### 1. Introduction

### 2. Method

### 2.1 Limit state function for the bearing capacity of breakwater foundation under wave loading

*w*' and

*w*are the effective weight and total weight of the comprising slice segment, respectively.

*q*is the vertical surcharge load distributed on the slice.

_{Ex}*F*and a correspond to the horizontal wave force and its lever arm about the failure center.

_{h}*R*is the radius of the considered failure surface.

*c*is the apparent cohesion of the soil (or undrained shear strength in the case of cohesive soil), and

*ϕ*is the friction angle of the soil along the base surface of each slice.

*s*is the width of each slice. Finally,

*is the inclination angle horizontal to the base surface.*

*θ**g*based on the safety factor is defined by Eq. (2). The limit state function divides the input variable space into the safety and failure zones. In this study, the safety zone is depicted by the zone that includes the non-negative values of

*g*and the failure zone consists of the negative values of g. The limit state function is defined implicitly because the

*FS*term is implicit, as presented in Eq. (1).

### 2.2 Reliability analysis

*g*is determined. The probability of failure

*PF*is then calculated as the ratio of the number of failure events N Fails and the total size of MCS

*N*as shown in Eq. (3). A failure event locates in the failure zone and is counted whenever the value of its limit state function

_{MCS}*g*is negative. The cycles of MCS is set as 100,000 in this study. The COV of the PF from MCS,

*COV*, can be determined using Eq. (4) (Haldar and Mahadevan, 2000).

_{Pf}*Φ*in Eq. (5) is the cumulative probability density of the standard normal distribution.

### 2.3 Calibration of load and resistance factors

*LF*and resistance factor

*RF*can respectively be determined using Eq. (6) (Salgado and Kim, 2014).

*R*and

*Q*are resistance and load components.

*n*stands for the nominal value, and the asterisk stands for the limit values. It should be noted that the nominal values and the limit values used in Eq. (6) are assessed from the successive calibration process for each safety level. The limit points are defined as those that lie close to the limit surface. In this study, the cases with the safety factors between 0.99 and 1.01 are counted as limit points. This definition of limit points was employed in our previous study (Doan et al., 2021). The load and resistance factors are determined as the average using all considered limit state points.

### 3. Results and Discussions

### 3.1 Results of reliability analysis for the existing structures

### 3.2 Results of the calibrated load and resistance factors

*RF*determined by the ratio between RF and LF are shown in Fig. 7 for the two target RIs of 2.5 and 3.0.

_{n}