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International Journal of Pressure Vessels and Piping 88 (2011) 19e25

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International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

Probabilistic safety assessment of components

E. Roos*, G. Wackenhut, R. Lammert, X. SchulerMaterials Testing Institute, University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germany

a r t i c l e i n f o

Article history:Received 24 March 2010Received in revised form23 November 2010Accepted 29 November 2010

Keywords:Partial safety factorsFORMSORMProbabilistic fracture mechanicsR6-methodProbability of failureLimit state function

* Corresponding author. Tel.: þ49 (0)711/685-62604E-mail address: Eberhard.Roos@mpa.uni-stuttgart

0308-0161/$ e see front matter � 2010 Elsevier Ltd.doi:10.1016/j.ijpvp.2010.11.005

a b s t r a c t

Obviously, precautions against damage of components by operational and exceptional loads have to betaken. This is performed according to the procedures and standards of the nuclear regulations, e.g. byusing the strength and fracture mechanical parameters (warranty parameters) of the involved materials.Within this integrity proof by calculation, however, the effects of the uncertainties in the inputparameters cannot be evaluated quantitatively. By considering the possible parameter scatter in thegeometrical dimensions and loads while performing the integrity proof, the failure probabilities, andhence the caused uncertainties, become quantifiable. In addition, postulated flaw sizes can be consideredand the effects of operational measures can be evaluated. This requires, on one hand, exact knowledgeabout the scatter of the decisive parameters, and on the other hand knowledge about their effects on theemployed methods, and hence on the result of the calculation.

In this paper the applied reliability theory is described. Furthermore, the failure probabilities, espe-cially of the weld joints of a feed water line, were calculated and the available partial safety factors wereanalysed.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Concerning the manufacturing and operation of safety relevantpower plant components, failure must be excluded reliably. For thisreason, conservative assumptions are made and adequate safetyfactors are chosen for the design.

Even in cases inwhich component design is subjected to a purelydeterministic approach, a probabilistic analysis is of special value, asit allows the actual safety reserves to be estimated quantitatively.

As soon as the statistical uncertainties are identified, the failureprobability or reliability of a component can be calculated. Thus,instead of using subjective evidence (empirically determined safetyfactors) to ensure structural integrity, one can now rely on objectiveevaluation criteria based on a statistical analysis of the values ofinterest, i.e. on partial safety factors.

Based on a flawless pipeline system, the failure probability ofa component can be simplified and expressed as a function ofthe material state and the applied loads. A component fails if thematerial resistance R is smaller than the applied load. Due to thestochastic character of the parameters R and S it is necessary toqualify objectively the required safety distance between the cor-responding distributions. For this purpose, the so-called partial

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safety factors must be determined, which can be used to guaranteeproper operation.

On calculating the partial safety factors using standard proce-dures based on FORM/SORM, [1], one has to take into account thedrawbacks of the mathematical procedure behind: determining thedesign point as the minimum distance in the parameter space maynot find the global, but rather a local optimum. Our contributionshows, that the qualitative differences between solutions are notnegligible, even if the computed quantity is nearly the same. Inorder to decide, which solution is the right one, additional infor-mation and knowledge about the underlying system is needed.Although the concepts from FORM/SORM are well-known, alreadyestablished and the topic of many other recent publications ([2e4]),we emphasize, that the pure application of these methodsmay leadto unexpected results. Of course, the issue of multiple MPPs (mostprobable points) is known and there are efforts towards solving thisproblem in a general manner, but finding the correct solution stillneeds expert (engineering) knowledge, especially when solutionshave to be evaluated and compared together.

2. Probability of failure and safety index

In our further considerations to structural safety (see e.g. [5]), it isassumed that the load S and material resistance R can be describedby the stochastic independent normal distributions N(mS, sS) and N(mR, sR). Since the respective density functions are unbounded, an

Nomenclature

aR, aS weighting factors for sR and sS w.r.t. sG, resp.b safety indexgR, gS partial safety factor of the applied load and resistance,

resp.mR, mS, mG mean/expected value of R, S, G resp.F(x) cumulative density function (cdf) of the standard

normal distributionsR, sS, sG standard deviation of R, S, G, resp.su ultimate tensile strength (MPa)sy yield stress; the average of the 0.2% offset yield

strength sY, and the ultimate tensile strength (MPa)a crack depth (mm)c crack half-length (mm)Do pipe outher diameter (mm)

fG(g), FG(g) probability density function, cumulative distributionfunction of G, resp.

G stochastic variable, G R�S, limit state definitionJi Rice Integral at crack initiation (kJm2)KV impact energy ( J)Kp,R, Kp,S quantile factors of R, S, resp.Lr,max plastic collapse limit loadMb (Level D) bending moment, service level DMb,max maximum bending moment, (kNm)P internal pressure, (MPa)Pf, PoF probability of failureR, S stochastic variable for resistance, applied load, resp.r, s resistance, applied load, resp.rd, sd values of resistance, load at the design point, resp.rQ, sQ Q -quantile of R, S, resp.t wall thickness (mm)t0 nominal wall thickness (mm)

Fig. 1. Definition of the safety zones and the effect of distribution adjustment on theprobability of failure, (a), Safety zones, (b), Distribution adjustment on the probabilityof failure.

E. Roos et al. / International Journal of Pressure Vessels and Piping 88 (2011) 19e2520

overlapping of the density functions cannot be avoided. Thus, thereis always an area for which R�S 0 is true, so that for this particularcase an absolute operational safety cannot be guaranteed. As shownin Fig. 1, the overlapping area can be adjusted to fit the requirementsby reducing the variances and by modifying the average values ofloading or load capacity. Within this context, a differentiation ismade between the central and the nominal safety zone.

The central safety zone (see Fig. 1) is described as the distancebetween the average values mS and mR.

In practice, the interest rather lies on the nominal safety zone,i.e. the distance between the quantiles sQ and rQ, since this valuereflects the real distributions and genuinely describes the existingsafety reserves. Reducing the sQ and rQ uncertainties can increasethese safety reserves. As can be deduced from Fig. 1, the failureprobability decreases in case of sðIIÞS < sðIÞS . The stochastic variablesR and S can be put into a functional relationship, which makes itpossible to define the concept of a limit state.

G ¼ R� S ð1ÞThe area G < 0 thereby describes the failure region and G¼0

corresponds to the limit curve (surface). Since loading S andmaterialresistance R are independent and normally distributed (as assumedabove), their difference is also normally distributed and it is:

mG ¼ mR � mS; sg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2R þ s2S

q(2)

The central safety zone thereby corresponds to the distancebetween the average value mG and g¼ 0. This distance can bespecified as a multiple of the standard deviation sG, the so-calledsafety index b (see Fig. 2). This safety index is equivalent to theinverse variation coefficient of the limit state function:

b ¼ mGsG

¼ mR � mSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2R þ s2S

q (3)

The area beneath the density function for G � 0 corresponds tothe failure probability for the operating load and is thereforedenoted as the operating failure probability:

Pf ¼ PG<0 ¼ FGð0Þ ¼Z0

�N

fGðgÞdg ¼ F

��mGsG

�¼ Fð�bÞ (4)

The reliabilitynowcorresponds to the survival probabilityandcanbe calculated as the complement of the operative failure probability:

Pr ¼ 1� Pf (5)

The operating failure probability Pf and the safety index b areparameters, which are tied together by the standard normaldistribution, being hence equally adequate for reliability consider-ations. Thereby it has to be considered that the failure probabilitiesare only valid over a specific time period. Table 1 shows the changeof the safety index b and the operating failure probability Pf overa period of time. The values of b presented here are the requiredvalues needed in order to fulfill the given probability of failure for

Fig. 2. Definition of the failure probability Pf and safety index b

Table 2Classification of protection goals and event classes with limit values of b for lightwater reactors.

Protection goal/Event class 1 2 3 4 5Safety index b 4.7 4.2 2.5 1.0Primary system/Service level A B C D EIntegrity requirements/KTA 3201.2 X X X X X

E. Roos et al. / International Journal of Pressure Vessels and Piping 88 (2011) 19e25 21

the specified time period (time of operation). Based on Table 1, onecan see, that achieving safe operation (low probability of failure)over a longer period of time enforces a low value of b, which againimposes some restrictions on the involved parameter distributions/scatterings.

The parameter b is used in standards and regulations in order tostipulate limit values for failure probabilities. These limit values arealso used by KTA for a classification into event classes [6,7],whereby 5 event classes were defined (see Table 2)

3. Partial safety factors

For a statistical safety evaluation, the quantile values of the loadS and the material resistance R must be known or defined accord-ingly. Using the corresponding quantile factors Kp,S and Kp,R allowus to write

sQ ¼ mS þ Kp;S$sS;.rQ ¼ mR þ Kp;R$sR (6)

The position of the zero point g ¼ 0 (see Fig. 2), and hence thedistance to mG is determined by the variances of S and R. Anyway,the standard deviation sG is non-linearly related to the standarddeviations sS and sR. By introducing the so-called weighting factorsaS and aR, which represent the proportion of the respective variablewithin the probability of failure, one can define a linear relationshipas follows:

sG ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2R þ s2S

q¼ aS$sR þ aR$sS (7)

Table 1Required safety index b depending on the postulated operating failure probability Pfand the related period of operation.

Pf 10�1 10�2 10�3 10�4 10�5 10�6 10�7

b (1 Year) 1.28 2.32 3.09 3.72 4.27 4.75 5.20b (25 Years) e 0.76 1.97 2.81 3.48 4.06 4.56b (50 Years) e 0.21 1.67 2.55 3.21 3.83 4.41

The introduced weighting factors (see Figs. 2 and 3, Fig. 4 fora two-dimensional approach) can be calculated by using the stan-dard deviations of S and R.

aS ¼ sSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2R þ s2S

q ;aR ¼ sRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2R þ s2S

q (8)

From this it follows: a2R þ a2S ¼ 1.The choice of the operative probability of failure Pf, in fact of the

safety index b, defines the central safety zone mR � mS, for which itholds:

mR � mS ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2R þ s2S

q¼ baRsR þ baSsS (9)

Rearranging the formula above shows that the rated value sd andthe rated value rd (see equation below) will meet in a limit state,which is denoted as the design point (see e.g. Fig. 3). This approachhas a more general character and can always be used to derive anequation for the limit state.

rd ¼ mR � baRsR ¼ mS þ baSsS ¼ sd (10)

The quotients of the rated values and the related characteristicvalues provide the so-called partial safety factors corresponding tothe safety level defined by the safety index b:

gR ¼ rQrd

¼ mR þ Kp;RsRmR � baRsR

¼ 1þ Kp;RVR

1� baRVrand

gR ¼ sQsd

¼ ms þ Kp;sssms þ basss

¼ 1þ Kp;RVR

1þ baRVr

(11)

Hence it follows, that the rated values can be determined bymultiplication or division of the partial safety factors and thecharacteristic values, whereby it always holds that gR � 1 and gS �1 :

rd ¼ rQgR

; sD ¼ sQ $gS (12)

The partial safety factors are hence values, which are connectedby means of the weighting factors aS and aR with the variances ofeither distribution S and R. As they are interconnected, there areonly some particular combinations of S and R, for which a definitionis possible.

Fig. 3. Two-dimensional probability density function with design point.

Fig. 4. Projections of the two-dimensional probability density function in the standard-normal distributed state space.

E. Roos et al. / International Journal of Pressure Vessels and Piping 88 (2011) 19e2522

4. Probabilistic fracture mechanics evaluation of a feed waterline

Below, the application of the probabilistic analysis for a fracturemechanics evaluation is demonstrated on a feed water line ofa nuclear power plant. In doing so, the analysis was concentratedon the circumferential welds between the containment penetrationand the feed water nozzle of the piping. The fracture mechanicscalculations were made using the R6-method [8], [9].

4.1. Component geometry and postulated flaws

For the probabilistic integrity evaluation, a crack on the innersurface of the weld joint (20%wall thickness and 30� circumferentialangle)was postulated [10]. The flawarea chosen in the vicinity of theweld joint takes into account the grinding in these areas (induced bymanufacturing), andhence the associatedhigher variationof thewallthickness. Therefore, thewall thickness t is included in the calculationas a distributed parameter, Table 3. In doing so, it is assumed that thegrindings along the pipe circumference, as well as at the differentweld joints, are statistically independent and normally distributed.

Table 3Components and defects.

Parameter Value ormean value m

Standarddeviation s

Pipe diameter Do [mm] 406.4Nominal wall thickness t0 [mm] 12.5Wall thickness t [mm] 12.36 1.49Crack length 2c [mm] 99.85Crack depth a [mm] 2.5

The pipe diameter is supposed to be constant according to themanufacturing dimensions.

4.2. Component and material data

The segments of the feed water line consist mainly of thematerial 17MnMoV64 (WB 35, material code DIN 1.5403). The valuesprovided by a number of tests [11], were used for the determinationof the statistical distributions of some of the parameters (Table 4).The distribution of the crack initiation values Ji, [12], was taken fromthe correlation between impact energy KV and Ji, as described in[11,13]. This approach is advisable, because the experimentaldetermination of Ji is very expensive and the correlation is providingits value based on impact energy experiments, which are easier toperform.

It is known that the material 17MnMoV64 shows a significantorientation dependence concerning the impact energy. Only theLS-orientation is meaningful for the chosen defect orientation(circumferential defect with radial crack growth direction on theinner surface). Along this orientation, the basic material possessesconsiderably higher impact energy than the weld metal. In order to

Table 4Parameters of the material 17MnMoV64 at 300 �C

Parameter Value ormean value m

Standarddeviation s

Yield strength sy [MPa] 455.8 48Ultimate strength su [MPa] 660 33.9Impact energy KV�T [J] 151J-Integral Jiunitfrac [kJ/m2] 148 32.1

Fig. 5. Limit surface and determined design points.

Table 5Determined design points and PoF’s

Design point b PoF

FORM SORM

(1) 4.4310 4.6900 � 10�6 3.8657 � 10�6

(2) 4.4759 3.8040 � 10�6 3.5353 � 10�6

E. Roos et al. / International Journal of Pressure Vessels and Piping 88 (2011) 19e25 23

determine the distribution of the Ji -values, the impact energyvalues of the weld material were used.

In terms of a conservative approach, the strength characteristicsof the base material were taken, which were of a lesser magnitudecompared to the weld material. The yield strength sy was assumedto be a stochastic parameter. The ultimate tensile strength su wassupposed to be a constant value (mean value), because the ultimatestrength is only used to determine the plastic collapse limit loadLr,max in the R6-Method, and therefore it has no further impactwithin this safety analysis.

4.3. Load characteristics

The bending moments used herein were yielded by 3D-FE-analyses, covering operational and exceptional loads along thepiping as well. The calculations were done for stresses belonging toservice level D. In doing so, the maximum bending momentamounts toMb,max¼356.2 kNm. The value of P 7.9MPa bar was usedfor the internal pressure.

Fig. 6. Basins of attraction of two solutions of FORM depending on the start values.

4.4. Partial safety factors

In general, the determination of a design point (and designdistance b, as presented in the theoretical part of this paper), yieldsan acceptable approximation of the partial safety factors. Thedetermination of b was performed by using two dedicated reli-ability analysis methods, namely FORM and SORM ([14,15]), andusing the R6 failure assessment curve for building the limit statefunction like [16]. Fig. 5 shows the limit surface in the three-dimensional space (t, Ji, sy ), corresponding to the crack initiationstates. Subsequently, the minimal distance(s) from the mean valuesto this 3d-surface was determined (design points in Figs. 5 and 6,Table 5), finally resulting in the partial safety factors depicted below(Table 6).

As can be seen in Fig. 5, the computation of b yields two possiblepoints (local minima), depending on the start values (see Fig. 6)used within the iterative methods FORM and SORM, respectively[17]. For further considerations, we propose to use the pointdenoted here as (1) (see Table 5). The partial safety factors and thelocation of point (1) are presented in Table 6.

The decision to take the design point (1) instead of design point(2), see Table 6, is based on material engineering knowledge, andnot only on the value of b being smaller, as the difference in thereliability index is not so significant (and could even be a matter ofcomputational accuracy). In our case, the parameter values ofdesign point (1) are considered to be more appropriate because theresulting J-integral at crack initiation Ji is more consistent with theimpact energy KV from Table 5 (acc. to e.g. [18e20], there isa correlation between these two entities).

Besides, the fact that the design point (1) has the smaller valueof b implies a higher PoF, so this approach is also the moreconservative one.

5. Probability of failure on distinct evaluation points

In order to quantify the safety of the whole pipe system, thefailure probability was calculated for a subset of 8 circumferentialwelds (1e7 with D0 406.4 mm, 8 with D0 440 mm) along the pipe(circumferential welds with higher bending moments, [10]) byusing the input parameters mentioned above. The distributedparameters were the yield strength and the crack initiation value.Constant input values were the wall thickness, the piping diameter,the ultimate strength and the applied load. Thereby, the wallthickness was not regarded as a stochastic parameter, but the

Table 6Partial Safety Factors of the randomly distributed parameters considered.

Wall thickness Yield strength J-Integral

t [mm] sy [MPa] Ji [kJ/m2]

Mean values 12.36 455.8 148Design point (1) 5.91 415.39 134.17Partial safety factor (1) 2.09 1.10 1.10Design point (2) 11.99 455.16 4.55Partial safety factor (2) 1.03 1.00 32.53

Fig. 7. Probability of failure depending on the bending moment.

Fig. 8. Comparison of the calculated bending moments for PoF 50% and 1% with the maximum bending moments for load level D.

Table 7PoF for selected weld seam points along the pipe

Weld Bending momentservice level D [kNm]

Wallthickness [mm]

PoF

FORM SORM

1 289.6 9.8 7.07 � 10�6 7.01 � 10�6

2 312.7 9.6 9.50 � 10�6 9.35 � 10�6

3 312.8 9.3 1.15 � 10�5 1.12 � 10�5

4 334.3 9.4 1.35 � 10�5 1.31 � 10�5

5 304.5 10.3 6.41 � 10�6 6.36 � 10�6

6 297.1 11.6 4.27 � 10�6 4.25 � 10�6

7 337.7 10.7 6.87 � 10�6 6.80 � 10�6

8 356.2 14.4 3.49 � 10�6 3.48 � 10�6

E. Roos et al. / International Journal of Pressure Vessels and Piping 88 (2011) 19e2524

minimal thickness at the correspondingweld seam point was takeninstead. Fig. 7 shows the resulting failure probability curves.

Fig. 8 shows the bending moments corresponding to a proba-bility of failure of 50% and 1%, respectively. Also depicted are theassumed operational moments at the considered weld seam points,as provided by related FEM computations. The probability of failurerelated to these operational moments was calculated usingFORM and SORM, see Table 7, in the same way as described above(b selection based on expert knowledge). From Table 7, one canrecognize that the considered bending moments are very safe andprovide enough safety reserves for the investigated feed water line.

The integrity proof, assumed to entirely cover the loads andother constraints, was carried out for the failure momentMb 674 kNm (plastic limit load, PLL), Fig. 8, for the overall pipe. Asone can see, this limiting bending moment is here always slightly

E. Roos et al. / International Journal of Pressure Vessels and Piping 88 (2011) 19e25 25

below the values corresponding to the probability of failure of 50%presented before. The 50% quantile is regarded here because itcorresponds to crack initiationwithin the context of the R6-method[8]. Thus, the probabilistic analysis is in concordancewith the resultof the deterministic assessment of the pipe integrity, although thedeterministic approach is slightly more conservative. In fact, bothapproaches show that a failure of the pipe can be excluded witha significant reserve in safety. However, when doing a probabilisticsafety assessment using FORM/SORM, the relevance to materialsmechanics, as proposed in 4, has to be taken into account.

6. Conclusions

If one performs an integrity assessment proof while consideringthe possible scatter of the material parameters, the geometricaldimensions and the loads, then the probabilities of failure and therelated uncertainties become quantifiable.Moreover, postulated flawsizes can be considered and the operational effects can be evaluated.This requires an exact knowledge of the scatter of themost importantparameters, on the one hand, and of their impact on the usedprocedures, andhenceon the resultof the computations, on theother.

Within this paper, some basics of reliability theory were pre-sented and applied to a piping system. Using the example of a feedwater pipe, the probabilities of failure at some circumferential weldjoints were calculated. The partial safety factors corresponding tothe highest bending moment available were determined. Thereby,we have shown that safety reserves are present, but there must besome expert knowledge available in order to determine the correctsolution (the materials mechanics relations have to be considered).The results are also consistent with the deterministic integrityassessment procedure.

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