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 4305 design parameters are analyzed and the error equation of achieved performance probabilities based on ASCE/SEI 4305 is also obtained. It is found that the confidence levels for design results obtained using ASCE/SEI 4305 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 4305 criteria and the error values between risk model with target confidence level and mean risk model using ASCE/SEI 4305 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.
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 [
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 [
The mean seismic probability risk model has been accepted as a basis for riskinformed decisionmaking 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 decisionmakers. 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 4305 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.
The limit state probability
Modern seismic risk analysis, beginning with the seminal paper by Cornell [
The seismic hazard function approximation in closed form can be expressed as [
The seismic fragility can be expressed as [
Substituting (
There are some approximations for (
The interval function of the fragility model is defined as [
Equation (
Equation (
Cornell [
Substituting (
Equation (
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 (
When
It is assumed that all parameters of (
From the definitions of the variables
When there exists the epistemic uncertainty in the analysis or
Confidence levels of mean seismic probability risk model used in West United States sites.
Confidence levels of mean seismic risk probability model used in Central and Eastern United States sites.
The mean seismic risk model is obtained through calculating the mean of the distribution representing an epistemic uncertainty with the range of confidence levels. From (
The confidence levels of the mean probability risk model are calculated using the parameters
Confidence levels and error values of mean seismic probability risk model for the usual range of


Confidence levels and error results between risk model with 99% confidence  








1.5  5.6789  61.18% (2.1895)  71.49% (6.3685)  80.28% (11.3305)  87.20% (13.9460)  92.22% (12.1224)  95.58% (7.3452) 
1.75  4.1146  58.15% (1.3930)  65.96% (3.8346)  73.14% (7.2461)  79.47% (10.8745)  84.82% (13.4364)  89.15% (13.8175) 
2  3.3219  56.60% (1.0495)  63.01% (2.7618)  69.09% (5.1830)  74.68% (8.1008)  79.69% (10.9961)  84.05% (13.1603) 
2.25  2.8394  55.64% (0.8594)  61.18% (2.1895)  66.49% (4.0473)  71.49% (6.3685)  76.11% (8.9240)  80.28% (11.3305) 
2.5  2.5129  55.00% (0.7385)  59.92% (1.8374)  64.69% (3.3476)  69.24% (5.2539)  73.51% (7.4455)  77.45% (9.7071) 
2.75  2.2762  54.53% (0.6547)  59.00% (1.5998)  63.36% (2.8784)  67.55% (4.4937)  71.53% (6.3888)  75.27% (8.4360) 
3  2.0959  54.17% (0.5930)  58.30% (1.4286)  62.34% (2.5433)  66.25% (3.9476)  69.99% (5.6114)  73.53% (7.4551) 
3.25  1.9536  53.89% (0.5456)  57.74% (1.2993)  61.53% (2.2926)  65.20% (3.5383)  68.74% (5.0212)  72.11% (6.6895) 
3.5  1.8380  53.66% (0.5079)  57.29% (1.1981)  60.86% (2.0979)  64.34% (3.2210)  67.71% (4.5602)  70.93% (6.0810) 
3.75  1.7421  53.47% (0.4771)  56.91% (1.1166)  60.31% (1.9424)  63.62% (2.9681)  66.84% (4.1913)  69.94% (5.5885) 
4  1.6610  53.31% (0.4515)  56.60% (1.0495)  59.84% (1.8152)  63.01% (2.7618)  66.10% (3.8897)  69.09% (5.1830) 
4.25  1.5914  53.17% (0.4298)  56.32% (0.9933)  59.43% (1.7092)  62.49% (2.5903)  65.46% (3.6389)  68.35% (4.8439) 
4.5  1.5309  53.05% (0.4112)  56.08% (0.9453)  59.08% (1.6194)  62.03% (2.4455)  64.90% (3.4271)  67.70% (4.5566) 
4.75  1.4778  52.95% (0.3950)  55.87% (0.9039)  58.77% (1.5423)  61.62% (2.3215)  64.41% (3.2458)  67.12% (4.3101) 
5  1.4307  52.85% (0.3807)  55.69% (0.8677)  58.50% (1.4753)  61.26% (2.2141)  63.97% (3.0889)  66.61% (4.0963) 
5.25  1.3886  52.77% (0.3681)  55.52% (0.8358)  58.25% (1.4165)  60.94% (2.1201)  63.58% (2.9517)  66.15% (3.9093) 
5.5  1.3507  52.69% (0.3568)  55.37% (0.8075)  58.03% (1.3645)  60.65% (2.0372)  63.22% (2.8307)  65.73% (3.7442) 
5.75  1.3164  52.62% (0.3466)  55.24% (0.7821)  57.83% (1.3180)  60.38% (1.9633)  62.90% (2.7231)  65.35% (3.5974) 
6  1.2851  52.56% (0.3374)  55.11% (0.7592)  57.64% (1.2763)  60.14% (1.8971)  62.60% (2.6268)  65.01% (3.4660) 
In fact, the influence of epistemic uncertainty on risk results is not only related to confidence level
The relative distance between
The theoretical basis of ASCE/SEI 4305 riskconsistent seismic design is as follows [
For implementation purposes of ASCE/SEI 4305, the parameters in the risk equation are reformulated in terms of design terms which are familiar to engineers such as structural engineers [
Substituting the value
It is assumed that
Substituting (
Equation (
The design factor equations of ASCE/SEI 4305 are expressed as follows:
Summary of earthquake design provisions of ASCE/SEI 4305 [
Seismic Design Category (SDC)  Target performance goal ( 
Probability ratio ( 
Hazard exceedance probability ( 



3  1 × 10^{−4}  4  4 × 10^{−4}  0.8  0.40 
4  4 × 10^{−5}  10  4 × 10^{−4}  1.0  0.80 
5  1 × 10^{−5}  10  1 × 10^{−4}  1.0  0.80 
The mean probability risk model is the theoretical basis of ASCE/SEI 4305 riskconsistent seismic design methodology. For analyzing the confidence of the risk results based on ASCE/SEI 4305 design methodology, the interval risk estimation functions of ASCE/SEI 4305 are obtained. The confidence equations of ASCE/SEI 4305 riskconsistent seismic design are as follows.
Equation (
Substituting (
Equation (
Equations (
Solving (
Substituting (
Solving (
Substituting (
When neither
For three SDCs of ASCE/SEI 4305 listed in Table
Confidence levels of exact DF for SDCs 3, 4, and 5 for the range of
ASCE/SEI 4305 [
For fully considering the reliability of achieved performance probabilities using ASCE/SEI 4305 riskconsistent seismic design methodology, the error equation is obtained. The error equation, which is defined as the relative distance between point estimation function (see (
It is found that the error equation of achieved performance probabilities based on ASCE/SEI 4305 is the same as the mean seismic risk model, as the mean seismic probability risk model is the theoretical basis of ASCE/SEI 4305.
The typical normalized spectral acceleration hazard curves values are listed in Table
Typical normalized spectral acceleration hazard curve values of ASCE/SEI 4305 [

Eastern US  California  

1 Hz  10 Hz  1 Hz  10 Hz  
SA  SA  SA  SA  
5 × 10^{−2}  0.014  0.018  0.087  0.046 
2 × 10^{−2}  0.027  0.034  0.13  0.072 
1 × 10^{−2}  0.045  0.055  0.175  0.100 
5 × 10^{−3}  0.07  0.089  0.236  0.139 
2 × 10^{−3}  0.143  0.169  0.351  0.215 
1 × 10^{−3}  0.235  0.275  0.474  0.334 
5 × 10^{−4}  0.383  0.424  0.629  0.511 
2 × 10^{−4}  0.681  0.709  0.814  0.762 
1 × 10^{−4}  1.00  1.00  1.00  1.00 
5 × 10^{−5}  1.46  1.41  1.23  1.22 
2 × 10^{−5}  2.35  2.13  1.61  1.51 
1 × 10^{−5}  3.27  2.88  1.89  1.76 
5 × 10^{−6}  4.38  3.65  2.2  2.05 
2 × 10^{−6}  6.44  4.62  2.68  2.42 
1 × 10^{−6}  8.59  5.43  3.1  2.72 
5 × 10^{−7}  10.34  6.38  3.58  3.06 
2 × 10^{−7}  13.21  7.9  4.24  3.56 
1 × 10^{−7}  15.9  9.28  4.67  3.84 
In ASCE/SEI 4305, both of the following criteria need be obtained [
Less than 1% probability of acceptable performance for the DBE ground motion, which is defined as the Design Response Spectra (DRS)
Less than 10% probability of acceptable performance for a ground motion equal to 150% of the DBE ground motion, which is defined as the DRS.
For the two criteria in ASCE/SEI 4305, we could calculate the achieved performance probabilities
Seismic margin factors for different







0.30  1.10  1.35  1.5  2.2  2.58 
0.40  1  1.31  1.52  2.54  3.13 
0.50  1  1.41  1.69  3.2  4.16 
0.60  1  1.5  1.87  4.04  5.53 
When the achieved performance probability
Confidence levels of achieved performance probabilities for SDCs 3 and 4 based on the first and second criteria of ASCE/SEI 4305 for representative hazard curves.
Confidence levels of achieved performance probabilities for SDC 5 based on both the first and second criteria of ASCE/SEI 4305 for representative hazard curves.
The error analysis for achieved performance probabilities for the above example is conducted. The results are shown in Table
Error results for both the first and second criteria of ASCE/SEI 4305 for representative hazard curves.
SDC  Hazard curves  Error results between risk model with 99% confidence and mean risk model  











3  EUS 1 Hz  1.2163  1.8022  2.4889  3.2777 
EUS 10 Hz  1.4066  2.1043  2.9285  3.8777  
Calif 1 Hz  3.1584  4.9483  7.0248  9.2106  
Calif 10 Hz  2.3304  3.6001  5.1107  6.8066  


4  EUS 1 Hz  1.2163  1.8022  2.4889  3.2777 
EUS 10 Hz  1.4066  2.1043  2.9285  3.8777  
Calif 1 Hz  3.1584  4.9483  7.0248  9.2106  
Calif 10 Hz  2.3304  3.6001  5.1107  6.8066  


5  EUS 1 Hz  1.3967  2.0885  2.9056  3.8464 
EUS 10 Hz  1.6317  2.4652  3.4559  4.5957  
Calif 1 Hz  3.4400  5.4025  7.6479  9.9412  
Calif 10 Hz  4.1385  6.5117  9.1067  11.5157 
In this study, the confidence equation of the mean seismic risk model is derived. It could be found that the confidence levels of the mean probability risk model for different sites are changed, and the confidence levels are also low for most sites.
The error equation between
The confidence equation derivation of design factor DF and failure probability
The error equation of achieved performance probabilities based on ASCE/SEI 4305 riskconsistent seismic design methodology is obtained. It could be found that error results between target confidence level (the target confidence is taken as 0.99 in the paper) and achieved performance probabilities based on ASCE/SEI 4305 are changed for different sites and the error values are large for some sites.
It is suggested that 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.
The authors declare that they have no conflicts of interest.
The financial support received from the National Science Foundation of China (Grant nos. 51678209, 51378162, and 91315301), the Research Fund of Ministry of Science and Technology of China (2013BAJ08B01), and the Open Research Fund of State Key Laboratory for Disaster Reduction in Civil Engineering (SLDRCE12MB04) is gratefully appreciated.