Nonlinear Dynamic Analysis of Tall Bridge Piers Under Multidimensional Pulse Earthquakes Considering Varying Damping Ratios

Abstract

The dynamic response of tall bridge piers with varying damping ratios under three-dimensional pulse ground motion remains insufficiently understood. To control the pulse characteristic parameters accurately and eliminate interference from actual seismic records, this study uses the earthquake wave synthesis software to generate three pulse seismic waves and non-pulse seismic waves with varying seismic characteristic periods. The dynamic response analysis of tall bridge piers under one-dimensional, two-dimensional, and three-dimensional seismic input conditions is carried out. The influence mechanism of pulse effect, damping ratio and ground motion dimension on structural response is mainly discussed. The results show that the peak displacement and peak shear stress response of tall bridge pier structures under pulse ground motion are 0.0614 m and 0.1727 MPa larger than those under non-pulse ground motion, respectively. The responses of the displacement and shear stress of the tall bridge pier subjected to pulse ground motion exceed those under non-pulse ground motion. When the action time exceeds 18 s, the influence on the displacement and shear stress time history curve of the tall bridge pier is ranked as follows: pulse ground motion > damping ratio > non-pulse ground motion. Under multidimensional non-pulse ground motion, the maximum errors in peak displacement at the Z section and peak shear stress at the YZ section of a tall bridge pier are 0.05% and 5.27%, respectively. These errors increase to 0.67% and 1.68% under multidimensional pulse ground motion, respectively. Compared with one-dimensional seismic conditions, two-dimensional and three-dimensional ground motions result in smaller displacement and shear stress errors at the Z section, but larger errors at the X section, particularly for peak displacement and shear stress at the YZ section. This highlights the greater complexity of multidimensional seismic forces and their varying impacts on different sections of tall bridge piers.

1. Introduction

The accelerated pace of economic growth has led to a notable increase in the speed of bridge construction. Bridge structures have a significant impact on human activity, industrial processes, and regional economic development. However, an increasing number of areas are located in seismic zones. As a key component of transportation infrastructure, the seismic performance of bridges is directly related to life safety, economic resilience, and social stability. Therefore, it is paramount to consider the seismic performance of bridges during both the construction and subsequent service phases [1]. The damping ratio serves as a crucial internal parameter to characterize the energy dissipation behaviour of structures during vibratory motion. It is instrumental in defining the dynamic behaviour of a structure, which significantly influences its overall dynamic response [2]. Specifically, the damping ratio has a pronounced effect on the seismic performance of tall structures [3]. The earthquake performance of the bridge structure is contingent upon the earthquake performance of the constituent column structure. In previous studies, regardless of whether the effect of water bodies was considered, the influence of non-pulse ground motion on the response of the bridge has been primarily focused on the response of bridges. Lee et al. [4] analysed the seismic performance of a bridge subjected to the Kobe earthquake and identified the deformation law. Solberg et al. [5] devised a novel bridge pier, based on the test findings, which demonstrated a marked enhancement in seismic resilience compared with the classical bridge pier. Zhu et al. [6] calculated the random response of a continuous railway bridge under earthquake action with a primary focus on the estimation of uncertain factors and the identification of sensitive uncertain parameters. In the case of a significant earthquake, the beams and bearings of tall bridge piers are particularly vulnerable to sliding during strong earthquakes, which may lead to severe structural damage [7]. Soleimani [8] proposed a clustering algorithm for studying the earthquake performance of tall bridge piers. Qi et al. [9] conducted a shaking table test on a circular rectangular hollow tall bridge pier and reported that different cross-sections affect the earthquake features of the bridge pier.

Yun et al. [10,11] proposed an innovative testing approach to analyse the dynamic behaviour of rigid frame bridges under seismic loading, employing an underwater shaking table for simulation purposes. Li et al. [12] developed a coordinated model and validated its precision through experiments on an underwater shaking table, while simultaneously evaluating the dynamic behaviour of a bridge pier.

Han et al. [13] proposed a three-dimensional numerical model of a reinforced concrete cable-stayed bridge, and studied the response behaviour of its friction sliding bearings during bidirectional seismic actions. Their investigation revealed critical insights into the performance of these bearings under earthquake-induced forces. Lin et al. [14] adopted finite element analysis and shaking table experiments to examine the dynamic responses of long-span bridges. Their findings indicated that bridges subjected to multidimensional seismic actions will amplify their dynamic response. Meng et al. [15] adopted the pseudo-excitation approach to examine the dynamic properties of cable-stayed bridges when exposed to multidimensional seismic and wave actions. This approach provided valuable examinations of the structural behaviour under different loading conditions. Yun et al. [16] used diffraction wave theory to study the dynamic nonlinearity of bridge piers under the combined effects of multidimensional earthquakes and wave actions. These findings indicate that the dynamic response of bridge piers is significantly greater under multidimensional ground motion loading than under single-directional earthquake excitation, emphasizing the importance of pondering multidimensional ground motion effects in bridge design.

Pulse ground motions result in significantly higher destructive loadings than non-pulse ground motions. This increased intensity makes it particularly susceptible to structural damage and potential collapse. Zhou et al. [17] developed a detailed nonlinear analysis model that encompasses the full range of potential damage to components, ranging from minor damage to total failure. The causes of earthquake obliteration of bridge components and the mechanism of beam failure under near-fault earthquakes have also been studied. Chen et al. [18] employed a vulnerability analysis method to evaluate the seismic resilience of tall bridge piers under near-fault motion, and reported that damage was more likely to occur when the bridge pier height was greater. Srivastava et al. [19] studied the influence of various factors, including pier height, aspect ratio, and longitudinal reinforcement, on the behaviour of reinforced concrete bridges during earthquake events in near-fault regions. Chen et al. [20] conducted shaking table tests on tall bridge piers, revealing that the speed and frequency of seismic pulses influence their vulnerability. Chen et al. [21] found that near-field pulse earthquakes increase the interaction between piers and surrounding water, thereby intensifying structural risks. These studies underscore the critical need for advanced modelling and design strategies to mitigate the influences of near-fault seismic motions on bridge integrity.

At present, there are few studies on the damping ratio of tall bridge pier structures. Generally, only the dynamic response under a fixed damping ratio is concerned, and the damping is often limited to isolated parameters, ignoring the influence of high-order modes and low-order modes. There are few studies on pulse ground motion, which are often only analysed by unidirectional horizontal pulse input, ignoring the phase modulation effect of multidimensional input. Therefore, the earthquake features of tall bridge pier structures’ combination of damping variation and synthetic multidirectional pulse input have yet to be investigated. Accordingly, this study employs ground motion synthesis software to generate varying pulse ground motions and non-pulse ground motions with characteristic periods (T_g = 0.45 s; T_g = 0.40 s; T_g = 0.35 s) for analysis. For the first time, the differential effects of various damping ratios on the displacement and shear stress responses of tall bridge pier structures under different dimensions of pulse and non-pulse ground motions are systematically analysed, and the interaction between multidimensional ground motions and structural damping is revealed. The extreme influence of pulse ground motion on a tall pier structure is clearly quantified by time history analysis. It is found that the peak values of displacement and shear stress response under pulse ground motion are significantly higher than those under non-pulse ground motion. In the long-term, the pulse effect exceeds the dominant effect of the damping ratio on the dynamic response, and the response of different sections under multidimensional ground motion is quite different. Through multidimensional pulse seismic input, variable damping ratio control and nonlinear time history analysis, this study systematically reveals the failure mechanism of tall-pier bridges in near-fault earthquakes, proposes a new idea of ground motion direction sensitivity and damping ratio optimization, and provides a theoretical basis for the follow-up study of multifield coupling of bridge pier structures.

2. Dynamic Time History Analysis of the Structure

The reasons for damping are extremely complicated, and mathematical and physical characteristic languages are employed to describe the damping characteristics. The damping force $F_c$ is commonly described as the product of the damping coefficient and the particle velocity:

$$F_c=cv$$

(1)

where $c$ denotes the damping coefficient, and $v$ represents the relative velocity of the particle.

The damping ratio is a key parameter in characterizing the behaviour of oscillating systems:

$$\xi =c/c_{cr}$$

(2)

$$c_{cr}=2m{\omega }_0=\sqrt{2km}$$

(3)

where $c_{cr}$ denotes the critical damping coefficient; $m$ denotes the mass; ${\omega }_0$ represents the natural frequency; and $k$ represents the stiffness coefficient.

For the bridge pier structure system, considering the impact of viscous damping, the motion equation of the free vibration of the viscous damping system is [22]:

$$\left[M\right]\left\{a\right\}+\left[C\right]\left\{v_d\right\}+\left[K\right]\left\{x\right\}=\left\{F_e\right\}$$

(4)

where $\left[M\right]$ denotes the mass matrix of the structure; $\left[C\right]$ denotes the damping matrix of the structure; $\left[K\right]$ denotes the stiffness matrix of the structure; $\left\{x\right\}$ denotes the displacement of the structure; $\left\{v_d\right\}$ denotes the velocity of the structure; $\left\{a\right\}$ denotes the acceleration of the structure; and $\left\{F_e\right\}$ represents the external force of the structure.

In this study, the structural dynamic analysis adopts the Rayleigh damping method [16]:

$$\left[C\right]=\alpha \left[M\right]+\beta \left[K\right]$$

(5)

where $\alpha$ denotes the mass damping coefficient, and $\beta$ denotes the stiffness damping coefficient. The calculation formula is as follows:

$$\left.\begin{array}{l}\alpha ={\displaystyle \frac{2\xi f_m{f}_n}{f_m+f_n}}\\ \beta ={\displaystyle \frac{2\xi }{f_m+f_n}}\end{array}\right\}$$

(6)

where $\xi$ represents the damping ratio of the structural material, and $f_m$ and $f_n$ are the $m$ and $n$-order natural circular frequencies, respectively. It is recommended to take the first and fifth orders.

The error calculation formula is as follows:

$$E_M={\displaystyle \frac{R_2-R_1}{R_1}}\times 100\%$$

(7)

where $R_2$ and $R_1$ are the dynamic response values under different conditions.

3. Model of a Tall Bridge Pier with Multidimensional Ground Motion Input

C40 concrete was selected as the primary material for the bridge pier, with a bridge pier section size of 4 m × 4 m × 40 m. A mass of 6.0 × 10⁵ kg was added to the top of the bridge pier to simulate the impact of beams on the structure. According to the standard [23], the damping ratios of 3%, 5%, and 10% were selected.

Table 1. Physical parameters of the bridge pier material.

Physical Parameters	Values
Density	2430 kg/m3
Elastic modulus	3.2 × 1010 N/m2
Poisson’s ratio	0.191
Damping ratio	3%; 5%; 10%

displays the physical parameters of the concrete material utilized in constructing the bridge pier. Finite element modelling of the bridge pier was performed via ANSYS R21 software, where the bridge pier was represented by the Solide45 element and the added mass by the Mass 21 element. The boundary conditions constrain all degrees of freedom at the bottom of the bridge pier. The model included 46,529 nodes and 40,961 elements. This simulation aimed to assess the pier’s response to multidimensional ground motion, as shown in Figure 1. The technical route of finite element modelling method, finite element analysis, and concrete constitutive relation are strictly referred to from the literature [11,24,25,26]. The convergence of the mesh is analysed by comparing different mesh densities, and the errors of the first-order frequency and peak displacement under different mesh densities are compared. It is found that when the element number is 80,000, the error of the first-order frequency calculated by the finite element model with the element number of 40,961 calculated in this study is 0.03%, the error of the peak displacement is 3.5%, and the error value is less than 5%. Therefore, the mesh density used in this study ensures the balance of calculation accuracy and efficiency, and ensures the model is accurate and reliable. The shape is close to the finite element model in the literature [26], but the size parameters are very different.

4. Synthesis of Seismic Waves

To simulate earthquake action, real second and third site conditions are considered. This study employed the ground motion synthesis software EQsignal v1.2.1, taking GB50011-2010 as the target spectrum. The characteristic periods were varied (T_g = 0.45 s; T_g = 0.40 s; T_g = 0.35 s), and three kinds of non-pulse ground motions and pulse ground motions were synthesized. The peak acceleration of non-pulse ground motions with different character periods were 1.0 m/s², and the peak accelerations of pulse ground motions were 1.39 m/s², 1.22 m/s², and 1.40 m/s², respectively, with a time interval of 0.01 s and a total time length of 30 s [27,28,29].

This study considers varying damping ratios and examines the displacement and shear stress responses of a tall bridge pier subjected to one-, two-, and three-dimensional ground motions. The acceleration time history and spectrum curves of tall bridge piers under non-pulse and pulse ground motions are shown in Figure 2 and Figure 3, respectively. This investigation focuses mainly on the effect of multidimensional ground motions and different damping ratios on tall bridge pier behaviour, and provides valuable insights into the design of seismic-resistant bridge structures.

A comparative analysis of Figure 2 and Figure 3 was performed to study the significant differences between the acceleration time history and acceleration response spectrum curves for pulse and non-pulse ground motions. These differences underscore the contrasting spectral characteristics of the two types of seismic waves. Within 5 to 10 s, the acceleration time history curves for pulse and non-pulse ground motions exhibit distinct variations. This finding demonstrates that the spectral performance of earthquake waves undergoes alteration subsequent to the introduction of the pulse to the seismic wave time history. There are significant variations in these spectral characteristics of earthquake waves as a function of various characteristic periods.

5. Results Analysis

The vertical ground motion influences the horizontal and vertical dynamic behaviour of tall structures [30]. Numerical simulation analyses and shaking table test findings have indicated that the seismic design of tall structures must consider the effects of multidimensional ground motion [14]. The bidirectional interaction resulting from near-fault motion can increase damage to bridge piers [31]. Consequently, studying the effects of multidimensional pulse ground motions on the dynamic behaviour of tall bridge piers is essential for enhancing seismic performance and tall structural design.

5.1. Tall Bridge Pier Under Pulse Ground Motion

A damping ratio of 5% is adopted to simulate one-dimensional ground motion along the z-axis, characterized by a period of 0.45 s and a peak acceleration of 0.20 g. The dynamic behaviour of a tall bridge pier is studied under the influence of pulse and non-pulse ground motions. The displacement at the Z section (30 m) and shear stress at the YZ section (15 m) are selected for investigation. The time history curves of the displacement and shear stress under various earthquakes are shown in Figure 4, providing insights into the structural performance of the bridge pier under varying earthquake loading conditions.

Figure 4 illustrates that tall bridge piers subjected to pulse ground motion exhibit a peak displacement of 0.1448 m, which is greater than the 0.0834 m displacement observed under non-pulse conditions, representing a 73.62% increase. The largest difference in displacement responses occurs during the 5–10 s interval, where the displacement time history under pulse motion is more complex and exhibits greater variability than that under non-pulse motion. This period is critical, highlighting the distinct influence of pulse ground motion on structural behaviour. These findings underscore the importance of considering the ground motion type evaluation of bridge pier structures.

The peak shear stress response of a tall bridge pier subjected to pulse ground motion is greater than that under non-pulse ground motion. The peak shear stresses are 0.3987 MPa and 0.2260 MPa, respectively, with an error of 76.42%. In particular, at the 5–10 s interval, the peak shear stress error of the two earthquakes reaches a maximum. The shear stress response time history curves of tall bridge piers subjected to pulse ground motion are more complicated than those of tall bridge piers under non-pulse ground motion. In the range of 5–10 s, the time history curves of the two earthquakes exhibit notable dissimilarities. This is because the pulse signal applied in this interval alters the spectral characteristics of the original non-pulse ground motion.

The analysis indicates that the displacement and shear stress responses of tall bridge piers are significantly greater when they are exposed to pulse ground motions compared to non-pulse motions. Additionally, the time history curves for displacement and shear stress exhibit greater complexity and variability when the structure is under pulse ground motion. These findings indicate the significance of considering pulse ground motion effects in the earthquake evaluation of tall bridge piers, as ignoring these effects may lead to inaccurate assessments of the structural performance.

To investigate the underlying mechanisms by which pulsed ground motion affects the behaviour of bridge piers, this study used the fast Fourier transform to transform the time histories of the responses of tall bridge piers. This allowed these time-domain signals to be converted to frequency-domain representations, providing a clearer understanding of the structural dynamics under pulsed earthquake excitation, as illustrated in Figure 5.

Figure 5 illustrates the responses of a tall bridge pier under pulse and non-pulse ground motions, specifically in terms of displacement and shear stress in the frequency domain. When the frequency is f1, the displacement and shear stress amplitudes for the tall bridge pier under pulse ground motion are significantly greater than those in the non-pulse scenario. Furthermore, the amplitude curves corresponding to displacement and shear stress under pulse ground motion exhibit greater complexity and variability, reflecting the more complicated frequency spectral characteristics of pulse ground motions. This increased complexity results in a significantly more dynamic and unpredictable influence on the structural response of the tall bridge pier. These findings emphasize the need for advanced analytical models to capture the various effects of pulse ground motions on the structural integrity of tall bridge piers. The first-order frequency obtained by the pulse ground motion is small, close to the frequency of the bridge pier, which is more likely to cause the resonance effect and make the damage of the structure more significant.

5.2. Varying Damping Ratios Affect the Dynamic Response

This study investigated the dynamic behaviour of a tall bridge pier under different damping ratios (3%, 5%, and 10%) when subjected to synthetic earthquake waves. The ground motion, which was characterized by a 0.45 s period and a peak acceleration of 0.20 g, was modelled in the Z direction. The principal aim was to assess the influence of the damping ratio on the bridge pier’s response to both pulse and non-pulse seismic actions. The displacement at 35 m in the Z direction and the shear stress at 10 m in the YZ section of the tall bridge pier were identified as the key parameters for analysis.

Figure 6 illustrates the time histories of the displacement and shear stress for tall bridge piers with different damping ratios (3%, 5%, and 10%) under non-pulsed ground motion. The peak displacement values for the three damping ratios are 0.1301 m, 0.1070 m, and 0.0732 m, respectively, showing a 77.73% maximum variation in displacement. The corresponding peak values for the peak shear stress are 0.2931 MPa, 0.2412 MPa, and 0.1800 MPa, respectively, with a maximum variation of 62.83%. As the damping ratio increases, the peak displacement and shear stress of the tall bridge pier decrease.

When the time is within the range of 1–18 s, the time histories are nearly identical with three damping ratios. When the time exceeds 18 s, the time histories exhibit discrepancies. The results demonstrated that the influence of non-pulse ground motion on displacement and shear stress is relatively minor in comparison with the influence of the damping ratio. These findings demonstrate the more significant role of the damping ratio in controlling the structural responses of tall bridge piers under loading conditions.

Figure 7 presents the time histories for the displacement and shear stress of a tall bridge pier with different damping ratios subjected to pulse ground motions. As shown in Figure 7, the peak displacement values for the three different damping ratios of the tall bridge pier are 0.1973 m, 0.1836 m, and 0.1566 m, respectively, with the maximum error in peak displacement being 25.99%. For the peak shear stress, the corresponding values are 0.4558 MPa, 0.4307 MPa, and 0.3806 MPa, resulting in a maximum error of 19.76%. These findings indicate that as the damping ratio increases, both the peak displacement and peak shear stress of the bridge pier decrease gradually under pulse ground motion. The difference in peak displacement and peak shear stress for pulse ground motion is much smaller than that under non-pulse motions. This is mainly due to the frequency effect, because if the main frequency of the pulse is far from the sensitive frequency band of the structure, the energy input efficiency is reduced, resulting in a small difference between the dynamic stress amplitude and the static stress. When the time exceeds 18 s, the displacement and shear stress time histories converge for the three damping ratios, showing no significant differences. This suggests that the effect of the damping ratio is small compared to that of pulse earthquakes, and that pulse ground motion exerts a more substantial influence on the dynamic behaviour of tall bridge piers.

In summary, pulse ground motion substantially influences the dynamic behaviour of tall bridge piers, resulting in significant variations in displacement and shear stress, whereas the effects of different damping ratios are less pronounced over time. The results highlight the dominant role of pulse ground motion in shaping the structural response, with diminishing effects on the damping ratios.

5.3. 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/s² (Z direction), 1.6 m/s² (X direction), and 1.2 m/s² (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 piers 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.

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 piers 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 piers, 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 piers.

This study examines the dynamic features of a tall bridge pier under multidimensional pulse ground motion actions. To evaluate the different damping ratios, the variations in the peak displacement of the Z section and the peak shear stress of the YZ section of a tall bridge pier along the height subjected to multidimensional pulse ground motion are analysed. Figure 10 illustrates the trend in the displacement peak value of the Z section of a tall bridge pier structure with different damping ratios under multidimensional pulse ground motions.

Table 2. Peak displacement values of the Z section of the tall bridge pier structure with varying damping ratios under multidimensional pulse ground motion.

Pier Heights (m)	Displacement (m)
	$\mathit{\xi }$ = 3%			$\mathit{\xi }$ = 5%			$\mathit{\xi }$ = 10%
	1D	2D	3D	1D	2D	3D	1D	2D	3D
5	0.0062	0.0062	0.0037	0.0058	0.0058	0.0058	0.0049	0.0049	0.0049
10	0.0229	0.0229	0.0139	0.0213	0.0212	0.0212	0.0182	0.0181	0.0181
15	0.0480	0.0479	0.0297	0.0446	0.0444	0.0444	0.0381	0.0379	0.0379
20	0.0795	0.0793	0.0501	0.0740	0.0736	0.0736	0.0631	0.0628	0.0628
25	0.1159	0.1155	0.0743	0.1078	0.1072	0.1071	0.0920	0.0915	0.0915
30	0.1556	0.1549	0.1012	0.1448	0.1438	0.1437	0.1235	0.1228	0.1228
35	0.1973	0.1962	0.1299	0.1836	0.1823	0.1822	0.1566	0.1557	0.1556
40	0.2421	0.2408	0.1608	0.2248	0.2234	0.2233	0.1915	0.1904	0.1904

illustrates the Z section peak displacement values of tall bridge piers with different damping ratios under multidimensional pulse ground motions. Figure 11 illustrates the trends of peak shear stress for the YZ section of a tall bridge pier structure with different damping ratios under multidimensional pulse ground motions.

Table 3. Peak shear stress of the YZ section of the tall bridge pier structure with varying damping ratios under multidimensional pulse ground motion.

Pier Heights (m)	Shear Stress (MPa)
	$\mathit{\xi }$ = 3%			$\mathit{\xi }$ = 5%			$\mathit{\xi }$ = 10%
	1D	2D	3D	1D	2D	3D	1D	2D	3D
5	0.4928	0.4853	0.3200	0.4709	0.4635	0.4631	0.4198	0.4160	0.4158
10	0.4557	0.4458	0.2935	0.4307	0.4236	0.4233	0.3806	0.3759	0.3758
15	0.4240	0.4136	0.2761	0.3987	0.3901	0.3899	0.3484	0.3425	0.3424
20	0.3891	0.3830	0.2651	0.3659	0.3587	0.3587	0.3189	0.3121	0.3122
25	0.3608	0.3517	0.2666	0.3367	0.3295	0.3291	0.2887	0.2850	0.2849
30	0.3185	0.3118	0.2553	0.2970	0.2912	0.2907	0.2539	0.2496	0.2495
35	0.2698	0.2627	0.2280	0.2512	0.2455	0.2455	0.2135	0.2099	0.2098
39.75	5.2428	5.2090	4.8242	5.0922	5.0050	4.8623	4.6830	4.1019	3.8914

illustrates the YZ section peak shear stress of the tall bridge pier structure with different damping ratios under multidimensional pulse ground motions.

The analysis is shown in Figure 10, and Table 2 demonstrates that the peak displacement at the Z section of a tall bridge pier increases as the bridge pier height increases under multidimensional pulse ground motions. When the damping ratios are set to 5% and 10%, the peak displacements for three-dimensional ground motions are nearly identical, indicating that the structure behaves similarly under these conditions. However, at a lower damping ratio of 3%, the peak displacement under three-dimensional ground motion is lower than that under one-dimensional or two-dimensional ground motions. These findings indicate that the damping ratio of a tall bridge structure has a notable effect on its seismic response to multidimensional ground motion actions. Furthermore, the interactions between the different directional components of the ground motion also significantly affect the displacement behaviour.

As shown in Figure 11 and Table 3, the YZ section value of the peak shear stress of a tall bridge pier with different damping ratios under three-dimensional ground motion demonstrates a decrease with increasing bridge pier height. However, there is a sudden change point at the top, where the value abruptly increases. For damping ratios of 5% and 10%, the YZ section peak shear stress values of a tall bridge pier under three-dimensional ground motions are similar. When the damping ratio is 3%, the peak value of the YZ section shear stress of the tall bridge pier subjected to three-dimensional ground motions is less than that of one-dimensional and two-dimensional ground motions, indicating that the different damping ratios affect the input law for multidimensional ground motion of the tall bridge pier.

A comparison of Figure 10 and Figure 11 demonstrates that the multidimensional ground motions possess a negligible effect on the displacement of the Z section and the shear stress of the YZ section of the tall bridge pier under different damping ratios. This is because when the tall bridge pier is under multidimensional ground motion actions, the three directions of the earthquake interact with each other, which reduces both the displacement of the Z section and the shear stress of the YZ section. Under three-dimensional seismic input, the maximum peak acceleration in the Z direction (PGA = 2 m/s²) plays a predominant role in influencing the displacement in the Z-direction and shear stress in the YZ section of a tall bridge pier. The peak displacement values and time histories of the Z section, along with the shear stress responses in the YZ section, exhibit close trends, highlighting the substantial effect of the Z-direction ground motion on these structural parameters. In contrast, the accelerations in the X and Y directions have negligible influences on both the Z-direction displacement and the YZ section shear stress. This suggests that seismic design for tall bridge piers should prioritize considerations related to Z-direction accelerations for accurate performance evaluation of the tall bridge.

To gain further insight into the effect of multidimensional ground motion on the dynamic behaviour of tall bridge piers, the dynamic behaviour of such structures under different dimensions of pulse ground motion is studied. This is achieved through one-dimensional (Z direction), two-dimensional (X and Z directions), and three-dimensional (XYZ directions) analyses. When considering tall bridge piers with different damping ratios, the displacement of the X-direction section at a height of 20 m and the shear stress of the XY direction section at identical points are selected as the objects of examination. To illustrate, the time history curves of the tall bridge pier were generated using a damping ratio of 5%.

Figure 12 presents the displacement and shear stress responses of a tall bridge pier’s X section and XY section under multidimensional pulse ground motions, and a 5% damping ratio was selected. The peak displacements for the X section subjected to 1D, 2D, and 3D ground motions are 2.4497 × 10⁻¹² m, 0.0416 m, and 0.0416 m, respectively. The corresponding peak shear stresses at the XY section are 1.5259 × 10⁻¹¹ MPa, 0.2470 MPa, and 0.2468 MPa. These results highlight the significant increase in both peak displacement and shear stress in the XY section under multidimensional ground motions, and emphasize the need to consider multidimensional seismic effects in the service and analysis of tall bridge piers.

Figure 13 and Figure 14 show the variation trends of the X section displacement peak and XY section shear stress peak of the tall bridge pier structure with different damping ratios under multidimensional pulse ground motions, respectively.

Table 4. X section peak displacement values of tall bridge piers with different damping ratios under multidimensional pulse ground motion.

Pier Heights (m)	Displacement (m)
	$\mathit{\xi }$ = 3%			$\mathit{\xi }$ = 5%			$\mathit{\xi }$ = 10%
	1D	2D	3D	1D	2D	3D	1D	2D	3D
5	2.3783 × 10−13	0.0036	0.0036	1.8606 × 10−13	0.0032	0.0032	1.1612 × 10−13	0.0032	0.0032
10	8.8947 × 10−13	0.0135	0.0135	6.9469 × 10−13	0.0117	0.0117	4.9610 × 10−13	0.0119	0.0118
15	1.8826 × 10−12	0.0285	0.0284	1.4681 × 10−12	0.0248	0.0248	1.0479 × 10−12	0.0250	0.0250
20	3.1468 × 10−12	0.0473	0.0472	2.4497 × 10−12	0.0416	0.0416	1.7494 × 10−12	0.0419	0.0419
25	4.6151 × 10−12	0.0691	0.0690	3.5862 × 10−12	0.0612	0.0612	2.5625 × 10−12	0.0615	0.0615
30	6.2258 × 10−12	0.0931	0.0929	4.8297 × 10−12	0.0828	0.0828	3.4522 × 10−12	0.0831	0.0831
35	7.9231 × 10−12	0.1186	0.1183	6.1375 × 10−12	0.1057	0.1056	4.3875 × 10−12	0.1059	0.1059
40	9.6923 × 10−12	0.1454	0.1451	7.4841 × 10−12	0.1301	0.1301	5.3736 × 10−12	0.1300	0.1300

gives the X section peak displacement values of tall bridge pier structures with different damping ratios under multidimensional pulse ground motions.

Table 5. Peak shear stress of the XY section of tall bridge pier with varying damping ratios under multidimensional pulse ground motion.

Pier Heights (m)	Shear Stress (MPa)
	$\mathit{\xi }$ = 3%			$\mathit{\xi }$ = 5%			$\mathit{\xi }$ = 10%
	1D	2D	3D	1D	2D	3D	1D	2D	3D
5	3.0518 × 10−11	0.3232	0.3225	2.2880 × 10−11	0.3162	0.3153	1.5259 × 10−11	0.3054	0.3050
10	3.0518 × 10−11	0.2956	0.2952	1.5259 × 10−11	0.2908	0.2900	1.1444 × 10−11	0.2826	0.2821
15	1.9074 × 10−11	0.2751	0.2747	1.5259 × 10−11	0.2672	0.2667	1.5259 × 10−11	0.2624	0.2620
20	1.9074 × 10−11	0.2499	0.2496	1.5259 × 10−11	0.2470	0.2468	1.1444 × 10−11	0.2412	0.2412
25	1.7166 × 10−11	0.2311	0.2310	1.5259 × 10−11	0.2243	0.2240	1.5259 × 10−11	0.2178	0.2176
30	1.5259 × 10−11	0.2232	0.2231	1.5259 × 10−11	0.1978	0.1975	9.5367 × 10−12	0.1917	0.1914
35	1.4305 × 10−11	0.2017	0.2009	1.3351 × 10−11	0.1726	0.1722	9.5367 × 10−12	0.1601	0.1599
39.75	7.8125 × 10−9	4.6484	4.6849	7.8125 × 10−9	4.4148	4.1414	7.8125 × 10−9	3.3451	2.8067

shows the XY section peak shear stress of a tall bridge pier structure with different damping ratios under multidimensional pulse ground motions.

Figure 13 and Table 4 show that the peak displacement at the X section of tall bridge piers increases with pier height under three-dimensional ground motion. In addition, the displacement under two- and three-dimensional seismic action is significantly greater than that under one-dimensional motion action, emphasizing the need to consider multidirectional seismic effects in structural evaluations. In this study, torsional or biaxial bending effects in three-dimensional motion are not considered. According to the seismic design code, the load applied in the main load direction is large, and the structural symmetry makes the coupling effect negligible.

As illustrated in Figure 14 and Table 5, the peak shear stress of the XY section of a tall bridge pier structure with different damping ratios under three-dimensional ground motion decreases as the tall bridge pier height increases. However, a sudden point is observed at the top of the tall bridge pier. The peak shear stress of the XY section of a tall bridge pier structure with different damping ratios under two-dimensional and three-dimensional earthquake action is markedly greater than that under one-dimensional earthquake action. This suggests that the effects of two-dimensional and three-dimensional earthquakes must be considered for the shear stress response of the XY section of a tall bridge pier.

6. Conclusions and Discussion

This study employed EQSignal software to synthesize three ground motions, each with unique characteristic periods, to assess their effects on the displacement and shear stress behaviour of tall bridge piers subjected to multidimensional seismic loading. This research specifically considered the influence of damping ratios, pulse ground motion characteristics, and multidimensional seismic actions. The analysis focused on the factors that interact to affect the structural behaviour of the bridge pier. The results revealed several key findings regarding the combined impact of the variables on the displacement and shear stress distributions within the pier:

(1)

Compared with those under non-pulse ground motion, the displacement and shear stress behaviour of a tall bridge pier under pulse ground motion are significantly greater. In the time range of 0–5 s, the displacement and shear stress time histories exhibit more intricate variations when the ground motion is in the form of pulse ground motion. This is particularly evident in the case of tall bridge piers.

(2)

When under pulse ground motion, the peak displacement and shear stress errors of tall bridge pier structures, incorporating three different damping ratios, are notably smaller than those under non-pulse ground motion conditions. When the time t on the displacement and shear stress time history curves exceeds 18 s, the influence of pulse ground motion on the variations in the curves becomes more significant than the impact of the damping ratio itself. In contrast, for non-pulse ground motion, the influence of the damping ratio on the changes in the displacement and shear stress time histories is more pronounced than that of the ground motion.

(3)

When subjected to two-dimensional and three-dimensional non-pulse ground motions, the displacement at the Z section and the shear stress at the YZ section of a tall bridge pier are found to be smaller than those under one-dimensional ground motion. When the tall bridge pier is under the influence of two-dimensional and three-dimensional pulse ground motion, both the peak displacement at the Z section and the peak shear stress at the YZ section of the bridge pier are reduced compared with the response under one-dimensional motion. However, when the tall bridge pier is subjected to two-dimensional and three-dimensional pulse ground motion, the displacement at the X section and the shear stress at the XY section are considerably greater than the corresponding response under one-dimensional ground motion.

The present study primarily examines the impact of different damping ratios on the responses of tall bridge piers under non-pulse and pulse ground motions of varying dimensions. The synthetic pulse ground motion used in this paper does not fully reflect the directionality and spatial variability of the actual near-fault ground motion. In the future, the multidirectional velocity pulse ground motion with more representative spectral characteristics should be used to study the seismic performance of abridge, and a regional pulse spectrum prediction model should be constructed. The damping used in this paper may not fully reflect the contribution of higher-order modes. Combined with machine learning, an adaptive damping system is designed to dynamically adjust the influence of balanced damping on the seismic performance of the structure.