A11102 ITMSbE

NBS BUILDING SCIENCE SERIES 162

Ring-on-Ring Tests and Load Capacity

of Cladding Glass

"TA

'^^"^ 'ARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS

1G2

1934
^

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;.ATio;.'."'. r;Trr---'i OF •

NBS BUILDING SCIENCE SERIES 162

Ring-on-Ring Tests and Load Capacity

of Cladding Glass

Emil Simiu^

Dorothy A. Reed^'^

Charles W. C. Yancey

Jonathan W. Martin^

Erik M. Hendrickson^

Armando C. Gonzalez-^

Masayoshi Koike^

James A. Lechner'*

Martin E. Batts^

Prepared for:

National Science Foundation

Washington, DC 20550

Center for Building Technology, National Engineering Laboratory, National Bureau

of Standards, Gaithersburg, MD 20899

^Present address: Department of Civil Engineering, University of Washington,

Seattle, WA 98195

^Center for Materials Science, National Bureau of Standards, Gaithersburg, MD

20899

Center for Applied Mathematics, National Engineering Laboratory, National Bureau

of Standards, Gaithersburg, MD 20899

^Paratech Corporation, Chevy Chase, MD 20815

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1984
Library of Congress Catalog Card Number: 84-601098

National Bureau of Standards Building Science Series 162

Natl. Bur. Stand. (U.S.), BIdg. Sci. Ser. 162, 60 pages (Aug. 1984)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402

ill
ABSTRACT

Although ring-on-ring test results have been used in the past to obtain

information on the strength of glass, no methodology has so far been developed

in the literature explicitly relating such results to the load capacity of

cladding glass. The main purpose of this report is to propose such a methodology. The proposed methodology makes use of recent advances in the modeling

of the fracture mechanics behavior of glass and the calculation of stresses in

plates exhibiting geometric nonlinearity . Evidence is presented which strongly

suggests that the probability distribution of the load capacity of cladding

glass panels whose failure is due to surface flaws can be estimated reliably

on the basis of results of ring-on-ring tests used in conjunction with (a)

numerical methods for the analysis of stresses in plates, and (b) information

on the elastic and fracture mechanics behavior of glass currently available or that can be obtained routinely. Two interesting findings are noted. First,

owing to the way in which results of ring-on-ring tests are utilized, the

relatively large variabilities typical of fracture mechanics parameters, as well as the uncertainties with respect to the shapes of surface flaws, have a minor effect on the estimation of load capacities. Second, two-parameter

Weibull distributions, previously used in the literature to model the strength

of glass and the load capacity of cladding panels, are not consistent with

experimental results. On the other hand, three-parameter Weibull distributions

model the observed glass behavior credibly.

Keywords: buildings; engineering mechanics; failure; fracture mechanics;

glass; loads (forces); probability theory; ring-on-ring tests;

strength.

iii
ACKNOWLEDGMENTS

The writers would like to express their appreciation to S. M. Wiederhorn,

S. W. Freiman, E. R. Fuller, K. Jakus, and B. R. Lawn, of the Inorganic

Materials Division, NBS, for useful discussions concerning the research

presented in this report. Ring-on-ring tests and tests on indented specimens

reported on herein were carried out with the assistance of C. Mastal and B. Hu.

The valuable assistance of S. T. Wu, of the Structures Division, NBS,

J. J. Filliben and S. D. Leigh, of the Statistical Engineering Division, NBS,

and D. P. Bentz, of the Materials Division, NBS, is also gratefully acknowledged.

The cover photo is by NBS photographer, Mark Heifer.

This work was supported in part by the National Science Foundation under agreement

CEE 8308329, Division of Civil and Environmental Engineering. Dr. M. P. Gaus of

the National Science Foundation served as Program Director. Any opinion, findings,

and conclusions or recommendations expressed in this publication are those of

the authors and do not necessarily reflect the views of the National Science

Foundation.

iv
TABLE OF CONTENTS

Page

1. INTRODUCTION 2. 2. ESTIMATION OF LOAD CAPACITY OF CLADDING PANELS , 2

2.1 Fracture Mechanics of Glass 2 2.2 Relation Between Initial Strength and the Final Strength

Corresponding to a 60-sec Load ^ 2.3 Estimation of 60-sec Load Causing Failure of a Panel 5 2.4 Specification of Initial Strengths, S-^CAi^, Aajj,) 6

3. RING-ON-RING TESTS 8

3.1 Principle and Description of Ring-on-Ring Testing 8

3.2 Estimation of Initial Strengths from Ring-on-Ring Test

Results, and Influence of Uncertainties With Respect to

Parameters B and n Upon the Estimation of Design Loads 9 3.3 Results of Ring-on-Ring Tests 1*^

4. ESTIMATES OF THE LOAD CAPACITY OF GLASS PANELS 15

5. SUMMARY AND CONCLUSIONS 19

REFERENCES 20

APPENDIX I - MAXIMUM LIKLIHOOD ESTIMATION FOR THE THREE-PARAMETER

WEIBULL DISTRIBUTION BASED ON A SAMPLE WITH CENSORING

DUE TO COMPETING RISKS

APPENDIX II - DEPENDENCE OF ESTIMATED 60-SEC LOAD CAPACITY, p60 (0.008),

UPON NUMBER OF RING-ON-RING SPECIMENS BEING

TESTED

APPENDIX III - COMPUTER PROGRAM FOR ESTIMATING PROBABILITY DISTRIBUTION

OF LOAD CAPACITY p^g

APPENDIX IV - COMPUTER PROGRAM FOR ESTIMATING PARAMETERS OF GLASS

STRENGTH DISTRIBUTION

V

1. INTRODUCTION

For most common construction materials (e.g., steel or concrete), the load

capacity of structural members is determined on the basis of strength data

obtained from tests conducted on small size standard specimens, rather than

from the destructive testing of full-size members. The economy inherent in the use of small-size standard specimens is due not only to their lower cost as compared to the cost of full-size members, but also to the fact that strength

data obtained by testing a sufficiently large number of such specimens can be

used for designing a wide variety of structural members with different

configurations, types of loading, and sizes.

Procedures for determining the load capacity of glass panels on the basis of

strength data obtained by testing small standard specimens do not currently

exist. For this reason design charts issued by glass manufacturers have traditionally been based on destructive tests performed on full-size glass panels

(see, e.g., reference 1). It has been pointed out in the literature (see, e.g.,

reference 2), that such charts exhibit significant inconsistencies. Such

inconsistencies are due at least in part to the relatively small numbers of

panels (between 2 and 30—see references 4, 5, and 6) used in most of the

tests on which the charts were based. Since the charts cover a wide range of

panel sizes, the number of panels that would have to be subjected to destructive

tests in order to develop dependable data for design could in practice be

prohibitive.

It therefore is desirable to develop a methodology for estimating cladding

panel design loads from strength data obtained by testing small standard

specimens. The purpose of this report is to propose such a methodology. The

methodology utilizes recent advances [3, 14] , which enable the probability

distribution of the load capacity of glass panels to be estimated by numerical

methods on the basis of information on the strength, stiffness, and fracture

mechanics properties of glass.

Information characterizing the elastic behavior of glass is available or can

be obtained by well established test methods. The elastic parameters exhibit

relatively small variability, so that uncertainties pertaining to their actual

values do not significantly affect the estimation of design loads. In recent

years test methods have been developed to obtain fracture mechanics parameters

governing subcritical crack growth and strength degradation under load. Associated with the fracture mechanics parameters are large uncertainties, which

include uncertainties with respect to the shape of surface flaws. However, it

is shown in this work that the consequences of these uncertainties are minor

from an engineering point of view.

The critical question, then, is whether it is possible to obtain from tests of

small standard specimens the information on the strength of glass needed to

estimate reliably the load capacity of cladding panels whose failure is due to

surface flaws. Evidence presented in this work strongly suggests that, if the

ring-on-ring testing method is employed, the answer to this question is

affirmative.

1
2. ESTIMATION OF LOAD CAPACITY OF CLADDING PANELS

The purpose of this section is to review pertinent developments that make it

possible to estimate numerically the probability distribution of the load

capacity of cladding panels if the fundamental parameters characterizing the

behavior of glass are known.

2.1 FRACTURE MECHANICS OF GLASS

The basic criterion for fracture is derived from the Griffith equilibrium

expression, and may be written as

Ki = Kic (1)

where Kj = stress intensity factor, and Kjq = critical value of Kj. If

equation 1 holds, the system reaches the state of instability wherein the rate

of crack growth becomes for practical purposes infinite and failure occurs [8,

9]. Kj^Q is a property of the material and is determined experimentally. The

stress intensity factor, Kj, is proportional to the actual stresses in the

material in the presence of cracks causing stress concentrations. Kj can be

expressed as follows [8, 9]:

Ki(t) = Ya(t) v^TcIF (2)

where

Y = geometric shape factor,

a = nominal stress (i.e., stress calculated by assuming the absence of

cracks)

c = length of crack normal to the stress

t = time.

The geometric shape factor, Y, in equation 2 is a function of crack geometry

and plate dimensions [10]. For semi-elliptical surface flaws (figure 1) for

which c/h = 0 and a/b 0, values of Y calculated in reference 10 for various

ratios c/a are shown in table 2.1.

According to experiments reported in references 11, 12, and 13, the following

relationship holds for the rate of subcritical crack growth in the region

Ki < Kic:

lf=4K?(t) (3)

The parameters*^ and n depend upon ambient humidity and temperature and are obtained experimentally. Equation 3 expresses quantitatively the fact that the

cracks in an element of glass subjected to stress for some length of time will

grow - albeit not catastrophically - provided that the stress is contained

2
Table 2.1 Dependence of Factor Y Upon Ratio c/a for c/h 0 and a/b = 0

c/Y

0.00 1.985

0.25 1.827

0.50 1.581

0.75 1.353

1.00 1.163

1.10 1.157

1.20 1.149

1.30 1.139

1.40 1.127

1.50 1.116

1.60 1.103

1.70 1.090

1.80 1.077

1.90 1.064

2.00 1.050

within a certain range. This phenomenon is referred to as static or dynamic

fatigue according to whether the stress is constant or time-dependent.

It follows from equations 1 and 2 that the strength of glass, S, i.e., the

value of the nominal stress at which failure occurs, is

^IC

S(t) (4)

Y/^(tT

If Kj and c are eliminated from equations 2, 3, and 4 and the notation S(0) =

Sj^ is used (S-^ = initial strength), the following relationship is obtained:

S(t) = [S^ ^ - 1 / a" (t) dx] ^ (5)

B 0

where

1/B = -^Ay2 (5a)

[11]. If we consider an area, A\^, over which the tension stress, a(t), is uniform

and independent of direction, failure occurs within that area if

a(t) > S(Ak, t) (6)

3
where S(A]^, t) = strength calculated by equation 5 in which Sj^ corresponds to

the largest initial flaw within the area A^^, regardless of the direction of

that flaw, that is

(7)

and CjnaxC-^k, ~ length of largest flaw within Aj^ at time t = 0.

We now consider the case in which the state of stress is uniform over the area

A]^, but the principal stresses within the area are unequal. The effect of

normal stress is by far the strongest as far as crack propagation is concerned

[9, p. 54], and that the effect of shear stresses may therefore be neglected

(see, e.g., reference 14). The center of A^ is denoted by M]^, and the normal

tension stresses at time t parallel to direction are denoted by aCM^, ajj,, t).

We define the sector Aaj^,, centered on the direction a^, and such that normal

stresses at time t parallel to any radial direction contained within that

sector may be assumed to differ negligibly from a(Mk, aj^,, t) [figure 2].

Failure within the area Ay^ will not necessarily be initiated by a flaw normal,

or almost normal, to the largest principal stress, since the largest of these

flaws may well be relatively small. Neither will failure be necessarily initiated by the largest flaw within the area A^, since that flaw may well be

perpendicular to a relatively low normal stress. Rather, failure will be initiated by the largest of the flaws normal to any radius within the sector Aajj,

[whose length is denoted by c^-^ (A^, Aa^, t)] to which there corresponds a strength, S(A]^, Aa^, t), such that:

For any sector, Aa^, the strength at time t, S(A]^, Aa^, t), is calculated by

equation 5 in which ct(Mj^, a^, t) is substituted for aCx) and the initial

strength S^CA^^, Aa^) is substituted for Sj^. The following relation holds:

where Cu^x^^k* Aajj,, 0) = length of largest initial crack normal to any radius

within the sector Aaj2,.

2.2 RELATION BETWEEN INITIAL STRENGTH AND THE FINAL STRENGTH CORRESPONDING TO

A 60-SEC LOAD

A particular case of practical interest is that in which the load acting on the

panel has constant value over a 60-sec time interval and is zero outside that

t) > S(Ai^, Aa£, t) (8)

S^(A^^, Aaji) = 1/2

^<^max(^k' 0)

(9)

4
interval^. The 60-sec load induces at point Mj^ a normal stress parallel to

the direction a^, denoted by a5o(%> ot^), which is constant throughout the

duration of the load and equal to zero at all other times. Failure initiated

by a flaw normal or almost normal to a radius within the sector Aaji occurs at

time t = 60 sec if the stress <760(^k> equal to the strength SCAj^, Aajj,, t = 60). Substituting <75o(Mk» <^0 <^it) and S^CAj^, Aaj^,) for in equation 5,

it follows that the failure stress is given by the relation

^60 (Mk, a^) [olo (Ml,, a^) +

|q ] +

|q sJ ^Aj,, Aa;^) (10)

B/MD^ ^2 For soda-lime glass ^^^Q^^y^i ot^) is of the order of 10"^(MPa) or more, and B/60

is of the order of l(MPa) or less, so that the failure stress may be

written as n-2

^i^ (A^, Aaj^)

%0(%» ajl) =^ TT— (11)

1 1/n

(60 i)

2.3 ESTIMATION OF 60-SEC LOAD CAUSING FAILURE OF A PANEL

The relationship between the load acting on a cladding panel and the stresses

in the panel is generally nonlinear. This relationship can be obtained by

using, for example, a finite-difference program such as that developed by

Texas Tech University [15], and is commonly expressed in terms of the nondimensional quantities

LF = pb^/Dh (12)

SF = ab^h/D (13)

where LF and SF = loading and stress factors, respectively, p = uniform load

per unit area, a = stress, b = smaller side of rectangular plate, D = flexural

rigidity, defined by:

D = (14)

12(l-v2)

where E = modulus of elasticity, v = Poisson's ratio, and h = thickness of the

plate.

^ The 60-sec load is the standard reference load used in glass cladding design charts

in the United States.

5
Once the relationship between loads and stresses is known, it is possible to

obtain the 60-sec loads, P^q* corresponding to the failure stresses o^q (Mj^, a^)

calculated by equation 11 for each point }\ of a sufficiently dense grid and

for each of a sufficient number of directions aj^. The smallest of these loads,

denoted by P50> is the load causing failure (i.e., the load capacity) of the

panel characterized by the set of initial strengths Sj[(A]^, Aajj). Calculations

of the load capacity, P60> carried out for a large number, M, of panels.

The probability distribution of 60-sec load capacity can be estimated from the

M values so obtained. A computer program for estimating this distribution if

the initial strengths S-|[(A;^, Aajj) are specified is described in Appendix III.

In the case of heat-strengthened or tempered glass, the procedure for estimating

the probability distribution of the 60-second load capacity p^g is the same,

except that stresses cy5o(Mic>0'e^~'^R should be used in equation 11 in lieu of the

stress, 0()QiKy^,ai) ^ where Or denotes the residual thermal stress, which can be determined by routine experimental procedures.

2.4 SPECIFICATION OF INITIAL STRENGTHS SjCA]^, Aaj^)

The initial strength Sj^(A^) corresponding to the largest flaw within Aj^,

regardless of its direction, is commonly described by a Weibull distribution:

Si(Ak) - Ps TM

P[Si(Ak)] = 1 - exp { -[

g^(^^)

] } (15)

From the assumption that the number of flaws of any given size is on the average

proportional to the area A being considered, it follows that

Si(A) - Pg m

P[Si(A)] = 1 - exp { -[ —s-(^) ] } ~ (16)

where

m

S^(A) = (- (16a)

[25, pp. 5 and 10]. We will refer to equation 16 as the fundamental Weibull

distribution of the strength of glass.

6
Similarly, from the assumption that the flaw orientations are uniformly

distributed [i.e., that the number of flaws normal to the stress o^o^^k' ^l^^

is on the average equal to Aajj,/(iT/2 ) times the number of flaws parallel to any

direction], it follows that the probability distribution of the initial strength

Si(Ak, Aaji) is

S^CAj^ Aaj^ ) ~ l^s P[S,(A^. L.,n = 1 - exp 1 -l^J--^^ r] (17)

where

1/m

S^CA^, Aaji) = (2^) S^CAk) (18)

Assuming that equations 15 through 18 are valid, it follows that the initial

strength S-j^(A]^, Aajj) can be specified by the probability distribution given by

equation 17, provided that the parameters Sq(A), yg, and m are known.

These parameters can be obtained from ring-on-ring tests, which are described

in the following section.

^ Or to the stress o^qQA^, -aj^) = ct^qCMj^, a^^).

7
3. RING-ON-RING TESTS

3.1 PRINCIPLE AND DESCRIPTION OF RING-ON-RING TESTING

Ring-on-ring testing devices involve the creation of a state of uniform

axi symmetric tension stress in the central portion of one of the faces of a circular plate. This can be accomplished by placing the plate on a segmented

circular ring and by applying on its upper surface a load transmitted through a circular ring concentric with and having a smaller diameter than the segmented

support

As noted in reference 17 , because of the high elastic modulus and hardness of

glass, any nonperfect contact between rigid loading rings and the test sample

can lead to deviations from axisymmetry in the stress field. Reference 17

describes a ring-on-ring device designed to eliminate such nonuniformities.

Each ring consists of a closely wound coil. The load that each coil transmits

to the plate is applied by a rubber diaphragm which covers a circular groove

filled with fluid. The function of the fluid is to equalize the loading along

the coils. Strain gage measurements, and measurements of strength of indented

specimens, are reported in reference 17 for ringon-ring devices both of the

rigid type and of the type just described. According to the results of reference 17, errors in the measurement of strengths due to the use of rigid ringon-ring devices are less than 5 percent for annealed soda-lime float glass.

The errors are considerably larger (about 20 percent) for thermally tempered

crown glass.

The uniform stress in the central portion of the tension face of a circular

plate subjected to a ring-on-ring loading test is

a = -1^ [2(1 + V) An ^ + (l-v)(a2-b2) ajj

4irh2 b a2 r2

where p = load, h = plate thickness, a = radius of the support ring, b = radius

of the loading ring, R = radius of the disc, and v = Poisson's ratio [18].

For the case b/a = 1/2, the distribution of the stress, a, along a radius, r, is represented in figure 3 [19], Note that, owing to local effects, the

central portion of the plate within which the stress a is given by Eq. 19 is

defined by the relation r ^ 0.705b,

It follows from appendix I that the precision of strength estimates based on the

testing of any given sample of ring-on-ring specimens Increases with the number

of specimens within that sample that have failure origins in the central,

uniform stress area of the specimen. It would therefore be desirable to attempt

the development of test devices aimed at inhibiting stress corrosion on the

tension side of the nonuniform stress area of the specimen. This could be

accomplished by immersing that side in, or covering it with, an inert agent

such as water-free oil.

8
3.2 ESTIMTION OF INITIAL STRENGTHS FROM RING-QN-RING TEST RESULTS, AND

INFLUENCE OF UNCERTAINTIES WITH RESPECT TO PARAMETERS B AND n UPON THE

ESTIMATION OF DESIGN LOADS

Ring-on-ring tests yield the stress at the time of failure, a(tf), which is

equal to the strength at the time of failure, S(tf). To estimate the initial

strength corresponding to S(tf), equation 5 is used. In the case of a ramp

loading on a specimen that exhibits a linear load-stress relationship up to

failure [21, 22], steps similar to those that led to equation 11 yield the

relation

1/n n-2

a(tf) - [—^iip-] Si (20)

''f B

Equation 20 allows the estimation of Sj^ from measurements of a(tf) and tf. The

estimates of S^ depend upon the values assumed for n and B.

We now consider a specimen subjected to a ramp-like load, for which the time to

failure and the stress at the time of failure were found to be tf and a(tf),

respectively. We seek the stress, 059 » induced by a constant load with a 60-sec

duration that would have caused failure of that same specimen. It follows from

equation 11 (with Aa^ = it/2) and equation 20 that

^60== —r; ^^^f^ ^21)

[(n+l) 60

tf

Equation 21 shows that the ratio a(^Q/aitf) is independent of B. Thus, even

though the variability of B is fairly large, reflecting as it does uncertainties

with respect to ^jq, n. A, and Y (see equation 5a and table 2.1), the effect of

this variability upon the ratio 0(^Q/aitf) can be ignored.

Table 3.2 lists results of calculations which show that for ratios tf/60 of the

order of 0.5 to 2 the effect of the parameter n upon the ratio o^^q/ o(tf)

is small^. The calculations were based on two values: n=16 and n=19.7

[23]. These values are consistent with results of tests conducted within the

framework of this project on indented soda-lime specimens, in accordance with

the method described in reference 24. The tests yielded the value n=15.95

when measurement results were not corrected for residual stress effects at the

indented crack tips, and the value n=20.6 when the residual stress effects

were accounted for. The magnitude of the residual stresses at the crack tips

of a glass panel, and therefore the effective value of n that should be used

in determining fatigue effects for the panel, is unknown. However, it is

reasonable to assume that the latter will lie between the values n=15.95 or so

^ The choice of a proper loading rate can in practice ensure that the ratios t^/60

are indeed of that order - see tables 3.3 and 3.4.

9
and n=20.6 or so. Hence the choice of the values n=16 and n=19.7 for the

sensitivity calculations of table 3.2.

Table 3.2 Ratios a^^Q/oitf) for tf = 30 sec and tf = 120 sec,

Assuming n = 19.7 and n = 16.00

n = 19.7 n = 16 Difference

tf = 120 sec 0.888 0.875 1.5%

tf = 30 sec 0.828 0.802 3%

Equation 21 and table 3.2 (which show, respectively, that the ratio o^^q/ oitf)

Is Independent of B, and that for any given stress, a(tf), obtained by testing

a rlng-on-rlng specimen, the corresponding 60-sec strength, 059, depends weakly

upon n) suggest that estimates of the probability distribution P(p5q), of

the 60-sec load capacity of a panel, P6o» Inferred on the basis of results

of rlng-on-rlng tests are also Independent of B and weakly dependent upon n. This was confirmed by estimates of PCp^q) carried out for a 4 ft x 4 ft x

1/8 in. annealed glass panel in which several assumed sets of values of n and

1/B were used to convert strengths at time of failure, a(tf), into initial

strengths,

3.3 RESULTS OF RING-ON-RING TESTS

This section presents results of ring-on-ring tests conducted within the

framework of this project. The ring-on-ring testing device (figure 4) consisted

of rigid rings with radii a = 0.0603 m (support ring) and b = 0.0254 m (loading

ring). The device was employed in combination with a lOK ibf Universal Testing

Machine. All the specimens were subjected to ramp loads (i.e., loads Increasing

for practical purposes linearly with time). The glass used in the tests was new

and was obtained from the same manufacturer and batch.

Tests were conducted in air on a set of 56 annealed float glass square specimens

with side D = 7 in. (0.1792 m) and on a set of 29 annealed float glass circular

specimens with radius R = 3.5 in (0.889 m) . The nominal thickness was h = 1/4

in. (6 mm) for all specimens. The stresses at the center of the plates were

calculated by equation 19, in which h was the measured (rather than the nominal)

thickness for each specimen. Following reference 19, in the case of the square

plates the parameter R in equation 19 was assumed to be equal to one-half the

average of the edge and diagonal lengths. This was confirmed experimentally

to within a few percentage points by strain-gage measurements of stresses on circular and square rlng-on-rlng specimens. The test results and the calculated

values of the stresses at the center of the plates at the time of failure, 0(tf),

are listed in tables 3.3 and 3.4. It is noted that the influence of humidity

upon the results of tables 3.3 and 3.4 is negligible (see reference 14, p.

35).

10
If it is assumed that n = 19.7 and 1/B = 0.0738569(MPa)~2s~l (corresponding to

j» n <A0 = 1.08 (MPa)"" 2 s"^ [23], Kj^, = 0.75 MPa [23], and Y = 1.12 - see equation 5a), it follows from equation 20 that the initial strengths are

= 0.7272 t^l^''-'^ [a(tf)]l^-^/l^-^ (22)

Statistical analyses of the initial strengths calculated by equation 22 from

the stresses a(tf) were carried out as indicated in appendix I. It was assumed that the area A referred to in appendix I is a circle with radius

r ^ 0.705b = 0.705 in. (0.0179 m) , i.e., A = TTr2 1.56 in. 2 (0.001 m2)

(see figure 3).

Estimated parameters of the Weibull distributions of the initial strengths, Sj^,

from results of tests used in_conj unction with equation 22 are listed in table

3.5, where the sample means, S-j^, standard deviations, s(Sj^), coefficients of

variation, s(S£)/Sj^, sample maximum, Sj_jjjg2j, and sample minimum, Sj^^j^j^,

of the data are also shown.

The initial strengths Sj^ calculated by equation 22 are nominal, rather than

actual, since the values of B and n used therein are uncertain. However, it

follows from the form of equations 11 and 20 and from the results of table 3.2

that, if the values of B and n used in these equations are the same, the effect

of uncertainties with respect to the actual values of B and n largely cancels

out when estimating the load capacity of glass panels.

11
Table 3.3 Test Data for Circular Plates

h tf p r RH% a(tf)

1 0.5334 45 2358.5 0.76 67 47.32

2 0.5639 54 3453.2 0.76 67 61.99

3 0.5690 59 3419.9 1.27 71 62.06

4 0.5613 48 2839.1 1 .78 71 51.44

5 0.5537 73 4534.6 1.27 71 84.44

6 0.5613 43.5 2109.3 2.03 66 38.22

7 0.5080 36 1637 .6 1.27 66 36.23

8 0.5080 41.5 2073.7 2.29 66 45.87

9 0.5334 31 1713.3 2.03 60 34.38

10 0,5486 82 4694.8 2.29 60 89.05

11 0.5436 57 2536.5 0.00 67 49.00

12 0.5512 59 3924.9 1 .27 67 73.75

13 0.5334 47 3021 .6 1 .02 67 60.63

14 0.5359 70 3742.5 1 .52 67 74.39

15 0.5537 45 2460.9 1.27 72 45.82

16 0.5588 52 2963.7 1.27 72 54.18

17 0.5563 45 2336.3 2.29 72 43.10

18 0.5563 73 3898.2 2.03 70 71.91

19 0.5537 51 2269.5 1.52 70 42.26

20 0.5537 50 2216.1 2.29 70 41.26

21 0.5461 73 3804.8 1.52 70 72.83

22 0.5486 54 2594.4 2.03 70 49.21

23 0.5359 71 3675.7 0.51 70 73.07

24 0.5385 76 5255.5 2.54 60 103.46

25 0.5410 76 3960.5 2.54 72 77.25

26 0.5537 62 3956.1 2.79 71 73.66

27 0.5461 74 4000.6 2.79 67 76.58

28 0.5639 49 2523.2 3.30 66 45.30

29 0.5334 37 2055.9 3.30 67 41.25

where h = thickness in cm

tf = load time in seconds

p = failure load in Newtons

r = distance from origin of fracture to center in cm

RH = relative humidity

oXtf) = final strength in MPa

12
Table 3.4 Test Data for Square Plates

h f- £ r

1 0.5385 2647 .

D / 9 9Q 7 /< en no

2 0.5359 2287 .3 54 n 7 fx IL 'J .03

3 0.5563 2429.7 48 0U • 7f» / D 1

AT n7

4 0.5537 2621 .

61 2 .03 \j\> AAHO Qfl . 7U

5 0.5461 2242.8 49 1 7Q DO 1 . ZO

6 0.5461 391 1 .6 73 79 7 1 OS

7 0.5461 3773 .6 81 79 AQ07 mHLA 9 8 0.5436 2883.6 74 1 97 79

9 0.5461 4427.8 95 0.76 66 9<\ AS

10 0.5461 3017 .

73 0.51 66

11 0.5410 2451 .9 67 1 7Q A9DZ ASJ • QA70

12 0.5334 4561 .3 100 7n R7 Q S 13 0.5334 4525 .

1 nn 1 70 / u ft7 o 9(i / . ZD

14 0.5334 3163.9 80 1 97 70 O i • uu

15 0.5436 2309.6 63 1 .02 A7 8R

16 0.5359 2919.2 75 2 .29 62 SS 76

17 0.5537 2558.8 67 1.27 66 AS 7Q

18 0.5461 5157 .6 97 1 .27 66 94 .87

19 0.5486 5736.

106 1.52 66 1 OA SA 20 0.5436 5006.3 101 2.29 62 92.94

21 0.5410 3840.4 73 1.52 62 71 .98

22 0.5410 4583.5 93 2.29 62 85.91

23 0.5349 3119.5 73 1.52 62 59.59

24 0.5334 4401 .1 93 2 .03 68 84 8S

25 0.5512 3408.7 84 2.03 68 61 .55 26 0.5410 5402 .3 104 2 . 29 62 1 n 1 96

27 0.5512 4458.9 93 0.76 62 80.51

28 0.5359 4579.1 103 1 . 27 62 87 .47 29 0.5588 4570.

92 1.02 62 80.29

30 0.5512 1984.7 51 1.79 66 35.84

31 0.5461 4423.3 98 2.03 66 81.37

32 0.5436 5816.2 117 2.54 66 101 .98

33 0.5385 2558.8 56 2.54 62 48.41

34 0.5512 3399.8 82 2.54 70 61.39

35 0.5410 4783.8 103 2.54 62 89.66

36 0.5512 2656.7 73 2.54 68 47 .97 37 0.5461 5388.9 102 2.54 68 99.13

38 0.5537 5647 .107 2.54 68 101 .05 39 0. 5461 5660.4 111 2.54 68 104. 12 40 0.5436 4147 .4 89 2.54 66 76.99

41 0.5563 5024 .

101 2.54 62 89.06

42 0.5385 4104.2 101 2.54 62 96.56

43 0.5334 2233.9 65 2.54 62 43.07

44 0.5385 5526.9 110 2.54 68 104.56

45 0.5410 2710.1 55 2.79 66 50.80

46 0.5410 3088 .3 63 2.79 66 57.89

47 0.5359 3964.9 94 2.79 62 74.74

48 0.5385 3333.1 84 2.79 70 63.06

49 0,5461 5611.5 114 2.79 70 103.22

50 0.5563 5117.5 100 2.79 62 90.72

51 0.5359 4699.2 87 3.05 66 89.76

52 0.5358 4294.3 97 3.05 62 82.03

53 0.5486 3288.6 80 3.05 62 59.94

54 0.5334 2621.1 74 3.30 68 50.54

55 0.5486 2843.6 74 3.56 68 51.83

56 0.5334 4325.4 97 3.56 62 83.40

where h = thickness in cm

tf = load time in seconds

p = failure load in Newtons

r = distance from origin of fracture to center in

RH = relative humidity

a(tf) = final strength in MPa

13
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14
4. ESTIMATES OF THE LOAD CAPACITY OF GLASS PANELS

The methodology for estimating the 60-sec load capacity of glass panels

proposed in this work consists of using the procedure described in section 2.3

in conjunction with equations 17 and 18. The parameters SqCA]^), m, and Pg in

these equations are obtained by fitting a Weibull distribution to the nominal

initial strengths Sj^ calculated from the breaking strengths a(tf) by using

equation 20 (or a similar relation if the dependence of load on time is not

linear). The values of the parameters B and n used in equation 20 must be

the same as those of equation 11. Values of n used in the calculations should

be based on experimental results obtained, e.g., by techniques described in

reference 24.

This methodology was applied to a 4 ft x 4 ft x 1/8 in (1.22 m x 1.22 m x 3 ram)

annealed glass panel simply supported on four sides, using the values E = 68.9 GPa, V = 0.22, the fracture mechanics parameters listed in section 3.3,

and the parameters of the Weibull distribution of the initial glass strength

listed in table 3.5. The grid size and the angle Aa£ used in the numerical

calculations were 7.62 mm x 7.62 mm and 18°, respectively. For each of the

sets of two and three parameters listed in table 3.5 the load capacities, P60>

of 1000 of panels were estimated, and the values so obtained were fitted by

two- and three-parameter Weibull distributions, respectively. The parameters

of the best fitting distributions are listed in table 4.1, which also lists

mean values, P60 > standard deviations, s(p5o), coefficients of variation,

s(p5o)/p60» lo^ds corresponding to a probability of failure of 8 in 1000,

P50 (0*008), and loads corresponding to a probability of failure of 0.5.

Estimates of the load capacity corresponding to a probability of failure of 8 in

1,000, p5o(0.008), based on full-scale measurements are provided for design

purposes in references 1 and 3. According to reference 1, p5o(0.008) = 26 psf

(1 psf = 47.9 Pa). According to reference 3, P5o(0.008) - 23 psf. It is

noted that to account for strength degradation in service, the value of reference 3 corresponds to strengths reduced by a factor of 2/3 with respect to

those obtained in new glass [26]. Had this reduction not been effected, i.e.,

had the strength of new glass been used, to the value P5o (0.008) - 23 psf

there would have corresponded roughly the value P5q( 0.008) = (3/2) x 23 = 34.5 psf.

The values p^gCO.OOS) estimated in this report on the basis of three-parameter

Weibull distributions (col. 9 of table 4.1, Case I) are somewhat higher than

the corresponding value based on references 1 and 3 (35.5 psf to 42.1 psf versus

26 psf to 34.5 psf.) These differences could be explained by two factors.

First, the estimates obtained in this report do not take into account edge

failures, which can reduce the load capacity of panels considerably. (For

example, according to reference 7, 38 percent of the total number of panel

failures reported therein originated at the edges.) Second, sampling errors

in the estimation of the Weibull parameters of the strength may be present,

since the estimated values of p5o(0.008) are based on the testing of a limited

number of ring-on-ring test specimens (from 29 for case 1 to 85 for case 3).

Therefore, as shown in appendix II, the true values of p5o(0.008) could well

be as low as about 20 psf and as high as about 50 psf. Note also that the

15
estimated coefficients of variation of the load capacity p^Q obtained by

using three-parameter Weibull distributions (0.09 to 0,105) are lower than the

value indicated in reference 3 (- 0.22). On the other hand, as can be seen from table 4.2 which summarizes test results reported in reference 7, coefficients of variation of the load capacity of new glass panels obtained from any

one manufacturer are close in at least four out of eight cases to those estimated

herein.

It is concluded that the procedure proposed in this report provides credible

estimates of the load capacity of panels experiencing surface failures provided

that three-parameter Weibull distributional models are used. The precision of

these estimates depends upon number of ring-on-ring test results on which they

are based, as suggested by numerical experiments presented in appendix II.

A second conclusion of interest is that modeling the strength of glass and the

load capacity of cladding panels by two-parameter Weibull distributions leads

to a clear incompatibility between results obtained by testing ring-on-ring

specimens on the one hand, and the documented behavior of cladding panels on

the other. Indeed, the use of two-parameter distributions yields estimates of

P^q(0.008) of the order of 1 psf (see table 4.1). These are grossly incompatible

with the values quoted previously from references 1 and 3. A similar conclusion

was reached independently by Walker [27].

16
Table 4.1 Estimated Statistics and Weibull Distribution Parameters of the

1-min Load Capacity, p^q > ^o^^ a 4 ft x 4 ft x 1/8 in Annealed

Glass Panel Supported on Four Sides (Based on Tests in Air)

Case

P60

(psf)

s(p6o)

(psf)

s(P60)/P60 (P60)o

(psf)

1^60 (psf)

TM&0 P60(0.008)

(psf)

P60(0-5)

(psf)

(1) (2) (3) (4) (5) (6) (7) (8) (9)

1 42.8 3.86 0.090 9.80 34.14 2.41 35.5 42.6

2 50.9 5.00 0.098 11.04 41.08 2.05 42.1 50.3

3 45.4 4.78 0.105 12.20 34.57 2.42 36.2 45.1

1 7.3 3.43 0.471 8.23 0 2.26 0.98 7.0

lib 2 9.5 4.87 0.515 10.71 0 2.06 1.03 9.0

3 6.9 3.58 0.516 7.86 0 2.06 0.75 6.6

^ Based on three-parameter Weibull distributions of load capacity p^Q and

three-parameter Weibull distributions of the strength of glass.

Based on two-parameter Weibull distributions of load capacity p^Q and

two-parameter Weibull distributions of the strength of glass.

P50 ~ sample mean

s(P60^ ~ sample standard deviation

(P50)o ~ scale parameter

y^Q = location parameter

mgQ = shape parameter

P50(0«008) = 1-min load capacity corresponding to probability of failure of

panel of 8 in 1,000

P5q(0.5) = 1-min load capacity corresponding to probability of failure of 0.5.

17
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5. SUMMARY AND CONCLUSIONS

A methodology was proposed for estimating the probability distribution of the

load capacity of annealed glass panels whose failure is due to surface flaws.

The methodology requires the calculation of stresses induced on the panel

surface by the external loads and employs information on the modulus of elasticity, Poisson's ratio, the fracture mechanics parameters, and the probabilistic description of the glass strength. This description is obtained by using

the ring-on-ring testing method. It is shown that owing to the way in which

this method is used, errors in the estimation of the load capacity due to

uncertainties with respect to the fracture mechanics parameters of glass and

to the shape of surface flaws are largely cancelled. Results of ring-on-ring

tests and of calculations based on those tests show that credible predictions

of panel load capacities are obtained that are consistent with data available

in the literature, provided that the probability distribution of the glass

strength and the probability distribution of the panel load capacity are modeled

by the three-parameter Weibull distribution. The two-parameter Weibull distribution appears to provide an incorrect model of the strength of glass and of

the load capacity of glass panels.

The results obtained in this work strongly suggest that the proposed methodology

can provide reliable estimates of the load capacity of glass panels whose

failure is due to surface flaws. However, a definitive statement to this effect

would require validation based on (1) larger ring-on-ring test samples than

those used in this work, and (2) reliable statistics of the load capacity of

panels manufactured from the same batch of glass as the ring-on-ring test

specimens

The topic of glass panel failures due to edge imperfections was not addressed

in this report. As shown by the statistics in table 4.2, the ratio of such

failures to the total number of failures is highly variable. It is suggested

that the probability distribution of the load capacity of glass panels, regardless of type of failure, can be modeled from information on statistics of edge

failures on the one hand and of surface failures on the other.

In view of their relatively low cost, ring-on-ring tests could be more economical

than full-size panel tests, in spite of the relatively large number of specimens

that would have to be tested in order to attain acceptable precisions of the

estimates. The extent to which this is the case would have to be determined

by studies based on more extensive test data than have been obtained within

the framework of this project.

Ring-on-ring testing may be a desirable alternative to full-size panel tests

not only for economical reasons, but also in situations where the amount of

material available for testing is limited. This might be the case in studies

of in-service strength degradation in which the material being tested consists

of weathered window glass recovered from existing buildings.

19
REFERENCES

1. L.O.F Technical Information - Strength of Glass Under Wind Loads , 805-109,

Toledo, Ohio, January 1980.

2. Reed, D.A. and Simiu, E. , Wind Loading and Strength of Cladding Glass , BSS 154, National Bureau of Standards, May 1983.

3 . PPG Glass Thickness Recommendations to Meet Architects Specified 1-Minute

Wind Load , Technical Services/Flat Glass Division, PPG Industries, April

23, 1979.

4. Orr, L., Engineering Properties of Glass , Publication 478, Building

Research Institute, National Academy of Sciences, National Research Council,

Washington, D.C. , 1957.

5. Bowles, R. and Sugarraan, B. , "The Strength and Deflection Characteristics

of Large Rectangular Glass Panels Under Uniform Pressure," Glass Technology , Vol. 3, No. 5 pp. 156-170, 1962.

6. Hershey, R.L. and Higgins , T.H. , "Statistical Prediction Model for Glass

Breakage from Nominal Sonic Boom Loads," Booz-Allen Applied Research, Inc.,

Bethesda, Maryland (NTIS Accession No. AD-763-594), January 1973.

7. Ontario Research Foundation, Dynamic Fatigue of Flat Glass - Phase II , Final Report (67039), Ontario Research Foundation, Mississagua, Canada,

July 1982.

8. Hertzberg, R.W. , Deformation and Fracture Mechanics of Engineering Materials , John Wiley and Sons, New York, NY, 1976.

9. Lawn, B.R. and Wilshaw, T.R. , Fracture of Brittle Solids , Cambridge

University Press, Cambridge, England, 1970.

10. Newman, J.C. , Jr. and Riju, T.S., "Analyses of Surface Cracks in Finite

Plates Under Tension or Bending Loads," NASA Technical Paper 1578,

December 1979.

11. Fuller, E., Wiederhorn, S.M., Ritter, J.E., and Oates, P.B., "Proof

Testing of Ceramics: Part II: Theory," J. Materials Science , Vol. 15,

1980, pp. 2275-2281.

12. Ritter, J.E., Jr. et al., "Proof Testing of Ceramics: Part I: Experiment,"

J. Materials Science , 15, 1980, pp. 2282-2295.

13. Wiederhorn, S.M. , "Dependence of Lifetime Predictions of the Form of the

Crack Propagation Equation," Fracture 1979 , Vol. 3, ICF4, Waterloo, Canada,

June 19-24, 1979, pp. 893-901.

14. Beason, W.L. and Morgan, J.R, "Glass Failure Prediction Model," Journal

of Structural Engineering Vol. 110, No. 2, February, 1984, pp. 197-212

20
15. Vallabhan, C.V.G. and Wang, B.Y.-T., "Nonlinear Analysis of Rectangular

Glass Plates by Finite Difference Method," Institute for Disaster Research,

Texas Tech University, Lubbock, Texas, June 1981.

16. Weibull, W. , "A Statistical Theory of the Strength of Materials," Royal

Swedish Academy of Engineering, Proceedings No. 151, Stockholm, 1939.

17. Marshall, D.B., "An Improved Biaxial Flexure Test for Ceramics," Ceramic

Bulletin , Vol. 59, No. 5, 1980, pp. 551-553.

18. Roark, R.J., Formulas for Stress and Strain , 4 ed. , Ch. 10, McGraw-Hill,

New York, 1965.

19. Ritter, J.E., Jr. et al., "Appraisal of Biaxial Strength Testing,"

Proceedings of the XII International Commission of Glass, Alburquerque , NM,

June 1980.

20 . Standard Test Method for Structural Performance of Exterior Windows,

Curtain Walls, and Doors by Uniform Static Air Pressure Difference , ASTM

E 330-79, 1982 Annual Book of ASTM Standards, Part 18, ASTM, 1916 Race

Street, Philadelphia, PA, 19103, pp. 1037-1044.

21. Vitman, F.F. and Pukh, V.P., "A Method for Determining the Strength of

Sheet Glass," Industrial Laboratory , Vol. 29, 1963, pp. 863-867.

22. Kao, R. , Capps , W. , and Perrone, N. , A General Numerical Large Deflection

Solution of the Coaxial Ring Circular Glass Plate Flexure Problem , The

Catholic University of America, October 1970.

23. Weiderhorn, S.M., "Subcritical Crack Growth in Ceramics," In Fracture

Mechanics of Ceramics , Vol. 2, Bradt, R.C., Hasselman, D.P.H., and

Lange, F.F., eds.. Plenum Publishing Corp, New York, 1974, pp. 613-646.

24. Lawn, B.R. et al., "Fatigue Analysis of Brittle Materials Using Indentation

Flaws," J. of Materials Sicence , Vol. 16, 1981, pp. 2846-2854.

25. Batdorf, S. B. , "Fundamentals of the Statistical Theory of Fracture" in

Fracture Mechanics of Ceramics, Vol. 3, R. C. Bradt et al., eds.. Plenum

Press, N.Y., 1981.

26. Krall, W. R. , Siskos, W. R. , Spindler, R. G., and Stewart, R. A., The

Behavior of Float Glass Under Uniform Wind Loading, Proceedings , The Fourth

U.S. National Conference on Wind Engineering Research, B. J. Hartz, ed..

Department of Civil Engineering University of Washington, Seattle, Washington,

1981.

27. Walker, G. , (James Cook University of North Queensland, Australia), Personal

Communication, Nov. 1983

21
Figure 1. Notations

Figure 3. Tangential and Radial Stress Distribution (a

b/a = 1/2;b = radius of loading ring)

radius of support ring;


APPENDIX I. MAXIMUM LIKELIHOOD ESTIMATION FOR THE THREE PARAMETER WEI BULL

DISTRIBUTION BASED ON A SAMPLE WITH CENSORING DUE TO COMPETING

RISKS

Let X be a random variable representing the tensile strength of glass cladding

specimens within an area subjected to uniform axial bending. It is assumed

that the distribution of X is Weibull that is,

where a, 3>0, X>0, x>X , and a, 3, and X are the three parameters of the

distribution.

We assume that a number of specimens, say n, are tested and each of the specimens

has the same Weibull distribution of tensile strengths X. However, what we observe at each trial is either the tensile strength at which the specimen

fails or the tensile strength outside the area A. For a gien specimen, let Q

denote the strength at which failure occurs outside the area A. Then what we observe is one of the two events [Al-1 , Al-2]

[X=x]n[X<Q] or [Q=q]0[X>Q].

Then the likelihood, L, of the event

k n

F(x) = l-exp[-(

3

n [Xi < Qi] [Xi = Xi] n [Xj>Qj] [Qj=qj]

1=1 j=k+l

(Al.l)

where k < n is the random number of failures observed, has the form

k n

L = C n f(xi) n [l-F(qj)]

i=l j=k+l

(A1.2)

where f is the density function, F is the distribution function, and

k

n!

(n-k) !k!

The logarithm of the likelihood function for equation A1.2 is

k n in L = in [C n f(xi) E (l-F(qj))]

i=l j=k+l
and can be rewritten as

k

Jin L = An C + kilna - kciAng + (a-1) I iln(xi-X)

i=l

k x.-X ^ n q.-X ^

- 2 (^—) - S (-^)

i*l ^ j=k+l e

On differentiating equation A1.3 with respect to a, 3, and X in turn and equating

the resulting expressions to zero, one obtains the maximura likelihood estimation

equations for the three Weibull parameters [Al-3]. These three equations are nonlinear in the three parameters and can be solved iteratively by a NewtonRaphson method as discussed in [Al-1] and [Al-3].

References

Al-1 Barter, H. L., Moore, A. H. , "Maximum-Likelihood Estimation of the Parameters

of Gamma and Weibull Populations From Complete and From Censored Samples,"

Technometrics 7(4) 639-643 (1965).

Al-3 Lemon, G. H. "Maximum Likelihood Estimation for the Three Parameters

Weibull Distribution Based on Censored Samples," Technometrics 17(2)

247-254 (1975).

Al-2 Whittaker, I. C. Besunner, P. M. , A Reliability Analysis Approach to

Fatigue Life Variability of Aircraft Structures , Air Force Materials

Laboratory Tech. Report AFML-TR-69-85 (1969).
APPENDIX II. DEPENDENCE OF ESTIMATED 60- SEC LOAD CAPACITY, P50 (0-008), UPON NUMBER

OF RING-ON-RING SPEQMENS BEING TESTED

Estimates of the order of magnitude of the error in the estimation of the load

capacity p5o(0.008) were carried out as follows. It was assumed that the

probability distribution of the strength can be modeled by a three-parameter

Weibull distribution with scale, location, and shape parameters (corresponding

to an area A = 1.56 in'^) S^ = 118.1 MPa, \i^=U7 ,7 MPa, and m = 1.27 (see table

3.5, case 3). It is further assumed that the probability distribution of the

load capacity p^g is also Weibull with parameters to be determined.

The first step in obtaining the desired estimates consists of generating from

this distribution by Monte Carlo simulation a sample of N values of the glass

strength S. These N values can be viewed as strengths that would be measured

if N ring-on-ring specimens were tested.

The second step is to fit a three-parameter Weibull distribution to these N values.

The third step is to estimate from this distribution the corresponding 60-sec load

capacity p5o(0.008) as shown in section 2.3.

These three steps were carried out 20 times for each of the sample sizes N = 15, 100, 250, 500, and 1,000. The results of the calculations are summarized

in table A2.1, which shows for each value N the means of 20 estimates p5o( 0.008),

their standard deviation s[p5o(0.008) ] , and their minimum and maximum values,

min [p5o(0.008)] and max [(p5o(0.008)] . Table A2.1 suggests the order of

magnitude of the sample size needed to obtains estimates of P50 (0.008) with

various precisions. Note that these estimates are tentative, since (1) they do

not account for the effect of censoring discussed in appendix Al , and (2) they assume that the Weibull distribution parameters of the glass strength obtained

from the testing of only 89 specimens are the "true" parameters.
Table A2.1 Estimated Means, Standard Deviations, Minimum Values, and Maximum

Values Obtained from Sets of Estimated Values p5o(0.008). Corresponding

to Various Ring-on-Ring Test Sample Sizes n.

15 100 250 500 1000

P50(0.008)

s[p6o(0.008)

min [p5o(0.008)]

max [p5o(0.008)]

33.10

lA.lO

2.53

58.24

34.36

8.20

17.54

48.26

35.50

5.16

26.35

47.75

34.98

4.23

27.84

43.45

34.69

2.01

29.53

37.54
APPENDIX III. COMPUTER PROGRAM FOR ESTIMATING PROBABILITY DISTRIBUTION

OF LOAD CAPACITY p^Q
C Computer program PSIXTY is a modified and corrected version

C of the program listed in "Wind Loading and Strength of

C Cladding Glass" by D.A. Reed and E. Simiu (NBS BSS 154).

C It estimates the probability of failure of square glass

C panes subjected to a constant sixty second pressure. The

C initial strength values for each element on each pane are

C simulated from Weibull distribution input parameters. The

C stress at the center of each element is obtained as a nonC linear function of pressure by using the Texas Tech

C University program referred to in Section 2.3 of this

C report. The stress-pressure relationship is summarized

C in subroutine LF of this program.

C

C To obtain failure statistics, numerical experiments are

C conducted. A user-defined number of panes (=N in main program)

C are loaded, and a distribution of the breaking pressure is

C determined. The breaking pressure is different for each pane,

C since each pane has a random Weibull strength distribution.

C The pane is divided into square elemental

C areas, wherein each element has a random Weibull distribution

C strength assigned for each of the 20 directions. Symmetry is

C taken advantage of in this program.

C

C This program was developed for panes with size 48"xM8"x1/8".

C However, with small modifications, it can be used for panes

C of any specified size.

C

C The basic procedure is:

C

C FOR EACH OF N PANES ... C GENERATE THE RANDOM STRENGTHS FOR EACH DIRECTION IN EACH

C OF THE DISCRETE ELEMENTS OF THE PANE.

C

C FOR EACH LOCATION, FIND THE 60-SECOND PRESSURE THAT WILL

C CAUSE THE STRESS=STRENGTH.

C

C FIND THE MINIMUM 60-SECOND PRESSURE FOR THE PANE.

C

C FIT THE N MINIMUM 60-SECOND PRESSURES TO A WEIBULL DISTRIBUTION.

C

C INPUT: N=Number of panes to use.

C SSHAPE,SSCALE,SLOC= Input Weibull strength distribution

C parameters.

C SEEDsRandom seed for random # generator.

C

C OUTPUT: 60-second breaking pressure (psf ) parameters of the

C fitted Weibull distribution.

C 60-second breaking pressure statistics (mean,S. D. ,C. 0. V.

)

C

C Subroutines called:

C

C Internal: F

C WEIB

C PSIXTY

C FDMIN

C DIR
C LF

C

C External: GGUBS - an IMSL routine to generate uniform random #'3

C SRTAD - an IMSL routine to sort inplaoe the elements

C of a vector.

C

REAL MEAN

DOUBLE PRECISION SEED, SEEDS

COMMON /P60PAR/ PSHAPE, PSCALE, PLOC

READ (5,100) N

READ (5,110) SSHAPE,SSCALE,SLOC,AREA

READ (5,120) SEED

100 FORMAT (110)

110 FORMAT (4F10.0)

120 FORMAT (D20.0)

SEEDS = SEED

WRITE (6,130) SSHAPE, SSCALE,SLOCy AREA, N, SEEDS

130 FORMAT (//,

146H COMPUTER PROGRAM PSIXT Y,//,

2 37H GLASS STRENGTH WEIBULL DISTRIBUTION:,/.

3 21H SHAPE PARAMETER =, F8.3,/,

3 21H SCALE PARAMETER =, F7.2,/,

4 21H LOCATION PARAMETER =, F7.2,//,

5 21H TEST SPECIMEN AREA =, F8.3,//,

6 21H NUMBER OF PANELS =, 15,//,

7 21H RANDOM NUMBER SEED =, D20. 10)

CALL PSIXTY

+ ( N , AREA , SSHAPE , SSCALE , SLOC , PSHAPE , PSCALE , PLOC , MEAN , STDDEV , COV

,

+ SEED)

F8 = F(0.008)

FM = F(0.5)

WRITE (6,140) PSHAPE, PSCALE, PLOC, MEAN, STDDEV, COV, F8.FM

140 FORMAT (//,

1 19H PSIXTY PARAMETERS:,/,

2 21H SHAPE PARAMETER =, F10.3,/,

3 21H SCALE PARAMETER =, F 9.2,/.

4 21H LOCATION PARAMETER =, F 9.2,//,

5 21H MEAN =, F 9.2,/,

6 21H STANDARD DEVIATION =, F10.3,/,

7 21H COV =, FIO.3,//,

8 21H F(0.008) =, F 9.2,/.

9 21H F(0.5) (MEDIAN) =, F 9.2)

END

FUNCTION F(Z)

COMMON /P60PAR/ PSHAPE, PSCALE, PLOC
F = PLOC + PSCALE«(-ALOG(1.0 - Z) )••( 1. 0/PSHAPE)

RETURN

END

SUBROUTINE WEIB(N,M,MR,SSHP,SSCL,SL0C,E1 ,E2,E3,T,S1 ,S2,S3)

C PROGRAM WEIBULL—COMPUTES THREE PARAMETER WEIBULL

C MAXIMUM LIKELIHOOD ESTIMATES FOR BOTH LEFT AND

C RIGHT SINGLE CENSORING

C DATE: JUNE 31,1981

C DATE PROGRAM CHECKED OUT: JULY 9,1981

C PROGRAMMER: JONATHAN W. MARTIN

C REFERENCE: HARTER,H. L. ;MOORE, A. H. MAXIMUM LIKELIHOOD

C ESTIMATION OF THE PARAMETERS OF GAMMA AND WEIBULL

C POPULATIONS FROM COMPLETE AND CENSORED SAMPLES.

C TECHNOMETRICS 7: 639-643; 1965. ERRATA, 9: 195; 1967.

C NOTATION FOR INPUT DATA:

C N = SAMPLE SIZE (BEFORE CENSORING), N=1000 OR LESS

C AS DIMENSIONED

C SSCL= 0 IF THE SCALE PARAMETER THETA IS KNOWN

C SSCL = 1 IF THE SCALE PARAMETER THETA IS UNKNOWN

C SSHP = 0 IF THE SHAPE PARAMETER K IS KNOWN

C SSHP = 1 IF THE SHAPE PARAMETER K IS UNKNOWN

C SLOC = 0 IF THE LOCATION PARAMETER C IS KNOWN

C SLOC = 1 IF THE LOCATION PARAMETER C IS UNKNOWN

C T(I) = I-TH ORDER STATISTIC OF SAMPLE (I=1,N)

C (SUBSTITUTE BLANK CARDS FOR UNKNOWN CENSORED

C OBSERVATIONS)

C M = NUMBER OF OBSERVATIONS REMAINING AFTER

C CENSORING N-M FROM ABOVE

C C(1)=INITIAL ESTIMATE (OR KNOWN VALUE) OF

C THETA(I) = INITIAL ESTIMATE (OR KNOWN VALUE) OF THETA

C EK(1) = INITIAL ESTIMATE (OR KNOWN VALUE ) OF K

C MR = NUMBER OF OBSERVATIONS CENSORED FROM BELOW,

C NORMALLY ZERO INITIALLY

C NOTATION FOR OUTPUT DATA:

C N, SSCL, SSHP, SLOC, M,C(1),THETA(1),EK(1)—SAME AS FOR INPUT

C C(J) = ESIMATE FOR LOCATION PARAMETER AFTER J-1

C ITERATIONS (OR KNOWN VALUE ) OF

C THETA(J) = ESTIMATE OF SCALE PARAMETER AFTER J-1

C ITERATIONS (OR KNOWN VALUE) OF THETA

C EK(J) = ESTIMATE OF SHAPE PARAMETER AFTER J-1

C ITERATIONS (OR KNOWN VALUE) OF K

C (MAXIMUM VALUE OF J AS PRESENTLY DIMENSIONED

C IS 550)

C EL = NATURAL LOGARITHM OF LIKELIHOOD FOR C(J) ,THETA(J)

C EK(J)

DIMENSION THETA(5550),EK(5550)

DOUBLE PRECISION C ( 5550) ,T( 1000) ,SLK

DIMENSION X(600),Y(600)

REAL E1,E2,E3
C INTEGER PSCL(2),PSHP(2),PL0C(2)

C DATA PSCL/' YES',' NO '/,

C 1 PSHP/' YES',' NO '/,

C 2 PLOC/' YES',' NO '/

C WRITE(6,«)N,M,MR,SSHP,SSCL,SL0C

C WRITE(6,«)E1,E2,E3

C 4 READ(5,1)N,M,MR,SSHP,SSCL,SL0C

C IF(N) 66,66,77

C 77 READ(5,5) EK( 1 ) ,THETA( 1 ) ,C( 1

)

EK(1)=E1

THETA(1)=E2

C(1)=E3

C 5 FORMAT (3F5.0)

C READ(5,121)(DATAS(J),J=1,18)

ISCL = SSCL + 1 ISHP = SSHP + 1 ILOC = SLOC + 1 EN = N

C DO 2 1=1,

IF (M) 64,64,32

32 EM =M

31 ELNM = 0.

JCOUNT = 0

EMR = MR

MRP = MR + 1 NM = N - M + 1 DO 34 I=NM,N

EI = I

34 ELNM = ELNM + ALOG(EI)

IF(MR) 66,35,74

74 DO 75 1=1, MR

EI = I

75 ELNM = ELNM - ALOG(EI)

35 DO 30 J=1,550

JCOUNT = JCOUNT + 1 IF(J-I) 66.25,37

37 JJ = J-1

SK = 0.

SL = 0.

DO 6 I = MRP,M

6 SK = SK + (T(I)-C(JJ))«*EK(JJ)

IF(SSCL) 7,7,8

7 THETA(J) = THETA(JJ)

GO TO 9

8 IF(MR) 66,19,20

19 THETA(J) = ((SK +(EN-EM)»(T(M)-C(JJ))»«EK(JJ))/EM)

1»»(1./EK(JJ))

GO TO 9

20 X(1) = THETA(JJ)

LS = 0

DO 21 L=1,55

LL = L - 1 LP = L + 1 X(LP) = X(L)

ZRK = ((T(MRP)-C(JJ))/X(L))««EK(JJ)
Y(L) = -EK(JJ)»(EM-EMR)/X(L) + EK(JJ)»SK/X(L)»«(EK( JJ) + 1+EK(JJ)«(EN-EM)«(T(M)-C(JJ))««EK(JJ)/X(L)»»(EK(JJ)+1.

)

2-EMR«EK ( JJ

) »ZRK»EXP ( -ZRK ) / ( X ( L) • ( 1

. -EXP ( -ZRK ) )

)

IF(Y(L)) 53,73,54

53 LS = LS -1

IF(LS + L) 58,55,58

54 LS = LS + 1 IF(LS - L) 58,56,58

55 X(LP) = .5»X(L)

GO TO 61

56 X(LP) = 1.5»X(L)

GO TO 61

58 IF(Y(L)»Y(LL)) 60,73,59

59 LL = LL-1

GO TO 58

60 X(LP) = X(L) + Y(L)«(X(L)-X(LL))/(Y(LL)-Y(L))

61 IF(ABS(X(LP)-X(L))-1.E-4) 73.73,21

21 CONTINUE

73 THETA(J) = X(LP)

9 EK(J) = EK(JJ)

IF(SSHP) 12,12,11

11 DO 17 I = MRP,M

17 SL = SL + DLOG(T(I)-C(JJ))

X(1)=EK(J)

LS=0

DO 51 L=1,55

SLK = 0.

DO 18 I=MRP,M

18 SLKsSLK +(DLOG(T(I)-C(JJ))-ALOG(THETA(J)))»(T(I)-C(JJ))

1»«X(L)

LL = L-1

LP = L+1

X(LP)=X(L)

ZRK=((T(MRP)-C(JJ))/THETA(J))»«X(L)

Y ( L ) = ( EM-EMR ) « ( 1 . /X ( L

) -ALOG ( THETA ( J ) ) ) +SL-SLK/THETA ( J

)

1«»X(L)+(EN-EM)»(AL0G(THETA(J))-DL0G(T(M)-C(JJ)))

2»(T(M)-C(JJ))»»X(L)/THETA(J)««X(L)

3+EMR«ZRK» ( ALOG ( ZRK ) /X ( L )

) •EXP ( -ZRK ) / ( 1

. -EXP ( -ZRK )

)

IF(Y(L)) 43,52,44

43 LS = LS -1

IF(LS+L) 47,45,47

44 LS = LS + 1 IF(LS-L) 47,46,47

45 X(LP) = .5»X(L)

GO TO 50

46 X(LP) = 1.5«X(L)

GO TO 50

47 IF(Y(L)«Y(LL)) 49,52,48

48 LL = LL - 1 GO TO 47

49 X(LP) = X(L)+Y(L)«(X(L)-X(LL))/(Y(LL)-Y(L))

50 IF(ABS(X(LP)-X(L))-1.E-4) 52.52,51

51 CONTINUE

52 EK(J) = X(LP)

12 C(J) = C(JJ)

IF(SLOC) 25,25,14
14 IF(1.-EK(J)) 16,78,78

78 IF(SSCL+SSHP) 57,57,16

16 X(1) = C(J)

LS = 0

DO 23 L=1,55

SKI = 0.

SR = 0.

DO 15 I=MRP,M

SKI = SKI +(T(I)-X(L))»«(EK(J)-1. ) 15 SR = SR+ 1./(T(I)-X(L))

LL = L-1

LP = L + 1 X(LP) = X(L)

ZRK = ((T(MRP) -X(L))/THETA(J))»«EK(J)

Y(L) = (1.-EK(J))«SR+EK(J)»(SKU(EN-EM)«(T(M)-X(L))

1»«(EK(J)-1. ))/THETA(J)««EK(J)-EMR»EK(J)»ZRK»

2EXP(-ZRK)/((T(MRP)-X(L))«(1.-EXP(-ZRK)))

IF(Y(L)) 39,24,40

39 LS = LS-1

IF(LS+L) 70,41,70

40 LS = LS+1

IF(LS-L) 70,42,70

41 X(LP) = .5»X(L)

GO TO 22

42 X(LP) = .5»X(L)+.5»T(1)

GO TO 22

70 IF(Y(L)«Y(LL)) 72,24,71

71 LL = LL - 1 GO TO 70

72 X(LP) = X(L)+Y(L)»(X(L)-X(LL))/(Y(LL)-Y(L))

22 IF(ABS(X(LP)-X(L))-l.E-4) 24.24,23

23 CONTINUE

24 C(J) = X(LP)

GO TO 25

57 C(J) = T(1)

25 IF(MR) 66.38,69

38 DO 63 I = 1,M

IF(C(J)+1.E-4-T(I)) 68,67,67

67 MR = MR + 1 WRITE(6,201)

201 FORMAT (» ',» LINE 200 IN PROGRAM—INITIAL LOCATION PARAMETER',

1 'ESTIMATE GREATER'/ 'THAN FIRST OBSERVED FAILURE TIME')

63 C(1) = T(1)

68 IF(MR) 66,69,31

69 SK = 0.

SL = 0.

DO 36 I=MRP,M

SK = SK+(T(I)-C(J))»«EK(J)

36 SL = SL+DLOG(T(I)-C(J))

ZRK=((T(MRP)-C(J))/THETA(J))»«EK(J)

EL=ELNM+ ( EM-EMR ) • ( ALOG ( EK ( J )

) -EK ( J

) »ALOG ( THETA ( J ) ) )

1(EK(J)-1.)»SL-(SK+(EN-EM)«(T(M)-C(J))«»EK(J))/

2 ( THETA ( J

) »»EK ( J ) ) +EMR*ALOG ( 1

. -EXP ( -ZRK )

)

IF(J-3) 30.27,27

27 IF(ABS(C(J)-C(JJ))-1.E-4) 28.28,30
28 IF(ABS(THETA(J)-THETA(JJ))-1.E-4) 29.29,30

29 IF(ABS(EK(J)-EK(JJ))-1.E-4) 99.99,30

30 CONTINUE

C OUTPUT

99 CONTINUE

S1=THETA(JC0UNT)

S2=EK(JC0UNT)

S3=C(JC0UNT)

GO TO 66

64 CONTINUE . 66 RETURN

END

SUBROUTINE PSIXTY ( NPANEL , AREA , C 0 , SO , SU , MP , PEST , SV

1 .XMEAN,ZSAM,COV,DSEED)

C

C (SEE DATA STATEMENT) v C SIDEA = LENGTH OF THE SQUARE

C THE LENGTH OF THE SIDE OF THE SQUARE IS INPUT IN INCHES

C TH = THICKNESS OF THE PLATE

C THICKNESS INPUT IN INCHES

C MODULUS OF ELASTICITY (E) INPUT IN PSI

C

C INPUT:

C NPANELS = # PANELS TO 'BREAK* TO FIT WEIBULL DISTR. TO

C SO,CO,SU = WEIBULL STRENGTH PARAMETERS FOR AN ELEMENTAL AREA = C

C OUTPUT:

C XMEAN = MEAN P60

C ZSAM = S.D. OF P60

C COV = C.O.V. OF P60

C PEST = SCALE PARAMETER OF WEIBULL MLE FIT OF P60 FOR NPANELS

C MP = GAMMA " n n C SV = LOCATION PARAMETER " " C

C PROCEDURE:

C FOR EACH OF NPANEL PANES...

C GENERATE 216 R.V. STRENGTHS (FROM UNDERLYING WEIBULL

C DISTR. ) 36 LOCATIONS W/ EACH 6 DIRECTI0NS=216

C

C FOR EACH LOCATION, IF SIGMA=STRENGTH, FIND CORRESPONDING

C LOAD(P 60) WHICH GIVES THAT STRESS ( STRENGTH ) , I.E. THAT

C LOAD WHICH WILL BREAK IT FOR THAT LOCATIONS STRENGTH.

C

C FIND THE MINIMUM LOAD (P60) FOR THIS PANEL.

C

C FIT THE MINIMUM LOADS (P 60) FOR THE NPANEL PANES TO A WEIBULL

C DISTRIBUTION.

C

C NOTE THAT SYMMETRY IS USED IN THIS PROCEDURE, I.E. THE PANE IS

C DIVIDED INTO FORU QUADRANTS (BY SYMMETRY) AND THIS QUADRANT IS

C DIVIDED BY ITS AXIS OF SYMMETRY (THE DIAGONAL) IN TWO.

C

C THE NUMBERING IN SUBROUTINE DIR AND LF USES THE UPPER HALF OF

C LOWER LEFT QUADRANT. THE CENTROID OF THE PANE IS NUMBERED
C NODE #1. THE 24" SQUARE QUADRANT OF THE MS" SQUARE PANE IS

C DIVIDED INTO 8X8 3 INCH SQUARES. NODE # 1 IS THE CENTROID OF

C THE ENTIRE PANE AND IS THE CENTROID OF ELEMENT #1. NODE # 2 IS

C 3 INCHES LEFT OF NODE #1 AND IS THE CENTROID OF ELEMENT #2.

C ETC. TILL NODE # 8 IS 21 INCHES LEFT OF NODE #1 AND IS THE

C CENTROID OF ELEMENT #8. AGAIN, USING SYMMETRY, NODE #9 IS

C DIRECTLY UNDER N0DE#2 AND IS THE CENTROID OF ELEMENT #9.

C NODE #10 IS 3 INCHES LEFT OF NODE #9 ETC UNTIL NODE # 15

C WHICH IS 18 INCHES LEFT OF NODE #9. THEN NODE # 16 IS DIRECTLY

C UNDER NODE #10. ETC. THEN NODE #22 IS DIRECTLY UNDER NODE* 17.

C ETC. UNTIL THE LOWEST ROW WHICH HAS ONLY NODE #36.

C NOTE THAT EACH ELEMENT IS A THREE INCH SQUARE (THIS IS

C NECESSARY SINCE THE R.V. STRENGTH ASSUMES AN EQUAL ELEMENTAL

C AREA. THUS, ELEMENT #1 HAS NO COUNTERPART IN THE OTHER THREE

C QUADRANTS. LIKEWISE ELEMENT'S #2-8 EACH REPRESENT FOUR OTHER

C ELEMENTS IN THE PANE (ALSO WITH THE ELEMENTS ON THE DIAGONAL)

C THE REMAINING ELEMENTS ARE REPLICATES OF 8 OTHER ELEMENTS IN

C THE PANE.

C

DOUBLE PRECISION SIG(36 ,6) ,SI (36 ,6) ,PEQ( 36 ,6) , A1 (216)

,

•RSIG ( 36 , 6 )

, S2 , BP , M, RN , RY , RNM, KIC , ROOT , COEFS , COEFP

,

•SCALE , SMIN , SMAX , RF , DSEED , E . MIN ( 1 000

)

REAL F(216),ZSAM,C0V,XMEAN,MP

INTEGER IPOINT,NUPPER

DIMENSION ICHEK(216,2)

EQUIVALENCE (SI(36,6) ,PEQ(36,6)

)

COMMON //IPLACE(216,2),IC0UNT

C

DATA KIC , A , RN , RY/ 0. 75D0 , 1 . 08 , 1 9. 6 9D0 , 1 . 25D0/

DATA TH,SIDEA,PR/0. 125,48.0,0.21/

DATA E/1D07/

DATA RF,IPOINT/1.0D0,1/

C

C CALCULATIONS

C

M=1/C0

RNM=RN-2.

ROOTS 1./RN

C

C 1/B IS A PARAMETER USED IN EQN. 19 OF THE BSS REPORT

C RY=1.25; A=1.08; N=19.69 FROM [5] OF BSS 154

C SEE ABOVE DATA STATEMENT

C

BP=(RNM«A«RY»RY»(KIC«»RNM) )/2.

C

C FLEX=FLEXURAL RIGIDITY

C

FLEX=( E»TH»TH«TH )/( 12. »( 1. -PR«PR ))

C

S2=SIDEA«SIDEA

C

C CF. EQN (9) OF BSS 154

C THE AREA FOR WHICH WE ARE SIMULATING A STRENGTH IS THE

C ELEMENT SIZE, WHICH AS EXPLAINED ABOVE IS 1/16 OF SIDE

C THUS, AREA= (SIDEA/16)»»2

C HOWEVER, TAKING DIRECTION INTO ACCOUNT AS EXPLAINED BELOW,
C THIS AREA IS DIVIDED BY 10.0

C

SELEM=(SIDEA/16. )«»2

SNEW=SO» (AREA/(SELEM/10. ))*«M

C0EFS=(S2«TH)/FLEX

COEFP= ( S2»S2 ) / (FLEX«TH

)

C

C THESE COEFFICIENTS ARE USED FOR NONDIMNSIONALIZING

C PRESSURES AND STRESSES.

C

SCALES ( 1 . / ( 60. »BP )

) «*ROOT

R00T=RNM«R00T

C

C OBTAIN THE DIRECTIONAL MULTIPLIERS FROM [9] OF BSS 154

C

CALL DIR(RSIG)

C

C THEN CALCULATE THE STRESSES IN TERMS OF THE STRENGTHS...

C

C IF THE STRESSES ARE LESS THAN OR EQUAL TO

C ZERO, THEN OMIT THESE FROM CONSIDERATION...

C

C LOGICAL UNIT 9 CONTAINS A FILE WITH PSEUDO-RANDOM

C NUMBERS WHICH FOLLOW A UNIFORM DISTRIBUTION ON

C (0,1). AN ALTERNATIVE METHOD WOULD BE TO GENERATE THE NUMBERS

C DIRECTLY FROM A SUBROUTINE.

C

C THEN GENERATE INITIAL STRENGTH VALUES

C

DO 926KK=1,NPANEL

1=0

DO 12J=1,36

IF(J.EQ. 1) THEN

Nsl

ELSEIF(J.EQ.9 .OR. J.EQ. 16 .OR. J.EQ.22 .OR. J.EQ.27 .OR. + J.EQ. 31 .OR. J.EQ. 34 .OR. J. EQ. 36 .OR. ( J. GE. 2. AND. J. LE. 8) )THEN

N=4

ELSE

N=8

ENDIF

C

C SINCE THE STRENGTH IS RANDOMLY SIMULATED W/IN EACH ELEMENT

C WE CAN ARBITRARILY ORIENT THE COORD SYSTEM, S.T. THE MAXIMUM

C PRINCIPAL STRESS OCCURS § ALPHA=0

C DIVIDING THE CIRCLE INTO 20 = 18 DEGREE ARCS, THE FIRST ARC

C IS REPEATED TWICE, @ ARC#1 AND 180 DEGREES AROUND THE CIRCLE.

C SIMILARLY, ARC#6, CORRESPONDING TO THE MINOR PRINCIPAL STRESS

C ALSO OCCURS TWICE. THE STRESSES IN THE ARCS BETWEEN ARC # 1 AND

C ARC # 6 OCCUR FOUR TIMES (BY SYMMETRY). THUS STRESS @ ARC#1

C OCCURS 1/10; STRESS e ARC#6 OVVURS 1/10; REMAINING ARCS

C STRESS OCCURS 1/5 EACH; 1/10+1/10+4( 1/5)=1.

C USING THE ANALOGY OF EQN (11) IN BSS 154

C THE AREA CORRESPONDING TO SO IS 3 SQUARE INCHES DIVIDED

C BY 10. ALSO, THE STRESSES CORRESPONDING TO ARC#2-5,

C ARE SIMULATED TWICE, BECAUSE THEY OCCUR TWICE AS OFTEN AS

C THOSE CORRESPONDING TO THE MAJOR AND MINOR PRINCIPAL STRESSES
C THUS....

C

DO 12 K=1,6

IF (K.NE. 1 .AND. K.NE.6) N=N«2

C

C GGUBS is an IMSL routine to generate N random uniform #'s

C

CALL GGUBS (DSEED,N,F)

CALL FDMIN(F,N,FMIN)

IF ( FMIN .EQ. 0 ) THEN

FMIN=FMIN+0. 00001

ENDIF

C SI= STRENGTH (UNITS OF MPA)

C SIG=STRESS FACTOR (SF) (UNITS OF PSI) = F(SIGMA 60)

C SIG FROM EQN. (13) & (19) 0.00689 PSI PER MPA

C

C FOLLOWING LINE CHANGED FROM FMIN TO (1.-FMIN)

C ORIG. WAS OK SINCE ONLY ONE RANDOM #0-1 & A RANDOM UNIFORM #

C BETWEEN 0-1 IS EQUIVALENT TO RANDOM 1- RANDOM #

C C

SI(J,K)=SNEW»((-AL0G((1.-FMIN)))»»M) +SU

SIG(J,K)=SCALE»( SI(J,K)»»R00T ) SIG(J,K)= C0EFS«( SIG(J,K)/( . 00689»RSIG( J.K) )

)

IF( SIG(J,K) .GT. 0)THEN

1=1+1

C J==> POSITION

C K==> DIRECTION

ICHEK(I,1)=J

ICHEK(I,2)=K

ENDIF

12 CONTINUE

ICOUNT=I

C

C SORT THE REMAINING STRESSES....

C

DO 160 I=1,IC0UNT

J=ICHEK(I,1)

K=ICHEK(I,2)

A1(I)=SIG(J,K)

160 CONTINUE

C

IC=ICOUNT

Cx

C SRTAD is an IMSL routine to sort inplace the IC elements of vector

C A1 min to max.

C

CALL SRTAD(A1,1,IC)

SMIN=A1(1)

SMAX=A1(IC0UNT)

SMAX=RF»SMAX

C

C OMIT FROM CONSIDERATION THE FOLLOWING STRESS VALUES...

C

GO TO (10,10,30)IPOINT

C

C FOLLOWING LOOP IS UNNECESSARY IF RF=0.0!
10 12=0

DO 161 I=1,IC0UNT

J=ICHEK(I,1)

K=ICHEK(I,2)

IF( SIG(J,K) .LE. SMAX )THEN

12=12+1

IPLACE(I2,1)=J

IPLACE(I2,2)=K

ENDIF

161 CONTINUE

C

IC0UNT=I2

C

GO TO 15 - C

30 CONTINUE

DO 162I=1,IC0UNT

J=ICHEK(I,1)

K=ICHEK(I,2)

IF( SIG(J.K) .EQ. SMIN )THEN

IPLACE(1,1)=J

IPLACE(1,2)=K

ENDIF

162 CONTINUE

C

C CALCULATE LOAD FACTOR, LF ... PEQ=LF=F(SF).

C

15 CALL LF(SIG,IPOINT,PEQ)

C

GO TO (11,11,31)IP0INT

C

11 JK=0

DO 17 IV=1,IC0UNT

J=IPLACE(IV,1)

K=IPLACE(IV,2)

JK=JK+1

IF( PEQ(J,K) .LE. 0 )THEN

PEQ(J,K)=1.0D20

ENDIF

C A1=LF/C0EFP = PRESSURE (UNITS OF ?)

A1(JK)=PEQ(J,K)/C0EFP

17 CONTINUE

C

C SORT VALUES ....

C

CALL SRTAD ( A 1

, 1, ICOUNT

)

MIN(KK)=144.»A1(1)

GO TO 926

C

C IF ONLY THE MINIMUM STRESS IS CHECKED,

C CALCULATE ONLY ONE VALUE OF PEQ

C

31 J=IPLACE(1,1)

K=IPLACE(1,2)

MIN(KK)=144.»PEQ(J,K)/C0EFP

C
926 CONTINUE

C

C UNITS CHANGED TO PSF FOR CALCULATIONS

C MIN=P60(PSF)MIN OF PANEL'S P60

C

CALL SRTAD (MIN , 1

, NPANEL

)

C

SUM1=0.0

SUM2=0.

DO 30031=1, NPANEL

3003 SUM1=SUM1+MIN(I)

XMEAN=SUM1 /NPANEL

DO 30041=1, NPANEL

3004 SUM2=SUM2+( MIN (I) - XMEAN )«*2

ZSAM=SQRT(SUM2/( NPANEL - 1 ))

COV=ZSAM/XMEAN

C

N=NPANEL

C SSHP,SSCL,SL0C=1 ===>SHAPE, SCALE, LOCATION PARAMETER UNKNOWN

C ElsINITIAL ESTIMATE OF SHAPE PARAMETER

C E2= " n n SCALE " C E3= " " " LOCATION " C

SSHP=1.

SSCL=1.

SL0C=1.

E1=2.3

E2=:10.

E3=MIN(1) - 2.0

IF(E3 .LE, 0.0) E3=0.001

IF( SU .LE. 0.1) THEN

E3 = 0.001 @ SLOC = 0.0

ENDIF

311 MR=0

C WRITE (6, 305)

305 FORMAT ( IX, 3H305)

C

CALL WEIB ( N , N , MR , SSHP , SSCL , SLOC , E 1 , E2 , E3 . MIN . PEST , MP , SV

)

C

C

RETURN

END

SUBROUTINE FDMIN(F,N,FMIN)

DIMENSION F(N)

C

FMIN=1.E+10

DO 10 1=1,

FMIN=AMIN 1 ( FMIN , F ( I )

)

10 CONTINUE

RETURN

END

C

SUBROUTINE DIR(RSIG)

DOUBLE PRECISION PH,RSIG(36 ,6) ,C(6) ,SII(6) ,S2S1 (36)

DATA S2S1/1.0D0,.98D0,.94D0,.89D0,.79D0,.68D0..52D0,. 12D0,.97D0.

».93D0,.86D0,.77D0,.65D0,.47D0,.05D0,.89D0,.82D0,.71D0..57D0,.36D0,
09D0, 0. 73D0, 0. 61BO,iO.

». 21D0,-. 24D0,, 47D0,. 28D0, , O2D0,-. 38D0, » 83D0,-o 17D0,-« 52D0,

«-0.39D0,-0.66D0,-.8lD0/

DO 5K=1,6

PH=(K-1)«3. 14159/10.

C (K ) =DCOS ( PH ) »DCOS ( PH

)

5 SII(K)=DSIN(PH)«DSIN(PH)

DO 10J=1,36

DO 10K=1,6

RSIG(J,K)=C(K) + S2S1(J)»SII(K)

10 CONTINUE

RETURN

END \ ^ C

SUBROUTINE LF(SIG, IPOINT, PEQ)

DOUBLE PRECISION PEQ(36 ,6 ) ,SIG(36 ,6 ) ,LIF(25) ,L(36 ,25)

COMMON // IPLACE(2l6,2)sIC0UNT

C

DATA ((L(J,I),I=1,24),Jsl,6)/0,6,8,13.7,20.6,27.5,34.0,40.5,

«46. 6, 53. 0,58. 3, 63. 0,104. 0,129. 0,149, 167, 182, 195. 207, 218,

•228,310, 359, 407, 481, 0,6. 7, 13. 5, 20. 4, 27. 0,33.7, 40.4, 46.1,

•52,57.7,63,103,130,151,170,186,199,211,222,233,319,364,410,

•484.0,6.5,13,19.7,26.3,32.8,39,45,51,56.7,62,104.134,157,

•177,194,209,223.235,246,333.384.430,497,0,6,12.3,18.5,24.9.

•30.9,37.2,43,49,54.2,59,103,136,163,186,206,224,240,255,

•268.370,432,484,550,0,5.5,11.2,16.8,22.6,28.3,33.8,39,45,50,

•55,90,135,165,192,215.237,257,275,290,420,509,581,661,0,

•4.7,9.5,14.3,19.2,24.1,28.9,34,39,43,48,90,125.157,186,

•212,234,257,278,296,457,580,687,792/

C

DATA ((L(J,I),Is1,24),J=7,11)/0,3.6,7.2,10.8,14.5,l8.2,21.7,

•25.5,29.3,33,37,71,102,130,156,180,203,224.244,263,426,

•563,689,811,0,2.0,4,6.1,8.3,10,12,14,17,19,21,40,59,78,

•94,109,125,139,153,166,279.377,471,563,0,6.7,13.6,20.4,

•27. 1,33.7,40.0,45.9,52.2,57.5,62. 1,102.6,129.2,149.7,167.7,

•183,196.4,209.2,220.6,232.4,319.8,375.9,428.4,504.2,0,6.6,

•13.3,20,26.6,33,39.2,45. 1,51.2,56.6,61.3,102.8,131.9,154.8,

•174.9,192.3,207.5,221.3,233.7,245.7,337. 1,397.8,454.0,530.8,

•0,6.3,12.7,19,25.4,31.3,37.4,42.9,49,54.2,59.1,102.6,134.7,

•161.5,184.7,205. 1,223.3,239.8,254.6,268.3,373.5,441.5,

•500. 1,574.2/

C

DATA ((L(J,I),I=1,24),J=12,15)/0,5.8.11.6,17.3,23.2,28.6,

•34.4,39.6,44.9,50.4,54.9,98.9,134.2,164.5,191.5,215.4,

•237.3,257.3,275.8,291.8,424.9,516.9,593.1,672.8,0,5,

•10,15,20,24.7,29.6,34.4,39,43.6,47.9,89.4,125.1,156.9,

•186,212.5,236.9,259.8,281. 1,300.3,463. 1,587.7,697,800.7,

•0.3.9,7.9,10,15.5,19.2,23,26.6,30. 1,33.8.37.4,71.2,101.9,

•130.5,157.1,182,205.1,227.2,247.9,267.3,434.8,573.2,699.7,

•820.9,0,2.5,4.9,7.3,9.4,11.8,13.9,16,18,2,20.2,22.2,41.9,

•60.8,78.6,95.7,111.8,126.9,141.6,155.6,169. 1,285.7,385.8,

•479.8,571.5/

C

DATA ((L(J,I),I=1,24),J=16,20)/0,6.6.13.3,19.9,26.4,32.7,38.9,

•44.4,50.4,55.7,60.3,100.7,128.7,151.0,170.4,187,201.9,

•216. 1,230.2,242.8,349.3,423.7,490.2,572.4.0,6.5,12,8.19.3,
•25.6,31.6,37.7,^3. 1,48.7,53.9,58.5,99.7,130.8,157. 1,180.5,

»201. 1,219.9,236.9,252.8,266.6,381.3,463.6,537.5,621.6,0

•,6.1,12.1,18.1,23.9,29.5,35.2,40.2,45.6,50.5,55,96.8,131.2,

•161.7,189.4,214.4,237.3,258.3,278.2,295,435.7,534.2,620,

•705.3,0,5.5,10.8,16. 1,21.2,26.1,31.2,35.7,40.2,44.8,48.9,

•88.6,123.6,156. 1,186.3,214.4,240.5,264.7,287.8,308.5,

•481.8,612. 1,724.5,828.9,0,4.5,8.9,13.2,17.3,21.3,25. 1,

•28.9,32.7,36.2,39.5,72.1,102.4,131.5,159.3,186. 1,210.8,

•234. 7 , 257. 5 , 278. 9 , 46 0. 4 , 6 05. 2 , 735. 4 , 856. 9/

DATA ( (L( J, I), 1=1, 24), J=21, 24)70,3.3, 6. 2, 9. 3, 12. 1,14.7,17.2,

•19.5,21.9,24. 1,26. 1,45.6,63.7,81.5,99,116.2,132.2,148.2,

•163.5,178.5,307.3,413.9,510.9,603.4,0,5.7,6.5,12.9,19.2,

•25.3,31.2,37,42.4,47.8,52.7,57.3,97.3,127.3,153.0,175.0,

•195.213,229.244,391.7,493.4,583.5,676.3,0,5.6,6.3,12.4,

•18.4,24.2,29.8,35.3,40.4,45.6,50.4,54.7,94.6,126.9,156,

•184,209,233,255,276,444.6,556.9,655.6,750.8,0,5.3,5.8,

•11.4,16.9,22.2,27.3,32.3,36.9,41.6,45.9,50.2,88.1,121.6,

•154.184,214,241,268,294,507.1,648.2,769.1,876.5/

DATA ( (L( J, I), 1=1, 24), J=25, 28)70,4.7,5, 9. 9, 14. 6, 19. 1,23.4,

•27.6,31.5,35.2,39.1,42.4,74.6,103.9,133,161,190,217,

•244,269,501.7,662.8,802.8,929.4,0,3.6,3.9,7.7,11.3,14.7,

•17.8,20.9,23.6,26.4,28.9,31.3,52,70.2,88. 1,106,124,141,

•159,176,344.9,468.2,576.2,674.6,0,5.7,6.2,12.2,18.2,23.9,

•29.3,34.7,39.7,44.6,49.3,53.6,92.4,124.1,180,205.3,228.2,

•250. 1,270.8,288.7,436.2,569,687.7,798.2,0,5.6,5.9,11.8,

•17.3,22.7,27.9,32.9,37.6,42.3,46.6,50.9,88. 1,120.6,181,

•210.3,237.9,265.2,291.7,316.4,525.4,680.3,811.5,926.77

DATA ( (L( J, I), 1=1, 24), J=29, 32)70, 5. 1,5.4,10.7,15.8,20.5,25. 1,

•29.6,33.8,38.1,41.8,45.5,78.5,82.3,165,193.9,221.8,249.7,

•277.7,305.1,548.9,739.9,900.6,1041.0,0,4.2,4.5,8.9,12.9,

•17,20.9,24.4,27.8,31.2,34.2,37,61.8,76.8,119.9,138.6,

•156.9,175.5,194.2,213.2,395.9,553.2,688.4,806.5,0,5.4,

•11.6,17.3,22.6,27.8,32.7,37.2,41.9,46.1,50,87,120,151,

•172.9,210.2,237.9,265.6,292.3,318.1,538.5,700.9,827,923,0,

•5.4,11.2,16.5,21.5,26.4,31.2,35.5,39.9,43.9,47.9,83,114.

•144,172.8,201.7,230,258.286.4,314.7,582.5,809.3,998.4,

•1157.67

DATA ((L(J.I),I=1,24),J=33,36)70,4.7,9.9, 14.7, 19.2,23.6,27.8,

•31.8,35.7,39.2,40,73,99,122,143.9,165.4,186.3,206.9,227.3,

•248.2,456.5,658.9,843,1001.9,0,5.4,11.2,16.6,21.7,26.6,

•31.4,36,40.6,44.8,48.8,85.5,118.148.5,178.3,207.8,236.9,

•265.9,295.323.7,607.3,865.2,1085.7,1264.5,0,5. 1,10.6,

•15.8,20.7,25.6,30. 1,34.6,39.2,43.3,47.3,83.5,114.5,143.3,

•170.6,196.7,222.6,247.7,272.6,296.9,538.2,780.4,1015.6,

•1229.8,0. ,5. 1,10.8,16. 1,21.4,26.4,31.3,36,40.8,45.3,49.7,

•89.6,124.4,156.7,187.9,217.2,246.4,274.7,302.4,328.9,

•587.5,846.4,1113.9,1389.17

DATA (LIF(I), 1=1, 25)70, 24. 8, 49.7, 74. 5, 99, 124, 149, 174, 199,

•223,249,497,745,993,1242,1490,1738,1987,2235,2484,4967,

•7450,9934, 12417, 1000007
C APPROXIMATIONS FOR THE LARGEST VALUES OF SIGMA

C ARE TAKEN FROM THE FOLLOWING REPORT : C "PROPOSED METHOD FOR DETERMINING THE THICKNESS

C OF GLASS IN SOLAR COLLECTOR PANELS," BY

C DONALD MOORE, JET PROPULSION LABORATORY,

C CALIFORNIA INSTITUTE OF TECHNOLOGY, PASADENA,

C CALIFORNIA, MARCH, 1980.

C

DO 6000J=1,29

6000 L(J,25)=1500

DO 7000J=30,36

7000 L(J,25)=5000

C

C CALCULATE THE LOAD FACTOR FOR EACH STRESS

C

DO 1000 II=1,IC0UNT

IF( IPOINT .EQ. 3 )G0 TO 300

J=IPLACE(II,1)

K=IPLACE(II,2)

SSrSIG(J,K)

GO TO 500

300 JrIPLACE(1,1)

K=IPLACE(1,2)

SS=SIG(J,K)

500 IF(SS .GT. L(J,25)) THEN

PEQ(J,K)=LIF(25)

GOTO 1000

ENDIF

IF(SS.EQ.O)THEN

PEQ(J,K)=0

GOTO 1000

ENDIF

IF(SS.LT. L(J,2)) THEN

PEQ(J.K)=SS«(LIF(2)-LIF(1))/L(J,2)

GOTO 1000

ENDIF

DO 2001=2,25

IF(SS .GT. L(J.I-I) .AND. SS .LT. L(J,I)) THEN

R=(SS-L(J,I-1))«(LIF(I)-LIF(I-1))/(L(J,I)-L(J,I-1))

1 + LIF(I-I)

PEQ(J,K)=R

GOTO 1000

ENDIF

IF(SS .EQ. L(J,I)) THEN

PEQ(J,K)=LIF(I)

GOTO 1000

ENDIF

200 CONTINUE

IF( IPOINT .EQ. 3 )G0 TO 400

1000 CONTINUE

C

400 RETURN

END
APPENDIX IV, COMPUTER PROGRAM FOR ESTIMATING PARAMETERS OF GLASS

STRENGTH DISTRIBUTION
*BATCH

c

C PROGRAM CDMPUTES THE MAXIMUM LIKELIHOOD ESTIMATES

C FOR THE TWO OR THREE PARAMTER WE I BULL AND

C FOR THE TWO PARAMETER LOGNORMAL FOR PROGRESSIVE CENSORED

C SAMPLES

C

C OUTPUT INCLUDES PARAMETRIC ESTIMATES, NUMBER OF SAMPLES

C AND NUMBER OF FAILED SAMPLES, FOR THE PURPOSES OF THIS

C PROJECT, ONLY THE ESTIMATES OF THE WE I BULL DISTRIBUTION

C ARE USED.

C

C SUBROUTINE HAMLET IS USED TO OBTAIN TWO PARAMETERS

C WHILE THIRD PARAMETER IS OPTIMIZED WITHIN THE MAIN

C PROGRAM USING AN INCREMENTAL OPTIMIZING SEARCH (i.e.

C FIRST SEARCH IN TEN PERCENT INTERVALS AND THEN TAKE

C MOST PROMISING TWENTY PERCENT REGION AND SEARCH IN

C TWO PERCENT INTERVALS, ETC.)

C OPTIMUM OCCURS WHEN LOGARITHM OF LIKELIHOOD FUNCTION

C IS A MAXIMUM

C

C INPUT DATA:

C

C N-- NUMBER OF SAMPLES -

' C NF-- NUMBER OF FAILURES

C I PROG- FLAG WHICH INDICATES IF PROGRESSIVE

C CENSORING IS IN EFFECT ( 1-- YES; OTHER-NO)

C ICR- NUMBER OF PARAMETERS USED IN FIT

C (O-TWO5 < >0-THREE)

C T(I)-FAILURE STRENGTHS OF SAMPLES AS CALCULATED FOR THE

C CENTRAL PORTION OF THE RING

C G( I )--CENSORED VALUES OF STRENGTH (READ IN FROM SUBROUTINE

C IDC--FLAG USED TO DETERMINE STARTING POINT FOR

C OPTIMIZING THIRD PARAMETER (AS % OF 1ST

C STRENGTH VALUE)

C 4-85>:; OTHER-99/::

C LOGICAL UNIT ASSIGNMENTS:

C

C 5-DATAFILE ( INPUT)

C 9 CONSOLE

C 6- DATAPI LE (OUTPUT)

C /'-CONSOLE (USED BY SUBROUTINE)

C

c***+>*-* *****************

COMMON / PARAM / ALPHA , BE T A , X BAR 1 , S , N6 , X NF

D I MENS I ON T ( 500 ) , TLOG ( 500

REAL 6 ( 1 50 ) , T 1 (1 50 ) , G 1 (1 50 ) , LSTAR (10) , G2 ( 1 50 )

, JR

DATA T/500*0. 0/ , TL0G/500*0. 0/

GFLAG-0

READ (5,5) N,NF •

• - 5 FORMAT (213)

READ (5,5) I PROG, ICR

AN N

DO 500 I = 1 ,NF

READ (5,1) T ( I

1 FORMAT (F 15. 0)

' 500 CONTINUE

CALL HAMLET ( N , NF , T , I PROG , G , GFLAG

IF(ICR.EQ.O) GOTO 223

C

C BEGIN OPTIMIZING SEARCH FOR THIRD PARAMETER
WRITE (9, 15)

1 5 FORMAT (

' I NF'UT I DC ( 3- 857. , 4--95-/. ) ? ' ) READ (9,*) IDC

GFLAG=2

DO 64 IV=1 ,NF

64 Tl ( IV) =T (IV)

NPG=M-NF

DO 69 JV=1 ,NPG

69 Gl ( JV) =G ( JV)

XLHI--=99, 0

IF <IDC.EQ.3) XLHI-95.

IF (IDC.EQ.4) XLHI=85.

XLLO^O.

XL.STEP=--10.

84 DO 93 JL=1 , 10

93 LSTAR ( JL) =0.

15=0

C

C HERE IS THE OPTIMIZATION LOOP

C

DO 97 I R= 1 1, 1

,

- 1 JR=XLHH-I5*XLSTEP

15=15+1

C= (Tl ( 1 ) *JR/100.

)

IF(C.LT.O.O) C=0.0

C

C REEVALUATE FAILURE TIMES AND CENSORED TIMES

C

DO 87 JQ=1 ,NF

87 T ( JQ) =T1 ( JQ) -C

DO 77 JP1=1 ,NP6

77 G2 ( JPl ) =G1 (JPl ) -C C

C CALL HAMLET AGAIN TO COMPUTE NEW 1ST AND 2ND

C PARAMETERS

C

CALL HAMLET ( N , NF , T , I PROG , G2 , GFLAG

)

A=ALPHA

B=BETA

XNF=NF

SUM 1=0.

SUM2=0.

DO 74 JZ=1 , XNF

SUM 1 =SUM 1 +L06 ( T ( J Z ) ) 74 SUM2=SUM2+ (T (JZ) /BETA) **ALPHA

DO 76 J4=1,NP6

76 SUM2=SUM2+ (G2 ( J4) /BETA ) **ALPHA

C

C COMPUTE LOGARITHM OF LIKELIHOOD FUNCTION

C

154 LSTAR ( 15) =XNF*LOG (A) -XNF*A*L06 (B) + (A-1 . O ) *SUM 1 --SUM2

WR ITE ( 6 , * ) LSTAR (15) , A , B , C , SUM 1

, SUM2 , NPG

IF ( 15. LE. 1 ) GOTO 97

IF ( IDC. EQ. 1 ) GOTO 97

I F ( LSTAR (15). GT . LSTAR ( I 5- 1 ) ) GOTO 97

C

C OPTIMUM HAS BEEN LOCATED; REDEFINE SEARCH INTERVAL

C

XNLO=JR

XNHI = XNL0-i-2. *ABS (XLSTEP)

IF ( XNH I . GT. XLH I ) XNHI=XLHI

IF (XNLO.EQ. XNHI) XNLO=XNH I+XLSTEP

XLLO=XNLO

XLHI=XNHI

XLSTEP= ( XLLO-XLHI ) / 10.
CHECK FOR PRECISION OF .0001 PERCENT

97

C1=ABS (LSTAR ( 15) -LSTAR (15-

C2=ABS ( . 00000 1*LSTAR (15)

IF (C1.LE.C2) SOTO 223

GOTO 84

CONTINUE

OUTPUT RESULTS

1 ) )

223 WRITE (6, 6057) N,NF, BETA, ALPHA, C, XBARl ,S

6057 FORMAT(///// ' NUMBER OF SPECIMENS LOADED', 129//

1' NUMBER OF SPECIMENS FAILED ', 129//

WEIBULL SCALE PARAMETER ', F30. 5//

WEIBULL SHAPE PARAMETER ', F30 . 5/

/

6' LOCATION PARAMETER ' ,F30. 5//

4' LOBNORMAL SCALE PARAMETER ', F27 . 5/

/

5' LOBNORMAL SHAPE PARAMETER ', F27 . 5

WR I TE ( 6 , 6058 ) N , NF , ( T ( I ), I = 1, NF

6058 FORMAT (213,1 000 ( / F 1 0 . O )

STOP

END

SUBROUT I NE HAMLET ( N , NF , F , I PROG , G , GFLAG

COMMON /PARAM/ ALPHA , BETA , XBAR 1

, S , NB , XNF

REAL F ( 150) ,6( 150) , X ( 150)

REAL MUU

VUA =1.

X31 =9.

ZC=0

GIG =1.

IF ( IPROG. EQ. 1 ) GIG=0

NFF = NF

XNF = NF

XN = N

NG = N - NF

NGG=NG

I F ( NG . EQ . 0 ) GO TO 511

IF (GFLAG. NE. 0) GOTO 5110

DO 5 1 30 I = 1 , NG

IF (IPROG. EQ. 1 ) GOTO 5120

G ( I ) = F (NF)

GOTO 5130

READ (5,5121

FORMAT (FIO. 0)

CONTINUE

CONTINUE

5120 G ( I ) 5121

5130

5 1 1

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

c c C ORDER FAILURE AND CENSORED TIME DATA C

C C

cccccccccccccccccccccccccccccccccccccccccccccccccccccc

BAG = -10.

GNG = NG

IF (GNG-.Ol) 1100, 1100, 1109

1109 IF (BIG - .01) 1106, 1106, 1100

1106 NG1=NG-1

DO 1112 1=1, NGl

11=1+1

DO 1115 J=I 1 ,NG

IF (G(I) - G(J)) 1115, 1115, 1118

1118 GMID = G(J)

B(J) = G(I)

G(I) = GMID

1115 CONTINUE

WRITE(7.4887) G(I)

26.

1 02

134.

135.

136.

141

142.

149.

151

152.

153.

154.

1 55

156.

157.

158.

159.

160.

161

162

16

164.

165.

166,

167.

168.

1 A9.
4887 FORMAK'O',' G(I)',F10,3) 170.

1112 CONTINUE 171.

WRITE (7,4887) e(NG)

1100 NFMl =NF - 1 172.

DO 3 1=1 ,NFM1 173.

11=1+1 174.

DO 1 J=I1,NF 175.

IF (F<I)-F(J)) 1,1,2 176.

2 FMID=F(J) 177.

F(J)=F(I) 178.

F(I)=FMID 179.

1 CONTINUE 180.

X < I

) =AL0G10 (F ( I ) ) LOGGING

3 CONTINUE 182.

X(NF) = ALDG10(F (NF) ) 183.

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 1 84

C C 185.

C FIRST TWO ORDERED FAILURE TIMES C 186.

C C 187.

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 188.

FNl = F (1 ) 189.

FN2 = F<2) 190.

FNL = F<NF) 191.

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 200,

C C 201.

C UNBIASED ML POINT ESTIMATES FOR LOG NORMAL C 202.

C POP. PARA. ARE CALCULATED BELOW. C 203.

C C 204.

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 205

XSUM=0. SIGMA

X2SUM=0. XBAR

DO 4 1=1 , NF 208.

XSUM=XSUM+X ( I ) 209.

X2SUM=X2SUM+X ( I ) *X ( I ) 210.

4 CONTINUE 211.

XN=N 212.

XNF=NF 213.

FNF = XNF 214.

FN= N 215.

FN6 = FN - FNF 216.

XBAR=XSUM/XNF 217.

XBARL = XBAR 218.

S2=(X2SUM-XSUM*XBAR) / (XNF-1. ) 219.

S=SQRT(S2) 220.

S=S+. 00001 221.

cccccccccccccccccccccccccccccccccccccccccccccccccccccc

C C 255.

C UNBIASED ESTIMATES OF LOG NORMAL MODEL ARE C 256.

C GIVEN BELOW FOR CENSORED CASE C 257.

C C 258.

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC 259

C THIS SECTION FOR HAMLET CALLED INSERT C 260.

IF (FN - FNF - .01) 9100, 9100, 9103 261.

9107 SIG =.557 /ALPHA

MUU = BETA * .56 ** (1. / ALPHA) 263.

C AAA 264.

C BBB 265.

MUU = ALOGIO(MUU) 266.

C THE BELOW WORK IS COMMON TO HAMLET AND L0G7 268.

8048 IF (GIG - 0.1) 8020, 8020, 8024 269.

C 270.

C LOG NORMAL MLE — SINGLE CENSORING 271.

C 272.

8024 EPS = (ALOGIO (G ( 1 ) ) - MUU) / SIG 273.

ZE = CNORML (EPS, --150., 0.0, 1.0) 274.

FEZE = EXP (-(EPS *EPS/2.)) * 0.3989423 275.
IE = FEZE /

C

C

C

8020

3032

8028

4016

401.2

4008

4024

4020

4C>04

4005

4000

8040

8044 aose

8050

ZEE FMC3 *

( 1 . ZE

ZE)

EASY = ZEE

A

Bl

CI

ZEE * ZEE + EPS *

* EPS

<ZE -EPS)

EPS * A

(ZEE + BI)

BO TO 8028

L06 NORMAL MLE PROGRESSIVE CENSORING

ZEE = O.

EASY - O. AI = 0,

BI = O.

CI = 0,

DO 8032 I CE = 1 , NG

EPS = ( ALOG 1 0 ( 6 ( I CE ) ) - FEZE = EXP (-(EPS *EPS/2.

ZE - CNORML

ZE = FEZE/

ZEE = ZEE + EASY = EASY

AP=ZE * (ZE

ZE)

BP

CP

AI

BI

CI

ZE

EPS

A I + BI + BP

CI -I- CP

(EP

( 1

ZE

+ EPS *

- EPS)

EPS * AP

X- (ZE + BP)

AP

IbO,

MUU ) /

) ) * O.

, 0 . 0

SIG

3989423

1 . 0)

ZE

CONTINUE

TST

TST2

SR2

FL =

/ SIG

( 1

GL

PL

QL

RL

DL

EH= EK=

(XBAR - MUU)

= TST * TST

(S2 / (SIG * SIG)

TST + ZEE / FNF

TST2 + SR2 - 1. + EASY/FNF

AI/FNF) / SIG

TST + BI /FNF) /SIG

1 . /FNF)

( 1 - \ - (3

PL *

(GL * (FL *

RL

QL

QL

(TST2 + SR2)

-QL * QL

- FL * RL) / - GL * PL) /

1 + CI /FNF) / SIG

DL

DL

CONTINUE

IF (ABS(EK) - SIG/:

EK = EK/2.

GO TO 4016

IF (ABS(EH) - 0. EH == EH/ 2.

BO TO 4008

CONTINUE

BAB ~ -6.

TEl = SIG +EK

TE2 = MUU +EH

ZC = ZC + 1. IF (ZC-53,) 4000,

CONT I NUE

WRITE (7,4005)

FORMAT ( 'SEE LINE : GO TO 308

CONTINUE

IF (TEl) 8040, 8040

SIG = SIG/2.

BAB == 6.

GO TO 8058

SIG = TEl

IF(TE2) 8050, 8050,

MUU=MUU/2.

BAB=6.

02) 4008, 4008, 40 1:

) 4020 , 4020 , 4024

40(1)0 , 4004

29 IN HAMLET '

8044

3054

276

277

278

279

280

281

282

283

284

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

317

318

319

320

321

324

325

326

327

328

329

334

335

336

338

339

340

341
8054

8060

8049

8064

C THE

C

Cc

c9404

C

C

C

C

C

9 1 00

GO TO 8060

liUU == TE2

IF (BAB) 8049, 8048, 8048

ERROR = ABS (EH)+ ABS (EK)

AERR = 0.0004

IF (ERROR - AERR) 8064, 8064, 8048

XBAR = 10. ** MUU

S=SIG

ABOVE

XBARL

BNF = ANF =

WORK IS COMMON TO HAMLET AND L0G7

- MUU

XBAR

2. * SIG

IF (BAG) 9111, 9111, 9404

MLE WE I BULL

PARAMETERS

PARAMETERS TOO LARGE—USE DEFAULT

ALPHA = 0.557 / S

BEEL = XBARL + 0.2506

BETA --= 10. BEEL

GO TO 9111

/ ALPHA

133

COMPUTE NEW FANG PARAMETERS

AL = 0.557/S

BEEL=XBAR + 0.2506816/AL

BEE= 10. ** BEEL

XBAR= 10. ** XBARL

WUM=0.

WUMH=0.

DO 133 1=1, NF

WUM=WUM + F(I)

WUMH= WUMH + 1 . /F ( I

)

CONTINUE

R)

(ESS/BNF ~ 1 . )

)

SECTION CALCS SHAPEESS = WUM/XNF

REC = WUMH/XNF

R = l./REC

BNF = SORT (ESS * ANF = SORT (2. » CMLE OF WBL PARAS FOLLOW-FIRST

ALO= AL » VUA

60 TO 12

9103 ALO = F(l) / (F(2) - F(l) +5)

AL = 0.557/S

BEEL = XBAR + 0.2506 / AL

BEE = 10. ** BEEL

ALO = AL

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

C C

C UNBIASED ESTIMATES OF WEIBULL MODEL ARE C

C GIVEN BELOW FOR THE COMPLETE AND CENSORED

C CASE C

C C

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

12 IF (ALO*X (NF) -150. ) 207,207,208

207 SFA= 0.

SFAX= O.

SFAX2 = 0.

DO 5 1=1, NF

P = F(I) ** SFA= SFA + P

SFAX= SFAX + P

SFAX2= SFAX2 +

5 CONTINUE

LAST

ALO

*P

X ( I ) *2

* X ( I

)

302585

* X ( I ) 302585*2. 302585

342.

343.

344.

345.

346.

347.

348.

349.

350.

351

352.

353.

354.

355.

356.

357.

358.

359.

360.

361

362.

363.

364.

365.

366.

367.

368.

369.

BE

371 . NWFGLD

374

375.

376,

377.

378.

379.

380.

BNF

382.

SECTION FOR NEW FANG

384.

388.

389.

390

391

392.

393.

394.

395.

396.

397.

398.

399.

400

401

402.

403.

404.

405.

406.

407.

408.

409.

410.
SGA=0. 411.

SBAX=0. 412.

SGAX2=0. 413.

IF (GNG - .01) 1130, 1130, 1133 414.

1133 IF (GNG - 7.01) 1136, 1136, 1139 415.

1139 IF (GIG - .01) 1136, 1136, 1142 , 416.

1142 P== G(l) ** ALO 417.

GLOGG = ALDG (G(l)) 418.

SGA= GNG * P 419.

SGAX= SGA * GLOGG 420.

SGAX2= SGAX > GLOGG 421.

GO TO 1130 . 422.

1136 DO 1148 1=1, NG 423.

P= G(I) **ALO 424.

GLOGG= ALOG (G(I)) . 425.

SGA= SGA + P 426.

SGAX- SGAX + GLOGG * P 427.

SGAX2= SGAX2 + P * GLOGG * GLOGG 428.

1148 CONTINUE 429.

1130 SFA= SFA + SGA 430.

SFAX = SFAX + SGAX v 431.

SFAX2 = SFAX2 + SGAX2 432.

FALO= SFAX/SFA - 1 . /ALO - XBARL*2 . 302585 433.

SFA2-2*AL0G (SFA) 434

DALO 1 = ALOG ( SFA X 2

) -ALOG ( SFA

DAL02=2*AL0B (SFAX ) -SFA2

DFAL0= (10. **DAL01 ) - (10. **DAL02) +1 . / (AL0*AL0)

"32 IF (ABS(DFALO) - .00001) 30, 30, 31 ^ 436.

30 DFALO= 10. * DFALO 437.

GO TO 32 438.

31 CORR= -FALO/ DFALO 439.

IF (ABS(CORR) .0001) 8, 8, 9 440,

9 TRY= ALO + CORR 441.

IF (TRY) 10, 10, 11 442.

10 ALO= ALO/ 2. 443.

GO TO 12 444.

11 ALO= TRY 445.

GO TO 12 446.

S ALPHA= ALO + CORR 447.

BETA = (SFA/XNF) ** (1. /ALPHA) 448.

GO TO 223 449.

208 CONTINUE 450.

IF (FNG - 0.1) 209, 210, 210 451.

C 452.

C STATEMENT 12 POSITIVE—WE I BULL PARAMETERS ABORTED 453.

C 454.

210 WRITE (7, 211)

211 FORMAT ('SEE LINE 453 IN HAMLET')

GO TO 308

209 CONTINUE . 457.

ALPHA==AL 458.

BETA---BEE ^ 459.

BAC = 8. V 460.

223 CONTINUE 461.

I F ( FNG - .01) 9111,9111,91 07

C 463.

C DETERMINE IF COMPLETE SAMPLE: IF NOT FIND 464.

C CENSORED LOG NORMAL PARAMETERS 465.

C 466.

9111 CONTINUE 467.

ALPHA = ALPHA/ VUA 468.

NL=NFF 469.

NW=NFF 470.

XBARl = ALOG(XBAR) 488.1

S = S*AL0G(10.) 488.2

308 CONTINUE 504-
RETURN

END 505

FUNCTION CNORML. ( XH , XL. , XM , XS ) 506

C S00887 CNORML F"'S-497 CHIANG E. C. H. 661004 6600 507

P(X)=.5*(1.-(1. / 141 12821*X+. 0B864027*X**2 508

1-I-. 02743349*X**3-. 00039446* X**4-i- . 00328975*X**5 509

2)**8)) 510

C IF NM=1,X1 AND X2 SHOULD BE STANDARDIZED TO N(0,1) 511

Xl= <XH-XN) /XS 512

X2=(XL-Xri) /XS 513

Z1=X1/1 . 414213567 514

XI = Zl 515

Z2=X2/1 . 414213567 516

IF (Z1*Z2) 1 ,2,3 517

C WHEN Zl AND Z2 HAVE DIFFERENT SIGN 518

1 Z2=ABS(Z2) 519

CN0F^ML=P(Z2)+P(Z1) 520

GO TO 100 521

C TO FIND WHETHER Zl OR Z2 IS 0 522

2 CN0RML=P(ABS(Z1+Z2) ) 523

GO TO 100 524

C WHEN Zl AND Z2 HAVE THE SAME SIGN 525

3 Z2=ABS(Z2) 526

Z1=-ABS(Z1) 527

CN0RML-ABS(P(Z2)-P(Z1) ) 528

100 RETURN 529

END 530

t-BEND

NBS-n4A IREV. 2-8C) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

REPORT NO.

SHEET (See instructions) NBS BSS 162 August 1984

4. TITLE AND SUBTITLE

Ring-on-Rlng Tests and Load Capacity of Cladding Glass

5. AUTHOR(S) E. Slmiu, D. A. Reed, C. W. C. Yancey, J. w. Martin, E. M. Hendrickson,

A. C. Gonzalez, M. Koike, J. A. Lechner, M. E. Batts

6. PERFORMING ORGANIZATION (If joint or other thon NBS. see /nstructions;

National Bureau of Standards

Department of Commerce

Gaithersburg, MD 20899

7. Contract/Grant No. CEE 8308329

8. Type of Report & Period Covered

Final

• Partially sponsored by:

National Science Foundation

Washington, DC 20550

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 84-601098

I I Document describes a computer program; SF-185, PIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) Although ring-on-ring test results have been used in the past to obtain

infortnation on the strength of glass, no methodology has so far been developed

in the literature explicitly relating such results to the load capacity of

cladding glass. The main purpose of this report is to propose such a methodology. The proposed methodology makes use of recent advances in the modeling

of the fracture mechanics behavior of glass and the calculation of stresses in

plates exhibiting geonetric nonlinearity . Evidence is presented which strongly

suggests that the probability distribution of the load capacity of cladding

glass panels whose failv.re is due to surface flaws can be estimated reliably

on the basis of results of ring-on-ring tests used in conjunction v^7ith (a)

numerical methods for the analysis of stresses in plates, and (b) information

on the elastic and fracture mechanics behavior of glass currently available or

that can be obtained routinely. Two interesting findings are noted. First,

owing to the way in vjhich results of ring-on-ring tests are utilized, the

relatively large variabilities typical of fracture mechanics parameters, as well as the uncertainties with respect to the shapes of surface flaws, have a minor effect on the estimation of load capacities. Second, tv.'o-pararaeter

Weibull distributions, previously used in the literature to model the strength

of glass and the load capacity of cladding panels, are not consistent with

experimental results. On the other hand, three-parameter Weibull distributions

model the observed glass behavior credibly.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) buildings; engineering mechanics; failure; fracture mechanics; glass; loads (forces);

probability theory; ring-on-ring tests; strength.

13. AVAILABILITY

[X] Unlimited

I I For Official Distribution. Do Not Release to NTIS

[X] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Q]] Order From National Technical Information Service (NTIS), Springfield, VA, 22161

14. NO. OF

PRINTED PAGES

60

15. Price

USCOMM-DC 6043-P80

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