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Seismic failure probability evaluation of redundant fast breeder reactor piping system by probabilistic structural response analysis
Nuclear Engineering and Design 175 (1997) 237 – 245
Seismic failure probability evaluation of redundant fast breeder reactor piping system by probabilistic structural response analysis Akira Yamaguchi * O-arai Engineering Center, Power Reactor and Nuclear Fuel De6elopment Corporation, O-arai, Ibaraki 311 -13, Japan Received 1 December 1996; accepted 20 December 1996
Abstract The seismic failure probability and the correlation coefficient of the multiple failure mode of the heat transport system of a three-loop fast breeder reactor have been evaluated based on a probabilistic structural response analysis. It has been found that the most probable failure mode of the heat transport system has less impact on the core cooling capability than other modes. The correlation coefficient of the heat transport system loops is approximately 0.9. It is found that the correlation comes from the common structural properties rather than the common seismic input. The present approach is useful for quantifying the correlation coefficient and the seismic fragility of the redundant component failure that is used in the systems analysis. © 1997 Elsevier Science S.A.
1. Introduction Structural and equipment fragilities, i.e. seismic failure probabilities as a function of the seismic input intensity are evaluated in a Seismic Probabilistic Safety Analysis (SPSA) study. In the SPSA, one makes full utilization of the power plant systems logic models, developed in the internal events analyses. By making full use of the internal event models, an external event analysis becomes consistent in level of detail with the internal events analysis. Redundant components composing a safety system are often treated as dependent and the redundancy is conservatively * Tel.: + 81 292 674141; fax: + 81 292 677834; e-mail: [email protected]
accounted for in SPSAs. This simplified assumption is reasonable and verified as long as the neglection of the redundancy does not influence the core damage frequency. Otherwise the correlation of the multiple components failure should be quantified and the redundancy be taken into consideration. It seems the simplification used in the SPSA, in treating the multiple component failure is not consistent with the internal event analysis. As pointed out by Bohn and Lambright (1990), the common-cause failure possibility represents a potentially significant risk to the nuclear power plant during an earthquake. In the Seismic Safety Margins Research Program (SSMRP) (Smith et al., 1981), extensive multiple time history analyses have been performed and a distinct pattern was observed. On the basis of the findings, a set of
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A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245
rules was formulated, which predicted the correlation coefficients between components. These correlation coefficients are looking at the nature and location of the responses and is applicable to both PWR and BWR. The rule describes components on the same floor slab, and sensitive to the same spectral frequency range will be assigned response correlation= 1.0 (Bohn and Lambright, 1990). A SPSA study has been performed of a looptype Fast Breeder Reactor (FBR) (Nakai et al., 1993). It has been found that the structures of the heat transport system are relatively important from the viewpoint of the frequency of seismically-induced core damage. The following is an explanation of the importance of the coolant boundary structures. The FBR has preferable passive safety characteristics that emergency reactor shutdown and decay heat removal do not rely on alternating current power supply at all. Hence the coolant boundary structure of the FBR plays an important role for the safe shutdown during the seismic event, being compared with electrical equipment. The quantification of the correlation coefficient is not straightforward in general and the common cause failure is not easily evaluated under the seismic conditions. If one follows the rule mentioned above, the correlation coefficient of a heat transport system (HTS) of a three-loop FBR becomes 1.0. In this study, the correlation coefficient of the multiple failure mode and the seismic failure probability of the heat transport system has been evaluated based on probabilistic structural response analyses.
vessel is the cold leg. The pipe between the IHX and the pump is called crossover leg. The coolant temperature in the cold and crossover legs is approximately 400°C. The three heat transport loops are placed in every 120° direction each as shown in Fig. 1. The structure of the equipment in the system is thin-walled because the internal pressure is almost atmospheric and the maximum design temperature is beyond those of light water reactors. Therefore, the seismic load is one of the critical design consideration. It is noted that decay heat in the core of the FBR can be removed by only one of the three heat transport loops, i.e. the FBR system is triply redundant with regard to the decay heat removal. It is achieved by natural circulation and is not dependent on the electricity at all. Therefore, the diesel generator and off-site power are not essential for the seismic accident sequence in the FBR. Then the FBR plant is free from the alternating current power supply to achieve the emergency cold shutdown. It was found in the internal event probabilistic safety analysis, that the reliability of decay heat removal is extremely high. To take advantage of the preferable characteristics of the FBR, it is important to maintain the structural integrity of the system and to bear the coolant inside the structure. The seismic failure probability of the triply redundant heat transport system is a point of concern in the SPSA of the FBR.
2. System description The FBR plant analyzed in this study has three heat transport loops. The schematic drawing of the heat transport system is shown in Fig. 1. Major components of the system are a reactor vessel (RV), intermediate heat exchangers (IHXs) and coolant pumps. They are connected with the hot, crossover, and cold leg pipes. The hot leg piping in which coolant temperature is around 550°C connects the reactor vessel and the IHX. The piping running from the pump to the reactor
Fig. 1. Schematic drawing of primary heat transport system.
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Table 1 Median and lognormal standard deviation of membrane plus bending stress (kg mm−2) Loop 1
Hot leg Crossover leg Cold leg Union
Loop 2
Loop 3
Three loops
Median
b
Median
b
Median
b
Median
b
3.4 3.9 2.2 4.0
0.29 0.28 0.30 0.28
3.4 4.2 2.2 4.2
0.32 0.31 0.31 0.31
3.5 4.1 2.2 4.2
0.33 0.31 0.30 0.30
3.4 4.1 2.2 4.1
0.30 0.29 0.29 0.28
3. Probabilistic response analysis of heat transport system In a preliminary SPSA study, failure of the multiple heat transport loops may be assumed dependent of each other for simplicity because the design and qualification method is common to the three loops. However, the correlation should be quantified in the detailed analysis because the three loops are placed in every 120° direction each and the response is expected to be varied. Computer program, SMACS (Seismic Methodology Analysis Chain for Statistics) has been applied to the coupled analysis of reactor building structures and primary equipment such as the reactor vessel, IHX, pump, and piping system (Morishita, 1993). A total of 30 ensembles of artificial time histories of input earthquakes are generated so their statistical response spectrum is in accordance with a site-specific median-centered target spectrum. Variabilities also included in the analysis are soil and structural properties. The failure mode of the piping is assumed to be inelastic buckling of an elbow because the maximum load is observed in most cases at elbow sections. An applied external load S, i.e. the primary membrane plus bending stress is compared with the capacity of the piping material to evaluate the factor of safety. Each response analysis is performed with a set of random earthquake time history and soil and structure properties. From the individual response analysis, maximum load in j-th loop ( j =1, 2, or 3) is obtained for i-th simulation (i= 1, 2,…, 30). Here one denotes the maximum applied load in j-th loop as Sij for i-th simulation. If the maximum applied load is taken for cold, crossover and hot leg, respectively, one
obtains statistics of the maximumm load for each loop and each leg. Table 1 shows the median value and lognormal standard deviation b of the maximum seismic load for each loop and leg. Also included in Table 1 are those for the union event of the cold leg, crossover leg and hot leg failure. For the union event of the failures in three legs, median and the standard deviation of the largest load in a loop (not in each leg) for 30 simulations are used to evaluate the median and standard deviation of the maximum load. The last column denoted as total is the statistics for all the three loop data. It is seen that the difference of the median and b values among the three loops is small and they are practically in accordance with the numbers obtained from all the three loop data (the last column in Table 1). By fitting the 90 values (30 simulations times three loops) of Sij to a lognormal distribution, a cumulative probability function is obtained for the maximum seismic load in a single heat transport loop. Therefore, comparing the cumulative probability function of S with the strength of the material and multiplying other factors of conservatism or unconservatism, one obtains the seismic fragility of the heat transport piping. In each structural response analysis, the seismic loads in three loops vary from each other. One obtains the maximum load for each heat transport loop. The smallest one of the loads in the three loops for each simulation, Smin, is defined as: Smin = min{S1, S2, S3}
(1)
where Si is the maximum load in the i-th loop and min{} shows minimum one of S1, S2 and S3. If
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Table 2 Median and lognormal standard deviation of membrane plus bending stress for multiple failure mode (kg mm−2)
Hot leg Crossover leg Cold leg Union
Smin
Smax
Smid
3 loop failure
1 loop failure
2 loop failure
Median
b
Median
b
Median
b
3.2 3.8 2.0 3.8
0.31 0.28 0.30 0.28
3.7 4.4 2.3 4.4
0.31 0.31 0.29 0.30
3.5 4.1 2.2 4.1
0.30 0.29 0.30 0.28
the material strength is less than Smin, it is expected that three loops fail. Therefore, the smallest one corresponds to the multiple failure of three loops. On the other hand, the largest one, Smax, Smax =max{S1, S2, S3}
(2)
is equivalent to the applied stress that causes a single loop failure with survival of the remaining two loops. Here max{} gives the maximum one. Likewise, the midst value of {S1, S2, S3} Smid = mid{S1, S2, S3}
(3)
corresponds to the response level at which two loops fail. Table 2 shows the median and b values of Smin, Smax and Smid. The fitting of S to the lognormal distribution is excellent and the correlation is 0.96 or more. It should be noted that the present coupled dynamic analysis takes the partial correlation of the three loops into account automatically. If the three loops are totally dependent, the values derived from all the data in Table 1 are to be used.
seismic fragility. A total of three parameters are used, i.e. the median factor of safety FS, the uncertainty bU, and the randomness bR. bU is a variability that can be reduced by additional efforts, while bR is an intrinsic variability that cannot be reduced. In this study, the composite variability bC is used in the following. The composite variability is defined by: bC = b 2U + b 2R Thus the seismic fragility is expressed as:
Pf(a)= F
1 ln(a/FS) bC
The seismic fragility of equipment is defined as a failure probability on condition that an earthquake takes place. The fragility is usually defined as a function of the intensity of the earthquake, i.e. peak ground acceleration or local response. In this study, it is expressed in terms of a safety factor relative to the design earthquake, S2. Lognormal distribution is assumed to describe the
n
(5)
where a is the ratio of the intensity of an earthquake to the S2 earthquake level, F is cumulative normal probability distribution function, and Pf(a) is the conditional failure probability as a function of a. The seismic fragility is a product of factors of safety with regard to the equipment capacity FC, inelastic energy absorption capability Fm and system redundancy FSYS as follows: FS = FCFmFSYS
4. System level fragility of seismic failure
(4)
(6)
The system redundancy factor reflects that a complex piping system usually has potential for force and moment redistribution at failure threshold. Then FSYS is the ratio of the system collapse load to a single pipe element collapse load. A factor of safety with respect to the seismic capacity is defined as the ratio of strength to load: FC =
sC − sN S
(7)
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Table 3 Median safety factors and the lognormal standard deviation Total (piping fragility)
Hot leg Crossover leg Cold leg Union
Minimum (3 loop failure) Maximum (1 loop failure)
Median
b
Median
b
Median
b
Median
b
9.2 8.4 15.9 7.7
0.40 0.39 0.39 0.39
10.0 9.1 16.9 8.3
0.40 0.39 0.40 0.39
8.6 7.8 15.0 7.1
0.41 0.40 0.39 0.40
9.2 8.4 15.9 7.7
0.40 0.39 0.40 0.39
where sC and S are the allowable load for failure and the applied load of a component. sN is the normal load such as dead weight. It is conservatively assumed that sN =0, because the internal pressure is atmospheric in FBRs and the normal load is small in comparison with the seismic load. The allowable load is evaluated as follows. The operating temperature in the hot leg is around 550°C. From the median yield strength test data of type 304 stainless steel at 550°C is 13.25 kg mm − 2 with lognormal standard deviation of 0.095. Those for the crossover leg and cold leg, where the sodium temperature is 400°C are 14.36 and 0.083kg mm − 2, respectively. The design allowable load is expressed as: sC =1.5KSSm
Midst (2 loop failure)
(8)
where KS and Sm are shape factor and 90% of material yeild stress, respectively. According to Kennedy and Campbell (1985), the following values are recommended for the ductility factor and system redundancy factor: Fm = 1.96, bm =0.226, FSYS =1.22, and bSYS = 0.10. Using these values and Table 2, the factor of safety relative to the design earthquake can be calculated as shown in Table 3. Based on the probabilistic response analysis using SMACS, the seismic fragility for each of the heat transport equipment was obtained. The summary is shown in Fig. 2. It is seen that the IHX and the coolant pump are rugged and the failure probability is quite low. Within the three legs of the piping system, seismic failure of the cold leg is not probable. This fact is important because the cold leg pipe is just downstream of the primary pump as well as upstream of the reactor vessel.
This result suggests that the sudden decrease of coolant flow caused by the pumping of coolant out is less probable than the slower flowing out of the hot or crossover legs. Also, the flow reversal from the reactor vessel to the cold leg piping is not probable. The fragilities of the crossover leg and hot leg pipes are greater than or comparable to that of reactor vessel as seen from Fig. 2. The failure probability in the crossover leg is the largest. The three portions are the dominant contributors to the heat transport system failure. It is noted that the piping seems to be slightly more fragile than the reactor vessel. The reactor vessel is common to all the loops and its failure results in loss of core cooling capability. On the other hand, the piping is triply redundant because one loop natural circulation is sufficient for decay heat removal. The loss of decay heat removal results from the union of the reactor vessel failure and triple failure of the three loops. Therefore it is necessary to estimate the triple failure of the pipe legs.
Fig. 2. Seismic failure probability of HTS components.
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Fig. 3. Seismic failure probability of cold leg piping.
Fig. 5. Seismic failure probability of hot leg piping.
Figs. 3–5 show the fragilities for the three loop failure of the cold leg, crossover leg and hot leg piping, respectively. The piping system fragilities evaluated with totally dependent or independent assumptions and reactor vessel fragility are also included in Figs. 3 – 5. It can be said that from Fig. 3 the cold leg piping is not a dominant contributor to the heat transport system failure. If one assumes some dependency among the three loops, Fig. 4 suggests that the primary heat transport system failure is dominated by the piping failure. However, if the three loops are independent from each other, the reactor vessel would be the most critical component in the heat transport system. The fragilities of the heat transport piping system is shown in Fig. 6 with the reactor vessel fragility curve. The failure of the heat transport piping system is defined by the intersection of the
failure event in each loop. The failure event in each loop is evaluated by the union of the cold, crossover and hot legs failure. We can see the heat transport system of the FBR has sufficient safety margins to the design S2 earthquake level even if the perfect dependence among the three loops is assumed. If the three loops are independent, the fragilities of the piping system and the reactor vessel are very comparative. However, the most realistic assumption, i.e. the partial correlation case gives greater failure probability through the whole range of the earthquake level. Therefore, it can be said that the loss of decay heat removal capability is caused by piping failure rather than reactor vessel failure. In other words even if the decay heat cannot be removed by the heat transport piping, the reactor vessel is expected to be filled with coolant. Therefore, another decay heat removal system that is different in design, location
Fig. 4. Seismic failure probability of crossover leg piping.
Fig. 6. Seismic failure probability of HTS piping system.
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Fig. 7. (a) Scattergram for the seismic load between loop 1 and loop 2 (Case 1). (b) Scattergram for the seismic load between loop 2 and loop 3 (Case 1). (c) Scattergram for the seismic load between loop 1 and loop 3 (Case 1).
and direction from the primary heat transport system is effective for maintaining decay heat removal capability after the primary piping failure takes place. Such an example would be a direct reactor auxiliary cooling system (DRACS) that flow path directly comes out of the reactor vessel and does not rely on the primary heat transport system.
5. Correlation of seismic failure The correlation coefficients of the response among the three loops are obtained from the above-mentioned coupled analysis (it is named
Case 1). The scattergram of the seismic load between loops 1 and 2 are shown in Fig. 7(a). Fig. 7(b)–(c) show the seismic load scattergram for other combinations, i.e. between loops 2 and 3, and between loops 1 and 3, respectively. It was found that the structural response of the piping system to the same seismic input is different by each loop because the three loops are placed in different directions relative to the seismic input motion. According to Bohn and Lambright (1990), the correlation coefficient of 1.0 is recommended as mentioned in Section 1 for the heat transport piping in this case. However, this analysis suggests the correlation coefficient of 0.9 is the best estimate as shown in Table 4 and Fig. 7(a–c).
A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245
244 Table 4 Correlation coefficient Loop
1 –2
1–3
2–3
Average
Case 1 Case 2
0.91 0.81
0.92 0.76
0.90 0.69
0.91 0.75
Additional analysis is performed to see the change of the correlation coefficient by neglecting the variability of soil and structural properties (designated as Case 2). Comparing Cases 1 and 2, b can be separated into the variability deriving from seismic input and that from soil and structural properties. It is found that in Case 2, the
correlation coefficients lie around 0.75 (see Fig. 8(a)–(c); Table 4). Because r and b are evaluated by the response analyses (Tables 3 and 4), one can separate the variability into seismic input and soil and structural portions. Common portion of the lognormal standard deviation is calculated by the following equation (Reed et al., 1985): b 2Com = rb 2
(9)
The variability b is divided into the independent part bInd and common cause part bCom as: b= b 2Ind + b 2Com
(10)
Fig. 8. (a) Scattergram for the seismic load between loop 1 and loop 2 (Case 2). (b) Scattergram for the seismic load between loop 2 and loop 3 (Case 2). (c) Scattergram for the seismic load between loop 1 and loop 3 (Case 2).
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Table 5 Independent and common portion of b Source of b
Seismic input (Case 2)
Soil and structure
Total (Case 1)
r b bInd bCom
0.75 0.15 0.074 0.13
0.98 0.24 0.073 0.24
0.91 0.28 0.10 0.27
Hence the independent portion of b can be evaluated. Using these results one can easily perform the systems analysis of the heat transport system considering the partial correlation (Yamaguchi, 1991). The independent and common portions are evaluated as shown in Table 5. It is seen from this table that the correlation comes mostly from common structural properties rather than common seismic input.
6. Conclusions The fragilities for the primary heat transport system in the FBR have been evaluated by the probabilistic structural response analysis for all the failure modes of the system. It has been found that the heat transport system of the FBR has sufficient safety margins to the design earthquake level even if the perfect dependence among the three loops is assumed. Hence the perfect dependence is a reasonable assumption for estimating the core damage frequency from the viewpoints of conservatism and simplicity. The dominant failure mode of the system has found to be the crossover leg piping failure. This failure mode has less influence on the core cooling capability. The correlation coefficient of the heat transport loops is approximately 0.9. It is found that the
correlation comes from the common structural properties rather than the common seismic input. The present approach is useful for quantifying the correlation coefficient and the seismic fragility of the redundant component failure. With the fragility and the correlation coefficient, the systems analysis can be performed considering the partial correlation among the multiple equipment.
References Bohn, M.P., Lambright, J.A., 1990. Procedures for the External event core damage frequency analyses for NUREG1150. NUREG/CR-4840. Kennedy, R.P., Campbell, R.D., 1985. Reliability of pressure vessels and piping under seismic loads. ASME PVP Tech. Morishita, M., 1993. Fragility Development Based on Probabilistic Response Analysis, Annual Meeting of AESJ (in Japanese). Nakai, R., Yamaguchi A., Morishita, M., 1993. Seismic systems analysis for an LMFBR Plant. Proc. of PSA’93, Clearwater Beach. Reed, J.W. et al., 1985. Analytical techniques for performing probabilistic seismic risk assessment of nuclear power plants. Proc. of 4th ICOSSAR, Kobe. Smith, P.D. et al., 1981. Seismic safety margins research program — phase I final report. NUREG/CR-2015. Yamaguchi, A., 1991. Seismic fragility analysis of the heat transport system of LMFBR considering partial correlation of multiple failure modes. Proc. of SMiRT-11, Paper M04/2, Tokyo.
Seismic failure probability evaluation of redundant fast breeder reactor piping system by probabilistic structural response analysis Akira Yamaguchi * O-arai Engineering Center, Power Reactor and Nuclear Fuel De6elopment Corporation, O-arai, Ibaraki 311 -13, Japan Received 1 December 1996; accepted 20 December 1996
Abstract The seismic failure probability and the correlation coefficient of the multiple failure mode of the heat transport system of a three-loop fast breeder reactor have been evaluated based on a probabilistic structural response analysis. It has been found that the most probable failure mode of the heat transport system has less impact on the core cooling capability than other modes. The correlation coefficient of the heat transport system loops is approximately 0.9. It is found that the correlation comes from the common structural properties rather than the common seismic input. The present approach is useful for quantifying the correlation coefficient and the seismic fragility of the redundant component failure that is used in the systems analysis. © 1997 Elsevier Science S.A.
1. Introduction Structural and equipment fragilities, i.e. seismic failure probabilities as a function of the seismic input intensity are evaluated in a Seismic Probabilistic Safety Analysis (SPSA) study. In the SPSA, one makes full utilization of the power plant systems logic models, developed in the internal events analyses. By making full use of the internal event models, an external event analysis becomes consistent in level of detail with the internal events analysis. Redundant components composing a safety system are often treated as dependent and the redundancy is conservatively * Tel.: + 81 292 674141; fax: + 81 292 677834; e-mail: [email protected]
accounted for in SPSAs. This simplified assumption is reasonable and verified as long as the neglection of the redundancy does not influence the core damage frequency. Otherwise the correlation of the multiple components failure should be quantified and the redundancy be taken into consideration. It seems the simplification used in the SPSA, in treating the multiple component failure is not consistent with the internal event analysis. As pointed out by Bohn and Lambright (1990), the common-cause failure possibility represents a potentially significant risk to the nuclear power plant during an earthquake. In the Seismic Safety Margins Research Program (SSMRP) (Smith et al., 1981), extensive multiple time history analyses have been performed and a distinct pattern was observed. On the basis of the findings, a set of
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A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245
rules was formulated, which predicted the correlation coefficients between components. These correlation coefficients are looking at the nature and location of the responses and is applicable to both PWR and BWR. The rule describes components on the same floor slab, and sensitive to the same spectral frequency range will be assigned response correlation= 1.0 (Bohn and Lambright, 1990). A SPSA study has been performed of a looptype Fast Breeder Reactor (FBR) (Nakai et al., 1993). It has been found that the structures of the heat transport system are relatively important from the viewpoint of the frequency of seismically-induced core damage. The following is an explanation of the importance of the coolant boundary structures. The FBR has preferable passive safety characteristics that emergency reactor shutdown and decay heat removal do not rely on alternating current power supply at all. Hence the coolant boundary structure of the FBR plays an important role for the safe shutdown during the seismic event, being compared with electrical equipment. The quantification of the correlation coefficient is not straightforward in general and the common cause failure is not easily evaluated under the seismic conditions. If one follows the rule mentioned above, the correlation coefficient of a heat transport system (HTS) of a three-loop FBR becomes 1.0. In this study, the correlation coefficient of the multiple failure mode and the seismic failure probability of the heat transport system has been evaluated based on probabilistic structural response analyses.
vessel is the cold leg. The pipe between the IHX and the pump is called crossover leg. The coolant temperature in the cold and crossover legs is approximately 400°C. The three heat transport loops are placed in every 120° direction each as shown in Fig. 1. The structure of the equipment in the system is thin-walled because the internal pressure is almost atmospheric and the maximum design temperature is beyond those of light water reactors. Therefore, the seismic load is one of the critical design consideration. It is noted that decay heat in the core of the FBR can be removed by only one of the three heat transport loops, i.e. the FBR system is triply redundant with regard to the decay heat removal. It is achieved by natural circulation and is not dependent on the electricity at all. Therefore, the diesel generator and off-site power are not essential for the seismic accident sequence in the FBR. Then the FBR plant is free from the alternating current power supply to achieve the emergency cold shutdown. It was found in the internal event probabilistic safety analysis, that the reliability of decay heat removal is extremely high. To take advantage of the preferable characteristics of the FBR, it is important to maintain the structural integrity of the system and to bear the coolant inside the structure. The seismic failure probability of the triply redundant heat transport system is a point of concern in the SPSA of the FBR.
2. System description The FBR plant analyzed in this study has three heat transport loops. The schematic drawing of the heat transport system is shown in Fig. 1. Major components of the system are a reactor vessel (RV), intermediate heat exchangers (IHXs) and coolant pumps. They are connected with the hot, crossover, and cold leg pipes. The hot leg piping in which coolant temperature is around 550°C connects the reactor vessel and the IHX. The piping running from the pump to the reactor
Fig. 1. Schematic drawing of primary heat transport system.
A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245
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Table 1 Median and lognormal standard deviation of membrane plus bending stress (kg mm−2) Loop 1
Hot leg Crossover leg Cold leg Union
Loop 2
Loop 3
Three loops
Median
b
Median
b
Median
b
Median
b
3.4 3.9 2.2 4.0
0.29 0.28 0.30 0.28
3.4 4.2 2.2 4.2
0.32 0.31 0.31 0.31
3.5 4.1 2.2 4.2
0.33 0.31 0.30 0.30
3.4 4.1 2.2 4.1
0.30 0.29 0.29 0.28
3. Probabilistic response analysis of heat transport system In a preliminary SPSA study, failure of the multiple heat transport loops may be assumed dependent of each other for simplicity because the design and qualification method is common to the three loops. However, the correlation should be quantified in the detailed analysis because the three loops are placed in every 120° direction each and the response is expected to be varied. Computer program, SMACS (Seismic Methodology Analysis Chain for Statistics) has been applied to the coupled analysis of reactor building structures and primary equipment such as the reactor vessel, IHX, pump, and piping system (Morishita, 1993). A total of 30 ensembles of artificial time histories of input earthquakes are generated so their statistical response spectrum is in accordance with a site-specific median-centered target spectrum. Variabilities also included in the analysis are soil and structural properties. The failure mode of the piping is assumed to be inelastic buckling of an elbow because the maximum load is observed in most cases at elbow sections. An applied external load S, i.e. the primary membrane plus bending stress is compared with the capacity of the piping material to evaluate the factor of safety. Each response analysis is performed with a set of random earthquake time history and soil and structure properties. From the individual response analysis, maximum load in j-th loop ( j =1, 2, or 3) is obtained for i-th simulation (i= 1, 2,…, 30). Here one denotes the maximum applied load in j-th loop as Sij for i-th simulation. If the maximum applied load is taken for cold, crossover and hot leg, respectively, one
obtains statistics of the maximumm load for each loop and each leg. Table 1 shows the median value and lognormal standard deviation b of the maximum seismic load for each loop and leg. Also included in Table 1 are those for the union event of the cold leg, crossover leg and hot leg failure. For the union event of the failures in three legs, median and the standard deviation of the largest load in a loop (not in each leg) for 30 simulations are used to evaluate the median and standard deviation of the maximum load. The last column denoted as total is the statistics for all the three loop data. It is seen that the difference of the median and b values among the three loops is small and they are practically in accordance with the numbers obtained from all the three loop data (the last column in Table 1). By fitting the 90 values (30 simulations times three loops) of Sij to a lognormal distribution, a cumulative probability function is obtained for the maximum seismic load in a single heat transport loop. Therefore, comparing the cumulative probability function of S with the strength of the material and multiplying other factors of conservatism or unconservatism, one obtains the seismic fragility of the heat transport piping. In each structural response analysis, the seismic loads in three loops vary from each other. One obtains the maximum load for each heat transport loop. The smallest one of the loads in the three loops for each simulation, Smin, is defined as: Smin = min{S1, S2, S3}
(1)
where Si is the maximum load in the i-th loop and min{} shows minimum one of S1, S2 and S3. If
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Table 2 Median and lognormal standard deviation of membrane plus bending stress for multiple failure mode (kg mm−2)
Hot leg Crossover leg Cold leg Union
Smin
Smax
Smid
3 loop failure
1 loop failure
2 loop failure
Median
b
Median
b
Median
b
3.2 3.8 2.0 3.8
0.31 0.28 0.30 0.28
3.7 4.4 2.3 4.4
0.31 0.31 0.29 0.30
3.5 4.1 2.2 4.1
0.30 0.29 0.30 0.28
the material strength is less than Smin, it is expected that three loops fail. Therefore, the smallest one corresponds to the multiple failure of three loops. On the other hand, the largest one, Smax, Smax =max{S1, S2, S3}
(2)
is equivalent to the applied stress that causes a single loop failure with survival of the remaining two loops. Here max{} gives the maximum one. Likewise, the midst value of {S1, S2, S3} Smid = mid{S1, S2, S3}
(3)
corresponds to the response level at which two loops fail. Table 2 shows the median and b values of Smin, Smax and Smid. The fitting of S to the lognormal distribution is excellent and the correlation is 0.96 or more. It should be noted that the present coupled dynamic analysis takes the partial correlation of the three loops into account automatically. If the three loops are totally dependent, the values derived from all the data in Table 1 are to be used.
seismic fragility. A total of three parameters are used, i.e. the median factor of safety FS, the uncertainty bU, and the randomness bR. bU is a variability that can be reduced by additional efforts, while bR is an intrinsic variability that cannot be reduced. In this study, the composite variability bC is used in the following. The composite variability is defined by: bC = b 2U + b 2R Thus the seismic fragility is expressed as:
Pf(a)= F
1 ln(a/FS) bC
The seismic fragility of equipment is defined as a failure probability on condition that an earthquake takes place. The fragility is usually defined as a function of the intensity of the earthquake, i.e. peak ground acceleration or local response. In this study, it is expressed in terms of a safety factor relative to the design earthquake, S2. Lognormal distribution is assumed to describe the
n
(5)
where a is the ratio of the intensity of an earthquake to the S2 earthquake level, F is cumulative normal probability distribution function, and Pf(a) is the conditional failure probability as a function of a. The seismic fragility is a product of factors of safety with regard to the equipment capacity FC, inelastic energy absorption capability Fm and system redundancy FSYS as follows: FS = FCFmFSYS
4. System level fragility of seismic failure
(4)
(6)
The system redundancy factor reflects that a complex piping system usually has potential for force and moment redistribution at failure threshold. Then FSYS is the ratio of the system collapse load to a single pipe element collapse load. A factor of safety with respect to the seismic capacity is defined as the ratio of strength to load: FC =
sC − sN S
(7)
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Table 3 Median safety factors and the lognormal standard deviation Total (piping fragility)
Hot leg Crossover leg Cold leg Union
Minimum (3 loop failure) Maximum (1 loop failure)
Median
b
Median
b
Median
b
Median
b
9.2 8.4 15.9 7.7
0.40 0.39 0.39 0.39
10.0 9.1 16.9 8.3
0.40 0.39 0.40 0.39
8.6 7.8 15.0 7.1
0.41 0.40 0.39 0.40
9.2 8.4 15.9 7.7
0.40 0.39 0.40 0.39
where sC and S are the allowable load for failure and the applied load of a component. sN is the normal load such as dead weight. It is conservatively assumed that sN =0, because the internal pressure is atmospheric in FBRs and the normal load is small in comparison with the seismic load. The allowable load is evaluated as follows. The operating temperature in the hot leg is around 550°C. From the median yield strength test data of type 304 stainless steel at 550°C is 13.25 kg mm − 2 with lognormal standard deviation of 0.095. Those for the crossover leg and cold leg, where the sodium temperature is 400°C are 14.36 and 0.083kg mm − 2, respectively. The design allowable load is expressed as: sC =1.5KSSm
Midst (2 loop failure)
(8)
where KS and Sm are shape factor and 90% of material yeild stress, respectively. According to Kennedy and Campbell (1985), the following values are recommended for the ductility factor and system redundancy factor: Fm = 1.96, bm =0.226, FSYS =1.22, and bSYS = 0.10. Using these values and Table 2, the factor of safety relative to the design earthquake can be calculated as shown in Table 3. Based on the probabilistic response analysis using SMACS, the seismic fragility for each of the heat transport equipment was obtained. The summary is shown in Fig. 2. It is seen that the IHX and the coolant pump are rugged and the failure probability is quite low. Within the three legs of the piping system, seismic failure of the cold leg is not probable. This fact is important because the cold leg pipe is just downstream of the primary pump as well as upstream of the reactor vessel.
This result suggests that the sudden decrease of coolant flow caused by the pumping of coolant out is less probable than the slower flowing out of the hot or crossover legs. Also, the flow reversal from the reactor vessel to the cold leg piping is not probable. The fragilities of the crossover leg and hot leg pipes are greater than or comparable to that of reactor vessel as seen from Fig. 2. The failure probability in the crossover leg is the largest. The three portions are the dominant contributors to the heat transport system failure. It is noted that the piping seems to be slightly more fragile than the reactor vessel. The reactor vessel is common to all the loops and its failure results in loss of core cooling capability. On the other hand, the piping is triply redundant because one loop natural circulation is sufficient for decay heat removal. The loss of decay heat removal results from the union of the reactor vessel failure and triple failure of the three loops. Therefore it is necessary to estimate the triple failure of the pipe legs.
Fig. 2. Seismic failure probability of HTS components.
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Fig. 3. Seismic failure probability of cold leg piping.
Fig. 5. Seismic failure probability of hot leg piping.
Figs. 3–5 show the fragilities for the three loop failure of the cold leg, crossover leg and hot leg piping, respectively. The piping system fragilities evaluated with totally dependent or independent assumptions and reactor vessel fragility are also included in Figs. 3 – 5. It can be said that from Fig. 3 the cold leg piping is not a dominant contributor to the heat transport system failure. If one assumes some dependency among the three loops, Fig. 4 suggests that the primary heat transport system failure is dominated by the piping failure. However, if the three loops are independent from each other, the reactor vessel would be the most critical component in the heat transport system. The fragilities of the heat transport piping system is shown in Fig. 6 with the reactor vessel fragility curve. The failure of the heat transport piping system is defined by the intersection of the
failure event in each loop. The failure event in each loop is evaluated by the union of the cold, crossover and hot legs failure. We can see the heat transport system of the FBR has sufficient safety margins to the design S2 earthquake level even if the perfect dependence among the three loops is assumed. If the three loops are independent, the fragilities of the piping system and the reactor vessel are very comparative. However, the most realistic assumption, i.e. the partial correlation case gives greater failure probability through the whole range of the earthquake level. Therefore, it can be said that the loss of decay heat removal capability is caused by piping failure rather than reactor vessel failure. In other words even if the decay heat cannot be removed by the heat transport piping, the reactor vessel is expected to be filled with coolant. Therefore, another decay heat removal system that is different in design, location
Fig. 4. Seismic failure probability of crossover leg piping.
Fig. 6. Seismic failure probability of HTS piping system.
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Fig. 7. (a) Scattergram for the seismic load between loop 1 and loop 2 (Case 1). (b) Scattergram for the seismic load between loop 2 and loop 3 (Case 1). (c) Scattergram for the seismic load between loop 1 and loop 3 (Case 1).
and direction from the primary heat transport system is effective for maintaining decay heat removal capability after the primary piping failure takes place. Such an example would be a direct reactor auxiliary cooling system (DRACS) that flow path directly comes out of the reactor vessel and does not rely on the primary heat transport system.
5. Correlation of seismic failure The correlation coefficients of the response among the three loops are obtained from the above-mentioned coupled analysis (it is named
Case 1). The scattergram of the seismic load between loops 1 and 2 are shown in Fig. 7(a). Fig. 7(b)–(c) show the seismic load scattergram for other combinations, i.e. between loops 2 and 3, and between loops 1 and 3, respectively. It was found that the structural response of the piping system to the same seismic input is different by each loop because the three loops are placed in different directions relative to the seismic input motion. According to Bohn and Lambright (1990), the correlation coefficient of 1.0 is recommended as mentioned in Section 1 for the heat transport piping in this case. However, this analysis suggests the correlation coefficient of 0.9 is the best estimate as shown in Table 4 and Fig. 7(a–c).
A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245
244 Table 4 Correlation coefficient Loop
1 –2
1–3
2–3
Average
Case 1 Case 2
0.91 0.81
0.92 0.76
0.90 0.69
0.91 0.75
Additional analysis is performed to see the change of the correlation coefficient by neglecting the variability of soil and structural properties (designated as Case 2). Comparing Cases 1 and 2, b can be separated into the variability deriving from seismic input and that from soil and structural properties. It is found that in Case 2, the
correlation coefficients lie around 0.75 (see Fig. 8(a)–(c); Table 4). Because r and b are evaluated by the response analyses (Tables 3 and 4), one can separate the variability into seismic input and soil and structural portions. Common portion of the lognormal standard deviation is calculated by the following equation (Reed et al., 1985): b 2Com = rb 2
(9)
The variability b is divided into the independent part bInd and common cause part bCom as: b= b 2Ind + b 2Com
(10)
Fig. 8. (a) Scattergram for the seismic load between loop 1 and loop 2 (Case 2). (b) Scattergram for the seismic load between loop 2 and loop 3 (Case 2). (c) Scattergram for the seismic load between loop 1 and loop 3 (Case 2).
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Table 5 Independent and common portion of b Source of b
Seismic input (Case 2)
Soil and structure
Total (Case 1)
r b bInd bCom
0.75 0.15 0.074 0.13
0.98 0.24 0.073 0.24
0.91 0.28 0.10 0.27
Hence the independent portion of b can be evaluated. Using these results one can easily perform the systems analysis of the heat transport system considering the partial correlation (Yamaguchi, 1991). The independent and common portions are evaluated as shown in Table 5. It is seen from this table that the correlation comes mostly from common structural properties rather than common seismic input.
6. Conclusions The fragilities for the primary heat transport system in the FBR have been evaluated by the probabilistic structural response analysis for all the failure modes of the system. It has been found that the heat transport system of the FBR has sufficient safety margins to the design earthquake level even if the perfect dependence among the three loops is assumed. Hence the perfect dependence is a reasonable assumption for estimating the core damage frequency from the viewpoints of conservatism and simplicity. The dominant failure mode of the system has found to be the crossover leg piping failure. This failure mode has less influence on the core cooling capability. The correlation coefficient of the heat transport loops is approximately 0.9. It is found that the
correlation comes from the common structural properties rather than the common seismic input. The present approach is useful for quantifying the correlation coefficient and the seismic fragility of the redundant component failure. With the fragility and the correlation coefficient, the systems analysis can be performed considering the partial correlation among the multiple equipment.
References Bohn, M.P., Lambright, J.A., 1990. Procedures for the External event core damage frequency analyses for NUREG1150. NUREG/CR-4840. Kennedy, R.P., Campbell, R.D., 1985. Reliability of pressure vessels and piping under seismic loads. ASME PVP Tech. Morishita, M., 1993. Fragility Development Based on Probabilistic Response Analysis, Annual Meeting of AESJ (in Japanese). Nakai, R., Yamaguchi A., Morishita, M., 1993. Seismic systems analysis for an LMFBR Plant. Proc. of PSA’93, Clearwater Beach. Reed, J.W. et al., 1985. Analytical techniques for performing probabilistic seismic risk assessment of nuclear power plants. Proc. of 4th ICOSSAR, Kobe. Smith, P.D. et al., 1981. Seismic safety margins research program — phase I final report. NUREG/CR-2015. Yamaguchi, A., 1991. Seismic fragility analysis of the heat transport system of LMFBR considering partial correlation of multiple failure modes. Proc. of SMiRT-11, Paper M04/2, Tokyo.