differential operators defined in Appendix - M _x,M _y,M _xy bending and twisting moments per unit length - N _x,N _y,N _xy membrane forces per unit length - N _i differential operators defined in Appendix - P _mn,p _i constants defined in Eqs. (24) and (25) - Q _x,Q _y transverse shear forces per unit length - q, q ₀ nonuniformly and uniformly distributed loads per unit area - R _i tracing constant for rotatory inertia - R _mn,s _n,T _mn constants defined in Eqs. (24) and (21) - S _i differential operators defined in Appendix - S _ij ^(k) elastic stiffness of thek-th layer - T _s tracing constant for transverse shear - u, v inplane displacements - u ⁰,v ⁰ midsurface in-plane displacements - w ₀ maximum transverse displacement at centre of plate - w transverse displacement - , slope functions - coefficient defined in Eq. (10) - _ij ⁰ middle surface strain components - _ij total strain components - _ij Poisson's ratio - 0^(k) mass density ofk-th layer - fundamental frequency - ₀ fundamental linear frequency     

An exact approach to the elastic state of stress of shear deformable antisymmetric angle-ply laminated plates

The state space concept is used to develop Lévy-type solutions for anti-symmetric angle-ply laminated plates using the Yang, Norris and Stavsky (YNS) theory. The theory is a generalization of Mindlin's theory for isotropic plates applied to laminated anisotropic plates and includes shear deformation.

The solution is applicable to rectangular plates which are simply supported on two opposite edges and have arbitrary boundary conditions on the remaining ones. Numerical results are presented emphasizing the effects of length/thickness ratio, degree of anisotropy, number of layers, ply-angles and transverse shear correction coefficients on the elastic state of stress.     

Dynamic response of thick laminated annular sector plates subjected to moving load

The dynamic response of thick laminated annular sector plates with simply supported radial edges subjected to a radially distributed line load, which moves along the circumferential direction, is studied. A three-dimensional hybrid method composed of series solution, the layerwise theory and the differential quadrature method in conjunction with the finite difference method is employed. The fast rate of convergence and high accuracy of the method are demonstrated through different examples. Additionally, as a limit case, the out-of-plane dynamic responses of circular curved beams is obtained and compared with those of an unconstrained higher order shear deformation curved beam theory, which is formulated here. Then, the effects of different parameters such as the sector angle, thickness-to-outer radius ratio, ply lay out and the load velocity on the out-of-plane response of the symmetric and antisymmetric cross-ply laminated sector plates are investigated. The results can be used as benchmark solutions for future works.     

Elastic stability of skew laminated composite plates subjected to biaxial follower forces

The present study investigates the elastic stability of skew laminated composite plates subjected to biaxial inplane follower forces by the finite element method. The plate is assumed to follow first-order shear deformation plate theory (FSDPT). The kinetic and strain energies of skew laminated composite plate and the work done by the biaxial inplane follower forces are derived by using tensor theory. Then, by Hamilton's principle, the dynamic mathematical model to describe the free vibration of this problem is formed. The finite element method and the isoparametric element are utilized to discretize the continuous system and to obtain the characteristic equations of the present problem. Finally, natural vibration frequencies, buckling loads (also the instability types) and their corresponding mode shapes are found by solving the characteristic equations. Numerical results are presented to demonstrate the effects of those parameters, such as various inplane force combinations, skew angle and lamination scheme, on the elastic stability of skew laminated composite plates subjected to biaxial inplane follower forces.     

Nonlinear flexural analysis of laminated composite plates using RBF based meshless method

The paper presents the nonlinear flexural response of laminated composite plates. The mathematical formulation of the actual physical problem of the laminated composite plate subjected to mechanical loading is presented utilizing higher order shear deformation theory and von-Karman nonlinear kinematics. These nonlinear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. A meshfree technique based on multiquadric RBFs is used for analysis of the problems. Isotropic, orthotropic and laminated composite plates with immovable simply supported and clamped edges are analyzed.     

Nonlinear parametric instability of antisymmetrically laminated angle-ply plates

Parametric instability of antisymmetrically laminated angle-ply rectangular plates is considered in this paper. The boundaries of instability domains are determined using linear theory. Then the frequency-amplitude relationship of the nonlinear parametric vibration within the instability domains is obtained using an asymptotic method.     

Nonlinear parametric instability of antisymmetrically laminated angle-ply plates

Parametric instability of antisymmetrically laminated angle-ply rectangular plates is considered in this paper. The boundaries of instability domains are determined using linear theory. Then the frequency-amplitude relationship of the nonlinear parametric vibration within the instability domains is obtained using an asymptotic method.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading

The nonlinear dynamic behavior of simply supported tapered sandwich plates subjected to air blast loading is investigated theoretically and numerically. The plate is supposed to have both tapered core and tapered laminated face sheets and be subjected to uniform air blast load. The theory is based on a sandwich plate theory, which includes von Kármán large deformation effects, in-plane stiffnesses, inertias and shear deformations. The sandwich plate theory for plates with constant thickness which have one-layered face sheets found in the literature is developed to analyze the tapered sandwich plates with multi-layered face sheets. The equations of motion are derived by the use of the virtual work principle. Approximate solution functions are assumed for the space domain and substituted into the equations. The Galerkin method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The tapered sandwich plate subjected to air blast load is also modelled by using the finite element method. The displacement–time and strain–time histories are obtained. The theoretical results are compared with finite element results and are found to be in an agreement.     

复合材料层合板的非线性组合共振特性及分岔

考虑几何非线性项和阻尼的影响, 给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程, 利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解, 得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论, 得到了解的稳定性判定条件。通过数值算例, 分析了不同参数对系统组合共振及其分岔特性的影响。结果表明, 随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变, 系统存在多幅值现象、滞后现象和跳跃现象, 出现不稳定解, 且在某些参数点处具有运动性态发生变化的分岔特性, 表现出较为复杂的动力学特性。