Procedure Statistical Assessment of Sample Test Results

Transcription

1 BAM - 3.2: Sectio Pressure Equipmet Pressure Receptacles; Fuel Gas Storage Systems 26 th September 204 Techical Aex Statistical Assessmet Samples (SAS) of the Cocept Additioal Tests (CAT): Procedure Statistical Assessmet of Sample Test Results SAS Itroductio This documet describes a procedure for the statistical assessmet of burst ad load cycle tests iteded for evaluatio withi the cocept additioal tests (CAT). The mai purpose is to cosider mea ad scatter values as well ucertaities due to a limited sample size for safety assessmets. The described statistical procedure is based o sample test results from Slow Burst Tests (SBT) accordig to CAT 5 ad Load Cycle Test (LCT) accordig to CAT 6. The procedure icludes the calculatio of mea ad scatter values. These sample properties will be compared to a dedicated miimum survival rate SR mi accordig to CAT 2. The defied miimum survival rate is valid for the whole service life util ed of life (EoL). Composite pressure receptacles at begiig or mid of life have (BoL or MoL) to provide a proper reserve due to degradatio of their stregth properties over lifetime. A safety assessmet with limited sample size geerally icludes a ucertaity regardig the true values of the tested desig type. This ucertaity strogly depeds o the umber of tested specimes ad ca be reduced by a icrease of sample size. The followig procedure cosiders the sample size as part of the safety assessmet by implemetatio of a defied cofidece level. Recommeded sample size for a safety assessmet is 5 accordig to CAT 4. The procedure covers the assessmet of samples with a sample size differet from 5 as well. A higher sample size reduces ucertaity ad therefore leads to a reductio of required sample properties for exceedig the miimum survival rate. SAS 2 Scope The procedure is suitable for all types ad sizes of composite pressure receptacles produced i large umbers. I geeral this is assumed for desig types ot exceedig a water capacity of 450 litres. Desig types which are ot assessable by sufficiet sample testig due to their limited productio umbers shall be examied ad moitored with a equivalet procedure i accordace with the competet authority. SAS 3 Statistical Assessmet of SBT ad LCT The followig chapters describe a method to quatify the survival rate of samples tested i accordace with CAT 5 (Slow Burst Test) ad/or CAT 6 (Load Cycle Test). The method is based o the calculatio of sample mea ad scatter properties (SAS 3.) ad the compariso by graphical or aalytical assessmet as described i SAS 3.2. The resultig properties of a sample are iteded to be compared with the miimum survival rate SR mi (defied i CAT 2) for the relevat test. Note: The statistical descriptio ad assessmet of test results are elaborated employig examples i [] to [6]. page of

2 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 SAS 3. Calculatio of statistical sample properties I the followig, the sample size meas the umber of specime per sample ad is amed. The test result i =, 2, 3... for each specime is give. The sample shall be evaluated for their mea ad scatter value as defied i table SAS-. It is recommeded to perform a outlier test accordig to SAS 3.4 prior to the assessmet. Extreme outliers shall be assessed cocerig their relevace for safety of the relevat populatio of pressure receptacles (compare SAS 3.4). Coclusios shall be discussed with the competet authority. Table SAS-: Calculatio of sample parameters for SBT ad LCT Calculatio of sample parameters Slow Burst Test Load Cycle Test List of symbols MSP = Maximum Service Pressure MSP is depedet o the desig type approval: Either MSP meas the test pressure PH i case of a approval for geeral service or MSP meas the developed gas pressure at maximum allowable settled temperature (65 C) i case of a desig type approval for dedicated gas service. = sample size = sample size p i = burst pressure of a idividual specime (Result of oe burst test) N i = Number of load cycles to leakage of idividual specime (Result of a hydraulic load cycle test to PH or MSP) m pb = Mea value of burst pressure m N = mea value of log- LCT s pb = Stadard deviatio of burst pres. s N = Stadard deviatio of log-lct Ω 50% = Relative mea of burst pressure N 50% = Media of load cycle results Ω s = Relative scatter of burst pressure N s = Scatter of load cycle results Calculatio of sample mea m = pb p i i= m log N = log0( N i ) i= Ω 50 % = m pb MSP N 50 % = 0 m log N Calculatio of sample scatter s pb = ( pi m pb ) i= 2 s ( log0 ( N i mlog N ) log N = ) i= 2 Ω = s s pb MSP N s = 0 s log N page 2 of

3 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 SAS 3.2 Assessmet of survival rate After havig quatified the sample properties i accordace with SAS 3., they ca be draw ito the relevat performace chart: Either i Fig. SAS- ad Fig. SAS-2 for burst test results (SBT) or i Fig. SAS-3 ad Fig. SAS-4 for cycle test results (LCT). There each sample is represeted by oe dot, idepedet from the sample size. Note: These performace charts are supplemeted by lies of costat survival rates SR. These lies permit a estimatio of the SR of the whole populatio withi the ucertaity expressed by the cofidece level. The assessmet of cofidece depeds o the sample size. The survival rate SR mi as demaded i CAT 2 is represeted by a curve. SR mi eed to be separately determied for SBT ad LCT accordig to CAT 2. The safety of a populatio deems to meet requiremets as requested i CAT if the sample dot is foud to be above the curve, relevat for SR, sample size ad cofidece level. The performace charts Fig. SAS- (SBT) ad Fig. SAS-3 (LCT) already cosiders the ucertaity due to a limited sample size of 5 specimes. If the sample dot is foud above the relevat curve for SR mi, the survival rate of the tested desig type is higher tha SR mi with a probability of at least 95%. Note: A higher sample size usually reduces the values to be demostrated for mea ad scatter, due to a reductio of statistical ucertaity. I case of a sample size differet from = 5 the sample should be checked with a adjusted performace chart (e.g. Fig. SAS-2 or Fig. SAS-4) or has to fulfil the requiremet as show i table SAS-2. Values for k BT ad k LC for differet sample size ca be foud i table SAS-3 ad table SAS-4. Note: Miimum survival rates SR mi as defied i CAT ad deduced performace charts ad tables as preseted here focuses o ed of life (EoL). Cyliders at begi or mid of life have to provide a proper reserve to cover expected degradatio durig service. Table SAS-2: Check of sufficiet survival rate Check of sample survival rate i compliace with requiremets i CAT ad a cofidece level of 95% Requiremet Slow Burst Test Ω SR SBT > SRmi is met if k 50 % BT Ω s > Requiremet m Load Cycle Test SR LCT > SRmi is met if log N klc slog N > 0 page 3 of

4 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 Table SAS-3: values for k BT for differet sample sizes k BT for cofidece level 95% SR mi = Table SAS-4: values for k LC for differet sample size k LC for cofidece level 95% SR mi = SAS 3.3 Compariso of two samples by parameter test The parameter test as defied i table SAS-5 permits a rough compariso of mea ad scatter values of two samples. The statistical sigificace of this test is aroud 95%. That meas i case of a successful test, the true values of the tested pressure receptacles are with a probability of at least 95% differet from each other. The sample size for a compariso of mea values should be at least 5 specimes for each sample. For a sufficiet compariso of scatter values, a sample size of at least 0 specimes for each sample is recommeded. The parameter test ca be used for the compariso of e. g. two productio batch samples accordig to CAT 0. I this case, sample cotais the results out of the iitial batch ad sample 2 is a follow up sample represetig e.g. the aual productio batch test results. page 4 of

5 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 Table SAS-5: Similarity Test Similarity test of two samples Slow Burst Test Load Cycle Test Mea value of sample 2 is sigificatly lower tha sample if: Ω 50%, > Ω 50%,2 + T m Ω s 2 + Ω 2 s2 2 m logn, > m logn,2 + T m s 2 logn, + slogn,2 2 2 with: T m =.8 if T m =.7 if + 2 > 0 T m =.6 if Scatter of sample 2 is sigificatly higher tha sample if: T s < Ω s2 / Ω s T s < s N2 / s N with: T s = 2.5 if T s = 2.0 if + 2 > 0 T s =.8 if T s =.5 if SAS 3.4 Outlier test for extreme values The followig test procedure i accordace with GRUBBS (table SAS-6) is able to detect sigle outliers withi a sample of burst or load cycle test results. The statistical sigificace of this test is aroud 95%. That meas i case of a successful test the detected value is a outlier with a probability of at least 95%. The applicatio of the test requires to search the maximum or miimum value of the sample ad to calculate the absolute differeces betwee this value ad the sample mea. The result is divided by the stadard deviatio of the sample. If the result is higher tha the critical value G, the correspodig value ca be cosidered a outlier. The critical values are show i the followig Fig. SAS-0. I case of suspicio of sigificat deviatios which caot be detected by this method a graphical assessmet of sample results should be applied (e.g. ru chart, box plot, ormal probability plot or histogram). page 5 of

6 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 Table SAS-6: Outlier test accordig to GRUBBS Outlier Test for maximum ad miimum values Slow Burst Test Miimum burst pressure is a outlier if: m pb mi(p i ) > G s pb Load Cycle Test Miimum load cycle result is a outlier if: m logn log 0 mi(n i ) > G s logn Maximum burst pressure is a outlier if: max(p i ) m pb s pb > G Maximum load cycle result is a outlier if: log 0 max(n i ) m logn s logn > G Grubbs' outlier test for extreme values critical value, G [-] sample size Fig. SAS-0: Critical values for Grubbs outlier test for a statistical sigificace of 95% page 6 of

7 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 Literature [] Mair, G. W.; Hoffma, M.; Schöfelder, T.: The Slow Burst Test as a Method for Probabilistic Quatificatio of Cylider Degradatio proceedigs of the Iteratioal Coferece o Hydroge Safety ICHS 203, Brussels: Coferece Proceedigs 203; ID 02; [2] Mair, G. W., Hoffma, M., Scherer, F., Schoppa, A., Szczepaiak, M.: SLOW BURST TESTING OF SAMPLES AS A METHOD FOR QUANTIFICATION OF COM- POSITE CYLINDER DEGRADATION Elsevier: Iteratioal Joural of Hydroge Eergy; ad Vol. 39 (204), Issue NN, p. NN ISSN [3] Mair, G. W.; Hoffma, M.: Statistic Evaluatio of Sample Test Results to Determie Residual Stregth of Composite Gas Cyliders; MP Materials Testig, Volume 55 (203) No 0, p ; ISSN [4] Becker, B.; Mair G. W.: Eifluss der Stichprobegröße auf die Bewertug der Überlebeswahrscheilichkeit vo Druckgefäße aus Verbudwerkstoffe; Techische Sicherheit (TS) Bd. 4 (204), Nr. 7/8 (Juli/Aug.), S. NN - NN. [5] Mair, G. W.; Scherer, F.: Statistische Bewertug vo Prüfergebisse zur Restfestigkeitsbetrachtug vo Composite-Druckgefäße Techische Sicherheit (TS) Bd. 3 (203), Nr. 7/8 (Juli/Aug.), S ISSN page 7 of

8 Techical Aex Statistical Sample Assessmet from 26 th September 204 Fig. SAS-: Performace chart for a graphical assessmet of results from sample testig by SBT, sample size = 5 page 8 of 7

9 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September Slow Burst Test: Sample Evaluatio by Normal-Distributio survival rate SR = % ; cofidece level 95% sample size = Mea value of burst pressure related to MSP: Ω 50% [-] desig type idetificatio: B. Becker Stadard deviatio of burst pressure related to MSP: Ω s [-] Fig. SAS-2: Performace chart for a graphical assessmet of results from sample testig by SBT, survival rate SR = % page 9 of

10 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 Fig. SAS-3: Performace chart for a graphical assessmet of results from sample testig by LCT, sample size = 5 page 0 of

11 Techical Aex SAS Statistical Assessmet of Sample Test Results from 26 th September 204 Load Cycle Test: Sample Evaluatio by WEIBULL-Distributio survival rate SR = % ; cofidece level 95% 00,000 sample size = Media of residual hydraulic LCs: N50% [-] 0,000 desig type idetificatio: 30 B. Becker 204, Scatter of hydraulic LCs: N s [-] Fig. SAS-4: Performace chart for a graphical assessmet of results from sample testig by LCT, survival rate SR = % page of