Structural Finite Element Analysis of Bridge Piers with Consideration of Hydrodynamic Forces and Earthquake Effects for a Sustainable Approach

The demand for intra-urban transportation has been increasing, thus more and more long-span river and canal bridges are constructed [1]. These bridges have to be designed for long-term durability amid some fairly adverse environmental conditions. Typically, deep-water pile foundations would be required [2]. It is, however, currently unknown whether they have seismic resistance, especially if the ground motion is coupled with wave action during an earthquake. Though already proven effective against static and certain dynamic loads (e.g., due to wind and the water flow), their seismic capacity is not yet clear [3, 4].

Many investigations have been conducted both domestically and internationally to study the behavior of water structures in response to dynamic ground motions and waves. Yamada's team [5] employed the Kanai spectrum and waves derived from the Bretschneider power spectrum in order to produce ground shaking. Karadeniz [6] employed spectral analysis on three-dimensional structures and considered the occurrence of earthquakes and waves in water with a depth of more than 80 meters as a random process. In order to assess the seismic behavior of flexible structures taking into account fluid motion and structural stiffness, Fukusumi et al. [7] conducted simple numerical experiments. Etemad et al. [8] employed the Nogami model to study the response of pile foundations to seismic waves and the interaction of pile and soil in different directions of wave travel and ground motion [9].

Abbasi and Gharabaghi calculated the seismic response of a 3 dimensional model of an offshore drilling platform that considered the effects of nonlineality and wave direction. He and Li [10] intended to utilize wave theory and Morrison's formula in order to assess the seismic behavior of structures affected by both quakes and waves. Alsultani et al. [11] employed the added mass approach to simulate the entire pile-soil-column water system and investigated the dynamic pressure and water level changes. In this context, Song et al. [12] created a simple computational procedure to estimate the pressure differential between the water and the solid during an earthquake, this is of great importance to the engineering community. This analysis is crucial to ensuring the proper design and safety of bridges during earthquakes. Additionally, this type of modeling has a significant role in implementing early warning systems and dynamic response mitigation strategies that will ensure that structures can cope with the combined effects of hydrodynamic and seismic forces.

The common lack of consideration of hydrodynamics in the studies that are currently underway underscores the necessity of such analysis. As a result, their conclusions cannot be universally applied or to dynamic scenarios that involve the behavior of deep-sea foundations under seismic pressure. Effective predictions in three-dimensional space must accurately represent the response of the column, pile, soil, and water system to an earthquake. This response must be in terms of the relative displacement, acceleration, sheer, sensitive moment, and hydrodynamic pressure coefficient.

Recent investigations have identified various components of this molecule. In order to assess a cable-stayed bridge that is supported by the seabed from the perspective of earthquake-induced structural interaction, Li et al. [4] conducted underwater shake tables that flickered. This investigation aims to explore the difficulties associated with the interaction of multiple hazardous scenarios and their effect on the stability of the bridge. Other scientists [13] dedicated their research to the way waves and associated currents affect bridge piers and provided significant information regarding the hydrodynamic forces that occur and their association with structural design.

These researchers examined the effectiveness of bridge piers in adverse situations, one example of which was a dynamic rise in flood water caused by a dam failure [1]. Their results indicate that it's vital to consider the responses to piercing from different directions in order to create truly resilient structures that can handle these disasters. Another research by Huynh et al. They advocated the need to consider the long-term effects of environmental factors when predicting quakes. Huynh et al. [14] investigated the temporal evolution of the resistance of offshore structures to seismic action as a function of erosion and corrosion induced by chloride...

The study [15] has been made on hydrodynamic loading, specifically in the wave and flow coupling case of bridge decks. Her work increased our knowledge of complex fluid dynamics and forces resultant in bridge structures. Additionally, in the study by Lu et al. [16], more scopes on hydrodynamics have been done and introduced new designing concepts through the review of research progress and future prospects of column-type floating underwater tunnels.

Several studies have been carried out on the problems associated with high pile cap foundations of offshore bridges. This study brings out the hydrodynamic challenges and proposals for enhancement to the design of these most critical structural elements Dynamic response of bridges to extreme floods was investigated by Zhu et al. [17], which underlines the significance that fluid-soil-structure interaction has on the predictability of design performance in severe events.

Xu and Cai [18] studied the interactions between bridge decks and waves via numerical experiments and determined the importance of lateral stiffness in regards to the dynamic behavior of structures. This research demonstrated the capacity of the bridge's deck structure to resist wave impacts when dealing with a single wave. Other investigations have demonstrated that Qu et al. [19] studied the safety of coastal bridges in the New York metropolitan area during severe storms using the example of Hurricane Sandy as a guide. Their findings indicate the necessity of strong design and alteration strategies in order to preserve coastal infrastructure from future disasters.

The main objective of this study is to further knowledge of the behavior of deepwater pile foundations under extreme loading conditions, through detailed seismic ground motion and wave simulations. Therefore, we will provide engineers and researchers with the necessary information for enhancing the seismic performance of deepwater bridge piers dynamic response hydrodynamic pressure coefficients, relative displacement, acceleration, shear forces, as well as moments. The 3D finite element model to be developed herein will be applied in this work to be able to:

• Enhancing Simulation Accuracy: By accurately representing pile-soil interactions and fluid dynamics, our model improves the fidelity of seismic response predictions for deepwater pile foundations.

• Improving Design Practices: The findings of our research will lead to increased efficiency in the construction of bridges and make sure that the piers of these structures can withstand the effects of a combined earthquake and wave action.

• Long-term Structural Safety: In earthquake-prone areas, addressing the bridge piers' resistance to severe loading conditions helps ensure the durability and security of vital infrastructure. By using sophisticated computational techniques to model intricate interactions between structural elements and environmental forces, this study fills important knowledge gaps. Our goal is to provide engineers and decision-makers with the necessary resources to create and manage robust bridge infrastructure in demanding maritime settings.

Water-bridge interaction forms an important aspect of dynamic behavior studies regarding bridge structures in an earthquake. Motion of these constituents affects water flow and the force water applies on the bridge is known as hydrodynamic pressure. The governing equations for transient structural dynamics can be written in matrix format as:

${[M]\{\ddot{x}(t)\}+[C]\{\dot{x}(t)\}+[K]\{x(t)\}}=\left\{F_H(t)\right\}+[M]\left\{\ddot{x}_g(t)\right\}$                    (1)

The structural mass matrix is referred to as [M], the damping matrix is referred to as [C], and the stiffness matrix is referred to as [K]. The vector of relative displacement, velocity and acceleration of the structure is written as $\{x(t)\},\{\dot{\mathrm{x}}(t)\}$ and $\{\ddot{\mathrm{x}}(t)\}$ respectively. The vector of acceleration associated with the earthquake's ground motion is written as $\left\{\ddot{x}_g(t)\right\}$. The fluid pressure vector that acts on the bridge's structure, such as B.

The current and hydrodynamic force associated with the earthquake is represented by the symbol $\left\{F_H(t)\right\}$.

Mass Matrix: This matrix represents the distribution of mass within the structure. Each element of the matrix correlates to how mass is distributed across different parts of the structure, influencing how it responds to dynamic loads.

Damping Matrix: The damping matrix accounts for the energy dissipation mechanisms within the structure. These mechanisms could include material damping (energy loss due to internal friction) and external damping (energy loss due to friction at supports or connections).

• Stiffness Matrix: This matrix represents the structural stiffness, indicating the ability of the structure to resist deformation. The elements of the stiffness matrix are derived from the material properties and geometric configuration of the structure.

• Displacement Vector: This vector describes the relative displacement of different points in the structure at any given time.

• Velocity Vector: This vector indicates the rate of change of displacement with respect to time, representing the velocity of the structural points.

• Acceleration Vector: This vector denotes the rate of change of velocity with respect to time, representing the acceleration of the structural points.

• Ground Acceleration Vector: This vector represents the acceleration of the ground during an earthquake, which induces inertial forces in the structure.

• Hydrodynamic Force Vector: This vector includes forces exerted by water on the structure. These forces result from both the water current and wave action, as well as the additional hydrodynamic forces induced by the seismic activity.

The formula for the dynamic motion equation is as follows:

$[M]\{\ddot{x}(t)\}+[C]\{\dot{x}(t)\}+[K]\{x(t)\}=C_M \rho V\left(\ddot{u}-\ddot{x}_0\right)+\frac{1}{2} C_D \rho A\left|\dot{u}-\dot{x}_0\right|\left(\dot{u}-\dot{x}_0\right)+\rho V\left\{\ddot{x}_g(t)\right\}$           (2)

ρ is the density of water; $C_M$ and $C_D$ are the inertia and drag coefficient of water respectively; V and A are the amount of volume and area of the surface exposed respectively; $\dot{x}_0$ and $\ddot{x}_0$ are the absolute velocity and acceleration of water respectively; u ̇ and u ̈ are the velocity and acceleration of water relative to the water respectively.

Inertia force coefficient is a concept used to estimate added mass effects. The surrounding water has a major contribution to the effective inertia of the body during its acceleration.

• Drag force coefficient: This coefficient quantifies the drag force, which is the resistance force exerted by the water as the structure moves through it.

• Exposed volume: This is the volume of the structural elements exposed to the water, which influences the inertial forces.

• Exposed area: This is the surface area of the structure exposed to water, which affects the drag forces.

• Water velocity: The velocity of the water relative to the structure. It includes the effects of water currents and waves.

• Water acceleration: The acceleration of the water relative to the structure.

• Absolute velocity of structure: The velocity of the structure itself, considering both the motion induced by seismic activity and its inherent movement through the water.

• Absolute acceleration of structure: The acceleration of the structure due to seismic forces.

The hydrodynamic force per unit height, P, from Eq. (2) may be stated as illustrated in Eq. (3):

$P=C_M \rho A\left(\ddot{u}-\ddot{x}_0\right)+\frac{1}{2} C_D \rho B\left|\dot{u}-\dot{x}_0\right|\left(\dot{u}-\dot{x}_0\right)+\rho A\left\{\ddot{x}_g(t)\right\}$                   (3)

The second term is responsible for the drag force; the first component to the right of Eq. (3) is responsible for the inertia force, while structural effect is expressed in the third term.

Hydrodynamic drag force, which in many instances involves coupled current and wave impacts, considerably influences the dynamic response of offshore and coastal structures [12, 26, 27]. Thus, in a new formulation, let us replace "u ̇" by "C+W," where C signifies water current speed and W wave characteristics, such that the Morison equation is made current sensitive, as shown in Eq. (4):

$P=C_M \rho A\left(\ddot{u}-\ddot{x}_0\right)+\frac{1}{2} C_D \rho B\left|(C+W)-\dot{x}_0\right|\left((C+W)-\dot{x}_0\right)+\rho A \ddot{u}$                  (4)

The component of inertia mentioned in the first part of the equation (Eq. (4)) is reduced to the expression below to calculate the coefficient of inertia:

$C_M=\frac{M}{\rho H \frac{\pi}{4} D^2\left(\ddot{u}-\ddot{x}_0\right)}$                    (5)

where, D is the pile's diameter and H is the water depth. Eq. (5) demonstrates that the hydrodynamic force is independent of frequency and amplitude action and proportional to H and D².

The force of drag in the second half of the equation (Eq. (4)) can be approximately represented by the following equation (Eq. (6)) to calculate the drag coefficient:

$C_D=\frac{M}{\frac{1}{2} \rho H D\left|(C+W)-\dot{x}_0\right|\left((C+W)-\dot{x_0}\right)}$                 (6)

According to the formula, the hydrodynamic drag coefficient is independent of wave action and water flow and is proportional to both H and D.

Other than the drag, the buoyancy force (which acts in perpendicular to the velocity vector and around the axis of the component due to the orbital motion of the water particles) is also connected to the loads on the undersea components of the bridge. The magnitude, direction, and length of the lift force are still unknown and cannot be applied to the Morrison equation, which results in the appearance of noise in the measurement of the drag and inertial components [28, 29].

The model replicates the exact dynamic response of the bridge pier to the interaction of waves, the water current's velocity potential, and ground motion resulting from seismic waves. Table 3 provides data on the characteristics of currents and waves expected over a 100-year period at the location. It focuses solely on the transverse (y-axis) alignment of the bridge regarding water loads from currents and waves. For every simulation conducted, the initial water level was set at 12 meters. If the Environment Conditions and Load Standard indicate that CD=1.2, CM=2.0, and CA=1.0.

Table 3. Current-wave parameters

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Figure 3. The acceleration time history of November 2017, Halabjah earthquakes in Baghdad

In this research, the progression of acceleration at the Halabjah earthquake that occurred in Baghdad on November 12th, 2017 is employed as the seismic input to simulate the response of the model structure. The initial accelerometer demonstrates that the total time course of ground excitation is 200 seconds and has a peak value of 0.11 g at 41.5 seconds. Figure 3 illustrates the progression of the horizontal component of this seismic stimulation. To investigate the effect of different earthquake motions with different peak accelerations, three different frequencies (20 Hz, 40 Hz, and 60 Hz) are used.

While the results presented are extensive, the discussion lacks depth in interpreting these results in the context of existing knowledge. Here, we provide a more detailed analysis of the findings, compare them with empirical data and other numerical simulations, and discuss their broader implications.

7.1 Displacement and acceleration analysis

Two key engineering criteria for the structural performance of a bridge pier are the top displacement concerning the pier's bottom and the average acceleration at the pier's top. The top displacement in relation to the bottom is the most vital factor in the pier's seismic design, as it has a direct effect on structural integrity. Additionally, it governs how seismic excitation influences the movement of the bridge deck, rendering it equally important.

The response of the pier is affected by hydrodynamic pressure from current waves, which reaches its maximum near the top. Consequently, the relative displacement during both current-wave and seismic occurrences exceeds that experienced during seismic actions alone. This trend suggests that relative displacement is more responsive to lower levels of acceleration, decreasing from input acceleration values of 0.05 g to 0.20 g. These findings align with the research conducted by Moazam et al. [3], which demonstrated that hydrodynamic forces significantly impacted the structural responses to seismic events.

7.2 Internal force response

The shear force and moment at the pier base are an order of magnitude higher than those at the top, based on an internal forces study. Therefore, the bottom part becomes more prone to developing plastic hinges because of an earthquake shock. These values are again comparable with previous numerical calculations and real findings work such as in reference [29] where the bottom pier was proved very susceptible to seismic forces due to higher concentrations of stresses.

This is an indication that the hydrodynamic pressure does affect the shear force and moment responses, but it is more visible at the higher accelerations. This underscores the imperative need for rigorous seismic design that incorporates two impacts, hydrodynamic and seismic forces combined, especially when the structure is in deep water, where both forces play a more significant role.

7.3 Comparisons with empirical data

Comparison of empirical data from past earthquakes, including the 2011 Tohoku earthquake, shows similar responses of bridge piers under the dynamic impact of seismic and hydrodynamic loads. Also, findings [12] in empirical studies already exist, which reported increased displacements and internal stresses in bridge piers due to hydrodynamic forces.