Bsi bs en 12603 2002

30086955 pdf BRITISH STANDARD BS EN 12603 2002 Glass in building — Procedures for goodness of fit and confidence intervals for Weibull distributed glass strength data The European Standard EN 12603 20[.]

Glass in building —

Procedures for

goodness of fit and

confidence intervals for

Weibull distributed

glass strength data

The European Standard EN 12603:2002 has the status of a

British Standard

ICS 81.040.20

This British Standard, having

been prepared under the

direction of the Building and

Civil Engineering Sector Policy

and Strategy Committee, was

published under the authority

of the Standards Policy and

A list of organizations represented on this subcommittee can be obtained on request to its secretary

Cross-references

The British Standards which implement international or European

publications referred to in this document may be found in the BSI Catalogue

under the section entitled “International Standards Correspondence Index”, or

by using the “Search” facility of the BSI Electronic Catalogue or of British

— aid enquirers to understand the text;

— present to the responsible international/European committee any enquiries on the interpretation, or proposals for change, and keep the

Amendments issued since publication

EUROPÄISCHE NORM November 2002

ICS 81.040.20

English version

Glass in building - Procedures for goodness of fit and confidence intervals for Weibull distributed glass strength data

Verre dans la construction - Procédures de validité de

l'ajustement et intervalles de confiance des données de

résistance du verre au moyen de la loi de Weibull

Glas im Bauwesen - Bestimmung der Biegefestigkeit von Glas - Schätzverfahren und Bestimmung der Vertrauensbereiche für Daten mit Weibull-Verteilung

This European Standard was approved by CEN on 7 September 2002.

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 Management Centre 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 Management Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Greece, Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland and United Kingdom.

EUROPEAN COMMITTEE FOR STANDARDIZATION

C O M IT É E U R O P É E N D E N O R M A LIS A T IO N EUROPÄISCHES KOMITEE FÜR NORMUNG

page

Foreword 3

Introduction 4

1 Scope 5

2 Normative references 5

3 Terms and definitions 5

4 Symbols and abbreviated terms 5

5 Goodness of fit 6

6 Point estimators for the parametersββ and θθof the distribution 7

6.1 Censored sample 7

6.2 Uncensored (complete) sample 9

7 Assessment of data and tests 11

7.1 The Weibull diagram 11

7.2 Graphical representation of the estimated distribution function 11

7.3 Plotting of sample data in the Weibull diagram 11

7.3.1 Single values 11

7.3.2 Classified values 12

7.4 Assessment of sample data 12

8 Confidence intervals 12

8.1 Confidence interval for the shape parameter ββ 12

8.2 Confidence interval for the value of the distribution function G(x) at a given value of x , of the attribute X 15

8.3 Confidence interval for the scale parameter θθ 18

8.3.1 Method for all samples 18

8.3.2 Method for uncensored samples 18

8.4 Confidence interval for the value x of the attribute X at a given value G ( x) of the distribution function 21

8.4.1 Method for all samples 21

8.4.2 Method for uncensored samples 22

Annex A (informative) Examples 23

A.1 Uncensored sample 23

A.1.1 Data 23

A.1.2 Statistical evaluation 24

A.2 Censored sample 27

A.2.1 Data 27

A.2.2 Statistical evaluation 29

Annex B (informative) Weibull graph 32

Bibliography 33

This document (EN 12603:2002) has been prepared by Technical Committee CEN/TC 129 "Glass in building", thesecretariat of which is held by IBN

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 May 2003, and conflicting national standards shall be withdrawn at the latest byMay 2003

In this standard the annexes A, B and C are informative

According to the CEN/CENELEC Internal Regulations, the national standards organizations of the followingcountries are bound to implement this European Standard: Austria, Belgium, Czech Republic, Denmark, Finland,France, Germany, Greece, Iceland, Ireland, Italy, Luxembourg, Malta, Netherlands, Norway, Portugal, Spain,Sweden, Switzerland and the United Kingdom

This European Standard is based on the assumption that the statistical distribution of the attribute taken intoconsideration can be represented by one single Weibull distribution function, even where in certain cases (e.g lifetimemeasurements) mixed distributions have frequently been observed For this reason, the user of the standard has tocheck by a goodness of fit test whether the measured data of a sample can be represented by means of one singleWeibull function Only in this case can the hypothesis be accepted and the procedures described in this standard beapplied

The user decides on this question also considering all previous relevant data and the general state of knowledge in thespecial field Every extrapolation into ranges of fractiles not confirmed by measured values requires utmost care, themore so the farther the extrapolation exceeds the range of measurements

NOTE The three-parameter Weibull function is:

The calculation can be based either on an uncensored or a censored sample There are several methods of censoring

In this standard only the following method of censoring is considered:

- given a numberr < n of specimens of which attribute valuesxi were measured

ISO 2854:1976, Statistical interpretation of data - Techniques of estimation and tests relating to means and variances.

ISO 3534, Statistics - Vocabulary and symbols

3 Terms and definitions

For the purposes of this European Standard, the terms and definitions given in ISO 3534 apply

4 Symbols and abbreviated terms

X attribute taken into consideration;

x, xi, xr values ofX;

G(x) distribution function ofX = percentage of failure;

xo, β, θ parameters of the three-parameter Weibull function;

^ identification label for point estimators (e.g β ˆ, θ ˆ, Gˆ);

1-α confidence level;

i value used in the goodness of fit test;

L value used in the goodness of fit test;

r number of specimens of which attribute valuesxi were measured;

NOTE The sample is ordered, i.e.x1≤x2≤x3 ≤xrr ≤ n;

Cr;n factor used in estimating θ ˆ;

s int(0,84n) = largest integer < 0,84n ;

η,ξ ordinate and abcissa of the Weibull diagram;

χ2 chi-square distribution function;

y,v,γ auxiliary factors used in estimating the confidence limits of G(x);

A,B,C constants used in evaluating v ;

H(f2) variable used in evaluating γ;

Tn;α/2,Tn;1-α/2 coefficients used in estimating the confidence limits ofθ;

Subscripts:

un lower confidence limit;

ob upper confidence limit;

z confidence interval limited on two sides

5 Goodness of fit

Sort the r values ofx into rank ascending order

Compute for each value fromi = 1 to i = r - 1:

3 4

ln

1 4

3 ) 1 (

4 ln ln

) ln(

) ln( 1

n

i n n

i n

= / 2

1

1 1 2 /

2 /

2 /

1

r i

i

r r i

i

r

r

where the symbol   r / 2 is used to denote the largest integer less than or equal to r/2

Reject the hypothesis that the data is from a Weibull distribution at the α significance level if:

The values of the fractiles of the Fdistribution can be found for example in Table IV of ISO 2854:1976.

6 Point estimators for the parameters ββ and θθ of the distribution

6.1 Censored sample

x -

x r

r;n

ln ln

Asymptotic estimate for large n : Cr,n = cp + a1/n + a2/n2

6.2 Uncensored (complete) sample

x -

x s

n s

n + s

= i

n

ln ln

=

1

1 5772 , 0 ln

1 exp

7 Assessment of data and tests

7.1 The Weibull diagram

The probability diagram for the Weibull distribution is drawn up in such a way that the distribution function of a parameter Weibull distribution is represented by a straight line

two-The ordinate axis is graduated according to the function

=

1

1 ln

NOTE Such forms are available As a rule, diagrams should be used with a range ofG-values from G = 1 × 10-3 = 0,1 % to

G = 0,999 = 99,9 % The necessary range of x-values depends on the value β of the shape parameter

7.2 Graphical representation of the estimated distribution function

The point estimators of the shape parameterβ and the scale parameterθ define a straight line in the Weibull diagram; it

is appropriate to define this straight line though the following two points:

0

ˆ ×

=

This straight line shall be plotted into the diagram

7.3 Plotting of sample data in the Weibull diagram

Each value x i of the Ordered Sample shall be co-ordinated to an estimated value:

4 , 0

3 , 0 ˆ

+ n

i

-= ) x

7.4 Assessment of sample data

The straight line plotted according to 7.2 and the points which represent the measured values of the sample,plotted according to 7.3 can be compared visually

Systematic deviations can examined in detail taking into consideration the general knowledge of the basic technicaland scientific facts and the results of previous relevant research For instance, if the distribution of the attribute valuescan be approximated by segments of straight lines with different slopes, a mixed Weibull distribution may be assumed.This can be taken as a hint that several basic mechanisms determine the attribute values x i Such a detailedexamination is beyond the scope of this standard

8 Confidence in t ervals

The equations of the following sub-clauses are valid for the case that the confidence intervals are limited on two sides(subscriptz) Where the confidence intervals are limited only on one side, α/2 shall be replaced byα in the followingequations

The confidence level (1 -α) is to be chosen by the user of this standard

8.1 Confidence interval for the shape parameter ββ

The upper limit of the confidence interval for the shape parameterβ at the confidence level (1 -α) is:

f

z ob;

1

2 2 1

1

ˆ χ β

1

2 2

1

ˆ χ β

f1 shall be obtained by multiplication of the figures from Table 4 by the sample sizen

Table 4 — Values of the function f 1 / n

Asymptotic estimate for large n : f1/n = h0 + h1/n + h2/n2

For uncensored samples (r/n=1) a good approximation isf1/n = 3,085 - 3,84/n

The values χ 2 1 α 2

1; - /

1; / f

are quantiles of the chi-square distribution with f1 degrees of freedom Values are given in Table 5

Table 5 — The 2,5 % and 97,5 % quantiles of the χχ2 distribution

Degrees of freedom f

p

2,5 % 97,5 % 1

2 3 4

0,000982 0,0506 0,216 0,484

5,02 7,38 9,35 11,1

Table 5 (continued)

Degrees of freedom f

p

2,5 % 97,5 % 5

6 7 8 9

0,831 1,24 1,69 2,18 2,70

12,8 14,4 16,0 17,5 19,0 10

11 12 13 14

3,25 3,82 4,40 5,01 5,63

20,5 21,9 23,3 24,7 26,1 15

16 17 18 19

6,26 6,91 7,56 8,23 8,91

27,5 28,8 30,2 31,5 32,9 20

21 22 23 24

9,59 10,3 11,0 11,7 12,4

34,2 35,5 36,8 38,1 39,4 25

26 27 28 29

13,1 13,8 14,6 15,3 16,0

40,6 41,9 43,2 44,5 45,7

Table 5 (continued)

Degrees of freedom f

p

2,5 % 97,5 % 30

40 50 60 70 80 90 100

16,8 24,4 32,4 40,5 48,8 57,2 65,6 74,2

47,0 59,3 71,4 83,3 95,0 106,6 118,1 129,6

1 ln

ln

ˆ ln

ˆ

x G x

(19)

Equation for auxiliary factorν:

Cy - By + A

Table 6 — Constants A.n , B.n and C.n

n r/n

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 B.n

10 39,04 12,052 5,609 3,233 2,172 1,650 1,384 1,255 1,170

20 140,7 23,96 9,136 4,666 2,850 2,000 1,570 1,350 1,248 1,159

30 100,4 20,96 8,416 4,410 2,743 1,949 1,546 1,339 1,248 1,165

40 87,06 19,68 8,088 4,292 2,692 1,925 1,534 1,335 1,249 1,161

B, C and A are obtained by dividing the values in the table byn.

For uncensored samples (r/n=1) good approximations are:

where f2 and H(f2) are determined from Table 7

NOTE γ and f 2 depend on the value ofG ˆ x ( ), the sample size, n, and the ratio r/n γ and f2 are independent of β ˆ

Table 7 — f 2 and H ( f 2 ) as a function of νν

Then the limits of the confidence interval forG are:

8.3 Confidence interval for the scale parameter θθ

8.3.1 Method for all samples

The limits of the two-sided confidence interval for the scale parameter θ at the confidence level (1 - α) shall be

-=

j z;

ob;

z un;

j z;

ob;

+ j z;

ob;

θ

θ θ

-=

j z;

un;

z ob;

j z;

un;

+ j z;

un;

θ

θ θ

The iteration can be started with θ ob ; z ; 0 = θ un ; z ; 0 = θ ˆ

After each iteration, new values ofGob,z(x = θun;z;j) and Gun,z(x = θob;z;j) shall be calculated by the method in 8.2

The iteration shall be stopped when two successive values of both θob;z and θun;z are equal to within the requiredaccuracy For instance, for evaluation of strength test results, a difference less than 0,1 % of should give sufficientaccuracy

8.3.2 Method for uncensored samples

In the case of uncensored (complete) samples, the following simpler equations can be used:

with the coefficientsTn;α/2 and Tn;1-α/2 taken from Table 8.

Table 8 — Confidence factors T n;p

8.4.1 Method for all samples

The confidence interval forx at a given G(x) can be calculated by solving the transcendent equations:

chi-Graphical extrapolation allows for the immediate determination of the limits of the confidence interval of x from theWeibull diagram

To determine the confidence interval of a given value ˆx 2 of the point estimation numerically, a value x > ˆ 1 x ˆ 2 shall

be chosen, and for this value ˆx 1 the confidence interval of the distribution function G(x1) shall be calculated according

to 8.2, to obtain the confidence limits G ob ; z ( ) ˆx 1 and G un ; z ( ) ˆx 1

) ˆ ( ˆ 1 ln ˆ

1 2

1 1

;

;

x G - x

=

x

z un;

ln

) ˆ ( ˆ 1 ln ˆ

1 2

1 1

;

;

x G - x

=

x

z ob;

z un

z un;

β

(30)

8.4.2 Method for uncensored samples

For G≤ 0,632, the following simpler equations can be used

1 ln

1

x G -

1 ln

1

x G -

=

β

The values ofθob;z and θun;z shall be calculated according to 8.3.1 or 8.3.2

This simplified method of calculation results in a more conservative estimate for the confidence limits of x, comparedwith the more exact method of extrapolation as described in 8.4.1

In samples where n≥ 20 and β≥ 5, and for values of G < 0,1 the following equations shall be used These give abetter approximation to the exact method described in 8.4.1

1 ln ˆ

1

x G -

1 ln ˆ

1

x G -

=

β

Table A.1 — Results of an experiment to determine breakage stress

Table A.1(continued)

A.1.2 Statistical evaluation

A.1.2.1 Point estimations

The method is described in 6.2

From Table 3, forn = 24, κn = 1,4975

s = int(0,84 x 24) = 20

Hence β ˆ = 18,67 and θ ˆ = 49,26 N/mm2

A.1.2.2 Estimation of confidence intervals

For the 95 % confidence intervals, 1 -α/2 = 0,975 and α/2 = 0,025

a) The method for determining the confidence interval for the shape parameter is given in 8.1

From Table 4, by linear interpolation, f1/n = 2.918, sof1 = 70,03

From Table 5, χ2

70,03;0,975 = 95,05 and χ2

70,03;0,025 = 48,78

Hence, βob;z = 25,34 and βun;z = 13,01

b) The method for determining the confidence intervals for G ( x ) is given in 8.2 and the results of the computations

are given in Table A.2