Statically determinate and indeterminate trusses from the standpoint of the principle of stationary action

Abstract

Objective. The article considers the correspondence of statically definable and statically indefinable trusses to the general physical principle of stationary action and its specific manifestation in the mechanics of deformable solids – the generalized principles of Lagrange and Castigliano.

Method. The design tasks for the farms were solved on the basis of the theory of structural synthesis using the generalized Castigliano variational principle and the Lagrange multiplier method. Isoperimetric problems with a coupling equation that establishes a given volume of material are considered. Equations for determining the variable parameters are derived from the stationarity condition of the generalized functional of the potential strain energy. For a three–core statically indeterminate truss, its degeneration into a single core was found, and for a multi-span truss, complex numbers for the desired parameters were found.

Result. A generalization of Vasyutinsky's theorem, formed without taking into account the problem of stability of the equilibrium of the system elements, is given. It is established that for a statically definable truss, the minimum of the potential energy of deformation corresponds to the minimum of the volume of the material even in the presence of compressed rods. A generalization of Levy's theorem is given, taking into account the stability of the equilibrium of compressed rods, which confirmed the search for rational systems in the field of statically definable systems with global minima of the potential energy of deformation and the volume of the material. Devoid of an objective criterion of rationality, the task of synthesizing a statically indeterminate farm can be considered with a variety of subjective criteria with a local minimum of the objective function.

Conclusion. To ensure global indicators of the mechanical resistance qualities of structural systems, in particular trusses, ther analysis and synthesis should be carried out on a single methodological basis using the principle of stationary action.