A novel strategy to construct exact structural-property matrices for nonprismatic Timoshenko’s frame elements

•Timoshenko’s frame elements with generically variable rigidity.•Novel process for building Timoshenko’s shape functions.•“Exact” structural-property matrices for frame elements.•Reconstitution of the complete equilibrium path in geometric nonlinear analyses of frames.•General comments on several nonlinear algorithms. Assuming Timoshenko’s beam hypothesis, this paper proposes a unified strategy to derive exact finite-element (FE) matrices for framed structures having elements with variable rigidity. Its basic idea is to apply the principle of virtual forces (PVF), at the element level, to obtain a flexibility-based set of equations from which structural-property and nodal-load coefficients can be directly evaluated. The variable physical-geometric characteristics along the frame elements are approximated by polynomials of different orders. For evaluating structural-property coefficients that depend on the deformation of the structure, as e.g. the geometric stiffness coefficients, one employs Timoshenko’s consistent shape functions. A novel process for building them under the most general cases of rigidity variation is presented in this paper. In this study, we particularly apply the technique to effect second-order analyses of 2D frames with nonprismatic elements.