# Uncertainty Evaluation of Stochastic Structural Response with Correlated Random Variables

Authors: Qiang Fu; Jianjun Liu; Jiarui Shi; Xiao Li; Xueji Cai; Zilong Meng
Journal: Shock and Vibration
 (2022)

---

## Abstract

It has been realized that the influence of system parameter uncertainties may be very significant, even dominant, in stochastic response evaluation. Nevertheless, in reality, this evaluation process may be difficult to conduct due to these parameter variables (viz. structural property parameters, such as stiffness, damping, and strength, and excitation characteristics parameters, such as frequency content and duration) that are usually correlated with each other. Therefore, this study devotes to develop a method for evaluating stochastic response uncertainty involving correlated system parameter variables. In this method, the evaluation expression for the mean and standard deviation of the maximum response including uncertainty parameter variables are provided first; subsequently, a third-moment pseudo-correlation normal transformation is able to be performed for converting the correlated and non-normal system parameter variables with unknown joint probability density function (PDF) or marginal PDF into the mutually independent standard normal ones; ultimately, a point estimate procedure (PEP) based on univariate dimension reduction integration can be carried out for evaluating the structural stochastic response including uncertainty system parameters. Several numerical examples with an engineering background involving correlated system parameter variables are analyzed and discussed under stochastic excitation, and their results are compared with those yielded by Monte Carlo simulation (MCS) so as to demonstrate the effectiveness of the approach proposed. It indicated that the method proposed, in this study, provides an effective path to deal with uncertainty evaluation of stochastic structural response involving correlated random variables.


---

## Body

## 1. Introduction

Stochastic response analysis is crucially significant for structural performance evaluation and probabilistic risk assessment [1–6]. Nevertheless, to get a structural realistic and accurate evaluation in practical engineering is always challenging due to the uncertainties that are generally considered primarily to be limited to stochastic excitation, in fact, the effect of uncertainty system parameters, which include structural property parameters (such as stiffness, mass, and intensity) and excitation characteristic parameters (such as spectrum and duration) on response evaluation that is equally important, even dominate [7–12]. Besides, it also makes evaluation difficult owing to these parameters that usually are mutually correlated in reality [13–16]. Thus, what the present study mainly attempts is to evaluate stochastic uncertainty response involving correlated random variables.To evaluate structural stochastic uncertainty response, a lot of effort has been put into it by scholars and many celebrated works are able to be traced toward this topic. Generally, the uncertainty response evaluation can be classified into two categories: the time-domain analysis and the frequency-domain analysis [17–19]. In the frequency-domain analysis, the direct integration method and Monte Carlo simulation, two accurate evaluation methods, are adopted by some scholars in a linear single degree of freedom; however, when the dynamic system is complex or high safety, those methods are difficult to repeatedly apply on account of that plenty of complicated analysis, such as eigenvalue analysis, computation of participation factor, and solution of linear equations, which are necessary, especially nonlinear system [20, 21]. Additionally, the response surface method is also capable to be performed to evaluate the uncertainty response for avoiding the difficulties in the computation of derivatives; however, some studies found that accurate results yielded by this approximation method are mainly depending on the location of the fitting points [10, 15, 22]. Other scholars [23, 24] found the first-order approximate of mean response for uncertain linear systems, which is obtained by applying Taylor series expansion, which is coincident with the given result; thus, this mean solution is treated to equivalent uncertain stochastic response. However, other research studies [3, 10, 25] are realizing the greater the coefficient of variance, the greater the error between the mean and truth value of uncertainty response; therefore, the variance of response is taken into account for improving the mean result. In the time-domain analysis, their analysis procedure and conclusion of applying these methods mentioned above are fundamentally similar to the frequency domain except that the first two moments of the response are obtained by statistic counting all the response results, which are obtained by multiple time-dependent inputs acting on the structure [26].In summary, no matter what domain is used, the uncertainty response expressions are capable to be approximated with respect to the mean values of parameter variables by utilizing Taylor series expansion [1, 2]. Nevertheless, the Taylor series expansion method is requiring sensitivity analyses of response, which involves several complex calculation procedures, such as eigenvalue analysis, the computation of participation factor and spectral moment, and inverse distribution function; besides, the result is also not ideal when this method has been applied in the nonlinear system; what’s more, the method only considers the influence of structural property parameters on the uncertain response under determinate multiple inputs; however, the effect of excitation characteristic parameters on the uncertainty response is equally significant [10, 25].Specifically, Zhao et al. [10] propose a point estimation procedure (PEP), which is evaluating random response related to system uncertainty parameter by several repetitions of random response analysis under determination parameters, to directly evaluate the first two order moments of the structural maximum response distribution including uncertainty system parameters without any sensitivity analysis. The stochastic response of a 15-floor nonlinear structure, considering four structural property parameters and three characteristic parameters, in his paper, is steadily evaluated. What their result reveals is that PEP, a conceptually more straightforward and computationally effective method, can avoid the three main difficulties mentioned above form the series expansion method. However, in his paper, while applying the PEP, the system parameter variables are viewed as mutually independent ones. As a matter of fact, most of the system parameter variables, in actual engineering, are usually involving correlation [27, 28]. Therefore, prior to adopting the PEP, it is necessary to convert partially correlated system parameter variables into independent ones.In response to this difficulty, in general, the Rosenblatt transformation is a classical method to implement the aforementioned correlated transformation process [29]; however, different results may be yielded if the integral order with respect to system parameters variables is exchanged; furthermore, the premise of this method applied is that the information for joint PDF of system parameter variables needs to remain completeness [30]. Thus, the Rosenblatt transformation, in a certain situation, is just an ideal method that is not easy to be implemented. On the other hand, if partial information about joint PDF of system parameter variables is capable to be acquired, the Nataf transformation, which just requires the marginal PDFs and correlation matrix of system parameter variables, is a useful alternative approach available to handle those variables related to correlation [16, 30–32]. Nevertheless, in engineering practice, both the methods above may not be achieved, if encountering the entire information for PDF of system parameter variables can scarcely be obtained except their statistical moments and correlation matrix. Under the circumstances, Lu et al. [33] recently provided a third-moment pseudo-correlation normal transformation, with the aid of the first three statistic moments (mean, standard deviation, and skewness) and correlation matrix of system parameter variables, for realizing the conversion process about correlated and non-normal system parameter variables with unknown joint PDF and marginal PDFs to mutual independent standard normal ones. However, at present, applying this effective transformation technique for evaluating stochastic uncertainty response involving correlated system parameter variables has not been investigated yet.In this study, based on the studies above, the main objective is to extend the point estimation procedure based on a third-moment pseudo-correlation normal transformation for evaluating stochastic uncertainty response involving correlated system parameter variables. The remainder of this study is organized as follows: in Section2, the statistical analytical expression of stochastic response evaluations with determination parameters is reviewed first; on this basis, stochastic response with independent system parameters can be evaluated utilizing the PEP. In Section 3, a third-moment pseudo-correlation normal transformation is introduced into PEP, for solving the correlated and non-normal system parameter variables to mutual independent and standard normal ones. In Section 4, the flowchart and main calculative procedure for evaluating stochastic response with uncertainty parameters are illustrated. It is then followed by Section 5, in which several examples, which involve the response uncertainty evaluation with correlated system parameter variables, are analyzed and discussed utilizing the presented approach. Eventually, the conclusions of this study are presented in Section 6.


## 2. Stochastic Response Evaluation with Uncertainty Parameters

To evaluate the structural uncertainty response, in general, is not easy on account of the structural stochastic responseR(t) cannot be explicitly descripted. From another perspective, its maximum response value, i.e., Rmax, is a random variable; then, the structural uncertainty response can be conveniently evaluated on the condition that the statistical characteristic values of Rmax, several concrete analytical expressions, are able to be known.


### 2.1. Stochastic Response Evaluation under Deterministic Parameters

The PDF of maximum peak value for zero-mean normal stationary random processR(t), in time period T, does not exceed a certain threshold that is considered to approximately obey the Poisson distribution [34]. According to Davenport’s equation, its mean value and standard deviation can be obtained as follows:(1)μm=2InνT+0.57722InνTσR.(2)σm=π12InνTσR,where µm and σm denote the mean value and standard deviation of the maximum response, respectively; T denotes the duration of stochastic excitation; v denotes the mean cross ratio; and it can be calculated by the following equation:(3)ν=12πσR˙σR.In equations (1)–(3), σR denotes the standard deviation of the stochastic response, which can be acquired from the state-of-the-art techniques, e.g., those based on random process and vibration theory, and σR˙ is corresponding derivative. For a linear single or multiple degrees of freedom system, σR is capable to be obtained based on the mode decomposition method, in which its main calculation principles are as follows [1, 2]:(4)Syyω=Hω∗SxxωHωT.(5)Syxω=Hω∗Sxxω.(6)Sxyω=SxxωHωT.Equations (4)-(6) represent when the linear system is subjected to the stationary random excitation {x(t)}, whose power spectrum matrix is [Sxx(ω)]; then, the power spectrum matrix of system response {y(t)} can be expressed as [Syy(ω)], and its mutual power spectrum matrix is able to be expressed as [Syx(ω)] and [Sxy(ω)], respectively. [H(ω)] denotes the response matrix in the frequency domain. In addition, for a nonlinear single or multiple degrees of freedom system, the equivalent linearization method is recommended to calculate response standard deviation σR [3–5].


### 2.2. Stochastic Response Evaluation under System Parameter Uncertainties

In the structural stochastic response evaluation described in the previous section, these system parameters, including structural property parameters and excitation characteristics parameters, are premised to be given. While these system parameters are considered to treat as a group of random variablesX, then, the maximum uncertainty response of the structure, Rmax, can be described as a function of X:(7)Rmax=RmX.Generally, the maximum response variableRmax cannot be explicitly descripted by random vibration theory, except it is the mean μm and standard deviation σm. Therefore, this study’s primary aim is to evaluate structural uncertainty response by adopting the statistics of maximum response variable Rmax, i.e., equations (1)-(2).While the uncertainty system parameters are described by random variablesX, the expression of mean μm and standard deviation σm become a function about X, i.e., µm(X) and σm(X), respectively. For a group of determination values of X = x, the conditional mean and standard deviation of maximum response variables can be evaluated by directly substituting equations (1)-(2), i.e., µm(X = x) and σm(X = x). Subsequently, when system parameters X are considered as uncertainty random variables, the overall uncertainty response, i.e, mean µM and standard deviation σM, can be evaluated by integrating over the whole area of X. The entire evaluation procedure mentioned above is shown in the following equations.In the first step, the conditional mean of maximum responseµm(X) and its standard deviation σm(X) can be evaluated as follows [35]:(8)μmX=∫RmXfRm|XdRm.(9)σm2X=∫RmX−μmX2fRm|XdRm.Subsequently, integrating over the whole area of system parameter variablesX, the overall mean of maximum response µM and its standard deviation σM can be obtained [10]:(10)μM=∫μmXfXdX=EμmX.(11)σM2=∫RmX−μMX2fRm|XfXdRmdX=Eμm2X+Eσm2X−μM2.It is worth noting that the result ofE[μm2(X)] is not equal to μM2 owing to different integral variables; similarly, the result of E[σm2(X)] is not equal to σM2; in addition, the conditional probability density function f(Rm|X) in equations (8)-(11) is no need to be evaluated due to it is just facilitating description about the whole assessment process.As long as the results ofE[µm(X)], E[μm2(X)] and E[σm2(X)] can be obtained, the structural uncertainty response, in terms of equations (8)-(11), is able to be stably evaluated. Based on this view, in order to facilitate description and comprehension of PEP to be applied for uncertainty response evaluation in subsequent studies, supposing R is just a symbol, and R represents any one of µm, µm2, and σm2. Then, while E[µm(X)], E[μm2(X)] and E[σm2(X)] are capable to be written in an uniform form as shown in equation (12).(12)ERX=∫RXfXdX.in which R(X) denotes conditional maximum response statistics, i.e., any one of µm(X), µm2(X), and σm2(X); X denotes a set of arbitrary system parameter variables.


### 2.3. Point Estimation Procedure Based on Univariate Dimension Reduction Integration

In this section, the PEP can be conducted for evaluating stochastic uncertainty response. By observing equation (12), it can be found that the essence of assessing the uncertain stochastic response is to evaluate the expectation of R(X), in which X represents a set of arbitrary system parameter vectors.In general, evaluating this expectation process based on PEP is capable to be divided into two steps [36, 37]: To begin with, in order to avoid a large number of calculations, the univariate dimension reduction integration is first performed for approximating conditional maximum response statistics R(X) as a series of sum functions, in which function R(X) just includes one random variable Xi as follows:(13)RX≅∑i=1nRi−n−1Rμ,where(14)Ri=RXi=RT−1Ui.(15)Rμ=Rμ.in which Xi =  μ1X,…,μi−1X,Xi,μi+1X,…,μnXT; μiX denotes the mean of the ith term of R(X) in the original space. Note that standard vertical character Xi represents a vector, in which this vector is consisted of an independent random variable, i.e., italic character Xi and the mean of all the remaining arbitrary variables; similarly, Ui = μ1U,…,μi−1U,Ui,μi+1U,…,μnUT; μiU denotes the mean of the ith term of R(U) in the standard normal space; and standard vertical character Ui represents a vector, in which this vector is consisted of an independent standard normal random variable, i.e., italic character Ui and the mean of all the remaining standard normal variables. T−1 denotes the inverse of the standard normal space; μ denotes the vector whose system parameter variables take their corresponding mean value.In the second place, the mean of conditional maximum response statisticsR(X), i.e., overall maximum response statistics E[R(X)], can be evaluated by using direct PEP with regard to Ri in equation (14). It can be formulated as follows:(16)ERX≅∑i=1nμRi−n−1Rμ,in which μRi represents a function of R(X) with only one system parameter variable Xi, in which ith term of R(X) is independent parameter variable Xi, and the remaining terms of R(X) take the corresponding mean of parameter variables. R(μ) represents all system parameter variables of R(X) that take their corresponding mean values; then, μRi in equation (16) can be steadily directly evaluated by using PEP as follows:(17)μRi=ERXi=ERT−1Ui=∑k=1mPkRT−1ui,k,where ui,k denotes the estimation value for adopting k-point estimation at the ith coordinate position in the standard normal space, and Pk is their corresponding weight. They can be readily obtained by using the following equation (38):(18)ui,k=2xk,Pk=wkπ.in which xk and wk are the abscissas and weights for Hermite integration with the weight function exp(−x2) that can be found in Abramowitz and Stegum. In particular, five-point and seven-point estimates in standard normal space are commonly used in practice; thus, their corresponding estimating point and weight are given in Tables 1 and 2 respectively.Table 1
Five-point estimate corresponding the estimating point and weight.
Estimation pointWeightsFive-point estimateu0 = 0p0 = 0.53333u1+ = −u1− = 1.35563p1+ = p1− = 0.22208u2+ = −u2− = 2.85697p2+ = p2− = 1.12574 × 10–2Table 2
Seven-point estimate corresponding the estimating point and weight.
Estimation pointWeightsSeven-point estimateu0 = 0p0 = 0.45714u1+ = −u1− = 1.15441p1+ = p1− = 0.24012u2+ = −u2− = 2.36676p2+ = p2− = 0.03076u3+ = −u3− = 3.75044p3+ = p3− = 5.48269 × 10–4


## 2.1. Stochastic Response Evaluation under Deterministic Parameters

The PDF of maximum peak value for zero-mean normal stationary random processR(t), in time period T, does not exceed a certain threshold that is considered to approximately obey the Poisson distribution [34]. According to Davenport’s equation, its mean value and standard deviation can be obtained as follows:(1)μm=2InνT+0.57722InνTσR.(2)σm=π12InνTσR,where µm and σm denote the mean value and standard deviation of the maximum response, respectively; T denotes the duration of stochastic excitation; v denotes the mean cross ratio; and it can be calculated by the following equation:(3)ν=12πσR˙σR.In equations (1)–(3), σR denotes the standard deviation of the stochastic response, which can be acquired from the state-of-the-art techniques, e.g., those based on random process and vibration theory, and σR˙ is corresponding derivative. For a linear single or multiple degrees of freedom system, σR is capable to be obtained based on the mode decomposition method, in which its main calculation principles are as follows [1, 2]:(4)Syyω=Hω∗SxxωHωT.(5)Syxω=Hω∗Sxxω.(6)Sxyω=SxxωHωT.Equations (4)-(6) represent when the linear system is subjected to the stationary random excitation {x(t)}, whose power spectrum matrix is [Sxx(ω)]; then, the power spectrum matrix of system response {y(t)} can be expressed as [Syy(ω)], and its mutual power spectrum matrix is able to be expressed as [Syx(ω)] and [Sxy(ω)], respectively. [H(ω)] denotes the response matrix in the frequency domain. In addition, for a nonlinear single or multiple degrees of freedom system, the equivalent linearization method is recommended to calculate response standard deviation σR [3–5].


## 2.2. Stochastic Response Evaluation under System Parameter Uncertainties

In the structural stochastic response evaluation described in the previous section, these system parameters, including structural property parameters and excitation characteristics parameters, are premised to be given. While these system parameters are considered to treat as a group of random variablesX, then, the maximum uncertainty response of the structure, Rmax, can be described as a function of X:(7)Rmax=RmX.Generally, the maximum response variableRmax cannot be explicitly descripted by random vibration theory, except it is the mean μm and standard deviation σm. Therefore, this study’s primary aim is to evaluate structural uncertainty response by adopting the statistics of maximum response variable Rmax, i.e., equations (1)-(2).While the uncertainty system parameters are described by random variablesX, the expression of mean μm and standard deviation σm become a function about X, i.e., µm(X) and σm(X), respectively. For a group of determination values of X = x, the conditional mean and standard deviation of maximum response variables can be evaluated by directly substituting equations (1)-(2), i.e., µm(X = x) and σm(X = x). Subsequently, when system parameters X are considered as uncertainty random variables, the overall uncertainty response, i.e, mean µM and standard deviation σM, can be evaluated by integrating over the whole area of X. The entire evaluation procedure mentioned above is shown in the following equations.In the first step, the conditional mean of maximum responseµm(X) and its standard deviation σm(X) can be evaluated as follows [35]:(8)μmX=∫RmXfRm|XdRm.(9)σm2X=∫RmX−μmX2fRm|XdRm.Subsequently, integrating over the whole area of system parameter variablesX, the overall mean of maximum response µM and its standard deviation σM can be obtained [10]:(10)μM=∫μmXfXdX=EμmX.(11)σM2=∫RmX−μMX2fRm|XfXdRmdX=Eμm2X+Eσm2X−μM2.It is worth noting that the result ofE[μm2(X)] is not equal to μM2 owing to different integral variables; similarly, the result of E[σm2(X)] is not equal to σM2; in addition, the conditional probability density function f(Rm|X) in equations (8)-(11) is no need to be evaluated due to it is just facilitating description about the whole assessment process.As long as the results ofE[µm(X)], E[μm2(X)] and E[σm2(X)] can be obtained, the structural uncertainty response, in terms of equations (8)-(11), is able to be stably evaluated. Based on this view, in order to facilitate description and comprehension of PEP to be applied for uncertainty response evaluation in subsequent studies, supposing R is just a symbol, and R represents any one of µm, µm2, and σm2. Then, while E[µm(X)], E[μm2(X)] and E[σm2(X)] are capable to be written in an uniform form as shown in equation (12).(12)ERX=∫RXfXdX.in which R(X) denotes conditional maximum response statistics, i.e., any one of µm(X), µm2(X), and σm2(X); X denotes a set of arbitrary system parameter variables.


## 2.3. Point Estimation Procedure Based on Univariate Dimension Reduction Integration

In this section, the PEP can be conducted for evaluating stochastic uncertainty response. By observing equation (12), it can be found that the essence of assessing the uncertain stochastic response is to evaluate the expectation of R(X), in which X represents a set of arbitrary system parameter vectors.In general, evaluating this expectation process based on PEP is capable to be divided into two steps [36, 37]: To begin with, in order to avoid a large number of calculations, the univariate dimension reduction integration is first performed for approximating conditional maximum response statistics R(X) as a series of sum functions, in which function R(X) just includes one random variable Xi as follows:(13)RX≅∑i=1nRi−n−1Rμ,where(14)Ri=RXi=RT−1Ui.(15)Rμ=Rμ.in which Xi =  μ1X,…,μi−1X,Xi,μi+1X,…,μnXT; μiX denotes the mean of the ith term of R(X) in the original space. Note that standard vertical character Xi represents a vector, in which this vector is consisted of an independent random variable, i.e., italic character Xi and the mean of all the remaining arbitrary variables; similarly, Ui = μ1U,…,μi−1U,Ui,μi+1U,…,μnUT; μiU denotes the mean of the ith term of R(U) in the standard normal space; and standard vertical character Ui represents a vector, in which this vector is consisted of an independent standard normal random variable, i.e., italic character Ui and the mean of all the remaining standard normal variables. T−1 denotes the inverse of the standard normal space; μ denotes the vector whose system parameter variables take their corresponding mean value.In the second place, the mean of conditional maximum response statisticsR(X), i.e., overall maximum response statistics E[R(X)], can be evaluated by using direct PEP with regard to Ri in equation (14). It can be formulated as follows:(16)ERX≅∑i=1nμRi−n−1Rμ,in which μRi represents a function of R(X) with only one system parameter variable Xi, in which ith term of R(X) is independent parameter variable Xi, and the remaining terms of R(X) take the corresponding mean of parameter variables. R(μ) represents all system parameter variables of R(X) that take their corresponding mean values; then, μRi in equation (16) can be steadily directly evaluated by using PEP as follows:(17)μRi=ERXi=ERT−1Ui=∑k=1mPkRT−1ui,k,where ui,k denotes the estimation value for adopting k-point estimation at the ith coordinate position in the standard normal space, and Pk is their corresponding weight. They can be readily obtained by using the following equation (38):(18)ui,k=2xk,Pk=wkπ.in which xk and wk are the abscissas and weights for Hermite integration with the weight function exp(−x2) that can be found in Abramowitz and Stegum. In particular, five-point and seven-point estimates in standard normal space are commonly used in practice; thus, their corresponding estimating point and weight are given in Tables 1 and 2 respectively.Table 1
Five-point estimate corresponding the estimating point and weight.
Estimation pointWeightsFive-point estimateu0 = 0p0 = 0.53333u1+ = −u1− = 1.35563p1+ = p1− = 0.22208u2+ = −u2− = 2.85697p2+ = p2− = 1.12574 × 10–2Table 2
Seven-point estimate corresponding the estimating point and weight.
Estimation pointWeightsSeven-point estimateu0 = 0p0 = 0.45714u1+ = −u1− = 1.15441p1+ = p1− = 0.24012u2+ = −u2− = 2.36676p2+ = p2− = 0.03076u3+ = −u3− = 3.75044p3+ = p3− = 5.48269 × 10–4


## 3. The Third-Moment Pseudo-Correlation Normal Transformation for Correlated and Non-Normal Variables into Independent Normal Ones

In Section2.3, while PEP is conducted, the system parameter variables X require to be treated as a set of mutually independent variables. However, in reality, most of these are regarded as correlated variables with unknown joint and marginal PDFs [13–16]. Thus, the third-moment pseudo-correlated normal transformation is introduced to resolve the difficulty procedure of transforming correlated variables into the mutually independent ones [33]. This transformation procedure mainly includes the following two steps.


### 3.1. The Third-Moment Transformation for Correlated Non-Normal System Parameter Variables into Correlated Standard Normal Ones

Without loss of generality, an arbitrary system parameter variableXi is able to be standardized into Xis, whose mean and variance are 0 and 1, respectively. Meanwhile, according to third-moment transformation technology, Xis can be approximated by a second-order polynomial normal function. The whole process above can be expressed in the following equation [39, 40]:(19)Xis=Xi−μXiσXi=SuUi=ai+biUi+ciUi2,where µXi and sXi represent the mean and standard deviation of a set of system parameter variables X, respectively. Su(Ui) is a second-order polynomial about Ui; ai, bi, and ci represent polynomial relevant coefficients of Su(Ui), and these polynomial coefficients, in which their expression include the skewness α3X of Xi, can be formulated by equations (23)-(26) in detail.For the sake of facilitating derivation and comprehension hereinafter, the equation (19) is rewritten so as to reflect the relationship between Xi and Ui as follows:(20)Xi=μXi+σXiai+biUi+ciUi2.Observing the equation (20), the independent non-normal random variables Xi with unknown joint PDF and marginal PDFs, on the basis of the third-moment transformation, have been converted into the standard normal ones Ui.


### 3.2. The Cholesky Decomposition for Correlated Standard Normal System Parameter Variable into Independent Ones

According to third-moment transformation technology, i.e., equation (19), in view of the correlation effect between system parameter variables Xi and Xj, in which their correlative coefficient is ρij, then, their corresponding standardized variables Xis and Xjs, respectively, can be expressed as follows:(21)Xis=ai+biZi+ciZi2=ai,bi,ci⋅1,Zi,Zi2T.(22)Xjs=aj+bjZj+cjZj2=1,Zj,Zj2⋅aj,bj,cjT.in which the superscript T denotes the vector transpose; Zi and Zj, as those described hereinabove, are two correlated standard normal variables, and Zi and Zj, corresponding coefficients of ai, bi, and ci in equation (21) and aj, bj, and cj in (22) are capable to be determined as follows:(23)ci=−ai=Sgnα3Xi2cosSgnα3Xiθi−π3.(24)bi=1−2ci2;θi=arctan−8−α3Xi2α3Xi.(25)cj=−aj=Sgnα3Xj2cosSgnα3Xjθj−π3.(26)bj=1−2cj2;θj=arctan−8−α3Xj2α3Xj,in which α3Xi and α3Xj denote skewness of Xi (or Xis) and Xj (or Xjs), respectively, and by observing the equations (24) and (26), the range of α3X should be restricted in the efficient interval, i.e., −22≤α3X≤22. Fortunately, this bound is not restricted to general engineering applications.Letρ0ij be the correlation coefficient between Zi and Zj, after to relevantly convert and derivate, ρ0ij and ρij are able to be determined as [33].(27)ρ0ij=−1−2ci21−2cj2+1−2ci21−2cj2+8cicjρij4cicj.When obtained the equation (27), two correlated variables with known statistical moments and correlation coefficient can be converted into two correlated standard normal variables. Meanwhile, this procedure can be easily extended to n variables with known statistical moments and correlation matrix, and their equivalent correlation matrix of standard normal variables, Cz, can be summarized as follows:(28)CZ=1ρ012⋯ρ012ρ0121⋯ρ02n⋮⋮⋱⋮ρ012ρ0n2⋯1,in which ρ0ij represents the correlative coefficient of Zi and Zj, i, j = 1, 2, …, n. By utilizing Cholesky decomposition, Cz can be rewritten as follows [41]:(29)CZ=L0L0T,where L0 denotes the lower triangular matrix acquired from Cholesky decomposition, and L0T is its transpose matrix. Then, the correlated standard normal random vector Z can be transformed into the independent standard normal vector U = (U1, U2, …, Un) adopting the lower triangular matrix L0 as follows:(30)U=L0−1Z,where L0−1 is the corresponding inverse matrix of L0, which can be expressed as follows:(31)L0=l110⋯0l21l22⋯0⋮⋮⋱⋮ln1ln2⋯lnn.After relevant transforming the equation (30), Zi can be formulated by substituting the equation (31) into equation (30) as follows:(32)Zi=∑k=1ilikUk,i=1,2,…,n.Combining the equation (20), then, the correlated and non-normal system parameter variables Xi can be expressed by the independent standard normal ones Ui as follows:(33)Xi=μXi+σXiai+bi∑k=1ilikUk+cj∑k=1ilikUk2,i=1,2,…,n.Substituting this equation above into the expression of conditional maximum response statisticsR(X), the expression comes from equation (14) in Section 2.3. Then, evaluating uncertainty stochastic response is able to be expressed as follows:(34)ERX=ERX1,X2,…,Xn=ERU1,U2,…,Un=ERU.in which X = (X1, X2, …, Xn)T is a system parameter vector involving the correlated and non-normal coefficient matrix CX; E[R(U)] denotes the overall maximum response statistics, which includes independent standard normal system parameter variables, i.e., U = (U1, U2, …, Un).


## 3.1. The Third-Moment Transformation for Correlated Non-Normal System Parameter Variables into Correlated Standard Normal Ones

Without loss of generality, an arbitrary system parameter variableXi is able to be standardized into Xis, whose mean and variance are 0 and 1, respectively. Meanwhile, according to third-moment transformation technology, Xis can be approximated by a second-order polynomial normal function. The whole process above can be expressed in the following equation [39, 40]:(19)Xis=Xi−μXiσXi=SuUi=ai+biUi+ciUi2,where µXi and sXi represent the mean and standard deviation of a set of system parameter variables X, respectively. Su(Ui) is a second-order polynomial about Ui; ai, bi, and ci represent polynomial relevant coefficients of Su(Ui), and these polynomial coefficients, in which their expression include the skewness α3X of Xi, can be formulated by equations (23)-(26) in detail.For the sake of facilitating derivation and comprehension hereinafter, the equation (19) is rewritten so as to reflect the relationship between Xi and Ui as follows:(20)Xi=μXi+σXiai+biUi+ciUi2.Observing the equation (20), the independent non-normal random variables Xi with unknown joint PDF and marginal PDFs, on the basis of the third-moment transformation, have been converted into the standard normal ones Ui.


## 3.2. The Cholesky Decomposition for Correlated Standard Normal System Parameter Variable into Independent Ones

According to third-moment transformation technology, i.e., equation (19), in view of the correlation effect between system parameter variables Xi and Xj, in which their correlative coefficient is ρij, then, their corresponding standardized variables Xis and Xjs, respectively, can be expressed as follows:(21)Xis=ai+biZi+ciZi2=ai,bi,ci⋅1,Zi,Zi2T.(22)Xjs=aj+bjZj+cjZj2=1,Zj,Zj2⋅aj,bj,cjT.in which the superscript T denotes the vector transpose; Zi and Zj, as those described hereinabove, are two correlated standard normal variables, and Zi and Zj, corresponding coefficients of ai, bi, and ci in equation (21) and aj, bj, and cj in (22) are capable to be determined as follows:(23)ci=−ai=Sgnα3Xi2cosSgnα3Xiθi−π3.(24)bi=1−2ci2;θi=arctan−8−α3Xi2α3Xi.(25)cj=−aj=Sgnα3Xj2cosSgnα3Xjθj−π3.(26)bj=1−2cj2;θj=arctan−8−α3Xj2α3Xj,in which α3Xi and α3Xj denote skewness of Xi (or Xis) and Xj (or Xjs), respectively, and by observing the equations (24) and (26), the range of α3X should be restricted in the efficient interval, i.e., −22≤α3X≤22. Fortunately, this bound is not restricted to general engineering applications.Letρ0ij be the correlation coefficient between Zi and Zj, after to relevantly convert and derivate, ρ0ij and ρij are able to be determined as [33].(27)ρ0ij=−1−2ci21−2cj2+1−2ci21−2cj2+8cicjρij4cicj.When obtained the equation (27), two correlated variables with known statistical moments and correlation coefficient can be converted into two correlated standard normal variables. Meanwhile, this procedure can be easily extended to n variables with known statistical moments and correlation matrix, and their equivalent correlation matrix of standard normal variables, Cz, can be summarized as follows:(28)CZ=1ρ012⋯ρ012ρ0121⋯ρ02n⋮⋮⋱⋮ρ012ρ0n2⋯1,in which ρ0ij represents the correlative coefficient of Zi and Zj, i, j = 1, 2, …, n. By utilizing Cholesky decomposition, Cz can be rewritten as follows [41]:(29)CZ=L0L0T,where L0 denotes the lower triangular matrix acquired from Cholesky decomposition, and L0T is its transpose matrix. Then, the correlated standard normal random vector Z can be transformed into the independent standard normal vector U = (U1, U2, …, Un) adopting the lower triangular matrix L0 as follows:(30)U=L0−1Z,where L0−1 is the corresponding inverse matrix of L0, which can be expressed as follows:(31)L0=l110⋯0l21l22⋯0⋮⋮⋱⋮ln1ln2⋯lnn.After relevant transforming the equation (30), Zi can be formulated by substituting the equation (31) into equation (30) as follows:(32)Zi=∑k=1ilikUk,i=1,2,…,n.Combining the equation (20), then, the correlated and non-normal system parameter variables Xi can be expressed by the independent standard normal ones Ui as follows:(33)Xi=μXi+σXiai+bi∑k=1ilikUk+cj∑k=1ilikUk2,i=1,2,…,n.Substituting this equation above into the expression of conditional maximum response statisticsR(X), the expression comes from equation (14) in Section 2.3. Then, evaluating uncertainty stochastic response is able to be expressed as follows:(34)ERX=ERX1,X2,…,Xn=ERU1,U2,…,Un=ERU.in which X = (X1, X2, …, Xn)T is a system parameter vector involving the correlated and non-normal coefficient matrix CX; E[R(U)] denotes the overall maximum response statistics, which includes independent standard normal system parameter variables, i.e., U = (U1, U2, …, Un).


## 4. Point Estimation Procedure for Uncertainty Response Evaluation Involving Correlated System Parameter Variables Based on the Third-Moment Pseudo-Correlation Normal Transformation

Combining the PEP with the third-moment pseudo-correlation transformation, as those are described hereinabove, uncertainty evaluation of stochastic structural response involving correlated system parameter variables is capable to be readily conducted. The entire procedure is illustrated in Figure1, and its corresponding explanatory items are able to be described as follows:(1)
The analytical expression is acquired for uncertainty evaluation of mean and standard deviation of maximum stochastic response, i.e., equations (1) and (2), involving correlated system parameter variables X with unknown joint PDF and marginal PDFs.(2)
The expressions ofE[µm(X)], E[μm2 (X)] and E[σm2(X)] are unified, which can be derived by utilizing equations (8) and (9) into an integration form for easy describing conditional maximum response statistics R(X) with regard to arbitrary system parameter variables, i.e., equation (12).(3)
The expression ofE[R(X)], in which X is a set of arbitrary system parameter vectors, is converted to the expression of E[R(U)], in which U is a group of mutually independent ones, based on the third-moment pseudo-correlation normal transformation.(4)
Conditional maximum response statisticsR(U) are approximated as a series of sum functions, in which these functions just include one standard normal random variable Ui, based on the univariate dimension reduction integration.(5)
The ultimate overall maximum response statisticsE[R(X)] are obtained, i.e., the mean of conditional statistics of maximum response R(X), by directly utilizing PEP.Figure 1
Flowchart for uncertainty evaluation of stochastic response with correlated system parameter variables.


## 5. Numerical Examples and Investigations

To verify the simplicity, efficiency, and accuracy of the proposed method for uncertainty evaluation of stochastic response involving correlated system parameter variables, two numerical cases are presented. In example 1, a nonlinear single degree of freedom system, with known marginal PDF and correlation matrix of system parameter variables, subject to Gaussian white noise excitation is investigated, and their evaluation process of uncertainty response involving two correlated parameters by the presented method is illustrated step by step; in example 2, the third-moment pseudo-correlation normal transformation for a two degree of freedom linear simply isolated bridge, just known the first three moments of system parameter variables instead of joint PDF and marginal PDF, taking into account uncertainties of three correlated structural property parameters and two correlated excitation characteristic parameters, subjected to dynamic excitation, is first applied, and then, the effects of correlation and uncertainties are evaluated and discussed in detail. In these examples, a comparative analysis and discussion are, respectively, conducted between given parameter variables and uncertainty ones, and correlated parameter variables and independent ones. Meanwhile, MCS, a supplement approach, is used to investigate the accuracy of the presented method.


### 5.1. Example 1: A Nonlinear SDOF Subject to Gaussian White Noise Excitation

Considering an example of a nonlinear SDOF system under dynamic excitation, the excitation is modeled as Gaussian white noise, in which its intensityS0, the duration T, the natural frequency f, and damping ratio ξ are uncertain. Their probability models are known as marginal PDFs as shown in Table 3. Assuming that the correlation coefficient between the intensity S0 and the duration T is 0.3, the remaining variables are mutually independent.Table 3
Marginal PDFs and correlation matrix of system parameter variables.
ParameterDistributionMeanCOVSkewnessCorrelation matrixTLognormal10 s0.30.9310.60.61S0Type II extreme value0.65dm2/s30.61.14ξLognormal0.030.30.93fNormal2 Hz0.20


#### 5.1.1. Response Evaluation under Determination Parameters

When system uncertainty just is considered to be limited to stochastic excitation, to evaluate the uncertainty response is needed to substitute the means of these parameter variables into equations (1)-(2), respectively; then, the response evaluation results under given parameters can be obtained as equations (35)-(36). Noting v, in equations (1)-(2), represents the average number of times that threshold limit is exceeded per unit time, i.e., v is equivalent to natural frequency f:(35)μm=2InfT+0.57722InfTσR=2.683σR.(36)σm=π12InfTσR=0.524σR,where σR denotes the system displacement response standard deviation. While ignoring the higher-order terms of nonlinearity coefficient ε, its analytical expression can be approximately obtained based on the perturbation method, a nonlinear analysis method of random vibration theory, as follows:(37)σR≈πS022πf3ξ1−3επS022πf3ξ2.Supposing nonlinearity coefficientε is taken as 0.01; equation (37) is substituted into equations (35) and (36), respectively; then, the mean and standard deviation of maximum response, under the given parameter, are obtained as follows: μMdet = 4.602 × 10−2 and σMdet = 8.986 × 10−3.


#### 5.1.2. Response Evaluation with Uncertainty Independent Parameters

When uncertainties in system parametersS0, f, ξ, and T are considered, on condition that they will be treated as mutually independent random variables, then, the overall mean of the maximum response analytical expression, μΜ, can be expressed as follows:(38)μM=∫μmS0,f,ξ,TfS0,f,ξ,TdS0dfdξdT.in which(39)μmS0,f,ξ,T=0.088S0InTπ2f5/2ξ+0.025S0π2f3ξInfT1−0.03S02256π4f6ξ2.Univariate dimension reduction integration can be directly used for equation (39); then, the conditional mean of maximum response µm(X) can be expressed as follows:(40)μmX≅∑i=14μmi−4−1μmμ,where(41)μm1=μmS0=0.071S01−2.0×10−5S2,μm2=μmf=0.053f−31−8.355×10−5f−60.406In10f+2In10f,μm3=μmξ=5.311×10−4ξ−2ξ2−1.175×10−9,μm4=μmT=9.156×10−3+9.329×10−3InTIn2T,μmμ=μmμ=1.770×10−2.In this example, a five-point estimation in standard normal space is selected as given in Table1. For mutually independent system parameters, corresponding coordinates of estimation point μµmi in original space can be readily calculated by the Rosenblatt transformation. The result is shown in Table 4.Table 4
Five estimation point coordinates in original space for example 1.
Parameter variablesThe original coordinates corresponding to five estimated pointsX2−X1−X0X1+X2+T4.1406.4349.5781.426×1012.216 × 101S02.804 × 10−13.741 × 10−15.553 × 10−11.0393.349ξ1.242 × 10−21.930 × 10−22.873 × 10−24.278 × 10−26.665 × 10−2f8.572 × 10−11.4582.0002.5423.142Substituting the estimation point coordinates in original space from Table4 and its weight (see Table 1) into the (17), the evaluated result of univariate conditional mean response µmi(X) and its result from the corresponding five estimation points can be obtained in Table 5.Table 5
Estimation results forμmi(X) in original space.
Parameter variablesμmiThe estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T4.571 × 10−24.001 × 10−24.315 × 10−24.575 × 10−24.822 × 10−25.082 × 10−2S04.601 × 10−21.985 × 10−22.649 × 10−23.918 × 10−27.358 × 10−22.371 × 10−1ξ5.017 × 10−21.111 × 10−17.153 × 10−24.799 × 10−23.228 × 10−22.077 × 10−2f6.081 × 10−25.118 × 10−11.136 × 10−14.602 × 10−22.313 × 10−21.257 × 10−2Eventually, combining Table5 and equation (16), the overall mean of maximum response is able to be evaluated as follows:(42)μMInd=EμmX≅∑i=14μmi−4−1μmμ=6.463×10−2.According to equations (9) and (11), the analytical overall variance expression of the maximum response, σM2, in this example, can be expressed as follows:(43)σM2=T1+T2−T3,T1=Eμm2X=∫μm2S0,f,ξ,TfS0,f,ξ,TdS0dfdξdT,T2=Eσm2X=∫σm2S0,f,ξ,TfS0,f,ξ,TdS0dfdξdT,T3=μM2.in which(44)μm2S0,f,ξ,T=0.088S0InTπ2f5/2ξ+0.025S0π2f3ξInfT1−0.03S02256π4f6ξ22,σm2S0,f,ξ,T=S0323πf3ξInfT1−0.03S04256π4f6ξ22.The entire evaluation process is similar as the overall mean of maximum responseμM above is analyzed; thus, the evaluation results of μmi2(X) and σmi2(X), hereafter, are directly provided in Table 6.Table 6
Estimation results forμmi2(X) in original space.
Parameter variablesμmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T2.093 × 10−31.606 × 10−31.861 × 10−32.093 × 10−32.325 × 10−32.582 × 10−3S02.815 × 10−33.941 × 10−47.018 × 10−41.535 × 10−35.414 × 10−35.625 × 10−2Ξ2.743 × 10−31.236 × 10−25.118 × 10−32.309 × 10−31.041 × 10−34.315 × 10−4F7.063 × 10−32.620 × 10−11.290 × 10−22.118 × 10−35.353 × 10−41.582 × 10−4Combining Table6 and equation (16), the overall mean square of maximum response is able to be evaluated as follows:(45)Eμm2X≅∑i=14μmi2−4−1μm2μ=8.360×10−3.Combining Table7 and equation (16), the overall standard deviation square of maximum response is able to be evaluated as follows:(46)Eσm2X≅∑i=14σmi2−4−1σm2μ=4.373×10−4.Table 7
Estimation results forσmi2(X) in original space.
Parameter variablesσmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T8.277 × 10−51.144 × 10−49.470 × 10−58.193 × 10−57.220 × 10−56.381 × 10−5S01.073 × 10−41.502 × 10−52.676 × 10−55.853 × 10−52.064 × 10−42.141 × 10−3ξ1.046 × 10−44.710 × 10−41.951 × 10−48.802 × 10−53.971 × 10−51.644 × 10−5f3.851 × 10−41.813 × 10−26.021 × 10−48.076 × 10−51.773 × 10−54.661 × 10−6Then, substituting the result ofE[μm2(X)] and E[σm2(X)] into equation (11), the overall standard deviation square of maximum response can be evaluated as follows:(47)σMInd=Eμm2X+Eσm2X−μMind2=6.797×10−2.To investigate the evaluation process of response uncertainty in detail, the uncertainties of four cases, i.e., given parameters, structural property parameters (ξ, f), excitation characteristics parameters (T, S0), and all parameters (T, S0, ξ, f), are considered, respectively. Meanwhile, the accuracy of their results is able to be confirmed by utilizing the MCS with a size of 106. The results of these cases are listed in Table 8.Table 8
Comparison between the MCS and the presented method under different parameter cases.
Considered casesPEP methodMCSMethod error (%)μMIndσMIndμMMCSσMMCSΕμMΕσΜGiven parameters4.602 × 10−28.986 × 10−3————Structure parameters2.498 × 10−27.473 × 10−32.630 × 10−27.120 × 10−35.0164.957Excitation parameters4.570 × 10−28.001 × 10−44.792 × 10−27.777 × 10−44.6322.880All parameters6.463 × 10−26.797 × 10−26.221 × 10−27.151 × 10−23.8904.950From Table8, some conclusions, in this example, can be revealed: (I) the parameter uncertainties have a great influence on response evaluation relative to determination parameters, and the effect of structural parameters on the overall mean evaluation of maximum response is more obvious than that of excitation parameters. (II) All uncertainty parameters are simultaneously considered, and these uncertainties, relative to other cases, have the greatest effect on the response evaluation. (III) The results from PEP are basically in agreement with those acquired using MCS, in which the maximum error between them is just 5.016%.


#### 5.1.3. Response Evaluation Involving Uncertainty Correlated Parameters

To investigate the evaluation process of response uncertainty involving correlated parameter variables, the example, hereinabove, considering all uncertainty parameters (T, S0, ξ, and f), will be further analyzed.While considering the correlation between the durationT and the intensity S0, the third-moment pseudo-correlation transformation is applied first, so as to convert their correlated variables into independent standard normal variables. The correlated matrix including all parameters can be written as follows:(48)ρ=10.6000.610000100001.According to equations (23)-(30), the polynomial coefficients of each parameter can be calculated, and the result is provided in Table 9.Table 9
Polynomial coefficients of each parameter.
VariablesaibiciT−0.1560.9750.156S0−0.6750.2960.675ξ−0.1560.9750.156f010Combining equations (27)-(29) and Cholesky decomposition, the lower triangular matrix L0 can be obtained as follows:(49)L0=10000.6150.7880000100001.Substituting the lower triangular matrixL0 into equation (33), these correlation variables, utilizing the five-point estimation, were transferred as mutually independent estimation point coordinates Xi in the original space. These transformation results are shown in Table 10.Table 10
Five-point estimated coordinates in original space after correlation transformation.
Parameter variablesThe original space coordinates corresponding to five estimated pointsX2+X1+X0X1+X2+T1.4295.93310.00014.07018.571S0−9.13 × 10−1−9.208 × 10−26.500 × 10−11.3922.213ξ4.287 × 10−31.180 × 10−23.000 × 10−24.220 × 10−25.571 × 10−2f8.572 × 10−11.4582.0002.5423.143Substituting the estimation point coordinates in original space from Table10 and their weight (see Table 1) into equation (17), the evaluated result of univariate conditional mean, conditional mean square, and conditional standard variance square of response, i.e., μmi(X), μmi2(X), and σmi2(X), and their function value corresponding five estimation point in original space can be obtained in Tables 11-13.Table 11
Estimation results forμmi(X) in original space after correlation transformation.
Parameter variablesμmiThe estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T4.561 × 10−23.169 × 10−24.260 × 10−24.602 × 10−24.814 × 10−24.979 × 10−2S04.602 × 10−2−6.464 × 10−2−6.519 × 10−34.602 × 10−29.856 × 10−21.566 × 10−1ξ6.170 × 10−23.220 × 10−11.170 × 10−14.602 × 10−23.272 × 10−22.578 × 10−2f6.079 × 10−25.118 × 10−11.135 × 10−14.602 × 10−22.314 × 10−21.257 × 10−2Table 12
Estimation results forμmi2(X) in original space after correlation transformation.
Parameter variablesμmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T2.086 × 10−31.004 × 10−31.815 × 10−32.118 × 10−32.317 × 10−32.479 × 10−3S03.620 × 10−34.179 × 10−34.251 × 10−52.118 × 10−39.714 × 10−32.455 × 10−2ξ5.582 × 10−31.037 × 10−11.369 × 10−22.118 × 10−31.071 × 10−36.143 × 10−4f7.062 × 10−32.620 × 10−11.289 × 10−22.118 × 10−35.356 × 10−41.582 × 10−4Table 13
Estimation results forσmi2(X) in original space after correlation transformation.
Parameter variablesσmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T8.523 × 10−52.303 × 10−49.779 × 10−58.076 × 10−57.249 × 10−56.693 × 10−5S01.380 × 10−41.593 × 10−41.621 × 10−68.075 × 10−53.704 × 10−49.360 × 10−4ξ2.128 × 10−43.953 × 10−35.220 × 10−48.076 × 10−54.081 × 10−52.342 × 10−5f3.850 × 10−41.181 × 10−26.015 × 10−48.076 × 10−51.773 × 10−54.659 × 10−6Having taken into account the effect of correlation of durationT and spectral S0 of the excitation, the overall mean and standard deviation of maximum response, combing Tables 11–13 and equation (16), are able to be evaluated as follows:(50)μMCor=EEμmX≅∑i=14μmi−4−1μmμ=7.606×10−2,σMCor=Eμm2X+Eσm2X−μMcor2=0.869.In order to further investigate the uncertainty effect on stochastic response involving correlated parameter variables in detail, four cases involving correlated parameter variables, i.e., given parameters, structure parameters (ξ and f), excitation parameters (T and S0), and all parameters (T, S0, ξ, and f), are considered, respectively. Meanwhile, the accuracy of their results is able to be confirmed by utilizing the MCS with a size of 106. The results of these cases are listed in Table 14.Table 14
Result comparison between independent variables and correlation variables.
Considered casesμMIndσMIndμMCovσMCovError with MCS (%)EμMCovEσMCovGiven parameters4.602 × 10−28.986 × 10−3——Structural parameters2.498 × 10−27.473 × 10−37.647 × 10−27.202 × 10−23.7303.311Excitation parameters4.570 × 10−28.001 × 10−44.561 × 10−24.061 × 10−24.1242.895All parameters6.463 × 10−26.797 × 10−27.606 × 10−28.238 × 10−22.5783.193From Table14, some conclusions, in this example, can be revealed: (I) the correlation in this example has a great influence on response evaluation relative to the independent situation: if ignored the effect of correlation, the evaluation result in overall mean of maximum response will be underestimated, especially for structural property parameters and all parameters. When considering the excitation characteristic parameter, the overall standard deviation of the maximum result will be overestimated; when considering all parameters, the opposite conclusion will be obtained. (II)While the correlation is taken into account, the effect of excitation characteristic parameters on the overall mean of maximum response and the overall standard deviation is more obvious than in other cases. (III) The calculation results including correlation are basically in agreement with those obtained using MCS, in which the maximum error between them is just 4.124%.


### 5.2. Example 2: 2-DOF Linear Simply Isolated Bridge Subjected to Dynamic Excitation

Figure2 shows a 2-DOF linear isolated bridge in which one of the masses is subjected to dynamic vertical excitation P(t).Figure 2
The calculation diagrams of simply supported isolated bridges.Assuming their structure-property parameters, that is, massm, elastic modulus E, and initial moment I, are the same, the natural frequency ωi of the simply supported isolated bridge, based on linear vibration theory, can be provided as follows:(51)ω1=5.69EImL3,ω2=22EImL3,where E, I, and L are the elastic modulus, initial moment, and span of the bridge, respectively.In the frequency domain, the standard deviation of the displacement responses, based on the random vibration theory, is obtained as follows:(52)σy12=πS04m212ξ1ω13+12ξ2ω23+8ξ1ω1+ξ2ω2ω12−ω222+4ξ1ξ2ω1ω2ω12+ω22+ξ12+ξ22ω12ω22,σy22=πS04m212ξ1ω13+12ξ2ω23−8ξ1ω1+ξ2ω2ω12−ω222+4ξ1ξ2ω1ω2ω12+ω22+ξ12+ξ22ω12ω22,where σyi is the standard deviation of the displacement responses of each mass, S0 is the spectral intensity of the excitation, which is modeled as Gaussian white noise, and T is its corresponding duration νi≈2π/ωi is approximately the mean cross ratio corresponding to ith mass, and both masses have the same damping ratio ξ. The joint and marginal PDFs of these uncertainty system parameters are unknown, except for the first three moments and the correlation matrix, as listed in Table 15.Table 15
Statistic moments and correlation matrix of uncertain system parameters.
ParameterMeanCOVSkewnessCorrelation matrixT15 s0.40.4311.00.50.51.0S0780 cm2/s30.61.608ξ10.0430.30.431ξ20.0690.30.431m4.616 × 105 kg0.10.301L1.500 × 104 mm0.10.3011.00.30.20.31.00.10.20.11.0I7.087 × 1011 mm40.10.301E3.150 × 104 MPa0.10To investigate the evaluation influence of both masses on response uncertainty involving correlated parameter variables, the mean and standard deviation of the maximum response evaluation under the given parameters are contrasted with the correlation parameters. Meanwhile, for the uncertainty response evaluation, the following three cases were analyzed:Case 1.
Considering only uncertainties contained in excitation parameters,S0 and T.Case 2.
Considering only uncertainties contained in the structural parametersL, I, E, m, and ξ.Case 3.
Considering uncertainties of all uncertainty parameters,S0, T, L, I, E, m, and ξ.
Partial calculation results with regard to the correlation transformation process (third-moment pseudocorrelation normal transformation technology) are given, such as polynomial coefficients of each parameter (see Table16), the lower triangular matrix L0, and five estimation point coordinates, in the original space (see Table 17).
Combiningequations (27)–(29) and the Cholesky decomposition, thelower triangular matrix L0 can be obtained as follows:(53)L0=10.5360.839001000100001000001000000.3010.953000000.2010.0420.979.
Having been converted by the third-moment pseudonormal transformation technology, the five estimation point coordinates in the original space are listed in Table17.
Both masses’ calculation results of the response evaluation, including correlated parameter variables, are provided inTables18 and 19. Meanwhile, the accuracy of their results can be confirmed by utilizing an MCS with a size of 106, which are listed in Tables 18 and 19 as well.
After observing and contrasting the results in Table18, some conclusions on Mass 1 can be drawn as follows: (I) To consider the uncertain and correlated system parameter variables, relative to the determination parameter situation, the evaluation results of overall mean and standard deviation of the maximum response have a great influence; for the overall mean of maximum response, the evaluation results will be overestimated; for the overall standard deviation of maximum response, the evaluation result will be overestimated under considering structure-property and all parameters, while the evaluation results, under excitation-characteristics parameters considered, will be underestimated. (II) For the overall mean of maximum response, the effect of the excitation-characteristic parameters was more remarkable than structure-property parameters; for the overall standard deviation of maximum response, the opposite conclusion is obtained. (III) Their calculation results are in agreement with those obtained using MCS, in which the maximum error between them is merely 4.993%. Similarly, the conclusions of Mass.1 are suitable for Mass.2 inTable 19.Table 16
Polynomial coefficients of each parameter.
VariablesaibiciT−0.0720.9950.072S0−0.2830.9160.283ξ1−0.0720.9950.072ξ2−0.0720.9950.072E−0.05030.9970.050m−0.05030.9970.050L−0.05030.9970.050I010Table 17
Five estimation point coordinates in original space after correlation transformation.
Parameter variablesThe original space coordinates corresponding to five estimated pointsX2+X1+X0X1+X2+T32.1406.86615.00023.13432.142S02.625 × 105−9.557 × 1037.800 × 1041.656 × 1052.625 × 105ξ17.985 × 10−22.551 × 10−24.310 × 10−26.112 × 10−27.985 × 10−2ξ212.8114.093 × 10−26.930 × 10−29.710 × 10−21.281 × 10−1E5.934 × 1053.990 × 1054.616 × 1055.242 × 1055.935 × 105m1.928 × 1041.296 × 1041.500 × 1041.703 × 1041.929 × 104L9.263 × 10115.882 × 10117.087 × 10118.292 × 10119.627 × 1011I4.249 × 1042.629 × 1043.150 × 1043.671 × 1044.249 × 104Table 18
Response evaluation result comparison between presented method and MCS for Mass.1.
Considered casesPresented methodMCSMethod error (%)μMCovσMCovμMMCSσMMCSEμMEσMGiven parameters5.141 × 10−34.256 × 10−3————Structure parameters3.952 × 10−31.466 × 10−23.767 × 10−31.538 × 10−24.9114.681Excitation parameters3.657 × 10−35.061 × 10−33.493 × 10−35.327 × 10−34.5004.993All parameters2.466 × 10−31.452 × 10−22.553 × 10−31.412 × 10−23.4162.832Table 19
Response evaluation result comparison between presented method and MCS for Mass.2.
Considered casesPresented methodMCSMethod error (%)μMCovσMCovμMMCSσMMCSEμMEσMGiven parameters7.251 × 10−32.186 × 10−3————Structure parameters1.808 × 10−31.835 × 10−21.886 × 10−31.905 × 10−24.1363.675Excitation parameters6.403 × 10−34.512 × 10−36.379 × 10−34.370 × 10−3.7623.249All parameters9.595 × 10−41.850 × 10−29.255 × 10−41.768 × 10−23.6734.638


## 5.1. Example 1: A Nonlinear SDOF Subject to Gaussian White Noise Excitation

Considering an example of a nonlinear SDOF system under dynamic excitation, the excitation is modeled as Gaussian white noise, in which its intensityS0, the duration T, the natural frequency f, and damping ratio ξ are uncertain. Their probability models are known as marginal PDFs as shown in Table 3. Assuming that the correlation coefficient between the intensity S0 and the duration T is 0.3, the remaining variables are mutually independent.Table 3
Marginal PDFs and correlation matrix of system parameter variables.
ParameterDistributionMeanCOVSkewnessCorrelation matrixTLognormal10 s0.30.9310.60.61S0Type II extreme value0.65dm2/s30.61.14ξLognormal0.030.30.93fNormal2 Hz0.20


### 5.1.1. Response Evaluation under Determination Parameters

When system uncertainty just is considered to be limited to stochastic excitation, to evaluate the uncertainty response is needed to substitute the means of these parameter variables into equations (1)-(2), respectively; then, the response evaluation results under given parameters can be obtained as equations (35)-(36). Noting v, in equations (1)-(2), represents the average number of times that threshold limit is exceeded per unit time, i.e., v is equivalent to natural frequency f:(35)μm=2InfT+0.57722InfTσR=2.683σR.(36)σm=π12InfTσR=0.524σR,where σR denotes the system displacement response standard deviation. While ignoring the higher-order terms of nonlinearity coefficient ε, its analytical expression can be approximately obtained based on the perturbation method, a nonlinear analysis method of random vibration theory, as follows:(37)σR≈πS022πf3ξ1−3επS022πf3ξ2.Supposing nonlinearity coefficientε is taken as 0.01; equation (37) is substituted into equations (35) and (36), respectively; then, the mean and standard deviation of maximum response, under the given parameter, are obtained as follows: μMdet = 4.602 × 10−2 and σMdet = 8.986 × 10−3.


### 5.1.2. Response Evaluation with Uncertainty Independent Parameters

When uncertainties in system parametersS0, f, ξ, and T are considered, on condition that they will be treated as mutually independent random variables, then, the overall mean of the maximum response analytical expression, μΜ, can be expressed as follows:(38)μM=∫μmS0,f,ξ,TfS0,f,ξ,TdS0dfdξdT.in which(39)μmS0,f,ξ,T=0.088S0InTπ2f5/2ξ+0.025S0π2f3ξInfT1−0.03S02256π4f6ξ2.Univariate dimension reduction integration can be directly used for equation (39); then, the conditional mean of maximum response µm(X) can be expressed as follows:(40)μmX≅∑i=14μmi−4−1μmμ,where(41)μm1=μmS0=0.071S01−2.0×10−5S2,μm2=μmf=0.053f−31−8.355×10−5f−60.406In10f+2In10f,μm3=μmξ=5.311×10−4ξ−2ξ2−1.175×10−9,μm4=μmT=9.156×10−3+9.329×10−3InTIn2T,μmμ=μmμ=1.770×10−2.In this example, a five-point estimation in standard normal space is selected as given in Table1. For mutually independent system parameters, corresponding coordinates of estimation point μµmi in original space can be readily calculated by the Rosenblatt transformation. The result is shown in Table 4.Table 4
Five estimation point coordinates in original space for example 1.
Parameter variablesThe original coordinates corresponding to five estimated pointsX2−X1−X0X1+X2+T4.1406.4349.5781.426×1012.216 × 101S02.804 × 10−13.741 × 10−15.553 × 10−11.0393.349ξ1.242 × 10−21.930 × 10−22.873 × 10−24.278 × 10−26.665 × 10−2f8.572 × 10−11.4582.0002.5423.142Substituting the estimation point coordinates in original space from Table4 and its weight (see Table 1) into the (17), the evaluated result of univariate conditional mean response µmi(X) and its result from the corresponding five estimation points can be obtained in Table 5.Table 5
Estimation results forμmi(X) in original space.
Parameter variablesμmiThe estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T4.571 × 10−24.001 × 10−24.315 × 10−24.575 × 10−24.822 × 10−25.082 × 10−2S04.601 × 10−21.985 × 10−22.649 × 10−23.918 × 10−27.358 × 10−22.371 × 10−1ξ5.017 × 10−21.111 × 10−17.153 × 10−24.799 × 10−23.228 × 10−22.077 × 10−2f6.081 × 10−25.118 × 10−11.136 × 10−14.602 × 10−22.313 × 10−21.257 × 10−2Eventually, combining Table5 and equation (16), the overall mean of maximum response is able to be evaluated as follows:(42)μMInd=EμmX≅∑i=14μmi−4−1μmμ=6.463×10−2.According to equations (9) and (11), the analytical overall variance expression of the maximum response, σM2, in this example, can be expressed as follows:(43)σM2=T1+T2−T3,T1=Eμm2X=∫μm2S0,f,ξ,TfS0,f,ξ,TdS0dfdξdT,T2=Eσm2X=∫σm2S0,f,ξ,TfS0,f,ξ,TdS0dfdξdT,T3=μM2.in which(44)μm2S0,f,ξ,T=0.088S0InTπ2f5/2ξ+0.025S0π2f3ξInfT1−0.03S02256π4f6ξ22,σm2S0,f,ξ,T=S0323πf3ξInfT1−0.03S04256π4f6ξ22.The entire evaluation process is similar as the overall mean of maximum responseμM above is analyzed; thus, the evaluation results of μmi2(X) and σmi2(X), hereafter, are directly provided in Table 6.Table 6
Estimation results forμmi2(X) in original space.
Parameter variablesμmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T2.093 × 10−31.606 × 10−31.861 × 10−32.093 × 10−32.325 × 10−32.582 × 10−3S02.815 × 10−33.941 × 10−47.018 × 10−41.535 × 10−35.414 × 10−35.625 × 10−2Ξ2.743 × 10−31.236 × 10−25.118 × 10−32.309 × 10−31.041 × 10−34.315 × 10−4F7.063 × 10−32.620 × 10−11.290 × 10−22.118 × 10−35.353 × 10−41.582 × 10−4Combining Table6 and equation (16), the overall mean square of maximum response is able to be evaluated as follows:(45)Eμm2X≅∑i=14μmi2−4−1μm2μ=8.360×10−3.Combining Table7 and equation (16), the overall standard deviation square of maximum response is able to be evaluated as follows:(46)Eσm2X≅∑i=14σmi2−4−1σm2μ=4.373×10−4.Table 7
Estimation results forσmi2(X) in original space.
Parameter variablesσmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T8.277 × 10−51.144 × 10−49.470 × 10−58.193 × 10−57.220 × 10−56.381 × 10−5S01.073 × 10−41.502 × 10−52.676 × 10−55.853 × 10−52.064 × 10−42.141 × 10−3ξ1.046 × 10−44.710 × 10−41.951 × 10−48.802 × 10−53.971 × 10−51.644 × 10−5f3.851 × 10−41.813 × 10−26.021 × 10−48.076 × 10−51.773 × 10−54.661 × 10−6Then, substituting the result ofE[μm2(X)] and E[σm2(X)] into equation (11), the overall standard deviation square of maximum response can be evaluated as follows:(47)σMInd=Eμm2X+Eσm2X−μMind2=6.797×10−2.To investigate the evaluation process of response uncertainty in detail, the uncertainties of four cases, i.e., given parameters, structural property parameters (ξ, f), excitation characteristics parameters (T, S0), and all parameters (T, S0, ξ, f), are considered, respectively. Meanwhile, the accuracy of their results is able to be confirmed by utilizing the MCS with a size of 106. The results of these cases are listed in Table 8.Table 8
Comparison between the MCS and the presented method under different parameter cases.
Considered casesPEP methodMCSMethod error (%)μMIndσMIndμMMCSσMMCSΕμMΕσΜGiven parameters4.602 × 10−28.986 × 10−3————Structure parameters2.498 × 10−27.473 × 10−32.630 × 10−27.120 × 10−35.0164.957Excitation parameters4.570 × 10−28.001 × 10−44.792 × 10−27.777 × 10−44.6322.880All parameters6.463 × 10−26.797 × 10−26.221 × 10−27.151 × 10−23.8904.950From Table8, some conclusions, in this example, can be revealed: (I) the parameter uncertainties have a great influence on response evaluation relative to determination parameters, and the effect of structural parameters on the overall mean evaluation of maximum response is more obvious than that of excitation parameters. (II) All uncertainty parameters are simultaneously considered, and these uncertainties, relative to other cases, have the greatest effect on the response evaluation. (III) The results from PEP are basically in agreement with those acquired using MCS, in which the maximum error between them is just 5.016%.


### 5.1.3. Response Evaluation Involving Uncertainty Correlated Parameters

To investigate the evaluation process of response uncertainty involving correlated parameter variables, the example, hereinabove, considering all uncertainty parameters (T, S0, ξ, and f), will be further analyzed.While considering the correlation between the durationT and the intensity S0, the third-moment pseudo-correlation transformation is applied first, so as to convert their correlated variables into independent standard normal variables. The correlated matrix including all parameters can be written as follows:(48)ρ=10.6000.610000100001.According to equations (23)-(30), the polynomial coefficients of each parameter can be calculated, and the result is provided in Table 9.Table 9
Polynomial coefficients of each parameter.
VariablesaibiciT−0.1560.9750.156S0−0.6750.2960.675ξ−0.1560.9750.156f010Combining equations (27)-(29) and Cholesky decomposition, the lower triangular matrix L0 can be obtained as follows:(49)L0=10000.6150.7880000100001.Substituting the lower triangular matrixL0 into equation (33), these correlation variables, utilizing the five-point estimation, were transferred as mutually independent estimation point coordinates Xi in the original space. These transformation results are shown in Table 10.Table 10
Five-point estimated coordinates in original space after correlation transformation.
Parameter variablesThe original space coordinates corresponding to five estimated pointsX2+X1+X0X1+X2+T1.4295.93310.00014.07018.571S0−9.13 × 10−1−9.208 × 10−26.500 × 10−11.3922.213ξ4.287 × 10−31.180 × 10−23.000 × 10−24.220 × 10−25.571 × 10−2f8.572 × 10−11.4582.0002.5423.143Substituting the estimation point coordinates in original space from Table10 and their weight (see Table 1) into equation (17), the evaluated result of univariate conditional mean, conditional mean square, and conditional standard variance square of response, i.e., μmi(X), μmi2(X), and σmi2(X), and their function value corresponding five estimation point in original space can be obtained in Tables 11-13.Table 11
Estimation results forμmi(X) in original space after correlation transformation.
Parameter variablesμmiThe estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T4.561 × 10−23.169 × 10−24.260 × 10−24.602 × 10−24.814 × 10−24.979 × 10−2S04.602 × 10−2−6.464 × 10−2−6.519 × 10−34.602 × 10−29.856 × 10−21.566 × 10−1ξ6.170 × 10−23.220 × 10−11.170 × 10−14.602 × 10−23.272 × 10−22.578 × 10−2f6.079 × 10−25.118 × 10−11.135 × 10−14.602 × 10−22.314 × 10−21.257 × 10−2Table 12
Estimation results forμmi2(X) in original space after correlation transformation.
Parameter variablesμmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T2.086 × 10−31.004 × 10−31.815 × 10−32.118 × 10−32.317 × 10−32.479 × 10−3S03.620 × 10−34.179 × 10−34.251 × 10−52.118 × 10−39.714 × 10−32.455 × 10−2ξ5.582 × 10−31.037 × 10−11.369 × 10−22.118 × 10−31.071 × 10−36.143 × 10−4f7.062 × 10−32.620 × 10−11.289 × 10−22.118 × 10−35.356 × 10−41.582 × 10−4Table 13
Estimation results forσmi2(X) in original space after correlation transformation.
Parameter variablesσmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T8.523 × 10−52.303 × 10−49.779 × 10−58.076 × 10−57.249 × 10−56.693 × 10−5S01.380 × 10−41.593 × 10−41.621 × 10−68.075 × 10−53.704 × 10−49.360 × 10−4ξ2.128 × 10−43.953 × 10−35.220 × 10−48.076 × 10−54.081 × 10−52.342 × 10−5f3.850 × 10−41.181 × 10−26.015 × 10−48.076 × 10−51.773 × 10−54.659 × 10−6Having taken into account the effect of correlation of durationT and spectral S0 of the excitation, the overall mean and standard deviation of maximum response, combing Tables 11–13 and equation (16), are able to be evaluated as follows:(50)μMCor=EEμmX≅∑i=14μmi−4−1μmμ=7.606×10−2,σMCor=Eμm2X+Eσm2X−μMcor2=0.869.In order to further investigate the uncertainty effect on stochastic response involving correlated parameter variables in detail, four cases involving correlated parameter variables, i.e., given parameters, structure parameters (ξ and f), excitation parameters (T and S0), and all parameters (T, S0, ξ, and f), are considered, respectively. Meanwhile, the accuracy of their results is able to be confirmed by utilizing the MCS with a size of 106. The results of these cases are listed in Table 14.Table 14
Result comparison between independent variables and correlation variables.
Considered casesμMIndσMIndμMCovσMCovError with MCS (%)EμMCovEσMCovGiven parameters4.602 × 10−28.986 × 10−3——Structural parameters2.498 × 10−27.473 × 10−37.647 × 10−27.202 × 10−23.7303.311Excitation parameters4.570 × 10−28.001 × 10−44.561 × 10−24.061 × 10−24.1242.895All parameters6.463 × 10−26.797 × 10−27.606 × 10−28.238 × 10−22.5783.193From Table14, some conclusions, in this example, can be revealed: (I) the correlation in this example has a great influence on response evaluation relative to the independent situation: if ignored the effect of correlation, the evaluation result in overall mean of maximum response will be underestimated, especially for structural property parameters and all parameters. When considering the excitation characteristic parameter, the overall standard deviation of the maximum result will be overestimated; when considering all parameters, the opposite conclusion will be obtained. (II)While the correlation is taken into account, the effect of excitation characteristic parameters on the overall mean of maximum response and the overall standard deviation is more obvious than in other cases. (III) The calculation results including correlation are basically in agreement with those obtained using MCS, in which the maximum error between them is just 4.124%.


## 5.1.1. Response Evaluation under Determination Parameters

When system uncertainty just is considered to be limited to stochastic excitation, to evaluate the uncertainty response is needed to substitute the means of these parameter variables into equations (1)-(2), respectively; then, the response evaluation results under given parameters can be obtained as equations (35)-(36). Noting v, in equations (1)-(2), represents the average number of times that threshold limit is exceeded per unit time, i.e., v is equivalent to natural frequency f:(35)μm=2InfT+0.57722InfTσR=2.683σR.(36)σm=π12InfTσR=0.524σR,where σR denotes the system displacement response standard deviation. While ignoring the higher-order terms of nonlinearity coefficient ε, its analytical expression can be approximately obtained based on the perturbation method, a nonlinear analysis method of random vibration theory, as follows:(37)σR≈πS022πf3ξ1−3επS022πf3ξ2.Supposing nonlinearity coefficientε is taken as 0.01; equation (37) is substituted into equations (35) and (36), respectively; then, the mean and standard deviation of maximum response, under the given parameter, are obtained as follows: μMdet = 4.602 × 10−2 and σMdet = 8.986 × 10−3.


## 5.1.2. Response Evaluation with Uncertainty Independent Parameters

When uncertainties in system parametersS0, f, ξ, and T are considered, on condition that they will be treated as mutually independent random variables, then, the overall mean of the maximum response analytical expression, μΜ, can be expressed as follows:(38)μM=∫μmS0,f,ξ,TfS0,f,ξ,TdS0dfdξdT.in which(39)μmS0,f,ξ,T=0.088S0InTπ2f5/2ξ+0.025S0π2f3ξInfT1−0.03S02256π4f6ξ2.Univariate dimension reduction integration can be directly used for equation (39); then, the conditional mean of maximum response µm(X) can be expressed as follows:(40)μmX≅∑i=14μmi−4−1μmμ,where(41)μm1=μmS0=0.071S01−2.0×10−5S2,μm2=μmf=0.053f−31−8.355×10−5f−60.406In10f+2In10f,μm3=μmξ=5.311×10−4ξ−2ξ2−1.175×10−9,μm4=μmT=9.156×10−3+9.329×10−3InTIn2T,μmμ=μmμ=1.770×10−2.In this example, a five-point estimation in standard normal space is selected as given in Table1. For mutually independent system parameters, corresponding coordinates of estimation point μµmi in original space can be readily calculated by the Rosenblatt transformation. The result is shown in Table 4.Table 4
Five estimation point coordinates in original space for example 1.
Parameter variablesThe original coordinates corresponding to five estimated pointsX2−X1−X0X1+X2+T4.1406.4349.5781.426×1012.216 × 101S02.804 × 10−13.741 × 10−15.553 × 10−11.0393.349ξ1.242 × 10−21.930 × 10−22.873 × 10−24.278 × 10−26.665 × 10−2f8.572 × 10−11.4582.0002.5423.142Substituting the estimation point coordinates in original space from Table4 and its weight (see Table 1) into the (17), the evaluated result of univariate conditional mean response µmi(X) and its result from the corresponding five estimation points can be obtained in Table 5.Table 5
Estimation results forμmi(X) in original space.
Parameter variablesμmiThe estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T4.571 × 10−24.001 × 10−24.315 × 10−24.575 × 10−24.822 × 10−25.082 × 10−2S04.601 × 10−21.985 × 10−22.649 × 10−23.918 × 10−27.358 × 10−22.371 × 10−1ξ5.017 × 10−21.111 × 10−17.153 × 10−24.799 × 10−23.228 × 10−22.077 × 10−2f6.081 × 10−25.118 × 10−11.136 × 10−14.602 × 10−22.313 × 10−21.257 × 10−2Eventually, combining Table5 and equation (16), the overall mean of maximum response is able to be evaluated as follows:(42)μMInd=EμmX≅∑i=14μmi−4−1μmμ=6.463×10−2.According to equations (9) and (11), the analytical overall variance expression of the maximum response, σM2, in this example, can be expressed as follows:(43)σM2=T1+T2−T3,T1=Eμm2X=∫μm2S0,f,ξ,TfS0,f,ξ,TdS0dfdξdT,T2=Eσm2X=∫σm2S0,f,ξ,TfS0,f,ξ,TdS0dfdξdT,T3=μM2.in which(44)μm2S0,f,ξ,T=0.088S0InTπ2f5/2ξ+0.025S0π2f3ξInfT1−0.03S02256π4f6ξ22,σm2S0,f,ξ,T=S0323πf3ξInfT1−0.03S04256π4f6ξ22.The entire evaluation process is similar as the overall mean of maximum responseμM above is analyzed; thus, the evaluation results of μmi2(X) and σmi2(X), hereafter, are directly provided in Table 6.Table 6
Estimation results forμmi2(X) in original space.
Parameter variablesμmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T2.093 × 10−31.606 × 10−31.861 × 10−32.093 × 10−32.325 × 10−32.582 × 10−3S02.815 × 10−33.941 × 10−47.018 × 10−41.535 × 10−35.414 × 10−35.625 × 10−2Ξ2.743 × 10−31.236 × 10−25.118 × 10−32.309 × 10−31.041 × 10−34.315 × 10−4F7.063 × 10−32.620 × 10−11.290 × 10−22.118 × 10−35.353 × 10−41.582 × 10−4Combining Table6 and equation (16), the overall mean square of maximum response is able to be evaluated as follows:(45)Eμm2X≅∑i=14μmi2−4−1μm2μ=8.360×10−3.Combining Table7 and equation (16), the overall standard deviation square of maximum response is able to be evaluated as follows:(46)Eσm2X≅∑i=14σmi2−4−1σm2μ=4.373×10−4.Table 7
Estimation results forσmi2(X) in original space.
Parameter variablesσmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T8.277 × 10−51.144 × 10−49.470 × 10−58.193 × 10−57.220 × 10−56.381 × 10−5S01.073 × 10−41.502 × 10−52.676 × 10−55.853 × 10−52.064 × 10−42.141 × 10−3ξ1.046 × 10−44.710 × 10−41.951 × 10−48.802 × 10−53.971 × 10−51.644 × 10−5f3.851 × 10−41.813 × 10−26.021 × 10−48.076 × 10−51.773 × 10−54.661 × 10−6Then, substituting the result ofE[μm2(X)] and E[σm2(X)] into equation (11), the overall standard deviation square of maximum response can be evaluated as follows:(47)σMInd=Eμm2X+Eσm2X−μMind2=6.797×10−2.To investigate the evaluation process of response uncertainty in detail, the uncertainties of four cases, i.e., given parameters, structural property parameters (ξ, f), excitation characteristics parameters (T, S0), and all parameters (T, S0, ξ, f), are considered, respectively. Meanwhile, the accuracy of their results is able to be confirmed by utilizing the MCS with a size of 106. The results of these cases are listed in Table 8.Table 8
Comparison between the MCS and the presented method under different parameter cases.
Considered casesPEP methodMCSMethod error (%)μMIndσMIndμMMCSσMMCSΕμMΕσΜGiven parameters4.602 × 10−28.986 × 10−3————Structure parameters2.498 × 10−27.473 × 10−32.630 × 10−27.120 × 10−35.0164.957Excitation parameters4.570 × 10−28.001 × 10−44.792 × 10−27.777 × 10−44.6322.880All parameters6.463 × 10−26.797 × 10−26.221 × 10−27.151 × 10−23.8904.950From Table8, some conclusions, in this example, can be revealed: (I) the parameter uncertainties have a great influence on response evaluation relative to determination parameters, and the effect of structural parameters on the overall mean evaluation of maximum response is more obvious than that of excitation parameters. (II) All uncertainty parameters are simultaneously considered, and these uncertainties, relative to other cases, have the greatest effect on the response evaluation. (III) The results from PEP are basically in agreement with those acquired using MCS, in which the maximum error between them is just 5.016%.


## 5.1.3. Response Evaluation Involving Uncertainty Correlated Parameters

To investigate the evaluation process of response uncertainty involving correlated parameter variables, the example, hereinabove, considering all uncertainty parameters (T, S0, ξ, and f), will be further analyzed.While considering the correlation between the durationT and the intensity S0, the third-moment pseudo-correlation transformation is applied first, so as to convert their correlated variables into independent standard normal variables. The correlated matrix including all parameters can be written as follows:(48)ρ=10.6000.610000100001.According to equations (23)-(30), the polynomial coefficients of each parameter can be calculated, and the result is provided in Table 9.Table 9
Polynomial coefficients of each parameter.
VariablesaibiciT−0.1560.9750.156S0−0.6750.2960.675ξ−0.1560.9750.156f010Combining equations (27)-(29) and Cholesky decomposition, the lower triangular matrix L0 can be obtained as follows:(49)L0=10000.6150.7880000100001.Substituting the lower triangular matrixL0 into equation (33), these correlation variables, utilizing the five-point estimation, were transferred as mutually independent estimation point coordinates Xi in the original space. These transformation results are shown in Table 10.Table 10
Five-point estimated coordinates in original space after correlation transformation.
Parameter variablesThe original space coordinates corresponding to five estimated pointsX2+X1+X0X1+X2+T1.4295.93310.00014.07018.571S0−9.13 × 10−1−9.208 × 10−26.500 × 10−11.3922.213ξ4.287 × 10−31.180 × 10−23.000 × 10−24.220 × 10−25.571 × 10−2f8.572 × 10−11.4582.0002.5423.143Substituting the estimation point coordinates in original space from Table10 and their weight (see Table 1) into equation (17), the evaluated result of univariate conditional mean, conditional mean square, and conditional standard variance square of response, i.e., μmi(X), μmi2(X), and σmi2(X), and their function value corresponding five estimation point in original space can be obtained in Tables 11-13.Table 11
Estimation results forμmi(X) in original space after correlation transformation.
Parameter variablesμmiThe estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T4.561 × 10−23.169 × 10−24.260 × 10−24.602 × 10−24.814 × 10−24.979 × 10−2S04.602 × 10−2−6.464 × 10−2−6.519 × 10−34.602 × 10−29.856 × 10−21.566 × 10−1ξ6.170 × 10−23.220 × 10−11.170 × 10−14.602 × 10−23.272 × 10−22.578 × 10−2f6.079 × 10−25.118 × 10−11.135 × 10−14.602 × 10−22.314 × 10−21.257 × 10−2Table 12
Estimation results forμmi2(X) in original space after correlation transformation.
Parameter variablesμmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T2.086 × 10−31.004 × 10−31.815 × 10−32.118 × 10−32.317 × 10−32.479 × 10−3S03.620 × 10−34.179 × 10−34.251 × 10−52.118 × 10−39.714 × 10−32.455 × 10−2ξ5.582 × 10−31.037 × 10−11.369 × 10−22.118 × 10−31.071 × 10−36.143 × 10−4f7.062 × 10−32.620 × 10−11.289 × 10−22.118 × 10−35.356 × 10−41.582 × 10−4Table 13
Estimation results forσmi2(X) in original space after correlation transformation.
Parameter variablesσmi2The estimation results corresponding to five estimated pointsX2+X1+X0X1+X2+T8.523 × 10−52.303 × 10−49.779 × 10−58.076 × 10−57.249 × 10−56.693 × 10−5S01.380 × 10−41.593 × 10−41.621 × 10−68.075 × 10−53.704 × 10−49.360 × 10−4ξ2.128 × 10−43.953 × 10−35.220 × 10−48.076 × 10−54.081 × 10−52.342 × 10−5f3.850 × 10−41.181 × 10−26.015 × 10−48.076 × 10−51.773 × 10−54.659 × 10−6Having taken into account the effect of correlation of durationT and spectral S0 of the excitation, the overall mean and standard deviation of maximum response, combing Tables 11–13 and equation (16), are able to be evaluated as follows:(50)μMCor=EEμmX≅∑i=14μmi−4−1μmμ=7.606×10−2,σMCor=Eμm2X+Eσm2X−μMcor2=0.869.In order to further investigate the uncertainty effect on stochastic response involving correlated parameter variables in detail, four cases involving correlated parameter variables, i.e., given parameters, structure parameters (ξ and f), excitation parameters (T and S0), and all parameters (T, S0, ξ, and f), are considered, respectively. Meanwhile, the accuracy of their results is able to be confirmed by utilizing the MCS with a size of 106. The results of these cases are listed in Table 14.Table 14
Result comparison between independent variables and correlation variables.
Considered casesμMIndσMIndμMCovσMCovError with MCS (%)EμMCovEσMCovGiven parameters4.602 × 10−28.986 × 10−3——Structural parameters2.498 × 10−27.473 × 10−37.647 × 10−27.202 × 10−23.7303.311Excitation parameters4.570 × 10−28.001 × 10−44.561 × 10−24.061 × 10−24.1242.895All parameters6.463 × 10−26.797 × 10−27.606 × 10−28.238 × 10−22.5783.193From Table14, some conclusions, in this example, can be revealed: (I) the correlation in this example has a great influence on response evaluation relative to the independent situation: if ignored the effect of correlation, the evaluation result in overall mean of maximum response will be underestimated, especially for structural property parameters and all parameters. When considering the excitation characteristic parameter, the overall standard deviation of the maximum result will be overestimated; when considering all parameters, the opposite conclusion will be obtained. (II)While the correlation is taken into account, the effect of excitation characteristic parameters on the overall mean of maximum response and the overall standard deviation is more obvious than in other cases. (III) The calculation results including correlation are basically in agreement with those obtained using MCS, in which the maximum error between them is just 4.124%.


## 5.2. Example 2: 2-DOF Linear Simply Isolated Bridge Subjected to Dynamic Excitation

Figure2 shows a 2-DOF linear isolated bridge in which one of the masses is subjected to dynamic vertical excitation P(t).Figure 2
The calculation diagrams of simply supported isolated bridges.Assuming their structure-property parameters, that is, massm, elastic modulus E, and initial moment I, are the same, the natural frequency ωi of the simply supported isolated bridge, based on linear vibration theory, can be provided as follows:(51)ω1=5.69EImL3,ω2=22EImL3,where E, I, and L are the elastic modulus, initial moment, and span of the bridge, respectively.In the frequency domain, the standard deviation of the displacement responses, based on the random vibration theory, is obtained as follows:(52)σy12=πS04m212ξ1ω13+12ξ2ω23+8ξ1ω1+ξ2ω2ω12−ω222+4ξ1ξ2ω1ω2ω12+ω22+ξ12+ξ22ω12ω22,σy22=πS04m212ξ1ω13+12ξ2ω23−8ξ1ω1+ξ2ω2ω12−ω222+4ξ1ξ2ω1ω2ω12+ω22+ξ12+ξ22ω12ω22,where σyi is the standard deviation of the displacement responses of each mass, S0 is the spectral intensity of the excitation, which is modeled as Gaussian white noise, and T is its corresponding duration νi≈2π/ωi is approximately the mean cross ratio corresponding to ith mass, and both masses have the same damping ratio ξ. The joint and marginal PDFs of these uncertainty system parameters are unknown, except for the first three moments and the correlation matrix, as listed in Table 15.Table 15
Statistic moments and correlation matrix of uncertain system parameters.
ParameterMeanCOVSkewnessCorrelation matrixT15 s0.40.4311.00.50.51.0S0780 cm2/s30.61.608ξ10.0430.30.431ξ20.0690.30.431m4.616 × 105 kg0.10.301L1.500 × 104 mm0.10.3011.00.30.20.31.00.10.20.11.0I7.087 × 1011 mm40.10.301E3.150 × 104 MPa0.10To investigate the evaluation influence of both masses on response uncertainty involving correlated parameter variables, the mean and standard deviation of the maximum response evaluation under the given parameters are contrasted with the correlation parameters. Meanwhile, for the uncertainty response evaluation, the following three cases were analyzed:Case 1.
Considering only uncertainties contained in excitation parameters,S0 and T.Case 2.
Considering only uncertainties contained in the structural parametersL, I, E, m, and ξ.Case 3.
Considering uncertainties of all uncertainty parameters,S0, T, L, I, E, m, and ξ.
Partial calculation results with regard to the correlation transformation process (third-moment pseudocorrelation normal transformation technology) are given, such as polynomial coefficients of each parameter (see Table16), the lower triangular matrix L0, and five estimation point coordinates, in the original space (see Table 17).
Combiningequations (27)–(29) and the Cholesky decomposition, thelower triangular matrix L0 can be obtained as follows:(53)L0=10.5360.839001000100001000001000000.3010.953000000.2010.0420.979.
Having been converted by the third-moment pseudonormal transformation technology, the five estimation point coordinates in the original space are listed in Table17.
Both masses’ calculation results of the response evaluation, including correlated parameter variables, are provided inTables18 and 19. Meanwhile, the accuracy of their results can be confirmed by utilizing an MCS with a size of 106, which are listed in Tables 18 and 19 as well.
After observing and contrasting the results in Table18, some conclusions on Mass 1 can be drawn as follows: (I) To consider the uncertain and correlated system parameter variables, relative to the determination parameter situation, the evaluation results of overall mean and standard deviation of the maximum response have a great influence; for the overall mean of maximum response, the evaluation results will be overestimated; for the overall standard deviation of maximum response, the evaluation result will be overestimated under considering structure-property and all parameters, while the evaluation results, under excitation-characteristics parameters considered, will be underestimated. (II) For the overall mean of maximum response, the effect of the excitation-characteristic parameters was more remarkable than structure-property parameters; for the overall standard deviation of maximum response, the opposite conclusion is obtained. (III) Their calculation results are in agreement with those obtained using MCS, in which the maximum error between them is merely 4.993%. Similarly, the conclusions of Mass.1 are suitable for Mass.2 inTable 19.Table 16
Polynomial coefficients of each parameter.
VariablesaibiciT−0.0720.9950.072S0−0.2830.9160.283ξ1−0.0720.9950.072ξ2−0.0720.9950.072E−0.05030.9970.050m−0.05030.9970.050L−0.05030.9970.050I010Table 17
Five estimation point coordinates in original space after correlation transformation.
Parameter variablesThe original space coordinates corresponding to five estimated pointsX2+X1+X0X1+X2+T32.1406.86615.00023.13432.142S02.625 × 105−9.557 × 1037.800 × 1041.656 × 1052.625 × 105ξ17.985 × 10−22.551 × 10−24.310 × 10−26.112 × 10−27.985 × 10−2ξ212.8114.093 × 10−26.930 × 10−29.710 × 10−21.281 × 10−1E5.934 × 1053.990 × 1054.616 × 1055.242 × 1055.935 × 105m1.928 × 1041.296 × 1041.500 × 1041.703 × 1041.929 × 104L9.263 × 10115.882 × 10117.087 × 10118.292 × 10119.627 × 1011I4.249 × 1042.629 × 1043.150 × 1043.671 × 1044.249 × 104Table 18
Response evaluation result comparison between presented method and MCS for Mass.1.
Considered casesPresented methodMCSMethod error (%)μMCovσMCovμMMCSσMMCSEμMEσMGiven parameters5.141 × 10−34.256 × 10−3————Structure parameters3.952 × 10−31.466 × 10−23.767 × 10−31.538 × 10−24.9114.681Excitation parameters3.657 × 10−35.061 × 10−33.493 × 10−35.327 × 10−34.5004.993All parameters2.466 × 10−31.452 × 10−22.553 × 10−31.412 × 10−23.4162.832Table 19
Response evaluation result comparison between presented method and MCS for Mass.2.
Considered casesPresented methodMCSMethod error (%)μMCovσMCovμMMCSσMMCSEμMEσMGiven parameters7.251 × 10−32.186 × 10−3————Structure parameters1.808 × 10−31.835 × 10−21.886 × 10−31.905 × 10−24.1363.675Excitation parameters6.403 × 10−34.512 × 10−36.379 × 10−34.370 × 10−3.7623.249All parameters9.595 × 10−41.850 × 10−29.255 × 10−41.768 × 10−23.6734.638


## 6. Conclusions

This study focused on evaluating uncertainty response with correlated random variables, the primary conclusion, on the basis of two examples, which were able to be drawn:(1)
Without being any sensitivity analysis with regard to the maximum response, the first two statistical moments of the maximum response, including correlated system parameter variables, were able to be fast evaluated.(2)
With just needing the first three moments and correlation matrix of system parameter variables instead of knowing their joint PDF or marginal PDF, the mean value and standard deviation of the maximum response, including correlated system parameter variables, were capable to be steadily evaluated.(3)
While taking into account the correlated system parameter variables, the overall mean and standard deviation’s evaluation results of maximum response, relative to the determination parameter situation, have a great influence: ignoring the effect of these parameter variables will underestimate or overestimate the actual response evaluation.(4)
When considering the effect of correlation among system parameter variables, one can find the effect of statistic maximum response evaluation with respect to structural property parameters is relatively obvious than excitation characteristic parameters in example 1, whereas the opposite conclusion can be obtained in example 2.(5)
Compared with the ones using MCS, the accuracy of response evaluation by the presented method was trustworthy.



---

*Source: 1496358-2022-06-06.xml*