# SIST EN 843-5:2007

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

## 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 a 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

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### Standards Content (Sample)

SLOVENSKI STANDARD

SIST EN 843-5:2007

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

SIST EN 843-5:2007 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------

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.

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EN 843-5:2006 (E)

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

0

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

2

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EN 843-5:2006 (E)

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.

3

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EN 843-5:2006 (E)

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

4

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EN 843-5:2006 (E)

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.

5

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EN 843-5:2006 (E)

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

6

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EN 843-5:2006 (E)

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.

7

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EN 843-5:2006 (E)

NOTE 4 The population mean x is related to β by:

1

x=Γβ 1+

m

(8)

where

Γ is the gamma function.

The gamma function is sometimes represented by a non-integral factorial:

11

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.

0

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

0

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

8

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EN 843-5:2006 (E)

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

0

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

0

σ strength of test piece

f

9

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EN 843-5:2006 (E)

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

0

ˆ

σ maximum likelihood estimate of Weibull characteristic strength of test piece

0

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

0

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

0

ˆ

ˆ

European Standard, the values of m and σ for the two sets shall be deemed to be equivalent at the

0

same confidence level if the results of one lie within the confidence interval of the other, or vice versa.

10

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EN 843-5:2006 (E)

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

0

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

0

ˆ

functions to zero yielding, respectively, estimates m and σˆ , for m and σ :

0 0

N

mˆ

σσln

∑

fj fj

N

11

j=1

−−lnσ = 0 (13)

∑

fj

N

Nmˆ

mˆ =1

j

σ

∑ fj

j=1

and

1/mˆ

N

1

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

0

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.

11

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EN 843-5:2006 (E)

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

0

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.

0

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:

0

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

0

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

**...**

SLOVENSKI oSIST prEN 843-5:2005

PREDSTANDARD

februar 2005

Sodobna tehnična keramika - Monolitna keramika - Mehanske lastnosti pri

sobni temperaturi – 5. del: Statistična analiza

Advanced technical ceramics - Monolithic ceramics - Mechanical properties at room

temperature - Part 5: Statistical analysis

ICS 81.060.30 Referenčna številka

oSIST prEN 843-5:2005(en)

© Standard je založil in izdal Slovenski inštitut za standardizacijo. Razmnoževanje ali kopiranje celote ali delov tega dokumenta ni dovoljeno

---------------------- Page: 1 ----------------------

EUROPEAN STANDARD

DRAFT

prEN 843-5

NORME EUROPÉENNE

EUROPÄISCHE NORM

November 2004

ICS Will supersede ENV 843-5:1996

English version

Advanced technical ceramics - Monolithic ceramics. Mechanical

properties at room temperature - Part 5: Statistical analysis

Céramiques techniques avancées - Céramiques Hochleistungskeramik - Monolithische Keramik -

monolithiques - Propriétés mécaniques à la température Mechanische Eigenschaften bei Raumtemperatur - Teil 5:

ambiante - Partie 5 : Analyse statistique Statistische Analysis

This draft European Standard is submitted to CEN members for enquiry. It has been drawn up by the Technical Committee CEN/TC 184.

If this draft becomes a European Standard, 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.

This draft European Standard was established by CEN 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 Management Centre 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, Slovakia,

Slovenia, Spain, Sweden, Switzerland and United Kingdom.

Warning : This document is not a European Standard. It is distributed for review and comments. It is subject to change without notice and

shall not be referred to as a European Standard.

EUROPEAN COMMITTEE FOR STANDARDIZATION

COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

Management Centre: rue de Stassart, 36 B-1050 Brussels

© 2004 CEN All rights of exploitation in any form and by any means reserved Ref. No. prEN 843-5:2004: E

worldwide for CEN national Members.

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prEN 843-5:2004 (E)

Contents Page

Foreword. 3

1 Scope. 4

2 Normative references. 4

3 Definitions. 4

3.1 Flaws. 4

3.2 Flaw distributions. 5

3.3 Mechanical evaluation. 5

3.4 Statistical terms. 6

3.5 Weibull distribution. 7

4 Symbols. 8

5 Significance and use . 9

6 Basis of method . 11

6.1 Maximum likelihood method. 11

6.2 Bias correction. 11

6.3 Confidence interval. 12

7 Procedure. 12

7.1 Graphical representation of data. 12

7.2 Determination of Weibull parameters by maximum likelihood method. 13

7.3 Determination of limits of the confidence interval .14

8 Report. 14

Annex A (informative) Fortran program . 15

Annex B (informative) Pascal program . 21

Annex C (informative) Basic program . 26

ˆ

Annex D (normative) Unbiasing factors for m . 31

Annex E (normative) Confidence factors for σˆ . 32

0

ˆ

Annex F (normative) Confidence factors for m . 34

Annex G (informative) Worked examples. 36

Annex H (informative) Example test report. 42

Annex I (informative) Relationship between characteristic strengths of test pieces or

components of different size or shape, or with different stress fields applied. 44

Bibliography . 46

2

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prEN 843-5:2004 (E)

Foreword

This document (prEN 843-5:2004) has been prepared by Technical Committee CEN/TC 184

“Advanced technical ceramics”, the secretariat of which is held by BSI.

This document is currently submitted to the CEN Enquiry.

This document will supersede ENV 843-5:1996.

3

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prEN 843-5:2004 (E)

1 Scope

This standard describes 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 strictly to be 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 Standard permits the analysis of the general situation where crack growth or creep may have

occurred, provided that it is recognised 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 standard employs the same calculation procedures as ISO 20501:2003 (see annex J, Ref. [1]),

method A, but does not provide a method for dealing with censored data, Method B.

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, Advanced technical ceramics — mechanical properties of monolithic ceramics at room

temperature — Part 1: Determination of flexural strength

EN 1006, Advanced technical ceramics — methods of testing monolithic ceramics — guidance on the

selection of test-pieces for the evaluation of properties

EN ISO/IEC 17025, General requirements for the competence of testing and calibration laboratories

ISO 2602, Statistical interpretation of test results — estimation of the mean: confidence interval

ISO 3534-1, Statistics — vocabulary and symbols — probability and general statistical terms

3 Definitions

For the purposes of this standard the following definitions apply.

Definitions of additional statistical terms may be found in ISO 2602, ISO 3534-1, or other source literature

on statistics.

3.1 Flaws

3.1.1

flaw

inhomogeneity, discontinuity or structural feature, e.g. a grain boundary, large grain, pore, impurity or

crack, in a material which when loaded provides a stress concentration and a risk of mechanical failure

NOTE The term flaw should not be taken as meaning the material is functionally defective, but rather as

containing an inevitable microstructural inhomogeneity.

4

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prEN 843-5:2004 (E)

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, e.g. 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). An example is when every test piece contains flaw type A, and some contain

additionally a second independent type B

3.2.4

concurrent (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. Thus 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 result 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, and hence 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

5

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prEN 843-5:2004 (E)

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; in the present

case, 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. 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

the 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)

∫

−∞

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.

6

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prEN 843-5:2004 (E)

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, e.g.

strength measurements from a batch of similar test-pieces

3.4.8

sampling

process of selecting test-pieces for a test. For the purposes of this standard the guidance given in EN

1006 shall 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 Weibull distribution

The continuous random variable x has a two-parameter Weibull distribution if the probability density

function is given by:

m−1 m

m x x

f (x) = exp− x > 0 (4)

β β β

f (x) = 0 x ≤ 0 (5)

This corresponds with a cumulative distribution function:

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 1 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 standard is that the Weibull distribution will approximate to the true distribution of

strengths observed.

NOTE 2 This 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 Standard.

The population mean x is related to β by:

1

x=Γβ 1+ (8)

m

where Γ is the gamma function.

7

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prEN 843-5:2004 (E)

NOTE The gamma function is sometimes represented by a non-integral factorial:

11

Γ+1!= (8A)

mm

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

nom

strength σ = σ , is given by the cumulative distribution function:

f nom

m

σ

f

P = 1− exp − σ > 0 (9)

f f

σ

0

P = 0 σ ≤ 0 (10)

f f

where: P is the probability of failure

f

σ is the Weibull characteristic strength (at P = 0,6321), acting as the scale parameter

0 f

Defined in the above manner, the Weibull characteristic strength depends on the test piece geometry and

on the multiaxiality of the stress field applied.

Caution is therefore needed in the use of Weibull statistical parameters beyond the population from which

they are derived.

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 I for

0

information on the theoretical relationship between strengths of test-pieces of different stressed volumes or areas.

4 Symbols

The symbols used in this standard are defined below:

A component surface area

A effective component surface area

eff

b unbiasing factor for Weibull modulus estimate

C lower limit of confidence interval for

l 0

C upper limit of confidence interval for

u 0

D lower limit of confidence interval for

l

D upper limit of confidence interval for

u

f probability density function

F cumulative probability distribution function

g a 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

8

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prEN 843-5:2004 (E)

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

0

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

0

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

nom

for the 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

0

σˆ maximum likelihood estimate of Weibull characteristic strength of test piece

0

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 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.

9

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prEN 843-5:2004 (E)

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 standard.

NOTE 2 Use of other methods of estimating m and σ , such as least squares fitting of a straight line to the

0

ranked data points as performed for the visual inspection (see 7.1), is not permitted by this 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 must 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

ˆ

different values of m and σˆ due to the natural scatter in sampling from the population. For the

0

ˆ

purposes of this standard, the values of m and σˆ for the two sets shall be deemed to be

0

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 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.

10

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prEN 843-5:2004 (E)

6 Basis of method

6.1 Maximum likelihood method

Once it is determined that a valid two-parameter Weibull distribution can be fitted to the data set being

ˆ

evaluated (see 7.1), the maximum likelihood estimates of Weibull modulus, m , and characteristic

strength, σˆ , can be determined.

0

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 − (11)

∏

σ σ σ

j=10 0 0

where N is the number of fracture data.

This function is maximised by differentiating the log likelihood (ln(L)) with respect to m and σ , and setting

0

ˆ

these functions to zero yielding, respectively, estimates m and σˆ , for m and σ :

0 0

N

ˆ

m

σσln

∑

fj fj

N

j=1 11

−−lnσ = 0 (12)

∑

fj

N

ˆ

Nm

mˆ j=1

σ

∑

fj

j=1

and

ˆ

1/ m

N

1

mˆ

σˆ = σ (13)

0 ∑ fj

N

j=1

ˆ

Equation (12) must be solved numerically to obtain a solution for m , which can then be used to solve for

σˆ through equation (13). The required fractional accuracy of solution (ε) shall be ≤ 0,001, giving three

0

ˆ

significant digits in the value of m .

A computer may be used for this task. The proper implementation of any computer program shall be

checked by employing the example data in Annex G for the defined level of accuracy, ε.

NOTE 1 The computer programs in annexes A to C incorporate appropriate routines for the interval halving

ˆ

ˆ

method for numerically solving for m and σ . They may need to be modified to suit different computer systems.

0

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

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 D. This unbiasing

factor has been determined by a Monte Carlo method sampling randomly from a large population with a

ˆ ˆ

predetermined true value of m, allowing correction of the biased value m to the corrected value m :

cor

ˆ ˆ

m = m.b (14)

cor

11

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prEN 843-5:2004 (E)

where the unbiasing factor b is read from annex D.

ˆ ˆ

The bias in σ is minimal compared with that in m , and no bias correction is required.

0

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 as part of this Pre-

standard.

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

u

confidence interval limit factor t at (1 - α/2). The values of factors t and t are determined from the table in

l u l

annex E. The upper and lower values of σˆ corresponding to the upper and lower limits of the confidence

0

interval are determined respectively as:

t

u

C = σˆ exp − (15)

u 0

ˆ

m

t

l

C = σˆ exp − (16)

l 0

ˆ

m

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 annex F

ˆ

in a similar manner as for σ using the same value of α. For the given number of test-pieces, N, and the

0

same confidence level, (1 - α), the upper confidence interval limit factor l for a two-sided test is

u

determined at α/2, and the lower confidence interval limit factor l at (1 - α)/2. The upper and lower limits of

l

ˆ

the confidence interval for m are determined respectively as:

ˆ

D =ml/ (17)

uu

ˆ

D =ml/ (18)

ll

7 Procedure

7.1 Graphical representation of data

In order to provide visualisation of the distribution of strengths for determining the relevance of this

method of statistical analysis, prepare a graphical 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 = (19)

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:

12

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prEN 843-5:2004 (E)

yP=−lnln()1/(1 ) (20)

ifi

and for the abscissa:

x = ln(σ ) (21)

ifi

using scaling such that the range of data covers the whole ordinate and abscissa. This is termed a

"Weibull plot".

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 may become too steep for

effective judgement of the straightness of a fitted line. In particular, the abscissa may 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.

If with sufficient data points there is no trend to linearity a two-parameter Weibull distribution may not be

appropriate, and use of this analysis method may provide an incorrect as

**...**

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