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Review of the hydrodynamic challenges in the design of elevated pile cap foundations for seacrossing bridges
Advances in Bridge Engineering volume 1, Article number: 21 (2020)
Abstract
With the increasing demand for land transportation in coastal areas, the number of seacrossing bridges has increased rapidly. In the construction of seacrossing bridges, the elevated pile cap foundation is one of the most commonly used foundation types. This paper summarizes four main aspects of the research on the hydrodynamic effects on elevated pile cap foundations through timing analysis and keyword cooccurrence analysis. The relevant studies are reviewed from the aspects of waves, currents, and their interactions; the hydrodynamic load on the elevated pile cap foundation; fluidstructure interactions; and the structural responses of bridges that are supported by elevated pile cap foundations. Finally, the following future prospects for hydrodynamic studies of elevated pile cap foundations for the seacrossing bridge have been discussed, which include wavecurrent interaction mechanism, wavecurrent load, fully coupled fluidstructure interactions, and structural dynamic response under hydrodynamic combined with the other hazards.
1 Introduction
In recent decades, the demand for land transportation in coastal areas of various countries has continued to grow, which has led to a rapid development stage in the construction of seacrossing bridges (Ti et al. 2019; Zhu et al. 2019). In contrast to land bridges, seacrossing bridges are in a more complex and variable marine environment, which inevitably suffers from various harsh natural phenomena, such as strong waves, rapid currents, and earthquakes (Guo et al. 2015). This paper collects all the opened design schemes of seacrossing bridges worldwide in the past 100 years (since 1920) and identifies 121 bridges for which the full span exceeds 5 km. Statistics on their construction times are presented in Fig. 1. Before 2000, there were a total of 38 longspan seacrossing bridges worldwide. Most of them were located in the United States, Japan, and Denmark. The first seacrossing bridge was completed in China in 2005. Since then, the construction of seacrossing bridges in China has entered a rapid development stage. As of October 2020, 58 longspan seacrossing bridges had been built or were under construction in China, which accounted for 47.9% of the longspan seacrossing bridges globally.
In the construction and development of seacrossing bridges, pile and caisson foundations are two main foundation types that have been widely used. The application percentages of pile and caisson foundations in the longspan seacrossing bridges that are under construction are presented in Fig. 2. Among these bridges, 93% adopted pile foundations.
The commonly used pile foundations for seacrossing bridges include a pile group and a pile cap. They can be divided into two types, namely, low and elevated pile cap foundations, according to the elevation of the pile cap from the seabed. A low pile cap foundation is a pile foundation with a cap that is at or below the seabed (Fig. 3b). In contrast, an elevated pile cap foundation has a cap that is above the seabed and usually close to the sea level (Fig. 3c). It has the characteristics of satisfactory geological adaptability, low cost, and convenient construction (Huang et al. 2002). Among the seacrossing bridges with pile foundations that are under construction, 85% adopt elevated pile cap foundations, and 15% adopt low pile cap foundations. Many previously constructed seacrossing bridges also adopt elevated pile cap foundations, such as Hangzhou Bay Bridge (Cheng et al. 2008), Jiaozhou Bay Bridge (Tian 2010), Hong KongZhuhaiMacao Bridge (Qiu et al. 2013), Pingtan Straits RailcumRoad Bridge (Chen and Mei 2016), Donghai Bridge (Chen 2019), Zhoushan Mainland and Islands Link Project (Cui et al. 2019).
As the construction of seacrossing bridges gradually develops from offshore to deep water, hydrodynamic effects have become critical factors that control the design of seacrossing bridges (Chen et al. 2009) (Fig. 4). For example, the 100year recurrence wave of the Xihoumen RailcumRoad Bridge is as high as 7.8 m, and its 100year recurrence current velocity reaches 3.32 m/s. In the Taiwan Strait, waves may reach as high as 12 m (Chen 1981). In addition, the seacrossing bridge on the eastern coast of China is highly likely to suffer from earthquakes. Under the action of an earthquake, the hydrodynamic effects on the seacrossing bridge will significantly change the dynamic characteristics and seismic response of the structure, which will exacerbate the damage that is caused by the earthquake.
Damage to the seacrossing bridge by the extreme marine environment will cause huge economic losses and pose huge challenges in subsequent disaster relief work. Waves and storm surges that were caused by Hurricane Ivan in 2004, Hurricane Katrina in 2005 (Fig. 5a), and Hurricane Ike in 2008 successively destroyed many seacrossing bridges in the United States (Hang and Xiao 2009; Padgett et al. 2012; Murphy et al. 2009). In Japan, the HanshinAwajidaishinsai in 1995 damaged the Higashi Kobe Bridge and the Akashi Kaikyo Bridge (Chen 2016) (Fig. 5b). Previous accidents have shown that the hydrodynamic effect on the seacrossing bridge is one of the main factors that causes damage. Therefore, ensuring the structural safety of elevated pile cap foundations under hydrodynamic effects is an important challenge in the development of seacrossing bridges.
Seacrossing bridges have been constructed in China for 20 years. Many scholars have conducted detailed studies and reviews on the hydrodynamic effects on the superstructure of seacrossing bridges (Fang 2012; Xu and Cai 2015). However, few researchers have conducted indepth reviews on the hydrodynamic effects on elevated pile cap foundations. This paper uses the CiteSpace software to conduct a dynamic analysis of the literature on the hydrodynamic effects on elevated pile cap foundations over the past two decades. The development route and current scenario are examined, and the research focus and direction are summarized. Finally, prospects in the future investigation of hydrodynamic effects on elevated pile cap foundations are discussed.
2 Literature analysis of the research on hydrodynamic effects on elevated pile cap foundations
This paper used an information visualization tool, namely, CiteSpace (Chen 2006), to conduct a literature visualization analysis of the knowledge graph of the research on the hydrodynamic effects on elevated pile cap foundations over the past two decades. Relevant studies from January 1, 2000 to November 10, 2020 were collected from the databases of the Web of Science (website: http://apps.webofknowledge.com/) and the China National Knowledge Infrastructure (website: https://www.cnki.net/) using keywords such as wave, current, wavecurrent, pile, pile cap, dynamic response, seismic, and earthquake. After removing the duplicates, 283 papers were selected for the literature analysis.
2.1 Timing analysis
A timing analysis can identify the development trend of research on the hydrodynamic effects on elevated pile cap foundations. This paper counted the number of relevant publications each year in the literature, and the results are plotted in Fig. 6. The annual number of publications showed an overall upward trend and reached a small peak in 2010. The research on the hydrodynamic effects on elevated pile cap foundations can be divided into three stages: The first stage is from 2000 to 2008, which is the initial stage, and the studies focus mainly on waves, current forces, and pile group effects. The total number of papers in this stage is 77. The second stage is from 2008 to 2012, during which a small peak appeared. Waves, computational fluid dynamics (CFD), and wavecurrent interactions are the research focuses at this stage. The stage after 2012 is a period of rapid development in the research on hydrodynamic effects on elevated pile cap foundations. The research focus has gradually shifted to fluidstructure interactions, the finite element method, and the dynamic response. According to the timing analysis results, the hydrodynamic effects on elevated pile cap foundations are receiving increasing attention.
2.2 Keyword cooccurrence analysis
Keyword cooccurrence analysis can be used to identify the research focus and direction (Chen 2006). This paper analysed the relevant literature on the hydrodynamic effects of elevated pile cap foundations and identified the keyword cooccurrence network, which is illustrated in Fig. 7. Table 1 lists the highfrequency keywords and highcentrality keywords in the keyword cooccurrence network.
The keywords with a high frequency of cooccurrence include “Seacrossing bridges”, “Fluidstructure interaction”, “Wave”, and “Finite element method”, among other keywords. Among them, “Wave force” and “Current force” correspond to the research direction of hydrodynamic load; “Fluidstructure interaction” and “Dynamic response” correspond to the research direction of fluidstructure interaction; and “Finite element method” and “CFD” correspond to the research direction of numerical simulation. These keywords reflect scholars’ investigations of the hydrodynamic effects on elevated pile cap foundations from various aspects.
The betweenness centrality is an indicator that measures the importance of a node in the knowledge graph and indicates the significance of papers. According to Table 1, “Wave force”, “Finite element method”, “Seacrossing bridges”, “Wavecurrent interaction”, and “Wavecurrent force”, among other keywords, have high betweenness centrality, which are research focuses. Cluster analysis is applied to the keywords to obtain the cluster label words, such as “Wave”, “Fluidstructure interaction”, “Timedomain simulation”, “Wave force”, and “Seacrossing bridge”. By comprehensively analysing the results of the timing analysis, keywords cooccurrence analysis, and cluster analysis, the research on the hydrodynamic effects on elevated pile cap foundations can be reviewed from the following four aspects: waves, currents and their interactions; the hydrodynamic load on the elevated pile cap foundation; fluidstructure interaction; and the structural responses of bridges that are supported by elevated pile cap foundations.
3 Waves, currents and their interaction
Waves and ocean currents are common phenomena in nature (Fig. 8). The propagation of waves in the ocean is always accompanied by ocean currents (Wang and Li 1999). The interaction between them is a complex nonlinear process. When a wavecurrent acts on an elevated pile cap foundation, the hydrodynamic load cannot be calculated by simply summing the wave load and the current load linearly (Ren 1983). Thus, the investigation of waves, currents, and their interaction is highly necessary.
3.1 Wave theory
Wave theory can be divided into linear wave theory and nonlinear wave theory (Wang and Liang 1991). Linear wave theory assumes that the wave height is infinitely small relative to the wavelength and that the water particles move slowly. Therefore, the nonlinear effect of the free surface can be ignored, and the boundary conditions can be simplified to linear conditions. However, the wave height in the actual marine environment is often as high as several metres to tens of metres. The nonlinear effect of the free surface cannot be ignored.
Abandoning the assumption that the wave height is infinitely small relative to the wavelength, Stokes proposed the Stokes wave theory, which considers the nonlinear effect of the free surface. This theory considers the influence of the wave steepness and regards it as the main factor that determines wave propagation (Stokes 1847). Based on the study of Strokes, Michael et al. proposed the secondorder Stokes wave theory in 1945. Then, Skjelbreia successively proposed the thirdorder Stokes wave theory and the fifthorder Stokes wave theory in 1958 and 1961, respectively (Sarpkaya et al. 1982; Ma and Zhao 2002; Larmaei and Mahdi 2010).
When a wave propagates offshore, the Stokes wave theory no longer satisfies the accuracy requirements for describing wave propagation due to the frictional drag of the seabed. In 1895, Korteweg and de Vries used the Jacobian elliptic cosine function cn to describe the wave propagation and proposed the Cnoidal wave theory (Korteweg and de Vries 1895). This theory is one of the most important shallow water nonlinear wave theories. Based on the Cnoidal wave theory, Korteweg and de Vries derived the analytical solution for solitary waves (Korteweg and de Vries 1895). A solitary wave can be regarded as an extreme case of a Cnoidal wave. It is an advanced wave in which the water particles only move along the direction of wave propagation.
For the wave theories that are discussed above, Méhauté and Bernard (2013) used two dimensionless independent parameters, namely, H/gT^{2} and d/gT^{2}, to specify the applicable ranges of each wave theory, as illustrated in Fig. 9. In addition, the stream function wave theory (Dean and Dalrymple 1991) and many other modern theories that have been established and applied to numerical simulations (Andrzej et al. 2008; Kriebel 1998; Lewis 1997; Palma et al. 2006).
3.2 Current profile
The most common ocean currents are tidal currents, windgenerated currents, circulational currents, loop and eddy currents, soliton currents, and longshore currents. In engineering practice, a current can be regarded as a stable flow field in which the current velocity vector is only a function of the depth. According to the DNVRPC205 (Veritas 2010), the total current velocity at a position (x, y) should be expressed as the vector sum of each current component:
When detailed field measurements are not available, the current can be assumed to be unidirectional, and the variation of the tidal current velocity in shallow water with the depth can be modelled using a simple powerlaw function (Fig. 10a).
The variation of a windgenerated current can be regarded as a linear profile (Fig. 10b) from z = − d_{0} to the still water level or as a slab profile (Fig. 10c).
or
It can be assumed that windgenerated currents vanish at a distance below the still water level.
where V_{c}(z) is the total current velocity at level z, z is the distance from the still water level or the positive upwards distance, V_{c, tide}(0) is the tidal current velocity at the still water level, V_{c, wind}(0) is the windgenerated current velocity at the still water level, d is the water depth to the still water level (taken positive), d_{0} is the reference depth for the windgenerated current and d_{0} = 50 m, and α = 1/7.
In the deep water along an open coastline, the windgenerated current velocities at the still water level can be expressed as follows:
where U_{1 hour, 10 m} is the 1h sustained wind speed at height of 10 m above sea level.
3.3 Wavecurrent interaction
The interaction between waves and currents is highly complicated. Many scholars have investigated the wavecurrent interaction from the perspectives of flow field characteristics and wave parameter changes. Kemp and Simons (1982, 1983) examined the velocity distribution of the wavecurrent field. The results demonstrated that when the wave and current are in the same direction, the velocity of the upper water particles increases while the velocity of the lower water particles decreases. When the wave and current are reversed, the opposite is true. Ren (1983) conducted an experimental study to observe wave particles’ movement in which a liquid tracer was added to a wavecurrent flume. As shown in Fig. 11, the velocity of the wave particles in the current is highly consistent with the Airy wave theory in static water. It is suggested that if the wave propagates to a uniform current, only the wave parameters change, and the wave theory in static water remains applicable to the uniform current. Li and Ren further analysed the velocity field of wavecurrent interaction through physical experiment and theoretical derivation, respectively (Li 1983; Ren 1983). The results demonstrated that the horizontal velocity of the water particles in the wavecurrent field could be obtained by superimposing the original uniform current velocity and wave particle velocity in the current.
The current studies on the variation of wave parameters in currents can be divided into two categories: regular waves and irregular waves. In the 1860s, LonguetHiggins and Stewart (1961) derived the perturbation solution of a linear wave in an oppositely nonuniform current according to the principle of wave energy flux conservation. Then, Brevik conducted experiments on rough and smooth beds to examine the variation of the wavelength and wave height of a regular wave against a uniform current (Brevik and Bjørn 1979; Brevik and Luftkrigsskolen, Trondheim 1980). Considering the nonlinear effects of waves, Thomas investigated the interactions of an irrotational current with a linear wave and a nonlinear wave (Thomas 1981, 1990). Li (1982) proposed a calculation formula for the wavelength and wave height of a thirdorder Stokes wave in a uniform current based on wave energy flux conservation and wave action flux conservation. Considering the nonlinearity of the velocity distribution, Wang and Li (1996) proposed a fifthorder theory for solving for the interaction between a Stokes wave and an exponential profile current.
With the development of computers, numerical simulation is widely used in the study of wavecurrentstructure interactions. Accurate simulation of the wavecurrent field is the key problem that must be solved before conducting such investigations. As illustrated in Fig. 12, there are three methods for wavecurrent interaction simulation: The first is the velocity superposition method, which linearly superimposes the horizontal velocities of wave particles and the current at the velocity boundary (Fig. 12a) (Park et al. 2001; Li and Lin 2010). At this time, the wave parameters after the interaction of the wave and current should be used. In 2012, Wei (2012) established a threedimensional numerical flume based on the velocity superposition method and simulated the interaction between the secondorder Stokes wave and the current. Qin et al. (2013) used this method to simulate the interactions between waves and currents with various velocity profiles.
Via comparison with the experimental results, the accuracy of the numerical simulation is demonstrated. The second method is to create waves at the entrance boundary and add currents to the calculation domain to simulate wavecurrent interactions (Fig. 12b). Xiao et al. (2013) imitated the experimental method and added a current inlet and a current outlet at the bottom of the calculation domain for wavecurrent interaction simulation. In the third method, the current inlet is set at the entrance boundary, and a mass source is used to generate waves in the calculation domain (Fig. 12c). Zhang and Ding used this method to establish twodimensional and threedimensional models, respectively, of the interactions between regular waves and currents (Zhang et al. 2014; Ding et al. 2015).
These numerical simulation methods are limited mainly to the interactions between regular waves and currents. Few numerical simulation methods are available for the interactions between irregular waves and currents.
4 Hydrodynamic load on the elevated pile cap foundation
A seacrossing bridge is in a complex and variable marine environment, which inevitably suffers from various harsh natural phenomena, such as waves, currents, and their combined effect. As the construction of seacrossing bridges gradually develops from offshore to deep water, these hydrodynamic effects have become the critical factors that control the design of seacrossing bridges (Fig. 13). Therefore, the accurate calculation of the hydrodynamic load on elevated pile cap foundations is important for guaranteeing the safety of seacrossing bridges.
4.1 Wave load
4.1.1 Wave load on the pile group
For the calculation of the wave load on an offshore structure, various methods should be adopted according to the ratio of the structure size to the wavelength. The pile of the elevated pile cap foundation usually has a small geometric size and is a smallscale structure (D/L < 0.2, where D is the diameter of the pile and L is the wavelength). This structure has no readily observable effect on wave propagation. The Morison equation is usually used to calculate its wave load (Liu et al. 2020) (Fig. 14). According to the Morison equation, the horizontal wave load that acts at any height of the column includes two components: the drag force f_{D} and the inertial force f_{M}. The wave load per unit length of a vertical cylinder at any height can be expressed as:
where u and \( \frac{du}{dt} \) are the velocity and acceleration, respectively, of the water particle on the axis of the cylinder; C_{d} is the drag force coefficient; C_{m} is the inertial force coefficient; A is the projected area of the unit cylinder height perpendicular to the wave propagation direction; V_{0} is the volume of cylinder per unit height; ρ is the fluid density.
The calculation of the wave load on a pile group differs substantially from that of the wave load on a single pile. The mutual influence between component piles, such as the phase difference (Fig. 15a), interference effect, and shadowing effect (Fig. 15b), cannot be ignored (Andrzej and Ryszard 2008).
Before 2000, the study of the pile group effect focused only on four pile arrays at most. However, the pile group foundation of a seacrossing bridge is far more complicated than this. In 2009, Yao and Liu (2009) considered a ninepile array pile foundation as the study object to investigate the pile group effect that is caused by the phase difference. The results demonstrated that the pile group effect that is caused by the phase difference decreases with the increase of the pile spacing and fluctuates with the wave incident angle change.
The influences of the interference effect and the shadowing effect on the pile group are highly complicated. They are usually considered by the pile group coefficient (the ratio of the wave force of each component pile to that of a single pile). Through a physical model test, Lei (2013) investigated the pile group effect of a fivepile structure under the action of regular waves and irregular waves. This study focused on the load on each component pile and its proportion in the total load when the wave load on the pile group reached the maximum. In 2018, Zhang et al. (2018) examined the action of multidirectional irregular waves on nine piles in a sidebyside arrangement through physical model experiments. The relationships of the inline force and the transverse force with the KC number and the relative pile spacing are analysed. On this basis, Zhang et al. (2019a, b) investigated the variations of the pile group coefficients of the inline force and the transverse force with the relative pile spacing and the KC number. The pile group coefficient of each pile in the pile group was compared. Regarding the geometrically asymmetrically pilesupported structures as the research objects, Li et al. (2019) proposed two geometric parameters, namely, the asymmetry coefficient and the area ratio of the structure, which expanded the application scope of the empirical formula for wave force calculation. Subsequently, Li et al. (2020) investigated the wave load on geometrically asymmetrically pilesupported structures with six pile spacings and four alignment angles. The empirical formula for the dimensionless wave force was modified, and the pile group coefficient of geometrically asymmetrically pilesupported structures was obtained.
At present, the semiempirical and semitheoretical Morrison equation remains the main method for calculating the wave load on the pile group. The interference effect and shadowing effect are mainly considered by the pile group coefficient. However, the understanding of the interaction mechanism between the pile group and wave remains insufficient.
4.1.2 Wave load on the pile cap
The pile cap of a seacrossing bridge is usually a largescale structure (D/L > 0.2). The reflection and diffraction of waves around it cannot be ignored. Therefore, for the calculation of the wave load on the pile cap, diffraction theory is typically adopted. Li et al. (2017) used the Morison equation, diffraction theory, and the CFD method to calculate the wave loads on pile caps of various relative scales. The results demonstrated that diffraction theory and the CFD method are applicable to the calculation of wave loads on largescale structures. The interaction between waves and structures involves not only a diffraction force but also a radiation force that is generated by structural oscillations. Adopting the threedimensional potential flow theory in consideration of the effect of the radiation force, Qiu and Zhao (2019) investigated the variation of the wave load of a pile cap with the wave period, draft, and wave direction. Considering the influence of the fluid viscosity, Dong et al. (2020) established a threedimensional numerical wave flume with the Fluent software, conducted a numerical simulation of a largescale pile cap under wave action, and analysed the influence of the wave height, submerged depth, period, and pile cap size.
An elevated pile cap is located near the still water level and often suffers from significant wave impact. Wave impact at the pile cap involves wave breaking, strong nonlinearity, and other complicated factors (Xu et al. 2020) (Fig. 16). Diffraction theory is no longer applicable. In 1995, Suchithra and Koola (1995) proposed a formula for calculating the wave impact load that acts on offshore platforms based on experiments. It can be expressed as:
where F_{s} is the impact pressure, ρ is the fluid density, A_{s} is the stressed area, C_{s} is the impact coefficient, and u is the water particle velocity.
AASHTO, API, and DNV all use this method to calculate the wave impact load. The results demonstrated that the impact coefficient is related to the wave period and the clearance between the pile cap and the still water level. Considering various pile caps, Wang and Ren analysed the influences of the wave height, wave period, clearance, and other factors on the wave impact load and fitted an empirical formula for the peak pressure (Ren and Wang 1999, 2003; Ren et al. 2006). Experimentally, Ding et al. (2008) investigated the impact pressure distribution characteristics of the bottom of an open threedimensional structure and the variation of the impact pressure with related influencing factors. Lan et al. (2010) examined the distribution of the impact pressure on the bottom of a pile cap and determined the variation of the impact pressure with the wave period, wave steepness, and the relative clearance. Using the FLOW3D software, Xu and Wei (2019) established a threedimensional numerical model of the interaction between waves and an elevated pile cap and analysed the characteristics of wave loads. The pressure distribution on the bottom of the elevated pile cap and the influence of the clearance height on the wave load were examined.
Considering the timevarying characteristics of the wave impact load, Sihouette modified the calculation formula of the wave impact load to include time variables (Raaij and Gudmestad 2007). According to the experimental results and nonlinear potential flow theory, Wienke and Oumeraci (2005) divided the impact process into two stages and established calculation formulas for each stage. In 2007, Raaij and Gudmestad (2007) fitted numerical simulation results and proposed a threesegment model of the wave impact load time history of offshore platforms (Fig. 17a). Analysing the normalized wave impact load time history, Wei et al. (2020) established a time history model of the impact load in which the rising part is a composition of trigonometric functions and the falling part is a power function (Fig. 17b).
Although many scholars have investigated the timevarying characteristics of wave impact loads, these research results are not yet systematic. The research on the wave impact loads of pile caps with complex crosssectional shapes is insufficient.
4.2 Current load
In engineering practice, the current load of the structure is typically calculated via the following formula (Fig. 18):
where F_{D} is the standard value of the current load; C_{D} is the drag coefficient, which differs among shapes of structures; ρ is the fluid density; v is the design velocity of the current; and A_{c} is the projected area of the component on the plane that is perpendicular to the current direction.
The elevated pile cap foundation is composed of a complex pile group and a cap. The complex arrangement of piles will lead to complexity of the local flow, which renders highly difficult the accurate calculation of the current load of the elevated pile cap foundation using Eq. 9. The research on the current load and drag coefficient of the pile group has always attracted attention. In 2006, Deng (2006) conducted a detailed review of studies on the drag force characteristics of the flow around the pile. The previous results were summarized, and the deficiencies and gaps in the studies were identified. Then, Deng (2007) investigated the drag coefficient values of tandem and parallel double cylinders at various cylinder spacings and proposed a calculation formula for the current load of double cylinders. However, according to Lv (2013), the method and formula that were proposed by Deng for calculating the total current load on the pile group have various limitations. Considering tandem and parallel double cylinders as the research object, Deng (2010a) further investigated the shadowing effect and the transverse interference effect of double cylinders experimentally. After summarizing the established research results, Deng (2010b) examined the current load of tandem double cylinders under a uniform current at a limited water depth. The results demonstrated that the shadowing effect coefficient k_{z} of the cylinder is independent of the Reynolds number R_{e} and the current velocity v only if the ratio of the cylinder spacing to the cylinder diameter is S_{z}/D. As shown in Fig. 19, the shadowing effect decreases with the increase of S_{z}/D. If the ratio of the cylinder spacing to the cylinder diameter is sufficiently large enough, the rear cylinder is no longer affected by the front cylinder.
Based on the finite volume method (FVM), Zhuang et al. (2015) used the tide module TC2D of the COASTALTOOL software to simulate the flow around the pile. In the fine grid model, the influences of the pile section shape, pile group arrangement, and pile spacing on the flow around the pile are examined. In the coarsegrid model, a formula for calculating the drag force of the pile group inside an element based on the principle of drag force superposition was proposed. In addition, a simplification method was proposed for considering the shadowing effect of the pile group in the element. Considering the effects of a twophase flow, Wu (2019) regarded singlecolumn, doublecolumn, and threecolumn pile groups with various arrangements and column spacings as the research objects and used numerical simulation software, namely, Fluent, to establish a threedimensional and twophase flow model of the flow around columns in the subcritical region. According to the numerical flow field, the current force characteristics of the column, the generation mechanism, and the influence of the column spacing were analysed. In 2020, Yan and Xu (2020) established a threedimensional hydrodynamic numerical model for investigating the distributions of the longitudinal, transverse, and total current forces of a rectangular array pile group under various pile spacings. The longitudinal, transverse, and total current forces of rhomboidal, circular, and dumbbell cylindrical pile groups under various pile spacing were also examined.
Many studies have been conducted on the influences of the crosssectional shape, pile arrangement, and pile spacing on the current loads on pile groups. However, these studies have not deeply explored the local flow mechanism of complex pile groups.
4.3 Wavecurrent load
4.3.1 Wavecurrent load on the pile group
For the calculation of the wavecurrent load of a vertical cylinder, an extended form of the Morison equation (Ren 1983) should be utilized, which involves the following two problems: the calculation method for the water particle velocity and the selection coefficients C_{d} and C_{m} (Sarpkaya et al. 1984). The two problems are interrelated and can be examined from the following three perspectives:

1)
Universal hydrodynamic coefficients
According to the experimental results in the wavecurrent flume, Iwagaki and Li obtained a new KC^{∗} number, which correlates with C_{d} and C_{m} in the wavecurrent field (Iwagaki et al. 1983; Li and Zhang 1986). The KC^{∗} number can be expressed as:
where θ = arccos(u_{c}/u_{m}), u_{m} is the peak velocity of the wave particles, and u_{c} is the current velocity.
The KC^{∗} number is universally used not only for wave, current, and wavecurrent fields but also for regular waves and irregular waves (Li et al. 1991).

2)
The relationship between C_{d} and C_{m} in the wavecurrent field and the corresponding coefficients in the wave field
Ren (1985) used the least square method to identify the relationship between C_{d(wave − current)} and C_{d(wave)} based on experiments, which is plotted in Fig. 20. Taking C_{d(wave)} as 1.22, the calculation formula for C_{d(wave − current)} can be expressed as:
where V is the current velocity and \( \overline{u} \) is the averaged maximum horizontal velocity of the wave particles along the water depth.

3)
The relationship between C_{d}, C_{m} and KC_{2} = (u_{max} + U)T/D
Wang and Li (1995) analysed the physical mechanism of wavecurrent interaction and proposed a method for calculating C_{d} and C_{m} in the wavecurrent field from the weighted average of C_{d} and C_{m} in the wave field and the current field. Then, the relationship among C_{d}, C_{m} and KC_{2} = (u_{max} + U)T/D was identified.
The wavecurrent load on an inclined pile has yet to be determined. Lan and Huang investigated the wavecurrent loads of inclined piles and identified a correlation between the wave surface spectrum and the wavecurrent force spectrum (Lan et al. 2004; Huang et al. 2009).
Similar to waves, when a wavecurrent interacts with a pile group, the mutual influence among the piles cannot be ignored. Li et al. (1991) regarded tandem and parallel double piles as research objects and explored the variations of the pile group coefficient with the relative pile spacing and the KC number when regular waves and irregular waves interact with currents. Then, Li and Wang (1992a, b) analysed the timedomain and frequencydomain characteristics of the inline, lift, and resultant forces of tandem double piles with various pile spacings under the combined action of irregular waves and currents. HuangFu et al. (2004) used experimental methods to investigate the variations of the pile group coefficient with the KC number, current velocity, and water depth when irregular waves interact with currents. Regarding the pile foundation of the Donghai Bridge as the research object, Mei and Liu (2005) analysed the wavecurrent load of the inclined pile array that is induced by irregular waves and currents. They found that the variation of the pile group coefficient with the KC number is influenced by the current velocity and water depth. Tian (2019a) analysed the variations of wavecurrent loads of a single pile and a pile group with the current velocity. The results demonstrate that when the current velocity is small, the wavecurrent load of the component pile in the pile group is less than that of the single pile. As the current velocity increases, the wavecurrent load of the component pile in the pile group becomes larger than that of the single pile.
Under the combined action of waves and currents, the mutual influence among piles is stronger than the effect of the waves, and the research on the pile group coefficient under the wavecurrent action remains insufficient. Most of the studies focus on simple pile groups, and few studies have been conducted on complex pile groups.
4.3.2 Wavecurrent load on the pile cap
The wavecurrent loads for largescale structures must be calculated via various approaches according to the relative strength of the waves and currents (Hu et al. 2012). When the wave and current are both weak, their coupling effect can be ignored. The wave load and the current load can be calculated separately via potential flow theory, and the wavecurrent load can be obtained by linearly superimposing them (Hu et al. 2012). In the case of strong waves and weak currents, the weak current assumption and potential flow theory can be used to calculate wavecurrent loads. Teng (1996) proposed a new integral equation that is suitable for higherorder boundary element methods for solving the problem of the combined action of deepwater waves and weak currents on threedimensional structures. Under the condition of strong waves and strong currents, potential flow theory is no longer applicable. The studies on the wavecurrent load involve mainly physical model experiments and numerical simulations. Tao and You (1991) conducted a numerical simulation study on the nonlinear interaction between waves and currents and examined the loads that are induced by wavecurrent forces on largescale structures. Hu et al. (2012) summarized a calculation method for the wavecurrent load of a seacrossing bridge foundation. The problems to be solved were identified, and corresponding solutions were presented. Nie et al. (2013) and Mei et al. (2017) proposed a simplified calculation method for wavecurrent loads on largescale rectangular pile caps that was based on a physical model experiment. Liu (2017) used the CFX software to calculate the wavecurrent loads on the substructure of a seacrossing bridge and proposed a simplified calculation method. Ji et al. (2019) investigated the hydrodynamic pressure and wavecurrent force on a steel box cofferdam via field measurement and numerical simulation. Kang et al. (2019) established a numerical model for the interaction between a threedimensional wavecurrent field and a largescale pile cap to analyse the influence of the incident angle on the wavecurrent load.
Like waves, there will be an impact effect that cannot be ignored when the wavecurrent interacts with the elevated pile cap (Hu et al. 2012). As shown in Fig. 21, when the wave and the current are both strong, the time history curve of the wavecurrent load has a small peak before the maximum; hence, a substantial impact occurs during the interaction. The impact of the wavecurrent on elevated pile caps is a challenging topic in the construction of seacrossing bridges. However, related research is highly scarce. Experimentally, Tian (2019b) investigated the wavecurrent impact loads on pile caps of various shapes. The influences of the relative wave height and the relative current velocity were analysed. The results demonstrated that the wavecurrent load on a pile cap increases with the relative wave height, regardless of the shape.
Most scholars use experimental and numerical simulation methods to investigate the wavecurrent loads on elevated pile caps, and the theoretical research is lacking. No formula has been established for calculating the wavecurrent impact load on an elevated pile cap.
5 Fluidstructure interaction
The size of an elevated pile cap of a seacrossing bridge is large. When it vibrates, the fluidstructure interaction between the surrounding water and the vibrating pile cap will cause a large additional hydrodynamic force and change the dynamic characteristics of the structure (Jiang et al. 2017, Wei et al. 2019, Zhang et al. 2019). In 2010, Song et al. (2010) regarded the foundation of the South Tower of the Third Nanjing Yangtze River Bridge as the research object and conducted shaking table tests on the foundation using a steel box with and without water. The results of the shaking table tests were compared with those of the simple calculation method of hydrodynamic force under water and no water conditions. It was found that when lowfrequency harmonics were input, the results of the simple method were in satisfactory agreement with the test results. When highfrequency harmonics were input, the results of the simple method were 10% lower than the test results. Huang (2012) also used a steel box that was filled with water to conduct the shaking table test on a pier with a pile group foundation. The method he proposed, which was based on radiation wave theory, for calculating the dynamic interaction between water and the pier under an earthquake action was evaluated. Liu et al. (2012) conducted a shaking table test of a submerged pilepier system model and discussed the effects of hydrodynamic pressure on the dynamic characteristics and seismic response of the structure. They found that the natural vibration frequency of the model in water is lower than that without water. The magnitude of the hydrodynamic pressure is related to the peak acceleration and decreases along the height of the model. Wei et al. (2015) used analytical methods to obtain the frequencydomain solution of the dynamic response of deepwater piles in consideration of the combined action of surface waves and earthquakes. The effects of structural flexibility, highorder modal shapes, superstructural contribution, structuresoil interaction, and water compressibility on the dynamic response of the structure are explored.
Hydrodynamic forces can be represented as added masses. Many scholars have developed simplified equations for calculating the hydrodynamic added masses of submerged columns or piers with various crosssections (Jiang et al. 2017). Lai et al. (2004) proposed a semianalytical and seminumerical method for calculating the hydrodynamic pressure on cylindrical piers under an earthquake. Based on the Morrison equation, Gao and Zhu (2006) used an added mass to consider the effect of water and proposed the use of the relative water depth in seismic design to determine whether it is necessary to consider the hydrodynamic pressure. Wang et al. (2011) idealized an elevated pile cap as a truncated cylinder that was immersed in water and established a calculation method for the added mass matrix and added damping matrix of the pile cap. An approximate method for equating a noncylindrical pile cap to a cylindrical pile cap was proposed. The seismic motion equation of a deepwater elevated pile cap foundation was established in the frequency domain. This study only considered the pile cap and not the dynamic response of the pile group foundation. As shown in Fig. 22, Wei et al. (2013) conducted an experiment and used a threedimensional finite element model to investigate the fluidstructure interactions of a deepwater pile group during free vibration. The influences of water depth changes, the pile cap shape, and the pile foundation arrangement on the dynamic characteristics and hydrodynamic pressure distribution of deepwater pile foundations were explored. Pang et al. (2015) developed a numerical program for calculating the added mass of rectangular pile caps that were immersed in water by using the ratio of natural frequencies with and without the effect of fluidstructure interaction. Regarding rectangular pile caps as the research objects, Zhang et al. (2020) used Latin hypercube sampling (LHS) to sample pile caps with various physical parameters. The added mass coefficients of all pile cap samples were calculated via a threedimensional potentialbased numerical method. A Bayesian updated analytical model of the hydrodynamic added mass of a rectangular pile cap was proposed.
Although fluidstructure interaction analysis is an effective method for solving for the seismic response of elevated pile cap foundations of seacrossing bridges, many aspects of this method must be improved, such as the selection of control equations, the control of grid movement, the determination of initial conditions of ground motions, and the selection of fluidstructure interaction algorithms.
6 Structural responses of bridges that are supported by elevated pile cap foundations
The elevated pile cap of a seacrossing bridge has large upper weight and small lateral stiffness; hence, it is sensitive to impact loads. When the pile cap is subjected to an impact load, a highly complicated dynamic response will be induced (Guo et al. 2011). Researchers have often used static methods to determine the bearing capacities of offshore structures (Golafshani et al. 2011; Wei et al. 2014). However, the marine environment is constantly changing with time and space. Although the static analysis method is simple and fast, it cannot evaluate the real dynamic responses of these structures (Golafshani et al. 2011). Golafshani et al. (2011) compared the static and dynamic incremental wave analysis (SIWA and DIWA) results of the SPD2 and DS jacket platforms under wave action. As shown in Fig. 23, the SIWA method will underestimate the structural response under wave action and increase the risk of structural damage.
6.1 Wave and current responses
Waves and currents often impact the elevated pile cap foundations of seacrossing bridges. It is necessary to investigate the dynamic response of such a structure under waves and currents. Hallam et al. (1977) used the finite element method to discretize the structure. The wave load was calculated via an analytical method, and the dynamic equation under wave action was established. This method was used to examine the dynamic response of a fixed structure under the action of waves and currents. Laya et al. (1984) improved the above method by using the relative velocity of the wave and the structure to consider the effect of wavestructure interaction on the dynamic response of the structure. Guo et al. (2011) conducted an experimental study on the dynamic responses of marine structures under wave impact loads. The maximum acceleration response of the platform structure under wave impact load and its possible occurrence times were obtained. Zhang (2012) used the ABAQUS and FLUENT software to conduct fine finite element discretization of structures and fluids, respectively, and used the MPCCI software to solve the fluidstructure interaction vibration equation of doublecolumn piers under wavecurrent action. Regarding the buried pile cap of the Hong KongZhuhaiMacao Bridge as the research object, Wu et al. (2014) analysed the dynamic response of the pile foundation under wavecurrent action. Wang and Chen (2014) used the improved JONSWAP spectrum to simulate irregular waves and used the Morison equation to calculate the wave load on a steel pipe pile. The finite element method and the Newmark method were used to solve for the dynamic response of a trestle steel pipe pile under wavecurrent loads. Pan et al. (2020) used Abaqus to establish a threedimensional finite element model of a single pile and pile groups under the conditions of multilayer soil and singlelayer soil. The Morison equation was used to calculate the wavecurrent load, and the responses of each model under various load conditions were determined.
In a numerical simulation study of the dynamic response of the elevated pile cap foundation under a wavecurrent action, a fully coupled simulation of the fluidstructure interaction is difficult to realize. The wavecurrent force is applied to the structure as an external load. Studies on the fluidstructure interactions of elevated pile cap foundations and wavecurrents still rely mainly on physical experiments.
6.2 Multihazard response
In contrast to land bridges, seacrossing bridges are in a complex and variable marine environment. When an earthquake occurs, a seacrossing bridge will bear the combined action of the seismic excitation and hydrodynamic pressure that are generated by waves and currents. A seismic analysis of seacrossing bridges must consider the combined action of waves and currents. Li and Huang (2012) used diffraction theory to consider the effect of waves and used radiation theory to consider the hydrodynamic pressure. Using a selfcompiling program, they analysed the dynamic response of a bridge pier under the combined action of an earthquake and waves. Yuan and Liu (2013) conducted dynamic calculations on the piers of Zhanjiang East Island Bridge. They determined that when waves and earthquakes act simultaneously, the horizontal force and bending moment of the pile substantially exceed those when the earthquake is acting alone. Lin and Yuan (2015) established a method for analysing the seismic responses of deepwater bridge piers under wave action. The dynamic responses of a bridge pier under three working conditions were analysed: earthquake action only, wave action only, and earthquake and wave action simultaneously.
Waves and currents in the ocean often occur at the same time, and the dynamic responses of seacrossing bridges under the combined action of earthquakes and wavecurrents cannot be ignored. Chen et al. (2008) used the fifthorder Stokes wave theory and the Morison equation to calculate wave forces and applied the calculated wave forces on the piers in the form of distributed forces. Then, the nonlinear seismic response characteristics of deepwater bridge piers under the action of wavecurrents were analysed. Regarding the deepwater foundation of the Qiongzhou Strait Bridge as the research object, Chen et al. (2015) used finite element dynamic analysis methods to investigate the effects of various hazards such as earthquakes, ship collisions, and waves. The displacement response of a deepwater foundation was analysed, and its dynamic response characteristics were discussed. Liu and Zhang (2017) systematically reviewed the research progress on seacrossing bridges under the individual or combined action of waves, currents, and earthquakes and discussed future prospects. Based on the nonlinear Morison equation, Wu et al. (2018) established the structural dynamics equation under the combined action of waves, currents, and earthquakes. Through finite element discretization, the dynamic response of a deepwater pilecappier structure system was calculated, and the influences of various wave and current parameters were analysed. Wu et al. (2018) used a numerical simulation method to calculate the displacement and stress response of a pile foundation under the action of wavecurrents and earthquakes. The influences of the relative water depth, relative velocity ratio, wave steepness, period, direction angle, and phase angle were analysed. Using ADINA, Jiang et al. (2019) established a fluidstructure interaction model for investigating the combined action of waves, currents, and earthquakes on deepwater piers and discussed the influences of various wave and current parameters on the dynamic responses of deepwater piers. Yun and Liu (2020) regarded a deepwater highpier whole bridge as the research object. Under the combined action of waves, currents, and earthquakes, they conducted a shaking table test on a 1:220 geometric scale model (Fig. 24a). As shown in Fig. 24b, acceleration sensors, strain sensors, and hydrodynamic pressure sensors were installed on the model to record the structural response and the hydrodynamic pressure. According to the experimental results, the combined effects of earthquakes, waves, and currents are not simply superimposed effects. There are complex interactions among them.
Limited by the conditions of the underwater shaking table test, the study on the dynamic response of an elevated pile cap foundation under the combined action of waves, currents, and earthquakes uses mostly numerical simulation methods, which lack the support of reliable experimental data.
7 Conclusions and prospects
In the design and construction of elevated pile caps of seacrossing bridges, the influence of hydrodynamic effects such as waves, tidal currents, and earthquakes must be considered. The experimental and theoretical studies on these loads have continuously deepened, and the results have been referenced and applied in bridge designs. In this paper, the literature visualization analysis method is used to analyse the research results on the hydrodynamic effects of elevated pile cap foundations that have been obtained over the past 20 years. The development route and knowledge structure of the research are summarized. By reviewing the studies that have been conducted on the hydrodynamic effects of elevated pile cap foundations, the following unresolved problems are identified:

(1)
Physical model tests and numerical simulations are the main methods for investigating the interaction between a wavecurrent and an elevated pile cap foundation. However, they all have limitations in simulating wavecurrent interactions. The test conditions limit the applicability of the physical model test, and it is difficult to simulate a highReynoldsnumber environment. Numerical methods for simulating the interactions between irregular waves and currents have not been reported. The wavecurrent interactions mechanism requires indepth investigation.

(2)
The hydrodynamic loads on single piles and pile groups have been examined by many scholars. Some of these loads can be calculated by following the corresponding design standards. However, the complex local flow that is caused by the hydrodynamic effects on the pile group foundation requires indepth investigation.

(3)
With the increase of the wave height and the current velocity, the interaction of an elevated pile cap with waves and currents will gradually show a substantial impact effect, which involves complex factors such as wave breaking and strong nonlinearity. The mechanism of wave and current impact has yet to be elucidated, and no calculation method for the wavecurrent impact load has been reported.

(4)
Fluidstructure interaction analysis is an effective method for determining the seismic responses of elevated pile cap foundations of seacrossing bridges. However, fully coupled fluidstructure interactions under the combined action of waves, currents, and earthquakes is difficult to realize.

(5)
Research on the structural dynamic response under hydrodynamic with other hazards requires dynamic tests under multiple excitations of earthquakes, waves, and currents. Such tests can provide reliable data and information support for the theoretical investigation of the dynamic responses of elevated pile foundations under the combined action of earthquakes, waves, and currents.
Therefore, the combined application of the physical model test, numerical simulation method, and theoretical analysis is of substantial significance and value for research on the hydrodynamic effects on elevated pile cap foundations of seacrossing bridges.
Availability of data and materials
All data, models, and code used during the study appear in the published article.
Abbreviations
 H :

Wave height
 T :

Wave period
 g :

Acceleration of gravity
 L :

Wavelength
 f _{ H } :

The wave load per unit length of a vertical cylinder at any height
 f _{ D } :

The drag force
 f _{ M } :

The inertial force
 u :

The velocity of the water particle
 \( \frac{du}{dt} \) :

The acceleration of the water particle
 C _{ d } :

The drag force coefficient
 C _{ m } :

The inertial force coefficient
 ρ :

Fluid density
 KC :

Keulegan–Carpenter number
 R _{ e } :

Reynolds number
 CFD:

Computational fluid dynamics
 FVM:

Finite volume method
 LHS:

Latin hypercube sampling
 SIWA:

Static incremental wave analysis
 DIWA:

Dynamic incremental wave analysis
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Acknowledgments
The authors would thank the Southwest Jiaotong University and colleagues from China Railway Major Bridge Reconnaissance & Design Institute, for their assistance and discussions in proposing the research for the hydrodynamic effect on the elevated pile cap foundation.
Funding
This work was supported financially by the National Natural Science Foundation of China (Grant Nos. 51978578 and 51708455) and the Fundamental Research Fund for Central Universities (A1920502051907–2001).
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Conceptualization, K. W.; Formal analysis, M. Z., Z. T.; Investigation, C. Z., M. Z., Z. T.; Supervision, K. W., M. Z. and S. Q.; Writing—original draft, K. W., C. Z..; Writing – review & editing, K. W., and Z. C. All authors have read and agreed to the published version of the manuscript.
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Wei, K., Zhou, C., Zhang, M. et al. Review of the hydrodynamic challenges in the design of elevated pile cap foundations for seacrossing bridges. ABEN 1, 21 (2020). https://doi.org/10.1186/s43251020000209
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DOI: https://doi.org/10.1186/s43251020000209