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%.

To address the high bending stresses and potential structural failure risks caused by differential settlement at expansion joints during bridge widening projects of straight bridges, this paper proposes an “Adaptive Rebound Displacement Compensation Device”. Existing research primarily focuses on analyzing settlement patterns and passive control standards, with limited attention to active dynamic regulation. Notably, the bending stress induced by new pier settlements can reach 3–5 times that of vehicle loads, posing serious safety concerns. Through theoretical derivation, this study clarifies the relationship between superstructure loss of strength and factors such as pier settlement, device stiffness, friction coefficient, and L-shaped baffle angle, and a comprehensive design framework is established accordingly. Combining numerical simulations, laboratory tests, and field measurements from engineering practices, multiple validation approaches are employed. The simulation results demonstrate that the proposed device can limit deck subsidence to 10–20% of pier settlement height, and experimental outcomes align closely with theoretical predictions. This device has been successfully implemented in a bridge widening project on a highway section in Jiangxi Province. It should be noted that all data presented in the paper are derived from finite element method (FEM) numerical simulations, and there are currently no on-site measurements of the device’s performance. FEM analysis indicates that the device demonstrates certain feasibility for practical engineering applications. Compared to scenarios without the installation of this device, bridge deck displacements can be reduced by approximately 16.5%. By enabling adaptive rebound through self-adjustment mechanisms for settlement compensation, this device significantly alleviates bending stresses at expansion joints, breaking through traditional passive control limitations. This study provides an innovative approach for actively controlling settlement differences in the widening of straight bridges, offering significant implications both at the theoretical and practical levels.