Título/s: | Computation and Uncertainty Evaluation of Offset Yield Strength |

Autor/es: | Matusevich, Ariel; Mancini, Reinaldo; Massa, Julio |

Institución: | INTI-Córdoba. Laboratorio de Materiales. Córdoba, AR |

Editor: | ASTM |

Palabras clave: | Tensiones; Software; Ensayos; Incertidumbre; Sistemas computarizados; Métodos de cálculo; Computación |

Idioma: | eng |

Fecha: | 2013 |

Ver+/- Ariel E. Matusevich,1 Julio C. Massa,2 and Reinaldo A. Mancini3
Computation and Uncertainty Evaluation of Offset Yield Strength REFERENCE: Matusevich, Ariel E., Massa, Julio C., and Mancini, Reinaldo A., “Computation and Uncertainty Evaluation of Offset Yield Strength,” Journal of Testing and Evaluation, Vol. 41, No. 2, 2013, pp. 1–14, doi:10.1520/JTE20120100. ISSN 0090-3973. ABSTRACT: This paper presents computer procedures for the calculation of offset yield strength (Sy) and for the evaluation of the uncertainty in its computation. Offset yield strength is obtained from the plot of stress-strain data recorded in a tension test, as the stress that corresponds to the intersection between the stress-strain curve and a line parallel to its proportional region (offset by a prescribed strain). In the proposed method, the problem is reduced to finding the point of intersection between two straight lines, one that fits the curve in the neighborhood of the intersection and the offset line. For the fitting of each line, we propose the use of a weighted total least-squares algorithm that takes into account uncertainties in both ordinates and abscissas. The evaluation of the uncertainty associated with Sy, in accordance with the Guide to the Expression of Uncertainty in Mea- surement, considers the correlation between the parameters involved in its calculation. The implementation of these procedures motivated the develop- ment of dedicated software for the computation of tensile parameters from tension-test raw data and for the estimation of their associated uncertainties. To validate the program, developed in MATLAB as a standalone application, we used a set of ASCII data curves that have agreed val- ues for the tensile parameters and which are publicly available at the web site of the National Physical Laboratory of the United Kingdom. Using these curves we demonstrate the validity of the proposed method for the computation of Sy; to validate the uncertainty-evaluation procedure, we use the law of propagation of probability distributions through Monte Carlo simulation. The computational tool, whose capabilities are presented in this work, is currently being used at the Laboratory of Mechanical Testing of the National Institute of Industrial Technology (INTI), in Co´rdoba, Argentina. KEYWORDS: yield strength, proof stress, uncertainty, software, tension test Introduction Offset yield strength (Sy), known as proof stress in European countries, is a measure of yielding widely used for design and specification purposes. This parameter is obtained from the plot of stress-strain data recorded in a tension test, as the stress that corre- sponds to the intersection between the stress-strain curve and a straight line parallel to another that fits the initial linear portion of the curve; the horizontal distance between both lines, referred to as offset, is prescribed as a percentage of the extensometer gauge length (typically 0.1 % or 0.2 % for metals) [1]. A computer method for the calculation of Sy involves the linear regression of the proportional region of the stress-strain diagram and the fitting of the non-linear zone in the neighborhood of the intersection. Because manufacturers of testing machines usually develop their own software for machine control and processing of test parameters, few algorithms for the computation of Sy are publicly available (see Ref 2, for instance). A result for Sy should be accompanied by a parameter that quantifies the accuracy in its determination. This parameter, which represents the uncertainty associated with the measurement, allows realistic comparison of results from different laboratories, within a laboratory, or with reference values given in specifica- tions or standards. In addition, to comply with the requirements of the International Standard ISO/IEC 17025 [3], accredited labora- tories shall estimate the uncertainty of measurement using accepted methods of analysis [3]. The need for an internationally accepted procedure for expressing measurement uncertainty led to the publication of the Guide to the Expression of Uncertainty in Measurement [4], hereinafter referred to as the GUM. The GUM proposes a standard procedure, known as the GUM uncertainty framework, mainly devoted to linear (or linearized) measurement models. This framework can be applied to a wide range of prob- lems, but has limitations. Supplement 1 to the GUM gives an al- ternative procedure based on a Monte Carlo method, that can be applied in cases where the GUM uncertainty framework is not ap- plicable or its validity is not clear [5]. We briefly introduce both alternatives in this work. To our knowledge, there are only two published approaches on the evaluation of the uncertainty associated with Sy: (i) a simplified methodology for evaluating uncertainties of measurements pro- posed by Loveday [6], which is addressed in an informative annex of the tensile standard EN-10002-1 [7], and (ii) a method published in the Manual of Codes of Practice for the Determination of Uncer- tainties in Mechanical Tests on Metallic Materials, developed within the European Project UNCERT [8]. Both methods are based on the GUM uncertainty framework. Manuscript received April 5, 2012; accepted for publication August 2, 2012; published online January 22, 2013. 1Research Engineer, INTI-Co´rdoba, and Assistant Professor, Departamento de Estructuras, Universidad Nacional de Co´rdoba, Av. Ve´lez Sarsfield 1561, Co´rdoba, X5000JKC, Argentina, e-mail: ariel.matusevich@gmail.com 2Professor, Departamento de Estructuras, Universidad Nacional de Co´rdoba, Av. Ve´lez Sarsfield 1611, Co´rdoba, X5016GCA Argentina, e-mail: jmassa@efn.uncor.edu 3Head of the Materials Division, INTI-Co´rdoba, and Assistant Professor, Departamento de Materiales y Tecnologı´a, Universidad Nacional de Co´rdoba, Av. Ve´lez Sarsfield 1561, Co´rdoba, X5000JKC, Argentina, e-mail: rmancini@inti.gob.ar Copyright VC 2013 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. 1 Journal of Testing and Evaluation, Vol. 41, No. 2, 2013 Available online at www.astm.org doi:10.1520/JTE20120100 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. The simplified methodology for evaluating uncertainties of
tension-test parameters is based upon an “error budget” concept that uses tolerances specified in testing and calibration standards. In such an approach, uncertainties are calculated as percentages of the parameters they are associated with (2.3 % for the case of Sy) and can be regarded as upper bound estimations [6]. Code of Practice 7 (CoP 7) of the UNCERT Manual [8] deals with the determination of uncertainties in tensile testing. The procedure given in CoP 7 for the evaluation of the uncer- tainty in the computation of Sy has flaws; the method takes into account the uncertainties in the estimation of the ordinate inter- cept and the slope of the straight line that fits the elastic part of the stress-strain diagram, but ignores the correlation between both parameters and does not consider the uncertainty associated with the approximation of the non linear part of the curve. If the tension test is carried out using a computer-controlled testing machine, test parameters are automatically processed by dedi- cated software; because typical testing-machine programs do not return fitting parameters and their standard deviations, which are essential for uncertainty estimation, the application of CoP 7 requires reanalysis of tension-test raw data. For this reason, CoP 7 also gives guidance on how to calculate Sy from tension-test data [8]. In this work, we present computer procedures for the computa- tion of Sy and for the estimation of the uncertainty in its computa- tion. The calculation of Sy is based on searching a portion of the raw-data curve in the neighborhood of its intersection with the off- set line, in which a linear fit is valid. Then, we determine Sy as the point of intersection between two straight lines; for the fitting of each line, we propose the use of a weighted total least-squares algorithm that considers uncertainties in both ordinates and abscis- sas [9]. To evaluate the uncertainty in the determination of Sy, in accordance with the GUM uncertainty framework, we take into account the uncertainties and correlations between the fitting pa- rameters involved in its calculation. We validate the proposed method following the guidelines of GUM S1, using the method of propagation of probability distributions through Monte Carlo sim- ulation [5]. The implementation of the procedures proposed in this article for computation and uncertainty evaluation of Sy motivated the de- velopment of a MATLAB standalone application to post process tension-test raw data, to calculate typical tensile parameters and to estimate their uncertainties. To validate the software, we used ref- erence tension-test curves from several materials, obtained through ASCII files that are publicly available at the website of the National Physical Laboratory of the United Kingdom (NPL) [10]. These datasets, developed for tensile-software validation, were originated in the European-Union-funded project TEN- STAND (tensile standard) and represent typical tensile character- istics of a variety of industrially important materials [11]. In this work, we present the validation of the procedure for the determi- nation of Sy. This paper is organized as follows. First, we present the pro- posed method for the computation of Sy. After giving an introduc- tion to the estimation of uncertainties according to the GUM, we describe the uncertainty-evaluation procedure for Sy. Then, we present the computational tool that implements the proposed methods. Next, we analyze validation exercises and numerical examples. Finally, we present the conclusions of the paper. Proposed Procedure for the Computation of Offset Yield Strength Offset yield strength is the quotient between yield force Fy and the original cross-sectional area A0 of the test specimen Sy ¼ Fy A0 (1) Yield force is obtained as the point of intersection between the load-extension curve (F – d) and a line parallel to its proportional region. In this section, we present a procedure for the computation of Fy, schematized in Fig. 1, that can be summarized as follows: • Determine the upper limit UL and the lower limit LL that define the proportional part of the load-extension curve. • Compute the parameters of a line I, FI ¼ b1 þ md; that best fits load-extension data in the region delimited by UL and LL. • Calculate the ordinate intercept of an offset line II, FII ¼ b2 þ md; drawn at a distance from line I whose hori- zontal projection is bLe, where b is typically 0.001 or 0.002 and Le is the initial gauge length of the extensometer used in the tension test. • Compute the parameters of a third line III, FIII ¼ b3 þ m3 d; that fits a small portion of the load-extension curve in the FIG. 1—Scheme for the computation of Fy. 2 JOURNAL OF TESTING AND EVALUATION Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. neighborhood of its intersection with line II, where a linear
fit is valid. • Obtain Fy as the point of intersection between line II and line III. In what follows, we detail each of the preceding items. Upper Limit and Lower Limit of the Proportional Region The points UL and LL, shown in Fig. 1, delimit the region of the load-extension curve where data points best seem to follow a straight line. The procedure for the computation of these points, schematized in Fig. 2, is based on a method given in CoP 7 of the UNCERT manual [8]. To reduce the number of calculations in this analysis, we exclude the portion of the F–d curve beyond the maximum load Fu. Upper Limit—By removing data pairs in the direction indi- cated in Fig. 2(a), we obtain datasets of decreasing number of ex- perimental points. Using ordinary linear regression, we compute the slopes m of straight lines that fit each set of F–d data m ¼ n Xn i¼1 diFi Xn i¼1 di Xn i¼1 Fi n Xn i¼1 d2i Xn i¼1 di !2 (2) where n is the number of data pairs. According to CoP 7 [8], we reach the upper limit UL when the following ratio results minimum: urel ¼ u mð Þ m (3) In Eq 3, m is the slope calculated by Eq 2 and u(m) is its standard deviation u mð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n Xn i¼1 F2i b Xn i¼1 diFi ! Xn i¼1 Fi b Xn i¼1 di !Xn i¼1 Fi n 2ð Þ nXn i¼1 d2i Xn i¼1 di !224 35 vuuuuuuuut (4) where b is the ordinate intercept of the fitted line. In some cases, when anomalies at the start of the tension test occur (these anoma- lies are typically associated with specimen straightening and ini- tial slackness in the load train), the minimization of Eq 3 leads to the initial segment of the F–d curve (see Figs. 1 and 2). To avoid this outcome, we propose the minimization of urel ¼ u mð Þ m2 (5) that discards lines of lower slope. Lower Limit—Once the upper limit has been determined, we use the same procedure to obtain the lower limit LL; in this case, we search for the LL in the opposite direction, as indicated in Fig. 2(b). Least-Squares Fitting of the Proportional Region Force-elongation data within the proportional region is usually fit- ted by ordinary linear regression. This easy-to-apply technique assumes that abscissas, extension data in our case, are known exactly; because uncertainty associated with extension measure- ment cannot be considered negligible, this assumption does not hold. On the other hand, the problem of fitting a straight line with errors in both coordinates is not straightforward; it consists in finding the parameters of the line Y ¼ bþ mX that minimize the following function [12]: v2 ¼ Xn k¼1 xk Xkð Þ2 u2x;k þ yk Ykð Þ2 u2y;k " # (6) where (yk, xk) denote n given data pairs with estimated standard deviations (uy,k, ux,k), whereas (Yk, Xk) are points of the straight line. In this work, we propose the use of a weighted total least- squares algorithm to solve Eq 6, referred to as WTLS [9], which treats x-and-y data symmetrically. In such a method, the two- dimensional minimization problem is reduced to the one- dimensional search of a minimum, using a different parametriza- tion of the straight line. Krystek and Anton, authors of the WTLS algorithm, have implemented this method as a MATLAB function that can be obtained through the MATLAB Central (File Exchange) [13]. Experimental points (yk, xk) and their associated standard deviations (uy,k, ux,k) are the input arguments of this func- tion; the program returns the parameters of the fitted line, the min- imum value of v2 found, and the complete uncertainty matrix, that is, variances and covariance of the fitting parameters. In the WTLS algorithm, the weights associated with data points depend on the standard uncertainties of experimental data; therefore, the effectiveness of the method depends on the correct evaluation of these uncertainties.FIG. 2—Determination of the points that delimit the proportional region. MATUSEVICH ETAL. ON OFFSET YIELD STRENGTH 3 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. Uncertainty in Load-Extension Data—The calibra-
tion certificates of the load cell and the extensometer used in the tension test list uncertainties for different ranges of force and exten- sion, respectively. From these certificates we can interpolate the uncertainties (uy,k, ux,k) that correspond to each data pair (yk, xk). In a simplified approach, we may assume proportional uncer- tainties for load-and-extension data and use the tolerances of measuring devices specified in standards. Current tensile stand- ards, ASTM E8/E8M-11 [14] and ISO 6892-1 [15], stipulate simi- lar accuracy requirements for the measuring devices involved in the determination of Sy. In this work, we adopt the classifications for testing machines and extensometers given in ISO 7500-1 [16] and ISO 9513 [17], respectively. Tables 1 and 2 list reference tol- erances for testing machines and extensometers, extracted from Section 3 of the UNCERT manual [8]. Parameters of the Offset Line The ordinate intercept of the offset line II, drawn at a horizontal distance b Le from line I, is obtained as follows (see the lower region of Fig. 1): b2 þ m bLeð Þ ¼ b1 ! b2 ¼ b1 m bLeð Þ (7) Fitting of the F–d Curve in the Neighborhood of the Intersection When the fracture point f lies below the offset line II (see Fig. 1), it is possible to calculate the point of intersection between the off- set line and the raw-data curve. By inspecting experimental points of increasing extension, we search for the first point below the offset line; this point, denoted by B ¼ dB;FBð Þ, satisfies the following condition: FB < b2 þ mdB (8) The point that immediately precedes point B is designated as A ¼ dA;FAð Þ; points A and B are shown in the enlarged zone of Fig. 1. We can approximate the nonlinear zone of the raw-data curve by the straight line that passes through points A and B. To improve this rough approximation, we use the WTLS algorithm to obtain the parameters of line III, that fits several points on the right of B and on the left of A. To determine the number of data pairs used for fitting line III, nIII, we use the underlying idea of an algorithm by Goodman et al. [2], that chooses straight line seg- ments by comparing the fits of a straight line and a parabola. Determination of nIII—We add np points on the right of B and np points on the left of A to obtain datasets of increasing number of points nIII ¼ 2þ 2np (9) Using ordinary least squares, we fit each of the resulting data- sets by a straight line and a parabola. To compare both fits, we cal- culate the mean square error of the linear fit, MSEl MSEl ¼ 1 n 2 Xn i¼1 bl þ mldið Þ Fi½ 2 (10) and the mean square error of the quadratic fit, MSEq MSEq ¼ 1 n 3 Xn i¼1 bq þ mqdi þ cqd2i Fi 2 (11) When the portion of the curve under analysis is close to linear, the gain in accuracy of fit by using a parabola will be small and the quadratic term cq in the fitted polynomial will be close to zero. Sometimes, a linear fit may be as good or better than a quadratic fit and consequently, MSEl MSEq ! MSEq MSEl 1 (12) which is mathematically possible because the denominator of MSEl is larger than the denominator of MSEq. If we denote R ¼ MSEq=MSEl, fitted data can be considered essentially linear whenever R 1 [2]. In the proposed method for the determination of nIII, we con- sider 2 np 15 and pick the value of np that gives the longer interval with R 1; otherwise, we choose np¼ 2 (nIII¼ 6). Point of Intersection To obtain yield extension dy, we set the equation of line II equal to the equation of line III and solve for d b2 þ mdy ¼ b3 þ m3 dy ! dy ¼ b3 b2 m m3 (13) After replacing Eq 7 into Eq 13, we calculate yield force Fy as follows: IIðdyÞ ¼ IIIðdyÞ ¼ Fy ! Fy ¼ mb3 m3b1 þ b mm3Le m m3 (14) As Eq 14 indicates, five parameters, b1, b3, m, m3, and Le are involved in the computation of Fy. Introduction to Uncertainty Estimation According to the GUM The standard uncertainty u(y) associated with the measurement result y of a quantity Y is a parameter that characterizes the TABLE 1—Uncertainty in force measuring devices (Ref 8). Class of Machine Expanded Uncertainty (k¼ 2), U % 0.5 60.44 1 60.88 2 61.75 3 62.61 TABLE 2—Uncertainty in strain measurement using extensometers (Ref 8). Class of Extensometer Expanded Uncertainty (k¼ 2), U % 0.2 60.2 0.5 60.5 1 61.0 2 62.0 4 JOURNAL OF TESTING AND EVALUATION Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. dispersion of the values that could reasonably be attributed to Y,
expressed as a standard deviation [4]. Uncertainty evaluation involves the use of a model to represent our knowledge about the measurement process. The model relates the quantity subject to measurement, Y, referred to as measurand, and input quantities X1, X2, …, XN Y ¼ fm X1; X2;…;XNð Þ (15) To express information, usually incomplete, about input contribu- tions, the model formulation involves assignment of probability density functions (PDFs) to input quantities gX1 n1ð Þ; gX2 n2ð Þ;…; gXN nNð Þ (16) where n1, n2, …, nN represent possible values of X1, X2, …, XN, respectively. In the GUM approach, the best estimates xi of input quantities are the expected values of Xi, where i¼ 1, 2, …, N. As part of the measurement process, we estimate standard uncer- tainties u(x1), u(x2), …, u(xN), and covariances u(xi, xj), i= j, associ- ated with input contributions. If uncertainties u(xi) are estimated by statistical means from a number of repeated observations of Xi, they are designated as Type A according to the GUM; if they are eval- uated by any other means (e.g., extracted from a calibration report, or estimated based on past experience) they are classified as Type B. In general, the aims of a measurement process are: (i) the esti- mation of the expected value y of Y, (ii) the evaluation of the standard uncertainty u(y) associated with the expected value, and (iii) the determination of the lower limit and the higher limit of an interval (expanded uncertainty) that can be expected to contain a large prescribed portion of the values that can reasonably be attributed to Y. To achieve these goals, the GUM proposes a framework based on the law of propagation of uncertainty, whereas its supplement 1 (GUM S1) uses the method of propaga- tion of probability distributions through Monte Carlo simulation; both approaches are briefly discussed next. Propagation of Uncertainty Because the model given by Eq 15 is also valid for estimated quantities, the measurement result y is obtained as y ¼ fm x1; x2;…; xNð Þ (17) To evaluate the uncertainty u(y) associated with y, the GUM proposes the use of the law of propagation of uncertainty u yð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN i¼1 @fm @xi u xið Þ 2 þ 2 XN1 i¼1 XN j¼iþ1 @fm @xi @fm @xj u xi; xj vuut (18) based on a first-order Taylor series expansion of the measurement model, valid when Eq 15 is either linear or can be approximated by a linear function. The partial derivatives in Eq 18 are usually referred to as sensitivity coefficients and are denoted by cxi , where i¼ 1, 2, …, N. The computation of expanded uncertainty requires the use of the PDF of the output quantity. Instead of calculating the output PDF explicitly, the GUM uncertainty framework, based on the Central Limit Theorem, assumes that the output PDF is either Gaussian or a t-distribution. The expanded uncertainty U is calcu- lated as U ¼ kuðyÞ (19) so that the interval [y U, yþU] has a prescribed coverage prob- ability of the output distribution. The parameter k in Eq 19, known as coverage factor, takes a value of 2 for the case of a normal out- put distribution and a coverage probability of 95.45 %. The GUM gives a procedure to calculate k, based on the estimation of the (effective) degrees of freedom of input quantities, through the Welch-Satterthwaite equation [4]; however, the procedure has inconsistencies and limitations [18–20]. To illustrate the GUM uncertainty framework we present the diagram of Fig. 3, extracted from the excellent work by Sommer and Siebert on modeling of measurements for uncertainty evalua- tion [21]. Propagation of Probability Distributions As Fig. 4 illustrates, the PDFs assigned to input quantities can be propagated through the measurement model to obtain the PDF of the output quantity Y, gY (g). Because the propagation of distributions can be carried out analytically only in special cases, it is often implemented using a Monte Carlo method (MCM) [22]. The MCM performs a charac- terization of the input quantities based on the random sampling of their associated probability density functions; GUM S1 provides specific details of the method and examples of its application [5]. A step-by-step procedure of the method can be summarized as follows [23]. FIG. 3—Illustration of the GUM uncertainty framework. MATUSEVICH ETAL. ON OFFSET YIELD STRENGTH 5 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. • Select the number M of Monte Carlo trials to be made.
• By performing a random sampling of the PDF of each input quantity, obtain a set of M vectors n11; …; n1N ; …; nM1 ; …; nMN g; where nji is the jth random sample of ni: • Use the model to compute a set g1 ¼ fm n11; …; n1N ; …; gM ¼ fm nM1 ; …; nMN g of independent random samples of the output PDF, gY (g). • Calculate the estimate y and the standard uncertainty u(y) of the output quantity Y as the arithmetic mean and the stand- ard deviation of g1; …; gM ; respectively. • Sort the model values g1; …; gM in increasing order, g 1ð Þ … g Mð Þ ; and use the sorted values to determine the (probabilistically symmetric or shortest) coverage interval gðLÞ; gðHÞ at a coverage probability p, where H – L equals the integer part of pM þ 12. Validation of the GUM Uncertainty Framework Although the GUM uncertainty framework works well in many sit- uations, the applicability of the method depends most notably on • a valid linear characterization of the model through a first order Taylor approximation, • the applicability of the Welch-Satterthwaite formula [4] for the estimation of effective degrees of freedom, and • the assumption that the probability distribution for the out- put quantity is either Gaussian or a scaled and shifted t- distribution. Because the method of propagation of distributions through Monte Carlo simulation does not have these limitations, it can be used to validate procedures for uncertainty evaluation that are based on the GUM uncertainty framework. In the validation procedure, we determine whether the interval y U ; yþ U½ ; calculated through the GUM uncertainty framework, agrees with the coverage interval gðLÞ; gðHÞ provided by a Monte Carlo method, to a stipulated com- parison accuracy e. When the following conditions are satisfied: dlow ¼ y Uð Þ g Lð Þ e (20) dhigh ¼ yþ Uð Þ g Hð Þ e (21) the comparison is successful and the GUM uncertainty framework has been validated in this instance [5]. The numerical tolerance e in Eqs 20 and 21 depends on the number ndig of significant digits regarded as meaningful in the numerical value of u(y); usually, ndig¼ 1 or ndig¼ 2. To determine e, we express the value of u(y) in the form c 10l, where c is an ndig-digit integer and l is an inte- ger; then, we calculate the comparison accuracy as follows [5]: e ¼ 1 2 10l (22) Uncertainty in the Computation of Offset Yield Strength Using Eqs 1 and 14, we obtain the following mathematical model for the computation of offset yield strength: Sy ¼ Fy A0 ; where Fy¼ mb3 m3b1 þ b mm3Leð Þ= m m3ð Þ (23) As Eq 23 indicates, the proposed model depends on the parame- ters of lines II and III, Le, and A0. Measurements of force and extension influence the computation of the parameters of lines II and III; in fact, the WTLS algorithm requires the estimation of the standard uncertainties associated with ordinates and abscissas, force and extension, respectively. Other sources that influence force-extension recordings do not appear explicitly in the model equation and are difficult to quan- tify; they include: • the alignment of the test specimen, that affects the resulting slope m of the proportional part of the force-extension dia- gram [24], • testing-machine characteristics (e.g., stiffness, method and control of operation), and • speed of testing (within the range allowed in the corre- sponding tensile standard). The application of Eq 18 for the evaluation of the standard uncertainty u(Sy) leads to u Sy ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cFy uðFyÞ 2þ cA0 uðA0Þ½ 2 q (24) because Fy and A0 are uncorrelated. The sensitivity coefficients in Eq 24 are cFy ¼ @Sy @Fy ¼ 1 A0 ; cA0 ¼ @Sy @A0 ¼ Fy A20 (25) If additional sources of uncertainty (that do not appear in the model equation) were evaluated, their squared standard uncertain- ties (variances) could be added inside the square root in Eq 24 [4]. To calculate expanded uncertainty U at a coverage probability of 95.45 %, we assume that the output PDF is Gaussian and use Eq 19 with k¼ 2. The remainder of this section is devoted to the evaluation of the uncertainties associated with the input quantities indicated in Eq 24. Uncertainty in the Computation of Fy To apply the law of propagation of uncertainty for the case of Eq 14, we must consider the mutual correlation between slope and ordinate intercept in line II and also in line III FIG. 4—Illustration of the propagation of probability distributions. 6 JOURNAL OF TESTING AND EVALUATION Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. u Fy
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cb1 u b1ð Þ½ 2þ cmu mð Þ½ 2 þ cb3 u b3ð Þ½ 2þ cm3 u m3ð Þ½ 2 þ cLe u Leð Þ½ 2þ2cb1 cmu b1;mð Þ þ 2cb3 cm3 u b3;m3ð Þ vuuuuut (26) The sensitivity coefficients in Eq 26 are given by the following expressions: cb1 ¼ @Fy @b1 ¼ m3 m m3 (27) cm ¼ @Fy @m ¼ m3 b1 b3 bLem3ð Þ m m3ð Þ2 (28) cb3 ¼ @Fy @b3 ¼ m m m3 (29) cm3 ¼ @Fy @m3 ¼ m b3 b1 þ bLemð Þ m m3ð Þ2 (30) cLe ¼ @Fy @Le ¼ b mm3 m m3 (31) Uncertainty in the Computation of the Parameters of Lines II and III—The uncertainty matrix associated with the fitting parameters of a straight line F ¼ bþ md is R ¼ u 2 mð Þ u m; bð Þ u m; bð Þ u2 bð Þ (32) where u2 denotes variance (square of standard uncertainty) and u(m, b) is the covariance of the fitting parameters m and b. For the case of the WTLS algorithm, the procedure for the evaluation of Eq 32 can be found in Ref 9. The MATLAB function pro- grammed by the authors of the WTLS algorithm, publicly avail- able at the MATLAB Central [13], returns the parameters of the fitted line with their variances and covariance. Our program for the computation of Sy and its uncertainty includes the cited func- tion as a subroutine. Uncertainty in Le—To estimate the standard uncertainty in the initial length of the extensometer we consider two contribu- tions: (i) a relative standard uncertainty of 60.5 % associated with a Class 1 extensometer (see Table 2), and (ii) a relative error of 61 % attributed to the positioning of the extensometer on the test specimen u Leð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:005Leð Þ2þ 0:01Leffiffiffi3p 2s (33) In Eq 33, we have assumed a rectangular PDF for the distribu- tion of the error associated with the positioning of the extensometer. Uncertainty in the Calculation of the Initial Cross-Sectional Area A0 Determination of the initial cross-sectional area requires the mea- surement of specific dimensions of the test specimen using appro- priate instruments. We can identify Type A sources of uncertainty involved in the measurement of the test-specimen dimensions and Type B contributions related to the calibration of measuring instruments. The computation of A0 and the evaluation of its standard uncertainty involve the following steps. I. Find the mathematical expression that relates specimen dimensions (input quantities) and A0 (measurand) A0 ¼ fm X1; X2; …; XNð Þ (34) For the case of a specimen with rectangular cross- sectional area, the required dimensions are the width w and the height h of the specimen reduced section; the relation between the input quantities and the measurand is A0 ¼ w h. II. To calculate cross-sectional area, determine the esti- mates of input quantities xi as mean values of n measurements xi ¼ xi ¼ 1 n Xn k¼1 xik (35) with n 3. III. Use the law of propagation of uncertainty given by Eq 18 to compute the combined standard uncertainty asso- ciated with the measurement of A0 u A0ð Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cx 1 uðx1Þ½ 2þ þ cxN uðxN Þ½ 2 q (36) In Eq 36, we have assumed that input quantities are independ- ent. However, if we use the same instrument to measure two or more dimensions of the test specimen, the resulting estimates are correlated; however, the degree of correlation is usually small and can be safely ignored [25]. To evaluate the standard uncertainties u xið Þ in Eq 36, we use the following procedure: I. For each input dimension, compute standard deviation s xið Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n 1 Xn k¼1 xi k xið Þ2 s (37) and experimental standard deviation s xið Þ ¼ t s xið Þffiffiffi n p (38) where t is the student t factor that corresponds to a level of confidence of 68.27 % (one standard deviation). II. Estimate the standard uncertainty uCAL of the instru- ment used in the measurement of xi, from its certificate of calibration. III. Combine the uncertainties obtained in steps (i) and (ii) to obtain u(xi). u xið Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2ðxiÞ þ uCALð Þ2 q (39) If no information about the measurement process is available, we assume that A0 has been determined with an accuracy of 61 % to comply with the requirements of ASTM E8-11 [14] or ISO 6892-1 [15]; considering a level of confidence of 95.45 % (k¼ 2) for this requirement, we obtain MATUSEVICH ETAL. ON OFFSET YIELD STRENGTH 7 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. u A0ð Þ ¼ 0:005A0 (40)
Computer Implementation In this section, we describe the main features of the program INcerTI, developed for the computation of tension-test parameters and for the estimation of their associated uncertainties. The name INcerTI is a play on the Spanish word for uncertainty (incertidum- bre) and INTI (Instituto Nacional de Tecnologı´a Industrial) [26], the institution that supported the project. This computational tool was developed in the MATLAB programming language [27] as a standalone application and does not require MATLAB to be in- stalled in the system. INcerTI is run through the graphical user interface shown in Fig. 5. FIG. 5—Main command window of INcerTI. FIG. 6—Input of the number of diameter estimates. FIG. 7—Input of diameter estimates. 8 JOURNAL OF TESTING AND EVALUATION Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. INcerTI enables the calculation and the uncertainty evaluation
of the following tensile parameters: (i) tensile strength, (ii) offset yield strength, (iii) yield points (if they were present), (iv) yield strength for a specified total extension (extension-under-load method), (v) percentage elongation after fracture, (vi) percentage reduction of area (for specimens of circular cross section), and (vii) tensile strain-hardening exponents (n values). In what follows we describe the stages required for the computation of Sy. Data Input Computer controlled testing machines typically employ a dedi- cated software that stores test data and force-extension recordings in text files. To process tensile parameters, INcerTI is capable of reading force-extension recordings from three types of files: (i) ASCII files produced by Series IX software from INSTRON (which controls the testing machine in our laboratory), (ii) stand- ard spreadsheets, and (iii) ASCII validation files, available at the web site of NPL [10]. We use the “Open” menu to locate and load files; the corresponding force-extension diagram is plotted in the command window once the file has been loaded. To examine dif- ferent regions of the diagram and interpret results, the tools “Zoom,” “Pan,” and “Limits” are very useful (see the “Plot- analysis” panel at the bottom of the command window in Fig. 5). Using the “Axes” menu, we can change the axes of the diagram between force–extension, force–strain or stress-strain. Determination of Cross-Sectional Area To determine the cross-sectional area and its uncertainty, we select the type of specimen using the “Type” pop-up menu (see Fig. 5). Available options include: unmachined test pieces, machined specimens whose cross-sectional area may be circular, annular, or rectangular, and tension specimens taken from large-diameter tu- bular products. When we click the “Area” button, the program asks for information about the measurement process in an interac- tive way, then computes cross-sectional area and its uncertainty. Figures 6–8 show the interactive input of data for the case of a test piece of circular cross section whose diameter has been measured four times using a caliper. When cross-sectional area is given with no accompanying in- formation about its estimation, we choose the “Given area” option available in the “Type” pop-up menu; then, after clicking the “Area” button we input cross-sectional area and an estimate of the accuracy in its determination, as Fig. 9 illustrates. Computation of Offset Yield Strength The proposed method for the computation of Sy requires the esti- mation of the standard uncertainties associated with force- extension data; INcerTI provides two methods for the evaluation of these uncertainties, which can be selected through the “Options” menu in the main command window. One method interpolates the uncertainties for each extension-force data pair, FIG. 8—Uncertainty of the measuring instrument. FIG. 9—Input of a given cross-sectional area. FIG. 10—Data input for the computation of offset yield strength. TABLE 3—Premium quality ASCII datasets. TENSTAND Dataset Material Le(mm) A0 (mm2) 1 Nimonic 75, CRM 661 50 78.46 6 Nimonic 75, CRM 661 50 78.54 10 13 % Mn steel 50 77.55 17 316L Stainless steel 50 78.65 22 Tin coated packaging steel 80 3.97 30 Sheet steel—DX56 80 14.17 38 Aluminum sheet—soft AA5182 80 4.62 42 Aluminum sheet—soft AA1050 80 14.48 46 Aluminum sheet—soft AA5182 80 29.59 50 Sheet steel—DX56 50 8.77 57 Synthetic digital curve—zero noise 50 78.54 61 Synthetic digital curve—0.5 % noise 50 78.54 63 Synthetic digital curve—1 % noise 50 78.54 MATUSEVICH ETAL. ON OFFSET YIELD STRENGTH 9 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. by processing text files that contain actual calibration data from
the extensometer and from the testing machine. The other method assumes, in a simplified approach, that uncertainties associated with extension and force are proportional to specified uncertainty tolerances of the extensometer and the testing machine, respec- tively. When we use the latter method, we specify the class of ex- tensometer and the class of testing machine using the third and fourth menus in the command window (see Fig. 5). The dialog window shown in Fig. 10 appears when we click the “Calculate” button of the offset-yield-strength panel; the percentage uncertain- ties associated with force and extension correspond to tolerances listed in Tables 1 and 2 for the classes of instruments we have selected, though we can type different values. Figure 5 illustrates a typical output calculation for offset yield strength. INcerTI shows the resulting values for Sy and its expanded uncertainty in designated boxes in the main screen; in addition, the program draws a graph that includes the following features: (i) the original F–d diagram, (ii) the line that fits the pro- portional region of the diagram (line I), (iii) two circles that high- light the points UL and LL that delimit the proportional region, (iv) the offset line (line II), the line that approximates the non- proportional region (line III), and (v) a cross that indicates the point of intersection between II and III. Analysis of Case Studies To study the validity of the procedure for the computation of Sy, we use a set of ASCII datasets with agreed values for the tensile parameters, developed as part of the TENSTAND project [11]. The set of files covers a range of industrially important materials that include: structural steels, stainless steels, aluminum alloys, tin coated packaging steels, BCR Nimonic 75 tensile reference mate- rial (CRM 661), and synthetic datafiles with different levels of noise in force data (0 %, 0.5 %, and 1 %). TABLE 4—Procedure validation for 0.2 % offset yield strength. Case (a) Case (b) TENSTAND Dataset Sy 0.2 % (MPa) Agreed Values Sy 6 U Difference (%) Sy 6 U Difference (%) 1 309.6 – 310.1 309.6 6 3.2 0 310.0 6 3.1 0 6 308.0 – 308.6 308.4 6 3.1 0 308.8 6 3.1 0.07 10 337.1 – 337.2 337.0 6 3.6 0.02 337.3 6 3.4 0.02 17 261.0 – 261.2 260.1 6 2.8 0.36 261.1 6 2.7 0 22 562.5 – 564.6 562.7 6 5.8 0 562.6 6 5.7 0 30 162.7 – 162.9 162.8 6 1.7 0 162.8 6 1.6 0 38 396.4 – 397.1 396.5 6 4.0 0 396.6 6 4.0 0 42 30.01 – 30.05 30.08 6 0.32 0.09 30.08 6 0.31 0.08 46 134.5 – 134.8 134.4 6 1.4 –0.09 134.4 6 1.3 0.06 50 163.9 – 164.0 163.9 6 1.7 0 163.9 6 1.6 0 57 434.3 434.1 6 4.6 0.04 434.4 6 4.4 0.02 61 438.1 – 441.6 434.8 6 4.4 0.75 435.0 6 4.4 0.70 63 446.5 – 448.2 434.7 6 4.4 2.65 434.9 6 4.4 2.60 TABLE 5—Procedure validation for 0.1 % offset yield strength. Case (a) Case (b) TENSTAND Dataset Sy 0.2 % (MPa) Agreed Values Sy 6 U Difference (%) Sy 6 U Difference (%) 1 303.4 – 304.5 303.3 6 3.3 0.04 304.2 6 3.1 0 6 300.5 – 301.8 301.3 6 3.3 0 302.2 6 3.1 0.14 10 334.5 – 334.9 334.0 6 3.5 0.14 335.0 6 3.4 0.02 17 244.7 – 245.2 243.0 6 2.7 0.70 245.0 6 2.5 0 22 525.6 – 530.6 525.5 6 6.0 0.03 525.3 6 5.6 0.05 30 157.2 – 157.6 157.5 6 1.6 0 157.5 6 1.6 0 38 385.2 – 386.8 385.7 6 4.0 0 385.7 6 3.9 0 42 26.48 – 26.55 26.56 6 0.3 0.06 26.56 6 0.28 0.05 46 133.4 – 133.9 133.7 6 1.4 0 134.0 6 1.3 0.07 50 158.6 –158.7 158.8 6 1.6 0.05 158.8 6 1.6 0.06 57 432.4 432.3 6 4.6 0.03 432.5 6 4.4 0.03 61 431.8 – 434.1 432.0 6 4.4 0 432.2 6 4.3 0 63 429.6 – 432.7 432.7 6 4.4 0 432.7 6 4.3 0 10 JOURNAL OF TESTING AND EVALUATION Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. Table 3 lists details of the thirteen datasets used in the present
validation analysis. TENSTAND WP2 Report [11] provides a range of agreed values for Sy (for typical offsets, 0.1 % and 0.2 %) for all the datasets, with the exception of the synthetic file with 0 % noise (dataset 57), where an absolute value could be deter- mined. Results for Sy that lie within the interval or coincide with its boundaries are considered true values. The methods proposed in this article for the computation of Sy and for the evaluation of its uncertainty require the estimation of the standard uncertainties associated with force-and-elongation FIG. 11—Analysis of dataset 63, synthetic file with 1 % noise. TABLE 6—Comparison between the GUM uncertainty framework and a Monte Carlo method. TENSTAND Dataset Sy 0.2 % (MPa) U (Sy) (MPa) [Sy U, SyþU] (MPa) Sy (MCM) (MPa) [g(L), g(H)] (MPa) dlow dhigh 1 309.552 3.151 [306.401, 312.703] 309.560 [306.412, 312.717] 0.01 0.01 6 308.386 3.145 [305.241, 311.531] 308.393 [305.252, 311.543] 0.01 0.01 10 337.039 3.587 [333.452, 340.625] 337.049 [333.460, 340.644] 0.01 0.02 17 260.057 2.840 [257.217, 262.897] 260.059 [257.196, 262.912] 0.02 0.01 22 562.667 5.752 [556.916, 568.419] 562.681 [556.917, 568.441] 0.00 0.02 30 162.813 1.653 [161.160, 164.466] 162.817 [161.163, 164.470] 0.00 0.00 38 396.546 4.030 [392.516, 400.576] 396.558 [392.525, 400.602] 0.01 0.03 42 30.0764 0.317 [29.7594, 30.3934] 30.0771 [29.7599, 30.3942] 0.001 0.001 46 134.385 1.367 [133.019, 135.752] 134.388 [133.024, 135.758] 0.01 0.01 50 163.891 1.664 [162.227, 165.554] 163.894 [162.235, 165.563] 0.01 0.01 57 434.137 4.633 [429.504, 438.770] 434.143 [429.531, 438.804] 0.03 0.03 61 434.806 4.402 [430.404, 439.209] 434.818 [430.408, 439.217] 0.00 0.01 63 434.681 4.401 [430.280, 439.082] 434.690 [430.294, 439.099] 0.01 0.02 MATUSEVICH ETAL. ON OFFSET YIELD STRENGTH 11 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. recordings and information about the measurement of the speci-
men cross-sectional area. Because TENSTAND documentation does not provide this information, to carry out analyses we make the following considerations based on the fact that the tension tests that produced the datasets were carried out according to EN 10002-1 [7]: • We evaluate the standard uncertainty associated with the determination of cross-sectional area using Eq 40, which assumes the minimum accuracy required by EN 10002-1 and a normal probability distribution. • To estimate the standard uncertainties associated with load- and-elongation data we use the tolerances of measuring devices listed in Tables 1 and 2. From all the possible com- binations of measuring instruments that comply with EN 10002-1, we analyze two cases, designated as (a) and (b). In combination (a), both the extensometer and testing machine are Class 1 (minimum requirement), whereas in combination (b) the extensometer is Class 1 and the testing machine is Class 0.5. Tables 4 and 5 list results and agreed values for Sy when offsets are 0.2 % and 0.1 %, respectively. In these tables, relative differ- ences between computed results and reference values have been calculated with respect to the nearest limit of the reference inter- val. All results are accompanied by their expanded uncertainties, for a level of confidence of 95.45 % (k¼ 2). Computation of 0.2 % Offset Yield Strength If we exclude from Table 4 the results that correspond to the syn- thetic files with noise (files 61 and 63), we observe that for combi- nation (a) of measuring devices, six values lie within the agreed intervals whereas a maximum difference of 0.36 % is displayed in the case of file 17; for combination (b), six results agree with ref- erence values (including file 17) whereas differences in the remaining cases are less than 0.07 %. Although results for synthetic files 61 and 63 differ signifi- cantly from agreed values (almost 3 % for file 63), calculations are very close to the reference value with 0 % noise (file 57); how- ever, the agreed ranges for these curves do not contain the exact result (0 % noise). In Fig. 11, we examine the region of the inter- section between the offset line and raw data curve for the case of file 63; data pairs used for the fitting of line III are highlighted. Because line II crosses a cloud of raw-data points, it is not clear which criterion must be adopted for defining the point of intersec- tion. As the program gave accurate values in all the cases that cor- respond to actual tension-test recordings, we may conclude that results provided by INcerTI for 0.2 % yield strength are reliable. Computation of 0.1 % Offset Yield Strength Table 5 demonstrates the validity of the proposed procedure for the computation of 0.1 % yield strength. Calculated values are accurate in all cases, with the exception of dataset 17; this case exhibits a 0.70 % variation for combination (a) of measuring instruments, but the result lies in the middle of the agreed interval for combination (b). Small differences in the results can be expected because of the lack of information about the testing machine and extensometer used in each case. Note that agreed values for synthetic files with noise (datasets 61 and 63) do contain the exact result with 0 % noise; in contrast with the agreed intervals that correspond to these datasets for 0.2 % yield strength. Validation of the Uncertainty-Evaluation Procedure Through Monte Carlo simulation, we examine the validity of the proposed uncertainty-evaluation procedure; we follow the guide- lines that are addressed in a previous section of this paper (Intro- duction to Uncertainty Estimation According to the GUM). The MCM requires the random sampling of the PDFs associ- ated with input quantities; for the model of Eq 23, the PDFs include normal distributions assigned to A0 and Le and bivariate normal distributions associated with the parameters of lines II and III. The implementation of the MCM is straightforward in the MATLAB environment, using built-in functions available in the Statistics Toolbox [27]. In the present validation exercise, we consider the computation of 0.2 % yield strength and assume that both the extensometer and the testing machine are Class 1 (combination (a) of measuring instruments). We computed the shortest coverage intervals at p ¼ 0:9545 for the thirteen datasets, using M ¼ 9 106 trials in each case. Table 6 lists the coverage intervals given by both meth- ods, the GUM uncertainty framework and the MCM; results are dis- played with a high number of significant digits for comparison purposes only. To compare the coverage intervals, we consider ndig¼ 2 to obtain comparison accuracies according to Eq 22; we use e ¼ 0:005 MPa for the case of file 42, and e ¼ 0:05 MPa for the remaining datasets. As Table 6 indicates, the comparison between the GUM uncertainty framework and the MCM is successful for the 13 case studies, because the conditions given by Eq 20 and Eq 21 are satisfied by a safe margin. Although the validation exercise does TABLE 7—Fractional uncertainty contributions, file 22, u(A0)¼ 0.005A0. Percentage Contributions to u(Sy) Offset (% Le) Sy (MPa) U (%) rA0 rLe rI rIII 0.01 353.4 1.43 48.99 4.46 5.95 40.60 0.02 405.7 1.15 75.08 6.62 3.08 15.22 0.05 477.2 1.08 85.30 5.26 0.57 8.87 0.10 525.3 1.06 89.38 2.93 0.10 7.59 0.20 562.6 1.01 97.80 1.35 0.01 0.83 TABLE 8—Fractional uncertainty contributions, file 22, u(A0)¼ 0.0025A0. Percentage Contributions to u(Sy) Offset (% Le) Sy (MPa) U (%) rA0 rLe rI rIII 0.01 353.4 1.14 19.36 7.05 9.41 64.18 0.02 405.7 0.76 42.96 15.16 7.04 34.84 0.05 477.2 0.65 59.19 14.61 1.59 24.61 0.10 525.3 0.61 67.78 8.89 0.30 23.03 0.20 562.6 0.52 91.76 5.08 0.05 3.11 12 JOURNAL OF TESTING AND EVALUATION Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. not guarantee the validity of the proposed uncertainty-evaluation
procedure in all possible cases, the method can be expected to work satisfactorily in situations similar to the ones studied here. Uncertainty Analysis To study uncertainty variability in the calculation of Sy, we con- sider dataset 22 as a case study. To analyze situations where the offset line intersects the raw-data curve in steeper zones, we examine results for five offsets distances from 0.01 % to 0.2 %. To assess the influence of uncertainty components in the evalua- tion of overall standard uncertainty uðSyÞ, we compute the frac- tional contributions of A0, Le, line I, and line III rA0 ¼ cA0 u A0ð Þ½ 2 u2 Sy (41) rL e ¼ cLe u Leð Þ 2 u2 Sy (42) rI ¼ cb1 u b1ð Þ½ 2þ cmu mð Þ½ 2þ2cb1 cmu b1;mð Þ u2 Sy (43) rIII ¼ cb3 u b3ð Þ½ 2þ cm3 u m3ð Þ½ 2þ2cb3 cm3 u b3;m3ð Þ u2 Sy (44) that represent variance contributions to total variance. Using options in the “Results” menu of INcerTI, we can visualize frac- tional uncertainty contributions in the form of pie charts. Table 7 lists (for five offset distances), results for Sy, their per- centage uncertainties U %, and percentage contributions of all components of the uncertainty budget. All cases were calculated using combination (b) of measuring instruments (Class 0.5 testing machine and Class 1 extensometer) and assuming the minimum required accuracy in the determination of cross-section area, according to Eq 40. As expected, higher uncertainties are observed for shorter offsets (U varies from 1.43 % to 1.01 %); the calculation of Sy is more sensitive to variations in the parameters of lines I and III when the offset line intersects the raw-data curve in steeper regions. As Table 7 indicates, the uncertainty in cross- sectional area dominates the uncertainty components. For typical offsets, 0.1 % and 0.2 %, uncertainty in A0 represents 89.38 % and 97.80 % of the uncertainty budget, respectively. For this reason, the expanded uncertainties listed in Tables 4 and 5 correspond to per- centages of calculated values that are between 1 % and 1.1 %. If we examine the results listed in Table 8, where we have assumed that u(A0) is one half of the tolerance value, percentage uncertainties for Sy vary from 1.14 % to 0.53 % (from the shortest to the longest off- set distance); in addition, the fitting of line III is the main uncer- tainty component for the shortest offset, but uncertainty in cross- sectional area dominates for longer offset distances. Concluding Remarks We have presented a method for the computation of offset yield strength that adapts features of two published procedures and incorporates the novel use of a weighted total least-squares algo- rithm for the required fits of force-extension data. Even though the method requires the input of the uncertainties associated with force and extension recordings, we have obtained reliable results for TENSTAND validation files, assuming tolerance values for these uncertainties. The use of the WTLS algorithm, which takes into account uncertainties in both force and extension, enables a thorough esti- mation of the uncertainty associated with the calculation of offset yield strength. The proposed uncertainty-estimation procedure, developed according to the GUM uncertainty framework, is expected to give comparable results to those given by a Monte Carlo method. We emphasize that the proposed procedure evalu- ates the uncertainty associated with the calculation process that leads to the value of Sy, for a given force-extension curve. Although several sources that affect tension-test recordings, and consequently the computed value of Sy, are very difficult to quan- tify, tensile standards continuously improve test methods to mini- mize the influence of these sources. We have shown some capabilities of INcerTI, a dedicated pro- gram for post-processing tension-test raw data that integrates, in a novel approach, calculation and uncertainty evaluation of tensile parameters. Acknowledgments The authors thank Professors Laura Felicia Matusevich and Michael Anshelevich of Texas A&M University for their help in improving this article. The authors also thank the technicians of the Laboratory of Mechanical Testing of INTI–Co´rdoba, Julio Costa and Julio Helale, who use the program INcerTI daily, and whose comments have helped us improve this computational tool significantly. References [1] Dieter, G., Mechanical Metallurgy, McGraw-Hill, New York, 1986, pp. 275–324. [2] Goodman, M., Jorgensen, J., and Wonsiewicz, B., “Computer-Aided Interpretation of Stress-Strain Curves,” J. Test. Eval., Vol. 2, 1974, pp. 361–369. [3] ISO/IEC 17025, 2005, “General Requirements for the Competence of Testing and Calibration Laboratories,” International Organization for Standardization, Geneva, Switzerland. [4] JCGM 100, 2008, “Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement (GUM 1995 with minor corrections),” http://www.bipm.org/utils/ common/documents/jcgm/JCGM_100_2008_E.pdf (Last accessed 28 June 2012). [5] JCGM 101, 2008, “Evaluation of Measurement Data—Supple- ment 1 to the ‘Guide to the Expression of Uncertainty in Mea- surement’—Propagation of Distributions Using a Monte Carlo Method,” http://www.bipm.org/utils/common/documents/jcgm/ JCGM_101_2008_E.pdf (Last accessed 28 June 2012). [6] Loveday, M. S., “Room Temperature Tensile Testing: A Method for Estimating Uncertainties of Measurement,” http://publications.npl.co.uk/npl_web/pdf/cmmt_mn48.pdf (Last accessed 28 June 2012). MATUSEVICH ETAL. ON OFFSET YIELD STRENGTH 13 Copyright by ASTM Int'l (all rights reserved); Thu Mar 21 14:23:26 EDT 2013 Downloaded/printed by Texas A M UnivEbsco pursuant to License Agreement. No further reproductions authorized. [7] EN 10002-1, 2001, “Metallic Materials. Tensile Testing. Part
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