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Experimental study of wave loads on elevated pile cap of pile group foundation for sea-crossing bridges
Ocean Engineering 197 (2020) 106896
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Experimental study of wave loads on elevated pile cap of pile group foundation for sea-crossing bridges Bo Xu a, Kai Wei a, *, Shunquan Qin a, b, Jie Hong a a b
Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, China China Railway Major Bridge Reconnaissance & Design Institute Co., Ltd., Wuhan, 430050, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Wave force Elevated pile cap Sea-crossing bridges Wave flume experiment Pile group Clearance height
Elevated pile cap, which transits the load from sea-crossing bridge pylon to pile group, is always close to the sea level and suffering from extreme wave loadings. This paper describes a wave flume experimental program to investigate wave loads on the elevated pile cap of pile foundation for sea-crossing bridges. Two specimen con figurations including isolated cap and cap with pile group were set up with different clearance heights. Wave forces on the cap, including horizontal and vertical force and moment, were measured simultaneously under five nonlinear regular wave trains. The effects of pile group, clearance height and wave height on wave forces on the cap were investigated. The experimental results show that the clearance height and wave height affect wave forces significantly. The largest horizontal force occurs when the cap is fully submerged, while the largest vertical force occurs when the bottom of the cap locates at still water level (SWL). As the wave height increases, the horizontal forces increase except for the fully submerged case and the vertical forces increase firstly but decrease when the wave height exceeds 0.18 m. Pile group increases the vertical uplift forces on the cap when the bottom of the cap locates above SWL.
1. Introduction With the advent of booming demand in ground transportation across straits and channels, development of long-span sea-crossing bridge is being pursued to connect the mainland and islands all over the world. Pile group foundations are widely used to support contemporary longspan sea-crossing bridges, such as China East Sea Bridge (Liu et al., 2007), San Francisco–Oakland Bay Bridge (Frick, 2008), Hangzhou Bay Bridge (Ye, 2010) and Pingtan Rail-cum-road Sea-crossing Bridge (Ti et al., 2019), due to its good applicability in deep water site and complex seabed terrain. However, the elevated pile cap, which transits the load from bridge pylon to pile group, is always close to the sea level and suffering from extreme wave loadings (Ti et al., 2018). In recent years, catastrophes like hurricanes Ivan in 2004, Katrina in 2005, Ike in 2008, Indian Ocean tsunami in 2004, Tohoku tsunami in 2011 and Typhoon Haiyan in 2013 caused extensive damage to infrastructures and led a large number of coastal bridges to be damaged or collapsed (Douglass et al., 2004; Robertson et al., 2007; Stearns and Padgett, 2011; Unjoh, 2006; Kosa, 2011; Akiyama et al., 2013; Mas et al., 2015). Considering the risk of structural failure of sea crossing bridges due to extreme waves (Ti et al., 2019; Wei et al., 2019), it is hence important to estimate wave
loads on the elevated pile cap for assessments of structure safety and construction feasibility of pile foundation for sea-crossing bridges. Many experiments were carried out to investigate the wave force and pressure on offshore elevated structures. Lan et al. (2006) carried out an experiment to investigate the vertical force on a slab of sea-crossing bridge piers under regular and irregular waves. Ding et al. (2008) designed an experiment to investigate unidirectional random wave slamming on a platform structure and effects of the wave direction. Cuomo et al. (2009) conducted large-scale flume experiments to inves tigate the dynamics of wave loading on coastal highway bridges. Based on the experimental results, Cuomo et al. (2010) introduced a prediction formula for quasi-static and breaking wave-induced impact forces and overturning moments at vertical seawalls and breakwaters. Bradner et al. (2010) performed laboratory tests on 1:5 scale reinforced concrete specimen of coastal bridge superstructure to investigate wave-in-deck loads under regular and random waves considering the flexibility and dynamic properties of the supporting bridge substructure. Wei et al. (2013) conducted an experiment to study the dynamic response of bridge pile foundations submerged in water. Guo et al. (2015) con ducted a hydrodynamic experiment to investigate the wave force acting on the superstructures of coastal bridge and compared to Douglass et al.
* Corresponding author. E-mail addresses: [email protected] (B. Xu), [email protected] (K. Wei). https://doi.org/10.1016/j.oceaneng.2019.106896 Received 21 August 2018; Received in revised form 24 October 2019; Accepted 23 December 2019 Available online 7 January 2020 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. The testing facility: (a) photo and detail of the wave flume; (b) dimension of the flume (all dimensions are in meter).
(2006) and Model of AASHTO guidelines (2008). Seiffert et al. (2014) experimentally studied the horizontal and vertical forces acting on a two-dimensional horizontal plate by solitary waves. Wave loads under different wave heights, water depths as well as submergence depths and elevations above still water level were discussed in detail. Chuang et al. (2018) investigated the effect of aeration and air compressibility on the wave slamming forces of offshore platform experimentally. However, these studies paid little attention to the elevated pile cap of pile foun dation for sea-crossing bridges. Moreover, Ti et al. (2018) conducted a field measurement of random wave pressure on a real cofferdam for sea-crossing bridge during typhoon and observed the secondary peaks in the pressure spectrums. Although the first order diffraction theory is commonly used to assess the wave force on the large-scale marine structures, such as elevated pile cap, it is not sufficient to deal with waves with nonlinear kinematics and caps locates above the mean sea level (Molin, 2002). The blocking and shading effect of pile group on the wave force of the pile cap has yet been well understood. This study investigates the wave forces acting on the elevated cap of pile foundation for sea-crossing bridges experimentally. The 5th-order Stokes regular waves with five different heights and the associated lower bound of periods according to testing breaking limits were used to simulate the extreme waves. Two configurations of the specimen including isolated cap and cap with pile group were set up considering five clearance heights. The following two objectives are set: (1) to discern the characteristics of the time history of wave forces under different clearance and wave height conditions; and (2) to understand effects of clearance height, wave height and pile group on the wave loads acting on the pile cap.
2. Experimental setup 2.1. Wave testing facilities The experiments of this study were conducted in the wave flume at Southwest Jiaotong University. The flume is equipped with a piston-type wave-generating system. The flume is 60 m in length, 2 m in width and 1.8 m in height with side walls comprised of steel frames and toughened glass, which provides a clear and wide visual range during the testing. As shown in Fig. 1, the piston-type wave maker designed by Dalian Uni versity of Technology is installed at the right end of the flume and able to generate various one-dimensional wave environments, such as linear and nonlinear regular wave, irregular wave, focused wave, breaking wave, etc. In the process of making waves, the motion file of the wave maker could be updated automatically according to the feedback from actual wave condition obtained by the data acquisition system installed in the flume until the expected waves are made. The wave-absorbing beach, which is made of porous polymer material, is positioned at the left end of the flume to absorb the wave energy and to reduce the wave reflection. The testing bridge locates and operates on the stainless-steel tracks placed at the top of the side walls. In the following testing, we define the Cartesian coordinate of the flume as illustrated in Fig. 1 (a). Xdirection is the longitudinal direction from the wave maker to the waveabsorbing beach, Y-direction is the transverse direction pointing from in-plane to out-of-plane against the flume and Z-direction is the vertical direction with positive direction upward. Four wave gauges were installed in the flume to measure the water surface elevations, as illustrated in Fig. 1 (b). Wave gauges WG2, WG3 and WG 4 were installed at 10 cm left, 10 cm right and 10 cm rear of the tested cap specimen, respectively. WG1 was installed at 2 m front of the WG4 to measure the wave conditions in the empty wave flume. After the testing specimen was mounted in the flume, WG1 was relocated at 40 cm 2
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Fig. 2. Sketch of pile group foundation for bridge that inspired the specimen: (a) Sea-crossing bridge supported by a group-pile cap foundation; (b) the photo of wave acting on an elevated pile cap under construction.
Fig. 3. Dimensions and installation detail of the tested specimen: (a) dimensions of the specimen; (b) detail of the connection between specimen and testing bridge (all dimensions are in centimeter).
front of the cap in order to measure the reflection of the wave induced by the specimen. The sampling frequency of wave gauges was set to 100 Hz that is sufficient to record the fluctuation of water surface through a subsequent comparison for validity. The F/T (force and torque) transducer was used to measure both force and torque acting on the specimen along X-, Y- and Z-directions. Although the sampling frequency of the transducer can be set as high as 7000 Hz, it was found in previous experiments (Tomiczek et al., 2017; Guo et al., 2015) that a sampling frequency of 200 Hz is sufficient to capture the impulsive forces and structural responses signals. In this study, a sampling frequency of 1000 Hz, which is higher than 200 Hz, was chosen here to ensure the accuracy and efficiency of the measure ments during wave impacts on the cap. All experimental processes were recorded visually through HD camera with a sampling speed of 60 fps.
bridge that crosses Pingtan Strait in the southeast coast of China. As shown in Fig. 2 (a), the prototype foundation is composed of twelve concrete piles and one concrete pile cap. The pile cap is 36 m in length, 24 m in width and 13.5 m in height, elevated on the piles with diameter of 4.5 m. Fig. 2 (b) shows a photo of the wave acting on an elevated pile cap under construction. The reduced scale tested specimen, shown in Fig. 3 (a), with di mensions inspired from the prototype foundation, was composed of a group of 12 circular piles and a rectangular pile cap. Considering the flume dimensions and abilities of experimental facilities, the tested specimen was designed according to the Froude similarity using a 1:90 geometric scaling. The tested specimen was made of acrylic material, which has light weight and high rigidity. The pile was manufactured with external diameter of 5 cm and thickness of 5 mm to ensure its strength. The dimension of the cap specimen was a rigid box with length of 40 cm, width of 26.67 cm and height of 15 cm. In order to study the effect of the pile group on the wave force of the pile cap, two configurations of tested specimen, (1) the isolated pile cap and (2) the pile cap with pile group, were set up. For the isolated pile cap
2.2. Configuration of tested specimen The design of the reduced scale specimens tested in this study was inspired by a pile cap of the group-pile foundation for a sea-crossing 3
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Fig. 4. Specification of clearance height for specimen: pile cap with pile group (all dimensions are in centimeter).
configuration, the F/T transducer was fastened firmly on the testing bridge through the lifting rod. The tested cap specimen was suspended on the bottom of the F/T transducer by the rigid alloy frame, as illus trated in Fig. 3 (b). For the pile cap with pile group configuration, the piles were fully fixed on the floor of the wave flume. Considering the major objective of this study is to address the wave force acting on the pile cap, a tiny gap (approximately 1 mm) was kept between the bottom of the cap and the top of the pile group to exclude the wave force on the pile group from the force measured by the F/T transducer.
Table 1 Testing regular wave parameters. Wave height H (m)
Wave period T (s)
Wave length L(m)
H/gT2
d/gT2
0.04 0.06 0.09 0.12 0.15 0.18 0.21 0.30 0.40 0.50
0.59 0.73 0.91 1.04 1.15 1.25 1.35 1.60 1.88 2.06
0.570 0.872 1.350 1.762 2.147 2.517 2.896 3.942 5.110 5.959
0.0117 0.0115 0.0111 0.0113 0.0116 0.0117 0.0117 0.0119 0.0115 0.0120
0.2636 0.1722 0.1108 0.0848 0.0694 0.0587 0.0503 0.0358 0.0260 0.0216
2.3. Clearance height setting In the actual practice of the pile group foundation, one important design parameter is the clearance height between the bottom of the cap and Still Water Level (SWL). Five different clearance heights s were investigated experimentally for both configurations of tested specimen.
Fig. 5. Comparison of the water elevation between the measured data and the theoretical solutions of 5th order Stokes wave theory under different wave conditions: (a) H ¼ 0.09 m T ¼ 0.91 s; (b) H ¼ 0.12 m T ¼ 1.04 s; (c) H ¼ 0.15 m T ¼ 1.15 s; (d) H ¼ 0.18 m T ¼ 1.25 s; (e) H ¼ 0.21 m T ¼ 1.35 s. 4
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Table 2 The first order natural vibration frequencies of the tested specimen.
Fig. 6. Comparison of water surface measured by different wave gauges under wave condition with H ¼ 0.12 m and T ¼ 1.04 s.
s (cm)
f1first (Hz)
f1second (Hz)
f1third (Hz)
~f (Hz) 1
4 2 0
15.69 15.45 15.16 13.09 10.32
15.78 15.51 15.26 13.15 10.31
15.66 15.47 15.16 13.13 10.33
15.71 15.48 15.19 13.12 10.32
7.5 15
measured by wave gauge WG2 in the flume and the theoretical eleva tions calculated by the 5th order Stokes wave theory were compared in Fig. 5. Good agreements can be found between the measured and theoretical water elevation for all five experimental wave conditions, which demonstrates that the wave maker generates satisfactory regular wave trains in the flume. Moreover, water surface elevations measured by wave gauges WG1, WG2 and WG4 are compared as well to verify the stability of the wave along the flume. Because WG2 and WG3 locate in the same section, only WG2 is used. It is clear from Fig. 6 that there are few deviations among them in terms of wave heights and periods neglecting the phase differ ence caused by the position. It can be concluded from the results that the wave train was stable while it was propagating in the flume.
As shown in Fig. 4, s ¼ 2 and 4 cm were two cases that the cap bottom was above the SWL, s ¼ 0 cm referred to the case that the cap bottom coincided with the SWL, and the latter two cases s ¼ 7.5 cm and 15 cm related to the cases that the cap was partially and fully submerged in the water, respectively. In the experiment, the depth of the water d in the flume was fixed at 0.9 m during the whole experimental study, which was conducive to comparison among the cases with different clearance heights and wave conditions. Therefore, the clearance height s was increased or decreased by changing the length of the lifting rod. The pile consists of five segments to meet five different clearance height re quirements. Upper segment was connected to the lower segment by acylic. When the clearance height was changed, the height of the pile group was changed accordingly by adding or removing the segments to keep the air gap between cap and pile group constant.
3.2. Natural vibration testing Although the support system of the cap specimen is expected to be designed as rigid as possible, the natural vibration of the tested speci mens is still difficult to be eliminated, especially when it vibrates along X direction under wave impact. Moreover, the extending length of the lifting rod changes with the clearance height and causes the change of natural vibration frequency of the specimen. It is hence necessary to obtain the first order natural vibration frequency f1 of the specimen installed on the testing bridge with different clearance heights. The free vibration testing was conducted as soon as the specimen was installed in the flume. An impact hammer was used to excite the specimen along the X-axis. Free vibration testing of the specimen was repeated three times to guarantee the accuracy of the testing results. Fig. 7 shows timedomain and frequency-domain results of X-direction reaction force re cord from the F/T transducer for the cap specimen with s ¼ 4 cm. Natural frequencies of cap specimens with five different clearance heights are listed in Table 2 with range from 10.32 to 15.71 Hz. It should be noted that the smaller natural frequency is associated to smaller clearance height. It is because, 1) smaller clearance height relates to longer free length of the lifting rod, and 2) when s becomes negative, the cap specimen is submerged in water, which reduces the natural fre quency of the specimen (Zhang et al., 2019).
2.4. Experimental wave conditions In this study, the water depth in the flume was set to be 0.9 m. The wave parameters of ten regular extreme wave trains with wave height H from 0.04 m to 0.5 m were extracted through wave conditions calibra tion tests in the flume. The period of the calibrated extreme wave train is close to the lower bound of period, which keeps the wave as steep as possible and exactly unbroken in the flume. Table 1 lists the wave pa rameters of the calibrated extreme wave trains in the flume. Five extreme wave trains with wave height H from 0.09 m to 0.21 m with increments of 0.03 m in Table 1 were selected as the experimental wave conditions for the following study. 3. Experimental procedure 3.1. Validation of regular wave train in the flume According to the wave parameters listed in Tables 1 and i.e., H/gT2 and d/gT2, all five selected extreme wave conditions were found in the range of the 5th order Stokes wave theory (Le Mehaute, 1976). To verify the quality of the generated regular waves, water surface elevations
Fig. 7. Time domain and frequency domain results of X-direction reaction force for the cap specimen with s ¼ 4 cm during natural vibration testing: (a) time history; (b) Fourier spectrum. 5
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Fig. 8. Results of horizontal force on the isolated cap specimen: (a) time history; (b) Fourier spectrum (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s).
Fig. 9. Results of vertical force on the isolated cap specimen: (a) time history; (b) Fourier spectrum (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s).
Fig. 10. The inundation of the cap during wave propagation (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s). (a) The cap elevated over static water (b) The cap submerged in wave crest.
3.3. Post-processing of measured data Figs. 8 and 9 present a sample of horizontal and vertical force on the isolated cap specimen with s ¼ 2 cm under wave conditions with H ¼ 0.15 m and T ¼ 1.15s. Slamming force can be clearly found in the time histories of both horizontal and vertical force. According to the Fourier spectrum, the first peak has the highest power and its frequency equals to the frequency of the regular wave. Except the first peak, there are still many higher order peaks, of which the frequencies are integer N times the first peak frequency. Similar phenomenon was also observed and reported by Guo et al. (2015) in their study. The nonlinear waves and higher-order velocity potentials induced by incident and diffracted waves are the reasons for the N-times frequency wave forces. It is interesting to notice that the amplitude of the Fourier spectrum of the horizontal wave force FX increases in the range from 12.44 Hz to 15.48 Hz that equals the average value of the natural frequency of the tested specimen (highlighted by the dashed black frame in Fig. 8b). This
Fig. 11. Comparison of raw data and filtered data with different cutoff fre quency fcutoff (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s). 6
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Fig. 12. Normalized time series of horizontal wave forces: subfigure (a), (c), (e), (g) and (i) shows the wave force on the isolated cap under five wave trains, respectively; and subfigure (b), (d), (f), (h) and (j) shows the wave force on the cap with pile group under five wave trains, respectively.
is mainly caused by the disturbance from structural natural vibration. However, the lower bound of the range is smaller than natural frequency of the tested specimen. Fig. 10 presents the inundation of the cap during wave propagation by images. The frequency we tested in Table 2 is structural frequency under still water (Fig. 10a). However, the extent of inundation changes significantly due to the fluctuation of water surface during the propagation of a wave train. Previous experiment carried out by Wei et al. (2013) has shown that structural natural frequency of the cap varies with the inundation volume of the cap. The larger the
inundation volume is, the lower the structural frequency is. It can explain why the lower bound of the frequency range for the structural vibration becomes smaller. The low-pass 3rd-order Butterworth filter is used to filter the raw data. The comparison of filtered data and raw data of horizontal force is illustrated in Fig. 11. When the cutoff frequency is set to be the exact natural frequency of specimen given in Table 2, significant noise still exists in the curve. However, most of the noise can be filtered, when the lower bound of structural vibration range (e.g. 12.44 Hz according to 7
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specimen are analyzed to obtain more insight into the characteristics of wave forces. And the effects of pile group, clearance height and wave height on the wave loads acting on the cap are investigated based on the experimental data. Five incident wave trains times five clearance heights constitute 25 experimental working conditions for each configuration of tested specimen, e.g. the isolated pile cap and the pile cap with pile group. 4.1. Horizontal wave forces on the cap In order to show and compare all clearance height conditions in one figure, the time axis of the time history of the wave force was normalized by the wave period corresponding to the given wave height. The time histories for all working conditions use the same phase. The duration of the time history is four times wave period. Fig. 12 shows the normalized time series of the horizontal wave forces FX on the isolated cap and the cap with pile group. The positive and negative value refer to forward and backward force along X-direction, respectively. Compared with the horizontal wave forces on the isolated cap, the force on the pile cap with pile group does not show obvious dissimilarity in the shape of the curve. For s � 0, the horizontal wave force arises abruptly reaching to the crest in a short time, then diminishes below zero, and finally returns to zero. With increase of submerged depth of the cap for s � 0, the slamming nature of the horizontal wave force disappears gradually and the peak value of the force become more stable compared with the former cases with s � 0. Fig. 13 shows the process of the wave acting on the cap with pile group obtained by images for case with s ¼ 2 cm and under wave condition with H ¼ 0.18 m and T ¼ 1.25 s. The front of incoming wave breaks due to the reflection of last wave. Wave breaking combined with wave run-up before the front of the cap occurs when horizontal wave force climbs to the peak. Finally, the wave passes away with the wave force settling down and the slamming process is over. The horizontal backward wave force in this case is small because the cap is hung above SWL so that it is the wave crest that mainly acts on the cap and the wave trough gets through below the cap without any contact. 4.2. Vertical wave forces on the cap Fig. 14 shows the normalized time series of vertical wave forces FZ on the isolated cap and the cap with pile group. Positive and negative value in the figure mean the uplift and downward force along Z-direction, respectively. It can be seen that vertical wave forces on the cap with pile group varies more intensely than those of isolated cap, specifically when the bottom of the cap locates at or above SWL. The shading and blocking of pile group aggravates the movement of water particles below the cap specimen, provokes more complex state of trapped air and leads to the larger vertical force, compared with the case without pile group. The vertical wave forces on the cap with s � 0 generally go through three phases. Initially, slamming force occurs with considerable magnitude but exceptionally short duration, followed by quasi-static uplift force dwindling down in a relatively long duration. Secondary peak appears sometimes and depends on the clearance height of the cap. Finally, the vertical downward force occurs and turns to zero when the waves depart from the cap. This three-phase mode of vertical wave-in-deck force has been presented in the previous experiments reported by Ren and Wang (2003), Cuomo et al. (2007) and Ren et al. (2016). The vertical uplift force is significantly larger than the downward force when the bottom of the cap locates at or above SWL. However, the vertical downward forces increase with the increase of the submerged depth and become larger than uplift forces when s ¼ 7.5 and 15 cm. Fig. 15 shows the process of the wave acting on the cap for case with s ¼ 2 cm and s ¼ 15 cm. Compared Fig. 15 (a) with Fig. 13 (b), it can be found that the vertical wave force reaches its crest earlier than the horizontal wave force. Ac cording to Fig. 15 (b), the crest of the wave has been higher than the top of the cap, and the wave crest climbed over the top of the cap with a lot of water overtopping it, which caused larger vertical downward force. It
Fig. 13. Images of process of wave acting on the cap with pile group (s ¼ 2 cm, incident wave propagates from right to left with H ¼ 0.18 m, T ¼ 1.25 s). (a) Impending wave slamming (48.28 s) (b) Horizontal wave force reaches its crest (48.48 s) (c) Horizontal wave force falls to the trough (48.85 s) (d) Wave de parts from the specimen (48.98 s).
Fig. 8) is used as the cutoff frequency. Therefore, the lower bound of structural vibration range is used as the cutoff frequency to filter the raw data of the horizontal force in the following sections. For the vertical force, since the natural frequency of the specimen in Z-direction is pretty high and no obvious contribution of the high frequency component is observed, the raw data is directly used for the following analysis. 4. Results and discussions In this section, the horizontal and vertical wave forces on the cap 8
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Fig. 14. Normalized time series of vertical wave forces: subfigure (a), (c), (e), (g) and (i) shows the wave force on the isolated cap under five wave trains, respectively; and subfigure (b), (d), (f), (h) and (j) shows the wave force on the cap with pile group under five wave trains, respectively.
can be found from Figs. 12 and 14 show that the vertical uplift force is larger than the horizontal forward force for elevated conditions, while the vertical uplift force turns smaller than the horizontal forward force for submerged conditions.
regular wave trains were measured in time-domain by F/T transducer simultaneously. Because the time history of wave force includes several crest and trough values. In order to investigate different effects on the wave force acting on the cap, the following representative values of the wave forces are defined for each working condition: horizontal forward force (positive), horizontal backward force (negative), vertical uplift force (positive) and vertical downward force (negative). The represen tative value of horizontal forward force and backward force is equal to
4.3. Effect of pile group The horizontal, vertical forces and moment on the pile cap under 9
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Fig. 15. Images of wave acting on the cap with a clearance height of (a) s ¼ 2 cm and (b) s ¼ 15 cm under the following wave condition: H ¼ 0.18 m and T ¼ 1.25 s (the incident wave propagates from right to left).(a) Vertical wave force reaches to its crest (s ¼ 2 cm, 48.38 s). (b) Wave overtopped the top of the cap (s ¼ 15 cm, 47.29 s).
Fig. 17. Measurement and relationship of representative wave forces: (a) elevation sketch of measurement setup, and (b) wave induced moment as a function of the horizontal and vertical force.
makes some of the forward moving particles turn into upward moving and slam on the bottom of the cap. In order to understand the contribution of horizontal and vertical force to the moment, as illustrated in Fig. 17 (a), the moment -MY is described as a function of horizontal force FX and vertical force FZ by Eq. (1). Considering the absolute value of the peak moment (positive) is much greater than that of the trough moment (negative), only the largest peak moment was selected for one working condition. The horizontal and vertical wave force components which occur simultaneously with the largest moment were then extracted. Fig. 17 (b) depicts three com ponents of force on the isolated cap and the cap with pile group in the three-dimensional (3D) coordinates. There are 25 points for each configuration of the specimen. Space plane was used to fit the data by least square method. It is clear that the horizontal wave force contributes far more to the moment than the vertical wave force regardless of the specimen configuration. The difference in the three components of force between two configurations of specimen are inconspicuous.
Fig. 16. Comparison of wave forces on the cap between the two configurations of the specimen.
the mean value of the crest and trough values in the time history of horizontal wave force, respectively. Likewise, the representative value of vertical uplift force and vertical downward force is equal to the mean value of the crest and trough values in the time history of vertical wave force, respectively. Comparison of four representative values of hori zontal and vertical wave forces between the two configurations of specimens is illustrated in Fig. 16. Because the wave used in the test is nonlinear, the positive and the negative representative values are not equal. Therefore, there are four points for each working condition and a total of 100 points for 25 working conditions without repeatability testing data. It can be seen from Fig. 16 that the horizontal force of both speci mens has negligible inequalities, while the vertical uplift force of the cap with pile group is slightly larger than that of the isolated cap. It is because pile group blocks the water particles passing under the cap and
MY ¼ FZ ⋅ x0 þ FX ⋅ðl0
z0 Þ
(1)
4.4. Effect of clearance height The standard deviation of the representative values of wave forces are calculated according to the experimental data to provide informa tion about the repeatability of the data. Fig. 18 illustrates the 10
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Fig. 18. Representative values of wave forces versus clearance heights under different wave height: (a) horizontal wave force; (b) vertical wave force.
Fig. 19. Representative values of wave forces versus wave heights with different clearance heights: (a) horizontal wave force; (b) vertical wave force.
representative values and error bar of the wave forces on the cap with pile group varying with clearance heights. Because the incident wave in the study is regular and stable, the deviation of the data is small for most of the working conditions. It is clear from Fig. 18 (a) that both horizontal forward and backward wave forces decrease with increase of clearance height, and the forward forces under all wave conditions occur when the
cap is fully submerged in water. It is because larger inundation area of the cap receives more pulling force during the propagation of wave. While the vertical uplift force firstly increases until the case with the bottom of the cap located at SWL, then decreases with increasing clearance height according to Fig. 18 (b). In other words, the largest vertical uplift force occurs when the clearance height is zero. This 11
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phenomenon has also been found in previous studies (Park et al., 2017; Hayatdavoodi et al., 2014; Seiffert et al., 2015). The vertical downward force is almost constant under either s � 0 or s � 0. The absolute value of the force for s � 0 is larger than that for s � 0. Contrary to the uplift force, the absolute value of downward force is small for the clearance height of s ¼ 0. It should be noted that the tiny air gap in this study (Fig. 3) influences the vertical wave force especially for setup of s ¼ 0 cm. Two CFD models were established to evaluate the effect of air gap on the wave forces of the pile cap. One is that the pile group is firmly attached to the cap without any air gap, and the other one includes a tiny air gap of 0.001 m between cap and pile. Refined grid size, k-ω turbulent model and the volume of fluid method (VOF) are adopted. According to the CFD sim ulations, the air gap increases the peak value and causes a 15% positive mean shift of the vertical wave force on the cap with s ¼ 0. This is because the water particles entrapped inside the air gap when the wave crest passed through the cap with s ¼ 0. The water inside the gap in creases not only the bottom wetted area of the cap, but also the distri bution of uplift pressure, and hence gives birth to the increase and mean shift of vertical wave forces. Therefore, the value of vertical uplift force for s ¼ 0 could be overestimated due to the existence of air gap.
forces. The increase of wave height induces a growth in horizontal wave force, except for the fully submerged case. The vertical wave force in creases firstly but exhibits a decreasing tendency when the wave height exceeds 0.18 m due to overtopping waves. The horizontal wave force contributes more to the moment than the vertical force on the cap. Pile group increases the vertical uplift force slightly when the bottom of the cap locates above SWL. It should be noted that the conclusions are drawn based on some assumptions. Firstly, the waves used in this experiment are merely unidirectional waves and the incident direction of the wave is perpen dicular to the front of the specimen. The directionality of the specimen installation is worthy to be investigated experimentally to attain some more generalizable results. Secondly, irregular waves which are more paralleled to the reality should also be considered and it is interesting to compare the results of regular waves with that of irregular waves. Furthermore, the tiny air gap, which was originally set up between pile and cap to exclude the force on the pile group from the measured force, is found to influence the vertical wave forces significantly when s ¼ 0. Some waterproof countermeasure, such as the rubber socket scheme (Lan et al., 2010), should be applied in the future to keep the water out of the gap and eliminate the experimental error.
4.5. Effect of wave height
Acknowledgements
Fig. 19 presents the representative values and error bar of the wave forces on the cap with pile group as a function of wave heights. Fig. 19 (a) indicates that the horizontal forward force increases nearly linearly with increasing wave height when the bottom of the cap locates at or above water, while the horizontal backward force varies slowly as the wave height increases from 0.09 m to 0.15 m and increases distinctly as wave height increases from 0.15 m to 0.21 m. In this study, larger wave has longer wave length and can diffract around the cap much easier. The movement of water particle in the opposite direction induces apparent horizontal backward force. Nevertheless, both absolute values of hori zontal forward and backward wave forces firstly increase then decrease with the increase of wave height for clearance height of 15 cm, with the largest value occurring under wave height of 0.15 m. It is because the water particle of larger wave is much easier to pass above the cap than that of smaller wave. Fig. 19 (b) shows that the vertical uplift force increases with an increment in wave height from 0.09 m to 0.18 m but decreases when the wave height is up to 0.21 m, because large wave would overtop the top of the cap and diminish the vertical uplift force. As s � 0, the variation of vertical downward force is not obvious under various wave heights. However, when the cap is submerged in water, the vertical downward force is significantly affected by the wave height, and the wave with large height would induce large vertical downward force.
The authors would like to acknowledge the financial support of National Natural Science Foundation of China (Grants No. 51708455) and the Fundamental Research Fund for Central Universities (A1920502051907-2-001). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.oceaneng.2019.106896. References AASHTO, 2008. Guide Specifications for Bridges Vulnerable to Coastal Storms. American Association of State Highway Transportation Officials, Washington, D.C. Akiyama, M., Frangopol, D.M., Arai, M., Koshimura, S., 2013. Reliability of bridges under tsunami hazards: emphasis on the 2011 Tohoku-oki earthquake. Earthq. Spectra 29 (s1), 295–314. Bradner, C., Schumacher, T., Cox, D., Higgins, C., 2010. Experimental setup for a largescale bridge superstructure specimen subjected to waves. J. Waterw. Port, Coast. Ocean Eng. 137 (1), 3–11. Chuang, W.L., Chang, K.A., Mercier, R., 2018. Kinematics and dynamics of green water on a fixed platform in a large wave basin in focusing wave and random wave conditions. Exp. Fluid 59 (6), 100. Cuomo, G., Tirindelli, M., Allsop, W., 2007. Wave-in-deck loads on exposed jetties. Coast Eng. 54 (9), 657–679. Cuomo, G., Shimosako, K.I., Takahashi, S., 2009. Wave-in-deck loads on coastal bridges and the role of air. Coast Eng. 56 (8), 793–809. Cuomo, G., Allsop, W., Bruce, T., Pearson, J., 2010. Breaking wave loads at vertical seawalls and breakwaters. Coast Eng. 57 (4), 424–439. Ding, Z.Q., Ren, B., Wang, Y.X., Ren, X.Z., 2008. Experimental study of unidirectional irregular wave slamming on the three-dimensional structure in the splash zone. Ocean. Eng. 35 (16), 1637–1646. Douglass, S.L., Hughes, S., Rogers, S., Chen, Q., 2004. The Impact of Hurricane Ivan on the Coastal Roads of Florida and Alabama: A Preliminary Report. Coastal Transportation Engineering Research & Education Center, University of South Alabama. Douglass, S.L., Chen, Q., Olsen, J.M., Edge, B.L., Brown, D., 2006. Wave Forces on Bridge Decks. Final Report for U.S. Department of Transportation, Federal Highway Administration. Office of Bridge Technology, Washington, D.C. Frick, K.T., 2008. The cost of the technological sublime: daring ingenuity and the new san francisco–oakland Bay bridge. In: Premius, H., Flyvbjerg, B., Van Wee, B. (Eds.), Decision-making on Megaprojects: Cost-Benefit Analysis, Planning and Innovation. Edward Elgar, Cheltenham, UK, pp. 239–262. Guo, A.X., Fang, Q.H., Bai, X.D., Li, H., 2015. Hydrodynamic experiment of the wave force acting on the superstructures of coastal bridges. J. Bridge Eng. 20 (12), 04015012. Hayatdavoodi, M., Seiffert, B., Ertekin, R.C., 2014. Experiments and computations of solitary-wave forces on a coastal-bridge deck. Part II: deck with girders. Coast Eng. 88, 210–228.
5. Conclusions This paper presents an experimental program of extreme wave forces on the elevated pile cap of pile group foundation for sea-crossing bridge. The reduced scale pile cap specimens, including the isolated cap and pile cap with pile group, were set up in the wave flume. The results are presented and discussed as a function of clearance height, wave height and pile group. The wave force on the cap is a function of both clearance height and wave height. The largest horizontal wave force occurs when the cap is fully submerged, while the largest vertical uplift force occurs when the bottom of the cap locates at SWL. Higher clearance height reduces the horizontal wave force but induces larger vertical wave force on the cap. Wave slamming force occurs when the bottom of the cap locates above SWL. Compared with horizontal slamming force, the vertical slamming force has more than two times magnitude and shorter rising time. Therefore, the clearance height of the cap should be given considerable attention to balance the horizontal and vertical components of wave 12
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Ocean Engineering 197 (2020) 106896 Seiffert, B.R., Hayatdavoodi, M., Ertekin, R.C., 2015. Experiments and calculations of cnoidal wave loads on a coastal-bridge deck with girders. Eur. J. Mech. B Fluid 52, 191–205. Stearns, M., Padgett, J.E., 2011. Impact of 2008 hurricane Ike on bridge infrastructure in the houston/galveston region. J. Perform. Constr. Facil. 26 (4), 441–452. Ti, Z., Wei, K., Qin, S., Mei, D., Li, Y., 2018. Assessment of random wave pressure on the construction cofferdam for sea-crossing bridges under tropical cyclone. Ocean Eng. 160, 335–345. Ti, Z., Zhang, M., Li, Y., Wei, K., 2019. Numerical study on the stochastic response of a long-span sea-crossing bridge subjected to extreme nonlinear wave loads. Eng. Struct. 196, 109287. Tomiczek, T., Prasetyo, A., Mori, N., Yasuda, T., Kennedy, A., 2017. Effects of a macroroughness element on tsunami wave amplification, pressures, and loads: physical specimen and comparison to Japanese and US design equations. Coast Eng. J. 59 (1), 1750004-1. Unjoh, S., 2006. Damage investigation of bridges affected by Tsunami during 2004 north Sumatra Earthquake, Indonesia. In: Proceedings of the Fourth International Workshop on Seismic Design and Retrofit of Transportation Facilities, San Francisco, USA, 10. Wei, K., Liu, Q., Qin, S., 2019. Nonlinear Assessment of Offshore Steel Trestle Subjected to Wave and Current Loads. Ships Offshore Struct. 1–13. Wei, K., Yuan, W.C., Bouaanani, N., 2013. Experimental and numerical assessment of the three-dimensional modal dynamic response of bridge pile foundations submerged in water. J. Bridge Eng. 18 (10), 1032–1041. Ye, J., 2010. Construction technologies of super-long pile foundation in Hangzhou Bay Bridge. In: Deep Foundations and Geotechnical in Situ Testing, pp. 232–237. Zhang, J., Wei, K., Qin, S., 2019. An efficient numerical model for hydrodynamic added mass of immersed column with arbitrary cross - section. Ocean. Eng. 187, 106192.
Kosa, K., 2011. Damage analysis of bridges affected by tsunami due to Great East Japan Earthquake. In: 27th US-Japan Bridge Engineering Workshop, pp. 55–65. Lan, Y.M., Liu, H., Xue, L.P., Chen, G., 2006. An experimental study on vertical hydrodynamic force on a circular slab near free surface by water waves. J. Hydrodyn. Ser. B 18 (2), 184–191. Lan, Y.M., Guo, W.H., Liu, H., Song, Q.H., Yuan, J.T., 2010. Experimental study on hydrodynamic loads on a horizontal slab and pile in regular wave. Chin. J. Hydrodyn. 25 (4), 184–558. Le Mehaute, B., 1976. Introduction to Hydrodynamics and Water Waves. SpringerVerlag, New York. Liu, S.X., Li, Y.C., Li, G.W., 2007. Wave current forces on the pile group of base foundation for the East Sea Bridge, China. J. Hydrodyn. 19 (6), 661–670. Mas, E., Bricker, J., Kure, S., Adriano, B., Yi, C., Suppasri, A., Koshimura, S., 2015. Field survey report and satellite image interpretation of the 2013 Super Typhoon Haiyan in the Philippines. Nat. Hazards Earth Syst. Sci. 15 (4), 805–816. Molin, B., 2002. Hydrodynamique des structures offshore. Editions Technip. Park, H., Tomiczek, T., Cox, D.T., van de Lindt, J.W., Lomonaco, P., 2017. Experimental specimening of horizontal and vertical wave forces on an elevated coastal structure. Coast Eng. 128, 58–74. Ren, B., Wang, Y., 2003. Experimental study of irregular wave impact on structures in the splash zone. Ocean. Eng. 30 (18), 2363–2377. Ren, B., Liu, M., Li, X.L., Wang, Y.X., 2016. Experimental investigation of wave slamming on an open structure supported elastically. China Ocean Eng. 30 (6), 967–978. Robertson, I.N., Yim, S., Riggs, H.R., Young, Y.L., 2007. September. Coastal bridge performance during hurricane Katrina. In: Third International Conference on Structural Engineering, Mechanics and Computation, SEMC-2007, Cape Town, South Africa, Millpress, Rotterdam, The Netherlands, pp. 1864–1870. Seiffert, B., Hayatdavoodi, M., Ertekin, R.C., 2014. Experiments and computations of solitary-wave forces on a coastal-bridge deck. Part I: flat plate. Coast Eng. 88, 194–209.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental study of wave loads on elevated pile cap of pile group foundation for sea-crossing bridges Bo Xu a, Kai Wei a, *, Shunquan Qin a, b, Jie Hong a a b
Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, China China Railway Major Bridge Reconnaissance & Design Institute Co., Ltd., Wuhan, 430050, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Wave force Elevated pile cap Sea-crossing bridges Wave flume experiment Pile group Clearance height
Elevated pile cap, which transits the load from sea-crossing bridge pylon to pile group, is always close to the sea level and suffering from extreme wave loadings. This paper describes a wave flume experimental program to investigate wave loads on the elevated pile cap of pile foundation for sea-crossing bridges. Two specimen con figurations including isolated cap and cap with pile group were set up with different clearance heights. Wave forces on the cap, including horizontal and vertical force and moment, were measured simultaneously under five nonlinear regular wave trains. The effects of pile group, clearance height and wave height on wave forces on the cap were investigated. The experimental results show that the clearance height and wave height affect wave forces significantly. The largest horizontal force occurs when the cap is fully submerged, while the largest vertical force occurs when the bottom of the cap locates at still water level (SWL). As the wave height increases, the horizontal forces increase except for the fully submerged case and the vertical forces increase firstly but decrease when the wave height exceeds 0.18 m. Pile group increases the vertical uplift forces on the cap when the bottom of the cap locates above SWL.
1. Introduction With the advent of booming demand in ground transportation across straits and channels, development of long-span sea-crossing bridge is being pursued to connect the mainland and islands all over the world. Pile group foundations are widely used to support contemporary longspan sea-crossing bridges, such as China East Sea Bridge (Liu et al., 2007), San Francisco–Oakland Bay Bridge (Frick, 2008), Hangzhou Bay Bridge (Ye, 2010) and Pingtan Rail-cum-road Sea-crossing Bridge (Ti et al., 2019), due to its good applicability in deep water site and complex seabed terrain. However, the elevated pile cap, which transits the load from bridge pylon to pile group, is always close to the sea level and suffering from extreme wave loadings (Ti et al., 2018). In recent years, catastrophes like hurricanes Ivan in 2004, Katrina in 2005, Ike in 2008, Indian Ocean tsunami in 2004, Tohoku tsunami in 2011 and Typhoon Haiyan in 2013 caused extensive damage to infrastructures and led a large number of coastal bridges to be damaged or collapsed (Douglass et al., 2004; Robertson et al., 2007; Stearns and Padgett, 2011; Unjoh, 2006; Kosa, 2011; Akiyama et al., 2013; Mas et al., 2015). Considering the risk of structural failure of sea crossing bridges due to extreme waves (Ti et al., 2019; Wei et al., 2019), it is hence important to estimate wave
loads on the elevated pile cap for assessments of structure safety and construction feasibility of pile foundation for sea-crossing bridges. Many experiments were carried out to investigate the wave force and pressure on offshore elevated structures. Lan et al. (2006) carried out an experiment to investigate the vertical force on a slab of sea-crossing bridge piers under regular and irregular waves. Ding et al. (2008) designed an experiment to investigate unidirectional random wave slamming on a platform structure and effects of the wave direction. Cuomo et al. (2009) conducted large-scale flume experiments to inves tigate the dynamics of wave loading on coastal highway bridges. Based on the experimental results, Cuomo et al. (2010) introduced a prediction formula for quasi-static and breaking wave-induced impact forces and overturning moments at vertical seawalls and breakwaters. Bradner et al. (2010) performed laboratory tests on 1:5 scale reinforced concrete specimen of coastal bridge superstructure to investigate wave-in-deck loads under regular and random waves considering the flexibility and dynamic properties of the supporting bridge substructure. Wei et al. (2013) conducted an experiment to study the dynamic response of bridge pile foundations submerged in water. Guo et al. (2015) con ducted a hydrodynamic experiment to investigate the wave force acting on the superstructures of coastal bridge and compared to Douglass et al.
* Corresponding author. E-mail addresses: [email protected] (B. Xu), [email protected] (K. Wei). https://doi.org/10.1016/j.oceaneng.2019.106896 Received 21 August 2018; Received in revised form 24 October 2019; Accepted 23 December 2019 Available online 7 January 2020 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. The testing facility: (a) photo and detail of the wave flume; (b) dimension of the flume (all dimensions are in meter).
(2006) and Model of AASHTO guidelines (2008). Seiffert et al. (2014) experimentally studied the horizontal and vertical forces acting on a two-dimensional horizontal plate by solitary waves. Wave loads under different wave heights, water depths as well as submergence depths and elevations above still water level were discussed in detail. Chuang et al. (2018) investigated the effect of aeration and air compressibility on the wave slamming forces of offshore platform experimentally. However, these studies paid little attention to the elevated pile cap of pile foun dation for sea-crossing bridges. Moreover, Ti et al. (2018) conducted a field measurement of random wave pressure on a real cofferdam for sea-crossing bridge during typhoon and observed the secondary peaks in the pressure spectrums. Although the first order diffraction theory is commonly used to assess the wave force on the large-scale marine structures, such as elevated pile cap, it is not sufficient to deal with waves with nonlinear kinematics and caps locates above the mean sea level (Molin, 2002). The blocking and shading effect of pile group on the wave force of the pile cap has yet been well understood. This study investigates the wave forces acting on the elevated cap of pile foundation for sea-crossing bridges experimentally. The 5th-order Stokes regular waves with five different heights and the associated lower bound of periods according to testing breaking limits were used to simulate the extreme waves. Two configurations of the specimen including isolated cap and cap with pile group were set up considering five clearance heights. The following two objectives are set: (1) to discern the characteristics of the time history of wave forces under different clearance and wave height conditions; and (2) to understand effects of clearance height, wave height and pile group on the wave loads acting on the pile cap.
2. Experimental setup 2.1. Wave testing facilities The experiments of this study were conducted in the wave flume at Southwest Jiaotong University. The flume is equipped with a piston-type wave-generating system. The flume is 60 m in length, 2 m in width and 1.8 m in height with side walls comprised of steel frames and toughened glass, which provides a clear and wide visual range during the testing. As shown in Fig. 1, the piston-type wave maker designed by Dalian Uni versity of Technology is installed at the right end of the flume and able to generate various one-dimensional wave environments, such as linear and nonlinear regular wave, irregular wave, focused wave, breaking wave, etc. In the process of making waves, the motion file of the wave maker could be updated automatically according to the feedback from actual wave condition obtained by the data acquisition system installed in the flume until the expected waves are made. The wave-absorbing beach, which is made of porous polymer material, is positioned at the left end of the flume to absorb the wave energy and to reduce the wave reflection. The testing bridge locates and operates on the stainless-steel tracks placed at the top of the side walls. In the following testing, we define the Cartesian coordinate of the flume as illustrated in Fig. 1 (a). Xdirection is the longitudinal direction from the wave maker to the waveabsorbing beach, Y-direction is the transverse direction pointing from in-plane to out-of-plane against the flume and Z-direction is the vertical direction with positive direction upward. Four wave gauges were installed in the flume to measure the water surface elevations, as illustrated in Fig. 1 (b). Wave gauges WG2, WG3 and WG 4 were installed at 10 cm left, 10 cm right and 10 cm rear of the tested cap specimen, respectively. WG1 was installed at 2 m front of the WG4 to measure the wave conditions in the empty wave flume. After the testing specimen was mounted in the flume, WG1 was relocated at 40 cm 2
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Fig. 2. Sketch of pile group foundation for bridge that inspired the specimen: (a) Sea-crossing bridge supported by a group-pile cap foundation; (b) the photo of wave acting on an elevated pile cap under construction.
Fig. 3. Dimensions and installation detail of the tested specimen: (a) dimensions of the specimen; (b) detail of the connection between specimen and testing bridge (all dimensions are in centimeter).
front of the cap in order to measure the reflection of the wave induced by the specimen. The sampling frequency of wave gauges was set to 100 Hz that is sufficient to record the fluctuation of water surface through a subsequent comparison for validity. The F/T (force and torque) transducer was used to measure both force and torque acting on the specimen along X-, Y- and Z-directions. Although the sampling frequency of the transducer can be set as high as 7000 Hz, it was found in previous experiments (Tomiczek et al., 2017; Guo et al., 2015) that a sampling frequency of 200 Hz is sufficient to capture the impulsive forces and structural responses signals. In this study, a sampling frequency of 1000 Hz, which is higher than 200 Hz, was chosen here to ensure the accuracy and efficiency of the measure ments during wave impacts on the cap. All experimental processes were recorded visually through HD camera with a sampling speed of 60 fps.
bridge that crosses Pingtan Strait in the southeast coast of China. As shown in Fig. 2 (a), the prototype foundation is composed of twelve concrete piles and one concrete pile cap. The pile cap is 36 m in length, 24 m in width and 13.5 m in height, elevated on the piles with diameter of 4.5 m. Fig. 2 (b) shows a photo of the wave acting on an elevated pile cap under construction. The reduced scale tested specimen, shown in Fig. 3 (a), with di mensions inspired from the prototype foundation, was composed of a group of 12 circular piles and a rectangular pile cap. Considering the flume dimensions and abilities of experimental facilities, the tested specimen was designed according to the Froude similarity using a 1:90 geometric scaling. The tested specimen was made of acrylic material, which has light weight and high rigidity. The pile was manufactured with external diameter of 5 cm and thickness of 5 mm to ensure its strength. The dimension of the cap specimen was a rigid box with length of 40 cm, width of 26.67 cm and height of 15 cm. In order to study the effect of the pile group on the wave force of the pile cap, two configurations of tested specimen, (1) the isolated pile cap and (2) the pile cap with pile group, were set up. For the isolated pile cap
2.2. Configuration of tested specimen The design of the reduced scale specimens tested in this study was inspired by a pile cap of the group-pile foundation for a sea-crossing 3
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Fig. 4. Specification of clearance height for specimen: pile cap with pile group (all dimensions are in centimeter).
configuration, the F/T transducer was fastened firmly on the testing bridge through the lifting rod. The tested cap specimen was suspended on the bottom of the F/T transducer by the rigid alloy frame, as illus trated in Fig. 3 (b). For the pile cap with pile group configuration, the piles were fully fixed on the floor of the wave flume. Considering the major objective of this study is to address the wave force acting on the pile cap, a tiny gap (approximately 1 mm) was kept between the bottom of the cap and the top of the pile group to exclude the wave force on the pile group from the force measured by the F/T transducer.
Table 1 Testing regular wave parameters. Wave height H (m)
Wave period T (s)
Wave length L(m)
H/gT2
d/gT2
0.04 0.06 0.09 0.12 0.15 0.18 0.21 0.30 0.40 0.50
0.59 0.73 0.91 1.04 1.15 1.25 1.35 1.60 1.88 2.06
0.570 0.872 1.350 1.762 2.147 2.517 2.896 3.942 5.110 5.959
0.0117 0.0115 0.0111 0.0113 0.0116 0.0117 0.0117 0.0119 0.0115 0.0120
0.2636 0.1722 0.1108 0.0848 0.0694 0.0587 0.0503 0.0358 0.0260 0.0216
2.3. Clearance height setting In the actual practice of the pile group foundation, one important design parameter is the clearance height between the bottom of the cap and Still Water Level (SWL). Five different clearance heights s were investigated experimentally for both configurations of tested specimen.
Fig. 5. Comparison of the water elevation between the measured data and the theoretical solutions of 5th order Stokes wave theory under different wave conditions: (a) H ¼ 0.09 m T ¼ 0.91 s; (b) H ¼ 0.12 m T ¼ 1.04 s; (c) H ¼ 0.15 m T ¼ 1.15 s; (d) H ¼ 0.18 m T ¼ 1.25 s; (e) H ¼ 0.21 m T ¼ 1.35 s. 4
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Table 2 The first order natural vibration frequencies of the tested specimen.
Fig. 6. Comparison of water surface measured by different wave gauges under wave condition with H ¼ 0.12 m and T ¼ 1.04 s.
s (cm)
f1first (Hz)
f1second (Hz)
f1third (Hz)
~f (Hz) 1
4 2 0
15.69 15.45 15.16 13.09 10.32
15.78 15.51 15.26 13.15 10.31
15.66 15.47 15.16 13.13 10.33
15.71 15.48 15.19 13.12 10.32
7.5 15
measured by wave gauge WG2 in the flume and the theoretical eleva tions calculated by the 5th order Stokes wave theory were compared in Fig. 5. Good agreements can be found between the measured and theoretical water elevation for all five experimental wave conditions, which demonstrates that the wave maker generates satisfactory regular wave trains in the flume. Moreover, water surface elevations measured by wave gauges WG1, WG2 and WG4 are compared as well to verify the stability of the wave along the flume. Because WG2 and WG3 locate in the same section, only WG2 is used. It is clear from Fig. 6 that there are few deviations among them in terms of wave heights and periods neglecting the phase differ ence caused by the position. It can be concluded from the results that the wave train was stable while it was propagating in the flume.
As shown in Fig. 4, s ¼ 2 and 4 cm were two cases that the cap bottom was above the SWL, s ¼ 0 cm referred to the case that the cap bottom coincided with the SWL, and the latter two cases s ¼ 7.5 cm and 15 cm related to the cases that the cap was partially and fully submerged in the water, respectively. In the experiment, the depth of the water d in the flume was fixed at 0.9 m during the whole experimental study, which was conducive to comparison among the cases with different clearance heights and wave conditions. Therefore, the clearance height s was increased or decreased by changing the length of the lifting rod. The pile consists of five segments to meet five different clearance height re quirements. Upper segment was connected to the lower segment by acylic. When the clearance height was changed, the height of the pile group was changed accordingly by adding or removing the segments to keep the air gap between cap and pile group constant.
3.2. Natural vibration testing Although the support system of the cap specimen is expected to be designed as rigid as possible, the natural vibration of the tested speci mens is still difficult to be eliminated, especially when it vibrates along X direction under wave impact. Moreover, the extending length of the lifting rod changes with the clearance height and causes the change of natural vibration frequency of the specimen. It is hence necessary to obtain the first order natural vibration frequency f1 of the specimen installed on the testing bridge with different clearance heights. The free vibration testing was conducted as soon as the specimen was installed in the flume. An impact hammer was used to excite the specimen along the X-axis. Free vibration testing of the specimen was repeated three times to guarantee the accuracy of the testing results. Fig. 7 shows timedomain and frequency-domain results of X-direction reaction force re cord from the F/T transducer for the cap specimen with s ¼ 4 cm. Natural frequencies of cap specimens with five different clearance heights are listed in Table 2 with range from 10.32 to 15.71 Hz. It should be noted that the smaller natural frequency is associated to smaller clearance height. It is because, 1) smaller clearance height relates to longer free length of the lifting rod, and 2) when s becomes negative, the cap specimen is submerged in water, which reduces the natural fre quency of the specimen (Zhang et al., 2019).
2.4. Experimental wave conditions In this study, the water depth in the flume was set to be 0.9 m. The wave parameters of ten regular extreme wave trains with wave height H from 0.04 m to 0.5 m were extracted through wave conditions calibra tion tests in the flume. The period of the calibrated extreme wave train is close to the lower bound of period, which keeps the wave as steep as possible and exactly unbroken in the flume. Table 1 lists the wave pa rameters of the calibrated extreme wave trains in the flume. Five extreme wave trains with wave height H from 0.09 m to 0.21 m with increments of 0.03 m in Table 1 were selected as the experimental wave conditions for the following study. 3. Experimental procedure 3.1. Validation of regular wave train in the flume According to the wave parameters listed in Tables 1 and i.e., H/gT2 and d/gT2, all five selected extreme wave conditions were found in the range of the 5th order Stokes wave theory (Le Mehaute, 1976). To verify the quality of the generated regular waves, water surface elevations
Fig. 7. Time domain and frequency domain results of X-direction reaction force for the cap specimen with s ¼ 4 cm during natural vibration testing: (a) time history; (b) Fourier spectrum. 5
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Fig. 8. Results of horizontal force on the isolated cap specimen: (a) time history; (b) Fourier spectrum (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s).
Fig. 9. Results of vertical force on the isolated cap specimen: (a) time history; (b) Fourier spectrum (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s).
Fig. 10. The inundation of the cap during wave propagation (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s). (a) The cap elevated over static water (b) The cap submerged in wave crest.
3.3. Post-processing of measured data Figs. 8 and 9 present a sample of horizontal and vertical force on the isolated cap specimen with s ¼ 2 cm under wave conditions with H ¼ 0.15 m and T ¼ 1.15s. Slamming force can be clearly found in the time histories of both horizontal and vertical force. According to the Fourier spectrum, the first peak has the highest power and its frequency equals to the frequency of the regular wave. Except the first peak, there are still many higher order peaks, of which the frequencies are integer N times the first peak frequency. Similar phenomenon was also observed and reported by Guo et al. (2015) in their study. The nonlinear waves and higher-order velocity potentials induced by incident and diffracted waves are the reasons for the N-times frequency wave forces. It is interesting to notice that the amplitude of the Fourier spectrum of the horizontal wave force FX increases in the range from 12.44 Hz to 15.48 Hz that equals the average value of the natural frequency of the tested specimen (highlighted by the dashed black frame in Fig. 8b). This
Fig. 11. Comparison of raw data and filtered data with different cutoff fre quency fcutoff (s ¼ 2 cm H ¼ 0.15 m T ¼ 1.15s). 6
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Fig. 12. Normalized time series of horizontal wave forces: subfigure (a), (c), (e), (g) and (i) shows the wave force on the isolated cap under five wave trains, respectively; and subfigure (b), (d), (f), (h) and (j) shows the wave force on the cap with pile group under five wave trains, respectively.
is mainly caused by the disturbance from structural natural vibration. However, the lower bound of the range is smaller than natural frequency of the tested specimen. Fig. 10 presents the inundation of the cap during wave propagation by images. The frequency we tested in Table 2 is structural frequency under still water (Fig. 10a). However, the extent of inundation changes significantly due to the fluctuation of water surface during the propagation of a wave train. Previous experiment carried out by Wei et al. (2013) has shown that structural natural frequency of the cap varies with the inundation volume of the cap. The larger the
inundation volume is, the lower the structural frequency is. It can explain why the lower bound of the frequency range for the structural vibration becomes smaller. The low-pass 3rd-order Butterworth filter is used to filter the raw data. The comparison of filtered data and raw data of horizontal force is illustrated in Fig. 11. When the cutoff frequency is set to be the exact natural frequency of specimen given in Table 2, significant noise still exists in the curve. However, most of the noise can be filtered, when the lower bound of structural vibration range (e.g. 12.44 Hz according to 7
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specimen are analyzed to obtain more insight into the characteristics of wave forces. And the effects of pile group, clearance height and wave height on the wave loads acting on the cap are investigated based on the experimental data. Five incident wave trains times five clearance heights constitute 25 experimental working conditions for each configuration of tested specimen, e.g. the isolated pile cap and the pile cap with pile group. 4.1. Horizontal wave forces on the cap In order to show and compare all clearance height conditions in one figure, the time axis of the time history of the wave force was normalized by the wave period corresponding to the given wave height. The time histories for all working conditions use the same phase. The duration of the time history is four times wave period. Fig. 12 shows the normalized time series of the horizontal wave forces FX on the isolated cap and the cap with pile group. The positive and negative value refer to forward and backward force along X-direction, respectively. Compared with the horizontal wave forces on the isolated cap, the force on the pile cap with pile group does not show obvious dissimilarity in the shape of the curve. For s � 0, the horizontal wave force arises abruptly reaching to the crest in a short time, then diminishes below zero, and finally returns to zero. With increase of submerged depth of the cap for s � 0, the slamming nature of the horizontal wave force disappears gradually and the peak value of the force become more stable compared with the former cases with s � 0. Fig. 13 shows the process of the wave acting on the cap with pile group obtained by images for case with s ¼ 2 cm and under wave condition with H ¼ 0.18 m and T ¼ 1.25 s. The front of incoming wave breaks due to the reflection of last wave. Wave breaking combined with wave run-up before the front of the cap occurs when horizontal wave force climbs to the peak. Finally, the wave passes away with the wave force settling down and the slamming process is over. The horizontal backward wave force in this case is small because the cap is hung above SWL so that it is the wave crest that mainly acts on the cap and the wave trough gets through below the cap without any contact. 4.2. Vertical wave forces on the cap Fig. 14 shows the normalized time series of vertical wave forces FZ on the isolated cap and the cap with pile group. Positive and negative value in the figure mean the uplift and downward force along Z-direction, respectively. It can be seen that vertical wave forces on the cap with pile group varies more intensely than those of isolated cap, specifically when the bottom of the cap locates at or above SWL. The shading and blocking of pile group aggravates the movement of water particles below the cap specimen, provokes more complex state of trapped air and leads to the larger vertical force, compared with the case without pile group. The vertical wave forces on the cap with s � 0 generally go through three phases. Initially, slamming force occurs with considerable magnitude but exceptionally short duration, followed by quasi-static uplift force dwindling down in a relatively long duration. Secondary peak appears sometimes and depends on the clearance height of the cap. Finally, the vertical downward force occurs and turns to zero when the waves depart from the cap. This three-phase mode of vertical wave-in-deck force has been presented in the previous experiments reported by Ren and Wang (2003), Cuomo et al. (2007) and Ren et al. (2016). The vertical uplift force is significantly larger than the downward force when the bottom of the cap locates at or above SWL. However, the vertical downward forces increase with the increase of the submerged depth and become larger than uplift forces when s ¼ 7.5 and 15 cm. Fig. 15 shows the process of the wave acting on the cap for case with s ¼ 2 cm and s ¼ 15 cm. Compared Fig. 15 (a) with Fig. 13 (b), it can be found that the vertical wave force reaches its crest earlier than the horizontal wave force. Ac cording to Fig. 15 (b), the crest of the wave has been higher than the top of the cap, and the wave crest climbed over the top of the cap with a lot of water overtopping it, which caused larger vertical downward force. It
Fig. 13. Images of process of wave acting on the cap with pile group (s ¼ 2 cm, incident wave propagates from right to left with H ¼ 0.18 m, T ¼ 1.25 s). (a) Impending wave slamming (48.28 s) (b) Horizontal wave force reaches its crest (48.48 s) (c) Horizontal wave force falls to the trough (48.85 s) (d) Wave de parts from the specimen (48.98 s).
Fig. 8) is used as the cutoff frequency. Therefore, the lower bound of structural vibration range is used as the cutoff frequency to filter the raw data of the horizontal force in the following sections. For the vertical force, since the natural frequency of the specimen in Z-direction is pretty high and no obvious contribution of the high frequency component is observed, the raw data is directly used for the following analysis. 4. Results and discussions In this section, the horizontal and vertical wave forces on the cap 8
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Fig. 14. Normalized time series of vertical wave forces: subfigure (a), (c), (e), (g) and (i) shows the wave force on the isolated cap under five wave trains, respectively; and subfigure (b), (d), (f), (h) and (j) shows the wave force on the cap with pile group under five wave trains, respectively.
can be found from Figs. 12 and 14 show that the vertical uplift force is larger than the horizontal forward force for elevated conditions, while the vertical uplift force turns smaller than the horizontal forward force for submerged conditions.
regular wave trains were measured in time-domain by F/T transducer simultaneously. Because the time history of wave force includes several crest and trough values. In order to investigate different effects on the wave force acting on the cap, the following representative values of the wave forces are defined for each working condition: horizontal forward force (positive), horizontal backward force (negative), vertical uplift force (positive) and vertical downward force (negative). The represen tative value of horizontal forward force and backward force is equal to
4.3. Effect of pile group The horizontal, vertical forces and moment on the pile cap under 9
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Fig. 15. Images of wave acting on the cap with a clearance height of (a) s ¼ 2 cm and (b) s ¼ 15 cm under the following wave condition: H ¼ 0.18 m and T ¼ 1.25 s (the incident wave propagates from right to left).(a) Vertical wave force reaches to its crest (s ¼ 2 cm, 48.38 s). (b) Wave overtopped the top of the cap (s ¼ 15 cm, 47.29 s).
Fig. 17. Measurement and relationship of representative wave forces: (a) elevation sketch of measurement setup, and (b) wave induced moment as a function of the horizontal and vertical force.
makes some of the forward moving particles turn into upward moving and slam on the bottom of the cap. In order to understand the contribution of horizontal and vertical force to the moment, as illustrated in Fig. 17 (a), the moment -MY is described as a function of horizontal force FX and vertical force FZ by Eq. (1). Considering the absolute value of the peak moment (positive) is much greater than that of the trough moment (negative), only the largest peak moment was selected for one working condition. The horizontal and vertical wave force components which occur simultaneously with the largest moment were then extracted. Fig. 17 (b) depicts three com ponents of force on the isolated cap and the cap with pile group in the three-dimensional (3D) coordinates. There are 25 points for each configuration of the specimen. Space plane was used to fit the data by least square method. It is clear that the horizontal wave force contributes far more to the moment than the vertical wave force regardless of the specimen configuration. The difference in the three components of force between two configurations of specimen are inconspicuous.
Fig. 16. Comparison of wave forces on the cap between the two configurations of the specimen.
the mean value of the crest and trough values in the time history of horizontal wave force, respectively. Likewise, the representative value of vertical uplift force and vertical downward force is equal to the mean value of the crest and trough values in the time history of vertical wave force, respectively. Comparison of four representative values of hori zontal and vertical wave forces between the two configurations of specimens is illustrated in Fig. 16. Because the wave used in the test is nonlinear, the positive and the negative representative values are not equal. Therefore, there are four points for each working condition and a total of 100 points for 25 working conditions without repeatability testing data. It can be seen from Fig. 16 that the horizontal force of both speci mens has negligible inequalities, while the vertical uplift force of the cap with pile group is slightly larger than that of the isolated cap. It is because pile group blocks the water particles passing under the cap and
MY ¼ FZ ⋅ x0 þ FX ⋅ðl0
z0 Þ
(1)
4.4. Effect of clearance height The standard deviation of the representative values of wave forces are calculated according to the experimental data to provide informa tion about the repeatability of the data. Fig. 18 illustrates the 10
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Fig. 18. Representative values of wave forces versus clearance heights under different wave height: (a) horizontal wave force; (b) vertical wave force.
Fig. 19. Representative values of wave forces versus wave heights with different clearance heights: (a) horizontal wave force; (b) vertical wave force.
representative values and error bar of the wave forces on the cap with pile group varying with clearance heights. Because the incident wave in the study is regular and stable, the deviation of the data is small for most of the working conditions. It is clear from Fig. 18 (a) that both horizontal forward and backward wave forces decrease with increase of clearance height, and the forward forces under all wave conditions occur when the
cap is fully submerged in water. It is because larger inundation area of the cap receives more pulling force during the propagation of wave. While the vertical uplift force firstly increases until the case with the bottom of the cap located at SWL, then decreases with increasing clearance height according to Fig. 18 (b). In other words, the largest vertical uplift force occurs when the clearance height is zero. This 11
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phenomenon has also been found in previous studies (Park et al., 2017; Hayatdavoodi et al., 2014; Seiffert et al., 2015). The vertical downward force is almost constant under either s � 0 or s � 0. The absolute value of the force for s � 0 is larger than that for s � 0. Contrary to the uplift force, the absolute value of downward force is small for the clearance height of s ¼ 0. It should be noted that the tiny air gap in this study (Fig. 3) influences the vertical wave force especially for setup of s ¼ 0 cm. Two CFD models were established to evaluate the effect of air gap on the wave forces of the pile cap. One is that the pile group is firmly attached to the cap without any air gap, and the other one includes a tiny air gap of 0.001 m between cap and pile. Refined grid size, k-ω turbulent model and the volume of fluid method (VOF) are adopted. According to the CFD sim ulations, the air gap increases the peak value and causes a 15% positive mean shift of the vertical wave force on the cap with s ¼ 0. This is because the water particles entrapped inside the air gap when the wave crest passed through the cap with s ¼ 0. The water inside the gap in creases not only the bottom wetted area of the cap, but also the distri bution of uplift pressure, and hence gives birth to the increase and mean shift of vertical wave forces. Therefore, the value of vertical uplift force for s ¼ 0 could be overestimated due to the existence of air gap.
forces. The increase of wave height induces a growth in horizontal wave force, except for the fully submerged case. The vertical wave force in creases firstly but exhibits a decreasing tendency when the wave height exceeds 0.18 m due to overtopping waves. The horizontal wave force contributes more to the moment than the vertical force on the cap. Pile group increases the vertical uplift force slightly when the bottom of the cap locates above SWL. It should be noted that the conclusions are drawn based on some assumptions. Firstly, the waves used in this experiment are merely unidirectional waves and the incident direction of the wave is perpen dicular to the front of the specimen. The directionality of the specimen installation is worthy to be investigated experimentally to attain some more generalizable results. Secondly, irregular waves which are more paralleled to the reality should also be considered and it is interesting to compare the results of regular waves with that of irregular waves. Furthermore, the tiny air gap, which was originally set up between pile and cap to exclude the force on the pile group from the measured force, is found to influence the vertical wave forces significantly when s ¼ 0. Some waterproof countermeasure, such as the rubber socket scheme (Lan et al., 2010), should be applied in the future to keep the water out of the gap and eliminate the experimental error.
4.5. Effect of wave height
Acknowledgements
Fig. 19 presents the representative values and error bar of the wave forces on the cap with pile group as a function of wave heights. Fig. 19 (a) indicates that the horizontal forward force increases nearly linearly with increasing wave height when the bottom of the cap locates at or above water, while the horizontal backward force varies slowly as the wave height increases from 0.09 m to 0.15 m and increases distinctly as wave height increases from 0.15 m to 0.21 m. In this study, larger wave has longer wave length and can diffract around the cap much easier. The movement of water particle in the opposite direction induces apparent horizontal backward force. Nevertheless, both absolute values of hori zontal forward and backward wave forces firstly increase then decrease with the increase of wave height for clearance height of 15 cm, with the largest value occurring under wave height of 0.15 m. It is because the water particle of larger wave is much easier to pass above the cap than that of smaller wave. Fig. 19 (b) shows that the vertical uplift force increases with an increment in wave height from 0.09 m to 0.18 m but decreases when the wave height is up to 0.21 m, because large wave would overtop the top of the cap and diminish the vertical uplift force. As s � 0, the variation of vertical downward force is not obvious under various wave heights. However, when the cap is submerged in water, the vertical downward force is significantly affected by the wave height, and the wave with large height would induce large vertical downward force.
The authors would like to acknowledge the financial support of National Natural Science Foundation of China (Grants No. 51708455) and the Fundamental Research Fund for Central Universities (A1920502051907-2-001). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.oceaneng.2019.106896. References AASHTO, 2008. Guide Specifications for Bridges Vulnerable to Coastal Storms. American Association of State Highway Transportation Officials, Washington, D.C. Akiyama, M., Frangopol, D.M., Arai, M., Koshimura, S., 2013. Reliability of bridges under tsunami hazards: emphasis on the 2011 Tohoku-oki earthquake. Earthq. Spectra 29 (s1), 295–314. Bradner, C., Schumacher, T., Cox, D., Higgins, C., 2010. Experimental setup for a largescale bridge superstructure specimen subjected to waves. J. Waterw. Port, Coast. Ocean Eng. 137 (1), 3–11. Chuang, W.L., Chang, K.A., Mercier, R., 2018. Kinematics and dynamics of green water on a fixed platform in a large wave basin in focusing wave and random wave conditions. Exp. Fluid 59 (6), 100. Cuomo, G., Tirindelli, M., Allsop, W., 2007. Wave-in-deck loads on exposed jetties. Coast Eng. 54 (9), 657–679. Cuomo, G., Shimosako, K.I., Takahashi, S., 2009. Wave-in-deck loads on coastal bridges and the role of air. Coast Eng. 56 (8), 793–809. Cuomo, G., Allsop, W., Bruce, T., Pearson, J., 2010. Breaking wave loads at vertical seawalls and breakwaters. Coast Eng. 57 (4), 424–439. Ding, Z.Q., Ren, B., Wang, Y.X., Ren, X.Z., 2008. Experimental study of unidirectional irregular wave slamming on the three-dimensional structure in the splash zone. Ocean. Eng. 35 (16), 1637–1646. Douglass, S.L., Hughes, S., Rogers, S., Chen, Q., 2004. The Impact of Hurricane Ivan on the Coastal Roads of Florida and Alabama: A Preliminary Report. Coastal Transportation Engineering Research & Education Center, University of South Alabama. Douglass, S.L., Chen, Q., Olsen, J.M., Edge, B.L., Brown, D., 2006. Wave Forces on Bridge Decks. Final Report for U.S. Department of Transportation, Federal Highway Administration. Office of Bridge Technology, Washington, D.C. Frick, K.T., 2008. The cost of the technological sublime: daring ingenuity and the new san francisco–oakland Bay bridge. In: Premius, H., Flyvbjerg, B., Van Wee, B. (Eds.), Decision-making on Megaprojects: Cost-Benefit Analysis, Planning and Innovation. Edward Elgar, Cheltenham, UK, pp. 239–262. Guo, A.X., Fang, Q.H., Bai, X.D., Li, H., 2015. Hydrodynamic experiment of the wave force acting on the superstructures of coastal bridges. J. Bridge Eng. 20 (12), 04015012. Hayatdavoodi, M., Seiffert, B., Ertekin, R.C., 2014. Experiments and computations of solitary-wave forces on a coastal-bridge deck. Part II: deck with girders. Coast Eng. 88, 210–228.
5. Conclusions This paper presents an experimental program of extreme wave forces on the elevated pile cap of pile group foundation for sea-crossing bridge. The reduced scale pile cap specimens, including the isolated cap and pile cap with pile group, were set up in the wave flume. The results are presented and discussed as a function of clearance height, wave height and pile group. The wave force on the cap is a function of both clearance height and wave height. The largest horizontal wave force occurs when the cap is fully submerged, while the largest vertical uplift force occurs when the bottom of the cap locates at SWL. Higher clearance height reduces the horizontal wave force but induces larger vertical wave force on the cap. Wave slamming force occurs when the bottom of the cap locates above SWL. Compared with horizontal slamming force, the vertical slamming force has more than two times magnitude and shorter rising time. Therefore, the clearance height of the cap should be given considerable attention to balance the horizontal and vertical components of wave 12
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