# Reliability-Based Design Optimization of Structures Considering Uncertainties of Earthquakes Based on Efficient Gaussian Process Regression Metamodeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. RBDO Problem and EGO-EGRA Approach

#### 2.1. RBDO Formulation

**d*** where the objective function takes the extreme value subjected to the constraints, which can be described [28,29] as follows:

**d**) is the objective function, g

_{i}(

**d**) ≤ 0 denotes the constraint, I represents the number of constraints,

**d*** is called the optimal solution, and the superscripts ‘L’ and ‘U’ denote the lower and upper bounds of variables, respectively.

**d**represents the vector of deterministic design variable,

**Z**and

**P**denote the vectors of random design variable and random parameter, and their mean values are

**μ**and

_{Z}**μ**, respectively;

_{P}**D**represents the vector of design variable; $\mathrm{Pr}\{G(\mathbf{d},\mathbf{Z},\mathbf{P})<0\}$ is the failure probability corresponding to the performance function $G(\mathbf{d},\mathbf{Z},\mathbf{P})$; and P

_{T}denotes the target failure probability of the reliability constraint. If the performance function value is positive, the structure is safe; otherwise, it fails. There can be multiple reliability constraints in an RBDO problem. Another form of RBDO is to optimize structural reliability [24], which can be expressed as follows:

#### 2.2. Efficient Global Optimization

_{best}is the current best objective value, and $\Phi (\cdot )$ and $\varphi (\cdot )$ are the probability distribution function and probability density function of the standard normal distribution, respectively. The basic steps of EGO are as follows: (1) select a small number of samples in the design space and calculate their function values; (2) establish the metamodel; (3) search for the point

**d**′ with the maximum EI value; and (4) if the convergence condition EI < 0.01·f

_{best}is satisfied at point

**d**′, stop infill-sampling and take the design point

**d*** corresponding to f

_{best}as the optimal solution of the optimization problem, otherwise add point

**d**′ to the DoE and return to step (2).

#### 2.3. EGO-EGRA Approach

**x**of the metamodel consists of

**d**,

**Z**, and

**P**. Then, in order to obtain new samples that can refine the metamodel at the limit state surface, the global optimization algorithm is used to search the

**x**space for the point ${\mathbf{x}}_{new}$ with the maximum EF function value. The EF function is defined as follows

**D**can be calculated without calling the real performance function. Then, by transforming RBDO problem (1) or (2) into the following:

## 3. Proposed RBDO Method for Structures Subjected to Earthquakes

#### 3.1. Metamodel of the EDP

**d**,

**Z**and the structural random parameter

**s**), EDP R is a random variable, for which its statistical properties can be expressed by the mean and variance [11]. Based on this assumption, the record-to-record variation can be implicitly incorporated in a set of ground motion records [18]. In this study, EDP is considered to be lognormally distributed [35]. At each sample in DoE, the mean ${\mu}_{\mathrm{ln}R}$ and standard deviation ${\sigma}_{\mathrm{ln}R}$ of the logarithms of EDP can be obtained by performing NLTHA for all selected records [20].

**x**) to be fitted, the GPR model is built based on a DoE composed of a group of experiments

**X**= [

**x**

^{(1)},

**x**

^{(2)}, …,

**x**

^{(m)}] and their outputs

**y**= [y

^{(1)}, y

^{(2)}, …, y

^{(m)}]

^{T}, where

**x**

^{(i)}represents a n-dimensional input vector and y

^{(i)}represents its output. At a test point ${\mathbf{x}}_{*}$, the joint prior distribution of

**y**and $F({\mathbf{x}}_{*})$ is as follows:

_{N}represents the standard deviation of the normally distributed noise; K(

**X**,

**X**) = [k

_{ij}]

_{m}

_{×m}is the covariance matrix and k

_{ij}= k(

**x**

^{(i)},

**x**

^{(j)}) is the covariance between

**x**

^{(i)}and

**x**

^{(j)}; $K(\mathbf{X},{\mathbf{x}}_{*})=K{({\mathbf{x}}_{*},\mathbf{X})}^{\mathrm{T}}$ is the m × 1 covariance matrix between ${\mathbf{x}}_{*}$ and X; and

**I**

_{m}is the m-dimensional identity matrix. The prediction $\widehat{F}({\mathbf{x}}_{*})$ and predicted variance ${\sigma}_{\widehat{F}}^{2}({\mathbf{x}}_{*})$ of ${\mathbf{x}}_{*}$ are obtained by the following formulas:

**d**,

**Z**,

**s**, and im are fixed, the value of the EDP is expressed as follows.

**x**of the EDP’s metamodel consists of

**d**,

**Z**and

**P**, i.e.,

**x**= [

**d, Z, P**]. Variable u related to the randomness of seismic demand is regarded as a random parameter, and structural random parameter

**s**is also included in

**P**. By choosing samples in

**x**space and calculating their corresponding R values according to Formula (16), DoE can be generated. Based on the DoE, the GPR model of EDP is constructed, which can provide the prediction $\widehat{R}(\mathbf{x})$ and prediction error ${\sigma}_{\widehat{R}}(\mathbf{x})$ of the response. The schematic of modeling for the EDP is shown in Figure 1.

**D**= [

**d**

**,**

**μ**], by combining

_{Z}**d**with N points randomly generated according to the distribution parameters of

**Z**and

**P**, MCS samples

**X**

^{MC}= [${\mathbf{x}}_{1}^{\mathrm{MC}}$, ${\mathbf{x}}_{2}^{\mathrm{MC}}$, …, ${\mathbf{x}}_{N}^{\mathrm{MC}}$] are obtained. Then, the failure probability that the response exceeds the threshold z can be estimated by the following:

#### 3.2. Refinement of the Metamodel

_{i}in

**P**can be taken as ${F}_{i}^{-1}[\Phi (\pm 4)]$, where ${F}_{i}^{-1}(\cdot )$ is the inverse function of the probability distribution function of P

_{i}. The sampling ranges of

**d**and

**Z**can be the bounds in the optimization problem. The size of the initial DoE can be close to the number of input variables of the metamodel.

**X**

^{c}= [${\mathbf{x}}_{1}^{\mathrm{c}}$, ${\mathbf{x}}_{2}^{\mathrm{c}}$, …, ${\mathbf{x}}_{r}^{\mathrm{c}}$] for refining the metamodel needs to be first generated without calculating the corresponding responses. The accuracy of the model at the location where MCS random points are very sparse has little influence on the reliability results; thus, there is no need to sample many candidate points with a small joint probability density. The candidate set is chosen by LHS according to the statistical characteristics of the variables, which is obtained as follows: a set of points

**Q**= [

**q**

_{i}]

_{r}=[q

_{ij}]

_{r}

_{×n}is uniformly generated in the space of (0, 1)

^{n}using LHS, and candidate set

**X**

^{c}is obtained by transforming [9]

**Q**as follows.

**Z**and

**P**can be treated as uniformly distributed variables.

#### 3.3. Computational Procedure of RBDO

- (1)
- Select a group of seismic records and establish a structural analysis model.
- (2)
- Generate an initial DoE: sample n
_{s}(e.g., the number of the input variables) initial experiments in the space of**x**= [**d, Z, P**] by LHS, perform NLTHA for all selected records at each sample, and calculate the seismic demand according to Formula (16). - (3)
- In terms of Formula (18), use LHS to choose a candidate set containing r points (r = 10,000 in this research).
- (4)
- Construct the GPR model of EDP with DoE.
- (5)
- Pick the point ${\mathbf{x}}_{new}$ with the minimum value of U function from the candidate set according to Formula (20). If ${p}_{U<2}$ < ζ (ζ is taken as 0.02), proceed to step (6); otherwise, add the new sample ${\mathbf{x}}_{new}$ with its seismic demand to the DoE, and return to step (4).
- (6)
- By Formula (17), transform RBDO problems (1) or (2) into the form in Formulas (8) or (9) and search for the optimal solution using the EGO algorithm.

## 4. Numerical Studies

_{c}of calls to FEA. N

_{c}is equal to the number of samples for constructing the metamodel of the EDP. In an FEA, the NLTHA is carried out for all selected earthquake records.

^{2}.

#### 4.1. Example 1: A Steel Frame

^{3}. The uniformly distributed load q on beams is 30.0 kN/m. A wide flange cross-section is adopted for the beam. The section depth is 0.3 m, the web thickness is 0.016 m, and the flange width and thickness are 0.22 m and 0.016 m, respectively. A square tube section with a width of b

_{c}and wall thickness of t

_{c}is adopted for the column. The beam and column members are modeled using displacement-based beam–column elements. The bilinear model is employed to simulate the nonlinear properties of steel [53], as shown in Figure 4b. The ratio of post-yield to initial stiffness B is 0.02. The initial elastic modulus and yield strength of the column steel are E

_{1}and f

_{1}, respectively. The initial elastic modulus and yield strength of the beam steel are E

_{2}and f

_{2}, respectively.

_{b}. With reference to the evaluation standard for moderate damage of steel frames in GB50011-2010, deformation limit φ

_{b}is taken as 1/100 in this example.

_{1}, E

_{2}, f

_{1}, f

_{2}, and t

_{c}is also considered in this optimization problem. E

_{1}, E

_{2}, f

_{1}, and f

_{2}are random parameters; t

_{c}is a random design variable with a mean of ${\mu}_{{t}_{\mathrm{c}}}$. Their distribution information is shown in Table 2. Since the column parameters have a great impact on seismic resistance, the width b

_{c}and the thickness mean value ${\mu}_{{t}_{\mathrm{c}}}$ are taken as design variables. The bounds of b

_{c}and ${\mu}_{{t}_{\mathrm{c}}}$ are [0.22, 0.5] m and [0.01, 0.02] m, respectively. Generally, the failure probability of a structural component is required to be less than 0.05 in civil engineering [7]. In this optimization problem, the failure probability of the structure is limited to less than 0.05, and the column section area S

_{c}is selected as the optimization objective to minimize the costs. The RBDO problem can be described as follows.

_{c}and ${\mu}_{{t}_{\mathrm{c}}}$ in the initial scenario are 0.25 m and 0.016 m, respectively.

_{1}-f

_{2}-E

_{1}space is shown in Figure 5. A total of 52 samples were added during AL sampling, and the iteration history of sampling is shown in Figure 6a. It can be observed that the response predictions of the new samples were mostly located in the vicinity of the threshold φ

_{b}. With the expansion of the size of DoE, the difference between $\phi ({x}_{new})$ and $\widehat{\phi}({x}_{new})$ gradually decreased, and the value of ${p}_{U<2}$ decreased with fluctuations. After establishing the metamodel, the failure probability corresponding to design variable

**D**could be obtained. The failure probability values obtained by the proposed method were compared with those by MCS and EGO-EGRA. The simulation samples of MCS were selected by LHS. According to Ref. [10], 200 samples generated using LHS are sufficient to ensure the accuracy of the MCS results for seismic reliability calculation; thus, 1000 simulation points were selected in this work. The reliability values of 25 groups of design variables were calculated, and the number of calls to FEA in MCS is 25,000 and that number in EGO-EGRA is 120. The reliability results obtained by different methods are plotted in Figure 7. The three surfaces are very close to each other, which verifies the accuracy of the reliability information provided by the proposed method.

^{2}to 0.0128 m

^{2}, and the failure probability was less than 5%. The result of the proposed method is close to that of EGO-EGRA, but the calculation cost is less, which validates the efficiency and accuracy of this method.

_{I}, were 3, 6, and 17, respectively. The changes in $\phi ({\mathbf{x}}_{new})$, $\widehat{\phi}({\mathbf{x}}_{new})$, and ${p}_{U<2}$ in the sampling processes are shown in Figure 6b–d. The difference between the four solutions of the proposed method is small, which indicates that this method is insensitive to the number of initial samples and can obtain optimization results stably.

#### 4.2. Example 2: A Reinforced Concrete Frame

^{3}. The uniformly distributed load q on the structure is 35 kN/m. The cross-section of the member is composed of steel reinforcement, cover concrete, and core concrete. The diameter of rebar is 28 mm. The thickness of the cover concrete is 0.03 m. The cross-section of beam is a rectangle with a size of 0.4 m × 0.6 m. A square cross-section is adopted for the column. The width of the column section of the first floor is b

_{1}; the column cross-sections of the second, third, and fourth floor are the same, the width of which is b

_{2}. The material properties of the steel bar are simulated using a bilinear model. The initial elastic modulus and yield strength of the steel are E

_{s}and f

_{s}, respectively, and the ratio of the post-yield to the initial stiffness is 0.05. The Kent–Scott–Park model is employed to simulate the nonlinear characteristics of concrete [53], as shown in Figure 9. The compressive strength of the cover concrete, f

_{c,cover}, is 2.758 × 10

^{7}Pa, and the strain at maximum strength ε

_{c,cover}is 0.002; the crushing strength f

_{u,cover}is 0, and the strain at crushing strength ε

_{u,cover}is 0.06. The crushing strength of the core concrete, f

_{u,core}, is 2.413 × 10

^{7}Pa [2], and the strain at maximum strength ε

_{u,core}is 0.02; the compressive strength and strain at maximum strength are f

_{c,core}and ε

_{c,core}.

_{b}is taken as 1/120. In this RBDO problem, b

_{1}and b

_{2}are regarded as deterministic random variables, and the bounds are ${b}_{1},{b}_{2}\in \left[0.3,0.5\right]\mathrm{m}$. The randomness of material property parameters f

_{s}, E

_{s}, f

_{c}, and ε

_{u,core}are considered, and the distribution information of the random parameters is shown in Table 5. In the initial scenario, b

_{1}and b

_{2}are 0.33 m and 0.43 m, respectively. The optimization objective is to maximize the reliability of the structure, and the total volume V

_{c}of columns is required not to exceed the column volume V

_{c0}of the initial scenario. The RBDO problem can be described as follows.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- FEMA-356; Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Federal Emergency Management Agency: Washington, DC, USA, 2000.
- Ni, P.; Li, J.; Hao, H.; Yan, W.; Du, X.; Zhou, H. Reliability analysis and design optimization of nonlinear structures. Reliab. Eng. Syst. Saf.
**2020**, 198, 106860. [Google Scholar] [CrossRef] - Yazdani, H.; Khatibinia, M.; Gharehbaghi, S.; Hatami, K. Probabilistic performance-based optimum seismic design of RC structures considering soil–structure interaction effects. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civ. Eng.
**2017**, 3, G4016004. [Google Scholar] [CrossRef] - Zou, X.; Wang, Q.; Wu, J. Reliability-based performance design optimization for seismic retrofit of reinforced concrete buildings with fiber-reinforced polymer composites. Adv. Struct. Eng.
**2018**, 21, 838–851. [Google Scholar] [CrossRef] - Mishra, S.K.; Roy, B.K.; Chakraborty, S. Reliability-based-design-optimization of base isolated buildings considering stochastic system parameters subjected to random earthquakes. Int. J. Mech. Sci.
**2013**, 75, 123–133. [Google Scholar] [CrossRef] - Hadidi, A.; Azar, B.F.; Rafiee, A. Reliability-based design of semi-rigidly connected base-isolated buildings subjected to stochastic near-fault excitations. Earthq. Struct.
**2016**, 11, 701–721. [Google Scholar] [CrossRef] - Peng, Y.; Ma, Y.; Huang, T.; De Domenico, D. Reliability-based design optimization of adaptive sliding base isolation system for improving seismic performance of structures. Reliab. Eng. Syst. Saf.
**2021**, 205, 107167. [Google Scholar] [CrossRef] - FEMA P695; Quantification of Building Seismic Performance Factors. Federal Emergency Management Agency: Washington, DC, USA, 2009.
- Su, L.; Li, X.L.; Jiang, Y.P. Comparison of methodologies for seismic fragility analysis of unreinforced masonry buildings considering epistemic uncertainty. Eng. Struct.
**2020**, 205, 110059. [Google Scholar] [CrossRef] - Dolsek, M. Incremental dynamic analysis with consideration of modeling uncertainties. Earthq. Eng. Struct. Dyn.
**2009**, 38, 805–825. [Google Scholar] [CrossRef] - Perotti, F.; Domaneschi, M.; De Grandis, S. The numerical computation of seismic fragility of base-isolated nuclear power plants buildings. Nucl. Eng. Des.
**2013**, 262, 189–200. [Google Scholar] [CrossRef] - Lagaros, N.D.; Fragiadakis, M. Fragility assessment of steel frames using neural networks. Earthq. Spectra
**2007**, 23, 735–752. [Google Scholar] [CrossRef] - Khatibinia, M.; Salajegheh, E.; Salajegheh, J.; Fadaee, M.J. Reliability-based design optimization of reinforced concrete structures including soil–structure interaction using a discrete gravitational search algorithm and a proposed metamodel. Eng. Optim.
**2013**, 45, 1147–1165. [Google Scholar] [CrossRef] - Pang, Y.; Dang, X.; Yuan, W. An artificial neural network based method for seismic fragility analysis of highway bridges. Adv. Struct. Eng.
**2014**, 17, 413–428. [Google Scholar] [CrossRef] - Towashiraporn, P. Building Seismic Fragilities Using Response Surface Metamodels. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, CA, USA, 2004. [Google Scholar]
- Park, J.; Towashiraporn, P. Rapid seismic damage assessment of railway bridges using the response-surface statistical model. Struct. Saf.
**2014**, 47, 1–12. [Google Scholar] [CrossRef] - Saha, S.K.; Matsagar, V.; Chakraborty, S. Uncertainty quantification and seismic fragility of base-isolated liquid storage tanks using response surface models. Probabilistic Eng. Mech.
**2016**, 43, 20–35. [Google Scholar] [CrossRef] - Ghosh, S.; Ghosh, S.; Chakraborty, S. Seismic reliability analysis of reinforced concrete bridge pier using efficient response surface method–based simulation. Adv. Struct. Eng.
**2018**, 21, 2326–2339. [Google Scholar] [CrossRef] - Zhang, Y.; Wu, G. Seismic vulnerability analysis of RC bridges based on Kriging model. J. Earthq. Eng.
**2019**, 23, 242–260. [Google Scholar] [CrossRef] - Xiao, Y.; Ye, K.; He, W. An improved response surface method for fragility analysis of base-isolated structures considering the correlation of seismic demands on structural components. Bull. Earthq. Eng.
**2020**, 18, 4039–4059. [Google Scholar] [CrossRef] - Datta, G.; Bhattacharjya, S.; Chakraborty, S. Efficient reliability-based robust design optimization of structures under extreme wind in dual response surface framework. Struct. Multidiscip. Optim.
**2020**, 62, 2711–2730. [Google Scholar] [CrossRef] - Liu, H.; Ong, Y.S.; Cai, J. A survey of adaptive sampling for global metamodeling in support of simulation-based complex engineering design. Struct. Multidiscip. Optim.
**2018**, 57, 393–416. [Google Scholar] [CrossRef] - Zhuang, X.; Pan, R. A sequential sampling strategy to improve reliability-based design optimization with implicit constraint functions. J. Mech. Des.
**2012**, 134, 021002. [Google Scholar] [CrossRef] - Bichon, B.J.; Eldred, M.S.; Mahadevan, S.; McFarland, J.M. Efficient global surrogate modeling for reliability-based design optimization. J. Mech. Des.
**2013**, 135, 011009. [Google Scholar] [CrossRef] - Chen, Z.; Qiu, H.; Gao, L.; Li, X.; Li, P. A local adaptive sampling method for reliability-based design optimization using Kriging model. Struct. Multidiscip. Optim.
**2014**, 49, 401–416. [Google Scholar] [CrossRef] - Meng, Z.; Zhang, Z.; Zhang, D.; Yang, D. An active learning method combining Kriging and accelerated chaotic single loop approach (AK-ACSLA) for reliability-based design optimization. Comput. Methods Appl. Mech. Eng.
**2019**, 357, 112570. [Google Scholar] [CrossRef] - Li, G.; Yang, H.; Zhao, G. A new efficient decoupled reliability-based design optimization method with quantiles. Struct. Multidiscip. Optim.
**2020**, 61, 635–647. [Google Scholar] [CrossRef] - Wang, C.; Qiu, Z. Improved numerical prediction and reliability-based optimization of transient heat conduction problem with interval parameters. Struct. Multidiscip. Optim.
**2015**, 51, 113–123. [Google Scholar] [CrossRef] - Wang, C.; Qiu, Z.; Xu, M.; Li, Y. Novel numerical methods for reliability analysis and optimization in engineering fuzzy heat conduction problem. Struct. Multidiscip. Optim.
**2017**, 56, 1247–1257. [Google Scholar] [CrossRef] - Chen, Z.; Qiu, H.; Gao, L.; Li, P. An optimal shifting vector approach for efficient probabilistic design. Struct. Multidiscip. Optim.
**2013**, 47, 905–920. [Google Scholar] [CrossRef] - Jones, D.R.; Schonlau, M.; Welch, W.J. Efficient global optimization of expensive black-box functions. J. Glob. Optim.
**1998**, 13, 455–492. [Google Scholar] [CrossRef] - Qian, J.; Cheng, Y.; Zhang, J.; Liu, J.; Zhan, D. A parallel constrained efficient global optimization algorithm for expensive constrained optimization problems. Eng. Optim.
**2021**, 53, 300–320. [Google Scholar] [CrossRef] - Lv, Z.; Lu, Z.; Wang, P. A new learning function for Kriging and its applications to solve reliability problems in engineering. Comput. Math. Appl.
**2015**, 70, 1182–1197. [Google Scholar] [CrossRef] - Vamvatsikos, D.; Cornell, C.A. Incremental dynamic analysis. Earthq. Eng. Struct. Dyn.
**2002**, 31, 491–514. [Google Scholar] [CrossRef] - Lagaros, N.D.; Papadrakakis, M. Robust seismic design optimization of steel structures. Struct. Multidiscip. Optim.
**2007**, 33, 457–469. [Google Scholar] [CrossRef] - Pozzi, M.; Wang, Q. Gaussian Process Regression and Classification for Probabilistic Damage Assessment of Spatially Distributed Systems. KSCE J. Civ. Eng.
**2018**, 22, 1016–1026. [Google Scholar] [CrossRef] - Su, G.; Peng, L.; Hu, L. A Gaussian process-based dynamic surrogate model for complex engineering structural reliability analysis. Struct. Saf.
**2017**, 68, 97–109. [Google Scholar] [CrossRef] - Dixon, M.; Ward, T. Information-Corrected Estimation: A Generalization Error Reducing Parameter Estimation Method. Entropy
**2021**, 23, 1419. [Google Scholar] [CrossRef] [PubMed] - Šinkovec, H.; Geroldinger, A.; Heinze, G. Bring more data!—A good advice? Removing separation in logistic regression by increasing sample size. Int. J. Environ. Res. Public Health
**2019**, 16, 4658. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Preuss, R.; Von Toussaint, U. Global optimization employing Gaussian process-based Bayesian surrogates. Entropy
**2018**, 20, 201. [Google Scholar] [CrossRef] - Rasmussen, C.E.; Williams, C.K.I. Gaussian Processes for Machine Learning; MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- Xiao, Y.J.; Yue, F.; Zhang, X.A.; Shahzad, M.M. Aseismic Optimization of Mega-sub Controlled Structures Based on Gaussian Process Surrogate Model. KSCE J. Civ. Eng.
**2022**. [Google Scholar] [CrossRef] - Lelièvre, N.; Beaurepaire, P.; Mattrand, C.; Gayton, N. AK-MCSi: A Kriging-based method to deal with small failure probabilities and time-consuming models. Struct. Saf.
**2018**, 73, 1–11. [Google Scholar] [CrossRef] - Yun, W.; Lu, Z.; Zhou, Y.; Jiang, X. AK-SYSi: An improved adaptive Kriging model for system reliability analysis with multiple failure modes by a refined U learning function. Struct. Multidiscip. Optim.
**2019**, 59, 263–278. [Google Scholar] [CrossRef] - Forrester, A.; Sobester, A.; Keane, A. Engineering Design via Surrogate Modelling: A Practical Guide; John Wiley & Sons Ltd.: Chichester, UK, 2008. [Google Scholar]
- Jäntschi, L.; Bolboacă, S.D. Computation of probability associated with Anderson–Darling statistic. Mathematics
**2018**, 6, 88. [Google Scholar] [CrossRef] [Green Version] - Cornell, C.A. Engineering seismic risk analysis. Bull. Seismol. Soc. Am.
**1968**, 58, 1583–1606. [Google Scholar] [CrossRef] - Chen, G.X.; Zhang, K.X.; Xie, J.F. A simple calculation method for site liquefaction risk analysis. Earthq. Resist. Eng. Retrofit.
**1992**, 1, 26–29. (In Chinese) [Google Scholar] [CrossRef] - Gao, X.W.; Bao, A.B. Probabilistic model and its statistical parameters for seismic load. Earthq. Eng. Eng. Vib.
**1985**, 1, 13–22. (In Chinese) [Google Scholar] [CrossRef] - Chen, G.X.; Zhang, K.X.; Xie, J.F. A Theory Study Reliability on the Ground Aseismic Analysis Method. J. Harbin Univ. Civ. Eng. Archit.
**1996**, 29, 36–43. (In Chinese) [Google Scholar] - GB 50011-2010; Code for Seismic Design of Buildings. Ministry of Construction of the People’s Republic of China: Beijing, China, 2010. (In Chinese)
- Wang, X.; Shahzad, M.M.; Shi, X. Research on the disaster prevention mechanism of mega-sub controlled structural system by vulnerability analysis. Structures
**2021**, 33, 4481–4491. [Google Scholar] [CrossRef] - Haukaas, T.; Scott, M.H. Shape sensitivities in the reliability analysis of nonlinear frame structures. Comput. Struct.
**2006**, 84, 964–977. [Google Scholar] [CrossRef] - Tavakoli, R.; Kamgar, R.; Rahgozar, R. Optimal location of energy dissipation outrigger in high-rise building considering nonlinear soil-structure interaction effects. Period. Polytech. Civ. Eng.
**2020**, 64, 887–903. [Google Scholar] [CrossRef]

**Figure 6.**The response of the added samples in Example 1. (

**a**) N

_{I}= 9. (

**b**) N

_{I}= 3. (

**c**) N

_{I}= 6. (

**d**) N

_{I}= 17.

No. | Earthquake Name | Year | Station Name | Arias Intensity (m/s) | Magnitude |
---|---|---|---|---|---|

1 | Cape Mendocino | 1992 | Eureka—Myrtle and West | 0.3 | 7.01 |

2 | Cape Mendocino | 1992 | Fortuna—Fortuna Blvd | 0.3 | 7.01 |

3 | Landers | 1992 | Yermo Fire Station | 0.9 | 7.28 |

4 | Northridge-01 | 1994 | Downey—Co Maint Bldg | 0.6 | 6.69 |

5 | Northridge-01 | 1994 | Hollywood—Willoughby Ave | 0.9 | 6.69 |

6 | Northridge-01 | 1994 | LA—Baldwin Hills | 0.7 | 6.69 |

7 | Northridge-01 | 1994 | Moorpark—Fire Sta | 0.9 | 6.69 |

8 | Kocaeli_ Turkey | 1999 | Iznik | 0.4 | 7.51 |

9 | Cape Mendocino | 1992 | College of the Redwoods | 0.6 | 7.01 |

10 | Chuetsu-oki_ Japan | 2007 | Joetsu Ogataku | 0.7 | 6.8 |

11 | Chuetsu-oki_ Japan | 2007 | Sanjo Shinbori | 2 | 6.8 |

12 | Chuetsu-oki_ Japan | 2007 | Nakanoshima Nagaoka | 2.1 | 6.8 |

13 | Chuetsu-oki_ Japan | 2007 | Yan Sakuramachi City watershed | 0.7 | 6.8 |

14 | Iwate_ Japan | 2008 | Kami_ Miyagi Miyazaki City | 0.4 | 6.9 |

15 | Iwate_ Japan | 2008 | Matsuyama City | 1.2 | 6.9 |

16 | Iwate_ Japan | 2008 | Iwadeyama | 1.8 | 6.9 |

17 | Iwate_ Japan | 2008 | Misato_ Miyagi Kitaura—B | 0.7 | 6.9 |

18 | Iwate_ Japan | 2008 | Minamikatamachi Tore City | 1.4 | 6.9 |

19 | Iwate_ Japan | 2008 | Yokote Masuda Tamati Masu | 0.3 | 6.9 |

20 | Iwate_ Japan | 2008 | Yokote Ju Monjimachi | 0.4 | 6.9 |

Variable | Distribution Type | Mean | Standard Deviation |
---|---|---|---|

f_{1} (Pa) | Lognormal | 3 × 10^{8} | 3 × 10^{7} |

f_{2} (Pa) | Lognormal | 3 × 10^{8} | 3 × 10^{7} |

E_{1} (Pa) | Lognormal | 2 × 10^{11} | 2 × 10^{10} |

E_{2} (Pa) | Lognormal | 2 × 10^{11} | 2 × 10^{10} |

u | Normal | 0 | 1 |

a (m/s^{2}) | Type II extreme value | 1.17 | 1.376 |

t_{c} (m) | Lognormal | ${\mu}_{{t}_{\mathrm{c}}}$ | 0.05·${\mu}_{{t}_{\mathrm{c}}}$ |

No. | Input Variable Vector x | φ | |||||||
---|---|---|---|---|---|---|---|---|---|

f_{1}(10 ^{8} Pa) | f_{2}(10 ^{8} Pa) | E_{1}(10 ^{11} Pa) | E_{2}(10 ^{11} Pa) | u | a (m/s ^{2}) | b_{c}(m) | t_{c}(m) | ||

1 | 2.614 | 4.143 | 2.339 | 1.636 | 1 | 2.59 | 0.395 | 0.01875 | 0.00785 |

2 | 4.143 | 3.837 | 1.837 | 1.837 | −2 | 4.86 | 0.5 | 0.01125 | 0.00697 |

3 | 3.837 | 2.309 | 2.439 | 2.138 | 3 | 9.39 | 0.465 | 0.015 | 0.03796 |

4 | 2.003 | 3.532 | 1.636 | 2.038 | 4 | 13.92 | 0.36 | 0.01625 | 0.28035 |

5 | 4.449 | 2.92 | 2.138 | 1.937 | 0 | 18.45 | 0.29 | 0.02 | 0.06172 |

6 | 3.532 | 4.449 | 2.038 | 2.339 | 2 | 7.12 | 0.22 | 0.01375 | 0.12321 |

7 | 2.92 | 2.003 | 1.937 | 1.736 | −1 | 11.65 | 0.255 | 0.01 | 0.03365 |

8 | 2.309 | 3.226 | 2.239 | 2.439 | −3 | 16.18 | 0.43 | 0.0125 | 0.01047 |

9 | 3.226 | 2.614 | 1.736 | 2.239 | −4 | 0.33 | 0.325 | 0.0175 | 0.00025 |

Method | N_{c} | Design Variable Vector D | S_{c} (m^{2}) | ${\hat{\mathit{p}}}_{\mathit{f}}$ | |
---|---|---|---|---|---|

b_{c} (m) | ${\mathit{\mu}}_{{\mathit{t}}_{\mathbf{c}}}\left(\mathbf{m}\right)$ | ||||

EGO-EGRA | 120 | 0.3322 | 0.01 | 0.01289 | 0.0481 |

Proposed method | 9 + 52 | 0.3300 | 0.01 | 0.01280 | 0.0497 |

3 + 61 | 0.3302 | 0.01 | 0.01281 | 0.0496 | |

6 + 58 | 0.3304 | 0.01 | 0.01282 | 0.0496 | |

17 + 55 | 0.3321 | 0.01 | 0.01288 | 0.0483 | |

Initial scenario | - | 0.25 | 0.016 | 0.01498 | 0.0914 |

Variable | Distribution Type | Mean | Standard Deviation |
---|---|---|---|

f_{s} (Pa) | Lognormal | 3.07 × 10^{8} | 3.07 × 10^{7} |

f_{c} (Pa) | Lognormal | 3.447 × 10^{7} | 3.447 × 10^{6} |

E_{s} (Pa) | Lognormal | 2.01 × 10^{11} | 2.01 × 10^{10} |

ε_{u,core} | Lognormal | 5 × 10^{−3} | 5 × 10^{−4} |

u | Normal | 0 | 1 |

a (m/s^{2}) | Type II extreme value | 1.17 | 1.376 |

No. | Input Variable Vector x | φ(x) | |||||||
---|---|---|---|---|---|---|---|---|---|

f_{s}(10 ^{8} Pa) | f_{c}(10 ^{7} Pa) | E_{s}(10 ^{11} Pa) | ε_{u,core}(10 ^{−3}) | u | a (m/s ^{2}) | b_{1}(m) | b_{2}(m) | ||

1 | 2.363 | 4.76 | 1.342 | 5.377 | 3 | 6.18 | 0.35 | 0.375 | 0.1648 |

2 | 4.24 | 4.409 | 2.571 | 7.415 | 1 | 4.23 | 0.425 | 0.3 | 0.02113 |

3 | 2.988 | 5.112 | 2.981 | 4.867 | −3 | 10.08 | 0.45 | 0.425 | 0.00566 |

4 | 3.614 | 3.707 | 1.957 | 3.338 | −4 | 2.28 | 0.3 | 0.4 | 0.00139 |

5 | 3.927 | 2.301 | 2.776 | 4.357 | 4 | 8.13 | 0.375 | 0.475 | 0.28316 |

6 | 4.553 | 4.058 | 1.547 | 6.396 | −2 | 12.03 | 0.4 | 0.5 | 0.01429 |

7 | 3.301 | 3.355 | 1.752 | 3.848 | 2 | 13.98 | 0.5 | 0.325 | 0.22224 |

8 | 2.675 | 2.653 | 2.366 | 6.905 | −1 | 15.93 | 0.325 | 0.35 | 0.04925 |

9 | 2.05 | 3.004 | 2.161 | 5.886 | 0 | 0.34 | 0.475 | 0.45 | 0.00066 |

Method | N_{c} | Design Variable Vector D | V_{c} (m^{3}) | ${\hat{\mathit{p}}}_{\mathit{f}}$ | |
---|---|---|---|---|---|

b_{1} (m) | b_{2} (m) | ||||

EGO-EGRA | 196 | 0.448 | 0.386 | 9.02 | 0.0424 |

Proposed method | 9 + 101 | 0.442 | 0.388 | 9.02 | 0.0423 |

Initial scenario | - | 0.33 | 0.43 | 9.02 | 0.096 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xiao, Y.; Yue, F.; Wang, X.; Zhang, X.
Reliability-Based Design Optimization of Structures Considering Uncertainties of Earthquakes Based on Efficient Gaussian Process Regression Metamodeling. *Axioms* **2022**, *11*, 81.
https://doi.org/10.3390/axioms11020081

**AMA Style**

Xiao Y, Yue F, Wang X, Zhang X.
Reliability-Based Design Optimization of Structures Considering Uncertainties of Earthquakes Based on Efficient Gaussian Process Regression Metamodeling. *Axioms*. 2022; 11(2):81.
https://doi.org/10.3390/axioms11020081

**Chicago/Turabian Style**

Xiao, Yanjie, Feng Yue, Xinwei Wang, and Xun’an Zhang.
2022. "Reliability-Based Design Optimization of Structures Considering Uncertainties of Earthquakes Based on Efficient Gaussian Process Regression Metamodeling" *Axioms* 11, no. 2: 81.
https://doi.org/10.3390/axioms11020081