Seismic failure probability evaluation of redundant fast breeder reactor piping system by...

Nuclear Engineering and Design 175 (1997) 237–245

Seismic failure probability evaluation of redundant fastbreeder 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 transportsystem 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 corecooling capability than other modes. The correlation coefficient of the heat transport system loops is approximately0.9. It is found that the correlation comes from the common structural properties rather than the common seismicinput. The present approach is useful for quantifying the correlation coefficient and the seismic fragility of theredundant component failure that is used in the systems analysis. © 1997 Elsevier Science S.A.

1. Introduction

Structural and equipment fragilities, i.e. seismicfailure probabilities as a function of the seismicinput intensity are evaluated in a Seismic Proba-bilistic Safety Analysis (SPSA) study. In theSPSA, one makes full utilization of the powerplant systems logic models, developed in the inter-nal events analyses. By making full use of theinternal event models, an external event analysisbecomes consistent in level of detail with theinternal events analysis. Redundant componentscomposing a safety system are often treated asdependent and the redundancy is conservatively

accounted for in SPSAs. This simplified assump-tion is reasonable and verified as long as theneglection of the redundancy does not influencethe core damage frequency. Otherwise the correla-tion of the multiple components failure should bequantified and the redundancy be taken into con-sideration. It seems the simplification used in theSPSA, in treating the multiple component failureis not consistent with the internal event analysis.

As pointed out by Bohn and Lambright (1990),the common-cause failure possibility represents apotentially significant risk to the nuclear powerplant during an earthquake. In the Seismic SafetyMargins Research Program (SSMRP) (Smith etal., 1981), extensive multiple time history analyseshave been performed and a distinct pattern wasobserved. On the basis of the findings, a set of

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A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245238

rules was formulated, which predicted the correla-tion coefficients between components. These cor-relation coefficients are looking at the nature andlocation of the responses and is applicable to bothPWR and BWR. The rule describes componentson the same floor slab, and sensitive to the samespectral frequency range will be assigned responsecorrelation=1.0 (Bohn and Lambright, 1990).

A SPSA study has been performed of a loop-type Fast Breeder Reactor (FBR) (Nakai et al.,1993). It has been found that the structures of theheat transport system are relatively importantfrom the viewpoint of the frequency of seismi-cally-induced core damage. The following is anexplanation of the importance of the coolantboundary structures. The FBR has preferable pas-sive safety characteristics that emergency reactorshutdown and decay heat removal do not rely onalternating current power supply at all. Hence thecoolant boundary structure of the FBR plays animportant role for the safe shutdown during theseismic event, being compared with electricalequipment.

The quantification of the correlation coefficientis not straightforward in general and the commoncause failure is not easily evaluated under theseismic conditions. If one follows the rule men-tioned above, the correlation coefficient of a heattransport system (HTS) of a three-loop FBR be-comes 1.0. In this study, the correlation coefficientof the multiple failure mode and the seismic fail-ure probability of the heat transport system hasbeen evaluated based on probabilistic structuralresponse analyses.

2. System description

The FBR plant analyzed in this study has threeheat transport loops. The schematic drawing ofthe heat transport system is shown in Fig. 1.Major components of the system are a reactorvessel (RV), intermediate heat exchangers (IHXs)and coolant pumps. They are connected with thehot, crossover, and cold leg pipes. The hot legpiping in which coolant temperature is around550°C connects the reactor vessel and the IHX.The piping running from the pump to the reactor

vessel is the cold leg. The pipe between the IHXand the pump is called crossover leg. The coolanttemperature in the cold and crossover legs isapproximately 400°C. The three heat transportloops are placed in every 120° direction each asshown in Fig. 1. The structure of the equipment inthe system is thin-walled because the internalpressure is almost atmospheric and the maximumdesign temperature is beyond those of light waterreactors. Therefore, the seismic load is one of thecritical design consideration.

It is noted that decay heat in the core of theFBR can be removed by only one of the threeheat transport loops, i.e. the FBR system is triplyredundant with regard to the decay heat removal.It is achieved by natural circulation and is notdependent on the electricity at all. Therefore, thediesel generator and off-site power are not essen-tial for the seismic accident sequence in the FBR.Then the FBR plant is free from the alternatingcurrent power supply to achieve the emergencycold shutdown. It was found in the internal eventprobabilistic safety analysis, that the reliability ofdecay heat removal is extremely high. To takeadvantage of the preferable characteristics of theFBR, it is important to maintain the structuralintegrity of the system and to bear the coolantinside the structure. The seismic failure probabil-ity of the triply redundant heat transport systemis a point of concern in the SPSA of the FBR.

Fig. 1. Schematic drawing of primary heat transport system.

A. Yamaguchi / Nuclear Engineering and Design 175 (1997) 237–245 239

Table 1Median and lognormal standard deviation of membrane plus bending stress (kg mm−2)

Three loopsLoop 3Loop 1 Loop 2

Median Median bMedian bb Median b

3.5 0.33 3.4Hot leg 0.303.4 0.29 3.4 0.320.294.10.31Crossover leg 3.9 4.10.28 4.2 0.31

2.2 0.30 2.2Cold leg 2.2 0.290.30 2.2 0.314.2 0.28Union 4.0 0.300.28 4.14.2 0.31

3. Probabilistic response analysis of heattransport system

In a preliminary SPSA study, failure of themultiple heat transport loops may be assumeddependent of each other for simplicity because thedesign and qualification method is common to thethree loops. However, the correlation should bequantified in the detailed analysis because thethree loops are placed in every 120° direction eachand the response is expected to be varied. Com-puter program, SMACS (Seismic MethodologyAnalysis Chain for Statistics) has been applied tothe coupled analysis of reactor building structuresand primary equipment such as the reactor vessel,IHX, pump, and piping system (Morishita, 1993).A total of 30 ensembles of artificial time historiesof input earthquakes are generated so their statis-tical response spectrum is in accordance with asite-specific median-centered target spectrum.Variabilities also included in the analysis are soiland structural properties.

The failure mode of the piping is assumed to beinelastic buckling of an elbow because the maxi-mum load is observed in most cases at elbowsections. An applied external load S, i.e. the pri-mary membrane plus bending stress is comparedwith the capacity of the piping material to evalu-ate the factor of safety. Each response analysis isperformed with a set of random earthquake timehistory and soil and structure properties. Fromthe individual response analysis, maximum loadin j-th loop ( j=1, 2, or 3) is obtained for i-thsimulation (i=1, 2,…, 30). Here one denotes themaximum applied load in j-th loop as Sij for i-thsimulation. If the maximum applied load is takenfor cold, crossover and hot leg, respectively, one

obtains statistics of the maximumm load for eachloop and each leg.

Table 1 shows the median value and lognormalstandard deviation b of the maximum seismicload for each loop and leg. Also included in Table1 are those for the union event of the cold leg,crossover leg and hot leg failure. For the unionevent of the failures in three legs, median and thestandard deviation of the largest load in a loop(not in each leg) for 30 simulations are used toevaluate the median and standard deviation of themaximum load. The last column denoted as totalis the statistics for all the three loop data. It isseen that the difference of the median and b

values among the three loops is small and they arepractically in accordance with the numbers ob-tained from all the three loop data (the lastcolumn in Table 1).

By fitting the 90 values (30 simulations timesthree loops) of Sij to a lognormal distribution, acumulative probability function is obtained forthe maximum seismic load in a single heat trans-port loop. Therefore, comparing the cumulativeprobability function of S with the strength of thematerial and multiplying other factors of conser-vatism or unconservatism, one obtains the seismicfragility of the heat transport piping.

In each structural response analysis, the seismicloads in three loops vary from each other. Oneobtains the maximum load for each heat trans-port loop. The smallest one of the loads in thethree loops for each simulation, Smin, is definedas:

Smin=min{S1, S2, S3} (1)

where Si is the maximum load in the i-th loop andmin{�} shows minimum one of S1, S2 and S3. If


Table 2Median and lognormal standard deviation of membrane plus bending stress for multiple failure mode (kg mm−2)

Smin Smax Smid

2 loop failure1 loop failure3 loop failure

bMedianMedian bMedian b

0.31 3.5Hot leg 3.2 0.31 0.303.70.31 4.1Crossover leg 3.8 0.28 4.4 0.29

2.20.29 0.30Cold leg 2.32.0 0.300.284.1Union 3.8 0.28 4.4 0.30

the material strength is less than Smin, it is ex-pected that three loops fail. Therefore, thesmallest one corresponds to the multiple failure ofthree loops. On the other hand, the largest one,Smax,

Smax=max{S1, S2, S3} (2)

is equivalent to the applied stress that causes asingle loop failure with survival of the remainingtwo 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 twoloops fail. Table 2 shows the median and b valuesof Smin, Smax and Smid. The fitting of S to thelognormal distribution is excellent and the corre-lation is 0.96 or more. It should be noted that thepresent coupled dynamic analysis takes the partialcorrelation of the three loops into account auto-matically. If the three loops are totally dependent,the values derived from all the data in Table 1 areto be used.

4. System level fragility of seismic failure

The seismic fragility of equipment is defined asa failure probability on condition that an earth-quake takes place. The fragility is usually definedas a function of the intensity of the earthquake,i.e. peak ground acceleration or local response. Inthis study, it is expressed in terms of a safetyfactor relative to the design earthquake, S2. Log-normal distribution is assumed to describe the

seismic fragility. A total of three parameters areused, i.e. the median factor of safety FS, theuncertainty bU, and the randomness bR. bU is avariability that can be reduced by additional ef-forts, while bR is an intrinsic variability thatcannot be reduced. In this study, the compositevariability bC is used in the following. The com-posite variability is defined by:

bC=b2U+b2

R (4)

Thus the seismic fragility is expressed as:

Pf(a)=F� 1

bC

ln(a/FS)n

(5)

where a is the ratio of the intensity of an earth-quake to the S2 earthquake level, F is cumulativenormal probability distribution function, andPf(a) is the conditional failure probability as afunction of a.

The seismic fragility is a product of factors ofsafety with regard to the equipment capacity FC,inelastic energy absorption capability Fm and sys-tem redundancy FSYS as follows:

FS=FCFmFSYS (6)

The system redundancy factor reflects that a com-plex piping system usually has potential for forceand moment redistribution at failure threshold.Then FSYS is the ratio of the system collapse loadto a single pipe element collapse load.

A factor of safety with respect to the seismiccapacity is defined as the ratio of strength to load:

FC=sC−sN

S(7)


Table 3Median safety factors and the lognormal standard deviation

Minimum (3 loop failure) Maximum (1 loop fail-Total (piping fragility) Midst (2 loop failure)ure)

b MedianMedian b Median b Median b

9.20.41 0.40Hot leg 8.69.2 0.40 10.0 0.407.8 0.40 8.4 0.39Crossover leg 8.4 0.39 9.1 0.39

0.4015.90.39Cold leg 15.9 15.00.39 16.9 0.400.40 7.7Union 7.7 0.39 8.3 0.39 7.1 0.39

where sC and S are the allowable load for failureand the applied load of a component. sN is thenormal load such as dead weight. It is conserva-tively assumed that sN=0, because the internalpressure is atmospheric in FBRs and the normalload is small in comparison with the seismic load.

The allowable load is evaluated as follows. Theoperating temperature in the hot leg is around550°C. From the median yield strength test dataof type 304 stainless steel at 550°C is 13.25 kgmm−2 with lognormal standard deviation of0.095. Those for the crossover leg and cold leg,where the sodium temperature is 400°C are 14.36and 0.083kg mm−2, respectively. The design al-lowable load is expressed as:

sC=1.5KSSm (8)

where KS and Sm are shape factor and 90% ofmaterial yeild stress, respectively.

According to Kennedy and Campbell (1985),the following values are recommended for theductility 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 ofsafety relative to the design earthquake can becalculated as shown in Table 3.

Based on the probabilistic response analysisusing SMACS, the seismic fragility for each of theheat transport equipment was obtained. The sum-mary is shown in Fig. 2. It is seen that the IHXand the coolant pump are rugged and the failureprobability is quite low. Within the three legs ofthe piping system, seismic failure of the cold leg isnot probable. This fact is important because thecold leg pipe is just downstream of the primarypump as well as upstream of the reactor vessel.

This result suggests that the sudden decrease ofcoolant flow caused by the pumping of coolantout is less probable than the slower flowing out ofthe hot or crossover legs. Also, the flow reversalfrom the reactor vessel to the cold leg piping isnot probable.

The fragilities of the crossover leg and hot legpipes are greater than or comparable to that ofreactor vessel as seen from Fig. 2. The failureprobability in the crossover leg is the largest. Thethree portions are the dominant contributors tothe heat transport system failure. It is noted thatthe piping seems to be slightly more fragile thanthe reactor vessel. The reactor vessel is commonto all the loops and its failure results in loss ofcore cooling capability. On the other hand, thepiping is triply redundant because one loop natu-ral circulation is sufficient for decay heat removal.The loss of decay heat removal results from theunion of the reactor vessel failure and triple fail-ure of the three loops. Therefore it is necessary toestimate the triple failure of the pipe legs.

Fig. 2. Seismic failure probability of HTS components.


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 loopfailure of the cold leg, crossover leg and hot legpiping, respectively. The piping system fragilitiesevaluated with totally dependent or independentassumptions and reactor vessel fragility are alsoincluded in Figs. 3–5. It can be said that fromFig. 3 the cold leg piping is not a dominantcontributor to the heat transport system failure. Ifone assumes some dependency among the threeloops, Fig. 4 suggests that the primary heat trans-port system failure is dominated by the pipingfailure. However, if the three loops are indepen-dent from each other, the reactor vessel would bethe most critical component in the heat transportsystem.

The fragilities of the heat transport piping sys-tem is shown in Fig. 6 with the reactor vesselfragility curve. The failure of the heat transportpiping system is defined by the intersection of the

failure event in each loop. The failure event ineach loop is evaluated by the union of the cold,crossover and hot legs failure. We can see the heattransport system of the FBR has sufficient safetymargins to the design S2 earthquake level even ifthe perfect dependence among the three loops isassumed. If the three loops are independent, thefragilities of the piping system and the reactorvessel are very comparative. However, the mostrealistic assumption, i.e. the partial correlationcase gives greater failure probability through thewhole range of the earthquake level. Therefore, itcan be said that the loss of decay heat removalcapability is caused by piping failure rather thanreactor vessel failure. In other words even if thedecay heat cannot be removed by the heat trans-port piping, the reactor vessel is expected to befilled with coolant. Therefore, another decay heatremoval 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.


Fig. 7. (a) Scattergram for the seismic load between loop 1 and loop 2 (Case 1). (b) Scattergram for the seismic load between loop2 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 transportsystem is effective for maintaining decay heatremoval capability after the primary piping failuretakes place. Such an example would be a directreactor auxiliary cooling system (DRACS) thatflow path directly comes out of the reactor vesseland does not rely on the primary heat transportsystem.

5. Correlation of seismic failure

The correlation coefficients of the responseamong the three loops are obtained from theabove-mentioned coupled analysis (it is named

Case 1). The scattergram of the seismic loadbetween loops 1 and 2 are shown in Fig. 7(a). Fig.7(b)–(c) show the seismic load scattergram forother combinations, i.e. between loops 2 and 3, andbetween loops 1 and 3, respectively. It was foundthat the structural response of the piping system tothe same seismic input is different by each loopbecause the three loops are placed in differentdirections relative to the seismic input motion.According to Bohn and Lambright (1990), thecorrelation coefficient of 1.0 is recommended asmentioned in Section 1 for the heat transportpiping in this case. However, this analysis suggeststhe correlation coefficient of 0.9 is the best estimateas shown in Table 4 and Fig. 7(a–c).


Table 4Correlation coefficient

Average1–2 2–31–3Loop

0.90 0.91Case 1 0.91 0.920.76 0.69Case 2 0.81 0.75

correlation coefficients lie around 0.75 (see Fig.8(a)–(c); Table 4). Because r and b are evaluatedby the response analyses (Tables 3 and 4), one canseparate the variability into seismic input and soiland structural portions.

Common portion of the lognormal standarddeviation is calculated by the following equation(Reed et al., 1985):

b2Com=rb2 (9)

The variability b is divided into the independentpart bInd and common cause part bCom as:

b=b2Ind+b2

Com (10)

Additional analysis is performed to see thechange of the correlation coefficient by neglectingthe variability of soil and structural properties(designated as Case 2). Comparing Cases 1 and 2,b can be separated into the variability derivingfrom seismic input and that from soil and struc-tural properties. It is found that in Case 2, the

Fig. 8. (a) Scattergram for the seismic load between loop 1 and loop 2 (Case 2). (b) Scattergram for the seismic load between loop2 and loop 3 (Case 2). (c) Scattergram for the seismic load between loop 1 and loop 3 (Case 2).


Table 5Independent and common portion of b

Seismic input (Case 2) Soil and structure Total (Case 1)Source of b

0.75 0.98r 0.910.15 0.24 0.28b

0.100.073bInd 0.0740.270.24bCom 0.13

Hence the independent portion of b can be evalu-ated. Using these results one can easily performthe systems analysis of the heat transport systemconsidering the partial correlation (Yamaguchi,1991).

The independent and common portions areevaluated as shown in Table 5. It is seen from thistable that the correlation comes mostly from com-mon structural properties rather than commonseismic input.

6. Conclusions

The fragilities for the primary heat transportsystem in the FBR have been evaluated by theprobabilistic structural response analysis for allthe failure modes of the system. It has been foundthat the heat transport system of the FBR hassufficient safety margins to the design earthquakelevel even if the perfect dependence among thethree loops is assumed. Hence the perfect depen-dence is a reasonable assumption for estimatingthe core damage frequency from the viewpoints ofconservatism and simplicity. The dominant failuremode of the system has found to be the crossoverleg piping failure. This failure mode has less influ-ence on the core cooling capability.

The correlation coefficient of the heat transportloops is approximately 0.9. It is found that the

correlation comes from the common structuralproperties rather than the common seismic input.

The present approach is useful for quantifyingthe correlation coefficient and the seismic fra-gility of the redundant component failure. Withthe fragility and the correlation coefficient, thesystems analysis can be performed consideringthe partial correlation among the multiple equip-ment.

References

Bohn, M.P., Lambright, J.A., 1990. Procedures for the Exter-nal event core damage frequency analyses for NUREG-1150. NUREG/CR-4840.

Kennedy, R.P., Campbell, R.D., 1985. Reliability of pressurevessels and piping under seismic loads. ASME PVP Tech.

Morishita, M., 1993. Fragility Development Based on Proba-bilistic Response Analysis, Annual Meeting of AESJ (inJapanese).

Nakai, R., Yamaguchi A., Morishita, M., 1993. Seismic sys-tems analysis for an LMFBR Plant. Proc. of PSA’93,Clearwater Beach.

Reed, J.W. et al., 1985. Analytical techniques for performingprobabilistic seismic risk assessment of nuclear powerplants. Proc. of 4th ICOSSAR, Kobe.

Smith, P.D. et al., 1981. Seismic safety margins researchprogram—phase I final report. NUREG/CR-2015.

Yamaguchi, A., 1991. Seismic fragility analysis of the heattransport system of LMFBR considering partial correla-tion of multiple failure modes. Proc. of SMiRT-11, PaperM04/2, Tokyo.

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