e guide

Computing Uncertainty

for Charpy Impact

Machine Test Results

J. D. Splett

C.N. McCowan

H. K. Iyer

C.-M. Wang

Nisr

National Institute of

Standards and Technology

U.S. Department of Commerce

Special

Publication

960-18

NIST Recommended Practice Guide

Special Publication 960-18

Computing Uncertainty for

Charpy Impact Machine Test

Results

J. D. Splett

H. K. Iyer

C. -M.Wang

Statistical Engineering Division

Information Technology Laboratory

C. N. McCowan

Materials Reliability Division

Materials Science and Engineering Laboratory

National Institute ofStandards andTechnology

Boulder, Colorado 80305

October 2008

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

National Institute of Standards andTechnology

Patrick D. Gallagher, Deputy Director
Certain commercial entities, equipment, or materials may be identified in this document

in order to describe an experimental procedure or concept adequately. Such identification

is not intended to imply recommendation or endorsement by the National Institute of

Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Special Publication 960-18

Natl. Inst. Stand. Technol. Spec. Publ. 960-18, 42 pages (October 2008)

CODEN: NSPUE2

U. S. GOVERNMENT PRINTING OFFICE

Washington: 2008

For sale by the Superintendent of Documents, U. S. Government Printing Office

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Mail: Stop SSOP, Washington, DC 20402-0001
Contents

1. Introduction . 2. Uncertainty of a Test Result

3. Type B Uncertainty Evaluation

4. Machine Bias

5. Direct Verification 1 6. Temperature

7. Expanded Uncertainty

8. Examples 1 8.1 Both <ssystematic and 5 systematlc and Their Uncertainties

are Negligible

8-2 £systematlc has One Component 1 9. Closing Remarks 2

10. References 2

Appendix A. Uncertainty Details 2

A.l Test Material 2

A.2 Indirect Verification Test 2

A.3 NIST Reference Value 2

A.4 Estimating Machine Bias 2

Appendix B. Direct Verification Components of Uncertainty 2

B. l Anvils and Supports, A 2

B.2 Height of Pendulum Fall, h 2

B.3 Potential Energy, E 2

B.4 Impact Velocity, v 3

1 B.5 Center of Percussion, L 3 B.6 Friction Loss, D 3 B.7 Scale Accuracy, r 3 Appendix C. /-Table 3< Appendix D. Glossary of Terms 3

NIST Recommended Practice Guide:

Computing Uncertainty for Charpy

Impact Machine Test Results

J. D. Splett, 1 C. N. McCowan,2 H. K. Iyer,

1 and C.-M. Wang1

National Institute of Standards and Technology

325 Broadway

Boulder, CO 80305

This recommended practice guide demonstrates how to determine the uncertainty

associated with mean absorbed energy of specimens tested on a Charpy impact

machine. We assume that the Charpy machine has successfully met the

requirements for both direct and indirect verification as described in the ASTM E

23, Standard Test Methods for Notched Bar Impact Testing of Metallic Materials.

We follow the recommendations and procedures in the "Guide to the Expression of

Uncertainty in Measurement" for computing uncertainty. We assume the reader is somewhat familiar with the Charpy machine verification program administered by

the National Institute of Standards and Technology.

Keywords: absorbed energy; Charpy V-notch; impact test; pendulum impact test; uncertainty; verification testing

1. Introduction

The absorbed energy of a test material, measured using a Charpy impact machine,

is often reported as the mean absorbed energy of a set of specimens. However, the

sample mean does not account for known sources of bias, including machine bias,

which can be substantial. We address the estimation of a test result for the case in which the test result is corrected for known biases and the case in which it is not.

It is left to the user's discretion whether or not to correct a test result. Computing the reported test result is straightforward; however, computing the

uncertainty associated with the test result requires more consideration. The

purpose of this document is to clarify the concept of uncertainty and to provide

Charpy laboratories with a procedure for computing the uncertainty of a test result.

Statistical Engineering Division, Information Technology Laboratory

2 Materials Reliability Division, Materials Science and Engineering Laboratory

1
Computing Uncertainty for Charpy Impact MachineTest

Before valid Charpy measurements can be made in the laboratory, the machine

needs to pass both direct and indirect verification tests as specified in ASTM E 23

[1]. Even if a Charpy machine has passed the indirect verification test, it is likely that results for the verification specimens differ from the certified value. This

difference can be used to quantify machine bias. Thus, the indirect verification

results and the certified reference value (along with their uncertainties) play key

roles in the calculation of uncertainty of a test result. We express uncertainty according to the accepted criteria described in the "Guide

to the Expression of Uncertainty in Measurement," or GUM [2], taking into

consideration both random and systematic sources of error. The procedure we

recommend for computing uncertainty is very general and can accommodate any

number of random or systematic error sources including the following:

Anvils and supports

Center of percussion

Center of strike Friction loss Height of pendulum fall Impact velocity

Material inhomogeneity

Operator

Potential energy

Repeatability

Scale accuracy

Test temperature

The uncertainty contributions from individual error sources can be estimated if they are identified as significant, but generally these errors are assumed to be

minimized by adjustments made to the machine during direct verification and by

following the standard test procedure. As will become apparent, the calculation

of uncertainty is greatly (and often) simplified by assuming that direct verification

contributions are zero, and only contributions from indirect verification are considered. This is a widely accepted approach to the calculation of uncertainty

for Charpy impact tests, and is used in standards such as ISO 148-1 [3]. We

present more detail here, because understanding the individual contributions to uncertainty, and how to quantify them, leads to better control of the test. We

encourage the users to consider these, and other relevant details. We present an example in Section 2 that provides instructions for calculating

the uncertainty of a test result. Section 3 provides details regarding the Type B

evaluation of errors, Section 4 addresses the computation of machine bias, Section

5 discusses direct verification sources of error, Section 6 addresses temperature

measurement errors, Section 7 provides some information about expanded

uncertainty, and Section 8 gives some example uncertainty calculations. Complete

details regarding the justification of the uncertainty procedures are given in Appendix A.

2
Uncertainty of a Test Result #

2. Uncertainty of a Test Result

In this section, we provide details for computing the uncertainty of a test result within the context of an example. A Charpy laboratory will typically compute the

sample mean and sample standard deviation of n specimens of the test material

using the following two equations:

2>,

(i)

i n-\

(2)

The degrees of freedom (df) associated with the sample standard deviation, s , are w-1. It is important to note that s includes all sources of random error, including

machine variability, material variability, and the typical variations expected when

following the standard test procedure. The individual components of the random

error cannot be estimated separately in the case of destructive impact testing

(multiple measurements on the same specimen are needed to do this). In addition,

random errors (unlike systematic errors) do not remain constant during the

measurement of n specimens, so these errors do not result in a bias. The data given in Table 1 are used to illustrate the calculations needed to assess

the uncertainty associated with the result for our example. Table 1 lists test results and summary statistics for n = 5 observations of a particular test material

measured at 80 °C.

Suppose we are also given the values in Table 2. (We provide details for computing the quantities in Table 2 in subsequent sections.) Our best estimate of

the machine bias is b, which is defined as the difference between the verification

result for the test machine and the certified value of the verification specimens.

Systematic errors due to all other factors that are not already included in the

machine bias are denoted by <?systematlc . The values u{b) and w(esystematlc ) are the

uncertainties associated with b and esystematlc , while dfb and df represent degrees of

freedom for u(b) and w(<?

c ).

3
>> Computing Uncertainty for Charpy Impact MachineTest

Table 1. Measurement results for a test material.

Observed data, J Summary statistics 58.0 n = 5 62.0

y = 57.6 J 54.0 5 = 3.6 J 54.0 df=4

60.0

Table 2. Example quantities required to compute uncertainty of a test result. Machine bias Systematic error

Estimate

Uncertainty

Degrees of freedom

b = -4.2 J

!!(£) = 2.8 J

df = 84

b

^systematic ^.0 J

W(£Systemanc) = - 6 J df =2

To compute the uncertainty of the test result, we first compute a mean of the test material that is corrected for machine bias and all other systematic effects:

^corrected ^systematic '

Substituting the values from Tables 1 and 2 into Eq. (3) gives

(3)

y,corrected 57.6 J - (-4.2 J) - (3.0 J) = 58.8 J

Next, we calculate the uncertainty of the corrected mean, ycorrected . Assuming all of

the terms in ycorrected are independent, the combined standard uncertainty of vcorrected is

"(^corrected) = \ ~+ U (b) + U (^systematic )

Substituting the appropriate values from Tables 1 and 2 into Eq. (4) gives

(4)

"(jWced ) = + < 2 ' 8 + <

- 6 =3-3 J.

4
Uncertainty of a Test Result

Typically, the standard uncertainty is multiplied by a coverage factor that expands

the uncertainty to form an "uncertainty" interval about the measurement result. The interval is expected to encompass a large fraction of possible values of

the result. Thus, the expanded uncertainty is defined as the combined standard

uncertainty multiplied by a coverage factor. The coverage factor is often set equal to two for simplicity, but this approximation can be problematic, so it is recommended that the degrees of freedom be used to obtain the appropriate

coverage factor. An uncertainly interval with 100(1 -a) % coverage probability (a is 0.05 for 95 %

coverage) is given by

^corrected — h-a/z .ditS

' U (^corrected ) ' (5)

where t

x_a/M^ is found in a /-table (see Appendix C). The degrees of freedom

associated with u( vcorrected )

,

^ _

li corrected) iC(b)

| u (gsystematlc ) (6)

df n +

dfb dfe

are determined from the Welch- Satterthwaite approximation as described in the

GUM. [2]

Substituting appropriate values from Tables 1 and 2 into Eq. (6) gives

dfeff = « = 47.9,

lf(3.6) 2 Y

| (2.8)

4

| (0.6)

4 5 84

which rounds down to 47. Using a /-table we get a coverage factor of / 975 47 =

2.012. Thus, a 95 % interval for our example is

58.8 J±/0975 47

- 3 . 3 J 58.8 J±2.012-3.3 J 58.8 J±6.6 J.
Computing Uncertainty for Charpy Impact MachineTest

The expanded uncertainty U is 6.6 J, indicating that 95 % of possible

measurement results lie within the uncertainty interval (52.2 J, 65.4 J). If we compute the interval based on the uncorrected value, we can express our

uncertainty interval as (y±U)-(b + esystawAc ) (57.6 J±6.6 J) -(-4.2 J + 3.0 J) (51.0 J, 64.2 J) + 1.2 J

and just report the uncorrected interval (5 1 .0 J, 64.2 J) along with the correction

(1.2 J). The decision to report a corrected test result is left to the user. However,

if the corrected test result is reported, we recommend that the report clearly state how the correction was computed and include pertinent information such as the

magnitude and sign of the correction, the test standard used, and the source of the

indirect verification specimens tested. The remainder of this document is dedicated to providing additional details regarding the computation of individual components needed to compute the

uncertainty of a Charpy test result.

3. Type B Uncertainty Evaluation

Typically, direct estimates of systematic errors based on actual measurements

are difficult to obtain and even harder to quantify because the required data are not generally available. In such cases, uncertainties due to systematic errors are estimated based on past experience, engineering knowledge, information from

published literature, and so on. An uncertainty evaluation that does not involve

actual measurements is called a Type B uncertainty evaluation. Type A uncertainty

evaluations are based on data obtained under repeatability conditions. Type B

uncertainty evaluations can be associated with either random or systematic errors, but

are most commonly used with systematic errors. Type B uncertainty evaluations utilize assumptions regarding distributions of errors. For example, instrument manufacturer's specifications can be thought of as limits

to a rectangular distribution. From this, the standard uncertainty associated with

measurements by that instrument can be deduced. These types of uncertainties can be

highly subjective, but are sometimes useful. The following example (also shown in Section B.7) illustrates how to use a manufacturer's specification for a Type B uncertainty evaluation. Suppose r is the random error in the Charpy machine scale mechanism and ± Ar represents the

manufacturer's specified error bounds of the measurement instrument. Assuming that the error can be anywhere within the ± Ar bounds, a rectangular distribution is used

6
to describe the distribution of possible biases, and in this case bounds are already

expressed in the proper units (joules). The standard uncertainty of r is

u{r)~

A rectangular distribution is often used in the absence of specific information about

the error distribution; however, other distributions can be used if more is known

about the errors. (See Reference [2] for details regarding Type B uncertainty

evaluations.) It is also necessary to provide an estimate of degrees of freedom

for each uncertainty component. We will assume dfr = oo , which implies that we

know u (r) exactly. The GUM provides a method for assigning a df value to Type B

estimates of uncertainty, which will be demonstrated shortly.

In the previous scale-error example, the distribution of possible errors was defined

by the interval (-r,r) , which is centered on zero. Sometimes the distribution of

a systematic error is centered on a value other than zero, resulting in a nonzero

systematic error estimate. For example, an operator might be consistently reading

the scale too high, so that the distribution of errors is described by a rectangular

distribution defined by (a,b) , where a and b are both greater than zero (0 < a<b).

In this case, the estimated systematic error is (a + b)/2 and the associated standard

uncertainty is (b-a)/2yj3 .

There are also systematic errors associated with the test procedures that can be

approximated using a Type B uncertainty evaluation. Suppose an operator notices

that the lengths of fractured specimen halves are uneven and determines that the specimens were all impacted off-center (striker impact is not aligned with the

notch). In addition, the operator knows that the 1 mm to 2 mm offsets observed

for the broken specimens result in an increase in the absorbed energy between 2 J and 4 J based on extensive experience with this particular material. To estimate the systematic error and its uncertainty, we assume that the 2 J and 4 J limits to error represent bounds of a rectangular distribution so that 2J + 4J .t a t- . 4J-2J = " = 3J TM& tt(gsy5tematic ) = = 0-6 J .

To determine the degrees of freedom associated with w(esystematlc )

, we employ a useful

relationship from the GUM (Eq. (G.3)). In general,

df=i

2 Anu where the quantity in square brackets represents the relative uncertainty, or the uncertainty of the uncertainty. In our example, we judge the uncertainty of

7
Computing Uncertainty for Charpy Impact MachineTest

to be 50 % or 0.50. so that

dfe =|[0.50]-

2 =2.

In general, the degrees of freedom provide information regarding the quality of the uncertainty estimate. For Type A uncertainty evaluations, the degrees of freedom

provide an objective measure of quality, while degrees of freedom associated with

Type B uncertainty evaluations provide a subjective measure of quality.

We can also combine several sources of systematic error to determine esystaal6c and

its uncertainty. For example, suppose we would like to combine three independent

sources of systematic error: friction loss, potential energy, and impact velocity, so

that

^tematic + £ +

Then the combined standard uncertainty of e

"(^syste^c ) - ^ir(D) + ir(E) + ir(v) ,

with effective degrees of freedom from the Welch- Satterthwaite approximation.

if. ^ ( systematic) df, = - : ; u\D) u\E) u\v)

dfD +dfE +dfv This type of procedure can be applied to any number of independent systematic

errors.

4. Machine Bias

To estimate the machine bias, we assume that the machine bias for the material

under test is the same as the machine bias based on the indirect verification. This

is an important assumption that allows us to estimate machine bias for all test materials. We use the results of an indirect verification test and the associated

reference value for our best estimate of machine bias.

where

F = ^—

(8)
Mac as

is the sample mean absorbed energy from the indirect verification test based on

nv = 5 test results, ^v, tematic represents errors due to all systematic effects associated

with indirect verification test, and R represents the certified reference value for the

batch of verification specimens.

To illustrate the computation of machine bias and its uncertainty, we will return

to the example from Section 2. Table 3 lists quantities provided by the National

Institute of Standards and Technology (NIST) with the high-energy verification test specimens that were used for the most recent high-energy indirect verification of

the Charpy machine of interest. We use the high-energy indirect verification test results because the nominal value

of the absorbed energy of the test material is closest to the high-energy verification

material. The uncertainty associated with the certified reference value (z/(R))

is provided by NIST with the results of the indirect verification test (or by request).

Table 4 displays the indirect verification data that were observed when the

verification set was broken on the machine of interest.

Table 3. Information provided by NIST for high energy verification

specimens.

Reference value, R

Reference value standard uncertainty, u(R)

Degrees of freedom, df

R

109.9 J

2.6 J 102

Table 4. High energy indirect verification test results. Verification set data, J Summary statistics 108.0

nv = 5 104.0 V = 106.2 J 109.0 Sy = 2.3 J 106.0 dfv = 4

104.0

9
Computing Uncertainty for Charpy Impact MachineTest

Table 5. Systematic error associated with the indirect verification.

Estimate, £systematic 0.5 J Standard uncertainty, u(S) 0.2 J Degrees of freedom, df 10

The "V" subscript is used to distinguish the indirect verification results from the

test material results. The sample standard deviation associated with the indirect

verification specimens ( Sv ) is calculated as s was calculated previously in Section 2,

As was the case for s , Sv also includes all sources of random error related to both

machine variability and material variability, and the individual contribution of

errors cannot be determined.

Suppose we are given Systematic its associated uncertainty, and degrees of freedom,

as shown in Table 5. We will not elaborate on the origin of the systematic error in Table 5; however, the same general procedures used to estimate esystematlc , discussed

in detail in Section 3, can also be used to estimate Systematic- Although hematic = 05 J in this illustrative example, typically Somatic

is assumed to be zero because errors that are well understood and could be

corrected for are minimized during direct verification of the machine. So,

neglecting contributions to the bias from 8 f , the estimated machine bias O o ^systematic ' is calculated as the difference between the mean of the specimen tested in the

indirect verification test and the certified value of the specimens tested. For our example, in which Ss stematlc is not assumed to be zero, the machine bias is

Assuming independent input quantities, the standard uncertainty of the machine

bias is

(9)

b = 106.2 J - 0.5 J - 1 09.9 J = -4.2 J .

(10)

10
D"; act Verification

Substituting the appropriate values from Tables 3 through 5 into Eq. (10) provides

the following estimate of the standard uncertainty of the machine bias:

'(2 3 J)"

*/(&) = J——^- + (0.2 J)" +(2.6 J)" =2.8 J

The degrees of freedom associated with the uncertainty estimate.

u

4 (b)

dfv l«ry

| "Vsystemanc)

| U* (R) (H)

df8 ~ dfR are determined from the Welch- Satterthwaite approximation. In our example, the

degrees of freedom are w =8Mi

1 (2.3)' 4{ 5 which rounds down to 84.

{ (0.2)

4 (2.6y

10 102

In the examples presented here, the "bias compared to what?" issues are clear. Machines verifying to ASTM E 23 requirements are all compared with a single

target for impact energy, defined by ASTM E 23. However, when considering the

performance of an ASTM E 23 machine to machines not tested under ASTM E

23 requirements, the comparison is less direct because bias can exist between the

various verification systems used around the world (multiple certified values for absorbed energy). We encourage the users to understand this issue, and how it might affect them. Users should also know that the various national measurement

institutes distributing impact verification specimens are working to minimize

biases among them, and make the quantification of bias for impact testing more

transparent to users around the world.

5. Direct Verification

Direct-verification uncertainty sources are related to physical properties of the

Charpy machine including: anvil and supports, center of strike, potential energy, impact velocity, center of percussion, friction loss, and scale accuracy. With the

possible exception of friction loss, all direct verification sources of uncertainty

are Type B evaluations. We provide information regarding calculation of the

individual sources of direct verification uncertainty in Appendix B. While it is relatively easy to compute each individual source of uncertainty, it is difficult to quantify the uncertainty components in terms of the effect on Charpy

measurements in joules.

11
Computing Uncertainty for Charpy Impact MachineTest

The recognized sources of uncertainty for our problem are minimized during

the direct verification of an impact machine and by following the standard test procedure. So, it is general practice to estimate the uncertainty of impact tests from the results of indirect verifications and the variations associated with repeat

measurements on the material being tested. However, it is also of interest, and

part of the exercise in calculating uncertainty, to better understand your machine

and process so that it might be better controlled and quantified. It is left up to individual laboratories to identify and include the appropriate uncertainty sources. Although it is common for laboratories to ignore the uncertainty due to direct- verification bias, it is important to acknowledge the potential for error due to these

sources. Thus, it is informative for laboratories to document their reasons for

either including or excluding direct verification sources of error. If possible, the

uncertainty associated with direct verification should be re-examined each time the

machine is verified directly.

6. Temperature

Although systematic error due to temperature probably exists to some extent for all Charpy measurements, it is difficult to quantify the sign (direction) and magnitude

of the error. Thus, we typically assume the estimated error is zero, but there is some uncertainty associated with the estimate. This section outlines a procedure

that can be used to estimate the uncertainty due to systematic temperature errors. The uncertainty due to temperature does not depend on machine properties;

however, it is highly dependent on the material being tested. For example, steels undergo a transition in fracture behavior from brittle to ductile with increasing

temperatures. Supplemental data can be collected for a particular steel of interest,

and used to estimate the uncertainty associated with temperature. If later measurements are taken in stable regions defined by the lower shelf or upper shelf

(Figure 1), then the uncertainty associated with temperature is probably negligible.

However, the uncertainty due to temperature can be significant if measurements are being taken in the transition region of the curve.

12
Upper Shelf

Lower Shelf

Temperature

Figure 1. A temperature transition curve.

u 1 1

1

1

1

1

1 11

1

1

1

1

1 f 35 37 39 41 43 45 47 49

Temperature, C

Figure 2. Mapping oftemperature error into energy error in the temperature transition region.

Assuming we have data for a particular material that have been collected across a range of temperatures, we can fit a straight line to the data within the temperature

transition region (ignoring the shelf data). The information from the regression

fit can be used to quantify the effect of the temperature error on impact energy

(Figure 2) for future samples of the same material.

For example, suppose we are interested in collecting some new data at 43 °C,

but our temperature can be measured only to within ±2 °C. The true temperature

could be anywhere in the range of 41 °C to 45 °C. Using the regression equation,

we can compute the value of impact energy for both 41 °C and 45 °C, thus

13
Computing Uncertainty for Charpy Impact MachineTest

providing a range of potential impact energy values AE. Assuming the true impact

energy has a rectangular distribution within AE , we can use the range of impact

energy to compute the uncertainty as follows:

2V3

The degrees of freedom are dft =n-2, where n is the number of observations

used in the regression fit.

Optionally, multiple measurements could be made at each temperature (which is how the original curve is obtained) and define the uncertainty as the maximum

uncertainty observed in the region. This procedure can also be applied in cases where upper or lower shelf regions have more gradual slopes.

7. Expanded Uncertainty

Sometimes we need to calculate an expanded uncertainty, U , which is just the

combined standard uncertainty multiplied by a constant, or coverage factor, so that

The coverage factor k is determined by looking up the appropriate value in a /-table

(Appendix C) based on the degrees of freedom associated with w(ycorrected ). The

expanded uncertainty associated with a 95 % interval is U95 = k95

• w(_ycorrected ) = /

.9 75 . dfcfr

• w(.ycorTected ) . ^12)

The expanded uncertainty is interpreted as an uncertainty interval encompassing a

large fraction of possible measurement results. The degrees of freedom can be difficult to determine if there are many sources of

uncertainty within w(ycorrected ) . Fortunately, we can compute the effective degrees of

freedom from the Welch-Satterthwaite approximation [2] ^Greeted)

9 (13)

df

f 2 s dfb dfe where df = n-\ and dfe are from the Type B uncertainty evaluation (see Section 3). We will also need to calculate dfb from

i4
1 dfV

u

4 (b) 14)

dt; where dfv = nv -1 . dfR

is provided by NIST with the indirect verification

specimens, and df5 is from Type B uncertainty evaluation (see Section 3).

In general, an uncertainty interv al for vcorrected is

^ corrected ~ ^\—a or

^corrected ~ A-^<.df,ff (^corrected ) (15)

Typically a is 0.05, which corresponds to a 95 % interv al. If a Charpy lab does

not report results corrected for machine bias and systematic errors, they may want

to indicate the magnitude of the estimated biases for informational purposes,

(y-b-i^+U^ or (y±C/1 .J-(6 + Vematlc ) • (16)

Thus, the interv al would be shifted by b + esysbaoatic if the laboratory wished to report

the corrected mean absorbed energy; however, the expanded uncertainty would not be affected by the machine bias and systematic error corrections.

In practice, k = 2 is often used to compute the expanded uncertainty to approximate a 95 % interval, and the effective degrees of freedom are never

calculated. However, if the effective degrees of freedom are small, then the level of confidence is thought to be less than 95 %.

8. Examples

All examples utilize the data displayed in Tables 1,3, and 4.

15
Computing Uncertainty for Charpy Impact MachineTest

8.1 Both esystematic and SsyMic and Their Uncertainties Are Negligible

In the general case,

Corrected = y ~ b - &systematic '

but if the systematic error associated with the material variation esystematic is assumed

to be negligible, then

^corrected b "

If the systematic J error for the test machine variation § is assumed to be

systematic negligible,

and

b = V-5syM^c -R = V-R = 106.2 J - 109.9 J = -3.7 J

ycoirattd =y-6 = 57.6J-(-3.7J) = 61.3 J.

The combined standard uncertainty of b is

^- + ir(R)

(2.3 jy + (2.6 J)'

2.8 J,

with effective degrees of freedom

df

( C 2 V 4 df df, 4

(2.8)

4

"(2-3)

2 Y

r 5

84.5

(2-6)

4 102

16
Examples

which rounds down to 84. Thus, the uncertainty of the corrected mean value is

W ( ^corrected) + U (b) + U (£systematlc )

+ u

2 (b)

(3.6 J)

: V 5 3.2 J,

+ (2.8 J)

:

with effective degrees of freedom

df

i<\yc (3.2)

4

df f2 +ll \b) 1 \ n J 4 V 5 J

43.5 (2.8V

84

which rounds down to 43. A 95 % interval for >

; corTected is

^corrected — t\-%dfe{[

' U(^corrected ) 61.3 J±f 97M3 -3.2 J 61.3 J±2.017-3.2 J 61.3 J±6.5 J. The expanded uncertainty, associated with a 95 % level of confidence is 6.5 J. The 95 % uncertainty interval is (54.8 J, 67.8 J).

If the value reported is not corrected for machine bias, we can express our

interval as (y±U)-b

(57.6 J ±6.5 J) -(-3.7 J) (51.1 J, 64.1 J) + 3.7J.

The Charpy laboratory may or may not wish to disclose the estimated machine

bias, however the information is available if needed. Notice that the interval for

the uncorrected parameter is shifted just by the amount of the correction and the

expanded uncertainty is the same regardless of whether or not the reported value is corrected.

17
Computing Uncertainty for Charpy Impact MachineTest

8.2 esystematic Has One Component

Suppose <?systematic contains the error due to temperature so that £systemalic = i . The

temperature error is systematic because it is likely to be in the same direction

(always warmer or always cooler than the target temperature) for a single set of

measurements. However, we do not typically estimate the temperature error, so we will we assume the value of e tematic is zero. The uncertainty associated with

£ IS

systematic

^systematic) = W (0

The procedure outlined in Section 5 will be used to determine the uncertainty due

to temperature «(?) . Figure 3 displays temperature data for the material of interest along with the regression line fit to the data in the transition region (ignoring the

data on the "shelves").

Suppose our test specimens from Table 1 were broken using a temperature of 80° C,

which is within the temperature transition region. A regression line was fit to the

21 data points in the transition region, resulting in the following equation:

E(J) = -0.03973(J) + 0.74084(J/°C) • T(°C) .

-20 20 40 60 80 100 120 140 160

Temperature, C

Figure 3. Temperature datafor the test material. The straight line in the plot represents a

regressionfit to the data in the transition region only. We ignore data on the "shelves " at -10°Cand 141 °C.
Examples

If our temperature can be measured to within ±1 °C, then the true temperature is between 79 °C and 8 1 °C, and the energy range is defined by the following:

E(J) = -0.03973(J) + 0.74084(J/°C)-79(°C) = 58.5 J E(J) = -0.03973(J) + 0.74084(J/°C)-81(°C) = 60.0 J AE = 60.0 J-58.5 J = 1.5 J .

Next, the energy range is converted to a standard uncertainty based on a

rectangular distribution,

^= ^= 04,,

with degrees of freedom dft = «-2 = 21-2 = 19. Thus the uncertainty of <?systematlc is

with 19 degrees of freedom.

The mean absorbed energy corrected for machine bias and other systematic effects

is

^corrected =5^"Wc = 57.6 J -(-3.7 J)-0 J = 61.3 J ,

where b , its uncertainty, and degrees of freedom have not changed from example

7.1. The combined standard uncertainty is

with degrees of freedom

df =

»

4 (.vcomcKd ) = QSf = 492

df

s~

I

| u\b)

| u (gsystematlc ) 1 dfb dfe 4

(3.6)

2 V | (2.8)

4

| (0.4)

4 5

J 84 19

which rounds down to 49. A 95 % uncertainty interval for ycorrected is

^corrected — ^l-^,dfeff

' U (^corrected ) 61.3 J±? 975 49

- 3 . 3 J 61.3 J + 2.010-3.3 J 61.3 J±6.6 J.
Computing Uncertainty for Charpy impact MachineTest

The expanded uncertainty is 6.6 J, and the uncertainty interval encompassing 95 %

of possible measurement results is (54.7 J, 67.9 J). If the value reported is not

corrected for bias, we can express our interval as

(y±U)-(b+esyMk) (57.6 J±6.6 J) — (—3.7 J + J) (51.0 J, 64.2 J) + 3.7 J.

9. Closing Remarks

We have developed a procedure for estimating the uncertainty associated with

a reported mean absorbed energy from a Charpy test. The procedure is flexible

enough to account for several systematic error sources, if necessary, and allows the user the choice of correcting the reported mean or not. The uncertainty procedure

in this document applies to measurements completed in a Charpy laboratory.

Occasionally, there is some confusion about the NIST reference value, the

reference value uncertainty, and Charpy verification limits with respect to results obtained in a Charpy laboratory. The reference value is the measured mean absorbed energy of a batch of reference specimens. The reference value

uncertainty describes the variability of the reference value and includes material,

system, and machine variability. The reference value uncertainty does not describe

the variability of a single verification specimen or the variability in the verification

specimens (specimen variation cannot be estimated separately from machine

variation). In the Charpy laboratory, the reference value and its uncertainty are used only to estimate the bias of a Charpy machine and the uncertainty of the bias; they provide no information regarding Charpy measurements for other materials.

It is also important to remember that the reference value uncertainty is associated

with a specific measurement result, while the verification limits describe the

acceptable variation among means for a test method. These two items are not

necessarily related.

20
nonces #

10. References

[1] ASTM E 23-02a, "Standard Test Methods for Notched Bar Impact Testing of

Metallic Materials," ASTM International, West Conshohocken, PA.

[2] International Organization for Standardization, Guide to the Expression of

Uncertainty in Measurement, International Organization for Standardization,

Geneva Switzerland, 1993 (corrected and reprinted 1995).

[3] ISO 148-1, "Metallic materials - Charpy pendulum impact, Part 1: Test method",

ISO, Geneva, Switzerland.

[4] Yamaguchi, Y., Takagi, S., and Nakano, H., "Effects of Anvil Configurations on Absorbed Energy," Pendulum Impact Testing: A Century of Progress, STP

1380, T. A. Siewert and M. P. Manahan, Sr., Eds., American Society for Testing

and Materials, West Conshohocken, PA, 2000.

[5] NIST/SEMATECH e-Handbook ofStatistical Methods,

http://www.itl.nist.gov/div898/handbook/, July 18, 2006.

21
Computing Uncertainty for Charpy Impact MachineTest

Appendix A. Uncertainty Details

The following information is provided for completeness and to document the

justification for the recommended uncertainty procedures.

A.l Test Material

We define a single measurement for a test material measured in a Charpy

laboratory as

V = Li + h + £ -\- € + 6. + f? si r*Y Y /.inhomogeneity i. repeatability /'. other random systematic

for i" = l,2,...,/i measurements. Terms on the right side of the equation having the

"f subscript denote random errors that change from measurement to measurement.

ju

} represents the true mean breaking energy of the test material if the material

could have been tested on the three NIST reference machines.

bY represents the true machine bias for the test material. This term includes

all machine differences that are constant for the duration of the set of n measurements.

homogeneity represents the material inhomogeneity.

^.repeatabii.ty represents the machine repeatability.

ei other random represents all other sources of error due to random effects.

Systematic represents errors due to all other systematic effects that are not already

included in the machine bias (for example, operator error). Systematic errors remain constant for the duration of the set of n measurements.

The mean of n measurements of the test material is

v = u -\- h ~\~ p ~\~ p ~\~ p ~\~ p J r*Y Y inhomogeneity repeatability other random systematic 5

and the true variance of y is

22
A

2 2 2 /—x inhomogeneity , ^repeatability , other random var(j/) = —- + — + , n n n

S

2 which is estimated by —, with df = n - 1 degrees of freedom. The three random

n

errors cannot be estimated separately. The corrected value is

-^corrected 3 ^systt

The indirect verification results will be used to estimate bY and its uncertainty, and

we will assume that <?systematlc is zero. There is uncertainty associated with each of

the estimated systematic errors. The combined standard uncertainty of corrected

value is

^corrected ) = \ ~+ (bY ) + (^systematic ) The effective degrees of freedom based on the Welch- Satterthwaite approximation

are

^ \ ycorrected )

eff 2 « +

u

4 (bY ) t n\esystematlc ) df n J dfb dfe The expanded uncertainty associated with vcorrected is U = tl-%dleS

- ^corrected)- The corrected value reported by the Charpy laboratory has the form ycorTected ± U . If a Charpy laboratory does not report results corrected for machine bias, they may

want to indicate the magnitude of the estimated bias for informational purposes as

(y ~ K - hematic ) ±U OT (y±U)~ (by + <?systematlc )

.

A.2 Indirect Verification Test

The Charpy laboratory's indirect verification test will be used to estimate

machine bias in conjunction with the associated NIST reference value. A single

measurement in the indirect verification test is defined as

23
Computing Uncertainty for Charpy Impact MachineTest

TT 1 C* C* C* C*

/' /fz V /, inhomogeneity /.repeatability /', other random systematic '

where i = l,2,...,nv measurements (nv is usually five). The "/" subscript denotes

errors that change from measurement to measurement.

juz represents the true mean breaking energy of the reference material if the

material could have been tested on the NIST reference machines.

bv represents the machine bias for the reference material. This term includes

all machine differences that are constant for the duration of the set of nv measurements.

8 t, • , represents the reference material inhomogeneity. /.inhomogeneity * o j 8i repeatabillty represents the machine repeatability.

8i other random represents all other sources of error due to random effects.

Systematic represents errors due to all other systematic effects that are not already

included in the machine bias. Systematic errors remain constant for the duration

of the set of nv measurements.

The mean of ny measurements is

^ f^Z fy' + ^inhomogeneity ^repeatability ^other random ^systematic '

and the variance of V ,

2 2 2 inhomogeneity repeatability other random Vcir( v ) — 1 1

S

is estimated by —, with dfv = nv - 1 degrees of freedom. The three random

nv errors cannot be estimated separately.

24
Appendix A

A.3 NIST Reference Value

The NIST reference value will be used to estimate machine bias in conjunction

with the customer's associated verification test.

According to ASTM E 23-06, the reference value of Charpy indirect verification

specimens is established using three master machines maintained by NIST.

In the NIST Charpy verification program, the reference value and its associated

uncertainty are based on two sets of measurements. The first set of measurements

involves breaking 75 verification specimens (25 on each master machine) from

a "pilot" lot to determine if the material meets the rigid specifications of the

verification program. If the material is acceptable, the remaining verification

specimens in the lot are machined and a second set of measurements are performed from the full "production" (25 on each master machine). Assuming

the production lot has not changed significantly from the original pilot lot, the

material is sold to the public in sets of five specimens as a Standard Reference

Material. The reference value R is established using the 75 verification lot and 75

production lot specimens.

We make the following assumptions when determining the reference value and its uncertainty.

1 . The reference value is defined to be the "truth," so there is no bias associated

with the reference value.

2. There is no difference between pilot lot specimens and production lot specimens. (Differences are evaluated using a Mest for means and an F-test

for variances.) In the event that the verification lot and production lot have

significantly different means and/or variances, the reference value will be

based solely on the production lot data.

A.3.1 Reference Machine

We define a single measurement taken on a NIST reference machine as

^\k Yk. inhomogeneity Yk, repeatability Y

k

, other random ^systematic '

where k = l,2,...,n1 measurements («, is usually 50). The "&" subscripts on the

right hand side of the equation denote errors that change from measurement to measurement.

M\ represents the true mean breaking energy of the reference material as measured

by the NIST reference machine.

25
( ompul ing Uncei tainty for Charpy impact MachineTest

//t,inhomogenaty represents the reference material inhomogeneity.

^repeatability represents the machine repeatability.

7k, other random represents all other sources of errors due to random effects.

/systematic represents the errors due to all systematic effects. Systematic errors remain constant for the duration of the set of nx measurements. Although we

assume /systematic is zero, it does have some uncertainty.

The mean of nx measurements taken on a NIST reference machine is

^1 Ml /mhomogcneity Yrepeatability Yother random ~^ Ysystematic ' and the variance associated with the mean,

\ ^"inhomogeneity ^"repeatability ^"other random var(Z, ) = —- + — + j nx w, w, S

2 is estimated by — , withdf, =nl -1 degrees of freedom. The three random errors nx cannot be estimated separately.

The corrected value for the NIST reference machine is Z1, corrected = Z1 -Y/ systematic ' The combined standard uncertainty of the corrected value is

W(^l , corrected ) ~ \ \ ^ U (/systematic )

which has effective degrees of freedom

U (^.corrected) dfz, df

^ (/systematic

)

based on the Welch-Satterthwaite approximation.

26
Appendix A

The procedure for computing Zlcorrected , w(Zl corrected )

, and dfz for one reference

machine also applies to the remaining two NIST reference machines so that we

obtain, Z2 corrected , u(Z2 corrected )

, and dfZi for the second reference machine, and

Z3 corrected » U ( Z3 corrected) » and dfZi f°r me mW reference machine. The results from

all three reference machines are needed to compute the NIST reference value, as we discuss below.

A.3.2 NIST Reference Value

The NIST reference value based on data observed for the three reference machines

is defined as

r-z ~

where ^ , /j2

, and //3 denote the respective true mean breaking energies for each of

the three reference machines. The NIST reference value defines the true breaking

energy of the material.

We estimate the NIST reference value using

Zl, corrected Z2, corrected Z3, corrected

which has combined standard uncertainty

"( R) =

yj^l"

2 (Zl, corrected ) + ^ ( Z2, corrected ) + " ( Z3,corrected )] , and effective degrees of freedom

u\R)

df,R / 1 \4

corrected , corrected )

| (j) » (Z3,co.rected) dfz, df, "

df.

based on the Welch-Satterthwaite approximation. The reference value expanded

uncertainty is

27
Computing Uncertainty for Charpy Impact MachineTest

A.4 Estimating Machine Bias

Assume the machine bias is the same for the new material ( bY

) and the reference

material ( bv ), so that bY = bv = b . The best estimate of the machine bias b is

which has combined standard uncertainty

systematic

The effective degrees of freedom based on the Welch-Satterthwaite approximation

are

$v)

^(Systematic) , u\ff)

dfs dfR dfx Appendix B. Direct Verification Components of Uncertainty

B.l Anvils and Supports, A

A paper by Yamaguchi, Takagi, and Nakano [4] provides some information

regarding the uncertainty associated with anvil configurations. Assuming that other

Charpy machines behave similarly to the machine tested in the paper, we can use

the uncertainties listed in the paper (Table 9) as ballpark estimates. Table B.l lists the uncertainty estimates for low, high, and super-high energies.

Table B.l. Estimated uncertainties due to the anvil and support bias.

Standard uncertainty Low energy High energy Super-high energy

U(A) 0.05 J 0.29 J 0.77 J

Since degrees of freedom are not provided in the paper, we will also assume that dfA = oo , which implies that we know u{A) exactly.

B.2 Height of Pendulum Fall, /i The height of the pendulum fall is

h = S(l-cosj3%

28
Appendix B

where 5 is the measured length of the pendulum, and j3 is the measured fall angle. Letting AS and Ap denote the manufacturer's stated error bounds,

respectively, and assuming a rectangular distribution bounded by ±AS and ±AJ3 , the uncertainties for S and J3 are

u(S) = ^L and = ^§ . V3 V3

Then the uncertainty of h is

ic(h)

dh

dS

ir(S) +

dh_

dp MSf

c] ir (S) + clu

2 (P) + 2cs c

p u(S, P)

,

where —dh , _ dh . = l-cos>ff and c

fi = —= S-(smfi).

dS op

li(S, P) are independent, then only the first two terms are needed to determine

the uncertainty. We can assume that dfs = x and df

p = oc , which implies that we know u(S) and u(fl) exactly. (See the ISO-GUM, G.4.3 [2] for details.) The

effective degrees of freedom associated with u(h) are

df

cju\S)

| c\u\P)

dfs dfB

based on the Welch- Satterthwaite approximation.

B.3 Potential Energy, E

The potential energy is E-h F

,
Computing Uncertainty for Charpy Impact MachineTest

where F is the measured supporting force exerted by the pendulum in horizontal

position, and h is the height of the pendulum fall defined in Section B.2. Letting

AF denote the manufacturer's stated error bound of the measurement instrument,

and assuming a rectangular distribution bounded by ±AF , the uncertainty of F is

Then the uncertainty of E is

^dh ){dF

u(h,F)

c\ u

2 (h) + c

2F u

2 (F) + 2ch cF u{K F)

,

where

dE Z7 a dE 1 c,= —= F and c c —= h

dh dF

The uncertainty associated with h is defined in Section B.2. If (h, F ) are independent, then only the first two terms are needed for the uncertainty. We can assume dfF = oo , which implies that we know u(F) exactly. The effective degrees

of freedom associated with u(E) are

dft c

A

h u\h)

| c

AF u\F)

df

based on the Welch-Satterthwaite approximation.

B.4 Impact Velocity, v The impact velocity is v

v = yl2-g-h , where g is the local acceleration of gravity, and h is the height of the pendulum

fall defined in Section B.2. Letting Ag denote the manufacturer's stated error bound of the measurement instrument (0.001 m/s2 according to ASTM E 23), and

30
assuming a rectangular distribution bounded by ±Ag , the uncertainty for g is

The uncertainty of v is

u\v)

dv_ 8g

- Q "

{ln

where

= c

2 u2 (g) + c2h u

2 (h).

dv h\fl dv gyfl

c? = —= i—

f

and ch -

dg 2yjg-h

1 dh 2y]g-h

The uncertainty associated with h is defined in Section B.2. We can assume

df = oo, which implies that we know u(g) exactly. The effective degrees of

freedom associated with w(v) are

df

^ c

Ag u\g)

| c\u\h)

' df +

df

based on the Welch-Satterthwaite approximation.

B.5 Center of Percussion, L

The center of percussion is

Ax

where g is the local acceleration of gravity defined in Section B.4, and p is the mean period of the swing of the pendulum from three measurements for 100

swings. (There may be some systematic error associated with P that should

taken into account.) The standard deviation of three p measurements is sp

, so the

uncertainty of the mean period is

with df

p =3-1 = 2 degrees of freedom.

31
Computing Uncertainty for Charpy Impact MachineTest

The uncertainty of L is

u\L) u (g) +

dL

dp

clu\g) + c

2 u\p).. 2

u (p)

where

dL p dL g • p

c

* = Tg

= J? and c

" = ^

= ^'

The uncertainty associated with g is defined in Section B.4. From the WelchSatterthwaite approximation, the effective degrees of freedom associated with

u(L) are

dfL = 4 4

1< {L

\ 4 c>4 (g) c%/ (/7) df df

B.6 Friction Loss, D

The friction loss is D = E -El ,

where E is the potential energy due to the combined indicator and pendulum, and

Ex is the potential energy due to the pendulum. The uncertainty of D is

n\D) = u

2 (E ) + ir (£, ) + 2u(E ,El ).

Assuming perfect correlation between E and Ex , a conservative estimate of the

covariance u{EQ ,Ex ) is

u(E0i El ) = yJifiE^ + u

2 ^) .

The effective degrees of freedom associated with u(D) are

32
Appendix B

u\D)

u\EQ ) + u\Ex )

based on the Welch-Satterthwaite approximation.

B.7 Scale Accuracy, r Let r represent the bias in the scale mechanism and ± Ar be the specified error bounds of the measurement instrument. Assuming a rectangular distribution, the

uncertainty of r is

We will assume dfr = x , which implies that we know u(r) exactly.
Computing Uncertainty for Charpy Impact MachineTest

Appendix C. f-Table

The following /-table values were taken from NIST/SEMATECH e-Handbook of

Statistical Methods [5]. Table C.l Upper critical values of Student's t distribution with degrees of freedom, df. df 0.900 0.950 0.975 0.990 0.995 0.999

1 3.078 6.314 12.706 31.821 63.657 318.313

2 1.886 2.920 4.303 6.965 9.925 22.327

3 1.638 2.353 3.182 4.541 5.841 10.215

4 1.533 2.132 2.776 3.747 4.604 7.173

5 1.476 2.015 2.571 3.365 4.032 5.893

6 1.440 1.943 2.447 3.143 3.707 5.208

7 1.415 1.895 2.365 2.998 3.499 4.782

8 1.397 1.860 2.306 2.896 3.355 4.499

9 1.383 1.833 2.262 2.821 3.250 4.296

10 1.372 1.812 2.228 2.764 3.169 4.143

11 1.363 1.796 2.201 2.718 3.106 4.024

12 1.356 1.782 2.179 2.681 3.055 3.929

13 1.350 1.771 2.160 2.650 3.012 3.852

14 1.345 1.761 2.145 2.624 2.977 3.787

15 1.341 1.753 2.131 2.602 2.947 3.733

16 1.337 1.746 2.120 2.583 2.921 3.686

17 1.333 1.740 2.110 2.567 2.898 3.646

18 1.330 1.734 2.101 2.552 2.878 3.610

19 1.328 1.729 2.093 2.539 2.861 3.579

20 1.325 1.725 2.086 2.528 2.845 3.552

21 1.323 1.721 2.080 2.518 2.831 3.527

22 1.321 1.717 2.074 2.508 2.819 3.505

23 1.319 1.714 2.069 2.500 2.807 3.485

24 1.318 1.711 2.064 2.492 2.797 3.467

25 1.316 1.708 2.060 2.485 2.787 3.450

26 1.315 1.706 2.056 2.479 2.779 3.435

27 1.314 1.703 2.052 2.473 2.771 3.421

28 1.313 1.701 2.048 2.467 2.763 3.408

1^11 i.jii 1 6QQ ? 04S 1 46? 1 7S6

30 1.310 1.697 2.042 2.457 2.750 3.385

40 1.303 1.684 2.021 2.423 2.704 3.307

50 1.299 1.676 2.009 2.403 2.678 3.261

60 1.296 1.671 2.000 2.390 2.660 3.232

70 1.294 1.667 1.994 2.381 2.648 3.211

80 1.292 1.664 1.990 2.374 2.639 3.195

90 1.291 1.662 1.987 2.368 2.632 3.183

100 1.290 1.660 1.984 2.364 2.626 3.174

00 1.282 1.645 1.960 2.326 2.576 3.090

34
Appendix D

Appendix D. Glossary of Terms

ny

sdf

dfvR

u(R)

dfR

"(^systematic) die

systematic

systematic) df.

/corrected "0* corrected) dfeff U

Number of test material samples measured

Mean absorbed energy of test material samples

Standard deviation of test material samples

Degrees of freedom for test material standard deviation Number of indirect verification samples measured

Mean absorbed energy of verification samples

Standard deviation of verification samples

Degrees of freedom for verification material standard deviation NIST reference value Standard uncertainty of NIST reference value Degrees of freedom for reference value standard uncertainty

Systematic error estimate associated with test material Standard uncertainty of test material systematic error Degrees of freedom for standard uncertainty of test material systematic error Systematic error estimate associated with verification material Standard uncertainty of verification material systematic error Degrees of freedom for standard uncertainty of verification material systematic

error Corrected test result Combined standard uncertainty of corrected test result Degrees of freedom for combined standard uncertainty of corrected test result Expanded uncertainty of corrected test result

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