Science and Technology of Nuclear Installations

Volume 2017 (2017), Article ID 2480940, 9 pages

https://doi.org/10.1155/2017/2480940

## Study on the Confidence and Reliability of the Mean Seismic Probability Risk Model

^{1}School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China^{2}Key Lab of Structures Dynamic Behavior and Control of China Ministry of Education, Harbin Institute of Technology, Harbin 150090, China^{3}State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Da-Gang Lu; nc.ude.tih@gnagadul

Received 26 November 2016; Revised 26 January 2017; Accepted 30 July 2017; Published 10 October 2017

Academic Editor: Eugenijus Ušpuras

Copyright © 2017 Xiao-Lei Wang and Da-Gang Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The mean seismic probability risk model has widely been used in seismic design and safety evaluation of critical infrastructures. In this paper, the confidence levels analysis and error equations derivation of the mean seismic probability risk model are conducted. It has been found that the confidence levels and error values of the mean seismic probability risk model are changed for different sites and that the confidence levels are low and the error values are large for most sites. Meanwhile, the confidence levels of ASCE/SEI 43-05 design parameters are analyzed and the error equation of achieved performance probabilities based on ASCE/SEI 43-05 is also obtained. It is found that the confidence levels for design results obtained using ASCE/SEI 43-05 criteria are not high, which are less than 95%, while the high confidence level of the uniform risk could not be achieved using ASCE/SEI 43-05 criteria and the error values between risk model with target confidence level and mean risk model using ASCE/SEI 43-05 criteria are large for some sites. It is suggested that the seismic risk model considering high confidence levels instead of the mean seismic probability risk model should be used in the future.

#### 1. Introduction

The mean seismic probability risk model has widely been used in seismic design and safety evaluation of critical infrastructures, such as nuclear power plants. Seismic probability risk assessment is one of seismic safety evaluation methodologies for nuclear power plants [1, 2]. Nowadays, the U.S. Nuclear Regulatory Commission has adopted the mean seismic probability risk model as the basis for risk-informed decision-making. ASCE/SEI 43-05, whose basis was also the mean seismic probability risk model, was a risk-consistent seismic design criterion in the United States [3]. Kennedy [4] provided the technical basis for the performance-goal based approach presented in the American Society of Civil Engineering Standard ASCE/SEI 43-05. In China, the newest modified seismic design code of nuclear power plants [5] suggested using the seismic evaluation methodology of ASCE/SEI 43-05, whose basis was also the mean seismic probability risk model. In some engineering domains other than nuclear engineering, the mean seismic risk model has also been widely studied and implemented. Lu et al. [6] conducted the seismic risk assessment for a reinforced concrete frame designed according to Chinese codes based on the mean seismic probability risk model.

However, the confidence of the mean seismic probability risk model is unknown. In other words, the mean seismic probability risk model could not convey the sense of confidence directly. Ellingwood and Kinali [7] proposed that some decision-makers may not be comfortable with the mean as a point estimation and prefer a more conservative fractile of the risk distribution, especially if consequences are severe. Many studies on the interval model of seismic risk and structural reliability have been conducted. For the 2000 SAC Federal Emergency Management Agency (FEMA) steel moment frame guidelines [8], the format based on quantitative confidence statements regarding the likelihood of the performance objective was provided. Lü and Yu [9] studied the interval model of seismic risk based on the mean hazard curve and capacity and demand models. Katona [10] studied the uncertainty in seismic safety analysis, including seismic probability risk assessment and seismic margin assessment, based on p-box theory. De Leon and Ang [11] conducted confidence bounds analysis on structural reliability estimations for offshore platforms, and they suggested that the estimation of percentile values, instead of mean values, of the calculated risk should be specified to ensure sufficient low risk levels for decisions-making purposes. Ang [12] proposed that the confidence level of seismic reliability model considering epistemic uncertainties should be as high as 90% to 95% for reducing the influence of epistemic uncertainties on reliability analysis results in some critical systems, while the mean safety index (obtained with the total of the combined aleatory and epistemic uncertainty) has a confidence level of only around 50%. In nuclear engineering domain, high confidence of low probability of failure (HCLPF), which was defined as 95% confidence of less than 5% probability of failure and commonly used as the seismic capacity of structures, systems, and components (SSCs) of nuclear power plants, has been an important concept, which showed that SSCs of nuclear power plants should have low failure risk with high confidence facing earthquake disasters.

The mean seismic probability risk model has been accepted as a basis for risk-informed decision-making by U.S. Nuclear Regulatory Commission. However, for nuclear power plants as critical infrastructure, whose accident consequences are severe, the less failure probability risk with higher confidence should be required by decision-makers. In this paper, the confidence and error equations derivation of the approximate mean seismic probability risk are conducted. Meanwhile, the theoretical basis of ASCE/SEI 43-05 code based on the approximate mean seismic probability risk model is extended to the approximate interval model. Confidence levels and error values based on these equations are then calculated. It is suggested that the seismic risk model considering high confidence levels instead of the mean probability risk model should be used for seismic design and safety evaluation of critical infrastructures such as nuclear power plants in the future.

#### 2. Seismic Probability Risk Models and Confidence and Error Equations

##### 2.1. The Approximate Point Estimation Model of Seismic Probability Risk

The limit state probability of seismic risk can be determined as the convolution of the seismic hazard curve and the fragility curve by either of the following two mathematically equivalent equations [13]:where is the hazard curve (the annual frequency of exceeding amplitude ) and is the fragility curve; (1) is equivalent to (2).

Modern seismic risk analysis, beginning with the seminal paper by Cornell [14] and some later studies by Ellingwood [7, 15], shows that the ground motion intensity can be represented by a type II distribution of extreme values, while the fragility function can be modelled by lognormal distribution.

The seismic hazard function approximation in closed form can be expressed as [7] where is the scale parameter, is the shape parameter, and is the constant; that is, .

The seismic fragility can be expressed as [16–18] in which represents the standard normal probability integral, is the median capacity, and represent the logarithmic standard deviation in capacity of aleatory uncertainty and epistemic uncertainty, respectively, is the square root of the sum of the squares (SRSS) of and .

Substituting (3) and (4) into (1), the so-called “risk equation” can be obtained:

There are some approximations for (6): (1) the seismic hazard function, which is approximated by (3), is assumed to be linear on a log-log scale; (2) the seismic fragility function is assumed to follow the lognormal distribution.

##### 2.2. The Approximate Interval Estimation Model of Seismic Probability Risk

The interval function of the fragility model is defined as [16–18]where is the confidence parameter.

Equation (7) can be transformed to

Equation (8) can be furtherly expressed as

Cornell [19] has pointed out that epistemic uncertainty can be modelled: (1) by assigning a lognormal (epistemic) distribution to and (2) by an epistemic lognormal distribution on median . McGuire et al. [20] suggested that, given the need to express seismic hazard with a single curve, the mean seismic hazard was the preferred single curve. In this paper, the mean seismic hazard curve is used, and the epistemic uncertainty is modelled only by the lognormal distribution on the median . Substituting (3) and (9) into (1), the interval estimation of the risk equation can be obtained:

Substituting (10) into (11), the risk equation considering the confidence of the fragility model can be obtained:

Equation (12) can be furtherly expressed as

##### 2.3. The Confidence and Error Equations of the Mean Seismic Risk Model

The mean seismic probability risk model has been widely used in seismic design and safety evaluation of some critical infrastructures, such as nuclear power plants. However, the mean seismic risk model has no direct information of confidence level, which the analyst has in the risk assessment. In order to analyze the confidence of the mean probability risk model, a new approach is proposed below in this paper.

Equation (6) represents the mean probability risk model, whose confidence level is unknown. Meanwhile, (13) is an interval risk model considering confidence level of the fragility model. The risk results could be, respectively, obtained from (6) and (13). Equation (6) represents the point estimation value of seismic risk, while (13) represents the interval estimation values of seismic risk with confidence . When is taken as a value , obtained from (13) would be equal to obtained from (6). In this paper, is regarded as the confidence of obtained from (6). Following this idea, this paper considers that when all parameters except of (6) and (13), which is taken as a value , are the same, the value shall be regarded as the associated confidence level of the mean probability risk model. Ang [12] has discussed the confidence level of mean reliability model and also expressed the same idea with this paper. Ang [12] studied the design of an underground tunnel to resist a strong-motion earthquake and calculated the mean safety index and safety index with different confidence levels, and it was found that when the mean safety index was used in the design of the tunnel lining, the confidence for its safety was only about 50%. In other words, the mean safety index was equal to a safety index with about 50% confidence level of interval safety index with different confidence levels for the given case study.

When of (6) is equal to of (13) and is taken as the value , the equation can be expressed as

It is assumed that all parameters of (6) and (13) except , which is taken as the value , are the same. When neither nor is equal to 0, the confidence of the mean probability risk model can be obtained as follows: where , which represents the slope of the seismic hazard curve, can be furtherly expressed as [3]where represents the ratio of spectra acceleration of probability levels 0.1 and , in which represents the probability of exceedance at uniform hazard response spectrum (UHRS).

From the definitions of the variables and , we could know that is larger than 1, so is always larger than 0, while is no less than 0. When is equal to 0, which means that the perfect knowledge is obtained for the model, (15) would be not suitable, and then the confidence equation of the mean probability risk model would be furtherly expressed as

When there exists the epistemic uncertainty in the analysis or is larger than 0, the confidence level of the mean probability risk model would be obtained using (15). Equation (15) shows that , which is the confidence of the mean probability risk model, is proportional to . From (15), it can be found that the confidence of the mean probability risk model in the regions with steep slopes of mean seismic hazard curves is higher than the regions with shallow slopes of the mean seismic hazard curves. So, for some critical infrastructures, such as nuclear power plants, which are located in the region with shallower slopes of mean seismic hazard curves, the mean probability risk model might be not appropriate. Typical values of in the West United States would be in the range of 3 to 6, while in the Central and Eastern United States, is typically 2.5 or less [13]. The confidence of the mean seismic risk model for the West United States and Central and Eastern United States is, respectively, shown in Figures 1 and 2. The results show that for the West United States the confidence of the mean seismic hazard model is in the range of nearly 55% to 96%, while for the Central and Eastern United States the confidence of the mean seismic hazard model is in the range of nearly 52% to 78%. So, when seismic risk is conducted in the Central and Eastern United States, the reliability of risk results based on the mean seismic risk model should be carefully examined and validated because of lower confidence levels.