Expected sliding distance of vertical slit caisson breakwater

Evaluating the expected sliding distance of a vertical slit caisson breakwater is proposed. Time history for the wave load to a vertical slit caisson is made. It consists of two impulsive wave pressures followed by a smooth sinusoidal pressure. In the numerical analysis, the sliding distance for an attack of single wave was shown and the expected sliding distance during 50 years was also presented. Those results were compared with a vertical front caisson breakwater without slit. It was concluded that the sliding distance of a vertical slit caisson may be over-estimated if the wave pressure on the caisson is evaluated without considering vertical slit.


Introduction
Performance based design has been one of the most emerging topics to engineers and researchers in coastal engineering during last decade. There have been a number of leading studies aiming to evaluation of the structural performance based on the reliability theory which considers stochastic variability of design parameters. The expected sliding distance of caisson type breakwater (Shimosako et al., 1998), the expected damage of breakwater armor blocks (Kim and Park, 2005), and the failure probabilities of caisson type quay-wall may be such an area of interest.
Especially, the method calculating expected sliding distance of a caisson breakwater has been frequently applied to the design practices for their simplicity and reasonable algorithm. Sliding distance calculation has been developed for decades (Goda and Takagi, 2000;Kim and Takayama, 2003;Esteban et al., 2007). Usually the calculated displacement within 0.3 m is accepted for the life time of 50 years (Burcharth, 2006). But the method was developed only for the vertical front caisson with no perforation on the front wall. Therefore, it cannot be directly applied to those caissons with dissipating walls. If the sliding distance is calculated based on the method, the results may be over or under-estimated.
To solve this problem, a new method of the sliding distance calculation for a vertical slit caisson is proposed. The algorithm of the proposed method is the same as the previous one but the time history of the pressure load is newly composed based on the pressure load formula by Takahashi et al. (1991).
In the numerical example, two types of caisson breakwaters with the same safety factors were compared by their sliding distance. It was found that the sliding distance of a vertical slit caisson breakwater should be calculated by considering the dissipated pressure time history to avoid overestimation.

Phase and wave load
Wave load on a vertical slit caisson was first formulated by Takahashi et al. (1991). They measured the wave pressure on the caisson from the large scale laboratory and defined three different phases to explain three local maximum pressures in the time domain as shown in Fig. 1.
The wave pressure to each phase can be calculated by Goda formula (Goda, 1974) with modified λ 1 and λ 2 as given in Table 1 where those subscripts of λ are referred to corresponding parts of the caisson as given in Fig. 2 λ Si is used to calculate wave pressure on vertical slit; λ Li for the lower part of the vertical slit; λ Mi for the bottom slab of the wave chamber; λ Ri for the rear wall; λ Ui for the uplift pressure. Subscripts 1 and 2 are for the slowly varying and impulsive wave pressures, respectively. In the calculation of α * for the rear wall, α 1 should be replaced by which is obtained with the parameters d', L', and instead of d, L, and B M respectively, where d' is the depth in the wave chamber, L' is the wavelength at water depth d, , and l is the width of the wave chamber including the thickness of the perforated vertical wall (Takahashi et al., 1991).

Time history
Wave load time history on a vertical slit caisson can be constructed by considering the crest shape and the calculated maximum pressure by using modified factors in Table 1. In Phases I and IIa, wave exerts impact load on the structures but in Phase IIb the smooth one. The wave load time histories are drawn in Fig. 3 where the solid line stands for Phase I, the thick solid line for Phase IIa, and the dotted line for Phase IIb.
In Fig. 3, time histories for Phases IIa and IIb are the same as that of the vertical front caisson without slit. Therefore, it can be calculated by Takahashi's method (Shimo-sako and Takahashi, 1998Takahashi, , 1999. Then, the time history by the Phase I wave is added in front of that. Since the Phase I wave attacks the caisson first, the time history of Phase I starts first and those of Phases IIa and IIb begin after a short time delay. The delay time (τ d ) is due to the passage of wave from the vertical slit to the rear wall. Therefore, it can be calculated by using the wave speed.
By summarizing the above physics, the pressure time histories on the vertical slit caisson can be calculated as follows:  (Takahashi et al., 1991) Phase I Phase IIa Phase IIb λ S1 0.85 0.7 0.3 Wave pressure on a vertical slit caisson (Takahashi et al., 1991). where denotes the wave pressure by Goda formula for each phase, t d and can be found by using wave height (H), water depth (d) and period (T) as follows: (5) The above equations were derived by replacing the water depth (h) in front of the structure by water depth (d) on the mound and d by d′ in the Takahashi's method.
Uplift force time history also can be calculated as: U max i where denotes the uplift pressure by Goda formula for each phase.
The downward pressure on the bottom slab of the wave chamber was assumed to be linear as: where P M1 and P M2 denote the downward pressure by wave crest IIa and IIb, respectively and were obtained by applying λ M1 and λ M2 instead of λ 1 and λ 2 in Goda's formula; a M and t m can be calculated as: (14) 3 Sliding distance

Single wave attack
The dynamic motion equation of a caisson under wave attack can be written as: where W, W a , W', μ, and x denote respectively the weight of the caisson, added mass, submerged weight of the caisson, friction coefficient between the caisson and rubble mound, and displacement of the caisson. In this equation, the damping force due to the wave-structure interaction is neglected. Damping is dominant when the caisson moves back and forth with the sides being exposed to water. But, it is assumed that the caisson is kept in contact with another caisson module side by side. Sliding distance can be found by solving the above equation. In solving the equation only the positive displacement is considered though there may be found sometimes negative displacement in the field measurement.
3.2 Expected sliding distance In Shimosako's method, it is assumed that 1000 waves attack a breakwater during a storm and one significant storm occurs in one year Takahashi, 1998, 1999). Then the sliding distance of a caisson is accumulated during lifetime. The evaluated sliding distance shows random in nature according to the wave height distribution. Therefore, the expected value of sliding distance should be calculated through the so-called Monte Carlo simulation. Fig. 4   Fig. 4. Flowchart of Monte Carlo simulation for sliding distance calculation.
Dong Hyawn KIM China Ocean Eng., 2017, Vol. 31, No. 3, P. 317-321 shows the flowchart of Monte Carlo simulation for evaluation of the expected sliding distance.

Single wave attack
For numerical simulation, a caisson breakwater with the width of 21.4 m and the crest height of 6.8 m was analyzed. The height of the front wall with the vertical slit is 3.0 m. the water depth in front of the breakwater is -24.5 m and the depth of the mound is -15.9 m.
When a single wave with the height of 15.0 m and the period of 11.0 s attacks the breakwater, a time series for the total horizontal force which equals the right hand side of Eq. (7) is calculated and compared with Shimosako's time series in Fig. 5. When the total horizontal force becomes positive, the caisson will slide in the positive direction. But it does not move when the force becomes negative. Fig. 6 shows the resultant caisson motion due to the wave attack. It tells us that the caisson begins to slide at about 3.0 s when the total horizontal force becomes positive. In other words, sliding of a vertical slit caisson mainly occurs at the tail of the wave while that of a caisson without slit at the front of wave. Table 2 compares the sliding distance of the caisson due to different single wave attack. Sliding distance of a vertical slit caisson is much smaller than that a caisson without slit.

Expected sliding distance
To calculate the expected sliding distance of a vertical slit caisson by the proposed method, the significant wave height of a year was assumed to follow the Weibull distribution of Eq. (8) with B, C, and k of 2.005, 1.595, and 1.525, respectively.
Randomness of the generated significant wave height was considered by applying the bias of -0.06 and the coefficient of the variance (COV) of 0.09. The friction coefficient between the caisson bottom and the mound was assumed to have the COV of 0.10 without bias.
Then, the expected sliding distances during 50 years were obtained through 5000 simulations. Table 3 shows the expected sliding distances for some different safety factors. The safety factor was controlled by putting additional mass to the caisson. It was found from the table that the vertical slit caisson sliding distance is much smaller than that of the caisson without slit. Except the case of SF=1.9, it is reduced to below 50%.

Conclusion
Wave force time history on a vertical slit caisson by wave attack was proposed by applying the static wave pressure of a slit caisson to the conventional wave pressure time history. Then the expected sliding distance of vertical slit caissons were calculated. It was concluded that the sliding distance of the vertical slit caisson is much smaller than that of a caisson without slit. Therefore, the sliding distance of vertical slit caisson may be over-estimated if the conventional approach without considering wave pressure dissipation is applied.