Theodore H. H. Pian

Finite Element Solutions for Laminated Thick Plates

In view of the increasing interest in using composite materials for aerospace structures, the analysis of layered anisotropic plates becomes essential. The so-called classical laminated-plate theory (CPT) does not include effects of shear deformation. Recent elasticity solutions [ 1, 2, 3] of laminated plates show the importance of the transverse shear effects. While analytical solutions are usually restricted to problems with simple loading and boundary conditions, the finite-element method is found attractive in dealing with complicated problems. To include the effects of transverseshear deformation, the rotation of surface normal for different layers of a laminated plate must be considered as different. This has been found to be not feasible by the conventional finite-element displacement method [4]. The above-mentioned difficulties can be avoided if a hybrid-stress finite-element model is used [5, 6, 7]. In this paper, a general quadrilateral multilayer plate element is derived using the hybrid-stress method. The transverse shear effects, as well as the coupling effects of stretching and bending, are included. A comparison of results with known solutions is made and excellent accuracy in predicting both displacements and stresses is observed.

On the convergence of the finite element method for problems with singularity

Abstract For problems with singularities, the convergence rate for the finite element method are often controlled by the nature of the solution near the points of singularity. Unless the singularities are properly handled, the regular so-called high-accuracy element will not be able to improve the rate of convergence.

A VARIATIONAL PRINCIPLE AND THE CONVERGENCE OF A FINITE-ELEMENT METHOD BASED ON ASSUMED STRESS DISTRIBUTION.

Abstract A variational principle is formulated as the foundation of the finite-element method proposed by Pian. Through this formulation an extension of Pians original proposal is made and the convergence of the approximate solution to the exact one is proved.

State-of-the-art development of hybrid/mixed finite element method

Abstract A brief review of multifield variational principles for the formulation of finite element methods in solid mechanics and an account of the evolution of hybrid/mixed finite element methods are presented first. Discussions of recent applications of the hybrid/mixed finite element methods include: (1) formulation of Loof and semi-Loof elements, (2) special elements based on a priori satisfaction of equilibrium and compatibility conditions, (3) analysis of heterogeneous materials with randomly distributed microstructures and (4) finite element and 3-D finite strip analyses of free edge problems in laminated composites.

The convergence of finite element method in solving linear elastic problems

Abstract This paper presents a theoretical development to show the sufficient conditions that will insure a finite element displacement analysis to converge to the exact displacement solutions when the size of the elements are progressively reduced. The order of such convergence is also estimated. The development is in connection with the three dimensional elasticity problem and the plate bending problem. A study is made to determine the possible means of evaluating the merits of different stiffness matrices to be used in the finite element analysis.

A new formulation of hybrid/mixed finite element

A new formulation of finite element method is accomplished by the Hellinger-Reissner principle for which the stress equilibrium conditions are not introduced initially but are brought-in through the use of additional internal displacement parameters. The method can lead to the same result as the assumed stress hybrid model. However, it is more general and more flexible. The use of natural coordinates for stress assumptions leads to elements which are less sensitive to the choice of reference coordinates. Numerical solutions by 3-D solid element indicate that more efficient elements can be constructed by assumed stresses which only partially satisfy the equilibrium conditions.

Consistency condition and convergence criteria of incompatible elements: General formulation of incompatible functions and its application

Abstract A consistency condition for the incompatible elements is suggested, and then various convergence criteria are obtained naturally. Starting with a strong patch test, the paper establishes a general formulation to generate incompatible element functions. Finally, an incompatible plane isoparametric element and an incompatible 3-D hexahedral solid element are presented.

Finite elements based on consistently assumed stresses and displacements

Abstract Finite element stiffness matrices are derived using an extende Hellinger-Reissner principle in which internal displacements are added to serve as Lagrange multipliers to introducethe equilibrium constraint in each element. In a consistent formulation the assumed stresses are initially unconstrained and complete polynomials and the total displacements are also complete such that the corresponding strains are complete in the same order as the stressesSeveral examples indicate that resulting properties for elements constructed by this consistent formulation are ideal and are less sensitive to distortions of element geometries. The method has been used to find the optimal stress terms for plane elements, 3-D solids, axisymmetric solids, and plate bending elements.

Mode shapes and frequencies by finite element method using consistent and lumped masses

Abstract The rate of convergence of the mode shapes and frequencies by the finite element method using consistent and lumped mass formulations has been established. Simple examples are given to demonstrate the results. It has been found that for a system of differential equations of second order such as the equations of equilibrium in terms of displacement in the theory of elasticity, membrane etc., a proper lumped mass formulation will not suffer any loss of rate of convergence utilizing simple elements. However, in the case of higher order differential equations or when the use of more complicated elements is required or desired, a consistent mass formulation often will provide a better rate of convergence.

Finite Element Methods in Continuum Mechanics

Publisher Summary The chapter presents a brief introduction to the different finite element formulations for linear elastic solids and discusses similar formulations for several other field problems. The chapter presents detailed illustrations for several typical finite element formulations. In the finite element formulation, displacement and stress fields are assumed to be continuous within each discrete element. This formulation calls for modified variational principles for which the continuity or equilibrium conditions along the interelement boundaries are introduced as conditions of constraint and appropriate boundary variables are used as the corresponding Lagrangian multipliers. The chapter presents the several variational principles and the corresponding models used in the finite element formulation. The large majority of the existing finite element formulations are based on the assumed displacement approach. The chapter discusses equilibrium problems of linear elastic solids. There are several other problems in solid mechanics, which can be formulated by means of variational principles and hence can be solved by finite element methods. The finite element methods have also been extended to nonlinear problems resulting from elastic-plastic material properties or from large deflections or finite strains.