STRUCTURAL ANALYSIS BASED ON APPLICATION OF PROPERTY OF THE GENERALIZED INVERSE MATRIX

Such a case that the condition for compatibility of strain is required in stress method

Abstract

 Question of whether it is necessary for stress method to use the condition for compatibility of strain or not might be brought up from the fact that the stress can derive from a formulation based on stress method by means of application of the property of the Moore-Penrose generalized inverse without the help of the compatibility condition.

 One answer to the question might be found in an analysis of the situation where the reciprocal theorem is not satisfied because of the reason that the transposed relation between the stress-force matrix and the displacement-strain matrix is disregarded. The situation might be able to find in the FEM analysis as rare case where the stress-nodal force matrix of element is independently made up without considering the corresponding relation of the nodal displacement- strain matrix.

 When the reciprocal theorem is not satisfied, we will face two kinds of problem to solve. One is to clear up the definition and the role of what is called the condition for compatibility of strain which is usually used to decide redundant force. Another is how to derive displacement from stress without the help of the principle of energy like the Castigliano’s theorem. Answer of each problem and the explicit forms of stress solution and displacement solution are represented in this paper.

 

 Results can summarize as follows,

 (i) What is called the condition for compatibility of strain, which is re-defined as the condition for existence of strain to the compatibility equation relating displacement to strain in paper, is equivalent to the orthogonal condition between strain and self-stress when the reciprocal condition is guaranteed. Which means it is necessary to clear up the distinction between the existence condition and the orthogonal condition when the reciprocal condition is not satisfied. The distinction is available to obtain stress as shown in the section 5.1.

 (ii) Displacement is obtained by the inverse operation of the generalized inverse matrix as the particular solution of the compatibility equation as shown in the section 5.2.

 (iii) Merit of the explicit forms of stress solution and displacement solution obtained in this paper is easily provable to coincide with the usual solution forms of displacement method. Origin of the merit is easy operation of the orthogonal projection matrix expressed by the generalized inverse. Then stress variables can treat as a set instead of a choice of the independent component of stress variables such as redundant force needed to make up the regular matrix.

 (iv) There is nothing to change the formulation of getting displacement and stress in case of displacement method even if the reciprocal theorem is satisfied or not. On the other hand, when the reciprocal theorem is kept in stress method the computational complexity of getting the solutions drastically decreases in less level than the corresponding displacement method Therefore, the reciprocal theorem should be kept to the stress method. How to apply the solution forms to the FEM analysis based on stress method is illustratively shown in example 2 where a square structure of the state of plane stress is divided into two triangular elements. The same structure composed of single rectangular element where the stiffness matrix is not regular is also treated as shown in example 3, and so on.