Song Cen

A new twelve DOF quadrilateral element for analysis of thick and thin plates

Abstract A simple quadrilateral 12 DOF plate bending element based on Reissner–Mindlin theory for analysis of thick and thin plates is presented in this paper. This element is constructed by the following procedure: 1. the variation functions of the rotation and shear strain along each side of the element are determined using Timoshenkos beam theory; and 2. the rotation, curvature and shear strain fields in the domain of the element are then determined using the technique of improved interpolation. nThe proposed element, denoted by ARS-Q12, is robust and free of shear locking and, thus, it can be employed to analyze very thin plate. Numerical examples show that the proposed element is a high performance element for thick and thin plates.

A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates

A new 4-node quadrilateral finite element is developed for the analysis of laminated composite plates containing distributed piezoelectric layers (surface bonded or embedded). The mechanical part of the element formulation is based on the first-order shear deformation theory. The formulation is established by generalizing that of the high performance Mindlin plate element ARS-Q12, which was derived based on the DKQ element formulation and Timoshenko’s beam theory. The layerwise linear theory is applied to deal with electric potential. Therefore, the number of electrical DOF is a variable depending on the number of plate sub-layers. Thus, there is no need to make any special assumptions with regards to the through-thickness variation of the electric potential, which is the true situation. Furthermore, a new “partial hybrid”-enhanced procedure is presented to improve the stresses solutions, especially for the calculation of transverse shear stresses. The proposed element, denoted as CTMQE, is free of shear locking and it exhibits excellent capability in the analysis of thin to moderately thick piezoelectric laminated composite plates.

Area co-ordinates used in quadrilateral elements

The area co-ordinate method has been successfully applied to construct triangular elements. In this paper, this method is generalized to construct quadrilateral elements, and the area co-ordinate theory for quadrilateral elements is systematically developed: (i) the area co-ordinates (L1 , L2 , L3 , L4) of any point in a quadrilateral are defined and two identical equations, which the four area co-ordinates should satisfy, are formulated and proved; (ii) two characteristic parameters for quadrilateral elements are defined and the degeneration conditions under which a quadrilateral degenerates into a parallelogram or a trapezoid or a triangle are given; (iii) transformation relations between the area co-ordinates and the Cartesian or isoparametric co-ordinates are presented, and several important formulae for quadrilateral area co-ordinates are listed. Copyright

Analytical trial function method for development of new 8‐node plane element based on the variational principle containing Airy stress function

Purposeu2009−u2009The purpose of this paper is to propose a novel and simple strategy for construction of hybrid‐“stress function” plane element. Design/methodology/approachu2009−u2009First, a complementary energy functional, in which the Airy stress function is taken as the functional variable, is established within an element for analysis of plane problems. Second, 15 basic analytical solutions (in global Cartesian coordinates) of the stress function are taken as the trial functions for an 8‐node element, and meanwhile, 15 unknown constants are then introduced. Third, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of the displacements along element edges, which are interpolated by element nodal displacements. Finally, the whole system can be rewritten in terms of element nodal displacement vector. Findingsu2009−u2009A new hybrid element stiffness matrix is obtained. The resulting 8‐node plane element, denoted as analytical trial function (ATF‐Q8), possesses excellent...

A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates

A simple displacement-based, quadrilateral 20 DOF (5 DOF per node) bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented in this paper. This element is constructed by the following procedure: (i) the variation functions of the rotation and the shear strain along each side of the element are determined using Timoshenkos beam theory; and (ii) the shear strain, rotation and in-plane displacement fields in the domain of the element are then determined using the technique of improved interpolation. Furthermore, a simple hybrid procedure is also proposed to improve the stress solutions. The proposed element, denoted as CTMQ20, possesses the advantages of both the displacement-based and hybrid elements. Thus, excellent results for both displacements and stresses, especially for the transverse shear stresses, can be obtained.

Some basic formulae for area co-ordinates in quadrilateral elements

The area co-ordinate method, which is an efficient tool in the construction of triangular elements, has been recently generalized to construct straight-sided quadrilateral elements. In this paper, the differential and integral formulae for area co-ordinates in quadrilateral elements are presented. Consequently, the numerical integration method that must be used in the formulation of the stiffness matrix in isoparametric elements can now be discarded. Copyright

Generalized Neumann Expansion and Its Application in Stochastic Finite Element Methods

An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertain systems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easy to implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator is proposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solution is sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationship between the GNE method, the perturbation method, and the standard Neumann expansion method is also discussed. Based on the links between these three methods, quantitative error estimations for the perturbation method and the standard Neumann method are obtained for the first time in the probability context.

Developments of Mindlin-Reissner Plate Elements

Since 1960s, how to develop high-performance plate bending finite elements based on different plate theories has attracted a great deal of attention from finite element researchers, and numerous models have been successfully constructed. Among these elements, the most popular models are usually formulated by two theoretical bases: the Kirchhoff plate theory and the Mindlin-Reissener plate theory. Due to the advantages that only continuity is required and the effect of transverse shear strain can be included, the latter one seems more rational and has obtained more attention. Through abundant works, different types of Mindlin-Reissener plate models emerged in many literatures and have been applied to solve various engineering problems. However, it also brings FEM users a puzzle of how to choose a “right” one. The main purpose of this paper is to present an overview of the development history of the Mindlin-Reissner plate elements, exhibiting the state-of-art in this research field. At the end of the paper, a promising method for developing “shape-free” plate elements is recommended.

A novel joint diagonalization approach for linear stochastic systems and reliability analysis

Purpose – The purpose of this paper is to present a novel gradient‐based iterative algorithm for the joint diagonalization of a set of real symmetric matrices. The approximate joint diagonalization of a set of matrices is an important tool for solving stochastic linear equations. As an application, reliability analysis of structures by using the stochastic finite element analysis based on the joint diagonalization approach is also introduced in this paper, and it provides useful references to practical engineers.Design/methodology/approach – By starting with a least squares (LS) criterion, the authors obtain a classical nonlinear cost‐function and transfer the joint diagonalization problem into a least squares like minimization problem. A gradient method for minimizing such a cost function is derived and tested against other techniques in engineering applications.Findings – A novel approach is presented for joint diagonalization for a set of real symmetric matrices. The new algorithm works on the numerica...

Method of Area Coordinate — From Triangular to Quadrilateral Elements

In this paper, the area coordinate method is generalized to formulate quadrilateral elements. The general theory of area coordinate for quadrilateral elements is presented and new quadrilateral elements for membrane and plate bending are formulated by using the area coordinate method.