# Reliability Analysis of Serviceability Limit State for Braced Excavation Considering Multiple Failure Modes in Spatially Variable Soil

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Random Finite Element Analysis Method

_{i}, y

_{i}) is the central coordinate ith of the random field element. δ

_{h}and δ

_{v}are the horizontal and vertical scales of fluctuation, respectively.

_{1}and L

_{2}can be obtained (Equation (2)).

_{i}and σ

_{i}are the mean value and standard deviation of ith random variables, ${\mu}_{{i}_{\mathrm{ln}}}$ and ${\sigma}_{{i}_{\mathrm{ln}}}$ are the corresponding value in lognormal space. ${E}_{(n\times 1)}^{i}$ and ${G}_{(n\times 1)}^{i}$ are the part of ${E}_{(n\times m)}$ and ${G}_{(n\times m)}$, respectively. $\left({E}_{(n\times m)}=\left({E}_{(n\times 1)}^{1},{E}_{(n\times 1)}^{2},\cdots {E}_{(n\times 1)}^{m}\right),{G}_{(n\times m)}=({G}_{(n\times 1)}^{1},{G}_{(n\times 1)}^{2},\cdots ,{G}_{(n\times 1)}^{m})\right)$.

#### 2.2. Deterministic Finite Element Model

#### 2.3. Modeling of Braced Excavation in Spatial Variability Soil

_{u}and friction angle φ) of clay soil were considered, as two negative correlated random fields, the relationship factor between c

_{u}and φ was equal to −0.5 [9]. For clay soil, there is a high linear correlation between elastic modulus and cohesion of soil [4], which can be approximately represented by empirical formulas. In addition, the high correlation between two random fields would disturb the Cholesky decomposition, it is hard to calculate Equation (2). Therefore, it is necessary to separate random variables with high correlation from the correlation matrix. Consequently, in this study, elastic modulus (E) is set as E = 300c

_{u}[4,40]. In other words, E is considered a random field that is related to c

_{u}. Other parameters are treated as constant.

- (1)
- Establish a sample model of the random finite element method. Different from the deterministic model shown in Figure 1, the clay layer was divided into 560 regions, corresponding to random field elements. Meanwhile, the statistical characteristics of random fields are determined, such as the mean value (μ), Coefficient of variation (COV), correlation matrix R, and fluctuation of scale in horizontal and vertical directions (δ
_{x}and δ_{y}). - (2)
- Based on the statistical characters in step 1, generate two negative correlated random fields of c
_{u}and φ via MATLAB codes, from Equations (1)–(4). The random fields of c_{u}or φ can be expressed as strength cloud charts (Figure 4). - (3)
- Via batch file and command codes in PLAXIS 2D, automatically import the information of random fields into finite element mesh. Similarly, calculate the random finite element models automatically by batch files and PLAXIS codes.
- (4)
- Export response data of excavations into EXCEL by Python codes [37], such as wall deflection, ground settlement, basal heave, and bending moment of the diaphragm wall.
- (5)
- Using the response data in step 5, calculate single mode failure probability and system failure probability.

_{u}and φ obey the three-deviation criterion. With the increase of fluctuation of scale, the deviation of c

_{u}or φ will decrease, and the random finite element models gradually transform into random variable models.

_{u}= 10%, 20%, 30%), three levels of friction angle (COVφ = 10%, 15%, 20%), six levels of vertical distance δ

_{y}= 1 m, 2.5 m, 5 m, 10 m, 25 m, 50 m are considered. Cases 1~9 aimed to evaluate the effect of spatial variability of soil properties on responses of excavation. Cases 3, 10~14 aimed to evaluate the effect of the scale of fluctuation on responses.

## 3. Effect of Spatial Variability on Responses of Braced Excavation

#### 3.1. The Influence of COVc_{u} and COVφ

_{u}, the COV of responses of excavation increased. For example, in case 3 (COVc

_{u}= 30%, COVφ = 20%), the range of basal heave was 46–114 mm, while in case 9 (COVc

_{u}= 10%, COVφ = 20%), the range of basal heave was 53–96 mm. It is noticeable that the COV of basal heave was 15.657% in case 1, while the COV of ground settlement and wall deflection was 6.644% and 4.778%, respectively. Basal heave was more easily affected by spatial variability of soil properties.

_{u}on the responses of excavation is shown in a boxplot chart. In Figure 7, the boxplot is divided into three parts, which represent COVc

_{u}= 10%, 20%, and 30%, respectively. Each part contains three cases, representing COVφ = 10%, 15%, 20%. In each boxplot, the black is the mean value of each case, which is close to the median values. It is proved that the excavation responses approximately follow a log-normal distribution. Compared with soil cohesion (c

_{u}) and elastic modulus (E), friction angle (φ) makes a lower influence on responses of excavation. There are few changes for median value with different COVφ, only the standard deviation slightly increases with the growth of COVφ. The mean value of wall deflection and basal heave slightly increases with the growth of COVc

_{u}. However, COVc

_{u}has no effect on the mean value of the ground settlement.

#### 3.2. The Influence of Vertical Fluctuation of Scale

_{v}) on excavation responses. Similarly, the curves of the fitting probability density function of each case are shown in Figure 9. The distribution is wider with an increase of δ

_{y}when δ

_{y}ranges from 1 m~25 m. When δ

_{y}is larger than 25 m, the distributions of excavation responses basically remain convergence. Compared with COVc

_{u}and COVφ, δ

_{y}has a larger impact on excavation responses. Based on the statistical data, δ

_{y}of soil usually ranges from 1 m~6 m, and the excavation depth is usually at the level of 5 m~50 m. Therefore, it is necessary to determine the value of δ

_{y}via field observation data and geological statistic method.

#### 3.3. Correlation Analysis of Excavation Responses

_{d}and δ

_{s}(0.77 for Spearman matrix). In addition, there is also a middle level of correlation between δ

_{d}and δ

_{b}, δ

_{s}and δ

_{b}, M and V (0.53, 0.47, 0.51 for Spearman matrix). The axial force of struts has a low level of correlation with other responses. However, the correlation between responses of excavation plays an important role in system reliability analysis, which is introduced in Section 4 in detail.

## 4. System Reliability Model Based on Serviceability Limit State

_{i}) is the probability density function of log-normal distribution, $\begin{array}{cc}f\left({x}_{i}\right)=\frac{1}{\sqrt{2\pi}{\sigma}_{i}{x}_{i}}\cdot \mathrm{exp}\left[\frac{-{\left(\mathrm{ln}{x}_{i}-{\mu}_{i}\right)}^{2}}{2{\sigma}_{i}^{2}}\right]& {x}_{i}\in (-\infty ,+\infty )\end{array}$.

^{−2}, and 0 (the value is lower than 10

^{−16}in MATLAB, which is considered 0). With different values of ${\delta}_{s\mathrm{max}}$, there is a great difference between failure probabilities, which is shown in Figure 11 and Figure 12.

_{u}, the gradient of the curve gradually decreases. For COVc

_{u}= 10%, the failure probability dramatically changes at 70~76 mm. However, for COVc

_{u}= 10%, the failure probability dramatically changes at 65~85 mm, which contains more uncertainty. In other words, for the same level of deformation thresholds, the larger COVc

_{u}is, the larger the failure probability is. Similar conclusions can be reached on basal heave failure mode from Figure 11b. Figure 12 shows the effect of δ

_{y}on failure probability. The notable change of failure probability based on wall deflection limitation occurs when δ

_{b}ranges from 55 mm to 105 mm, which is larger than COVc

_{u}(Figure 11a). At the same level of wall deflection limitation, the failure probability grows with δ

_{b}. The reason is that the fit distribution function becomes wider with the increase of δ

_{b}, which is mentioned in Section 3.2.

_{d}, δ

_{s}, δ

_{d}) is the three-dimensional log-normal distribution model.

**x**is the vector of random variables,

**μ**is the vector of the mean value of random variables,

**C**is the covariance matrix of random variables.

**C**. In this study, the Pearson matrix was adopted to calculate system failure probability. There are two main reasons:

- (1)
- Compared with the Spearman correlation matrix, the Pearson matrix only describes the linear correlation of random variables. However, in the system reliability analysis of the underground pipe gallery, Fu et al. adopted the Pearson correlation factor to characterize the correlation between multiple failure modes [44,45].
- (2)
- On the other hand, there is no doubt that the Pearson correlation factor is available for normal distribution. Other distributions can be transformed into standard normal space via Nataf transformations. One important assumption of Nataf transformation is that the correlation between random variables will not change during Nataf transformation. Moreover, Nataf transformation is only available for the Pearson correlation factor. Therefore, the Pearson correlation factor is still the main method to characterize the correlation of random variables in system reliability analysis.

_{u}and δ

_{y}. Based on the statistical characters of responses to excavation, similar conclusions are reached in Section 3. For the same level of COVc

_{u}, in design level III, system failure probabilities have a tendency to grow with δ

_{y}when δ

_{y}ranges from 1~25 m. For design level I, there is a negative correlation between ${p}_{{f}_{1}},{p}_{{f}_{3}}$ and δ

_{y}, while the correlation between ${p}_{{f}_{2}}$ and δ

_{y}is positive. For design level II, there is a positive correlation between ${p}_{{f}_{1}},{p}_{f2}$ and δ

_{y}, while ${p}_{{f}_{3}}$ become larger then smaller with the increase of δ

_{y}. The reason is that δ

_{y}has an impact on the mean value of excavation responses. Therefore, sometimes, the correlation between δ

_{y}and ${p}_{f}$ is not monotonic.

^{−3}. In general, the number of MCS should be larger than $10/{p}_{f}$ [41,46]. Therefore, the 2000-times MCS cannot verify the accuracy of ${p}_{{f}_{1}}$ and ${p}_{{f}_{2}}$ in design level III. However, the results prove that the probability density function is an efficient method to calculate ${p}_{f}$.

## 5. Discussion

- (1)
- For the Cholesky decomposition method, it was quicker and more accurate to generate random field samples compared to K-L expansion and local averaging. However, it is hard for Cholesky to generate random field samples with high correlation variables. For high correlation variables, coupling the variables with high correlation is an effective method.
- (2)
- It is noticeable that the function integral3 was adopted to calculate the triple integral. If the number of random variables is more than three, the function integral3 will not be available. However, the function mvncdf is available for more than three random variables.
- (3)
- With considering spatial variability of soil parameters, different failure modes will occur during deep excavation, such as wall deformation dominating and basal heave dominating. Different from the deterministic model, the deformation will be subject to elements with lower stiffness in the random finite element model [8]. The dominate deformation depends on the location of the lower stiffness elements. However, for geotechnical engineering, the real distribution of element stiffness is unknown. It is necessary to quantify the effect of uncertainty of soil parameters via reliability analysis.
- (4)
- From Table 4 and Figure 7, friction angle had little effect on failure probability and distribution of deformation responses. Similar conclusions were drawn from probabilistic analysis of vault settlement of tunnel [47]. Therefore, if the deformation index was chosen to evaluate failure probability, the spatial variability of elastic modulus should be considered first.
- (5)
- For system reliability analysis, a key point is how to characterize the correlation between each failure mode. From Section 3.3 and Section 4, it is concluded that there is a certain correlation between failure modes. Due to the limitations of the Pearson correlation matrix and Nataf transformation, the multiple-dimensional log-normal distribution model is used to calculate system reliability, which is a convenient method.
- (6)
- In this study, it is verified that the spatial variability of soil has a great influence on excavation deformations. For practical engineering, it is necessary to decrease the negative influence. However, the information from the geological survey is not taken to good use. Conditional random field model is an efficient way to integrate the geological data into a random finite element model. How to develop a conditional random finite method and establish a comprehensive system reliability model will be significant for the next studies.

## 6. Conclusions

- (1)
- The spatial variability of soil parameters has a negative effect on the safety of deep excavation. Basal heave is more subjected to soil spatial variability than wall deflection and ground settlement. The distribution of deformations induced by excavation becomes wider with the growth of the scale of fluctuation.
- (2)
- Different from ultimate limit state analysis, the deformation responses induced by excavations are more sensitive to elastic modulus and soil cohesion than friction angle. In addition, the high uncertainty of soil properties would entail different failure modes such as wall deflection dominating and basal heave dominating.
- (3)
- The responses of excavation, such as the deflection of the wall, ground settlement, and basal heave, basically follow log-normal distribution via KS-test and SW-test. The fitted probabilistic density functions can be used to carry out reliability analysis, which is an efficient method. Via Latin hypercube sampling technique and K-S test, fewer samples are needed to estimate failure probabilities than in Mote Carlo simulations.
- (4)
- The multiple lognormal probabilistic density function is a convenient method to describe the correlation between failure modes and calculate the system reliability of deep excavation. System failure probability is usually lower than single failure probabilities. The system failure probability is sensitive to the design level of excavation. It is necessary to determine the design level based on the geometry size of excavation, geological conditions, and surrounding building environment.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Case Number | Wall Deflection (mm) | Ground Settlement (mm) | Basal Heave (mm) | |||
---|---|---|---|---|---|---|

μ | COV | μ | COV | μ | COV | |

1 | 74.91 | 4.778% | 39.38 | 6.644% | 77.95 | 15.657% |

2 | 74.94 | 4.858% | 39.30 | 5.559% | 77.97 | 15.308% |

3 | 74.97 | 5.068% | 39.32 | 5.618% | 78.18 | 15.258% |

4 | 74.24 | 3.249% | 38.78 | 3.735% | 74.02 | 10.157% |

5 | 74.28 | 3.446% | 38.79 | 3.757% | 74.15 | 10.132% |

6 | 74.36 | 3.702% | 38.83 | 3.880% | 74.39 | 10.119% |

7 | 73.86 | 1.838% | 38.49 | 1.933% | 71.63 | 5.026% |

8 | 73.92 | 2.139% | 38.50 | 2.148% | 71.78 | 5.031% |

9 | 73.99 | 2.481% | 38.54 | 2.477% | 71.98 | 5.048% |

10 | 74.79 | 3.434% | 39.12 | 3.767% | 77.51 | 11.188% |

11 | 75.21 | 6.789% | 39.54 | 7.437% | 78.77 | 18.279% |

12 | 75.45 | 8.697% | 39.76 | 9.355% | 79.05 | 20.696% |

13 | 75.76 | 11.001% | 39.97 | 11.564% | 79.38 | 22.888% |

14 | 75.91 | 12.127% | 40.08 | 12.734% | 79.57 | 23.388% |

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**Figure 2.**Deformation curve of deterministic model: (

**a**) ground settlement curve; (

**b**) wall deflection curve.

**Figure 6.**Statistical distribution of excavation responses: (

**a**) histogram of wall deflection in case 3; (

**b**) Quantile-Quantile plot of wall deflection; (

**c**) histogram of wall deflection with different COVc

_{u}; (

**d**) histogram of ground settlement with different COVc

_{u}; (

**e**) histogram of basal heave with different COVc

_{u}.

**Figure 7.**The effect of COVc

_{u}and COVφ on excavation responses: (

**a**) Boxplot of wall deflection; (

**b**) Boxplot of ground settlement; (

**c**) Boxplot of basal heave.

**Figure 8.**Effect of spatial variability on failure mode: (

**a**) basal heave dominate; (

**b**) wall deflection dominate.

**Figure 9.**Effect of δ

_{y}on distribution of excavation responses: (

**a**) histogram of wall deflection with different δ

_{y}; (

**b**) histogram of ground settlement with different δ

_{y}; (

**c**) histogram of basal heave with different δ

_{y}.

**Figure 10.**Spearman correlation matrixes of responses (Case 3). Notes: δ

_{d}, δ

_{s}, δ

_{b}are the vector of wall deflection, ground settlement and basal heave, respectively. M, V are the vector of bending moment and shear force of diaphragm wall, respectively. F

_{Ni}is the vector of axial force of strut in ith layer.

**Figure 11.**Spatial variability effects on failure probability: (

**a**) different thresholds of wall deflection; (

**b**) different thresholds of basal heave.

**Figure 12.**Effects of ${\delta}_{y}$ on failure probability at different thresholds of: (

**a**) wall deflection; (

**b**) basal heave.

Stage | Construction | Parameter of Retaining Structural System | |
---|---|---|---|

EA (kN) | EI (kN·m^{2}) | ||

1 | Initial confined consolidation | - | - |

2 | Activate diaphragm wall | 3 × 10^{6} | 2.5 × 10^{6} |

3 | Lower the ground water to GL −9 m | - | - |

4 | Excavate to −4 m and activate the first strut at GL −1 m | 2 × 10^{6} | - |

5 | Excavate to −8 m and activate the second strut at GL −5 m | 2 × 10^{6} | - |

6 | Lower the ground water to GL −17 m | - | - |

7 | Excavation to GL −12 m and activate the third strut at GL −9 m | 2 × 10^{6} | - |

8 | Excavate to GL −16 m and activate the fourth strut at GL −13 m | 2 × 10^{6} | - |

Number | Soil Layer | T | γ | ν | E | c_{u} | φ | Ψ | ${{\mathit{E}}_{50}^{\mathit{r}\mathit{e}\mathit{f}}}^{}$ | ${{\mathit{E}}_{\mathit{o}\mathit{e}\mathit{d}}^{\mathit{r}\mathit{e}\mathit{f}}}^{}$ | ${{\mathit{E}}_{\mathit{u}\mathit{r}}^{\mathit{r}\mathit{e}\mathit{f}}}^{}$ | m |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(m) | (kN/m^{3}) | - | (MPa) | (kPa) | $(\xb0$) | $(\xb0)$ | (MPa) | (MPa) | (MPa) | - | ||

1 | Clay | 20 | 17 | 0.3 | 9 | 30 | 20 | - | - | - | - | - |

2 | Sand | 40 | 19 | - | 0 | 32 | 2 | 40 | 40 | 120 | 0.5 |

_{u}, φ is elastic modulus, soil cohesion and friction angle for MC model.

Case Number | COVc_{u} | ${\mathit{\sigma}}_{{\mathit{c}}_{\mathit{u}}}/\mathbf{kPa}$ | COVφ | ${\mathit{\sigma}}_{\mathit{\phi}}/\xb0$ | δ_{y} (m) | δ_{x} (m) |
---|---|---|---|---|---|---|

1 | 30% | 9 | 10% | 2 | 2.5 | 25 |

2 | 30% | 9 | 15% | 3 | 2.5 | 25 |

3 | 30% | 9 | 20% | 4 | 2.5 | 25 |

4 | 20% | 6 | 10% | 2 | 2.5 | 25 |

5 | 20% | 6 | 15% | 3 | 2.5 | 25 |

6 | 20% | 6 | 20% | 4 | 2.5 | 25 |

7 | 10% | 3 | 10% | 2 | 2.5 | 25 |

8 | 10% | 3 | 15% | 3 | 2.5 | 25 |

9 | 10% | 3 | 20% | 4 | 2.5 | 25 |

10 | 30% | 9 | 20% | 4 | 1 | 25 |

11 | 30% | 9 | 20% | 4 | 5 | 25 |

12 | 30% | 9 | 20% | 4 | 10 | 25 |

13 | 30% | 9 | 20% | 4 | 25 | 25 |

14 | 30% | 9 | 20% | 4 | 50 | 25 |

Case Number | Design Level | ${\mathit{p}}_{{\mathit{f}}_{1}}$ | ${\mathit{p}}_{{\mathit{f}}_{2}}$ | ${\mathit{p}}_{{\mathit{f}}_{3}}$ | ${\mathit{p}}_{\mathit{f}}^{\mathit{s}\mathit{y}\mathit{s}}$ |
---|---|---|---|---|---|

1 | I | 9.983 × 10^{−1} | 3.916 × 10^{−1} | 9.467 × 10^{−1} | 9.991 × 10^{−1} |

II | 8.060 × 10^{−2} | 1.787 × 10^{−2} | 5.682 × 10^{−1} | 5.757 × 10^{−1} | |

III | 2.874 × 10^{−7} | 9.487 × 10^{−5} | 1.571 × 10^{−1} | 1.571 × 10^{−1} | |

2 | I | 9.981 × 10^{−1} | 3.649 × 10^{−1} | 9.490 × 10^{−1} | 9.989 × 10^{−1} |

II | 8.533 × 10^{−2} | 6.780 × 10^{−3} | 5.705 × 10^{−1} | 5.771 × 10^{−1} | |

III | 4.475 × 10^{−7} | 6.327 × 10^{−6} | 1.555 × 10^{−1} | 1.555 × 10^{−1} | |

3 | I | 9.974 × 10^{−1} | 3.699 × 10^{−1} | 9.513 × 10^{−1} | 9.986 × 10^{−1} |

II | 9.535 × 10^{−2} | 7.510 × 10^{−3} | 5.778 × 10^{−1} | 5.857 × 10^{−1} | |

III | 1.26 × 10^{−7} | 8.15 × 10^{−6} | 1.592 × 10^{−1} | 1.592 × 10^{−1} | |

4 | I | 1.000 | 1.988 × 10^{−1} | 9.779 × 10^{−1} | 1.000 |

II | 1.022 × 10^{−2} | 3.134 × 10^{−5} | 4.288 × 10^{−1} | 4.294 × 10^{−1} | |

III | 1.343 × 10^{−14} | 4.362 × 10^{−12} | 2.441 × 10^{−2} | 2.441 × 10^{−2} | |

5 | I | 1.000 | 2.013 × 10^{−1} | 9.791 × 10^{−1} | 1.000 |

II | 1.495 × 10^{−2} | 3.537 × 10^{−5} | 4.355 × 10^{−1} | 4.366 × 10^{−1} | |

III | 3.890 × 10^{−13} | 6.022 × 10^{−12} | 2.511 × 10^{−2} | 2.511 × 10^{−2} | |

6 | I | 9.999 × 10^{−1} | 2.170 × 10^{−1} | 9.809 × 10^{−1} | 9.999 × 10^{−1} |

II | 2.300 × 10^{−2} | 6.758 × 10^{−5} | 4.481 × 10^{−1} | 4.501 × 10^{−1} | |

III | 1.538 × 10^{−11} | 3.260 × 10^{−11} | 2.676 × 10^{−2} | 2.676 × 10^{−2} | |

7 | I | 1.000 | 2.248 × 10^{−2} | 9.998 × 10^{−1} | 1.000 |

II | 6.651 × 10^{−6} | 3.331 × 10^{−16} | 1.744 × 10^{−1} | 1.744 × 10^{−1} | |

III | 0.000 | 0.000 | 2.576 × 10^{−6} | 2.576 × 10^{−1} | |

8 | I | 1.000 | 3.731 × 10^{−2} | 9.998 × 10^{−1} | 1.000 |

II | 1.037 × 10^{−4} | 1.922 × 10^{−13} | 1.847 × 10^{−1} | 1.847 × 10^{−1} | |

III | 0.000 | 0.000 | 3.105 × 10^{−1} | 3.105 × 10^{−1} | |

9 | I | 1.000 | 6.539 × 10^{−2} | 9.998 × 10^{−1} | 1.000 |

II | 7.863 × 10^{−4} | 1.955 × 10^{−10} | 2.010 × 10^{−1} | 2.011 × 10^{−1} | |

III | 0.000 | 0.000 | 4.393 × 10^{−1} | 4.393 × 10^{−1} | |

10 | I | 9.999 × 10^{−1} | 2.696 × 10^{−1} | 9.667 × 10^{−1} | 1.000 |

II | 2.001 × 10^{−2} | 1.245 × 10^{−3} | 4.240 × 10^{−1} | 4.274 × 10^{−1} | |

III | 4.775 × 10^{−12} | 1.104 × 10^{−7} | 3.237 × 10^{−2} | 3.237 × 10^{−2} | |

11 | I | 9.832 × 10^{−1} | 4.237 × 10^{−1} | 9.188 × 10^{−1} | 9.908 × 10^{−1} |

II | 1.716 × 10^{−1} | 3.765 × 10^{−2} | 5.704 × 10^{−1} | 5.885 × 10^{−1} | |

III | 2.408 × 10^{−4} | 6.932 × 10^{−4} | 2.064 × 10^{−1} | 2.065 × 10^{−1} | |

12 | I | 9.540 × 10^{−1} | 4.555 × 10^{−1} | 8.930 × 10^{−1} | 9.740 × 10^{−1} |

II | 2.354 × 10^{−1} | 8.504 × 10^{−2} | 5.613 × 10^{−1} | 5.893 × 10^{−1} | |

III | 3.342 × 10^{−3} | 6.223 × 10^{−3} | 2.312 × 10^{−1} | 2.324 × 10^{−1} | |

13 | I | 9.113 × 10^{−1} | 4.746 × 10^{−1} | 8.747 × 10^{−1} | 9.468 × 10^{−1} |

II | 2.897 × 10^{−1} | 1.402 × 10^{−1} | 5.570 × 10^{−1} | 5.882 × 10^{−1} | |

III | 1.662 × 10^{−2} | 2.339 × 10^{−2} | 2.489 × 10^{−1} | 2.534 × 10^{−1} | |

14 | I | 8.905 × 10^{−1} | 4.807 × 10^{−1} | 8.701 × 10^{−1} | 9.308 × 10^{−1} |

II | 3.097 × 10^{−1} | 1.659 × 10^{−1} | 5.578 × 10^{−1} | 5.850 × 10^{−1} | |

III | 2.713 × 10^{−2} | 3.632 × 10^{−2} | 2.557 × 10^{−1} | 2.620 × 10^{−1} | |

MCS ^{a} | I | 9.965 × 10^{−1} | 3.931 × 10^{−1} | 9.499 × 10^{−1} | 9.970 × 10^{−1} |

II | 7.823 × 10^{−2} | 1.752 × 10^{−2} | 5.667 × 10^{−1} | 5.747 × 10^{−1} | |

III | 0.000 | 1.000 × 10^{−3} | 1.580 × 10^{−1} | 1.590 × 10^{−1} | |

Error (%) ^{b} | I | 0.19 | 0.3 | 0.33 | 0.21 |

II | 3.03 | 3.05 | 0.27 | 0.17 | |

III | - | 90 | 0.53 | 1.16 |

^{a}The aim of this case is to verify the accuracy of failure probabilities obtained by probabilistic density functions (PDF).

^{b}The error of p

_{f}between probabilistic density functions and Monte Carlo simulations (Take case 1 as an example).

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## Share and Cite

**MDPI and ACS Style**

Hong, L.; Chen, L.; Wang, X. Reliability Analysis of Serviceability Limit State for Braced Excavation Considering Multiple Failure Modes in Spatially Variable Soil. *Buildings* **2022**, *12*, 722.
https://doi.org/10.3390/buildings12060722

**AMA Style**

Hong L, Chen L, Wang X. Reliability Analysis of Serviceability Limit State for Braced Excavation Considering Multiple Failure Modes in Spatially Variable Soil. *Buildings*. 2022; 12(6):722.
https://doi.org/10.3390/buildings12060722

**Chicago/Turabian Style**

Hong, Li, Longlong Chen, and Xiangyu Wang. 2022. "Reliability Analysis of Serviceability Limit State for Braced Excavation Considering Multiple Failure Modes in Spatially Variable Soil" *Buildings* 12, no. 6: 722.
https://doi.org/10.3390/buildings12060722