An Exact Geometrically Nonlinear 3D Truss Finite Element with Variable Axial Rigidity Based on Positional Formulation

Abstract

This paper introduces an exact geometrically nonlinear 3D truss finite element based on the positional formulation, capable of accounting for arbitrarily varying axial rigidity along the element’s length. The proposed formulation accurately captures large-displacement behavior using a single element per member, without requiring mesh refinement. A key advantage is the elimination of local-to-global coordinate transformations, thereby reducing computational cost. The formulation derives the axial flexibility using the flexibility method and the Principle of Virtual Forces, avoiding the need for exact shape functions. Numerical examples, including 2D and 3D trusses with polynomial axial rigidity variations, show perfect agreement with analytical solutions in terms of nodal displacements, nodal positions, axial forces, and limit loads. The method also exhibits faster convergence compared to existing approaches, confirming its accuracy, robustness, and efficiency as a powerful technique for nonlinear analysis of truss structures with arbitrarily varying axial rigidity.