Abstract
The Conventional Finite Element Method for the analysis of plate bending based on a the principle of stationarity of total potential energy which is well known as Stiffness Formulation , using a variety of conformal and no-conformal elements was widely applicable in early of numerical solutions for such problems , later in order to improve the efficiency of such solutions , a more general and flexible formulation called Mixed Formulation based on variational principles which can be regarded as an extension of the Stiffness principle become important alternative. The explicit stiffness matrix for plate bending was given in the most if it`s not in all literatures so the reader can easy follow the solving procedure for such problems and verifying any published results , however such like-matrix (Augmented Matrix) in case of mixed formulation not given , at least for simple elements in the literatures as well as concerned researches . This quietly leads to some difficulties concerning the verification as well as understanding the published results. The main objective of this paper is to introduce this matrix in abbreviated size followed by a applicability through a some detailed examples for some bending models. The derived matrix will be helpful subject of research work in addition to , reveals a very good feeling with understanding and verification the published results in plus to comparing with the analytical solutions of different plate bending problems as the reader can do that.
Highlights
In the Conventional Finite Element Method (CFEM), which is based on the Stiffness Formulation, yields a very accurate displacement for most cases, while the stresses, which are derived as post-calculation from the displacement field, are not continuous across the boundaries
Many researches for Finite Element Method as well as other techniques, for plate bending problems, but the explicit forms of the Augmented Matrix were not given for some reasons [1]
This paper describes the construction of this augmented matrix and presents its explicit form through systematic way in order to overcome the size difficulty for finite element plate bending models
Summary
In the Conventional Finite Element Method (CFEM), which is based on the Stiffness Formulation, yields a very accurate displacement for most cases, while the stresses (or moments), which are derived as post-calculation from the displacement field, are not continuous across the boundaries. This drawback of (CFEM) may be considered as the major one for solving various engineering problems up to date, and in order to have stress continuity across the element boundary, the moments should be included as independent variables in the formulation Such an approach is referred to as Mixed Formulation, which is a more general and flexible formulation utilizing variational principles that can be regarded as an extension of the stiffness principle taking significant place in modern field of engineering analysis. This paper describes the construction of this augmented matrix and presents its explicit form through systematic way in order to overcome the size difficulty for finite element plate bending models In absence of such explicit matrices for all, the correctness and fineness of the published results so obtained remains in doubt due to many interpretations, such as the mathematical expressions of matrix itself, threating of boundary conditions, typographical errors. It can indicate the reason for this summarized detailed work in order to be simple, but a brief recall revision for many of basic concepts may be necessarily convenient for the reader while some detail of proofs was omitted, for the sake of brevity
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