# Structural reliability analysis

\begin{equation} \small

Y = f \left ( \boldsymbol{X} \right )
\tag{1}
\end{equation}

The probability of failure $ P_{f} $ is defined as the probability of structural response being in a domain of unacceptable operation, such as response exceeding a critical threshold, $ y_{c} $, and it is typically defined in terms of a limit state function

\begin{equation} \small g \left ( \boldsymbol{X} \right ) = y_{c} - f \left ( \boldsymbol{X} \right ) \quad {\rm such\ that}\ P_{f} = {\rm Pr}\left [ g \left ( \boldsymbol{X} \right ) \leq 0 \right ] \tag{2} \end{equation}

There are a number of reliability analysis method, like Monte Carlo simulation (MCS)[1],first-order reliability method (FORM)[2],second-order reliability method[3] and the moment methods[4]. One of the main challenges is to balance the accuracy and efficiency of the reliability analysis, that is

- minimize the function evaluations
- find the most appropriate probability distribution or explicit approximation of the implicit limt state function

**FIG.1:**The model of numerical example

and the results obtained by MCS, proposed method, RSM with only quadratic term (QP) and complete quadratic term (CQP) are shown in Table 1,where $ \beta,\ k_final,\ and\ N $ is the Hasofer-Lind Rackwitz-Fiessler(HL-RF) reliability index, the number of iterations and function evaluations. It can be observed that the proposed method is able to obtain accurate result with less function evaluations

**Table 1**Results summary of numerical example

Method | $ k_{final} $ | $ N $ | $ \beta $ | Error(%) |

MCS | -- | $ 10^7 $ | 2.725 | -- |

RSM with QP | 3 | 23 | 2.710 | 0.6 |

RSM with CQP | 3 | 39 | 2.619 | 3.9 |

proposed method | 3 | 25 | 2.710 | 0.6 |

However, the method only consider component reliability analysis within linear elasticity case. In the future I would like to dig more about structural reliability analysis in system or dynamic case.

Reference

[1] Rubinstein RY (1981) Simulation and the Monte Carlo method.Wiley,New York

[2] Hohenbichler M, Rackwitz R (1982) First-order concepts in system reliability. Structural safety 1(3): 177-188.

[3] Breitung K (1984) Asymptotic approximations for multinormal integrals. Journal of Engineering Mechanics 110(3): 357-366.

[4] Xi Z, Hu C, Youn BD (2012) A comparative study of probability estimation methods for reliability analysis. Struct Multidiscip Optim 45:33–52.

[5] Fan W, Shen W, Zhang Q, et al. A new response surface method based on the adaptive bivariate cut-HDMR. Engineering Computations.

[6] Fan W, Wei J, Ang AHS, Li Z (2016) Adaptive estimation of statistical moments of the responses of random systems. Probabilistic Eng Mech 43:50–67.

[7] Chowdhury R, Rao BN (2009) Hybrid High Dimensional Model Representation for reliability analysis. Computer Methods in Applied Mechanics & Engineering 198(5 :753-765.