Exact static analysis of helicoidal structures of arbitrary shape and variable cross section

This paper concerns the formulation of a new element-based method for the static behavior of a spatial bar, with variable cross section of an elastic and isotropic material, under arbitrary loads. Using only one element it is possible to derive the exact stiffness matrix and equivalent loads (up to any desired accuracy), for any continuous polynomial variation of axial, shear, torsional, and bending stiffnesses, and load along the member. Both the cross-section dimensions and the shape of the bar can vary along the curved member as polynomial expressions. The problem is described by six differential equations. These are second-order equations with variable coefficients, with six unknown displacements, three translations, and three rotations at every point along the member. In the proposed method the exact shape functions of the member are computed, then the terms of the stiffness matrix are found from the shape functions. Several examples are solved and compared to results from the relevant literature. Only a few parameters are needed to identify the model. In the proposed method the computational time is reduced and the results are exact. The proposed method has practical application for the design of concrete stairs and ramps.