## 5 Results Analysis
### Tall Bridge Pier Under Multidimensional Ground Motions  
This study examines the seismic response of a tall bridge pier subjected to synthetic earthquake motions, with damping ratios set at 3%, 5%, and 10%. The earthquake waves adopted characteristic periods of 0.45 s (Z direction), 0.40 s (X direction), and 0.35 s (Y direction) according to the seismic design standard [32], resulting in peak accelerations of 2 m/s\({}^{2}\) (Z direction), 1.6 m/s\({}^{2}\) (X direction), and 1.2 m/s\({}^{2}\) (Y direction), respectively. This study focuses on the displacement and shear stress responses under pulse and non-pulse ground motions in 1D (Z direction), 2D (Y and Z directions), and 3D (XYZ directions) settings. The primary variables considered are the displacement at 40 m in the Z direction and the shear stress at 5 m in the YZ direction. This investigation aims to provide insight into the structural performance of tall bridge pier under different seismic conditions and multidimensional input configurations. In this study, 1D represents one-dimensional earthquake action, 2D represents two-dimensional earthquake action, and 3D represents three-dimensional earthquake action. Figure 7: Time history curves of the response of tall bridge pier with varying damping ratios under pulse ground motion: (**a**) displacement and (**b**) shear stress. The study target is set with a damping ratio of 5%. Figure 8 illustrates the time histories of the Z section displacement and YZ section shear stress response of tall bridge pairs subjected to multidimensional non-pulse ground motions. As illustrated in Figure 8, the tall bridge pier exhibits consistent dynamic behaviour across multidimensional non-pulse ground motions (1D, 2D, and 3D) under identical damping conditions. The displacement time history for the Z section shows negligible variation across the varying input dimensions, with peak displacements of 0.13225 m, 0.13221 m, and 0.13219 m for 1D, 2D, and 3D ground motions, respectively. The maximum error between these values is 0.05%. The shear stress time histories for the YZ section reveal a close trend, with peak shear stresses of 0.2639 MPa, 0.2507 MPa, and 0.2623 MPa under 1D, 2D, and 3D ground motions, respectively, resulting in a peak stress maximum error of 5.27%. The dynamic behaviour of the Z section and YZ section for two-dimensional and three-dimensional non-pulse ground motions is less pronounced than that for one-dimensional motion. This reduction in response can be explained by the interference between different earthquake waves when subjected to 3D ground motion, which reduces the peak dynamic effects. These findings highlight the significance of considering multidimensional effects in the seismic analysis of tall bridge pairs, particularly in regions prone to complex ground motion characteristics. The analysis was conducted with a damping ratio of 5%. As shown in Figure 9, the displacement-time history of the Z section and shear stress-time history of the YZ section of a tall bridge pier subjected to multidimensional pulse ground motion were assessed. Under a consistent damping ratio, the peak displacements for the Z section under 1D, 2D, and 3D ground motion are 0.2248 m, 0.2234 m, and 0.2233 m, respectively, with a maximum deviation of 0.67%. For the YZ section, the peak shear stresses are 0.4709 MPa, 0.4636 MPa, and 0.4631 MPa, respectively, with a maximum error of 1.68%. These results indicate that two-dimensional and three-dimensional ground motions produced lower peak displacements and shear stresses than do one-dimensional ground motion, suggesting that the interaction between the components of multidimensional earthquake actions influences the structural response by altering the displacement and shear stress behaviour of the tall bridge pairs. This study examines the dynamic features of a tall bridge pier under multidimensional pulse ground motion actions.