Experimental study on the cyclic behavior of monopiles in fine sandy beds under regular waves

This paper presents an experimental study on the wave-induced behavior of monopiles. Laboratory experiments were conducted at the constant initial state of the sandy beds in a wave flume with a soil trench. The responses of the pilehead displacement, the pile strain and the pore water pressure on regular waves were investigated. The experimental results show that the monopiles lean along the direction of the wave progression and the inclination increases with the duration of wave actions. The pile-head displacement (consisting of the permanent displacement and cyclic displacement) increases as the wave height increases, especially more significantly for the permanent displacement. The head-fixed pile suffers from larger wave load than that on the head-free pile under the same wave condition. Increasing pile diameter or fixing fins on the monopile is effective in reducing the pore water pressure in the upper part of the bed and the permanent displacement.


Introduction
Monopiles have been widely used as foundations for offshore wind turbines in the recent decades, because of their suitability in different soils and convenient construction. Compared with onshore wind power projects, the offshore foundations suffer from more complex loads in the offshore environment. A number of factors need to be considered in the design and construction of monopiles. Except for the effects of cyclic horizontal loads (wind and wave) on the change in the horizontal stiffness of the pile-soil system (Swane, 1983;Poulos, 1988;Levy et al., 2009), the pore water pressure in a porous seabed may also be built up and could lead to liquefaction (Sumer et al., 1999;Zhang et al., 2013;Liu et al., 2014) causing instability for adjacent structures (Passon and Branner, 2014;Sui et al., 2016). Obviously, as for the monopile-supported structures in sandy seabeds, a comprehensive understanding of the monopile's behavior under wave loading is important for engineering construction and maintenance.
The cyclic behavior of laterally-loaded piles has obtained growing attention since the 1960s and 1970s motivated by the development of the offshore petroleum industry (Wichtmann et al., 2008). For the actual offshore structures using pile foundations, the failure rate is very low, which seems to indicate that the early design approaches are relatively conservative. A better design idea may allow for a lateral foundation displacement to some extent as long as the offshore wind turbines can be operated normally (Randolph et al., 2005;Grabe, 2008). Unlike the cases of static laterally loading, both the rotation angle and displacement of the pile increase remarkably with the increasing numbers of load cycles because of the soil degradations (Swane, 1983;Poulos, 1988). This phenomenon has been reported in model test studies (Verdure et al., 2003;Rosquoët et al., 2007;von Pablo, 2011), and the degradation effect is considered to be related with the cycle number and cyclic load. Furthermore, the pile-type optimization is one of the main approaches to improve the lateral cyclic behavior of piles. For example, the pile with "wings" attached to its outer surface near the mud line (fin pile) is used to acquire more soil resistance in the upper weak seabed (Peng et al., 2004(Peng et al., , 2011Bienen et al., 2012).
The pore water pressure showed a weakening effect on soil mechanical properties (Seed et al., 1975), which further aggregates the displacement response of piles greatly. The relationship between the pore water pressure and the pile displacement is investigated with respect to piles in the offshore environment (Taiebat and Carter, 2000;Tasan et al., 2010;Hansen, 2012). It needs to be pointed out that the pore water pressure induced by the pile vibration is merely concentrated in these studies based on the assumption of a cyclic point wave load. Moreover, the pile diameter also has an influence on the pore water pressure, especially for the monopile with a diameter usually larger than 3 m (Hansen, 2012). The studies conducted by Swane (1983) and von Pablo (2011) revealed that the larger-diameter piles could lead to the weak dissipation due to longer drainage path for the pore water pressure around a laterally-loaded pile. Furthermore, the wave flume experimental results indicated that the wave action is also a key factor for the buildup of the pore water pressure (Pastor et al., 2006;Jeng et al., 2013).
For the actual monopiles in offshore environment, the effects of the pile displacement and wave actions on the seabed are of great significance for the analysis of the cyclic behavior. However, the relevant studies concerning these effects are still little. And in the meanwhile, the effectiveness of fins fixed on the monopiles has not been clear yet when being subjected to a real wave load. Therefore, this study aims to experimentally investigate the response of monopiles and fine sandy beds under wave loading and to understand the influence of the wave height, pile diameter, pile type and pile rigidity.

Wave flume and instrumentation
The experiments were conducted in a wave flume (50 m in length, 1 m in width and 1.3 m in height). The wave flume is equipped with a piston-type wave generator on one end to generate regular waves and a gravel beach on the other end to reduce the wave reflection. Fig. 1 shows the details of this facility. Fig. 1a provides the schematic diagram of the wave flume. The instrumentation of the monopile and the fin pile are presented in Figs. 1b and 1c, respectively.
The soil trench was 2 m in length, 0.75 m in width and 0.33 m in depth as shown in Fig. 1a. A sloping false floor (1 m in width and 0.25 m in height) which was used together with the soil trench, were applied to contain the sand to simulate more pile-soil interaction in the experiments. The design of the sloping false floor was according to the experiments conducted by Tzang and Ou (2006). Experimental observations indicated that the waveform on the sandy bed could keep stable by utilizing the transition sections. Finally, the sandy bed thickness and the model pile (including monopiles and fin piles) embedded depth reached 0.58 m and 0.3 m, respectively. Model piles were vertically located at the center of the soil trench (27 m away from the wave generator). In addition, the water depth was constant at 0.55 m (0.3 m above the mud line) in all experiments.
In the present experiments, the model piles had two values of the outer diameter, D=3 and 5 cm. The minimum distances between the model piles and the soil trench sidewall or the flume sidewall were 7.5D or 10D (for the case of D=5 cm), respectively. In general, there are two types of the boundary effects in the experiments. The first is about the impact of the trench boundary on mutual interaction of pile and soil. As discussed by Nogami and Novak (1977), the boundary effect may be ignored for the model of dynamic pile-soil interaction with small deformation when the pile is 3D away from the sidewalls. Davisson (1970) also con- firmed that the group effect becomes negligible with a pile spacing more than 8D even though a large deformation of the pile takes place. Given that the pile displacement observed was smaller than that resulted from the full horizontal bearing capacity, the minimal distance between the pile axis and the sidewall being 7.5D leads to a slight boundary effect. The second is about the impact of the sidewall on the mutual interaction of the pile and waves. According to Li et al. (2015), the boundary effect on the wave-structure interaction can be ignored when the structure is 3D away from the flume sidewall. In this experiment, the pile was 10D away from the flume sidewall, so it could be considered slight and ignored.
The following four parameters were recorded: the pore water pressure, water surface elevation, pile-head displacement, and pile strain. As seen from Fig. 1a, the pore water pressure measurements were made at the points that are 1.5D away from the model pile axis at three different depths (z) below the mud line (z=5, 15, and 25 cm). The water surface elevation over the pore water pressure was measured. A laser displacement sensor, fixed at 2 cm below the pile head, was used to measure the pile-head displacement for avoiding the disturbance of the testing equipment on the model pile's behavior. Two columns of strain gauges were attached to the outer surfaces of some typical model piles. The locations of the strain gauges were 0, 5, 10, 20, 30 and 40 cm above the mud line respectively. The strain gauges and their wires were covered by 1 mm thick epoxy coating to meet the demand of waterproofing. Based on the measured strain of cross sections, the pile moment and wave load can be calculated.
Pore water pressure sensors (PS-MPS-25), which had a 2-cm-long and 0.6-cm-diameter head sections, were first fixed at a rigid stent and then placed at the design position in the soil trench. The signal of the pore water pressure was amplified through the signal processing device (USB-2533) and recorded with DAQ-pro data acquisition software. The water surface elevation was measured with the conventional resistant-type wave gauge (YWS200-XX) with the maximum sample rate of 100 Hz. The laser displacement sensor (optoNCDT1402-20) used in the experiments had a measurement range of 20 mm, with the resolution of 0.002 mm and the maximum sample rate of 1.5 kHz. The resistance and sensitivity coefficients of the strain gauges (2.8 mm in the gate length and 2.0 mm in the gate width) were respectively 120.2±0.1 Ω and (2.22±1)%. The strain signal was collected through the signal conditioning device (DH-3810) and recorded with a dynamic strain acquisition instrument (DH5920). The sample rates of all test items were set at 50 Hz for simplification.

Soil and monopiles
The used mixture mass proportion of the soil for the sandy bed was fine sand:silt =3:1. The fine sand and silt had the grain sizes of 0.15 mm and 0.06 mm, respectively, which were prepared by manual grinding and screening. Note that d 50 (the grain size at which 50% of the soil) was 0.15 mm and the soil could be generally classified as fine sand. The basic soil properties are listed in Table 1. As tabulated in Table 1, γ s is the specific weight of soil grains; γ t , the specific weight of soil; and γ, the specific weight of water. The permeability coefficient, k, of the soil was determined by the standard constant head method (Lambe and Whitman, 1969) and its value given in Table 1 corresponded to the pre-test value of the void ratio e=0.53. The maximum void ratio (e max ) and the minimum void ratio (e min ) were determined by the Chinese standard (Ministry of Water Resources of the People's Republic of China, 1999).
A typical filling process shown below is used to prepare the seabed to ensure a constant initial state of soil filled in different experiment groups. First, the fine sand and silt were mixed well in a tank according to the mass proportion of 3:1 in the dry state. Then, the soil trench was filled with some water before the mixed soil rained down to ensure the water surface was always above the mud line. Later, nearly 3 kPa static pore water pressure has been applied on the bed for 24 h. The relative density (D r ) and void ratio (e) of the soil (z=15 cm) were approximately 0.73 and 0.53 respectively based on the aforementioned process.
Four model piles including tube and fin piles were used in the experiments, as listed in Table 2. The variation parameters were the rigidity and diameter. Except for the aluminum tube pile with D=3 cm, other three model piles were made from plexiglass tube to obtain relatively large pile strain. All model piles had the wall thickness of 2 mm and length of 1 m. As for the fin pile shown in Fig. 1c, the upper sides of the fins were designed at the mud line, according to the study conducted by Irvine et al. (2003). The fin pile had two fins fixed symmetrically on its outer surface. The fin plane was perpendicular to the direction of the wave propagation to obtain a larger pile-soil interaction area. Consideration for the size of the model piles, the fin was de-  2.3 Experimental procedure In general, the experimental procedure was as follows.
(1) The flume and the soil trench were firstly cleaned up. Then, the sloping false floor was installed at the specific locations. Next, the rigid stent used for fixing pore water pressure sensors and the instrument model pile were installed in the soil trench, as shown in Fig. 1a. (2) The soil trench was then filled with clean water to a certain depth. The bed was carefully prepared with the mixed uniform soil via the sand-raining technique. During the preparation, the surface of the sandy bed was smoothly leveled off with a scraper, and the water surface was kept 10 cm higher than the mud line. (3) When the preparation of the bed was finished, the flume was slowly filled with water up to the design depth. Subsequently, the sandy bed was allowed to rest for 24 h and the installation work for the experiment equipment was conducted. (4) The wave maker was switched on to generate the desired waves. Testing items were continuously collected until the experiments were completed.
A total of nine group tests were conducted under wave loading. Tests 1 to 5 simulated the condition of the headfree monopile embedded in the bed, Tests 6 to 8 simulated the condition of the head-fixed monopile but free at the bottom end, and Test 9 was used as a counterpart condition of the fine sandy bed. The regular wave with the wave period of 1 s and wave height of 8 or 10 cm was used in the experiments. The recording time for data presented in this paper is 200 s, and the significant wave loading lasted approximately 190 s. Note that the model-prototype similarities were difficult to be fully considered when the tests related with the pile-soil interaction were performed in 1g condition. However, the moderate scale effects were acceptable for the purpose of the mechanism study (Heller, 2011).

Experimental results and discussions
3.1 Typical dynamic response 3.1.1 Pile-head displacement Fig. 2 illustrates the displacement of the monopile for Test 1, in which the forward direction of the vertical coordinate axis is the orientation of the wave progression. Among them, the measured pile-head displacement (y) within 190 s wave period is shown in Fig. 2a. The maximum displacement at the head is 0.55 mm under the cyclic wave loading (Fig. 2a), while its value in the mud line is naturally smaller which is due to the rotation angle.
The pile-head displacements of Test 1 could be separated into two parts named the permanent displacement (y p ) in Fig. 2b and the cyclic displacement (y c ) in Fig. 2c, which is based on the linear moving average scheme (LMAV) provided by Foda and Tzang (1994). For the cyclic behavior of the laterally-loaded piles presented in the experiments, the permanent displacement increases with time (t) and eventually reaches 0.14 mm (≈0.005D) in Fig. 2b. Furthermore, the permanent displacement in Fig. 2b is positive, which means that the monopole leans towards the wave incident direction. This is because the regular waves generated in the flume were not linear waves. Actually, the nonlinearity induced by the shallow water effect will change the wave transformation pattern, resulting in steeper wave crests and gentler wave troughs. Thus, the wave load on the monopile will be similar to the effect of asymmetrical cyclic load. Regarding this point, the observations of von Pablo (2011) from the model experiments (cycle number>4×10 6 cycles) also support the view that permanent displacement will occur at the larger-amplitude side of the cyclic load. Fig. 2b also demonstrates that the permanent displacement (or inclination) increases stably during wave loading. This may be related to the load duration, because Peng et al. (2011) reported that the growth trend of the permanent displacement may be slow in the later stage based on the experimental result with 10 4 load cycles.
From Fig. 2c, it can be seen that the cyclic amplitude owns a minor change under waves with H=8 cm. This result shows that the plastic deformation of the soil around the monopile is limited and also indicates that the soil degradation is not obvious in Test 1. That means the pore water pressure induced by the waves and pile vibrations was relatively small at the most parts of the surrounding soil, compared with the effective stress. And this point will be later verified through the data collected by the pore water pressure sensors.

Pore water pressure
Figs. 3 and 4 demonstrate the pore water pressure (p) varying with time at three different depths from Tests 9 and 1, respectively. The periodic oscillation of the pore pressure could be found although the soil used in the experiment is prepared by mixing the fine sand and silt, which is due to the large mass proportion of the fine sand by artificial man-ufacture.
With the same method as proposed by Foda and Tzang (1994), the pore water pressure measured in Tests 9 and 1 can be divided into two components, i.e., the residual pore water pressure and oscillating pore water pressure. As for Test 9, the residual pore water pressure is 0.015 kPa at z=5 cm and t=200 s, which only occupies 5.44% of the average amplitude of the oscillating pore water pressure. The result indicates that the residual pore water pressure in the fine sandy bed with H=8 cm is limited. Fig. 5 illustrates the vertical distribution in the average amplitude of the pore water pressure, where p aa /p 0 is the ratio of the average amplitude to that at the bed surface, after t=25 s when the response of the monopile and bed becomes relatively stable, and d s is the thickness of the bed. As the depth z increases from 5 to 25 cm, p aa /p 0 decreases from 0.65 to 0.12 in Test 9 and from 0.70 to 0.15 in Test 1, as presented in Fig. 5. Moreover, p aa /p 0 increases by 2.81% to 5.55% based on the comparison between the results of Test 9 and Test 1. Two reasons can be responsible for this result. One is the compression effect on the surrounding soil due to the pile displacement response (von Pablo, 2011;Hansen, 2012), the other is the wave reflection caused by the impeding effect of the monopile presence on the wave propagation (Sui et al., 2016).

Wave load
The wave load on the model pile can be calculated by the measured pile strain based on the pure bending theory for beams. Pure bending refers to the flexure of a beam under a constant bending moment, which only occurs in regions of a beam consisting of homogeneous, linearly elastic materials where the shear force is zero (Gere, 2004). Generally, these assumptions are not completely valid in reality. However, detailed investigations show that the normal stresses calculated from the theory are not significantly altered by the presence of the shear stresses (Timoshenko and Goodier, 1970), and these assumptions are just inapplicable near the beam supports or a concentrated load. Based on the normal stress in pure bending, the moment at the cross section i1 can be calculated from: where M i1 (t) is the moment of the cross section i1 at time t, E is the Young's modulus of elasticity, Δε i1 (t) is the strain difference of cross section i1 at time t, and W z is the section modulus.  To simplify calculations, the wave load distributed on pile segment i is set equivalent to a point load on the top of pile segment i, as illustrated in Fig. 6. The point load is defined as: where F i (t) is the point load at the top of pile segment i at time t; M i1 (t) and M i2 (t) are the moments at the top and bottom of pile segment i, respectively; l i is the length of pile segment i. Finally, the total wave load on the pile, F T , can be proximately obtained by accumulating all point loads on pile segments. Although the wave load on the pile segment is not a point load actually, the effect of the load distribution on the pile-soil interaction remains limited on the basis of Saint-Venant's principle. Fig. 7 presents the test results of F T on the monopile changing with time in Test 1, while that of the head-fixed pile in Test 6 is shown in Fig. 8. Because F T is deduced from the pile strain, its period is close to the wave period (T=1 s). This result also indicates that the monopile is in the state of typical forced vibration. In addition, the wave load is mainly related to the response of the monopole and wave parameters. For example, as shown in Fig. 7, F T does not significantly change with time t in Test 1 partly because of the relatively stable cyclic amplitude (Fig. 2c). Moreover, the average total wave load in Test 1 is 4.89% lower than that in Test 6 after 25 s. This result suggests that the wave load was influenced by the displacement response of the monopile.

Effect of the wave height
Wave height, one of the major parameters of waves, directly affects the pile displacement, the pore water pressure in beds and the wave load on piles. In the present study, two wave heights including H=8 and 10 cm were chosen to study its effects on the soil and monopiles. The variation of the head displacement of the monopile with time in Test 1 (H=8 cm) and Test 2 (H=10 cm) are presented in Fig. 9. As the results show, when the wave height increases from 8 to 10 cm, the permanent displacement (y p ) increases by 73.59% at t=200 s. Meanwhile, the average value of the cyclic amplitude (y ca ) increases by 14.54% from t=25 to 200 s. This situation is closely related to the velocity and acceleration of water particles acting on the monopile. The larger wave height means the bigger velocity and acceleration, which will lead to a larger wave load on the monopile. Therefore, the permanent displacement and cyclic amplitude both increase with the wave height.
The development of the amplitude of the pore water pressure (p a ) at three different depths is presented in Fig. 10.    Fig. 9. Comparison of the pile-head displacement between Tests 1 and 2.

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HUANG Ting et al. China Ocean Eng., 2017, Vol. 31, No. 5, P. 607-617 After 25 s, the average of p a separately increases by 28.22%, 24.69% and 11.21% at the depths of 5, 15 and 25 cm when the wave height increases from 8 to 10 cm. On the other hand, the influence of the wave height gradually decreases as the depth of soil increases. Moreover, the ratio of the pore water pressure to the effective stress almost reaches 1 at the depth of 5 cm when H=10 cm (Test 2). This result suggests that the momentary liquefaction within the upper part of soil has occurred for the silty grains in the bed. Beside the effect of wave height, the loss of the soil strength caused by the liquefaction also enhances the increase of the pile-head displacement. When the pile diameter/wave length (D/L) ≤0.15, the wave force on a unit length of the pile can be obtained from the Morison equation (Morison et al., 1950), as follows: uẋ where C M and C D are the inertia and drag coefficient, respectively; ρ is the water density; u is the horizontal component of water particle velocity; is the horizontal component of water particle acceleration; is the horizontal velocity of the pile. For comparison, the tested amplitude values of the total wave load (F Ta ) and the value (D3H10, D3H8) calculated with the Morison equation are shown in Fig. 11. With the thickness (1 mm) of the epoxy coating on the model piles, the diameter of the pile D=3.2 cm was used in the calculation. For the pile with the circular cross section, the following parameters were adopted: the inertia coefficient, C M =2, and the drag coefficient, C D =1.2. In addition, the third-order Stokes wave was used according to Le Méhauté's (1976) proposal. As described in Fig. 11, compared with the difference between the experimental and the calculated res-ults for H=8 cm (Test 1, Test 6), that for H=10 cm (Test 2, Test 7) is more significant, which may be related to the nonlinearity of the waveform. Furthermore, F Ta on the headfree pile (Test 1, Test 2) is smaller than that on the headfixed pile (Test 6, Test 7). For example, the average F Ta in Test 2 is 6.74% lower than that in Test 7 after 25 s. Because the monopile is in a stable state of typical forced vibration, the relative velocity and relative acceleration between the water particles and the monopole decline with the increasing displacement response.

Effects of the pile diameter
Monopiles with the diameter of 5 and 3 cm are used to study the effects of diameters. The permanent displacement ratio (y p-D5 /y p-D3 ) and the cyclic amplitude ratio (y ca-D5 /y ca-D3 ) of the monopile with D=5 cm to that with D=3 cm from 25 to 200 s are presented in Fig. 12. The results show that the pile-head displacement of the larger-diameter monopile is smaller, especially the permanent displacement. Compared with the variation of cyclic amplitude ratio (Fig. 12b), the permanent displacement ratio shows an increase trend (Fig. 12a)   Fig. 10. Comparison of the amplitude of pore water pressure between Tests 1 and 2.  HUANG Ting et al. China Ocean Eng., 2017, Vol. 31, No. 5, P. 607-617 613 as time extends. This result means that the increase rate of the permanent displacement of the lager-diameter monopole is larger even the smaller-diameter one has the larger permanent displacement. Besides that, the displacement response of monopoles is related to the pore water pressure and the wave load, which will also be analyzed. The time series of the amplitude ratios of the pore water pressure at different depths are displayed in Fig. 13. The amplitude of the pore water pressure in Tests 4 and 1 are represented by p 4a and p 1a , respectively. The average amplitude ratio is about 0.95 at the depth of 5 cm after 25 s, which indicates that the pore water pressure in the surrounding soil of the monopole with D=5 cm is smaller than the one with D=3 cm. This situation may be related to two conflicting causes. One is that the larger-diameter monopile has less compression effect on the surrounding soil for the smaller displacement response, which will induce relatively smaller pore water pressure. The other cause is that the wave reflection behind the pile and the longer drainage path effectively lead to the buildup of the pore water pressure. The former effect is more evident than the latter one in the present study. As a result, the pore water pressure related to the larger-diameter monopile is smaller. On the other hand, the fluctuation of the amplitude ratios is more significant as the depth increases. The high average ratio (>1) is possibly related to large rigid displacement of the larger-diameter monopile and weak local drainage characteristics in the deeper bed. Fig. 14 gives the time series of the amplitude of the total wave load (F Ta ) and the calculated values. The calculated values for the monopiles with D=5 and 3 cm (the 1-mm thickness of the epoxy coating is also considered) under the wave of H=8 cm by the Morison equation is defined as D5H8 and D3H8, respectively. All coefficients adopted in the calculation are the same as those in Subsection 3.2.1. As shown in Fig. 14, the average value of the total wave load in Test 4 (head-fixed pile) is 1.83% smaller than that in Test 8 (head-free pile) with D=5 cm. Compared with the result of the cases with D=3 cm, the effect of the wave nonlinearity is more obvious for the larger-diameter monopiles.

Effects of the pile type
Two kinds of model piles including the monopile (Test 1) and the fin pile (Test 3) were used to study the effects of the pile type. The time variation of the pile-head displacement under the wave of H=8 cm from t=25 to 200 s is shown in Fig. 15. The permanent displacement and cyclic amplitude ratios between the fin pile and the monopile are defined as y p-FD3 /y p-D3 and y ca-FD3 /y ca-D3 , respectively. The measured data indicate that the displacement response of the fin pile is less than that of the monopile. Additionally, the difference in the permanent displacement between two pile types is relatively larger than that of the cyclic amplitude, such as the average values of y p-FD3 /y p-D3 and y ca-FD3 /y ca-D3 are about 0.69 and 0.96, respectively, as shown in Fig. 15. In consideration of the minor difference of the wave load between the monopile and the fin pile, fixing fin is verified as an effective method to obtain more soil resistance based on the present flume experiments. Compared with the method of enlarging diameter, the pile-soil system of the fin pile is more likely to maintain an elastic state due to the small wave load.   Fig. 15. Ratios of the pile-head displacement in Test 3 to that in Test 1. Fig. 16 gives the amplitude ratios of the pore water pressure in Test 3 to that in Test 1, and p 3a is the amplitude of the pore water pressure in Test 3. Investigation results show that the pore water pressure near the fin fixed depth (z=5 cm) is smaller than that of monopile. This phenomenon indicates that the longer drainage path caused by fixing fin has little effect on the dissipation of the pore water pressure, or says that the effect of the displacement response is more significant. In addition, the average ratios at deeper test points are also larger than 1, which are similar to the results observed in Fig. 13.

Effects of the pile rigidity
To investigate the effect of the pile rigidity (EI, I is the moment of inertia), model piles made from aluminum and plexiglass tubes were respectively adopted in Tests 5 and 1. For this parameter study, the diameter of the model piles is 3 cm, and the wave height is 8 cm. The permanent displacement ratio (y p-AlD3 /y p-D3 ) and the cyclic amplitude ratio (y p-AlD3 /y p-D3 ) between the aluminum tube pile and the plexiglass tube pile are shown in Fig. 17. Generally speaking, the decrease in the pile deflection corresponds to the increase in the pile rigidity. The cyclic amplitude of the aluminum tube pile is significantly smaller than that of the plexiglass tube pile in Fig. 17b, such as the average value of y ca-AlD3 /y ca-D3 is only 0.46. The smaller cyclic displacement leads to the lower compression effect on the surrounding soil, which eventually results in the smaller permanent displacement of the aluminum tube pile (Fig. 17a), although the decrease of the permanent displacement is minor.
It is noteworthy that the rigidity ratio between the aluminum tube pile and the plexiglass tube pile is about 22:1 in the experiments. The increasing space of the rigidity ratio should be smaller for monopiles under the most conditions. Since the insignificant decrease of the permanent displacement, it can be predicted that the effects of the pile rigidity on the monopole displacement response may be slighter in practical engineering. Enlarging the diameter or fixing the fins may be the effective method to decrease the wave-induced displacement of monopiles.

Conclusions
In the present study, a series of laboratory experiments were performed to investigate the response of monopiles and fine sandy beds under regular waves. Moreover, the influences of the wave height, pile diameter, pile type and pile rigidity on the response were discussed. Based on the experimental results presented, the following conclusions can be drawn.
(1) The wave-driven pore water pressure in the fine sandy bed mainly oscillates periodically. The monopiles lean towards the direction of the wave progression due to the wave nonlinearity, and its inclination increases with time. Compared with the tests without piles, the average amplitudes of the pore water pressure increase by 2.81% to 5.55% due to the response and the presence of the monopile.
(2) As the wave height increases from 8 to 10 cm, the permanent displacement increases by 73.59%. The effect of the wave height on the pore water pressure weakens with the increase of the soil depth. Owing to the smaller relative velocity and acceleration, the average amplitude of the total wave load on the head-free pile is 6.74% smaller than that on the head-fixed pile under 8-cm-high waves.
(3) Compared with the smaller-diameter monopile, the monopile with larger diameter shows a much smaller permanent displacement but a larger increase rate. The pore water pressure reduces by 5% with the increasing pile diameter from 3 to 5 cm due to a decreasing compression effect on the surrounding soil.
(4) The fin pile shows a lower displacement response than the monopile under regular waves. Furthermore, the Fig. 16. Amplitude ratios of the pore water pressure in Test 3 to that in Test 1. Fig. 17. Ratios of the pile-head displacement in Test 5 to that in Test 1.
HUANG Ting et al. China Ocean Eng., 2017, Vol. 31, No. 5, P. 607-617 average permanent displacement ratio between the fin pile and the monopile, with the value of about 0.69, is relatively smaller than the average cyclic amplitude ratio whereas the increasing tendency is similar.
(5) The cyclic amplitude of the monopile decreases with the increasing pile rigidity. However, the rigidity ratio up to 22:1, mostly larger than the ratio in actual engineering, cannot result in an obvious decrement in the permanent displacement. Therefore, the use of increasing the rigidity of monopiles may not effectively control the displacement response.