Three-dimensional semi-analytical model for the static response and sensitivity analysis of the composite stiffened laminated plate with interfacial imperfections

For the stiffened composite laminated plates with interfacial imperfections, the problem of static response and sensitivity analysis was investigated in Hamilton system. Firstly, the meshfree formulation of Hamilton canonical equation for the composite laminated plate with interfacial imperfections was deduced by the linear spring-layer and the state-vector equation theory. And then, based on the equation of plates and stiffeners, governing equation of the composite stiffened laminated plate was assembled by using the spring-layer model again to ensure the compatibility of stresses and the discontinuity of displacements at the interface between plate and stiffeners. At last, a three-dimensional hybrid governing equation was developed for the static response analysis and sensitivity analysis.     

Nonlinear static response of laminated composite plates by discrete singular convolution method

In this paper, large deflection analysis of laminated composite plates is analysed. Nonlinear governing equation for bending based on first-order shear deformation theory (FSDT) in the von Karman sense is presented. These equations have been solved by the method of discrete singular convolution (DSC). Regularized Shannon’s delta (RSD) kernel and Lagrange delta sequence (LDS) kernel are selected as singular convolution to illustrate the present algorithm. The effects of plate aspect ratio, fiber orientation, boundary conditions, thickness-to-side ratio, and applied load on the nonlinear static response of the laminated plate are investigated.     

An enriched macro finite element for the static analysis of thick general quadrilateral laminated composite plates

This article presents the formulation of an enriched macro finite element based on the trigonometric shear deformation theory for the static analysis of symmetrically laminated composite plates. Shear correction factor is not required because this theory accounts for tangential stress-free boundary conditions on the plate boundary surfaces. The macro element is obtained using the principle of virtual work and Gram-Schmidt orthogonal polynomials as enrichment functions. The implementation of the obtained algorithm is simple and efficient, and allows studying general quadrilateral plates with a single macro element. Several examples are presented to show the capability and applicability of the developed formulation.     

Analysis of the nonlinear dynamic response of viscoelastic symmetric cross-ply laminated plates with transverse matrix crack

Based on the Schapery’s 3-D constitutive relationship and the Von Karman’s plate theory, the constitutive equations and the nonlinear dynamic governing equations for viscoelastic symmetric cross-ply laminated plates with transverse matrix cracks are derived. By using the finite difference method and Newmark-β scheme, the unknown functions are separated and these equations are iterated to seek solutions. Numerical calculating results show that the effect of damage on the nonlinear dynamic responses of structures is significant.     

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.     

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     

Panel flutter analysis of general laminated composite plates

The problem of nonlinear aeroelasticity of a general laminated composite plate in supersonic air flow is examined. The classical plate theory along with the von-Karman nonlinear strains is used for structural modeling, and linear piston theory is used for aerodynamic modeling. The coupled partial differential equations of motion are derived by use of Hamilton’s principle and Galerkin’s method is used to reduce the governing equations to a system of nonlinear ordinary differential equations in time, which are then solved by a direct numerical integration method. Effects of in-plane force, static pressure differential, fiber orientation and aerodynamic damping on the nonlinear aeroelastic behavior of the plate are studied. Results show that the fiber orientation has significant effect on dynamic behavior of the plate and the asymmetric properties, changes the behavior of the limit cycle oscillation.     

Analytical solution for bending of cross-ply laminated plates under thermo-mechanical loading

The static thermo-elastic response of symmetric and anti-symmetric cross-ply laminated plates has been investigated by the use of a unified shear deformation plate theory. The present plate theory enables the trial and testing of different through-the-thickness transverse shear-deformation distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. The validity of the present theory is demonstrated by comparison with solutions available in the literature. A wide variety of results is presented for the static response of simply supported rectangular plates under non-uniform sinusoidal mechanical and/or thermal loadings. The influence of material anisotropy, aspect ratio, side-to-thickness ratio, thermal expansion coefficients ratio and stacking sequence on the thermally induced response is studied.     

Transverse stress profiles reconstruction for finite element analysis of laminated plates

The present work focuses on a posteriori, equilibrium based, reconstruction of transverse stress profiles in the finite element analysis of FSDT laminated plates. The accuracy of this reconstruction depends on accuracy of the first and second-order derivatives of the plate stress resultants, which is not guaranteed by most available low-order plate finite elements. To cure this trouble, two different strategies, based on the Recovery by Compatibility in Patches procedure, are here proposed and compared. Numerical results of typical reconstructed transverse stress profiles are presented showing the effectiveness of the proposed approach.     

Comparative study of finite element based response evaluation methods for laminated plates

In this paper, four methods for the through the thickness response evaluation of laminated composite and sandwich plates are comparatively evaluated in terms of their characteristics and numerical accuracy. The laminated composite and sandwich plates are subjected to thermo-mechanical loading. The four methods include two post-processor-type methods and two re-analysis-type methods. The post-processor-type methods based on the three-dimensional stress equilibrium equations and the three-dimensional thermo-elasticity equations. And the re-analysis methods are based on the incorporation of the nonlinear through the thickness displacements recovered using post-processing technique in the re-analysis phase. The accuracy of the proposed methods for the evaluation of the through the thickness responses of the laminated plates are assessed via numerical examples of various configurations of laminated plates. Also, applicability and computational efficiencies of the methods are discussed. This research was supported in part by a grant from the BK-21 program for Mechanical and Aerospace Engineering Research at Seoul National University. The authors also gratefully acknowledge the financial support from the Ministry of Science and Technology through the National Laboratory Programs.     

A review of meshless methods for laminated and functionally graded plates and shells

This review focuses mainly on the developments of element-free or meshless methods and their applications in the analysis of composite structures. This review is organized as follows: a brief introduction to shear deformation plate and shell theories for composite structures, covering the first-order and higher-order theories, is given in Section 2. A review of meshless methods is provided in Section 3, with main emphasis on the element-free Galerkin method and reproducing kernel particle method. The applications of meshless methods in the analysis of composite structures are discussed in Section 4, including static and dynamic analysis, free vibration, buckling, and non-linear analysis. Finally, the problems and difficulties in meshless methods and possible future research directions are addressed in Section 5.     

Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates

This paper presents a generalized hybrid quasi-3D shear deformation theory for the bending analysis of advanced composite plates such as functionally graded plates (FGPs). Many 6DOF hybrid shear deformation theories with stretching effect included, can be derived from the present generalized formulation. All these theories account for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces not requiring thus a shear correction factor. The generalized governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FGP subjected to transverse load for simply supported boundary conditions. Numerical examples of the new quasi-3D HSDTs (non-polynomial, polynomial and hybrid) derived by using the present generalized formulation are compared with 3D exact solutions and with other HSDTs. Results show that some of the new HSDTs are more accurate than, for example, the well-known trigonometric HSDT, having the same 6DOF.     

A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory

A study of bending deformations of sandwich plates using a layerwise theory of laminated or sandwich plates is presented. The analysis is based on a wavelet collocation technique to produce highly accurate results. Numerical results for symmetric laminated composite and sandwich plates are presented and discussed.     

HSDT-layerwise analytical solution for rectangular piezoelectric laminated plates

The present paper develops a formulation for laminated plates with extensional distributed piezoelectric sensors/actuators. This formulation is based on linear electroelasticity, and an equivalent single layer is used for the mechanical displacement field, applying a Higher-Order Shear Deformation Theory (HSDT), whereas a layerwise discretization is used in the thickness direction for the electric potential. The electric and mechanical local equilibrium equations and local constitutive equations for the problem are identified. The Principle of Virtual Work is used to derive the dynamic equilibrium equations in terms of generalized forces and the consistent boundary conditions. The piezoelectric laminate constitutive equations are built and used to write the equations of motion in terms of generalized displacements. Finally, analytical solutions for simply supported square laminates with piezoelectric layers are developed. The entire laminate, composed of the base structure and piezoelectric layers, can be arbitrary orthotropic. The solution is adequate for an arbitrary number of piezoelectric layers and stacking positions. Moreover, the solution takes into account all material coefficients, whether mechanical, piezoelectric or dielectric. Analytical results are obtained for static bending, both in sensor and actuation modes, and for free vibration of symmetric cross-ply laminates with piezoelectric layers externally bonded to the plate.     

Re-analysis procedure for laminated plates using FSDT finite element model

 A new method is proposed for effective analysis of laminated plates incorporating accurate through-the-thickness distribution of displacements, strains and stresses in the finite element formulation. It is a two-step analysis procedure. In the first step, displacements are obtained using a post-processing procedure based on the three-dimensional stress equilibrium equations and the thermoelasticity equations, from the results of FSDT finite element analysis. In the second step, the higher-order through-the-thickness distribution of displacements are reflected on the subsequent finite element analysis. The effectiveness of the present approach for the analysis of laminated plates is shown by numerical examples. Received: 13 September 2001 / Accepted: 23 May 2002     

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.     

The elasto-plastic impact analysis of functionally graded circular plates under low-velocities

In this study, three-dimensional elasto-plastic analysis of functionally graded (FG) circular plates under low-velocity impact loads is presented. The FG circular plate is composed of ceramic (SiC) and metal (Al) phases, varying in a predetermined fashion through the plate thickness. The elasto-plastic behaviour of the FG circular plate is described by the TTO model. In the analyses, the ceramic phase is taken to be an isotropic elastic material whereas the metal phase is taken to be elasto-plastic material in accordance with the TTO model. The locally effective material properties were evaluated using homogenization method which is based on the Mori–Tanaka scheme. The effects of compositional gradient exponent and impactor velocity on the elasto-plastic impact response of the FG circular plates are investigated, and results are presented graphical form. The compositional gradient exponent and impactor velocity have significant effects on the elasto-plastic impact response of the FG circular plates. The elasto-plastic impact response of FG circular plates is similar to those of homogeneous plates. Therefore, the TTO model can be used to describe the mechanical behaviour of FG plates beyond the elastic range, assuming that the material response is essentially governed by the spreading of plasticity in the metal phase.     

简谐激励下复合材料加筋板的基体微裂纹损伤演化

研究了在简谐激励作用下复合材料加筋板基体微裂纹损伤的演化行为及其对加筋板动力特性的影响。基于平均微裂纹密度和断裂力学方法, 建立了复合材料加筋板基体微裂纹演化的刚度退化准则。由于该准则考虑了载荷作用周期数的影响, 从而能够更合理地分析周期性动载荷作用下基体微裂纹损伤演化规律。采用Mindlin一阶剪切理论和复合材料模态阻尼模型, 建立了复合材料加筋板动力分析的有限元方法, 研究了在简谐激励作用下, 含分层损伤复合材料加筋板振动过程中诱发的基体微裂纹损伤的演化、 刚度退化, 频率折减和动力响应。      

Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory

A new higher order shear deformation theory for elastic composite/sandwich plates and shells is developed. The new displacement field depends on a parameter “m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The present theory accounts for an approximately parabolic distribution of the transverse shear strains through the shell thickness and tangential stress-free boundary conditions on the shell boundary surface. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved using Navier-type, closed form solutions. Static and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Shells and plates are subjected to bi-sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The accuracy of the present code is verified by comparing it with various available results in the literature.