Distorsion in Structural Mechanics

The behaviour of elastic perfectly plastic structures under given loadings and distorsion is examined until plastic failure (analysis problem). “Distorsion” is here for the strain which does not satisfy the compatibility equations. A distorsion can be caused by prestressing, supports displacements or lack of precision during fabrication stage (the plastic strain is the example of distorsion, too). The mathematical models for analysis problems are created by the help of extremum energy principles—principles of complementary energy and total potential energy minimum. The emphasis is put on traditional to the structural mechanics bar and plate bending systems. The main unknowns are the self-equilibrium forces, displacements and strains. In the case of bar systems the analysis problem is formulated by means of linear yielding conditions as a quadratic programming problem. The analysis of Lagrange's generalised problem shows that the displacements and forces from given distorsions and plastic strains should be obtained by the same expressions. For the algoritmisation of Lagrange's problem different methods can be used: finite difference or finite element methods. Very convenient are well-known procedures of the equilibrium finite elements (with forces as main unknowns; the conditions of equilibrium satisfied) for building of stiffness and flexibility matrices. In the present paper only on the estimation of distorsion in the self-equilibrium finite elements and in the structural shakedown problems are given. The numerical example of plate analysis is presented.

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