Stiffness and transfer matrices of a non-naturally curved 3D-beam element

This article deals with the 3D-curved beam element with non-naturally curved differential geometry governing equations, i.e. which arc length is a function of a parameter, as for example the classical case of the ellipse. The derivative of the arc length with respect to a parameter chosen, meaning the Frenet frame velocity along the central-line, is not unitary. A twelve linear ordinary differential equation system governing its mechanical behaviour is presented. Exact analytical and numerical approximate procedures are proposed. Both transfer and stiffness matrices are provided to determine the internal forces and displacements in a 3D-curved beam element defined by its parametric equations with varying cross-section area and generalized loads applied. Both matrix methods offer directly the load transmission and equivalent vectors, respectively. The semi-circular arch subjected to a distributed load in and out of its plane, a semi-elliptical arch and an elliptic–helical beam examples are given for verification.