Reliability analysis and design optimization of nonlinear structures

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Reliability analysis and design optimization of nonlinear structures Pinghe Ni , Jun Li , Hong Hao , Weimin Yan , Xiuli Du , Hongyuan Zh PII: DOI: Reference:

S0951-8320(19)31049-X https://doi.org/10.1016/j.ress.2020.106860 RESS 106860

To appear in:

Reliability Engineering and System Safety

Received date: Revised date: Accepted date:

18 August 2019 23 January 2020 8 February 2020

Please cite this article as: Pinghe Ni , Jun Li , Hong Hao , Weimin Yan , Xiuli Du , Hongyuan Zh , Reliability analysis and design optimization of nonlinear structures, Reliability Engineering and System Safety (2020), doi: https://doi.org/10.1016/j.ress.2020.106860

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Highlights: 

Reliability-based design optimization (RBDO) of nonlinear structures are conducted.



Kriging based method and First-order Reliability Method are used.



Numerical studies on nonlinear structures are performed.



Kriging based method is more accurate and efficient for RBDO.

1

Reliability analysis and design optimization of nonlinear structures Pinghe Ni1, Jun Li 2, Hong Hao 2,*, Weimin Yan1, Xiuli Du1, Hongyuan Zhou1 1

Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China

2

Centre for Infrastructural Monitoring and Protection, School of Civil and Mechanical Engineering, Curtin University, Kent Street, Bentley, WA 6102, Australia Emails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]

Abstract: Reliability analysis and design optimization of structures have been gaining a significant amount of attention in recent decades. Most of the current studies are based on linear structural analysis. The study on reliability analysis and design optimization for nonlinear structures has not been well explored. This paper presents studies on reliability analysis and design optimization for nonlinear structures, by using the Kriging based method and First-Order Reliability Method (FORM). Numerical studies on nonlinear reinforced concrete structures and steel frame structures are carried out to verify the accuracy and efficiency of the proposed methods. The results demonstrate that the FORM and Kriging based methods have the same accuracy as those from Monte Carlo Simulation (MCS) method. Reliability-based design optimization (RBDO) is conducted for nonlinear structures, in which the dimensions of structures can be optimized and the target occurrence probability can be achieved. Compared with FORM based RBDO method, the Kriging based method is more accurate and efficient. The response sensitivity is not required in the Kriging based method, which makes it more versatile.

Keywords: Reliability analysis; Reliability-based design optimization; First-order reliability method; Kriging method; Nonlinear structure; Design optimization.



Corresponding Author, John Curtin Distinguished Professor, Centre for Infrastructural Monitoring

and Protection, School of Civil and Mechanical Engineering, Curtin University, Kent Street, Bentley, WA 6102, Australia. Email: [email protected] 2

1 Introduction The objective of structural design optimization is to find out the optimal structural dimensions and configurations by minimizing the specific volume/weight/cost of structures under certain constraints on structural stress, deformation and/or natural frequencies, etc. In deterministic structural design optimization, the uncertainties in loadings and system parameters are taken into account by using safety factors and considering the worst-case design scenario [1]. These procedures overestimate the effect of uncertainty and lead to conservative designs, which is cost ineffective. In the reliability-based design optimization (RBDO), the structural failure probability taking account of the uncertainties is directly considered as probability constraints in the optimization procedure. RBDO provides not only cost-effective results of structural design, but also a high-confidence solution. RBDO has been successfully used for the design of tuned mass damper system [2], composite structures [3, 4], bridge structures [5-7], and reinforced concrete structures [8-10]. For example, in earthquake engineering, the seismic response calculation of structures involves a number of uncertain parameters, i.e. material properties and site characteristics. Following the guidelines [11], the performance of a structure should satisfy a set of reliability criteria under frequent and rare seismic events. Therefore, RBDO can be considered as an ideal approach for structural seismic design. In the reliability-based optimization, a double-loop method is usually conducted to find out the optimization design solution for a structure, considering the probabilistic constraints. The constraints are implicit functions of design variables and random variables. In the outer loop, the design variable is optimized in each iteration, while in the inner loop, reliability analysis based on finite element model is performed to estimate the failure probability. RBDO requires to determine the failure probability in the optimization process, which could take a long time for calculation. It is necessary to choose an efficient approach for reliability analysis. Many reliability analysis methods, such as the First-Order Reliability Method (FORM) [12], the Second-Order Reliability Method [13] and Monte Carlo Simulation (MCS) method, have been proposed to estimate the failure probability of structures. However, numerical simulation techniques may not be suitable for reliability-based optimization 3

problems, because it requires a large number of simulations for reliability analysis and therefore it is computational intensive. FORM is a highly efficient method for finite element model based reliability analysis, which has been extensively studied in different engineering problems, i.e., probabilistic design of earthquake-resistant structures [14], reliability assessment of steel industrial structures [15] and uncertainty analysis of nonlinear systems [16]. When using FORM, the performance function is approximated by the first-order Taylor expansion [17]. The failure probability can be obtained from the inverse transformation of the reliability index at the most probable failure point (MPP). The calculation of response sensitivity matrix with respective to the random variables is a necessary component in FORM. When the response sensitivity matrix is unavailable, the iterative search of MPP cannot proceed and FORM cannot estimate the failure probability [18]. Such situations widely occur in the nonlinear finite element analysis. Many efforts have been made to improve FORM [18]. The finite difference method and the direct differentiation method are the two widely used methods for the calculation of the response sensitivity [19]. Finite difference methods may suffer from the accuracy problem in the estimation of response sensitivity with different incremental values in the perturbed parameters. The direct differentiation methods can provide more consistent results than finite difference methods. However, in some cases, the performance function cannot be differentiated analytically and the response sensitivity is unavailable. It is also worth mentioning that FORM cannot provide adequately accurate results for highly nonlinear performance functions [20]. Such challenges restrict the application of FORM for nonlinear finite element reliability analysis. In recent years, surrogate model techniques gain a large amount of attention, due to their high computational efficiency for uncertainty analysis. The surrogate model is an analytical model to represent the output of a physical system, using a limited number of design of experiments (DOE). The surrogate model techniques, such as the responses surface method [21], artificial neural networks [22], polynomial chaos [23-26] and Kriging method [27-30], have been proposed for engineering problems, i.e., reliability analysis of aircraft structures, stability analysis of soil slope, optimization design of composite structures and stochastic dynamic analysis of bridge structures. The response surface method is efficient, yet the accuracy is relatively low. The artificial neural networks require a large number of data samples to train and construct the surrogate model. 4

Polynomial chaos can be used to predict the output of the physical model with a high accuracy. However, when the number of random variables is large, it takes a long time to evaluate the coefficients of polynomial chaos expansions. Kriging method is an interpolation method which was first proposed for the estimation of gold in geostatistics [31]. Kriging method was also developed for reliability analysis with active learning techniques [32-36]. For example, Ling et al. [32] estimated time-dependent failure probability by using the combination of Kriging and U-learning function. The important sampling and subset simulation techniques are also considered to reduce the computational cost. Yuan et al. [33] proposed a new active learning function for system reliability analysis. The sampling point with the maximum probability on the system performance function was selected to update the Kriging model. Other learning functions, such as, expected learning function [37], expected improvement function [30], reliability-based expected improvement function [30] and least improvement function [38], have also been proposed. Kriging method has been successfully used in many complex engineering problems [29, 39-41]. For example. Abdallah et al. [42] proposed Hierarchical Kriging approach to predict the bending moment of a large wind turbine. The uncertainties in the wind turbulence, wind speed, and wind shear were considered. Yi et al. [43] proposed slope reliability analysis method with Kriging method and particle swarm optimization algorithm. The hyperparameters in the Kriging model were estimated by particle swarm optimization algorithm. Compared with pattern search method, the swarm optimization algorithm is more robust and provides stable results. Papadimitriou et al. [44] developed a method to find out the optimal sensor placement of an elastic beam and a plate, in which Kriging method was conducted to estimate the response at unmeasured points. Ling et al. [45] estimated the global reliability sensitivity indices with Kriging method. It has been demonstrated that the Kriging is an efficient method for structural reliability analysis. However, there is limited studies on using surrogate model techniques for nonlinear structural reliability analysis and design optimization. Most of current studies on nonlinear finite element reliability analysis are based on the first-order second-moment (FOSM) technique [18, 46-49], which takes a lot of efforts on deriving the responses sensitivity. This paper presents studies on reliability analysis and design optimization of nonlinear structures with Kriging method and FORM. The uncertainties in the material parameters are considered. A 5

double loop method is used for RBDO of nonlinear structures. In the outer loop, the design variables, i.e. geometry parameters of nonlinear structures, are optimized with sequential quadratic programming method. In the inner loop, the reliability analysis is performed by using the Kriging method and FORM, respectively. Numerical studies on reinforced concrete structures and steel industrial frame are conducted to verify the accuracy and efficiency of using the proposed method for RBDO of nonlinear structures. Results show that the failure probability of nonlinear structures can be estimated accurately. With the proposed design optimization method, the optimized cross-section area can be obtained and the performance of the structure can be improved. Results also demonstrate that the Kriging based design optimization method is more efficient than FORM.

2 Reliability analysis and design optimization 2.1 Introduction of reliability analysis In reliability analysis, the limit state function is introduced to represent the performance of a structure. Let x be the vector of random variables, the limit state function can be expressed as y  g  x

(1)

When g  x   0 , the structure is safe. g  x   0 means that the structure is unsafe. The value of the limit state function is closely related to the random variables x of the physical model. Reliability analysis aims to determine the probability when the structural failure occurs. The failure probability is defined as the probability of limit state function being a negative value, which can be written as

P  F   P  g  x   0

(2)

In practice, the analytical solution of Eq. (2) is often unavailable. Numerical techniques, such as MCS, importance sampling [50, 51] and subset simulations [34, 52], have been developed to obtain the failure probability. However, such methods may take a long time to perform reliability analysis and obtain an accurate results, especially when the failure probability is very low. Usually it may take more than 10,000 runs to obtain the failure probability of a civil engineering structure. 2.2 Reliability analysis with FORM FORM was originally developed by Neil and Hasofer [53], which utilizes the first-order Taylor 6

expansion to approximate the limit state function. Such approximation is made in the standard normal space. A point is defined as MPP, which has the shortest distance from the limit state surface to the origin. It also contains the highest probability density on the performance function. The failure probability can be estimated from the MPP. In the FORM, the random variables x are transformed into standard normal variables u . This transformation is to make the continuous distribution functions (CDFs) of the random variables remain the same, which is known as the Nataf transform [54] u  T  x

(3)

where T is generally a nonlinear mapping that is dependent on the type of random distribution of x . After the transformation, the limit state function can be expressed as y  g  u

(4)

In the standard normal space, the MPP can be located by solving an optimization problem with one equality constraint expressed as

min  = u (5)

subject to g  u   0

The iteration is defined to starts at u0. In the k-th step, the limit state function can be approximated with the first-order Taylor expansion as g  u   g  uk   g  uk  u  uk 

T

 

k where uk is the design point of the k-iteration, and g u

(6)

is the responses sensitivity of g  u 

with respect to the random variables uk. The distance  k 1 and the design point uk+1 are updated in a recursive format as



u

k 1

k 1

  k

 

k 1

g  uk 

g  uk  g  uk 

g  uk 

(7)

(8)

The iteration continues until the convergence criterion is met. The reliability index is defined as the shortest distance from the limit state surface to the origin, which can be obtained as 7

*  u*

(9)

The failure probability can be obtained from the inverse standard normal transformation, which is expressed as Pf      *

(10)

The accuracy and efficiency of FORM has been studied by using explicit limit state functions [47]. It should be noted that, in the nonlinear finite element reliability analysis, the explicit limit state function and the analytical response sensitivity are usually not available. Therefore the accuracy of FORM is usually relatively low.

2.3 Reliability analysis with Kriging method Kriging is an interpolation method, which provides the best linear unbiased estimation of a function. The basic idea of Kriging method is that the output response of a physical model consists of two parts with the first one based on a linear regression model and another based on a stochastic process model. The model output can be express as y  x  f T  x β  z  x

where f  x    f1  x  ,

, f m  x 

T

is the known basis function, β   1 ,

(11)

, m 

T

is the vector of

the regression coefficient, and z  x  is assumed to be a Gaussian stationary process with zero mean. Different basis functions can be chosen to represent the linear regression. Usually the first- or second- order polynomials-based functions are selected. The covariance matrix of the stochastic part of the i-th and j-th DOE points ( xi and x j ) can be expressed as





Cov z  xi  , z  x j    2 R  xi , x j , λ  i, j=1, 2, …, n



where R xi , x j , λ



(12)

denotes a correlation function, n is the number of the experiments (sampled

points),  2 is the variance of the stochastic process model and λ   1 ,

,  p  is the vector of

hyper-parameters. The most widely used correlation function is generalized exponential model that is expressed as [55]

8

2  p R  xi , x j , λ   exp   k  xik  x kj    k 1 

(13)

where xik and x j denote the k-th component of the i-th and j-th DOE points, k is the k-th k

hyper-parameters, and p is the number of the design variables.



To construct a kriging surrogate model, a set of DOE points x  x1 , x2 ,

xn  are generated

first, where n is the number of experiments. The outputs of model are given as Y  x    y  x1  , y  x2  ,

y  xn  . Since Y is assumed to be a Gaussian process, the observation T

vector can be written as

Y~

 Fβ , σ R  2 Y

(14)

where Fβ is the mean value and σY2 R is the variance. The component of Fij can be obtained from Fij  f j  xi  , i=1, 2, …, n, j=1, 2, …, m

(15)

The unknown parameters σY , β and λ can be obtained from the experimental points

x   x1 , x2 , xn  and observation points Y  x    y  x1  , y  x2  , y  xn T . Since Y  x  follows Gaussian distribution, the likelihood function can be expressed as





L Y  x  β, σ , λ  2

det  R 



1 2

N 2 2

 2 σ 

T  1  exp   2 Y  x   Fβ  R1 Y  x   Fβ   2σ 

(16)

The regression coefficient β and the variance σY can be obtained from the least square method as 1 βˆ   F T R1F  F T R1Y

σˆY2 



1 Y  Fβˆ n



1



R 1 Y  Fβˆ

(17)



(18)

and the hyperparameters λ is obtained as

n n 1 λˆ  arg min  log  det  R    log  2 σY2    2 2 2

(19)

When the unknown parameters in the Kriging surrogate model are obtained, the unknown points can be estimated. yˆ  x  is defined as the predicted output from Kriging surrogate model. The 9

augmented observation vector Y T , yˆ  x  

T

also satisfies the Gaussian distribution, which can be

written as  F   R ˆ ˆ2  T  β , σY  T  f  x   r  x   

 Y  ˆ ~  y  x

r  x    1  

(20)

where r  x  is the correlation function between an unknown point x and the design points in the experimental set as





r  x   R x, x1 , λˆ ,



R x, xn , λˆ



(21)

The best linear unbiased prediction of the model output yˆ  x  can be obtained as



T T yˆ  x   f  x  βˆ  r  x  R1 Y  Fβˆ

and variance is expressed as





(22)

σ 2yˆ  x   σˆY2 1  r  x  R1r  x   u  x   F T R1F  u  x  T

T

1



(23)

where u  x   F T R1r  x   f  x 

(24)

The Kriging surrogate model can be combined with the MCS method to estimate the failure probability. A small number of design experiments are generated to construct the Kriging surrogate model. Then, a large number of experimental points are generated from MCS method. The predicted output from the Kriging surrogate model yˆ  x  is used to replace those generated from the physical model y(x), which reduces the computational complexity. Then the failure probability can be obtained from Eq. (2). A more detailed review of the Kriging method can be found in [24]. The Kriging surrogate model can be also combined with the importance sampling algorithms [50] and subset simulation to estimate the rare event failure probability.

2.4 Formulation of RBDO The problem of reliability-based design can be written as Minimize: Cost  θ  Subject to: P  gi  x , θ   0  =Pi  Qi , i  1, 2, , nc

(25)

θlower  θ  θupper 10

where θ is the vector of design variables, Cost(θ) is the objective function, Pi is the probability of the event i and Qi the maximum occurrence probability, gi  x, θ  is the limit state function of event i, nc is the number of probability constraints; θlower and θupper are the lower and upper bounds of the design variables, respectively. The RBDO problem is usually solved by a two-level approach, where the outer loop aims at solving the optimization problem in terms of design variable θ, and the inner loop aims at performing the reliability analysis in terms of random variable x. Eq. (25) can be solved by using the sequential quadratic programming (SQP) method [56]. In the SQP method, the forward different method is conducted to estimate the gradient of failure probability. The different step is choose to be 1% of the initial design variables, according to the suggestion in Ref. [57]. The final results from Kriging method and FORM method can be used to verify the accuracy of the gradient estimation.

3 Numerical studies Three numerical examples are studied in this section to demonstrate the accuracy and efficiency of the proposed approach for RBDO of nonlinear structures.

3.1 Nonlinear reinforced concrete (RC) beam Reliability based design of a nonlinear cantilever RC column is performed to investigate the efficiency of the proposed method. The dimension of the RC column is shown in Figure 1(a). The total length of the column model is 10.97 m. The cross-section dimension of the column is 1.52× 1.52 m2, as shown in Figure 1(b). The column model is discretized into ten displacement-based beam-column elements. The stress–strain behavior of the concrete material is represented by the uniaxial smoothed Popovics-Saenz model [58], while the reinforcement is represented by the uniaxial Giuffre-Menegotto-Pinto model [48]. The gravity load of 1360 kN is applied at the top of the column. The pushover analysis is performed to evaluate of the maximum capacity of the column. The lateral load is linearly increased with displacement control strategy. The type of the force-deformation curve of the column is shown in Figure 1(c).

11

Stochastic response analysis, reliability analysis and design optimization of nonlinear structures are computational intensive and may have the convergence issues. This paper mainly includes the uncertainties in materials properties, such as stiffness and mass. The limit state function is defined as the maximum deformation of structures being smaller than a preset allowable value under specific loading. To define the scope and investigate the uncertainty effect clearly, the uncertainties in the applied load is not considered in this paper. The random variables are shown in Table 1, with their marginal probability distribution, mean value, and coefficient of variation (COV). The uncertainties in the material models of concrete and reinforcement are considered in this example. The studies with mixed random distributions are worthy exploring. However, the realistic experimental data for such random variables are limited. Using the lognormal random distribution to represent some properties of random material parameters of civil engineering structures is more promising and a usual practise from a practical point of view, since they are naturally positive valued [44, 45]. In this paper, the distributions of random variables and their coefficients of variation (COVs) are carefully selected according to existing references and experimental data [24, 36, 43, 46]. The random variables are generated from Nataf transform [54]. The performance function is defined as g  x   Fmax  x   b

(26)

where Fmax  x  is the maximum horizontal load carrying capacity and b is a preset threshold value. The reliability analysis is performed to obtain the probability of P  g  x   0  . Reliability analysis with Kriging method, MCS and FORM are performed to verify the accuracy of the proposed method. In the Kriging method, a group of design experiments are generated from Latin hypercube sampling method and the total number of simulations is 400. The linear regression part is modeled by the quadratic polynomial basis function, and the correlation function of the stochastic part is modeled by Gaussian form. Genetic algorithm is used to obtain the hyper-parameters and the Leave-one-out cross-validation error is 0.051%. To verify the accuracy and feasibility of the developed surrogate model, another validation set is generated with 1000 samples. The scatter plot of the predicted maximum carrying capacity from the surrogate model against the output from the physical model is shown in Figure 2. These results demonstrate the accuracy of the Kriging method for output prediction. The comparison of probability distribution function (PDF) 12

obtained with MCS method and Kriging method is given in Figure 3. In MCS method, 100,000 simulations are used to estimate the PDF. The PDF from Kriging method matches well with that from MCS method. The estimated probability values from Kriging method, MCS and FORM methods are shown in Figure 4. In FORM, the random variables are projected into standard normal space and the response sensitivities are obtained from the forward difference method. The threshold values b in Eq. (26) are set as 1000, 1050, 1100, 1150, 1200 and 1250. The results from FORM are slightly different from those obtained from MCS method and Kriging method. The typical failure of civil engineering structures has usually small failure probability. However, for such scenarios, the analysis of nonlinear structures may become unstable and non-convergence issues occur frequently in the finite element analysis. In the FORM algorithm, the finite element analysis is performed to estimate the limit-state function and its gradient in each iteration. Once the FE analysis fails to converge during this iteration, then the FORM algorithm cannot proceed. This is the main challenge problem for reliability analysis of nonlinear structures with FORM [18]. The proposed Kriging method does not suffer from such a problem, because new samples can be generated to replace the non-converged simulations. The primary objective of this paper is to verify the efficacy and accuracy of the developed RBDO method of nonlinear structures. The small failure probability case is not considered, but the analysis can be conducted for such scenarios. Reliability based design with Kriging method and FORM method are carried out to improve the performance of this RC beam. The optimization problem is defined as Minimize:  2 Subject to: P  Fmax  x   1200  0   35%,

(27)

1.52    1.82

where  2 is the area of cross-section, and the lower and upper bounds are 1.52 and 1.82, respectively. The initial value of  is 1.52 and the probability of P  Fmax  x   1200  is 5.93% generated from MCS method. In the outer loop, the sequential quadratic programming method is performed to find out the optimized solution of  . The inner loop of reliability constrain is estimated by FORM and Kriging method. Figure 5 shows the value of the objective function and the occurrence probability with iterations is shown in Figure 6. The converged results can be obtained with only several iterations. The final optimized cross-section area as well as the probability and 13

computational time from FORM and Kriging surrogate methods are listed in Table 2. Results show that when the width of column increases 7.9% from 1.52 m to 1.64 m, the probability of P  Fmax  x   1200  will increases 29% from 5.93% to 35%. Both FORM and Kriging methods can

lead to accurate results. All the computations are conducted on a workstation with an Intel Core i9-9900K Coffee Lake Processor (8-Core, 16-Thread, 3.6 GHz, 5.0 GHz Turbo) and 32 GB memory. Table 2 shows the computational time with FORM and Kriging method. The FORM takes more time and iterations than Kriging method to obtain the optimized solution.

3.2 Three-storey steel frame with nonlinear material properties Reliability based design of a three-bay, three-storey steel frame structure is performed to investigate the accuracy and efficiency of the proposed method. The structure has been studied in [48]. The dimension of the steel frame is shown in Figure 7(a). The height of each floor is 4 m and the width of each bay is 5 m. Beam and column members are modeled by using four displacement-based elements. The cross-section of beam and column members is the same, as shown in Figure 7(b). The cross-section is further discretized into fibers to represent the nonlinear behavior (ten fibers for the web and two for each flange). The stress–strain behavior of steel fibers is represented by the Giuffre-Menegotto-Pinto model [48], as shown in Figure 7(c). The uncertainty in the material model is considered in this example. Table 3 provides the marginal probability distribution, mean value and COV of each random variable. The material parameters such as elastic modulus, steel yield strength and stiffness ratio of each column/beam at each floor is assumed as a random variable. For example, the elastic modulus, E, of the column at each floor is assumed as a lognormal random variable with a mean value of 200,000MPa and 5% COV. The random variable has a correlation coefficient of 0.6 with the elastic modulus of the other floor. The random variables are generated from Nataf transform [54]. The gravity loads and lateral loads are applied on the frame structure, that is, 5×104 N for the gravity loads at the external nodes and 105 N at the internal nodes. The lateral loads are linearly decreasing with the height as shown in Figure 7(a). The maximum lateral load is 400 kN at the top. The loading are the same as those presented in a previous study [48]. Reliability analysis is conducted to obtain the occurrence probability P of the horizontal deformation of Node 4 when it is 14

smaller than a preset value. The performance function is defined as

g  x   Dmax  x   b

(28)

where Dmax  x  is the maximum horizontal deformation of Node 4 and b is a preset threshold value. The reliability analysis is conducted to obtain the probability P  Dmax  x   b  0  . In the first study, the preset threshold value is set to be 0.2. Different numbers of DOEs are generated to build the Kriging surrogate model, and the effect on the predictability of surrogate model is studied. The Kriging models constructed from 400, 600 and 800 of DOEs are defined as Model 1, 2 and 3. Figure 8 shows the predicted value from different Kriging models against the actual value from the finite element model. The leave-one-out error is a statistical index to estimate the accuracy and performance of the surrogate model, which can be calculated using the predicted value from Kriging models and the actual value from the finite element model [59]. The leave-one-out error of Model 1, 2, and 3 are 0.0152, 0.0096, and 0.0087, respectively. The predicted results of Model 3 match well with the actual value, as observed in Figure 8(c), which indicates that 800 of DOEs are sufficient to build up an accurate model to predict the deformation of steel frame. In the following studies, 800 of DOEs are used to construct the Kriging model for reliability analysis and design optimization. Figure 9 shows the comparison of PDF from Kriging based method against the MCS method. The results of MCS method is generated from 100,000 simulations. These results can verify the accuracy of the proposed Kriging based method. Figure 10 shows the occurrence probability from Kriging method, FORM and MCS methods. The results of FORM are obtained from the same procedure described in Example 1. The results from Kriging method are the same as those obtained from MCS, while the results of FORM are slightly different. Again, the accuracy of Kriging method for reliability analysis is verified. The Kriging and FORM based design optimizations are carried out to improve the performance of the steel frame. The reliability based design problem is defined as Minimize: A   Subject to: P  Dmax  x   0.2  0   90%,

(29)

lower    upper 15

where A   is the area of wide flange steel section. In this example, the initial values of depth tb and web thickness tw are set as 250mm and 20mm, respectively. The width and thickness of the flange (bf and tf) are 250mm and 20mm, respectively. Different lower and upper bounds are applied in the RBDO. The lower bounds for these three models are the same. tb, tw, bf and tf are set as 200 mm, 16 mm, 200 mm, and 16 mm, respectively. The values of lower bounds are equal to 80% of the initial values of design variables. The upper bounds are equal to 150%, 120% and 110% of the initial values of design variables for Model 1, 2, and 3, respectively. The results of Kriging based method are shown in Figure 11 and Tables 4-6. The proposed Kriging based design optimization takes several iterations to obtain the final solution. In all these scenarios, the cross-section area is reduced while the occurrence probability is increased to the target value. Tables 4-6 summarize the final design optimization results. The results of FORM are almost the same as those from Kriging based method, which can be used to verify the accuracy of both methods. In Scenario 1, the cross-section area is reduced from 15000mm2 to 11779.74 mm2, and the probability reaches the target value. The results also show that increasing the depth of the wide flange section tb is the most efficient approach to increase the probability P  Dmax  x   0.2  0  . The second and third important factors are the flange width bf and flange thickness tf, as shown from the results in Scenario 2 and 3. The cross-section is reduced to 12771.42 mm2 and 137495.46 mm2, respectively. Tables 4, 5 and 6 also indicate that the FORM takes more time in the design optimization than Kriging based method. The reason is explained as follows. In the FORM, the sequential optimization is used. If the nonlinear finite element analysis cannot satisfy a preset threshold values (10-8 in the Newton method), the FORM will not continue. To ensure the optimization processing, the increment step will be reduced and it will take more time to obtain the reliability analysis results. In the Kriging method, the model outputs of each DOE are generated independently. If one sample fails to obtain the convergence result in the nonlinear finite element analysis, new sample is generated to replace the old one.

3.3 Three-dimensional nonlinear RC frame 16

The third example is a three-dimensional nonlinear RC frame structure with concrete slabs at each floor, as shown in Figure 12(a). The height of each story is 3.66m, and the span of each bay is 6.1m. Beams and columns of the structure are modeled by using displacement-based Euler-Bernoulli elements, and each component is further discretized into five finite elements. The cross-sections of beam and column elements are shown in Figures 12(b) and (c), respectively. The cross section is modelled with different materials including confined concrete, unconfined concrete and steel reinforcement. The bilinear hysteretic model [60] is selected to represent the nonlinear behavior of the reinforcement steel, while the Kent-Scott-Park model [60] is used to represent the concrete. The tension stiffening of concrete is not considered in this study. The concrete slabs are modeled through a diaphragm constraint at each floor to enforce rigid in-plane behavior. The vertical loads q=8kN/m2 are uniformly distributed at each floor. When finishing the static load analysis, the structure is subjected to a quasi-static pushover analysis. The horizontal force is applied at the center of each floor. The force at the top of the structure is 400 kN, while the force at the second and the first floors are 800/3 kN and 400/3 kN, respectively. In the pushover analysis, the uncertainties in the material parameters are taken into account. Eight parameters are used to characterize the various structural materials involved in the structure, namely three parameters for the confined concrete (fc,core: peak strength, εc,core: strain at peak strength, fcu,core: crushing strength), two parameters for the unconfined concrete (fc,cover: peak strength, εc,cover: strain at peak strength) and three parameters for the reinforcement steel (fy: yield strength, E0: initial stiffness, B: hardening ratio). Table 7 shows the marginal PDFs of these material parameters. These material parameters are modeled as independent random variable and their characteristics are obtained from studies reported in the literature. Reliability analysis is conducted to obtain the occurrence probability P of the horizontal deformation of Node 19 with the performance function defined as

g  x   Dmax  x   b

(30)

where Dmax  x  is the maximum horizontal deformation of Node 19. In the Kriging method, the model is built from 600 DOEs. Figure 13 shows the predicted deformation from Kriging method against validation set. Figure 14 shows the PDF from Kriging 17

method against those from MCS method. In the MCS method, 100,000 simulations are used. These results can be used to verify the accuracy of the Kriging method. Figure 15 shows the occurrence probability with different threshold values b. The results of FORM and Kriging method match well with those from MCS method. Reliability based design are carried out to improve the performance of the RC structure with the objective function defined as Minimize: A   Subject to: P  Dmax  x   0.2  0   95%,

(31)

lower    upper where A   is the cross-section area of column. In this example, the initial occurrence probability is 39.13% and the initial value of cross-section area is 0.2116 m2. The design variables are the width and length of column. The initial values of the width and length of column are the same and set as 0.46m. The lower bound is 0.368m and the upper bound is 0.69m. Figure 16 shows the optimization processing of using Kriging and FORM based methods. With the proposed design optimization methods, the cross-section area reduces from 0.2116m2 to 0.1857m2, while the occurrence probability increases from 39.13% to 95%. The optimized results of Kriging based method are almost the same as those from FORM. It is time consuming to perform RBDO of nonlinear structures. As shown in Table 8, it takes 12 hours and 7 iterations to the find the optimal design solution with the proposed Kriging method. The computational time with FORM is about 23 hours. For real engineering problems, when more design variables and more complicated finite element model are involved in the RBDO, more significant computational demand saving and improved efficiency will be obtained to evaluate the gradient of the failure probability and to achieve the optimal design solution. The high-performance computational techniques [49, 50] can also be explored to improve the computational efficiency of RBDO in further studies.

4 Conclusions The study of nonlinear finite element reliability analysis has not been well explored. In this paper, FORM and Kriging method are applied for reliability analysis and design optimization of 18

nonlinear structures. The uncertainties in the material parameters such as peak strength of concrete, yield strength and initial stiffness of reinforcement are considered. Numerical studies on reliability analysis of RC structures and steel frame structures are conducted to verify the accuracy and efficiency of the proposed methods. The occurrence probabilities of nonlinear structures under pushover loading can be estimated. Results show that both Kriging method and FORM can estimate the occurrence probabilities of nonlinear structures as accurately as those from MCS method. RBDO is carried out to improve the performance of nonlinear structures. Results show that the optimized cross-section area and the target probabilities can be obtained with both methods. Results also indicate that the Kriging method is more feasible for complex engineering problems than FORM, because the gradient of the limit state function is not required in the Kriging method. Most of existing studies are based on analytical limit state functions, which may not be applied for complex engineering problems. RBDO of RC structures and steel frame structures are conducted in this paper, and the results of design optimization show that with the proposed approach, the cross-section of the structures is reduced, while the failure occurrence probability is significantly increased. These results demonstrate the efficiency of the proposed method for complex engineering problems.

Acknowledgements The authors gratefully acknowledge the support for this research by the research project of Beijing Municipal Education Commission (No. IDHT20190504).

19

References [1] Allen M, Maute K. Reliability-based design optimization of aeroelastic structures. Structural and Multidisciplinary Optimization. 2004;27:228-42. [2] Mrabet E, Guedri M, Ichchou M, Ghanmi S. Stochastic structural and reliability based optimization of tuned mass damper. Mechanical Systems and Signal Processing. 2015;60:437-51. [3] Antonio CC, Hoffbauer LN. Reliability-based design optimization and uncertainty quantification for optimal conditions of composite structures with non-linear behavior. Engineering Structures. 2017;153:479-90. [4] Savran M, Aydin L. Stochastic optimization of graphite-flax/epoxy hybrid laminated composite for maximum fundamental frequency and minimum cost. Engineering Structures. 2018;174:675-87. [5] Kusano I, Baldomir A, Jurado JÁ, Hernández S. The importance of correlation among flutter derivatives for the reliability based optimum design of suspension bridges. Engineering Structures. 2018;173:416-28. [6] Wang W, Morgenthal G. Reliability analyses of RC bridge piers subjected to barge impact using efficient models. Engineering Structures. 2018;166:485-95. [7] Chen H-P, Mehrabani MB. Reliability analysis and optimum maintenance of coastal flood defences using probabilistic deterioration modelling. Reliability Engineering & System Safety. 2019;185:163-74. [8] Castaldo P, Mancini G, Palazzo B. Seismic reliability-based robustness assessment of three-dimensional reinforced concrete systems equipped with single-concave sliding devices. Engineering Structures. 2018;163:373-87. [9] Nasrollahzadeh K, Aghamohammadi R. Reliability analysis of shear strength provisions for FRP-reinforced concrete beams. Engineering Structures. 2018;176:785-800. [10] Chen H-P. Residual Flexural Capacity and Performance Assessment of Corroded Reinforced Concrete Beams. Journal of Structural Engineering. 2018;144:04018213. [11] FEMA-356. Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Washington, DC: Federal Emergency Management Agency; 2000. [12] Hohenbichler M, Rackwitz R. First-order concepts in system reliability. Structural Safety. 1982;1:177-88. [13] Der Kiureghian A, Dakessian T. Multiple design points in first and second-order reliability. Structural Safety. 1998;20:37-49. [14] Austin M, Pister K, Mahin S. Probabilistic design of earthquake-resistant structures. Journal of Structural Engineering. 1987;113:1642-59. [15] Dudzik A, Radoń U. The reliability assessment for steel industrial building. Advances In Mechanics: Theoretical, Computational and Interdisciplinary Issues. 2016:163-6. [16] Choi CK, Yoo HH. Uncertainty analysis of nonlinear systems employing the first-order reliability method. Journal of Mechanical Science and Technology. 2012;26:39-44. [17] Ditlevsen O. Narrow reliability bounds for structural systems. Journal of Structural Mechanics. 1979;7:453-72. [18] Koduru S, Haukaas T. Feasibility of FORM in finite element reliability analysis. Structural Safety. 2010;32:145-53. [19] Haukaas T, Der Kiureghian A. Finite element reliability and sensitivity methods for performance-based earthquake engineering: Pacific Earthquake Engineering Research Center, College of Engineering, University of California, Berkeley; 2004. 20

[20] Lim J, Lee B, Lee I. Sequential optimization and reliability assessment based on dimension reduction method for accurate and efficient reliability-based design optimization. Journal of Mechanical Science and Technology. 2015;29:1349-54. [21] Alibrandi U. A response surface method for stochastic dynamic analysis. Reliability Engineering & System Safety. 2014;126:44-53. [22] Izquierdo J, Márquez AC, Uribetxebarria J. Dynamic artificial neural network-based reliability considering operational context of assets. Reliability Engineering & System Safety. 2019;188:483-93. [23] Xiu D, Karniadakis GE. The Wiener--Askey polynomial chaos for stochastic differential equations. Siam J Sci Comput. 2002;24:619-44. [24] Ni P, Xia Y, Li J, Hao H. Using polynomial chaos expansion for uncertainty and sensitivity analysis of bridge structures. Mechanical Systems and Signal Processing. 2019;119:293-311. [25] Sudret B. Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety. 2008;93:964-79. [26] Ni P, Li J, Hao H, Xia Y, Du X. Stochastic dynamic analysis of marine risers considering fluid-structure interaction and system uncertainties. Engineering Structures. 2019;198:109507. [27] Kaymaz I. Application of kriging method to structural reliability problems. Structural Safety. 2005;27:133-51. [28] Dubourg V, Sudret B, Deheeger F. Metamodel-based importance sampling for structural reliability analysis. Probabilistic Engineering Mechanics. 2013;33:47-57. [29] Li D, Tang H, Xue S, Guo X. Reliability analysis of nonlinear dynamic system with epistemic uncertainties using hybrid Kriging-HDMR. Probabilistic Engineering Mechanics. 2019;58:103001. [30] Zhang X, Wang L, Sørensen JD. REIF: A novel active-learning function toward adaptive Kriging surrogate models for structural reliability analysis. Reliability Engineering & System Safety. 2019;185:440-54. [31] Matheron G. A simple substitute for conditional expectation: the disjunctive kriging. Advanced geostatistics in the mining industry: Springer; 1976. p. 221-36. [32] Ling C, Lu Z, Zhu X. Efficient methods by active learning Kriging coupled with variance reduction based sampling methods for time-dependent failure probability. Reliability Engineering & System Safety. 2019;188:23-35. [33] Yuan K, Xiao N-C, Wang Z, Shang K. System reliability analysis by combining structure function and active learning kriging model. Reliability Engineering & System Safety. 2020;195:106734. [34] Zhang J, Xiao M, Gao L. An active learning reliability method combining kriging constructed with exploration and exploitation of failure region and subset simulation. Reliability Engineering & System Safety. 2019;188:90-102. [35] Gaspar B, Teixeira A, Soares CG. Adaptive surrogate model with active refinement combining Kriging and a trust region method. Reliability Engineering & System Safety. 2017;165:277-91. [36] Cadini F, Santos F, Zio E. An improved adaptive kriging-based importance technique for sampling multiple failure regions of low probability. Reliability Engineering & System Safety. 2014;131:109-17. [37] Wang Z, Shafieezadeh A. REAK: Reliability analysis through Error rate-based Adaptive Kriging. Reliability Engineering & System Safety. 2019;182:33-45. [38] Sun Z, Wang J, Li R, Tong C. LIF: A new Kriging based learning function and its application to 21

structural reliability analysis. Reliability Engineering & System Safety. 2017;157:152-65. [39] Dong Y, Teixeira A, Soares CG. Application of adaptive surrogate models in time-variant fatigue reliability assessment of welded joints with surface cracks. Reliability Engineering & System Safety. 2020;195:106730. [40] Menz M, Gogu C, Dubreuil S, Bartoli N, Morio J. Adaptive coupling of reduced basis modeling and Kriging based active learning methods for reliability analyses. Reliability Engineering & System Safety. 2020;196:106771. [41] Zhang J, Taflanidis A. Bayesian model averaging for Kriging regression structure selection. Probabilistic Engineering Mechanics. 2019;56:58-70. [42] Abdallah I, Lataniotis C, Sudret B. Parametric hierarchical kriging for multi-fidelity aero-servo-elastic simulators—Application to extreme loads on wind turbines. Probabilistic Engineering Mechanics. 2019;55:67-77. [43] Yi P, Wei K, Kong X, Zhu Z. Cumulative PSO-Kriging model for slope reliability analysis. Probabilistic Engineering Mechanics. 2015;39:39-45. [44] Papadimitriou C, Haralampidis Y, Sobczyk K. Optimal experimental design in stochastic structural dynamics. Probabilistic Engineering Mechanics. 2005;20:67-78. [45] Ling C, Lu Z, Cheng K, Sun B. An efficient method for estimating global reliability sensitivity indices. Probabilistic Engineering Mechanics. 2019;56:35-49. [46] Wen Y. Reliability and performance-based design. Structural Safety. 2001;23:407-28. [47] Haukaas T. Unified reliability and design optimization for earthquake engineering. Probabilistic Engineering Mechanics. 2008;23:471-81. [48] Haukaas T, Scott MH. Shape sensitivities in the reliability analysis of nonlinear frame structures. Computers & structures. 2006;84:964-77. [49] Mínguez R, Castillo E. Reliability-based optimization in engineering using decomposition techniques and FORMS. Structural Safety. 2009;31:214-23. [50] Echard B, Gayton N, Lemaire M, Relun N. A combined importance sampling and kriging reliability method for small failure probabilities with time-demanding numerical models. Reliability Engineering & System Safety. 2013;111:232-40. [51] Papaioannou I, Geyer S, Straub D. Improved cross entropy-based importance sampling with a flexible mixture model. Reliability Engineering & System Safety. 2019;191:106564. [52] Dubourg V, Sudret B, Bourinet J-M. Reliability-based design optimization using kriging surrogates and subset simulation. Structural and Multidisciplinary Optimization. 2011;44:673-90. [53] Hasofer AM. An Exact and Invarient First Order Reliability Format. J Eng Mech Div, Proc ASCE. 1974;100:111-21. [54] Qin Q, Lin D, Mei G, Chen H. Effects of variable transformations on errors in FORM results. Reliability Engineering & System Safety. 2006;91:112-8. [55] Martin JD, Simpson TW. Use of kriging models to approximate deterministic computer models. AIAA Journal. 2005;43:853-63. [56] Boggs PT, Tolle JW. Sequential quadratic programming. Acta numerica. 1995;4:1-51. [57] Ni P, Xia Y, Law S-S, Zhu S. Structural damage detection using auto/cross-correlation functions under multiple unknown excitations. International Journal of Structural Stability and Dynamics. 2014;14:1440006. [58] Gu Q. Finite element response sensitivity and reliability analysis of soil-foundation-structure-interaction (SFSI) systems: UC San Diego; 2008. 22

[59] Elisseeff A, Pontil M. Leave-one-out error and stability of learning algorithms with applications Stability of Randomized Learning Algorithms Source. International Journal of Systems Science IJSySc. 2002;6. [60] Quan G, Barbato M, Conte JP. Handling of Constraints in Finite-Element Response Sensitivity Analysis. Journal of Engineering Mechanics. 2009;135:1427-38.

23

Table 1 Marginal PDFs of material parameters for the RC beam Material

Probability

Mean

distribution

value

Parameter (unit)

COV (%)

Confined

peak strength fc,core (MPa)

lognormal

34.47

20

concrete

strain at peak strength εc,core

lognormal

0.005

20

Unconfined

peak strength fc,cover (MPa)

lognormal

27.58

20

concrete

strain at peak strength εc,cover

lognormal

0.002

20

yield strength fy (MPa)

lognormal

307

10

initial elastic tangent E0 (GPa)

lognormal

201

3.3

strain-hardening ratio B

lognormal

0.02

20

Reinforcement

Table 2 Design optimization results of the beam 

2

(m)

(m2)

P  Fmax  x   1200 

Computational time

Initial

1.52 2.31

5.93%

-

FORM based method

1.65 2.72

35%

90 min

Kriging based method

1.64 2.69

35%

60 min

24

Table 3 Marginal PDFs of material parameters for the three-storey steel frame i=1, 2, 3, …, 6

Distribution

Mean value

COV

Correlation coefficient

fy,i

lognormal

300 MPa

10%

0.6

Ei

lognormal

200,000 MPa

5%

0.6

Bi

lognormal

0.02

10%

0.6

Depth (mm) Lower bound 200 upper bound 375 Initial value 250 FORM based 336.2340 method Kriging 336.9975 based method

Depth (mm) Lower bound upper bound Initial value FORM based method Kriging based method

200 300 250

Table 4 Scenario 1 of example 2 web flange flange Area thickness width thickness (mm2) (mm) (mm) (mm) 16 200 16 30 375 30 20 250 20 15000

Probability

Computational time

32.91%

-

16

200

16

11779.74

90%

14 hours

16

200

16

11791.96

89.96%

8 hours

Probability

Computational time

32.91%

-

Table 5 Scenario 2 of example 2 web flange flange Area thickness width thickness (mm2) (mm) (mm) (mm) 16 200 16 30 300 30 20 250 20 15000

300

16

249.1070

16

12771.42

90%

18 hours

300

16

250.3713

16

12811.88

89.95%

10 hours

25

Depth (mm) Lower bound upper bound Initial value FORM based method Kriging based method

200 275 250

Table 6 Scenario 3 of example 2 web flange flange Area thickness width thickness (mm2) (mm) (mm) (mm) 16 200 16 22 275 22 20 250 20 15000

Probability

Computational time

32.91%

-

275

16

275

17.0826

13795.46

90%

24 hours

275

16

275

17.1886

13853.75

89.97%

16 hours

Table 7 Marginal PDFs of material parameters for the RC structure Parameter (unit)

Probability distribution

Mean value

COV (%)

peak strength fc,core (MPa)

lognormal

34.47

10

strain at peak strength εc,core

lognormal

0.005

10

crushing strength fcu,core (MPa)

lognormal

24.13

10

Unconfined

peak strength fc,cover (MPa)

lognormal

27.58

10

concrete

strain at peak strength εc,cover

lognormal

0.002

10

yield strength fy (MPa)

lognormal

307

10

initial elastic tangent E0 (GPa)

lognormal

201

10

strain-hardening ratio B

lognormal

0.02

10

Material

Confined concrete

Reinforcement

Table 8 Design optimization result of Example 3

Lower bound upper bound Initial value FORM based method Kriging based method

Width (m)

Length (m)

Area (m2)

Probability

0.368 0.69 0.46 0.5046 0.5045

0.368 0.69 0.46 0.368 0.368

0.2116 0.1857 0.1857

39.13% 94.99% 94.92%

Computational time 23 hours 12 hours

26

Cover concrete Reinforcement

Deformation

Core concrete

0.13m Weight =1360kN

Force

1.52m

(b) Cross-section dimension Length =10.97m

(a) Dimension of the reinforced

(c) The force-drift curve

concrete column Figure 1 Numerical example of a cantilever beam column

Figure 2 Predicted maximum carrying capacity from Kriging method against validation set

27

Figure 3 PDF of maximum carrying capacity from MCS and Kriging methods

Figure 4 Comparison of probability values with MCS, Kriging method and FORM method

28

Figure 5 Cross-section area with iterations

Figure 6 Occurrence probability with iterations

29

5×3m 4

8 (6)

12 (6)

(5)

7 (4)

(6)

11 (4)

(3)

(3)

2

6 (2) (1)

1

5

15 (4) (3)

(3)

10 (2)

(1)

(5)

(5)

(5)

3

16

4×3m

14 (2) (1)

(1)

9

13

(a) Dimension of a numerical three-storey frame structure

tf

tw

tb+2tf

Stress

fy

1

B× E

E

1

bf

Strain

(b) Cross-section

(c) Material model

Figure 7 Three-storey steel frame structure

30

(a) Kriging surrogate model built from 400 DOEs

(b) Kriging surrogate model built from 600 DOEs

(c) Kriging surrogate model built from 800 DOEs Figure 8 Prediction from Kriging method against validation set

31

Figure 9 PDF of deformation from MCS and Kriging methods

Figure 10 Comparison of occurrence probabilities with MCS, Kriging and FORM methods

32

(a) Cross section area and occurrence (b) Normalized value of Design probability of Scenario 1 variables in Scenario 1

(c) Cross section area and occurrence (d) Normalized value of Design probability of Scenario 2 variables in Scenario 2

(e) Cross section area and occurrence (f) Normalized value of Design probability of Scenario 3 variables in Scenario 3 Figure 11 Design optimization of steel frame

33

13 16

Cover concrete

19

P

Reinforcement Core concrete

15 9

12

8

Length=0.4

14

10 18

2P/3

Width =0.6

11 5 P/3

6

(b) Cross-section of beam

17 Length=0.46

7

Cover concrete

1

2 3

4

6.1 (a) Dimension of RC frame

Reinforcement

Width =0.46

Core concrete

6.1 (c) Cross-section of column

Figure 12 Three-story RC structure (Unit: m)

Figure 13 Predicted maximum carrying capacity from Kriging method against validation set

34

Figure 14 PDF of deformation from MCS and Kriging methods

Figure 15 Comparison of occurrence probability with MCS, Kriging and FORM methods

35

(a) Cross section area

(b) Occurrence probability

(c) Width of column

(d) Length of column

Figure 16 Design optimization of a RC frame with Kriging and FORM methods

36

Conflict of Interest and Authorship Conformation Form Please check the following as appropriate: o

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

o

This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

o

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

37

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

38