Transient response of composite sandwich plates

The transient response of orthotropic, layered composite sandwich plates is investigated by using two new C⁰ four and nine node finite element formulations of a refined form of Reddy's higher-order theory. This refined third order theory accounts for parabolic variation of the transverse shear stresses, and requires no shear correction factors. The assumed strain approach is employed to model both thin and thick plates without any major defects like shear locking and parasitic spurious zero energy modes. A consistent mass matrix formulation is adopted. The Newmark direct integration scheme is used to solve the governing equilibrium equations. The parametric effects of plate aspect ratio, length to thickness ratio, boundary conditions and lamination scheme on the transient response are investigated. The present results are in very close agreement with earlier published results in the literature and can serve as a benchmark for future investigators.     

Dynamic (transient) analysis of layered anisotropic composite-material plates

A shear-flexible finite element is employed to investigate the transient response of isotropic, orthotropic and layered anisotropic composite plates. Numerical convergence and stability of the element is established using Newmark's direct integration technique. Numerical results for deflections and stresses are presented for rectangular plates under various boundary conditions and loadings. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the transient response are investigated. The present results agree very closely with the results available in the literature for isotropic plates, and the results for composite plates should serve as bench marks for future comparisons by other investigators.     

A design of laminated composite plates using graded orthotropic fiber-reinforced composite plies

A new lamination scheme is proposed through the design of a graded orthotropic fiber-reinforced composite ply for achieving continuous variations of material properties along the thickness direction of laminated composite plates. First, a micro-structure of graded unidirectional fiber-reinforced composite ply is designed and its effective graded elastic properties are estimated using finite element procedure. Next, the new lamination scheme is demonstrated through the conversion of a conventional laminated composite plate (CLCP) into a conventional-graded laminated composite plate (CGLCP) utilizing presently designed graded orthotropic composite ply. The suitability of this conversion/proposed lamination scheme is substantiated through the bending analysis of both the plates (CLCP and CGLCP).     

Non-homogeneous response of cross-ply laminated elastic plates using a higher-order theory

A consistent formulation for the bending of cross-ply laminated composite plates that possess non-homogeneous elastic properties is presented. Based on a third-order shear deformation plate theory, the governing equations are obtained using the principle of virtual work. With the help of the small parameter method, a wide variety of results are presented for the symmetric and antisymmetric analysis of non-homogeneous rectangular laminated plates. The influence of non-homogeneity, lamination schemes, aspect ratio and material anisotropy on the deflections and stresses is investigated. The new results for non-homogeneous response of composite plates should serve as bench marks for future comparisons.     

Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order theory

A higher-order shear deformable C° continuous finite element is developed and employed to investigate the transient response of isotropic, orthotropic and layered anisotropic composite plates. The governing ordinary linear differential equations are integrated using the central difference explicit time integration scheme. A special mass matrix diagonalization scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. Numerical results for deflections and stresses are presented for rectangular plates under various boundary conditions and loadings. The parametric effects of the time step, finite element mesh, lamination scheme and orthotropy on the transient response are investigated. The numerical results are compared with those available in the literature, and with the results obtained by solving the same problems using the Mindlin plate element.     

Nonlinear Deflection Control of Laminated Plates Using Third-Order Shear Deformation Theory

In this paper, the third-order shear deformation theory is employed to study static and dynamic deflection control of laminated composite plates. The effects of shear deformation and geometric nonlinearity (in the von Kármán sense) on the bending and transient response are investigated using the finite element method. Magnetostrictive material, Terfenol-D, layers are used to actively control the deflection via simple negative velocity feedback control in a closed loop. The effects of the lamination scheme, types of load, and boundary conditions on the deflection are investigated.     

A generalized higher-order theory for buckling of thick multi-layered composite plates with normal and transverse shear strains

The onset of buckling in square laminated multi-layered composite plates, subject to unidirectional in-plane loads, is investigated within the framework of a generalized higher-order shear deformation theory suitable to capture significant transverse shear and thickness-wise deformation effects. The displacement field is expanded in a Taylor series of the thickness coordinate with arbitrary polynomial degree; in turn, the series coefficients, expressed as a superposition of admissible functions, are determined according to the Rayleigh–Ritz method. Truly higher-order polynomial terms, along with a sufficient number of in-plane admissible functions, are shown to be necessary for convergence towards the fundamental buckling load multiplier. As a by-product, reduced-order models are identified for various plate geometries and lamination schemes. The sensitivity of the lowest buckling load with respect to the nondimensional parameters (the thickness ratio, the ratio between the elastic moduli, the ply angle) is investigated. In particular, the attention is focused on the cross-over phenomenon between the lowest two buckling eigenvalues in multi-layered composite square plates with different lamination schemes. The presented results shed light onto the buckling behavior of thick shear-deformable multi-layered plates.     

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.     

Transient dynamic and damping analysis of laminated anisotropic plates using a refined plate theory

A refined model is presented for the linear transient dynamic and damping analysis of laminated anisotropic composite plates. Experimental measurements of specific damping capacity of unidirectional composite beams are used to predict the specific damping capacity of laminated composite plates in various modes of vibration. A finite element idealization is adopted, and the quadratic Lagrangian element is used together with selective/reduced integration. A viscous damping approximation is then employed to calculate the damped transient response of laminated plates. The effects of transverse shear deformation, symmetry condition, boundary conditions, anisotropy, aspect ratio, fibre orientation and the lamination scheme on specific damping capacity and damped transient response are investigated. Realistic examples illustrate the importance of these parameters. The present results agree very closely with experimental results available in the literature and can serve as a benchmark for future comparison by other investigators.     

A penalty plate-bending element for the analysis of laminated anisotropic composite plates

A C⁰ (penalty) finite element is developed for the equations governing the heterogeneous laminated plate theory of Yang, Norris and Stavsky. The YNS theory is a generalization of Mindlin's theory for homogeneous, isotropic plates to arbitrarily laminated anisotropic plates and includes shear deformation and rotary inertia effects. The present element can also be used in the analysis of thin plates by appropriately specifying the penalty parameter. A variety of problems are solved, including those for which solutions are not available in the literature, to show the material effects and the parametric effects of plate aspect ratio, length-to-thickness ratio, lamination scheme, number of layers and lamination angle on the deflections, stresses, and vibration frequencies. Despite its simplicity, the present element gives very accurate results.     

On the response of MMC laminated plates to non-uniform temperature loading: the effect of temperature-dependent material properties

Thermal response of antisymmetrically laminated metal matrix composite (MMC) plates subjected to non-uniform temperature field is analysed. Temperature dependence of both elastic and viscoplastic properties of the metallic matrix is taken into account; this suggests that a non-uniformly heated plate should be considered as a non-homogeneous structure. A micro-to-macro approach is employed to establish the instantaneous thermo-inelastic constitutive law at each point of the plate and to perform the structural analysis.

Results are presented for simply-supported and clamped graphitealuminium plates. The effects of boundary conditions, lamination angle, length-to-thickness ratio and different types of spatial temperature distributions are illustrated. Comparisons with the results obtained using an approach that treats the effect of temperature-dependent material properties in a simplified manner are shown. Comparisons with the corresponding elastic solutions (which neglect the inelastic effects in the metallic matrix) are given.     

Buckling of shear deformable laminated composite plates

A shear deformable finite element is developed for the buckling analysis of laminated composite plates. The finite element formulation is based on Mindlin's theory in which shear correction factors are derived from the exact expressions for orthotropic materials. A variety of problems on uniaxial and shear bucklings of laminated composite plates are solved. The effects of material properties, plate aspect ratio, length-to-thickness ratio, number of layers and lamination angle on the buckling loads of symmetrically and antisymmetrically laminated composite plates are investigated. Optimal lamination arrangements of layers for maximizing the buckling loads of the plates are determined.     

Large deformation analysis of moderately thick laminated plates on nonlinear elastic foundations by DQM

In this paper, the nonlinear behavior of symmetric and antisymmetric cross ply, thin to moderately thick, elastic rectangular laminated plates resting on nonlinear elastic foundations are studied using differential quadrature method (DQM). The first-order shear deformation theory (FSDT) in conjunction with the Green’s strain and von Karman hypothesis are assumed for modeling the nonlinear behavior. Elastic foundation is modeled as shear deformable with cubic nonlinearity. The differential quadrature (DQ) discretized form of the governing equations with the various types of boundary conditions are derived. The Newton–Raphson iterative scheme is employed to solve the resulting system of nonlinear algebraic equations. Comparisons are made and the convergence studies are performed to show the accuracy of the results even with a few number of grid points. The effects of thickness-to-length ratio, aspect ratio, number of plies, fiber orientation and staking sequence on the nonlinear behavior of cross ply laminated plates with different boundary conditions resting on elastic foundations are studied.     

Damping optimization of symmetrically laminated plates with transverse shear deformation using lamination parameters

This paper deals with the damping characteristics of symmetrically laminated plates with transverse shear deformation. First, the effect of laminate configuration on the damping characteristics is investigated for cantilevered laminated plates based on the Reissner–Mindlin’s first-order shear deformation theory. To examine the effect of laminate configuration, the concept of specific damping capacity is introduced and the damping characteristics are represented on the lamination parameter plane, where the damped stiffness invariants in transverse shear are newly proposed in this paper. Next, the optimal laminate configurations for the cantilevered laminated plates with maximal damping are determined taking into account the transverse shear effect by using differential evolution in which lamination parameters are used as intermediate design variables. The relation between the laminate configurations and the damping characteristics is discussed based on the concept of lamination parameters.     

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.     

An extension to classical lamination theory for use with functionally graded plates

An extension to classical lamination theory is presented for the improved analysis of thin to moderately thick functionally graded plates. The method results in an explicit formulation that accommodates any through-thickness variation in the elastic, hygrothermal and piezoelectric properties of each layer. Additionally, variations in the material rotation angle, temperature, moisture content and electric field strength through each layer are taken into account. The method relies on representing with polynomial series the variation in both the properties of each ply and the hygrothermal and piezoelectric loading. Validation problems are presented that demonstrate the application and accuracy of the method.     

Dynamic stability of stiffened laminated composite plates and shells subjected to in-plane pulsating forces

The dynamic stability of laminated composite stiffened or non-stiffened plates and shells due to periodic in-plane forces at boundaries is investigated in this paper. A three-dimensional (3-D) degenerated shell element and a 3-D degenerated curved beam element are used to model plates/shells and stiffeners, respectively. The characteristic equations to find the natural frequencies, buckling loads and their corresponding mode shapes are obtained from the finite element equation of motion. Then, the method of Hill's infinite determinants or the method of multiple scales is applied to analyse the dynamic instability regions. Numerical results are presented to demonstrate the effects of various parameters, such as skew angle, lamination scheme, stiffened scheme, in-plane force type and curvature of cylindrical shell, on the dynamic stability of stiffened and non-stiffened plates and shells subjected to in-plane pulsating forces at boundaries.     

Buckling analysis of composite plates using moving least squares differential quadrature method

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.     

Dynamic analysis of laminated composite plates traversed by a moving mass based on a first-order theory

The dynamic response of angle-ply laminated composite plates traversed by a moving mass or a moving force is investigated. For this purpose, a finite element method based on the first-order shear deformation theory is used. Stationary and adaptive mesh techniques have been applied as two different meshing schemes. The adaptive mesh strategy is then used to avoid off-nodal position of moving mass. In this manner, the finite element mesh is continuously adapted to follow and comply with the path of moving mass. A Newmark direct integration method is employed to solve the equations of motion. Parametric study is directed to find out how different parameters like mass of the moving object as well as the type of the angle-ply laminated composite plates affect the dynamic response. Numerical results show the significant effects of the stacking order on the dynamic responses of the composite structures under a moving mass. It is found that although [30/−60/−60/30] lamination shows the highest maximum vertical deflection but [−45/45/45/−45] lamination has the highest value of the dynamic amplification factor. The dynamic amplification factor for different stacking orders and mass velocities is less than 1.25.     

广义函数论在厚板动力分析中的应用

本文运用广义函数论处理δ函数的导数,对集中力作用下的厚板进行动力分析。基本方程采用由本文作者推演得到的改进的Donnell厚板振动方程。用样条配点法求解广义坐标,对集中力作用下的厚板弹性动力响应给出解的一般表达式。文中以能形板为例给出了数值结果,对按改进理论和经典理论所预示的结果进行了比较。本文方法可以用于厚板物理非线性和几何非线性动力分析和厚板与连续介质共同作用的场合,文中给出了有关表达式。