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NATIONAL BUREAU OF STANDARDS REPORT

2355

A HISTORICAL NOTE ON THE APPLICATION OF THE

"WEAKEST-LI NKH IDEA TO TENSILE STRENGTHS

by

Julius Lieblein

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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Errata

for

a National Bureau ©f Standards Report 23SS

?*A Historical Not© on the Application

of the iWeakest^Link 9 ld©a to

Tensile Strengths®9

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-11-1107 12 March 1953 2355

A HISTORICAL NOTE ON THE APPLICATION OF THE

*WEAKES T-LI NK **[ DEA TO TENSILE STRENGTHS

by

Julius Lieblein

PREPRINT

The publication, rep Approved for public release by the i part, Is prohibited

unless permission is i Director of the National Institute of ndards,Washlngton 25 ,d,c, Suchpern Sta nda rds and Technology ( N 1ST) ort has been specifically prepared If th; On October 9 2015 mrt for Its own use.

A Historical Note on the Application

of the "Weakes t-LinkH Idea to Tensile Strengths

by

A*

4% Julius Lieblein

ABSTRACT

It appears to be commonly believed that the use of

the ®weakes t-linkw hypothesis and extreme value methods

in connection with strength of test specimens originated

with F. T. Peirce in an article published in 1926.

The discovery by the writer of a pair of longforgotten articles in two engineering journals of the

l880*s shows that we must push the date of first application

of extreme values back by nearly £0 years, if not more.

These articles also show a surprisingly modern approach,

and it is therefore felt worthwhile to discuss them to

some extent, particularly since they may not be readily

accessible

.

National Bureau of Standards, Washington, D. C

INTRODUCTION

Practically every writer (e.g. Epstein [1]) who

makes a historical reference to the use of extreme values

in tensile strength testing dates such application from

Peirce's paper in 1926 [2], which in turn is based on

Griffith's theory of flaws enunciated in 1920 [3 l* In

the Interests of historical accuracy, it therefore seems

important to report any material that comes to light

which would significantly affect the widely-held belief

concerning priori ty 0 The purpose of this note is to

discuss excerpts from two articles written by W. S,

Chaplin, Professor of Civil Engineering, University of

Tokyo, that appeared in American engineering journals over

80 years ago and recently came to the attention of the

writer s ^ The first [[|J was published in Van Nostrand's

Engineering Magazine for December 1880 and was entitled

w The Relation Between the Tensile Strengths of Long and

Short Bars:'' the second [5] appeared in Proceedings of the

Engineers 1 Club for 1882 and was entitled ”0n the Relative

Tensile Strengths of Long and Short Bars,"

IT The writer is indebted to Dr. Churchill Eisenhart for

showing him an old engineering text [6] in which these

articles were cited.

- 2 -

CHAPLIN ' S ARTICLE OF 1880

This was a brief article which explained the simple

theoretical points involved in the nweakes t-linkn procedure* The author began by mentioning the applicability

of the (Gaussian) "law of errors" to the variation in

strengths of individual specimens or pieces of a given size,

and showed that p is the chance that each of n pieces

of the same size would exceed a certain tensile strength

if p is the probability that each piece exceeds it» He

then considered the key question that was the purpose of

the article (p. 44^) ‘

” Suppose that many pieces of cross

section c 5 and length one inch have

been tested for tensile strength

with an average result S

q

, and a probable variation in one piece of

P

q

; what will be the probable average

strength, S. , of pieces of the same

n

cross section and a length of n

Inches ?

- 3 -

n Knowing the probable variation

in a piece one inch long, we are

able to construct the curvp showing the probability of ahy and all

variations in a piece of this

length. From this curve we can

obtain the probability that the

piece one inch long will break

between any limits of variation.

The probability that an inchpiece will break above a negative

variation, -x is + A , in which

x

A represents the probability that

X

the piece will break between 0 and

-x. In a piece n inches long there

are n pieces one inch long; the probability that any one of these will

break above -x being «5 + A 3 the

probability that all of . them will

break above this limit, or that

the strength of the whole piece

will be at least S [emphasis

Q

;

supplied] will be

(.5+A

x

) n

.

-k

As Sx

is an average, it is as

probable that a piece n inches

long will break above i t as below

it; hence the probability that

a piece n inches long will break

above it is We have then

(.5+Ax ) n

= .5 ,

in which A is the unknown quantity.

We easily obtain

a -•£; or = 2(vTF -. 5 ).

"

In (not very different) modern symbols and terminology this stated essentially that if |)(x) is the

(Gaussian) probability that the tensile strength of

a piece of given length is greater than x, then the

probability that a piece n times as long has strength

greater than y is [])(y)]

n

. Putting [<£(y) ] n

= §• and

using the appropriate normal tables yields the average

(i.e. median) value of the strength y of a piece n

inches long (of similar cross-section).

The underlined portion above is the essence of the

wweakest-link" theory— that a chain is no stronger than

its weakest link. It is rather interesting that, unlike

we of today, the author managed to express the idea

quite clearly without the use of this overworked phrase*

The author then converted the relation [J)(y) ] = "k into a useful table and chart, and explained that if ^ were replacedby 1/4 in this relation the corresponding

value of y would give the probable error for variation

in strengths of the long specimen,, This theory was then

compared with the results of tests on 5 specimens of

annealed Japanese copper wire of lengths 1, 4 j> 12,

16 inches, and agreement of theory with experiment was

found to be very satisfactory* Other tests were cited,

and one of these sources [ 7 ] provided a clue for a still

earlier reference (I 87I) to the idea of a material break!

at its weakest part. The entire passage involved is

interesting, and is here quoted in full (emphasis

supplied )

1

n The idea was at one time prevalent,

that the tensile strength of a

bar of iron, etc., became greater

by the bars being broken; for it

was found that a piece of a bar

broken by tension would bear more

than the entire bar did; and that

if a piece of a piece were broken
.

'

Wm:
” 6 ”

in the same way s it would in turn

support more than before 0 But this

proved nothing more than that the

bar broke at its weakest part first s and at the next weakest in succession

The article concluded (p 0 khk) with a note of

advice that is not without interest in the light of

modern experimental procedures?

n It is to be hoped that those who

have testing machines and occasion

to make numerous tests will publish

either all their individual results s or will give the probable variation

as well as the average strength of

the materials which they study 0 It

really tells but little about a

material to give only the average

breaking weight | uniformity of

strength, or a small probable

variation, is a very valuable quality s and without knowing whether

a material has a small probable

variation or not, no engineer can

properly decide what factor of safety

shall be used in designing a structure,/*

7

CHAPLIN'S ARTICLE OF 1882

This later paper 9 which went into greater detail

than the one in 1880 5 had for Its purposes (p„ l8)s

H 1 o To prove that long bars are on

the average weaker to resist

tensile stress than short ones

of the same material and cross»

sec tion;

M 2. To show how the reduction in

strength may be found when the

proper experiments have been

made on the short bars o'*

The author's first point was carried out In a few brief paragraphs,, After explaining that iron

is not a uniform material and that tensile strength

must vary from bar to bar s he stated (p„ 19 )°

® A bar under tensile stress

yields and breaks at one points

it is more or less extended at

all points,, but it breaks at only

one s and It is the strength at

this point which determines the

strength of the bar to resist

tensile stress

- 8 -

M Suppose now that pieces a, a , a^,

etc., have been cut from a bar

A, and tested in the usual way„

The average strength of the pieces

a, a^, s.^ 9 etc., will necessarily

be greater than the strength of

the weakest piece; or, as it is the

weakest piece which determines the

strength of A ? [emphasis supplied]

the average tensile strength of

the pieces a, a^, a^, etc OJ will be

greater than the strength of A. Or

long bars are on the average weaker

to resist tensile stress than short

of the same material and cross-section . It would have seemed unncessary to

point out and insist on this fact, if

it had not been denied by as great an

authority as Fairbairn. w

- 9 -

Apparently this doctrine of the "weakest piece" must

have been promulgated sometime earlier^ in order for

to have been considered with sufficient attention to

elicit a denial by a top worker in the field.

The author's line of reasoning concerning his

second point, calculation of the reduction in strength

due to length, was essentially an elaboration of his

method in the first article. It is also interesting to

note his lament on the paucity of experimental data of

the kind

that his

necessary for testing the theory. He indicated

theory would have important practical application

in showing that the usual factor of safety was always an

overstimate. Thus for the Niagara suspension bridge,

he found that the safety factor of 3.07 should really be

only 2.76, a reduction of 10 percent.

In spite of the promising outlook for the theory

the article seems to have ended on a pessimistic note

and a hope that is still far from being realized (p. 23)

V Other evidence for this is provided by the quotation

from [7] given above

- 10

M> Although I do not hope to see any

practical application of this law,

yet I consider it of some importance

as one of those steps which will

finally give us such a knowledge of

the strength of materials that

engineers will no longer be obliged,

as they all now are, to make

structures much stronger in practice than they are in theory. w

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