The Fundamentals of the Theory of Calculation of Elastic Systems and Variation Principles of Structural Mechanics

This chapter deals with the basic fundamentals of the theory of calculation of elastic systems and variation principles of structural mechanics. This chapter discusses the following: potential field of forces; potential energy of masses and definition of force; virtual work of external and internal forces; the principle of virtual displacements; virtual work of external forces or potential deformation energy; total potential energy; extreme properties of potential energy; the principle of virtual displacements; the methods of drawing up equations of motion; Lagrange variation principle, and Lagrange formula; the principle of virtual changes in the stress state and formulation of the principle; Castigliano variation principle; the principle of least work; Hamilton principle; and the principle of reciprocity of work. The variation principles of structural mechanics discussed in this chapter present a conceptually new approach to studying the course of dynamics of structures. Thorough knowledge of the basic theorems of elastic systems and variation principles of structural mechanics are the bases for a successful study of general and special courses in the dynamics of structures. From a methodological point of view, the methods and materials presented in the second chapter allow a smooth and organic transition to the study of energy calculation methods, that are very effectively used in structural and applied mechanics, in the theory of elasticity, as well as in the dynamics of structures.