Study on the Hydrodynamic Effects of Bridge Piers Under Velocity-Type Pulse Ground Motion Based on Different Characteristic Periods

Abstract

This study was based on a target spectrum (GB183062015) and synthesized different characteristic periods, pulse ground motions, and non-pulse ground motions utilizing EQsignal v1.2.1 software. It also investigated the dynamic behaviors of bridge piers in seismic motions with varying characteristic periods, pulse and non-pulse effects, and the influence of 0 m and 10 m water depths. The findings indicated that the peak acceleration and stress behaviors vary significantly under different characteristic periods of ground motion. The maximum error in peak acceleration behavior of a bridge pier under ground motions of varying characteristic periods is 19.25%, while the maximum error in peak stress response is 11.35%. The acceleration and stress behaviors of a bridge pier under pulse ground motion action are more considerable than those under non-pulse seismic motion action. When the characteristic period is 0.40 s, the maximum error in peak acceleration of the bridge pier structure under pulse seismic motion and non-pulse seismic motion action is 86.08%, with the maximum error of the peak stress reaches 80.68%. The existence of water serves to minimize the natural frequency of the bridge pier. The pulse effects result in a maximum error of 40.49% for the peak acceleration and a maximum discrepancy of 323.08% for the peak stress of the bridge pier. The hydrodynamic effects result in a maximum error of 33.51% for the acceleration peak and 12.90% for the stress peak of the bridge pier. The effect of the pulse symptoms on the dynamic behavior of the bridge pier is considerably more pronounced than that of the hydrodynamic effects, with an intricate and complex influencing mechanism. In bridge flood protection and seismic design and optimization, it is essential to consider the impact of pulse seismic motion with varying characteristic periods.

1. Introduction

As a lifeline project, a bridge plays a significant role in disaster prevention and mitigation. Consequently, the seismic motion of bridges has consistently constituted a focal point and a subject of intense scrutiny among scholars. Nevertheless, the primary investigation is centered on the dynamic analysis of bridges under non-pulse seismic motion, specifically examining the influence of water on a bridge’s dynamic behaviors and characteristics, while not including the effect of some structural design aspects on the hydrodynamic effects [1]. Based on the finite element solution, a simplified hydrodynamic added mass model and time domain model were adopted to examine the hydrodynamic pressure and dynamic behaviors of cylinder structures or bridge piers with varying cross-section dimensions under non-pulse ground motion [2,3]. In examining the behaviors of piers under non-pulse seismic motion action, some effective and convenient methods are employed, including the simplified hydrodynamic pressure expression [4], effective calculation methods [5,6], and so forth. Chen et al. [7] studied the seismic presentation of sea-crossing bridges and found that the size of the pier structure has a striking on the hydrodynamic force of the bridge. Kouhdasti et al. [8] integrated the hydrodynamic effect caused by earthquakes into the new evaluation method and studied the influence of strong coupling caused by hydrodynamic pressure and structural flexibility changes on the seismic performance of the structure. Zhang et al. [9] proposed a correction approach for the calculation of added mass based on the dress of the added mass coefficient, and studied the influence of the added mass of a bridge structure on the hydrodynamics. Zhang et al. [10] examined the effects of hydrodynamic forces on the dynamic behavior of deep-water bridge piers by using displacement and internal force influence coefficients. Their findings verified that the coefficients accurately predict structural behavior, offering valuable tools for assessing the property of water flow on bridge pier stability in dynamic conditions.

In comparison to non-pulse seismic waves, pulse ground motion causes more serious damage to a structure. Consequently, a greater number of scholars have examined the dynamic response of pulse seismic motion action to structures. For the influence of pulse ground motions, some structures may require measures such as strengthening steel bars to control displacement and other responses [11]. In light of the corroboration of the impact of pulse seismic motion through observational evidence, it is imperative to accord due consideration to the influence of pulse seismic motion when evaluating the seismic resilience of structures [12]. The characteristic period affects the magnitude, acceleration, velocity, and other parameters of the pulse seismic motion. The striking of the characteristic period on the structural behavior is discrete [13]. Brown et al. [14] employed a test to examine the determinant of the near-fault seismic motion on the columns and piers of a bridge. The findings indicated that the near-fault ground motion accumulated the strain, curvature, and drift ratio of the bridge. Zhong et al. [15] examined the risk of using simply supported beam bridges under pulse earthquakes. It was established that the directional pulse effect poses a greater risk of damage to the bridge structure. Chen et al. [16] employed both a test and numerical simulation to examine the seismic behavior of high-pier bridges under pulse ground motion action. Their findings indicated that the characteristics of pulse ground motion have an evidential consequence on the seismic presentation of high-pier bridges. Jia et al. [17] investigated the dynamic behavior of cable-stayed bridges under pulse seismic excitation. The accuracy of their finite element model was confirmed through experimental validation. The study determined that near-fault pulse ground motion would result in an increase in the peak values of the bridge’s dynamic behaviors and an increase in the requirement for damping force. Xu et al. [18] designed a tall-pier continuous rigid-frame bridge scale test model, and employed the test to examine the determinant of pulse parameters on the seismic behavior of the model bridge. Jia et al. [19] using the ANSYS software platform to form a bridge finite element model, and studied the influence of pulsation effect characteristics on the consequence of sea-crossing bridges. Jia et al. [20] carried out multi-directional pulse seismic motion on cable-stayed bridges, and found that pulse ground motion is more sensitive to the direction of the ground motion than non-pulse ground motion, while the seismic force generated is more apparent. Shi et al. [21] adapted near-fault pulse ground motion to examine the influence of a three-dimensional input on the non-linear time history and the seismic wreck and crack evolution of the girder.

The hydrodynamic behavior of a bridge pier under pulse ground motion has yet to be investigated in water environments. This study utilized the ground motion signal processing software EQSignal [22,23,24] to synthesize three non-pulse seismic motions and pulse seismic motions with varying characteristic periods (Tg = 0.45 s; Tg = 0.40 s; Tg = 0.30 s). The dynamic behavior of a bridge pier was investigated under two varying water levels (0 m and 10 m) and studied using non-pulse seismic waves and pulse seismic waves. It was indicated that water had a striking effect on the acceleration and stress behavior of the bridge pier. The outcome of the pulse seismic motion on the acceleration and stress of the bridge pier was more pronounced than that of non-pulse seismic waves. Furthermore, the influence of the pulse seismic motion on the dynamic behavior of the bridge pier was highly complex. The water influenced the action law of pulse seismic motion but the overall dynamic behavior of pulse seismic motion was considerably more significant than the hydrodynamic effects associated with the water itself. In this paper, the velocity-type pulse seismic motion, different characteristic periods, the joint effect of the bridge pier and water, and the mechanism of interaction between the pulse effect and hydrodynamic effect are discussed in depth. The result is a compounded effect used to optimize the earthquake design of bridges and improve the earthquake disaster prevention ability.

2. Hydrodynamic Added Mass Theory

The Morison equation [25] is the fundamental theory for evaluating the added mass of hydrodynamics. In a seismic event, it is often necessary to disregard the impact of the cylindrical mass on the surrounding water body. Consequently, this equation is particularly suitable for small-diameter column structures.

Here, we modify, simplify, and linearize the Morison equation and obtain the following equation [26,27]:

$$f_e=-{\displaystyle \frac{1}{2}}{C}_{D\mathrm{e}}\rho d\sqrt{{\displaystyle \frac{8}{\pi }}}{\sigma }_{v_e}{v}_e-\left(C_{Me}-1\right)\rho {\displaystyle \frac{\pi d^2}{4}}{a}_e$$

(1)

where $C_{D\mathrm{e}}$ denotes the coefficient of friction; $C_{Me}$ denotes the inertia coefficient; $\rho$ represents the water density; $d$ represents the structure diameters; $v_e$ and $a_e$ the absolute velocity and acceleration of the structure, respectively.

Here, $M_{we}=\left(C_{Me}-1\right)\rho {\displaystyle \frac{\pi d^2}{4}}$ is called the hydrodynamic added mass.

If the cross-section of the bridge pier is rectangular, it is necessary to modify the expression of the hydrodynamic added mass [28], whereby the correction coefficient is obtained [29,30,31]:

$$K_{ce}=1.51{\left(d/b\right)}^{-0.17}$$

(2)

where $d$ denotes the length of the rectangle perpendicular to the seismic wave direction; $b$ denotes the length of the rectangle parallel to the direction of the seismic wave.

According to European norms [32], the hydrodynamic mass in the unit of a circular pier with a diameter of $d$ is:

$$m_{we}=\rho {\displaystyle \frac{\pi d^2}{4}}$$

(3)

The hydrodynamic added mass in the unit of the rectangular section pier is:

$$m_e=K_{\mathrm{ce}}\rho {\displaystyle \frac{\pi d^2}{4}}$$

(4)

The expression of the hydrodynamic effect is:

$$E_f={\displaystyle \frac{D_{water}-D_{nowater}}{D_{nowater}}}\times 100\%$$

(5)

where $D_{water}$ denotes the dynamic behavior in water environments; $D_{nowater}$ denotes the dynamic behavior without water environments.

The expression of the pulse effect is:

$$E_p={\displaystyle \frac{D_p-D_{np}}{D_{np}}}\times 100\%$$

(6)

Here, $D_p$ denotes the dynamic behavior of the bridge under pulse seismic motion action; $D_{np}$ denotes the dynamic behavior of the bridge under non-pulse ground seismic action.

3. Pier–Water Finite Element Model

The material of the bridge pier is C45 concrete. The length, width, and height of the pier section are 1.2 m × 1.2 m × 12 m; the water depth is 10 m; the elastic modulus is 3.35 × 10¹⁰ Pa; the density is 2600 kg/m³; the Poisson’s ratio is 0.195; and the concentrated added mass at the top of the bridge pier is 5 × 10⁵ kg [33,34,35,36]. We utilized ANSYSR21 for the finite element modeling and analysis, while the SOLIDE45 element was utilized to simulate the bridge pier and the MASS21 element was used to simulate the added mass. The hydrodynamic added mass method is still an accurate approach for calculating the causation of water on a structure [37]. The water application method reference [36] was also used, while the pier–water finite element model was established as shown in Figure 1, whereby the number of nodes was 20,449 and the number of elements was 17,280.

For the finite element model of the bridge pier in Figure 1, a modal analysis was carried out. In the anhydrous environment (d = 0 m), the first-order frequency of the bridge pier was 0.53604 Hz; in the water environment (d = 10 m), the first-order frequency of the bridge pier was 0.53451 Hz, indicating that the existence of water will reduce the frequency of the pier, thereby increasing the period of the bridge pier.

4. Select Ground Motion

In this study, the ground motion synthesis software EQsignal was employed to synthesize three non-pulsed seismic waves, targeting the GB183062015 [22,23,24]. The time interval was 0.01 s, with a total duration of 30 s.

Figure 2 shows the acceleration time histories of non-pulsed seismic waves and pulse seismic waves. Figure 3 shows the response spectrum curves of non-pulsed seismic motion and pulse seismic motion with varying characteristic periods.

From Figure 2 and Figure 3, it can be observed that the acceleration time histories of the different characteristic periods (Tg = 0.45 s; Tg = 0.40 s; Tg = 0.30 s) are notably different, and the response spectrum curves are also quite different.

5. Response of Pulse Seismic Waves to Bridge Piers

5.1. The Responses of Earthquakes with Different Characteristic Periods to Bridge Piers

The non-pulse seismic motion was adopted for an examination. The acceleration at a height of 8 m and the stress at a height of 5 m of the bridge pier were selected as the research objects. All calculations pertaining to errors and hydrodynamic effects were calculated according to Formula (5). For example, when the structure acts on the earthquake, the dynamic behavior peak value in the anhydrous environment is A, the dynamic behavior peak value in the water environment is B, and the hydrodynamic effect is (B − A) × 100%/A.

Figure 4 shows the acceleration and stress time histories of bridge piers subjected to varying characteristic period ground motions in an anhydrous environment. From Figure 4, the peak accelerations of the bridge piers under the three characteristic period ground motions actions are 0.7739 m/s², 0.8798 m/s², and 0.9229 m/s², respectively. When Tg = 0.30 s, the peak acceleration value is the greatest. When Tg = 0.45 s, the peak acceleration value is the lowest, and the error between the two values is 19.25%. The peak stress values of the bridge pier under the three characteristic period seismic motions are 8.3838 MPa, 8.4858 MPa, and 9.9335 MPa, respectively. When Tg = 0.30 s, the stress peak is the largest, while when Tg = 0.45 s, the stress peak is the smallest, and the error between the two reaches 11.35%. The results demonstrate that the acceleration and stress time histories of the bridge piers under different characteristic periods of seismic action are extremely complicated, with significant differences in both the peak acceleration and the peak stress. As can be observed from the diagram, there are significant discrepancies between the acceleration time histories of the various characteristic periods, as well as between the stress time history curves. As the characteristic period decreases, both the peak acceleration and peak stress behavior increase.

5.2. Response of a Bridge Pier Under Pulse Ground Motion

The acceleration at the bridge pier height of 10 m and the stress at 3 m were taken as the investigation items. Figure 5 illustrates the acceleration and displacement time histories of the bridge pier under various characteristic periods of pulse earthquake and non-pulse earthquake actions. As can be observed in Figure 5, the acceleration time histories of bridge piers subjected to varying characteristic period seismic actions are extremely different, and the displacement response time histories are also extremely different. When subjected to identical characteristic periods, the bridge pier’s acceleration and displacement responses differ markedly between pulse and non-pulse motions. Pulse-type ground motions produce greater acceleration and displacement responses compared to non-pulse motions. This disparity highlights the substantial symptom of the motion type on the dynamic behavior of the structure, with pulse motions inducing higher peak forces and displacements. Such findings stress the paramount need to adopt motion characteristics in seismic events to ensure the resilience of infrastructure, particularly in regions prone to pulse-like ground motions.

When Tg = 0.45 s, the peak acceleration values of the bridge pier under non-pulse seismic motion and pulse ground motion are 1.0694 m/s² and 1.3252 m/s², respectively, and the discrepancy between the two earthquake conditions is 23.92%. The peak stress values are 10.7879 MPa and 20.2414 MPa, respectively, with an error of 87.63%. When Tg = 0.40 s, the peak acceleration values of the bridge pier under non-pulse seismic motion and pulse ground motion are 1.1844 m/s² and 1.4778 m/s², respectively, with an error of 24.77%. The peak stress values are 10.8770 MPa and 19.6423 MPa, respectively, and the error is 80.59%. When Tg = 0.30 s, the peak acceleration values of the bridge pier in response to non-pulse seismic motion and pulse ground motion are 1.1566 m/s² and 1.4228 m/s², respectively, and the discrepancy between the two conditions is 23.02%. The peak stress values are 12.7339 MPa and 20.0871 MPa, respectively, with an error of 57.75%.

Among the three characteristic periods, the errors in the peak acceleration and peak stress of a bridge pier are different under pulse and non-pulse seismic motions. The largest peak stress error occurs at Tg = 0.45 s, while the highest peak acceleration error is observed at Tg = 0.40 s. The smallest errors for both peak acceleration and peak stress are found at Tg = 0.30 s. This indicates that under varying characteristic periods of ground motion, the non-pulse seismic motion and pulse ground motion have a considerable effect on the acceleration and stress responses of the bridge piers. This is mainly because different characteristic periods are associated with distinct ground motion spectrum characteristics.

To substantiate the conclusion with greater clarity, a fast Fourier transform was applied to analyze the acceleration and stress time histories presented in Figure 5 and to obtain the frequency domain representation of the acceleration and the stress response time histories, as illustrated in Figure 6. From Figure 6, when the period of the seismic motion Tg = 0.45 s, the peak acceleration amplitudes of the bridge pier under non-pulse seismic motion actions and pulse ground motion actions are 0.1915 m/s² and 0.2572 m/s², respectively, showing a discrepancy of 34.31%. The peak values of stress are 3.8756 MPa and 4.9359 MPa, respectively, showing a discrepancy of 27.36%. When the period of the seismic motion Tg = 0.40 s, the peak acceleration values of the bridge pier under non-pulse seismic motion and pulse seismic motion are 0.2701 m/s² and 0.5026 m/s², respectively, showing a discrepancy of 86.08%. The peak stress values are 5.6364 MPa and 10.1839 MPa, respectively, with an error of 80.68%. When the period of seismic motion Tg = 0.30 s, the peak acceleration values of the bridge pier in response to non-pulse seismic motion and pulse ground motion are 0.2896 m/s² and 0.4300 m/s², respectively, with an error of 48.48%. The peak values of the stress amplitude are 5.9665 MPa and 8.6089 MPa, respectively, with an error of 44.29%.

The acceleration and stress response of the first-order frequency f1 under pulse seismic motion and non-pulse seismic motion actions are identical. In the frequency domain diagram of the stress response, the stress amplitude curve obtained under pulse seismic motion action exhibits notable fluctuations within the range of 0–0.5 Hz. The acceleration and stress amplitudes of the bridge pier under pulse ground motion action are higher than those under non-pulse seismic motion. This demonstrates that the acceleration and displacement of the bridge pier under pulse ground motion action are significantly larger than those under non-pulse seismic motion action. Additionally, the spectrum characteristics of pulse seismic waves are extremely complicated.

5.3. Influence of Pulse Ground Motion on the Hydrodynamic Effect of Bridge Piers

This study investigates the effects of water levels on the dynamic behavior of bridge piers, focusing on acceleration at a 5 m height and stress at a 1 m height. By comparing water depths of 0 m and 10 m, the analysis reveals that the existence of water, especially at higher levels, generally amplifies the dynamic behavior of the pier under non-pulse seismic motion. This consequence is primarily attributed to the low height-to-size ratio of the bridge piers [36]. However, various factors influence the dynamic behavior, including the spectral characteristics of the seismic motion [38], the dimensions of the bridge, and the water level. Higher water levels are found to increase the forces acting on the bridge, altering the natural frequency and damping properties. Consequently, a thorough understanding of these dynamic interactions is crucial for assessing the seismic resilience of bridges, particularly in areas with fluctuating water levels [39].

Figure 7 illustrates the acceleration and stress behavior time histories of bridge piers at different water levels under non-pulse ground motions with distinct characteristic periods. Figure 8 shows the acceleration and stress behavior time histories of bridge piers with varying water levels subjected to pulse ground motion with distinct characteristic periods.

From Figure 7, showing that when the bridge pier is under non-pulse seismic motion action, when Tg = 0.45 s, the peak acceleration rates of the bridge pier in the anhydrous environment and the water environment are 0.4629 m/s² and 0.5129 m/s², respectively. The hydrodynamic effect of acceleration is 10.80%; the peak stress values are 13.2089 MPa and 12.9822 MPa, and the hydrodynamic effect of stress is −1.72%.

When Tg = 0.40 s, the peak acceleration values of the pier structure in an anhydrous environment and the water environment are 0.4941 m/s² and 0.5126 m/s², respectively, and the hydrodynamic effect of the acceleration is 3.74%. The peak stress values are 13.3561 MPa and 13.3728 MPa, respectively, and the hydrodynamic effect of stress is 0.13%. When Tg = 0.30 s, the peak acceleration values of the pier structure in the anhydrous environment and the water environment are 0.5903 m/s² and 0.5612 m/s², respectively, and the hydrodynamic effect of the acceleration is −4.93%. The peak stresses are 15.5085 MPa and 15.0731 MPa, respectively, with a corresponding hydrodynamic effect of the stress of −2.81%.

From Figure 8, where the bridge pier is under pulse ground motion action, when Tg = 0.45 s, the peak acceleration values of the bridge pier in an anhydrous environment and a water environment are 0.5422 m/s² and 0.6074 m/s², respectively. The hydrodynamic effect of acceleration is 12.03%; the peak stress values are 21.2677 MPa and 21.3105 MPa, and the hydrodynamic effect of stress is 0.2%. When Tg = 0.40 s, the peak acceleration values of the bridge pier structure in the anhydrous environment and the water environment are 0.6392 m/s² and 0.6616 m/s², respectively, and the hydrodynamic effect of the acceleration is 3.5%. The peak stress values are 21.1582 MPa and 21.1707 MPa, respectively, and the hydrodynamic effect of stress is 0.06%. When Tg = 0.30 s, the peak acceleration values of the bridge pier in the anhydrous environment and the water environment are 0.6951 m/s² and 0.6358 m/s², respectively, and the hydrodynamic effect of the acceleration is, thus, calculated to be −8.53%. The peak stress values are 21.2333 MPa and 21.2878 MPa, respectively, with a hydrodynamic effect of stress of 0.26%.

From Figure 7 and Figure 8, the water has a relatively minor effect on the acceleration and displacement time histories. However, the water has an achiever causation magnitude effect on the values, irrespective of whether the bridge pier is under non-pulse seismic motion action or pulse ground motion action. When the bridge pier structure is under non-pulse ground motion action conditions, with a characteristic period of Tg = 0.45 s, the existence of water leads to a reduction in the stress response. When the characteristic period Tg = 0.30 s, the existence of water will reduce the acceleration and stress behavior. When the bridge pier structure is under pulse seismic motion action, when the characteristic period Tg = 0.30 s, the existence of water leads augments the acceleration behavior. In other conditions, the existence of water intensifies the acceleration and stress behavior.

To investigate the hydrodynamic effect of the bridge pier more clearly, the acceleration peak trend diagrams for varying water levels under non-pulse ground motion were obtained and are presented in Figure 9 and Figure 10; the peak acceleration and peak stress for different water levels under non-pulse ground motion are shown in

Table 1. Acceleration peak values of different water levels under non-pulse ground motions.

Pier Heights (m)	Acceleration (m/s2)
	Tg = 0.45 s			Tg = 0.40 s			Tg = 0.30 s
	0 m	10 m	Error (%)	0 m	10 m	Error (%)	0 m	10 m	Error (%)
1	0.0376	0.0502	33.51	0.0395	0.0465	17.72	0.0514	0.0549	6.81
3	0.2280	0.2654	16.40	0.2324	0.2529	8.82	0.2897	0.2973	2.62
5	0.4629	0.5126	10.74	0.4941	0.5126	3.74	0.5903	0.5612	−4.93
6	0.5652	0.6318	11.78	0.6178	0.6362	2.98	0.7239	0.7053	−2.57
8	0.7739	0.8428	8.90	0.8798	0.8426	−4.23	0.9229	0.9394	1.79
10	1.0694	1.0465	−2.14	1.1844	1.1626	−1.84	1.1566	1.1863	2.57
12	1.3977	1.3363	−4.39	1.4725	1.4711	−0.10	1.5263	1.5194	−0.45

and

Table 2. Stress peak values of different water levels under non-pulse ground motions.

Pier Heights (m)	Stress (MPa)
	Tg = 0.45 s			Tg = 0.40 s			Tg = 0.30 s
	0 m	10 m	Error (%)	0 m	10 m	Error (%)	0 m	10 m	Error (%)
1	13.2089	12.9822	−1.72	13.3561	13.3728	0.13	15.5058	15.0731	−2.79
3	10.7879	10.6136	−1.62	10.877	10.8807	0.03	12.7339	12.3408	−3.09
5	8.3838	8.2299	−1.84	8.4858	8.4699	−0.19	9.9335	9.6055	−3.30
6	7.1876	7.0582	−1.80	7.2298	7.2108	−0.26	8.4513	8.1749	−3.27
8	4.7929	4.6996	−1.95	4.8314	4.8205	−0.23	5.7057	5.4977	−3.65
10	2.4628	2.4125	−2.04	2.4842	2.4679	−0.66	2.9353	2.8225	−3.84
12	0.0083	0.0078	−6.02	0.0086	0.0086	0.00	0.0155	0.0135	−12.90

.

From Figure 9 and Table 1, the bridge pier is subjected to non-pulse ground motion action under three different characteristic period conditions; irrespective of whether the water level is anhydrous or normal, the peak acceleration increases with the increase in bridge pier height. For the characteristic periods of Tg = 0.45 s and Tg = 0.40 s, the peak acceleration behavior obtained at the top of the bridge pier is greater for the anhydrous water depth than for the normal water level. When the characteristic period Tg = 0.30 s, the peak acceleration response obtained under anhydrous water is greater than the normal water level at any position of the bridge pier. The acceleration hydrodynamic effect demonstrates a declining trend with an increase in bridge pier height, indicating that the striking of water on the bottom of the bridge pier is extremely significant and warrants attention.

From Figure 10 and Table 2, it can be seen that the bridge pier is subjected to non-pulse ground motion action, under three different characteristic period conditions, despite of the water depth is o m or 10 m, the peak stress decreases along with the uptick of the height of the bridge pier, but under the three different characteristic period conditions, the presence of water reduces the stress behavior of the bridge pier structure.

Figure 11 and Figure 12 respectively illustrate the peak acceleration trend diagram and stress peak trend diagram of bridge piers at varying water levels in response to pulse ground motion action.

Table 3. Acceleration peak values of different water levels under pulse ground motions.

Pier Heights (m)	Acceleration (m/s2)
	Tg = 0.45 s			Tg = 0.40 s			Tg = 0.30 s
	0 m	10 m	Error (%)	0 m	10 m	(%)	0 m	10 m	Error (%)
1	0.0438	0.0524	19.63	0.0511	0.0529	3.52	0.0581	0.0615	5.85
3	0.2600	0.2927	12.58	0.3096	0.3181	2.75	0.3414	0.3284	−3.81
5	0.5422	0.60744	12.03	0.6392	0.6616	3.50	0.6951	0.6358	−8.53
6	0.6839	0.759	10.98	0.8028	0.8428	4.98	0.8632	0.7959	−7.80
8	0.9755	1.0557	8.22	1.1471	1.1838	3.20	1.148	1.0779	−6.11
10	1.3252	1.3489	1.79	1.4777	1.489	0.76	1.4228	1.4488	1.83
12	1.724	1.7133	−0.62	1.8368	1.7945	−2.30	1.9187	1.9365	0.93

and

Table 4. Stress peak values of different water levels under non-pulse ground motions.

Pier Heights (m)	Stress (MPa)
	Tg = 0.45 s			Tg = 0.40 s			Tg = 0.30 s
	0 m	10 m	Error (%)	0 m	10 m	Error (%)	0 m	10 m	Error (%)
1	21.2677	21.3105	0.20	21.1582	21.1707	0.06	21.2333	21.2878	0.26
3	20.2414	20.208	−0.17	19.6423	19.5614	−0.41	20.0871	20.1301	0.21
5	16.5325	16.472	−0.37	15.8153	15.693	−0.77	16.3772	16.4029	0.16
6	14.3072	14.254	−0.37	13.664	13.5505	−0.83	14.1611	14.1746	0.10
8	9.5704	9.5094	−0.64	9.1408	9.0422	−1.08	9.5256	9.5153	−0.11
10	4.7636	4.7294	−0.72	4.5588	4.5036	−1.21	4.7181	4.7062	−0.25
12	0.0344	0.0330	−4.07	0.0310	0.0270	−12.9	0.0280	0.0310	10.71

respectively present the peak acceleration and peak stress values of bridge piers with distinct water levels under pulse ground motion action.

Figure 11 and Table 3 illustrate that when a bridge pier is under pulse seismic motion, three distinct characteristic periods, and varying water levels, the acceleration peak as the height of the bridge pier rises and the peak acceleration is enhanced. When the characteristic periods are Tg = 0.45 s and Tg = 0.40 s, the existence of water generally amplifies the acceleration behavior of the bridge pier, except for the bridge pier top. When the characteristic period Tg = 0.30 s, the presence of water may increase the acceleration behavior of the bridge pier and may also reduce the acceleration behavior of the bridge pier structure.

From Figure 12 and Table 4, the peak stress experienced by a bridge pier subjected to pulse ground motion decreases with increasing bridge pier heights under three distinct characteristic periods and water levels. The presence of the water exerts a negligible influence on the stress response.

To examine the relationship between pulse ground motion and the hydrodynamic effect in more detail, according to Formula (6), Figure 13 and Figure 14 present a trend of acceleration and stress pulse effects for different water levels under varying characteristic periods of ground motion.

As shown in Figure 13, the existence of the pulse enhances the acceleration response of the bridge pier. For a characteristic period of Tg = 0.45 s, the pulse effect intensifies as the height of the bridge pier rises. Conversely, for Tg = 0.40 s, the acceleration pulse effect initially increases with height but then begins to decrease as the pier’s height continues to rise. In the case of Tg = 0.30 s, the acceleration pulse effect consistently increases with the bridge pier’s height. Additionally, the data for a water level of 0 m reveal a stronger acceleration pulse effect compared to that observed at a 10 m water depth, suggesting that the existence of water at higher levels mitigates the consequence of the acceleration pulse effect.

It can be seen from Figure 14 that in three distinct characteristic periods and with varying water levels, the stress pulse effect exhibits an increasing trend as the height of the bridge pier rises. When the characteristic periods are Tg = 0.45 s and Tg = 0.40 s, the trends of the stress pulse effect are almost identical under both the 0 m and 10 m water depths. When the characteristic period Tg = 0.30 s, the stress pulse effect of the 10 m water depth surpasses that of the 0 m water level, indicating that the existence of the water increases the stress pulse effect.

When comparing Table 1, Table 2, Table 3 and Table 4, under varying characteristic period conditions, it can be found that the maximum error of the acceleration pulse effect is 40.49%, while the maximum error value of the acceleration hydrodynamic effect is 33.51%. The maximum error of the stress pulse effect is 323.08%, while the maximum error of the stress hydrodynamic effect is 12.90%. This demonstrates that the pulse effect is significantly more noticeable than the hydrodynamic effect. However, the presence of water also influences the features of pulse ground motion.

In summary, the presence of water will inevitably reduce the frequency of the bridge pier and increase the natural vibration period of the bridge pier. Nevertheless, when enhancing the seismic resilience of the bridge pier, the key is the determinant of the spectral characteristics associated with varying characteristic periods of seismic events. However, the behavior of a bridge pier that is not submerged in the water is complicated and variable when the bridge pier is under a pulse of ground motion. In comparison to the regularity of non-pulse ground motion, the dynamic behavior of a bridge pier under pulse seismic motion is complicated, and the pulse effect has a greater influence on the dynamic behavior of the bridge pier structure than the hydrodynamic water effect. These factors are crucial and should not be overlooked.

6. Conclusions and Discussion

In this paper, non-pulse ground motion and pulse ground motion with varying characteristic periods (Tg = 0.45 s; Tg = 0.40 s; Tg = 0.30 s) were synthesized based on EQsignal software. A dynamic response analysis was conducted on the bridge pier at 0 m and 10 m water depths. This study investigated the influence of ground motion, pulse seismic motion, and the presence of water with different characteristic periods on the dynamic behavior of a bridge pier. The following conclusions were drawn:

(1) A bridge pier is under ground motion actions with varying characteristic periods. This results in a distinct acceleration response, peak value, and stress peak value for the stress response. As the characteristic period Tg decreases, both the peak acceleration and the resulting peak stress in the pier structure increase progressively.

(2) When subjected to identical characteristic periods, the acceleration and displacement response time histories of a bridge pier exhibit notable differences under non-pulse ground motion and pulse seismic motion actions. The peak acceleration and displacement under pulse seismic motion are notably higher than those under non-pulse motion, and the corresponding curves are more complex.

(3) The existence of water around a bridge pier reduces its natural frequency, although whether this increases the acceleration and stress response primarily depends on the spectral characteristics of the ground motion in varying characteristic periods. While the pulse effect typically dominates the dynamic behavior of the pier, the hydrodynamic effect is less significant. The water surrounding the pier mitigates the intensity of the pulse effect. The interaction between water and the structure is complex, as the pulse effect tends to induce more significant acceleration and stress responses compared to the hydrodynamic forces. Nevertheless, the existence of water will mitigate the impact of the pulse effects.

This study mainly investigates the dynamic behaviors subjected to pulse ground motion action with varying characteristic periods of action. Previous research has established that the significant and complex effects of pulse ground motion on bridge pier structures are both significant and complex, although this is based on a single dimension. Future research should investigate the determinants of pulse seismic motion, including the striking of the multi-dimensional pulse seismic motion input on the dynamic behaviors of a bridge pier, including multi-dimensional models.

In comparison to non-pulse seismic motion, pulse ground seismic motion exerts a more pronounced destructive influence on a bridge pier. An examination of the mechanism of pulse ground motion will facilitate the identification of the destructive potential of pulse ground motion, which is a crucial consideration in the design of a bridge pier. Through the implementation of an appropriate design, timely maintenance, and continuous evaluation, the sustainability of bridge piers in the context of pulse ground motion can be enhanced, thereby ensuring the continued safe and stable utilization of the bridge following an earthquake.