The Determination of Uncertainties in Creep Testing to European Standard pren 10291

Transcription

1 UNCERT COP 1: Manal of Codes of Practice for the Determination of Uncertainties in Mechanical Tests on Metallic Materials Code of Practice No. 1 The Determination of Uncertainties in Creep Testing to Eropean Standard pren 191 D J Foster Mitsi Babcock Technology Centre High Street, Renfrew PA4 8UW Scotland UNITED KINGDOM Isse 1 September

2 UNCERT COP 1: CONTENTS 1. SCOPE. SYMBOLS 3. INTRODUCTION 4. PROCEDURE FOR THE ESTIMATION OF UNCERTAINTY IN CREEP TESTS Step 1 - Identification of the parameters for which ncertainty is to be estimated Step - Identification of all sorces of ncertainty in the test Step 3 - Classification of sorces of ncertainty according to Type A or B Step 4 - Estimation of sensitivity coefficient and ncertainty for each sorce of ncertainty Step 5 - Comptation of the measrands combined ncertainties c Step 6 - Comptation of the expanded ncertainty U e Step 7 - Presentation of reslts 5 REFERENCES APPENDIX A1: Calclation of ncertainties in measred qantities APPENDIX A: Formlae for the calclation of measrands combined ncertainties APPENDIX B: A worked example for calclating ncertainties in creep tests APPENDIX C: Vales of stress exponent n, and (activation energy Q) / R APPENDIX D: Derivation of some formlae for ncertainties

3 UNCERT COP 1: CODE of PRACTICE FOR THE DETERMINATION OF UNCERTAINTIES IN CREEP TESTING TO ASTM E SCOPE This procedre covers the evalation of ncertainties in the determination of the reslts of creep tests carried ot according to the following testing []: Eropean Standard pren 191: Metallic Materials - Uniaxial Creep Testing In Tension - Method of Test (Jan. 1998) This test method is also covered by ASTM E [3], which has more detailed instrctions and discssion of procedre and calclation. Recommendations of Eropean Creep Collaborative Committee Working Grop 1 [4] have also been incorporated. This Code of Practice is aimed at determination of ncertainties in the reslts of a single creep test. The effect of repeating a test nder nominally the same conditions, or at a series of stresses or temperatres, is discssed in [1] Section 4. Repeatability is also discssed in [5].. SYMBOLS a, b initial width & thickness of rectanglar specimen a, b minimm width & thickness of rptred rectanglar specimen d initial diameter of parallel length d minimm diameter after rptre e extensometer elongation reading ε strain å& strain rate L, L initial and after-rptre gage lengths L c initial parallel length L e, L e initial and after-rptre extensometer lengths L e dring-test extensometer elongation L r initial reference length S, S initial, and minimm after-rptre, cross-sectional areas s machining tolerance for specimen diameter or width Loads and Stresses P load on test piece initial stress on test piece σ Page 1 of 35

4 UNCERT COP 1: Test Reslts A percent elongation after rptre å& min minimm strain rate in secondary creep t rptre life of specimen t fx time to x% creep strain percent redction of area after rptre Z Other Parameters A constant in strain rate vs. stress / temperatre eqation n stress exponent for creep Q creep activation energy R gas constant t time when rptre first recorded T test temperatre 3. INTRODUCTION There are reqirements for test laboratories to evalate and report the ncertainty associated with their test reslts. Sch reqirements may be demanded by a cstomer who wishes to know the bonds within which the reported reslt may be reasonably assmed to lie; or the laboratory itself may wish to nderstand which aspects of the test procedre have the greatest effect on reslts so that this may be monitored more closely or improved. This Code of Practice has been prepared within UNCERT, a project fnded by the Eropean Commission s Standards, Measrement and Testing programme nder reference SMT4 - CT97-165, in order to simplify the way in which ncertainties are evalated. It is hoped to avoid ambigity and provide a common format readily nderstandable by cstomers, test laboratories and accreditation athorities. This Code of Practice is one of seventeen prepared and tested by the UNCERT consortim for the estimation of ncertainties in mechanical tests on metallic materials. These are presented in [1] in the following Sections: 1. Introdction to the evalation of ncertainty. Glossary of definitions and symbols 3. Typical sorces of ncertainty in materials testing 4. Gidelines for the estimation of ncertainty for a test series 5. Gidelines for reporting ncertainty 6. Individal Codes of Practice (of which this is one) for the estimation of ncertainties in mechanical tests on metallic materials This CoP can be sed as a stand-alone docment. Nevertheless, for backgrond information on measrement ncertainty and vales of ncertainties of devices sed commonly in Page of 35

5 UNCERT COP 1: material testing, the ser may need to refer to the relevant section in [1]. Several sorces of ncertainty, sch as the reported tolerance of load cells, extensometers, micrometers and thermocoples are common to several mechanical tests and are inclded in Section of [1]. These are not discssed here to avoid needless repetition. The individal procedres are kept as straightforward as possible by following the same strctre: The main procedre Fndamental aspects for that test type A worked example This docment gides the ser throgh several steps to be carried ot in order to estimate the ncertainties in creep test reslts. The general process for calclating ncertainty vales is described in [1]. 4. PROCEDURE FOR ESTIMATING UNCERTAINTIES IN CREEP TESTS Step 1 - Identification of the Measrands for Which Uncertainty is to be Determined The first Step consists of setting the measrands, i.e. the qantities which are to be presented as reslts of the test. According to reqirements, Table 1 lists the measrands and the intermediate qantities sed in their derivation. Not all the qantities will be reqired in a particlar test. For example strains dring a stress-rptre test might not be measred. Extensive investigations may be ndertaken in addition to these reslts, to determine activation energy, stress exponent, and parameters in modelling strain, strain rate and time and dctility at rptre, as fnctions of stress, temperatre, composition, heat-treatment etc. Evalation of ncertainties of these parameters is beyond the scope of the present procedre. Table 1. Intermediaste reslts and measrands, their nits, and symbols within pren 191 Qantity Unit Symbol Intermediate reslts initial stress MPa σ original cross-sectional area of parallel section mm S minimm cross-sectional area after creep rptre mm S Test measrands percentage redction of area after creep rptre Z percentage elongation after creep rptre A creep rptre time h t creep elongation time (time to x% creep strain) h t fx minimm strain rate in secondary creep h -1 å& min Page 3 of 35

6 UNCERT COP 1: The qantities in Table 1 are calclated from the following formlae Initial Cross-Sectional Area S, and Stress σ For cylindrical specimens and for rectanglar In both cases S = π d / 4 S = a * b σ = P / S (1a) (1b) (1c) 4.1. Final Cross-Sectional Area S, and Percentage Redction of Area Z For a cylindrical specimen with minimm diameter d, S is given by the formla corresponding to Eq. (1a). In the case of rectanglar specimens, a method may have to be devised for a fractre srface, which is not of an easily measrable shape. According to ASTM E139 paragraph 1.4, redction of area is only reported for rond specimens. Z = 1(S - S ) / S = 1(1 - (S /S )) () Percentage Elongation After Creep Rptre, A When initial and after-rptre lengths are measred between gage length marks on the parallel section: A = 1 (L - L ) / L = 1 (L / L - 1) (3a) In some cases the specimen has raised ridges (pren 191 Fig. type c or e), or V-notches in enlarged ends (type d) for extensometer attachment. These can be sed for rptre elongation measrement. The reference length is the parallel length pls a calclated addition for the radised sholders (see A1.3), and the elongation is the increase of length between the attachment points. Then: A = 1 (L e - L e ) / L r = 1 L e / L r (3b) Creep Rptre Time, t Page 4 of 35

7 UNCERT COP 1: If specimen load or displacement is logged at intervals of time t, a rptre recorded at time t is eqally likely to have occrred at any time within the interval (t - t, t ), with the expectation t = t - t/ (4) This calclation will not arise if the specimen load or extension is recorded continosly Creep Elongation Time, t fx t fx represents the time to x% creep strain, e.g. t f. for.% strain. Provided the data sampling freqency is sfficiently high, extensometer readings near the reqired elongation will be obtained. Becase of extensometer ncertainty, these will not lie exactly on a straight line, bt linear regression analysis then gives the estimated time for the reqired strain, its error, and the strain rate. See Appendix B for an illstration of this method. ASTM E139 paragraph 1.6. reqires strain measrements at.1h intervals p to 1h, then at 1h intervals, if times at given percentage strains are to be taken from a single reading. Higher freqency of measrement may be needed in the initial stages. For low strain rate tests, the target elongation may be shown at several times. The mean of these gives the reqired reslt Minimm Strain Rate in Secondary Creep, &ε min Recorded strain/time data will sally not lie on a smooth crve, and derived strain rate over each recording interval will be erratic. Strain rate has to be averaged over a period in which it appears to be constant, and this can be carried ot manally from the strain/time or strainrate/time crves. å& min cold also be determined by compting a best-fit crve for the strain/time data. Step - Identification of all Sorces of Uncertainty In the second Step, the ser identifies all possible sorces of ncertainty which may have an effect on the test. This list cannot be exhastively identified beforehand as it is niqely linked to the laboratory s test method and the apparats sed. Therefore a new list shold be drafted each time one test parameter changes (when a plotter is replaced by a compter and printer Page 5 of 35

8 UNCERT COP 1: for example). To help the ser list all sorces of ncertainty, five categories have been defined. The following table (Table ) gives the five categories and examples of sorces. It is important to note that this table is NOT exhastive. Other sorces can contribte to ncertainties depending on specific testing configrations. Users are encoraged to draft their own list corresponding to their own test facilities. Table. Sorces of ncertainty and their likely contribtion to ncertainties in measrands (1 = major contribtion, = minor contribtion, blank = insignificant or no contribtion) Sorce Affected Measrand S S s Z A t t fx å& min Test Piece specimen initial dimensions minimm diameter after rptre 1 1 extmtr./gage length after rptre 1 Apparats load cell or lever / weights specimen temperatre bending stresses extensometer 1 Environment control of ambient temperatre Method data logger time interval estimation of creep strain time estimation of å& min (graphical/statistical) reference length sholder integral Operator Step 3 - Classification of all Sorces According to Type A or B In accordance with ISO TAG 4 'Gide to the Expression of Uncertainties in Measrement' [6], sorces of ncertainty can be classified as type A or B, depending on the way their inflence is qantified. If a sorce's inflence is evalated by statistical means (from a nmber of repeated observations), it is classified type A. If a sorce's inflence is evalated by any other mean (manfactrer's docments, certification,...), it is classified type B. Attention shold be drawn to the fact that one same sorce can be classified as type A or B depending on the way it is estimated. For instance, if the diameter of a cylindrical specimen is Page 6 of 35

9 UNCERT COP 1: measred once, or taken from the machining specification, that parameter is considered type B. If the mean vale of ten consective measrements is taken, then the parameter is type A. Step 4. Estimation of Sensitivity Coefficient and Standard Uncertainty for each Sorce of Uncertainty In this step the ncertainty of the measrand, i, for each inpt sorce x i, is calclated according its inpt qantity vale, the probability distribtion and the sensitivity coefficient, c i. Appendix A1 describes the derivation of the ncertainties for the primary measred qantities and sorces of ncertainty in a creep test, as listed in Tables 1 and. The ncertainty ( (x i ) ) of the inpt qantity x i is defined as one deviation and is derived from the ncertainty of the inpt qantity. For a Type A ncertainty, the ncertainty vale is not modified. For Type B, it is divided by a nmber, d v, associated with the assmed probability distribtion of the parameter within its ncertainty range. The divisors for the distribtions most likely to be encontered are given in Chapter of [1]. The contribtion ( i ) of the individal inpt qantity s ncertainty ( (x i ) ) to the ncertainty of the measrand is fond by mltiplying (x i ) by the sensitivity coefficient c i. This is derived from the relationship between otpt (measrand, y) and inpt qantities(x i ), and is eqal to the partial derivative, i.e. if y = F(x 1, x,...) where F denotes some fnction then c i = y / x i and i = c i (x i ) Step 5: Calclation of Combined Uncertainties of Measrands Assming that the N individal ncertainty sorces are ncorrelated, the measrand's combined ncertainty, c (y), can be compted in a root sm sqares manner: ( y) = [ c. ( x )] c N i= 1 i i This ncertainty corresponds to pls or mins one deviation on the normal distribtion law representing the stdied qantity. It will be seen that, in some cases, the ncertainty of an inpt qantity is itself a combined ncertainty from an earlier stage in the calclations. For example, ncertainty of rptre time (t ) depends on (inter alia) the ncertainty in stress (σ ). This in trn depends on load and Page 7 of 35

10 UNCERT COP 1: cross-sectional area, and the latter again on initial diameter, which is one of the primary measred qantities. In the detailed procedre for calclating combined ncertainties (in Appendix A) and the illstrative worked example (Appendix B), these sccessive ncertainties are calclated separately in seqence. Step 6: Comptation of the Expanded Uncertainty U e This Step is optional and depends on the reqirements of the cstomer. The expanded ncertainty U e is broader than the combined ncertainty U c, bt in retrn, the confidence level increases. The combined ncertainty U c has a confidence level of 68.7%. Where a high confidence level is needed (aerospace and the electronics indstries), the combined ncertainty U c is broadened by a coverage factor k to obtain the expanded ncertainty U e. The most common vale for k is, which gives a confidence level of approximately 95%. If U c is tripled the corresponding confidence level rises to 99.73%. Standard worksheets can be sed for calclation of ncertainties, and an example is shown in Table 3 below. In creep tests, the sensitivity coefficients giving the ncertainty in a measrand de to the ncertainties in the primary measred qantities (diameter, temperatre, load etc.) are rather complicated, and there has to be a series of intermediate steps, as illstrated in the worked example in Appendix B. The formla for calclating the combined ncertainty in Table 3 is given in Appendix A Eq. (13c). Table 3. Typical worksheet for ncertainty bdget calclations in estimating the ncertainty in percentage elongation A sorce of ncertainty symbol vale ncer- prob. divisor c i i (type) tainty distribn parallel length (B) L c a rect. 3 1 =a/ 3 end correction v not inclded reference length L r +v =a/ 3 initial extensometer length (A) L e x b normal 1 1 b final extensometer length (A) L e y c normal 1 1 c elongation L e y-x = (b +c ) percentage elongation A L e / L r combined ncertainty c normal expanded ncertainty U e normal Eq.(13c) Page 8 of 35

11 UNCERT COP 1: Step 7: Presentation of reslts Once the expanded ncertainty has been chosen, the final reslt can be given in the following format: V = y ± U with a confidence level of X% where V is the estimated vale of the measrand, y is the test (or measrement) mean reslt, and U is the expanded ncertainty. The reslts wold therefore be presented in the form shown below. These are the basic test reslts, and the Standards list other information reqired in the test report. The reslts of a creep test condcted according to ASTM E on sample XYZ13, with a confidence interval of 95% are: Test Conditions: Initial stress σ Temperatre T Test Reslts: Rptre time t + U e (t ) Time to x% strain t fx + U e (t fx ) Minimm strain rate å& min + U e (å& min ) Percentage elongation A + U e (A ) Percentage redction of area Z + U e (Z ) 5. REFERENCES (1) Manal of Codes of Practice for the determination of ncertainties in mechanical tests on metallic materials. Project UNCERT, EU Contract SMT4-CT97-165, Standards Measrement & Testing Programme, ISBN , Isse 1, September. () Eropean Committee for Standardisation: Eropean Standard pren 191: Metallic Materials - Uniaxial Creep Testing In Tension - Method of Test (Jan. 1998). (3) American Society for Testing and Materials: Designation E193-95: Standard Practice for Condcting Creep, Creep Rptre and Stress Rptre Tests of Metallic Materials. (4) Brite Eram BE554: Eropean Creep Collaborative Committee Working Grop 1 Recommendations. (5) DG Robertson Development and Validation of a Code of Practice for a Low Cost Method of Strain Measrement from Interrpted Creep Testing, Final Report, ERA Report 99-84, Febrary 1999, ERA Project Eropean Commission Standards Measrements and Testing Programme, CEC No. MAT1-CT9465. (6) BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML: Gide to the Expression of Uncertainty in Measrement. (ISO 1993) (Known as the TAG4 Gide ). (7) American Society for Metals: Atlas of Creep and Stress-Rptre Crves (ASM International1988). Page 9 of 35

12 UNCERT COP 1: A1.1 Introdction APPENDIX A1 Calclation of Uncertainties in Measred Qantities The reason for classifying sorces as type A or B is that each type has its own method of qantification. By definition, a type A sorce of ncertainty is already a prodct of statistical comptation. The calclated inflence is ths left as is. Type B sorces of ncertainty can have varios origins: a manfactrer's indication, a certification, an expert's estimation or any other mean of evalation. For type B sorces, it is necessary for the ser to estimate for each sorce the most appropriate (most probable) distribtion (frther details are given in Section of [1]). According to the distribtion model chosen, a correction factor is reqired in order to compte the ncertainty s for each type B sorce. TYPE A : Statistically compted inflence A Type A ncertainty is most often compted from a set of N repeated measrements of the reqired qantity x. This then gives a mean vale µ for x, and Type A ncertainty estimate: ncertainty s (x) = ( deviation) / N, = 1 N i= 1 ( x ) N µ i N 1 The deviation is denoted by STDEV in Microsoft Excel, and often by σ n-1 on hand calclators. (Note the se of σ here for deviation, and elsewhere for stress). A Type A ncertainty can also be calclated when the vale of a dependent variable y is measred at a series of vales of an independent variable x, and an estimate of Y, the vale of y when x = X, is derived by regression analysis. In the worked example in Appendix B, the time at 1% strain and its ncertainty are calclated from a series of strain/time data points. TYPE B : Standard ncertainty s made of the given ncertainty, divided by a factor d v given in Section of [1] : s = / d v The most common distribtion model for Type B ncertainties is the rectanglar, which means that the tre vale of a measred qantity is eqally likely to have any vale within the range (µ + ), where µ is the mean vale. The probability of the vale lying between x and (x+δx) is δx/(), i.e. independent of x, and from the definition of Page 1 of 35

13 UNCERT COP 1: deviation for a continos variable, it can be shown that: s = / 3 i.e. in this case, d v = 3. Trning now to the Sorces of Uncertainty listed in Table (page 7), we demonstrate how the typical ncertainties in individal measred qantities, differences in apparats etc. lead to each sorce s inflence. It is assmed that measring eqipment has been appropriately calibrated, and that a test procedre has been written following the which minimises typical measrement errors, and that the procedre is being applied by a trained operator Important Note: Some Type B errors have been calclated in the following discssion and the worked example in Appendix B. It will be self-evident to the capable laboratory how Type A errors on individal measrements can lead to smaller ncertainties for an individal test piece, or how repeat measrements on several test pieces can lead to the calclation of Type A errors on the evalated qantities directly (see also Section 4 of [1]). A1. Uncertainties in Test Specimen Initial Diameter / Width & Thickness The specimen diameter d, or width and thickness a, b, and their ncertainties, may be taken from the specimen drawing if the specimen has been checked and certified to be within the machining tolerance of ± s. For example, in pren 19, Table.1 specifies a maximm difference between any two transverse dimensions, i.e. s, of.3mm for specimens 6 to 1 mm in diameter. The ncertainty is then Type B with rectanglar distribtion, i.e. ncertainty, (d ), (a ) or (b ), is eqal to s / 3. Alternatively, specimen dimensions can be measred at different positions with a micrometer or calliper and a Type A estimate of ncertainty 1 (x) made as shown on the previos page (x denotes d, a or b ). The diameter vales mst also comply with the reqirements of the Standard for the difference between maximm and minmm vale. The measring instrment will also have its own ncertainty m, available from the manfactrer s specification or periodic calibrations. If this is given as a maximm error, a rectanglar distribtion may be assmed, giving a Type B ncertainty: sm = m / 3 This is combined with the ncertainty of the set of measrements ( 1 (x) ) by qadratic smmation: (x) = [ ( 1 (x)) + ( sm ) ] (5) Page 11 of 35

14 UNCERT COP 1: Dimensional ncertainties affect the ncertainty of initial stress throgh Eqs. (1a), (1b) and (1c), and redction of area throgh Eq. (). Uncertainty in stress also contribtes to the ncertainties in rptre time, time to given creep strain, and strain rate. A1.3 Uncertainties in Measring Initial Gage, Extensometer and Reference Lengths The length may be between gage length marks, as on a stress-rptre specimen. The ncertainty for diameter can be taken from the machining specification tolerance (Type B), or measred several times (Type A) and combined with the ncertainty of the calliper. When an extensometer length between raised ridges or machined grooves is sed as the basis for elongation calclation, the reference length is the sm of parallel length and a correction for the radised sholders of the form: L s d = d( x) n dx where n = stress exponent for creep rate, d(x) = diameter at position x, and the integral is taken for x over the radised length. When n is nknown, it is assmed to be 5 ([] pp., 3). Sholder strain corrections are also discssed in ASTM E [3]. The ends of the parallel section may be difficlt to locate for measrement prposes, and parallel length is best taken from the machining specification. Its ncertainty is Type B, with vale = tolerance/ 3, assming rectanglar distribtion within the permitted range. The ncertainty of the end correction is difficlt to assess, particlarly if an assmed vale for stress exponent is sed. However, calclations with typical specimen sizes have indicated that a +1 variation in n gives +. mm change in the integral, or.5% of a 5mm parallel length. The worked example has a 4.7% ncertainty in the extension to rptre ( L e ). This is therefore the main factor in ncertainty of percentage elongation. Even thogh the validity of the expression integrated is also open to qestion, it is sggested that the end correction ncertainty contribtion be considered negligible. Several measrements shold be made, at different circmferential positions (Type A) for an extensometer length between raised ridges or machined grooves. Alternatively, a type B estimate can be made from the machining specification. Page 1 of 35

15 UNCERT COP 1: A1.4 Uncertainty in Measring Load (Type B) This is a Type B estimate derived from the calibration certificate of the load cell or lever system and weights. A rectanglar distribtion is assmed again, so the ncertainty (P) is s / 3, where ±s is the certified maximm error. If the error is given as a percentage of the applied load, it is a relative ncertainty. For a load cell, it may be presented as a percentage of the maximm load which is an absolte ncertainty in the applied load. Load ncertainty affects ncertainties in stress (Eq. (1)) and times to rptre or given creep strain and minimm strain rate (Appendix D). A1.5 Uncertainty in Measring After-Rptre Minimm Cross-section (Type A) The ncertainty in after-rptre minimm diameter of cylindrical specimens is type A, eqal to ( deviation of n measrements)/ n (n ideally abot 1). For each measrement, the specimen shold be separated and re-assembled, and measred on a different diameter. This procedre may be jdged to be too time-consming, and also will case damage to fractre srfaces, which might be needed for scanning electron microscopy. Monting the two pieces in rotating centres will speed p the operation. One or two measrements will not allow an estimate of ncertainty in redction of area. If the fractre is not circlar, several measrements of minimm diameter mst be made (ASTM E ). The error of the measrements of d, if obtained, is combined with the error of the measring instrment, for initial diameter as shown in Eq. (5). The calclation of area ncertainty from diameter ncertainty is shown below in A.3. This ncertainty affects redction of area throgh Eq. (). After-rptre minimm cross-sectional area is not normally reqired on a rectanglar specimen. A1.6 Uncertainties in Measring After-Rptre Gage and Extensometer Lengths (Type A) The gage length between marks on the parallel section, or extensometer length between raised ridges, and its ncertainty, are measred on the re-assembled rptred specimen. As for minimm diameter, it is preferable to separate and re-assemble the specimen for each measrement, to give independent estimates. Page 13 of 35

16 UNCERT COP 1: The ncertainty of the after-rptre length based on n measrements is ( deviation)/ n. It is a major contribtor to ncertainty in percentage elongation (Eqs. 3a, 3b). A single measrement of after-rptre length will not allow calclation of percentage elongation ncertainty. A1.7 Uncertainty in Measring Rptre Time (Type B) If specimen load or displacement are logged at intervals of time t, a rptre recorded at time t is eqally likely to have occrred at any time within the interval (t - t, t ), with expectation t = t - t/ and the initial determination of ncertainty of t is m (t ) = t / ( 3) (6) i.e. type B. The ncertainty in t is assmed to be negligible. The sbscript m in m (t ), and in sbseqent eqations below, denotes that the expression is an initial estimate based on ncertainties in the data sed in the direct measrement. The inflences of ncertainties in load and temperatre mst then be added, as shown in Appendix A. A1.8 Uncertainty in Measring Creep Elongation Time (Type A or B) As noted in Section 4 Step 1.5, three sitations are possible, depending on strain rate and data recording freqency. (a) interpolation between data points above and below the target elongation (b) the target elongation may be recorded at more than one time with case (b) tending to occr in long-term tests with very low strain rates (c) a single reading is taken where strain is nearest the target vale, as per ASTM E In case (a), a nmber of data points will normally cover a period of constant elongation rate and linear regression can be sed to determine the time for the reqired elongation, the error of the estimate, and the strain rate. In case (b), sppose the elongation is indistingishable from the target vale over a time period t 1. The tre elongation time is more likely to be near the centre of the ncertainty band than at an edge. Page 14 of 35

17 UNCERT COP 1: If a normal distribtion is assmed, with the range t 1 covering deviations, the initial ncertainty is given by: m (t fx ) = t 1 / (7) The vale chosen for the divisor will not be important, since the ncertainty in case (a) or (b) is normally mch less than that de to temperatre and stress ncertainties. In case (c), the ncertainty is Type B, with vale eqal to half the interval between strain measrements. A1.9 Uncertainty in Temperatre (Type B) Temperatre ncertainty (T) will normally be type B, combining maximm errors of e m (T) in the measrement thermocople e (T) for along-specimen niformity e c (T) for the measring system by the sal qadratic smmation method. The ncertainty of temperatre is then (T) = [( e m (T) + e (T) + e c (T) ) ] / 3 (8) Temperatre affects strain rate, rptre and creep times in a non-linear manner, and the derivation of reslting ncertainties in t is given in Appendix D. A1.1 Uncertainty in Extensometer Reading of Elongation (Type B) The extensometer shold be calibrated, and certified to have a maximm readot error over its working range. If this is +s mm, the ncertainty (e) of the extensometer reading is s/ 3, assming a rectanglar distribtion. Depending on the system employed, the effect of long-term amplifier drift may have to be inclded in extensometer ncertainty. A1.11 Effect of Environment Variation of ambient temperatre may affect specimen temperatre control and extensometer amplifiers. The extent of these shold be determined, and added to the relevant ncertainty if significant. Page 15 of 35

18 UNCERT COP 1: Close control of laboratory temperatre is essential for reslts of low ncertainty. A1.1 Effect of Bending Stresses Non-coaxiality of gage length, threads and straining rods cases bending stresses. Laboratories shold examine this factor in ncertainty of their own reslts. For each test machine, bending stresses can be measred on a set of strain gages arond the circmference of a specimen. The effect of bending stresses will be related to the inherent dctility of the material at that stress, temperatre and time (the latter bringing microstrctral changes into the eqation). Therefore predicting their effect pon rptre time is extremely difficlt, as the dctility (and hence capacity to accommodate bending) will change with time and temperatre. ECCC WG1 Isse (Vol. 3 Annex 1 ch..6) [4] sggests that bending stresses be less than. σ. The WG recommend that their inflence be investigated, bt assme that within this limit, rptre time and dctility will not be affected. ASTM E139 [3] reqires that strain gage measrements be made at room temperatre, and bending strain shold not exceed 1% of axial strain. A lower limit may be set for brittle materials. For the prposes of this Code of Practice, bending stresses shold be monitored and accrate axiality maintained, bt no method is proposed to calclate the effect of bending on ncertainty of reslts. A1.13 Effect of Method The inflence of data logging interval on rptre time ncertainty has been considered above. According to ASTM E , the time interval between data records shold be not more than 1% of the anticipated test dration, or 1 hor, whichever is longer. Minimm strain rate may be estimated by laying a straightedge along a strain/time plot, and it will be possible to obtain an ncertainty from repeated measrements. Crve fitting, or regression analysis if linear, can give a error for the minimm slope. Page 16 of 35

19 UNCERT COP 1: APPENDIX A Formlae for the Calclation of the Measrands Combined Uncertainties A worked example is presented in Appendix B to illstrate the following operations. A.1 Calclation of Uncertainty in Initial Cross-Sectional Area For a cylindrical specimen relative ncertainty: ( S ) ( d ) S absolte ncertainty: ( S ) and for a rectanglar specimen = (9a) d ( d ) ðd = (9b) ( ) ( a ) ( b ) + or ( S ) b (a ) a (b ) S S a b + = (9c) A. Calclation of Uncertainty in Stress From Eq. (1c), this is the reslt of ncertainties in load and cross-sectional area: ( σ ) ( P) ( S ) c σ = + P (1) S Relative area ncertainty (S )/S, is derived from Eq. (9a) or (9c). A.3 Calclation of Uncertainty in Final Minimm Cross-Sectional Area For a cylindrical specimen, the formla corresponds to Eqs. (9a, b) above: ( ) ( d ) = or ( S ) S S d ( d ) ðd = (11) A.4 Calclation of Uncertainty in Redction of Area The treatment below considers only ncertainties in specimen measrements. In general, increasing temperatre or stress gives shorter rptre time and higher dctility, and in principle, Page 17 of 35

20 UNCERT COP 1: an ncertainty component for redction of area (or elongation) de to ncertainties in stress and temperatre cold be calclated. However, the effect coefficients Z / σ etc. are not generally known or theoretically modelled, bt are believed to be relatively small. This contrasts with creep rates and rptre times, which have a high stress exponent and exponential temperatre dependence. Since Eq. () is not a simple seqence of terms added or mltiplied together the formla for ncertainty is a little more complex (see Appendix D): ( Z ) S ( S ) ( S ) 1 = + S S c (1) The cross-sectional area ncertainties are given in Eqs. (9) and (11). If the information is available, the effects of ncertainties in stress and temperatre can be added as terms ( σ (Z )) and ( T (Z )). A.5 Calclation of Uncertainty in Percentage Elongation Uncertainties of the varios lengths are covered in Appendix A1.3 and A1.6. Based on L and L from Gage Length Marks The expression for ncertainty in elongation is eqivalent to Eq. (1): ( A ) L ( L ) ( L ) 1 = + L (13a) L c Based on Extensometer Length and Reference Length The elongation and its ncertainty are L e = L e - L e [( L e )] = [(L e )] + [(L e )] (13b) Percentage elongation and its ncertainty are then given by A = 1 L e / L r c ( A ) ( L ) ( L ) A e r = + L L e r (13c) Page 18 of 35

21 UNCERT COP 1: The remarks in A..4 on inclding effects of ncertainties in stress and temperatre on ncertainty of redction of area also apply here. A.6 Calclation of Uncertainty in Rptre Time The ncertainty m (t ) de to the length of the data logging period t, if applicable, has been given by Eq. (6). m (t ) = t / ( 3) The rptre time t is related to stress σ and absolte temperatre T (see Appendix D), and the additional components of ncertainty in t are then given by: s T ( t ) n ( σ ) t = σ ( t ) Q ( T) t for stress (14a) = for temperatre (14b) RT The vales of n and Q can be obtained from a series of rptre tests, or taken from existing data on a metallrgically similar material. Appendix C lists typical vales for for classes of alloys, and these may be sed if no other data are available. n and Q can be determined from a series of rptre tests at different stresses and temperatres, and performing linear regression analysis of log(t ) against log(σ net ), and log(t ) against (1/T). The sensitivity coefficients for stress and temperatre can be fond by performing for extra tests, two at temperatre T and stresses above and below σ, and two at σ and temperatres above and below T. The varied parameter shold differ from the central vale by at least five times its ncertainty. The ratio of the change in rptre time to change of parameter (stress or temperatre) gives the sensitivity coefficient c() for the parameter, and s (t ) = c(σ ) * (σ ) T (t ) = c(t) * (T) (14c) (14d) The combined ncertainty in rptre life is then the combination of the three components: c (t ) = { [ m (t )] ² + [ s (t )] ² + [ T (t )] ² } (15) Normally, the ncertainty of the initial rptre time vale, i.e. m (t ), will be mch less than s (t ) and T (t ). Page 19 of 35

22 UNCERT COP 1: A.7 Calclation of Uncertainty in Creep Elongation Time t fx This receives the same treatment as rptre time. An initial vale m (t fx ) was obtained above (A1.8), reflecting ncertainty in the time measrement which is sally negligible. The additional ncertainties de to stress and temperatre ncertainties are ( t ) and ( t ) n ( ó ) s fx = t fx for stress (16a) ó ( ) Q T = t fx for temperatre (16b) RT T fx Note that the factors mltiplying t fx on the right hand sides of these expressions are the same as in Eqs. (14a, b). Finally there is a component de to ncertainty in strain measrement. In A1.1, the ncertainty in extensometer reading (e) was obtained. Extension L e is the difference between initial and crrent vales, and has ncertainty: Strain ε is given by: ( L e ) = (e). and its ncertainty by ε = L e / L r These expressions give: ( å ) ( L ) ( L ) å = L e e + L r (ε) = [ (e) + (ε (L r ) ) ] / L r (17) If the strain rate in h -1 at the target strain x is å&, then the ncertainty in time at this point de to strain ncertainty is r e (t fx ) = (ε) / å& (17a) and finally c (t fx ) = [ ( m (t fx )) ² + ( S (t fx )) ² + ( T (t fx )) ²+ ( e (t fx )) ²] (18) Page of 35

23 UNCERT COP 1: A.8 Calclation of Uncertainty in Minimm Strain Rate å& min Graphical estimation of å& min does not readily provide an ncertainty m (å& min ), bt a Type A estimate may be made from repeated measrements, or analysis of strain/time data. As for t and t fx, this will be mch less than the effects of ncertainties in temperatre and stress. As noted in Appendix D, the same expression applies for all three of these parameters, i.e. ncertainty contribtions are: and then and s T ( å& ) min ( å ) n ( σ ) & = å min for stress (19a) σ ( T) Q & min = å& min for temperatre (19b) RT c (å& min ) = [ ( m (å& min )) + ( s (å& min )) + ( T (å& min )) ] (19c) Page 1 of 35

24 UNCERT COP 1: B1. Test Data APPENDIX B Worked Example of Creep Test Uncertainties Calclation This example is based on the reslts of a test on.5%cr 1% Mo weld metal. The specimen was machined according to a drawing with the following dimensions and tolerances (nits mm): diameter 8.98/9. parallel length extensometer length The diameter of the extensometer attachment position and blending radis to the parallel section gave a calclated sholder correction of.8mm. Test temperatre 565ºC main thermocople error e m (T) =.5ºC specimen niformity error e (T) = 1.5ºC measring system error e c (T) = ºC Load 179 N (stress = 17MPa). Weights and lever arm certified to give specified load +1%. Extensometer accracy +.mm After the initial period of primary creep, specimen parameters were recorded by compter at intervals of 8 hors. The specimen had failed at the 8h recording time. Time for 1% strain was reqired. Strain at 6 times arond this point are given: Time h Strain % Linear regression gives the reslts: time to 1% strain = 1181h ( error h): strain rate =.1%/h = 1.E-5h -1 Minimm strain rate was determined graphically to be 9.7E-6h -1 at arond 1h. The broken specimen gave the following 1 measrements of minimm diameter d (mm) 7.5, 7.65, 8.6, 7.79, 8.11, 8., 7.87, 8.11, 7.95, 7.99 Mean = 7.9 Standard deviation =.3 Standard ncertainty of mean d =.3 / 1 =.7mm Page of 35

25 UNCERT COP 1: Digital calliper error =.mm error =./ 3 =.1mm Total ncertainty of d = ( ) =.7 mm. The following are the 1 measrements of final extensometer length L e (mm) 64.5, 64.4, 64.4, 64., 64.53, 64.36, 64.8, 64.56, 64.54, Mean = Standard deviation =.19 Standard ncertainty of mean L e =.19 / 1 =.6 mm For this material and temperatre, Appendix C gives vales of 5.4 for the creep exponent n and 5K for Q/R. These data and the calclation reslts are shown in the calclation sheets Tables B1 to B9, at the end of this Appendix, together with references to the eqations sed in each ncertainty evalation. A copy of a spreadsheet is provided in Table B1, which does most of the calclating atomatically. B. Presentation of Reslts To obtain the sal 95% confidence, a coverage factor of shold be applied to the ncertainties in the table, as follows..5% Cr 1% Mo steel weld metal tested at 565ºC and 17MPa. Redction of area % Elongation % Rptre time h Time to 1% strain h Minimm strain rate (9.7+.)E-6 h -1 B3. Notes Absolte and Relative Uncertainties The left hand side of some eqations for ncertainties of vales consists of the ratio of the ncertainty to the vale, i.e. (x) / x = expression. In some cases, this ratio can be sed in a sbseqent calclation withot evalating the ncertainty itself. For example, ncertainty in stress (Eq. (1)) ses (S ) / S derived in Eq. (9a) or (9c). (S = original cross-sectional area). Page 3 of 35

26 UNCERT COP 1: In the worksheet table, ncertainties are given as absolte vales or ratios (%) as appropriate, or both. In the final reslts table, ncertainties shold be given in the same absolte nits as the reslt. Relative Importance of Uncertainty Components It can be seen that the ncertainty of initial time measrement for rptre or given creep strain is insignificant compared with ncertainties de to stress and temperatre variation. In this example, type B ncertainties for parallel length and extensometer length (both.9mm) were the main factors in ncertainty of % elongation. Type A evalation wold have given a figre similar to the ncertainty of final extensometer length (.6mm). The lower ncertainty will be particlarly valable for low-dctility materials. Linear Regression Hand Calclation Many hand calclators have programmed procedres to obtain the intercept and slope parameters (a and b) of the best fit straight line throgh a set of n (x, y) data points (for example (log(σ ), log(t ) ) vales). The estimated vale of y when x = X, and its ncertainty, are given by Y = a + bx ( Y) = 1 nσ y Σy n( n ) The latter fnction is available in Microsoft Excel as STEYX. Spreadsheet Calclation Aid ( ) ( ) [ nσ( xy) ΣxΣy] nσ x Σx ( ) ( ) Tables B1 to B9 have been incorporated into an Excel 5. spreadsheet, which does most of the calclation atomatically. It has been pasted into a Word Table containing the data of the worked example, and is given in Table B1. A working spreadsheet can be obtained from the athor, or made p by entering the text and formlae into a blank spreadsheet. All formatting has to be set as reqired. The spreadsheet formlae are given on the page following Table B1. New data needs to be entered only in the highlighted cells. In some places, an option is given for ncertainty as Type A or B, and in relative or absolte vales. In some sitations this is not available, for example temperatre errors will sally be absolte and Type B. However, the sheet can easily be modified as reqired. Uncertainties for initial measred creep strain time and minimm creep rate, if available, can be entered in G51 and G61. If not, zeros mst be entered. This version only provides for cylindrical specimens. Page 4 of 35

27 UNCERT CoP 1: Creep Test Uncertainties Calclation Sheets Some qantities which are sed later are represented by pper case letters, e.g. vale (Z) or (= Z). The symbols are sed later in explanatory comments in the right hand colmn. A % sign following an ncertainty vale indicates that this is a relative ncertainty, as a percentage of the vale to which it applies. Otherwise, ncertainties are absolte. The eqations referred to in the last colmn are in Appendix A. Table B1. Uncertainty in Initial Cross-Sectional Area Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) vale type vale prob. distrbn. divisor initial diameter d (mm) 8.99 B.1 rect. 3 ncertainty.6% (A).6 measrand vale S (mm ) ncertainty.1% (B).8 eqation B=A (9a) Table B. Uncertainty in Initial Stress Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) vale type vale prob. distrbn. divisor ncertainty load P (N) 179 B 1% rect. 3.6% (C) initial cross-sectional area S (mm ) B -.1% (=B) measrand vale ncertainty eqation σ (MPa) 17.6% (D) D= (B² + C²) (1) Page 5 of 35

28 UNCERT CoP 1: Table B3. Uncertainties in Final Minimm Cross-Sectional Area and Percentage Redction of Area Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) vale type vale prob. distrbn. final minimm diam. d mm 7.9 A.7 1 divisor ncertainty.9% (E).7 measrand vale final minimm crosssectional area S (mm²) 49.6 redction in area ncertainty 1.8% (F).9 eqation F=E (11a) Z (%) (1) Table B4. Uncertainty in Percentage Elongation Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) vale type vale prob. distrbn. divisor parallel length L c (mm) 5 B.5 rect. 3 end correction.8 not inclded ncertainty.58%.9 (G) measrand vale initial reference length L r (mm) 5.8 ncertainty.58%.9 (=G) eqation extensometer length L e (mm) 58 B.5 rect. 3.9 (H) ref. length elongation final ext meter length L e (mm) A.6.6 (I) L e (mm) 6.43 percent elongation.3 (J) (4.7%) J= (H²+I²) (13b) A (%) (13c) Page 6 of 35

29 UNCERT CoP 1: Table B 5. Uncertainty in Temperatre Sorce of Uncertainty Uncertainty in Sorce Affected Measrand Sorce qantity symbol vale type vale (nit) measrement thermocople T m (K) B.5 specimen niformity T (K) B 1.5 prob. distrbn. divisor ncertainty measrand vale ncertainty eqation controller error T c (K) B. total temperatre error (K) B.55 rect T (K) Table B6. Data from Appendix C, and Factors for Effects of Stress and Temperatre Uncertainties on Uncertainties of Rptre Time, Creep Time and Strain Rate. n = 5.4 Q / R = 5K n (σ ) / σ = n D =.3 (L) (T) Q / RT² =.15 (M) (14a, b) Table B7. Uncertainty in Rptre Time Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) vale type vale prob. distrbn. divisor ncertainty measrand vale ncertainty rptre time t (h) 4 B 4 rect. 3.3 (N) m (t )=.3 = N eqation initial stress σ (MPa) 17.6% s (t )=7 = P = L. * t temperatre T (K) rptre time T (t )=31 = S = M * t t (h) 4 41 = (N +P² +S²) (15) Page 7 of 35

30 UNCERT CoP 1: Table B8. Uncertainty in Time at 1% Strain Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) vale type vale prob. distrbn. divisor ncertainty measrand vale ncertainty time at 1% strain t f1 (h) 1181 A 1 m (t f1 )= = U eqation initial stress σ (Mpa) 17.6% s (t f1 )=38 = V = L.* t f1 (16a) temperatre T (K) T (t f1 )=14 = W = M * t f1 (16b) extensometer e (mm) B. rect. 3.1 ε.1 (1%) (ε) = 3E-4 (17) ε& =1E-5h -1 e (t f1 )=3 = X (17a) time at 1% strain t f1 (h) = (U +V²+W²+X ) (18) Table B9. Uncertainty in Minimm Strain Rate Sorce qantity Sorce of Uncertainty Uncertainty in Sorce Affected Measrand symbol (nit) Minimm strain rate ε& min (h -1 ) 9.7E-6 vale type vale not determined prob. distrbn. divisor ncertainty measrand vale ncertainty eqation initial stress σ (MPa) 17.6% 3.1E-7 (Y) = L * ε& min (19a) temperatre T (K) Minimm 1E-6 (Z) = M * ε& creep rate min (19b) ε& min (h -1 ) 9.7E-6 1.1E-6 = (Y +Z²) (19c) Page 8 of 35

31 UNCERT CoP 1: Table B1. Copy of Excel Spreadsheet for Calclation of Measrands and Uncertainties A B C D E F G H I J K L M N O Measrand Sorce of Uncertainty Uncertainty in Sorce Affected Measrand for ncertainty evalation Sorce qantity symbol (nit) vale type vale absolte relative prob. distrbn. divisor ncer'ty measrand vale ncertainty 4 5 Initial cross- initial diameter d (mm) 8.99 B.1 rect sectional area micrometer error m (mm) B rect d (mm) % 7 S (mm ) % Initial stress Load P (N) 179 B 1% rect % 11 initial cross-sect. area S (mm ) A.13% % σ (MPa) % Final minimm final min. diam. d (mm) 7.9 A cross section & micrometer error m (mm) B. rect d (mm) % 16 Redction of S (mm²) % area initial cross-sect. area S (mm ) Z (%) from gage length marks Percentage initial gage length L (mm) 3 elongation final gage length L (mm) Table contines Page 9 of 35

32 UNCERT CoP 1: Table B1 contination A B C D E F G H I J K L M N O Measrand Sorce of Uncertainty Uncertainty in Sorce Affected Measrand for ncertainty evalation Sorce qantity symbol (nit) vale type vale absolte relative prob. distrbn. divisor ncer'ty measrand vale ncertainty 4 5 Enter data in from extensometer length 6 one place only parallel length Lc (mm) 5 B.5 rect initial reference length 7 end correction.8 L r (mm) extensometer length L e (mm) 58 B.5 rect ref. length elongation 9 final ext meter length L e (mm) A L e (mm) percent elongation 3 A (%) Temperatre measring th'cople T m (K) B.5 36 specimen niformity T (K) B controller T c (K) B 38 total temp. error T (K) 838 B.55 rect T (K) Data from Appendix C, and Factors for Effects of Stress and Temperatre Uncertainties on Uncertainty of Rptre Time 4 stress exponent n 5.4 Q/R (K) 5 n(σ )/σ.3 (T) Q / RT² Table contines Page 3 of 35

33 UNCERT CoP 1: Table B1 final section A B C D E F G H I J K L M N O Measrand for ncertainty evalation Sorce of Uncertainty Uncertainty in Sorce Affected Measrand Sorce qantity symbol (nit) vale type vale absolte relative prob. distrbn. divisor ncer'ty measrand vale ncertainty Rptre time rptre time t (h) 4 B 4 rect (t ) 46 initial stress σ (MPa) % 7.5 (t ) 47 temperatre T (K) (t ) 48 t (h) Time at time at x% strain t fx (h) 1181 A 1. m (t fx )=. 5 x% strain initial stress σ (MPa) % s (t fx )= temperatre T (K) T (t fx )= extensometer e (mm) B. rect e (t fx )= strain rate at x% strain å& (h -1 ).1 56 reference length L r (mm) strain ε.1.31 t fx (h) Minimm minimm strain rate å& min (h -1 ) 9.7E-6. 1 m ( å& min ) 6 strain rate initial stress σ (MPa) % s ( å& min ) 3.1E-7 63 temperatre T (K) T ( å& min ) 1.E-6 64 å& min (h -1 ) 9.7E-6 1.1E-6 Page 31 of 35