Acceptable values of collapse margin ratio with different confidence levels

Structural Safety 84 (2020) 101938

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Acceptable values of collapse margin ratio with different confidence levels a

Yantai Zhang , Zheng He a b

a,b,⁎

T

Department of Civil Engineering, Dalian University of Technology, Dalian 116024, China State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China

ARTICLE INFO

ABSTRACT

Keywords: Collapse margin ratio Mean estimates approach Confidence interval method Aleatory randomness Epistemic uncertainty

Determining a reasonable seismic safety margin target (acceptable value) for a structure is an important first step in preventing structural collapse and ensuring life safety. Based on the confidence interval method, expressions for collapse fragility-based and risk-based acceptable collapse margin ratios (CMRs) with different confidence levels are proposed; additionally, the abovementioned expressions are derived by the mean estimates approach. Subsequently, acceptable CMRs applicable to Chinese specifications are provided. Compared with the mean estimates approach, the confidence interval method can be used to provide acceptable CMRs with different confidence levels to satisfy specific seismic targets. When the structural parameters are set to median values, the acceptable values calculated by the mean estimates approach exhibit only a 50% confidence level. However, these values are higher than the acceptable CMRs calculated using the confidence interval method at 50% confidence. As the confidence level increases, the acceptable CMRs calculated using the confidence interval method become much higher than those calculated using the mean estimates approach. Under the same conditions (same uncertainties and confidence levels), the collapse fragility-based acceptable CMRs are lower than the risk-based acceptable CMRs.

1. Introduction The main goal of earthquake engineering is to provide an appropriate safety margin to prevent collapse. Anti-collapse is not an absolute goal, because the constraints of uncertainty and economic conditions cause the structure to inevitably produce a small collapse probability [1]. FEMA P695 [2] established a consistent method for assessing building code design requirements to ensure minimal structural safety to resist collapse and a provided collapse margin ratio (CMR) as a simple and intuitive indicator to assess such a capability, wherein the CMR is defined as the ratio of the median collapse intensity to the intensity level of the maximum considered earthquake (MCE). Although the CMR-based evaluation method may require a larger amount of calculation [3,4] and cannot distinguish the possible difference in the safety level in structures with the same CMR but different record-torecord variabilities [5], the method is still favored by researchers worldwide [6–9]. Structural seismic designs should exhibit an appropriate confidence level to ensure that the structure does not collapse. The recommended minimum confidence level for immediate occupancy (IO) and collapse prevention (CP) performance levels should be 50% and 90%, respectively, based on the global behavior of structures in FEMA 350 [10].

⁎

Yun et al. [11] proposed a performance-based seismic evaluation method for steel frames that allows designers to estimate confidence levels, thereby satisfying the performance goals for a given hazard. By applying this evaluation method, the confidence level in achieving different performance objectives can be further specified in the design and analysis process [12–14]. Other researchers have proposed a method to calculate the annual collapse probability or the collapse probability at a certain hazard level of structures with different confidence levels [15–18]. However, a method to calculate an acceptable value of the CMR at a specific confidence level does not exist. The random nature of ground motions and the characteristics of structures complicate the prediction of structure collapse. When determining the acceptable CMRs, it is highly important to consider the uncertainty in predicting structural collapse. Uncertainty can be categorized into two types: aleatory randomness and epistemic uncertainty [19,8]. Aleatory randomness is primarily caused by record-to-record variability, denoted as βRTR. Epistemic uncertainty (βU) reflects the lack of knowledge regarding the physical mechanism of the building being modeled. Total epistemic uncertainty can be obtained by combining multiple uncertainties through the square root of the sum of squares (SRSS) method. In FEMA P695 [2], the collapse uncertainty is categorized into four types, i.e. record-to-record uncertainty (βRTR), design

Corresponding author at: Department of Civil Engineering, Dalian University of Technology, Dalian 116024, China. E-mail address: [email protected] (Z. He).

https://doi.org/10.1016/j.strusafe.2020.101938 Received 2 December 2018; Received in revised form 21 January 2020; Accepted 22 January 2020 Available online 08 February 2020 0167-4730/ © 2020 Elsevier Ltd. All rights reserved.

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requirements uncertainty (βDR), test data uncertainty (βTD), and modeling uncertainty (βMDL), in which the latter three can be classified as epistemic uncertainty. The process of determining the appropriate uncertainty value is complicated. Generally, aleatory randomness can be obtained by incremental dynamic analysis (IDA) [20], truncated IDA method [21], and multiple stripes analysis [22,23] using the sufficient number of seismic waves for a structure. Epistemic uncertainty generally refers to modeling uncertainty, and its determination is relatively complex and can be determined using reliability methods, such as the first-order second-moment [24–27], Monte Carlo [28,29] and Latin hypercube sampling methods [30–33]. Typically, two methods are used to incorporate the effect of aleatory randomness and epistemic uncertainty in collapse fragilities and risk assessment, i.e., the mean estimates and confidence interval methods [15,16,28]. In both methods, it is assumed that aleatory randomness and epistemic uncertainty are independent [15]. The direct substitution of epistemic uncertainty into capacity analysis through Monte-Carlobased methods is not considered herein [28,30]. The mean estimates approach combines two types of uncertainty by the SRSS method, which only changes the variance of the collapse fragility without changing the median collapse capacity. The confidence interval method considers aleatory randomness and epistemic uncertainty separately and can obtain the collapse median at different confidence levels by assuming a distribution function for the collapse median. Compared with the mean estimates approach, the confidence interval method can convey the confidence level used in the analysis [16,17,28]. The acceptable CMRs given by FEMA P695 [2] are determined based on the target acceptable values of the collapse probability assuming that the collapse fragility curve is lognormal, in which the mean estimates approach is adopted to consider various uncertainties. For simplicity, the acceptable values determined in this manner are referred to as the collapse fragility-based acceptable CMRs. Xian et al. [34] proposed a method to determine acceptable CMRs based on annual collapse probability, which considers the effect of each uncertainty through the mean estimates approach. For simplicity, the acceptable values determined by this process are referred to as risk-based acceptable CMRs. If the confidence interval method can be used in the calculations of collapse fragility-based and risk-based acceptable CMRs, the acceptable values at a certain confidence level will be obtained when the structure performance satisfies the collapse prevention criteria, i.e., the limits of collapse probability at the hazard level of the MCE or the annual collapse probability. Currently, the principles of decision making and determining the seismic design level have switched from uniform hazard-targeted to uniform risk-targeted ones [35]. It is particularly important to determine the annual collapse probability or 50-year collapse probability of structures, after which the ground motion level can be inversely analyzed. Hence, it is more applicable and practical to use risk-based acceptable CMRs combined with the annual collapse probability than the collapse fragility-based acceptable CMRs. Luco et al. [35] estimated the mean annual collapse probability of 2 × 10−4 for the conterminous United States. ASCE 7-16 [8] uses this recommended value to generate a risk-targeted ground motion zoning map. Currently, no relevant design codes or guidelines exist in Europe, but some relevant surveys have been conducted in recent years, and various annual collapse probabilities have been proposed [36]. This paper presents a conservative probability of annual collapse based on Chinese norms. Herein, the collapse fragility-based acceptable CMRs calculated by the mean estimates and confidence interval methods are provided first, in which the calculation expressions for the collapse fragility-based acceptable CMRs with different confidence levels are derived. Subsequently, the relationship of acceptable CMR to annual collapse probability and seismic hazard is established by the total probability theory, in which the mean estimates approach and confidence interval method are used to present the risk-based acceptable CMRs. The expression for the risk-based acceptable CMRs with different confidence

levels is proposed and the acceptable values of the annual collapse probability of Category-B and -C buildings are provided in consideration of Chinese specifications. 2. Collapse fragility-based acceptable CMRs FEMA P695 [2] defines the CMR as the ratio of the median collapse intensity (i.e., the intensity when 50% earthquake waves cause structural collapse, denoted as ηC) to the intensity level of MCE (2% probability of exceedance in 50 years, denoted as IMMCE), see Eq. (1).

CMR =

C

(1)

IMMCE

The connection between the median collapse and the MCE intensities can be illustrated in the collapse fragility curve; subsequently, the calculation expression for the acceptable CMRs can be recognized. The collapse fragility curve indicates the structural collapse probability distribution with respect to different ground motion levels. If the collapse intensity conforms to the lognormal cumulative distribution function, the collapse probability under a certain intensity measure (IM) (denoted as P(C|IM)) can be expressed as follows:

ln(IM )

P (C| IM ) =

ln(

C)

(2)

RTR

where Φ(.) is the standard normal cumulative distribution function; ηC and βRTR are the median collapse intensity and logarithmic standard deviation, respectively, of which both can be obtained by IDA or similar methods with the structural properties set to their median value. βRTR reflects the record-to-record variability, i.e., the aleatory randomness. According to the Chinese Standard for Seismic Fortification Classification of Buildings [37], all buildings are classified into four groups, i.e., Category -A, -B, -C, and -D, by their functional significance or socioeconomic effect. Category-A buildings are major buildings whose failure would result in the most severe secondary disaster or socioeconomic effect, e.g., nuclear power plants. Category-B buildings are buildings that are expected to be functional after experiencing earthquakes. Category-D buildings are temporary or unimportant buildings that do not require seismic performance assessment. Category-C buildings are typical buildings and do not belong to other categories. Herein, only Category-B and -C buildings are considered. The Code for Anti-collapse Design of Building Structures [38] specifies the acceptable values of collapse probability (denoted as x) at the MCE intensity level for Category-B and -C structures, as shown in Table 1. The probability x for Category-B and Category-C buildings is 1% and 5%, respectively. When the probability of collapse calculated by Eq. (2) is the probability x , the corresponding acceptable CMRs can be obtained (see Eqs. (3) and (4)).

ln(IMMCE )

ln(

C)

=

ln(CMR)

RTR

CMR = exp(

=x

(3)

RTR

RTR

1 (x ))

= exp(

(4)

RTR Kx )

where Kx is the standardized Gaussian variate associated with the 1 (x ) , see Table 2). For Category-B probability of x not exceeded (Kx = buildings, K1% = −2.326, for Category-C buildings, K5% = −1.645. In the above-mentioned acceptable CMR derivation, only aleatory randomness βRTR is considered, and the epistemic uncertainty βU is incorporated using the mean estimates and confidence interval methods, separately. Table 1 Acceptable maximum seismic collapse probability (%).

2

Earthquake

Category-C buildings

Category-B buildings

MCE

5

1

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Y. Zhang and Z. He

conforms to a lognormal distribution with median C and dispersion βU. Subsequently, the median of the collapse capacity with confidence level Y can be expressed as CY = C e U KY for CY < C (see Eq. (7)), where KY is the standardized Gaussian variate associated with the probability 1 (Y ) = 1 (1 Y ) , which can be calcuof Y not exceeded (KY = lated from Table 2). Furthermore, KY = 2.326 when Y = 99% and KY = 1.282 when Y = 90%. The collapse fragility curve with confidence level Y is lognormally distributed with median CY and standard deviation βRTR, which can be expressed by Eq. (8).

Table 2 Values of Kx for different probability x. x

Kx =

1% 5% 10% 20% 50%

−2.326 −1.645 −1.282 −0.842 0

1 (x )

2.1. Mean estimates approach

Y=

P (C| IM ) =

ln(

PY (C| IM ) =

(7)

TOT Kx )

ln(IM )

ln(

Y C )

(8)

RTR

As shown in Eqs. (9) and (10), the acceptable CMRY with confidence level Y can be derived from Eq. (8):

ln(IMMCE )

C)

TOT

CMR = exp(

ln( ^C )

Y C) U

The mean estimates approach combines the two uncertainties by the SRSS method to obtain the total collapse uncertainty, i.e., 2 2 TOT = RTR + U [15,39]. Subsequently, when the mean estimates approach is used, the collapse fragility curve becomes a lognormal distribution function of median value ηC and standard deviation βTOT. The collapse probability formula of Eq. (2) and the acceptable CMR formula of Eq. (4) become Eqs. (5) and (6), respectively.

ln(IM )

ln(

ln( ^C ) +

U KY

=

RTR

ln(CMRY )

U KY

=x

RTR

(9)

(5)

CMRY = exp(

(6)

RTR Kx

+

U KY )

(10)

When KY = 0, the effect of epistemic uncertainty is ignored, and the acceptable CMRY obtained by Eq. (10) has only a confidence level Y of 50%. Fig. 2 shows the acceptable CMRY values considering the two confidence levels of 90% and 95% for the two structure categories. As shown in Fig. 2, a higher confidence level results in a higher acceptable CMRY, and the acceptable CMRY increases with the uncertainty coefficients βRTR and βU. Eqs. (6) and (10) present the calculation expressions of acceptable CMRs from two perspectives, i.e., mean estimates and confidence interval methods. Unlike Eq. (6), Eq. (10) can provide the acceptable CMRs with confidence level Y required to satisfy the limit of collapse probability under the MCE intensity level. Fig. 3 illustrates the acceptable values of CMR calculated using both equations, using Category-B and Category-C buildings as examples. In both methods, βU was set to 0.4, and the confidence levels of 50%, 90%, and 95% were considered. As shown in Fig. 3, when the confidence level is 50%, the confidence interval method cannot consider the effect of epistemic uncertainty, and the calculated acceptable CMRs are significantly lower than those obtained using the mean estimates approach. When the confidence level is above 50%, the acceptable CMRs calculated by the mean estimates approach are significantly lower than those calculated using the confidence interval method, with an increase in aleatory randomness βRTR. Eq. (6) yields the acceptable CMRs based on the total collapse uncertainty βTOT and the collapse probability limit x , while Eq. (10) yields the acceptable CMRY with the confidence level Y based on the aleatory randomness βRTR, epistemic uncertainty βU, collapse probability limit x , and confidence level. However, the acceptable CMR calculation formulas of the two methods cannot consider seismic hazard in the structural design.

Eq. (6) is an expression for calculating the acceptable CMRs, in which uncertainties are considered by the mean estimates approach. This process is used in FEMA P695 [2] to establish acceptable CMRs. Fig. 1 illustrates the acceptable CMR values for Category-B and -C buildings. As the total uncertainty coefficient βTOT increases from 0.275 to 0.950, the acceptable CMRs for Category-B buildings increase from 1.896 to 9.116 and those for Category-C buildings increase from 1.572 to 4.771. As shown in Fig. 1, when the collapse probability x decreases and the collapse uncertainty increases, the acceptable CMR values increase. A smaller collapse probability criterion x indicates the need for a higher collapse resistance and hence a higher acceptable CMR value. A higher uncertainty indicates more uncertain factors in the new design and hence a higher acceptable CMR to compensate for the required structural collapse resistance. 2.2. Confidence interval method The confidence interval method is used to establish a collapse fragility curve with a certain confidence level (denoted as Y) to consider both aleatory randomness and epistemic uncertainty [16]. In the confidence interval method, it is assumed that the random median of ηC

3. Risk-based acceptable CMRs A risk-based structural performance assessment can provide a mean annual frequency exceeding the limit state (e.g., collapse) of a structure by considering a certain seismic hazard curve. In the following, the relationship between median collapse intensity and annual collapse probability is established using the total probability formula. Subsequently, the relationship between CMR and seismic hazard curve is established by substitution. Finally, risk-based acceptable CMRs can be obtained, which can be used to facilitate the structural anti-collapse design effectively.

Fig. 1. Collapse fragility-based acceptable CMRs for Category-B and -C buildings calculated using the mean estimates approach. 3

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Fig. 2. Collapse fragility-based acceptable CMRs for Category -B and -C buildings calculated using the confidence interval method.

The seismic hazard curve λIM can be expressed as a power function as follows: IM (IM ) = k 0 IM

Furthermore, the acceptable CMRs can be obtained using Eq. (15).

1 CMR = ¯IM , MCE exp k2 2

(11)

k

where k0 and k are empirical constants that can be determined by the intensity of a design-basis earthquake (10% probability of exceedance in 50 years) and the MCE intensity level [22], or by fitting to multihazard data. Based on the total probability theory, the collapse fragility curve (see Eq. (2)) is further combined with the seismic hazard curve λIM (see Eq. (11)) to obtain the annual collapse probability λC of the structure (see Eq. (12)). It is noteworthy that the mean hazard curve, denoted as IM , was used in this study to consider the epistemic uncertainty in the hazard curve [15,16]. C

=

P(C| IM )·|d ¯IM (IM )|

1 = ¯ IM ( C ) exp k 2 2

2 RTR

C

k

exp

1 2 k 2

2 RTR

As detailed in the foregoing description, the mean estimates approach combines two types of uncertainties by the SRSS method: 2 2 TOT = RTR + U . When the mean estimates approach is used, the collapse fragility curve becomes a lognormal distribution function with median ηC and standard deviation βTOT (see Eq. (5)). Subsequently, the annual collapse probability and the acceptable CMRs can be obtained by Eqs. (16) and (17), respectively.

(13)

The intensity level of the MCE is expressed as k 0 = substituting it into Eq. (13) yields: C

= ¯IM ,MCE IMMCE k

C

k

exp

1 2 k 2

2 RTR

= ¯IM ,MCE (CMR)

k exp

IM , MCE IMMCE

k;

C

1 2 k 2

(15)

3.1. Mean estimates approach

(12)

= k0

1 k C

Eq. (15) is the calculation expression for the risk-based acceptable CMRs, which combines the constant k of the hazard curve, mean annual exceedance probability of the MCE IM , MCE , aleatory randomness βRTR, and annual collapse probability λC. It is noteworthy that only aleatory randomness βRTR is considered in Eq. (15); the epistemic uncertainty βU will be further considered using the mean estimates and confidence interval methods, separately.

Furthermore, by substituting and simplifying Eq. (12), Eq. (13) can be obtained [40,39]. C

2 RTR

2 RTR

1 = ¯IM ( C ) exp k 2 2

2 TOT

1 CMR = ¯IM ,MCE exp k 2 2

(14)

= k0 2 TOT

C

k

exp

2 TOT

(16)

1 k C

Fig. 3. Acceptable CMRs calculated using the mean estimates and confidence interval methods. 4

1 2 k 2

(17)

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Eqs. (16) and (17) are expressions that consider the total uncertainty βTOT for the annual collapse probability and acceptable CMRs, respectively. Eq. (17) was first proposed by Xian et al. [34]. Eq. (17) combines the constant k of the hazard curve, mean annual exceedance probability of the MCE IM , MCE , total uncertainty βTOT, and annual collapse probability λC, where k and IM , MCE are generally specified by the code. Currently, the relevant Chinese specifications do not provide an acceptable λC value. In this study, acceptable values for the annual collapse probability λC for Category-B and -C buildings were determined based on the current specifications, which preserve different derivations and final results from the annual collapse probability determined by Xian et al. [34]. C = IMMCE exp( TOT Kx ) can be obtained by Eq. (5); substituting it into Eq. (16) yields: C

= ¯IM ,MCE exp(k

TOT Kx )

exp

1 2 k 2

2 TOT

seismic hazard is more stringent. However, this difference is closely related to the acceptable values of the annual collapse probability established in Fig. 5. Determining a reasonable annual collapse probability is an important research direction. 3.2. Confidence interval method In the confidence interval method, it is assumed that the random median of ηC conforms to a lognormal distribution with median of C and dispersion of βU. The collapse fragility curve with confidence level Y is lognormally distributed with median CY and standard deviation βRTR (see Eq. (8)), in which CY = C e U KY . The annual collapse probability CY with confidence level Y and the acceptable CMRY with confidence level Y can be obtained by Eqs. (19) and (20), respectively. Among them, Eq. (19) was first proposed by Zareian and Krawinkler [16].

(18)

Y C

These peak ground accelerations (PGAs) for frequent earthquakes (63.2% probability of exceedance in 50 years), design-basis earthquakes (10% probability of exceedance in 50 years) and the MCE (2% probability of exceedance in 50 years) specified in the Code for Seismic Design of Buildings [41] for various areas with different seismic precautionary intensities are used in this study. Every building, which is situated on the zones of precautionary intensity 6 or above, must be designed to resist the effects of earthquake motions. Assuming that the ground motion intensity level is Poisson distributed, the mean annual probability of exceedance ¯IM for the three PGA levels can be obtained. By fitting Eq. (11) with these three discrete points (PGA, ¯IM ), the seismic hazard curves, in other words k and k0, for each seismic precautionary intensity can be obtained (see Fig. 4). According to Eq. (18), Fig. 5 shows the annual collapse probability λC for Category-B and -C buildings under various precautionary intensities. For Category-B buildings (Fig. 5(a)), λC decreases with the increase in the total collapse uncertainty βTOT. For Category-C buildings (Fig. 5(b)), when the total collapse uncertainty βTOT is less than 0.75, λC decreases with the increase in βTOT. The annual collapse probability λC under each precautionary intensity reaches the minimum value near βTOT = 0.75. When βTOT is greater than 0.75, λC shows an opposite trend. For structural safety, the acceptable values of the annual collapse probability for Category-B and -C buildings were conservatively set to 2.5 × 10−5 and 1 × 10−4, respectively (see Fig. 5). Fig. 6 compares the risk-based acceptable CMRs calculated according to Eq. (17) and the collapse fragility-based acceptable CMRs calculated according to Eq. (6). As shown in Fig. 6, when the total collapse uncertainty is small, the acceptable CMRs calculated according to Eq. (17) is significantly larger than that calculated according to Eq. (6). With the increase in the total collapse uncertainty, the gap between them tends to decrease. Overall, the acceptable CMRs considering

=

PY (C| IM )·|d ¯IM (IM )| = k 0 ^C

k

exp

1 2 k 2

2 RTR

exp(KY k

U)

(19)

1 CMRY = ¯IM ,MCE exp k 2 2

2 RTR

exp(KY (k

U ))

Y C

1 k

(20)

The acceptable CMRY with different confidence levels can be obtained using Eq. (20). It is equivalent to ignoring the effect of aleatory randomness when KY = 0, and the current acceptable CMRY exhibits a confidence level of only 50%. As shown in Eq. (20), the acceptable CMRY depends on the annual exceedance probability of the MCE, k value of seismic hazard curve, aleatory randomness βRTR, epistemic uncertainty βU, and annual collapse probability CY with confidence level Y. To determine the acceptable values for CMRY at different confidence levels, the following attempts were conducted to determine the acceptable values for annual collapse probability CY with confidence level Y for Category-B and -C buildings based on existing specifications. When the collapse probability of a structure under the MCE ground motion satisfies the requirement of the acceptable value of collapse probability x corresponding to the MCE, it is clear from Eq. (8) that the median collapse intensity ^C can be expressed by the intensity level of the MCE IMMCE as follows (Y = 50% in Eq. (8)):

^ =IMMCE exp( C

(21)

RTR Kx )

where Kx is the standardized Gaussian variate associated with the 1 (x ) , see Table 2). probability x of not being exceeded (Kx = Substituting (21) into (19) yields: Y C

1 = ¯IM ,MCE exp k 2 2

2 RTR

exp(KY k

U

+ Kx k

RTR)

(22)

Eq. (22) presents an expression for the annual collapse probability with confidence level Y, in which IM , MCE can be obtained from Fig. 4. Using βU = 0.4 as an example, Fig. 7 shows the annual collapse probability with different confidence levels for Category-B and -C buildings. As the confidence level increases, the annual collapse probability increases significantly. With an increase in aleatory randomness βRTR, the annual collapse probability decreases. Additionally, Y = 50% is equivalent to not considering the effect of epistemic uncertainty, and the annual collapse probability has the lowest value. In this case, the Category-B and -C buildings’ C50% approach 2.5 × 10−5 and 1 × 10−4, respectively. Figs. 8 and 9 show the variation in the annual collapse probability at 95% and 90% confidence levels for Category-B and -C buildings, respectively. As the βRTR increases, the annual collapse probability decreases; the smaller the βU considered, the lower is the annual collapse probability. To ensure safety, the acceptable value of annual collapse probabilities with 95% and 90% confidence levels of Category-B buildings was set to 2.5 × 10−5, as shown in Fig. 8. Additionally, the

Fig. 4. Mean hazard curves for each precautionary intensity. 5

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Fig. 5. Annual collapse probability of Category-B and -C buildings calculated using the mean estimates approach.

Fig. 6. Risk-based acceptable CMRs for Category-B and -C buildings calculated using the mean estimates approach.

Fig. 7. Annual collapse probability with different confidence levels for Category-B and -C buildings - using βU = 0.4 as an example.

acceptable value of the annual collapse probabilities with 95% and 90% confidence levels of Category-C buildings was set to 1 × 10−4, as shown in Fig. 9. Both values are consistent with the annual collapse probability with the 50% confidence level determined from Fig. 7 and the annual collapse probability calculated using the mean estimates approach shown in Fig. 5. Furthermore, Figs. 10 and 11 show the acceptable CMRY with different confidence levels for Category-B and -C buildings, respectively, according to Eq. (20). As βRTR and βU increase, the acceptable CMRs increase; moreover, higher confidence levels require higher acceptable

CMRs. Using βU = 0.4 as an example, Fig. 12 illustrates the collapse fragility-based (Eq. (10)) and risk-based acceptable CMRs (Eq. (20)) with 95%, 90% and 50% confidence levels calculated by the confidence interval method, and the collapse fragility-based (Eq. (6)) and risk-based acceptable CMRs (Eq. (17)) calculated by the mean estimates approach for Category-B and -C buildings. Using the same method to consider uncertainty, the risk-based acceptable CMR is greater than the collapse fragility-based acceptable CMR, especially when βRTR is small; for example, the acceptable CMR obtained using Eq. (20) is larger than that 6

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Fig. 8. Annual collapse probability with different confidence levels for Category-B buildings.

Fig. 9. Annual collapse probability with different confidence levels for Category-C buildings.

Fig. 10. Acceptable CMRY with 90% and 95% confidence levels for Category-B buildings.

obtained using Eq. (10), and that obtained from Eq. (17) is larger than that obtained using Eq. (6). As shown in Fig. 12, when the confidence level is 50%, the effect of epistemic uncertainty is excluded, and the acceptable CMRs calculated using the confidence interval method (Eqs. (20) and (10)) are low. However, at this time (Y = 50%), the risk-based acceptable CMRs calculated using Eq. (20) are still higher than the collapse fragility-based acceptable CMRs calculated using Eq. (10). This may be caused by the acceptable value of annual collapse probability used in the calculation of Eq. (20). As the confidence level increases, the acceptable CMRs calculated using the confidence interval method (Eqs.

(20) and (10)) are greater than those calculated using the mean estimates approach (Eqs. (6) and (17)). 4. Conclusions Based on the confidence interval method, the expressions for collapse fragility-based and risk-based acceptable CMRs with different confidence levels were proposed, and the expressions for collapse fragility-based and risk-based acceptable CMRs obtained by the mean estimates approach were derived. The conclusions obtained are as follows: 7

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Fig. 11. Acceptable CMRY with 90% and 95% confidence levels for Category-C buildings.

Fig. 12. Comparison of acceptable CMRs for Category-B and -C buildings obtained using the four methods.

• The collapse fragility-based acceptable CMR calculated by the con-

•

•

based acceptable CMR significantly. From the perspective of structural safety, the acceptable value of annual collapse probability for Category-B and -C buildings could be set conservatively as 2.5 × 10−5 and 1 × 10−4, based on the Chinese code.

fidence interval method depended on the specified collapse probability at the MCE intensity level, confidence level, aleatory randomness, and epistemic uncertainty. Compared with the expressions of collapse fragility-based acceptable CMRs, those of risk-based acceptable CMRs further incorporated the annual collapse probability and seismic hazard curve. It is important to determine a reasonable annual collapse probability acceptable value for risk-based acceptable CMRs. The acceptable CMRs calculated using the mean estimates approach exhibited a confidence level of only 50%, while the confidence interval method could be used to calculate the acceptable CMRs at different confidence levels. When the confidence level was 50%, the acceptable CMR calculated by the confidence interval method was lower than that calculated by the mean estimates approach. This was because the effect of epistemic uncertainty could not be considered in the confidence interval method when the confidence level was set as 50%. As the confidence level increased, the acceptable CMR calculated using the confidence interval method was much higher than that using the mean estimates approach. The acceptable CMR increased with the enhancement of the requirements of seismic goals (i.e., the decrease in the probability of collapse of the MCE hazard level or annual collapse probability), as well as with the confidence level and uncertainty. Under the same uncertainties and confidence levels, the collapse fragility-based acceptable CMR was lower than the risk-based acceptable CMR. The acceptable value of annual collapse probability affected the risk-

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51878123) and the Fundamental Research Funds for the Central Universities (Grant No. DUT19G208). References: [1] Krawinkler H, Zareian F, Lignos DG, Ibarra LF. Significance of modeling deterioration in structural components for predicting the collapse potential of structures under earthquake excitations. In: Fardis M, editor. Advances in Performance-Based Earthquake Engineering. Geotechnical, Geological and Earthquake Engineering. Dordrecht: Springer; 2010. p. 173–81. [2] Federal Emergency Management Agency. Quantification of building seismic performance factors (FEMA P695). Washington, D.C.: FEMA; 2009. [3] He Z, Ou XY, Cheng ZH. Determination of structural collapse margin ratio using pushover analysis. Eng Mech 2014;31(6). https://doi.org/10.6052/j.issn.10004750.2013.01.0002. 197-202+249. (in Chinese). [4] Hardyniec A, Charney F. A new efficient method for determining the collapse margin ratio using parallel computing. Comput Struct 2015;148:14–25. https://doi. org/10.1016/j.compstruc.2014.11.003. [5] Miranda E, Eads L, Davalos H, Lignos D. Assessment of the probability of collapse of structures during earthquakes. Proceedings of 16th World Conference on Earthquake Engineering. Santiago, Chile. 2017. [6] Haselton CB, Liel AB, Deierlein GG, Dean BS, Chou JH. Seismic collapse safety of reinforced concrete buildings. I: Assessment of ductile moment frames. J Struct Eng

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