# Machine learning based reliability analysis of uncertain structures under stochastic excitation

Alexandre Saidi, Weizhen You, Abdelmalek Zine, Mohamed Ichchou

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Alexandre Saidi, Weizhen You, Abdelmalek Zine, Mohamed Ichchou. Machine learning based reliability analysis of uncertain structures under stochastic excitation. The 11th International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, QR2MSE2021 and ICRSE2021, Aug 2021, Chengdu, Sichuan, China. hal-05146374

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# Machine learning based reliability analysis of uncertain structures under stochastic excitation

Weizhen YOU*

Alexandre SAIDI

Abdel-malek ZINE

College of Mechanical and Electrical
Engineering

LIRIS

ICJ

Wenzhou University

Ecole Centrale de Lyon

Ecole Centrale de Lyon

Wenzhou, China

Lyon, France

Lyon, France

youweiz@sina.com

alexandre.saidi@ec-lyon.fr

abdel-malek.zine@ec-lyon.fr

Mohamed ICHCHOU
LTDS

Ecole Centrale de Lyon
Lyon, France

mohamed.ichchou@ec-lyon.fr

**Abstract**—Reliability assessment of structures can be treated as uncertainty quantification (UQ) problem due to the excitation uncertainties. When the structural properties become uncertain, the problem becomes more complex. In this research, the uncertainty space is seen as the joint distribution of structural uncertainty and excitation uncertainty. Assuming that the two kinds of uncertainties are relatively independent, a conditional failure probability (CFPs) can be estimated for each sample of structural properties. In this aspect, machine learning (ML) theories can be used to solve the regression problem between the structural uncertainties and CFPs. Once a ML process is conducted, it is ready to predict CFPs for other structural samples outside the training set. In numerical simulations, several ML models are studied and their performances are compared and analyzed. The simulation results show that ML helps greatly improve the efficiency of reliability assessment.

**Keywords**—structural reliability analysis (SRA), stochastic excitation, machine learning

## I. INTRODUCTION

The design and improvement of industrial products (e.g. automobiles, bridges, airplanes...) have made computational software unavoidable. Even though the mathematic models become more and more accurate and the computers runs much faster, the actual problems faced are still uneasy to analyze. In fact, the calculations always rely on some simplified assumptions about the system in order that the computational models are possible to build and analyze. Generally, model responses are determined by running this model that takes the parameter values as input. However, due to the limits of manufacturing process as well as the influence of complex environmental factors, the input values are becoming difficult to stay deterministic. In other words, they are characterized with uncertainties. The input uncertainties finally induce the uncertainties in the system responses.

Assume that a mechanical system has its behavior that is expressed by the simulation model $y = K(x) \in R$, where

$x \in D_x \subset R^M$. In this model, the input $x$ consists of $M$ variables; the output $y$ is the quantity of interest, such as load carrying capacity, a performance function; $D_x$ is a special space that contains all possible states of $x$ in assumed conditions. Once the probabilistic input (material properties, geometric loading...) is determined, the uncertainty propagation can be applied [1]. In theory, the failure probability calculation is accomplished directly by solving a multivariate integral analytically or numerically within the failure domain [2]. But this is only practical for low-dimension problems. MCS (Monte Carlo simulation) is a well-known technique for estimating failure probabilities. Realizations of the input $x$ are sampled according to its probabilistic distribution, then the model $M$ is run for each sample and the results are collected and post-processed [3]. A shortcoming of general MCS is that it suffers from a low efficiency. Thus much efforts have been devoted to developing alternative approaches, such as advanced simulation methods and surrogate models.

Importance sampling (IS) was proposed to improve the computational efficiency by reducing the samples analyzed. To apply IS technique, an importance sampling density (ISD) function is firstly developed such that more samples can be obtained from the failure region, meanwhile ensuring a low variability of the estimated probability. Response surface method (RSM) was also developed by researchers [4]. Support vector machine (SVM) attracted more attention recently ([5], [6]). Artificial neural networks (ANN) [7] and Gaussian process-based models [8] also have been introduced by many authors. Some studies focus on developing more efficient analysis by combining MCS with surrogate methods. However, they cannot avoid disadvantages in all situations. The disadvantages in ANNs mainly include the complex parameter optimization, low robustness as well as huge computational burden [9]. SVM is low efficient for complex structural systems and always induces large errors in sensitivity analysis of reliability values.
ML becomes more and more attractive in the domain of uncertainty quantifications and reliability assessment. In theory, any ML method designed for classification or regression problems can be applied in reliability analysis. However, the ML domain has not been extensively explored by the reliability community. In this respect, this contribution aims at developing a general but efficient framework that incorporates ML theories for reliability evaluation. In this framework, a training data is firstly generated by MC procedures. The input of the training data is a set of structural samples; the outputs are conditional failure probabilities (CFPs) corresponding to the specific input samples. Then a ML regression model is developed from this data set. This model will be employed to predict CFPs on new samples of structural properties and the overall failure probability can be estimated immediately. Notice that for very small CFPs, an IS method is introduced. Within this framework, this study focus on two ML models, i.e. Random forest (RF) [10] and Stacked generalization (Stacking) [11]. The two methods are all powerful ML tools in classification and regressions. Besides, they are based on different principles, so they can be compared in failure probability estimations.

In this paper, section II presents a general reliability analysis framework via MCS that gives a basic idea. Section III presents the detailed information about the proposed framework. Section IV presents some typical application examples in the field of structural dynamics to examine the performance of the proposed method.

## II. SRA BY GENERAL MCS

### 2.1 Deterministic case

In deterministic case, the structural properties are assumed certain and unchanged. To facilitate the simulation, the excitation process is described as a vector that consists of n independent elements, i.e. $z = [z_1, z_2, ..., z_n]$. Theoretically, the failure probability can be analytically expressed as a multi-dimension integral that is computationally impractical, but it can be approximated by MCS

$$P_F^D = p(F) = p(g(\mathbf{z}) \le 0) = \int_{g(\mathbf{z}) \le 0} f_Z(\mathbf{z})d\mathbf{z} = \int_{z \in \Omega_x} I_F(\mathbf{z})f_Z(\mathbf{z})d\mathbf{z} = E_{z \in \Omega_x}[I_F(\mathbf{z})] \quad (1)$$

where $g(\mathbf{z})$ is a predefined performance function of the structure; negative values of $g(\mathbf{z})$ constitute the failure region (i.e. F). $f_Z(\mathbf{z})$ denotes the probability density function (PDF) of stochastic excitation $\mathbf{z}$. $\Omega_x$ denotes the distribution zone of $\mathbf{z}$; $I_F(\mathbf{z})$ is a binary function that takes the value 1 for a negative value of $g(\mathbf{z})$; 0 otherwise; $E[\cdot]$ is mathematical expectation. This failure probability is called conditional failure probability (CFP).

### 2.2 Uncertain case

As a result of different factors (manufacture, environmental...), structural properties are found to be uncertain. Thus the evaluation of FP involves structural uncertainties. In this case, the uncertainty space is the joint distribution of structural parameters and excitations. In this research, the structural uncertainties and the excitation uncertainties are assumed relatively independent. Therefore, the general MCS to estimate FP becomes

$$P_F^I = p(F) = \int_{g(\mathbf{x}, \mathbf{z}) \le 0} f_{X,Z}(\mathbf{x}, \mathbf{z})d\mathbf{x}d\mathbf{z} = \int_{\mathbf{x}, \mathbf{z} \in \Omega_x} I_F(\mathbf{x}, \mathbf{z})f_{X,Z}(\mathbf{x}, \mathbf{z})d\mathbf{x}d\mathbf{z} = E_{\mathbf{x}, \mathbf{z} \in \Omega_x}[I_F(\mathbf{x}, \mathbf{z})] \quad (2)$$

## III. MACHINE LEARNING BASED SRA

### 3.1 The proposed framework

In this article, a basic assumption is that the uncertainties in the excitations as well as those in the properties of the object structure are relatively independent. Consequently, the estimation of FP is expressed in the following form

$$ 
\begin{aligned}
P_F &= \int_{\mathbf{x} \in \Omega_x} I_F(\mathbf{x}, \mathbf{z}) f_X(\mathbf{x}) f_Z(\mathbf{z}) d\mathbf{x} d\mathbf{z} \\
&= \int_{\mathbf{x} \in \Omega_x} \left[ \int_{\mathbf{z} \in \Omega_x} I_F(\mathbf{z} | \mathbf{x}) f_Z(\mathbf{z}) d\mathbf{z} \right] p_X(\mathbf{x}) d\mathbf{x} \\
&= \int_{\mathbf{x} \in \Omega_x} P_F^C(\mathbf{x}) f_X(\mathbf{x}) d\mathbf{x} \\
&= E_{\mathbf{x} \in \Omega_x} [P_F^C(\mathbf{x})]
\end{aligned} 
\quad (3) $$

where $P_F^C(\mathbf{x})$ is the CFP that can be seen as the 'weight' of the PDF $p_X(\mathbf{x})$ at $\mathbf{x}$. Therefore, the overall failure probability equals to the expectation of $P_F^C(\mathbf{x})$ that has the PDF $f_X(\mathbf{x})$. Fig. 1 illustrates the distribution of CFP in one dimension case of $\mathbf{x}$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Illustration of PDF of CFP in one dimension case</figcaption></figure>

ML aims to build a model based on a training data and make predictions via the rules encoded in the model. In this study, a few CFPs are firstly obtained via MCS for samples of the object structure according to its uncertainty. Then a specific ML model is learned that takes these samples as input and the CFPs as output. It will be used to make predictions on the samples outside the training data. In this study, this ML model is a surrogate that aims to map structural uncertainties into CFPs. Fig. 2 presents the framework developed in this study. N is the number of structural samples taken randomly.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Theoretical framework of ML-based SRA</figcaption></figure>
## 3.2 IS to estimate small probabilities

It is known that the crude MCS generally provides estimations that are seen as high-reliable reference values. But it always demands high computational burden to reach a convergence value especially when the estimated value is very small, i.e. they have mathematical relationship of inverse proportion. Due to this situation, IS method has been developed. Basically, an IS technique aims to construct a new density function from the old one so that the samples tend to be obtained from the failure region. The structural responses can be calculated by state space equations. It is assumed that the object structure is linear, so a response process is approximated by convolution integral that involves structural impulse response and the excitation. Briefly speaking, for an m-DOF structure, the system response can be evaluated by the following convolution integral,

$$y_i(t, \mathbf{x}) = \int_0^t h_i(t - \tau, \mathbf{x})z(\tau)d\tau \quad (4)$$

where i=1, 2, ..., m, are ordinal numbers of DOFs. $h_i(t - \tau, \mathbf{x})$ denotes unit impulse response of the ith DOF. $\mathbf{x}$ is a sample of structural uncertainty. Due to the linearity of the structure, $y_i(t, \mathbf{x})$ is actually a linear combination of the values $z(\tau)$ where $\tau$ ranges from 0 to t. The Gaussian stochastic excitation is a Gaussian process approximated as K-L expansions [13],

$$z(t) = \mu(t) + \sum_{i=1}^{n_{KL}} \sqrt{\lambda_i} u_i \phi_i(t) \quad (5)$$

here $\mu(t)$ denotes the mean; $u_i$, $i \in N^+$ is a variable that follows standard normal distribution. $n_{KL}$ denotes the number of kept terms. $\{\lambda_i, \phi_i\}$ is the solution of eigenvalue problem of Fredholm integral of the 2nd kind. Substituting $z(t)$ in eq.(4) for the expression in eq.(5) and make some changes of the calculation order, this gives

$$y_i(t, \mathbf{x}) = \int_0^t h_i(t - \tau, \mathbf{x}) \sum_{i=1}^{n_{KL}} u_i f_i(\tau) d\tau = \sum_{i=1}^{n_{KL}} u_i a_i(t, \mathbf{x}) = \mathbf{a}(t, \mathbf{x})^T \mathbf{u} \quad (6)$$

here $a_i(t, \mathbf{x}) = \int_0^t h_i(t - \tau, \mathbf{x}) f_i(\tau) d\tau$ and $f_i(\tau) = \sqrt{\lambda_i} \phi_i(t)$.

Assume there are some realizations of $z(t)$ which result in the event $y_i(t_j) \ge y_i^*$ at time $t = t_j$, where $y_i^*$ is a designed threshold for the ith DOF. The failure region is defined as follows

$$F_{i,j}: g_{i,j}(\mathbf{u}) = y_i^* - \|\mathbf{a}(t_j, \mathbf{x})^T \mathbf{u}\| \le 0 \quad (7)$$

here i is the ordinal number of DOF, j is the time point. Actually, this region consists of two subspaces that are symmetric and divided by a hyper-plane $y_i^* = \|\mathbf{a}(t_j, \mathbf{x})^T \mathbf{u}\|$. The two subspaces share a same distance from the origin, that is $\beta_{i,j}(\mathbf{x}) = y_i^* / \|\mathbf{a}(t_j, \mathbf{x})\|$. According to the theory of First-Order Reliability method, the failure probability for ith DOF at $t_j$ becomes

$$P(F_{i,j}) = 2 * \Phi(-\beta_{i,j}(\mathbf{x})) = y_i^* / \|\mathbf{a}(t_j, \mathbf{x})\| \quad (8)$$

here $\Phi(\cdot)$ is standard normal CDF. $F_{i,j}$ is the elementary failure region. Based on the elementary failure regions, an IS technique by S.K. Au [14] is employed to improve sampling efficiency.

## 3.3 ML models

A RF consists of several random trees that are induced from bootstrap replicates of the original training set, and it makes final predictions by aggregating the individual predictions of these trees. This method reduces the variance of predictions by introducing randomization into its construction process. A random tree inducing process can be seen as a recursive procedure of dividing a dataset. A random tree has a few nodes, each of which corresponds to a sub-dataset. The optimal splitting point of the dataset is determined by minimizing the sum of squared errors (SSE), i.e. $\min_{i,j} \left[ \sum_{x \in R_i} (y_i - c_i)^2 + \sum_{x \in R_j} (y_j - c_j)^2 \right]$, where $R_i = \{x | x_i \le x_{i,j}\}$ is the region that satisfies the condition $x_i \le x_{i,j}$; $R_j = \{x | x_j > x_{j,i}\}$ is a subset which satisfies $x_j > x_{j,i}$. Besides, $c_m$ is the averaged output value in the region, $c_m = E(y | x_i \in R_m)$, $m = 1, 2$. See more in [10].

Stacking is an ensemble learning technique that learns a meta-learner based on the output multiple base learners. To implement the procedure of Stacking, some base learners are firstly developed from the training dataset by employing different learning algorithms. Then a meta-learner is induced from the base learners. See Fig. 6. Once the meta-learner is generated, for a new example $\mathbf{x}$, a prediction is made as $H(h_1(\mathbf{x}), ..., h_T(\mathbf{x}))$, where $h_1(\mathbf{x}), ..., h_T(\mathbf{x})$ are the predictions by base learners.

# IV. NUMERICAL SIMULATIONS

## Simulation 1:5-DOF shear frame

In this example, 10 random variables are involved in modeling the uncertainties in the structural parameters and a large number of random variables (of the order of hundred) for stochastic excitation. The objective is to study the overall performance of the proposed approach for assessing first excursion probabilities. In particular, this example involves a 5-story linear shear frame (see Fig. 3) excited by a series of stochastic ground accelerations. It is assumed that the columns supporting each floor are perfectly clamped and that the beams are rigid. The bending stiffness of the different DOFs are given by

$$K_i = 2k_i = 2 \times 12 E_i I / h^3 = 24 E_i I / h^3 \quad (9)$$

where $i=1,2,...,5$. $E_i$ is the modulus of elasticity at the ith DOF; $I$ is the moment of inertia of column's section given by $I=0.0108m^4$; $h=0.4m$ is the height of each column. Assume the uncertainties (Gaussian) exist in the masses and the modulus of elasticity at different DOFs, see TABLE I.

<figure><img src="image_4.png" /><figcaption>Fig. 3. 5-DOF shear frame</figcaption></figure>
<table><caption>TABLE I. Parameter properties of a shear frame</caption><thead><tr><th>Variables</th><th>Mean(μ)</th><th>SD</th><th>Range scope</th></tr></thead><tbody><tr><td>m<sub>1</sub>, ..., m<sub>5</sub></td><td>4.5e4 kg</td><td>4.5e3 kg</td><td>μ ± μ/2</td></tr><tr><td>E<sub>1</sub>, ..., E<sub>5</sub></td><td>2.0e10 N/m<sup>2</sup></td><td>2.0e9 N/m<sup>2</sup></td><td>μ ± μ/2</td></tr></tbody></table>

In this test, the excitation is modeled as filtered Gaussian white noise that is realized by Kanai-Tajimi (KT) model [1, 5]. The input of KT model is Gaussian white noise with power spectral density $S_o = 0.031 m^2 / s^3$, natural frequency $\omega = 8\pi \text{ rad/s}$ and damping ratio $\xi_f = 0.4$. In applying IS method, the KL expansion terms are kept $n_{KL} = 300$; besides, 5050 points are produced as importance samples. A failure occurs only when the distance between two successive DOFs reaches a defined value within 20 seconds. Time is observed with a step $\Delta t = 0.05$ second.

<table><caption>TABLE II. Failure probability estimations ($y_{lim} = 0.15m$)</caption><thead><tr><th>Index</th><th>General MCS</th><th>RF-based</th><th>Stacking-based</th></tr></thead><tbody><tr><td>FP value</td><td>3.45e-2</td><td>3.36e-2</td><td>3.51e-2</td></tr><tr><td>n_sample</td><td>3500</td><td>40</td><td>40</td></tr></tbody></table>

Two different limit values in estimating FPs are assumed, i.e. $y_{lim} = 0.15m$ and $y_{lim} = 0.18m$. In the RF based approach, 20 trees are generated in a RF. In the Stacking-based approach, the Stacking model the base learners and meta-learner are {RF, SVM}, GB respectively. Both the two base learners consist of 20 random trees. It is known that each sample in the training set has 10 variables, by default 3 variables are randomly chosen for each node splitting.

TABLE II and TABLE III show the results from different methods. Notice that the estimated values are convergence values, and ‘n_sample’ is the number of samples analyzed to reach the convergence. It is found that the general MCS consumes many more samples than the other two methods before reach the convergence. The RF-based approach and Stacking-based approach just need 40 samples to achieve close estimations to those achieved by general MCS. Besides, the two ML-based methods have similar estimation results.

<table><caption>TABLE III. Failure probability estimations ($y_{lim} = 0.18m$)</caption><thead><tr><th>Index</th><th>General MCS</th><th>RF-based</th><th>Stacking-based</th></tr></thead><tbody><tr><td>FP value</td><td>5.01e-3</td><td>5.82e-3</td><td>5.93e-3</td></tr><tr><td>n_sample</td><td>10000</td><td>40</td><td>40</td></tr></tbody></table>

### 3.4 Simulation 2: 10-DOF Duffing oscillator

The 10-DOF Duffing type oscillator introduced by [16] has been widely studied as a benchmark problem in this domain. This part focus on this structure and test the proposed model. TABLE IV presents the uncertainty characteristics of the object structure. The stochastic excitation is described as a processed white noise signals:

$$z(t) = \omega_1^2 v_1(t) + 2\zeta_1\omega_1 v_2(t) - \omega_2^2 v_3(t) - 2\zeta_2\omega_2 v_4(t) \quad (10)$$

Here the state space expression related to the state vector $v(t) = [v_1(t), v_2(t), v_3(t), v_4(t)]^T$ of the filter is,

$$\hat{v}(t) = \begin{bmatrix} 0 & 1 & 0 & 0 \\ -\omega_1^2 & -2\zeta_1\omega_1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \omega_1^2 & 2\zeta_1\omega_1 & -\omega_2^2 & -2\zeta_2\omega_2 \end{bmatrix} v(t) + [0, \omega(t), 0, 0]^T \quad (11)$$

where $\omega(t)$ denotes the specific white noise that has an auto-correlation expressed as $E[\omega(t)\omega(t+\tau)] = I / \delta(\tau)h^2(t)$; $I$ is the excitation intensity value. $h(t)$ is a designed signal function that is denoted as

$$h(t) = \begin{cases} 0, & t=0 \\ t/2, & t \in [0, 2s] \\ 1, & t \in [2s, 10s] \\ \exp(-0.1(t-10)), & t \in [10s, 20s] \end{cases} \quad (12)$$

$\delta(\tau)$ is the Dirac delta function that equals to $+\infty$ at $t=0$ and $0$ at $t \le 0$. The values are set $\omega_1 = 15.0 \text{ rad/s}$, $\zeta_1 = 0.8$, $\omega_1 = 0.3 \text{ rad/s}$, $\zeta_1 = 0.995$ and $I = 0.08 \text{ m}^2/\text{s}^3$. For the input excitation, this study uses the same parameter settings as those presented in [16]. A failure occurs only when the distance between two successive DOFs reaches a limit within 20 seconds. Time is observed with a step $\Delta t = 0.05$ second. FPs are assessed under four assumed thresholds, see details in TABLE V. In this simulation, the first excursion probability is studied. For the first DOF, the thresholds $0.057m$ and $0.073m$ are considered respectively; for the 10th DOF, the thresholds 1.3cm and 1.7cm are considered.

<table><caption>TABLE IV. Characteristics of structural properties</caption><thead><tr><th>Variables</th><th>Mean(μ)</th><th>SD</th><th>Range scope</th></tr></thead><tbody><tr><td>m<sub>1</sub>, ..., m<sub>10</sub></td><td>10e3 kg</td><td>1.0e3 kg</td><td>μ ± μ/2</td></tr><tr><td>k<sub>1</sub>, k<sub>2</sub>, k<sub>3</sub></td><td>40e6 N/m</td><td>4.0e6 N/m</td><td>μ ± μ/2</td></tr><tr><td>k<sub>4</sub>, k<sub>5</sub>, k<sub>6</sub></td><td>36e6 N/m</td><td>3.6e6 N/m</td><td>μ ± μ/2</td></tr><tr><td>k<sub>7</sub>, k<sub>8</sub>, k<sub>9</sub>, k<sub>10</sub></td><td>32e6 N/m</td><td>3.2e6 N/m</td><td>μ ± μ/2</td></tr><tr><td>ζ<sub>1</sub>, ..., ζ<sub>10</sub></td><td>6.2e6 Ns/m</td><td>6.2e5 Ns/m</td><td>μ ± μ/2</td></tr></tbody></table>

In this simulation, the failure probabilities to be estimated are very small (below $10^{-4}$), so IS scheme is introduced to estimate CFPs. To apply K-L expansions, it is set $n_{KL}=290$. The RF regressor consists of 30 random trees generated from a training data with 300 samples, each of which contains 30 variables. By the proposed framework in Fig. 5, the FPs are estimated under four assumed thresholds. Comparisons are made between these results and those provided by a few representative papers, details are presented in TABLE V. To make it clear, the results coming from G. I. Schuëller [16] are labeled by '*'; case1.1 denotes "$y_{1,lim}=0.057m$"; case1.2 denotes "$y_{1,lim}=0.073m$"; case2.1 denotes "$y_{10,lim}=0.013m$"; case2.2 denotes "$y_{10,lim}=0.017m$".

<table><caption>TABLE V. Estimations of FPs in different cases</caption><thead><tr><th>Method</th><th>case1.1</th><th>case1.2</th><th>case2.1</th><th>case2.1</th></tr></thead><tbody><tr><td>General MCS</td><td>1.06e-4</td><td>8.07e-7</td><td>4.88e-5</td><td>2.52e-7</td></tr><tr><td>n_sample</td><td>2.98e7</td><td>2.98e7</td><td>2.98e7</td><td>2.98e7</td></tr><tr><td>SubsetSim*</td><td>1.2e-4</td><td>1.00e-6</td><td>6.60e-5</td><td>4.70e-7</td></tr></tbody></table>
<table><tr><td>n_sample</td><td>1850</td><td>2750</td><td>2300</td><td>2750</td></tr><tr><td>LineSam*</td><td>9.8e-5</td><td>9.7e-7</td><td>6.00e-5</td><td>4.60e-7</td></tr><tr><td>n_sample</td><td>360</td><td>3600</td><td>360</td><td>360</td></tr><tr><td>RF-based</td><td>7.5e-5</td><td>1.1e-6</td><td>4.3e-5</td><td>1.2e-7</td></tr><tr><td>n_sample</td><td>800</td><td>1000</td><td>800</td><td>900</td></tr><tr><td>Stacking-based</td><td>1.1e-4</td><td>9.1e-7</td><td>4.2e-5</td><td>1.2e-7</td></tr><tr><td>n_sample</td><td>800</td><td>1000</td><td>800</td><td>900</td></tr></table>

From TABLE V, the results tell that the Stacking based approach outperforms RF model in case1.1, case1.2 and case2.1. In case2.2, they give similar estimations. Considering the order of the estimated FPs, the two models achieve similar performances.

<table><caption>TABLE VI. Estimations of FPs with multi-criteria</caption><thead><tr><th>Method</th><th>case1</th><th>case2</th></tr></thead><tbody><tr><td>General</td><td>1.12e-4</td><td>1.00e-6</td></tr><tr><td>n_sample</td><td>1.0e6</td><td>2.0e7</td></tr><tr><td>RF-based</td><td>1.11e-4</td><td>1.10e-6</td></tr><tr><td>n_sample</td><td>800</td><td>1100</td></tr><tr><td>Stacking-</td><td>1.20e-4</td><td>1.18e-6</td></tr><tr><td>n_sample</td><td>800</td><td>1000</td></tr></tbody></table>

It is noticed that single threshold values are assumed to define the failure in TABLE V. To make further study, multi-thresholds are considered. The failure criterions are defined as case1: $y_{1,lim} = 0.057m \cup y_{10,lim} = 0.013m$, this means a failure occurs when the first DOF response is larger than 5.7cm or the tenth DOF response is larger than 1.3cm; case2: $y_{1,lim} = 0.073m \cup y_{10,lim} = 0.017m$, this means a failure occurs when the first DOF response is larger than 7.3cm or the tenth DOF is larger than 1.7cm. The FP estimations in the two cases are shown in TABLE VI. It is found that the two methods result in similar estimations, both of them achieve accurate estimations with a small number of samples.

<table><caption>Table VII. Estimations of FPs with different uncertainty types</caption><thead><tr><th rowspan="2">Method</th><th colspan="2">Lognormal</th><th colspan="2">Gamma</th><th colspan="2">Mix</th></tr><tr><th>case1</th><th>case2</th><th>case1</th><th>case2</th><th>case1</th><th>case2</th></tr></thead><tbody><tr><td>General<br>MCS</td><td>8.8e-5</td><td>4.0e-5</td><td>9.8e-6</td><td>3.7e-6</td><td>3.1e-5</td><td>4.0e-6</td></tr><tr><td>n_sample</td><td>1.0e6</td><td>1.0e6</td><td>2.0e7</td><td>2.0e7</td><td>2.0e7</td><td>2.0e7</td></tr><tr><td>RF-based</td><td>7.1e-5</td><td>4.0e-5</td><td>9.7e-6</td><td>3.6e-6</td><td>3.0e-5</td><td>4.1e-6</td></tr><tr><td>n_sample</td><td>1600</td><td>1600</td><td>500</td><td>800</td><td>1000</td><td>900</td></tr><tr><td>Stacking-<br>based</td><td>8.0e-5</td><td>4.1e-5</td><td>9.6e-6</td><td>3.7e-6</td><td>3.2e-5</td><td>4.1e-6</td></tr><tr><td>n_sample</td><td>1600</td><td>1600</td><td>500</td><td>800</td><td>1000</td><td>900</td></tr></tbody></table>

Notice that the test results in TABLE III and TABLE V all take assumptions that structural parameters are Gaussian

random. In this part, non-Gaussian variables are assumed. Three situations are studied here. 1) Lognormal: all random parameters are lognormal; 2) Gamma: all parameters are Gamma; 3) Mix: parameters in 1st-5th DOFs are Lognormal and those in 6th-10th DOFs are Gamma. Table VII shows the results in all three situations. To make it easier to read, case1 denotes "$y_{1,lim}=0.057m$"; while case2 denotes "$y_{10,lim}=0.013m$". As to the definition of failure, the 1st DOF or 10th DOF are considered. Notice that in Table VII, the two techniques provide estimations of high accuracy. In the 1st situation, Stacking behaves better than RF; while in the second one, both methods give estimations that are very similar to standard data provided by general MCS. In the second and third situations, the two methods all give estimations that are very close to the reference values.

## CONCLUSIONS

In this research, the reliability analysis of linear uncertain structures is carried out through a ML based framework. In this framework, a ML model is employed to solve the regression problem between structural uncertainties and conditional failure probabilities. Based on a training data, the ML is implemented, then a model is generated that is able to make predictions on the structural samples outside the training data. Then the overall failure probability of the structure is immediately estimated that considers both uncertainties in structural properties and the excitations. Within the proposed framework, the system response generally calculated by running finite element codes can be immediately determined by a ML model. For very small failure probabilities that are computationally impractical, a very efficient importance sampling method is employed. Two kinds of ML models, i.e. Random forest and Stacking, are studied in the proposed framework. Numerical simulations are performed on two typical structures, i.e. a 5-DOF shear frame and a 10-DOF Duffing oscillator. Two simulation cases conclude that, with a much reduced computational burden, the proposed framework achieves failure probability estimations that are very close to those achieved by general MCS.

## ACKNOWLEDGMENT

This work is supported by China Scholarship Council (CSC) and Wenzhou University, with professor Mohamed ICHCHOU as program director. All help and guidance from other colleagues are acknowledged.

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