EN 843-5:2006

Advanced technical ceramics - Mechanical properties of monolithic ceramics at room temperature - Part 5: Statistical analysis

This part of EN 843 specifies a method for statistical analysis of ceramic strength data in terms of a two-parameter Weibull distribution using a maximum likelihood estimation technique. It assumes that the data set has been obtained from a series of tests under nominally identical conditions.
NOTE 1   In principle, Weibull analysis is considered to be strictly valid for the case of linear elastic fracture behaviour to the point of failure, i.e. for a perfectly brittle material, and under conditions in which strength limiting flaws do not interact and in which there is only a single strength-limiting flaw population.
If subcritical crack growth or creep deformation preceding fracture occurs, Weibull analysis can still be applied if the results fit a Weibull distribution, but numerical parameters may change depending on the magnitude of these effects. Since it is impossible to be certain of the degree to which subcritical crack growth or creep deformation has occurred, this European Standard permits the analysis of the general situation where crack growth or creep may have occurred, provided that it is recognized that the parameters derived from the analysis may not be the same as those derived from data with no subcritical crack growth or creep.
NOTE 2   This European Standard employs the same calculation procedures as method A of ISO 20501:2003, but does not provide a method for dealing with censored data (method B of ISO 20501).

Hochleistungskeramik - Mechanische Eigenschaften monolithischer Keramik bei Raumtemperatur - Teil 5: Statistische Auswertung

Dieser Teil von EN 843 legt ein Verfahren für die statistische Auswertung von Festigkeitsdaten von Keramik in Form einer Zweiparameter-Weibull-Verteilung mit einem Schätzwert nach der Maximum-Likelihood-Methode fest. Es wird vorausgesetzt, dass der Datensatz aus einer Prüfreihe unter nominell identischen Bedingungen gewonnen wurde.
ANMERKUNG 1   Prinzipiell ist die Weibull-Analyse nur streng gültig bei linear-elastischem Bruchverhalten bis zum Punkt des Versagens, d. h. für einen vollkommen spröden Werkstoff, und unter Bedingungen, unter denen die Festigkeit begrenzende Fehler nicht in Wechselwirkung stehen und unter denen es nur eine einzige Festigkeit begrenzende Fehler-Grundgesamtheit gibt.
Tritt vor Brüchen subkritisches Risswachstum oder Kriechverformung auf, kann die Weibull-Analyse immer noch ange¬wendet werden, wenn die Ergebnisse einer Weibull-Verteilung folgen, numerische Parameter können sich aber abhängig von der Größe dieser Effekte verändern. Da es unmöglich ist, mit Sicherheit das Ausmaß des Auftretens von subkri¬tischem Risswachstum oder Kriechverformung anzugeben, lässt diese Europäische Norm die Analyse der allgemeinen Situation zu, in der Risswachstum oder Kriechen aufgetreten sein können, vorausgesetzt, dass berücksichtigt wird, dass die aus der Analyse abgeleiteten Parameter nicht die gleichen sein können wie die aus den Daten ohne subkritisches Risswachstum oder Kriechen abgeleiteten.
ANMERKUNG 2   Diese Europäische Norm wendet die gleichen Verfahrensweisen bei der Berechnung an wie Verfahren A von ISO 20501:2003 [1], liefert jedoch kein Verfahren, das sich mit zensierten Daten beschäftigt (Verfahren B von ISO 20501).

Céramiques techniques avancées - Propriétés mécaniques des céramiques monolithiques à température ambiante - Partie 5: Analyse statistique

La présente partie de l'EN 843 décrit une méthode d'analyse statistique des données de résistance mécanique des céramiques, sous forme d'une répartition de Weibull à deux paramètres en utilisant une technique d'estimation du maximum de vraisemblance. Cette méthode suppose que l'ensemble de données a été obtenu à partir d'une série d'essais effectués dans des conditions nominalement identiques.
NOTE 1   En principe, l'analyse de Weibull n'est jugée strictement valable que dans le cas d'un comportement élastique linéaire à l'instant de la rupture, c'est-à-dire pour un matériau parfaitement fragile, et dans des conditions où les défauts limitant la résistance mécanique n'interagissent pas et où n'existe qu'une seule population de défauts limitant la résistance mécanique.
Si une propagation sous-critique de la fissure ou une déformation par fluage précède la rupture, l'analyse de Weibull peut encore être appliquée si les résultats s'adaptent à une répartition de Weibull, mais les paramètres numériques peuvent varier selon l'amplitude de ces effets. Étant donné qu'il est impossible d'être certain de l'amplitude de la propagation sous-critique de la fissure ou de la déformation par fluage, la présente Norme européenne permet d'analyser la situation générale dans laquelle une propagation de la fissure ou un fluage a pu se produire, sous réserve qu'il soit reconnu que les paramètres déduits de l'analyse peuvent ne pas être identiques aux paramètres déduits des données obtenues sans propagation sous-critique de la fissure ou sans fluage.
NOTE 2   La présente Norme européenne utilise les mêmes procédures de calcul que celles de la méthode A de l'ISO 20501:2003 [1], mais ne fournit aucune méthode traitant des données tronquées (méthode B de l'ISO 20501).

Sodobna tehnična keramika - Monolitna keramika - Mehanske lastnosti pri sobni temperaturi – 5. del: Statistična analiza

General Information

• Status: Withdrawn
• Publication Date: 12-Dec-2006
• Withdrawal Date: 13-Apr-2025
• ICS: 81.060.30 - Advanced ceramics
• Technical Committee: CEN/TC 184 - Advanced technical ceramics
• Drafting Committee: CEN/TC 184/WG 3 - Monolithic ceramics
• Current Stage: 9960 - Withdrawal effective - Withdrawal
• Start Date: 21-Dec-2022
• Completion Date: 14-Apr-2025
• Mandate: M/076 - Mandate to CEN/CENELEC for standardization in the field of advanced ceramics
• Ref Project: SIST EN 843-5:2007 - Advanced technical ceramics - Mechanical properties of monolithic ceramics at room temperature - Part 5: Statistical analysis

Relations

• Replaces: ENV 843-5:1996 - Advanced technical ceramics - Monolithic ceramics - Mechanical tests at room temperature - Part 5: Statistical analysis
• Effective Date: 22-Dec-2008
• Replaced By: EN ISO 20501:2022 - Fine ceramics (advanced ceramics, advanced technical ceramics) - Weibull statistics for strength data (ISO 20501:2019)
• Effective Date: 28-Dec-2022

Standards Content (Sample)

SLOVENSKI STANDARD
01-maj-2007
1DGRPHãþD
SIST ENV 843-5:2000
6RGREQDWHKQLþQDNHUDPLND0RQROLWQDNHUDPLND0HKDQVNHODVWQRVWLSULVREQL
WHPSHUDWXUL±GHO6WDWLVWLþQDDQDOL]D
Advanced technical ceramics - Mechanical properties of monolithic ceramics at room
temperature - Part 5: Statistical analysis
Hochleistungskeramik - Mechanische Eigenschaften monolithischer Keramik bei
Raumtemperatur - Teil 5: Statistische Auswertung
Céramiques techniques avancées - Propriétés mécaniques des céramiques
monolithiques a température ambiante - Partie 5: Analyse statistique
Ta slovenski standard je istoveten z: EN 843-5:2006
ICS:
81.060.30 Sodobna keramika Advanced ceramics
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN 843-5
NORME EUROPÉENNE
EUROPÄISCHE NORM
December 2006
ICS 81.060.30 Supersedes ENV 843-5:1996
English Version
Advanced technical ceramics - Mechanical properties of
monolithic ceramics at room temperature - Part 5: Statistical
analysis
Céramiques techniques avancées - Propriétés mécaniques Hochleistungskeramik - Mechanische Eigenschaften
des céramiques monolithiques à température ambiante - monolithischer Keramik bei Raumtemperatur - Teil 5:
Partie 5: Analyse statistique Statistische Auswertung
This European Standard was approved by CEN on 11 November 2006.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the Central Secretariat or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the Central Secretariat has the same status as the official
versions.
CEN members are the national standards bodies of Austria, Belgium, Cyprus, Czech Republic, Denmark, Estonia, Finland, France,
Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania,
Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36  B-1050 Brussels
© 2006 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN 843-5:2006: E
worldwide for CEN national Members.

Contents Page
Foreword.3
1 Scope.4
2 Normative references.4
3 Terms and definitions .4
3.1 Flaws.4
3.2 Flaw distributions.5
3.3 Mechanical evaluation.5
3.4 Statistical terms.6
3.5 The Weibull distribution.7
4 Symbols.8
5 Significance and use .10
6 Principle of calculation .11
6.1 Maximum likelihood method .11
6.2 Bias correction.12
6.3 Confidence interval.12
7 Procedure.13
7.1 Graphical representation of data .13
7.2 Determination of Weibull parameters by maximum likelihood method.13
7.3 Determination of limits of the confidence interval.14
8 Test report.14
Annex A (informative) Relationship between characteristic strengths of test pieces or
components of different size or shape, or with different stress fields applied .15
Annex B (informative) FORTRAN program for calculating Weibull parameters.17
Annex C (informative) PASCAL program for calculating Weibull parameters.23
Annex D (informative) BASIC program for calculating Weibull parameters .28
ˆ
Annex E (normative) Unbiasing factors for estimation of Weibull modulus, m .33
Annex F (normative) Confidence factors for characteristic strength, σˆ .34
ˆ
Annex G (normative) Confidence factors for Weibull modulus, m .36
Annex H (informative) Worked examples.38
Annex I (informative) Example test report .43
Bibliography .45

Foreword
This document (EN 843-5:2006) has been prepared by Technical Committee CEN/TC 184 “Advanced
technical ceramics”, the secretariat of which is held by BSI.
This European Standard shall be given the status of a national standard, either by publication of an identical
text or by endorsement, at the latest by June 2007, and conflicting national standards shall be withdrawn at
the latest by June 2007.
This document supersedes ENV 843-5:1996.
EN 843 Advanced technical ceramics — Mechanical properties of monolithic ceramics at room temperature
comprises six parts:
Part 1: Determination of flexural strength
Part 2: Determination of Young’s modulus, shear modulus and Poisson’s ratio
Part 3: Determination of subcritical crack growth parameters from constant stressing rate flexural strength
tests
Part 4: Vickers, Knoop and Rockwell superficial hardness
Part 5: Statistical analysis
Part 6: Guidance for fractographic investigation
At the time of publication of this Revision of Part 5, Part 6 was available as a Technical Specification.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Cyprus, Czech Republic,
Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden,
Switzerland and United Kingdom.
1 Scope
This part of EN 843 specifies a method for statistical analysis of ceramic strength data in terms of a two-parameter
Weibull distribution using a maximum likelihood estimation technique. It assumes that the data set has been
obtained from a series of tests under nominally identical conditions.
NOTE 1 In principle, Weibull analysis is considered to be strictly valid for the case of linear elastic fracture behaviour to
the point of failure, i.e. for a perfectly brittle material, and under conditions in which strength limiting flaws do not interact
and in which there is only a single strength-limiting flaw population.
If subcritical crack growth or creep deformation preceding fracture occurs, Weibull analysis can still be applied if the
results fit a Weibull distribution, but numerical parameters may change depending on the magnitude of these effects.
Since it is impossible to be certain of the degree to which subcritical crack growth or creep deformation has occurred, this
European Standard permits the analysis of the general situation where crack growth or creep may have occurred,
provided that it is recognized that the parameters derived from the analysis may not be the same as those derived from
data with no subcritical crack growth or creep.
NOTE 2 This European Standard employs the same calculation procedures as method A of ISO 20501:2003 [1], but
does not provide a method for dealing with censored data (method B of ISO 20501).
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated references,
only the edition cited applies. For undated references, the latest edition of the referenced document (including any
amendments) applies.
EN 843-1:2006, Advanced technical ceramics — Mechanical properties of monolithic ceramics at room
temperature — Part 1: Determination of flexural strength
EN ISO/IEC 17025, General requirements for the competence of testing and calibration laboratories (ISO/IEC
17025:2005)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in EN 843-1:2006 and the following apply.
NOTE Definitions of additional statistical terms can be found in ISO 2602 [2], ISO 3534-1 [3], or other source
literature on statistics.
3.1 Flaws
3.1.1
flaw
inhomogeneity, discontinuity or structural feature in a material which when loaded provides a stress concentration
and a risk of mechanical failure
NOTE 1 This could be, for example, a grain boundary, large grain, pore, impurity or crack.
NOTE 2 The term flaw should not be taken as meaning the material is functionally defective, but rather as containing
an inevitable microstructural inhomogeneity.
3.1.2
critical flaw
flaw acting as the source of failure
3.1.3
extraneous flaw
type of flaw observed in the fracture of test pieces manufactured for the purposes of a test programme which will
not appear in manufactured components
NOTE For example, damage from machining when this process will not be used in the manufacture of components.
3.2 Flaw distributions
3.2.1
flaw size distribution
spread of sizes of flaw
3.2.2
critical flaw size distribution
distribution of sizes of critical flaws in a population of tested components
3.2.3
compound critical flaw distribution
flaw distribution which contains more than one type of strength controlling flaw not occurring in a purely concurrent
manner (3.2.4)
NOTE An example is when every test piece contains flaw type A and some contain additionally a second
independent type B.
3.2.4
concurrent critical flaw distribution
competing critical flaw distribution.
Multiple flaw distribution where every test piece contains representative defects of each independent flaw type
which compete with each other to cause failure
3.2.5
exclusive critical flaw distribution
multiple flaw distribution created by mixing and randomizing test pieces from two or more versions or batches of
material where each version contains a single strength-controlling flaw population
NOTE For example, each test piece contains defects exclusively from a single distribution, but the total data set reflects
more than one type of strength-controlling flaw.
3.2.6
competing failure mode
distinguishably different type of fracture initiation event that results from concurrent (competing) flaw distributions
(3.2.4)
3.3 Mechanical evaluation
3.3.1
fractography
analysis of patterns and features on fracture surfaces, usually with the purpose of identifying the fracture origin
and hence the flaw type
3.3.2
proof test
application of a predetermined stress to a test piece or component over a short period of time to ascertain whether
it contains a serious strength-limiting defect
NOTE This enables the removal of potentially weak test pieces or components from a batch. This procedure modifies the
failure statistics of the survivors, such that the two-parameter Weibull distribution is typically no longer valid.
3.3.3
population mean
average of all strength results in a population
3.3.4
sample mean
average of all strength results from a sample taken from the population
3.3.5
strength population
ensemble of fracture strengths
3.4 Statistical terms
3.4.1
bias
consistent numerical offset in an estimate relative to the true underlying value, inherent in most estimating
methods
NOTE For the maximum likelihood method of estimation, the magnitude of the bias decreases with increasing
sample size.
3.4.2
confidence interval
interval for which it can be stated with a given confidence level that it contains at least a specified proportion of the
population of results, or estimates of parameters defining the population
NOTE For example, estimates of Weibull modulus and characteristic strength from a batch of test pieces.
3.4.3
confidence level
required probability that any one estimate will fall within the confidence interval
3.4.4
estimate
well-defined value that is dependent on the variation of strengths in the population
NOTE The resulting value for a given population can be considered an estimate of a distribution parameter associated
with the population as a whole.
3.4.5
probability density function
function f(x) is a probability density function for the continuous random variable x if:
f (x)≥ 0 (1)
and:
∞
f (x)dx= 1 (2)
∫
−∞
such that the probability, P, that the random variable x assumes a value between a and b is given by:
b
P(a< x≤b)= f (x)dx= F(b)−F(a) (3)
∫
a
where F is the cumulative distribution function
3.4.6
ranking estimator
means of assigning a probability of failure to a ranked value in a collection of strength values
3.4.7
sample
collection of measurements or observations on test pieces selected randomly from a population
NOTE For example, strength measurements from a batch of similar test pieces.
3.4.8
sampling
process of selecting test pieces for a test
NOTE For the purposes of this European Standard the guidance given in ENV 1006 [4] should be noted.
3.4.9
unbiased estimate
estimate of a distribution parameter which does not contain a bias or which has been corrected for bias
3.5 The Weibull distribution
3.5.1
Weibull distribution
The continuous random variable x has a two-parameter Weibull distribution if the probability density function (see
3.4.5) is given by:
m−1 m
 
m x  x
f (x)=   exp−   x> 0 (4)
    
β β β
      
 
f (x)= 0 x≤ 0 (5)
NOTE 1 This corresponds with a cumulative distribution function as follows:
m
 
 x
F(x)= 1− exp−   x> 0 (6)
 
β
 
 
 
F(x)= 0 x≤ 0 (7)
where
m is the Weibull modulus or shape parameter (> 0);
β is the scale parameter (> 0).
NOTE 2 The random variable representing the fracture strength of a ceramic test piece will assume only positive
values, and the distribution is asymmetric about the mean. These characteristics rule out the use of the normal distribution
amongst others and point to the use of the Weibull distribution or similar skewed distributions. The assumption made in
this European Standard is that the Weibull distribution will approximate to the true distribution of strengths observed.
NOTE 3 This European Standard is restricted to the use of the two-parameter Weibull distribution. Other forms, such
as the three-parameter method which assumes the existence of a non-zero minimum value for x, are outside the scope of
this European Standard.
NOTE 4 The population mean x is related to β by:

x=Γβ 1+

m

(8)
where
Γ is the gamma function.
The gamma function is sometimes represented by a non-integral factorial:
 
Γ+1 =

mm
 
(9)
3.5.2
Weibull modulus
measure of the width of the Weibull distribution defined by parameter m in Equation (4)
3.5.3
Weibull characteristic strength
strength value at a probability of failure of 0,632
NOTE 1 If the random variable representing the strength of a ceramic test piece is characterized by the above
equations, then the probability that a test piece will not sustain a nominal stress σ , i.e. has a nominal strength σ = σ ,
nom f nom
is given by the cumulative distribution function:
m
 
σ
 
f
 
P = 1− exp−  σ > 0 (10)
f f
 
σ
 
 0
 
P = 0 σ ≤ 0 (11)
f f
where
P is the probability of failure;
f
σ is the Weibull characteristic strength.
NOTE 2 Defined in the above manner, the Weibull characteristic strength depends on the test piece geometry and on
the multiaxiality of the stress field applied.
NOTE 3 When testing three-point and four-point bend test pieces from the same population, different values of σ will
be derived, reflecting different stressed volumes or surface areas in the two geometries. See Annex A for information on
the theoretical relationship between strengths of test pieces of different stressed volumes or areas.
NOTE 4 Caution is needed in the use of Weibull statistical parameters beyond the population from which they are
derived.
4 Symbols
For the purposes of this document, the following symbols apply.
A component surface area
A effective component surface area
eff
b unbiasing factor for Weibull modulus estimate
 t 
l
ˆ
C lower limit of confidence interval for σ , i.e. C =σ exp− 
l 0
l 0
mˆ
 
t
 
u
ˆ
C upper limit of confidence interval for σ , i.e. C =σ exp− 
u 0
u 0
ˆ
m
 
ˆ ˆ
D lower limit of confidence interval for m , i.e. D = m /l
l l l
ˆ ˆ
D upper limit of confidence interval for m , i.e. D = m /l
u u u
f probability density function
F cumulative probability distribution function
g function describing the normalised variation of stress over the volume (or area) of a component
i number assigned to an individual strength value of the sample in ascending ranked order
ˆ
l , l factors for determining respectively the upper and lower limits of the confidence interval of m
u l
m Weibull modulus for the population
ˆ
m estimate of m found by the maximum likelihood method
ˆ ˆ
m value of m corrected by factor b to provide an unbiased estimate of m
cor
N number of tested test pieces
P failure probability of the test piece
f
t , t factors for determining respectively the upper and lower limits of the confidence interval of σˆ
u l 0
V volume
V unit volume
V effective volume
eff
x random variable
x population mean of random variable x
α confidence level
β scale parameter
ε fractional accuracy required in determining maximum likelihood estimates of m and σ
σ strength of test piece
f
th
σ strength of the i ranked test piece in a population
fi
th
σ strength of the j un-ranked test piece in a population
fj
σ nominal stress in test piece at instant of failure, usually taken to be equal to the fracture strength for the
nom
purposes of strength assessment
σ maximum stress in a component against which the stress distribution is referenced
max
σ Weibull characteristic strength of test pieces
ˆ
σ maximum likelihood estimate of Weibull characteristic strength of test piece
5 Significance and use
The strength of advanced technical ceramics is not usually a deterministic parameter. It depends on the nature,
size and orientations of the flaws within the test piece relative to the stress field being applied. This European
Standard applies to most monolithic advanced technical ceramics.
NOTE 1 The Weibull formalism can also be applied successfully in most cases to particulate and whisker reinforced
ceramics which fracture in a catastrophic mode. However, in many cases the failure mechanisms in fibre-reinforced
ceramic matrix composites preclude its use.
The purpose of this European Standard is to provide unbiased estimates of the parameters of the underlying
strength distribution of a population of ceramic test pieces in order to assess numerically the scatter in strengths of
the population. There are a number of ways of determining such estimates, including least squares, moments,
and maximum likelihood methods. The maximum likelihood method has been found to be the most efficient
estimator for small sample numbers based on producing a smaller coefficient of variation of Weibull modulus, m,
and for this reason it is chosen in this European Standard.
NOTE 2 Use of other methods of estimating m and σ , such as least squares fitting of a straight line to the ranked data
points as performed for the visual inspection (see 7.1), is not permitted by this European Standard because they provide
less reliable estimates of m.
Many factors affect the numerical values characterising the distribution of fracture strengths. These include:
1. The number of tests taken as an indicator of the population. The reliability of the estimates increases with
increasing size of the sample, but there are practical limits to the number of tests that might be employed
for cost reasons to be balanced against the improvement in accuracy this produces. It is recommended
that the sample size should not be less than 30.
2. The assumption is made that the sample of test pieces can describe the population by having critical
flaws representative of the population. It should be recognised that the sampling made from the
population shall be on a random basis to reflect fully the true distribution. For example, rejection of part of
the population, e.g. by proof-testing, may modify the applicability of two-parameter Weibull statistics.
3. The method of preparation of test pieces for testing. Most test pieces contain more than one inherent flaw
type and preparing the surfaces of the test pieces prior to testing, e.g. surface grinding, can add another
type of flaw which may change the dominance of the inherent flaws. Concurrent flaw distributions result in
competing failure modes which vary in dominance depending on preparation methods.
4. Under identical conditions of testing, two data sets derived from the same population will result in different
ˆ
ˆ
values of m and σ due to the natural scatter in sampling from the population. For the purposes of this
ˆ
ˆ
European Standard, the values of m and σ for the two sets shall be deemed to be equivalent at the
same confidence level if the results of one lie within the confidence interval of the other, or vice versa.
It is often the case that concurrent, compound or exclusive flaw distributions exist in a population. These can lead
to a bimodal or multimodal distribution of strengths, perhaps with some test pieces failing from one type of flaw,
and others from a second type. In such cases a single two-parameter Weibull distribution cannot validly be fitted
to the data. This European Standard incorporates a visual inspection method (see 7.1) based on simple data
plotting to make the decision whether a Weibull analysis can usefully be made.
NOTE 3 Method B of ISO 20501 [1] deals with the case of ‘censored statistics’, e.g. where it has been possible
fractographically to identify several competing flaw distributions within a batch of test pieces, such that each test can be
assigned to a given flaw type. To compute the Weibull parameters associated with each flaw type, it is necessary
effectively to suspend the tests which failed prematurely from other flaw types, but include them in the computation on the
basis that they contained the flaw type being analysed, but at an unknown strength level. This is known as ‘right censoring’
(higher data become unknown quantities). An alternative approach is needed in the mathematical analysis.
6 Principle of calculation
6.1 Maximum likelihood method
Once it is determined that a valid two-parameter Weibull distribution can be fitted to the data set being evaluated
ˆ
(see 7.1), the maximum likelihood estimates of Weibull modulus, m, and characteristic strength, σˆ , can be
determined.
The likelihood function L for a single critical flaw distribution is given by the expression:
m−1 m
N  
σ σ
 m   
fj fj
    
L = exp−  (12)
∏
    
σ σ σ
 
j=1
 0 0  0
 
where
N is the number of fracture data.
This function is maximized by differentiating the log likelihood (ln(L)) with respect to m and σ , and setting these
ˆ
functions to zero yielding, respectively, estimates m and σˆ , for m and σ :
0 0
N
mˆ
σσln
∑
fj fj
N
j=1
−−lnσ = 0 (13)
∑
fj
N
Nmˆ
mˆ =1
j
σ
∑ fj
j=1
and
1/mˆ
N
 
 
ˆ
m
 
σˆ = σ (14)
 
0 ∑ fj
 
N
 j=1 
 
 
ˆ
Equation (13) is solved numerically to obtain a solution for m , which can then be used to solve for σˆ through
Equation (14). The required fractional accuracy of solution (ε) shall be ≤ 0,001, giving three significant digits in the
ˆ
value of m .
A computer may be used for this task.
NOTE 1 The computer programs provided as examples in Annexes B to D incorporate appropriate routines for the
ˆ
interval halving method for numerically solving for m and σˆ . They may need to be modified to suit different computer
systems.
NOTE 2 As an alternative to the interval halving method, a Newton-Raphson method of solution may be employed.
These two methods are known to provide equivalent results within the accuracy requirements of this European Standard.
6.2 Bias correction
ˆ
The estimate m provided by this method has a bias which gives an overestimate of the true Weibull modulus m.
It is necessary to correct it using an unbiasing factor tabulated in Annex E. This unbiasing factor has been
determined by a Monte Carlo method sampling randomly from a large population with a predetermined true value
ˆ ˆ
of m, allowing correction of the biased value m to the corrected value m :
cor
mˆ =mˆ⋅b (15)
cor
where the unbiasing factor b is read from Table E.1.
ˆ
The bias in σˆ is minimal compared with that in m and no bias correction is required.
6.3 Confidence interval
A measure of the uncertainty of the parameters determined from the data according to this method is given by the
corresponding confidence interval which is determined for each data set.
The first step is to determine the required confidence level, 1 - α. It is common practice to set it to 90 % or 95 %
according to the requirements of the parties to the calculation. For a given number of test pieces, N, the upper
confidence interval limit factor t for a two-sided test is determined at α/2, and the lower confidence interval limit
u
factor t at (1 - α/2). The values of factors t and t are determined from Table F.1. The upper and lower values of
l u l
σˆ corresponding to the upper and lower limits of the confidence interval are determined respectively as:
 t 
u
ˆ
C =σ exp−  (16)
u 0
mˆ
 
 t 
l
ˆ
C =σ exp−  (17)
l 0
mˆ
 
ˆ
where the value of m used is the biased value before correction.
ˆ
The upper and lower limits of the confidence interval for the parameter m are determined from Table G.1 in a
similar manner as for σˆ using the same value of α. For the given number of test pieces, N, and the same
confidence level, (1 - α), the upper confidence interval limit factor l for a two-sided test is determined at α/2, and
u
ˆ
the lower confidence interval limit factor l at (1 - α)/2. The upper and lower limits of the confidence interval for m
l
are determined respectively as:
D = mˆ /l (18)
u u
D = mˆ /l (19)
l l
NOTE The proper implementation of any computer program devised to follow this calculation principle may be
checked by employing the example data in Annex H for the defined level of accuracy, ε.
7 Procedure
7.1 Graphical representation of data
In order to provide visualization of the distribution of strengths for determining the relevance of this method of
statistical analysis, prepare a graphical or Weibull plot as follows:
1. Rank the strength data in ascending order. Assign a probability value to each data point in accordance with
the ranking estimator:
i− 0,5
 
P = (20)
 
fi
N
 
th
for the i ranked data point of strength σ .
fi
2. Plot a graph of the ranked data points using for the ordinate:
yP=−lnln()1/(1 ) (21)
ifi

and for the abscissa:
x = ln(σ ) (22)
ifi
using scaling such that the range of data covers the whole ordinate and abscissa.
NOTE 1 Commercially available Weibull probability graph paper providing non-linear axes is in most cases not
appropriate for the representation of strength data since the plot of the data points can become too steep for effective
judgement of the straightness of a fitted line. In particular, the abscissa can be too contracted and requires expansion of
scale to the range covered by the values of strength plotted logarithmically in order to make a useful inspection of the data.
Censoring of the data set (removal of individual results on the basis of a predetermined criterion) shall not be
done without clear fractographic evidence that the censored values originate from particular types of flaw that can
be excluded from the analysis by agreement between parties.
NOTE 2 This European Standard does not provide a definitive criterion for judging the satisfactory nature of linearity of
fit. The judgement of the fit in relation to the calculated limits of the confidence interval corresponding with the chosen
confidence level (see 7.3 and Annex H, Figures H.1 and H.2) is left to agreement between parties to the analysis. In the
first example given in Annex H, a two-parameter Weibull analysis in accordance with this European Standard is deemed to
be valid within the overall confidence bands estimated by the maximum likelihood method. In the second example,
outlying data points are close to the limits of the 90 % confidence interval, and thus the two-parameter fit can be deemed
to be only marginally valid. Two particular cases merit further comment:
1. If there are more than one or two low strength outliers, perhaps resulting from different types of flaw present in only a
small proportion of the sample, the Weibull plot will have the appearance that separate fits to upper and lower parts
might be appropriate. A single two-parameter Weibull distribution as described in this European Standard is thus
inappropriate. Separation of the test data into two or more populations for further analysis should not be done without
fractographic confirmation that the separation is justified. (See Note 3 in Clause 5.)
2. A general curvature of the plot suggesting a non-zero minimum strength may result from previous rejection of
samples due to damage or defects, or to breakage during preparation, or to prior proof testing. Such a curvature is
deemed to invalidate the analysis method in this European Standard when data points lie outside the limits of the
confidence interval of the analysis.
7.2 Determination of Weibull parameters by maximum likelihood method
Calculate the Weibull parameters using the maximum likelihood method.
NOTE If appropriate, one of the computer programs given in Annexes B to D can be used according to suitability or
availability.
Employ all data points and use the level of accuracy in the iteration process, ε, of 0,001.
ˆ ˆ
Using the table in Annex E, determine the factor b for unbiasing the value of m , and then calculate m as the
cor
unbiased estimate.
7.3 Determination of limits of the confidence interval
Using Equations (16) to (19) and tabulated values for the appropriate value of α in Annexes F and G, calculate the
ˆ ˆ
upper and lower limits of the confidence interval of m and σˆ using the biased value of m .
8 Test report
The report shall be in accordance with the reporting requirements of EN ISO/IEC 17025 and shall contain the
following information:
a) name and address of the testing establishment;
b) date of the evaluation, unique identification of the report and of each page, name and address of the
customer and the signatory of the report;
c) reference to this method, i.e. determined in accordance with EN 843-5;
d) listing of the fracture strength data employed and any other information concerning its origin, including
test material, test geometry, test conditions, fractographic evidence etc., relevant to the data set;
e) graph relating ranked strength data to corresponding estimates of probability of failure (a Weibull plot)
and the conclusions drawn from the plot as to the suitability of two-parameter Weibull statistical analysis
in accordance with this method;
ˆ
f) biased and unbiased estimates of Weibull modulus, m , to three significant digits;
ˆ
g) estimate of Weibull characteristic strength, σ , expressed in MPa to three significant digits;
ˆ ˆ
h) estimates of the values of m and σ corresponding to the upper and lower limits of the confidence
interval, to three significant digits.
NOTE An example test report is given in Annex I.
Annex A
(informative)
Relationship between characteristic strengths of test pieces or
components of different size or shape, or with different stress fields
applied
One approach to the relationship between strengths of test pieces of different sizes or stress fields is based on the
stressed volume (appropriate for volume-distributed flaws) or stressed surface area (where strength-determining
flaws are distributed at the surface only). This concept is developed, for example, in references [5 to 10]

If the Weibull distribution is to be related to the volume V of a uniaxially stressed test piece, Equations (10) and
(11) are rewritten:
m
 
σ
 
f
m
 
 
()
P = 1− exp − g V dV , σ > 0 (A.1)
f
f
 ∫
 V σ 
0 0V
 
V
 
P=≤0, σ 0 (A.2)
ff
where
σ is the material scale parameter related to a unit stressed volume V , in Equation (10);
0v 0
g(V) is a dimensionless function for the manner in which the stress field varies over the volume of the
component. The local stress is typically normalised by the maximum stress, e.g. the nominal flexural
stress, i.e.
σ (V )=σ g(V )
max
where
σ = σ at failure.
max f
The integral in Equation (A.1) is the effective stressed volume of the component V , which is generally rather
eff
different to the geometric volume.
For surface flaws, the equivalent formulation for area A is used to calculate σ , the scale parameter for unit
0a
stressed area.
If it is required to determine σ for tensile tests in which σ is the measured characteristic strength and V is the
0v 0
gauge section volume this is obtained as follows:
1/m
σ =V σ (A.3)
0v 0
For σ for a uniformly stressed area, an analogous calculation using area A is employed.
0a
For other uniaxially stressed geometries where the stress level is not uniform throughout the stressed volume, it is
necessary to replace V (or A) with the value determined from the integral in Equation (A.1), the effective volume,
V (or area A ). For example, for four-point bending in accordance with the method given in EN 843-1:
eff eff
1/m


 
 
L
i
 
V m+ 1

 
 
L
  
0 
  
σ =σ (A.4)
0 0 
v
2(m+ 1)
 
 


where
L and L are the inner and outer loading spans;
i o
V is the stressed volume over the outer span, i.e. L multiplied by cross-sectional area.
o
NOTE Similar formulations for biaxial or triaxial stress are available. However, this is outside the simple approach in
this European Standard.
This method has been found to match adequately experimental results under conditions where the strength-
determining flaw type in different sized test pieces or components is the same. For example, it explains the
difference between results in three and four-point bending tests. However, it clearly would not provide a true
assessment for concurrent or extraneous flaw distributions, i.e. if between test piece or component types there is
a change in dominance of strength-determining flaw type with a different size distribution which might yield
different values of m and σ for each test piece or component type.
0v
Annex B
(informative)
FORTRAN program for calculating Weibull parameters
B.1 General
The FORTRAN program (see B.2) based on an interval halving method for determining the minimum of the log
ˆ
likelihood function, can be used to compute the maximum likelihood estimate of m, m , and of σ , σˆ .
0 0
NOTE The program requires double precision variables, and may have to be modified according to the compiler
employed and the output requirements.
B.2 FORTRAN program
C  *           PROGRAM MAXL               * C
C
IMPLICIT REAL *8(A-H, O-Z)
DOUBLE PRECISION ST (120)
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
OPEN(UNIT=8,FILE='MAXLT4.DAT',STATUS='OLD')
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C  Reads NIa, number of results to be analysed, and then ST(I) the
C  individual values
C
READ (8,*) NIA
DO 5  I = 1,NIA
READ (8,*) ST(I)
5 CONTINUE
EPS2=0.001
C
C  EPS2 is the accuracy requirement for the repeatability of
C  estimates of Weibull modulus made
C
CALL INTHAL(ST,NIA,EPS2,WEIB2)
PRINT *,WEIB2
CALL CHARSTR(ST,NIA,WEIB2,SIGHAT)
PRINT *,SIGHAT
STOP
END
C
C
C*************************************************************************

SUBROUTINE INTHAL (SIG, NI, EPS, WEIB)

C*************************************************************************
C
C  This subroutine is an interval halving method for obtaining the
C  iterative calculation of Weibull modulus
C
PARAMETER (ITMAX=1000)
IMPLICIT DOUBLE PRECISION (A-H, O-Z)

C  Input the initial value bounds of the Weibull modulus between
C  0.1 and 100 to begin the procedure

11  CONTINUE
PRINT 900, 'START INTERVAL INPUT (A,B):'
PRINT *, '(RECOMMENDED VALUES: 0.1, 100)'
READ *, A, B
X1=A
X2=B
C
C  Iteration loop using two log likelihood functions F1 and F2
C
ITER=0
F1=FUNCT (X1, SIG, NI)
F2=FUNCT (X2, SIG, NI)
IF (F1*F2.GE.0.0) THEN
PRINT 900, 'INTERVAL CONTAINS NO OR MANY NIL POINTS!'
PRINT *, 'PLEASE GIVE A NEW START INTERVAL!'
GOTO 11
ENDIF
100  CONTINUE
ITER=ITER+1
IF (ITER.GT.ITMAX) THEN
PRINT *
PRINT *, 'REQUIRED ACCURACY NOT REACHED AFTER', ITMAX,
PRINT *, 'ITERATIONS'
WEIB=X3
GOTO 500
ENDIF
X3=(X1+X2)/2.
F3=FUNCT(X3,SIG,NI)
IF (F3.EQ.0.0) THEN
WEIB=X3
GOTO 200
ENDIF
IF (F2*F3.LT.0.0) THEN
X1=X2
X2=X3
F1=F2
F2=F3
ELSE
X2=X3
F2=F3
ENDIF
C
C  Testing for the accuracy criterion
C
IF (ABS(X2-X1).GT.ABS(X2)*EPS) THEN
GOTO 100
ELSE IF (ABS(F2).LE.ABS(F1)) THEN
WEIB=X2
ELSE
WEIB=X1
ENDIF
200  CONTINUE
C
C  Results
C
PRINT *
PRINT *, 'RELATIVE ACCURACY ',EPS,' REACHED WITH'
PRINT *,  ITER,' ITERATIONS'
PRINT *,  'START INTERVAL WAS: ', A, B
PRINT *
500  CONTINUE
900  FORMAT('0' ,A)
RETURN
END
C
C ****************************************************************************

DOUBLE PRECISION FUNCTION FUNCT(X, SIG, NI)

C ****************************************************************************
C  ->  SIG: Array of strength values
C  ->  NI:  Array size
C  ->  X:  New position of the evaluation
C  This subroutine calculates the summations in Equation (12) based on an
C  estimate of Weibull modulus X
C
IMPLICIT DOUBLE PRECISION (A-H, O-Z)
C
DOUBLE PRECISION SIG(NI)
C
SUM1=0.
SUM2=0.
SUM3=0.
DO 10 I=1, NI
SUM1=SUM1+LOG(SIG(I))
SUM2=SUM2+LOG(SIG(I))*SIG(I)**X
SUM3=SUM3+SIG(I)**X
10  CONTINUE
DNI=DBLE(NI)
FUNCT=DNI/X+SUM1-DNI*SUM2/SUM3
C
RETURN
C
C **************************************************************************
C
DOUBLE PRECISION FUNCTION CHARSTR(SIG, NI, WEIB, SIGH)
C
C **************************************************************************
C
C  Subroutine for calculating sigmahat from Equation (13) from estimated value of m
C
IMPLICIT DOUBLE PRECISION (A-H, O-Z)
C
DOUBLE PRECISION SIG(NI)
C
SUM4 = 0.
DO 20 I=1,NI
SUM4 = SUM4+(SIG(I)**WEIB)
DNI = DBLE(NI)
SIGH = (SUM4/NI)**(1/WEIB)
RETURN
C
END
Annex C
(informative)
PASCAL program for calculating Weibull parameters
C.1 General
The PASCAL program (see C.2) based on an interval halving method can be used for determining the minimum
ˆ
of the log likelihood function, and hence the maximum likelihood estimate of m, m , and of σ , σˆ .
0 0
NOTE It is written in a similar way to the Fortran program but strength values have been normalised to minimize the
risk of floating point overflow. It has been check-run using a Pascal compiler. The program may have to be modified
according to the compiler employed and the output requirements.
C.2 PASCAL program
PROGRAM MAXL;
{** This program should work properly under a suitable version of Pascal. **}
{** If no coprocessor present then use 8087 emulation or change type of **}
{** MAXLREAL from EXTENDED to REAL or DOUBLE **}
{** This program requires the following data in file "input.dat" :   **}
{** Number of strength results                   NI  **}
{** Individual strength results                  ST
...