Backfilling configuration to improve tenacity of composite-type breakwaters

ABSTRACT Composite-type breakwaters are reinforced by piling rubble stones and constructing counterweight fillings (known as reinforcing embankments) behind caissons. Important performance requirements for breakwaters include minimal damage and high strength, even when the external forces exceed the design forces. In this study, the failure process and final state of breakwaters with reinforcing embankments are investigated via centrifuge model tests, and the cross-sectional configuration of the reinforcing embankment for improving the tenacity of breakwaters is determined. The results show that, in the overturning mode, when the number of rubble stones decreases, the reinforcing embankment deforms in accordance with the inclination of the caisson; subsequently, the caisson overturns and mounts onto the embankment. When balance is not maintained at that position, the caisson slides down the slope surface, resulting in catastrophic failure. A series of centrifuge model tests qualitatively show that placing more rubble stones adjacent to the caisson is less likely to result in such catastrophic failure. Furthermore, the stability of the breakwaters is evaluated via circular slip analyses, which demonstrate the importance of increasing the volume of the reinforcing embankment adjacent to the caisson in terms of the stability.


Introduction
Breakwaters are constructed to protect port areas from high waves, storm surges, and tsunamis. Typically, composite-type breakwaters are used when the water level at the breakwater is high. Composite-type breakwaters feature concrete caissons on a foundation mound composed of rubble stones (Figure 1). Forming a rubble-mound breakwater using only rubble stones in deep water requires a significant number of stones, which is uneconomical. Furthermore, constructing a rubble-mound breakwater that can withstand high wave forces is difficult. The ground at which breakwaters are constructed is typically soft, which renders it difficult to construct a vertical breakwater using only concrete or steel structures. Compositetype breakwaters combine the advantages of both rubble-mound and vertical breakwaters, and can distribute load to the underlying original ground while requiring less rubble stones. In Japan, most breakwaters are composite-type breakwaters, whose effectiveness has been recognized. However, during the 2011 Great East Japan Earthquake (hereinafter referred to as the 2011 earthquake), an unexpectedly large tsunami affected the Tohoku coastal area of Japan, which damaged several port facilities (Kazama and Noda 2012;Mase et al. 2013;Mori et al. 2013;Sugano et al. 2014;Takahashi et al. 2011). An 11.8 m tsunami affected the Port of Kamaishi in Iwate Prefecture and resulted in a significant difference in water level in front of and behind the breakwater at the bay mouth (Tomita and Yoen 2012). The surge wave lasted for approximately 10 min, followed by a large backwash, which exerted wave forces on the harbor side of the breakwater. These hydraulic conditions caused severe damage to the composite-type breakwater, resulting in the overturning and sliding of the caissons off the mound and the collapse of the mound.
After the 2011 earthquake, the tsunami heights assumed in the design of breakwaters were increased for many areas, and existing breakwaters must be reinforced to withstand such tsunamis. In addition, the recent global climate change has resulted in more intense typhoons in Japan, which necessitates measures for managing with high waves. A promising method to reinforce compositetype breakwaters is to pile up rubble stones behind the caisson to form a counterweight filling, as shown in Figure 2, of which the embankment is known as a reinforcing embankment. This method has been previously proposed as a countermeasure against high waves and has since been adopted (Kikuchi, Shinsha, and Eguchi 1998;MLIT et al. 2009). At the time, the behavior of rubble stone ground was not fully understood, and the safety degree of the design was questionable. In addition, only the effect of reducing the caisson sliding was considered, not the effect of preventing overturning and increasing the bearing capacity. Since the 2011 earthquake, the authors and other research groups have investigated the effects of reinforcing embankments (Arikawa et al. 2013;Miyamoto et al. 2014;Shinsha et al. 2014;Takahashi et al. 2015). The results showed that reinforcing embankments increased the bearing capacity of the mound and reduced the sliding of caissons. Takahashi (2021) conducted a detailed failure characterization of reinforcing embankments and mounds, compared experimental and analytical results, and developed a stability assessment method for composite-type breakwaters with reinforcing embankments. The abovementioned studies demonstrated that reinforcing embankments enhances the stability of composite-type breakwaters; as such, they are currently being used in the field.
The most important performance requirement for breakwaters is that they must be resistant to the design wave forces at least; moreover, even if the breakwater is damaged by external forces exceeding the design wave forces, the degree of damage should be minimal, and the integrity of the breakwater should be preserved to the maximum extent. The increased strength of breakwaters owing to reinforcing embankments has been investigated in previous studies; however, reduced damage owing to the implementation of reinforcing embankments has not been investigated. In other words, the failure process and final state of a composite-type breakwater with a reinforcing embankment, i.e. the process by which it ruptures when subjected to external forces exceeding its limit, have not been clarified. A better understanding of the above will enable the design of reinforcing embankments that retain their performance and demonstrate their tenacity.
In this study, the failure process and final state of a composite-type breakwater with a reinforcing embankment are investigated using centrifuge model tests, and the cross-sectional configuration of the reinforcing embankment that can improve the tenacity of composite-type breakwaters under a high external force is investigated. Specifically, model tests with different cross-sectional configurations of the reinforcing embankment are systematically conducted using the centrifuge model tests, which can reproduce the stresses in the ground of a prototype composite-type breakwater by applying a centrifugal force to the model. Based on the results, the failure characteristics of a breakwater with a reinforcing embankment are investigated, and the cross-sectional configuration of the embankment that does not result in complete performance loss, even when subjected to damage, is discussed. Furthermore, the stability of the breakwaters is also evaluated via circular slip analyses. Reinforcing embankments are originally used to reinforce existing composite-type breakwaters; however, the authors believe that they are similarly applicable to newly constructed composite-type breakwaters. If breakwaters can be further strengthened, as attempted in the present study, then the use of composite-type breakwaters with reinforcing embankments will undoubtedly increase.

Failure modes of composite-type breakwaters backfilled with rubble stones
First, the failure modes of normal composite-type breakwaters and those with reinforcing embankments were considered. As assumed in the design of composite-type breakwaters, composite-type breakwaters without reinforcing embankments exhibit three failure modes: sliding and overturning of the caisson, and mound failure due to insufficient bearing capacity (see Figure 3a). In addition, the bar arrangement of the caisson was designed by considering the reaction forces exerting on the bottom near the rear toe. In practice, sliding, overturning, and mound failures do not occur alone, as assumed in the design; in fact, the failure modes are combined and diverse in many cases. For example, even if the sliding mode dominates, a small number of rubble stones on the mound surface near the toe of the sliding caisson will be moved, and the caisson will tilt. Furthermore, even if the sliding mode is dominant in the early stages of displacement, if the caisson tilts and the horizontal distance between the center of gravity and rear toe becomes closer, then the moment of resistance to overturning decreases, and the overturning mode may be dominant from the middle of the displacement. However, the basic failure modes when considering the failure process and final state are sliding, overturning, and mound failures. Even if a composite-type breakwater comprises a reinforcing embankment, the three basic failure modes are the same as those for a composite-type breakwater without a reinforcing embankment: sliding and overturning of the caisson, and mound failure due to insufficient bearing capacity, as presented by Takahashi (2021) and shown in Figure 3b. When the caisson slides on the mound, an area of shear strain begins to appear near the rear toe of the caisson and expands into the slope of the reinforcing embankment. This is not described as a slip surface because the grain size of rubble stones is relatively large relative to the dimensions of the mound and reinforcing embankment; therefore, a slip surface such as that in sandy or clayey ground is unlikely to develop. Figure 4a shows the shear failure of the reinforcing embankment during the sliding of the caisson, as presented by Takahashi (2021). If the top of the reinforcing embankment is low and its top width is wide, then a shear strain area develops from the rear toe of the caisson toward the top of the reinforcing embankment. When the top of the reinforcing embankment is high and its top width is narrow, then an area of shear strain develops from the rear toe of the caisson to the slope of the reinforcing embankment. In any case, when the caisson slides, the reinforcing embankment elevates and its horizontal reaction force increases. If the caisson slides sufficiently far until it shifts up to the slope of the reinforcing embankment and the embankment collapses, the horizontal reaction force from the reinforcing embankment vanishes; however, sliding is unlikely to remain dominant, and the caisson will unlikely shift to that extent. Next, considering the case of mound failure, an area of shear strain extending from the bottom of the caisson to the reinforcing embankment appeared. An example in which a slip surface is intentionally developed using boulders is shown in Figure 4b, as presented by Takahashi (2021). Although the caisson was buried within the mound and reinforcing embankment and its performance as a breakwater deteriorated because of the reduced top height of the caisson, the caisson should not slide down. By contrast, the characteristics of caisson overturning differ significantly from the   above. The installation of reinforcing embankments alongside the caisson renders it less likely to overturn; however, once the caisson envelopes the reinforcing embankment, it will slide down the slope of the reinforcing embankment. In this case, the caisson cannot function as a breakwater. Among the three failure modes, sliding and mound failures are naturally tenacious; however, overturning is not tenacious in some cases. Therefore, the failure process and final state must be considered while prioritizing the overturning failure.
When assuming that a tsunami strikes a breakwater, the possibility of overflows beyond the top of the breakwater must be considered. If overflow occurs, then the reinforcing embankment may be scoured. However, as the installation of reinforcing embankments is expected to be effective, one must prevent scour caused by overflows or minimize scour such that the function of the breakwater can be maintained. Therefore, a prerequisite of this study is that the reinforcing embankment must be generally intact/undamaged. To prevent scour, covering must be provided; to minimize scour, the weight of rubble stones used for reinforcing embankments must be increased or embankments should be installed in a position that prevents flow generated by overflow. If scour caused by overflow can be prevented or scour can be minimized, then the failure mode that can catastrophically damage a breakwater with a reinforcing embankment is caisson overturning, as described above. Based on these considerations, the present study focuses on the overturning mode of a caisson to investigate its failure characteristics.

Similarity law
Hydraulic model tests to reproduce waves have generally been conducted in a gravitational field. This is likely because the motion of the water is, to some extent, clear, and the motion of prototype-scale water can be estimated using a model. In hydraulic model tests, the ground is typically simplified using a fixed bed, and the behavior of waves is prioritized. However, model tests to reproduce ground deformation and failure are often conducted in centrifugal acceleration fields. This is because the deformation and failure behavior of soils is complex owing to the variety of soil types and because they contain soil particles, pore water, and pore air. Because the behavior of soil depends significantly on stress, researchers have attempted to directly reproduce the strain in a prototype-scale ground by applying centrifugal force to a model, such that the same stress in the ground can be exerted in the prototype. However, as ground behavior is prioritized, the external forces from water are simplified and replaced by loads from motordriven devices.
Some problems cannot be simplified into either of these two problems. One such example is the phenomenon considered in this study. To maintain the external forces from water on the caisson as the latter shifts, and to reproduce overflow and seepage forces in the mound, one must use water in the model. Meanwhile, granular rubble stones must be used in the model to reproduce the behavior of geomaterials, such as reinforcing embankments and mounds. The thickness of the reinforcing embankment was not small, for example 5 m, and the confining pressure was sufficiently large to affect the stiffness and strength of the ground. Therefore, the model tests were performed in a centrifugal acceleration field containing water and stones, which is a distinct feature of the present study. Combined wave and ground model tests have been performed previously in a centrifugal acceleration field. The earliest tests were performed by Sekiguchi and Phillips (1991) and Phillips and Sekiguchi (1992). Subsequently, the researchers such as Baba et al. (2002), Cheng (2003), Gao and Randolph (2005), Takahashi et al. (2010), Takahashi et al. (2014), Takahashi, Morikawa, and Kashima (2019), and Takahashi et al. (2022) conducted the same tests; however, this centrifuge model test has yet to be generalized.
In this section, a brief summary of the similarity laws is provided. Froude's law is often used for the similarity laws of waves. The ratio of inertia to gravity is known as the Froude number Fr, which is typically used to reveal the behavior of waves. Fr is expressed as follows: where U is the flow velocity, g the gravitational acceleration, and L the characteristic length. In the model tests using centrifugal acceleration, gravity g and length L in Equation (1) are multiplied by N and 1/N, respectively, where N is the ratio in a model scale. In this case, the flow velocity U in the numerator need not be multiplied by the scale ratio. As the behavior of water depends on the flow velocity, its reproduction is vital to centrifuge model tests. Moreover, N was set to 50 in this study, which is sufficiently small to discount the viscous forces and surface tension of water against inertia. The most significant advantage of the ground similarity law is that it can reproduce prototype-scale stresses in a centrifugal acceleration field, as mentioned earlier. Alternatively, the strain can be reproduced by reproducing the stress, and the similarity ratios of the other parameters that describe the ground state can be determined automatically. Model tests performed on the ground in a gravity field are disadvantageous as they require assumptions regarding the stress -strain properties of the soil, and the similarity ratios between dimensions and displacements are inconsistent. Basically, the displacements relative to the dimensions will be small, and it indicates that the deformation of the reinforcing embankment and the overturning behavior of the caisson cannot be properly simulated in a gravity field. Meanwhile, model tests performed in a centrifugal acceleration field do not exhibit such problems. Actual sand and clay were used in the model tests because the scale effects regarding grain size are negligible, and the physical properties of soil change if the grain size is reduced to match the model size. However, in the model tests using rubble stones, the scale effect was not negligible because the grains were relatively large relative to the structure size, but the physical properties of the rubble stone ground, such as stiffness and strength, did not change significantly even when the grain size was reduced to match the model size. Therefore, model tests using rubble stones with a grain size corresponding to the model size should be conducted.
The similarity law for combined water -soil is difficult to establish. For behaviors exhibited near the interface between water and ground, such as scour, similarity laws have not yet been established, partly because those behaviors are yet to be elucidated. However, as the focus of this study is not on phenomena near the interface, such as scour, the similarity law was relaxed. Seepage forces are exerted on rubble stones owing to the difference in water levels between the front and back of breakwaters, and the similarity law for this phenomenon has been discussed comprehensively by Takahashi et al. (2014). In the pores within rubble stones, the flow of water becomes turbulent, and the similarity ratio between the seepage force and other forces is the same, even when viscous fluids or other approaches are not used. The similarity ratio between water flow and the velocity of the flow within rubble stones is the same as well. Notably, the rubble stones and pore water behave as a single mixed flow.
In the present study, such an intense ground flow is unlikely to occur; however, if it does occur, then the similarity law is as follows. A phenomenon similar to it is sand-gravel-type debris flow, and the similarity law of this model test is used as a reference. The most dominant parameter in sand-gravel-type debris flows is the frictional force between the gravel particles. The equation for this frictional force includes density, gravitational acceleration, area, and water depth, which results in the Froude's law when the ratio of the frictional force to the inertia force is considered. Although other derivations can be performed, the final result is Froude's law. Hence, the similarity law for sand-graveltype debris flows is typically regarded as Froude's law. As mentioned above, Froude's law is satisfied in a centrifugal acceleration field, where the flow velocity is preserved, and is consistent with other flow velocities.

Test preparation and procedure
A schematic view of the cross section is shown in Figure 5a, and a photograph of the prepared model is shown in Figure 5b. In the model test, a water level difference was generated between the front and back of the breakwater model to simulate a tsunami. This composite-type breakwater was a hypothetical section for the model test. The centrifugal acceleration was set to 50 g; therefore, the ratio of the dimensions of the model to the prototype was 1/50, which implies that the model simulated a 50× larger composite-type breakwater. The inner dimensions of the specimen container model were 1400 mm (length) × 600 mm (depth) × 80 mm (width). The ground preparation method was as follows: First, a grease-coated membrane was attached to the sides of the container to reduce the friction between the sides of the container and the ground. Iide silica sand No. 7 (average grain size: 0.18 mm) was used to prepare the foundation. A dense foundation with a relative density of   approximately 90% was prepared using the air pluviation method (see Takahashi et al. 2006). More realistic modeling was used preparing the foundation because the foundation deformation can affect the deformation of the reinforcing embankment and mound. Rubble ground was created on the surface to simulate a mound. The rubble stones were deposited with almost zero fall height, and the relative density was adjusted by lightly tapping the rubble stones using a rod for compaction two to three times every time the layer thickness reached approximately 40 mm. This method has been confirmed to increase the relative density to approximately 50%. Three types of rubble stones were used, and their photographs are shown in Figure 6. The rubble stones featured a grain size of 5-7 mm (250-350 mm in a prototype scale) and an average mass of 0.14 g per rubble stone (172 N in a prototype scale). Although these rubble stones were smaller and lighter than those used in actual fields, they were employed to increase the deformation of the rubble mound and to observe the failure of the compositetype breakwater. Rubble stones of various sizes were used for the reinforcing embankments, i.e. 2-3 mm (100-150 mm in a prototype scale) rubble stones, which were smaller than the typical rubble stones; and ~10 mm (500 mm in a prototype scale) rubble stones, which were larger than the typical rubble stones. The average mass of each type mentioned above was 0.11 g (135 N) and 2.34 g (2867 N), respectively. The rubble stones were not flattened, elongated, or broken; therefore, their shapes were similar to those of rubble stones used in actual breakwaters. A model caisson was installed after the mound was prepared. The caisson comprised a box constructed using glued acrylic plates filled with sand and lead balls, and its weight was adjusted such that its density was similar to that of an actual caisson; its density was 2100 kg/m 3 . The height of the caisson was increased relative to its width such that that among sliding, overturning, and mound failures, overturning would be dominant. After the caisson was installed, the reinforcing embankment was prepared in the same manner as the mound. A water pressure gauge was installed to measure the water level in front of the caisson, and rubble stones to reduce the impact of falling water during water supply were installed to complete the model.
Model tests were conducted systematically by varying the cross-sectional configuration of the reinforcing embankment (19 cases) and the type of rubble stone (four cases), as shown in Table 1. Figure 7 shows the cross sections for all cases. Table 1a shows the cases featuring various cross-sectional configurations: some cases feature various heights and top widths of the reinforcing embankment, whereas others feature a step in the middle of the slope of the reinforcing embankment at the same height as the top of the mound, as well as various widths. The cross-sectional area was not fixed such that the effect of the reinforcing embankment scale can be assessed. Additionally, model tests were performed under extreme crosssection conditions, e.g. Case E0 without a reinforcing embankment, and Case E0-W* with an extended mound width. Case E0-W1.5-R is the same condition as Case E0-W1.5, in which the caisson lay on the extended mound without sticking into the mound. In Case E8.7-0, the top width of the reinforcing embankment was 0 m, the shoulder of the reinforcing embankment was aligned with the height of the caisson top, and the cross-sectional configuration of the embankment was triangular. Case E6-12-C is a continuation of Case E6-12, in which the caisson did not shift, even though the reinforcing embankment was scoured, and the model test was continued based on the state in which the embankment was scoured. Table 1b lists the test conditions for the cases in which the material of the rubble stones used for reinforcing the embankment was varied. The rubble stones used were those described above, and the configuration of the reinforcing embankment was varied with several representative ones. The cross-sectional area shown in Table 1 is the area of the entire reinforcing embankment; v is the area of the reinforcing embankment, which is larger than the mound top in a prototype scale; w represents the reinforcing embankment width from the rear toe of the caisson at the height of the mound top in a prototype scale.
The prepared model was placed on a centrifuge platform, and a centrifugal acceleration of 50 g was applied to the model. Centrifuge Mark II-R owned by the Port and Airport Research Institute (see  for details) was used. Under the applied centrifugal force, water flowed from the water tank at the top of the specimen container to the front of the caisson and drained through a hole at the back of the caisson, thereby maintaining the water level difference between the front and back of the caisson for a certain duration. In other words, a dam-break test was conducted to generate a water-level difference and simulate a tsunami. For more information regarding this apparatus, please refer to . The difference in the water levels between the front and back of the caisson in Case E6-12, where the caisson did not shift, is shown in Figure 8 in a prototype scale.
The initial water level and the water level reference height were at the bottom of the caisson. The water level difference was determined from the water pressure difference measured using the water pressure gauges installed at the front and back of the caisson shown in Figure 5a. As shown in the figure, the water level difference increased once the water supply was started; subsequently, it reached a peak value of 9 m in a prototype scale, decreased slightly, and stabilized at 7 m. This continued for approximately 8 s (400 s in  a prototype scale), after which the water tank was empty and no more water could be supplied, and the water level difference returned to 0 m thereafter. Although some tsunamis last longer than 400 s, in all cases except Case E6-12, the scour of the reinforcing embankment and mound and the movement of the caisson could be determined to have reached the final state. In Case E6-12, where the final state was not reached, the continuation was carried out as Case E6-12-C. The water level difference peaked at 0.5 s because the water level at the back of the caisson remained at a low level before the water supply was started, and the water level increased to a steady state as water flowed from the front to the back of the caisson owing to overflow and seepage. The water level in front of the caisson was controlled by the overflow from the top of the caisson and the seepage in the embankment and mound. The water levels in all cases were almost the same regardless of the configuration of the reinforcing embankment. The water level at the back was fixed by overflow into the drainage tank. These test settings resulted in almost the same water level difference in all cases.

Failure process and final state of breakwaters
Photographs captured during the test are shown in Figure 9. They depict the typical failure process and final states for four cases: Case E4, which featured the height and top width of the reinforcing embankment, i.e. 1/3 H (H is the height at the front of the caisson) (see Figure 9a); Case E4-W2, which featured a step of width H in the middle of the slope of the reinforcing embankment of Case E4 (see Figure 9b); Case E6, which featured the height and top width of the reinforcing embankment, i.e. 1/2 H (see Figure 9c); and Case E8.7-0, where the top width of the reinforcing embankment was 0 m, the shoulder of the embankment was aligned with the top of the caisson, and the cross-sectional configuration of the embankment was triangular, as shown in Figure 9d. In the top right-hand corner of the photographs, the time elapsed since the water supply was started is shown in a model scale.
In Case E4, the slope surface of the reinforcing embankment was scoured by the overflow, the reinforcing embankment was deformed, the caisson tilted and climbed over the embankment, and the caisson slid down to the foundation ground. Meanwhile, in Case E4-W2, although a slight overflow occurred, it was observed that the rubble stones on the upper slope of the reinforcing embankment flowed out owing to seepage and overflow, thus deforming the reinforcing embankment and causing the caisson to overturn. The overturned caisson stopped on the step, and the top level of the caisson after the overturn was higher than that in Case E4. This suggests that the caisson developed greater tenacity. The deformation of the reinforcing embankment in Case E4-W2 is discussed. The displacement of the reinforcing embankment was calculated via particle image velocimetry using photographs captured by a high-speed camera. Figure 10 shows the deformations at time periods 0.50-0.71, 0.71-0.76, 0.76-0.79, 0.79-0.81, 0.81-0.87, and 0.87-1.00 s as vectors, with the photographs at the beginning of each period. Based on the figure, the reinforcing embankment deformed as the caisson was tilted. In particular, the deformation of the upper section of the reinforcing embankment was more significant than that of the mound itself, and the size of the upper section was assumed to be a key to sustain the caisson. In Case E6 (see Figure 9c), the slope of the reinforcing embankment was scoured by the overflow, and the caisson tilted as the scouring of the embankment continued, causing it to climb over the embankment. After approximately 5 s (250 s in a prototype scale) of water descending from the top of the overtopped caisson onto the embankment and then scouring it, the caisson climbed over the embankment and stopped near the slope of the embankment before scouring occurred; more tenacious than that in Case E4-W2. In Case E8.7-0, the caisson required a longer time to begin tilting compared with the other three cases because the rubble stones near the top of the slope of the reinforcing embankment remained stationary, even after the slope of the reinforcing embankment had been scoured. The caisson did not climb over the reinforcing embankment after overturning but stopped on top of it. In other words, this caisson was the most tenacious among the caissons considered in the four cases shown in Figure 9. Thus, the time until the caisson overturned and the height and position of the caisson after overturning differed depending on the size and configuration of the reinforcing embankment; additionally, the failure process and final state of the caisson were different. Figure 11 summarizes the final states after the caisson stopped shifting in all the cases. The white line indicates the position of the composite-type breakwater before a tsunami is introduced. The classification of the height and position of the caisson after overturning, as described below, is shown in the topright corner of each photograph. Table 2 shows the test results of the overflow and reinforcing embankment scour in addition to the classification of failure, which is also discussed below. Because the differences in the failure states owing to the size and configuration of the reinforcing embankment have been described above, the effects of scour, seepage flow, and stone grain size on the failure state are shown in Figure 11 and Table 2. In the cases with a small cross-sectional area v of the reinforcing embankment and Case E4-FN using 2-3 mm rubble stones, the overflow on the caisson and scouring of the embankment did not occur or, if they occurred slightly, the extent was insignificant, the rubble stones in the embankment were outflown by seepage flow instead of overflow, and the caisson overturned. In Case *-CS, where 10 mm rubble stones with a larger grain size were used, the rubble stones remained stationary during overflow, and the reinforcing embankment was not scoured. In other cases, the reinforcing embankment overflowed and scoured.
Among the test conditions for rubble stones of varying sizes, Cases E4-FN (2-3 mm), E4 (5-7 mm), and E4-CS (approximately 10 mm) featured the same configuration and cross-sectional area of the reinforcing embankment. In Case E4-FN, which featured rubble stones with small grain sizes, the rubble stones in the embankment flowed out via seepage instead of overflow; the caisson overturned, slid down the slope, and then stopped shifting immediately above the foundation ground. In Case E4, the overflow scoured the slope of the reinforcing embankment, causing the caisson to tilt over and slide down from the embankment to the foundation ground. In Case E4-CS, which feature large grain sizes, seepage flow occurred from the slope of the reinforcing embankment when the water level difference between the front and back of the caisson began to increase; however, the slope of the reinforcing embankment did not outflow. The gradual tilting of the caisson owing to the water level difference caused a few rubble stones on the slope to roll; however, scouring of the reinforcing embankment after the overflow was not indicated. After the caisson stopped tilting, overflow continued because of the difference in water levels between the front and back of the caisson; however, the reinforcing embankment did not scour, and the caisson did not climb over the embankment as it tilted. The test results for the cases in which rubble stones of various materials were used for the embankment showed that for embankments with the same configuration and cross-sectional area, by using rubble stones with larger grain sizes, the embankment was less likely to be washed out by seepage flows, less likely to be scoured by overflow, and less likely to collapse even when the caisson tilted.

Discussions regarding backfilling configuration
The height and position of the caisson after it overturned are classified, as shown in Figure 12, and the post-failure performance of each case is discussed. The height of the caisson is considered to be an important indicator of the performance of the composite-type breakwater after the caisson overturns, and the postfailure performance is classified using the height of the caisson top after overturning occurs. The mound top height is expressed in percentage. In particular, the height of the caisson front before overturning is 100%. After overturning, the height is expressed as follows: I = more than 90%; II = 70%-90%; III = 40%-70%; IV = 0%-40%. Meanwhile, the position of the caisson after overturning is defined as follows: S = no overturning; A = the caisson tilted, and the caisson bottom is at the top of the mound; B = the caisson climbs over the reinforcing embankment, and the caisson bottom is above the top of the mound; C = the caisson slides down to a position above the foundation ground; F = the caisson slides down to the foundation ground. For example, in the case shown in Figure 9, Case E4 features height IV/position F; Cases E4-W2 and E6, III/C; and Case E8.7-0, I/B. The classification of all cases is shown in Figure 11. Using this classification, the position of the caisson after overturning is shown in Figure 13a, and the height is shown in Figure 13b for the series with various cross-sections of the reinforcing embankment. The horizontal axis in the figures represents the width of the reinforcing embankment from the rear toe of the caisson at the height of the mound top w, and the vertical axis represents the crosssectional area of the embankment higher than the mound top v. The reasons for using w and v will be explained later; however, they are important parameters that represent the failure process and final state. The dotted lines in the figure show the relationship between w and v, where the entire reinforcing embankment has the same cross-sectional area, and the dotted line from the bottom left increases by 25 m 2 in terms of the cross-sectional area of the entire reinforcing embankment in a prototype scale. In Figure 13b, the domain is categorized into three sections that define the height of the caisson after overturning: (I) 0-40%, (II and III) 40%-90%, and (IV) more than 90%. Figure 13(a,b) show that in the cases where the cross-sectional area v of the rubble stones adjacent to the caisson exceed 30 m 2 , the caisson remained stationary before it climbed over the reinforcing embankment. Even in cases where the number of rubble stones in the embankment was low, the caisson did not slip down to the foundation ground. The caisson is unlikely to slide down the slope of the reinforcing embankment to the foundation ground, regardless of the presence of a step on the slope. For breakwaters with a cross-sectional area v of less than 30 m 2 , the caisson slid down to the foundation ground when the width w of the reinforcing embankment was less than 14 m, and the caisson was prevented from sliding down to the foundation ground when w was greater than 14 m. To prevent the caisson from sliding off the step, the reinforcing embankment width w had to be increased to 14 m or more in the cross section, which was the threshold value in this study. Hence, one can conclude that the cross-sectional area v of the reinforcing embankment higher than the mound top, as shown on the vertical axis, determines whether the caisson overtops the embankment; meanwhile, the width w at the mound top, shown on the horizontal axis, determines whether the overturned caisson can be prevented from sliding down to the foundation ground. Hence, v and w are two important parameters that control the progression of breakwater failure.
Next, the differences in the failure states when the same number of rubble stone is used for the entire reinforcing embankment are discussed. In an actual design, when the number of rubble stones is the same, it would be desirable for breakwaters to be tenacious. Cases in which the cross-sectional areas of the entire reinforcing embankment were approximately 100 m 2 and a step was absent were compared.
In Case E6, where the height and top width of the reinforcing embankment was 6/12 H, the caisson remained on the reinforcing embankment and then climbed over it, whereas in Case E8.7-0, where the top width was 0 m and the reinforcing embankment was triangular, the area near the top of the reinforcing embankment was less scoured and the rubble stones remained in the vicinity of the caisson; therefore, the caisson required a longer duration to tilt and did not climb over the reinforcing embankment. Although the configurations of the reinforcing embankments in Cases E6 and E8.7-0 were close, these cases were in the transition zone of the post-failure caisson position as shown in Figure 13b, which may cause a significant difference in the results. Subsequently, cases where the cross-sectional areas of the entire reinforcing embankment were approximately 75 m 2 were compared. In Case E6-3, where the height of the reinforcing embankment was 6/12 H and the top width was 3/12 H, the caisson climbed over the reinforcing embankment and stopped halfway up the slope of the embankment; meanwhile, in Case E5, where the height and top width were 5/12 H, the caisson climbed over the reinforcing embankment and slid down to the foundation ground. Comparisons between Cases E8.7-0 and E6 and Cases E6-3 and E5 showed that for the same cross-sectional area of the entire reinforcing embankment, a triangular configuration with a larger  height of the reinforcing embankment was more likely to cause the stones of the embankment to remain near the caisson when the caisson overturned, and was less likely to cause the caisson to slide down. Cases involving a step are discussed next. The test results for a cross-sectional area of approximately 100 m 2 of the entire reinforcing embankment showed that a smaller v and a wider w with a step resulted in deteriorated post-failure performance, with the classification of failure on height changing from I to II or III and the classification on position changing from B to C. For the same number of stones, piling them up adjacent to the caisson was more effective in promoting the performance of the breakwater than expanding the width of the mound, and using a configuration without a step was preferable. In the cases where the cross-sectional areas of the entire reinforcing embankment were less than 75 m 2 , widening w rather than increasing v prevented the caisson from sliding down to the foundation ground, thus avoiding a catastrophic failure. However, this will reduce the stability of the composite-type breakwater against sliding of the caisson and mound failure due to insufficient bearing capacity, and it is better to place a certain amount of stones beside the caisson. To summarize the failure states when the number of rubble stones used for the entire reinforcing embankment was fixed, the best configuration was triangular, which can be achieved by increasing the number of rubble stones adjacent to the caisson, instead of introducing a step. However, in practice, considering the stability against overtopping waves and overflows and the fact that the ground near the water surface is prone to collapse due to waves, the reinforcing embankment should be configured in a trapezoidal shape (similar to a triangle) and should not be extremely near the water surface.
Similarly, for the cases with rubble stones of different sizes, the position of the caisson after overturning is shown in Figure 14a, and its height is shown in Figure 14b. As the amount of scour decreased and the strength of the ground increased with an increase in the rubble stone size, the breakwater failure due to overturning was less progressive. In the small crosssectional area of the entire reinforcing embankment, the same tendency as that shown in the cases with various cross-sectional configurations of the reinforcing embankment was observed (Figure 13b). The results in Figure 13b can be described as a relationship in which the quantitative threshold varies with the sizes of the rubble stones and compositetype breakwater, but not qualitatively. Therefore, the design of a reinforcing embankment should focus on the stability of composite-type breakwaters, as shown in the next section, and the result of Figure 13b should be considered in the design. In other words, one should assume that a caisson is likely to climb over  and slide down unless a certain number of stones are provided adjacent to it; if the number of stones is fixed for the entire reinforcing embankment, then piling up rubble stones adjacent to the caisson without providing a step is preferable.

Verification of stability
The results of the centrifuge model tests and the discussions of the results indicate that, for a compositetype breakwater with a reinforcing embankment designed to resist external forces greater than the design wave force, placing more rubble stones adjacent to the caisson is more effective than increasing the width of the mound. However, the stability against mound failure could be reduced if more rubble stones are placed adjacent to the caisson. One should verify whether increasing the number of rubble stones along the side of the caisson will increase the stability of the breakwater against the design wave. In this study, circular slip analysis, which can be performed to assess the stability of composite-type breakwaters, was used to investigate the change in the stability when the cross-sectional configuration of the reinforcing embankment was varied.
Circular slip analysis based on the simplified Bishop method (Bishop 1955) was performed for the calculations. In the design standard for port facilities in Japan (MLIT et al. 2020), the stability of the bearing capacity of composite-type breakwater mounds and breakwaters with reinforcing embankments must be evaluated via circular slip analysis. Takahashi (2021) indicated that the results of circular-slip analysis underestimated the resistance force at the limit state of the ground. This implies that the reinforcing embankment or mound will not immediately fail, even if the design wave force is exerted on it. According to Takahashi (2021), a caisson typically displaces by approximately 0.3 m, even when an external force with a resistance-to -action ratio of 1.0 is applied. This displacement may represent a limit similar to the usability limit of the breakwater as a structure. The circular slip analysis qualitatively represented the reaction forces from the embankment when the cross-sectional configuration of the reinforcing embankment was varied. These results suggest that circular slip analysis is effective for assessing stability.
The cross-sectional configuration of the reinforcing embankment used in the calculations is illustrated in Figure 15. Additionally, the figure shows the results of the calculations presented below. The dimensions of the caisson and mound were determined by referring to an existing composite-type breakwater, and various cross-sectional configurations of reinforcing embankments were provided without changing the crosssectional area. The shear strength of the mound and embankment used in the calculations was c = 20 kN/ m 2 and ϕ = 35°, respectively, as provided in the Japanese design standard (MLIT et al. 2020). The effective unit volume weight was set to γ' = 9.8 kN/m 3 by assuming that all rubble stones were underwater. If some of the stones were above the water surface, they will be more stable because their weight would be heavier than their underwater weight. Two types of slips were considered in the calculations: shallow slip from the rear toe of the caisson, assuming that the caisson slid or overturned, and deep slip from the bottom of the caisson, assuming mound failure due to insufficient bearing capacity (Figure 16). For the former shallow slip, the horizontal load exerting on the reinforcing embankment was varied to obtain a horizontal load with a resistance-to-action ratio of 1.0. The friction force was calculated by multiplying the horizontal load with the coefficient of friction (tan 15°). The working height of the horizontal load was 1/3 the height of the reinforcing embankment. The horizontal load with a resistance-to-action ratio of 1.0 was the resistance force of the embankment in the shallow slip failure mode. For the latter deep slip, the ratio between the reaction force from the embankment and the   frictional force on the mound surface was set to 0.5, and a horizontal load of 843 kN/m was applied. However, in Case C4, in which a step was provided and the configuration of the reinforcing embankment featured two steps, the limit value of the horizontal load at the shallow slip was 597 kN/m; therefore, the friction force was shared by adding 246 kN/m (843 kN/ m minus the limit value) to the friction force on the mound surface. A vertical load was applied to the reinforcing embankment by multiplying the horizontal load from the embankment by the coefficient of friction (tan 15°). Pressure was applied to the mound surface by dividing the vertical load, which was the weight of the caisson minus its friction force, by the loading width. The loading width of the vertical load was determined using the equilibrium equation between the external force exerting on the caisson and the gravitational force. The relationship between these forces is illustrated in Figure 16. The calculation results for each cross-sectional configuration are shown in Figure 15. In addition to the circular slip surface, which indicated the lowest stability, the figure shows the critical horizontal load at shallow slip and the resistance-to-action ratio at deep slip. The resistance-to-action ratio was obtained by dividing the resistance force by the action force. The higher the number, the greater is the margin of the resistance force; therefore, the breakwater with the higher number was more stable. For both shallow and deep slips, the stability was the highest at the section of the reinforcing embankment with a triangular cross-sectional configuration, followed by the reinforcing embankment with 1/3 the height of the caisson, 1/4 the height of the caisson, and two-stage reinforcing embankment. In other words, if the same cross-sectional area of the reinforcing embankment is to be provided in a composite-type breakwater, then more rubble stones should be placed adjacent to the caisson instead of increasing the width of the mound. This is consistent with the tendency of the caisson, which exhibits tenacity when overturned by high external forces. Hence, the cross-sectional area of the reinforcing embankment at the side of the caisson must be increased to both improve the stability of the breakwater with the reinforcing embankment and to demonstrate the tenacity of the breakwater when the caisson overturns. However, as mentioned above, considering other aspects, using a trapezoidal instead of a triangular cross-sectional configuration for actual breakwaters is preferable.

Conclusions
In this study, the failure process and final state of a composite-type breakwater with a reinforcing embankment were investigated via centrifuge model tests, and the cross-sectional configuration of the reinforcing embankment that can improve the tenacity of the composite-type breakwater even under high external forces was determined. The results obtained are as follows: (1) Based on a discussion of the failure modes of composite-type breakwaters with reinforcing embankments, the failure mode that resulted in performance loss under high external forces was caisson overturning. Hence, model tests were conducted in this study under conditions where the caisson overturned.
(2) Similarity laws were organized for water, ground, and combined problems, and centrifuge model tests were performed. In the model tests performed in this study, the cross-sectional configuration of the reinforcing embankment and the type of rubble stone were varied. When a significant number of rubble stones were used in the reinforcing embankment, the caisson was less likely to overturn and slide. However, when significantly less rubble stones were used, the reinforcing embankment deformed in accordance with the tilt of the caisson, causing the caisson to overturn, climb onto the reinforcing embankment, and slide down the slope of the reinforcing embankment when its balance was not maintained at that position.  (3) The results of a series of model tests showed that the number of rubble stones on the side of the caisson, and not the configuration of the reinforcing embankment, determined whether the caisson would climb over the reinforcing embankment. Additionally, a wider mound prevented the caisson from sliding down the mound slope; a threshold for effectiveness existed. For a certain amount of stones and the same number of stones, piling them up adjacent to the caisson was more effective in promoting the performance of the breakwater than expanding the width of the mound.
Circular slip analyses confirmed the stability of composite-type breakwaters with reinforced embankments. In terms of the stability, the cross-sectional area of the reinforcing embankment on the side of the caisson was effective. However, in actual composite-type breakwaters, considering other aspects, a trapezoidal cross-section with a horizontal surface at the top of the reinforcing embankment is preferable.