Probabilistic Analysis for Structures with Hybrid Uncertain Parameters

In practical engineering problems, the distribution parameters of random variables cannot be determined precisely due to limited experimental data.)e hybrid uncertain model of interval and probability can deal with the problem, but it will produce extensive computation and it is difficult to meet the requirement of the complex engineering problem analysis. In this scenario, this paper presents a vertex method for the uncertainty analysis of the hybrid model. By combining the traditional finite element method, it can be applied to the structural uncertainty analysis. )e key of this method is to demonstrate the monotonicity between expectation and variance of the function and distribution parameters of random variables. Based on the monotonicity analysis, interval bounds of the expectation and variance are directly calculated by means of vertex of distribution parameter intervals. Two numerical examples are used to evaluate the effectiveness and accuracy of the proposed method. )e results show the vertex method is computationally more efficient than the common interval Monte Carlo method under the same accuracy. Two practical engineering examples are to show that the vertex method makes the engineering application of the hybrid uncertain model easy.


Introduction
Uncertainties are widely encountered in practical engineering problems, and traditional analysis techniques for uncertainties are based on probability theory. In probability theory, uncertain parameters are represented by random variables, which are quantified by distribution function and correlation coefficient. Many probabilistic analysis methods that are based on probability models have been well developed, such as the Monte Carlo method [1,2], full factorial numerical integration [3], univariate dimension reduction [4], first-order reliability method [5,6], and Bayesian approach [7,8]. For the uncertainty analysis of structures, the probabilistic theory and the finite element method are combined to generate stochastic finite element analysis (SFEM) [9]. SFEM techniques include Monte Carlo simulation, perturbation method [10,11], and spectral stochastic finite element method [12,13]. However, constructing the precise probability model requires a large amount of experimental data which is impractical, especially for the complex engineering problem.
To overcome the disadvantages of the probabilistic method, hybrid uncertain models are developed. e hybrid uncertain models integrate the classical probabilistic method and interval analysis, which are commonly divided into two types [14]. For the models of the first type, uncertain model parameters with sufficient experimental data are represented by the probability model, and uncertain parameters with limited information are quantified by intervals. Several analysis techniques for this model have been developed [15][16][17][18][19]. Hybrid models of the second type are also called probability-box or interval random variable models in the literature [20][21][22][23]. All of the uncertain parameters are quantified by the probability model, including some interval distribution parameters, due to the limited information. In the traditional probability analysis, this model can be directly constructed by interval estimation with a confidence level; hence, it is considered as an extension of the probability model. Since this model was proposed by Elishakoff and Colombi [24], it has been applied in the multifield of structural analysis, and several analysis methods for this model have been developed. Based on the monotonicity analysis for probability transformations of random variables, Jiang et al. [25,26] proposed two efficient algorithms for solving the second hybrid reliability model. By combining Monte Carlo simulation and finite element method, Zhang et al. [27,28] proposed an interval Monte Carlo method for computing the failure probabilistic intervals and the response probability bounds of structures with interval probability variables. Xia and Yu [13] developed an interval random perturbation method for obtaining the interval expectations and variances of the responses of acoustic fields and structural-acoustic systems with interval probability variables. Xiao and Lu [29] presented a sampling method for the structural response analysis with dependent interval random variables. Xiao et al. [30] improved the double-loop sampling method by using monotonicity analysis for the interval random variables. Liu et al. [21] used the optimized univariate dimension reduction method and the Johnson distributions fitting method to calculate the probability bounds of the response function. Liu et al. [31] proposed a new uncertainty propagation method that is based on the sparse grid technique and saddle point approximation.
From the overall perspective, significant successes have also been realized with the hybrid interval probability model. However, studies on interval random variables are still in their preliminary stage. Especially for the second type of hybrid uncertainty model, some important issues that are associated with its response analysis remain unsolved. One of the very challenging tasks is to improve the computational efficiency. e second hybrid model can be regarded as a set of many probability models that have the same distribution function and different distribution parameters; hence, uncertainty analysis of the hybrid model typically is of products with higher cost than that of the corresponding probability model. e Monte Carlo method is the simplest and the most versatile method for uncertain problems with random variables and can be combined with the finite element method to solve practical engineering problems. However, two-layer nesting sampling is commonly involved in the hybrid model analysis [32], which will lead to extremely low computational efficiency especially for complex problems. Various modified methods that are based on the perturbation technique are provided for improving the computational efficiency [33]. ese methods use Taylor series expansion to calculate the relative matrix; thus, the available deterministic finite element analysis (FEA) cannot be directly utilized. To apply these methods to the practical problems, new and more complex FEA codes must be developed. To use the hybrid uncertain model in practical applications, the efficient and practical computational techniques must be developed.
In this paper, a new methodology is presented for the static structural response analysis with uncertain parameters, which is represented by the second type of hybrid model. is hybrid model is named the interval random variable model in this paper. For obtaining the intervals of expectation and standard variance of the response vector of the static structural system with interval random variables, a vertex method is proposed. A monotonicity analysis is conducted for the expectation and variance of the response vector with respect to interval variables. On the basis of the monotonicity analysis, the intervals of expectation and variance of the response vector are calculated by the classical probabilistic method. e response vector of the complex structural system is calculated by the determined FEA. e remainder of this paper is organized as follows. e mathematical background regarding the interval random arithmetic is discussed in Section 2. e method for the static structural system with interval random variables is proposed in Section 3. In Section 4, the effectiveness and efficiency of the proposed method are evaluated by four numerical examples. In Section 5, the conclusions of this study are presented.

Expectation and Variance of Interval
Variables. Let X denote a random vector with n independent random variables: (1) Assuming a distribution, the parameters of X i are interval variables. e interval variables that are associated with the random parameter X i can be represented as an interval vector or in component forms where x L i and x R i are the lower and upper bounds, respectively, of the interval variable vector x I i ; j is the number of interval variables in the interval vector x I i ; and x L i,j and x R i,j are the lower and upper bounds, respectively, of the interval variable x I i,j . us, the interval random vector with interval parameters x I i can be denoted as e expectation E(X(x I )) and variance var(X(x I )) of the independent interval random vector X(x I ) can be expressed as

Calculation of the Expectation and Variance Based on
Monotonicity Analysis. Consider the following linear function of n independent interval random variables: f X x I � a 0 + a 1 X 1 x I 1 + a 2 X 2 x I 2 + · · · + a n X n x I n , (6) where X i (x I i ) is an interval random variable and a i is a coefficient of the linear function. Based on classical statistical theory, the expectation and variance of the response distribution of f(X(x I )) are To obtain the lower and upper bounds of expectation and variance of the interval random function f(X(x I )), a new arithmetic will be proposed in this section. e spatial derivative of E(f(X(x I ))) with respect to x i,j can be expressed as A monotonicity analysis of the expectation for commonly used PDFs is conducted, and the results are presented in Table 1. According to Table 1 is monotonic, and extreme points of E(f(X(x I ))) exist on the boundaries of x I . In this case, the vertex method can be used to calculate the extreme points.
minimizes E(f(X(x I ))). In other words, the optima of E(f(X(x I ))) are the vertex of the interval x I . According to the monotonicity analysis, the values that minimize and maximize E(f(X(x I ))) can be determined. Subscripts L and R represent the minimum and maximum values, respectively, of E(f(X(x I ))). For example, (x L i,j ) L is the value at which the minimum is attained and (x L i,j ) R is the value at which the maximum is attained. e extreme minimum value vector and the extreme maximum value vector are denoted as x L and x R . us, extreme values of the expectations can be expressed as Hence, where E I (f(X(x I ))) is the interval of expectation of f(X(x I )). e partial derivative of var(f(X(x I ))) with respect to which is always negative or positive for the commonly used PDFs in Table 1. Similar to the analysis above, the interval of variance of f(X(x I )) can be expressed as where var I (f(X(x I ))) is the interval of variance of f(X(x I )). According to the above analysis, the expectation and variance of equation (6) with the uncertain distribution parameters are monotonous. Based on the monotonicity analysis, the vertex method is presented to quickly calculate the interval bounds of expectation and variance by equations (11) and (12).
is vertex method can avoid the optimal iterative search process.

Structural Uncertainty Analysis Static
System with Interval Random Variables

Calculation of the Expectation and Variance of a Structural
Static System. In the previous section, an interval random vector is defined. An interval random matrix is constructed in this section, which is a matrix whose elements are the interval random variables. Consider the real linear equations that are solved by a displacement-based linear static FEA.
In this paper, it focuses on the large finite element (FE) model with a load vector that includes interval random variables. Hence, the real linear equations can be expressed as where K is the real stiffness matrix, F(X(x I )) is the interval random structure load vector, and U is the interval random displacement response vector of the system. e solution of (13) that is considered in this paper is expressed in terms of the intervals of expectation and standard variance of U.
Solving the matrix equation (13), the interval random displacement response vector can be expressed as where en, the nodal displacement u i (x I ) can be obtained as Mathematical Problems in Engineering Merging the same interval random X i (x I i ), equation (16) can be expressed as where (a 1 , a 2 , . . . , a n ) is the coefficient of the linear function. According to equations (10) and (12), the interval expectation and variance of u i (X(x I )) are

Solving the Coefficient of Uncertainty Function.
To obtain equation (17), it must solve equation (14) to obtain the inverse of the stiffness matrix, namely, K − 1 . However, the computation of K − 1 is time-consuming, and K − 1 , which is typically nonsingular, requires a large amount of memory for storage, especially for large-scale engineering problems.
In this section, a new method is proposed for computing the coefficient in equation (16), which avoids the computation of K − 1 .
Consider n independent interval random variables X i (x I i )(i � 1, 2, ..., n). Sampling n sets from X i (x I i ) and substituting them into equation (13), the node displacements are calculated by the following equation: u i,1 � a 1 X 1,1 + a 2 X 1,2 + · · · + a n X 1,n , u i,2 � a 1 X 2,1 + a 2 X 2,2 + · · · + a n X 2,n , . . . u i,n � a 1 X n,1 + a 2 X n,2 + · · · + a n X n,n , (19) where (X i,1 , X i,2 , . . . , X i,n )(i � 1, 2, . . . , n) is the i-th sampling data from (X 1 (x I 1 ), X 2 (x I 2 ), . . . , X n (x I n )),(a 1 , a 2 , . . . , a n ) are coefficient that are associated with inverse of the stiffness matrix, and (u i,1 , u i,2 , . . . , u i,n ) is the i-th node displacement under n sampling data. By solving the linear equation (19), the coefficient in (17) can be obtained. In engineering application, the node displacements can be calculated by the commercial FEA software. erefore, the present algorithm avoids to solve the inverse matrix of the stiffness matrix and improves the computation efficiency of the vertex method.

Static Response of a Truss Structure.
To evaluate the efficiency of the approach that is presented in this paper, a linear elastic plane truss is considered, as illustrated in Figure 1. e design values of the structural parameters for all members are the cross-sectional areas for elements A 1 ∼A 6 � 10.32 cm 2 , the cross-sectional areas for elements A 7 ∼A 15 � 6.45 cm 2 , and Young's modulus E � 2 × 10 11 Pa.
Suppose that loads P 1 , P 2 � 0.3, and P 3 � 0.1 m are interval random variables; their parameters are listed in Table 2.
In the following, the interval Monte Carlo method (IMC) [34] and the vertex method (VM) are used to calculate the lower and upper bounds of the expectation and variance of the structural displacement response at node 5. In every IMC method, the first step is to arbitrarily generate intervals via the inverse transform method, and the second step is to use interval FEA to compute the interval structural system. en, the lower and upper bounds of the expectation and variance can be determined. is model has been investigated with the IMC method in [34]; hence, the results with 5000, 50000, and 500000 samples are presented in Table 3. e intervals of the expectation and variance for the proposed method are also specified in Table 3. To compare the two methods in terms of efficiency and accuracy, relative errors are calculated and are presented in Table 4.
According to Table 4, the accuracy of the results is influenced by the simulation times for IMC. To improve the accuracy, more computational time is required. us, the results of IMC 500,000 are closer to the exact values. e intervals of expectation and variance that are calculated by the proposed method and IMC 500,000 are close to each other. e IMC is time-consuming as large simulations and interval arithmetic are required. For realizing the same level of accuracy with the proposed method, the total number of simulations for IMC 500,000 is 5 × 10 5 . Because of the monotonicity analysis, the proposed method saves much time.

Static Response of a Cantilever Plate.
Consider the analysis of a cantilever plate, as illustrated in Figure 2. Suppose that there is no preload in any element, and only a force is applied at the top-right corner of the plate. e material of this plate is steel. e deterministic values of the plate are a Poisson ratio of m � 0.3 and a thickness of H � 0.1 m. Suppose that Young's modulus E, the concentrated loads P, and the density ρ are interval random variables. as presented in Table 5. e finite element method is used to compute the displacements in the plate A. To demonstrate the analysis technique, the 3-node triangular element idealization in Figure 2(b) is considered.
In this example, the finite element model includes 8 nodes and 6 elements. e self-weights of the triangular elements are considered and are equivalent to the element nodal load matrix. As Young's modulus E is an interval Mathematical Problems in Engineering random variable, the inverse of the stiffness matrix K cannot be calculated directly. Instead, it can obtain K � EK 1 and K − 1 � 1/EK − 1 1 , where K 1 is a new matrix and K − 1 and K − 1 1 are the inverse matrices of K and K 1 , respectively. e node displacement vector is U � K − 1 F � 1/EK − 1 1 F. e displacement of node 4 can be expressed as u 4 � (1/E)f(P, ρ), where f(P, ρ) is a linear function of P and ρ. e expectation and variance can be computed via the following equations: where the interval bounds of E(1/E) and var(1/E) can be calculated by classical numerical integration. As F(P, ρ) is a linear function, the proposed method is used to be compute E I (F(P, ρ)) and var I (F(P, ρ)). Synthesizing the above method, the interval expectation E I (u 4 ) and the interval variance var I (u 4 ) are obtained: From this example, the proposed method can be applied to FE models in which the stiffness matrix includes common interval random variables.

Automobile Frame Structure.
e proposed method is applied to the uncertainty analysis of a practical automobile frame, which is modified from the numerical example in [35]. As illustrated in Figure 3, the frame is composed of two side beams and eight cross beams, and four equivalent distributed forces, namely, Q 1 , Q 2 , Q 3 , and Q 4 , are applied to the frame, which result from the operator cabin, engine assembly, gasoline tank, and goods, respectively. Because the vertical stiffness affects the performance of the automobile, the real maximum displacement d of the frame in the Ydirection is analysed. In this application, Q 1 , Q 2 , Q 3 , and Q 4 represent four uniformly distributed forces that act on the frame. Young's modulus E is 2.07 × 10 5 MPa. e density ρ and the four external forces Q 1 ∼ Q 4 are treated as interval random variables, which are listed in Table 6.
In this example, FEA is used to compute the real displacement d of the frame in the Y-direction. FE model of the automobile frame structure includes 30854 nodes and 29038 shell elements, and it has 92562 degrees of freedom. So the inverse matrix of the stiffness matrix is a 92562 × 92562 nonsingular matrix. It is hard to solve and save the inverse matrix. However, it is very easy to solve the displacement vector of the automobile frame structure with commercial FE software. For this problem, the solve process is given as follows. By the method in Section 3.2, the function of uncertain parameter and output response can be calculated. Firstly, 5 sets of data sampling from the five interval variables are denoted as (ρ i , 1, 2, ..., 5). Secondly, using the 5 sets of data solves the nodal displacement with the FEA and will get six nodal displacements of d i (i � 1, 2, . . . , 5). en, the linear coefficient of the interval random variables associated with displacement response d is solved by equation (19). Lastly, the interval expectation and variance of d can be computed by the present method. If it is to compute the other nodal interval expectation and variance, result from 5-time computations of FEA can be utilizable.
To evaluate the efficiency, multiple nodal interval expectations and variances and their computation times with the proposed method are listed in Table 7. Simulation of this FE model and solution of the expectation and variance are conducted using ANSYS 12.1 and MATLAB 7.1, respectively, on a 3.10 GHz Intel (R) Core (TM) i3-2100. e runtime for a single FE model with the ANSYS software is 14.1082 s, and 5 rounds of FEA require 70.5 s. e computation time of a nodal interval of expectation and variance with MATLAB code is 0.004 s. Hence, the total computing time for the expectation and variance of a node is 70.5 s. Point A in Figure 3 is the most likely point for generating the maximum displacement in the Y-direction. Via the    Figure 2: Finite element idealization of the cantilever plate: (a) a cantilever plate; (b) 3-node triangular element model.     Table 7. e acquisition of all nodal expectations and variances only requires 208.066 s, which is the sum of 6 times the computation time of FE model and the product of 30854 nodes and 0.004 s. e double Monte Carlo method with 10000 samples is used to calculate the displacements of 30854 nodes [30]. e calculation requires 4.3529 × 10 5 s, which is equivalent to 120.9151 h, which is more than 2000 times the computation time of the proposed method (Table 8). According to this numerical example, the proposed method is highly computationally efficient for this complex engineering problem.

A Tall Wind Turbine Tower.
Tower is mainly bearing structure of a large wind turbine. It supports the gravitational and inertial loads of nacelle and wind rotor as well as aerodynamic loads, actuation loads, and so on [36]. Tower reliability and safety is key factor in ensuring the normal operation of the wind turbine. It is important to acquire the accurate loads under the limit state. However, tower loads are difficult to calculate. Load calculation involves multisource uncertainties that include variations in wind tunnels measurements, geometric distortions of the blade under loading, 3D rotational correction, and so on [37]. erefore, tower loads under the extreme wind speed cannot quantify by the probability model. e interval random variables are an effective means of quantifying the loads. In this section, the loads are represented by interval random variables. e vertex method is used to calculate the displacement of tower.
In this example, tower is from a 2 MW horizontal-axis wind turbine. e diameter of the wind rotor is 100 m, and the high of hub center is 80 m. e geometry dimensions of tower are shown in Figure 4. ANSYS software is used to analyze the structural response. SOLD45 solid elements and hexahedron mesh are applied to establish the finite element model, which includes 153686 nodes and 190742 elements. Loads and the material parameters are given in Table 9. FE model is plotted in Figure 5. e displacement counter is also given in Figure 6, which is used to validate the FE model. ere are 6 interval random variables in the FE model of wind turbine tower. According to equation (22), 6 times of random sample is performed to acquire the from the interval random variables F x , F y , F z , M x , M y , and M z . Substituting them into the finite element model, 6 groups of nodal displacement are calculated by the AYSYS software. Solving equation (19), coefficient a i (i � 1, 2, 3, . . . , 6) is computed. Based on theoretical analysis, the maximum displacement locates the top of the tower. So, nodes of top tower are selected and their coefficient a i (i � 1, 2, 3, . . . , 6) is calculated.
ere is monotonicity between loads and displacement, so the vertex method can be used to calculate the interval expectation and variances. e nodal displacement of 121 nodes located in the top of tower is computed and shown in Figure 7. In Figure 7, red points and black points are the lower and upper bounds of expectation, respectively. Standard deviations of displacement are calculated and plotted in Figure 8. Because standard deviations of the interval random variables are constants, standard deviation of a nodal displacement is also a constant. In practical engineering design of wind turbines tower, the maximal displacements are concerned. So, the maximal upper bounds of expectation and the maximal standard deviations are selected as the design values, which are shown in Table 10. ese parameters could be used for reliability analysis and structural design.
is example is to show the analysis and design produce using the interval random variables. e traditional uncertainty analysis methods for the interval random variables are time-consuming, which makes it difficult to solve the complex engineering problem. e proposed method takes full advantage of the monotonicity and makes the engineering application of the hybrid model possible.

Conclusion
In this paper, a vertex method is proposed for the static analysis of structures that have interval random variables. Expressions for calculating the expectation and variance of interval random responses are developed. A monotonicity analysis of the response expectation and variance with respect to interval distribution parameters and variables is conducted. On the basis of the monotonicity, the intervals of the expectation and the standard variance of the response vector are obtained by the traditional probability method. e efficiency and accuracy of the proposed method are demonstrated based on four examples. e proposed method is simple. e developed FEA code utilizes a highly popular method that has been widely used for analysis of practical engineering structures. Solving for the expectation and variance of a linear function with a random vector, traditional probability theory yields an accurate mathematical expression. Monotonicity is applied to compute the extreme values of the function. ese traditional methods are integrated to form the vertex method. erefore, the proposed method and the traditional method are compatible. Although the proposed method can be used only for a specified type of uncertain structures, it can address large-scale engineering problems with a traditional technique. In the future work, it will extend the proposed technique to the uncertain analysis of more general structural models.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.