# Probabilistic Assessment of Structural Integrity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Deterministic Finite Element Modeling of the Structure

_{iI}is a diagonal mass matrix; ü

_{iI}is the nodal acceleration (${\dot{u}}_{iI}$ and ${u}_{iI}$ are the velocity and displacement, respectively) of node I in the ith direction; ${f}_{iI}^{int}$ and ${f}_{iI}^{ext}$ are the internal and external nodal forces, respectively; i refers to the coordinate direction (x, y, z); I refers to the node number; and n is the time step number.

#### 1.2. Evaluation of the Aging and Degradation Uncertainty

## 2. Probabilistic Methods for Structural Reliability and Uncertainty Analysis

- What failure probability is expected?
- How long does a deterministic analysis take?
- How many random variables does the problem have?
- What computational resources are available?

#### 2.1. Monte Carlo Simulation Method

#### 2.2. First-Order Reliability Method

- Transforming the space of the basic random variables x
_{1}, x_{2}, …, x_{n}into a space of standard normal variables. - Exploring, within this transformed space, the point with the minimum distance from the origin on the limit-state surface (referred to as the design point).
- Approximating the failure surface in the vicinity of the design point.
- Calculating the failure probability associated with the approximated failure surface.

_{o}= Φ(−β),

#### 2.3. Response Surface/Monte Carlo Simulation Method

**x**) = Y(x

_{1}, …, x

_{n}) ≈ a

_{0}+ a

_{1}x

_{1}+…+a

_{n}x

_{n}+ a

_{n}

_{+1}x

_{1}

^{2}+ … + a

_{2n}x

_{n}

^{2}+ a

_{2n+1}x

_{1}x

_{2}+…

## 3. Integration of Deterministic and Probabilistic Methods

- Importing a Deterministic Model (Finite Element Model).

- 2.
- Defining Random Variables with Initial Set Screened Using Sensitivity Analysis.

- 3.
- Describing the Failure Criterion Based on Deterministic Criteria.

- 4.
- Running Deterministic Analysis to Obtain Response for Each Set of Random Variables.

- 5.
- Analyzing and Reviewing Results, Considering Probabilistic Estimates.

#### Methodology for Integrated Analysis of Failures

## 4. Example of the Applications of Integrated Methods for the Structural Integrity

#### 4.1. Introduction Regarding the Case Study

#### 4.2. Model for the Analysis of Damage to the Adjacent Piping

#### 4.3. Data for Probabilistic Analysis

- Mechanical properties.

- Geometry data.

#### 4.4. Selected Limit States for Analysis

- The structural integrity of adjacent GDHs after impact.
- The structural integrity of the GDH-supporting wall.

- Limit State 1: contact between the broken group distribution header and the adjacent pipe.
- Limit States 2, 3, 4, 5, and 6: The concrete adjacent to the group distribution header fixity in the support wall reaches the ultimate strength for compression and loses resistance to further loading. The same limit states at all five integration points through the wall thickness were checked.
- Limit State 7, 8, 9, 10: The strength limit of the first layer of rebars in the concrete support wall at the location of the group distribution header fixity is reached, and the rebars can fail. The same limit states at all four layers were checked in the analysis using the MCS method. In the case of the FORM, the computational effort is proportional to the number of random variables and limit states. The probabilities of the failure of all rebar layers were received as ridiculously small and similar to the MCS analysis. Therefore, Limit State 7 (the strength limit of the first layer of rebars) was used in the analysis using the FORM.
- Limit State 11: The impacted GDH pipe element reaches the ultimate strength of pipe steel, and the pipe will be destroyed.

- System Event 1—Comprising Limit State 2, Limit State 3, Limit State 4, Limit State 5, and Limit State 6. This system event is considered true if all the limit states are true, evaluating the probability of concrete failure at all integrated points.
- System Event 2—Encompassing Limit State 7, Limit State 8, Limit State 9, and Limit State 10. Similar to System Event 1, this event is true if all the corresponding limit states are true. It assesses the probability of rebar failure across all layers.

#### 4.5. Probabilistic Analysis Results

#### 4.5.1. Probabilistic Analysis Using MCS Method

- The rebar input variable of the supporting wall (2 in Figure 3)—input random variable 15.
- The yield stress of the reinforcement rebars of the supporting wall (2 in Figure 3)—input random variable 16.
- The thickness of the broken GDH pipe (3 in Figure 3)—input random variable 20.
- The mid-surface radius of the broken GDH pipe (3 in Figure 3)—input random variable 21.
- The Young’s modulus of the impacted GDH pipe (4 in Figure 3)—input random variable 22.
- The yield stress of the impacted GDH pipe (4 in Figure 3)—input random variable 24.

- The yield stress of the broken GDH pipe (3 in Figure 3)—input random variable 19.
- The rebar area of the supporting wall (2 in Figure 3)—input random variable 12.
- The Poisson’s ratio of the impacted GDH pipe (4 in Figure 3)—input random variable 23.
- The mid-surface radius of the impacted GDH pipe (4 in Figure 3)—input random variable 26.
- The rebar area of the supporting wall (2 in Figure 2)—input random variable 14.
- The Poisson’s ratio of the concrete of the supporting wall (2 in Figure 3)—input random variable 1.
- The Young’s modulus of the concrete of the supporting wall (4 in Figure 3)—input random variable 2.

#### 4.5.2. Probabilistic Analysis Applying the FORM

^{−9}to 0.485 (Table 3). This limit state indicates that the ultimate stress of the rebars will be reached in layers 8, 9, and 10, and that these rebars may fail. The probability of failure for the first layer of concrete rebar is ridiculously small—close to zero (2.092 × 10

^{−9}). The system event was used for analyzing the probability of failure during the same computational run at all integration points of the concrete rebar element. The calculated probability of “System Event 2” is approximately 0 (Table 4). Thus, the ultimate stress of the concrete rebar has a ridiculously small probability of being reached, and the rebars in the support wall may fail with an equally small probability.

#### 4.5.3. Probabilistic Analysis Using the RS/MCS Method

^{7}”—is as follows:

^{7}) + (18.6745 × L1

_{1-1})+ (9.04392 × L1

_{1-3}) + (5.85753 × 10

^{7}× P4) + (−0.000360327 × Y4) +

(−7.24406 × 10

^{9}× re1) + (−1.03215 × 10

^{9}× re3) + (1.22263 × 10

^{10}× re4) + (0.0081826 × r5) +

(0.00255908 × Yi7) + (6.88826 × 10

^{8}× t7) + (1.22627 × 10

^{8}× m7) + (−0.000177385 × Y8) +

(8.63534 × 10

^{7}× P8) + (0.0239527 × Yi8) + (5.77211 × 10

^{7}× m8).

^{7}. L1

_{1-1}is the Load Unit 1-1, and L1

_{1-3}is the Load Unit 1-3 (Load Unit 1-1 and Load 1-3 are loading points at different times). P4 is the Poisson’s ratio of wall 2 (Figure 3), Y4 is the Young’s modulus of the concrete of wall 2, re1 is the rebar 1 area of wall 2, re3 is the rebar 3 area of wall 2, re4 is the rebar 4 area of wall 2, r5 is the yield stress of the reinforcement bar in wall 2, Yi7 is the yield Stress of pipe 3, t7 is the thickness of pipe 3, m7 iks the mid-surface radius of pipe 3, Y8 is the Young’s modulus of pipe 4, P8 is the Poisson’s ratio of pipe 4, Yi8 is the yield Stress of pipe 4, and m8 is the mid-surface radius of pipe 4.

^{8}) + (−177.237 × L1

_{1-1}) + (−204.367 × L1

_{1-3}) + (−4.85967 × 10

^{8}× P4) + (0.00103062 × Y4) +

(5.94666 × 10

^{10}× re1) + (−4.37383 × 10

^{10}× re3) + (−3.91527 × 10

^{9}× re4) + (−0.357241 × r5) +

(0.0785711 × Yi7) + (−7.64796 × 10

^{8}× t7) + (−9.9672 × 10

^{7}× m7) + (0.00315964 × Y8) +

(−5.69678 × 10

^{8}× P8) + (0.051506 × Yi8) + (−1.04162 × 10

^{9}× m8).

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The top view of the GDHs within the compartment (

**a**) and the integrated model comprising GDH pipes and concrete walls (

**b**) for examining the impact on piping is depicted as follows: 1—adjacent wall; 2—GDH’s support wall; 3—broken GDH; 4—impacted GDH pipe; and 5—contact element between GDH pipes.

**Figure 4.**Significant random variables (362, 201, 202, 203, and 204) for Element Response Stress Equivalent.

**Figure 5.**Significant random variables (342, 301, 302, and 303) for Element Response Stress Equivalent.

**Figure 8.**The failure probability of impacted GDH pipe at dynamic loading due to guillotine rupture.

Material/ Location | Parameter | Distribution | Mean | Unit | COV |
---|---|---|---|---|---|

Reinforced Concrete | |||||

Concrete (walls 1 and 2 in Figure 3) | Poisson’s ratio | Log-normal | 0.2 | - | 0.10 |

Young’s modulus | Log-normal | 2.7 × 10^{10} | Pa | 0.10 | |

Uniaxial tensile strength | Log-normal | 1.5 × 10^{6} | Pa | 0.10 | |

Reinforcement bars (walls 1 and 2) | Yield stress | Log-normal | 3.9 × 10^{8} | Pa | 0.03 |

GDH Pipe | |||||

Austenitic Steel (pipes 3 and 4 in Figure 3) | Young’s modulus | Log-normal | 1.8 × 10^{11} | Pa | 0.03 |

Poisson’s ratio | Log-normal | 0.3 | - | 0.03 | |

Yield stress | Log-normal | 1.8 × 10^{8} | Pa | 0.03 | |

Contact of Pipes | Contact modulus | Normal | 1.8 × 10^{11} | Pa | 0.10 |

Material/Location | Parameter | Distribution | Mean | Unit | COV |
---|---|---|---|---|---|

Reinforced Concrete | |||||

Reinforcement bars | Reinforcement layer thickness (data are presented for 1 rebar) | Log-normal | 0.0031 | m | 0.05 |

GDH Pipe | |||||

Austenitic steel | Wall thickness | Log-normal | 0.0015 | m | 0.05 |

The mid-surface radius of the pipe | Log-normal | 0.1550 | m | 0.05 |

Name * | Definition * | Probability | Beta, See (6) |
---|---|---|---|

LS 1 | NR (572) DD 2 (Y) < −0.425 | 0.506254 | 0.0156768 |

LS 2 | ER (362) SE < −1.7 × 10^{7} | 0.500205 | 0.0005131 |

LS 3 | ER (201) SE < −1.7 × 10^{7} | 0.503351 | 0.0083994 |

LS 4 | ER (202) SE < −1.7 × 10^{7} | 0.474443 | −0.0641049 |

LS 5 | ER (203) SE < −1.7 × 10^{7} | 0.499217 | −0.0019636 |

LS 6 | ER (204) SE < −1.7 × 10^{7} | 0.498017 | −0.0049704 |

LS 7 | ER (342) SE > 5.9 × 10^{8} | 2.092 × 10^{−9} | −5.8772800 |

LS 8 | ER (301) SE > 5.9 × 10^{8} | 0.476667 | −0.0585204 |

LS 9 | ER (302) SE > 5.9 × 10^{8} | 0.473746 | −0.0658572 |

LS 10 | ER (303) SE > 5.9 × 10^{8} | 0.484998 | −0.0376143 |

LS 11 | ER (529) SE > 4.12 × 10^{8} | 0.287865 | −0.5596310 |

Name | Probability | Beta, See (6) |
---|---|---|

LS 2 & LS 3 & LS 4 & LS 5 & LS 6 | 0.050190 | −1.64337 |

LS 7 & LS 8 & LS 9 & LS 10 | ~0 | −4.01317 |

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**MDPI and ACS Style**

Alzbutas, R.; Dundulis, G.
Probabilistic Assessment of Structural Integrity. *Axioms* **2024**, *13*, 154.
https://doi.org/10.3390/axioms13030154

**AMA Style**

Alzbutas R, Dundulis G.
Probabilistic Assessment of Structural Integrity. *Axioms*. 2024; 13(3):154.
https://doi.org/10.3390/axioms13030154

**Chicago/Turabian Style**

Alzbutas, Robertas, and Gintautas Dundulis.
2024. "Probabilistic Assessment of Structural Integrity" *Axioms* 13, no. 3: 154.
https://doi.org/10.3390/axioms13030154