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Some investigations into the finite element method with special reference to plane stress Khanna, Jitendra

Abstract

Plane stress stiffness matrices are derived explicitly for square Isotropic elements under different assumptions on the stress distribution. An explicit (8 x 8) matrix is obtained under the assumption of uniform σx, σy, linear τxy and thus it is shown that the Gallagher matrix belongs to the class of parametric matrices. Two (10 x 10) matrices are obtained under the assumption of linear σx, σy, τxy using interior nodal translations and corner edge rotations respectively as additional generalized displacements. These two matrices do not appear suitable for general usage but will perform as well as the Turner matrix under the same nodal loads. A (12 x 12) matrix is derived under the assumption of hyperbolic σx, σy, and parabolic τxy, again exemplifying the use of corner edge rotations as additional generalized displacements. This matrix behaves unexpectedly with varying Boisson's ratio. A method of evaluating stiffness matrices, which reduces the necessity of comparing finite element solutions with analytical ones, is formulated. In this method a comparison is made of the strain energy of deformation produced within a finite element by the different matrices under the same nodal loads. It is shown that such comparisons require the study of special matrices i.e. the stiffness difference matrix and the inverse difference matrix which are obtained from the matrices under comparison. It is proved that the results of the element matrix comparisons apply to the structure. It is shown that the strain energy of a finite element under normalised loads is bounded between the maximum and minimum eigenvalues of the inverse matrix. The strain energy comparison criterion is used in the study of parametric matrices. An explicit parametric inverse is obtained. Explicit parametric eigenvalues are obtained for the inverse difference matrix and the stiffness difference matrix, and it is verified that they give identical results for the matrix comparisons. It is proved that the parametric matrices produce the exact strain energy under uniform nodal loads. It is shown that the stiffness matrix parameter and the inverse matrix parameter represent a measure of the strain energy under non-uniform nodal loads so that the strain energy can always be bounded by varying the parameter. It is proved that if strain energy curves are drawn with respect to structure sub-division then no two curves will intersect. It is proved that all parametric strain energy curves will converge towards the true solution with progressive structure subdivision. A strain energy ordering is obtained for the parametric matrices and the following conclusions are drawn. The Pian matrix is the best displacement matrix. The Gallagher matrix is inferior to the Turner, Pian, and Argyris-Melosh matrices. Constant stress tri-nodal triangles are generally inferior to the use of square elements. Matrices satisfying microscopic equilibrium or capable of representing uniform stresses will not necessarily yield good results. A method is proposed for obtaining upper bounds on the strain energy of a region under plane stress by replacing the continuum with a psuedo-truss system, the bar forces of which provide the equilibrium and self-straining solutions. Two examples of its application are presented, and an indication is obtained that upper bounding by varying the matrix parameter will give better results for the same structure subdivision.

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