Three-Node Torsional Spring Element Formulation for the Analysis of Reconfigurable Bar-Linked Structures

Abstract This technical note presents the derivation, validation, and application of a three-node torsional spring (3NTS) element for the analysis of bar-linked, reconfigurable structures. The 3NTS element assigns rotational stiffness to a joint (node) of two axial force members (bars) in truss-like assemblies. This element avoids the use of rotational degrees-of-freedom in the model by recasting its resisting moment into equivalent nodal forces, which are consistent with global equilibrium, thereby keeping the model size compact and computationally efficient. The 3NTS is integrated into standard nonlinear solvers to simulate large-displacement response and validated against analytical solutions of two benchmark examples: the simplest 3NTS structure and the buckling of a vertical column. We further apply the framework to a reconfigurable truss structure from our previous work to illustrate potential functional use cases and outline its broader applicability to metamaterials, kirigami systems, and biomechanical assemblies. An open-source matrix structural analysis tool implementing the 3NTS and axial force members is made available with this note.