Numerical study of pile group effect on the hydrodynamic force on a pile of sea-crossing bridges during earthquakes

Ocean Engineering 199 (2020) 106999

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Numerical study of pile group effect on the hydrodynamic force on a pile of sea-crossing bridges during earthquakes Wu An-jie a, *, Yang Wan-li b a b

School of Civil Engineering, Guizhou Institute of Technology, Guiyang, Guizhou, 550003, China School of Civil Engineering, Southwest Jiaotong University, Chengdu, Sichuan, 610031, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Pile group effect Fluid-structure interaction Numerical method Earthquake Wave Current

Pile group effect is crucial for correct estimation of the hydrodynamic force on a pile within a pile-group foundation of sea-crossing bridge during earthquakes unlike the case of single isolated pile. A numerical calculation model considering fluid-structure interaction under the combined loads of earthquakes, waves and currents is established, based on the second development on ANSYS software. A detailed analysis of pile group effect in the earthquake-induced oscillatory flow field, is carried out by the presented method that is validated in advance. Pile group effect in the earthquake-current combined flow field, as well as in the earthquake-wave combined flow field are further discussed. Pile group effect has almost disappeared when relative spacing S/ D ¼ 4.5 and all piles behave like a single isolated pile in terms of earthquake-induced hydrodynamic forces, yet, in the coexistence field, the range of S/D influencing pile group effect has been expanded due to the presence of current and wave. For a sea-crossing bridge foundation, Pile group effect coefficient K is equal to 0.69, implying noticeable interference effects, and results show the hydrodynamic force on corner pile is greater than that on centre pile, strengthening section reinforcement pertinently so as to improve overall seismic performance.

1. Introduction In order to utmostly shorten the traffic distance between major cities in coastal regions, in recent years, many countries have built many seacrossing bridges at coastal and offshore locations, and are planning to build more sea-crossing bridges. Most of these bridges are located in zones characterized by high seismic hazard levels, which are not only subjected to common wave-current loads, but also are affected by earthquakes. Earthquake excitation can cause additional hydrodynamic pressure on the submerged part of bridge besides structural inertia force (Li, 2013; Yang, 2017). Therefore, the working conditions of the sea-crossing bridges are more complex than that of the conventional bridges. Unlike the case of wave forces (Morion et al., 1950; Sarpkaya and Isaacson, 1981; Guo et al., 2015; Huang et al., 2018) and wave-current forces (Li, 1991; Malenica et al., 1995; Cokgor, 2002; Song, 2006; Nie et al., 2013; Zhou et al., 2015) as the most frequently used loads on ocean engineering structures, less attention has been paid to earthquake-induced hydrodynamic force that affects potentially struc­ tural safety. The original Morison equation (Morion et al., 1950) was used only to calculate the wave force on pile standing in water. Later,

Morison equation was developed to calculate the earthquake-induced hydrodynamic pressure by Penzien and Kaul (1972). Yang and Li (2013) expanded Morison equation to calculate the hydrodynamic pressure during earthquakes caused by inner water and outer water simultaneously for circular and rectangular hollow piers. Li et al. (2019) investigated the coefficients in Morison equation for earthquake induced hydrodynamic pressure of cylinders. Huang and Li (2011) proposed hydrodynamic pressure formula based on the radiation wave theory for Newtonian incompressible fluid flow, and analyzed the influence of hydrodynamic pressure on the seismic response of bridge piers. The results indicated that the dynamic characteristics of pier changed and the seismic response is augmented because of the hydrodynamic pres­ sure effect. The aforementioned hydrodynamic pressure calculations during earthquakes usually assume that the fluid is originally in quiescence without consideration of wave or current motion. Actually, earthquake, wave and current may be simultaneously applied to bridges in the ma­ rine environment. So far, the understanding of the coupling mechanism of multi-disaster loads is not clear enough. Li and Huang (2012) estab­ lished a method of dynamic response analysis for bridges in deep water under combined actions of earthquake and wave using the linear

* Corresponding author. E-mail address: [email protected] (W. An-jie). https://doi.org/10.1016/j.oceaneng.2020.106999 Received 28 July 2018; Received in revised form 23 October 2019; Accepted 19 January 2020 Available online 7 February 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 1. The flowchart of the CFX-ANSYS two-way FSI analysis.

radiation wave theory and diffraction wave theory. The results indicated dynamic responses of bridge in deep water under separate earthquake and wave actions are nonlinear and cannot be combined by super­ position. Yuan and Liu (2013) studied the load effect of bridge pier under combined wave and earthquake actions by the extended Morison equation. It is noticed that both linear radiation wave theory and diffraction wave theory are suitable for dealing with inviscid incom­ pressible flows. For the large-scale pier or platform members, this assumption is reasonable because of the insignificant viscous effect. However, for the small-scale piles, namely slender piles, the viscous effect is important and cannot be ignored, thereupon, the prerequisites of these two theories are no longer valid. Although the semi-empirical Morison equation has high accuracy for the small-scale component calculation, it is not suitable for the closelyspaced pile group used widely in sea-crossing bridges, due to no consideration of interference effects that affect the flow field around the oscillating piles. Currently, it is still difficult to conduct large-scale un­ derwater shaking table test, especially for the coexistence field, e.g. earthquake-current combined flow field, earthquake-wave combined flow field, earthquake-wave-current combined flow field and so on. Therefore, the seismic analysis method for pile-group foundations of seacrossing bridges is poor, which is a problem that needs to be solved urgently. Current literatures about pile group effect mainly focus on currents and waves, lack of concern about the earthquake-induced flow field. Obviously, earthquake and wave are two different conditions (Li et al., 2019). Furthermore, pile group effect in the coexistence field such as the

earthquake-current field, the earthquake-wave field as well as the earthquake-wave-current field is seldom discussed. In the present paper, a systematical study on pile group effect in the earthquake-induced flow field, the earthquake-current field and earthquake-wave field is carried out by the numerical method, based on the second development on ANSYS procedure. In addition, a reasonable approximate formula for pile group effect is proposed. 2. Numerical method of pile-water coupling vibration analysis 2.1. Method of fluid-structure interaction (FSI) analysis When a bridge is in the ocean and suffers from wave-currents and earthquakes, and the hydrodynamic force acted as an external force on bridge, the governing equation of transient structural dynamics can be refined as: � � ½M�f€ xðtÞg þ ½C�fxðtÞg _ þ ½K�fxðtÞg ¼ ½M� x€g ðtÞ þ fFH ðtÞg (1) where [M], [C] and [K] represent the structural mass matrix, damping matrix, and stiffness matrix, respectively. {x(t)}, {ẋ(t)} and {ẍ(t)} represent the structural relative displacement, velocity and acceleration vectors, respectively. {ẍg(t)} is the acceleration vectors of seismic ground motion. {FH(t)} is the fluid force vectors exerted on the bridge structure, including wave-current force and earthquake-induced hy­ drodynamic force. The structural governing equation can be discretized using the finite 2

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 2. Coordinates and boundaries setting for numerical wave flume.

element method (FEM). Meanwhile, the general governing equations of fluid dynamics (e.g., Navier-Stokes equations) are discretized using the finite volume method (FVM). In the numerical model, the Navier-Stokes equations are complemented with the volume-of-fluid (VOF) method to model the gas-liquid interface, which can be written in the vector as

ρ

du ¼ ρf dt

μ

rp þ rðdivuÞ þ μr2 u 3

(2)

In which, u is the fluid velocity, f is the body acceleration, ρ is the fluid density, p is the fluid pressure, and μ is the dynamic viscosity. r represents Hamilton operator, namely, differential operator, and div denotes divergence operator characterized by fluid volume expansion rate in unit time, obviously, this term of div u is equal to zero for the incompressible fluid. According to pressure and velocity boundary conditions, the mutual real-time feedback of the calculation results between fluid field and structure is implemented by the interface. This coupling process can be described in detail as the structure in water will generate deformations because of fluid pressure, meanwhile the fluid field will be affected by the motional structure with rigid body displacement and deformation, and the distribution and magnitude of fluid pressure will be changed accordingly. The final results are obtained by multiple iterations with an advanced numerical method, which is conducted by ANSYS combined with CFX in ANSYS Workbench platform. Where, the simulation is performed between Transient Structural (ANSYS) and Fluid Flow (CFX), and both modules are not only developed independently but also con­ nected through the system coupling module to make a two-way fluidstructure coupling computational framework. The flow of the CFXANSYS two-way FSI analysis is introduced in Fig. 1.

Fig. 3. Three-dimensional regular linear wave generated by numerical method for case H ¼ 3 m, T ¼ 6s, d ¼ 25 m.

Wave generation adopts the boundary generating wave method by defining analytical expressions of the velocity and the wave surface which vary with time at the inlet boundary. This method has strong operability and wide application range, and is particularly convenient for simultaneous generation of wave and current. For the case, it only needs to superimpose current velocity components on the basis of wave velocity components at the inlet boundary. Fluids are forced to make periodic alternating movements in a specified form at the inlet and outlet boundary defined by opening-velocity, which can eliminate wave reflections from the outlet boundary. And the relative static pressure is set as zero at the top boundary defined by opening-pressure. Both side surfaces of the flume are defined as symmetric boundary conditions, while the bottom of the flume is defined as no slip wall. The pile-water coupling surface is used to transfer the real-time data between bridge pile and flow field. The initial conditions of the transient calculation can be set up by compiling CEL codes according to the preset wave velocity, wave surface profile and corresponding hydrostatic pressure, so as to save the itera­ tive calculation time for forming a stable wave field. In the calculation model for a water-air two-phase flow of compressible viscous fluids, the water density is ρf ¼ 1000 kg/m3, the dynamic viscosity of water is μf ¼ 0.8899 � 10 3 Pa s, the air density is ρa ¼ 1.185 kg/m3, the dynamic viscosity of air is μa ¼ 1.831 � 10 5 Pa s, the reference pressure is 1atm ¼ 101325Pa (i.e., a standard atmospheric pressure), and the surface tension coefficient is 0.0725N/m. Taking a three-dimensional regular linear wave as an example (wave height H ¼ 3 m, wave period T ¼ 6s, water depth d ¼ 25 m), the wave profile generated by the numerical method can be seen clearly, as shown in Fig. 3.

2.2. Wave and current generation The VOF method is used to track the free-surface profile by intro­ ducing a fluid volume fraction aq. aq denotes the ratio of the volume occupied by the q-phase fluid to the total volume, whose value is be­ tween 0 and 1 (including 0 and 1). When aq ¼ 1, the q-phase substance is full, when aq ¼ 0, the q-phase substance is empty, and when 0
aq ¼ 1

(4)

q¼1

In which u, v, and w are the fluid velocity components in x, y and z directions, respectively. The sketch of a CFX numerical flume model is depicted in Fig. 2, where the length of flume is usually greater than 3 times wavelength, and the width is not less than 5 times pile diameter to reduce boundary interference. 3

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Table 1 Comparison between the water flow forces obtained by numerical method and experimental test by Deng and Zhang (2007) for currents past pile groups of two piles in tandem arrangement. relative spacing S/D

pile position

experimental values (N)

simulated values(N)

relative error (%)

5

front pile back pile front pile back pile front pile back pile front pile back pile front pile back pile front pile back pile

0.0603 0.0341 0.0616 0.0433 0.0656 0.0492 0.0656 0.0557 0.0656 0.0590 0.0656 0.0656

0.0642 0.0362 0.0671 0.0451 0.0689 0.0536 0.0689 0.0602 0.0689 0.0632 0.0689 0.0689

6.47 6.16 8.93 4.16 5.03 8.94 5.03 8.08 5.03 7.12 5.03 5.03

7 10 12 18 20

Notes: water depth d ¼ 0.225 m, pile diameter D ¼ 0.02 m, water flow velocity v ¼ 0.182 m/s, Reynolds number Re ¼ 3640, S denotes centre to centre spacing of piles.

Fig. 4. Hydrodynamic force on a circular pile versus time during earthquake.

2.3. Earthquake simulation

equation k-epsilon model is chosen as turbulence calculation model, the scalable wall function is used to model near wall region, the solver’s discrete format is set to high resolution, and the second order backward Euler scheme is used for time integration. Fig. 4 shows the time history of hydrodynamic forces on a circular vertical pile (length l ¼ 30 m, diameter D ¼ 1.8 m and water depth d ¼ 25 m), excited by the north-south component of the 1940 El-Centro earthquake. The peak value of total hydrodynamic force generated by both rigid motion and elastic vibration is 140.14 kN, corresponding to a maximum absolute displacement of 14.0 cm, and the peak value of hydrodynamic force generated by elastic vibration is 125.62 kN, cor­ responding to a maximum elastic displacement of 12.5 cm. It can be seen that the elastic vibration-induced hydrodynamic force accounts for a large proportion in total hydrodynamic force, noting that the afore­ mentioned two peaks do not occur at the same time, in other words, the maximum value of the elastic vibration-induced hydrodynamic force and the maximum value of the rigid motion-induced hydrodynamic force do not occur at the same moment, therefore the maximum value of total hydrodynamic force can’t be simply added by the two ones.

At the bottom of flume, a non-slip horizontal wall boundary char­ acterized by sand grain roughness with length dimension (hs ¼ 8 mm) is used to approximate a seabed, where seabed movement caused by earthquake is regarded as rigid motion ignoring seabed -water interac­ tion and, consequently, the focus is rather put on bridge structure-water interaction, aimed at promoting analysis efficiency of the problems involved. The bottom of pile fixed with seabed, moves synchronously with the seabed, and earthquake excitation is conducted by the userdefined function in ANSYS-CFX module. If a seismic excitation input to the seabed tied up with the bottom of pile is the displacement time history of seismic wave, the structural absolute displacement (including rigid and elastic displacement) will be obtained, then the hydrodynamic force exerted on the structure consists of two parts, namely the one generated by rigid motion and the other one generated by elastic vi­ bration. If a seismic acceleration time history is applied to the entire structure and the bottom of pile remains stationary, the hydrodynamic force caused only by the structural elastic vibration can be obtained. In the numerical model, the fluid domain is discretized using the FVM on structured non-uniform hexahedral grids. Meanwhile, the finite element discretization for bridge structure is performed using 20-node hexahedron element SOLID 186. Within CFX, the standard two-

3. Numerical model validation An earthquake motion can be decomposed into a series of harmonic

Fig. 5. Validation of the numerical model by the physical experimental data in Yang et al.(2018). 4

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 6. Comparison of response results given by the present method and Morison method for a circular pile with different d/l subjected to the combined loads of wave, current and earthquake.

motions with different amplitudes, frequencies and phases, and the su­ perposition law can be used to add up hydrodynamic force of each harmonic motion to form the total hydrodynamic force under the earthquake (Yang et al., 2017). Harmonic motion was chosen as exci­ tations exerted on a cylinder. The physical experimental data of a single cylinder vibrating in still water by Yang et al. (2018) are used to confirm the correctness of modeling method and the related parameters. The numerical model has the same dimensions and suffers the same external excitation as the physical experimental model. The comparison of the time histories of hydrodynamic forces between the physical experi­ mental model and the numerical model is shown in Fig. 5. It can be observed that the amplitudes of the hydrodynamic forces change from 0 to a constant value as the amplitudes of the harmonic waves change from 0 to 0.025 m gradually during the first several cycles and the two curves coincide well, indicating the solution of the present numerical model is trustable. Table 1 lists results of the water flow forces given by the present numerical method and the experimental test by Deng and Zhang (2007) for currents past pile groups of two piles in tandem arrangement. The comparison between experimental and simulated results shows

numerical simulation has good agreement to physical experiment, where the relative error is less than 9%. Morison formula (Penzien and Kaul, 1972) characterized by simple expression and clear physical meaning has preferable accuracy and reliability for small-scale components such as piles, and is chosen as the validation tool. Fig. 6 shows the response time history of a pile with circular crosssection (l ¼ 30 m and D ¼ 1.8 m) subjected to the combined loads of wave, current and earthquake at different relative water depth d/l for case H ¼ 3 m, T ¼ 6s, current velocity uc ¼ 2 m/s, obtained by the present method (two-way FSI analysis) and the Morison method. Wave and current are in the positive x-direction, the north-south, east-west and vertical components of El-Centro wave selected for seismic excita­ tion are in the direction of x, z and y respectively, and the coordinate directions are shown in Fig. 2. The drag coefficient CD and inertial co­ efficient CM affected by current in Morison equation are taken as 1.21 and 1.38 respectively, specified in the Code of Hydrology for Harbour and Waterway published by China Communications Press (2015). Here, the maximum structural displacement (absolute displacement) occurs at the top of pile, and the maximum equivalent stress (von-Mises) occurs at 5

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Table 2 The peak values of responses and their relative errors for a circular pile with different d/l subjected to the combined loads of wave, current and earthquake. d/l

Maximum displacement (cm)

15/30 ¼ 0.5 25/30 ¼ 0.83 30/30 ¼ 1

R0 (%)

The present method

Morison method

35.2 37.0 41.4

35.3 35.5 38.9

Maximum equivalent stress (MPa)

0.28 4.23 6.43

The present method

Morison method

21.0 21.8 22.8

21.1 20.8 21.2

R0 (%) 0.47 4.81 7.55

4. Analysis of hydrodynamic forces on pile groups during earthquakes The circular cross-section piles with rectangular arrangement as depicted in Fig. 7 are widely used as the foundation of sea-crossing bridge. Therefore, the following analysis mainly focuses on this form. For a given pipe group arrangement, pile group effect is quantita­ tively reflected by two indexes of K and Ki, which are defined respec­ tively as: Let

Fig. 7. Sketch of the circular piles in rectangular arrangement.

Fgroup ðtÞ t ¼ 0⋯tmax Fsingle ðtÞ � m � n

(6a)

fi ðtÞ ¼

Fi ðtÞ Fsingle ðtÞ

(6b)

t ¼ 0⋯tmax

If at some point t0, t0 2 ½0; tmax �,f (to) and fi (to) always meet the re­ quirements of the following formula respectively

the bottom of pile. It can be observed from Fig. 6 that the response curves obtained by the two methods agree well with each other, espe­ cially when d/l is relatively small. Table 2 lists the peak values of re­ sponses and their relative errors represented by R0, where R0 is defined as Response ​ peak ​ value Response ​ peak ​ value ​ ​ ​ by ​ the ​ present ​ method by ​ Morison ​ method R0 ¼ � 100% Response ​ peak ​ value ​ by ​ Morison ​ method

f ðtÞ ¼

jf ðt0 Þ

1j � jf ðtÞ

1j

(7a)

jfi ðt0 Þ

1j � jfi ðtÞ

1j

(7b)

Then K ¼ f (to)and Ki ¼ fi (to). In Eqs. (6a) and (6b), Fgroup(t), Fsingle(t) and Fi(t) are the hydrody­ namic force on a pile group, a single isolated pile and any a pile within a pile group during earthquake respectively (subscript i ¼ 1 to total number of piles in a group), m and n denote column and row number of piles in a group respectively, t represents time, and tmax is the end time of the earthquake. In fact, the vibration paces of all piles among a pile group during earthquakes are almost consistent and, consequently, the peak values of earthquake-induced hydrodynamic forces occur at the same time, due to no significant seismic wave passage effect as the result of short distance between piles and the assumption of seabed as a rigid body motion, which are also confirmed by the simulated results like that. In other words, the time of attaining maximum response for each pile is consis­ tent and there is no obvious phase difference. Hence, K and Ki can be simplified and rewritten in the other form

(5)

Given that R0 values are small and acceptable, it can be concluded that the numerical simulation method in this paper is of high accuracy and reliability as an alternative for deep-water pile seismic analysis. It can be seen that the value of R0 changes with d/l from negative to positive, noting that Morison equation’s hydrodynamic coefficients, i.e. CD and CM, are taken as constants without changing with water depth, which imply that the calculation results given by the Morison method are conservative when d/l is small, and unsafe when d/l is large.

Fig. 8. Curves of Fourier spectrum of El-Centro wave. (a) North-south component (b) Vertical component. 6

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 9. Variation of KH with respect to S/D for two piles in side by side arrangement with different d/l (l ¼ 30 m and D ¼ 1.8 m).

K¼

Fgroup ðtÞmax Fsingle ðtÞmax � m � n

(8a)

Fi ðtÞmax Fsingle ðtÞmax

(8b)

Ki ¼

KH ¼ K ¼

Fgroup ðtÞmax 1 Fgroup ðtÞmax ¼ Fsingle ðtÞmax � 2 � 1 2 Fsingle ðtÞmax

(9)

By symmetry, KH is equal to Ki, i ¼ 1, 2. The relationships between pile group effect KH and relative spacing S/D (namely the ratio of centre to centre spacing of piles to diameter) are demonstrated in Figs. 9 and 10. As seen, KH deceases gradually from >1 to 1 with increasing S/D, and it is stated that pile group interaction, i.e. lateral interference, en­ hances as the distance between two piles decreases. KH values are greater than 1 for S/D < 4.5, and its maximum value can reach 1.27 for E#1, d/l ¼ 1 and S/D ¼ 1.5 which represents the smallest calculated relative spacing, implying the resulting hydrodynamic force on each one of two piles in side by side arrangement increases up to 27% in com­ parison with that on single isolated pile, and ignoring the pile group effect will lead to unsafe results for this configuration. It can be pre­ dicted that the KH value will further increase by decreasing S/D until it reaches its minimum possible value S/D ¼ 1. For S/D � 4.5, KH is equal to 1, indicating that there is no interaction between piles and each pile behaves like a single isolated pile. It can be also observed from Fig. 9 that the lateral interference in­ creases with the increase of earthquake strength while decreases with the increase of relative frequency fe/fs, where fe/fs is defined as the ratio of the earthquake predominant frequency to the structure fundamental frequency. The introduction of relative frequency fe/fs is aimed at eliminating effects of structural property, noting that the fe values of both E#1 and E#2 wave are 1.13 Hz, the one of E#3 wave is 8.38 Hz (in Fig. 8), and the fs value of a pile given by theoretical calculation is 1.01. For the E#2 and E#3 waves with same strength while different fre­ quency, the KH value under the E#2 wave is greater than that under the

where Fgroup(t)max, Fsingle(t)max and Fi(t)max represent the peak values of earthquake-induced hydrodynamic force on a pile group, a single iso­ lated pile and any a pile within a pile group, respectively. Meanings of other symbols are same as aforementioned. As we all known, the north-south and vertical components of ElCentro wave have significant difference in frequency spectrum, as shown in Fig. 8, and the former strength is stronger than that of the latter, so these two waves and the amplitude modulated north-south wave whose amplitude is equal to the vertical component are designed as three comparative testing seismic waves, numbered as E#1, E#2 and E#3 respectively, to investigate the influence of earthquakes with different strength and frequencies on pile group effect in the flow field. 4.1. Interference characteristics of two piles in side by side and tandem arrangements 4.1.1. Side by side arrangement From Eq. (8a), the pile group effect K for two piles in side by side arrangement where the seismic wave direction is orthogonal to the centre connexion line of the piles located next to each other, in a more concrete form, can be expressed as

7

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 10. Variation of KH with respect to S/D for two piles in side by side arrangement under the action of different seismic waves (l ¼ 30 m and D ¼ 1.8 m).

E#3 wave, as a result of the implicit resonance, where the predominant frequency of E#2 is high closer to the structure fundamental frequency than E#3. It can be further seen from Fig. 10 that lateral interference increase as d/l increases. The reason is attributed to greater hydrody­ namic pressure if in larger water depth. In addition, an important implication from Fig. 10 is that pile group effect is more significantly affected by water depth when the pile spacing is relatively small. The careful compared results illustrate that pile group interaction KH (lateral interference) for the side by side arrangement is noticeably affected by relative spacing S/D and it depends primarily on relative spacing S/D, which is consequently regarded as the most relevant influencing parameter for pile group effect.

between two piles, which results in a reduction of hydrodynamic force on each one of two piles in tandem arrangement compared with that on single isolated pile. KH values are less than 1 for S/D < 4.5, and its minimum value can reach 0.74 for E#1, d/l ¼ 1 and S/D ¼ 1.5 which represents the smallest calculated relative spacing, indicating longitu­ dinal interference could achieve the average hydrodynamic force on two piles in tandem arrangement decrease of up to 26% over that on single isolated pile, i.e. the hydrodynamic force of pile group with two piles is 26% smaller than the resultant force of two isolated piles. And neglecting pile group effect may cause overestimation of the resulting hydrodynamic force for this configuration. It can be predicted that the KZ value will further decrease by decreasing S/D until it reaches its minimum possible value S/D ¼ 1. Pile group effect becomes negligible when S/D ¼ 4.5 and all piles behave like a single isolated pile in terms of the earthquake-induced hydrodynamic force. It can be also observed from Fig. 11 that the greater earthquake strength and relative frequency fe/fs, the more noticeable interference effect is. It can be further seen from Fig. 12 that longitudinal interference increase as d/l increases. In addition, an important implication is that pile group effect is more significantly affected by water depth when S/D is relatively small. Overall, pile group interaction KZ for the tandem arrangement is noticeably affected by S/D and it depends primarily on S/D, which is consequently regarded as the most relevant influencing parameter for pile group effect just like the case of side by side arrangement.

4.1.2. Tandem arrangement From Eq. (8a), pile group effect K for two piles in tandem arrange­ ment where the angle of the centre connexion line of the piles relative to the seismic wave direction is 0� , in a more concrete form, can be expressed as KZ ¼ K ¼

Fgroup ðtÞmax 1 Fgroup ðtÞmax ¼ Fsingle ðtÞmax � 1 � 2 2 Fsingle ðtÞmax

(10)

where for this configuration, generally, KZ 6¼ K1 6¼ K2, as the result of the asymmetry of seismic wave, characterized by unequal positive and negative peak levels. The relationships between pile group effect KZ and relative spacing S/D are shown in Figs. 11 and 12. As seen, KZ increases gradually from <1 to 1 with increasing S/D, that is, pile group interaction (named longitudinal interference) enhances with decreasing the distance 8

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 11. Variation of KZ with respect to S/D for two piles in tandem arrangement with different d/l (l ¼ 30 m and D ¼ 1.8 m).

4.2. Interference characteristics of more than two piles

For a given three piles in side by side arrangement where l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m, Table 3 lists the resulting hydrodynamic forces on single isolated pile, pile group and each pile of this group. It is apparent from the listed data that the hydrodynamic force on the middle pile is greater than that on the two neighbouring side piles. The amplification of the hydrodynamic force on the middle pile in side by side arrange­ ment is more noticeable than the side pile due to the influence of two neighbouring piles from both sides. The values of K obtained by both simulation and formula are also listed in Table 3. By comparison, the formula calculated value is relatively close to the simulated value, where the maximum error of both is 3.7%. For the cases of different seismic waves and different water depths, the numerical difference of K is not significantly large, where the maximum relative variation rate is 9%, indicating that group pile effect K is not noticeably affected by the seismic wave type and water depth. But it is worth noting that the seismic excitation frequency has a great influence on hydrodynamic force on whether single pile or pile group. Hydrodynamic force is so closely related to water acting area that it is also significantly affected by the water depth. The results illustrating the relationship between Ki and S/D for three piles in side by side arrangement with S/D changing from 1.5 to 5 are shown in Fig. 13. As seen, Ki decreases with increasing S/D until to reach constant value 1 at about S/D ¼ 5. Obviously, S/D is the most relevant influencing parameter. For different S/D, K2 > K1 and K3 (K1 and K3 are almost equal), i.e. the hydrodynamic force on the middle pile is still highly greater than that on the two side piles, especially S/D in the range of relative small values. The middle pile in this arrangement is, in fact, affected by two neighbouring piles while only one pile influences the side pile. Therefore, the lower hydrodynamic force amplification for side

4.2.1. Three piles in side by side arrangement In the previous section, we discussed the interference characteristics of two piles, and obtained KH (in Section 4.1.1) and KZ (in Section 4.1.2). For a pile group with more than two piles in rectangular arrangement, the hydrodynamic force of a pile in the group is affected by the neigh­ bouring piles in side by side and tandem arrangement unless the relative spacing S/D is large enough. It can be concluded from the previous analysis that the lateral interference will amplify the hydrodynamic force on a pile within a group in side by side arrangement while the longitudinal interference will reduce the hydrodynamic force on a pile within a group in tandem arrangement, so the pile group interaction of the rectangular arrangement is clearly a combination of both pile group interaction observed in side by side and tandem arrangement. The pile group effect K might as well be calculated approximately by a formula, which can be expressed as K¼

½1 þ ðm

1ÞKH �½1 þ ðn m�n

1ÞKZ �

(11)

It should be noted that the Equation (11) is inspired by total flow resistance coefficient (also called drag coefficient) in the literature of Deng and Xiu (2009), which is expressed as X CD ¼ ½1 þ ðm 1ÞKH �½1 þ ðn 1ÞKZ �CD (12) P where, CD is drag coefficient of a pile, CD denotes total drag coeffi­ cient of a pile group, m and n denote column and row number of piles in a group respectively, KH and KZ represent lateral drag interference co­ efficient and sheltering effect coefficient. 9

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 12. Variation of KZ with respect to S/D for two piles in tandem arrangement under the action of different seismic waves (l ¼ 30 m and D ¼ 1.8 m). Table 3 Calculation results of hydrodynamic force and pile group effect K for three piles in side by side arrangement under action of earthquakes (l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m).

Seismic wave E#1 E#2 E#3

Relative water depth 0.5 0.83 0.5 0.83 0.5 0.83

Hydrodynamic force/kN

K

Single isolated pile

No.1 pile

No.2 pile

No.3 pile

Pile group

Simulation value

Formula value

70.73 138.52 45.63 85.54 41.58 46.18

73.45 138.75 48.93 88.44 43.94 49.50

76.25 142.31 50.67 95.45 45.74 51.72

73.27 138.58 48.81 87.74 43.80 49.33

222.97 419.64 148.41 271.61 133.48 150.55

1.05 1.00 1.08 1.06 1.07 1.09

1.05 1.05 1.04 1.05 1.03 1.05

pile in side by side arrangement as compared to that obtained for the middle pile is physically justified.

sides (front and back). The same characteristics exist for different S/D. Ki (i ¼ 1, 2, and 3) increases as S/D increases until to reach constant value 1 at about S/D ¼ 5. It is similar to the side by side arrangement. Merely, pile group effect makes hydrodynamic force reduced here. The detailed description figures are omitted here for brevity. Comparison from the listed data in Table 3 shows that the formula calculated K value is relatively close to the simulated value, where the maximum error is 9.6%. For the cases of different seismic waves and different water depths, the values of K are relatively close, where the maximum relative variation rate is 9.4%. Consequently, it is still concluded that S/D is the most relevant influencing parameter for pile group effect.

4.2.2. Three piles in tandem arrangement For a given three piles in tandem arrangement where l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m, the resulting hydrodynamic forces on single isolated pile, pile group, and each pile of this group are listed in Table 4. It can be seen from the table that the hydrodynamic force on each pile within the group is slightly smaller than that on single isolated pile, which is attributed to pile group effect. Note that the reduction of hydrodynamic force on the middle pile in tandem arrangement is more noticeable than the side pile owing to the influence of two neighbouring piles from both 10

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Ocean Engineering 199 (2020) 106999

Fig. 13. Variation of Ki with respect to S/D for three piles in side by side arrangement with different d/l under the action of different seismic waves (l ¼ 30 m and D ¼ 1.8 m).

4.2.3. Four piles in 2 � 2 arrangement For a given pile group with four piles in 2 � 2 arrangement (l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m), the resulting hydrodynamic forces on single isolated pile, pile group, and each pile of this group are listed in Table 5. It can be informed from the table that the hydrodynamic force on each pile in the group is slightly smaller than that on single isolated pile as the result of pile group effect. For the symmetrical arrangement of S/D ¼ 2.78, each pile in the group is exposed to both lateral and longitudinal interference from the symmetric neighbouring piles, so that K values are grouped around K ¼ 1 indicating that there is inconspicuous pile group effect. For seismic wave E#2 and E#3, K value hardly varies with water depth. In general, for the cases of different seismic waves and different water depths, the values of K are relatively close, where the maximum relative variation rate is 10%. Also the formula calculated K value is

relatively close to the simulated value, where the maximum error is 9.1%. Therefore, it is proved that the proposed approximate formula is reasonable, but it is only suitable for the rectangular arrangement with orthogonal angler to seismic wave direction. 4.2.4. Nine piles in 3 � 3 arrangement For a given pile group with nine piles in 3 � 3 arrangement (l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m), the resulting hydrodynamic forces are shown in Fig. 14. It is observed from Fig. 14 that hydrodynamic force on each pile is significantly affected by the combination of seismic wave and water depth. For a given seismic wave and water depth, the scatter between the hydrodynamic force values of piles within group and single isolated pile is very small and the data points are concentrated around the optimal line. The variation range of Ki is 0.93–1.04, and K is very close to 11

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Table 4 Calculation results of hydrodynamic force and pile group effect K for three piles in tandem arrangement under action of earthquakes (l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m).

Seismic wave E#1 E#2 E#3

Relative water depth 0.5 0.83 0.5 0.83 0.5 0.83

Hydrodynamic force/kN

K

Single isolated pile

No.1 pile

No.2 pile

No.3 pile

Pile group

Simulation value

Formula value

70.73 138.52 45.63 85.54 41.58 46.18

63.26 117.64 43.14 78.19 38.94 43.44

62.22 117.50 40.01 77.35 37.23 41.92

63.41 118.21 43.21 78.60 39.15 43.66

188.85 353.35 126.36 234.14 115.33 129.02

0.89 0.85 0.92 0.91 0.92 0.93

0.95 0.94 0.98 0.96 0.96 0.95

Table 5 Calculation results of hydrodynamic force and pile group effect K for four piles in 2 � 2 arrangement under action of earthquakes (l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m).

Seismic wave

Relative water depth

E#1

0.5 0.83 0.5 0.83 0.5 0.83

E#2 E#3

Hydrodynamic force/kN

K

Single isolated pile

No.1 pile

No.2 pile

No.3 pile

No.4 pile

Pile group

Simulation value

Formula value

70.73 138.52 45.63 85.54 41.58 46.18

68.49 123.60 44.62 83.39 40.61 45.57

69.36 126.85 45.02 85.54 41.52 46.52

68.16 123.91 44.20 84.01 40.64 45.49

69.57 125.37 45.32 84.77 41.40 46.46

275.51 498.68 179.11 337.64 164.11 183.95

0.97 0.90 0.98 0.98 0.99 0.99

1.00 0.99 1.01 1.01 0.99 0.99

Fig. 14. Calculation results of hydrodynamic force and pile group effect K for nine piles in 3 � 3 arrangement under action of earthquakes (l ¼ 30 m, S ¼ 5 m and D ¼ 1.8 m).

1. The results show pile group effect K is not highly sensitive to seismic wave attributes and d/l. Pile group effect on hydrodynamic force on whole pile group with square arrangement or approximate square arrangement generally can be ignored, yet whether pile group effect on hydrodynamic force on single pile within the group can be neglected or not mainly depends on relative spacing S/D. In fact, the hydrodynamic force of single pile in pile group is more concerned than that of pile

group in design. The important implication to be observed alertly from Fig. 14 in the manuscript is that the hydrodynamic force on corner piles numbered 1, 3, 7 and 9 is greater than that on centre pile numbered 5. In order to make the multiple curves drawn in a picture easier to distinguish and identify, the piles numbered 2, 5, and 8 in the group are selected as the representative, and the hydrodynamic force time-history curves are 12

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Ocean Engineering 199 (2020) 106999

Fig. 15. Hydrodynamic force on piles versus time during earthquake for d/l ¼ 0.5.

Fig. 16. Hydrodynamic force on piles versus time during earthquake for d/l ¼ 0.83.

Fig. 17. Sketch of pile foundation (units: elevation in m, rest in cm).

plotted in Figs. 15 and 16. It can be seen that the vibration paces of each pile in the group during earthquake are almost consistent and, conse­ quently, the peak values of earthquake-induced hydrodynamic forces occur at the same moment corresponding to the maximum value of seismic wave displacement time history, due to no significant seismic wave passage effect as the result of short distance between piles and the assumption of seabed as a rigid body motion. Therefore, the time of

attaining maximum response for each pile is consistent and there is no obvious phase difference. It is further verified that the simplified defi­ nition of pile group effect coefficient (Formula 8) is reasonable. 4.3. Interference characteristics of pile group of a sea-crossing bridge A foundation of sea-crossing cable-stayed bridge adopts 31Ф2.5 m 13

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Ocean Engineering 199 (2020) 106999

Fig. 18. Analysis model for pile group foundation: (a)geometric model and (b)finite element model.

interlaced features reveals the apparent mutual interference effect be­ tween piles. Generally, the stiffness of a whole pile group-cap structure is much greater than that of an isolated pile group structure without cap. Therefore, the influence of the pile cap on hydrodynamic force on piles is mainly because the pile cap strengthens the overall stiffness of pile group structure, thereby reducing the hydrodynamic force generated by elastic deformation. The second important implication from Fig. 20 is that the direction of seismic wave action will change the hydrodynamic force of single pile in pile group while it is noted from Table 6 that it has no obvious effect on the hydrodynamic force on pile group. As seen from Table 6, for different action directions of seismic waves, there is no significant dif­ ference in the uplift force, while the horizontal force has a great change as the result of different hydrodynamic action areas on the side of cap. In view of the large bottom area of the cap, the uplift force is much greater than the horizontal force. Note that the uplift force is greater than the hydrostatic buoyancy, due to the water fluctuation caused by the earthquake. The third important implication to be observed alertly from Fig. 20 is that the hydrodynamic force on corner piles numbered 1, 5, 27 and 31 is greater than that on centre pile (No. 16). Therefore, the former deserves a key concern, and it is necessary to strengthen pertinently the stiffness of piles in the group, so as to promote overall seismic performance of sea-crossing bridge.

Fig. 19. Two modes of the input seismic wave in the transverse and longitu­ dinal direction of bridge.

bored piles. Sketch of pile foundation is depicted in Fig. 17. To simplify analysis, it is assumed that the bored piles are fixed to the seabed at the mud level, and free pile length, namely vertical distance between mud level and bottom of cap is unified to 25 m, where the cap is partially submerged in water at the ordinary water level, regardless of the interaction between water and soil, as well as the influence of the structure above cap. The analysis model is shown in Fig. 18. As seen in Fig. 19, two action modes, namely transverse and longi­ tudinal excitation, are discussed below, where E#1 is chosen as the excitation seismic wave. The first noticeable implication to be drawn from Fig. 20 is that most of the data points are below the red dashed line for different cases, i.e. the hydrodynamic force of most piles within pile group is smaller than that of single isolated pile. And for the case with cap, the hydrodynamic force of all piles is less than that of single isolated pile due to the double contribution of pile group effect and cap action. The calculation results show that the hydrodynamic force on the piles with cap is reduced by 20.2%~34.9% compared to the hydrodynamic force on the piles without cap, pile group effect K changes from 0.96 in the absence of cap to 0.69 in the presence of cap, as listed in Table 6, where S/D is 2.56 (<4.5), and the range of Ki is 0.88–1.05 for the case of no cap, indicating that the hydrodynamic force on a pile is significantly affected by the cap, but the relative pile spacing is also the important influencing parameter. It is stated from the streamline distribution nephogram in Fig. 21 that streamline construction with curved and

5. Pile group effect of hydrodynamic forces during earthquakes considering current influence 5.1. Interference characteristics of two piles in side by side arrangement For a given pile group with two piles in side by side arrangement ((l ¼ 30 m, d ¼ 25 m and D ¼ 1.8 m)), assume that seismic wave (E#1) is consistent with the direction of current, and both are perpendicular to the direction of pile arrangement. Table 7 lists the values of lateral interference KH, and the results illustrating the relationship between KH and relative spacing S/D ¼ 1.5–6 for different current velocity of v are plotted in Fig. 22. As seen in Fig. 22, the value of KH deceases gradually from >1 to 1 with increasing S/D, and it is stated that lateral interference, reduces as the distance between two piles increases, indicating that pile group ef­ fect in the earthquake-current combined flow field results in the amplification of hydrodynamic force on each pile within the group compared with single isolated pile. This magnification has not dis­ appeared until S/D � 6. It is noted that pile group effect is affected by current velocity, where 14

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Ocean Engineering 199 (2020) 106999

Fig. 20. Calculation results of hydrodynamic force on piles in pile group of sea-crossing bridge under E#1 excitation. Table 6 Calculation results of hydrodynamic force on sea-crossing bridge foundation and pile group effect under E#1 excitation. Cases First action mode for pile group without cap Second action mode for pile group without cap First action mode for pile group with cap Second action mode for pile group with cap

Hydrodynamic force on pile group/kN 10152.08 10240.48 7399.82 7355.48

the influence law relative to current velocity is not pronounced. But one thing is for sure, compared with the case of only earthquake-induced flow field under the same conditions, the highest magnification slightly decreases and the range of S/D influencing pile group effect has been expanded due to current driving effect. The upper limit value of S/ D influencing pile group effect changes from 4.5 where only earthquake acts to 6 where earthquake and current act together.

Hydrodynamic force on cap/kN

Pile group effect

Buoyancy

Horizontal force

Uplift force

K

Ki

– – 20535.60 20535.60

– – 174.11 292.09

– – 21244.70 21263.51

0.95 0.96 0.69 0.69

0.88–1.05 0.91–1.02 0.64–0.74 0.65–0.72

combined flow field. 6. Pile group effect of hydrodynamic forces during earthquakes considering wave influence Table 9 lists the results of KH for two piles in side by side arrangement under the combined action of earthquake (E#1) and wave, assuming that seismic wave is consistent with the direction of wave, and both are perpendicular to the direction of pile arrangement, where l ¼ 30 m, d ¼ 25 m, D ¼ 1.8 m, H ¼ 3 m, and T ¼ 6s. It can be seen from the table that the value of KH decreases with the increase of S/D, that is, the lateral interference between two piles decreases as S/D increases. The KH values are greater than 1 for S/D < 5, indicating that pile group effect in the coexistence field of earthquake and wave makes the magnification of hydrodynamic force on piles within the group compared to single iso­ lated pile. This magnification has almost vanished for S/D ¼ 5, and all piles behave like a single isolated pile in terms of the hydrodynamic force. The characteristics of pile groups in the coexisting field of earthquake and wave are different from those in the earthquake-induced flow fields, but not large. On balance, the effect of wave on earthquakeinduced hydrodynamic force is weaker than that of current. The wave field and the earthquake-induced flow fields are essentially same as the oscillating flow, but the boundary layer separation is not noticeable under the seismic excitation, where the drag force is gener­ ally smaller than the inertial force. In addition, the dominant frequency of seismic wave excitation is often higher than the loading frequency of regular wave. And the Keulegan–Carpenter (KC) number (KC ¼ umaxT/ D, where umax is the maximum horizontal wave-induced flow velocity, T is wave period, D is pile diameter) in the earthquake-induced flow field tends to be smaller than that in the wave field, so the earthquake-

5.2. Interference characteristics of two piles in tandem arrangement For a given pile group with two piles in tandem arrangement ((l ¼ 30 m, d ¼ 25 m and D ¼ 1.8 m)), assume that seismic wave (E#1) is consistent with the direction of current, and both are parallel to the direction of pile arrangement. Table 8 lists the values of longitudinal interference KZ, and the results illustrating the relationship between KZ and relative spacing S/D ¼ 1.5–6.5 for different v are plotted in Fig. 23. It is observed from Fig. 23 that the value of KZ increases gradually from <1 to 1 with increasing S/D, i.e. longitudinal interference reduces as the distance between two piles increases, indicating that pile group effect in the earthquake-current combined flow field results in the reduction of hydrodynamic force on each pile within the group compared with single isolated pile. This reduction has almost dis­ appeared when about S/D ¼ 6. Like the case of side by side arrangement, pile group effect is affected by current velocity, where the influence law relative to current velocity is also not pronounced. One thing is still certain that the range of S/D influencing pile group effect has been expanded due to current driving effect, compared with the case of only earthquake-induced flow field under the same conditions. When KZ ¼ 1, S/D ¼ 4.5 for the earthquakeinduced flow field while about S/D ¼ 6 for the earthquake-current 15

W. An-jie and Y. Wan-li

Ocean Engineering 199 (2020) 106999

Fig. 21. Streamline distribution nephogram of group piles for two action models at t ¼ 2.2s.

Table 7 Calculation results of KH for two piles in side by side arrangement at d/l ¼ 0.83, under E#1 excitation considering current influence.

v (m/s)

S/D

Fgroup (kN)

KH

v (m/s)

S/D

Fgroup (kN)

KH

0.5

0 1.5 3 5 6 0 1.5 3 5 6

130.23 320.75 280.21 262.37 260.47 149.94 365.24 314.83 305.68 303.35

– 1.23 1.08 1.01 1.00 – 1.22 1.05 1.02 1.01

1.5

0 1.5 3 5 6 0 1.5 3 5 6

179.17 427.98 377.40 368.87 364.81 218.80 539.54 461.88 450.13 445.96

– 1.19 1.05 1.03 1.02 – 1.23 1.06 1.03 1.02

1

2

Notes: S/D ¼ 0 denotes the case of single isolated pile, v is current velocity, and Fgroup is hydrodynamic force on pile group.

Fig. 22. Variation of KH with respect to S/D for two piles in side by side arrangement at d/l ¼ 0.83, under E#1 excitation considering current influence.

induced flow field is not exactly equivalent to the wave field. Table 10 shows the results of KZ for two piles in tandem arrangement under the combined action of earthquake (E#1) and wave, supposing that seismic wave is the same as the direction of wave, and both are parallel to the direction of pile arrangement, where l ¼ 30 m, d ¼ 25 m, D ¼ 1.8 m, H ¼ 3 m, and T ¼ 6s. As seen in Table 10, the value of KZ increases with the increase of S/ D, i.e. the longitudinal interference between two piles decreases as relative spacing increases. The KZ values are less than 1 for S/D < 5.5, indicating that pile group effect in the coexistence field of earthquake and wave makes the reduction of hydrodynamic force on piles within the group compared to single isolated pile. This reduction has almost van­ ished for about S/D ¼ 5.5. It is noted that compared with the case of only earthquake-induced flow field under the same conditions, the highest

reduction at S/D ¼ 2 decreases while the range of S/D influencing pile group effect has been expanded due to the presence of wave. 7. Conclusions Deep-water pile groups exposed to earthquake as well as to earthquake-current and earthquake-wave were investigated, but the focus was put on the former. To study pile group effect on the hydro­ dynamic force on a pile during earthquakes, different pile arrangements including single, side by side, tandem, 2 � 2, 3 � 3 and a sea-crossing bridge foundation were performed by the proven numerical method based on the second development on ANSYS software. The obtained key results may be summarized as follows:

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Ocean Engineering 199 (2020) 106999

Table 8 Calculation results of KZ for two piles in tandem arrangement at d/l ¼ 0.83, under E#1 excitation considering current influence.

Table 10 Calculation results of KZ for two piles in tandem arrangement at d/l ¼ 0.83, under E#1 excitation considering wave influence where H ¼ 3 m, and T ¼ 6s.

v (m/s)

S/D

Fgroup (kN)

KZ

v (m/s)

S/D

Fgroup (kN)

KZ

Relative spacing S/D

2

3

4

5

5.5

0.5

0 1.5 3 6 6.5 0 1.5 3 6 6.5

130.23 201.30 244.83 258.05 260.45 149.94 225.53 291.75 303.35 303.66

– 0.77 0.94 0.99 1.00 – 0.75 0.97 1.01 1.01

1.5

0 1.5 3 6 6.5 0 1.5 3 6 6.5

179.17 263.16 334.56 358.05 359.76 218.80 304.66 402.11 439.85 439.89

– 0.73 0.93 1.00 1.00 – 0.70 0.92 1.00 1.00

Pile group effect KZ in the coexisting field of earthquake and wave Pile group effect KZ in the earthquakeinduced flow field

0.91

0.97

0.96

0.98

1.00

0.82

0.94

0.96

1.00

1.00

1

2

(2) Earthquake-induced hydrodynamic force consists of two parts: namely one part generated by rigid motion and other part generated by elastic vibration. As these two peaks do not occur at the same time, in general, the maximum value of total hydrody­ namic force can’t be simply added by the two ones. (3) Under the action of earthquake, the lateral interference between two piles in side by side arrangement results in the magnification of hydrodynamic force on each pile within the group compared to single isolated pile, while the longitudinal interference between two piles in tandem arrangement causes the reduction of hydro­ dynamic force on each pile within the group compared to single isolated pile. And both decrease by increasing relative spacing S/ D, which have not disappeared until S/D ¼ 4.5, but until S/D ¼ 6 in the earthquake-current combined field, and until about S/D ¼ 5.5 in the earthquake-wave combined field. Compared with the case of only earthquake-induced flow field under the same con­ ditions, the highest magnification or reduction in the coexisting field slightly decreases while the range of S/D influencing pile group effect has been expanded due to the presence of current and wave. (4) Pile group effect is more or less affected by earthquake strength, relative frequency fe/fs, relative water depth d/l, and relative spacing S/D. However, S/D is the most relevant influencing parameter. (5) It is proved that the proposed approximate formula for pile group effect K is reasonable, but it is only suitable for the rectangular arrangement with orthogonal angle to seismic wave direction. The pile group effect on hydrodynamic force on whole pile group with square arrangement can be ignored, yet its influence on hydrodynamic force on a single pile within the group may not be neglected, especially at small relative pile spacing. (6) For a sea-crossing bridge foundation, the hydrodynamic force on single pile is affected by both pile group effect and cap action, K is equal to 0.69, implying that failure to take these influencing factors into account, may seriously overestimate the hydrody­ namic force of the single pile, and consequently resulting in a significant increase in project cost. Results show the hydrody­ namic force on corner pile is greater than that on centre pile. Therefore, the former deserves a key concern, and it is necessary to strengthen pertinently the stiffness of piles in the group, so as to improve overall seismic performance of sea-crossing bridge.

Notes: S/D ¼ 0 denotes the case of single isolated pile, v is current velocity, and Fgroup is hydrodynamic force on pile group.

Fig. 23. Variation of KZ with respect to S/D for two piles in tandem arrange­ ment at d/l ¼ 0.83, under E#1 excitation considering current influence.

Table 9 Calculation results of KH for two piles in side by side arrangement at d/l ¼ 0.83, under E#1 excitation considering wave influence where H ¼ 3 m, and T ¼ 6s.

Relative spacing S/D

2

3

4

5

Pile group effect KH in the coexisting field of earthquake and wave Pile group effect KH in the earthquake-induced flow field

1.09

1.06

1.03

1.00

1.15

1.07

1.04

1.00

Based on the results of this study, a substantial improvement of the understanding of pile group effect on the hydrodynamic force of a pile has been achieved which might lead to a safer design of sea-crossing bridge structures exposed to earthquakes. Because of high complexity of the coexistence fields, e.g. earthquake-current combined flow field, earthquake - wave combined flow field, and earthquake - wave-current combined flow field, further research is needed in these aspects.

(1) Comparison between the proposed numerical simulation method and Morison method, shows the presented method has a high accuracy and reliability, which can be used as an alternative for hydrodynamic analysis. One point to be clarified is that Morison equation’s hydrodynamic coefficients, i.e. CD and CM, are taken as constants without changing with water depth, consequently, by the Morison method a conservative result may be given when relative water depth is small, while a unsafe result may be ob­ tained when relative water depth is large.

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Ocean Engineering 199 (2020) 106999

Acknowledgements

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This work is supported by the National Natural Science Foundation of China (Grant No.51678491) and the Startup Project for High-level Talents of Guizhou Institute of Technology (Grant No. 0203001019008). The authors wish to thank the three anonymous re­ viewers for comments on the manuscript. References Cokgor, S., 2002. Hydrodynamic forces on a partly buried cylinder exposed to combined waves and current. Ocean Eng. 29 (7), 753–768. Deng, S.Y., Xiu, Q.H., 2009. Test study on flow resistance of cylindrical piles in rectangular arrangement under uniform current. Yangtze River 40 (19), 79–81. Deng, S.Y., Zhang, J.L., 2007. Study on experiment of testing for drag force of water flow around group of piles. J. North China Ins. Water Conserv. Hydroelectr. Power 28 (2), 86–90. Guo, A., Fang, Q., Li, H., 2015. Analytical solution of hurricane wave forces acting on submerged bridge decks. Ocean Eng. 108, 519–528. Huang, X., Li, Z.X., 2011. Influence of hydrodynamic pressure on seismic response of bridge piers in deep water. China Civ. Eng. J. 44 (1), 65–73. Huang, B., Zhu, B., Cui, S., et al., 2018. Experimental and numerical modelling of wave forces on coastal bridge superstructures with box girders, partI: regular waves. Ocean Eng. 149, 53–77. Li, Y., 1991. Wave-current forces on slender circular cylinders. China Ocean Eng. 5 (3), 287–310. Li, Z.X., Xuang, X., 2012. Dynamic response of bridge in deep water under combined earthquake and wave actions. China Civ. Eng. J. 45 (11), 134–140.

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