Fuzzy Sensitivity Analysis of Structural Performance

Abstract

Despite the versatility and widespread application of fuzzy randomness in structural and mechanical engineering, less attention has been paid to the formulation of sensitivity analysis for this uncertainty model. In this research, a brief review of the application of sensitivity analyses in structural engineering is provided, and then the concept of local sensitivity analysis is developed for the fuzzy randomness theory. Several sensitivity tests based on the classical probability theory are extended to this uncertainty model, namely, Monte Carlo simulation (MCS), tornado diagram analysis (TDA), and first-order second-moment method (FOSM). The multidisciplinary application of these methods in engineering is shown using a numerical example, a truss structure, and finally, seismic performance evaluation of a framed structure from a full-scale experimental test. The way of visualizing the results is also provided, which helps the interpretation and better understanding. The results show that the established tools can provide detailed insight into the uncertainty of fuzzy random models. The formulated fuzzy local sensitivity can show how the output uncertainty is affected by the uncertainty of input parameters and the effectiveness of each parameter on the output variability. The provided visualization technique can show variability, the fuzziness of variability, and the order of most influential parameters. Furthermore, efficient methods such as TDA and FOSM can substantially reduce the computational time compared to the MCS while providing an acceptable trade-off for accuracy.

1. Introduction

There are various sources of uncertainty in structural engineering that need to be accounted for during analysis, such as construction errors, geometric imperfections, load variability, etc. Generally, uncertainties for engineering parameters are divided into two main sources, which are called stochastic and informal uncertainties. Stochastic uncertainty refers to the inherent variability or random nature, and the informal or epistemic uncertainty is due to vague information and lack of knowledge. Many uncertainty models have been developed to quantify these uncertainties, including probability theory [1], interval modeling [2,3,4,5], fuzzy set theory [6,7], chaos theory [8,9], fuzzy randomness [10,11], etc.

Fuzzy randomness is an uncertainty model that has been developed in the past two decades and properly tailored for structural engineering and computational mechanics to take into account both stochastic and informal uncertainties [10,12,13]. In this model, stochasticity is treated using the classical probability theory and the informal uncertainty resulting from subjective influence is treated using the fuzzy set theory. This approach is suitable when there is insufficient data for probability distributions of the model, such as mean and standard deviation or when the model involves non-statistical characteristics in the mapping from the input space into the output space.

Besides the widespread application of fuzzy randomness and considering the abovementioned uncertainties [14,15,16,17], less attention has been paid to the formulation of sensitivity analyses for this uncertainty model. Sensitivity analyses are tools to quantitatively measure how the input uncertainties lead to the output uncertainties; these tests are utilized in scientific models or risk management for different purposes such as control of output variability, identifying influential factors or less important parameters for simplification, etc. [18,19].

There are different methods for sensitivity analysis, and depending on whether it takes the interaction of input parameters into consideration, they can be divided into two major categories, local and global sensitivity analyses. Monte Carlo simulation (MCS) and Tornado Diagram analysis (TDA) methods [20], and first-order second-moment (FOSM) [21] are among the most well-known local sensitivity analyses. Generally, the uncertainty is propagated due to one variable at a time in these methods, while other variables are fixed at their mean or median value. The only difference in the aforementioned methods is in the propagation and reduction of the computational effort. On the other hand, the global sensitivity analysis methods try to consider the whole input space, thus the interaction of input variables. The derivative-based method or the Morris method [22], and variance-based methods [18] are the most common global sensitivity techniques. Regional sensitivity analysis aims at indicating regions in the input space, which affect a specific class of the output [23,24].

The application of sensitivity analysis in civil engineering, computational mechanics, and other fields have increased in recent years. Some applications of these methods in structural and mechanical engineering are listed briefly in

Table 1. Different applications of sensitivity analysis in civil and structural engineering.

No.	Year	Method	Application	Reference
1	2005	FOSM	Seismic performance evaluation	Lee and Mosalam [21]
2	2009	Variance-based analysis	Imperfection of slender members	Kala [27]
3	2010	Variance-based analysis	Collapse analysis	Arwade et al. [28]
4	2011	MCS, FOSM, and TDA	Collapse analysis	Kim et al. [20]
5	2011	Variance-based analysis	Imperfection of steel frames	Kala [29]
6	2013	FOSM and TDA	Seismic performance evaluation	Kim and Han [30]
8	2014	FOSM and TDA	Seismic analysis of offshore structures	Kim and Nour Eldin [31]
9	2016	Variance-based analysis	Stability of steel frames	Kala [32]
10	2016	Fuzzy gradient-based analysis	Structural failure probability	Jafari and Jahani [25]
11	2017	TDA	Collapse analysis of RC frames	Yu et al. [33]
12	2017	TDA	Impact analysis of steel columns	Kang and Kim [34]
13	2018	Variance-based analysis and TDA	Impact analysis of structures	Javidan et al. [35]
14	2019	TDA	Progressive Analysis	Parisi et al. [36]
15	2019	Fuzzy variance-based analysis	Collapse analysis	Javidan and Kim [26]
16	2020	Derivative-based analysis	Viscoelasticity properties of dampers	Javidan and Kim [19]
17	2021	TDA	Seismic response of reinforced masonry shear walls	Elmeligy et al. [37]
17	2022	Derivative-based analysis	Seismic vulnerability	Chisari et al. [38]

. Sensitivity analyses are carried out within the framework of the probability theory, and as mentioned earlier, little or no effort has been paid to the formulation of sensitivity analysis for the fuzzy random theory. In recent years, there could be only two research works pointed to this. Jafari and Jahani [25] formulated a gradient-based sensitivity analysis to determine the sensitivity of the failure probability to the fuzzy random input variables. However, the aim of this research is to quantify the effects of input uncertainties on any considered output uncertainty and also sensitivity to fuzzy input variables. Javidan and Kim [26] developed a variance-based global sensitivity analysis for models with fuzzy probabilistic variables. The global sensitivity method used in that study is extremely time-consuming, and it is only related to probabilistic variables, and the effects of fuzzy variables are not quantified.

In this research, a local sensitivity analysis is proposed for models with fuzzy and fuzzy probabilistic variables. The fundamentals of fuzzy randomness and mapping onto the fuzzy output space are described first to provide the required insight and define the nomenclature. Then the main framework for the local sensitivity analysis based on the fuzzy input space is described. Three uncertainty propagation methods for sensitivity analysis from the probability theory are extended to the fuzzy randomness theory. To broaden the multidisciplinary understanding of the established sensitivity analyses, their applications are shown using three different simple to complicated and time-consuming examples. Then the results and implications of the sensitivity analyses are discussed.

2. Fuzzy Randomness and Sensitivity Analysis

2.1. Fundamentals

A model in an engineering problem is a set of rules and mathematical descriptions that are applied to map the variables from the input space onto the output space. In practice, there are uncertainties in both input variables (e.g., applied forces) and model parameters (e.g., structural properties), which in turn lead to uncertain output responses. Generally, there are two sources of uncertainty, which are stochastic uncertainty attributable to the intrinsic random nature of phenomena, and informal or epistemic uncertainty due to the imprecision or lack of data. Stochastic uncertainty has been properly handled using the classical probability theory; however, lack of data and imprecise probability distribution parameters are considered using fuzziness in the fuzzy random theory.

The fuzzy parameter $\tilde{A}$ is a set of ordered pairs that can be explained as,

$$\tilde{A}=\left\{\left(a,\lambda \left(a\right)\right)|a\in U\right\}$$

(1)

where the first entry $a$ is defined over the considered domain called fundamental set $U,$ and the second entry is the corresponding membership value $\lambda \left(a\right)$. The membership function $\lambda$ shows the uncertainty level and is a piecewise, convex, continuous function normalized to $\left[0,1\right]$. Technically, a parameter that is not fuzzy is called crisp in the nomenclature.

Another important parameter that is used in the fuzzy random theory is called α-level ${\alpha }_k$. The α-level set $A_{ {\alpha }_k}$ is a subset of fuzzy number elements, which has membership values greater than or equal to ${\alpha }_k$,

$$A_{{\alpha }_k}=\{a\in U|\lambda \left(a\right)\ge {\alpha }_k\}$$

(2)

Hence, each α-level of the fuzzy input parameters characterizes the range of variability for that parameter associated with that level of informal uncertainty. When the uncertainty level is higher and ${\alpha }_k$ value is smaller, the variability range of a fuzzy parameter is wider.

In the fuzzy randomness theory, there are two types of variables, fuzzy and fuzzy probabilistic variables. Uncertain parameters with non-statistical characteristics are quantified using fuzzy variables, as mentioned earlier. On the other hand, the stochastic parameters with imprecision are taken into account using fuzzy probabilistic variables. Fuzzy probabilistic variables are stochastic parameters whose distribution parameters, i.e., mean or standard deviation, are defined using fuzzy numbers for accounting for the imprecise probability distribution. In this study, the $j$th fuzzy variable is shown by ${\tilde{M}}_j$ and the $i$th fuzzy probabilistic variable and its fuzzy distribution parameter are denoted by ${\tilde{X}}_i$ and $\tilde{P}\left({\tilde{X}}_i\right)$, respectively.

Defining a distribution parameter as a fuzzy parameter in a fuzzy probability density function (PDF) $\tilde{f}\left(x_i\right)$ and the related fuzzy cumulative distribution function (CDF) $\tilde{F}\left(x_i\right)$ are shown in Figure 1. Each possible distribution is called trajectory and is included between the dashed and the solid lines, which are related to the membership values of $\lambda \left(x_i\right)=0$ and $\lambda \left(x_i\right)=1$, respectively. The membership function $\lambda \left(a\right)$ is usually estimated by a triangular function represented as $〈a_{min},a_{mean},a_{max}〉$, where $a_{min}$ and $a_{max}$ are the minimum and maximum values at the base of the triangle where $\lambda \left(a_{min}\right)=0$ and $a_{mean}$ corresponds to the maximum membership value $\lambda \left(a_{mean}\right)=1$.

The whole fuzzy analysis can be formulated as,

$$\widetilde{\underset{\_}{K}}\left({\tilde{M}}_j,\dots ,\tilde{P}\left({\tilde{X}}_i\right)\right)\to \widetilde{\underset{\_}{Z}}\left({\tilde{Z}}_1,\dots ,{\tilde{Z}}_n\right)$$

(3)

where $\widetilde{\underset{\_}{K}}$ is the fuzzy input vector consisting of fuzzy variables ${\tilde{M}}_j$ and fuzzy distribution parameters $\tilde{P}\left({\tilde{X}}_i\right)$ and it is mapped to the fuzzy result vector $\widetilde{\underset{\_}{Z}}$ with $n$ fuzzy output results ${\tilde{Z}}_n$. The mapping procedure $\to$ is considered as a general uncertain computational model and its fuzzy parameters herein are included in the input vector.

The most basic mapping procedures use Zadeh’s extension principle [39], which is explained first to provide a better insight. For this purpose, first the fuzzy input space $\tilde{K}$ is built as depicted in Figure 2a. The fuzzy input space is obtained by assigning the fundamental sets of fuzzy variables ${\tilde{M}}_j$ and fuzzy distribution parameters $\tilde{P}\left({\tilde{X}}_i\right)$ to the axes of the Cartesian coordinate system. Hence, each point in the fuzzy input space is a possible combination of elements of the fuzzy input parameters. The membership value for each point is the minimum membership value of its components.

Each point in the fuzzy input space gives one combination of considered fuzzy input variables. Each combination defines a common probability problem leading to the PDF of the output response, as depicted in Figure 2a, which is a trajectory of the fuzzy output PDF. By choosing a virtually infinite number of combinations from the fuzzy input space, the fuzzy PDF and fuzzy failure probability ${\tilde{P}}_f$ of the considered output response are determined. The membership value for each output result in the fuzzy output space is equal to the maximum membership of combinations leading to that response value. Failure probability is defined as the probability of exceeding an engineering demand parameter, which can be obtained using the cumulative distribution function. When there are fuzzy input parameters, the failure probability is also obtained as a fuzzy parameter, which is called fuzzy failure probability in this study.

The extension principle is computationally inefficient in high-dimensional spaces and the precision of the membership function is also highly dependent on the number of combinations from the fuzzy input space [10]. Thus, the α-level optimization is usually conducted, which is shown in Figure 2b. In the α-level optimization method, the range of output results at α-level ${\alpha }_k$ is obtained by considering the corresponding α-level ${\alpha }_k$ of input parameters as the search domain and find the maximum and minimum output response. The α-level subspace is built upon the intervals of fuzzy input parameters corresponding to that α-level. Hence, the fuzzy output response can be approximated by evaluating the fuzzy input parameters for a certain number of α-levels and finding their maxima and minima. By using the search algorithm, the maximum and minimum output responses related to that α-level are obtained, which is shown in Figure 2b. Unlike the extension principle, which is a brute-force method without any control over the characteristics of input parameters, the α-level optimization gives control to find the solution with the desired accuracy. In the extension principle, one needs to literally fill the input space with samples. On the contrary, it is possible to choose the number of α-levels in the α-level optimization, and their boundaries are obtained using an optimization algorithm, which reduces the computational effort. Further information about α-level optimization can be found in the literature [10,17].

2.2. Framework for Sensitivity Analysis of Fuzzy Probabilistic Models

The goal of this research is to quantify how the stochastic and informal uncertainties of fuzzy input parameters affect output responses. Based on the objective, computational effort, and the type of model, different sorts of sensitivity analyses can be performed. In classical probability theory, local sensitivity analyses usually fix all sources of stochasticity except one input variable at a time, and the output response is evaluated while propagating uncertainty from that parameter. Other parameters are usually fixed at their best estimates, which are mean or median. The range of the output response due to the stochasticity of each parameter is obtained as an interval. After finding the variability range of the output response due to each input parameter, the results of sensitivity analysis can be reported as the ordered obtained intervals or their relative influence.

Inspired by probability-based sensitivity analyses, a local sensitivity analysis for the fuzzy randomness theory is developed here. To this end, the informal uncertainty of fuzzy variables and fuzzy distribution parameters can be fixed to their best estimations using defuzzification. Defuzzification can be interpreted as a procedure, which reduces the dimension of fuzziness and gives the best crisp estimate for further use [10]. If a fuzzy variable is defuzzified, it works as a constant number at its best estimate. Thus, to evaluate the output uncertainty due to one parameter of interest at a time, the other fuzzy input variables can be fixed to their best estimates using defuzzification. On the other hand, fuzzy probabilistic variables can be fixed to their best estimates by defuzzifying distribution parameters and considering them equal to their mean values, similar to local sensitivity methods in probability theory.

In this research, the informal and stochastic uncertainties of all parameters except the one of interest are omitted, and the fuzzy response of the model is evaluated. Fuzzy variables are defuzzified, and fuzzy probabilistic variables are considered equal to defuzzified mean values and the analysis is done using one uncertain parameter at a time. Thus, the output variability and effects of stochastic and informal uncertainties from the considered parameter are obtained. This procedure is repeated for all input parameters and the influence of each parameter is determined and compared to each other.

2.3. Formulation of Sensitivity Analyses

To obtain the sensitivity indices, the aforementioned procedure can be formulated as follows. All fuzzy input parameters except the one of interest are defuzzified using the level rank method [10] and also fixed at their defuzzified means if they are fuzzy probabilistic variables. In this way, all fuzzy input parameters are treated such as deterministic variables and the fuzzy input space described earlier is built only using the fuzzy input parameter of interest. The same mapping procedure is applied and the output result will be a vector of sensitivity indices ${\tilde{S}}_r$ instead.

If the input variable of interest is a fuzzy variable ${\tilde{M}}_j$, the fuzzy input space is the same as the considered fuzzy variable. Each combination in this fuzzy input space leads to one deterministic problem. Then the effects of the fuzzy variable on the output response of the model is obtained as a fuzzy response (see Figure 3a). This can be expressed as,

$$\widetilde{\underset{\_}{K}}\left({\tilde{M}}_j,\dots ,p_i\right)\to {\widetilde{\underset{\_}{S}}}_j\left({\tilde{S}}_{1,j},\dots ,{\tilde{S}}_{n,j}\right)$$

(4)

where ${\tilde{S}}_{n,j}$ is sensitivity index of the $n$th output result ${\tilde{Z}}_n$ due to the $j$th fuzzy variable ${\tilde{M}}_j$ while other input parameters are fixed and defuzzified. This index shows the variability of the $n$th fuzzy output due to the uncertainty of the $j$th fuzzy input variable.

If the input parameter of interest is a fuzzy probabilistic variable ${\tilde{X}}_i$ as shown in Figure 3b, then the fuzzy input space is built upon its fuzzy distribution parameters $\tilde{P}\left({\tilde{X}}_i\right)$. Each point in the fuzzy input space defines a stochastic problem and it is a combination of the considered fuzzy probabilistic variable. The stochastic uncertainty due to this random variable is propagated and the output range and variability are obtained $\left[s_{n,i,l},s_{n,i,r}\right]$. Therefore, each combination in the fuzzy input space leads to an interval of the output response. By analyzing a virtually infinite number of combinations from the fuzzy input space and applying Zadeh’s extension principle [39], fuzzy sensitivity intervals are determined. This can be formulated as,

$$\widetilde{\underset{\_}{K}}\left(m_j,\dots ,\tilde{P}\left({\tilde{X}}_i\right)\right)\to {\widetilde{\underset{\_}{S}}}_i\left({\tilde{S}}_{1,i},\dots ,{\tilde{S}}_{n,i}\right)$$

(5)

which means that given the $i$th fuzzy probabilistic variable ${\tilde{X}}_i$ while others are fixed and defuzzified, the $n$th output result ${\tilde{Z}}_n$ is obtained, and it is considered as sensitivity index ${\tilde{S}}_{n,j}$. This index shows the variability of the $n$th output due to the uncertainty of the $i$th fuzzy probabilistic variable.

It should be emphasized that these fuzzy sensitivities are considered as local sensitivity analysis, which means that the interactions between the input variables is not taken into consideration. Each time, one uncertain parameter is considered and the output uncertainty is obtained using the MCS or is estimated using the FOSM or TDA. On the contrary, fuzzy global sensitivity [26] considers the interaction between the variables using the MCS, which is very time consuming.

2.4. Application of α-Level Optimization

The flow chart of local sensitivity analysis using α-level optimization is shown in Figure 4, which can be used along with α-levels shown in Figure 3 to better understand the procedure. If the input parameter is a fuzzy variable ${\tilde{M}}_j$, the interval $\left[m_{j,{\alpha }_k,l},m_{j,{\alpha }_k,r}\right]$ of the input parameter corresponding to the α-level ${\alpha }_k$ is considered as the search domain, as shown in Figure 3a. The maximum and minimum output responses $\left[s_{n,j,{\alpha }_k,l},s_{n,j,{\alpha }_k,r}\right]$ are determined while other parameters are defuzzified and fixed. If the input parameter is a fuzzy probabilistic variable, the intervals of its distribution parameters related to an α-level are considered as the search domain $\left[p_{i,{\alpha }_k,l},p_{i,{\alpha }_k,r}\right]$. Each point or combination gives one stochastic problem. The uncertainty is propagated for each combination and maximum and minimum output responses $\left[s_{n,i,l},s_{n,i,r}\right]$ for each combination are determined; these intervals are obtained for all combinations in the considered α-level. The lower bound and upper bounds of these intervals define the output variability for the corresponding α-level as,

$$\left[s_{n,i,{\alpha }_k,l},s_{n,i,{\alpha }_k,r}\right]=\left[min\left\{s_{n,i,l}\right\},max\left\{s_{n,i,r}\right\}\right]$$

(6)

3. Uncertainty Propagation Methods

To obtain the variability range of an output response $\left[s_{n,i,l},s_{n,i,r}\right]$ in a stochastic problem, there are different methods for uncertainty propagation or estimation; these methods are needed for obtaining the output variability range for each combination of fuzzy probabilistic variables. In this section, three well-known methods are explained and used in the development of fuzzy sensitivity analysis, namely, Monte Carlo simulation (MCS), tornado diagram analysis (TDA), and first-order second-moment method (FOSM). By using these methods, the variability range and interval of the output response $\left[s_{n,i,l},s_{n,i,r}\right]$ due to each combination of the considered fuzzy probabilistic variable can be obtained.

3.1. Monte Carlo Simulation

The simplest and the most commonly used method is MCS, which can easily handle different kinds of problems. Nevertheless, because of time-consuming analyses and simulations, this method is not always affordable and other methods such as FOSM or TDA are used.

As mentioned, each combination of the fuzzy input space defines a stochastic problem. Based on the distribution parameters of each combination in the fuzzy input space, random samples are generated and the output variability is measured in this method. The minimum number of samples is ensured by the convergence of the mean output response. Mean output response is obtained for different numbers of samples. The variability of this mean value by increasing the number of samples can be calculated using the coefficient of variation (CoV). By increasing the number of samples there will be a negligible change in the mean response, and the CoV of these mean values will be reduced. The convergence of the mean output response is ensured here with a CoV lower than 5% [20].

3.2. Tornado Diagram Analysis

TDA is the simplest method and can be efficiently used compared to more sophisticated methods such as global sensitivity analysis [35]. Instead of running MCS to find the output variability, the output response is directly estimated at low and high percentiles.

To this end, it is assumed that the low and high percentiles of the probabilistic input variable yield the two desired low and high percentiles of the output response. The range of the output response is determined given that the input variable equals ${\mu }_i\pm 1.5 {\sigma }_i$ or ${\mu }_i\pm 2 {\sigma }_i$, or the 10th and 90th percentiles, or any better estimation. Although this approximation is not always correct, this method is easy-to-use and can give reasonable estimates, which are completely justifiable by the significant reduction of the computational effort. In the present study the ranges of output responses are obtained using ${\mu }_i\pm 1.5 {\sigma }_i$ for each input parameter.

3.3. First-Order Second-Moment Method

In this method, the range of the output response for each combination of the fuzzy input space is estimated using the first-order Taylor series [21] instead of direct uncertainty propagation such as MCS. An output response $z_n=g\left(x_i\right)$, which is a function of a random variable can be approximated using its gradient ${\left(\frac{dg}{d{x}_i}\right)}_0$ and its value $g_0$ at $x_{i,0}$ as

$$z_n=g_0+{\left(\frac{dg}{d{x}_i}\right)}_0\left(x_i-x_{i,0}\right)$$

(7)

where $x_{i,0}$ is usually considered as the mean value of the random parameter $x_{i,0}={\mu }_{i,}$ and the distribution of $z_n$ is approximated by

$${\mu }_n\approx g\left({\mu }_i\right)$$

(8)

$${\sigma }_n\approx {\left(\frac{dg}{d{x}_i}\right)}_0{\sigma }_i$$

(9)

The gradient of the function is determined using the finite difference method and a minute perturbation about the mean value. Perturbation is defined as a small change in one input parameter, as in numerical differentiation. This can lead to a small change in an output result, which, divided by the perturbation size, gives the gradient or the numerical partial derivative. The perturbation size is usually indicated using a small fraction of the standard deviation ${\sigma }_i$, which can be in the order of 0.001. After estimating the mean and standard deviation of the output response, the output variability can be approximated by assuming a distribution function and the 10th and 90th percentiles for example, or by directly calculating ${\mu }_n\pm 2 {\sigma }_n$, ${\mu }_n\pm 1.5 {\sigma }_n$, or any other better approximation. In this research, the perturbation is considered to be $0.01$ and the output variability is obtained directly using ${\mu }_n\pm 1.5 {\sigma }_n$. Only in the last example is it considered that the output response follows a lognormal distribution and the 10th and 90th percentiles are calculated using the estimated mean and standard deviation from the FOSM method.

4. Examples

In this section, three examples are presented to show the application of the proposed sensitivity analyses in a multidisciplinary framework. By programming in MATLAB [40] required functions for fuzzy sensitivity analysis are provided to handle any type of problems. In these examples, the sensitivity analyses are performed for different levels of α-levels, and a mesh search technique is applied to each α-level subspace. To provide faster convergence, the α-level optimization and the MCS are conducted using the low-discrepancy Halton sampling method [41].

4.1. Numerical Example

The first example is a general mathematical problem from Saltelli et al. [18] based on the probability theory, which is slightly modified here for fuzzy probabilistic analysis. The output response ${\tilde{Z}}_1$ is the sum of three fuzzy probabilistic variables ${\tilde{X}}_i$ with uniform distributions. The problem is formulated as

$${\tilde{Z}}_1={\displaystyle \sum }_{i=1}^3{\tilde{X}}_i$$

(10)

$${\tilde{\mu }}_i=〈0.9\times 3^{i-1},3^{i-1},1.1\times 3^{i-1}〉$$

(11)

$${\tilde{\sigma }}_i=〈0.45, 0.50, 0.55〉\times 3^{i-1}$$

(12)

$$X_i ~ \mathrm{Uniform}\left({\mu }_i-{\sigma }_i,{\mu }_i+{\sigma }_i\right)$$

(13)

This fuzzified problem is the same as the one in the literature for the distribution parameters corresponding to the α-level of ${\alpha }_k=1$. Due to the linearity of the function, the output variability caused by each input parameter is evaluated at two α-levels ${\alpha }_k=0$ and $1$ using the three sensitivity analysis methods.

The fuzzy PDF and CDF of input parameters are demonstrated in Figure 5. By checking the convergence and CoV of the mean output response corresponding to $\lambda =1$, five hundred samples are chosen to conduct the MCS using the Halton low-discrepancy sampling method. The accuracy test results are shown in Figure 6. The fuzzy PDF and CDF of the output result are estimated using the Epanechnikov kernel function by 500 combinations and 500 samples from the fuzzy input space at $\lambda =1$ and they are demonstrated in Figure 7. Using the interval analysis, the closed-form range of the output result is between $\left[6.5, 19.5\right]$ at $\lambda =1$ and $\left[4.55, 21.45\right]$ at $\lambda =0$. The obtained results based on MCS are between $\left[6.88, 18.99\right]$ and $\left[5.16, 20.73\right]$ at $\lambda =1$ and 0, respectively, which are as expected.

For each input parameter fuzzy sensitivity analysis gives a fuzzy interval, which shows the output variability due to that input parameter; these results are shown for the MCS method in Figure 8. In conventional probabilistic analyses, the ranges of output variability due to the stochasticity of each parameter are arranged in descending order from top to bottom. Here the informal uncertainty can affect these ranges. In order to generalize the previous method, the variability in the boundaries of these ranges due to the informal uncertainty is visualized using a contour with respect to the membership value $\lambda$ as shown in Figure 9. This visualization method can help interpretation and a better understanding of the results. Thus, the current graph of fuzzy sensitivity analysis adds a second dimension to the conventional probability-based sensitivity analysis, depicting the effects of informal uncertainty on the output variability due to each parameter.

When there is no uncertainty and all input variables are defuzzified and fixed at their means, the output response is 13 and is shown in the graph by a vertical line. The output variability due to each parameter and their uncertainties can be seen as a range on both sides of this line. It can be observed that the output variability has a wider range when the informal uncertainty is increased. Parameter ${\tilde{X}}_3$ has the highest impact on the output variability and its informal uncertainty also has a considerable effect compared to the effects of the other parameters. The next significant parameters are ${\tilde{X}}_2$ and ${\tilde{X}}_1$, respectively. This is due to the fact that the model is sum of the three fuzzy probabilistic input parameters and the stochasticity is defined using the CoVs.

The results of the three uncertainty propagation methods are reasonably consistent in denoting the significance of input parameters. The MCS method directly propagates the uncertainty and then the variability is evaluated while TDA and FOSM estimate the range of variability. Compared to the results from MCS, TDA and FOSM method demonstrate a good accuracy. The results from the MCS method, which is the reference for the precision, are determined with a computational cost, 250 times higher than the TDA and FOSM methods. For each combination, the variability is calculated using 500 simulations for the MCS and 2 simulations using the TDA. For the FOSM method, one simulation is used to calculate the mean output response and one for perturbation. Considering the trade-off between computational efficiency and accuracy, FOSM and TDA can provide a good approximation. This can be quite efficient in terms of computational cost, especially for time-consuming nonlinear analysis. It is worthwhile mentioning that this problem is linear based on the closed-form solution, and the accurate output variability can be obtained by ${\mu }_i\pm \sqrt{3}{\sigma }_i$ as input parameters for TDA and ${\mu }_z\pm \sqrt{3}{\sigma }_Z$ for the FOSM method.

4.2. Ten-Bar Truss Example

The second example is an elastic plane truss structure shown in Figure 10, which consists of ten elements and has been frequently used in previous studies on reliability-based design [42,43,44]. The truss is an aluminum structure with six nodes and is subjected to two-point loads. The variables are elastic modulus $\tilde{E}$ and areas ${\tilde{A}}_i$ of the truss elements, and the point loads, ${\tilde{P}}_3$ and ${\tilde{P}}_5$. In the original version of the problem, all variables are statistically independent with a normal distribution and a CoV of $5\%$. The truss elements were designed optimally in such a way that the vertical downward deflection at node 5 is limited to $50.8 \mathrm{mm}$ and the failure probability corresponds to the reliability index of $\beta =3$ while the weight of the structure is minimized. Reliability index is defined as $\beta =-{\Phi }^{-1}\left(p_f\right)$ for the desired performance, where $p_f$ is the failure probability and ${\Phi }^{-1}$ is the inverse of the standard normal cumulative distribution function. In general, structures are designed for a reliability index of 3.0, i.e., $p_f=1.3\times {10}^{-3}$, while it is recommended to design common structures with a minimum reliability index of 3.8, i.e., $p_f=7.2\times {10}^{-5}$, according to the Eurocode [45].

Elements 2, 5, 6, and 10 were designed with the minimum area. The problem is fuzzified and accordingly modified in this study to demonstrate the application of the proposed sensitivity analysis for fuzzy probabilistic variables. The spatial variability is considered for different parameters except for the elastic modulus. The structural properties and the loads are listed in

Table 2. Nominal values of structural properties and loads for the ten-bar truss example.

Parameter	Fuzzy Mean Values	Distribution
Elastic modulus $\tilde{E}$ (MPa)	${\tilde{\mu }}_E=6.8\times {10}^4\times 〈0.95, 1.0, 1.05〉$	Normal
Area ${\tilde{A}}_i ({\mathrm{mm}}^2\times {10}^2)$	${\tilde{\mu }}_{A_i}=\left(221.6, 0.6, 191.5, 169.5, 0.6, 0.6, 21.5, 182.9, 168.6, 0.6\right)\times 〈0.9, 1.0, 1.1〉$
Loads ${\tilde{P}}_3$ and ${\tilde{P}}_5 \left(\mathrm{kN}\right)$	$444.8\times 〈0.95, 1.0, 1.05〉$

.

Fuzzification of parameters for real cases and existing structures shall be done based on the assessment of objective information to take into account the subjectivity. Herein, the parameters are considered as fuzzy probabilistic variables and they are hypothetically fuzzified to show the application of the proposed methods. The value corresponding to the membership value of $\lambda =1$ is the nominal value of the parameter. The output response is the vertical deflection at node 5 and the standard deviation of the parameters is defined as

$$nominal value of parameters\times 〈0.04, 0.05, 0.06〉$$

(14)

To show the distribution of the fuzzy probabilistic variables the elastic modulus, area of element 1, and load on node 5 are chosen as an example and are depicted in Figure 11.

To propagate the uncertainty to the numerical model and determine the deflections, the finite element code of the plane truss is provided in MATLAB. Similar to the previous example, the proper number of samples is determined first by running the MCS and increasing the number of samples when $\lambda =1$. The variation of the mean output value is evaluated and its convergence is checked. The Halton sampling method is applied and 500 samples are chosen based on the convergence test shown in Figure 12. The fuzzy PDF of the output response and the fuzzy CDF are estimated using 100 combinations from the fuzzy input space and the MCS with 500 samples, which are depicted for $\lambda =1$ and $\lambda =0$ in Figure 13. It is observed that the CDF at $\lambda =1$, consistent with the original version of the problem, confirms the optimal design of the truss, and the probability of vertical deflections exceeding $50.8 \mathrm{mm}$ is $p_f=0.002$ corresponding to $\beta =2.88$. Nevertheless, if the informal uncertainty exists, it can be quantified using the fuzzy randomness in the reliability-based design, and the fuzzy CDF provides a range of probabilities associated with the membership value.

Using the three mentioned methods and the mesh search technique with $100$ combinations at five α-levels, the fuzzy sensitivities are determined for each input parameters and compared in Figure 14. It is seen that the most influential parameter in the output uncertainty is the elastic modulus. Next influential parameters are, respectively, the load on node 5, the cross-sectional areas of elements 1, 8, 9, and 3, followed by the load on node 3 and the areas of elements 4 and 7. The areas of elements 2, 5, 6, and 10 designed with the minimum area have little or no influence in the response uncertainty. The effects of informal uncertainty can be seen on the range of the output response as contours. It is observed that this uncertainty has a considerable influence on the elastic modulus $\tilde{E}$ and the load at node 5, ${\tilde{P}}_5$.

The results from all three methods are consistent and comparable. The MCS shows a wider range of output response compared to the results from the FOSM and TDA analyses. FOSM and TDA show almost identical results yet with less computational effort compared to MCS. The MCS is conducted for the 13 variables and the sensitivities are determined at 5 α-levels using 100 combinations and 500 samples per combination, except when $\lambda =1$, which is obtained with one combination. Therefore, the total computational cost for the fuzzy sensitivities using the MCS is $2,606,500$ simulations; i.e., $13\times \left(4\times 100\times 500+1\times 1\times 500\right)=2,606,500$. On the other hand, TDA and FOSM require only 2 simulations per variable for each combination to estimate the output variability ranges of that variable. Therefore, the total computational cost for the TDA and FOSM methods are $10,426$ simulations, which is 250 times lower.

Owing to the computational efficiency of TDA and FOSM, the calculation of the fuzzy sensitivities using the MCS on a PC with the Intel ^® Core i7-7700k 4.2 GHz processor with a parallel pool of 30 workers is reduced from 42 h to only 5 min. To reduce the computational time drastically, it could also be possible to estimate the membership function of fuzzy sensitivity indices linearly from the interval at $\lambda =0$ to the one at $\lambda =1$ and conduct the analysis only for these two membership values.

4.3. Nonlinear Seismic Response of a Building Structure

To apply the proposed fuzzy sensitivity analysis to a real-case engineering problem, a case study structure is chosen from the full-scale experimental test of Negro and Verzeletti [46]. The structure was a four-story reinforced concrete moment frame shown in Figure 15 and its seismic performance was evaluated under a series of pseudo-dynamic tests. An artificial ground motion was imposed to the structure, which was generated using the Friuli earthquake record and the response spectrum with the peak ground acceleration (PGA) of $0.3$ g based on Eurocode 8 [47]. Since the test was conducted in one direction, the reference ground motion was multiplied by 1.5 to account for other sources of overstrength such as those for bi-directional actions and irregularity. Normal weight concrete C25/30 and the steel reinforcement with the characteristic yield strength of 500 MPa were used. The additional dead load of $2.0 \mathrm{kN}/{\mathrm{m}}^2$ representing the floor finishing and partitions, and the live load of $2.0 \mathrm{kN}/{\mathrm{m}}^2$ were considered. A more detailed explanation of the test, cross-sectional dimensions, and reinforcement details can be found elsewhere [48].

The structure is modeled using the Performance-based Earthquake Engineering (PBEE) toolbox [49], which uses the MATLAB environment for pre- and post-processing of models in conjunction with the OpenSees solver [50] for simulation. The beam-column elements are modeled by an elastic element and two inelastic rotational hinges at both ends defined by the moment-rotation relationship.

To have a precise simulation for sensitivity analysis, the accuracy of the analysis model is verified by comparing the analysis results with the experimental data. The compressive strength of concrete and the reinforcement detail of each element is considered identical to the as-built details. Zero viscous damping can be assumed in the nonlinear dynamic simulation of the pseudo-dynamic test [49]. However, to dissipate the free vibration after the seismic excitation and to keep the results of the simulation close to the experimental results, a low viscous damping ratio of 0.3% is considered. The reference ground motion along with the displacement of the top story from the experimental test and the analysis model is shown in Figure 16. It is seen that the analysis model of the structure can sufficiently simulate the nonlinear behavior of the structure under seismic loads.

The structural parameters of the analysis model are considered to be probabilistic based on the details of the experimental test and they are fuzzified to show the numerical procedure of the developed sensitivity analyses. The parameters are as listed in

Table 3. List of uncertainty parameters for the four-story structure.

Category	Parameters	$\mathit{\mu }$	$\mathit{\sigma }$	Distribution
Beam	${\tilde{f}}_{sy,b}$	$〈541, 570, 599〉$ MPa	$〈21, 28, 35〉$ MPa	Normal [51,52]
	${\tilde{f}}_{cm, b}$	$〈44, 46.8, 49〉$ MPa	$〈7.3, 7.5, 7.9〉$ MPa
Column	${\tilde{f}}_{sy,c}$	$〈541, 570, 599〉$ MPa	$〈21, 28, 35〉$ MPa
	${\tilde{f}}_{cm,c}$	$〈44, 46.8, 49〉$ MPa	$〈7.3, 7.5, 7.9〉$ MPa
Mass	$\tilde{M}$	$〈81.1, 85.4, 89.7〉$ ton	$8.5$ ton	Normal [21,52]
Seismic excitation	Bi-directional effects	---	---	$〈1.35, 1.5, 1.57〉$
	PGA	$〈0.167, 0.176, 0.185〉$ g	$〈0.095,0.099,0.102〉\mathrm{g}$	Lognormal [21]
	GM profile	GM record used in test $\lambda =1$	---	MCS [21]
		GM record suite $\lambda =0$

. All parameters are fuzzy probabilistic except the coefficient accounting for the bi-directional actions and the ground motion profile, which are considered as fuzzy variables. The compressive strength of concrete and the yield strength of steel reinforcement for the beams and the columns are considered without spatial variability.

The yield strength of steel and compressive strength of concrete are denoted by ${\tilde{f}}_{sy}$ and ${\tilde{f}}_{cm}$, respectively. Next indices b and c show whether they belong to beams or columns. Based on the literature, the yield strength of rebars follows a normal distribution with a CoV between $0.035$ and $0.065$ and it is recommended to use $0.05$ [51]. Thus, the standard deviation is fuzzified in such a way that the CoV of yield strength combinations are between $0.035$ and $0.065$ at $\lambda =0$ and it is equal to $0.05$ at $\lambda =1$. The standard deviation of the concrete compressive strength is fuzzified so that the variation is between $0.15$ and $0.18$ at $\lambda =0$ according to the previous studies [52] and $0.16$ at $\lambda =1$.

The mass $\tilde{M}$ is also considered as a fuzzy probabilistic variable without spatial variability whereas the loads are assumed to be constant [21]. The standard deviation of the mass was assumed as a crisp number equal to $8.5 \mathrm{tons}$, which corresponds to the CoV of $0.1$ at $\lambda =1$.

The PGA used for generating the target spectrum is considered as the intensity measure in this study. The mean value and the standard deviation of the PGA with the lognormal variability is determined in such a way that $0.3 \mathrm{g}$, the reference peak ground acceleration for the design basis earthquake (DBE) level, corresponds to the $10\%$ exceedance probability in 50 years and $3/2$ times the DBE corresponds to the maximum considered earthquake (MCE) level with a $2\%$ exceedance probability in 50 years. The fitted annual exceedance rate of different seismic intensities with this assumption is shown in Figure 17a which is called the hazard curve.

The ground motion (GM) profile is also an important aspect in the seismic performance of structures. The FEMA 273 guideline [53] for the seismic rehabilitation of buildings recommends that if seven earthquake records or more are utilized, using the average response for performance evaluation is permitted. To account for the frequency content of ground motions, six additional earthquake records are obtained from the PEER NGA database [54] to have seven ground motion records in total, considering the reference ground motion. It is possible to scale the records so that the mean acceleration spectrum of them matches the response spectrum of Eurocode 8 with the PGA of $0.3 \mathrm{g}$. However, in order to treat and evaluate the seismic intensity and the ground motion variability separately, six artificial earthquake records are generated and matched individually to the reference response spectrum using the SeismoMatch [55] software. Hence, the variability in the spectral acceleration is kept constant and the ground motion profiles only account for the frequency content as far as possible. The reference spectrum for the design of the structure according to Eurocode 8 and the response spectra of the other seven earthquakes are shown in Figure 17b.

Since the simulation is nonlinear time-history analysis and much more time-consuming compared to the previous examples, the membership functions of the fuzzy sensitivity indices are estimated using a triangular function as pointed out at the end of the second example. Hence, the ground motion profile is defined as a discrete fuzzy input parameter in such a way that when $\lambda =1$ the ground motion corresponds to the Friuli earthquake used in the test and when $\lambda =0,$ there are 7 earthquake records. Therefore, it should be noted that when $\lambda =1$, the parameters with the design basis earthquake intensity level correspond exactly to the condition of the experimental test. For the defuzzification of the ground motion profile effect, the benchmark structure is analyzed in a deterministic manner under the seven ground motion records. The RSN719-Superstition Hills earthquake record, which leads to the median MIDR of 2.43% is considered as the defuzzified ground motion profile. Except for the ground motion profile, the distributions of the input variables are depicted in Figure 18.

The number of samples is determined using the convergence test such as the previous examples. The convergence test is conducted at $\lambda =1$ and the results are shown in Figure 19. The parameter of interest for the performance evaluation of structures is usually the maximum inter-story drift ratio (MIDR) which is the maximum relative displacements between two consecutive stories divided by the height of the story. The convergence of the mean MIDR for the combination at $\lambda =1$ can be noticed in Figure 19. The CoV is limited to $5\%$ using 278 samples and the mean MIDR converges at the onset of this number of samples. Beyond this number of samples, it leads to considerable computational effort with slight improvements in the overall distribution of the output response.

Using nonlinear time history analysis in uncertainty quantification is quite time-consuming, and therefore a PC with Intel^® 12-core Xeon E5-2670 2.3 GHz processor with 80 parallel simulations is employed to reduce the computational time. Unlike previous examples with a simulation in a fraction of a second, in this example it takes from $60 \mathrm{s}$ to $511 \mathrm{s}$ with an average value of $199 \mathrm{s}$, depending on the earthquake record. In order to further reduce the computational time, 20 combinations are also used for the mesh search technique in the α-level subspace corresponding to $\lambda =0$. Additionally, each combination in the fuzzy input subspace of $\lambda =0$ corresponds randomly to one of the earthquake records using the Halton sampling method, rather than using a combinatorial approach.

By applying the previous techniques, the fuzzy probabilistic output response is estimated and the fuzzy CDF of MIDRs is presented in Figure 20, which demonstrates the distribution of the MIDRs and the seismic performance of the structure in its 50-year lifetime. The exceedance probability of MIDR can be obtained from the fuzzy CDF. Hence, it is possible to calculate the reliability index.

Sensitivity analysis of the 6 fuzzy probabilistic input parameters using the MCS is performed with 20 combinations at $\lambda =0$, one combination at $\lambda =1$, and 278 samples per combination. This requires a total number of $6\times \left(20+1\right)\times 278=35,028$ nonlinear time history analyses. The two other fuzzy parameters need only one simulation per combination. Overall, the fuzzy sensitivity analysis using MCS takes around 144 h on a parallel pool of 80 workers with the aforementioned processor, which means 80 simultaneous simulations running. Without the application of the parallel computing technique, the MCS could take one year. On the contrary, the TDA and the FOSM methods require around 16 and 12 h, respectively, without any parallel simulation. The results of the sensitivity analyses are depicted in Figure 21.

When all parameters are defuzzified and fixed at their means, the MIDR is 1.1%. The sensitivities are ordered based on the MCS results as the most accurate reference. As mentioned in Section 3.3, the output variability in this example is obtained by considering a lognormal distribution for the output response and calculating the 10th and 90th percentiles using the estimated mean and standard deviation from the FOSM method. By using the provided tool, it is conceived how the informal uncertainty in the input parameters can affect the output uncertainty corresponding to that variable. The effects of informal uncertainty are shown using contours affecting the range of the output stochasticity due to each input variable.

It can be observed that the TDA results are closer to the MCS in both stochastic and informal uncertainties compared to the FOSM. The most influential parameter is the seismic intensity PGA followed by the mass $M$ and the compressive strength of concrete in beams ${\tilde{f}}_{cm,b}$ and columns ${\tilde{f}}_{cm,c}$. The effects of ground motion profile, yield strength of beam reinforcements ${\tilde{f}}_{sy,b}$, bi-directional effects coefficient, and yield strength of column reinforcements ${\tilde{f}}_{sy,c}$ come next.

The results from the sensitivity analysis are compatible with the current practice in the seismic engineering where probabilistic analysis of structures, such as fragility analysis, is performed considering only variability in ground motions and their intensities. Given the seismic intensity, fragility analysis determines the failure probability of the structure meeting an objective performance. Fuzzy fragility analysis [21] is the same analysis considering fuzzy parameters, which leads to fuzzy failure probabilities under different seismic intensity. This is due to the much smaller effects of other uncertain parameters compared to the PGA. Herein, the effects of ground motion profile are small because the earthquake records are matched to the same reference spectrum. The output variability due to the PGA is $\left[0.2\%, 8.2\%\right]$ at $\lambda =0$ while it is $\left[0.7\%, 1.5\%\right]$ for the next most influential parameter, which is the mass. This shows the quantitative effects of uncertainty in these parameters.

Except for the large difference in the sensitivity of PGA, the TDA and FOSM methods can capture the output variability in a reasonable way, while the effects of informal uncertainty are better approximated using the TDA. The difference in sensitivities of PGA is due to the large deformations and partial collapse of the structure for some realizations, which in turn shows a large MIDR. Generally, drift ratios larger than 3% to 5% in analysis models are larger than collapse limit states and the number itself may not be an accurate estimate or meaningful. It can be smaller or much larger due to the partial collapse. As a result, it is hard to precisely estimate the output variability for highly nonlinear problems using the efficient methods. Nevertheless, the relatively large MIDRs from the FOSM and TDA imply this fact, which can be taken into consideration beforehand.

Overall, it is observed that the formulated fuzzy sensitivity analyses and the visualization method can give a detailed insight into the behavior of fuzzy random models. Furthermore, the provided efficient methods can drastically reduce the computational time which is of paramount importance for time-consuming nonlinear analysis in structural engineering.

5. Conclusions

The aim of this research is twofold; developing sensitivity analysis procedure for models with fuzzy and fuzzy probabilistic variables to quantify the effects of stochastic and informal uncertainties in input variables on the output response, meanwhile providing computational efficiency for the highly time-consuming sensitivity analyses in real-case engineering problems.

To this end, the concept of fuzzy sensitivity analysis was established first and three methods based on the conventional probability theory were extended to fuzzy probabilistic models; these methods are namely, Monte Carlo simulation (MCS), Tornado diagram analysis (TDA), and first-order second-moment (FOSM). The multidisciplinary applications of the developed analyses were carried out using three examples, a simple numerical example, a benchmark elastic truss structure, and finally a real-case full-scale experimental test of a four-story structure under seismic loads.

It was observed that the formulated methods could give a detailed insight into quantification of output uncertainty due to the fuzzy random nature of input parameters. The visualization method inspired by the classical local sensitivity analyses were extended to the provided methods. It was demonstrated that the efficient TDA and FOSM methods required significantly lower computational effort compared to the MCS while there was a trade-off between the computational efficiency and accuracy. The accuracy of efficient methods could be lower for nonlinear cases; however, it is a quite acceptable trade-off for reducing the analysis time from several days to couple of hours.