Bridges are vital transportation infrastructure in our country, with bridge piers being a critical component. Damage to bridge piers can pose significant safety hazards, potentially leading to bridge collapse. Verticality serves as a crucial indicator for assessing pier damage. In this study, we employed terrestrial 3D laser point clouds to evaluate bridge pier verticality. However, owing to curvature features on pier surfaces, the point clouds exhibit nonuniformity. Moreover, external factors such as collisions contribute to the presence of shadowed texture area on the pier surfaces. Therefore, in this paper, we propose the use of (1) interpolation-matrix transformation for point cloud surface curvature calculation, in order to solve the problem of low curvature calculation accuracy caused by the nonuniform distribution of the point clouds. (2) Shadow texture area identification and removal are achieved through changes in curvature features, aiming to enhance the bridge pier verticality change detection accuracy. (3) The verticality of the bridge piers is computed and monitoring procedures are established for detecting changes. Experimental results showed that the proposed method can detect bridge pier verticality.