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

Authors: Xiao-Lei Wang; Da-Gang Lu
Journal: Science and Technology of Nuclear Installations
 (2017)

---

## 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.


---

## Body

## 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 probabilityPF 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]:(1)PF=∫0+∞HadFCadada(2)PF=-∫0+∞FCadHadada,where H(a) is the hazard curve (the annual frequency of exceeding amplitude a) and FC(a) 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] (3)Ha=1-exp⁡-au-KH≈au-KH=kIa-KH,where u is the scale parameter, KH is the shape parameter, and kI is the constant; that is, kI=uKH.The seismic fragility can be expressed as [16–18] (4)FCa=Φln⁡a/aCβR2+βU2=Φlna/aCβCin which Φ[] represents the standard normal probability integral, aC is the median capacity, βR and βU represent the logarithmic standard deviation in capacity of aleatory uncertainty and epistemic uncertainty, respectively, βC is the square root of the sum of the squares (SRSS) of βR and βU.Substituting (3) and (4) into (1), the so-called “risk equation” can be obtained:(5)PF,mean=HaCexp⁡KHβC22(6)PF,mean=kIaC-KHexp⁡KHβC22.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](7)FCa=Φln⁡a/aC+βUΦ-1QβR,where Q is the confidence parameter.Equation (7) can be transformed to(8)FCa=Φln⁡a/aCexp⁡-βUΦ-1QβR.Equation (8) can be furtherly expressed as (9)FCa=Φln⁡a/aC,QβR(10)aC,Q=aCexp⁡-βUΦ-1Q.Cornell [19] has pointed out that epistemic uncertainty can be modelled: (1) by assigning a lognormal (epistemic) distribution to H(x) and (2) by an epistemic lognormal distribution on median aC. 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 H(x) is used, and the epistemic uncertainty is modelled only by the lognormal distribution on the median aC. Substituting (3) and (9) into (1), the interval estimation of the risk equation can be obtained: (11)PFQ=HaC,Qexp⁡KHβR22.Substituting (10) into (11), the risk equation considering the confidence of the fragility model can be obtained: (12)PFQ=kIaCexp⁡-βUΦ-1Q-KHexp⁡KHβR22.Equation (12) can be furtherly expressed as(13)PFQ=kIaC-KHexp⁡12KHβR2+βUKHΦ-1Q.


### 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 Q. When Q is taken as a value Qmean, PFQmean obtained from (13) would be equal to PF,mean obtained from (6). In this paper, Qmean is regarded as the confidence of PF,mean obtained from (6). Following this idea, this paper considers that when all parameters except Q of (6) and (13), which is taken as a value Qmean, are the same, the value Qmean 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.WhenPF,mean of (6) is equal to PF(Q) of (13) and Q is taken as the value Qmean, the equation can be expressed as(14)kIaC-KHexp⁡KHβC22=kIaC-KHexp⁡12KHβR2+βUKHΦ-1Qmean.It is assumed that all parameters of (6) and (13) except Q, which is taken as the value Qmean, are the same. When neither KH nor βU is equal to 0, the confidence of the mean probability risk model can be obtained as follows: (15)Qmean=ΦKHβU2,where KH, which represents the slope of the seismic hazard curve, can be furtherly expressed as [3](16)KH=1log⁡AR,where AR represents the ratio of spectra acceleration of probability levels 0.1HD and HD, in which HD represents the probability of exceedance at uniform hazard response spectrum (UHRS).From the definitions of the variablesAR and βU, we could know that AR is larger than 1, so KH is always larger than 0, while βU is no less than 0. When βU 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(17)Qmean=100%.When there exists the epistemic uncertainty in the analysis orβU is larger than 0, the confidence level of the mean probability risk model would be obtained using (15). Equation (15) shows that Qmean, which is the confidence of the mean probability risk model, is proportional to KH. 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 KH in the West United States would be in the range of 3 to 6, while in the Central and Eastern United States, KH 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.Figure 1
Confidence levels of mean seismic probability risk model used in West United States sites.Figure 2
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 (15) and (17), it is found that the confidence levels of the mean seismic risk model are always larger than 50%. The physical interpretation of this phenomenon could be expressed as follows: the mean seismic risk model is more sensitive to more severe alternative models considered in the analysis; in other words, the mean seismic risk model is more sensitive to the alternative models with higher confidence levels considered in the analysis.The confidence levels of the mean probability risk model are calculated using the parametersβU, which is usually in a range from 0.1 to 0.6, and AR, which is usually in a range from 1.5 to 6. The results are listed in Table 1. It is found that, for different values of βU and AR, the values of the confidence Q are different, so for different sites, the seismic risk probability model has different confidence levels, or it can also be said that the confidence levels of the seismic risk probability model are changed for different sites. The confidence levels of the seismic probability risk model are not high, the lowest and the highest values of which are 52.56% and 95.58%, respectively. We could also find that most confidence levels of the mean seismic risk model in the usual ranges of βU and AR are less than 90%, which is low for nuclear power plants.Table 1
Confidence levels and error values of mean seismic probability risk model for the usual range ofβU and KH values.
A
R
K
H
Confidence levels and error results between risk model with 99% confidenceand mean risk model (in brackets)
β
U
=
0.1
β
U
=
0.2
β
U
=
0.3
β
U
=
0.4
β
U
=
0.5
β
U
=
0.6
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 levelQ. For fully considering the reliability of the mean seismic risk model, the error equation is provided and analyzed. The error equation, which is defined as the relative distance between PF(Qtarget) and PF,mean, is expressed as follows: (18)Err=PFQtarget-PF,meanPF,mean=PFQtargetPF,mean-1=exp⁡Φ-1Qtarget-KHβU2KHβU-1,where Qtarget is the target confidence level; Err represents the relative error of PF,mean with respect to PF(Qtarget).The relative distance betweenPFQ0.99 (the target confidence level is taken as 0.99 in this paper) and PF,mean is listed in Table 1 for usual range of βU and KH values. It is found that the error between PFQ0.99 and PF,mean is related not only to the confidence level, but also to the product of βU and KH. For some sites with large KH, even if the confidence level is high, the error is still large. So at sites with high KH the mean seismic risk model should be more carefully examined and validated because of their larger error.


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

The limit state probabilityPF 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]:(1)PF=∫0+∞HadFCadada(2)PF=-∫0+∞FCadHadada,where H(a) is the hazard curve (the annual frequency of exceeding amplitude a) and FC(a) 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] (3)Ha=1-exp⁡-au-KH≈au-KH=kIa-KH,where u is the scale parameter, KH is the shape parameter, and kI is the constant; that is, kI=uKH.The seismic fragility can be expressed as [16–18] (4)FCa=Φln⁡a/aCβR2+βU2=Φlna/aCβCin which Φ[] represents the standard normal probability integral, aC is the median capacity, βR and βU represent the logarithmic standard deviation in capacity of aleatory uncertainty and epistemic uncertainty, respectively, βC is the square root of the sum of the squares (SRSS) of βR and βU.Substituting (3) and (4) into (1), the so-called “risk equation” can be obtained:(5)PF,mean=HaCexp⁡KHβC22(6)PF,mean=kIaC-KHexp⁡KHβC22.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](7)FCa=Φln⁡a/aC+βUΦ-1QβR,where Q is the confidence parameter.Equation (7) can be transformed to(8)FCa=Φln⁡a/aCexp⁡-βUΦ-1QβR.Equation (8) can be furtherly expressed as (9)FCa=Φln⁡a/aC,QβR(10)aC,Q=aCexp⁡-βUΦ-1Q.Cornell [19] has pointed out that epistemic uncertainty can be modelled: (1) by assigning a lognormal (epistemic) distribution to H(x) and (2) by an epistemic lognormal distribution on median aC. 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 H(x) is used, and the epistemic uncertainty is modelled only by the lognormal distribution on the median aC. Substituting (3) and (9) into (1), the interval estimation of the risk equation can be obtained: (11)PFQ=HaC,Qexp⁡KHβR22.Substituting (10) into (11), the risk equation considering the confidence of the fragility model can be obtained: (12)PFQ=kIaCexp⁡-βUΦ-1Q-KHexp⁡KHβR22.Equation (12) can be furtherly expressed as(13)PFQ=kIaC-KHexp⁡12KHβR2+βUKHΦ-1Q.


## 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 Q. When Q is taken as a value Qmean, PFQmean obtained from (13) would be equal to PF,mean obtained from (6). In this paper, Qmean is regarded as the confidence of PF,mean obtained from (6). Following this idea, this paper considers that when all parameters except Q of (6) and (13), which is taken as a value Qmean, are the same, the value Qmean 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.WhenPF,mean of (6) is equal to PF(Q) of (13) and Q is taken as the value Qmean, the equation can be expressed as(14)kIaC-KHexp⁡KHβC22=kIaC-KHexp⁡12KHβR2+βUKHΦ-1Qmean.It is assumed that all parameters of (6) and (13) except Q, which is taken as the value Qmean, are the same. When neither KH nor βU is equal to 0, the confidence of the mean probability risk model can be obtained as follows: (15)Qmean=ΦKHβU2,where KH, which represents the slope of the seismic hazard curve, can be furtherly expressed as [3](16)KH=1log⁡AR,where AR represents the ratio of spectra acceleration of probability levels 0.1HD and HD, in which HD represents the probability of exceedance at uniform hazard response spectrum (UHRS).From the definitions of the variablesAR and βU, we could know that AR is larger than 1, so KH is always larger than 0, while βU is no less than 0. When βU 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(17)Qmean=100%.When there exists the epistemic uncertainty in the analysis orβU is larger than 0, the confidence level of the mean probability risk model would be obtained using (15). Equation (15) shows that Qmean, which is the confidence of the mean probability risk model, is proportional to KH. 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 KH in the West United States would be in the range of 3 to 6, while in the Central and Eastern United States, KH 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.Figure 1
Confidence levels of mean seismic probability risk model used in West United States sites.Figure 2
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 (15) and (17), it is found that the confidence levels of the mean seismic risk model are always larger than 50%. The physical interpretation of this phenomenon could be expressed as follows: the mean seismic risk model is more sensitive to more severe alternative models considered in the analysis; in other words, the mean seismic risk model is more sensitive to the alternative models with higher confidence levels considered in the analysis.The confidence levels of the mean probability risk model are calculated using the parametersβU, which is usually in a range from 0.1 to 0.6, and AR, which is usually in a range from 1.5 to 6. The results are listed in Table 1. It is found that, for different values of βU and AR, the values of the confidence Q are different, so for different sites, the seismic risk probability model has different confidence levels, or it can also be said that the confidence levels of the seismic risk probability model are changed for different sites. The confidence levels of the seismic probability risk model are not high, the lowest and the highest values of which are 52.56% and 95.58%, respectively. We could also find that most confidence levels of the mean seismic risk model in the usual ranges of βU and AR are less than 90%, which is low for nuclear power plants.Table 1
Confidence levels and error values of mean seismic probability risk model for the usual range ofβU and KH values.
A
R
K
H
Confidence levels and error results between risk model with 99% confidenceand mean risk model (in brackets)
β
U
=
0.1
β
U
=
0.2
β
U
=
0.3
β
U
=
0.4
β
U
=
0.5
β
U
=
0.6
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 levelQ. For fully considering the reliability of the mean seismic risk model, the error equation is provided and analyzed. The error equation, which is defined as the relative distance between PF(Qtarget) and PF,mean, is expressed as follows: (18)Err=PFQtarget-PF,meanPF,mean=PFQtargetPF,mean-1=exp⁡Φ-1Qtarget-KHβU2KHβU-1,where Qtarget is the target confidence level; Err represents the relative error of PF,mean with respect to PF(Qtarget).The relative distance betweenPFQ0.99 (the target confidence level is taken as 0.99 in this paper) and PF,mean is listed in Table 1 for usual range of βU and KH values. It is found that the error between PFQ0.99 and PF,mean is related not only to the confidence level, but also to the product of βU and KH. For some sites with large KH, even if the confidence level is high, the error is still large. So at sites with high KH the mean seismic risk model should be more carefully examined and validated because of their larger error.


## 3. Confidence and Error Analysis of ASCE/SEI 43-05 Seismic Risk-Consistent Design Methodology

### 3.1. Theoretical Basis of ASCE/SEI 43-05 Risk-Consistent Seismic Design Methodology

The theoretical basis of ASCE/SEI 43-05 risk-consistent seismic design is as follows [13].For implementation purposes of ASCE/SEI 43-05, the parameters in the risk equation are reformulated in terms of design terms which are familiar to engineers such as structural engineers [13]. First, thep-percentile value of the mean fragility function is expressed as the design capacity:(19)aP=aCexp⁡-XPβC,where XP is the (1-p) percentile of the standard normal deviate.Substituting the valueaC of (19) into (6), the following is obtained:(20)PF,mean=kIaP-KHexp⁡KHβC22-XPKHβC.It is assumed thataP is equal to the design response spectrum (DRS) times a seismic margin factor FP:(21)aP=FP×DRS=FP×DF×UHRS,where FP is the factor of safety representing the conservatism introduced by the seismic design criteria such as NUREG-0800, and DF is the seismic design factor that is related to the uniform hazard response spectra (UHRS): (22)HD=HUHRS=kIUHRS-KHin which HD is the annual probability of exceeding the UHRS.Substituting (21) into (20) and making use of (22), we can obtain(23)PF,mean=HDFPDF-KHexp⁡KHβC22-XPKHβC.Equation (23) can be transformed to(24)DFmean=FP-1RPexp⁡KHβC22-XPKHβC1/KH.The design factor equations of ASCE/SEI 43-05 are expressed as follows:(25)DF=MaximumDF1,DF2DF2=0.6ARα,where DF1 and α are defined in Table 2, respectively, for SDCs 3–5.Table 2
Summary of earthquake design provisions of ASCE/SEI 43-05 [3].
Seismic Design Category (SDC)
Target performance goal (PF)
Probability ratio (RP)
Hazard exceedance probability (HD)
D
F
1
α
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


### 3.2. Interval Estimation Functions of ASCE/SEI 43-05 Risk-Consistent Seismic Design Methodology

The mean probability risk model is the theoretical basis of ASCE/SEI 43-05 risk-consistent seismic design methodology. For analyzing the confidence of the risk results based on ASCE/SEI 43-05 design methodology, the interval risk estimation functions of ASCE/SEI 43-05 are obtained. The confidence equations of ASCE/SEI 43-05 risk-consistent seismic design are as follows.Equation (13) represents the risk equation considering the confidence of the fragility model. Substituting the value aC of (19) into (13), we can obtain (26)PFQ=kIaP-KHexp⁡KHβR22-XPKHβC+βUKHΦ-1Q.Substituting (21) into (26) and making use of (22), we can obtain(27)PFQ=HDFPDF-KHexp⁡KHβR22-XPKHβC+βUKHΦ-1Q.Equation (27) can be transformed to(28)DFQ=FP-1RPexp⁡KHβR22-XPKHβC+βUKHΦ-1Q1/KH.Equations (27) and (28) represent the functions of failure probability PF and design factor DF considering confidence, respectively.Solving (27) for confidence Q of the mean failure probability PF,mean, which could be calculated using (23), the following can be obtained:(29)QPF,mean=Φln⁡PF,mean/HDFPDF-KH-KHβR2/2+XPKHβCβUKH.Substituting (23) into (29), we could furtherly obtain(30)QPF,mean=ΦKHβU2,where neither KH nor βU is equal to 0. According to the definitions of KH and βU, KH is positive, while βU is no less than 0. When βU is equal to 0, (30) would not be suitable, so the confidence equation of PF could be furtherly expressed as follows:(31)QPF,mean=100%.Solving (28) for confidence Q of the mean design factor DFmean, which could be calculated using (24), the following can be obtained: (32)QDFmean=Φln⁡DFmean/FP-1KH/RP-KHβR2/2+XPKHβCβUKH.Substituting (24) into (32), we could furtherly obtain(33)QDFmean=ΦKHβU2,where neither KH nor βU is equal to 0. According to the definitions of KH and βU, KH is positive, while βU is no less than 0. When βU is equal to 0, (33) would not be suitable, so the confidence equation of DF could be furtherly expressed as(34)QDFmean=100%.When neitherKH nor βU is equal to 0, the confidence functions of PF and DF of ASCE/SEI 43-05 could, respectively, be expressed as (30) and (33). It is found that the confidence levels of the two parameters are proportional to KH and βU, which are the same as the confidence levels of the mean seismic risk model. This conclusion could be easily understood, as the mean seismic probability risk model is the theoretical basis of ASCE/SEI 43-05.For three SDCs of ASCE/SEI 43-05 listed in Table2 [3], the exact design factor (DF) and the confidence of DF could be, respectively, calculated using (24) and (33). When the exact design factor (DF) is calculated, βC is usually taken as the range of 0.3 to 0.6, as βC lies in the range of 0.3 to 0.6 for most structures, systems, and components (SSCs) of nuclear power plants (NPPs) [13]. The DF in ASCE/SEI 43-05 could be calculated using (25). The confidence levels of the exact DF are shown in Figure 3. For analyzing the confidence of ASCE DF, βC is still taken as the range of 0.3 to 0.6 and it is assumed that βU is equal to βR in this paper. So actually βU and βR are, respectively, taken as the range of 0.3×2/2 to 0.6×2/2 in Figure 3.Figure 3
Confidence levels of exact DF for SDCs 3, 4, and 5 for the range ofAR and βC values.ASCE/SEI 43-05 [3] and Braverman et al. [13] have shown that ASCE DF approximation values approach the exact values. However, it is shown that the confidence levels of the exact DF are low, which are less than 90% from Figure 3.For fully considering the reliability of achieved performance probabilities using ASCE/SEI 43-05 risk-consistent seismic design methodology, the error equation is obtained. The error equation, which is defined as the relative distance between point estimation function (see (23)) and interval estimation function for the target confidence level (see (27)) of achieved performance probabilities based on ASCE/SEI 43-05 risk-consistent seismic design methodology, is expressed as follows:(35)Err=exp⁡Φ-1Qtarget-KHβU2KHβU-1.It is found that the error equation of achieved performance probabilities based on ASCE/SEI 43-05 is the same as the mean seismic risk model, as the mean seismic probability risk model is the theoretical basis of ASCE/SEI 43-05.


### 3.3. Confidence and Error Analysis of a Design Example Based on the ASCE/SEI 43-05 Risk-Consistent Seismic Design Methodology

The typical normalized spectral acceleration hazard curves values are listed in Table3 [3].Table 3
Typical normalized spectral acceleration hazard curve values of ASCE/SEI 43-05 [3].
H
S
A
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
3.84In ASCE/SEI 43-05, both of the following criteria need be obtained [3]:(1)
Less than 1% probability of acceptable performance for the DBE ground motion, which is defined as the Design Response Spectra (DRS)(2)
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 43-05, we could calculate the achieved performance probabilitiesPF using (23) and the confidence levels of the achieved performance probabilities PF using (30). For different SDCs, the values of HD are listed in Table 2. AR can be obtained from Table 3, while KH can be calculated using (16). Xp means (1-p) percentile of the standard normal deviate (when p is equal to 1%, Xp is 2.326, and when p = 10%, Xp=1.282). The seismic margin factors Fp for different β values are calculated by Kennedy [4], as listed in Table 4.Table 4
Seismic margin factors for differentβ values calculated by Kennedy [4].
β
F
1
%
F
5
%
F
10
%
F
50
%
F
70
%
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
5.53When the achieved performance probabilityPF is calculated, βC is usually taken as the range of 0.3 to 0.6, as βC lies in the range of 0.3 to 0.6 for most structures, systems, and components (SSCs) of nuclear power plants (NPPs) [13]. Kennedy [4] compared the results from the approximation risk equation with the precise convolution risk results, and it was found that the use of approximate power law hazard curve could give acceptable risk results. It could also be found that the failure probability designed according to ASCE/SEI 43-05 can meet two criteria [3, 4, 13]. However, it is shown that the confidence levels of failure probability PF are changed for different sites from Figures 4 and 5. The confidence levels of two example sites are low, which are less than 80%. For analyzing the confidence of failure probability PF, βC is still taken as the range of 0.3 to 0.6 and it is assumed that βU is equal to βR in this paper. So actually βU and βR are, respectively, taken as the range of 0.3×2/2 to 0.6×2/2 in Figures 4 and 5. We could also find that the confidence levels of the achieved performance probabilities in the region of California with steep slopes of the mean seismic hazard curves are larger than the confidence levels of the achieved performance probabilities in the regions of Eastern United States with shallow slopes of the mean seismic hazard curves. In a word, the uniform risk could be obtained using ASCE/SEI 43-05 criteria, but uniform high confidence level of the specified target risk could not be achieved based on ASCE/SEI 43-05 criteria.Figure 4
Confidence levels of achieved performance probabilities for SDCs 3 and 4 based on the first and second criteria of ASCE/SEI 43-05 for representative hazard curves.Figure 5
Confidence levels of achieved performance probabilities for SDC 5 based on both the first and second criteria of ASCE/SEI 43-05 for representative hazard curves.The error analysis for achieved performance probabilities for the above example is conducted. The results are shown in Table5. It could be found that error results between risk probabilities with target confidence level (the target confidence level is taken as 0.99 in this paper) and mean achieved performance probabilities based on ASCE/SEI 43-05 are changed for different sites and the error values are large for some sites.Table 5
Error results for both the first and second criteria of ASCE/SEI 43-05 for representative hazard curves.
SDC
Hazard curves
Error results between risk model with 99% confidence and mean risk model
β
C
=
0.3
β
C
=
0.4
β
C
=
0.5
β
C
=
0.6
F
1
%
=
1.1
F
1
%
=
1.0
F
1
%
=
1.0
F
1
%
=
1.0
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


## 3.1. Theoretical Basis of ASCE/SEI 43-05 Risk-Consistent Seismic Design Methodology

The theoretical basis of ASCE/SEI 43-05 risk-consistent seismic design is as follows [13].For implementation purposes of ASCE/SEI 43-05, the parameters in the risk equation are reformulated in terms of design terms which are familiar to engineers such as structural engineers [13]. First, thep-percentile value of the mean fragility function is expressed as the design capacity:(19)aP=aCexp⁡-XPβC,where XP is the (1-p) percentile of the standard normal deviate.Substituting the valueaC of (19) into (6), the following is obtained:(20)PF,mean=kIaP-KHexp⁡KHβC22-XPKHβC.It is assumed thataP is equal to the design response spectrum (DRS) times a seismic margin factor FP:(21)aP=FP×DRS=FP×DF×UHRS,where FP is the factor of safety representing the conservatism introduced by the seismic design criteria such as NUREG-0800, and DF is the seismic design factor that is related to the uniform hazard response spectra (UHRS): (22)HD=HUHRS=kIUHRS-KHin which HD is the annual probability of exceeding the UHRS.Substituting (21) into (20) and making use of (22), we can obtain(23)PF,mean=HDFPDF-KHexp⁡KHβC22-XPKHβC.Equation (23) can be transformed to(24)DFmean=FP-1RPexp⁡KHβC22-XPKHβC1/KH.The design factor equations of ASCE/SEI 43-05 are expressed as follows:(25)DF=MaximumDF1,DF2DF2=0.6ARα,where DF1 and α are defined in Table 2, respectively, for SDCs 3–5.Table 2
Summary of earthquake design provisions of ASCE/SEI 43-05 [3].
Seismic Design Category (SDC)
Target performance goal (PF)
Probability ratio (RP)
Hazard exceedance probability (HD)
D
F
1
α
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


## 3.2. Interval Estimation Functions of ASCE/SEI 43-05 Risk-Consistent Seismic Design Methodology

The mean probability risk model is the theoretical basis of ASCE/SEI 43-05 risk-consistent seismic design methodology. For analyzing the confidence of the risk results based on ASCE/SEI 43-05 design methodology, the interval risk estimation functions of ASCE/SEI 43-05 are obtained. The confidence equations of ASCE/SEI 43-05 risk-consistent seismic design are as follows.Equation (13) represents the risk equation considering the confidence of the fragility model. Substituting the value aC of (19) into (13), we can obtain (26)PFQ=kIaP-KHexp⁡KHβR22-XPKHβC+βUKHΦ-1Q.Substituting (21) into (26) and making use of (22), we can obtain(27)PFQ=HDFPDF-KHexp⁡KHβR22-XPKHβC+βUKHΦ-1Q.Equation (27) can be transformed to(28)DFQ=FP-1RPexp⁡KHβR22-XPKHβC+βUKHΦ-1Q1/KH.Equations (27) and (28) represent the functions of failure probability PF and design factor DF considering confidence, respectively.Solving (27) for confidence Q of the mean failure probability PF,mean, which could be calculated using (23), the following can be obtained:(29)QPF,mean=Φln⁡PF,mean/HDFPDF-KH-KHβR2/2+XPKHβCβUKH.Substituting (23) into (29), we could furtherly obtain(30)QPF,mean=ΦKHβU2,where neither KH nor βU is equal to 0. According to the definitions of KH and βU, KH is positive, while βU is no less than 0. When βU is equal to 0, (30) would not be suitable, so the confidence equation of PF could be furtherly expressed as follows:(31)QPF,mean=100%.Solving (28) for confidence Q of the mean design factor DFmean, which could be calculated using (24), the following can be obtained: (32)QDFmean=Φln⁡DFmean/FP-1KH/RP-KHβR2/2+XPKHβCβUKH.Substituting (24) into (32), we could furtherly obtain(33)QDFmean=ΦKHβU2,where neither KH nor βU is equal to 0. According to the definitions of KH and βU, KH is positive, while βU is no less than 0. When βU is equal to 0, (33) would not be suitable, so the confidence equation of DF could be furtherly expressed as(34)QDFmean=100%.When neitherKH nor βU is equal to 0, the confidence functions of PF and DF of ASCE/SEI 43-05 could, respectively, be expressed as (30) and (33). It is found that the confidence levels of the two parameters are proportional to KH and βU, which are the same as the confidence levels of the mean seismic risk model. This conclusion could be easily understood, as the mean seismic probability risk model is the theoretical basis of ASCE/SEI 43-05.For three SDCs of ASCE/SEI 43-05 listed in Table2 [3], the exact design factor (DF) and the confidence of DF could be, respectively, calculated using (24) and (33). When the exact design factor (DF) is calculated, βC is usually taken as the range of 0.3 to 0.6, as βC lies in the range of 0.3 to 0.6 for most structures, systems, and components (SSCs) of nuclear power plants (NPPs) [13]. The DF in ASCE/SEI 43-05 could be calculated using (25). The confidence levels of the exact DF are shown in Figure 3. For analyzing the confidence of ASCE DF, βC is still taken as the range of 0.3 to 0.6 and it is assumed that βU is equal to βR in this paper. So actually βU and βR are, respectively, taken as the range of 0.3×2/2 to 0.6×2/2 in Figure 3.Figure 3
Confidence levels of exact DF for SDCs 3, 4, and 5 for the range ofAR and βC values.ASCE/SEI 43-05 [3] and Braverman et al. [13] have shown that ASCE DF approximation values approach the exact values. However, it is shown that the confidence levels of the exact DF are low, which are less than 90% from Figure 3.For fully considering the reliability of achieved performance probabilities using ASCE/SEI 43-05 risk-consistent seismic design methodology, the error equation is obtained. The error equation, which is defined as the relative distance between point estimation function (see (23)) and interval estimation function for the target confidence level (see (27)) of achieved performance probabilities based on ASCE/SEI 43-05 risk-consistent seismic design methodology, is expressed as follows:(35)Err=exp⁡Φ-1Qtarget-KHβU2KHβU-1.It is found that the error equation of achieved performance probabilities based on ASCE/SEI 43-05 is the same as the mean seismic risk model, as the mean seismic probability risk model is the theoretical basis of ASCE/SEI 43-05.


## 3.3. Confidence and Error Analysis of a Design Example Based on the ASCE/SEI 43-05 Risk-Consistent Seismic Design Methodology

The typical normalized spectral acceleration hazard curves values are listed in Table3 [3].Table 3
Typical normalized spectral acceleration hazard curve values of ASCE/SEI 43-05 [3].
H
S
A
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
3.84In ASCE/SEI 43-05, both of the following criteria need be obtained [3]:(1)
Less than 1% probability of acceptable performance for the DBE ground motion, which is defined as the Design Response Spectra (DRS)(2)
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 43-05, we could calculate the achieved performance probabilitiesPF using (23) and the confidence levels of the achieved performance probabilities PF using (30). For different SDCs, the values of HD are listed in Table 2. AR can be obtained from Table 3, while KH can be calculated using (16). Xp means (1-p) percentile of the standard normal deviate (when p is equal to 1%, Xp is 2.326, and when p = 10%, Xp=1.282). The seismic margin factors Fp for different β values are calculated by Kennedy [4], as listed in Table 4.Table 4
Seismic margin factors for differentβ values calculated by Kennedy [4].
β
F
1
%
F
5
%
F
10
%
F
50
%
F
70
%
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
5.53When the achieved performance probabilityPF is calculated, βC is usually taken as the range of 0.3 to 0.6, as βC lies in the range of 0.3 to 0.6 for most structures, systems, and components (SSCs) of nuclear power plants (NPPs) [13]. Kennedy [4] compared the results from the approximation risk equation with the precise convolution risk results, and it was found that the use of approximate power law hazard curve could give acceptable risk results. It could also be found that the failure probability designed according to ASCE/SEI 43-05 can meet two criteria [3, 4, 13]. However, it is shown that the confidence levels of failure probability PF are changed for different sites from Figures 4 and 5. The confidence levels of two example sites are low, which are less than 80%. For analyzing the confidence of failure probability PF, βC is still taken as the range of 0.3 to 0.6 and it is assumed that βU is equal to βR in this paper. So actually βU and βR are, respectively, taken as the range of 0.3×2/2 to 0.6×2/2 in Figures 4 and 5. We could also find that the confidence levels of the achieved performance probabilities in the region of California with steep slopes of the mean seismic hazard curves are larger than the confidence levels of the achieved performance probabilities in the regions of Eastern United States with shallow slopes of the mean seismic hazard curves. In a word, the uniform risk could be obtained using ASCE/SEI 43-05 criteria, but uniform high confidence level of the specified target risk could not be achieved based on ASCE/SEI 43-05 criteria.Figure 4
Confidence levels of achieved performance probabilities for SDCs 3 and 4 based on the first and second criteria of ASCE/SEI 43-05 for representative hazard curves.Figure 5
Confidence levels of achieved performance probabilities for SDC 5 based on both the first and second criteria of ASCE/SEI 43-05 for representative hazard curves.The error analysis for achieved performance probabilities for the above example is conducted. The results are shown in Table5. It could be found that error results between risk probabilities with target confidence level (the target confidence level is taken as 0.99 in this paper) and mean achieved performance probabilities based on ASCE/SEI 43-05 are changed for different sites and the error values are large for some sites.Table 5
Error results for both the first and second criteria of ASCE/SEI 43-05 for representative hazard curves.
SDC
Hazard curves
Error results between risk model with 99% confidence and mean risk model
β
C
=
0.3
β
C
=
0.4
β
C
=
0.5
β
C
=
0.6
F
1
%
=
1.1
F
1
%
=
1.0
F
1
%
=
1.0
F
1
%
=
1.0
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


## 4. Conclusions

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 betweenPF(Qtarget) and PF,mean is given and analyzed. It could be found that the error between PF(Qtarget) (the target confidence is taken as 0.99 in the paper) and PF,mean is related not only to the confidence level, but also to the product of βU and KH. So for some sites with large KH, even if the confidence level is high, the error is still large. So at sites with high KH the mean seismic risk model should be more carefully examined and validated because of their larger error.The confidence equation derivation of design factor DF and failure probabilityPF of ASCE/SEI 43-05 code are conducted. It has been found that the confidence levels of DF and PF of ASCE/SEI 43-05 are changed, while the confidence levels are low. It is also found that the uniform risk could be obtained, but the uniform high confidence level of the specified target risk could not be achieved using ASCE/SEI 43-05 criteria.The error equation of achieved performance probabilities based on ASCE/SEI 43-05 risk-consistent 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 43-05 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.



---

*Source: 2480940-2017-10-10.xml*