Automatic Calculation Method for Effective Length Factor of Bridge Piers Considering Shear Deformation

The effective length factor (ELF) of bridge piers, a critical design parameter, is determined by solving the transcendental equation governing stability. Efficient and accurate solutions to these equations under various constraints are essential for automating bridge design software. In this paper, the bridge pier is simplified as an elastically restrained column based on the Timoshenko beam model, and the pier stability equation under general elastic constraints considering shear deformation is derived. By analyzing the distribution patterns of the solutions to the transcendental equations with and without considering shear deformation, a novel two-stage Adaptive Sequential Root Search Method based on bisection algorithm (ASRSBM2s) is proposed to calculate the ELF. In the first stage, the smallest positive root of the transcendental equation without considering shear deformation is first calculated, and the obtained positive root is used to restrict the solution domain of the transcendental equation considering shear deformation in the second stage. Compared with the results of the finite element method (FEM), the proposed algorithm can accurately determine the correct roots of the transcendental equation for various bridge scenarios, and the maximum relative error of the calculated ELF of bridge piers is below 2.5%.