# Investigation on the Mathematical Relation Model of Structural Reliability and Structural Robustness

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## Abstract

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## 1. Introduction

## 2. Mathematical Expression of Structural Reliability Index and Structural Robustness Evaluation Index

#### 2.1. Mathematical Expression of Structural Reliability Index

#### 2.2. Mathematical Expression of Structural Robustness Evaluation Index

## 3. Mathematical Relation Model Deduction of Structural Reliability and Structural Robustness

#### 3.1. Establishment of Mathematical Relation Model

#### 3.2. Effect of Structural Damage, Random Variation Factor, and Sensitivity

## 4. Two Case Studies

#### 4.1. Verification for Mathematical Relation Model

#### 4.2. Effect of Structural Damage and Random Variation Factor

#### 4.3. Sensitivity Analysis

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Mathematical relation model of structural robustness evaluation index ($R$) and structural reliability index ($\beta \in \left[{\beta}_{1},{\beta}_{0}\right]$).

**Figure 4.**Structural robustness evaluation index ($R$), structural reliability index ($\beta $), and the fitting curve of the cantilever beam (

**Left**: $R$ vs. Element No.,

**middle**: $\beta $ vs. Element No., and

**right**: $R$ vs. $\beta $).

**Figure 5.**Structural robustness evaluation index ($R$), structural reliability index ($\beta $), and the fitting curve of the truss beam (

**Left**: $R$ vs. Member No.,

**middle**: $\beta $ vs. Member No., and

**right**: $R$ vs. $\beta $).

**Figure 6.**Different variation trends of structural robustness evaluation index ($R$) and structural reliability index ($\beta $) of the cantilever beam subjected to damage effect ($\lambda D$) and variation factor ($VM$) (

**Left**: $R$ vs. $\lambda D$,

**middle**: $\beta $ vs. $\lambda D$,and

**right**: $\beta $ vs. $\lambda D$).

**Figure 7.**Structural robustness evaluation index ($R$) and structural reliability index ($\beta $) of the truss beam subjected to damage effect ($\lambda D$) and variation factor ($VF$) (

**Left**: $R$ vs. $\lambda D$,

**middle**: $\beta $ vs. $\lambda D$, and

**right**: $\beta $ vs. $\lambda D$).

**Figure 8.**Different sensitivities of the structural robustness evaluation index ($\partial R$) and structural reliability index ($\partial \beta $) of the cantilever beam subjected to damage effect ($\lambda D$) and variation factor ($VM$) (

**Top Left**: $\partial R$ and $\partial \beta $ vs. Element No.,

**top right**: $\partial R$ vs. $\lambda D$.

**Bottom left**: $\partial \beta $ vs. $\lambda D$,

**bottom right**: $\partial \beta $ vs. $\lambda D$ and $VM$).

**Figure 9.**Different sensitivities of the structural robustness evaluation index ($\partial R$) and structural reliability index ($\partial \beta $) of the truss beam subjected to damage effect ($\lambda D$) and variation factor ($VM$) (

**Top left**: $\partial R$ and $\partial \beta $ vs. Member No.,

**top middle and right**: $\partial R$ vs. $\lambda D$.

**Bottom left**: $\partial \beta $ vs. $\lambda D$,

**bottom middle and right**: $\partial \beta $ vs. $\lambda D$ and $VM$).

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**MDPI and ACS Style**

Jin, Q.-W.; Liu, Z.; He, S.-H. Investigation on the Mathematical Relation Model of Structural Reliability and Structural Robustness. *Math. Comput. Appl.* **2021**, *26*, 26.
https://doi.org/10.3390/mca26020026

**AMA Style**

Jin Q-W, Liu Z, He S-H. Investigation on the Mathematical Relation Model of Structural Reliability and Structural Robustness. *Mathematical and Computational Applications*. 2021; 26(2):26.
https://doi.org/10.3390/mca26020026

**Chicago/Turabian Style**

Jin, Qi-Wen, Zheng Liu, and Shuan-Hai He. 2021. "Investigation on the Mathematical Relation Model of Structural Reliability and Structural Robustness" *Mathematical and Computational Applications* 26, no. 2: 26.
https://doi.org/10.3390/mca26020026