# Probabilistic Risk Evaluation for Overall Stability of Composite Caisson Breakwaters in Korea

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Selection of Reliability Approaches

## 3. Sliding and Overturning Failure Modes

#### 3.1. Breakwater Structures and Wave Force Model

_{c}and H

_{c}represent the width and the heights of the caisson, respectively. h

_{r}indicates the height of the rubble mound. The ocean conditions are represented by the water depth at the site h, the water depth from the base of the caisson h′, the water depth over the berm d, and the variation in the tide level WL. Finally, the wave conditions are represented by the design wave height H

_{D}.

_{h}and F

_{u}) acting on the upright section are calculated using Goda’s formulae [30]. In Goda’s model, wave forces are determined from the pressures distributed over the height and width of the upright section, which are estimated based on the design wave height, determined by the mean of the 0.4% largest wave heights in the recorded wave data. The pressure is distributed triangularly on the bottom of the caisson (maximum pressure p

_{u}) and has a trapezoidal shape on the front face of the upright section (represented by pressure p

_{1}, p

_{2}, and p

_{3}). Figure 3 illustrates the wave pressures following Goda’s model on the caisson section.

#### 3.2. Performance Functions and Considered Random Variables

_{c}, L

_{rc}, and L

_{f}are the lever arms of the moments caused by the weights of the plain concrete structure (D

_{c}), the reinforced concrete structure (D

_{rc}), and the filling material (D

_{f}), respectively. The total weight of the materials comprising the caisson is represented as a summarized random input variable with a mean density of 2100 kg/m

^{3}in [19]. In this work, these dead loads are considered as uncertainties with different variations, which are dependent on not only the construction stage but also the construction method. For instance, the caisson body is a precast reinforced concrete structure, so it would have better quality control compared with the concrete cover slab or filling material built on sites. The friction coefficient between the concrete caisson and the rubble surface, f

_{c}has been well studied and empirically obtained through various studies [13,18]. Here, its bias is presumed as a normal distribution with the mean value of 1.06 and coefficient of variation (COV) of 0.15. B stands for buoyancy force that is highly dependent on the submerged volume of the structure below the water level. Additionally, because tidal variations influence not only the buoyancy force but also wave pressures, the water level is studied as an uncertainty based on the tide fluctuations in this study. Tide levels rely on astronomical and meteorological tides and the increased water level caused by nearshore waves [28,29]. In the present study, different BRWs on the eastern, northern, or western coasts, as shown in Figure 2, are dissimilarly explored to consider the different variations in the tide levels.

_{u}, and horizontal wave force, F

_{h}, estimated using Goda’s expression, are the random input variables in the study. It is proved that the wave forces obtained following the Goda’s formulae are conservative in the studies of Van der Meer et al. [13,31]. In other words, when Goda’s model is applied, the bias factor, λ, must be taken into account, which is generally less than unity [13], as shown in Equation (3).

#### 3.3. Reliability Analysis Using MVFOSM and FORM

_{f}, is converted from the result of the reliability index, β, using an approximation of the cumulative probability density of the standard normal distribution, Φ, as shown in Equations (4) and (5) [25,27].

_{g}, and the standard deviation, σ

_{g}, of the PF, g, as shown in Equation (6). Thereafter, the reliability index, β, is estimated at the mean values of all random variables, µ

_{X}, as shown in Equations (7) and (8) [27].

_{X}, σ

_{X}):

_{i}, that affects the results of the FS is estimated as cosine direction at the most probable failure point as shown in Equation (11).

_{i}represents the input random variable i in the initial variable coordinates, and U

_{i}stands for the variable in the normalized/reduced space. The subscript * shows the value of variables at the design point. The sensitivity factor is the cosine direction of the PF at the design point with for each variable.

_{i}shows the positive effect of the variable X

_{i}on the PF and vice versa. Furthermore, more substantial sensitive factors show more significant effects on the PF. The design point in the normalized space U* can be expressed as the product of the sensitivity factor and RI by using Equation (12).

^{2}= 0.9933) indicates a good approximation of the fitting function. The intercept of the fitting function is then set to zero, and the new equation β = 1.304FS can be obtained.

#### 3.4. Reliability Analysis Using MCS

_{f}, is determined using Equation (14).

_{MCS}and N

_{fail}are the simulation size and the number of failures occurring in the simulation, respectively. I(g(X) < 0) is an indicator function that reaches the value of unity if g(X) is less than zero and reaches the value of zero in other cases.

_{Pf}, is approximated from the number of simulation, N

_{MCS}, and the failure probability, P

_{f}, as shown in Equation (15). Generally, the number of simulations, N

_{MCS}, should be greater than ten times the reciprocal of P

_{f}(N

_{MCS}> 10/P

_{f}) so that the COV of P

_{f}estimated from the different MCS runs is less than 0.3 [33]. The relative error with a 95% confidence interval of binomial distribution of each simulation cycle is estimated with Equation (16) [27].

^{5}for the sliding reliability analysis. This size of simulation is also applied to other BRWs because the RI based on the FORM of the Onsan BRW is the largest among the nine BRWs. Meanwhile, based on the FORM results, the RIs for the overturning conditions are much higher than those of sliding, which is discussed in detail in Section 3.5. Consequently, the failure probability against overturning might be minor. By assuming the normal distribution, Figure 6 depicts the correlation between the smallest number of realizations to encounter at least 10 overturning failure events in the simulation and the expected RI. Particularly, if it is needed to reach a failure probability less than 10

^{−6}(β > 4.75), 10

^{7}should be the smallest number of realizations applied. This example demonstrates one of the disadvantages of MCS in comparison with other methods like FORM and MVFOSM. Moreover, the results for overturning are not critical. Therefore, the RIs for overturning are assumed as the same as the results from FORM and MVFOSM in this work.

_{Sliding}) of the Onsan BRW compared with unity. The histogram shows a long tail on the right that declares that the distribution of the results is not symmetric even though the involved variables are considered as symmetric distributions. In these cases, the procedure proposed by Allen et al. [34], in which the sliding PF is plotted with the standard normal variable, can be applied to estimate the RI from the PF based on the MCS, as shown in Figure 8. The point at which the PF takes the zero value shows the negative RI. On the other hands, the point where performance line intersects with the horizontal axis illustrates the mean value of the PF. With the normal distributions, the slopes of the lines indicate the standard deviation of the PF. Finally, the RIs based on the three methods are plotted in Figure 9 for the sliding and overturning conditions. The results are discussed in detail in Section 3.5.

#### 3.5. Discussion on Sliding and Overturning modes

^{−8}. The probability of the overturning of a caisson around its heel is considerably small, as expected, implying that the failures would occur in the subsoil before the caisson itself would overturn [19]. The bearing capacity of the foundation is, therefore, examined in the next section.

## 4. Bearing Capacity of Foundation by Circular Slip Failure Analysis

#### 4.1. Selection of Bishop’s Simplified Method

#### 4.2. Performance Function and Involved Random Variables

_{h}, at the top surface of the rubble mound; vertical force, F

_{u}; and the distance of the vertical force to the harbor side heel, B

_{z}. The equivalent resultants are then transmitted as reaction forces to the slope including the rubble mound and subsoil to determine the critical failure surface.

_{Ex}is the surcharge load distributing on the slice in the vertical direction. F

_{h}and a correspond to the horizontal wave force and its lever arm about the failure center. c is the apparent cohesion of soil (or undrained shear strength in case of cohesive soil) while φ is the friction angle of soil along the base surface of each slice. s is the width of each slice. Finally, θ is the inclination angle, horizontal to the base surface. The studied variables, X, is shown as a vector of the involved variables in Equation (18). In the BSM, the FS is defined by a nonlinear and implicit equation; therefore, the FS is approximated with several iterative steps to reach the acceptable tolerance.

^{2}for the apparent cohesion and 35° for the internal friction angle for typical rubble materials supporting the eccentric and inclined loads as standard values. Various geotechnical parameters of rubble mounds and the subsoil were investigated in PROVERB (2001), where the soil strength properties were assumed as either the normal or lognormal distribution. However, the unit weights of the rubble mound and the subsoil were treated as deterministic variables [13]. In this work, when analyzing the bearing capacity of the foundation, both the unit weight of the rubble mound and subsoils are considered as random variables, similar to the sliding and overturning states.

#### 4.3. Result and Discussion of Bearing Capacity Analysis

_{Pf}≈ 0.18) in the Gunsan case proves that the chosen size of simulation is suitable. Although all studied variables are assumed as normal distributions, the outcome of the FSs reveals a skewed distribution, as in the example shown in Figure 13 for the Gamcheon BRW. The skewness implies the asymmetry of the probability density function of the PFs, while a normal distribution (totally symmetric) would show a skewness value. The positive skew coefficients in all the studied cases imply that the PF’s distribution have a longer tail on the right compared with that on the left. This phenomenon reflects the nonlinear relationship of the safety factor with all the involved variables in the BSM. Additionally, the positive skewness instances are more densely located on the left side of the mean value. This result demonstrates that the safety factors of the reliability analysis are predominantly distributed below the mean value.

_{f}≈ 0.0417) and 3.24 (P

_{f}≈ 0.0028), corresponding to the sliding state and failure occurrence in the foundation. Thereby, under similar service conditions, sliding failures might be observed much more frequently (about 15 times) than foundation failures. However, lower failure probability for sliding state in comparison with failures in foundation does not suggest that sliding damages will always take place before failures occurring in the rubble mound or subsoil—sliding failures are merely more common among the three overall stability conditions.

## 5. Conclusions

^{−8}).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overall failure mode of breakwaters (BRWs) studied: (

**a**) Sliding, (

**b**) Insufficient foundation bearing capacity, (

**c**) Overturning.

**Figure 6.**Relationship between the smallest number of simulations and the expected reliability index in MCS.

**Figure 10.**Mechanism applied to estimate the PF against bearing capacity using a combination of Bishop’s simplified method and MCS.

**Figure 12.**Cumulative density of the bearing capacity safety factor of foundation in the standard normal space.

No. | Breakwater | Geometry Parameter | Ocean Condition | |||||||
---|---|---|---|---|---|---|---|---|---|---|

B_{c} | H_{c_Sea} | H_{c_Har} | h_{r} | h | h′ | d | H_{D} | WL | ||

1 | Donghae | 20.0 | 16.0 | 16.0 | 6.1 | 16.99 | 10.89 | 8.29 | 10.8 | 0.20 |

2 | Pohang | 14.0 | 12.0 | 12.0 | 3.5 | 11.75 | 8.25 | 7.25 | 8.58 | 0.12 |

3 | Ulsan | 19.0 | 21.5 | 17.5 | 3.0 | 18.66 | 15.66 | 14.16 | 9.9 | 0.30 |

4 | Onsan | 14.0 | 18.5 | 16.0 | 9.0 | 19.84 | 14.11 | 13.11 | 5.76 | 0.30 |

5 | Busan | 24.0 | 21.5 | 20.7 | 8.5 | 25.44 | 16.94 | 15.39 | 10.8 | 0.72 |

6 | Gamcheon | 20.0 | 18.0 | 17.5 | 3.0 | 17.28 | 14.28 | 12.08 | 10.8 | 0.64 |

7 | Jeju Outer | 24.0 | 25.8 | 22.8 | 1.3 | 24.83 | 17.83 | 16.03 | 11.7 | 1.42 |

8 | Jeju Aewol | 27.4 | 23.5 | 20.5 | 1.5 | 13.86 | 13.36 | 12.36 | 14.04 | 1.43 |

9 | Gunsan | 18.0 | 19.0 | 18.0 | 2.0 | 15.75 | 13.75 | 13.15 | 10.08 | 3.62 |

No. | Notation | Mean of Bias | COV of Bias | Distribution | Random Variable |
---|---|---|---|---|---|

1 | f_{c} | 1.06 | 0.15 | Normal | Friction coefficient |

2 | W_{c} | 1.02 | 0.02 | Normal | Weight of concrete |

3 | W_{rc} | 0.98 | 0.02 | Normal | Weight of reinforced concrete |

4 | W_{f} | 1.02 | 0.04 | Normal | Filling material |

5 | F_{u} | 0.77 | 0.260 | Normal | Vertical wave force |

6 | F_{h} | 0.90 | 0.222 | Normal | Horizontal wave force |

7 | W | 1.00 | COV ^{1} | Normal | Tidal level |

^{1}COV: 0.05/0.12/0.20 correspond to the West/South/East locations.

Parameter | Van der Meer et al. [31] | Vrijling [32] | CEM 2011 [28] | |
---|---|---|---|---|

No Model Tests | Model Test Performed | |||

Horizontal force, F_{h} | N(0.90, 0.20) | LN(0.90, 0.20) | N(0.90, 0.25) | N(0.90, 0.05) |

Horizontal moment, M_{h} | N(0.81, 0.37) | LN(0.72, 0.37) | N(0.81, 0.40) | N(0.81, 0.10) |

Vertical force, F_{u} | N(0.77, 0.20) | LN(0.77, 0.20) | N(0.77, 0.25) | N(0.77, 0.05) |

Vertical moment, M_{u} | N(0.72, 0.34) | LN(0.72, 0.34) | N(0.72, 0.37) | N(0.72, 0.10) |

**Table 4.**Results of mean value first-order second-moment (MVFOSM) and first-order reliability method (FORM) for sliding failure mode.

No. | Port | Sensitivity | β | |||||||
---|---|---|---|---|---|---|---|---|---|---|

f_{c} | W_{c} | W_{rc} | W_{f} | F_{u} | F_{h} | WL | FORM | MVFOSM | ||

1 | Donghae | 0.700 | 0.042 | 0.025 | 0.132 | −0.158 | −0.682 | −0.016 | 1.671 | 1.692 |

2 | Pohang | 0.693 | 0.039 | 0.026 | 0.141 | −0.164 | −0.686 | −0.013 | 1.608 | 1.624 |

3 | Ulsan | 0.714 | 0.032 | 0.026 | 0.172 | −0.121 | −0.667 | −0.021 | 1.843 | 1.873 |

4 | Onsan | 0.837 | 0.015 | 0.032 | 0.151 | −0.060 | −0.521 | −0.022 | 3.325 | 3.295 |

5 | Busan | 0.670 | 0.042 | 0.026 | 0.181 | −0.183 | −0.694 | −0.033 | 1.441 | 1.557 |

6 | Gamcheon | 0.681 | 0.038 | 0.033 | 0.153 | −0.169 | −0.692 | −0.050 | 1.510 | 1.616 |

7 | Jeju Outer | 0.738 | 0.032 | 0.029 | 0.160 | −0.117 | −0.642 | −0.052 | 2.125 | 2.302 |

8 | Jeju Aewol | 0.702 | 0.029 | 0.022 | 0.162 | −0.160 | −0.672 | −0.051 | 1.755 | 1.936 |

9 | Gunsan | 0.697 | 0.025 | 0.029 | 0.176 | −0.147 | −0.675 | −0.068 | 1.705 | 1.799 |

Notation | Mean of Bias | COV of Bias | Distribution | Random Variable |
---|---|---|---|---|

tanϕ | 1.00 | 0.10 | Normal | Internal friction angle |

c | 1.00 | 0.10 | Normal | Cohesion force |

γ | 1.00 | 0.10 | Normal | Saturated soil density of Armor |

0.03 | Saturated soil density of Sand | |||

0.02 | Saturated soil density of Clay | |||

0.02 | Saturated soil density of Rock |

No. | Port | P_{f} (×10^{−3}) | β | μ_{FS} | σ_{FS} | COV_{FS} | Skewness | Error of P_{f} (%) |
---|---|---|---|---|---|---|---|---|

1 | Donghae | 6.53 | 2.805 | 1.506 | 0.299 | 0.187 | 0.688 | 0.06 |

2 | Pohang | 10.19 | 2.658 | 1.495 | 0.321 | 0.201 | 0.788 | 0.05 |

3 | Ulsan | 0.81 | 3.419 | 1.897 | 0.484 | 0.236 | 0.885 | 0.18 |

4 | Onsan | 2.51 | 3.101 | 1.553 | 0.293 | 0.178 | 0.619 | 0.10 |

5 | Busan | 1.09 | 3.339 | 1.704 | 0.341 | 0.188 | 0.662 | 0.16 |

6 | Gamcheon | 1.70 | 3.214 | 1.727 | 0.425 | 0.228 | 1.001 | 0.13 |

7 | Jeju Outer | 1.28 | 3.294 | 1.622 | 0.304 | 0.177 | 0.628 | 0.14 |

8 | Jeju Aewol | 0.54 | 3.527 | 1.894 | 0.519 | 0.252 | 1.046 | 0.22 |

9 | Gunsan | 0.20 | 3.769 | 1.928 | 0.472 | 0.227 | 0.920 | 0.37 |

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## Share and Cite

**MDPI and ACS Style**

Doan, N.S.; Huh, J.; Mac, V.H.; Kim, D.; Kwak, K.
Probabilistic Risk Evaluation for Overall Stability of Composite Caisson Breakwaters in Korea. *J. Mar. Sci. Eng.* **2020**, *8*, 148.
https://doi.org/10.3390/jmse8030148

**AMA Style**

Doan NS, Huh J, Mac VH, Kim D, Kwak K.
Probabilistic Risk Evaluation for Overall Stability of Composite Caisson Breakwaters in Korea. *Journal of Marine Science and Engineering*. 2020; 8(3):148.
https://doi.org/10.3390/jmse8030148

**Chicago/Turabian Style**

Doan, Nhu Son, Jungwon Huh, Van Ha Mac, Dongwook Kim, and Kiseok Kwak.
2020. "Probabilistic Risk Evaluation for Overall Stability of Composite Caisson Breakwaters in Korea" *Journal of Marine Science and Engineering* 8, no. 3: 148.
https://doi.org/10.3390/jmse8030148