This paper proposes an efficient beam-column element formulation for the second-order analysis of members with arbitrary geometry. The proposed element accommodates cross-sectional variability through analytically derived cross-section varying ratios. An automatic mesh generation algorithm is developed to facilitate efficient discretization based on the variation of geometric properties. The influence of cross-section centroid eccentricity is incorporated into the element formulation via equivalent eccentricities and transformation matrices. Equivalent nodal forces for commonly encountered member loads are derived using the principle of complementary virtual work and equilibrium equations. Large deformation behaviour is captured using the co-rotational method within the updated Lagrangian framework, enabling efficient and accurate simulation of both P-Δ and P-δ effects in the second-order analysis. The proposed free-form element is validated against benchmark solutions using stepped elements through five examples, including nonlinear analysis of complex-shaped members, eccentric beams, members subjected to distributed loads, a planar toggle frame and a 3D hexagonal pavilion structure with free-form members. Notably, a free-form element provides accuracy comparable to that of 25–50 conventional stepped elements. Moreover, under uncertainty evaluation using the Monte Carlo Simulation (MCS), the distributions of structural responses of a curve-shaped component are generated by considering random variations in cross-section dimension. In the MCS, the proposed element achieves the same accuracy as stepped elements while requiring only 20% of the computation time (reducing runtime from 5 hours to 1 hour), which further validates its efficiency.
Liang et al. (Thu,) studied this question.