A STUDY OF THE DISTR!&1VTION AND MOMENTS OF BUNDLE S'rRENGTH
IN SEQUENTIAL :BREAKAGE OF PARALLEL FILA1'4IJ'TS
by
MOON WON SUH
Institute of Statistics
Mimeograph Series No. 622
May 1969
iv
TABLE OF CONTEN'rS
Page
. . ..
..
LIST OF FIGURES
019.".01;8
INTRODUCTION
THEORY ......
..
o
'"
~
ll)
$.
0
•
f>
(I
...
•
..
. ..
.
LIST OF TABLES ..
•
•
..
•
C!'
. .. . .
QI
•
••
0'.0."11"
o
•
•
•
0
• • •
0
"
•
0
~
•
•
· ..
..
••••••••
.·..
o
o
•
•
·..
Restricted Model for Breakage of Bundles •
Distribution Function of Bundle Strength .. • • •
0
. . . ..
0'
•
•
•
~
0
•
•
•
•
6
"
•
"
•
• • •
•
•
0
•
0
c
•
· .. .
Derivation . . . . ~ ..
0Glt.4)o.oo.elt9.00
••••
Geometrical Expression of Sn (x).. • • • • • • • • • • • • .. • •
•
e
e
_
eGo
.00
4
•
•
•
•
•
Properties of S (x) ...
Examples with ~ll n o " * 0 . 9 . • 0 •
G
•
•
•
•
•
•
•
•
0
~
~
~
~
~
Asymptotic Properties of Strength for Large Bundles
o
•
•
•
•
•
vi
vii
1
5
5
8
8
15
15
17
22
B
Asymptotic Distribution of nn o . .. • .. • • .. .. • • • • • • •
Wasted Fraction of Filaments in a Bundle
• •
It
$
•
•
•
•
.
.. · ..
Inequ.alities Concerning the Moments of the Bundle
Strength Distribution • .. •
• • • • • • ..
Bn k
Monotonieity of E(H ) with Unknown F(x) • • • • • .
n k
Monotonicity of E(n-) in Finite Populations • • • o • • • •
0
•
B k
Upper and Lower Bounds of E(...11)
n
•
•
•
0
•
•
..
tJ
•
•
..
•
..
•
Effect of Filament Length on Bundle strength
(I
f)
41
o
•
•
•
•
44
•
0
•
•
•
•
•
•
49
0
Effect of Length on the Wasted Fraction of filaments
in a B'undle
Probability of Strength Retention in Bundles of
Augmented Length "
lJ
•
•
• ..
n
35
0
••••
49
•
•
•
•
•
52
B(.~)
Asymptotic Properties of -!! for Large Bundles
31
(l
Generalization of the Weakest-Link Theory in a Bundle
..
•
22
•
(Jo
$
"
•
t)
6)
•
•
•
•
•
..
Q
0
0
It
•
•
•
•
55
0
..
..
•
..
"
..
•
..
•
•
•
•
..
0
•
•
..
..
..
•
55
•
G
•
•
•
e
_
•
•
•
•
•
•
59
Relaxed Model .for Breakage of Bundles
Distribution of B~le Load
• .. .. .. .. .. • .. .. • .. •
An Estimate of E("'!!)
"
Asymptotic Proper~ies of L (y) and B
n
n
" • .. .. .. ..
0
0
..
..
..
..
60
65
67
v
TABLE OF CONTENTS (continued)
Page
SUMMARY AND CONCLUSIONS
LIST
OF REFERENCES •
APPENDIX • • • • •
0
0
•
0
•
•
•
0
0
•
0
0
• • • • • • • • • • • •
69
Q
..
•
0
0
0
0
0
0
• • • • • • • • • • • •
73
• • • • • • • • • • • • • • • •
An Alternative Analysis of Ln(Y) with Relaxed Model
o.
•
•
•
•
&
•
•
. .... .
74
74
vi
LIST OF TABLES
Page
1.
E(B)
and Var(Bn ) for n
n
2.
E(Bn ) and Var(Bn ) for n
= 1,
=1,
2 and 3 when f(x) == CIe-ax , x -> O.
2 and 3 when f(x) = 1, 0 ~ x ~ 1.
20
21
vii
LIST OF FIGURES
Page
1.
Load-extension ourve of a four-filament bundle based
on filament loads
0
,.,
2.
c~
0
..
•
•
•
•
..
•
..
•
•
•
•
..
•
•
Geometrical expression of breakage region for bundles
of two and three filaments • • • .. • • • • • • • • •
0
0'
•
'4
..
0
0
•
3.. Distribution functions of bundle strength for n = 1, 2 and 3
based on filament strengths distributed as N(4,1)
'to.
.......
= 1, 2 ani 3
based on filament strengths distributed as N(4,1) • ..
4. Densi ty functions of bundle strength for n
The step function 1 - Fn (x)
.........
•
0
•
•
•
•
0
•
•
•
•
•
6. Graphical determination of x~.l) based on filament
strengths distributed as
7.
.
•
N~3,1)..
• •
•
•
•
e-
•
•
Load-extension curve ofa filament with relaxed model
..
·. .
· .. .
·. .
...
7
16
18
19
23
54
61
INTRODUCTION
The ver,r nature of a group of any kind in the physical universe
is that some of its quantitative properties relTJB.in j.ndeterminate until
the corresponding inherent properties of the elements in the group are
properly redefined within the group.
In particular, when a group
functions as an assembly of elements the complerlty of the group
....
structure often demands more than the simple rule of additivity for
expressing a group property.
Consider a group of straight and parallel filaments sampled from
an infinite population formed into a bundle by fixing or clamping the
filaments together at both ends.
When the prime objective of forming a
bundle is to have an increased strength, the natural question arises
as to how much and in what fashion the increase is brought about by
having a designated number of filaments in a bundle.
The engineering
significance of this problem has long been recognized in as many fields
of material science as there exist problems of breakage, failure or
fracture.
The filaments which constitute the elements of a bundle can
be, for example, textile fibers\) yarns, electric wires, rubber cords
or even polYmer chains.
The strength of a bundle can be defined, though
it may seem arbitrary Il as the minimum axial load above which every
filament in the bundle breaks, or the maximum load attainable in the
bundle throughout its load-extension history which extends to the final
filament break.
Much of the analYtical difficulties in theory of bundle strength
are directly attributed to the variation of tensile properties among the
filaments that constitute a bundle.
In particular, the difficulties are
multipled when such variation is of a statistical nature and has to be
2
imbedded into the physical system of sequential filament breakage that
is hon-stochastic in its nature.
Consequently 9 the system of bundle
failure has to be converted to probabilistic events in order to facilitate
a statistical approach.
The initial analYtical approach to the problem of bundle strength
was made by Peirce (6) in 1926.
By assuming an underlying distribution
of filament strength t he was able to provide a likelihood estimate of
bundle strength if the number of filaments in the bundle were very large.
Though his result was found to be correct ina later work (2), it was
based on a crude approximation with little mathematical rigor.
Moreover,
Peirce's results were obtained in the absence of the distribution of
bundle strength, and consequently he was unable to provide the expected
strength of a small bundle.
A method of deriving the probability distribution of bundle strength
is credited to Daniels (2).
On consi.dering successive filament breakages
in a bundle with a given number of filaments, he was able to convert a
set of conditions under which the bundle breaks into a multiple integral
of the joint density of filament strengths in order to obtain the
distribution function of bundle strength.
From the general form of
the distribution of bundle strength, he was successful in providing the
asymptotic form for large bundles which follows a normal distribution
with mean and variance that can be determined from the underlying
distribution of the filament strengths.
It was shown that the estimate
of mean bundle strength obtained by Peirce coincided with the asymptotic
mean of Daniels.
Although the mathematical context in the work of
Daniels carries certain vh-tues in its rigQr 9 it c6:J:,.Ica.inly cannot escape
a criticism. for being too complex and extremely lengthy in his proof
of asymptotic normality ~
In addition p the frequent expansions and
approximations employed throughout his analysis obscure the essential
physical significance attached to each stage.
other remarks on Daniels'
work will be made in the major parts of this study as necessities arise.
Other statistical works on bundle strength include one by Coleman
(1).
Under the premise that the strengths of filaments obey an Weibull
distribution he showed p by utilizing the major result of Danie1s p that
the efficiency of a large bundle decreases as the variance of filament
strength increases.
Coleman's studies on time dependence of mechanical
bre·akdown of bundles p which are referred to in the foregoing paper pare
irrelevant to the interest of this study and hence are excluded from
discussion.
The scope of this study embraces not only those aspects explored by
the previous workers, but also several other important areas formerly
untreated.
First p the distribution function of bundle strength is
derived based on a simple yet precise event description of bundle breakage.
This process eliminates the complex multiple integration of a joint
density presented in the work of Daniels
0
Secondly, the asymptotic
properties of strength for large bundles are deduced directly from the
definition of bundle strength without involving the distribution function.
Utilizing several measure theoretic modes of convergence the analytical
difficulties found in Daniels' work are avoided.
Next p emphasis is
placed on the efficiency of bundles with respect to the number of their
constituent filaments.
Major inequalities concerning the moments of
4
the bundle strength distribution are proved in order to compara bundles
of different sizes.
Also~
the lower and upper bounds of bundle
efficiencies are established in accord with the inequalities.
Another
part of this stuQy is devoted to the generalization of Peircegs 'weakestlink theory' (6) for filament bundles.
Changes in bundle strength are
examined with relation to the changes in filament length.
Finally.
a statistical stuQy is made under a newly proposed breakage model.
The
load being functionally dependent on extension. this model is intended
to relax one of the postulates imposed on the idealized model on which
the first part of this stuQy is based.
5
THEORY
Restricted Model for Breakage of Bundles
In view of the fact that the breakage of a bundle can occur only
through the breakage of the filaments constituting the bundle, it is
imperative to characterize the tensile properties of the filaments in
order to properly define the strength of a bundle.
Whatever the
properties of the filaments are, however 9 it is conceivable that a
sequential breakage of filaments results when a bundle is subjected to
a monotonically increasing load.
Also, for
either breaks or sustains the load.
a~
given load the bundle
If it sustains the load, the total
load is shared by the surviving filaments in a certain fashion.
As the
total load increases again gradually, there comes a point where one of
the surviving filaments breaks.
At this point, the load given up by
that particular filament is distributed to the remaining filaments, thus
increasing their shares.
Consequently, further breakages may occur
until another equilibrium is reached, or else successive breakages may
result in the breakage of the entire bundle.
It is perhaps important to visualize the bundle breakage as a
result of gradual extension of the constituent filaments.
Observing
that every filament is extended the same amount during the loading of
a bundle, the load in a bundle at a given extension can be considered
as the resultant load of the filaments at the extension.
Obviously,
the bundle load does not necessarily increase for an increase in bundle
extension as soon as filaments start to break.
Hence, the maximum
bundle load attainable in the load-extension history of a bundle is a
reasonable criteria for defining the strength of a bundle.
On
the
6
other hand p if an extension is considered as a mere consequence of
applying load p the minimum load above which every filament breaks
constitutes another side of the dual definition of bundle strength.
It is trivial to observe that the two foregoing definitions are
identical.
The breakage model to be applied for most part of this study is
founded on two postulates:
1) the load in every single filament
monotonically increases along with the increase in its extension before
it breaks, 2) for any given extension. the load in a bundle is equally
shared by the filaments in the bundle surviving at the particular
extension.
These two postulates bring two.fold simplification to the
algebraic expression of bundle strength.
The first postulate implies
that the bundle strength has to be found among the sums of filament
loads just prior to the point of first, second, • •
f
•
and the last
filament break and at these points the sums achieve their local maxima,
followed by the breaks.
sums:
The second postulate further simplifies these
when the filament strengths are arranged in an increasing order
of magnitude as Yl • Y2 , ••• , Yntl the load in an n filament bundle at the
first, second, ••• , and the nth filament break can now be expressed
as nY , (n-l) Y2' ••• , and Yn. Therefore, according to the given
l
definition, Bn II the strength of a bundle with n filaments is defined as
the following:
Bn
or,
:=
max {nYl' (n-l) Y29 ....
= ~<'"k:> n {<n-k+~)
Yk}
Y~
,0,::
Ii < Y2 <
... < Yn
(1)
7
_ _ _ _ • BUNDLE
10
• FILAMENT. LOAD
9
~
8
.....
II I
I II
7
0
«
0
-'
I
I
I
6
5
4
LOAD
l
11
/ I
...
I ~~ I
I,~
o
=0
POINT OF
FILAMENT
BREAK
I
I
I
I
I
I
Y4
I
I
I
I
I
I
EXTENSION
Figure 1.
Load-extension curve of a four-filament
bundle based on filament loads
The load-extension curve of a four-filament bundle is constructed
in Figure 1 on the basis of its filament curves.
The maximum bundle
load is achieved at e Z9 the extension at which the second breakage
occurs, and is shown to be 3YZ in this example.
The monotonicity assumption in the above restricted model is valid
when a bundle is subjected to a dead load or to a constant rate of
loading.
The assumption, of course, is not compatible with the constant
rate of extension experiments performed by the strain gauge type testers.
The possible decrease in filament load due to stress-relaxation can be
observed if the rate of extension is low.
However p such possibility
8
is remote even with a strain gauge tester pince the rate of extension
is quite high in ordinar,y strength tests so that no stress relaxation
is observable.
In the absence of the monotonicity assumption, the
coefficients to be attached to Y , Y , ••• , Y are absolutely
k
l
2
indeterminate and an analytic study of Bn hardly seems feasible.
The
second assumption of equal filament load is a fair approximation when
the filaments are more or less uniform in their load-extension
properties.
otherwise, this model may not be of value.
For this
reason a relaxed breakage model is examined in the last part of thi's
study.
Distribution Function of Bundle Strength
Derivation
SUppose the filaments in a bundle are randomly sampled from a
population of filaments for which the strength distribution is known.
Let the strengths of the constituent filaments be Xl' X , ••• , X
n
2
for a bundle of n filaments, and assume the underlYing distribution
function F(x) to be absolutely continuous with the corresponding density- - .
f(x).
Instead of working with the definition of B given by Eq. (1),
n
the following consideration of events leads to the distribution function
of B ina simple
n
w~.
Let the event Ei = (Xi ~; , Bn _l (with all but Xi) ~ x)
i ..!. II
the filament with strength Xi is breakable· under load
.:s ;
and the
bundle of (n-l) filaments without Xi is also breakable under load
.:s x.
It is noticed, for the event E , that Xi does not have to be the
i
weakest strength.
Existence of some Xj ~ ~ (j
I:
i) is allowed in the
9
event Ei .
The:refore t the event Ei simply provides a. sufficient
condition for a bundle to break rather than the actual breakage
However t since any breakage can be described b.Y one or more
sequence.
of E'sg the union of EV s exhausts all possible contingencies.
Therefore,
•.
by an elementary rule of inclusion and exclusion of events,
n
==
+
n
I: Fr(E.) -
i=l
~
l: Fr(E.lEj)
i#j
.L
n
) - ••• + (-1 )n+l P ( E..E ••• E ) •
l: Pr(EiE.E..
r ~ Z
J~
n
~.
In obtaining Pr (EiE ) above, it is necessary to show that the following
j
two events EiEj and Eij are equivalent.
~ Ej
-
[x,.:s i.
Bn _l (llith all but
x,.) :s x.
Xj
:s ~ •
Bn-1 (with all but X.)
< xJ
J -
~j = [x,.:s i.
Xj
:s i.
To prove the identity EiEj
Bn_2 (llita all but
= Eij ,
x,.
andXj )
:s xJ
it is sufficient to show that
It is trivial to see that Ei E =::::? Eij since the strength of a
j
bundle cannot possibly be increased by deleting a filament.
[Bn_l (llith all but
x,.):s xJ ===='>
Therefore,
[Bn_2 (llith all but Xi and Xj):S xJ
10
~n-l
(with aU but Xj )
XJ ~
:0
[Bn-2 (with all but JS. and Xj ) :0
xJ
For completeness. the above is to be shown by using the definition of
B (x) given in Eq. (1).
n
....
rn-l (with all but Xi) :0 x
where. X(k) is the k
..
filaments excluding
• [(n-l) X(l) :0
No:w suppose X
j
=:
th
J · <~:o
1
n-l {(n-k) X(k) :0 x}
order filament strength among the n-l
~.
x, ~n-2)
X(2) :0
x, ..., X(n-l) < xJ
XCi) and consider the strength of the n-2 filament
bundle with X(l)' X(2)9 ••• , X(~_l)' X(£+l)' .•• , and X(n_l) by deleting
tl1re filament corresponding to Xjt i'!..t Bn_ 2 (with all but ~ and Xj ).
Then
[Bn-2 (with all but
• [(n-2) X(l)
~ and Xj ) :0 xJ
< x, ••• , (n-p) X(£_l) :0 x, (n-1-1) XU+l )
:0 x, ••• , X(n-l):O x
::::> [(n-l) X(l)
:o~,
... ;
J
U_l ) :0 x,
(n-1+1 ) X
(n-1-1) X(.I+l) :0 x, ••• , X(n-l) :0 x
• [Bn-l (with all butJS.) :0
CO :0 x,
(n- p) X
J
x] ·
Similarly,
[Bn-2 (with all but JS. and Xj ) ::f x
J
::::> [ Bn-l (with all but Xj ) :0 x
J
11
Therefore ?
[ Bn-l (with all but
c:
..
,
lit) ::; x,
[Bn-2 (with all but
~Ej
Bn_1 (with all but Xj ) ::; x
J. and
lit
and Xj ) ::; x
J
hen.e
E:tj or ~Ej ~ Eij •
C
Conversely? suppose E.. is true, i.e.,
~J
[lit ::; ~ • Xj
::;
~
--
, Bn-2 (with
all but
lit
J.
and Xj ) ::; x
and designate the strengths of the n-2 filaments exoluding Xi and X
j
by Z(l)? Z(2)' ••
q
Z(n_2) in inoreasing order.
Then, when
Zu) < Xj < ZU+l) ,
[Xj ::;
~
, Bn-2 (with
= [Xj ::;~.
all but
lit
and Xj ) ::; xJ
(n-2) Z(l) ::; x, ... , (n-£-1) Zu) ::; x.
(n-~-2)
.... Z(n-2)::; xJ
nZ(l) ~ x, ••• , nZ(£) ~ x,
::; x..... Z(n_2) ::; xJ
c:
~n-l)
::: x.
:: !.:n=
rB . 1
Z(l)::;
x, •••• (n-£) Z(R) ::;x,
(n-f-2) Z(I+l):::
(n-~-1)
x, .... Z(n_2)::; xJ
(with all but X. ) < xJ
~
-
<.2£, Bn- 2 (with all but -~
X. and X.) < xJ
[ X.J-n
J-
=
rn-l (with all hut Xi) ::; xJ •
Xj
Zu +1)
12
Similar1Yli
rx,
l
x B 2 (with all but
. .
< -."',
<
n...
. X.J. and X.)
J-
~-n
c= lB . 1
-In-
(with all but X.)
J
<
-
J
x
xlJ ·
Therefore, from above,
=[ ~ :;;~.
E;.j
Xj :;;
=[xt :; ~.
~.
Bn-2 (with all but
Bn_l (with all but
~)
<x
~
J=
an:! Xj ) :;; xJ
];;i
and also
[x.J -< 2f,
1.J -c:::
E..
n
B. 2 (with all but X ) <
j n-
xJ
:=
E.
J
E
c:. Ei Ej , or Eij ~ E Ej
ij
i
and hence
It was shown, however, that
c:::
Ei Ej
Eij • 01' °EiE
j
==;;>
ij •
E
Therefore. E. E. = E~ .• or EiE. ~ E..•
1. J..L.J
J
1.J
From this,
P (E.lE.)
r
. L. J
= Pr (E..
)
1.J
== p2(.!.) S
n
n-
2(x).
In general, it can be shown for k ~ n that
E:tE2
where
E.-
~2 ••• k
==
[x...--:I. -<.!.
n
B"':'k (with
X
2
~ Xn ,
••• ,
x.
-1<:
~ nx.
'
x.:+1' x.:+2' .... x,,) :;; xJ·
...
!1.c
== il2. uk '
The proof for the sJrJove can be made simple by using mathematical induction
as followsg
E:tE2
Suppose
0 00
~-1
E1E2 • Ek = E12••• (k-l)
~2 ••• (k-l) Bk = ~2.ook
E12••• (k-l)
[1Il:::~. ~:::~ ..... ~-l:::;;'
~ :5. :.
holds.
Then,
~, and it is sufficient to prove
u
1·!!-·.
Ii:
Bn-k+l
(with~.....
Xn ) oS x.
Bn_l (with all but ~) ::: xJ
x
:: rL:lS..:s i'
X2 ~
x
x
x
ii'
... 0' ~-l ~ i'
~ ~ i'
Bn_k
(
witi} ~+l' .... , Xn
)
By the same argument applied to the two event case,
[Bn-k+l (with
~ .....
Hence, ~2 .... (k-l) ~
is true, then
E.tZ...k
===?
[~ oS~.
===?
fn-k+l (wi
xJ
E12••• k
==?
E12.... k
Xn) oS
Bn-k (with
th~.
:::::::>
[Bn_k (with
is immediate.
~+l'
.... Xn )
Conversely if
xJ
also by the same argument applied to the two event case.
hand,
E.tZ ...k
~
[lil : : ji.
~rn-k+l
Fro.. above, [Xl:::
Bn-k (wi th
(with
ii. ~ :f i.
===> [~ ::: i·
~+l'
.. ;.
x,,) oS
Bn-k (with 11.:+1' ••••
applying the same logic,
1
1Il. 11.:+1' .... x,,)::: xJ
Bn-k+1 (with
I:L.
.... J[n) :::
x,,) ::: xJ
....
oS
~+l'
11.:+1' ••••
x,,) ::: xJ
x,,) :::
1
On the other'
J.
J
~ x
xl
14
=> [Bn-k+2
(with
X:t
9
X2 '
~+l'
.... , In) oS xJ •
Continuing the above prooess, it can be shown that
ll:tz••. k::::::> [X:t ~ ~. .... x.:-l ~ ~.
==> [x.:-l ~;.
==> rn-l
Bn-Z (with
(with all but
B....k (with
lk+l' ...•
Xn )
~
1
X:J.. Xz· .... lk-z' lk+l' ... ,Xn)::; xJ
lk) ~~.
Now, observing the terms in the event El2 ... (k-l)\' it is obvious,
by combining the information above, that
e.
and, Pr (ElE2
0 ••
Ek) = Pr
=
(El2 ••• k)
p-k(~)
Sn_k(x) •
Finally, from the foregoing expression for Sn (x),
,
x S lex) - (2)
n
Sn (x) ~-nF(-)
n
rl--
[ F(-)
xJ2 S 2(x)
_
n
n-
+ (3) [F(i)] :3 Sn_:3(x) ;.. ... + (_l)n+l
S (x)
n
I:
~
k=l
(_l)k+l
where So(x)
=1
(~)
[F(x )] k S 'k(x),
n, n -
and Sl(x)
0
[F(~)J n
<x
= F(x).
Daniels' (2) Eq. (9.2) can be reduced to the above expression for
Sn (x) by a substitution, but the above derivation is free of the complex
multiple integration am Taylor expansion present in his study.
1.5
Geometrical Expression of Sn(x)
The breakage regions of two and three filament bundles $re easily
expressed as an area and a volume respectively by assigning each filament
strength to an axis as shown in Figure 2.
The relevance of this figure
to 52 (x) ,ani 5 (X) becomes obvious when the scale x is converted to
3
'.
F(x) ..
Properties of 5n (x)
The recurrence relation in Eq. (2) satisfies all the requirements
for 8 ex) to be a probability distribution function when 8
n
n-lex),
8 _ (x), ••• t F(x) are assumed to be well defined probability distri-
n 2
bution functions.
Clearly,
8 (0) = 0
n
n
k+l n
8 (00) = I: (-1)
(k) = 1
n
k=l
and the right continuity of 8 (x) is also preserved.
n
By successive
,
substitutions, 8n (x) can be rewritten in terms of F(~), F(n~l)' ••• , F(x)
l
for n = 1, 2, 3 and 4 as follows.
~ (x)'::
F(x)
x F(x) 82 (x) = 2F(2)
[ F('2)
x ] 2
x F('2)
x F(x) - 3'.
8 (x) ='-6F(3)
3
-
J2
[ F(3").
x
F(x)
3F(~) [F(~)] 2
+
[F(3)] 3
IThe formulae following the Eq. (10.3) in Daniels' (2) paper are
entirely erroneous. His formulae for B2 , B and B~ which correspond to
8 (x), 8 (x) and 84 (x) should read as at50ve3when his ~ is replaced
2
3
by F(~) for k = 1, 2, 3 and 4.
e
e
~
.
e
X3
X2
xI
,
.
~L:::'
21 .".
I
2
t-4-------/-1
I
..
//
.x
~
.
..
.
'''''''1
<
,/'/
i
/
",
I
",/"
I
I
I
I
//
I
/
I;.c--_ - - -
I
.
I
I
"
., . '
I
~
2
I
X
--XI
I
I
I
,.
/
~
FILAMENT BUNDLE
• X,
V
I
_ _ _ _ _ _ _ . J - / //
//
X2
3
Figure 2.
FILAMENT BUNDLE
Geometrical expression of breakage region for bundles
of two and. three filaments
I-'
~
17
S4(X)
~ 24F(~) F(~) F(~)
-
12F(~) [F(~) ]
+ 6 [
..,
-
F(x) -
2 F(x) -
F(~)J2 [F(~)J2
+4
12TF(~) J2
F(!) , l(x)
12F(~) F(~) [F(~)
[F(~)J3 F(~)
+
J
2
4F(~) [F(~~ 3
[F(~)J4
As n increases, the expression for 5n (x) in terms of the F(;),
k = 1, 2, ••• , n, rapidly inflates beyond a manageable limit.
Conse-
quently, for a large bundle, utilization of 5 (x) for obtaining the
n
expectation and varianoe of B is hardly feasible in praotioe.
However,
n
for relatively SIi1a11 n, the mean and varianoe oan be readily found with
the aid of a modern oomputing faoili ty.
complexity of 5n (x),
. -.the asymptotic
~oause
prope~ties
of the analytic
of Bn are of espeoial
value for approximating large bundle charaoteristios.
ExB.!Ples with Small n
The effect of bundle size n on the properties of bundle strength
Bn oan be better visualized by examining 5 (x) and its corresponding
n
densit,y S
n
'(x)
for a known F(x) and f(x).
Far the case n
= 1,
2 and 3,
a normal distribution with mean 4 and unit varianoe is assumed for
the filament strength, and the distribution and density fu.1'iction of
B2 and B are obtained in Figures 3 ani 4. The curves are based on
3
values of f(x) and F(x) in normal probability table. It is shown in
the figures that the expectation as well as the variance of B increases
. n
as n inoreases.
Interestingly, 52 t (x) and 53 I (x) remain almost
symmetrio although their algebraic expressions hardly imply any such
tendency~
Of course, the skewness may rapidly develop as n increases.
e
e
.,
-
.
,,
'
1.0
0.9
-..
-..
-)(
0.8
0.7
If)
C/)
0.6
)(
l'I
C/)
0.5
)(
0.4
u.
0.3
0.2
0.1
2
3
4
5
6
8
7
9
10
II
12
13
14
15
x
Figure 3.
Distribution functions of bundle strength for n = 1 9 2 ani 3
based on filament strengths distributed as N(4,1)
!-J
(Xl
e
.
0.5
I
I
'
e
..,
I
I
I
,
I
I
I
I
I
5
6
7
8
9
I
~
e
"
I
I
I
I
II
12
13
14
I
n=I
0.4
-)(
- ",
-.
(J)
0.3
)(
- t\!
en
--..
)(
0.2
~
O. I
2
3
4
10
X
F.i.gure 4.
Denaity functions of' bundle strength f'or n =: 1, 2 and :3
based on filament strengths distributed a.s N(4,1)
F'
'"
20
For an exponential ditrtribution, unlike the nomal case, the
expeeta:tion and varianee of B can be obtained by simple integrations.
n
For n
f(x)
=1, 2 and
=a.e-ax
3,
the major resUlts are given in Table 1 based on
3
Table I.
E(Bn ) and Var(B ) for n == 1, 2 and :3 when f(x) == eten
n
E(B )
n
1
(1)
2
1.667 (~)
3
2.223
ax , x > 0
Var(B )
n
2
(1.)
a.
a.
1.444 (1.)
a.
(~)
1. 761
(~)
2
2
In obtaining E(B ) and Var(Bn ) the following identities are
n
=
j
o
X
8' (x) dx
=j
rl
nOt.:
- S (x)l dx
n
J
It is noted in the table that the increase in expected bundle strength
is not proportional to the increase in bundle size.
It seems that the
efficiency of a bundle decreases as its size increases.
The general
assertion of this phenomenon will be made:-in a later section.
For the eases where Xi are positive, but bounded random varia,bles,
special care has to' be placed in the evaluation of Sn (x).
when
f(x)
=1
== 0
, otherwise
For example,
21
the maximum possible value of B is n..
n
~
x 2
== 2(~) .. x - (-)
2
=="
4x
2
Therefore, .for n == 2,
,O<x<l
, 2 < x
== 1
and .for n = 3,
S3(x)
= 3F(;)
=3
(~) (2. x 2 )
3
+[F(~)J3
S2(x) - 3 [F(;)J 2 F(x)
4
_ 3
(~)2
3
.. x +
(~)3
O<x<l
3
_ 3 (~) (x _ 1 x 2 ) _ 3 (~)2 .. 1 + (~)3
3
4
3
3
,1<x<2
=1
,3<x
and so on.
In Table 2, E(Bn ). and Var(Bn ) are obtained, .for n == 1, 2 and
3, using the above results.
Table 2. E(Bn ) and Var(B ) .for n
n
= 1,
2 and 3 when .f(x)
n
E(E )
n
Var(E )
n
1
1
2
1
12
2
g
2-
3
41
36
36
lli.(36)2
= 1,
0 < x
<:
1
22
Asymptotic Properties of Strength for Large Bundles
Due to the complex form. of the probability distribution function
of bundle strength as expressed by Eqo (2), the function Sn(x) provides
no simple operational ground which may lead to the asymptotic properties
of bundle strength.
The information as to how a particular choice of
F(x), the distribution of filament strength affects the properties of
a large bundle is indeed difficult to be extracted from Sn(x), regardless
whether or not F(x) is specified.
Therefore, the mode of attack in this seotion is to base the general
reasoning on the basic definition of bundle strength and gradually unveil
its inherent statistioal nature.
B
A.symptotio Distribution of
-!
Instead of working with B , the strength of an n filament bundle,
n
B
the asymptotic properties are to be examined for
tha.t is the
rf!-,
average oontribution of eaoh oonstituent filament to the strength of the
bundle.
From Eq. (1),
Bn =
n
Define
1
r
[n-k+l
.:smax
k ~ n
n
y }
k
,0
.:s Yl
< Y2 < ...
(3)
F (x) = 0
n
k
= -n
, Yk < x < Yk+l , k
= 1,2,
... , n-l
=1
i.!.., Fn (x) is the empirioal distribution funotion oorresponding to F(x),
and it is defined with left oontinuity instead of the usual right continuity.
Then, 1 - Fn(x) is the step .funotion as shown in Figure 5.
23
1----,1
I
I
I
6~-----"
I
I
n-1
n
I
c>---,
n-2
n
I
2
I
6
i
n
I
I
I
01-------..
I
I
1
n
I
...
o
Figure
x
Y
Y
Y
n-l
n-2
n
5. The step function 1 - Fn (x)
According to the above definition, Eq. (3) is reduced to the following:
:n
= 1<";"< n {[I
~ Fn(ykD Yk}
=1::o"'f..x::o n [Yk~l ~::o Y
{[I ~
k
= 0<ma:< Y
{[I
n
= 0 ::oma:'< ~
since
Fn(X>]
x}]
(where Yo = 0)
-Fn(X~ x}
{[I -
(4)
Fn(X>] X}
[1 - Fn (x)J x
=0
for
x = 0
and for x > Y •
n
The second equality above holds since [1 - Fn (x) ] x
increases within each interval Ik-l < x ~ Y
k
monotonically
(k = 1, 2, ••• , n).
24
B
Thus 9 the original expression nn, which was the maximum of
[ 1 - Fn ex) ] x
over the discrete points Y , Y , ••• , Yn ' is now
l
2
expressed in Eg. (4) as the maximum over the entire range of strengths
B
for x ~ O. Therefore, the asymptotic properties of nn is to be
examined based on Eq. (4) from now on.
Before oonsidering the
B
n
n'
asymptotic behavior of
the following theorem needs to be proved.
Theorem 1
The random variable
[1 - Fn(X)] x
oonverges uniformly in
[1 - F(X)] x, !.!..,
probability to
Pr {o <"':'< ~ [1 - Fn(x)] x - [1 - F(X)] xl >E} ---- t1
as n ---+
00
for an arbitrary £ > 0 if the second moment of
2
filament strength E(X ) is finite, arxl F(x) and f(x) are
absolutely continuous.
Proof:
[
Pr
- P
-
l'
sup
LO ~ x <
00
{max [O<x<Y
.sup
-
-
n
because, for x > Y , 1 - F (x)
nn
.s Pr
{o :/:
<Y
n
=0
~n(x) - F(X~X >E} + r {x·~.\~. - F(xHx >E}
P
(5)
2;
The first term to the right of the above inequality gives:
p
f0
r'L < x~ n ~n(X) -
< P {
-
r
= Pr
0
sup
Y
F(X)] x
I >E}
>£}
sup
<x.:s: Yn IFn(X) - F(X)! Yn
s.s~pS. Yn !Fn(X) -
{Yn • 0
P {Yn • 0 <S~Ps. \
+ r
F(X)!
>E •
Yn S.
IFn(X) - F(X)!
>E.
Yn >
l. rn}
l. rn}
where g A is a positive constant.
In Fq. (6)9
Pi' {Yn • 0
s. Pr
S. Pr
{l. rn·
{l. rn·
= Pr { In Dn
s.S:P<Yn IFn (x)
0
0
>
I >E.
Yn S.
l. rn}
I >E}
/~P< ~ /Fn(x) - F(x) I >E.}
s.S:Ps. Yn
!Fn(X) - F(x)
t-}
where p Dn = 0 <s~p<
B,y
- F(x)
00
IFn(X) - F(x)
I·
Kolmogorov's Theorem (4) proved in 1933,
00
•
(_l)k e- 2k
I::
2 2
Z
for z > 0
k=-oo
= 0
as n
InDn -4Z.
for z < 0
~oo
(6 )
26
Therefore it is possible to -write, for any z > 0
Pr {
rn D" > .}
- Pr {Z >
z} ~ !
for
n2 "J.
and also z can be chosen to satisf,y
o ~"'40
•
• •
for n ~ ~ •
+ '8"
Letting z :::
f- '
P {rnD
r
n >
f} *
<
fom 2
(6a)
"J.
The last term in Eq. (6) is evaluated as follows:
Pr {In • 0
~ Pr
~s~p< IJFn(X) -
F(x) I> E • In> A Iil}
{ In > AID} .
To prove Pr { In
:>
AID} - - 0 as n - -
~•
a theorem on p. 125 of Doob (3) is utilized here:
Theorem: Let ZIt Z2' ••• be mutually independent
random variables an:r let c be any positive constant.
If
1
-
n
1::
n j=l
Z.~O
J
then. ~~ j!l Pr {IZjl > en} : 0
The condition, E(X2 ) <
large numbers, that
00,
given in Theorem 1 implies, by the weak law of
27
1'1
I:
2
x.
1;:.1 J
p _
'""""->
1'1
1'1
:£
or t
j=l
2
eX. J
O<a<oo
a
a)
-40.
1'1
Letting X~ - a == Zjt and applying the theorem by Doob,
a
Let no be a number such that A2 - -1'1
> 0
(1'1
o
=1
if A2 > a).
Then, from above,
for
1'1
>
1'1
o
no '
2 a
letting c == A -
•
i~l Pr{I X~ - aI > en} -
0
as n- ~
28
(6b)
Combining the results given by Eqso (6a) and (6b), Eq. (6) is now
rewritten as
'.
for n
~ max
~,
{no'
n 2}
Finally, the second term to the right of the inequalityin Eq. (.5)
For this, the finiteness of E(X2 ) as an assumption
has to be evaluated.
has to be slightly modified.
2
E(X ) ==
:=
00
J x2
o
2
f(x) dx
J [1
o
- F(x)] x dx ,
the condition E(r) <:
lim
Due to the equa;Lities
G
x~ooLl-F(x)
00
immediately implies
] x==o
Le., there exists a point x' such that fOr x> ~,[l- F(x)] x < C
for an arbitrary £. > O.
Pr{x":\
[~ -
F(x)]
Utilizing this,
X
> €}
:: P {suPy [1 - F(x)] x > E, , Y > x'1
n
rx> n
J
+
Pr{x";Y [1- F(x)Jx
n
.::: 0
+ Pr
{ Yn .:::
Xl}
>1:. •
Yn:::x]
== [F(x') ] n
<
~
(8)
29
Now» Eq.
(7)
and
114. (8)
are substituted in Eq.
(5)
to yield ~he fina!!.
inequality~
Pr{o ~s~p< ~ 1[1 - Fn(x)] x for
~ max
II
[1 -
{no' nl' n 2 , n 3 }
F(x)] xl
>£}
<! +!.
a
(9)
•
Since E. and 6 are arbitrary positive values, it follows that
p
{
I'
as n
sup
0.sx<oo
-+'00
am thts completes the proof.
In the light of Theorem 1, the asymptotic properties of
deduced by noting the following.
that the modal value of
modal value of
[1 -
[1 -
~ are
First, it is true fr.om Theorem 1
Fn (x)
Jx converges in probability to the
•
F(x) ] x due to the well known inequality
<
sup
-O<x<oo
which leads to
Pr {
I
0 / ; (<
~ [1 -
Fn (x)] x - 0
~ Pr {o ~S~P< ~I~ - Fn(X)] x ~ [1 -
or,
~S;(< ~ [1 F(x)] x
F(x)] x
I,.
E}
I > c:}- 0
(10)
30
where,
XC)
[1 - Fex)
is the value of x at which [1 - F(x) ] x is maximi~ed. Here,
Jx is assumed to be a
unimodal .functiono
On the other hand,
directly from Theorem 1,
(11)
..
Therefore, conibining Eqs. (10) am (11), it is concluded that
Bn _ [1 _ F (x ) ]
n
n
x
0
The distribution of
0
~0
•
[1 - Fn (xo ) ] x o '
(12)
however, depends on the basic
properties of the sample distribution Fn (x).
Due to the fact that
nFn(x) is same as the number of observations that fall below x among n
total observations, nFn(x) follows a binomial distribution with F(x)
as the occurrence probability.
Moreover, it is well known that the
distrioution of Fn(x) is asymptotically normal with mean F(x), and
variance·; F(x) [1
distribution of
-
F(x)]
for any x >
[1 - Fn(Xo )] Xo is
and variance 1.
F(x0 )
n
[1 - F(x
0
)]
o.
Therefore, the aSyMptotic
normal with mean
[1 - F(xo )]
X
o
2
x 0 as x 0 is a constant, i.e. ,
[1 - F",(xo )] Xo ~A.Il. ( [1 - F(Xo )] x o '
~ F(xo ) [1 -
F(Xo )]
x~
I. (13)
The major conclusion given in Daniels' (2) work that
:n
~A.Il. (
[1 - F(Xo )] x o '
~ F(xo ) [1 -
F(xo )]
is plausible when the asymptotic distribution of
combined with the result given by Eq. (12).
x~ )
[1 - Fn (xo ) ]
(14)
Xo is
This is not to say that
Eq. (12) is a sufficient condition for concluding that the asymptotic
e·
31
B
distribut.."lon of nn is exactly same as that of [1 - Fn(Xo )] x •
o
However, it is th.e view of this author that Eq. (12) provides a better
ground to approximate the asymptotic properties pf :n by the above
....
compared to the various approximation factors utilized in Daniels' (2)
study•
Wasted Fraction of Filaments in a Bundle
Recalling the definition of B given by Eq. (1), the filaments
n
'Wi th strength Y , Y , ••• , Yj-l would oontribute no part to the strength
l
2
of a bundle if B is defined at the jth filament breakage, and the
n
maximum bundle load B
n
= (n-j+l)
with strength Yj , Yj +P
Y. would be supported by the filaments
••• , Y •
n
J
It is of interest, therefore, to
estimate" the quantity of wasted filaments in a bundle insofar as the
strength of the bundle is concerned.
For this, the order j of Y that
j
corresponds to
is a reasonable likelihood estimate of the number wasted.
Noting that
the event Bn == (n-j+l) Yj is converted to the following set of inequalities
which leads to Eq. (16);
u.,
n-j+2
n.;.j
n-j+l Yj-l' n-.j+l Yj+l '
n-j-l
1
n-j+l Yj + 2 , ••• , n-j+l Yn •
32
.,.
••• dYj+ZdYj +1 ] dY j
(16)
where Z = (n-j+l) Y.
J
e
For a bundle with small n, Eq. (16) can be used to find the order j
satisfying Eq. (1.5).
difficulty of Eq
0
When n becomes larger, however, the analytic
(15) prevents the practical use of the cnteria given
by Eq. (1.5).
An alternative way of estimating the wasted fraction is to utilize
the value
B
....!
to
n
X
o
at which
[1 - F(x
that n
0
[1 -
)] x
0
[1 - F(x) ] x is maximized o
The convergence of
in probability, as shown by Eq. (10), suggests
F(x )] is close to the number of filaments surviving at the
o
particular order of filament breakage where the bundle load is maximized
to represent B *
n
Therefore, the number wasted is estimated simply by
nF(x ), and the wasted fraction by F(x )' when n is very large.
o
o
a small
n~
For
one of the following two methods can be used to estima.te the
wasted fraction:
a) If
X
o
is readily obtainable, find the order j for which E(Y )
is the nearest to x
j
o
among all j = 1, 2,
estimate of the wasted fraction.
e •• ,
n.
Then, .J.
n· is an
"
33
b) Ifx is diffiGul t to obtain, utilize the fact that Xo is
o
detennined from
~x [1 -
F(x) ] x ==
0
:::::? f(x)x
= 1 - F(x )
000
Setting
X
•
o == Yj and taking expectations on both sides, it is shown that
E [f(Yj)Yj ]
==
== E
[1 -
io (j-l)~(n-j)~
n~
F(Yj )]
[F(Y.)] j-l
J
[1 - F(Y')Jn-j~l
J
dY
j
_ n-,j+l
- 1'1+1
_ n-.j+l
- 1'1+1
(17)
In solving Fq. (17) for j, the integer value j iE$the approximate one
that satisfies Eq. (17) best.
Of course, the left side of the equality
is difficult to obtain for certain classes of distributions.
It can be shown, in the following example, that the methods
a) and b) are consistent with the large
1'1
case where the wasted fraction
is estimated by F(x ):
o
For f(x) == CIe-ax ,
X
o ==
a:1 '
F(xo ) == 1 -
1
e
l:
Method (a) gives" by setting
X
0.6321 •
o
== E(Y ),
j
• 1 n
::= -
J
1
'+l:X
ex. n-J
---"'-...
----;;y e
dx
1
I: -
or.
n
log n-j+l
for large n
.n
== n-J
. '+1
...
.• .
l
lim
n~
00
n = 0.6321 •
Also, method (b) gives, by setting
[ ( ) .J .
n-j+l
=n+l '
E f Yj Yj
~rLl1+11 + 1.n +
==:>
••• + . n-J~+2J = 1
Similarly as above,
j ; 0.6321 n + 1.6321
..
lim
n-->
00
.sin =
0.6321
and the results of (a) and (b) agree with F(x ).
o
Inequa1ities Concerning the Moments of the
Bundle Strength Distribution
In order to answer the question as to how the size of a bundle governs
its strength, the moments of Bn can be compared for different n.
For
this, the distribution function Sn(x) has to be utilized and F(x) has
to be known exactly.
AIthough this is the only way to obtain the
moments of Bn , the results for limited n and a particular F(x) hardly
assert a general law applicable to other underlYing distributions.
Comparisons of moments are made in this study on the basis of the
definition of Bn given by Eq. (1).
For a general unspecified F(x),
a few useful inequalities are obtained concerning the moments of B •
n
35
B
and upper a.nd lower bounds are established for the moments of -n .
n
The
same inequa.lities are proved for a finite population case under the
soheme of simple random sampling.
'
..
Bn k
with Unknown F(x)
n
Monotonicity of E(-)
The following lemma is essential in order to prove the theorem on
B k
n
monotonicity of E(-)
n
in n.
Lemma. 1
For any set of n+l f'ilaments (n ~ 1), let the filament strengths be
Y , Y ,
l
2
e •• ,
Y + in inoreasing order, and define
n l
B 1 = strength of the n+l filament bundle with all yr s (filaments)
n+
B
ni
= strength of the n filament bundle with all yt s but Yi
i = 1, 2, ••• , n+l
! ..!.,
Bnl , B ,
n2
0 H,
Bn(n+l) are the strengths of all possible n filament
bundles that oan be generated by the original n+l filament bundle.
ThEm, the following inequality holds between B + and the B •
n l
ni
n, k
~
1
Proofz
a) k :: 1 oase
From the definition of B given by Eq. (1), the inequality
n
n+l
n Bn+l
~ i:l Bni
oan be rewritten as followsg
(18)
36
n •
max {<n+l)Y1P nYzp (n-l)Yy
<
max
+
max {nY1 , (n-l)Yy (n-2)Y4 ,
+
max {nYl' (n-l)Y2 ,
{nYZp
(n-l)Y3 P (n-2)Y4 ,
.411
e ,.
•• • t
3Yn- l' 2Yn , Yn+1}
3Yn- l , 2Yn , Yn+l}
·.. , 3Yn_l , 2Yn , Yn+~
(n-2)Y4 , ·.. , 3Y l' ZYn , Yn+~
n-
•
0
+
max {nYl' (n-l)YZ' (n-2)Y ,
3
+
max {nYl' (n-l)Y2 , (n-2)Y ,
3
+
max {nYl' (n-l)Y2 , (n-2)Y ,
3
• •• t
3Y _ , 2Y ,
n 2
n yn+J
·.. , 3Yn-2' 2Yn• •• t
l' yn+J
3Yn_ , 2Y l' Yn}
2
n-
•
(19)
If the inequality is proved for ever,y possible choioe of
Bn+l = (n- j +2)Y j , j = 1, 2, ••• , n+l, then the proof 'Will be oomplete.
(j = 1, 2, ••• , n+l). Then, it is noted
Suppose B 1 = (n-j+2)Y,
n+
J
for the terms to the right of the inequality in Eq. (19) that
B. ' >
nJ -
(n-j+l)Y. 1 > (n-j+l)Y.
J+ -
J
n+l
•••
~
~
B , > (j-l) (n- j +2)Y , + (n-j+2) (n-j+l)Y,
m-
J
= n(n-j+2)Y,J
1:
n Bn+1 by definition
n+l
i.e. ,
1 < ~ B .•
n+ - '~=
1 m
n B
J
37
b) k ~ 2 case
k
First note that Bn+.
1
:=
:max
1 ~ j ~ n+l
{(n- J'+4)k
~J'}
and similarly for every B~ (i :: 1, 2, ••• , n+l) the terms inside the
'.
brackets are raised to the kth power.
ever,y possible choice of
Suppose
B~+l
BKn+l'
= (n_j+2)k ~
If the inequa,lity is proved for
then the proof will be complete.
(j :: 1, 2, ••• , n+l).
Then similar to the
case with k = 1, it is noted in Eq. (19) that
_k
k
k
k uk
~ril' Bn2 , ••• , Bn(j_l) ~ (n-j+2) Jrj
k
k __k
k
Bnj > (n-j+l) rj+l ~ (n-j+l)
B~(j+l)' B~{j+2)'
•• Of
~
B~(n+l) ~
j
(n_j+l)k
~
•• •
Therefore, to prove Eq. (18) one has to show
[(j-l) (n_j+2)k + (n-j+2) (n-j+l)kJ
~ >
J -
k
n'
(n+l)k -
1 (n_j+2)k
~
J
38
sL
dx
sL
c(x) •
dx
= 1. ...
[<X-I)
J
+ (n-x+l)k
(n_x+Z)k-l
k(n_x+l)k-l (n_x+Z)-k+l + (k-l)(n-x+l)k(n-x+Z)-k
letting y • n-%+l •
yk-l
k
(y+l)
=:..
for y
~
[
k 1
k
1 ]
(y-l) + (2) y + ••• + . k-Z + k
T~
1 or equivalently for x
~
Y
>0
n •
i •..!•• C(x) is a monotone increasing fUnction of x for 1 ~ x < n.
k
min)
n
Therefore, 1 _< x _< n C(x = eel)
•
.
(n+l) k- 1
and
e(x) >
n
k
. k-l for 1 < x
(n+l)
~ n •
Combining the result for the case j • n+l, it is concluded that
k
k
(j-l) + (n...j+l)
>
n
k-l
(
(n-j+Z )
n+1 )k-l
for every j
:=
1, 2, '•••• n+l
-
(IS) is true for k > 2, ani this completes the proof of Lemma
1.
.
~:
Proof (b) above also applies to the case k • 1.
However (a) is
given for the simplicity of the proof available for that ease.
39
Theorem 2
B k
Under the restricted model for bundle breakage,
(k = 1, 2,
n
E(--)
n
3, ••• ) decreases monotonicallY in n under any continuous
distribution F(x) of filament strength provided the strengths of the
constituent filaments
~,
X ,
2
.'~.,
~
are independent and interchangeable
random variables.
Proof:
Observe in Eq. (18) of Lemma. 1 that B + , B , B , ... and
nl
n l
n2
Bn(n+l) are functions dependent on
~+r
Let the joint density of
and the joint density of
Xr,
X:!.'
a~l
~,
or all but one of
••• ,
Xr,
X2 , ••• ,
~+l be f(Xr' ~, •• . , Xh +l )
X , ••• , X _ , X + , ••• X + be
j l
j l
n l
2
=
fj(XJ.' x 2 ' •.•• , x '-l ' x'+l ' ••• , x n+l ) for j
1,2, ••• , n+l.
J
J.
The inequality in Eq. (18) is preserved by multiplYing both sides by
f('1.' x 2 ' ••• , ~+l) and integrating over '1.' x 2 ' •• ., and xn+l •
Therefore, utilizing the symmetry
n+l
I: El. (Ylf Y2' ••• , Yi-l' Yi+l' ••• , Y 1)
i=l m.
n+
n+l
= j~ Bnj (XJ., x2 ' •• ., xj_l' x j +l' ••• , ~+l)
Eq. (18) provides the inequality
n+l
• f (XJ.t x 2 '
eo., ~+l) i~l ~
40
n+1
0"
Itt
9'
x j +1 ' • u, x n+1 )
x j _r
where, R is the proper region of integration for
and Rj for
{XI?
x ' .. S9 x _
j r
2
... ,
x +
j r
X
i~j
dXi
{Xl' x2 '
••• ,
Xn+1}
,
n+1 } , j = 1, 2, ••• , n+l.
Since the Xi are interchangeable, the above inequality is reduced
to the following:
= (n+1)k E(~)
B
k
E(...1!±!)
n+1
B
< E(""!!)
-
n
or,
k
for k, n > 1
for k
=1,
2, 3, • u
and 1
~
m <: n •
(21)
As indicated by Theorem 2, the efficiency of a bundle, measured by
B
E(nnL is likely to decrease as n increases.
This in fact is in accord
with the same general trend often revealed by actual experiments.
Unfortunately? the theorem does not provide any evidence as to whether
B
Var(nn) also will decrease monotonically in n.
Although the following theorem may be intuitively obvious, it is
added here for completeness.
Theorem
3~
Under the same a.ssumptions a.nd definitions given in Theorem 2, the
inequality
E(B~) ~ E(~+l)
holds for k, n = 1, 2,
(22)
3, •••
•
•
41
Proo!':
Observing the terms Bl'~' BnZ , ••• , Bn(n+1) of the right side of
Eq. (19), the coefficients of the Yj 's in a particular column are
common whereas the orders of Yj's can decrease from top to bottom.
Therefore i t is evident that
Bnl ?: BnZ ?: ... ?: Bn(n+l) •
But, Bn+l = max {(n+l)Yl , Bnl} ·
and
•
n+l
t
i=l
k
B.
< (n+l).
ro. -
_k
If 1 •
n+
The same integration method used in the proof of Theorem 2 leads from
the above inequality to
(n+l)
E(~) ~
i.~., E(~) ~
(n+l)
E(~+l)
E(Jt<n+l)'
B k
Monotonicity of E(..2!)
n
in Finite Populations
Consider a rinite population of N filaments and the all possible
bundles of 1, 2, 3, •.. , and N filaments that can be generated from the
population.
If the probability of selecting a particular n filament
bundle (n < N) is determined under the scheme of simple random sampling.
the k th moment of bundle strength with n filaments is expressed by
•
E(~)
Jt< .•
nJ
42
With tb.is definl,tion, the following theorem is given in parallel
with Theorem 20
Theorem 4
B k
In a finite population of N filaments, E(nn)
in n for 1
~
decreases monotonically
n oS N under the scheme of simple random sampling.
Proof:
Consider all possible bundles of nand n+l filaments that can be
generated from the population of N filaments for 1
~
n oS N-l, and
designate their strengths by
and
Bnj
j = 1, 2,
B(n+1H
.R.
= 1,
2,
••• 9
(N)
n
., • • 9
(n~l) •
Note that each of the (n~l) bundles of n+l filaments can generate
bundles of n filaments in exactly n+l different ways, and that a
particular bundle of n filaments can be generated by exactly N-n different
bundles of n+l filamentso
Define
iBm
= strength
of a particular bundle (called i, say)
of n filaments that is generated by a particular
bundle (called
-e,
say) of n+l filaments among the
N
(n+l) bundles of n+l filaments, where
o eo _ 9
n+1
••• t
(n+l)
N
Then, from the
~rray
and .R. = 1, 2, ••• ,
of all possible
en~l)
~Bni's
for i = 1, 2, ••• , n+l
a particular bundle of n filaments, say Bnj ,
will be found exactly N-n times for every j = 1, 2, ••• , eN).
n
This
43
is true from the preceding observation and the fact that ,,(B
ni
belong to one of the B • f 13 9 j
nJ
follows that
N
1:
1 9 2 ••••• (n ).
must
Therefore, it
k = 1, 2, 3, •••
..
However. Eq. (18) of Lemma 1 gives for airy i= 1, 2,
lLk
n--B(n+1)1~ (n+1)
Summing both sides on
k-1 n+1
k
E .eBn:t
1=1
.u, (n~l)
•
1.,
( N)
k
n
n+1
k
-f~1 B(n+1).R:5 (n+1)
k-1
(N)
k-1
= (n+1)·
(N-n)
or
N
(n+l).
E
1=1
[B<n+ll£]
n+1
<
k
(N )
n~l
Il-n
- n+l
[B(n+1
1. = 1 n+l
U] k
i •.!. ,
a.nd this proves the theorem.
(N)
n
E
j=l
n
k
E B .
j=l nJ
[!-t
by Eq. (23)
r
< N-n
1
- n+l ( N )
n+l
k
= 1,
2, 3, ••.
(23)
tha.t
QQrollary
Bk
(....E)
'"
,..
lll
<-
-
'In
"
__k
~ X:1 -'1
1=
0
and, in particular, for k
B
.
:= 1~
k
~
2, 3, •••
=1
n
..Eo
n
-< XnOl'o B = E X.
n
•
i=l
(24)
1
Le., the strength of an n filament bundle is less than or equal to the
sum of its constituent filament strengths.
Proof:
Let N = n in Theorem 4.
Then,
B k
Bl k
1 n uk
= (..!1) < E(-)
= - I: x:n
1
n . I 1
1=
and for k = 1,
B
...!l < X
n - n
The implication of this corollary is not startling at all.
interesting, however, that such an assertion can be made based
It is
On
the
particular assumptions with which the breakage model was established.
B k
Upper and Lo.ver Bounds of E<rf)
B k
Having established the monotonioity of E(nn) , it will be of
B k
practical value if there exists a lower bound of E(nn) that
is greater than the trivial lower bound zero.
lemma is given.
For this, the following
-Lell1llla 2
Z3~
If Z19 Zzw
••• is
til.
sequence of non-negative random variables
that con1rerges to a constant c ~
then, !Ii
E(Z~) ~
ck
, c
~ 0,
a
in probability t
i..!..
t
if Zn ~ c
3, •••
k = 1, 2,
Proof:
.
•
By the Markov inequality,
P {Z > c r
n -
€}
<
E{Z~)
- (c- E.)
E{Z )k > (c _ E)k P {Z
n
-
r
k
>
n-
for
~
0
£ < c , or
E.} .
C -
Taking lim on both sides,
lim
E(Z~) ~
{c - £)k lim Pr {Zn
~
~
(c - t )k lim Pr { c - E
c - £}
~
= {c - E.)k lim Pr {IZn - c
Zn
~
c + E.}
I ~ E.}
By letting So ---+ 0 it follows that
Iini
E{Zk) > c k •
n
-
B k
The lower bound of E(..1l) is easily obtainable by using the above
n
B
lemma. with Zn = nn and c =
_
. 'D nBn ~
[
1 - F(x ) ] x •
o
o
B k
.
lim E(_.n)
>
n
-
[1 - F(xo )]
X
o
From Eq. (10),
Therefore,
[1 -F(x)J k
0
x
k
0
.
46
B k
But p becau.se E(..1l)
11
is a 1'I1o:notone increasing function of n,
B k
l'
B k
-1' E(~) =.:un
E(..22)
~m
n
~<X)
n
• Combining tbi s with the trivial upper bound
Bn k
Bl k
~lc
E(l) :: E(A~), an upper and lower bound of E(n-) can be given as the
following:
..
[ 1 - F(x)J
o
k
k
~_k
Bk
x· < E(_.n) < E(r-)
0
n
-
for k, n = 1, 2, 3, ••. _ (25)
B
Applying the bounds given above, an upper bound of Var(nn) is now
available;
o -<
B
B 2
B
Var(..2:.) = E(...2l) _ E2 (-!!)
n
B
n
n
2
~ E(mm) - [1 - F(xo )]
B
2
x;,
2
1 < m< n
2
+ E (X) - o
[1 - F(x.)J
x
-< Var(..l!!)
m
o
2
or, wi th m. = 1 it becomes simply
B
0.::s Var(nn) :5 E(:i) - [1 - F(xo )]
2
:: VareX) + E (X) -
2
2
X
o
[1 - F(xo )]2 x o
2
•
B
The upper bound of Var(nn) above can be improved by the inequality
given in Eq. (24);
B
...!! < X
n
-
n
Bn 2
_2
1
2
which leads to E(--) < E(X ) = - Var(X) + E (X).
-n
n
n
Therefore,
. (26)
47
B
It is of interest to know if the lower bound of E(..22)k given in
B k
n
E'q. (25) is the limit of E(nn) itself. It canbe shown to be so by
163 of Loeve (5);
utilizing a result on p.
If
IYn I -< Un with
U ~
I
n
I
Un
~
· U and
U finite, the~ Y ...E....;.. Y
implies that
By letting Y
n
I Yn
~
I Y.
n
B
== - n and U == -X ,
n
n
n
Y < U from Eq. (24).
n -
n
Further·, Un
-4,M
= E(X) =fJ.
and E(Un )
since Yn ~ [1 - F(x0
)J x
0
, finite (assume).
Therefore,
,
E(Y ) ~ [1 - F(x )] X
o
o
n
1·2.·
9
lim
B
n->ooE(!) = [1 - F(xo )] X o
B k
For k > 2, let Y = (...!!)
-
n
Since g(l ) =
n
xkn -4
xJ:cn
n
n
(= Var (X»
n
is a continuous fUnction of
f1k -Le.,
Un -E....;. f4k.
-
When k == 2, E(U )
t:l-
n
k
and U = X •
I:
EOc2)
n
is finite.
= rLJ2
I n , Xn -4fl
implies that
Now it remains to show.· E(U )
n
~ ,.Ll k •
2
+ SL ~ p.2 finite as n ~
n B
Hence, E(nn/
~
00
[1 - F(Xo )]2 x o2 •
In general,
E(Un )
= E(~) = lk
following reason:
n
E(JS. + X + ••• + Xn)k
2
~
p. k from the
if
48
Observe that
(X:t
+ X2 + •• + Xn)k can be considered as the sum of nk
t
.
terms of order k, and there exist exactly ( n. )' terms eaoh of which is
n-k •
the produot of k dii'ferent If s such as lJ.X ••• Xk • The other
2
t
k
n.
n T terms contain less than k different xts such as
(n-k) •
xiX
ilk_I' xi~X3
2 ....
E(xk) =
n
where
R~,
1[
k
n
•••
n:
~_29
I' k +
f
(n-k).
.. q
~
~ E(Ri
l]
so on.
,
Therefore,
(28)
f
k
i = 1, 2, ••• , n -
n. , ' are the terms with less than
(n-k).
).
k different Xv s in the expansion of
(JS.
+ X + •• + Xn)k.
2
Under the
assumption that E(Xk ) is fi,nite, E(R ) is finite for every i.
i
M= ~x {E(~)} , Mis certainly finite,
n.Y
k
[
n
(n-k)~
-
and
JM
l]
=
[1 - n(n-l )(n-2~k • •• (n"k+l
=
12k-I)]
[ 1 - 1(1 - i) (1 - i) ... (1 - -n- M~ 0
as n
lITh ereas,
Letting
t
k
I
n.
n
(n-k).
v
f
k
M
~fJ
~oo
k
as n --+
00
•
Therefore, using the results above, Eq. (28) yields
E(xk) .-.,. ffk as n --+
00
•
lim
En k
k
Hence, ~ooE(n) = [1 - F(xo )]
k
X
o
(29)
49
for every k :: 1, 2, """ if' E(xk) < 00.
Moreover, this result can be
used to prove that
B
Var(nn) ~ 0
as n ---;. 00 •
Effect of Filament Length on Bundle Strength
Generalization of the Weakest-Link Theory in a Bundle
It is a known phenomenon that the strength of a filament is likely
to decrease as its length is increased.
The probabilistic expression
for this, called "the weakest-link" theory by Peirce (6), is based on
the assumption that a filament of length
of
1
consists of a continuum
1 independent segments of unit length. Therefore, when the strengths
of the 1 segments are independently and identically distributed with
F(l)(x) as their common distribution function, the distribution function
of strength; F(P) (x), for the filament of length £ has the following
expression:
(30)
Clearly p Fen (x) is a monotonically increasing function of
x
E
i
for
(0, 00) and corresponds to the distribution function of the first
order statistic among
f
samples.
A generalization of the weakest-link theory in a bundle of n filaments
is immediate since the breakage of a bundle is realized by sequential
breakages of the n weakest segments of the filaments.
distribution function of strength for a bundle of n
.R,
Hence, 'the
fi~aments
of length
S(£)(x), is given in the following with F(x) replaced by F(J)(x)
n
in Fq. (2).
50
A slightly more general relation than Eq. (30),
g
F(R)(x) = 1 _
[1 _ F(m)(X)] m ,
(32)
can be used in order to obtain s(P)(x) in terms of F(m)(x) for the case
n
m ~ 2.
Although it seems intuitively obvious that s(R)(x) is likely to be
n
an increasing function of 1. 9 unlike the case for F(f·) (x), no indication
is given in Eq. (31) to support such intuition.
For this, the proof
of the following theorem utilizes a different way of expressing Sn(x)
which is due to Daniels (2).
Theorem
s~R)(x),
of
5
given by Eq. (3l)p is a monotonically increasing function
1..
Proof:
In deriving Sn(x), Daniels (2) starts 'With the multiple integral;
•••
.51
....
where
(33)
°~ F(~) ~ F(n~l) S
Similarly, by letting Zi
~ F(x) oS 1.
u.
= F(~)(Yi)'
i
=1,
2, ••• , n,
(34)
In comparing Eqs. (33) and (34), the difference in the definitions
of Zi dissipates as Zi become dwmny variables..
by letting F(l) (x)
F (jx) < F
= F(x),
co (x)
j'
In Eq. (30) it is noted,
that F(x) < F(R) (x) for x E (0, 00).
j = 1, 2, ••• ,
n.
Therefore,
(3.5)
Since the upper bound of the integration range monotonically decreases
along with the order of integration,it is assured that
Z
. < F(~j)
n-J
,j
= 1,
2,
.0-,
n
in Eq. (32)
This implies the integrand is always positive at every stage of integration, and hen.ce the higher the upper bound of the integra.tion range.
the greater the value after 1,ntegration.
Consequently Eq. (35) leads to
52
Sn (x) <
s~,n (:x)
for x E (OQ 00) and
i. ~
Alsop from Eg,. (32)9 F(i'(::;::) < F(m)(x) for 1
2.
~i <
Therefore p by a
m.
similar argument to the previous case, it can be shown that
S (i)(x)< S(m)(x)
n
n
for x
E
(Op (0) and I <
-
1<
mp
S(R) (x) is a monotonically increasing function of 1.
n
Corollary
The expected bundle strength with n filaments of length f ,
E(B~n)
monotonically decreases in
1..
Proof:
for I
s.! < III
E(B(~»
<n
since I -
s~,O (x)
> I -
monotonically decreases in
S~Ill) (x)
by Theorem
5.
Therefore,
.i .
Bee>
Asymptotic Properties of ~ for Large Bundles
B
The asymptotic properties previously derived for..1!
n
Ben
nn.
by assuming the distribution of filament strength is F("q) (x)
instead of F(x).
[I -
apply to
F U ) (x)
Defining
xCi)
as the value of x at which
o
Ben
Jx is maximized, it can be shown that ~ tends to follow,
asymptotically p a normal distribution with mean [I - F<R) (x CO)]
o
X
o
(R)
53
and variance 1
n
[1 -
[1 - FU)(:~: (R»
(')
F(x~Q»JR X o U)
]
and variance
[xU)
]2
0
FCO(x~ (Q»
or mean
$
~ [1 - F(x~,O)Jl {I - [1 - F(X~.f»Je}
• [x (Q )J2 in terms of F(x) which is assumed to be F(1) (x).
o
The value of
x(Q) satisfies the relation;
o
(36)
which reduces to
I-F(x )
o
when
i =1
..
B(l)
It is worth noting that the asymptotic mean of
Ben
n
that of - ..
n
This is true since
[1 -
-!-- is greater than
F(x) ] x > [1
-
R
F(x)] x for
every x E. (0, (0) and i?, 2, i.~"1J one is uniformly dominated by the other,
and. therefore t
o.<~x<
or,
00
[1 - F(x)] x >
[1 - F(xo )]
X
o >
° <~x<
00
[1 - F(x)]ix
[1 - F(x~R»Ji x~R)
However, no general assertion can be made as to whether or not the
B(l)
B(R)
asymptotic variance of ...2'L- is greater than that of -E- .
n
n
Using a normal distribution, an example is given in F.i.gure 6 to
show the change in x(,Q) as
o
1.
is increased.
The points x(l), x(2) t
.
0
0
are determined in the Figure by the absissae corresponding to the
intersecting points of y
=Rx
and y
=l(F~X}
f x
•
The decrease in the
....
--- .....,
1.0
-)(
.....
0.8
",,- F (x)
, 0.6
"
lA-
-.
)(
"",,
"-
0.2
12
--
I-F(x)
f(x)
X(I) -
o
6x 5x
F(X~»
)(
--
2.335
= 0.253
4x
"-
.---.
"
-
8
)(
lA-
6
I
.
~
)(
~
4
2
X~)
2 X~)
3
4
5
X
Figure 6.
Graphical determination of x (~) based on
filament strengths distribut~d as N(3,1)
6
55
value of
x~R)
in
also with almost
very slow as
i
1 is
obvious not only in the parlicmlar example but
a~ continuou~
f(x).
The rate of decrease becomes
gets larger.
Effect of Length on the Wasted Fraction of Filaments in a Bundle
It was shown in a previous section that the wasted fraction oan be
estimated by F(x ) when the distribution of filament strength is given
o
by F(x).
Similarly, when the distribution is given by F(R)(x) for the
filaments of length
by 1 -' [1 - F(l)(x
o
~,
the wasted fraction is estimable by F(Q) (:x:~.Q)" or
(2» J.R.
Though one may suspect that the wasted
fraction will increase along with the increase in 1., it can be false
in general.
The exact oondition for the above to be true is given by
the wasted fraction is invariant in
Bee>
n
asymptotic variance of -
n
b y -1 e -l(1 n
-1) n
(1 )2
e
,
",(l
1 for
large n.
Incidentally, the
for the above exponential case is given
,,'
and it decreases monotonically
in )(n •
Probabili ty of Strength Retention in Bundles of Augmented Length
Consider a case where a single filament or a bundle has to be
augmented by an extra length of filament or bundle to meet certain
purposes.
The length consumed in the augmenting process, such as
knotting, is excluded from consideration..
One practical problem in
56
this 5i tuation is to maintain an adequate strength after augmentation,
and it is of particular interest to examine the probability of retaining
the original strength after the length is augmented.
Furthermore, the
changes in such probability can be examined relative to the length to
be added as well as to the number of filaments in the bundle.
First, it is asserted that the strength of a filament or a bundle
cannot be increased after the length is augmented.
For a single
filament, the weakest-link theory provides the trivial proof' for the
assertion.
Precisely, when a filament of length m is addea. to a
filament of length fl, the strength lowest among the i + m segments is
necessarily lower than or equal to that among the original'£' segments.
For a bundle of n filaments 9 the argument is as follows:
Let
{Y~~)} • i·~ 1.
2..... n bo tho ordor statistics corl'E>sponding
to the strengths of the original n filaments of length;', and let the
be X~H+m) after
J
J
augmentation. Then, from the single filament case, yjCe> > X~R+Il1)
- J
for every j := 1, 2, ••• , n. Therefore, when the order statistics
strength of the particular filament 'With
corresponding to
(Q)
that Yj
~
{xiR+m)} is called
y~~)
{yiR-tm)}
(R+m)
.
Y
for every J = 1, 2, ••• , n.
j
.
. . =1 . 2 , ... ,.
j
(n-J·+l)y(.f() > (n_j+l)y(..e+m)
J
J"
max
{nY(~)
l'
or,
~~.£} ~
(n..
l)Y(.~)
2'
..• ,
B;;-tm) p where
y(..
£)} -> ' "
Therefore p
n and •
•
max {ny(£+m)
n
B~R+m)
, it is trivial to show
1
'
(n l)y('£'+m)
-
2
is understood as the strength of the
bundle augmented to the bundle 'With
B~-e)"
Furth&rmore',
J
y(.J.-tm)l
, ••• , n
57
Pr [
xce ) =
=P
r
x(R+m)] = 1 - Pr
[y(1) ~
[xCe>
>
x(f+m)]
z(m']
where 9 y(R> is the first oroer statistic among the original
R segments,
and Z(m) is that of the m segments to be added.
Letting the distribution functions of Y(£} ani Z(m) be G(j ) and
H(
~)
respectivelY9 since y(1) and Z(m) are 'independent..
==
J G( ~ )dR( 'l )
o
= Z{l- [1=m
1
F(7)i} · m[l- F(?)y-lf(j?)d'i'
F(~ )Jm-l f ( ~)d~-
[1 -
m
1
[1 -
F(Y')J..e+m-lf(~)
df
= l+m
~
(37)
for an;y F(x)
~
P1" [X(l) oS xJ and f{x) = F' (x) whioh are oontinuous.
For a bundle of n filaments, the. event
B~R) = B~R+ll1)
becomes
mOre complex. Vdth the foregoing definition of y~R), let Z~m) be the
".
"
J
,J
strength of the particular filament of length m to be added to the
filament with
y3R).
Then, it is noted that
is an ordered set whereas
for ever,y j
= 1,
{zi >}
2, ••• , n p
m
Y~~)
J
9
and
{YiR~
,
i = 1,2,
i = 1, 2, ••• , n is not.
Z~m)
J
.u, n
Also,
are understood as the first
58
order st.atistics among the j and
III
segments respectively.
Then the
following identities are easily understood;
P ·[B{Q)
rn
~
= B{-,?+m)l.
~~
t=..
.
k=l
• Pr
P {
[B{Q)
rk=ln
J
n
P .[y(. R)<
r
k -
= B{R+m)
mi· n(z.{.m)
-k
n
Bee>
'n
z.(m)
'-k+l' ••• ,
[B~~) = (n-k+l)y~R)J
= {n-k+l)y(Q)J}.
k
z(m»]
n
(38)
•
In other words, the event B(n = B(X'+m) requires the condition that if
n
B~R)
were represented by
y~~), y~~i, 00" y~O
n
(n-k+l)y~R),
the filaments corresponding to
should not decrease their strengths below
y~R)
after the lengths are augmented, whereas no such restriction is
• . '
(Q)
CV
0)
necessary for the filaments correspond~ng to YI ,YZ ' ••• , Y - •
k l
This is to say that the filament corresponding to
Y~£)
must retain
its strength as well as the order of strength k after the length is
augmented~
Observe
independent.
The proof of this is elementary and excluded here.
. .
~n
Fq.
( 8)
3
(Q)
em)
em)
em)
tha. t Yk ,Zk 'Zk+l' ••• , Zn
are all
Further, they are the first order statistics based on
samples with a common strength distribution F{x) of a segment.
Therefore, utilizing the notation for the single filament case,
P [y{R) < min (7_{m) z.{m) ••• , Zn{.m»J
r
k . -k '-k+I'
= PI' [y{R) :5 Z {(n-k+l)m)]
.
{(n-k+l )m)
where, Z
represents the
statistic among {n-k+I)IIl samples,
=
1 +(n!k+l)m
.
f~rst
by analogy to Eq. (37).
order
59
(39)
where
p
= P [B (.,() = (n-k+1 )ykCO ]
'nk
r
n
n
and
1:: Pnk = 1.
k=l
But,
(40)
for ever,y n = 2,
3,
4, ••••
The inequality above indicates that a bundle is mOJ:'e prone to
strength loss than a filament when their augmented lengths are same.
The changes in probability of strength retention are well reflected in
Eq s. (37) and (9) for the change in relative size of
.1
to m.
Relaxed Model for Breakage of Bundles
The breakage model introduced at the beginning of this study was
called a 'restricted model' in the sense that the assumptions used in
the model impose certain restrictions in its use.
th
It was assumed that
filament break is exactly (n-k+l)Y , or
k
th
the load in arty surviving filament at the k
break is Yk on the average.
the bundle load at the k
60
This assumption is valid when the variation of load among the- surviving
filaments is low or the value Yk is always olose to the median of the
filament loads at the
moment of k th filament b reak. Therefore. as soon
as the load-extension oharaoteristics of filaments deviate from the above
ideal situation. the validity of
th~
foregoing analysis beoomes
questionable.
Distribution of Bundle Load
The model to be given in the following is to overcome the deficienoy
of the previous one.
In thi s model. the bundle load at any- given
extension point is expressed by the sum of the loads in the surviving
filaments at that particular point. and the variation of load among
filainents at a given extension is justified by considering the filament
load as a random variable f'unotionally dependent on extension.
Let
I..i. (y)
= load on a filament at extension y
Si
= a positive random variable representing the hypothetical
asymptote of breaking load of a filament
Y
i
= a random variable which defines the breaking extension
of a filament
= a positive constant inherent to the material
and
li (y)
, i
= 1,
if Y. < y
~-
=1
if Yi > y •
2, ••• , n
(41)
61
Xi (Y)
(jj
~~
-- ---- .... ---
Q
Jt.
«
0
-'
y.
Y
I
EXTENSION
Figure
7.
Load-extension curve of a filament
with relaxed model
Then, the load-extension curve for a single filament is as given in
F.i.gure 7 based on Eq. (itlh
The shape of the load-extension curve shown
in Figure 7 is not at all an arbitrarily chosen one.
Experimental
results with quasi-elastic filaments often indicate the validity of
using an exponential curve as above with an asymptote.
shape of an
experiment~l
Although the
curve may differ at the portion pnor to the
so called 'yield point', the inaccuracy there can be ignored simply
because the actual breakage occurs far beyond the yield point.
Now, the strength of a bundle can be defined as the maximum
load attainable in a bundle as before.
Since the bundle load at a
62
given extension is the sum of the filament loads at that point,
the bundle strength; B , is given by
n
=
n
max
I;
O<y<oo'l
J.=
-.
S. (1
-
J.
~)
• 1
[Yi > YJ
(42)
•
n
t X1<Y) resembles that
i=l '
given in Figure 1 except that there exists no regularity that governs
The load-extension curve of a bundle based on
the size of the jumpso
= 1,
Under the assumption that Xi (y), i
are independently and identically distributed and that
~i
n
statistically independ.ent, the distribution function of
and Y are
i
t
i=l
n-fold convolution of the distribution function of Xi (y).
Define
H(u)
X. (y) is the
J.
= Pr (~.J.-< u)
F(v,y) = Pr [X.'J. (y) -< vJ
~S(m)
00
=I
o
•
eJ.mu dH(u)
00
aXe. )(5) =
Y.
I
0
e
isv
dF(v,y)
L.!_. 'P13(m) and aX(y)(x) are the characteristio funotions
corresponding to the distribution of 13 and X.J..(y) respeotively.
1
Then,
2, • u, n
:: Pr [~i (1. - eay) ~
l
'IT
[1 ~ G(y
>J
-I
+Pr(O~v) G(y)
=Pr
:: H (
1
=
V_ay) [1 - G(y)J + G(y)
e
Note that F(O, y) :: G(y), !~.!'"
•
(43)
F(v 9 y) has a jump at v ,. 0, but
otherwise it has a corresponding continuous dens!ty of the following
for v :> 0
~
and therefore 9 F(v 9 y) is a mixed-type distribution function.
:ay
00
:: I
o l=e
h(
letting u
==
[1
=
~) [1
1-e
=
'IT
l-ecr:r
G(y:)J (1 -
=
G(y)] dv
,
eo
ecr:r). J0 '!
h(u) du
(44)
64
Similarly,
E~~(Y)J
Ther~fore,
var[xi (y)J
= [1 - G(y)]
:=
(1 - eO¥)2
2
(1 - eO¥)2 E(13 )
[1 - G(y)]
2
- [1 -
G(y)]
=[1 -
G(y)J
E(~2)
(1- ;/J;J')2 E2 (13)
(1 - ~>2 [E(a2) - {1 - G(Y)} Ef(a>] .
(45)
It is noted in Eq. (4.5) that
Var [Xi (y)J
~
(1 - eO¥)2 Var(13) •
[1 - G(y)J
The characteristic function eX(y)(s) corresponding to F(v, y) can be
obtained as follows:
co
exy
( )(s) = I0
e
isv
dF(v, y)
00
= G(y) + I
o
e
isv
letting u
(
h(~Ct\Y)
1 )
l-eO¥
=:
v
l-;O¥
00
= G(y) + [ 1 - G(y)]
I e
o
[1 - G(y)J
dv
l-e
,
-O¥
is(l-e)u
()
h u du
by the definition of lfJ13 (m),
= G(y) + [1 - G(y)]
l.IJ13 [s(l
- ;;O¥)J
n
By letting Ln (y) =
~
i=l
X,)(y),
the bundle load at extension y,
. -i.e.,
-
(46)
65
Var [Ln<Y)]
=:
(1_eO'Y,2
n [l-G(y)]
¢
and the characterist.ic flIDctiem
L (y)
{E(~2)
(w)
.
n
-
[l-G(Y>]
E2(13~
(48)
corresponding to the
distribution of Ln(Y) is given in terms of ElX(y)(w) as;
ep
~(y)
(w)::
[€lX( ) ( w ) ]
1'1
Y
= {G(Yl
+
[l-G(Yl]
1jJ~
[W(l-e"'Y l]}
n
AI though the distribution of L (y) is understood as the n-fold convolution
n
of F(vgy)g it can be characterized by Eq. (49) as well~
The results given by Eqa. (47)J (48) and (49) can be obtained by an
alternative approach in which the physical meaning of the sum
n
E Xi (y) is more appealing than the previous method. The analysis given
i=l
in the Appendix explicitly defines the bundle load with relation to
the order of breakage extension.
The distribution of bundle strength defined by Eq. (42) is in fact
n
the distribution of L (y"') = I: Xi (y"') , where y '" is the point which
n
i=l
g
maximizes L (y). Therefore all major properties of L (y"" inherit
n .
tha.t of L (y) if y* can be known..
n
n
Howeverf) nothing is known about y'"
other than the fact that y'" occurs at one of the breakage points
Yp
Y2 9
...
0
and Yno
This is so because Ln(Y) monotonioally increases
within the intervals Y(k-l) < y .$ Y(k)' k
=:
If) 2 9
.... 9
n with
66
Yeo)
=:
0 and 0 S. Y(l) < Y(2) < ••• < Yen) <
ordered set of {Yi }
< y < 00 1n (y)
-
1
{Y(iJ
are the
max
n
)
1 < k < n I: Xi (Y (k)
i=l
II:
n
<k <
()
n . I: Xi Yk
1.=1
-
Therefore,
max
~x
=:
!.!...
,i:=l I, 2, .... , n, and hence Ln (y) achieves its
local maxima. at Y(l)' Y(2)' ... , and Y(n).
Bn :: 0
00,
•
The distribution of the above random variable is hardly obtainable under
the definition of
B
I.t (y).
If' the interest is in getting an approximate
estimate of E(..1l), however, the following oriteria is of value:
n
(50)
where, y
o
is determined from
dE [Ln(Y)]
dy
with
d~¥)
:I
:=
0
whioh leads, when G(x) is continuous
g(y) ,
•
In order to examine if the Yo that satisfies Eq. (51) provides the
maximum for E
[1n (Y)]
, observe first that
67
Theref'ore t when
the marl:m:wn
th~
solution Yf) is unique it necessarily provides
:f'o:r-'E~nbr}lwhen Eqo
(.50) is valido
If there exist two
solutions Yol and "'02 with Yol <: Y02 t then E [Ln(Yol ']
since
~
E [Ln(Y)]
is the maximum
ca.nnot be zero having started from E [Ln(O)J :: 0
0
In case there exist more than two solutions that satisfy Eqo (51), the
criteria of selection is reduced to those Yop i:: It 2, ooq k,
say 9 a. t which
or, equivalently,
Then» Yo is selected among the y . to satidy
o~
Asymptotic Properties of L (y) and B
1'1
1'1
It was shown in EqSe (44) ani (45) that
E [Xi (y)J
Va"
[xi (YlJ
:: [1 - G(y)]
(1 - ;pY) E(a)
= [1 - G(yl] (1 - _""l2
and
{E(~)
- [l-G(Yl]
E2(~l}
Therefore» since Xi are independently and identically distributed, by
the central limit theorem»
d
~ N (Ot l )
(52)
.
68
as n
--+~ c.o
for every y ::> 0, am the above is also true with y == y
o
SUMMARY AND CONCLUSIONS
In defim.ng the strength of a. bundle of n para.llel filaments, two
breakage models were constructed based on the tensile properties of the
constituent filaments.
Under each model, studies were carried out in
order to examine the statistical nature of the bundle strength.
The restricted model for bundle breakage was based on the postulates
that the load in a filament monotonically increases along with the
increase in its extension and that the total load in a bundle at
a~
extension point is equally shared by the surriving filaments at the
The distribution function of bundle strength, S (x), was
extension.
n
derived under the restricted model by employing a probabilistic argument
of events associated with the breakage of a bundle, and it was given in
terms of S
n-
l(x)p S
n-
2(x)p ••• and F(x), the distribution function of
the filament strength as well as in terms of F(x.), F( Xl)' ••• , and F(x).
n
n-
Although 3n (x) fa.cilitates computation of the moments of the randoJll
variable B p the strength of a bundle of n filaments, the complex
n
feature of Sn (x) lind ts its pra.ctical usefulness to relatively small n.
B
The a.symptotic properties of ....!!. were derived directly from the
n
definition of B
n
(Eq. (1» without utilizing S (x).
n
.
B
n
By converting -
n
to a function of the empirical distribution of the filament strength,
.
B
it was found that nn converges in probability to a constant determined
from F(x).
For a large n p it was shown also that the distribution of
Bn
- tends to follow a normal di atribution with mean and variance
n
dependent on F(x) (Irq. (14».
Further, the mean coincides the lindting
constant and the variance approaches zero as n becomes large.
70
By defining the wasted fraction of filaments in a bundle as the
ratio of the number of fila.ments broken before the load in the bundle
reaches its maximum to the total nU1l1ber of filaments in the bundle. two
different methods were proposed for its estimation.
Theoretically 9 the
strength of a burrlle is not affected by eliminating the filaments
that belong to the wasted fraction.
The average contribution of the individual filament to the strength
B
of its bundle. nn 9 tends to decrease as n increases. It was proved
for any distribution function F(x) of filament strengths. where the
constituent filament strengths are interohangeable random variables,
B k
n
tha t E(il)
decreases monotonically in n for every k = 1, 2, 3., ••• •
B k
It was shown also that the lower bound of E(-!!) coincides its limit
n
k
k
X
9 where X
is the mode of the function x [1 - F(x)]
o [1 = F(xo )]
o
assuming it is unique. whereas the upper bOl.l1"l.d can be given by either
E(ik,
or by
E(xk,.
n
B
From above, the upper bounds of Var(-1!) were
n
established (Fqs .. (26) and (27» and its limit was shown to approach
zero as n becomes large.
B k
In a finite population of filaments, E(..2!)
n
was shown to decrease
monotonioally in n when the expectation was defined under· the scheme
of simple random sampling.
A trivial, but useful case of the above was
B k
vk
that (n1'1) S, ~ for any k :> 0 if the n filaments were connnon in the
definitions of B and
1'1
X..
n
In any event, the monotonici ty of
B k
n
E(rr-' above implies the decrease of bundle efficiency along with the
increase in its size n 9
L~. 9
the increase in bundle strength is not
proportional to the inorease in n, but always less on the average.
71
The effeCiit 'Of filament length on the strength of a bundle was
examined by generalizing the weakest-link theory applicable to the
strength of a filament"
As expected, the probability of breakage in
a bundle of n filaments was shown to increase along with the uniform
increase of filament length within the bundle at any fixed load level,
i"e.,
--
sU
)(x) was shown to increase monotonically in i for every
n
positive
Xo
Hence, the expected bundle strength was shown to decrease
for a uniform increase in filament length.. The asymptotic mean am
BeQ )
variance of
were found for bundles of length .R by relating FCe)(x),
if--
the strength di stribution for filaments of length
i,
with F(l) (x),
that for the filaments of unit length. The results were such that the
B
asymptotic mean of nn was shown to deorease for an inorease in
filament length, however the same was not true in general for its
asymptotic variance"
No assertion oould be made as to whether or not
the wasted fraction is increased along with the inorease in filament
length.
In a bundle where the filaments of length
~
are augmented by
filaments of length m, the probability was obtained for the event that
the strength of the augmented bundle with length .R+m is same as the
strength of the original bundle with length
.R (Eq" (39».
The results
indicate that the probability of strength retention is decreased by
eithe l' increasing m or decreasing
i, or by both. Also, such probability
in a bundle was shown to be less than that in a single filament for fixed
The relaxed model for bundle breakage was designed to allow the
variation of loads among the surviving filaments at any particular
72
eJ.."tension of a. bundle ~
var.iables
Considerlng the filament loads as random
.fun~tio:nally dependent
on extension g the model defined the
bundle load at a particular extension to be the sum of n independently.
and identically distributed random variables representing the filament
loads at the extension.
The distribution function of bundle load was
easily defined based on the distribution of filament load at any
ext.ension.
The distribution function of bundle strength. however. was
hardly definable in a closed form.
Hence 9 the mean and variance of
bundle strength were estimated by utilizing the mode of the expected
bundle load.
The asymptotic properties of bundle strength were deduced
by using the central limit theorem.
Although only a particular load
function was given in the studyg the modification will be immediate
with other load functions which are similar to the one given in this
study.
73
LIST OF REFERENCES
L
Coleman, B. D. 19.58
On the strength of classical fibers and
fiber bundles" J. Mech o Phys. Solids 7:60-70.
2"
Daniels, H. E" 194.5. statistical theory of the strength of
bundles of threads. Proc. Roy. Soc. (London) Al83:40.5-43.5.
3.
Doob, J" Lo 19.53. stochastic Processes.
Inc., New York.
4.
Kolmogorov, A. No 1933. Sul1a determinizione empirica d1 una.
legge d1 distrlbuzione.. Giorn. dell' Inst. ItaL deg1i
Attuarl 4g 83-91.
..5.
Lo~ve, M.
6.
Peirce, F. T. 1926
Theorems on the strength of long and of
composite specimens.. J. Text. Inst. 17:3..55-368.
0
1963. Probability Theory.
Inc., Princeton, New Jersey.
0
John Wiley and Sons,
Third Ed. Van Nostrand Co.,
74
APPENDIX
.An AlteI.r.native Analysis of L (y) with Relaxed Model
n
The bundle load L (y) at extension y was previously given by the
n
sum of the filament loads at that extension. and the definition of
Xi (y) given by Eq. (37) indicates that the contribution of
tor y > Y., i.e. after the filament is broken.
~
--
is.. (y)
is nil
Therefore, L (y) is
n
contributed by all n filaments at 0 S y ~ Y(l)' by (n-l) filaments
at Y(l) < Y S Y(2)' ••• 9 by a single filament at Y(n..;,l) < Y S Yen)
and finally Ln(Y) is zero for Yen) -< y since no filament survives
in that region.
i
:=
Here,
{y
(i ~
are the ordered set of {Yi }
,
1, 2, ••• , n, the breaking extensions of the n filaments.
Without loss of generality, let the
···9
Y(2)' ••• , Yen) be ~, ~29
SIS
corresponding to Y(l)'
~n' respectively.
Then from the
argument above, Ln(Y) can be defined as follows:
n
L (y):: t
n
i=1
n
Xi(y)
==
t
i=l
== 0
S· (l-eay) ,0 S y S Y(l)
~
75
The1"e:tore,
[~(y) ~~J = Pl"[~(y) 5,~
Pr
/ 0
~ y ~ Y(l)]
~~
/ Y(l) <: y
+ Pr [L",(Y)
~~
/ Y(n-l) <: Y ~ Yen)] Pr(Y(n-l) <: Y ~
:I
P
~/
r[·:.i=l~ ~i (l-~) ~ %J.
Let
ani
~
==
Pr [Tk(y) <'
(~)
J
Yen) <: Y
•
III
~ Y(2)]
~ y ~ Y(l»
+ Pr [Ln(Y)
+ Pr [Ln(Y) S
Fk ( ~ ,y)
Pr(O
'J
[G(y)]k [l-G(y)] n-k •
Pr(Y(l) <: Y
Pr (Y(n) <: y)
[1 - G(y)]
n
~ Y(2»
Yen»~
76
Note that X:(y) are independently and identically distributed since
J.
l3i are
SOp
and that the distribution of Tk(y) is the k-fold convolution
of the distribution of X~(y)o
When the distribution of
Hence,
13.J. has a continuous density, so does the
Ln (y) has a continuous density for 0
l~
00
Because
dFk
(? ,y)
<' '
Therefore, Pr [Ln(Y)
distribution of Tk(y).
= E [Tk(Y)]
=.[
n-l
]
I: a. (n-i)
i=O J.
=n
[l-G(y)]
= OJ = an'
of the form;
= k(l-eClY ) E(I3) ,
(53)
The last equality above is due to the identities
n-l
I: (n-i) a.
i=O
J.
=
n-l
i
I:
n.
I [G(y)Ji [l_G(y)]n-i
i=O i~(n-i-l).
Similarly,
2
E [Ln (y)l
J
n=l
= I: a.
i=O
J.
00
J ~
0
but otherw.:i.se
2
dF . ( ~ ,y) + a • 0
n-J. ?
n
77
Therefore,
E
..
[L~(y)lj
«[ n~l.
i=O
(1_e~)2 E(~2)
<:8-i)a.J
1.
+ [ n-l
~ (n-i) (n-i-l)a.
. 0
1.=
1.
J(l-eay)
. 2 E2(~)
since
n-l
~ (n-i)a = n [l-G(Y~
i
i=O
and
n-l
n-2.'
i
n-i
~ (n-1)(n-i-l)a =
~
t n.
t [G(y)J
[l-G(y)]
1
1=0
i=O i. (n-i-2) •
a.s before
= n{n-l)
[l-G(y)] 2
Hence,
Var [Ln{Y)]
= E
[L~(Y)J
-
= n [l-G(y)]
..
E?- [Ln(Y)]
(1_;0.>,.,2
[E(~2)
- {l-G(Y)}
E?-(~)]
(54)
•
The characteristic function corresponding to the distribution of Ln (y).
is obtained as follows:
rh
r 1 wL (Y)J
(w) = E Le
n
TLn(Y)
n-l
= i.O
~
00
a.
1.
iW
!e
;
0
dF .(';,y)+ae
n-1. ?
n
o
.-
78
But,
ie
o
iW
,%
dFk(~'Y)=
fie
L0
= ik
.
•
00
where
and
1t'13(m) ==
,
.
J
•
dH(u)
H(u) == P (13. < u).
r
dFl(~,y)lk
[eiu.qa(l-eo;r)]
J e~mu
o
iW
~-
Therefore,
(5.5)
The results given by Eqs. (53)1/ (54) ani (.55) are same as that given
by Eqs. (47), (48) and ,(49) •
•
•