On equivalence of algebraic and finite element formulations of prestressed bar structures

Abstract

Abstract The equivalence of the finite element method and the algebraic formulation of the equations of moderately thick and thin elastic frames, beams, trusses, and grillages within the Timoshenko and Euler-Bernoullie theory derived earlier by Pełczyński, Gilewski (2019) is presented. The proposed algebraic formulas lead to the same systems of algebraic equations as in the finite element method approach with the use of exact shape functions. Furthermore, in order to extend the applicability of the formulation, derivations of equations for prestressed structures are presented. The formulations are supplemented with algorithms for building selected matrices, which are crucial for performing calculations using the derived formulation. In addition to the analyses performed, the benefits of the existence of a global deformation matrix $$\textbf{B}$$ are shown.