Free and forced vibration analysis of laminated composite plates and shells using a 9-node assumed strain shell element

The natural frequencies of isotropic and composite laminates are presented. The forced vibration analysis of laminated composite plates and shells subjected to arbitrary loading is investigated. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To develop a laminated shell element for free and forced vibration analysis, the equivalent constitutive equation that makes the computation of composite structures efficient was applied. The Mindlin-Reissner theory which allows the shear deformation and rotary inertia effect to be considered is adopted for development of nine-node assumed strain shell element. The present shell element offers significant advantages since it consistently uses the natural co-ordinate system. Results of the present theory show good agreement with the 3-D elasticity and analytical solutions. In addition the effect of damping is investigated on the forced vibration analysis of laminated composite plates and shells.     

A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells

 The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells. Received: 6 February 2002 / Accecpted: 6 January 2003 ID=" Present address: School of Civil Engineering, Asian Institute of Technology     

基于共旋三角形厚薄通用壳元的几何非线性分析

基于共旋列式方法发展了一种用于复合材料层合板结构几何非线性分析的简单高效的三结点三角形平板壳元。该壳元由具有面内转动自由度的广义协调膜元GT9与假设剪切应变场和假设单元转角场的广义协调厚薄通用板元TMT组合而成。为避免薄膜闭锁而采用单点积分计算与薄膜应变有关的项, 同时增加一个稳定化矩阵以消除单点积分导致的零能模式。基于层合板一阶剪切变形理论, 给出了考虑层合板具体铺层顺序的修正的横向剪切刚度, 使该壳元可用于中厚层合板结构的分析。由于共旋列式大转动小应变的假设, 共旋列式内核的几何线性的单元刚阵可仅计算一次而保存下来用于整个几何非线性求解的过程以提高计算效率。数值算例表明提出的壳元进行包括复合材料层合板结构的厚薄壳结构的几何非线性分析的精度高且效率高。     

An accurate shear-deformable six-node triangular plate element for laminated composite structures

A six-node triangle plate/shell element is developed for the analysis of laminated composite structures. This model is formulated using Hamilton's principle along with a first-order (Reissner/Mindlin) shear deformation theory. The element is based upon an isoparametric representation along with an interdependent interpolation strategy; bicubic polynomials for the transverse displacement and biquadratic polynomials for the element geometry, in-plane displacements and rotations. The resulting element, which is evaluated using exact numerical integration, has correct rank and is free of shear ‘locking’. Numerical results are presented that validate the new element and prove its outstanding convergence capabilities in comparison to existing triangular elements using standardized test problems (elastic eigenvalue analysis, patch test, static simply supported square-plate solutions) and experimentally measured vibration data of cantilevered isotropic and composite plates.     

A Continuum Based Thick Shell Element for Large Deformation Analysis of Layered Composites

A new continuum based thick shell model is presented for modeling orthotropic laminated shell structures undergoing large elastic deformations. An equivalent single-layer model involving seven nodal degrees of freedom is used. In that layered model, there are no restrictions on the number of layers, their thickness and their stacking sequence. The shell model accounts explicitly for the thickness change in the shell, as well as the normal stress and strain states through its thickness. Shear locking is avoided using an assumed natural strain formulation, while thickness locking is avoided using modified displacement interpolation functions. The performance of the layered shell element is tested using several linear and non-linear composite plate and shell problems involving anisotropic, angle and cross-ply laminates, cylindrical and spherical shells.     

Extended rotation-free shell triangles with transverse shear deformation effects

The paper extends recent work of the authors to include transverse shear effects on rotation-free triangular element for plates (O?ate and Zárate in Int J Numer Methods Eng 83(2):196–227, 2010). Two new shell triangular elements are presented, the EBST+ and the EBST+1. Transverse shear deformation effects are important for thick shells, as well when the shell is laminated or formed by composite material. The ingredients for the element formulation are: a Hu-Washizu type mixed functional and linear interpolation for the displacement field. In both elements presented a finite volume approach is used for computing the bending moments and the curvatures over a patch of elements. The nodal translational degrees of freedom of the original enhanced basic shell triangle (EBST) are extended with the two shear deformation angles via two different approaches. The first one uses a linear interpolation of the rotation angles inside the element (EBST+) and the second one assumes a constant field for the rotation angles (EBST+1). For the thin shell case the shear angles vanish and the new elements reproduce the good behaviour of the original thin EBST element. As a consequence the elements can reproduce the solutions for thick to thin shells situations without exhibiting shear locking. The numerical solution for the thick shell case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin EBST element. Examples of the good performance of the new rotation-free shell triangles are given.     

A new shear flexible cubic spline plate element for vibration analysis

Here, a new cubic B‐spline plate element is developed using field consistency principle, for vibration analysis. The formulation includes anisotropy, transverse shear deformation, in‐plane and rotary inertia effects. The element is based on a laminated refined plate theory, which satisfies the interface transverse shear stress and displacement continuity, and has a vanishing shear stress on the top and bottom surfaces of the plates. The lack of consistency in the shear strain field interpolations in its constrained physical limits produces poor convergence and results in unacceptable solutions due to locking phenomenon. Hence, numerical experimentation for the evaluation of natural frequencies of plates is carried out to check this deficiency with a series of assumed shear strain functions, redistributed in a field consistent manner. Copyright © 2002 John Wiley & Sons, Ltd.     

An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations

A formulation for 36‐DOF assumed strain triangular solid shell element is developed for efficient analysis of plates and shells undergoing finite rotations. Higher order deformation modes described by the bubble function displacements are added to the assumed displacement field. The assumed strain field is carefully selected to alleviate locking effect. The resulting element shows little effect of membrane locking as well as shear locking, hence, it allows modelling of curved shell structures with curved elements. The kinematics of the present formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the present element is free of small angle assumptions, and thus it allows large load increments in the geometrically non‐linear analysis. Various numerical examples demonstrate the validity and effectiveness of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

基于一阶剪切变形理论的新型复合材料层合板单元

基于一阶剪切变形理论(FSDT),本文构造一种新型的20自由度(每结点5个自由度),四边形复合材料层合板单元,适合于任意铺设情形的层合板的计算。它是按如下方式构造的:(1) 单元每边的转角和剪应变由Timoshenko层合厚梁理论来确定;(2) 对单元域内的转角场和剪应变场进行合理的插值;(3) 引入平面内双线性位移场来体现层合板面内与弯曲的耦合作用。本文单元,记为TMQ20,不存在剪切闭锁现象,在计算单层的各向同性板时可以退化为文[1]中优质的中厚板单元TMQ。在文[2]中将给出本文单元对于层合板问题的详细数值算例。     

Analysis for buckling and vibrations of composite stiffened shells and plates

This paper deals with development of triangular finite element for buckling and vibration analysis of laminated composite stiffened shells. For the laminated shell, an equivalent layer shell theory is employed. The first-order shear deformation theory including extension of the normal line is used. In order to take into account a non-homogeneous distribution of the transverse shear stresses a correction of transverse shear stiffness is employed. Based on the equivalent layer theory with six degrees of freedom (three displacements and three rotations), a finite element that ensures C⁰ continuity of the displacement and rotation fields across inter-element boundaries has been developed. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Results of vibration and buckling analysis of stiffened plates and shells are discussed.     

Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control

In this paper, we present an optimal low‐order accurate piezoelectric solid‐shell element formulation to model active composite shell structures that can undergo large deformation and large overall motion. This element has only displacement and electric degrees of freedom (dofs), with no rotational dofs, and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out‐of‐plane bending). The combination of the present optimal piezoelectric solid‐shell element and the optimal solid‐shell element previously developed allows for efficient and accurate analyses of large deformable composite multilayer shell structures with piezoelectric layers. To make the 3‐D analysis of active composite shells containing discrete piezoelectric sensors and actuators even more efficient, the composite solid‐shell element is further developed here. Based on the mixed Fraeijs de Veubeke–Hu–Washizu (FHW) variational principle, the in‐plane and out‐of‐plane bending behaviours are improved via a new and efficient enhancement of the strain tensor. Shear‐locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. We also present an optimal‐control design for vibration suppression of a large deformable structure based on the general finite element approach. The linear‐quadratic regulator control scheme with output feedback is used as a control law on the basis of the state space model of the system. Numerical examples involving static analyses and dynamic analyses of active shell structures having a large range of element aspect ratios are presented. Active vibration control of a composite multilayer shell with distributed piezoelectric sensors and actuators is performed to test the present element and the control design procedure. Copyright © 2005 John Wiley & Sons, Ltd.     

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 free vibration analysis of laminated composite shells in hygrothermal environments

The nonlinear free vibration behaviour of laminated composite shells subjected to hygrothermal environments is investigated using the finite element method. The present finite element formulation considers doubly curved shells, and the Green–Lagrange type nonlinear strains are incorporated into the first-order shear deformation theory. The analysis is carried out using quadratic eight-noded isoparametric elements. The validity of the model is demonstrated by comparing the present results with the solutions available in the literature. A parametric study is carried out varying the curvature ratios and side to thickness ratios of composite cylindrical shell, spherical shell and hyperbolic paraboloid shell panels with simply supported boundary conditions.     

Bending and vibration responses of laminated composite plates using an edge-based smoothing technique

In this paper, bending and vibration analysis of laminated composite plates is carried out using a novel triangular composite plate element based on an edge-based smoothing technique. The present formulation is based on the first-order shear deformation theory, and the discrete shear gap (DSG) method is employed to mitigate the shear locking. The smoothed Galerkin weak form is adopted to obtain the discretized system equations, and edge-based smoothing domains are used for the numerical integration to improve the accuracy and the convergence rate of the method. The present formulation is coded and used to solve various example problems of bending and free vibration of laminated composite plates. It is found that the present method can provide excellent results with a wide range of thickness and is free of shear locking.     

An eight‐node hybrid‐stress solid‐shell element for geometric non‐linear analysis of elastic shells

This paper presents eight‐node solid‐shell elements for geometric non‐linear analysis of elastic shells. To subdue shear, trapezoidal and thickness locking, the assumed natural strain method and an ad hoc modified generalized laminate stiffness matrix are employed. A selectively reduced integrated element is formulated with its membrane and bending shear strain components taken to be constant and equal to the ones evaluated at the element centroid. With the generalized stresses arising from the modified generalized laminate stiffness matrix assumed to be independent from the ones obtained from the displacement, an extended Hellinger–Reissner functional can be derived. By choosing the assumed generalized stresses similar to the assumed stresses of a previous solid element, a hybrid‐stress solid‐shell element is formulated. Commonly employed geometric non‐linear homogeneous and laminated shell problems are attempted and our results are close to those of other state‐of‐the‐art elements. Moreover, the hybrid‐stress element converges more readily than the selectively reduced integrated element in all benchmark problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely, Voigt model and Mori-Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM shells. Numerical results demonstrate that in most cases the natural frequencies of the FGM shells are increased but the nonlinear to linear frequency ratios of the FGM shells are decreased with increase in foundation stiffness. The results confirm that in most cases Voigt model and Mori-Tanaka model have the same accuracy for predicting the vibration characteristics of FGM shells.     

A four‐noded quadrilateral element for composite laminated plates/shells using the refined zigzag theory

A new bilinear four‐noded quadrilateral element (called quadrilateral linear refined zigzag) for the analysis of composite laminated and sandwich plates/shells based on the refined zigzag theory is presented. The element has seven kinematic variables per node. Shear locking is avoided by introducing an assumed linear shear strain field. The performance of the element is studied in several examples where the reference solution is the 3D finite element analysis using 20‐noded hexahedral elements. Copyright © 2013 John Wiley & Sons, Ltd.     

An Assumed Strain Triangular Solid Element for Efficient Analysis of Plates and Shells with Finite Rotation

A simple triangular solid shell element formulation is developed for efficient analysis of plates and shells undergoing finite rotations. The kinematics of the present solid shell element formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the kinematics of deformation is free of the limitation of small angle increments, and thus the formulation allows large load increments in the analysis of finite rotation. An assumed strain field is carefully selected to alleviate the locking effect without triggering undesirable spurious kinematic modes. In addition, the curved surface of shell structures is modeled with flat facet elements to obviate the membrane locking effect. Various numerical examples demonstrate the efficiency and accuracy of the present element formulation for the analysis of plates and shells undergoing finite rotation. The present formulation is attractive in that only three points are needed for numerical integration over an element.     

Finite element analysis of laminated composite plates using zeroth-order shear deformation theory

In this paper a generalized finite element model is developed for static and dynamic analyses of laminated composite plates using zeroth-order shear deformation theory (ZSDT). The theory ensures the parabolic distribution of transverse shear stresses across the plate thickness. A four-noded plate element is considered in this model and the generalized nodal variables are expressed using Lagrangian linear interpolation functions and Hermitian cubic interpolation functions. The solutions of the finite element model have been compared with the existing solutions for symmetric and antisymmetric laminated composite plates. The comparison confirms that the ZSDT can be efficiently used for finite element analysis of both thin and thick plates with high accuracy.     

Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method

In the present study, the thermal stability of laminated functionally graded (FGM) circular plates of variable thickness subjected to uniform temperature rise based on the first-order shear deformation plate theory is presented. Furthermore, two models for FGM plates with variable thickness, corresponding with two manufacturing methods, are proposed. The laminated FGM plate with variable thickness is considered as a sandwich plate constituted of a homogeneous core of variable thickness and two constant thickness FGM face sheets whose material properties are assumed to be graded in the thickness direction according to a simple power law. In order to determine the distribution of the prebuckling thermal load along the radius, the membrane equation is solved using the shooting method. Subsequently, employing the pseudo-spectral method that makes use of Chebyshev polynomials, the stability equations are solved numerically to evaluate the critical temperature rise. The results demonstrate that the thermal stability is significantly influenced by the thickness variation profile, aspect ratio, the volume fraction index, and the core-to-face sheet thickness ratio.