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Título:	*Computation and Uncertainty Evaluation of Offset Yield Strength*
Fuente:	Journal of Testing and Evaluation, Vol. 41, No. 2, 2013, p.1–14
Autor/es:	Matusevich, Ariel; Mancini, Reinaldo; Massa, Julio
Materias:	Tensiones; Software; Ensayos; Incertidumbre; Sistemas computarizados; Métodos de cálculo; Computación
Editor/Edición:	ASTM; 2013
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Ariel E. Matusevich,1 Julio C. Massa,2 and Reinaldo A. Mancini3

Journal of Testing and Evaluation, Vol. 41, No. 2, 2013 Available online at www.astm.org doi:10.1520/JTE20120100

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 Measurement, considers the correlation between the parameters involved in its calculation. The implementation of these procedures motivated the development 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 values 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 corresponds 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).

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 Tecnolog´ıa, Universidad Nacional de Co´rdoba, Av. Ve´lez Sarsﬁeld 1561, Co´rdoba, X5000JKC, Argentina, e-mail: rmancini@inti.gob.ar

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 speciﬁcations or standards. In addition, to comply with the requirements of the International Standard ISO/IEC 17025 [3], accredited laboratories 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 problems, but has limitations. Supplement 1 to the GUM gives an alternative procedure based on a Monte Carlo method, that can be applied in cases where the GUM uncertainty framework is not applicable 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 proposed 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 Uncertainties in Mechanical Tests on Metallic Materials, developed within the European Project UNCERT [8]. Both methods are based on the GUM uncertainty framework.

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2 JOURNAL OF TESTING AND EVALUATION

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 uncertainty in the computation of Sy has ﬂaws; the method takes into account the uncertainties in the estimation of the ordinate intercept 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 dedicated 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 computation of Sy and for the estimation of the uncertainty in its computation. The calculation of Sy is based on searching a portion of the raw-data curve in the neighborhood of its intersection with the offset 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 abscissas [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 parameters 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 simulation [5].

The implementation of the procedures proposed in this article for computation and uncertainty evaluation of Sy motivated the development 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 reference 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 TENSTAND (tensile standard) and represent typical tensile characteristics of a variety of industrially important materials [11]. In this work, we present the validation of the procedure for the determination of Sy.

This paper is organized as follows. First, we present the proposed method for the computation of Sy. After giving an introduction 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

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 m d; 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 m d; drawn at a distance from line I whose horizontal 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.

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

Xn Xn

n diFi À di Fi

m ¼ i¼1 Xn

i¼1

i¼!1 2

(2)

n di2 À

i¼1

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

uðmÞ m

(3)

In Eq 3, m is the slope calculated by Eq 2 and u(m) is its standard deviation

FIG. 2—Determination of the points that delimit the proportional region.

MATUSEVICH ET AL. ON OFFSET YIELD STRENGTH 3

uðmÞ ¼ u u v u u u u t u uﬃnﬃﬃﬃﬃﬃﬃX ﬃiﬃ¼ﬃnﬃﬃ1ﬃﬃﬃFﬃﬃﬃi2ﬃﬃﬃÀﬃðﬃﬃnﬃﬃbﬃÀﬃﬃX ﬃiﬃ¼ﬃn2ﬃ1ﬃﬃÞﬃ42ﬃdﬃﬃiﬃnFﬃﬃﬃX iﬃ!ﬃnﬃﬃﬃﬃÀﬃdﬃﬃﬃi2ﬃﬃﬃÀﬃﬃX ﬃi¼ﬃnﬃﬃ1ﬃﬃX ﬃFﬃﬃnﬃiﬃﬃÀﬃﬃdﬃﬃiﬃbﬃ!ﬃﬃX ﬃi2ﬃ¼ﬃn35ﬃﬃ1ﬃﬃﬃdﬃﬃﬃiﬃ!ﬃﬃﬃﬃX ﬃiﬃ¼ﬃnﬃﬃ1ﬃﬃﬃFﬃﬃﬃiﬃ

i¼1

(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 anomalies are typically associated with specimen straightening and initial 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

uðmÞ 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 b þ mX that minimize the

following function [12]:

Xn v2 ¼

k¼1

ðxk

À Xk Þ2 u2x;k

ðyk

À Yk Þ2 uy2;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 leastsquares algorithm to solve Eq 6, referred to as WTLS [9], which treats x-and-y data symmetrically. In such a method, the twodimensional minimization problem is reduced to the onedimensional search of a minimum, using a different parametrization 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 function; the program returns the parameters of the ﬁtted line, the minimum 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.

4 JOURNAL OF TESTING AND EVALUATION

TABLE 1—Uncertainty in force measuring devices (Ref 8).

Class of Machine

Expanded Uncertainty (k ¼ 2), U %

0.5

60.44

60.88

61.75

62.61

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 extension, 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 uncertainties for load-and-extension data and use the tolerances of measuring devices speciﬁed in standards. Current tensile standards, ASTM E8/E8M-11 [14] and ISO 6892-1 [15], stipulate similar 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 tolerances 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 mð bL eÞ ¼ b1 ! b2 ¼ b1 À mð bL eÞ

(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 offset 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 ðdB; FBÞ, satisﬁes the following condition:

FB < b2 þ m dB

(8)

The point that immediately precedes point B is designated as A ¼ ðdA; FAÞ; 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

TABLE 2—Uncertainty in strain measurement using extensometers (Ref 8).

Class of Extensometer

Expanded Uncertainty (k ¼ 2), U %

0.2

60.2

0.5

60.5

61.0

62.0

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 segments 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 2 þ 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

1 À

Xn ½ðbl

i¼1

ml di Þ

Fi2

(10)

and the mean square error of the quadratic ﬁt, MSEq

MSEq

1 À

i¼1

ÂÀ bq

mqdi

cqd2i Á

FiÃ2

(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 ! 1 MSEl

(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 consider 2 np 15 and pick the value of np that gives the longer interval with R ! 1; otherwise, we choose np ¼ 2 (nIII ¼ 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

mdy

m3 dy

b3 À b2 m À m3

(13)

After replacing Eq 7 into Eq 13, we calculate yield force Fy as follows:

II ðdyÞ

III ðdyÞ

mb3

m3b1 þ b m À m3

mm3L e

(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

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 fmðX1; X2; …; XN Þ

(15)

To express information, usually incomplete, about input contributions, the model formulation involves assignment of probability density functions (PDFs) to input quantities

gX1 ðn1Þ; gX2 ðn2Þ; …; gXN ðnN 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 i ¼ 1, 2, …, N.

As part of the measurement process, we estimate standard uncertainties u(x1), u(x2), …, u(xN), and covariances u(xi, xj), i = j, associated 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 evaluated 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 estimation 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 propagation 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 fmðx1; x2; …; xN Þ

(17)

To evaluate the uncertainty u(y) associated with y, the GUM proposes the use of the law of propagation of uncertainty

MATUSEVICH ET AL. ON OFFSET YIELD STRENGTH 5

uðyÞ ¼ u u t vﬃX iﬃ¼ﬃNﬃﬃ1ﬃﬃﬃﬃﬃﬃ@@ﬃﬃfﬃxﬃmﬃiﬃﬃuﬃﬃﬃðﬃﬃxﬃﬃiﬃÞﬃﬃ!ﬃﬃﬃ2ﬃﬃﬃþﬃﬃﬃﬃﬃ2ﬃﬃﬃNX ﬃiﬃ¼ﬃÀﬃ1ﬃ1ﬃﬃﬃjﬃX ¼ﬃﬃNiﬃþﬃﬃﬃ1ﬃﬃ@ﬃ@ﬃﬃfxﬃmﬃiﬃﬃﬃ@@ﬃﬃfﬃxﬃmﬃjﬃﬃuﬃﬃﬃÀﬃﬃxﬃﬃiﬃ;ﬃﬃxﬃﬃjﬃÁﬃﬃ

(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 i ¼ 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 calculated as

U ¼ kuðyÞ

(19)

so that the interval [y À U, y þ U] has a prescribed coverage probability 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 output 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 evaluation [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 characterization 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.

6 JOURNAL OF TESTING AND EVALUATION

regarded as meaningful in the numerical value of u(y); usually,

ndig ¼ 1 or ndig ¼ 2. To determine e, we express the value of u(y) in the form c Â 10l, where c is an ndig-digit integer and l is an integer; then, we calculate the comparison accuracy as follows [5]:

e ¼ 1 10l

(22)

FIG. 4—Illustration of the propagation of probability distributions.

• Select the number M of Monte Carlo trials to be made.

• •

ÀUBqgnuMysM1aepn¼;tethi…rtfefymo,;Àmrnmnoo1MNMibdn;Átegag…li;natow;ranahcnNMeodrsmÁeoegptmnuijootsifesfaimtanhMdpeseleijptntvhegÈencrgodat1ofennr¼dtshtoemfrÈmaPnÀÀsDndnaFo1111m;;mop……flese;;aoamnncf1N1NhpnÁÁliie:;;ns……pouft;;

the output PDF, gY (g).

• Calculate the estimate y and the standard uncertainty u(y) of

the output quantÈity Y as the Éarithmetic mean and the standard deviation of g1; …;ÈgM ; respecÉtively. • Sort the model values g1; …; gM in increasing order, gð1Þ … gðMÞ ; and use the sorted values to determine the

(ÂpgrðoLÞb;agbðiHliÞsÃticaat lalycosvyemramgeetrpircoboarbsihliotyrtpes,tw) hceorveeHrag–e

interval L equals

the integer part of pM þ 21.

Validation of the GUM Uncertainty Framework

Although the GUM uncertainty framework works well in many situations, 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 output quantity is either Gaussian or a scaled and shifted tdistribution.

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 ½y À U ; y þ U ; calculated through

tÂhgeðLGÞ;UgMðHÞuÃnpcreorvtaidinetdy

framework, agrees with the by a Monte Carlo method, to

coverage interval a stipulated com-

parison accuracy e. When the following conditions are satisﬁed:

dlow ¼ ðy À U Þ À gðLÞ e

(20)

dhigh ¼ ðy þ U Þ À gðHÞ 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

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:

Fy A0

;

where

Fy¼ ðmb3 À m3b1 þ b mm3L eÞ=ðm À m3Þ (23)

As Eq 23 indicates, the proposed model depends on the parameters 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 quantify; they include:

• the alignment of the test specimen, that affects the resulting slope m of the proportional part of the force-extension diagram [24],

• testing-machine characteristics (e.g., stiffness, method and control of operation), and

• speed of testing (within the range allowed in the corresponding tensile standard).

The application of Eq 18 for the evaluation of the standard

uncertainty u(Sy) leads to

ÀÁ u Sy

qÂﬃﬃcﬃﬃﬃFﬃﬃyﬃuﬃﬃﬃðﬃﬃFﬃﬃﬃyﬃÞﬃﬃÃﬃﬃﬃ2ﬃﬃþﬃﬃﬃﬃ½ﬃcﬃﬃAﬃﬃ0ﬃﬃuﬃﬃðﬃﬃAﬃﬃﬃ0ﬃﬃÞﬃﬃﬃﬃ2ﬃ

(24)

because Fy and A0 are uncorrelated. The sensitivity coefﬁcients in Eq 24 are

cFy

@Sy @Fy

1 A0

;

cA0

@Sy @A0

ÀFy A20

(25)

If additional sources of uncertainty (that do not appear in the model equation) were evaluated, their squared standard uncertainties (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 k ¼ 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

ÀÁ u Fy

u u u v u u tﬃ½ﬃﬃcﬃﬃbﬃþ1ﬃﬃuﬃﬃ½ﬃðcﬃbﬃbﬃﬃ13ﬃﬃuÞﬃﬃðﬃ2ﬃbﬃþﬃﬃ3ﬃﬃÞ½ﬃcﬃﬃ2mﬃﬃþﬃuﬃﬃﬃð½ﬃcﬃmﬃmﬃﬃÞﬃ3ﬃuﬃﬃ2ﬃðﬃﬃmﬃﬃﬃﬃ3ﬃﬃÞﬃﬃﬃ2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

þ ½cLe u ðLeÞ 2þ2cb1 cmu ðb1; mÞ þ 2cb3 cm3 u ðb3; m3Þ

(26)

The sensitivity coefﬁcients in Eq 26 are given by the following expressions:

cb1

@Fy @b1

Àm3 m À m3

(27)

@Fy @m

m3ðb1 À ðm

b3 À bLem3Þ À m3Þ2

(28)

cb3

@Fy @b3

m À m3

(29)

cm3

@Fy @m3

mðb3 À ðm

b1 þ b LemÞ À m3Þ2

(30)

cLe

@Fy @Le

b mm3 m À 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 b þ md is

R¼

u2ðmÞ uðm; bÞ

uðm; bÞ u2ðbÞ

(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 programmed by the authors of the WTLS algorithm, publicly available 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 function 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

uðLeÞ ¼ sðﬃﬃ0ﬃﬃﬃ:ﬃ0ﬃﬃ0ﬃﬃﬃ5ﬃﬃLﬃﬃﬃeﬃﬃÞﬃ2ﬃﬃþﬃﬃﬃﬃﬃﬃﬃﬃ0ﬃﬃ:ﬃp0ﬃﬃﬃ1ﬃﬃﬃﬃLﬃﬃﬃeﬃﬃﬃﬃﬃ2ﬃ

(33)

In Eq 33, we have assumed a rectangular PDF for the distribution 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 measurement of speciﬁc dimensions of the test specimen using appro-

MATUSEVICH ET AL. ON OFFSET YIELD STRENGTH 7

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 fmðX1; X2; …; XN Þ

(34)

For the case of a specimen with rectangular crosssectional 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 Á h. II. To calculate cross-sectional area, determine the estimates of input quantities xi as mean values of n measurements

1 Xn

xi ¼ xi ¼ n 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 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

uðA0Þ ¼ ½cx 1 uðx1Þ2þ Á Á Á þ ½cxN uðxN Þ2 (36)

In Eq 36, we have assumed that input quantities are independent. 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 uðxiÞ in Eq 36, we use the following procedure:

I. For each input dimension, compute standard deviation

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

sðxiÞ ¼

1 À

Xn 1 k¼1 ðxi k

xiÞ2

(37)

and experimental standard deviation

sðxiÞ

spðxﬃiﬃÞ n

(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 instrument 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).

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

uðxiÞ ¼ s2ðxiÞ þ ðuCALÞ2

(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 % (k ¼ 2) for this requirement, we obtain

8 JOURNAL OF TESTING AND EVALUATION

FIG. 5—Main command window of INcerTI.

uðA0Þ ¼ 0:005 A0

(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 (incertidumbre) and INTI (Instituto Nacional de Tecnolog´ıa 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 installed in the system. INcerTI is run through the graphical user interface shown in Fig. 5.

FIG. 6—Input of the number of diameter estimates.

FIG. 7—Input of diameter estimates.

MATUSEVICH ET AL. ON OFFSET YIELD STRENGTH 9

FIG. 8—Uncertainty of the measuring instrument.

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 dedicated 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) standard 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 different regions of the diagram and interpret results, the tools “Zoom,” “Pan,” and “Limits” are very useful (see the “Plotanalysis” 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-

FIG. 9—Input of a given cross-sectional area.

FIG. 10—Data input for the computation of offset yield strength.

bular products. When we click the “Area” button, the program asks for information about the measurement process in an interactive 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 information 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 estimation of the standard uncertainties associated with forceextension 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,

TABLE 3—Premium quality ASCII datasets.

TENSTAND Dataset

Material

Le (mm)

Nimonic 75, CRM 661

13 % Mn steel

316L Stainless steel

Tin coated packaging steel

Sheet steel—DX56

Aluminum sheet—soft AA5182

Aluminum sheet—soft AA1050

Aluminum sheet—soft AA5182

Sheet steel—DX56

Synthetic digital curve—zero noise

Synthetic digital curve—0.5 % noise

Synthetic digital curve—1 % noise

A0 (mm2)

78.46 78.54 77.55 78.65 3.97 14.17 4.62 14.48 29.59 8.77 78.54 78.54 78.54

10 JOURNAL OF TESTING AND EVALUATION

TENSTAND Dataset

1 6 10 17 22 30 38 42 46 50 57 61 63

TABLE 4—Procedure validation for 0.2 % offset yield strength.

Case (a)

Case (b)

Sy À 0.2 % (MPa) Agreed Values

309.6 – 310.1 308.0 – 308.6 337.1 – 337.2 261.0 – 261.2 562.5 – 564.6 162.7 – 162.9 396.4 – 397.1 30.01 – 30.05 134.5 – 134.8 163.9 – 164.0 434.3 438.1 – 441.6 446.5 – 448.2

Sy 6 U

309.6 6 3.2 308.4 6 3.1 337.0 6 3.6 260.1 6 2.8 562.7 6 5.8 162.8 6 1.7 396.5 6 4.0 30.08 6 0.32 134.4 6 1.4 163.9 6 1.7 434.1 6 4.6 434.8 6 4.4 434.7 6 4.4

Difference (%)

0 0 À0.02 À0.36 0 0 0 0.09 –0.09 0 À0.04 À0.75 À2.65

Sy 6 U

310.0 6 3.1 308.8 6 3.1 337.3 6 3.4 261.1 6 2.7 562.6 6 5.7 162.8 6 1.6 396.6 6 4.0 30.08 6 0.31 134.4 6 1.3 163.9 6 1.6 434.4 6 4.4 435.0 6 4.4 434.9 6 4.4

Difference (%)

0 0.07 0.02 0 0 0 0 0.08 À0.06 0 0.02 À0.70 À2.60

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, respectively. When we use the latter method, we specify the class of extensometer 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 uncertainties 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 proportional region of the diagram (line I), (iii) two circles that highlight the points UL and LL that delimit the proportional region, (iv) the offset line (line II), the line that approximates the nonproportional 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 material (CRM 661), and synthetic dataﬁles with different levels of noise in force data (0 %, 0.5 %, and 1 %).

TENSTAND Dataset

1 6 10 17 22 30 38 42 46 50 57 61 63

TABLE 5—Procedure validation for 0.1 % offset yield strength.

Case (a)

Case (b)

Sy À 0.2 % (MPa) Agreed Values

303.4 – 304.5 300.5 – 301.8 334.5 – 334.9 244.7 – 245.2 525.6 – 530.6 157.2 – 157.6 385.2 – 386.8 26.48 – 26.55 133.4 – 133.9 158.6 –158.7 432.4 431.8 – 434.1 429.6 – 432.7

Sy 6 U

303.3 6 3.3 301.3 6 3.3 334.0 6 3.5 243.0 6 2.7 525.5 6 6.0 157.5 6 1.6 385.7 6 4.0 26.56 6 0.3 133.7 6 1.4 158.8 6 1.6 432.3 6 4.6 432.0 6 4.4 432.7 6 4.4

Difference (%)

À0.04 0

À0.14 À0.70 À0.03

0 0 0.06 0 0.05 À0.03 0 0

Sy 6 U

304.2 6 3.1 302.2 6 3.1 335.0 6 3.4 245.0 6 2.5 525.3 6 5.6 157.5 6 1.6 385.7 6 3.9 26.56 6 0.28 134.0 6 1.3 158.8 6 1.6 432.5 6 4.4 432.2 6 4.3 432.7 6 4.3

Difference (%)

0 0.14 0.02 0 À0.05 0 0 0.05 0.07 0.06 0.03 0 0

MATUSEVICH ET AL. ON OFFSET YIELD STRENGTH 11

FIG. 11—Analysis of dataset 63, synthetic ﬁle with 1 % noise.

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

TENSTAND Dataset

1 6 10 17 22 30 38 42 46 50 57 61 63

TABLE 6—Comparison between the GUM uncertainty framework and a Monte Carlo method.

Sy À 0.2 % (MPa)

U (Sy) (MPa)

[Sy À U, Sy þ U] (MPa)

Sy (MCM) (MPa)

[g(L), g(H)] (MPa)

309.552 308.386 337.039 260.057 562.667 162.813 396.546 30.0764 134.385 163.891 434.137 434.806 434.681

3.151 3.145 3.587 2.840 5.752 1.653 4.030 0.317 1.367 1.664 4.633 4.402 4.401

[306.401, 312.703] [305.241, 311.531] [333.452, 340.625] [257.217, 262.897] [556.916, 568.419] [161.160, 164.466] [392.516, 400.576] [29.7594, 30.3934] [133.019, 135.752] [162.227, 165.554] [429.504, 438.770] [430.404, 439.209] [430.280, 439.082]

309.560 308.393 337.049 260.059 562.681 162.817 396.558

30.0771 134.388 163.894 434.143 434.818 434.690

[306.412, 312.717] [305.252, 311.543] [333.460, 340.644] [257.196, 262.912] [556.917, 568.441] [161.163, 164.470] [392.525, 400.602] [29.7599, 30.3942] [133.024, 135.758] [162.235, 165.563] [429.531, 438.804] [430.408, 439.217] [430.294, 439.099]

dlow

0.01 0.01 0.01 0.02 0.00 0.00 0.01 0.001 0.01 0.01 0.03 0.00 0.01

dhigh

0.01 0.01 0.02 0.01 0.02 0.00 0.03 0.001 0.01 0.01 0.03 0.01 0.02

12 JOURNAL OF TESTING AND EVALUATION

TABLE 7—Fractional uncertainty contributions, ﬁle 22, u(A0) ¼ 0.005A0.

Percentage Contributions to u(Sy)

Offset

(% Le) (MPa) (%)

rA0

rLe

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) (MPa) (%)

rA0

rLe

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

recordings and information about the measurement of the specimen 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 loadand-elongation data we use the tolerances of measuring devices listed in Tables 1 and 2. From all the possible combinations 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 differences between computed results and reference values have been calculated with respect to the nearest limit of the reference interval. All results are accompanied by their expanded uncertainties, for a level of conﬁdence of 95.45 % (k ¼ 2).

Computation of 0.2 % Offset Yield Strength

If we exclude from Table 4 the results that correspond to the synthetic ﬁles with noise (ﬁles 61 and 63), we observe that for combination (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 reference values (including ﬁle 17) whereas differences in the remaining cases are less than 0.07 %.

Although results for synthetic ﬁles 61 and 63 differ signiﬁcantly from agreed values (almost 3 % for ﬁle 63), calculations are very close to the reference value with 0 % noise (ﬁle 57); however, the agreed ranges for these curves do not contain the exact result (0 % noise). In Fig. 11, we examine the region of the intersection 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 correspond 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 À0.70 % 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 guidelines that are addressed in a previous section of this paper (Introduction to Uncertainty Estimation According to the GUM).

The MCM requires the random sampling of the PDFs associated 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 Â 106 trials in each case. Table 6 lists the coverage intervals given by both methods, the GUM uncertainty framework and the MCM; results are displayed with a high number of signiﬁcant digits for comparison purposes only. To compare the coverage intervals, we consider ndig ¼ 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

MATUSEVICH ET AL. ON OFFSET YIELD STRENGTH 13

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 consider 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 evaluation of overall standard uncertainty uðSyÞ, we compute the fractional contributions of A0, Le, line I, and line III

rA0

½cA0 uÀðA0ÁÞ2 u2 Sy

(41)

rL e

Â cLe

uÀðLeÁÞÃ2

u2 Sy

(42)

½cb1 uðb1Þ2þ½cmuðmÀÞ2Áþ2cb1 cmuðb1; mÞ u2 Sy

(43)

rIII

½cb3 uðb3Þ2þ½cm3 uðmÀ3Þ2Áþ2cb3 cm3 uðb3; m3Þ u2 Sy

(44)

that represent variance contributions to total variance. Using options in the “Results” menu of INcerTI, we can visualize fractional uncertainty contributions in the form of pie charts.

Table 7 lists (for ﬁve offset distances), results for Sy, their percentage 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 crosssectional 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 percentages 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 offset distance); in addition, the ﬁtting of line III is the main uncertainty component for the shortest offset, but uncertainty in crosssectional 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 algorithm 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 estimation 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 evaluates 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 quantify, tensile standards continuously improve test methods to minimize the inﬂuence of these sources.

We have shown some capabilities of INcerTI, a dedicated program 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—Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’—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).

14 JOURNAL OF TESTING AND EVALUATION

[7] EN 10002-1, 2001, “Metallic Materials. Tensile Testing. Part 1: Method of Test at Room Temperature,” European Committee for Standardization, Brussels, Belgium.

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ERRATUM

JTE20120100, Computation and Uncertainty Evaluation of Offset Yield Strength, Ariel E. Matusevich, Julio C. Massa, and Reinaldo A. Mancini, published online in the Journal of Testing and Evaluation (JTE), Volume 41, No. 2, March, 2013.

Page 9, TABLE 3, Row 7, Column 2, Aluminum sheet—soft AA5182 – should read: Aluminum sheet—hard AA5182.

Page 10, TABLE 5 heading of Column 2, Sy – 0.2 % (MPa) Agreed Values – should read: Sy – 0.1 % (MPa) Agreed Values.

Page 11, line 3, range of agreed values for Sy – should read: intervals of agreed values for Sy.

Page 12, sixth line from bottom, agreed ranges – should read: agreed intervals.

Page 13, Acknowledgments, lines 1 and 3 of the paragraph, The authors – should read: We.

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