# Error Evaluation and Suitability Assessment of Common Reliability Methods in the Case of Shallow Foundations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- test the hypothesis of whether the factor of safety (FS) follows normal or lognormal distribution;
- probability method error definition;
- defining the criteria for method suitability assessment.

#### Reliability Theory

## 2. Materials and Methods

- ${\mathsf{\varphi}}^{\prime}$ and ${\mathsf{\gamma}}^{\prime}$ are positive correlated, ${\mathrm{V}}_{\mathrm{g}}$ is uncorrelated with ${\mathsf{\varphi}}^{\prime}$ and $\mathsf{\gamma}$
- foundation soil is non-cohesive, homogeneous and isotropic
- no presence of groundwater in the soil

#### 2.1. Performance Function

_{1}, X

_{2}and X

_{3}are transformed from the space of real random variables (X space) to the standard normal space (U space) using the Rosenblatt transformation [29]. By applying the transformation, the performance function in U space yields:

#### 2.2. Reliabilty Integral

#### 2.3. Defining the Error of the Reliability Method

#### 2.4. The Criterion for the Estimation of the Suitability of Reliability Methods

#### 2.5. Short Overview of Reliability Methods

#### 2.5.1. Analytical First Order Second Moment (FOSM) and Taylor Series

^{th}parameter increased/decreased by one standard deviation from its mean value [6].

#### 2.5.2. Point Estimate Method (PEM)

^{N}, where N is the number of random variables. In these calculations, the value of the factor of safety is calculated with different combinations of random variables values, which are $\mathrm{E}\left(\mathrm{X}\right)+{\mathsf{\sigma}}_{\mathrm{x}}$ and $\mathrm{E}\left(\mathrm{X}\right)-{\mathsf{\sigma}}_{\mathrm{x}}$ (E(X) is the expected value, and ${\mathsf{\sigma}}_{\mathrm{x}}$ is the standard deviation of random variable X). The result is the reliability index $\mathsf{\beta}$, which is calculated based on the assumption of statistical distribution of the FS.

#### 2.5.3. First Order Reliability Method (FORM)

#### 2.5.4. Simplified FORM

#### 2.5.5. The Monte Carlo Method

_{f}values in the researched examples are in the order of magnitude ${10}^{-3}$; therefore, a relatively large number of simulations is required to achieve satisfactory result accuracy. The coefficient of variation is calculated according to the procedure defined in [34]. Random values were generated using the Mersenne Twister 19,937 algorithm from the open (MIT/x11 License) library of MathNet [35].

#### 2.5.6. Direct Integration Method

## 3. Results

#### 3.1. Statistical Distribution of Factor of Safety (FS)

^{5}data, along with the best-fit curves of normal and lognormal PDFs. All curves are skewed to the right with a single distinct peak. This type of histogram shape indicates that the data are unlikely to follow a normal distribution.

#### 3.2. The Results of Reliability Methods

#### 3.2.1. Monte Carlo Method

#### 3.2.2. FORM and Simplified FORM

#### 3.2.3. FOSM and Point Estimate Method (PEM)

#### 3.3. Reliability Methods Suitability Assessment

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Visualization of a simple reliability problem consisting of 2 random variables, X1 and X2, integrand ${\mathrm{f}}_{\mathrm{X}}\left(\mathrm{x}\right)$ and performance function $\mathrm{g}\left(\mathrm{X}1,\mathrm{X}2\right)$.

**Figure 4.**(

**a**) Geometrical representation of the reliability index in First Order Reliability Method (FORM); (

**b**) The iterative procedure for reliability index calculation using FORM [32].

**Figure 5.**The factor of safety histograms for different ${\mathsf{\varphi}}^{\prime}$, along with theoretical best-fit curves of normal and lognormal distribution.

**Figure 8.**Comparison of pf and $\mathsf{\beta}$ calculated with the assumption that FS is normal and lognormal along with the results of Direct Integration of reliability integral.

**Figure 10.**Relationship between the $\mathrm{C}\mathrm{O}{\mathrm{V}}_{{\mathrm{p}}_{\mathrm{f}}}$ and number of Monte Carlo (MC) runs.

**Table 1.**Results of the reliability analyses of embankments stability for Cases 1 and 2 [1].

Method | Case 1 | Case 2 | ||
---|---|---|---|---|

p_{f} [1] | $\mathsf{\beta}$ | p_{f} [1] | $\mathsf{\beta}$ | |

Direct Integration | 0.032 | 1.85 | $9\times {10}^{-5}$ | 3.75 |

Taylor Series, lognormal dist. of FS | 0.0036 | 2.69 | $3\times {10}^{-6}$ | 4.53 |

Taylor Series, normal dist. of FS | 0.01 | 2.33 | 0.003 | 2.75 |

PEM, normal dist. of FS | 0.017 | 2.12 | 0.0053 | 2.56 |

Simplified FORM | 0.024 | 1.98 | ${10}^{-4}$ | 3.72 |

**Table 2.**Results of reliability analyses of retaining wall [2].

Method | Sliding on Sand | Sliding in Clay | Bearing Capacity (Undrained) | |||
---|---|---|---|---|---|---|

p_{f} | $\mathsf{\beta}$ | p_{f} | $\mathsf{\beta}$ | p_{f} | $\mathsf{\beta}$ | |

Monte Carlo, 20,000 simulations | 0.024 | 1.98 | 0.022 | 2.01 | 0.018 | 2.10 |

Simplified FORM | 0.024 | 1.98 | 0.025 | 1.96 | 0.023 | 2.00 |

Taylor Series, normal dist. of FS | 0.055 | 1.60 | 0.028 | 1.91 | 0.015 | 2.17 |

PEM, normal dist. of FS | 0.045 | 1.70 | 0.05 | 1.64 | 0.043 | 1.72 |

PEM, lognormal dist. of FS | 0.026 | 1.94 | 0.016 | 2.14 | 0.011 | 2.29 |

Parameter | Designation | Distribution | Coefficient of Variation (COV) |
---|---|---|---|

${\mathsf{\varphi}}^{\prime}$ | ${\mathrm{X}}_{1}$ | Normal | 0.1 |

${\mathrm{V}}_{\mathrm{g}}$ | ${\mathrm{X}}_{2}$ | Normal | 0.1 |

${\mathsf{\gamma}}^{\prime}$ | ${\mathrm{X}}_{3}$ | Normal | 0.05 |

${\mathsf{\varphi}}_{\mathbf{m}}$${[}^{\xb0}]$ | ${\mathbf{V}}_{\mathbf{g},\mathbf{m}}$$\left[\mathbf{k}\mathbf{N}\right]$ | ${\mathsf{\gamma}}^{\prime}{}_{\mathbf{m}}$$[\mathbf{k}\mathbf{N}/{\mathbf{m}}^{3}]$ |
---|---|---|

28 | 184.1 | 17.8 |

32 | 276.3 | 18.5 |

36 | 424.1 | 19.3 |

40 | 669.6 | 20.1 |

44 | 1096.5 | 20.9 |

**Table 5.**Expected performance level according reliability indexes [16].

Expected Performance Level | β | Probability of Unsatisfactory Performance |
---|---|---|

High | 5.0 | 3 × 10^{−7} |

Good | 4.0 | 3 × 10^{−5} |

Above average | 3.0 | 0.01 |

Below average | 2.5 | 0.06 |

Poor | 2.0 | 0.023 |

Unsatisfactory | 1.5 | 0.07 |

Hazardous | 1.0 | 0.16 |

$\mathsf{\varphi}{[}^{\xb0}]$ | Quantiles to which the Lognormal Q-Q Plots are Linear (X) | Cumulative Density Function at X |
---|---|---|

28 | 7.0 | 0.98 |

32 | 7.3 | 0.95 |

36 | 9.2 | 0.94 |

40 | 11.9 | 0.93 |

44 | 17.1 | 0.93 |

Method | Mean Error [Absolute Value] | Mean Error [%] |
---|---|---|

FOSM analytical-LN | 0.16 | 6.1 |

PEM -LN | 0.30 | 11.3 |

Taylor Series-LN | 0.43 | 16.1 |

PEM-N | 1.57 | 59.2 |

FOSM analytical-N | 1.24 | 46.7 |

Taylor Series-N | 1.30 | 48.9 |

**Table 8.**Reliability methods suitability assessment for the case of ULS of shallow foundation according to the proposed criteria.

Method | Error (Absolute Value) | Acceptable Error | Meet the Criteria |
---|---|---|---|

Monte Carlo | $3\times {10}^{-3}$ | 0.25 | Yes |

FORM | $8\times {10}^{-3}$ | Yes | |

Simplified FORM | 0.05 | Yes | |

FOSM analytical-LN | 0.16 | Yes | |

PEM-N | 0.29 | No | |

Taylor Series-LN | 0.42 | No | |

PEM-N | 1.08 | No | |

FOSM analytical-N | 1.24 | No | |

Taylor Series-N | 1.35 | No |

**Table 9.**Reliability methods suitability assessment in the case of stability of embankments [1] according to the proposed criteria.

Method | CASE 1 ^{1} | CASE 2 ^{2} | ||||
---|---|---|---|---|---|---|

Error (Absolute Value) | Acceptable Error | Meet the Criteria | Error (Absolute Value) | Acceptable Error | Meet the Criteria | |

Taylor Series-LN | 0.84 | 0.25 | No | 0.78 | 0.5 | No |

Taylor Series-N | 0.48 | No | 1.00 | No | ||

PEM-N | 0.27 | No | 1.19 | No | ||

Simplified FORM | 0.13 | Yes | 0.03 | Yes |

**Table 10.**Reliability methods suitability assessment in the case of retaining wall [2] according to the proposed criteria.

Method | Sliding on Sand | Sliding in Clay | ||||
---|---|---|---|---|---|---|

Error (Absolute Value) | Acceptable Error | Meet the Criteria | Error (Absolute Value) | Acceptable Error | Meet the Criteria | |

Simplified FORM | 0.00 | 0.25 | Yes | 0.02 | 0.25 | Yes |

Taylor Series-N | 0.38 | No | 0.07 | Yes | ||

PEM-N | 0.28 | No | 0.34 | No | ||

PEM-LN | 0.04 | Yes | 0.16 | Yes |

**Table 11.**Reliability methods suitability assessment in the case of retaining wall [2] according to the proposed criteria.

Method | Bearing Capacity (Undrained) | ||
---|---|---|---|

Error (Absolute Value) | Acceptable Error | Meet the Criteria | |

Simplified FORM | 0.02 | 0.25 | Yes |

Taylor Series-N | 0.19 | Yes | |

PEM-N | 0.26 | No | |

PEM-LN | 0.31 | No |

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**MDPI and ACS Style**

Dodigović, F.; Ivandić, K.; Kovačević, M.S.; Soldo, B. Error Evaluation and Suitability Assessment of Common Reliability Methods in the Case of Shallow Foundations. *Appl. Sci.* **2021**, *11*, 795.
https://doi.org/10.3390/app11020795

**AMA Style**

Dodigović F, Ivandić K, Kovačević MS, Soldo B. Error Evaluation and Suitability Assessment of Common Reliability Methods in the Case of Shallow Foundations. *Applied Sciences*. 2021; 11(2):795.
https://doi.org/10.3390/app11020795

**Chicago/Turabian Style**

Dodigović, Filip, Krešo Ivandić, Meho Saša Kovačević, and Božo Soldo. 2021. "Error Evaluation and Suitability Assessment of Common Reliability Methods in the Case of Shallow Foundations" *Applied Sciences* 11, no. 2: 795.
https://doi.org/10.3390/app11020795