On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

Abstract

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.

Development of single function potential approach of orthotropic composite materials for the case of the plane strain condition

“…Many practical problems can be solved considering by the plane strain condition instead of three dimensional concepts. Using the existing elasticity formulation of isotropic material for the case of the plane strain condition, recently a single function formulation was developed by Deb Nath et al [11] for determining the elastic field of plane strain problems of isotropic materials in low computational cost. Here a single function formulation is proposed for determining the elastic field of orthotropic composite materials for the case of the plane strain condition.…”

“…It is noted that the recent research and developments in using the displacement potential boundary approach [20,[29][30]11] have generated renewed interest in the field of both analytical and numerical solutions of stress problems. Recently, Afsar et al [30] and Deb Nath et al [11] solved beam problems of orthotropic composite materials analytically and numerically considering the plane stress condition.…”

Analytical Solution of Mixed Boundary Value Problems Using the Displacement Potential Approach for the Case of Plane Stress and Plane Strain Conditions

“…It is noted that the recent research and developments in using the displacement potential boundary approach [26][27][28] have generated renewed interest in the field of both analytical and numerical solutions of stress problems. In this paper, firstly, a one end fixed duralumin plate having stiffening at rest of other edges subjected to a uniform tension at its right lateral edge is solved analytically for the case of plane stress and plane strain conditions and the solutions for both of the cases are discussed in a comparative fashion.…”

A Finite Difference Solution of a Simply Supported Beam of Orthotropic Composite Materials Using Displacement Potential Formulation

“…The displacement formulation, on the other hand, involves finding two displacement functions simultaneously from the two second-order elliptic partial differential equations of equilibrium, which is extremely difficult, and this problem becomes more serious when the boundary conditions are mixed [16]. The elastic field of some important structural elements made of isotropic and anisotropic materials is studied analytically and numerically using displacement potential formulation [1][2][3][4][5][6][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. It is noted that the recent research and developments in using the displacement potential boundary modeling approach [1][2][3][4][5][6][19][20][21][22][23][24][25][26][27][28][29][30][31]…”

“…The elastic field of some important structural elements made of isotropic and anisotropic materials is studied analytically and numerically using displacement potential formulation [1][2][3][4][5][6][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. It is noted that the recent research and developments in using the displacement potential boundary modeling approach [1][2][3][4][5][6][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]…”