ISO/TR 15263:2024

Technical
Report
ISO/TR 15263
First edition
Measurement uncertainties in
2024-01
mechanical tests on metallic
materials — The evaluation of
uncertainties in tensile testing
Incertitudes de mesure dans les essais mécaniques sur matériaux
métalliques — Évaluation des incertitudes pour les essais de
traction
Reference number
© ISO 2024
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ii
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions .1
3.2 Symbols .2
4 Steps for the evaluation of uncertainty . 4
4.1 General .4
4.2 Step 1 — Defining the measurands for which uncertainty is to be evaluated .5
4.3 Step 2 — Identifying all sources of uncertainty in the test .6
4.4 Step 3 — Estimating the standard uncertainty for each source of uncertainty.8
4.4.1 General .8
4.4.2 Type A evaluation of standard uncertainty .8
4.4.3 Type B evaluation of standard uncertainty.9
4.4.4 Uncertainty due to repeatability of measurement .9
4.5 Step 4 — Computing the combined uncertainty, u .10
c
4.6 Step 5 — Computing the expanded uncertainty, U .10
5 Reporting of results .10
Annex A (informative) Mathematical formulae for calculating uncertainties in tensile testing .12
Annex B (informative) Worked example for calculating uncertainty in the determination of
proof strength at ambient temperature .25
Annex C (informative) Interlaboratory scatter .29
Bibliography .35

iii
Foreword
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The procedures used to develop this document and those intended for its further maintenance are described
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This document was prepared by Technical Committee ISO/TC 164, Mechanical testing of metals, Subcommittee
SC 1, Uniaxial testing.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.

iv
Introduction
This document is based on a CEN Workshop Agreement which was issued in 2005 following a trilogy of
documents concerned with measurement uncertainties in mechanical tests on metallic materials. The
trilogy includes three documents, one concerned with the evaluation of uncertainties in low cycle fatigue
testing, one with creep testing and this document focused on tensile testing.
For a meaningful estimate of uncertainty, all primary sources of uncertainty are to be included and
their effects are to be properly quantified in the analyses. Reporting and interpreting the results of the
calculations is also of utmost importance.
The calculations given in this document only capture the uncertainty associated with the uncertainty of the
testing equipment’s sensors over a short time scale. The uncertainty of a test result is dependent on much
more than the uncertainty of the testing equipment’s sensors.
Contributions to uncertainty due to misalignment, long-term environmental effects, and intermittent
procedural errors, to name a few, are not included in these analyses. This is demonstrated by the results of
the interlaboratory reproducibility in Annex C compared to the example given in Annex B.
A more realistic value of the uncertainty of the properties of material can be estimated using reproducibility
data from laboratory intercomparisons involving several laboratories.
Results from reproducibility tests also include contributions to uncertainty from material inhomogeneity,
different testing machines, controlling, and processing software together with the influence of different
operators.
1) [1]
This document is based on CWA 15261-2 and UNCERT CoP 07 . It describes a method for evaluating the
uncertainty in tensile test results obtained from a series of tests that are performed in accordance with
[2] [3] [4] [5] [6] [7] [8]
ISO 3534-3, ISO 5725 series, ISO 6892-1, ISO 6892-2, ISO 9513, ISO Guides 33 and 35 . For a
general introduction on the subject of uncertainty in measurement and testing refer to References [12] and
[13].
1) Withdrawn
v
Technical Report ISO/TR 15263:2024(en)
Measurement uncertainties in mechanical tests on metallic
materials — The evaluation of uncertainties in tensile testing
1 Scope
This document describes how the evaluation of uncertainties in tensile tests can be obtained from tests at
room temperature (ISO 6892-1) or elevated temperature (ISO 6892-2).
This document reports how it can be applied to tests performed at ambient and elevated temperatures
under axial loading conditions with a digital acquisition of force and displacement.
2) [1]
NOTE 1 As CWA 15261-2 and UNCERT CoP 07 reports, the tests are assumed to run continuously without
interruptions on test pieces that have uniform gauge lengths.
NOTE 2 Annex C gives for information an indication of the typical scatter in tensile test results for a variety of
materials that have been reported during laboratory inter-comparison exercises.
2 Normative references
There are no normative references in this document.
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1.1
coverage factor
number that, when multiplied by the combined standard uncertainty, produces the expanded uncertainty
Note 1 to entry: It is dependent on the confidence level (e.g., 95 % probability). It also depends on the effective degrees
of freedom.
3.1.2
level of confidence
probability that the value of the measurand lies within the quoted range of uncertainty
3.1.3
measurand
specific quantity being reported as the measurement result
Note 1 to entry: A measurand can be a direct test reading or an estimate of a material property from other readings.
2) Withdrawn.
3.1.4
measurement
set of operations having the object of determining a value of the measurand
3.1.5
result
distinction is made between:
3.1.5.1
result of a measurement
value attributed to the measurand, obtained by measurement
3.1.6
standard deviation
positive square root of the variance
3.1.7
uncertainty of measurement
parameter, associated with the result of a measurement, that defines the range within which a specific
fraction of the distribution of values that could reasonably be attributed to the measurand is estimated to
fall (within a given confidence)
3.1.8
standard uncertainty
estimated standard deviation or estimated positive square root of the variance
3.1.9
expanded uncertainty
value obtained by multiplying the combined standard uncertainty by a coverage factor
3.1.10
variance
measure of the dispersion of a set of n measurement results. It is the sum of the squares of the deviations of
the measurement results from the average, divided by n-1
3.2 Symbols
The symbols used in this document and corresponding designations are given in Table 1.
Table 1 — Symbols and corresponding designations
Symbol Unit Designation
a mm Original thickness of a flat test piece
o
a mm Minimum thickness after fracture
u
b mm Original width of the parallel length of a flat test piece
o
b mm Minimum width after fracture
u
d mm Original diameter of the parallel length of a circular test piece
o
d mm Minimum diameter of a circular test piece after fracture
u
m MPa Slope of the elastic part of the stress-extension curve
E
a
E GPa Young’s modulus (modulus of elasticity)
F N Force
ΔF N Force increment
F N Force at R
eH eH
F N Force at R
eL eL
a 2 2
1 MPa = 1 N/mm ; 1 GPa = 1 kN/mm .
b
Depends on the property concerned.

TTaabbllee 11 ((ccoonnttiinnueuedd))
Symbol Unit Designation
F N Force at R
p p
F N Maximum force
m
L mm Original gauge length
o
L mm Extensometer gauge length
e
L mm Gauge length after fracture
u
n - Number of readings or results or evaluated data pairs in the linear regression or
numerical coefficient
NOTE This is also used for other parameters e.g., number of samples in a batch.
(See 3.4.2).
a
R MPa Stress
R MPa Upper yield strength
eH
R MPa Lower yield strength
eL
R MPa Tensile strength
m
R MPa Proof strength, plastic extension (e.g., 0,2 %, R )
p p0,2
A % Percentage elongation after fracture
S mm Original cross-sectional area
o
S mm Smallest cross-sectional area after fracture
u
Z % Percentage reduction of area after fracture
b
u Standard uncertainty (in general)
b
u(x ) Standard uncertainty on measurement x
i i
b
u Combined uncertainty (in general)
c
b
y Test (or measurement) mean result
b
u ( y) Combined uncertainty on the mean result of a measurement
c
b
Y Evaluated value of the measurand
d - Divisor associated with the assumed probability distribution
v
b
c Sensitivity coefficient (in general)
b
c Sensitivity coefficient associated with uncertainty on measurement x
i i
b
c Temperature sensitivity coefficient
T
b
x Individual value
i
b
Arithmetic mean
x
b
s Sample standard deviation
b
a , δ Mid-point value between the upper and lower limits
NOTE Subscripts corresponding to the concerned property, e.g., δ .
ao
k, k - Coverage factor
p
b
U Expanded uncertainty
ν - Effective degrees of freedom given by the Welsh-Satterthwaite method
eff
f - Degree of freedom (n – 1)
b
Y Evaluated value of the measurand
p % Coverage probability; confidence level
t - Factor of Student's distribution
b
m Slope of the regression line or strain rate sensitivity
NOTE Subscript corresponding to the concerned property, e.g., m .
E
b
b Intercept in the regression line
a 2 2
1 MPa = 1 N/mm ; 1 GPa = 1 kN/mm .
b
Depends on the property concerned.

TTaabbllee 11 ((ccoonnttiinnueuedd))
Symbol Unit Designation
b
S Empirical covariance in the linear regression
xy
b
S Standard deviation of x-values in the linear regression
x
b
S Standard deviation of y-values in the linear regression
y
r - Correlation coefficient in the linear regression
b
S Standard deviation of the slope of the regression line
m
b
S Standard deviation of the intercept of the regression line
b
S - Bound regarding the upper and the lower proportional limit for the determination
m(rel)
of Young’s modulus in the linear regression
ΔL mm Displacement increment based on initial gauge length
ΔL mm Plastic displacement
pl
ΔL mm Calculated zero-point
z
ΔL mm Elastic displacement
el
A % Elongation automatically given by an extensometer
(a)
A % Elongation determined manually
(m)
e - Strain / extension
-1
 s Strain rate
e
e - Plastic strain
pl
e - Strain at fracture
(rupt)
R MPa Stress at fracture
(rupt)
C % Correction in comparison with the percentage elongation value measured manual-
A(m)
ly
σ MPa True stress
ε - True plastic strain
-1
ε s True plastic strain rate
m´ Strain rate sensitivity
K' MPa Material flow stress at a true strain rate of unity
T °C Ambient temperature during testing
b
Numerical coefficients
n´, σ , T , C
1 o
yo
a 2 2
1 MPa = 1 N/mm ; 1 GPa = 1 kN/mm .
b
Depends on the property concerned.
4 Steps for the evaluation of uncertainty
4.1 General
Figure 1 shows the different steps for the evaluation of uncertainty.

Figure 1 — Steps for the evaluation of uncertainty
4.2 Step 1 — Defining the measurands for which uncertainty is to be evaluated
The measurands (quantities) for which the uncertainties are to be calculated are listed.
Table 2 shows the measurands that can be reported in tensile testing. These measurands are measured
directly or are not measured directly and are determined from other quantities (or measurements).
Table 2 — Measurands, measurements and their units and symbols
Measurand Unit Symbol
Original cross-sectional area mm S
o
Slope of the elastic part of the stress-extension curve MPa m
E
Proof strength at 0,2 % plastic extension MPa R
p0,2
Upper yield strength MPa R
eH
Lower yield strength MPa R
eL
Tensile strength MPa R
m
Percentage elongation after fracture % A
Percentage reduction of area % Z
Test piece original thickness (rectangular test piece) mm a
o
Test piece original width (rectangular test piece) mm b
o
Test piece original diameter (circular test piece) mm d
o
Original gauge length mm L
o
Force applied during test N F
Axial displacement during the test mm ΔL
Final gauge length mm L
u
Minimum diameter of a circular test piece after fracture mm d
u
a
The Young’s-modulus is not usually reported in tensile testing (only if ISO 6892-1:2019, Annex G is applied).

The measurands not measured are calculated with Formulae (1) to (9):
for rectangular test piece see Formula (1):
S = a ·b (1)
o o o
for circular test piece see Formula (2):
S = d ·π/4 (2)
o o
for slope of beginning of stress-extension curve see Formula (3):
m = ΔR/Δe = (ΔF· L )/(ΔL·S ) (3)
E o o
for proof strength see Formula (4):
R = F /S (4)
p p o
for upper yield strength see Formula (5):
R = F /S (5)
eH eH o
for lower yield strength see Formula (6):
R = F /S (6)
eL eL o
for tensile strength see Formula (7):
R = F /S (7)
m m o
for percentage elongation after fracture see Formula (8):
A = (L – L ) · 100/L (8)
u o o
for percentage reduction of area see Formula (9):
Z = (S –S ) · 100/S (9)
o u o
4.3 Step 2 — Identifying all sources of uncertainty in the test
All possible sources of uncertainty that can have an effect (either directly or indirectly) on the test are
identified.
The list cannot be identified comprehensively beforehand as it is associated uniquely with the individual test
procedure and apparatus used. A new list can be prepared each time a particular test parameter changes. To
help the user list all sources, four categories have been defined. Table 3 lists these categories and gives some
examples of the sources of uncertainty in each category.

Table 3 — An example of sources of uncertainty and their likely contribution to the uncertainties
in tensile testing measurands for a cold rolled steel (sheet type test piece) at ambient temperature
a
performed by a screw driven tensile testing machine
b
Source of uncertainty Type m R R R R A Z
E p0,2 eH eL m
1. Test piece
Variability of specimen dimensions B 1 1 1 1 1 1 1
Surface finish B 0 2 2 2 2 2 2
Residual stresses B 0 2 2 2 ? ? ?
Shape and size of test piece B 1 1 1 1 2 1-2 1-2
Shape of fracture B 0 0 0 0 0 1 1
Location of failure B 0 0 0 0 0 1 1-2
2. Test system
Cross-sectional area measuring unit B 1 1 1 1 1 0 1
Original gauge length B 1 1 0 0 0 1 0
Extensometer positioning B 1 1 0 0 0 1 0
Load train alignment B 1 1 1 1 1 1 2
Test machine stiffness B 1 1 1 1 2 2 2
Uncertainty in force measurement B 1 1 1 1 1 0 0
Uncertainty in displacement measure-
B 1 1 0 0 0 1 0
ment
3. Environment
Ambient temperature and humidity B 2 2 2 2 2 2 2
4. Test Procedure
Zeroing B 2 1 1 1 1 1 2
Stressing rate B 2 1 1 1 1 1-2 2
Straining rate B 2 1 1 1 1 1 1
Digitizing B 1 1 1 1 1 1 1
Sampling frequency B 1 1 1 1 2 2 0
Uncertainty in fracture area measure-
B 0 0 0 0 0 0 1
ment
Software B 1 1 1 1 2 2 0
NOTE This table is not exhaustive and is for guidance only; relative contributions can vary according to the material tested
and the test conditions. Individual laboratories are encouraged to prepare their own list to correspond to their own test facility
and assess the associated significance of the contributions.
a
1 = major contribution, 2 = minor contribution, 0 = insignificant or no contribution (zero effect), ? = unknown.
b
See 4.4 and References [1] and [9].
To simplify the uncertainty calculations, it can be advisable to regroup the significant sources affecting the
tensile testing results in Table 3 into one of the following four categories:
— Uncertainty due to the measurement of cross-sectional area
— Uncertainty due to the force measurement
— Uncertainty due to the displacement measurement
— Uncertainty due to evaluated quantities (e.g. proof strength)
The worked examples in Annex B use the above categorisation when assessing uncertainties.

4.4 Step 3 — Estimating the standard uncertainty for each source of uncertainty
4.4.1 General
Sources of uncertainty are classified as Type A or Type B, depending on the way their influence is quantified.
[1] [9]
, If the uncertainty is evaluated by statistical means from a number of repeated observations, it is
classified as Type A. If it is evaluated by any other means it is classified as Type B. The values associated
with Type B uncertainties can be obtained from a number of sources including calibration certificates,
manufacturers' information, or an expert's estimation. For Type B uncertainties, it is necessary for the user
to estimate for each source the most appropriate probability distribution.
The measurement process can usually be modelled by a functional relationship between the estimated input
quantities and the output in the form given by Formula (10):
y = f (x , x , …, x ) (10)
1 2 m
The standard uncertainty requires the determination of the associated sensitivity coefficient, c, which is
usually evaluated from the partial derivatives of the functional relationship between the output quantity
(the measurand) and the input quantities. The calculations required to obtain the sensitivity coefficients by
partial differentiation can be a lengthy process, particularly when there are many individual contributions
and uncertainty estimates are needed for a range of values. If the functional relationship for a particular
measurement is not known, the sensitivity coefficients can be obtained experimentally. In some cases, the
input quantity to the measurement might not be in the same units as the output quantity. For example, one
contribution to R is the test temperature. In this case the input quantity is temperature, but the output
p0,2
quantity is the stress that is in MPa. In such a case, a sensitivity coefficient, c (corresponding to the partial
T
derivative of the proof strength/ test temperature relationship), is used to convert from temperature to MPa
(for more information see Annex A).
Subsequent calculations will be made clearer if, wherever possible, all components are expressed in the
same way (e.g., either in the same units as used for the reported result or in relative terms, i.e., in percent.).
The standard uncertainty is defined as one standard deviation and is derived from the uncertainty of the
input quantity divided by a factor, d , associated with the assumed probability distribution. Divisors for the
v
typical probability distributions most likely to be encountered are shown in Table 4.
Table 4 — Typical values of the divisor d
v
Probability distribution d
v
Normal 1
Rectangular
Triangular
4.4.2 Type A evaluation of standard uncertainty
For a series of n repeated readings, the estimated standard uncertainty, u, of the arithmetic mean, x , is
calculated from the Formula (11):
s
u= (11)
n
where s is the sample standard deviation given by Formula (12):
n
s= (-xx (12)
)
i
∑
(-n 1)
i=1
4.4.3 Type B evaluation of standard uncertainty
The standard uncertainty of an input quantity that has not been obtained from repeated measurements can
be evaluated by scientific judgment based on all of the available information on the possible contributing
factors. The information can include:
— data provided in calibration and other certificates;
— manufacturer’s specification;
— previous measurement data;
— experience with or general knowledge of the behaviour of the relevant materials and instruments;
— uncertainties assigned to reference materials;
— uncertainties assigned to reference data taken from handbooks.
For most Type B evaluations:
— estimate the upper and lower limits of uncertainty, and,
— assume e.g., a rectangular probability distribution (i.e., the value is equally likely to fall anywhere in
between the upper and lower limits). The standard uncertainty for a rectangular distribution is by
Formula (13):
a
u =  (13)
where a is the mid-point value between the upper and lower limits. Rectangular distributions are quite
common but other distributions can occur. For example, the uncertainty, U, often stated on an instrument’s
calibration certificate is usually a normal distribution. In this case, the standard uncertainty is given by
Formula (14):
U
u =  (14)
k
where k is the coverage factor.
4.4.4 Uncertainty due to repeatability of measurement
Repeatability of measurement is a component that is included in the uncertainty calculations. Simply put,
repeatability represents the variability within a single laboratory (otherwise known as intra-laboratory
testing). In practice this can involve one or more operators (following the same test procedure) using one
or more sets of equipment over a reasonably short period of time during which neither the equipment
nor the environment is likely to change appreciably. The variability can be random in nature and due to
small changes in equipment, calibration, environment, and operator procedure. In material-dependent,
destructive tests such as the subject matter in this practice, this variability will inevitably be affected also
by some heterogeneity in the material tested. This can be kept to a minimum by the use of test pieces from
a careful choice of test material.
In this practice, in the absence of information on the repeatability of measurement on the particular material
or batch being tested, an estimate of the repeatability from a similar material or batch can be used in the
uncertainty calculations. This can be included in the uncertainty report.
For a reasonably large set of data (e.g. 10 or more), repeatability is represented by one standard deviation, s.
The associated uncertainty in the mean of n measurements is given by Formula (15):
s
u= (15)
n
NOTE For smaller sample sizes see 4.6.

4.5 Step 4 — Computing the combined uncertainty, u
c
Assuming that individual uncertainty sources are uncorrelated, the measurand's combined uncertainty,
u (y), can be computed using the root sum squares using Formula (16):
c
n
uy()= []cu()x (16)
c ∑ ii
i=1
where c is the sensitivity coefficient associated with x .
i i
This uncertainty corresponds to ± one standard deviation on the normal distribution law representing the
studied quantity. The combined uncertainty has an associated confidence level of 68,27 %.
4.6 Step 5 — Computing the expanded uncertainty, U
The expanded uncertainty, U, is obtained by multiplying the combined uncertainty, u , as calculated
c
in 4.5, by a coverage factor, k or k , which is selected on the basis of the level of confidence required
p
(ISO/IEC Guide 98-3). For a normal probability distribution, the most generally used coverage factor is 2
that corresponds to a confidence interval of 95,45 % (effectively 95 % for most practical purposes). Where
a higher confidence level is demanded by the customer (such as for aerospace and electronics industries), a
coverage factor of 3 is often used so that the corresponding confidence level increases to 99,73 %.
In cases where the probability distribution of u is not normal or where the number of data points used
c
in Type A analysis is small, a coverage factor k can be determined according to the effective degrees of
p
freedom, ν , given by the Welsh-Satterthwaite method in Formula (17) (see also ISO/IEC Guide 98-3:2008,
eff
Annex G for details):
uy()
c
ν = (17)
eff
n
uy()
i
∑
ν
i
i=1
Table 5 shows the values of the coverage factor k for a level of confidence of 95 %.
p
Table 5 — Student’s t-distribution table
ν 1 2 3 4 5 6 7 8 10 12 14 16
eff
k 13,97 4,53 3,31 2,87 2,65 2,52 2,43 2,37 2,28 2,23 2,20 2,17
p
ν 18 20 25 30 35 40 45 50 60 80 100 ∞
eff
k 2,15 2,13 2,11 2,09 2,07 2,06 2,06 2,05 2,04 2,03 2,02 2,00
p
Tables B.1 to B.4 show an example format of the calculation worksheets for estimating the uncertainty
in Young’s modulus and proof strength for a rectangular test piece. Annex A presents the mathematical
formulae for calculating uncertainty contributions.
5 Reporting of results
Once the expanded uncertainty has been evaluated, the results can be reported as in Formula (18):
Y = y ± U (18)
where
Y is the evaluated value of the measurand;
y is the test (or measurement) mean result;
U is the expanded uncertainty associated with y.

An explanatory note, such as that given in the following example can be added (change where appropriate):
EXAMPLE The reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage
factor, k = 2 (or k = state value), which for a normal distribution provides a level of confidence of approximately 95 %.
p
The uncertainty evaluation was carried out in accordance with ISO/TR 15263.
Details describing how the uncertainties were estimated can be appended to the test report. The extent of
the details can be agreed between the customer and the testing laboratory and can enable the customer to
reproduce the reported uncertainty calculations.

Annex A
(informative)
Mathematical formulae for calculating uncertainties in tensile testing
NOTE 1 To simplify matters Clauses A.1 to A.11 are limited to uncertainty affected by calibration, determination
of cross-sectional area, and evaluation procedure. With the exception of Clauses A.12 and A.13, it was not necessary
to study the mechanical behaviour of metallic materials under different conditions or to consult published analyses.
Basic concepts can be used. The methods of DOE (Design of Experiments) can be used for further studies to consider
the many parameters that affect results.
NOTE 2 The measurement uncertainties stated in calibration certificates of calipers or micrometers employed
for dimensional measurments are not representing the full uncertainty budget of their dedicated application and
therefore the stated uncertainties are not appropriate for direct use as standard uncertainties u in the above formulae.
A.1 Uncertainty of measurements
A.1.1 General
The x , the expectation or expected value of X is the midpoint of the range: xL=+()LUL /2 , with variance
i i i
given in Formulae (A.1) to (A.3):
()UL−LL
ux()= (A.1)
i
If the difference between two bounds, UL – LL, is denoted by 2δ, then Formula (A.1) can be written as
Formula (A.2):
δ
ux()= (A.2)
i
Example Variance of test piece original thickness (rectangular test piece) a can be written as Formula (A.3):
o
δ
a
2 o
ua()= (A.3)
o
If the thickness has been measured n times (and at least 5 times), the bounds can be estimated with
Formula (A.4) as follows:
— determine the mean value of a and the standard deviation s;
o
— determine the confidence region of the mean value:
st⋅ (,pf )
ua()= (A.4)
o
n
where
t is the factor of Student's distribution;
p is the confidence level;
f is the degrees of freedom (n – 1);
n is the number of measurements.
For p = 68,27 % and n = 5, the factor t = 1,15.
A.2 Uncertainty due to errors in the measurement of the cross-sectional area
A.2.1 Rectangular test piece
For a rectangular test piece, see Formula (A.5):
Sa=⋅b (A.5)
oo o
Sensitivity coefficients c associated with the uncertainty on the measurement x , see Formulae (A.6) and
i i
(A.7):
∂S
o
=b (A.6)
o
∂a
o
∂S
o
=a (A.7)
o
∂b
o
Uncertainty in S see Formula (A.8):
o,
2 2 2 2
uS()= bu ()aa+ ub() (A.8)
() ()
co oo oo
A.2.2 Circular test piece
For a circular test piece, see Formula (A.9):
π ⋅d
o
S = (A.9)
o
Sensitivity coefficients c associated with the uncertainty on the measurement x see Formula (A.10):
i i,
∂S π ⋅d
o o
= (A.10)
∂d 2
o
Uncertainty in S , see Formula (A.11):
o
22 2
duπ ()d
oo
uS()= (A.11)
co
A.3 Uncertainty in stress
For calculations of stress values, see Formula (A.12):
F
R= (A.12)
S
o
Sensitivity coefficients c associated with the uncertainty on the measurement x of stress values, see
i i,
Formulae (A.13) and (A.14):
∂R 1
= (A.13)
∂FS
o
∂R F
=− (A.14)
∂S
o S
o
Uncertainty in R , see Formula (A.15):
 
 
1 F
2 2
uR()= uF()+−  uS() (A.15)
 
c o
 2 
S
  S
o
 o 
A.4 Uncertainty in strain
For calculations of strain / extension values, see Formula (A.16):
ΔL
e= (A.16)
L
o
Sensitivity coefficients c associated with the uncertainty on the measurement x of strain / extension
i i,
values, see Formulae (A.17) and (A.18):
∂e
= (A.17)
∂ΔLL
o
∂e ΔL
=− (A.18)
∂L
L
o
o
Uncertainty in u see Formula (A.19):
e,
 
 1  ΔL
2 2
ue()= uL()Δ +−  uL() (A.19)
c   o
 2 
L
 o  L
 
o
A.5 Uncertainty in initial slope of the stress-extension curve m
E
A.5.1 Formulae for linear regression
For calculations of straight lines, see Formula (A.20):
ym=+xb (A.20)
Slope calculation, see Formula (A.21):
n n n
nx yx− y
∑ ii ∑ ii∑
i=1 i=1 i=1
m= (A.21)
n n
nx − ()x
∑ ii∑
i=1 i=1
Intercept calculation, see Formula (A.22):
n n
ym− x
∑ ii∑
i=1 i=1
b= (A.22)
n
Empirical covariance (S ), see Formula (A.23):
xy
n n
 
xy
 
∑ ii∑
n
1  
i=1 i=1
S = xy − (A.23)
xy ii
∑ 
n−1 n
i=1 
 
 
Standard deviation of x-values, see Formula (A.24):
 
n
 
 
 x 
∑ i
n  
 
i=1 
 
S = x − (A.24)
xi∑
n−1 n
 
i=1
 
 
 
Standard deviation of y-values, see Formula (A.25):
 n 
 
 
 y 
∑ i
n
 
 
2 i=1 
 
S = y − (A.25)
yi∑
n−1 n
 
i=1
 
 
 
Correlation coefficient (r), see Formula (A.26):
S
xy
r = (A.26)
SS⋅
xy
Standard deviation of the slope (S ), see Formula (A.27):
m
1−rS
()
y
S = (A.27)
m
()nS−2
x
Standard deviation of the intercept (S ), see Formula (A.28):
b
n
 
 x 
∑ i
 
2 i=1 
()nS−+1
x
2 n
SS= (A.28)
bm
n
Bound regarding the upper and the lower proportional limit for the determination of the initial slope of the
stress-extension curve, see Formula (A.29):
S
m
S =→ minimum (A.29)
m()rel
m
The data pair for the minimum of S represents the upper and the lower elastic limit.
m()rel
A.5.2 Combined uncertainty of m
E
The linear regression is used to determine the linear relationship between force and displacement, see
Formulae (A.30) and (A.31):
FL⋅ L
o o
m = =⋅m (A.30)
E
ΔLS⋅ S
o o
Fm=⋅ΔLb+ (A.31)
Therefore:
Fy= see Formula (A.20)
ΔLx= see Formula (A.20)
SS= see Formula (A.23)
ΔLF, xy
SS= see Formula (A.24)
ΔLx
SS= see Formula (A.25)
Fy
Sensitivity coefficients, c , associated with the uncertainty on the measurement, x , see Formulae (A.32) to
i i
(A.35):
∂m L
E o
= (A.32)
∂m S
o
∂m
m
E
= (A.33)
∂L S
oo
∂m mL⋅
E o
=− (A.34)
∂S
S
o
o
2 2
 
L mL
   m 
o 2 2 o 22
um()= um()+ uL()+−  u ()S (A.35)
c E     o o
 2 
S S
 oo   S
 
o
A.6 Uncertainty in the determination of proof strengths
The proof strengths are determined by methods using the series of values of force-displacement or force-
[4]
extension as defined in the relevant testing standards (e.g., ISO 6892-1 ) and as illustrated by Figure A.1,
see also Formulae (A.36) to (A.38).
ΔΔLL=−ΔΔLL− (plastic displacement) (A.36)
pl zel
where
ΔL
is the input data for displacement (e.g. recorded in ASCII-file);
ΔL is the calculated zero-point.
z
b
FL=⇒0 Δ =− (A.37)
z
m
F
ΔL = (elastic displacement) (A.38)
el
m
F is the input data of force (e.g., recorded in ASCII-file), see Formula (A.39):
bF−
bL≥⇒00ΔΔ≤⇒ LL=+Δ (A.39)
z pl
m
Figure A.1 — Determination of proof strengths

For calculations of plastic strains, see Formula (A.40):
ΔL bF−
e =+ (A.40)
pl
L mL⋅
oo
Sensitivity coefficients, c , associated with the uncertainty on the measurement x , of plastic strain values,
i i
see Formulae (A.41) to (A.45):
∂e
pl 1
== c (A.41)
∂ΔLL
o
∂e
bF−
ΔL ()
pl
=− − = c (A.42)
∂L
L mL⋅
o
oo
∂e
pl 1
= = c (A.43)
∂bm⋅L
o
∂e
pl
=− = c (A.44)
∂Fm⋅L
o
∂e
pl ()bF−
=− = c (A.45)
∂m
mL⋅
o
Uncertainty in plastic strain e , see Formula (A.46):
pl
22 22 22 22 22
ue()=+cu ()ΔLc uL()++cu ()bc uF()+cu ()m (A.46)
cplo1 2 3 4 5
From the recorded force-displacement diagram (see Figure A.1), we obtain a polynomial to determine
uF(), see Formulae (A.47) to (A.49):
e
pl
Fe=+αα e +α (A.47)
e 2 pl 1 pl o
pl
∂F
e
pl
=+2ααe (A.48)
21pl
∂e
pl
uF()=+()2ααeu ()e (A.49)
e 21pl pl
pl
Combined uncertainty in force at e , see Formula (A.50):
pl
uF()=+uF() uF() (A.50)
c ee
pl pl
Combined uncertainty in proof strength, see Formulae (A.51) and (A.52):
F
e
pl
R = (A.51)
p0,2
S
o
F
 
e
 1 
pl
2 2
uR()= uF()+−  uS() (A.52)
cp0,2   c e o
pl
 
S
 o  S
 o 
A.7 Uncertainty in determination of the tensile strength
For calculation of tensile strength values, see Formula (A.53):
F
m
R = (A.53)
m
S
o
and the sensitivity coefficients c associated with the uncertainty on the measurement of tensile strength,
i
see Formulae (A.54) and (A.55):
∂R 1
m
= (A.54)
∂FS
mo
∂R F
m m
=− (A.55)
∂S
o S
o
Uncertainty of R , see Formula (A.56):
m
 
  F
2 m 2
uR()= uF()+−  uS() (A.56)
 
cm m o
 2 
S
  S
o
 o 
A.8 Uncertainty in determination of the upper yield strength
The calculation of the uncertainty of R follows the same procedure as R , see Formula (A.57):
eH m
F
eH
R = (A.57)
eH
S
o
Sensitivity coefficients c associated with the uncertainty on the measurement x , see Formulae (A.58) and
i i
(A.59):
∂R
eH
= (A.58)
∂FS
eH o
∂R F
eH eH
=− (A.59)
∂S
S
o
o
Uncertainty of R , see Formula (A.60):
eH
 
F
 1 
2 eH 2
uR()= uF()+−  uS() (A.60)
ceH   eH o
 2 
S
 o  S
 
o
A.9 Uncertainty in determination of the lower yield strength
Similarly, for the lower yield strength R , see Formula (A.61):
eL
F
eL
R = (A.61)
eL
S
o
Sensitivity coefficients c associated with the uncertainty on the measurement are, see Formulae (A.62) and
i
(A.63):
∂R
eL
= (A.62)
∂FS
eL o
∂R F
eL eL
=− (A.63)
∂S
o S
o
Uncertainty of R , see Formula (A.64):
eL
 
  F
2 eL 2
uR()= uF()+−  uS() (A.64)
 
ceL eL o
 2 
S
 o  S
 
o
A.10 Uncertainty in the determination of the percentage elongation after fracture
A.10.1 Automatic extensometer
For an automatic extensometer, see Formula (A.65):
R
 
(RUPT)
Ae=− +C ⋅100 (A.65)
 
()a (RUPT) A
()m
m
 E 
The value of A depends on the location of the fracture within the parallel length of the test piece. C is
(a) A(m)
the correction in comparison with the percentage elongation value measured by hand.
Sensitivity coefficients c associated with the uncertainty on the measurement x , see Formulae (A.66) to
i i
(A.69):
∂A
()a
=100 (A.66)
∂e
()RUPT
∂A
a 1
()
=− ⋅100 (A.67)
∂Rm
()RUPT E
∂A R
()a ()RUPT
=− ⋅100 (A.68)
∂m
m
E
E
∂A
()a
=100 (A.69)
∂C
A
()m
Uncertainty of A , see Formula (A.70):
(a)
 
 
R
 1 
 ()RUPT 
2 2 22
 
uA()=+ue() − uR()+− um()+uC() ⋅100 (A.70)
cR()a ()UPTR  ()UPT EA
 
2 ()m
 
m
  m
E
E
   
 
A.10.2 Manual determination (e.g. vernier calliper)
For manual determination, see Formula (A.71):
LL−
 
uo
A = ⋅100 (A.71)
m  
()
L
 
o
Sensitivity coefficients c associated with the uncertainty on the measurement are, see Formulae (A.72) and
i
(A.73):
∂A
()m 1
=⋅100 (A.72)
∂LL
uo
∂A
L
()m
u
=− ⋅100 (A.73)
∂L
o L
o
Uncertainty of A , see Formula (A.74):
(m)
 
 
  L
 2 u 2 
uA()= uL()+−  uL() ⋅1100 (A.74)
 
c ()m u o
 
 2 
L
 o  L
 
o
 
 
A.11 Uncertainty in the determination of the percentage reduction of area
A.11.1 Determination of the reduced area — Rectangular test piece
For determination of the reduced area (rectangular test piece), see Formula (A.75):
Sa=⋅ b (A.75)
uu u
Sensitivity coefficients c associated with the uncertainty on the measurement x , see Formulae (A.76) and
i i
(A.77):
∂S
u
=b (A.76)
u
∂a
u
∂S
u
=a (A.77)
u
∂b
u
Uncertainty in S , see Formula (A.78):
u
2 2
2 2
uS()= ()bu ()aa+() ub() (A.78)
cu uu uu
A.11.2 Determination of the reduced area — Circular test piece
For determination of the reduced area (circular test piece), see Formula (A.79):
π ⋅d
u
S = (A.79)
u
Sensitivity coefficients c associated with the uncertainty on the measurement x , see Formula (A.80):
i i
∂S π ⋅d
u u
= (A.80)
∂d 2
u
Uncertainty of S , see Formula (A.81):
u
π d π
 
u
uS()= ud() =⋅du⋅ ()d (A.81)
cu   uu u
 
A.11.3 Determination of the percentage reduction of area
For determination of the percentage reduction of area, see Formula (A.82):
SS− 
ou
Z = ⋅100 (A.82)
 
...