The Determination of Uncertainties in Dynamic Young s Modulus

Transcription

1 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Manual o Codes o Practice or the Determination o Uncertainties in Mechanical Tests on Metallic Materials Code o Practice No. 3 The Determination o Uncertainties in Dynamic Young s Modulus C K Bullough ABB ALSTOM Power Technology Centre Cambridge Road Whetstone Leicester LE8 6LH UNITED KINGDOM Issue September 000

2 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 CONTENTS SCOPE SYMBOLS AND DEFINITIONS 3 INTRODUCTION 4 A PROCEDURE FOR ESTIMATING THE UNCERTAINTY IN DYNAMIC YOUNG S MODULUS Step - Identiying the parameters or which uncertainty is to be estimated Step - Identiying all sources o uncertainty in the test Step 3- Classiying uncertainty according to Type A or B Step 4- Estimating the sensitivity coeicient and standard uncertainty or each source Step 5- Computing the measurand s combined uncertainty u c Step 6- Computing the expanded uncertainty U Step 7- Reporting results 5 REFERENCES APPENDIX A Mathematical ormulae or calculating uncertainties in dynamic Young s modulus at ambient temperature APPENDIX B A worked example or calculating uncertainties in dynamic Young s modulus at ambient temperature

3 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000. SCOPE This procedure covers the evaluation o uncertainty in the determination o dynamic Young s modulus and related quantities in elastic materials at ambient temperatures according to the testing Standard: ASTM E , Dynamic Young s Modulus, Shear Modulus, and Poisson s Ratio by Impulse Excitation o Vibration, American Society or Testing and Materials, 997. ASTM E describes how the resonant requencies o elastic materials are excited by striking a rectangular or cylindrical bar which is ree to vibrate. A transducer and associated electronic equipment measure the requency which can be related through a knowledge o the bar s dimensions and mass, and the material s Poisson s ratio, to the dynamic Young s modulus. In the case o the rectangular bar, the undamental lexural requency can be excited and similarly used to calculate the dynamic shear modulus. Knowledge o both the dynamic Young s modulus and shear modulus can be used to determine Poisson s ratio where this is otherwise unknown.. SYMBOLS AND DEFINITIONS For a complete list o symbols and deinitions o terms on uncertainties, see Section o the main Manual []. It should be noted that not all the symbols and deinitions o terms on uncertainties used in this CoP are consistent with the GUM []. The ollowing list gives the symbols and deinitions used in this procedure. A empirical correction actor or dynamic shear modulus dependent on the width to cross-section ratio o a rectangular bar b width o the bar B empirical correction actor or dynamic shear modulus dependent on the width to cross-section ratio o a rectangular bar c i sensitivity coeicient COD coeicient o determination CoP Code o Practice D diameter o a cylindrical bar d v divisor used to calculate the standard uncertainty E dynamic Young s modulus F axial orce undamental resonant requency o a rectangular or cylindrical bar in lexure undamental resonant requency o a bar in torsion t This ASTM standard is commonly used in Europe as no suitable alternative ISO, EN, or national standard exists. Page o

4 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 G dynamic shear modulus k coverage actor used to calculate expanded uncertainty (usually corresponding to 95% conidence level) where a normal probability distribution can be assumed k p coverage actor used to calculate expanded uncertainty (usually corresponding to 95% conidence level) where a normal probability distribution cannot be assumed L length o the bar m mass o the bar n number o repeat measurements µ Poisson s ratio p conidence level q random variable q arithmetic mean o the values o the random variable x s experimental standard deviation (o a random variable) determined rom a limited number o measurements, n T nominal test temperature (in degrees Celsius or Kelvin, as indicated) t thickness o the bar T correction actor or dynamic Young s modulus, or undamental lexure mode o a rectangular bar T correction actor or dynamic Young s modulus, or undamental lexure mode o a cylindrical bar u standard uncertainty u c combined standard uncertainty U expanded uncertainty x i estimate o input quantity y test (or measurement) result v i degrees o reedom o standard uncertainty u v e eective degrees o reedom used to obtain k p 3. INTRODUCTION There are requirements or test laboratories to evaluate and report the uncertainty associated with their test results. Such requirements may be demanded by a customer who wishes to know the bounds within which the reported result may be reasonably assumed to lie; or the laboratory itsel may wish to understand which aspects o the test procedure have the greatest eect on results so that this may be monitored more closely or improved. This Code o Practice has been prepared within UNCERT, a project partially unded by the European Commission s Standards, Measurement and Testing programme under reerence SMT4 -CT97-65 to simpliy the way in which uncertainties in mechanical test on metallic materials are evaluated. The aim is to avoid ambiguity and provide a common ormat readily understandable by customers, test laboratories and accreditation authorities. Page o

5 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 This Code o Practice is one o seventeen prepared and tested by the UNCERT consortium or the estimation o uncertainties in mechanical tests on metallic materials. The Codes o Practice have been collated in a single Manual [] which has the ollowing sections:. Introduction to the evaluation o uncertainty.. Glossary o deinitions and symbols 3. Typical sources o uncertainty in materials testing 4. Guidelines or the estimation o uncertainty or a test series. 5. Guidelines or reporting uncertainty. 6. Individual Codes o Practice (o which this is one) or the estimation o uncertainties in mechanical tests on metallic materials. This CoP can be used as a stand-alone document. Nevertheless, or background inormation on measurement uncertainty and values o standard uncertainties o devices used commonly in material testing, the user may need to reer to the relevant section in Reerence. Several sources o uncertainty, such as the reported tolerance o load cells, extensometers, micrometers and thermocouples, are common to several mechanical tests and are included in Section 3 o the Manual []. These are not discussed here to avoid needless repetition. The individual procedures are kept as straightorward as possible by ollowing the same structure: The main procedure. Fundamental aspects and major contributions to the uncertainty or the speciic test type (Appendix A) A worked example (Appendix B) This document guides the user through several steps to be carried out in order to estimate the uncertainty in dynamic Young s modulus. The general process or calculating uncertainty values is described in section o the Manual []. 4. A PROCEDURE FOR ESTIMATING THE UNCERTAINTY IN DYNAMIC YOUNG S MODULUS Step. Identiying o the Parameters or which Uncertainty is to be Estimated The irst step is to list the quantities (measurands) or which the uncertainties must be calculated. Table shows the parameters that usually constitute the results o a test perormed to ASTM E Generally, dynamic Young s modulus, dynamic shear modulus and Poisson s ratio are considered to be the primary results o this testing standard. These measurands are not measured directly, instead they are determined rom other quantities (or measurements). Page 3 o

6 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Table. Measurands, Measurements, their Units and Symbols Measurands Units Symbol Dynamic Young s modulus Pa E Dynamic shear modulus Pa G Poisson s ratio dimensionless µ Measurements Units Symbol Mass o the bar Kg m Width o the bar m b Length o the bar m L Thickness o the bar m t Fundamental resonant requency o a Hz rectangular bar, or circular rod, in lexure Fundamental resonant requency o Hz t rectangular bar in torsion Diameter o rod m D Test temperature C T Step. Identiying all Sources o Uncertainty in the Test In Step, the user must identiy all possible sources o uncertainty that may have an eect (directly or indirectly) on the test. This list cannot be exhaustively identiied beorehand as it is uniquely linked to the test procedure and the apparatus used. This means that, a new list should be drated each time a particular test parameter changes (when a plotter is replaced by a computer or example). To help the user list all sources o uncertainty, our categories have been deined. Table lists the our categories and some examples o sources in each category. It is important to note that Table is NOT exhaustive and is or GUIDANCE only - relative contributions may vary according to the material tested and the test conditions. Individual laboratories are encouraged to prepare their own list to correspond to their own test acility, and assess the associated signiicance o the contributions. To simpliy the uncertainty calculations it may be advisable to regroup the signiicant sources o uncertainty in Table according to the ollowing categories: Uncertainty associated with the test piece measurements Uncertainty in associated with the apparatus 3 Uncertainty due to the environment 4 Uncertainty due to operator or procedure Appendix A presents the undamental aspects o the above categorisation. Page 4 o

7 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Table. Typical Sources o Uncertainty and Their Likely Contribution to Uncertainties on Dynamic Young s Modulus at Ambient Temperature [ = major contribution, = minor contribution, blank = insigniicant (or no) contribution] Source o uncertainty Type ) Measurand E m G. Test Piece Micrometer / calliper / operator errors in A or B measuring bar dimensions Shape tolerance, edge eects B Accuracy o balance A or B Not isotropic B. Apparatus Damping rom supports B Damping rom transducer B Accuracy o transducer/ B electronics 3. Environment Poor control o ambient temperature B Humidity - eect on test piece moisture B content 4. Method Incorrect calculation o T B Omission o A B Incomplete iterative solution o Poisson s ratio B Step 3. Classiying Sources o Uncertainty according to Type A or B. In accordance with ISO TAG4 Guide to the Expression o Uncertainties in Measurement', the sources o uncertainty can be classiied as Type A or B, depending on how their inluence is quantiied. I the uncertainty is evaluated by statistical means (rom a number o repeated observations), it is classiied Type A. I it is evaluated by any other means it should be classiied as Type B. The values associated with Type B uncertainties can be obtained rom a number o sources including calibration certiicates, manuacturer's inormation, or an expert's estimation. For Type B sources, it is necessary or the user to estimate or each source the most appropriate probability distribution (urther details are given in Section o Reerence ). Attention should be drawn to the act that one source can be classiied as either Type A or B depending on how it is estimated. For example, i the width o a rectangular test bar is measured once, that uncertainty associated with repeatability o measurements is considered Type B. I the mean value o two or more consecutive measurements is taken into account, then the uncertainty is Type A. Page 5 o

8 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Table 3 shows worksheets containing typical sources o uncertainty and their type or dynamic young s modulus, shear modulus and Poisson s ratio. Table 3a. Typical Worksheet or Uncertainty Budget Calculations or Estimating the Uncertainty in Dynamic Young's Modulus o a Rectangular Bar. Source o uncertainty Symbol Value Probability distribution Divisor d v Mass o the bar m rectangular 3 Width o the bar b rectangular 3 Length o the bar L rectangular 3 3 Thickness o the bar t rectangular 3 3 Fundamental requency in rectangular 3 lexure Mean value o E u(e) rep normal n- Combined standard u c normal u c (E) ν e uncertainty Expanded uncertainty U normal ν e c i u i (E) Pa ν i or ν e ) See Equation A5 ) The incomplete iterative solution o µ will have a computable but negligible eect on E through Equation A3. Table 3b. Typical Worksheet or Uncertainty Budget Calculations or Estimating the Uncertainty in Shear Modulus o a Rectangular Bar. Source o uncertainty Symbol Value Probability distribution Divisor d v Mass o the bar m rectangular 3 Width o the bar b rectangular 3 Length o the bar L rectangular 3 Thickness o the bar t rectangular 3 Fundamental requency in t rectangular 3 torsion Mean value o G u(g) rep normal n- Combined standard U c normal u c (G) ν e uncertainty Expanded uncertainty U normal ν e c i u i (G) Pa ν i or ν e ) See Equation A6 ) The omission o the correction actor A is or most practical purposes a bias, resulting in an overestimate o G. Page 6 o

9 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Table 3c. Typical Worksheet or Uncertainty Budget Calculations or Estimating the Uncertainty in Poisson's Ratio o a Rectangular Bar Source o uncertainty Symbol Value Probability distribution Divis or d v c I u i (µ) Dynamic Young's modulus E normal Dynamic Shear modulus G normal Mean value o µ u(µ) rep normal n- Combined standard U c normal u c (µ) ν e uncertainty Expanded uncertainty U normal ν e ν i or ν e see Equation A7 Step 4. Estimating the Standard Uncertainty and Sensitivity Coeicient or each Source o Uncertainty. In Step 4, the standard uncertainty, u, or each input source identiied in Table is estimated (see Appendix A). The standard uncertainty is deined as one standard deviation and is derived rom the uncertainty o the input quantity divided by the parameter, d v, associated with the assumed probability distribution. The divisors or the distributions most likely to be encountered are given in Section o Re. []. The standard uncertainty requires the determination o the associated sensitivity coeicient, c, which is usually estimated rom the partial derivatives o the unctional relationship between the output quantity (the measurand) and the input quantities. The calculations required to obtain the sensitivity coeicients by partial dierentiation can be a lengthy process, particularly when there are many individual contributions and uncertainty estimates are needed or a range o values. I the unctional relationship or a particular measurement is not known, the sensitivity coeicients may be obtained experimentally. In many cases the input quantity to the measurement may not be in the same units as the output quantity. For example, one contribution to dynamic Young s modulus, E, is the rectangular bar s width, b. In this case the input quantity, b, is measured in metres, but the output quantity, E, is in Pascals. In such case a sensitivity coeicient, c i (corresponding to the partial derivative o the relationship between E and b) is used to convert rom the width to the dynamic Young s modulus (or more inormation see Appendix A). The calculation o the sensitivity coeicients and review o the relative contributions o each source o uncertainty (Appendix A) leads to the ollowing general conclusions or measurements o a rectangular bar: Most o the uncertainty in dynamic Young's modulus is contributed by the uncertainty in measuring the undamental requency in lexure. Secondary contributions are made by Page 7 o

10 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 measuring the thickness o the bar, and its length (particularly i calipers are used or the latter). The uncertainty in dynamic Shear modulus in a rectangular bar is dominated by the uncertainty in measuring the undamental requency in torsion. Other contributions are generally negligible. The uncertainty in Poisson's ratio is ormed in almost equal proportions by uncertainties in dynamic Young's modulus, and by dynamic Shear modulus. Hence the major contributing sources are the measurement o requencies. To help in the process o calculation, it is useul to summarise the uncertainty analysis in a spreadsheet - or 'uncertainty budget'- as in Tables 3a, 3b and 3c. Appendix A includes the mathematical ormulae or calculating the uncertainty contributions and Appendix B contains a worked example. Step 5. Computing the Combined Uncertainty u c. Assuming that individual uncertainty sources are uncorrelated, the measurand's combined uncertainty, u c (y), can be computed using the root sum squares: u N c( y) = [ ci. u( xi )] () i= where c i is the sensitivity coeicient associated with measurement x i. This uncertainty corresponds to plus or minus one standard deviation on the normal distribution law representing the studied quantity. The combined uncertainty has an associated conidence level o 68.7%. This calculation o the combined uncertainty is also included in Table 3, or each o the measurands, E, G and µ. Uncertainties in E and G, contribute to the combined uncertainty in µ, since µ is wholly dependent on those two measurands. Step 6. Computing the Expanded Uncertainty U. The expanded uncertainty, U, is deined in Reerence as the interval about the result o a measurement that may be expected to encompass a large raction o the distribution o values that could reasonably be attributed to the measurand. It is obtained by multiplying the combined uncertainty, u c calculated in step 5, by a coverage actor, k, which is selected on the basis o the level o conidence required. For a normal probability distribution, the most generally used coverage actor is which corresponds to a conidence interval o 95.4% (eectively 95% or most practical purposes). The expanded uncertainty, U, is, thereore, broader than the combined uncertainty, u c. Where a high conidence level is demanded by the customer (such as or Page 8 o

11 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 aerospace industry, electronics), a coverage actor o 3 is oten used so that the corresponding conidence level increases to 99.73%. In cases where the probability distribution o u c is not normal (or where the number o data points use in a Type A analysis is small), the value o k should be calculated rom the degrees o reedom given by the Welsh-Satterthwaite method (see Reerence, Section 4 or more details). Step 7. Reporting results. Once the expanded uncertainty has been estimated, the results should be reported in the ollowing way: V= y ± U () where V is the estimated value o the measurand, y is the test (or measurement) result, U is the expanded uncertainty associated with y. An explanatory note, such as that given in the ollowing example should be added (change as appropriate): The reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage actor, k=, which or a normal distribution corresponds to a coverage probability, p, o approximately 95%. The uncertainty evaluation was carried out in accordance with UNCERT CoP 3: 000. Page 9 o

12 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: REFERENCES. Manual o Codes o Practice or the determination o uncertainties in mechanical tests on metallic materials. Project UNCERT, EU Contract SMT4-CT97-65, Standards Measurement & Testing Programme, ISBN , Issue, September BIPM, IEC, IFCC, ISO, IUPAC, OIML, Guide to the expression o uncertainty in measurement. International Organisation or Standardisation, Geneva, Switzerland, ISBN , First Edition, 993. [This Guide is oten reerred to as the GUM or the ISO TAG4 document ater the ISO Technical Advisory Group that drated it.] Identical documents: CEN, ENV BSI, Vocabulary o metrology, Part 3. Guide to the expression o uncertainty in measurement, PD 646: Part 3 : 995, British Standards Institution. 3. An Evaluation o Three Techniques or Determining Young s Modulus o Mechanically Alloyed Materials Smith JS, Wyrrick MD and Poole JM, Dynamic Elastic Modulus Measurement in Materials, ASTM STP 045, Wolenden A Ed, ASTM Philadelphia, PA, "Stahleisen-Sonderberichte Het 0: Physikalische Eigenschaten von Stahlen und ihre Temperaturabhangogkeit Polynome und graphische Darstellungen"; published by Verlag Stahleisen. ACKNOWLEDGEMENTS This document was written as part o project Code o Practice or the Determination o Uncertainties in Mechanical Tests on Metallic Materials. The project was partly unded by the Commission o European Communities through the Standards, Measurement and Testing Programme, Contract No. SMT4-CT Page 0 o

13 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 APPENDIX A MATHEMATICAL FORMULAE FOR CALCULATING UNCERTAINTIES IN DYNAMIC YOUNG S MODULUS AT AMBIENT TEMPERATURE In the irst part, A to A5, o Appendix A we summarise the main relationships rom the standard ASTM E which are used in the calculation o dynamic modulus. A to A3 cover rectangular bars; A4 and A5 cylindrical bars. There then ollows a discussion o how the uncertainties in individual measurements are likely to aect the uncertainties on reported measurands. A. Dynamic Young s modulus o a rectangular bar The dynamic Young's modulus is calculated rom the ollowing equation 3 3 E = ( m / b)( L / t ) T (A) where T is a geometrical correction actor or the undamental lexural mode to account or the bar s inite thickness and Poisson s ratio, and the other parameters are deined in Section o this CoP. I L/t > 0 the geometrical eects are considered negligible and T may be calculated directly rom Equation (A). T = [ ( t / L) ] (A) I L/t <0, then the value o T should be calculated rom Equation (A3). T = ( µ µ )( t / L) ( t / L) ( µ µ )( t / L) ( µ µ )( t / L) 4 (A3) I Poisson s ratio, µ, is unknown an initial value must be assumed. It may be subsequently estimated by an iterative procedure rom the calculation o shear modulus. A. Dynamic shear modulus o a rectangular bar. The dynamic Shear modulus is calculated rom the ollowing equation G = 4( Lm / bt)[ B / ( + A)] (A4) Where A, and B are empirical width to thickness correction actors t Page o

14 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 b / t + t / b B = 6 t( t / b) 5. ( t / b) + 0. ( t / b) (A5) A = 3 [ ( b / t) ( b / t) ( b / t) ] [. 03( b/ t) ( b / t) ] (A6) ASTM E allows A to be omitted, noting that errors o less than % will result. A3. Poisson s ratio. Assuming isotropic behaviour: A4. Dynamic Young s modulus o a cylindrical bar. For a cylindrical bar the dynamic Young's modulus is given by µ = ( E / G) (A7) 3 4 E = ( m )( L / D ) T (A8) 3 4 where E = ( m )( L / D ) T is a correction actor or the undamental lexural mode to account or the bar s inite thickness and Poisson s ratio. I L/t > 0, then 3 4 E = ( m )( L / D ) T may be calculated directly rom Equation (A9). T ' = [ ( D / L) ] (A9) I L/D <0, then the value o T should be calculated rom Equation (A0). T ' = ( µ µ )( D / L) ( µ +.73µ )( D / L) ( µ +.536µ )( D / L) ( D / L) 4 (A0) A5. Dynamic shear modulus o a cylindrical bar. For a cylindrical bar the dynamic Shear modulus is given by G =6m ( L / π D ) (A) No empirical correction ormulae are required or cylindrical bars. t Page o

15 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 A6. Sources o Uncertainty when Testing to ASTM E Turning now to the sources o uncertainty listed in the Main Procedure (Table ) it is demonstrated how the typical uncertainties in the individual measurements, dierences in apparatus etc. lead to each source s contribution to the error budget (sensitivity coeicient, c i, multiplied by that measurement's uncertainty, u). It is assumed that the measuring equipment has been appropriately calibrated, and that a test procedure has been written ollowing the standard which minimises typical measurement errors, and that the procedure is being applied by a trained operative. For simplicity, only the use o rectangular bars has been considered, and by way o illustration a mild steel bar o nominal dimensions 3mm x 5mm x 00mm is described. A7. Calculation o Partial Derivatives. The calculation o uncertainties in the reported values (E,G,µ) on rectangular test pieces is made by calculating the "sensitivity coeicients" rom the partial derivatives o equations (A, A4 and A7) containing the measurands. When a derived quantity, Y, is a unction o measurands x, x, x 3., and each x i is subject to uncertainty u(x i ), then the resulting uncertainty in Y is given by: Y U ( Y ) = U ( xi ). x i (A) The partial derivatives or equation A, which lead to the values o the sensitivity coeicients are given by: 3 3 E ( m / b)( L / t ) T 3 3 = = ( / b)( L / t ) T = E. ; m m m hence c i = (A3.) similarly: 3 3 E ( m / b)( L / t ) T = = E 3 3 E ( m / b)( L / t ) T E = = b b b 3 3 E ( m / b)( L / t ) T 3E = = L L L 3 3 E ( m / b)( L / t ) T 3E = = t t t ; c i = (A3.) ; c i = (A3.3) ; c i = 3 (A3.4) ; c i = 3 (A3.5) Page 3 o

16 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Thus, U ( E) = E U m m ( ) U( ) U( b) U( L) U( t) +. + b + 3. L + 3. t (A4) Strictly speaking, the correction actor, T, also contains the measurements t and L, and should have been dierentiated in Equations A3.4 and A3.5. However, with a typical value o (L/t) o 0, the contribution o the second term in Eq. A is Thus, the uncertainty in both t, and L is negligible in calculating the uncertainty in T, and has thereore been neglected. Simpliying Equation A4, by expressing the uncertainties as a percentage o the measurand (eg. u(t)% = 00 u(t)/t etc.) and acknowledging that the sensitivity coeicients will have absolute values, then we have: [ ] 0. 5 ( E )% = (.U( m )%) + (.U( )%) + (.U ( b) %) + ( 3.U( L )% ) ( 3.U( t).%) U + (A5) Equations A4 and A7 are similarly used to estimate the percentage uncertainties in G and m, as shown in equations A6 and A7, respectively. ( ) ( ) ( t ) ( ) ( ) U G % = (. U m %) +. U % + (. U b % ) + (. U( L) %) + ( U( t ).%) (neglecting the negligible contributions via A,B) (A6) [ ( ) ] U ( ) U ( E µ % (. )%). U ( G = + )% 0. 5 (A7) Equations A5, A6 and A7, thereore, orm the basis o calculating the combined uncertainty, U c, in Table 3 o the Main Procedure. A8. Uncertainty due to Measuring the Mass and Dimensions o the Test Bar. In general, it is to be expected that the width and thickness o the test piece will be measured using a micrometer. Typical uncertainties due to the accuracy o the micrometer, uniormity o the test piece and skill o the operator can amount to uncertainties o ±0.0000m, which with a minimum practical thickness o about 0.003m results in an uncertainty o 0.3% contributing through t -3 in Equation A. Some o this uncertainty can be reduced i larger thicknesses are used, since the errors in the micrometer readings and in the shape tolerance o machined test pieces are generally ixed. Typically, the width o a rectangular test piece is 5 times the thickness, and hence the uncertainties in its measurement are much less. Generally, calipers will be used to measure the length, and uncertainties o ±0.0005m are typical. This corresponds to 0.5% on a length o 0.00m, contributing through L 3 in Equation A Page 4 o

17 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Generally, on well-machined test bars the main sources o uncertainty in dimensional measurements are due to the tolerances o the micrometer and calipers, which are Type B errors, and cannot be reduced by repeated measurements. On the other hand, the laboratory should consider the need to perorm repeated measurements on test bars whose dimensions vary by more than tolerance o the measuring instruments. The shape tolerance will not only contribute to uncertainties in the measurement o dimensions, but also in the measurement o requency, as dierent requencies can be induced in non-uniorm bars. A round-robin study (Reerence 3) reported in ASTM E on a machined silicon nitride bar and an alumina bar which was not machined, and had a thickness variation o to m along its length, indicated that the non-uniormity o the alumina bar may have contributed about 0.4% to the uncertainty in requency. Similarly, rounded edges and chamers may cause changes in requencies. However, both these eects can usually be considered negligible or machined bars. Generally, the mass o a test piece can be measured with an uncertainty o ±0.000kg, which or a 0.070kg sample corresponds to 0.9%, contributing through m in Equation A. This uncertainty is almost entirely due to the tolerance in the scales, and is thereore Type B. A9. Uncertainty due to measuring requency. With proper experimental technique, damping rom the supports or the location o the transducer can be minimised, and can be considered negligible. (Damping causes a rise in requency, which biases the reported result.) On a undamental resonant requency o 6000Hz the uncertainty (5 repeated readings) is reported in ASTM E to be about 8Hz (or about.0% assuming a rectangular distribution on a typical requency o,000hz), which contributes through the term in Equation A. This uncertainty is comprised principally o tolerances in the measuring equipment, and sometimes its resolution, and is relatively diicult to quantiy except with calibrated signal generators and/or with round-robin investigations. Generally, the repeatability o requency measurements on a single test bar contributes a much smaller uncertainty than that quoted in the standard. This indicates that the combined uncertainty in measuring requency has as its major contribution tolerances in the test set-up, and it is thereore included as a Type B uncertainty in Table 3. A0. Uncertainty due to environment. Some materials such as certain composites and some rocks are aected by humidity which can lead to a change the undamental resonant requency, or a change in the mass. However, these inluences are considered negligible or most metallic materials. Similarly, ASTM E requires that the test temperature at measurement be recorded. In most laboratories the uncertainty in this measurement is likely to be better than +3 C. Its inluence is through variations in the test piece dimensions (during measurement o those dimensions, and during measurement o Page 5 o

18 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 requencies). Modulus values are also dependent on temperature, but small variations (+3 C) in temperature can usually be ignored. A. Uncertainty due to the operator, procedure and assumptions. As described above, the evaluated dynamic Young s modulus and shear modulus can be used to estimate Poisson s ratio. ASTM E describes an iterative procedure, stating that the Poisson s ratio is that when the dierence between the current and previous estimate is "less than %". This can give rise to a % uncertainty with a rectangular distribution. No guidance is given or the starting value o Poisson s ratio, and hence the inal value can be approached either rom below or above. Note also that i the correction actor A is omitted rom Equation A4, then a systematic error u A (most oten small and positive) must be added to the combined uncertainty in the ollowing ashion: ( ) ( ) A ( t ) ( ) ( ) ( ) ( ( ) ) ( ( ) ) U G % = U % + (. U m %) +. U % +. U b % +. U L % +. U t.% 05. (A8) The assumption o isotropic properties in calculating Poisson s ratio is also considered to be true. However, the test laboratory should always be alert to signiicant anisotropy in metals whose structure is heavily textured, in geological samples containing porosity etc. A. Uncertainty o the mean value o E (repeatability). Repeatability is a Type A uncertainty contribution. It is the standard deviation o the estimated mean value o a series o test results on dierent bars o the same material under the same conditions considered in the uncertainty analysis. u rep = ( E ) n i E n n (A9) where E is the mean dynamic Young s modulus. Obviously, i only one test is conducted then the repeatability should not be included in the calculations, and the reported uncertainty is that o a single test. The determination o dynamic Young s modulus is non-destructive, and a series o tests on the same test bar, say over a period o weeks with dierent operators, would give useul inormation about the repeatability o the test method without any inluence o material variability. However, the customer is generally more interested in determining the uncertainty in dynamic Young s modulus or several test bars, either rom the same batch or or dierent batches o material to the same grade. (The test laboratory should take due care to ensure that the test bars have been Page 6 o

19 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 sampled to be representative o the batch or grade.) In the irst case, the reported uncertainty is that o a mean value o a batch o material. In the second case, the reported uncertainty is that o a mean value o a grade o material. A study o dynamic Young's Modulus o 60 unalloyed and alloyed steel grades ound that repeatability was about +0.3% (thought to be with approximately a 95% coverage actor) over the temperature 0 to 600 C range, Re. [4]. Page 7 o

20 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 APPENDIX B A WORKED EXAMPLE FOR CALCULATING UNCERTAINTIES IN DYNAMIC YOUNG S MODULUS OF A RECTANGULAR BAR B. Introduction. A customer asked the testing laboratory to carry out dynamic modulus, shear modulus and Poisson s ratio measurements on a single, rectangular steel bar (marked: XYZ3), o nominal dimensions 0.003m x 0.05mx 0.00m, according to ASTM E The laboratory has considered the tolerances in its test acility, and ound that the sources o uncertainty were identical to those described in Table o this CoP B. Estimation o Input Quantities to the Uncertainty Analysis. All measurements were perormed according to the laboratory s own written test procedure, and were made within the controlled temperature (+ C) o the laboratory, using appropriately calibrated instruments. The laboratory has at its disposal a micrometer with a tolerance o +0.0mm, which was used to measure the bar s thickness and width, and a caliper with a tolerance o +0.5mm was used to measure the length. The laboratory made repeat measurements o the bar s thickness width and length, demonstrating that the shape tolerances were smaller than can be detected with the micrometer or caliper. A measuring scale with a tolerance o 0.g was used to measure the mass o the bar. 3 A piezoelectric transducer / oscilloscope system was used to measure the resonant requencies. The measuring system had been checked in a round-robin study with other laboratories, and the laboratories declared uncertainty o +8Hz on the measured requency was believed to apply. Repeat measurements each o the undamental requencies in tension and in torsion were made, and the mean taken. B3. Preliminary Calculations The ollowing calculations use mean values, except where stated. In the irst instance, a number o calculations were perormed to check the density, and provide measurements o length to thickness and width to thickness ratios, as ollows ρ= m / ( L. b. t) = kg m -3 (B) L/t = (B) b/t = (B3) Page 8 o

21 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Calculation o the area at various positions along the bar conirmed that the standard deviation divided by the mean is 0.9% and thereore the test bar was within the permitted shape tolerance. B4. Calculations o Young s Modulus, Shear Modulus and Poisson s Ratio. The Young s modulus was calculated rom Equation A. The length to thickness ratio was greater than 0, and the correction actor, T, estimated by Equation A3 (initial Poisson s Ratio o 0.3) and Equation. A, is and , respectively. Although the dierence was small, the value rom Equation A3 was speciied in the laboratory s procedure, and thereore the initial estimate o the Young s modulus was ound to be: E (initial) = 06. GPa Shear modulus was calculated rom Equation A4, using values or the correction actors B and A calculated rom Equations A5 and A6, (whilst ASTM E allows A to be omitted, a positive systematic error in G o.3% would have resulted.), giving G = GPa Poisson s ratio was then estimated using Equation A7, giving µ = 0.33, and compared with the initial value o 0.3. An iterative method was then used to adjust the initial value until it coincided with that estimated by Equation A7. In act, as the value o Young s Modulus is insensitive to Poisson s ratio when the length to thickness ratio is greater than 0 modulus and Poisson s ratio were ound to be: E (inal) = 06. GPa µ (i nal) = 0.3 Page 9 o

22 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Table B. Measurands, Measurements, their Units and Symbols within ASTM E Measurands Symbol Values Mean Dynamic Young s modulus, Pa E - - Dynamic shear modulus, Pa G - - Poisson s ratio, (dimensionless) µ Mass o the bar, kg m Width o the bar, m (Note ) b Length o the bar, m (Note ) L Thickness o the bar, m (Note ) t Fundamental resonant requency o a rectangular bar, or circular rod, in lexure, Hz Fundamental resonant requency o rectangular bar in torsion, Hz 4 3 t Temperature at time o measurements, C Notes:. Measurements were made at ive equally spaced positions along the length o the bar. The measurement positions or width and thickness correspond, so that an area calculation can be perormed to test or shape tolerance.. Young s modulus, shear modulus and Poisson s ratio are obtained by calculation Page 0 o

23 Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3: 000 Table 3a. Worksheet or Uncertainty Budget Calculations For Estimating the Uncertainty in Dynamic Young's Modulus o a Rectangular Bar Source o uncertainty Symbol Value Probability distribution Divisor d v c i u I (E) +GPa Mass o the bar m +0.g rectangular Width o the bar b +0.0mm rectangular Length o the bar L +0.5mm rectangular Thickness o the bar t +0.0mm rectangular Fundamental requency in 8Hz rectangular lexure Mean value o E u(e) rep 06. normal - n- Combined standard u c normal 4.34 ν e uncertainty Expanded uncertainty (k = ) U normal 8.68 ν e ν i or ν e ) Calculations based on Equation A5 ) The incomplete iterative solution o µ will have a computable but negligible eect on E through Equation A3. Table 3b. Worksheet or Uncertainty Budget Calculations For Estimating the Uncertainty in Shear Modulus o a Rectangular Bar Source o uncertainty Symbol Value Probability dis tribution Divisor d v Mass o the bar m +0.g rectangular 3 0. Width o the bar b +0.0mm rectangular Length o the bar L +0.5mm rectangular Thickness o the bar t +0.0mm rectangular Fundamental requency in t 8Hz rectangular 3.6 torsion Mean value o G u(g) rep 78.5 normal - n- Combined standard U c normal.66 ν e uncertainty Expanded uncertainty U normal 3.3 ν e c i u i (G) Pa ν i or ν e ) Calculations based on Equation A6 ) The omission o the correction actor A is or most practical purposes a bias, resulting in an overestimate o G. Page o

24 Page Standards Measurement & Testing Project No. SMT4-CT97-65 UNCERT COP 3:000 Table 3c. Worksheet or Uncertainty Budget Calculations For Estimating the Uncertainty in Poisson's Ratio o a Rectangular Bar Source o uncertainty Symbol Value Probability distribution Divisor d v c I u i (µ) Dynamic Young's modulus E 4.34GPa normal Dynamic Shear modulus G.66GPa normal Mean value o µ u(µ) rep 0.3 normal - n- Combined standard U c normal ν e uncertainty Expanded uncertainty U normal ν e ν i or ν e ) Calculations based on Equation A7 B5. Uncertainty Calculations and Reporting o Results. The mean values rom Table B and the tolerances described in the text were entered into the Uncertainty Calculation Worksheet, Tables Ba, Bb and Bc. Note that in perorming this test, the laboratory included an estimate o the correction actor, A, in its calculation o G, and also completed the iterative solution o µ, thereby removing two systematic sources o uncertainty. The repeated measures could, in theory, be used to estimate Type A errors or the measurands. However, the laboratory chose to use the Type B errors recorded in Table B, as the tolerances on the test devices exceeded those errors contributed by repeated measures. The test laboratory was supplied with a single test bar, and thereore could perorm the uncertainty calculation solely or that test bar, and not or a batch or grade o material. The laboratory presents the results in the orm shown below. The results o a test conducted according to ASTM E on a steel test piece reerence XYZ3 are: Dynamic Young s Modulus GPa Dynamic Shear Modulus GPa Poisson s Ratio The above reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage actor k=, providing a level o conidence o approximately 95%. The uncertainty evaluation was carried out in accordance with UNCERT COP 3: 000. Page o