The fatigue of metals : with chapters on the fatigue of wood and of concrete

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BOOK 620.1 12.M78 c. 1
MOORE # FATIGUE OF METALS

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THE FATIGUE OF METALS

August Wohleb,

THE outstanding pioneer figure in tlie experimental study of the
strength of materials under repeated stress, was born June 22,
1819, at Soltau, in the Kingdom of Hanover. He was the son of a
schoolmaster, and obtained his schooling at a trade school in
Hanover, where he had a scholarship. After his school days he
had practical experience in railway construction in the Kingdom
of Hanover, experience as a draftsman with the A. Borsig Loco-
motive Works in Berlin, and experience as an engine driver in
Hanover.

He entered the Prussian Railway Service in 1847 and in 1852 was
appointed a member of a commission to study causes of derail-
ments and details of locomotive construction. This work led him
to study axle failures in rolling stock.

In 1852 he succeeded in establishing in Berlin an experiment
station for tests of metals under repeated stress. It was in this
station that his great life work was done. This station was soon
enlarged by the addition of equipment for general materials testing,
became part of the Gewerbeakademie, and later a part of the
Technical High School.

Wohler's famous tests were made between the years 1852 and
1869. His machines are still preserved, and the commonest form
of repeated-stress testing machine in use today, the rotating-beam
machine is practically the same as the machine designed by Wohler.
In fact, the names "Wohler machine" and "Wohler test" are
frequently used in connection with rotating-beam fatigue tests.

Wohler did his work before the days of the metallurgical micro-
scope, and his concepts are the stress-strain concepts of Weisbach
and Rankine, but his critical value of stress, below which failure
wUl not occur even after an indefinitely large number of cycles of
stress, is the same value which today, under the names "endurance
limit" and "fatigue limit," is regarded as the most significant
index of the fatigue strength of a metal.

Wohler became manager of the Berlin Railway Car Works in
1869, and in 1874 was made one of the general directors of the
Alsatian State Railways. He retired from active professional life
in 1889. During his later professional years he advocated impact
tests for determining the acceptability of rails, axles, and tires.
It was on his initiative that the Prussian government in 1876
issued an official classification of iron and steel.

In 1896 the Verein Deutscher Ingenieure awarded Wohler their
highest honor, the Grashof medal. In 1901 the Technical High
School of Berlin-Charlottenberg conferred on him the degree of
Doctor Ingenieur.

Wohler died March 21, 1914, in the city of Hanover, a few
months before his ninety-fifth birthday. His work endures.

August Wohler
{Courtesy of Dr. Ing. C. Matschloss, Verein Deutscher Ingenieure.)

TEXTBOOK

OF THE

MATERIALS OF ENGINEERING
By HERBERT F. MOORE
With a Chapter on Concrete

By H. F. GONNERMAN
THIRD EDITION

325 pages, 6X9, Illustrated

T(\

THE FATIGUE OF METALS ^r

WITH CHAPTERS ON THE FATIGUE OF id-,

WOOD AND OF CONCRETE ^

BY

H. F. MOORE

Research Professor of Engineering Materials, University of Illinois

Member American Society for Testing Materials, In Charge,

Joint Investigation of the Fatigue of Metals

AND

J. B. KOMMERS

Professor of Mechanics, University of Wisconsin, Member American

Society for Testing Materials, Formerly Engineer of Tests, Joint

Investigation of the Fatigue of Metals

First Edition
Second Impression

McGRAW-HILL BOOK COMPANY, Inc.

NEW YORK: 370 SEVENTH AVENUE

LONDON: 6 & 8 BOUVERIE ST.; E. C. 4

1927

Copyright, 1927, by the
McGraw-Hill Book Company, Inc.

PHINTED IN THE UNITED STATES OF AMERICA

THE MAPLE PRESS COMPANY, YORK, PA.

T^o the many distinguished British investigators,
-*- who, especially during the present century,
have been foremost in forwarding our knowledge
of the fatigue phenomena of metals, this book
is dedicated.

Digitized by tine Internet Archive

in 2009 with funding from

Boston Library Consortium IVIember Libraries

http://www.archive.org/details/fatigueofmetalswOOmoor

PREFACE

The growing use of high-speed machinery during the last-
quarter century has greatly increased the necessity for experi-
mental knowledge of the behavior of materials, especially metals,
under repeated stress. Failure of metal parts under repeated
stress, ''fatigue" failure as it is commonly called, is usually
sudden, occurs without warning, and in many cases may be the
cause of a major structural disaster.

One of the purposes of the authors in writing this book is to
summarize the more important experimental facts concerning the
strength of metals under repeated stress. Another purpose is to
review, briefly the more important of the current theories of fatigue
of metals. A third purpose is to give a brief description of appa-
ratus and methods used in making an experimental study of the
fatigue of metals.

In compiling tables of results of tests of various materials, the
guiding principle has been to make available to the reader, in
compact and convenient form, typical results for various mate-
rials, giving in each case as complete a statement as possible of
the chemical composition and heat treatment of each metal.
This, it is believed, will enable the reader to form some judgment
as to what may normally be expected from these metals. The
results quoted are, for all the metals reported, obtained from
test data of tests covering a sufficient number of repetitions of
stress to render the results reliable.

It has been necessary to refer frequently to the use of the
ordinary formulas of mechanics of materials, or to the elaborate
mathematical methods of the theory of elasticity. For readers
who may wish to refresh their memory of such formulas and
analyses, reference is made to various standard texts on the
mechanics of materials (such as Boyd, Seely, Merriman, Poorman,
Maurer and Withey), and for those who have the time and the
inclination to study the theory of elasticity, to Love's great work,
"Mathematical Theory of Elasticity."

Frequent reference is also made to the metallographic study
of metals and related matters. For those readers interested in

X PREFACE

these phases of the study of metals, it is suggested that "The
Science of Metals" by Jeffries and Archer will be found an
excellent book to be read in connection with this.

Although data concerning the fatigue strength of non-metallic
materials are very few, two chapters have been given to a discus-
sion of such data as are available for wood and for concrete.

The authors acknowledge the assistance of many friends and
colleagues in the preparation of this book. Where photographs
or drawings have been contributed, acknowledgement is made by
a note. The authors have drawn very heavily on the published
results of tests by H. J. Gough, of the British National Physical
Laboratory, D. J. McAdam, Jr., of the U. S. Naval Engineering
Experiment Station, R. R. Moore, of the McCook Aviation
Field, Dayton, Ohio, T. M. Jasper, and others who have been
coworkers at the Illinois Engineering Experiment Station, and
the U. S. Forest Products Laboratory at Madison, Wis.

The authors wish to make special acknowledgement to Dr.
D, J. McAdam, Jr., who read the manuscript of this book and
made numerous valuable suggestions and constructive criticisms.

The Authors.

April, 1927.

CONTENTS

Page

Frontispiece (August Wohler) iii

Preface ix

Chapter

I. Stress and Strain in Metals — The Accuracy of the Ordi-
nary Concepts of Elastic Action 1

II. Historical Survey up to 1919 — Fundamental Concepts. . . 9

III. Slip, Overstrain, and Hysteresis 27

IV. Fracture under Repeated Stress 60

V. Testing Machines and Specimens for Fatigue Tests of

Metals 83

VI. Characteristic Results for Fatigue Tests 119

VII. The Effect of Range of Stress on Fatigue Strength . . . 173

VIII. "Stress Raisers" and Their Effect on Fatigue Strength — •

Stress and Corrosion 195

IX. Fatigue Failure under Service Conditions 226

X. Fatigue of Wood 244

XlT" Fatigue of Cement and Concrete 251

Appendix: Bibliography 290

Author Index 317

Subject Index 321

XI

THE FATIGUE OF METALS

CHAPTER I

STRESS AND STRAIN IN METALS— THE ACCURACY OF
THE ORDINARY CONCEPTS OF ELASTIC ACTION

Strain, Unit Strain. — Whenever a force is applied to any
member of a machine or a structure, the shape of the mem-
ber is altered. If the member has been properly designed
to withstand the force, the change of shape is small, usu-
ally not visible to the unaided eye, and on the removal of
the force, the member returns to its initial form as nearly
as can be determined by any ordinary measurements.
The change in any linear dimension of a member caused by
the application of a force is called the ''strain, " or deforma-
tion, and the change per unit of linear dimension is called
the ''unit strain" or "unit deformation."^

Stress, Unit Stress. — If a machine part or structural
member (Fig. 1) is acted on by forces P, P', there must be
set up within the body internal forces (measured in pounds
or kilograms) and called "stresses," which resist the tend-
ency of the external forces to tear apart or to crush the
member. Imagine the part of the body at one side of any
section mn to be cut away; then to balance the force P
(Fig. 1(6)) there must be stresses S. The summation of
these stresses makes up the total stress on the section mn.
If the stress over a small portion of the section be denoted
by A>S and the small area be denoted by AA, then for that

^ The definition used here is that common in American engineering texts
on mechanics of materials; physicists use the term "strain" for what is
defined above as unit strain (measured in inches per inch or miUimeters per
milUmeter). No confusion between the two systems of units need arise if
care is taken to keep in mind the units used for strain.

1

2 THE FATIGUE OF METALS

small area AiS/A^ is the intensity of stress, or the stress per
unit area, or more briefly, the unit stress.''-

If the stress is uniformly distributed over the whole area
of the cross-section mn, then the unit stress is P/A. Note
that this is true only if the stress is uniformly distributed.
In general, the unit stress will be different at different loca-
tions on the cross-section mn.

Fig. 1. — Machine part under stress.

Hooke's Law. — Under working conditions, for the mate-
rials commonly used to carry load in structures and
machines, it is very nearly true that stress is proportional
to strain. This statement is Hooke's law, and is named
after the English physicist who first stated it. Under
working loads Hooke's law agrees very closely with the
observed general action of rolled and forged metals and of
steel castings; it is a fairly close approximation for cast
metals in general, for concrete, for brick, and for wood; it
is a rough approximation for such materials as rubber,
leather, rope, and textile fabrics. For rolled and forged
metals and for steel castings there is a fairly well-defined
limiting unit stress, known as the proportional limit (or
the proportional elastic limit), above which Hooke's law
does not hold.

1 Physicists use the term "stress" for what is defined .tbove as "unit
stress."

STRESS AND STRAIN IN METALS 3

Formulas for Computation of Stress and Strain. — For

the ordinary computation of strain and stress in machine
and structural parts and in test specimens of various metals,
textbooks on the mechanics of materials give fairly simple
formulas. These formulas have been put into their pres-
ent form largely by Rankine, the famous Scotch engineer,
and form the skeleton of what is frequently known as the
Rankine mechanics of materials. Rankine mechanics of
materials neglects several factors assumed to be of minor
importance, notably, the strain which is at right angles to
any tensile or compressive stress, and which always accom-
panies that stress. For anything approaching complete
analysis of stress distribution in a machine or structural
part, Rankine mechanics of materials becomes inadequate,
and much more involved formulas become necessary. The
elaborate system of mathematical analysis which attempts
to take account of all stresses and strains in a structural
or machine member is known as the mathematical theory
of elasticity, although that name really includes the simple
Rankine mechanics of materials. Sometimes the more
elaborate system of stress analysis is known as the Saint
Venant mechanics of materials, from the name of the dis-
tinguished French mathematician and physicist who is the
outstanding figure in its development.

While there is not space in this book to show the deriva-
tion of the formulas for stress and strain in structural and
machine parts, some of the commoner formulas are given
for ready reference. All the formulas given are Rankine
formulas commonly used by structural engineers and
machine designers.

1. Axial Loading. — The stress (tension or compression)
is assumed to be uniformly distributed over the cross-
section of the piece loaded (rarely is this assumption an
exact one), and the resulting unit stress is given by the
formula

-I'

4 THE FATIGUE OF METALS

in which

S is the unit stress (pounds per square inch),

P is the axial load in pounds,

A is the area of the cross-section in square inches.

2. Direct Shear. — The stress on a rivet in a riveted joint
is a good illustration of direct shear. Uniform distribution
of stress over the cross-section of the piece is assumed,
although this is nearly always a very rough approximation,
and the formula for the unit stress is

s. = ^.

in which

Ss is the shearing unit stress (pounds per square inch),

P is the shearing load in pounds,

A is the area of cross-section in square inches.

3. Flexure. — ^Under working loads on machine parts and
structural members having cross-sections of symmetrical
shape and loaded in a plane containing an axis of symmetry
or at right angles to such an axis,^ the stress is assumed to
vary uniformly from a value of zero at a ' 'neutral axis"
passing through the centroid of a cross-section in a direction
perpendicular to the plane of loading, to a maximum value
at one side of the cross-section and to a minimum value
at the other (maximum value and minimum have opposite
signs). This assumption is a very close approximation for
flexural members with a span not less than about ten times
the depth. The maximum unit stress at the outside edge
of the piece is given by the formula

in which

S is the unit stress in tension or compression (pounds per
square inch),

1 For members with unsymmetrical cross-section (e.g., a Z-bar) or for
members not loaded in a plane of symmetry or at right angles thereto (e.g.,
an angle bar loaded parallel to one leg), the formulas given here do not
hold, and a much more elaborate analysis must be used. See Johnson, L. J.,
"An Analysis of General Flexure in a Straight Bar of Uniform Cross-section,"
Trans. Am. Soc. Civil Eng., vol. 56, p. 169, 1906.

STRESS AND STRAIN IN METALS 5

M is the bending moment at the section (inch-pounds),^

c is the distance in inches from the netural axis to the

extreme edge of the piece (there are two values of c,

one for tensile stress, one for compressive),

I is the moment of inertia of the cross-section in inches^. ^

4. Torsion. — Under working loads on round shafts and

shafts whose cross-section is a hollow circle, the shearing

stress is assumed to vary uniformly from zero at the axis

of the shaft to a maximum value at the surface, and the

shearing unit stress is given by the formula

Tr

in which

/Ss is the shearing unit stress (pounds per square inch),
T is the twisting moment in inch-pounds (for a shaft
transmitting H horsepower at iV r. p. m., T = 63,000
H/N),
J is the polar moment of inertia of the cross-section in
inches.^ (For a solid circular shaft J = 1.57 r^; for a
hollow circular shaft / = 1.57(r'^ — r'^) in which r
is the outside radius, and r' is the inside radius),
r is the outer radius of the shaft in inches.
This formula does not apply to shafts of non-circular cross-
section.

Assumptions Underlying the Mechanics of Materials. —
In both the simple Rankine system of stress analysis and
in the complicated Saint Venant system certain assump-
tions are made, among which are:

1. The material is homogeneous.

2. The material is isotropic (having equal elastic stiffness
in all directions).

3. The material is capable of being subdivided to. any
desired extent if the elementary areas approach zero in
magnitude, and the elastic properties of the elementary
particles are assumed to remain unchanged.

4. Hooke's law holds.

1 Values of bending moment for various loadings and of moment of
inertia for various cross-sections are given in texts on mechanics of materials
and in engineering handbooks.

6 THE FATIGUE OF METALS

Under the metallographic microscope, metals are seen
to be made up of mutually adhering crystalline grains with
occasional ''inclusions" of foreign matter. In general,
pure metals and solid solutions with only one phase have
only one kind of grain; but many structural metals, includ-
ing steel, are made up of two or more kinds of crystalline
grain, differing in strength. Moreover, by watching through
a microscope the action of a metal under stress, it is seen
that even in a pure metal the grains have planes of weak-
ness, and that long before any general yielding of the metal
has taken place, there is yielding in certain crystalline
grains whose planes of weakness are unfavorably oriented
to resist the stresses set up.

The above considerations make it evident that assump-
tions 1, 2, and 3, while they are true in a "statistical"
way for a mass of metal comprising a considerable number
of grains, are not true for individual grains, nor for an area
so small as to include cross-sections of only a few crystalline
grains. The exactness of Hooke's law has been discussed
in a previous paragraph.

Why is it that two systems of stress analysis which
depend on such inexact postulates have proved so reliable
a guide for practical stress analysis? One answer is that
the Rankine and the Saint Venant systems yield results
which are "statistically" true for the more important
stresses in structural members and machine parts. Just
as it is possible to use mortuary statistics to predict the
death rate of a community, so it is possible to use the
common elastic formulas to predict the behavior under
stress of a rather small group of crystalline grains. For
example, in applying the common flexure formula to deter-
mine the maximum unit stress in an I-beam, the I-beam
is pictured as divided into thin horizontal layers, and the
average unit stress in the outer layer is what is determined
by the formula. Just as it is impossible to use mortuary
statistics to predict the death of individuals, so it is impossi-
ble to use the formulas of even the elaborate mathematical
theory of elasticity to predict the failure under stress of a

STRESS AND STRAIN IN METALS 7

single crystalline grain of metal or the failure of a group of
a few grains. Even if the complex differential equations
involved could be solved, it is not at all likely that the use
of the mathematical theory of elasticity would permit the
accurate computation of unit stress at the root of a sharp
V screw thread.

In general, this ''statistical" correctness of the common
methods of stress analysis makes them satisfactory for
determining the significant unit stresses in structural parts
under dead load, and in many machine parts. In this con-
nection it may be noted that the significant unit stresses
for ductile metals can be predicted with higher degree of
accuracy than can the significant stresses in brittle materials.
This statistical correctness also makes the mathematical
theory of elasticity useful for indicating in a semiquantita-
tive way localized unit stress around rivet holes, at the
bottom of screw threads, and at other locations where the
maximum unit stress developed affects only a minute area.
The customary tacit assumption is that such localized
stresses are not important under static load, especially
for structural parts made of ductile metal. For example,
a localized overstress around a rivet hole produces no
noticeable general distortion in a water tank, nor does it
interfere with the functioning of the structure. As will be
more fully discussed in Chap. Ill, it must be recognized that
under repeated stress these ''negligible" localized stresses
may become of prime importance, owing to the tendency
to start a crack which spreads under successive cycles of
stress.

Reliability of Stress Computations. — The ordinary (Ran-
kine) formulas of mechanics of materials give a general
idea of the principal significant stresses in a structure, but
they do not give any idea of many localized stresses, fre-
quently of high intensity, which may cause failure under
repeated stress. These ordinary formulas may be used for
computing stresses in repeated-stress specimens if those
specimens are so designed as to be free from localized
stress.

8 THE FATIGUE OF METALS

The more elaborate formulas of the mathematical theory
of elasticity (Saint Venant's) afford a means of determining
these localized stresses, when the differential equations
can be solved;^ but when the areas concerned are minute,
the errors in underlying assumptions of homogeneity,
isotropy, and indefinite divisibility of the material render
the computed unit stresses inaccurate. It may be noted,
how^ever, that all test data available show that the use
of the mathematical theory of elasticity gives results for
localized stress which are on the safe side.

1 In some cases in which these differential equations cannot be solved,
recourse may be had to mechanical means of solution, such as the exami-
nation of transparent specimens under polarized light (see Coker, in
Engineering {London), p. 1, Jan. 6, 1911), and the soap-film method (see
Griffith and Taylor in Engineering {London), p. 546, Dec. 21, 1917).

Z'

CHAPTER II

HISTORICAL SURVEY UP TO 1919— FUNDAMENTAL
CONCEPTS

Introduction. — The three classes of stresses to which
engineering materials are commonly subjected are static or
steady stresses, repeated stresses, and impact stresses.
It is possible, of course, to have repeated impact stresses,
and these may be closely related to the simpler case of
ordinary repeated stresses. An I-beam in the floor of an
ordinary building is an example of a member subjected to
steady stress, the axle of a moving railway car is an example
of a member working under repeated stress, and the plunger
of a steam hammer in operation is an example of a member
subjected to impact stresses. To resist static stresses the
"elastic limit," or in ductile metals the "yield point,"
of the material is the most important criterion.^ To resist
repeated stresses it will be shown that the "endurance
limit" is the important criterion; while to resist impact
stresses the modulus of resilience, or the capacity to absorb
energy up to the elastic limit, is an important criterion,
although the ability of some materials to withstand occa-
sional extreme punishment without fracture is of great
practical importance. The present discussion will relate
especially to repeated stresses. These stresses may vary
from zero to a maximum value, from a positive minimum
to a positive maximum, or from a negative minimum to a
positive maximum. The last case is usually spoken of as
reversed stresses, and when the negative and positive
stresses are numerically equal, the term "alternating

1 The term "elastic limit" is used here to designate the lowest unit stress
at which there is observed appreciable inelastic action in a material. The
value obtained for elastic limit depends on the delicacy of apparatus and
methods used for detecting inelastic action.

9

10 THE FATIGUE OF METALS

stresses" is used. By the term ''range of stress" is meant
the algebraic difference between the maximum and mini-
mum stress. By the term ''endurance hmit " is meant that
unit stress which may be apphed to a given material for an
indefinitely large number of cycles without producing
rupture.

The term "fatigue" has been applied to the phenomenon
of fracture under repeated stresses, and while it is
admittedly not a proper descriptive term, it has become so
thoroughly established in the literature that it will be
adhered to in this book. It will be shown that the term
"progressive failure" is more precisely descriptive of the
action of repeated stresses on a member.

"Crystallization" of Metals. — It has often been observed
that when metals fracture due to repeated stresses, the
fracture has a decidedly crystalline appearance. It used
to be assumed that the metal had developed a crystalline
structure due to the action of the repeated stresses, and
even today this idea is quite commonly held. Many
experiments have shown that this idea is quite incorrect.
Metals are composed of crystals, and there is no change in
their inherent crystallinity due to the action of repeated
stresses. It will be shown that the action of repeated
stresses is highly localized and that if a bar which has
broken due to fatigue and shows a crystalline fracture is
nicked at some point away from the fracture and broken
by a single blow, it will be found that a crystalline fracture is
again revealed. The idea of "crystallization " undoubtedly
arose from the fact that many bars ruptured under fatigue
showed a coarsely crystalline fracture due to overheating,
defective chemical composition, or some maltreatment in
fabrication. The bars broke in many cases because these
defects made them particularly weak in resisting repeated
stresses.

Work of Albert. — The earliest tests on the effect of
repeated stresses of which the authors have seen any record
are those of Albert,^ made in Germany in 1829, on welded

1 Stahl u. Eisen, p. 437, 1896.

HISTORICAL SURVEY UP TO 1919 11

chain for mine hoists. In these tests a chain was held on
a 12-ft. disk, one end of the chain carrying a load. By
means of a crank coupling, the disk could be oscillated
through an arc, thus subjecting the chain links to repeated
bendings. The speed was 10 bends per minute, and tests
of 100,000 bendings were recorded.

Work of Fairbaim. — One of the earliest recorded experi-
ments on the effect of repeated stress is that of William
Fairbairn^ in 1864. He mentions previous experiments
performed by Captains James and Galton in which bars
were subjected to repeated loadings by means of cams.
One cam produced considerable vibration in applying the
load and the other released the load suddenly. By means
of the first cam three cast-iron bars were tested, one being
subjected to 10,000 bending repetitions at one-third of the
statical breaking load without failure. The other two bars
were subjected to one-half the statical breaking load
and failed at 28,602 and 30,000 repetitions, respectively.

By means of the second cam, five cast-iron bars were
subjected to deflections equal to those produced by one-
third of the breaking load. Three bars bore 10,000 repeti-
tions without failure, one failed at 51,538 repetitions, and
one bore 100,000 repetitions without failure. Three bars
subjected to deflections equal to those produced by one-
half the breaking load failed at 490, 617 and 900 repetitions,
respectively.

The first or vibratory cam next subjected a wrought-iron
bar, 2 in. square in section and 9 ft. between supports, to a
strain corresponding to five-ninths of the strain which per-
manently injured a similar bar. A permanent set of 0.015
in. was produced by 100,000 repetitions.

Some further tests were made on wrought-iron bars and a
small box girder of boiler plate.

The conclusions drawn from these experiments regarding
cast iron were that bars or girders were not safe when sub-
jected to deflections caused by one-half the breaking load,
that they were safe when the deflections were caused by

^Phil. Trans. Roy. Soc, p. 311, 1864.

12 THE FATIGUE OF METALS

one-third of the breaking load, and that these repeated
deflections did not seem to have any injurious effects on
the static properties of the metal.

With regard to the wrought-iron bars it was noticed that
they showed a progressive increase in the deflections and a
permanent set.

Fairbairn tested a wrought-iron girder 22 ft. long and
16 in. deep, made up of plates and angles. The web con-
sisted of a plate y^ by 15>^ in.; the top flange of a plate 3^^
by 4 in. and two 2- by 2- by He-in. angles, and the bottom
flange of a plate ^i by 4 in. and two 2- by 2- by ^f e-in.
angles. The load was applied to the beam by means of a
lever and dead load, and this load was lifted off and again
applied, causing more or less vibration.

The beam was subjected to 596,790 cycles of stress at
one-fourth of the ultimate load, then to 403,210 more cycles
at two-sevenths of the ultimate load, and then to 5,175 more
cycles at two-fifths of the ultimate load, when the beam
failed on the tension side near the middle of the span where
the load was being applied. The beam was repaired and
subjected to 3,150,000 cycles of stress at one-fourth of the
ultimate load, and then to 313,000 more cycles at one-third
of the ultimate load, when the beam failed on the tension
side.

Fairbairn concluded from these experiments that wrought-
iron girders when subjected to one-third of their ultimate
load, or a unit stress of about 15,700 lb. per square inch in
tension, are not safe, but that the unit stress of 11,000 lb.
per square inch fixed as the maximum allowable unit stress
by the Board of Trade was satisfactory.

Wohler's Experiments. — The first comprehensive series
of repeated-stress tests was that carried out by Wohler^
in repeated torsion, bending, and direct stress. These
tests included torsion between zero and a maximum stress,
torsion completely reversed, tension between various
limits of minimum and maximum, and rotating bendings

1 Zeit. fur Bauwesen, vols. 8, 10, 13, 16, and 20, 1860-1870; Engineering
{London), p. 199, Mar. 24, 1871, and following issues.

HISTORICAL SURVEY UP TO 1919 13

in which the stresses were completely reversed. Wohler
designed very ingenious machines for stressing his speci-
mens, but was forced to run his machines at slow speeds.
Since his rotating bending machine had a speed of only
72 r. p. m., and since this was undoubtedly the highest
speed used in any of his machines, it is not surprising that
it was necessary for him to spend 12 years at the work.
One of his rotating-beam specimens, for instance, was
given 132,250,000 cycles of stress without producing
fracture.

The materials used by Wohler for test specimens have
undergone such changes in manufacture and content that
it is not deemed desirable to reproduce his numerical results.
Only the conclusions to be drawn from Wohler's experi-
ments will be mentioned here, and they are as follows:

1. Wrought iron and steel will rupture at a unit stress not
only less than the ultimate static strength of the material,
but even less than the elastic limit, if the stress is repeated
a sufficient number of times.

2. Within certain limits the range of stress rather than
the maximum stress determines the number of cycles before
rupture.

3. For a given minimum or maximum unit stress an
increase of range of stress decreases the cycles necessary
for rupture.

4. For a given minimum or maximum unit stress there
appears to be a limiting range of stress which may be
applied indefinitely without producing rupture.

5. As the maximum applied unit stress increases, this
limiting range of stress decreases.

Wohler also studied the effect of abrupt changes in cross-
sections both under axial tension and under rotating bend-
ing, the effect of heat treatment, and the effect of the time
element in applying stress.

It is interesting to note the conclusions regarding limiting
stresses at which Wohler arrived from a study of his results.
He states that the unit stresses to which materials may be

14

THE FATIGUE OF METALS

subjected indefinitely under various conditions of stress
are those given in Table 1.

Table 1. — Limiting Repeated Stresses (Endurance Limits) Deter-

mined BY WOHLER

Ratio of

Ratio of

Maximum

Minimum

minimum

endurance

unit

unit

to

limit to ulti-

Material

stress,

stress,

maximum

mate tensile

pounds per

pounds per

unit stress

strength

square inch

square inch

(range
ratio '•)

(endurance
ratio)

Bars subjected to cycles of bending or tension-compression

Wrought iron.

Cast steel for axles ....

Untempered cast steel
for springs.

+ 17,100
+35,300
+47,100
+30,000
+51,400
+85,600
+53,500
+74,900
+85,600
+96,300

-17,100

0
+25,700
-30,000

0
+37,400

0
+26,800
+42,800
+64,200

-1.0

0
+0.54
-1.0

0
+0.44

0
+0.36
+0.50
+0.67

0.36
0.74
0.99
0.29
0.49
0.82

Bars subjected to cycles of torsional stress

Cast steel for axles.

+23,500
+40,700

-23,500
0

1.0
0

He concluded that the safe unit stress for wrought iron
under alternating stress might be 8,600 lb. per square
inch; that under tension it might vary from 3,300 to 19,300
lb. per square inch; and that the range of stress was to be
taken as constant if the minimum stress fell below 3,300
lb. per square inch.

For untempered cast steel the stress might be 12,800
lb. per square inch alternating, and from 11,500 to 35,000
lb. per square inch in one direction only.

For cast spring steel the stress might vary from 96,000
to 128,000 lb. per square inch.

HISTORICAL SURVEY UP TO 1919 15

Wohler expressed the opinion that railway axles might
occasionally be subjected to stresses equal to the endurance
limit without serious damage, and that tempered spring
steel might be subjected to three-fourths of the ultimate
strength if the play of the spring was small compared with
the total deflection.

The work done by Spangenberg^ in Germany and by
Baker^ in England confirmed Wohler's results.

Bauschinger's Researches. — The work done by Baus-
chinger^ in studying the action of repeated stresses and
allied matters is so important that it will be given in some
detail. His conclusions were as follows :

1. A tension stress above the yield point ^ increases the
yield point up to the applied stress, even if the bar is
immediately retested. Upon release of load and lapse of
time, the yield point increases even above the previous
maximum applied stress. This increase is noticeable
even after 1 day and may continue for weeks or even years.

2. A tension stress above the yield point reduces the
elastic limit (determined by delicate measurements of
deformation) often down to zero. Upon release of load
and lapse of time, the elastic limit increases, reaches the
applied stress after several days, and may rise above this
stress after suflScient lapse of time.

3. A tension stress which lies above the elastic hmit, but
below the yield point, increases the elastic limit immediately,
the more so the higher the initial stress. When the applied
stress approaches the yield point, the elastic limit reaches a
maximum, and is lowered when the yield point is exceeded.

4. As a rule, a stress above the yield point lowers the
modulus of elasticity. Upon release of stress and lapse of

1 Zeit. fur Bauwesen, 1874-1875.

2 Am. Soc. Mech. Eng., vol. 8, 1887.

3 Miti. Mech.-Tech. Lab. Kgl. Tech. Hochs., Heft 13, Miinchen, 1886;
Dinglers polytech. Jour., Bd. 224; Civil ingenieur, 1881.

* The technical definition of "yield point" is: that unit stress at which a
metal shows increase of deformation without additional stress. Practically,
yield point means that unit stress at which inelastic action can be detected
without the use of a delicate extensometer.

16 THE FATIGUE OF METALS

time, the modulus increases, and after several years is
found to be considerably above the original value.

5. Severe jars, as by cold hammering, lower the elastic
limit which has previously been raised by overstress and rest.
If the hammering produces tension, the elastic limit decreases
down to its original value, but otherwise it remains above it.

6. Heating followed by subsequent cooling will again
reduce the elastic limit and yield point which have been
increased by overstress and rest. For low-carbon steel the
effect becomes noticeable at 350°C. if the cooling is rapid,
and at 450°C. if the cooling has no effect on the two limits.

With wrought iron the effect is produced at about 400°C.
both for rapid and slow cooling.

7. Rapid cooling lowers the elastic limit and yield point,
especially the former, more effectively than slow cooling.
Rapid cooling usually decreases the elastic limit to zero or
nearly zero after heating up to 500°C., and surely by heating
up to a cherry-red heat. This is true for wrought iron,
low steel, and Bessemer steel. Slow cooling even after
heating to a cherry-red heat does not produce such a great
decrease.

8. A stress in tension (or compression) beyond the elastic
limit reduces considerably the elastic limit in compression
(or tension) the more the higher the applied stress is above
the original elastic limit. Even relatively small trans-
gressions of one elastic limit will reduce the opposite elastic
limit to zero. A period of rest will not again increase the
elastic limit, as is possible by loading in one direction
above the yield point.

9. Gradually increasing alternating stress in tension and
compression will not decrease the opposite elastic limit
unless one of the original elastic limits is exceeded.

10. An elastic limit in tension (or compression) which
has been lowered by previous stress above the elastic limit
in compression (or tension) can again be increased by
applying a gradually increasing alternating stress. Hut it
can be increased only up to a value which is considerably
below the original elastic limit.

HISTORICAL SURVEY UP TO 1919 17

11. Repeated stresses between zero and an upper limit
which coincides with or is close to the elastic limit will
not cause rupture. The elastic limit, however, must not
previously have been raised artificially by tension or cold
working, nor must there be any flaws in the material.
In the latter respect homogeneous material like low-carbon
steel is especially sensitive.

12. Repeated stresses between zero and an upper limit
in tension which coincides with or lies shghtly above the
elastic limit will increase the elastic limit, and the more so
the greater the number of repetitions, but not above a
certain limiting value.

13. If by the previous action (as in conclusion 12) the
elastic limit is increased above the applied stress, fracture
will not take place; but if the applied stress is so high that
the elastic limit cannot be augmented to this value, failure
must take place after a limited number of repetitions.

14. Millions of repetitions of stress do not alter the struc-
ture of a material, nor do they reduce the ultimate static
strength.

Bauschinger inferred from his experiments that a
material with an artificially raised elastic limit would have
its elastic limit reduced to a certain value by applying
alternating stresses below this artificial limit, and that this
new elastic limit would be the same as that obtained when
an elastic limit which has been reduced is gradually
increased by applying a slowly increasing alternating stress.
This new limit Bauschinger called the "natural elastic
limit," and he proposed the following principle:

15. If a material is to withstand an indefinite number of
repetitions of alternating stress, then the applied stress
must not exceed the natural elastic limit. ^

1 Bauschinger, working before the day of the metallurgical microscope,
very naturally emphasized "elastic limit" as a prime factor in determining
fatigue strength. Later, the idea of fatigue failure as a spreading fracture
became prominent. Practically, Bauschinger's "natural elastic limit"
and the modern "fatigue limit," or "endurance limit," may be regarded
as svnonomous terms.

18 THE FATIGUE OF METALS

Slip Bands. — The work of Ewing and Rosenhain and
Ewing and Humfrey did much to increase the available
information as to the possible mechanism of fatigue failure.
Ewing and Rosenhain^ observed that when a metal is
sufficiently stressed, the crystals of which the metal is
composed yield by slipping on certain gliding planes within
the crystal. This slipping has the effect of breaking up
the polished surface of a grain into elevations and depres-
sions in the nature of steps. Under vertical illumination
the steps show as dark lines, which Ewing and Rosenhain
called ''slip bands." The appearance of these bands was
straight in some metals, but in others the lines were wavy
and tended to fork or branch. After severe straining
there might be as many as four systems of intersecting slip
bands on the surface of the same grain.

Ewing and Humfrey^ carried on a similar study on slip
bands when specimens were subjected to reversed bending
stresses. Specimens were made of Swedish iron having an
ultimate tensile strength of 52,800 lb. per square inch and a
proportional elastic limit of about 29,100 lb. per square
inch. A reversed stress of about 20,000 lb. per square
inch produced slip lines on a few crystals after a few rever-
sals. When the stress was comparatively high, many
crystals were affected. With increase in the number of
cycles of stress, additional slip lines appeared which had
not been visible before, and the original ones showed a
tendency to broaden. As the number of cycles increased,
the broadening process continued, until some parts of the
surface became covered by groups of dark markings. At
this stage it was found that an actual crack had opened up
along some of the broadened slip bands. The cracks were
sometimes first seen on a single crystal, but they soon joined
from crystal to crystal, until a continuous crack ran across
the surface of the specimen, after which a few more
cycles of stress produced fracture.

^Phil. Trans. Roy. Soc, vol. 193^, p. 352, 1899. ^

2 Phil. Trans. Roy. Soc, vol. 200A, p. 241, 1902.

HISTORICAL SURVEY UP TO 1919 19

The specimens showed no sign of damage when subjected
to a stress of 11,200 lb. per square inch, but when the stress
was increased to 15,700 lb. per square inch, signs of fatigue
became visible after many cycles of applied stress. With a
stress of 20,100 lb. per square inch the damage was so
great that cracks were formed and the specimen failed.
The presumption is that with a sufficient number of applied
cycles the specimen would have failed at a stress of 15,700
lb. per square inch. It will be noted that this unit stress
was only a little greater than half of the proportional elastic
limit.

These experiments indicated that some crystals reach
their limit of elasticity sooner than others, which is no
doubt due to the fact that they are so oriented as to be in a
favorable position to permit slip on their gliding planes.
It is evident that a specimen built up in a complex manner
of many crystals will have a distribution of stress from
crystal to crystal which is by no means regular.

These experiments demonstrated that when a metal is
subjected to alternating stress of sufficient magnitude
certain crystals yield by slipping, as in other cases of non-
elastic strain. Ewing and Humfrey are of the opinion
that the surfaces on which slipping occurs continue to be
planes of weakness and that the effect of repeated sliding
and grinding results in the production of a burr, or rough
and jagged irregular edge, suggesting the accumulation of
debris. This repeated grinding tends to destroy the cohe-
sion of the crystal on the surfaces of slip, and in certain
cases actually develops into a crack. Once a crack is
formed, it develops rapidly because of concentration of
stress at the end of the crack. The tests show how a
crack may develop to failure under the action of repeated
stresses,^ even in sound, flawless metal.

The experiments help to explain why a fatigue fracture
shows no sign of local elongation, and why a specimen which

1 Recent researches, to be described later, have shown that slip may occur
without fatigue failure, and that repeated stresses may have a beneficial
as well as an injurious effect.

20

THE FATIGUE OF METALS

has been subjected to many reversals of stress shows no
deterioration which can be detected by a tensile test.
As long as cracks have not been formed, there is no reason
to suppose that a tensile test would detect any deterioration.
Bairstow's Experiments.— A very excellent piece of work,
which tends to throw much light on the behavior of metals
under the action of repeated stress, is that of Bairstow.^
He made tests on axle steel which had a yield point of
55,700 lb. per square inch and an ultimate strength of

0

10,000 20,000 50,000 20

Number of Repefi+ions of S+ress
(a)

Fig. 2. — Set and mechanical hysteresis under repeated stress. {Based o
Bairstow, in Phil. Trans. Roy. Soc.)

10 0 10

5+ress,fonspersq.in

(b)

20

85,500 lb. per square inch. When a specimen was subjected
to equal and opposite stresses of 31,600 lb. per square inch,
the cycle of extensions was represented at first by the
straight line shown in Fig. 2(6). As the number of
cycles of stress increased, the straight line was changed
into a loop, until after 18,750 cycles the width of the loop
was about 11 per cent of the original elastic extension.

Although the extensometer measured to about 0.000004
in., evidently it could not detect the fact that the specimen
was being subjected to stresses beyond its elastic limit,
which repeated stresses made apparent by developing a
hysteresis loop. ' Bairstow is of the opinion that at a slightly

^PMl. Trans. Roij. Soc, vol. 210^, p. 35, 1910.

HISTORICAL SURVEY UP TO 1919 21

smaller stress of about 29,000 lb. per square inch the speci-
men would have been perfectly elastic and that no number
of cycles would have developed a loop.

The stress on this specimen was increased to 33,600 lb.
per square inch, and an immediate increase in the width of
the loop was produced. At a stress of 45,200 lb. per square
inch and 29,280 cycles, the loop became very wide and had
the shape shown by EFGH in Fig. 2(6). In all these loops
the lines FG and HE were found to be parallel to the original
elastic line. The portion FG was obtained when tension was
reduced to zero, and HE when compression was reduced to
zero. At the lower stresses the width of the loop tended to
become constant as the number of cycles increased, and at
higher stresses the width of the loop actually decreased with
increase of cycles.

The observations were continued almost to the breaking
point, and since the extensometer gave no warning of the
deterioration which was going on, the actual damage must
have been extremely local. Since Bairstow is of the opin-
ion that the individual slips in the crystalline grains cannot
have increased in extent due to repetitions of stress, this
tends to give additional weight to the views of Ewing and
Humfrey regarding the mechanism of fatigue failure.

Bairstow found that when the stresses were not com-
pletely reversed, there was developed a '^ permanent exten-
sion" due to the repeated stresses in addition to the width
of the hysteresis loop, which width he called ^'cyclical
permanent set." The ''cyclical permanent set" is shown
in Fig. 2(a).

A specimen was subjected to a maximum stress of 41,000
lb. per square inch in tension and a minimum stress of
18,600 lb. per square inch in compression. The width of the
loop and the permanent extension gradually increased from
zero with increase of repetitions, and when the ''cyclical
permanent set" became constant, the rate of "permanent
extension" became small but not zero. Increase in range
of stress produced a wider loop and great permanent
extension.

22 THE FATIGUE OF METALS

Another specimen was subjected to a range of stress from
zero to 52,000 lb. per square inch in tension. For the
first 2,000 cycles the extensometer showed nothing, but
shortly afterwards a slow yield took place and at the same
time a hysteresis loop made its appearance. The width
of the hysteresis loop reached a maximum value at about
7,000 cycles and then remained constant for another 8,000
cycles. During this time the rate of permanent extension
was decreasing until it was very small.

When the range of stress for another specimen varied in
tension from 16,800 to 63,600 lb. per square inch, permanent
extension and a hysteresis loop occurred at the first load.
It will be noted that in this case the maximum stress was
greater than the yield point of the material.

The range of stress for a certain specimen was now made
from 42,300 to 77,000 lb. per square inch in tension. It will
be noted that this maximum stress was fairly near the
ultimate static strength. A permanent extension occurred
and a narrow hysteresis loop was formed, but this loop
decreased in width until at 6,000 cycles it was practically
a straight line. An increase of range did not increase the
permanent extension nor form a loop, but with a still
further increase of range a loop was formed and a slow
extension commenced.

One significant observation made by Bairstow was that
when the average unit stress of a given cycle of stress was
tension, then an extension occurred during the adjustment
of elastic limits to this cycle, and this extension was similar
to the extension observed in an ordinary tensile test when
the yield point is exceeded. This extension under repeated
stress occurred even when the maximum unit stress applied
was less than the static yield stress. The greater the
extension, the greater was the amount by which the elastic
limits were raised. There was no such extension when the
stresses applied were equal and opposite.

The practical conclusion to be drawn from this phenom-
enon is that under conditions in which elongation could
not be permitted, a unit stress somewhat less than the

HISTORICAL SURVEY UP TO 1919 23

yield point would determine the upper limit of any cycle of
stress in which the average unit stress was tension. It
will be shown later that this also applies to shortening when
the average unit stress of the cycle is compression.

Bairstow concludes from these experiments that iron and
steel are capable of adjusting themselves to cyclical varia-
tions of stress after a sufficient number of cycles have been
applied. When this adjustment is complete, the specimen
is perfectly elastic throughout the cycle and fatigue does
not occur, although slip may have occurred in the adjust-
ing process. This adjustment to a given cycle is possible
because the elastic limits are not fixed but can be raised or
lowered by cycles of stress.

The amounts by which the elastic limits may be adjusted
are limited, and if the range of stress is great enough, the
specimen becomes and remains inelastic, and a certain
amount of energy is expended in moving the portions of the
crystals with respect to each other, and this is probably
associated with the slip bands which Ewing and Humfrey
found, and which gradually develop into cracks.^

Bairstow determined the elastic ranges of Swedish iron,
axle steel of about 0.35 per cent carbon, and Bessemer steel
of about 0.46 per cent carbon, by determining the range
of stress which would not produce a hysteresis loop. He
also found the safe ranges for the Swedish iron and the
Bessemer steel by means of fatigue tests, using stresses
completely reversed. These values checked each other
fairly well.

Bairstow plotted range of stress as ordinates against
minimum stress as abscissae for these elastic ranges. He
also showed similar curves obtained from similar material
by Wohler's fatigue tests. These curves indicate that the
elastic ranges found by Bairstow and the safe ranges found
by Wohler are identical. Incidentally it may be remarked
that these curves indicate that for completely reversed

^ The exact connection between cracks and slip bands is not clear. It
will be shown that there certainly may be slip bands without cracks, but
whether there may be cracks without slip bands is not known.

24 THE FATIGUE OF METALS

stress the range is a maximum and that in general the range
decreases as the minimum stress approaches the value of
the maximum stress — in other words, as the ratio of mini-
mum stress to maximum stress approaches unity. ^

Other Work in Fatigue. — The formulas of Launhardt and
Weyrauch and the diagrams of Goodman and J. B. Johnson
will be discussed in Chap. VII.

In the United States a great amount of fatigue testing
was carried on by Howard^ at the Watertown Arsenal.

From 1896 to 1914 the British investigators of fatigue of
metals were particularly active, and the slip and hysteresis
of metals under repeated stress were given much study.
Several investigators made series of repeated-stress tests,
but no systematic series of long-time tests were found
feasible, though the need for such tests was pointed
out. Reynolds and Smith, Stanton and Bairstow, and J. H.
Smith developed the inertia type of fatigue-testing machine.
Kapp, Hopkinson, and Haigh developed the alternating-
current magnet type of fatigue-testing machine. Stro-
meyer made studies of thermal effects produced by repeated
stress, following lines suggested by Kelvin.

In 1900 Gilchrist^ put forward a picture of fatigue failure
which may be regarded as an early statement of the modern
picture of this problem, and which emphasizes localized
stress as a source of fatigue failure. In discussing Wohler's
results, Gilchrist sums up his views as follows :

1. The average stress in the bars broken in Wohler's machines did
not reach the statical breaking load.

2. The fracture was caused by the statical breaking limit being
exceeded at one point only, from which, when once started, rupture
.spread, at first rapidly and then more slowly, sometimes continuing
to complete separation of the two parts of the bar, but occasionally
stopping short of complete rupture.*

1 Elastic failure seems to be associated with slip, and the connection
between elastic failure and fatigue failure seems rather slight. Fatigue
failure seems to start in actual tearing apart of particles of metal.

- "Tests of Metals," 1888-1895, 1900-1909. ^

3 Gilchrist, J., "Wohler's Theories on Material under Repeated Stress,"
The Engineer {London), vol. 90, p. 203, 1900.

* Modern experiments make it rather doubtful if the crack spreads at first
rapidly and then slowly. Probably the reverse is true.

HISTORICAL SURVEY UP TO 1919 25

3. The raising of the stress at the point where the fracture com-
menced was due to an irregularity in the bar. This might be an
irregularity or discontinuity in the metal, either on the surface or in
the body of the bar.

4. A bar of uniform strength, whose surface was perfectly smooth,
with no sharp corners in the longitudinal configuration, and with a
perfectly homogeneous structure, would endure, without breaking, an
indefinite number of repetitions of a stress varying between zero and a
value near to the breaking strength.^

5. A bar similar to that under 4 could, under certain conditions,
endure an indefinite number of repetitions of a load varjang between
tension and compression of equal values both beyond the ordinary
primitive elastic limit.

In 1910 Basquin of Northwestern University presented
an important paper before the American Society for Testing
Materials.^ In that paper he pointed out that for the test
data available at that time the relation between stress and
number of cycles of stress to cause fatigue failure might be
expressed by the formula

S - KN-^
or

log S = log K — m log N,
in which

S is the maximum computed unit stress in the test

specimen,
N is the number of cycles of stress required for fracture,
K and m are constants depending on the material and on

the manner of making the test.
This formula fitted existing data fairly well, and where
differences were noted, this formula was on the side of
safety. It implied that there was no absolute endurance
limit for actual materials. Basquin's formula has not been
verified by later tests, but the discussion it caused was a
powerful factor in developing research in the fatigue of
metals, especially in the United States.

^ It may be pointed out that all machine parts made of any available
material fall far short of the ideal conditions here pictured.

2 "The Exponential Law of Endurance Tests," Proc, Am, Soc. Testing
Materials, vol. 10, p. 625, 1910.

26 THE FATIGUE OF METALS

In 1914 it was evident that there was need of a large
number of long-time tests to give more light on the ques-
tion of the existence of an endurance limit for metals used
in structural and machine parts. The development of the
airplane and the exigencies of the World War accentuated
this need, and several extensive series of investigations
have been carried out. The British investigations have
centered round the British National Physical Laboratory,
where Gough and Hanson have done noteworthy work.
In the United States three extensive investigations have
been the source of much data, and are still in progress:
(1) an investigation sponsored by the National Research
Council, the Engineering Foundation, and several com-
mercial firms, carried on at the University of Illinois under
the direction of H. F. Moore and his associates, J. B.
Kommers and T. M. Jasper; (2) an investigation carried
on at the U. S. Naval Engineering Experiment Station
under the direction of D. J. McAdam, Jr.; and (3) investi-
gations carried on at McCook Aviation Field, Dayton,
Ohio, under the direction of R. R. Moore.

This brief outline makes no pretense of giving anything
like a complete list of the investigators who have made
valuable contributions to the knowledge of the fatigue of
metals. The types of machines used and the important
results obtained will be discussed in succeeding chapters,
and a rather complete bibliography will be found in
Appendix A.

CHAPTER III

SLIP, OVERSTRAIN, AND HYSTERESIS

Slip under Static Stress. — The work of Ewing and Rosen-
hain on the behavior of metals under strain has been men-
tioned. They estabhshed the fact that the crystalhne
structure of metals is preserved even under severe plastic
strain, which might be supposed to destroy crystalline struc-
ture. They concluded that the distinction formerly drawn
between crystalline and non-crystalline metals was not
justified.

A c

I '/ V V '/ '/ >/ '/ '/ '/ '/ '/ '/ VV A A . ^ A A A \^ ,v A A \B

mmMMMl

Before Siraining
A g t C c

,:<m¥MMm^

Af+er S+raininq
■< >. ^

Fig. 3. — Intracrystalline slip in a metal. (Based on Ewing and Rosenhain in
Phil. Trans. Roy. Soc.)

Figure 3 represents a section through the upper part of
two adjacent grains, having cleavage planes as indicated by
the dotted lines, AB being a portion of the polished surface
and C being the junction between the two grains.

When the metal is strained beyond the elastic Hmit
parallel to AB, for instance, yielding takes place by finite
amounts of slip at a limited number of places as indicated
at a, b, c, d, and e. This breaks up the polished surface into
elevations and depressions in the nature of steps, which
under vertical illumination show the ''risers" of the steps
as dark lines or bands. That this explanation was correct
was proved by changing the incidence of the light, when the

27

28

THE FATIGUE OF METALS

bright areas became dark and some of the dark Unes became
bright. Figure 4 shows a micrograph of a pohshed and
etched surface of a steel sample which has been strained
and which shows the slip bands running across the crystals.
Apparently these slip bands occur in all metals as soon
as plastic deformation takes place, for they were found in
many pure metals as well as in alloys. In the case of iron

Fig. 4. — Slip bands in Armco iron. Magnification 75 X. {Micrograph by J. W.
Harsch at the University of Illinois.)

under tensile stress the bands appeared as soon as the yield
point was exceeded. Slip bands were developed under all
kinds of strain involving permanent deformation, and the
more severe the straining the more slip bands were formed.
The effect of a stress producing plastic strain is similar to
that of a force overcoming the static friction between
two surfaces. If the plastic strain takes place in this
manner by slipping, then the orientation of the parts of
one grain would remain uniform (except for the case of

SLIP, OVERSTRAIN, AND HYSTERESIS 29

twinning)^ no matter how much the outhne of the grain
might be changed by shps occurring within it. The
crystaUine structure of the metal would persist even after
the most drastic strains.

The later researches of Rosenhain^ showed that when a
tensile stress is applied to nearly pure iron, the soft ferrite
crystals of which iron is composed are elongated by the
process of slipping. With continued elongation the limit
of slip is reached and fracture takes place by tearing along
the cleavage planes on which slip has been taking place.

An ordinary carbon steel contains ferrite and pearlite.
Apparently the ferrite and pearlite are so closely inter-
locked that they are deformed about equally. The presence
of fissures near the actual fracture appeared to show that
for a while the pearlite was able to accommodate itself
to the deformation of the ferrite; but it gradually reached
its limit of deformation and then, being the stronger con-
stituent, started tearing fissures in the ferrite.

Slip under Repeated Stress. — The work of Ewing and
Humfrey on slip bands produced by repeated stresses has
been mentioned in Chap. II. Stanton and Bairstow^ later
obtained similar results. The former investigators found
that ordinary slip bands disappeared when the specimen
was repolished and reetched,^ but that when actual cracks
had been formed, these remained visible. When an incip-
ient crack had once formed across a certain set of crystals,
the effect of further repetitions was confined mostly to this
set of crystals, the other crystals changing very little.
During reversals of small stresses, slip lines were generally
found only in the central parts of crystals, not extending
out to the boundaries.

1 Sometimes grains are formed in such a manner that they are structurally
symmetrical with respect to a plane between them, one appearing to be the
mirrored image of the other. These are called "twinned crystals," and
twinning is of such a nature that if either part of the twin were revolved
through an angle, the two parts would possess the same orientation.

2 Jour. Iron and Steel Inst, No. 2, p. 189, 1906.

3 Jour. Inst. Civil Eng., No. 4, p. 78, 1905-1906.

* Later metallographists have developed methods of etching which do not
cover up all slip bands.

30 THE FATIGUE OF METALS

As is well known, one of the characteristic features of
fractures due to repeated stresses is the fact that they take
place suddenly, and, even with soft metals, show none of
the local drawing out and necking down which are associated
with ordinary tensile tests of ductile material. The
development of slip bands and the formation of cracks
explain why this type of fracture is to be expected under
the action of repeated stresses.

Later Work on Slip Bands. — Gough and Hanson^ report
some interesting results on Armco iron, which has the
surprising property, first reported for any metal by Moore
and Kommers^ in 1921, of having an endurance limit
greater than the static yield point of the material.
Gough and Hanson made careful metallographical examina-
tion of the stressed metal, both under static and alternating
stress. When the stress was below the limit of propor-
tionality in the static tests, they found no indications of
strain, but for various stresses above the proportional
elastic limit they found definite indications of plastic
strain similar to the usual slip-band markings.

Under alternating stresses just below the limit of pro-
portionality they found faint surface markings after 11,751,-
000 cycles. These were found on only a few crystals,
the remaining crystals showing no effect of stress. When the
stress was slightly above the proportional elastic limit, the
markings were of the same character as before, but more
crystals were affected.

When the stress was greater than the static yield point
but less than the endurance limit, the appearance was
different from that at the lower stresses. There now
appeared dark areas on certain crystals similar to those
found by Ewing and Humfrey. When these were examined
under a magnification of 1,400 diameters, they appeared
to consist of a series of roughly parallel lines which seemed
to be identical with slip bands. It is interesting to note
that this specimen withstood 40,000,000 cycles of the

1 Proc. Roy. Soc, vol. 104A, p. 538, 1923.

2 Univ. Illinois Eng. Exp. Sta., Bull. 124, 1921.

SLIP, OVERSTRAIN, AND HYSTERESIS 31

stress, which was only 900 lb. per square inch below the
endurance limit. This seems conclusive proof that slip
bands may form at stresses less than the endurance limit,
and that the production of slip bands is not a criterion of
ultimate failure by fatigue. When the stress was above
the endurance limit, the markings were similar to those
just described, but the dark areas were larger and more
numerous.

Gough and Hanson are of the opinion that the dark areas
consist of numerous slip bands rather than a few slip bands
which have been widened by attrition. They believe that
continuous slipping does not occur on the planes first
formed. It will be noted that these conclusions are
different from those of Ewing and Humfrey. Gough and
Hanson found that in mild steel and copper, as well as in
Armco iron, slip bands were formed at stresses less than the
endurance limit. In this connection it may be stated that
Moore and Kommers^ found endurance limits higher than
the proportional elastic limit for several steels; such cases
have also been reported for non-ferrous metals by R. R.
Moore,- Moore and Jasper,^ D. J. McAdam, Jr.,^ and
Lessells.^ McAdam has found that for all the pure metals
and solid-solution metals which he tested in the fully an-
nealed condition, the endurance limit is higher than the
proof stress or Johnson's elastic limit. This was found to be
true for both ferrous and non-ferrous metals. For annealed
copper and annealed aluminum he found an endurance limit
well above the highest value of stress which might be
designated as the yield point of the metal.

Effect of Overstrain.— It is a well-known fact that metal
which has been stressed beyond the yield point becomes
temporarily inelastic, but that it recovers its elasticity by
release of load and rest, and Muir^ has shown that it may

1 Univ. Illinois Eng. Exp. Sta., Bull. 124, 1921.

2 Proc. Am. Soc. Testing Materials, p. 547, 1924.

3 Univ. Illinois Eng. Exp. Sta., Bull. 152, 1925.
^ Amer. Soc. Steel Treating, p. 59, 1925.

* Proc. Am. Soc. Testing Materials, 1924.

6 Phil, Trans. Roy. Soc, vol. 193^, p. 1, 1900.

32 THE FATIGUE OF METALS

have its recovery greatly accelerated by immersion in
boiling water for a short time.

Beilby^ was of the opinion that when the surface of a
metal is polished, a thin layer of amorphous metal is
produced, and he believes that when a metal deforms by slip,
there are thin films of amorphous metal produced on the
surfaces of slip. This film may be in a temporarily mobile
condition. The hardening of metal due to overstrain is
accounted for by the fact that these layers of amorphous
metal harden.

Rosenhain^ has extended this theory and believes that
when a metal is subjected to mild deformation, there is
formed on the surfaces of slip a thin layer of disturbed and
temporarily mobile molecules, that these layers do not
remain permanently amorphous, but become reabsorbed
into the crystalline system from which they were formed.
When, however, the deformation is more severe, the layers
of amorphous material become too thick to be readily
reabsorbed during the short time of temporary mobility.
These layers, therefore, persist until the application of
heat produces sufficient mobility to permit their reabsorp-
tion into the crystalline system.

The question may then be asked, why, if no amorphous
material remains after slight straining, the properties of
the metal are changed, as is well known to be the case.
This is explained on two grounds, the first being that after
slip takes place on the planes of easy slip, the conditions
are no longer the same. These planes have lost their
tendency to easy slip, and a greater force than before is
needed to make them slip again, or else slip may take place
on other planes which were slightly stronger than the first
ones. This seems to be verified by the fact that when slip
has produced slip bands, a somewhat greater strain not
only produces new planes of slip but also deepens the old
ones, and this even after elastic recovery has taken place.

^Jour. Brit. Inst. Metals, No. 2, p. 5, 1911. ^

2 Jour. Brit. Iron and Steel Inst., No. 2, p. 189, 1906.

SLIP, OVERSTRAIN, AND HYSTERESIS

33

The second reason for the above view is that when slip has
taken place on several planes,

. . . the section of the original surface which was rectilinear to begin
with will be stepped at every intersection with other surfaces of slip
(see Fig. 5). Consequently further slipping on the original surface
of easiest slip must come to an end as soon as slip in other planes has
occurred; on further deformation the occurrence of slip is thus forced
upon surfaces not initially favorably situated for its occurrence, so that
increased force is required to bring it about.

I

Fig. 5. — Progressive slip. (Jeffries and Archer.)

Jeffries and Archer^ believe that Beilby's hypothesis,
that there is a production of amorphous metal at all planes
of slip and that slip can occur only once on each plane of
slip, is not consistent with experimental facts. They
believe that the greatest production of amorphous metal
occurs at the crystal boundaries, and that the total amount
formed is much less than has been assumed to be the case.

They sum up the causes for strain hardening as follows:^

1. Cold work produces a structure which simulates in many respects
that of a very fine-grained metal.

2. Because of the manner of origin of the cold-worked structure,
each grain fragment may have an orientation only slightly different

1 "The Science of Metals," p. 80.

2 "The Science of Metals," p. 209.

34 THE FATIGUE OF METALS

from that of its neighbors, so that a large number of grain fragments
may be so oriented as to be traversed b}^ a single slip. Slip through
such grain fragments is, however, interfered with by the disregistry at
the fragment boundaries, and therefore the hardness is increased.
In other words, the main cause of strain hardening is the slip interfer-
ence resulting from the disregistry of slip planes at the boundaries of the
grain fragments.

3. An additional cause of strain hardening is the disorganized layer of
atoms at self-stopping slip planes and the additional amorphous metal
generated at the old grain boundaries.

4. Since severe cold work tends to produce uniformity of orientation
among the grain fragments, it -is probable that there is a limit to the
hardness attainable by cold work. Judging from the hardness of
severely cold-worked iron and severely cold-worked aluminum, the
maximum hardness attainable by cold work is much lower than that
attainable by other methods, such as alloying and heat treating, and
hence much lower than the hardness corresponding to the absolute
cohesion of the metal.

Amorphous -cement Theory. — Bengough^ and Rosenhain
first suggested that the crystals of a metal were held
together by an intergranular cement. Rosenhain and
Ewen^ have developed this idea further. The fact that in
normal pure metals the intercrystalline boundaries are
surfaces of special strength rather than weakness, and the
further fact that a metal of fine-grained structure is stronger
than a metal of coarse-grained structure, suggest the pres-
ence of a material which has a special strength.

The idea suggested is that the cement is of the same
material as the metal itself and exists in an amorphous
condition. When a molten metal cools, the last portions
of the liquid are prevented from crystallizing in the regular
crystalline system, and these portions retain the amor-
phous condition of the liquid and fill the microscopic spaces
between the crystals in the body of the metal. This amor-
phous cement is essentially an undercooled liquid of
great viscosity.

At ordinary temperatures fractures occur across the crys-
tals of metal because of the strength of the cementing

1 Jour. Brit. Inst. Metals, No. 1, p. 123, 1912.

2 Jour. Brit. Insi. Metals, No. 2, p. 149, 1912.

SLIP, OVERSTRAIN, AND HYSTERESIS 35

material; at higher temperatures fractures occur at the
boundaries of crystals because of the greatly weakened
condition of the cement.

Mechanical Hysteresis. — When a member is loaded in
tension and then in compression, the stress-deformation
curve may form a loop, as has already been pointed out in
connection with Bairstow's work. This phenomenon is
called mechanical hysteresis, from analogy with magnetic
hysteresis. Bairstow expressed the opinion that when a
material acts in a purely elastic manner, without the pro-
duction of a loop, failure by fatigue would not occur. His
experiments made plain the fact that static tests in which
the material is carried through only a few cycles of stress
and strain cannot be of much utility in the field of repeated
stresses, for the reason that in some cases thousands of
cycles of stress were necessary to produce a loop. Hystere-
sis in general is of interest, however, because of the light
it may throw on the phenomenon of fatigue. It will be
shown later that the production of a hysteresis loop is not
necessarily a criterion of fatigue failure.

Ewing^ made some tests on long metal wires of various
kinds, loaded between two limits in tension below the
elastic limit, and found evidences of hysteresis in all cases.
He concluded that the work done in each of the cycles of
stress had an obvious bearing on the conclusions of Wohler
regarding the deteriorating effect of repeated stresses.

Hopkinson and Williams^ made some elastic-hysteresis
experiments on an 0.18 per cent carbon steel. They took
temperature readings with thermocouples accurate to about
0.05°C. and also measured the energy dissipated by elastic
hysteresis under cyclical variation of stress at a speed of
7,200 cycles per minute. The results showed that the dis-
sipation of energy increased about as the fourth power of the
stress range, there being evidences of energy dissipation
at a range of stress as low as about 24,600 lb. per square
inch.

1 Brit. Assoc, Repts., p. 502, 1889.

2 Proc. Roy. Soc, vol. 87^, p. 502, 1912.

36 THE FATIGUE OF METALS

The stress difference at the maximum width of the
hysteresis loop seemed to be somewhat greater in the static
tests than in the high-speed tests. The result obtained on
the maximum stress-width of the hysteresis loop was of the
same order of magnitude as found previously by Ewing, and
at a range of 38,000 lb. per square inch was about 0.59 per
cent of the maximum unit stress applied. The maximum
strain-width at the same range was about 0.000015 in. per
inch. The results indicated that there was probably a
decrease of hysteresis at speeds of 7,200 cycles per minute
as compared with very low speeds, but that the difference
could not be more than 30 per cent.

F. E. Rowett^ carried on similar experiments, but he
determined the area of the complete hysteresis loop more
exactly than did Hopkinson and Williams. The experi-
ments were made in torsion on thin tubes, and the high-
speed experiments were made at about 4,200 cycles per
minute. He found that the hysteresis was, probably within
5 per cent, the same at high speeds as at low speeds;
and further that if the results of Hopkinson and Williams
were calculated on the basis of the hysteresis-loop shape
which he determined, their results were almost in exact
agreement at high and low speeds.

The results indicated that for a hard-drawn tube of steel
of about 0.17 per cent carbon, the hysteresis at all stress
ranges was only about one-eighth of that for the same tube
after annealing. For the annealed tube the hysteresis loss
varied about as the cube of the stress range. The maxi-
mum stress-width of the hysteresis loop for the annealed
tubes was about 3.5 per cent of the maximum unit stress
applied at a range of 19,100 lb. per square inch. The
unannealed tube at a range of 19,100 lb. per square inch gave
a maximum stress-width of loop of 0.55 per cent of the
maximum unit stress. It will be noted that the latter result
is of the same order of magnitude as that of Hopkinson
and Williams, but that the result for the annealed tube is
very much greater.

1 Proc. Roy. Soc, vol. 89.4, p. 528, 1913-1914.

SLIP, OVERSTRAIN, AND HYSTERESIS 37

Guest and Lea^ determined some torsion hysteresis loops
on a mild-steel specimen about % in. in diameter and con-
taining about 0.15 per cent carbon. A mirror device was
used for measuring angle of twist and this permitted read-
ings of 0.000000291 radian. Hysteresis loops were obtained
at shearing unit stresses as low as ±1,500 lb. per square
inch. With increase in range of stress larger loops were
obtained. When a large loop had formed and the load was
increased and decreased at any stress point on the loop, small
hysteresis loops were obtained. At the lower stresses
there was no perceptible ''creep," by which is meant an
increase of strain at constant stress with increase of time.

The effect of slight overstrain was also studied. In
increasing the load for this test, creep was observed at a
unit stress of 18,200 lb. per square inch, and a slight over-
strain occurred at 22,800 lb. per square inch. After over-
strain, the width of the loop for a range of +9,100 lb. per
square inch was as great as before overstrain at + 15,200
lb. per square inch.

The effect of a rest of 10 days on overstrained material
was to reduce the width of the hysteresis loop about half,
and the range of stress without creep was considerably
increased by rest.

The effect of heating overstrained material to the tem-
perature of boiling water for 1 hour was to decrease the
width of the loop. This treatment was more effective
than 18 days of rest. The effect of heating to 330°C. after
overstrain was to reduce the width of the loop to about
three-fourths of the width after the operation of boiling in
water. After heating to 330°C., no creep took place up
to the load which caused yielding.

Guest and Lea say :

Since fatigue effects depend upon the gradual increase of the width
of the hysteresis loop with repetition, it would appear that boiling
and tempering at comparatively low temperatures remove initial strains,
and thus considerably increase the resistance of the steel to repetition
of stress.

^Proc. Roy. Soc, vol. 93^, p. 313, 1916-1917,

38 THE FATIGUE OF METALS

If hysteresis loops may be obtained in steel at stresses
as low as ± 1,500 lb. per square inch, as reported by Guest
and Lea, then it is obvious that the production of hysteresis
loops is not a criterion of final failure of the material in
fatigue. Moore and Kommers^ determined an endurance
limit for Armco iron in torsion at + 12,500 lb. per square
inch, and the material used by Guest and Lea would have
an endurance limit at least as great as this. A stress as
low as + 1,500 lb. per square inch would certainly not cause
failure even after billions of repetitions. Since, on the
other hand. Guest and Lea obtained yielding at 22,800 lb.
per square inch, which is certainly above the endurance
limit of this material, it seems clear that there is a certain
range of stress below the endurance limit and within the
ordinary static elastic limit of the material at which
hysteresis loops are formed but which will not cause
failure under repetition of stress. That a certain amount
of movement, permanent change of position, and adjust-
ment of metal particles may take place at stresses less than
the endurance limit seems to be further conclusively
demonstrated by the well-established fact that millions of
repetitions of stress below the endurance limit improve the
material, so that it is better able than before to withstand
repetitions at higher stresses.

Gough and Hanson^ report some very interesting results
on hysteresis loops found in stressing Armco iron in reversed
bending. For stresses less than the endurance limit, the
width of the hysteresis loops usually increased at first,
and then became constant or else actually diminished.
When they applied a stress above the endurance limit,
the loop steadily increased in width during 24,000 cycles.
After a rest of 18 hr., followed by 12,000 cycles, the loop
width had slightly increased, the effect of rest having been
completely obliterated by the subsequent cycles of stress.
The loop width remained constant during 111,200 more
cycles. During a rest of 72 hr. followed by 50,000 cycles,

1 Univ. Illinois Eng. Exp. Sta., Bull. 124.

2 Proc. Roy. Soc, vol. 104A, p. 538, 1923.

SLIP, OVERSTRAIN, AND HYSTERESIS 39

the width of the loop had diminished 46 per cent. After
this, the stress cycles caused rapid increase in width of loop,
and rupture took place after 250,000 more cycles.

It may be mentioned in passing that Gough and Hanson
in careful static tests in tension found permanent sets at all
stresses, indicating again that materials apparently do not
behave in a perfectly elastic manner even at very low
stresses.

Yield Stress and Yield Range. — J. H. Smith^ made
observations on the yielding of steel which led him to
believe that his yield ranges and Wohler's limiting ranges
were identical. He determined his yield ranges as follows:
A specimen was subjected to alternating stresses of zero
mean stress, at about 1,000 cycles per minute, and an incre-
ment of steady tension was then added and the extensometer
reading taken; the steady stress was then changed to the
same amount in compression, and the reading again taken.
This process was continued until a value of mean stress
was reached for which on reversal of the steady stress the
extensometer showed a yield. The maximum stress on
the specimen when yielding occurred was not, in general,
the ordinary static yield point as found in tension or
compression tests; it might be greater or less than these
values according to the range of stress employed. Yield-
ing seemed to take place on the tension side if the mean
stress was tension; on the compression side if the mean
stress was compression; while if the mean stress was zero,
the yield occurred in tension but was seldom obtained.

At this point it may be well to explain the synonymous
terms steady stress, average stress, and mean stress. Each
of these terms denotes the algebraic sum of the maximum
stress and minimum stress divided by two. If the stress
is ±10,000 lb. per square inch, then the range of stress is
20,000, and the mean stress is zero; if the maximum stress
is 15,000 lb. per square inch tension and the minimum
stress is 5,000 lb. per square inch compression, then the
range of stress is 20,000 lb. per square inch and the mean

1 Jour. Brit. Iron and Steel Inst., No. 2, p. 246, 1910.

40 ■ THE FATIGUE OF METALS

stress is 5,000 lb. per square inch; if the maximum stress is
15,000 lb. per square inch tension and the minimum stress
is 5,000 lb. per square inch tension, then the range of
stress is 10,000 lb. per square inch and the mean stress is
10,000 lb. per square inch. The mean stress plus half the
range of variable stress gives the maximum stress, while
the mean stress minus half the range of variable stress
gives the minimum stress.

In Smith's experiments the Wohler limiting ranges were
determined from tests most of which were at fewer cycles
than 1,000,000, so that they can hardly be considered very
reliable values. The experiments indicated that the yield
ranges could be varied within wide limits when the mean
stress was not zero, and that the Wohler limiting range was
not a fixed range even when the mean stress was zero.
When a yield range was raised, the modulus of elasticity
of the material appeared to be lowered.

One of the phenomena noted was that when specimens
were subject to ranges so that the maximum stress was
above the ordinary static yield point of the material, the
specimens showed very perceptible changes of diameter.
When the mean stress was compression, the diameter
increased; while when the mean stress was tension, the
diameter decreased. Furthermore, the same specimen
could have its diameter first increased and then decreased.
In one 0.63 per cent carbon steel in which the maximum
stress did not exceed the static yield point, the change in
diameter was not so marked as in the other cases, but still
existed.

Stress-strain Loops. — Smith and Wedgwood^ carried out
further experiments with the object especially of studying
the stress-strain loops formed under cychcal stress. They
found that the static yield point of a material was not
necessarily the upper hmit of the yield range, the upper
limit being in some cases greater and in some cases less.
When the lower Hmit of the yield range was zero, then the
upper limit was approximately equal to the ordinary static

1 Jour. Brit. Iron and Steel Inst., No. 8, p. 365, 1915.

SLIP, OVERSTRAIN, AND HYSTERESIS 41

yield stress. This means that when the upper hmit of the
yield range was greater than the static yield point, then the
lower limit of the range could not be of opposite sign.

The operations for getting the materials in the cycUc
state were as follows: An alternating range known to be
safe, with zero mean stress, was first applied. Mean stress
was then applied in tension and the strains noted, after
which the mean stress was gradually changed to the
same amount in compression. When these operations were
repeated a number of times, it was observed that the
maximum strains settled down to definite values, which
were repeated after each reversal of mean stress. These
operations were repeated with increasing mean stress and
in each case the strains settled down to fixed limits, until
finally yielding occurred at the tension limit, at the com-
pression limit, or at both limits. A material was considered
to have been brought into the ^'cyclic" state when its yield
stresses were equal in tension and compression. The speed
of the alternating range was varied in different experi-
ments, having values between about 500 and 1,000 cycles
per minute.

When a yield range had been determined, the material
could be brought back to a normal state by applying
gradually diminishing mean-stress ranges. This normal
state was not the primitive state, but a state in which there
was an elastic range which was apparently the Bauschinger
range.

When the material had been brought into the cyclic
state, it was found that the stress-strain diagrams were
complete loops as long as the stresses did not exceed the
equal yield stresses mentioned above. If the range of
stress was reduced, the loops diminished in size, and if the
diminishing range of stress was kept between equal and
opposite limits, the diminished loop became a straight line.

The first tests on loops were made with static loadings,
and the shape of the loop was similar to that found by
Bairstow and shown in Fig. 2. The loops for equal and
opposite stresses were symmetrical with respect to the

42 THE FATIGUE OF METALS

original elastic line of the stress-strain curve, and the
straight-line portions of the loops were always parallel to
the elastic straight line.

A rest of 24 hr. had no effect on the shape of the loop.
A rest of 14 days had the effect of giving a smaller loop at
first, but as the operations were continued, the loop became
larger and larger, and Smith and Wedgwood believe that
the material would finally have been brought into the condi-
tion of the original loop as obtained before the period of
rest.

In one test, after a certain loop had been traced, the
upper limit of stress was kept constant and the lower limit
decreased. When the lower limit was decreased for suc-
cessive loops, the loop diminished in width and finally
became a straight line. The elastic line so determined was
found to be of the same length for four different cases in
which one limit was kept constant at a point on the original
loop. This was true whether the upper limit was tension
or compression. These loops with unequal stresses in
tension and compression were not symmetrical with respect
to the elastic straight line.

The unloading portion of a symmetrical loop, which was
approximately a straight line down to zero stress, was
investigated by unloading to zero and loading again, under
which action a loop of narrow width was formed.

It may be of interest to note here that the original
material had a static yield point in tension of 35,000 lb.
per square inch, and the large symmetrical loop first traced
had a maximum unit stress in tension and compressioQ of
37,200 lb. per square inch. When the loop was successively
traced with diminishing maximum and minimum stresses,
the unit stress at which the loop became a straight line
was +22,400 lb. per square inch. This condition Smith
and Wedgwood called the Bauschinger state.

After the static loops had been studied, tests were made
in which a steady stress in tension or compression, p^us an
alternating stress, were obtained by means of revolving
unbalanced masses, and the strains due to the alternating

SLIP, OVERSTRAIN, AND HYSTERESIS 43

stresses were measured by a ray of light reflected from a
concave mirror mounted on a double knife-edge. An
illuminated line was used and not a spot of light, and to
specify the strains completely, the length of the illuminated
line and the change of position of its midpoint were noted.

The loops and diagrams which were obtained, therefore,
represented the mean stress and the position of the midpoint
of the illuminated line; that is, the deformation measured
was that of the steady stress and did not include the defor-
mation due to the alternating-stress cycle. It should be
noted that the maximum and minimum stresses and the
corresponding deformations for any one cycle of alternating
stress were not recorded by this method. In this respect
the tests are greatly different from those performed by
Bairstow and also from the static-loop tests of Smith and
Wedgwood, in which the strains of the specimen in going
through any particular cycle of stress could be studied in
detail.

It was found that loops traced in the manner described
above were almost exactly the same as the static loops
previously studied; but it must be borne in mind that while
the static loops represented maximum stresses and corre-
sponding maximum strains, the new loops represented mean
stresses and corresponding strains.

The tables of values given for yield ranges show that
apparently the static yield point cannot be greatly exceeded
without producing a yield of the material, although the
experiments did not determine how high the maximum
stress might be increased with a small alternating range of
stress before yield took place.

When a loop had been formed and the mean stress was
gradually reduced by an increment for each succeeding
loop, a condition was arrived at in which the mean stress-
strain diagram did not plot as a loop but as a straight line.
This condition was called by the authors the Bauschinger
state, and represented a range from a certain stress in ten-
sion to an equal stress in compression. The authors took
the mean-stress range represented by this straight line,

44 THE FATIGUE OF METALS

added to it the alternating-stress range, and called the
total range the Bauschinger range. The authors do not
state in which way they consider that the Bauschinger
range is related to the Wohler limiting range.

It should be clearly understood that what the authors
called the Bauschinger range is not related in any simple
way to the Wohler limiting range as commonly determined.
In an ordinary fatigue test a maximum- and minimum-
stress cycle is applied, and the steady stress, which may be
zero or not, is kept constant throughout the test. Smith
and Wedgwood, on the other hand, applied a constant
alternating range of stress and then varied the mean stress,
determining by diminishing loops a straight-line curve.
What they called the Bauschinger range, determined as
above described, does not help in answering the question
as to what range of stress, applied in the ordinary way,
could he withstood without failure.

The Bauschinger range as determined by Smith and
Wedgwood in the static tests corresponds to the ordinary
fatigue tests, and it is the opinion of the writers of this
book that the Bauschinger range so determined is more
likely to correspond to the Wohler limiting range than the
so-called yield range. The yield ranges produce loops and
the Bauschinger ranges do not produce loops. It has
already been pointed out, however, that the production of
loops does not necessarily mean fatigue failure. Whether
the Bauschinger ranges determined from the static tests,
or the yield ranges, correspond to the Wohler limiting range
can be satisfactorily answered only by recourse to fatigue
tests in which the endurance ranges are determined by long-
time tests.

"Creep" Phenomena. — The work of Gough and Hanson
in connection with slip bands and mechanical hysteresis
has already been mentioned. They believe that failure
under repeated stresses does not differ essentially from
failure under static stresses. If a stress sufficiently high is
applied, slip occurs on those crystals favorably oriented for
easy slip, which results in local strain hardening. The

SLIP, OVERSTRAIN, AND HYSTERESIS 45

amorphous metal formed on the plane of slip is hardened
immediately on completion of slip and resistance of the
metal to slip is strengthened on this plane. When the
stress is reversed, slip takes place, but not on the original
slip planes. If the process is repeated and the stress is not
too great, the metal may become so strengthened that it
will not fail under that range of stress. In other words,
the metal can be cold worked by repeated stress just as it
can be cold worked by static stress.

The amount of such overstraining is limited and ulti-
mately a point is reached at which a crack is formed. Their
experiments showed that the overstraining is localized in
certain areas, and they believe that it is probably localized
overstraining which causes a crack to be formed. This
conception has been put forward by a number of
investigators.

Gough and Hanson believe that when a metal is stressed
to a certain value, plastic yielding and '^ creep" occur in
certain unfavorably placed crystals. This will cause local
redistribution of the internal stresses, which may cause an
increase or decrease of stress in the immediate neighborhood.
A local increase of stress acting on a suitably placed cleav-
age plane may cause further slip, inducing further redistri-
bution. In certain ductile metals creep may cause sudden
yielding at a particular load, and in others it may continue
very gradually until it reaches a maximum. This creep
will increase as the stress is increased until finally a stress is
reached under which creep continues indefinitely and the
metal fails.

Assuming that creep has ceased under a certain stress,
the portions of the metal which have not suffered plastic
deformation will be under higher stress than those which
have slipped. When the stress is reduced, slip will occur
in those portions which were previously free from slip.
When the stress reaches zero, redistribution of stress con-
tinues in the so-called "elastic after- working." Creep and
elastic after- working are two aspects of the same process,
one being positive creep and the other negative creep.

46

THE FATIGUE OF METALS

Gough and Hanson refer to an experiment by Muir^
(see Fig. 6), in which two specimens of the same steel were
overstrained to the same extent, after which one was left
at no load for 40 days and the other was left loaded at
55,000 lb. per square inch for the same length of time.
In curve A the specimen was unloaded along abc, elastic

A„'

B,- 3 m ins.

Scale : 1 unif - Jqq-q of an inch IL

Fig. 6. — Cycles of loading and unloading for overstrained steel.
Muir in Phil. Trans. Roy. Soc.)

{Based on

after-working occurred at cd, the specimen was then loaded
along def, and unloaded a second time along a'b'c'. In
curve B the specimen was loaded along a^y, creep occurred
for 3 min., the specimen was unloaded along 5ef, and loaded
a second time along al3'y'. If curve A is rotated through
180 deg., it will fit quite well on curve B. The unloading

1 Phil. Trans. Roy. Soc, vol. 193^, p. 1, 1900.

SLIP, OVERSTRAIN, AND HYSTERESIS 47

part ab of curve A is straight and similar to the loading
part o;/? of curve B, and also be is similar to I3y. The elastic
after-working for curve A is similar to the creep for curve B.
Furthermore, the loop deja'b'c', representing approxi-
mately cyclic conditions, is similar to the loop Se^a'fi'y'.
In other words, this evidence indicates that the effects of
loading and unloading are similar, but of opposite sign.

When metal is stressed within the fatigue range, the cyclic
state is attained by plastic deformation and strain harden-
ing. The cyclic state is attained when plastic strain ceases,
and the metal can then withstand the cycle of stress inde-
finitely. When the stress is above the endurance limit,
the slipping action is the same as that which occurs below
that limit.

For cycles whose mean stress is not zero, the upper limit
of stress can be applied safely only if the lower limit is above
a certain stress. Since unloading causes plastic strain in a
manner similar to loading, the process of unloading cannot
be carried very far if the material is to withstand cycles of
stress indefinitely.

Mason ^ found in torsion tests that the strain became
greater when the speed of applying the cycles was reduced
from 200 to 2 per minute, and became smaller again when
the speed was again increased, the stress remaining the
same throughout. This effect was absent when the strains
were purely elastic. When the speed was reduced from
200 to 2 cycles per minute, the strain increased but imme-
diately started to decrease toward a certain asymptotic
value; while when the speed was increased to 200 cycles
per minute, the strain decreased but immediately started
to increase toward a certain asymptotic value.

This effect is explained by the writers of this book by
the action of creep, which in turn is related to the time
element. When the speed is 200 cycles per minute, there
is not enough time for large strain, but in every succeeding
cycle there is a readjustment of internal stress which is,
of course, influenced by the previous history of stressing.

^Proc. Roy. Soc, vol. 92A, p. 373, 1915-1916,

48 THE FATIGUE OF METALS

When the speed is reduced to 2 cycles per minute, there is
time for a greater strain, but in every succeeding cycle
there is a readjustment of internal stress, which is again
influenced by the history of stressing immediately preceding.

Mason considers as significant the hysteresis loops found
by Bairstow, in which the unloading part of the loop is
parallel to the original elastic curve. The writers of this
book would explain this as follows: If when the plastic
strain has occurred, the unloading is exactly similar to
loading, except for sign, then the curves obtained by Bair-
stow and by Smith and Wedgewood are to be expected.
Starting with the maximum stress in tension, the process
of unloading begins. Some of the elements of metal
have been stressed elastically, and they will return elasti-
cally to a lower stress. Some of the metal elements, how-
ever, were deformed plastically and hardened. These
elements, being in a new state, presumably will act elas-
tically. All the elements, therefore, for a certain range of
unloading, can behave elastically. Soon a range of strain is
reached which forces some of the crystals to slip plastically,
and the stress-strain diagram becomes curved. It must be
recalled that when these curves of Bairstow's were obtained,
there had usually preceded the measurement of strain a
considerable run at a constant stress, so that rather stable
conditions of cyclic straining had been obtained.

Mason ^ performed the following experiment: Running
at 200 cycles per minute, the strain range was 9.00 cm.
on the deformation scale, while after stopping and imme-
diately getting the range with dead weights, it was 11.65 cm.,
and the hysteresis loop was very far from being closed.
Running at 2 cycles per minute, the range was 9.90 cm.,
and immediately after stopping, it was 10.26 cm., with
dead weights, and the hysteresis loop was almost closed.
The writers of this book explain this action as follows:
The first change in strain was 2.65 cm. and was due to the
great change in speed ; the second change in strain ^^as only
0.36 cm., because of the much smaller change in speed.

1 Bril, Inst. Mech. Eng., 1917; Engineering (London), p. 211, Mar. 2, 1917.

SLIP, OVERSTRAIN, AND HYSTERESIS 49

The first range of strain at rest was indicated by 11.65 cm.,
and the second by 10.26 cm. In the first case the hysteresis
loop was far from being closed, and the specimen was not
adjusted to that range of strain. In the second case the
range of 10.26 cm. was smaller and the hysteresis loop was
almost closed, because there was so little difference between
2 cycles per minute and rest that little adjustment to this
range of strain was necessary.

The well-known fact that metal which is stressed below
its endurance limit is strengthened is in itself sufficient to
show that even at these lower stresses there must be an
action in the material which is not elastic. It is difficult
to conceive how elastic action could strengthen the material,
but it is easily understood how inelastic action could do this.
The evidence of slip bands and hysteresis loops at stresses
less than the endurance limit of the material is further evi-
dence that a material has the power of adjusting itself to
cycles of stress if these cycles of stress are within certain
limits. That a process of strain hardening is going on under
repeated stresses below the endurance hmit is evidently
quite as possible as it is under the action of static stresses
above the yield point.

Creep at High Temperatures. — The phenomenon of
creep at normal temperatures and also at higher tempera-
tures has been studied by Lea and his collaborators. Budgen
and Lea found ^ that a material had at a given tempera-
ture a 'limiting creep stress," that is, a stress above which
the material was progressively viscous. At ordinary tem-
peratures specimens kept under observation for many weeks
at stresses above the static yield point showed fairly steady
creep for some hours, but the creep eventually ceased if the
stress was below the ultimate strength. For each tem-
perature, also, there seemed to be a stress below which
creep ceased, but above which it was continuous.

Experiments were made on a 0.14 per cent carbon steel
having a breaking strength of 68,500 lb. per square inch at
15°C., and 62,700 lb. per square inch at 400°C. At ordinary

1 Brit. Assoc. Repts., 1924; Engineeritig (London), p. 500, Oct. 3, 1924.

50 THE FATIGUE OF METALS

temperatures the range of stress for 10,000,000 cycles
was +33,800 lb. per square inch, while at 400°C. it was
+ 39,200 lb. per square inch. When this material was
tested statically to determine the limiting creep stress at
400°C., it was found to be slightly greater than 31,400 lb.
per square inch. The range of stress, therefore, was greater
than that which would cause continuous creep, and the
half range was also greater.

This material was subjected (at 400°C.) to a maximum
stress of 49,300 lb. per square inch and a minimum stress of
17,900 lb. per square inch. With this range of stress, which
was equal to that which would cause continuous creep,
millions of cycles of stress could be applied without produc-
ing failure. Lea is of the opinion that when the range of
stress is above the limiting creep stress, fracture will prob-
ably occur ultimately. In some cases, however, 50,000,000
cycles of stress were withstood without fracture at such
ranges.^

At ordinary temperatures under 20,000,000 cycles of
equal and opposite stresses, the range of stress was about
equal to the ultimate strength of the steel. Since the ulti-
mate strength is the stress at which creep is continuous, it
would seem that there may be a relation between range
of stress and limiting creep stress.

Creep is apparently the criterion of slip, and persistent
creep is evidence of the inability of the material to resist
given shear stresses. Persistent creep implies an action
in which time plays an important part.

Lea is of the opinion that if the range of stress is below
that at which even for slowly applied loads there is no con-
tinuous creep, then the rate of application of stress is
apparently of little consequence, and it is probable that an
infinite number of cycles could be applied whatever the rate
of application of stress. He has shown that the range of
repetitions of stress can be raised more than 25 per cent by
slowly increasing the range of stress during applied cycles.

1 French has reported in the Proc. Am. Soc. Testing Materials for 1925
and 1926 much more exhaustive studies of creep under high temperatures.

SLIP, OVERSTRAIN, AND HYSTERESIS 51

This seems to indicate that very small centers of possible
creep can be healed by understraining. If, however, the
stress first applied exceeds a certain amount, then the dis-
placements are such as to prevent healing. Further, at
ordinary temperatures the viscosity coefficient is small com-
pared with what may be called the adhesive factor, and
thus speed of application has not so important an effect
as at high temperatures.

The limiting range of stress appears to be that range
below which molecular slips can take place in the material,
but after which new bonds may be established. This new
bonding is materially helped by raising the temperature,
and also by permitting slip to take place in very small incre-
ments during the application of cycles of stress. If the
applied stress exceeds a certain amount, then the relative
movement of the molecules is too great to permit rebonding,
and molecular separation occurs which results in the forma-
tion of a fatigue crack and final failure.

Yielding in Static Tests and in Fatigue Tests. — As has
been stated in Chap. II, Bairstow's experiments showed
that for equal and opposite stresses steel did not show a per-
manent extension. Figure 7 shows the experimental results
obtained by Bairstow when the mean stress of the cycle
was not zero. In the figure the curve OF E ABC is the
curve obtained in an ordinary static tension test. Under a
repeated stress equal to OG the first cycle did not show a
measurable extension of the specimen, but continued appli-
cations of stress which was slightly greater than the safe
range produced a slow yielding, represented by the line
GH. When the adjustment of elastic limits was complete,
there was no further extension beyond the point H due to
continued applications of stress. The point / on the curve
was obtained in a similar manner by repetitions of stress.
For the stress OE, which is considered to represent the
maximum non-destructive stress under completely reversed
stress, no extension of the specimen occurred.

When the maximum stress of a cycle was above the yield
point at AB, the extension was found to be due entirely to

52

THE FATIGUE OF METALS

the maximum stress and was not influenced by the range of
stress, which might be zero. Bairstow is of the opinion
that extensions such as GH would probably be caused even
by a range of stress which would not cause final failure.
For stresses below the yield point, therefore, iron and
steel appear to be able to maintain an unstable condition
for a considerable time under cyclical stress. The first
application may not show an extension which is measurable,
but this extension may increase thousands of times under
a constant cycle of repeated stress.

30

20

^

S^^^^

G yji

t

E

AXLE y

>TEEL

F

0 0.? 0,4- O.lo

Ex.+ens'ion,Tnillime+ers

Fig. 7. — Permanent extension under cycles of stress.
Phil. Trans. Roy. Soc.)

0.&

(Based on Bairstow in

The experiments showed that the fine EJDHB seemed to
join smoothly with the static curve BC. The region
EABDE seems, therefore, to be one which can be explored
by repeated stresses, but about which no information can
be obtained by a single application of stress as in a static
test. This seems to reinforce again the contention that
elastic hmits and yield points obtained in a static test can-
not be correlated with endurance limits.

It is well known that concrete and wood under d«^ad load
in long-duration tests yield gradually mth lapse of time.
It may be that this yielding phenomenon is similar to the

SLIP, OVERSTRAIN, AND HYSTERESIS 53

yielding for iron and steel under repeated stress in the
region EABDE in Fig. 7. Such yielding has not been
shown to occur in iron and steel at normal temperatures
below the static yield point.

Elasticity. — In an ordinary static tension test of steel the
increments of unit stress and the corresponding incre-
ments of unit deformation are determined. In the usual
test the value of unit stress is plotted as the ordinate and
the value of unit deformation as the abscissa, and the limit
of proportionality, or proportional elastic limit, is defined
as the maximum unit stress at which the unit deformation
remains proportional to the unit stress.

Sometimes a more tedious test is performed by going
back to zero load after each increment of stress and deter-
mining whether there is any permanent set. The unit
stress at which permanent set first appears is sometimes
defined as the "true" elastic limit. The time consumed
in making such a test is hardly justified, because extenso-
meter measurements have shown that the determination
of the stress at first permanent set is dependent on
the precision and sensitiveness of the measurements. If the
extensometer can detect very small deformations, then the
''true" elastic limit is found at a comparatively low value.
This evidence of inelastic action has been confirmed by
sensitive thermal measurements both in static tests and in
repeated-stress tests. It is evident, therefore, that the
true elastic limit, obtained by a static test on virgin metal,
can have little bearing on the phenomena of repeated
stresses.

Bairstow found in repeated-stress tests of steel that a
hysteresis loop was not developed in some cases until the
specimen had been subjected to thousands of repetitions.
On the other hand, tests of copper, a metal which has a
curved stress-deformation curve, have shown that copper
has a fairly well-defined endurance hmit. It is evident,
therefore, that neither initial apparent perfect elasticity,
nor initial inelastic action, is a criterion of the behavior
of materials under many applications of stress.

54 THE FATIGUE OF METALS

Perfect elasticity is sometimes defined as the quality
which permits a material to be stressed and then to return
to its original length without permanent set. Since a
material may do this, however, and in the process form a
loop which is closed at both ends, it does not seem that the
above definition is admissible. Perfect elasticity might
be defined as the quality which permits the stress-deforma-
tion curve under decreasing stress to coincide with the
curve under increasing stress. Such perfect elasticity is
evidently not common for engineering materials.

Bairstow was of the opinion that when the stresses were
low enough so that the hysteresis-loop width was zero,
then the specimen would not fail, but he also stated that
the presence of a hysteresis loop was not necessarily a
sign of failure. The experiments of Gough and Hanson
and of Moore and Kommers have shown that perfect
elasticity is not essential for indefinite endurance. The
development of heat at stresses less than the endurance
hmit has confirmed this result.

Elastic Hysteresis. — The term ''elastic hysteresis" is
found in engineering hterature and needs to be defined.
If by elastic hysteresis is meant the action under repeated
stress which may form a hysteresis loop, but which will not
result in final failure, it is evident that the term may be
used to describe a phenomenon which has been demon-
strated by experiment. Elastic hysteresis is associated in
fatigue tests with the attainment of stable conditions. A
hysteresis loop may exist; but as long as its width does
not increase under continued repetitions, the specimen will
not fail. If the loop width does continue to increase,
then the specimen will finally fail. A specimen of copper,
therefore, might show an initial loop of considerable
width, but if this loop reached and then maintained a
constant width under continued repetitions, the specimen
would not fail.

Concrete under repeated stress shows considerable initial
permanent set, but if the specimen under test succeeds in
reaching and maintaining a condition in which neither the

SLIP, OVERSTRAIN, AND HYSTERESIS 55

deformation nor permanent set keeps on increasing, then
apparently the specimen will not fail.

Temporary Effects. — These examples of the action of
materials under fatigue illustrate the presence of temporary
and transitory effects which tend to obscure the results
which are of real importance. Bairstow found, for instance,
that the width of the hysteresis loop at low, equal, and
opposite stresses tended to become almost constant,
but at higher stresses the width of the loop gradually
decreased as the number of cycles was increased. Yet
this decrease could not be looked upon as a sign that the
specimen would not fail under these stresses, because the
stress was known to be unsafe, and the effect was a tempo-
rary one which would not have continued indefinitely.
Had the test been continued to failure, the width of the
hysteresis loop would undoubtedly have decreased to some
fairly constant value and then increased again until failure
occurred.

This phenomenon of large initial hysteresis loops which
gradually decrease is undoubtedly associated with the
so-called ''heat bursts" which have been observed by a
number of experimenters. These heat bursts, as the name
implies, cause a temporary rise in temperature, after a
stress is first applied, followed by a subsequent fall in tem-
perature. They indicate the transitory plastic strains
which occur during the period when the specimen is adjust-
ing itself to a particular cycle of stress. Hankins^ has
shown that if a specimen is subsequently tested at the same
stress, heat bursts will not occur, the specimen being now
adjusted to that particular cycle of stress.

Recovery under Repeated Stresses. — Another phenom-
enon which has effects which are sometimes permanent
and sometimes temporary is the so-called ''recovery,"
which may occur under the action of repeated stresses,
and also that due to rest and mild heat treatment. The
recovery which consists in a decrease in the width of the
hysteresis loop after a stress is first applied is apparently

1 Brit. Research Comm. Aero., Repts. and Mem., No. 789, 1921.

56 THE FATIGUE OF METALS

permanent, provided the stress is below the endurance Hmit.
The metal seems to be cold worked due to the repeated
stresses, making it stronger not only under subsequent
static stresses but also under subsequent repeated stresses.
If the stresses are above the endurance limit, however,
and such recovery takes place, then the effect is only tem-
porary, and continued repetition will again begin to increase
the width of the hysteresis loop.

The effect of rest and mild heat treatment must be put
into the class of temporary effects. Both metals and con-
crete show smaller deformations for the same stress after a
period of rest, but experiments have shown that subsequent
repeated stresses soon bring the deformations back to the
value which they had before the rest period. Mild heat
treatment seems to be similar in effect to a long period of
rest, and as far as is known, it will produce only a temporary
effect in decreasing deformations. It should, however,
be noted that mild heat treatment may be very effective
in relieving internal stresses, and such action must not be
confused with the temporary effect which mild heat treat-
ment may have on a specimen subsequent to a period of
repeated stressing. Since the deformations after a period
of rest or after mild heat treatment may be smaller for a
time than just before the rest period, it is conceivable that
the total number of repetitions before failure might be
slightly increased by such treatment, but there is no evi-
dence that the endurance limit is changed in any way.

In static tests the ''healing" of overstrained metal by
rest or mild heat treatment has been observed by a number
of experimenters, and seems to be a well-established phenom-
enon. The strengthening of metal at ordinary tem-
peratures by repeated stressing below the endurance
limit also seems to be a well-established experimental fact.

Mason ^ made some tests to determine whether the tem-
perature of boiling water would have an effect under cyclical
stress similar to its effect under static stress. He found
that a 0.12 per cent carbon steel seemed to be more resist-

1 Advisory Comm. Aero., vol. 2, p. 569, 1923-1924.

SLIP, OVERSTRAIN, AND HYSTERESIS 57

ant to alternating shear at 60 than at 212°F., and that for
the same stress the range of strain was greater at 212 than
at 60°F. This was just the opposite of what might have
been expected. He concluded that either the healing which
might be concurrent with cyclic stressing was less at 212
than at 60°F., or else if the healing was more pronounced
at 212 than at 60°F., then evidently non-elastic strain was
more easily produced at the higher temperature than at
the lower.

On the other hand, as already mentioned in this chapter,
Lea and Budgen found that on a 10,000,000-cycle basis a
0.14 per cent carbon steel had a higher endurance Umit
under reversed axial stress at 752 than at 59°F. There
seemed to be no increase in endurance limit, however, until
a temperature of 392°F. had been passed. For two other
steels (chrome nickel) the endurance limit at normal tem-
peratures was higher than at elevated temperatures. Lea
and Budgen did not report the amounts of strain exhibited
by specimens at the different temperatures.

Moore and Jasper^ made static and fatigue tests on a
normalized 0.49 per cent carbon steel, cyclops metal, a
chrome-nickel steel with two heat treatments, and a heat-
treated 1.02 per cent carbon steel. In the case of the 0.49
per cent carbon steel the endurance limit increased with the
temperature up to about 900°F., and for one heat treat-
ment of the chrome-nickel steel the endurance limit
increased with the temperature up to about 500°F. For
the other steels, the endurance limit decreased slightly with
increase of temperature up to about 800 or 900°F. For all
the steels tested the endurance hmit fell off rapidly above
900°F.

Moore and Jasper determined the ultimate strength of
these materials under tests lasting some hours, and they
found that at high temperatures (about 1000°F.) the endur-
ance hmit at a speed of 1,500 cycles per minute approached,
and in the cases of the chrome-nickel steel slightly exceeded,
the ultimate tensile strength obtained from the tests lasting

1 Univ. Illinois Eng. Exp. Sta., Bull. 152, p. 9, 1925.

58 THE FATIGUE OF METALS

some hours. This result indicates, of course, that at
elevated temperatures the ultimate tensile strength falls
off much more rapidly than the endurance Umit does.
Since, therefore, the material is weaker at the elevated
temperatures and yet the ratio of endurance limit to
ultimate strength is higher at these temperatures than at
normal temperatures, the indication is that ''heaUng"
at elevated temperatures must be more effective than at
ordinary temperatures.

Since the resume of results given above as to the effect
of heat on specimens subjected to cyclical stress shows
more effective healing at some elevated temperatures but
no effect at others, compared with the effect at normal
temperatures, it is evident that at the present time no
general conclusion can be drawn as to the healing effect at
temperatures above the normal.

Bauschinger's Laws. — It may be of interest at this point
to recall the evidence which has been presented to see
whether or not it controverts the laws of Bauschinger
given in Chap. II. The last sentence of the eighth law
states that, after overstressing, a period of rest will not
again increase the elastic limit for the opposite kind of
stress, as is possible by loading in one direction only above
the yield point. If it is assumed that subjecting the
material to moderate heating has the same effect as a long
period of rest, then there is some evidence to controvert
this law. Moore and Kommers^ made some tests on hot-
rolled 0.18 per cent carbon steel which was cold stretched
so that its diameter was reduced from 0.50 to 0.44 in. This
material before cold stretching had an elastic limit of about
38,200, an ultimate strength of 61,500, and an endurance
limit of ± 28,000 lb. per square inch. After cold stretching
the endurance limit was +41,000 lb. per square inch. This
material had been heated to 260°C. (500°F.) after cold
stretching. This result would indicate that probably
the elastic limit in compression which was reduced to zero
by cold stretching must have been restored to something

1 Univ. Illinois Eng. Exp. Sta., Bull. 124, 1921,

SLIP, OVERSTRAIN, AND HYSTERESIS 59

like 41,000 lb. per square inch, and possibly a period of
rest would have had a similar effect, although the relative
effectiveness of rest and mild heat treatment is somewhat
uncertain.

It is believed that the remaining laws of Bauschinger
have not been disproved by experiments made since they
were formulated; but, on the other hand, a considerable
body of evidence reinforcing these laws has been collected
by various investigators.

Bauschinger's laws and the experimental evidence dis-
cussed in this chapter make it abundantly clear that the
results obtained from ordinary static tests cannot be
reUed upon in drawing conclusions as to fatigue strength.
It is hoped that the evidence thus far reviewed will help to
point out some of the factors that are of importance in
connection with fatigue strength.

CHAPTER IV
FRACTURE UNDER REPEATED STRESS

Introductory. — Although the microscope has shown that
metals are not homogeneous in structure, not isotropic,
and not capable of indefinite subdivision without change of
properties, the theory of elasticity has such a commanding
position and has proved so useful as a basis for design that
the idea of perfect elastic material and of an absolute
elastic limit below which no number of loadings can produce
any structural damage in the material still persists. This
explains, in part at least, the amount of attention paid
to the phenomena of inelastic action — slip and mechanical
hysteresis. In this study two facts have become apparent:
(1) Before fatigue failure occurs, a crack develops in the
metal, and (2) considerable slip may occur and considerable
energy may be lost in mechanical hysteresis without start-
ing a fatigue crack in some metals.

Recent tests have shown that for most metals the hmit-
ing stress for fatigue failure seems to be correlated with
the ultimate tensile strength or the ultimate shearing
strength rather than with any elastic limit. For some
metals, especially for annealed pure metals, the fatigue
limit is found above the elastic limit, and in some cases
above the yield point; for some metals, especially for cold-
drawn non-ferrous metals, the fatigue limit is found at a
stress lower than that at which there is the first evidence
of inelastic action.

Elastic failure of a machine part or of a test specimen
involves a quite general slip throughout a considerable
mass of metal; fatigue failure, on the other hand may
result from the spread of a crack at any cross-section.
It is believed that a separate chapter may well be devoted

m

FRACTURE UNDER REPEATED STRESS 61

to a study of the mechanism of the progressive fracture
which constitutes a fatigue failure.

Limitations of Elastic Theory as Applied to Structural
and Machine Parts. — The formulas of mechanics of mate-
rials have been and are of enormous use, but the assump-
tions on which they are founded are not strictly true.
Materials, at least all ordinary structural materials, are not
homogeneous and cannot be subdivided indefinitely with-
out change of properties. This means that the ordinary
formulas of mechanics of materials may be regarded as
giving results ''statistically" accurate, that is, accurate for
the general behavior of a group of, say, a few thousand
crystalline grains of metal, but not accurate for the behavior
of the metal in any one grain. In considering dead-load
strength of machine and structural parts made of ductile
metal, such a ''statistical" view is satisfactory. Unless a
considerable mass of metal is deformed beyond the yield
point, no serious structural damage is done. Around the
rivet holes in an I-beam there may be dozens of minute
areas stressed to the yield point, and no damage is done to
the beam as a whole so long as the load is steady.

If, however, the material is brittle (for example, cast
iron) then the case is different. The outstanding char-
acteristic of brittle metal is its inability to adjust itself to
local overstress without fracture. If an I-beam were made
of cast iron, then highly localized stress around rivet holes
probably would be a source of grave danger, even under
dead load.

The case is still different for repeated loading, and under
repetitions of loading, minute cracks tend to form at points
of highly localized stress and to spread. This is true both
for brittle and for ductile metal. The spreading of such a
crack, like a minute hacksaw cut, gradually diminishes
the area of sound metal remaining in any cross-section of a
piece; and the end of a spreading crack is in itself a point
of highly localized stress, so that there is a strong tendency
for the crack to be self-perpetuating. Under repeated load-

62

THE FATIGUE OF METALS

ing, localized stress in a structural or machine part cannot he
neglected even if parts are made of ductile metal.

Deformation and Slip from a Metallographic Viewpoint. —
Taking now the viewpoint of the metallographist, three
stages of deformation can be distinguished as metal is sub-
jected to increasing static stress: elastic deformation, sUp,
and fracture.

Elastic deformation, as the machine designer and the
structural engineer see it, consists in a very slight stretch-
ing, compressing, or sidewise shoving (shearing detrusion),
and this slight deformation disappears if the stress is

Compufed Siress
a+A-B

AciualSfress
ahngA-B

Fig. 8. — Nominal and actual stress in crystalline grained metal.

released. It has been noted that the engineer and the
elastician think of stress in a metal as a regularly distributed
internal force, but if metals be viewed through the metal-
lographist's eyes, they are seen to be made up of irregular
crystalUne grains, and between grains and within grains
unfavorably placed there must be many minute areas under
very high stress. This is illustrated in a rather crude way
in Fig. 8. If in addition to the irregularity of intergranular
stresses there is considered the effect of non-metallic
"inclusions" and of minute holes which are found in many
metals, the possibilities of still higher localized stresses are
evident. The stresses computed by the ordinary formulas

FRACTURE UNDER REPEATED STRESS

63

of mechanics of materials are much lower than the stresses
developed over many minute areas in the metal; perhaps
computed stresses are only a small fraction of the actual
localized stresses existing in structural and machine parts.
In recent years the X-ray spectroscope has given a pic-
ture of the atoms in a crystalline grain of metal, held
together by forces whose nature is as yet a mystery, and
arranged in some regular geometric pattern with a border
region at grain boundaries having a more or less irregular
atomic arrangement. The regular pattern of atoms which
is repeated to make up a crystal is known as the space
lattice of a metal, and from the viewpoint of the student of
atomic structure, elastic strain consists of a slight distortion

illllll.. JMi

V/////////////y/////y

Coi)

y/y//////////////////////////////y

side View End View

Cb)

Fig. 9. — Diagram of action of slip.

of the space lattice, which distortion disappears when stress
is released.

As stress is increased in ductile materials, there comes
about a state of affairs such that along certain planes of
weakness in crystalline grains atomic bonds are broken.
The divorced atoms slide over a few thousand other atoms,
after which most of them find new partners and form new
bonds with them. The remarkable thing is that the new
bonds seem to be stronger than the old, after a brief period
of restful adjustment. This action is known as ''slip"
and is shown under the metallographic microscope by a
series of "slip lines" or ''slip bands" such as are discussed
in Chap. Ill and shown in Fig. 4.

Slip may be pictured as an action analogous to that
shown by a pack of cards pressed together face to back and
subjected to slightly oblique endwise pressure, an arrange-

64 THE FATIGUE OF METALS

ment such as that shown in Fig. 9(a). Under a sufficiently
heavy push the pack would take a position like that shown
in Fig. 9(6). The end view of the pack is the end view
of a stepped surface; it is the edgewise view of these steps
that shows the slip lines through the microscope. If the
cards are slipped repeatedly, the faces and the backs would
become roughened and would offer increasing resistance
to further slip.

To the metallographist a major significance of slip is
the strengthening of planes of weakness within a crystal-
line grain. If slip could he brought about with no other effects
than the exchange and strengthening of atomic bonds, it would
be an entirely beneficial process so far as strength is con-
cerned. In some cases, e.g., cold-drawn steel, the process
is actually somewhat beneficial to the strength.

The Progressive Course of Fracture. — It is not possible,
however, for the process of slip to go on without there
being some locations where atomic bonds are broken and
new bonds are not formed; that is, minute, submicroscopic
cracks are developed. The earliest metallographic picture
of the mechanism of repeated stress of metals was a picture
of cracks developing at slipping surfaces, growing to
visibility under the microscope, and finally spreading to
failure.^ This is still a quite satisfactory picture, although
two other pictures of the origin and spread of cracks have
been recently presented — pictures which do not picture
cracks as necessarily originating at slipped surfaces. These
pictures will be shown in succeeding paragraphs.

If the strain on a metal is continually increased beyond
the original strain where general slip takes place, actual
fracture finally occurs. In the case of brittle materials
such fracture occurs before slip becomes widespread enough
to show a well-marked yield point. Under a single loading
in tension the final fracture of a metal, either ductile or
brittle, appears to take place simultaneously over the
whole section of a piece of metal. That section seems to

1 EwiNG and Humfrey, "Fracture of Metals under Repeated Alter-
nations of Stress," Phil. Trans. Roy. Soc, vol. 200^, p. 241, 1903.

FRACTURE UNDER REPEATED STRESS 65

act like the famous '' one-horse shay," which went to
pieces "all at once and nothing first, just as bubbles do
when they burst." A careful study of the bursting of
bubbles and of the failure of tension test pieces shows that
in both cases the actual fracture is progressive, not instan-
taneous. If the fracture of a tension test piece is examined,
there usually can be found evidence that the failure began
at some definite region and spread rapidly across the piece.
In some ductile metals fracture can be seen to progress
across the test specimen (especially in the case of thin
specimens) . Fracture under a single loading is a very rapid
progressive fracture. It may be safely stated that no
experimenter has ever loaded a test piece of metal so care-
fully and so accurately that all the atomic bonds on a
cross-section were broken at the same instant.

Under repeated loading a stress well below the ultimate
tensile strength will start a fracture in metal which spreads,
finally causing the failure of the entire cross-section of a
piece. This spread under repeated loading is very much
slower than the spread under a single increasing load.
Thousands or even millions of cycles of stress may be
required to develop the final failure of a machine part. Not
infrequently the spreading crack can be detected before
it has progressed to failure, and a disaster averted. This
repeated-stress fracture spreads slowly like a minute hack-
saw cut, but its rate of progress is accelerated, and just
before fracture, it is almost as rapid as is the spread of
fracture under a single increasing load. In fact, a typical
fatigue failure usually shows two distinct zones: (1) a
smooth surface where the crack has spread slowly and the
walls of the crack are battered smooth by repeated opening
and closing, and (2) a rough '^crystalline" surface indi-
cating the very much more sudden fracture of the core of
the piece.

Figure 10 shows the fracture of a rotating-beam test
specimen subjected to cycles of repeated flexure. Fracture
started at the outer circumference, and a crack gradually
spread inward. The walls of this crack were continually

66

THE FATIGUE OF METALS

shoved against each other as the crack opened and closed
under successive cycles of stress. The walls of this part
of the crack were worn smooth, and occasionally little
longitudinal breaks occurred, leaving steps in the surface
w^hich roughly resembled ripple marks left on sand by
flowing water. When the cracks had spread to the inner
circle shown in Fig. 10, the failure of the remaining metal

progressed so rapidly that a
rough ^/crystalline" surface
was left, such a surface as is
found when a steel specimen
with a sharp shoulder is frac- •
tured under a single very
heavy load.

Incipient Cracks. — Definite
knowledge as to the nature
of fatigue cracks in their ini-
tial stages is entirely lacking.
Attempts to use the micro-
scope to detect fatigue cracks
in their very early stages have
not met with much success.
If, however, a specimen is notched so as to localize
fractures at some definite cross-section, such fatigue cracks
in large crystalline grains of metal can be detected quite
early in the ''life" of the piece, and they can be seen to
multiply and to lengthen under successive cycles of stress.
Figure 11 shows three views of a specimen of Armco iron
subjected to violent reversals of flexure. The multiplica-
tion and lengthening of cracks is evident. It is, however,
exceedingly difficult to detect fatigue cracks in small-
crystalled metal, and it is exceedingly tedious to hunt for
microscopic cracks over any considerable area of surface
metal.

Figure 12 shows fatigue cracks in normalized 0.93 carbon
steel, in brass, and in Armco iron. In Fig. 12(a) the crack
is seen to traverse the ferrite of the steel, skirting the lam-
inae of cementite. It seems as if the junction of ferrite

Fig. 10. — Fracture of rotating shaft
under reversed bending.

FRACTURE UNDER REPEATED STRESS

67

o

(N -S b

to 6

68

THE FATIGUE OF METALS

t3 V

OS o

o X

o O

(

start

'of

Crack

•"^^ ' (

'*>'.- ■"*'<J^ ■^'

■^♦'

^

r'**^.

N

\

^

,K--

^p»fJV

7

---

. ^ ■ • e

^

' ■■ ' 9

'^i

"%%..

10

-^ '".

/

"<*.

End

°f

Crack

liM^

12

Fig. 12(d).— Course of fatigue crack in Armco iron. Magnification about 2,000 X, magnificatioa of original plates 3,500 X. {Uicrographshll F. F, Lucas at the Bell Telephone Laboratories.)

FRACTURE UNDER REPEATED STRESS 69

and cementite were the weak region in this steel. In
Fig. 12(6) the brass is seen to be made up of two
crystalhne ingredients, but they seem to differ but
httle in strength, since the fatigue crack goes straight
across both kinds of crystalhne grains. A crack typical
of fatigue cracks in large-grained pure metals is shown
in Fig 12(c) — Armco iron. This crack is seen to cross grain
boundaries, and to go out of a straight path to skirt
"inclusions." There seems to be a tendency for fatigue
cracks to pass through the boundary between an ''inclu-
sion" and the adjacent metal, as if the ''weld" or the
"cement" between inclusion and metal were a region of
special weakness.

Figure 12(d) shows a fatigue crack in Armco iron. The
magnification of the cut is 1,800 times, and at its
narrowest visible part the crack is about 500 atoms wide —
if the calculations of atom size by modern physicists are
accepted. This remarkable micrograph by F. F. Lucas
shows the crack seeking out some inclusions and avoiding
others, and at once suggests the presence of a multitude of
minute defects in the metal.

In spite of its difficulties, the microscopic study of fatigue
cracks in metals offers a very promising field for the investi-
gator. The question of initial stress in metals and of
whether such regions of stress are sources of fatigue cracks
also needs investigation. Some locations where there are
no cracks may be under high internal stress, stress so high
that but little additional stress is necessary to start a
crack.

In structural members and machine parts it is sometimes
possible to detect fatigue cracks before they have spread to
fracture. Some railroads and street railways make a
practice of inspecting axles of cars and locomotives at
regular intervals to see if such small cracks can be detected.
When this has been carefully done, there have been very
few disasters due to fatigue fractures in axles. In some
experiments now in progress at the University of Illinois, it

70 THE FATIGUE OF METALS

has been found possible to detect cracks^ in specimens l^i
in. in diameter cut from car axles when about one-half the
"life" of the specimen has passed, unless the applied stress
is very high.

Theoretical and Actual Strength of Metals. — The whole
question of fracture in metals brings up the relation of
theoretical cohesion and strength. From a determination
of the latent heat of fusion and the latent heat of vaporiza-
tion, physicists have computed the theoretical cohesion
of atoms for many metals, and if cohesion in solids at ordi-
nary temperatures is of the same order of magnitude as
cohesion in melting solids and vaporizing liquids, then the
tensile strength of most metals should be from fifteen to
twenty times as great as it is found to be in ordinary tension
tests.

The most obvious explanation of this great difference is
that in the common metals the system of atomic bonds is
far from perfect. It has already been noted that the
elastician's picture of continuous, homogeneous material
is not true for ordinary metals. If metals are considered
from the viewpoint of the metallographist, a rough picture
of the metals may be drawn by considering them as con-
tinuous but not homogeneous. If there are considered
imperfect bonds between atoms, minute cracks, and severe
internal stresses, which when slightly increased will produce
cracks, a picture may be drawn from the engineer's view-
point, a picture of metal which is homogeneous but not
continuous. Neither of these pictures can claim to be
complete. The metallographist's picture does not lend
itself to mathematical computations of strength. The
engineer's picture of metal which may be regarded as homo-
geneous but which has in it many small holes or many
irregularities of outline, does lend itself to such computa-

1 A method in successful use consists of applying oil to the surface of the
steel, rubbing off the free oil and then applying a coating of whiting and
alcohol. When this coating is dry, the specimen is rotated under load. Oil
which has penetrated the crack and was not removed when the surface was
wiped is forced out, if the crack is on the compression side and discolors the
whiting coating.

FRACTURE UNDER REPEATED STRESS 71

tion or, at least, to estimation. The mathematical theory
of elasticity, and the mechanical means which can be used
to solve some of its more complex equations can be employed
to determine approximately the effect of these supposed dis-
continuities. With this apologia for using a method of
analysis which is admittedly based on an incomplete picture,
but which is believed to be useful, the writers of this book
wish to present a discussion of two hypotheses of the
mechanism of fatigue failure.

The Internal-flaw Hypothesis. — In an extremely valu-
able paper^ the British physicist, A. A. Griffith, has devel-
oped a picture of the mechanism of the failure of materials
under stress. The paper treats the subject both from
the viewpoint of mathematical analysis of stress and strain,
and from the experimental viewpoint.

Griffith found that the computed unit stresses at rupture
existing at the ends of cracks in glass were of the order of
350,000 lb. per square inch, while the tensile strength of
the glass, as determined by an ordinary tension test, was
about 25,000 lb. per square inch. He also found, by drawing
this glass out into very fine fibers and by a series of tensile
tests of these fibers, in which the fragments of one test
were in turn tested, that a tensile strength of 491,000 lb.
per square inch was finally obtained. The above values
approach in magnitude the theoretical cohesive strength
for glass.

Griffith believes that the above-named results and also
the great difference between the theoretical cohesion of
solids and the actual values obtained in tension tests may
be best explained by the hypothesis that in all solids there
are, scattered throughout the mass of the solid, multitudes
of minute discontinuities or flaws, whose ruling dimensions
are large when compared with atomic dimensions and
distances. He believes that the effective strength of
engineering materials might be greatly increased, perhaps
ten to twenty times, if such flaws could be eliminated.

^"Phenomena of Rupture and Flow in Solids," Phil. Trans. Roy. Soc,
vol. 221A, p. 163, 1920.

72

THE FATIGUE OF METALS

Figure 12(d) supports the Griffith picture. Figure 13
is a cartoon of the Griffith idea. Metal is pictured as hav-
ing in it a multitude of minute cracks — cracks, say, 0.0002
in. long and a few score atoms wide, cracks which cannot
be detected by any present-day microscope. If such cracks
exist, they must be very numerous and must be scattered

Fig. 13. — Diagram to illustrate the Griffith theory of the structure of metals.

throughout the metal, else the metal could not be produced
with such dependable physical properties as is found to
be the case. These minute cracks weaken the metal
in two ways: (1) by diminishing the area of the cross-section,
and (2) by causing very high localized stress at the ends
of the crack. Attention must be called again to the limi-

AB= Mean Uni'i - Stress in Region near Crack
MN,PQ - Localized Unii-Siress aiends of Crack

Fig. 14. — Stress intensification at the ends of a crack.

tations of the theory of elasticity and to the improbability
that its formulas would apply with any high degree of
accuracy to such small areas of metal as are involved in
considering these cracks; however, the general qualitative
conclusions of the theory of elasticity may be expected to
furnish a useful guide for estimating the general effect of

FRACTURE UNDER REPEATED STRESS

73

such cracks. From such general conclusions it would seem
that if Fig. 14 represents such a small crack, the stress
intensification at the ends is a function of the direction
of the long axis of the crack with respect to the direction of
the stress and a function of the sharpness of curvature at
the end of the crack. It must be remembered that cracks

Fig. 15. — Fatigue crack in Armeo iron. Magnification 3,560 X.
by F. F. Lucas at the Bell Telephone Laboratories.)

{Micrograph

have rounded ends. High-power micrographs, of which
Fig. 15 is a sample, show this.

Griffith believes that the presence of such cracks may be
explained if it is supposed that a change in volume occurs
when the metal changes from the crystalline to the amor-
phous condition. Supposing a material contracts on decrys-

74 THE FATIGUE OF METALS

tallizing, then a stress cycle, which causes repeated shpping
in certain crystals, will produce amorphous material at
the crystal boundaries. The volume of amorphous material
will increase with repeated slipping, and if it fills less space
than the crystalline material did, the material in the imme-
diate neighborhood will be subjected to a tensile stress.
When this tensile stress exceeds a certain critical value, a
crack will form, and under further cycles of stress the crack
will spread and final rupture will occur.

It may be noted here that considerable evidence is avail-
able to show that the effect of overstrain is to decrease the
density of metals. If, then, amorphous material occupies a
larger volume than the crystalline material, it is quite
possible for this change in volume to produce both compres-
sive and tensile stresses in the immediate neighborhood and
thus again produce a crack. In the event of either decrease
or increase of volume due to the formation of amorphous
material, the damaging disturbance of internal structure
takes place, not immediately at the end of an internal flaw,
but some httle distance away from it.

Beilby^ has suggested that under alternating stresses a
film of '^ hard-phase" material is formed on a surface of slip,
and when the stress is reversed, slip occurs in the opposite
direction, but not on the same plane as before, because the
harder material is stronger. The second slip will occur on
an adjacent plane, producing two hard layers with a soft
layer between. If it be assumed that the production of the
hard layers has produced a tension normal to the layers,
then on further slipping it may be conceived that the hard
layers thicken at the expense of the soft crystalline material.
If this process is continued and the tension also, the
crystalline material wdll be used up in thickening the hard
sheets and an incipient crack will appear between them.

In considering this hypothesis in conjunction with the
fact that overstrain seems to produce an increase in volume
in the material, it is not clear why tension should be pro-
duced between the layers. It is conceivable, hci^ever,

1 Proc. Roy. Soc, vol. 79.4, p. 463, 1907.

FRACTURE UNDER REPEATED STRESS 75

how such an increase in volume of the hard layers could
act as a wedge producing tension on some other micro-
scopic portion of the body.

Whether or not either of the hypotheses mentioned repre-
sents what actually occurs, it is known that in fatigue some
mechanism is at work which either produces a microscopic
crack or else spreads a crack already existing in the virgin
material. The action of the repeated stresses is such as
to spread this crack until the member is so reduced in effec-
tive cross-section that complete failure results. This
spreading of a crack, even in a ductile metal, explains the
characteristic appearance of a fatigue fracture, which has
the features usually associated with a brittle material,
but none of the local elongation and reduction in cross-
section which accompanies the ordinary tensile failure of a
ductile material.

From the above pictures of failure an explanation is
found for the fact that while scratches do have an appreci-
able effect on the fatigue strength of a metal, they do not
have so serious an effect under the action of fatigue stresses
as theoretical stress calculations would indicate. If fatigue
failure is determined by the phenomena at the grain bounda-
ries, then the strength is determined not so much by the
stress range at the corner of a scratch as by the stress range
at a distance of about one crystal layer away. Since theory
indicates that the stress due to a scratch falls off very rapidly
with distance from the corner of the scratch, this explains
why fatigue tests show a smaller effect due to scratches and
sudden changes of cross-section than would be expected
from the calculation of maximum stresses based upon elastic
theory.

It is, of course, well known that under the action of
fatigue stresses failure occurs by the spreading of a crack.
Griffith's theory of failure would indicate that in many
cases the cracks are initially present in the material and
that the action of the repeated stresses has the effect of
spreading these cracks. Griffith's results show that the
inherent local strength of a solid is many times greater

76 THE FATIGUE OF METALS

than the average strength obtained in an ordinary tensile
test, and it is clear, therefore, why a certain minimum stress
is necessary to spread even those cracks which may exist
initially in a body. The theory also suggests how a crack
might be formed in material initially free from cracks, on
the assumption that material in the amorphous state has
a different volume from the same material in the crystal-
line state.

0. 13 Per Cenf- Carbon Si-eel
Tool Marks

Tool Si eel
^/ Tool Marks

0.33 Per Ceni Carbon She/
Ground Finish

Orounol Finish

0.33 Per Cenf Carbon She/
Reamer Finish

Fig. 16. — Surface irregularities of steel. Magnification 180 X. {Based on
micrographs of sectioned gelatin casts obtained by W. Norman Thomas in tests for
the British Aeronautical Research Com.mittee.)

The Surface -irregularity Hypothesis. — A second hypoth-
esis for explaining the start of fatigue cracks is that such
cracks start in the region of localized stress or in the adja-
cent region of structural damage, due to one of the many
minute hills and hollows which are found even on the most
carefully pohshed surfaces. Figure 16, which is based on
the work of the British experimenter, W. Norman Thomas
for the Aeronautical Research Committee,^ shows actual
surface irregularities magnified 180 times. Evid^tly

1 Brit. Research Comm. Aero., Repts. and Mem. 860, Vol. 2, p. 542, 1923-24.

FRACTURE UNDER REPEATED STRESS 77

there will be stress concentration at the bottom of the
minute notches in the surface of the metal, and the magni-
tude of the stress concentration at the root of any notch
depends on the depth of that notch and the sharpness of
curvature at the root of the notch.

The Russian physicist, Joffe, is inclined to feel that sur-
face irregularities rather than internal flaws are the starting
points for fatigue cracks, and he cites an interesting
experiment in which a single crystal of a salt was ground to
spherical shape and subjected first to thorough cooling in
liquid air and then to sudden immersion in molten lead.
Under such a change the surface of the sphere would
be free from stress, but the interior would be under heavy
tensile stress. Joffe calculated the stress set up to be nearly
equal to the theoretical cohesion of the salt; yet after the
test no evidence of any fracture external or internal was
found.

The two hypotheses given above are not contradictory,
but rather supplementary. For all we know, both internal
flaws and surface irregularities may be effective agents in
starting fatigue cracks. Moreover, the reasoning about
stress concentration, production of amorphous metal
at grain boundaries, splitting action due to '^wedges"
of strain hardened material, etc., is as applicable to the
surface-irregularity picture as to the internal-flaw picture.

The Mechanism of Progressive Fracture. — Whether the
origin of fatigue cracks is always at surface irregularities,
or whether they may originate at internal submicroscopic
flaws; whether they are always present, or whether they
originate as the result of internal stress plus stress due to
load; whether they always originate as the result of slip
in a metal, or whether they may start without previous
slip taking place; once they are started, they progress,
sometimes to fracture and sometimes to a state of equilib-
rium without fracture. The following somewhat specu-
lative picture of the progress of a crack is given in terms
of a combination of the internal-flaw and surface-irregu-
larity hypotheses.

78 THE FATIGUE OF METALS

In Fig. 17 the left-hand part of the figure refers to the
internal-flaw picture, while the right-hand part represents
the surface-irregularity picture. The reader is again
reminded that cracks and notches have roughly rounded
ends (see Fig. 15). Under increasing load at least three
things happen: (1) The crack shown at a and the notch
shown at h spread to the condition shown at a' and h' ,
respectively; (2) the curvature of the ends of the crack and
the curvature at the bottom of the notch become less sharp,

^^^^

In+ernal Flaw Surface IrreguIarHy

Hypo+he&is Hypo+hesis

note: Arrows f'r-^) show dlrec-fions of stresses

Fig. 17. — Diagram of growth of defects under repeated stress.

causing diminution of stress concentration and benefit
to the metal; and (3) slip as well as fracture occur at the
ends of the crack and at the bottom of the notch, probably
causing diminution of internal stress, and tending to
increase the strength of the material by cold working.
Under a moderate increase of loading, equilibrium is
reached, and the damage caused by the crack (or the notch)
spreading slightly and causing a reduced cross-section is
balanced by the diminished stress concentration at the
ends of the crack or the notch, and furthermore by the
tendency to strengthen the material at the points of highest

FRACTURE UNDER REPEATED STRESS 79

stress. Under a sufficiently great increase of load the
spread of the defects overbalances the beneficial effects
of lowered stress concentration; failure occurs, and its
final progress is very rapid. If the metal has low ductility,
the strengthening effect and the reduction of internal
stress due to slip are slight, since brittle material slips very
little before fracture.

To make this picture fit the case of repeated stress, a
third part must be added. If a load is applied not sufficient
to cause failure and that load is then released, the resulting
state of affairs may be pictured as at a" and b'\ The
crack a" is longer than the initial crack a, and the notch
6" is deeper than the initial notch b. However, the curva-
ture at the end of the crack and at the bottom of the notch
may be either sharper or less sharp than at first. For
example, if considerable slip has taken place at the end of
a crack or a notch, cold working the metal there, there would
be a tendency for the end to ''stay open" on release of load,
and the crack or notch would remain comparatively blunt
ended. If, on the other hand, the crack spreads with but
little slip accompanying its spread, on release of load there
will be less resistance to the ''closing up" of the end, and
ehe crack or the notch will tend to be sharp ended. Stress
concentration then may be either increased or diminished,
and the damage done by succeeding loads may be either
greater or less than that done by the first load, depending
on the magnitude of the load and upon the changing nature
of the material as the crack reaches different crystalline
grains. It is, then, not difficult to picture how the repeti-
tion of a load smaller than that required to fracture the
material at one application may fracture it under repeated
applications.

If the load is not merely repeated but is reversed, the
state of affairs may be pictured as at a'" and h'^'. The
reversed load might shorten the crack and the notch,
although this is by no means certain. It would
undoubtedly tend to make the curvature at the ends of the
defects more sharp than that after release of load (a" and

80 THE FATIGUE OF METALS

h") and thus tend to increase stress concentration for the
succeeding cycles of stress. It is then easy to see in a
general way how cycles of reversed stress are more likely to
spread a crack to failure than are cycles of one-direction
stress of the same maximum value.

To the user of material, the significance of slip seems to
lie in the location of some elastic limit or yield point which
marks the practical limit of retention of original form by a
machine part or a structural member. The significance of
spreading fracture lies in the location of an endurance
limit or fatigue limit, below w^hich repetition or reversal
of loading will not cause a crack to spread to failure. Both
limits are most conveniently measured in terms of computed
stress, that is, stress computed by the ordinary formulas of
mechanics of materials, which, as has been pointed out, is
really an average stress for a considerable number of
crystalline grains of metal.

Explanation of the Discrepancy between Theoretical
and Practical Effect of Holes, Scratches, Etc. — It is an
observed experimental fact that holes, screw threads,
notches, and other obvious defects in metals do not reduce
the strength of metals under repeated stress as much as is
indicated as probable by the theory of elasticity. In a
foregoing paragraph reference has been made to an explana-
tion of this fact on metallographic grounds, assuming the
formation of amorphous material as the result of slip, and
causing the damage to be done, not at the surface of the
hole or scratch, but at some appreciable distance there-
from, perhaps at the next grain boundary. There is here
offered an explanation based on considerations of stress
and strain under spreading fracture. This explanation is
beheved to be neither more nor less fanciful than the
explanation based on metallographic grounds, and is not
at all contradictory to it.

Holes, scratches, nicks, screw threads, and other similar
defects will be called ''imposed" defects, while small
internal flaws and irregularities of machined and rolled
surfaces will be called ''inherent" defects. First oi" all,

FRACTURE UNDER REPEATED STRESS 81

imagine the case of ideal inetal without any inherent defects,
metal homogeneous and continuous. An imposed defect
would then produce its full theoretical effect, as given by the
mathematical theory of elasticity. For the case of an
imposed defect consisting of a small circular hole, the
localized stress at the edge of the hole would be about three
times the average, and under a load of one-third the ulti-
mate of the ideal metal itself, a crack would form, the
stress concentration at its end would be high — higher than
that due to the small hole — and failure would take place
rapidly under repeated load.

Next, imagine a very defective metal in which the inher-
ent defects are of the same order of magnitude as
the imposed defects. For this metal an imposed defect
would produce no weakening except that due to the actual
area of metal removed. The inherent defects already have
set up stress concentrations as bad as those set up by the
imposed defect, and since they are of the same order of
magnitude, the area of influence round the inherent defects
is as great as the area of influence round the imposed
defect. If there is imagined a specimen with a hundred
small holes bored through it at points well scattered over
the surface, the tensile strength under repeated load is not
much lowered by boring one more hole.

In the third place, imagine metal in which the inherent
defects are of a smaller order of magnitude than the
imposed defect, which for purposes of illustration may
be a small hole. The inherent defects are spread through-
out the metal (or the inherent defects of surface are
spread around the edges of the hole), and under load the
stress-raising effects of inherent defects and of imposed
defect are added. If, however, under this additive effect
a crack starts and spreads, conditions change. The crack
itself may be considered as of the same order of magnitude
as the small inherent defects, and as it spreads, it soon
begins to get out of the area of influence of the hole. This
is crudely illustrated in Fig. 18, in which the dots represent
inherent defects. Initially the stress at a is the sum of the

9-n9:\%

82 THE FATIGUE OF METALS

stress S, the theoretical stress at the edge of the hole, and
a very high stress Q, due to stress concentration at the
inherent defect at a. If, however, the crack spreads to h,
the stress due to concentration at the end of the crack may
still be imagined to be of the order of magnitude of Q, but
that due to the stress concentration at the hole will be not
S> but a smaller value Sh, since the stress falls away very
rapidly as the distance from the hole is increased. Thus as
a crack spreads, stress concentrations tend to become
smaller. Hence when considering the spread of a crack to
failure, an imposed defect may be imagined to start cracks
as indicated by the theory of elasticity, but the imposed

Siress Disiribuiion ~
along OXofSiress /
due io Hole H V

Dois represent- minuie flows
or olisconiinui-f-ies, -
"Inheren-h "defects.

, , ^ The Hole His an
f*\' . " Imposed "defec-h

Fig. 18. — Diagram for stress as crack spreads from a large defect.

defect does not cause cracks to spread to failure as readily as
the theory of elasticity would indicate.

It seems reasonable to picture metals with many very
small inherent defects as approaching more closely to the
assumed conditions of the theory of elasticity than do
metals with fewer and larger inherent defects. Moreover,
the size of widespread inherent defects is, in general,
smaller for fine-grained than for coarse-grained metals.
Thus it is reasonable to find fine-grained metals, such as
heat-treated alloy steels, following more nearly the theory
of elasticity, when they have imposed defects, than do
coarser-grained metals ; that is, the effect of holes, scratches,
nicks, screw-threads, etc. might be expected to be relatively
more serious on fine-grained alloy steels than on ordinary
steels. This is found by experiment to be the case.

CHAPTER V

TESTING MACHINES AND SPECIMENS FOR FATIGUE
TESTS OF METALS

Importance of Fatigue Tests and Testing Apparatus. —

For any given metal the strength under repeated stress
seems to be a function of the ultimate tensile strength and
of the regularity of internal structure. The fatigue strength
is best measured by an '' endurance limit " or ''fatigue Umit,"
whose determination will be discussed in the next succeed-
ing chapter. The ratio of endurance limit under cycles of
reversed fiexural stress to ultimate static tensile strength
has been christened the "endurance ratio" by D. J.
McAdam, Jr. The endurance ratio varies for different
metals, and at least for a metal of unknown properties,
direct experimentation under cycles of known stress seems
to be the only way to determine fatigue strength satis-
factorily. Hence, the apparatus, specimens, test methods,
and methods of reducing test data for fatigue tests are of
prime interest.

Types of Repeated-stress Testing Machines. — Testing
machines for making fatigue tests under cycles of repeated
or reversed stress may be classified according to the type
of stress produced :

1 . Machines for cycles of axial stress (tension-compression) .

2. Machines for cycles of flexure.

3. Machines for cycles of torsion (shearing stress).
Another classification of testing machines for fatigue

tests would divide them as follows :

a. Machines producing for each cycle a definite load or
moment on the specimen, which remains constant through-
out the test.

h. Machines producing for each cycle a definite defor-
mation of the specimen, which remains constant throughout
the test.

83

84

THE FATIGUE OF METALS

c. Machines in which both the load or moment and the
deformation vary as the test proceeds.

In any repeated-stress testing machine, as in any ordi-
nary "static" testing machine, there must be provided a
mechanism for applying load or moment to the specimen,
and a mechanism for measuring the load or moment apphed
to the specimen. The load-applying mechanism and the
load-measuring mechanism may be combined in some
machines.

In any repeated-stress testing machine there must be
provided a counter for the number of cycles applied, and
some device by which, when the specimen breaks, this
counter automatically goes out of action. Frequently the
device which throws the counter out of action acts to stop
the testing machine itself.

Repeated-stress Testing Machines for Cycles of Axial
Stress. 1. Spring-type Machines. — Figure 19 shows in

FiQ. 19. — Diagram of axial-stress spring-type testing machine. (Jasper.)

diagram a typical axial-stress testing machine in which
the cycles of load are applied by means of a crank and
connecting-rod mechanism and in which the magnitude
of load is measured by the deformation of a spring. The
specimen S is directly attached to the heavy spring G.
The end of the spring away from the specimen is given a
reciprocating motion by means of the connecting rod K,
which is actuated by the variable-throw crank C. The
magnitude of tensile force or compressive force acting on
the specimen is measured by the extension or the compres-

TESTING MACHINES AND SPECIMENS 85

sion of the spring G, and this extension or compression is
measured by micrometers M' and M". Varying range of
load may be secured by adjusting the initial pressure on
the spring G by means of the screw R and the nuts N'N".
If with the cross-head H at midstroke this initial pressure
is zero, then the machine sets up cycles of completely
reversed axial stress in the specimen; during a cycle, load
changes from tension to compression of equal magnitude.
If the spring is so adjusted that the pressure is zero at the
end of a stroke, the machine sets up cycles of stress varying
from zero to a maximum, cycles of tensile stress for zero
adjustment at one end of the stroke, and cycles of com-
pression for zero adjustment at the other. The particu-
lar machine shown in Fig. 19 was designed by T. M. Jasper
at the University of Illinois.

In any repeated-stress testing machine of the spring
type it is necessary to limit the minimum time of one cycle
to a value well above that of the natural period of vibration
of the spring, else there will be set up interfering waves of
stress, and the measurement of stress will be very uncertain
in accuracy. With a spiral spring such as that shown in
Fig. 19 and a capacity of 4,000 lb. the maximum speed of
the machine was found to be about 200 r. p. m. Using a
flat spring of short natural period of vibration, spring-type
repeated-stress testing machines have been successfully
operated at speeds up to 1,000 cycles per minute.

The spring-type repeated-stress testing machine falls in
class c (p. 84), since any deformation of specimen or
loosening of grips during the test causes a reduced deforma-
tion of the spring and hence a falling off of the load. In
using a spring-type testing machine to produce cycles of
definite stress in the specimen, it is necessary to observe
spring deformations at frequent intervals, especially during
the early stages of the test, and to adjust the throw of the
crank so as to keep the stress constant.

The axial-stress testing machines used by Wohler in his
classic tests ^ were of the spring type, and that fact limited

1 A very full account of Wohler's tests including a description of hia
testing machines is found in Engineering {London), vol, 11, 1871,

86

THE FATIGUE OF METALS

the maximum speed to less than 100 cycles of stress per
minute.^

2. Inertia-type Machines. — In this type of machine a
mass of iron or other metal is given reciprocating motion
in which the maximum and minimum accelerations are
known. The alternating forces (+ and — ) accompanying
these accelerations are transmitted through the specimen.

Figure 20 shows in diagram an inertia-type machine
used at the British National Physical Laboratory by Stan-
ton and Bairstow. This machine has four reciprocating

Fig. 20. — Diagram of axial-stress inertia-type testing machine. {Stanton and

Bairstow.)

masses Wi, W2, Ws, and TF4 attached to two pairs of
opposed cranks Ci and C2, thus giving complete balance
in both horizontal and vertical directions. The maximum
acceleration of the reciprocating masses occurs at the ends
of the strokes of the cross-heads, and the maximum force
transmitted through the specimens ^1, ^2, Sz, Si is

W{2TnyR/. . R'

F = +

(■ - 1}

in which

F = the accelerating force at the end of a stroke in pounds ,
W = the weight of the reciprocating mass in pounds,
n = the speed of rotation in revolutions per second,

^ For other descriptions of spring-type axial-stress machines, see Univ.
Illinois, Eng. Exp. Sta., Bull. 142, p. 38.

TESTING MACHINES AND SPECIMENS 87

R = the radius of the crank in feet,
g = the acceleration due to gravity = 32.2 ft. per second

per second,
L = the length of the connecting rod in feet.

From the above equation it is evident that the machine
sets up in the specimens cycles of partially reversed stress.

An advantage of the inertia type of repeated-stress
testing machine is that it permits the use of high speeds.
It will be noted, however, that the force applied to the
specimen depends on the square of the speed of rotation.
This necessitates a very close control of the speed of the
line shaft or the motor which drives the machine. Ordinary
sources of power rarely will give sufficiently constant speed
for inertia-type machines, and rather elaborate speed-
regulating devices are usually necessary.^

3. Centrifugal-force-type Machines. — ^This type of machine
is really an inertia machine which, to set up cycles of
stress, utiUzes the centrifugal force of rotating unbalanced
masses instead of the inertia of reciprocating masses.
Figure 21 shows in diagram a machine of this type designed
and used by J. H. Smith of Belfast, Ireland.^ The
specimen S is fastened at one end to the framework of
the machine and at the other to the sliding cross-head C
which slides freely in guides G. This cross-head carries
a shaft on which are mounted disks Di, D2, on which are
fastened eccentric weights TFi, W2. The disks are driven
through a universal joint f7 by a drive disk D3, on which is
mounted an eccentric weight W3 which balances the com-
bination Wi W2. Disk D3 is driven by a shaft on which is
a drive pulley or a connection to a motor. As the disks
rotate, the centrifugal forces set up by the unbalanced
weights TFi and W2 cause cycles of alternate tension and

^ See Stanton, T. E., "Alternating Stress Testing Machine at the
National Physical Laboratory," Engineering (London), Feb. 17, 1905, for
details of an inertia-type repeated-stress testing machine.

2 See Smith, J.-H., "Testing Machine for Reversals of Stress," Engineering
(London), Mar. 10, 1905; and "Fatigue Testing Machine," Engineering
(London), July 23, 1909.

THE FATIGUE OF METALS

compression in the specimen, the maximum tensile (or
compressive) force being

W{2TnyR

F = +■

9

in which

F = maximum tensile force and maximum compressive

force in specimen in pounds,
W = combined weight of Wi and W2 in pounds,
n = number of revolutions per second,
R = radius to center of gravity of Wi and W2 in feet,
g = acceleration due to gravity = 32.2 ft. per second
per second.

Fig. 21. — Diagram of axial-stress centrifugal-force-type testing machine.

H. Smith.)

(/.

If it is desired to set up cycles of stress not completely
reversed, load is put on the specimen by tightening the
nut A'', thus putting a known steady load L on the speci-
men through the spring P. Then the cycle of load is from
a value of L -\- F to a value of L — F.

TESTING MACHINES AND SPECIMENS

89

Centrifugal-force testing machines require very close
speed regulation as do all inertia-type machines, the stress
set up varying as the square of the speed of rotation.

In all inertia-type machines (including centrifugal-force
machines) any deformation of specimen tends to increase
the throw of the reciprocating masses or of the cross-head
of the centrifugal force machine and hence tends to increase
the range of stress devel-
oped in a cycle. This action
is the reverse of that noted
in the case of spring-
type repeated-stress testing
machines.

4. Alternating-current
Magnet-type Machines. —
The general introduction of
alternating-current electric
circuits suggested to several
investigators the use of
alternating-current mag-
nets as a means of setting
up cycles of stress at a very
rapid rate. Hopkinson^
and Kapp^ both devised
such machines. A machine
of this type designed by
B. Parker Haigh of the ^ _. , ,

1 -\T 1 A 1 -TIG. 22. — Diagram of axial-stress alter-

Hoyal JNaval Academy at nating-current magnet-type testing ma-

Greenwich, England, ^«- "^^"'- ^^"''^^-^

/////7///n;;////;/////////

has

been developed commercially and is today, in spite of its
very high cost, the most widely used fatigue-testing machine
for repeated axial-stress tests. It is shown in diagram in
Fig. 22. The specimen & is attached at one end to the
framework of the machine and at the other to the armature
A , which is placed between two magnets M' and M" . These
magnets are energized by two-phase alternating current, one

^Proc. Roy. Soc, vol. 86^, November, 1911.
^Zeit. Ver. deut. Ing., Aug. 26, 1911.

90 THE FATIGUE OF METALS

phase being connected to each magnet. Thus the specimen
is alternately stretched and compressed by the action of
the magnets. If the air gap between armature and pole
pieces is the same above and below the armature, the current
in both magnets is the same, and after setting up a specimen
in the machine, the position of the armature is adjusted
until this equality of current is established as shown by
zero reading of a differentially wound ammeter connected
to both phases.

A measure of the force exerted during each cycle of
loading is obtained by the use of a voltmeter connected to
the secondary coils K' K" placed near the pole pieces of
the magnets. The readings of the voltmeter are calibrated
in terms of pounds pull or push by the use of a specimen
with a mirror extensometer attached. The specimen in
turn is standardized by means of a test in a static testing
machine. It is assumed that the modulus of elasticity of
the specimen is the same under cycHcal loading as under
static loading.

It is important that the machine be ''tuned " so that there
are introduced no unknown inertia forces. This tuning is
accomplished by means of the spring P. With no specimen
in place, the clamps Q are adjusted along the spring until
the armature oscillates in unison with the magnetic pull.
The magnetic pull does not vary greatly for slight variations
in frequency of supply current. It is, however, important
that the frequency of current be kept constant so that
the vibrating parts of the machine are kept ''in tune"
with the magnet pulls, and unknown inertia stresses are
avoided.

In this machine unequal deformation of the specimen in
tension and in compression would move the armature
nearer one pole piece than the other. It is usually neces-
sary to watch the machine rather closely for the first hour
of a test and, if necessary, adjust the position of armature
and the amount of current supplied, so that the armature
is kept midway between the pole pieces, and thpi^ forces
developed during a cycle remain constant.

TESTING MACHINES AND SPECIMENS

91

By adjusting the screw /, it is possible to superimpose a
known steady load L (either tension or compression) upon
the alternating load ±F set up by the magnetic pull,
causing a range of load during a cycle from L + F to
L -F.

This machine is usually operated at a speed of 2,000
cycles of stress per minute, and a special generator is
required to supply the low-frequency two-phase alternating
current necessary. Properly adjusted and calibrated,
this machine is capable of a high degree of precision. Its
principal drawback is its high cost.

,, ^Y

'''/////////////////////////////////////////////A

^ C li ll< Sec+iona+y-Y

W

Fig.

23. — Diagram of axial-stress rotating-specimen-type testing machine.
{Jasper.)

5. Rotating-specimen-type Machines. — Repeated-stress
testing machines operated by the application and removal of
dead weights have been used in a few special cases, but their
speed of operation is very slow indeed. A machine in
which the constant pull of dead weights causes cycles of
alternating axial-stress in a rotating specimen is shown in
Fig. 23.

This machine was designed by T. M. Jasper of the
University of Illinois and used in the investigation of
the fatigue of metals carried on at that institution. S is
the specimen whose outer end is fastened to the block J,
which can slide freely on guides G in the direction of the

92 THE FATIGUE OF METALS

axis of the specimen. This block / is supported by the
lever L, which rests on the knife-edge N. At the outer
end of the lever is hung a weight W. In the position shown,
the specimen is under compression, and the load on the
specimen equals the weight W X n/m. When the head H
rotates 90 deg., the upward push of the lever acts directly
against the guide G and produces no stress in the specimen.
When the head rotates 180 deg. from the position shown,
tensile stress is set up in the specimen. The machine
gives cycles of completely reversed axial stress.

The machine is run at a speed of 1,000 r. p. m. A dash
pot D minimizes surging of the weights. A revolution
counter K driven by a worm R gives the number of cycles
in a test. A screw V prevents excessive throw of the block
/ when a specimen breaks. A trigger device (not shown)
is actuated by the drop of the outer end of the lever L when
a specimen breaks, and this trigger device releases a spring
which opens the switch on the motor driving the machine.

This machine is not very expensive and, carefully
adjusted, is of a good degree of accuracy. It is necessary
to renew the heavy main ball bearings occasionally, and
this adds an appreciable item to the cost of upkeep of the
machine. This machine comes under class a (p. 83).
For each cycle there is applied a definite range of stress.

Repeated-stress Testing Machines for Cycles of Flexural
Stress. 1. Rotating-heam-type Machines. — Probably 90 per
cent of all repeated-stress tests made have been made on a
type of testing machine in which a transverse load is applied
to a rotating beam, either a cantilever beam or a beam
supported at the ends In this type of machine one side
of the rotating beam is under tensile stress and the opposite
side is under compressive stress, and as the beam rotates,
the stress on any longitudinal '^ fiber" changes from tension
to compression. This type of machine is inexpensive, can be
used at high speeds, and for stresses within the yield point
of the metal tested produces stresses which can be accu-
rately computed ; furthermore, a definite range of stress is
applied during each cycle.

TESTING MACHINES AND SPECIMENS

93

Figure 24 shows the rotating cantilever machine used by
Wohler in his classic tests. S denotes the specimens, a pair
to a machine. One end of each specimen is tapered as
shown at T and the tapered end driven into a tapered hole
in the axle C. The specimen is rotated by the drive pulley
D, which is driven by a belt. At the outer end of each
specimen is a gimbal G, and load is applied through a spring
balance P. A counter (not shown) is attached either to a
specimen or to the shaft driving the machine. From the
records of Wohler's tests it seems that this machine was
run at a speed not exceeding 100 r. p. m.

S =

and M = Pa,

'^^?^^777V77///y/yyyyy'/////////////////y////y////////y///////////7?^^77'^^
Fig. 24. — Diagram of rotating-cantilever-beam-type testing machine. {Wohler.)

In a rotating-cantilever fatigue test the upper surface
of the specimen is in tension, and the lower surface in com-
pression. Cycles of reversed stress are set up in any ' ' fiber"
as the specimen rotates, and the magnitude of the maximum
stress is

Mc
I
in which

M = bending moment at critical section of specimen in

inch-pounds,
P = load hung on specimen (see Fig. 24),
a = distance from line of load to critical section of

specimen in inches (a in Fig. 24),
/ = moment of inertia of cross-section of specimen at
critical section, for a circle I = 0.049 d^, in which
d = diameter.

94

THE FATIGUE OF METALS

c = distance in inches from neutral axis to outer
''fiber," for a circle c = one-half the diameter.

The above formula holds only within the proportional elas-
tic Umit of the metal. Up to the yield point of those metals
which have a yield point, the above formula is of fairly
satisfactory accuracy.

Later users of the cantilever machine have rather gen-
erally discarded the double specimen and have had one
specimen to a machine. Figure 25 shows in diagram the
cantilever machine used by McAdam at the U. S. Naval
Engineering Experiment Station at Annapolis, Md. The

.-B"

N 'Knur led Nuf
N"Hex.Nuf

S''

Fig. 25. — Diagram of rotating cantilever beam-type testing machine.
{McAdam.)

special features of McAdam's machine are the shape of
the specimen, which will be discussed in the section on speci-
mens for repeated-stress tests, and the chuck for holding
the specimen. This consists of a tapered split collet L
to which is attached a hexagonal nut N" . Against a
shoulder on this hexagonal nut iV" there bears a shoulder
of the knurled nut A"', and by means of the interaction of
nuts A"' and A"'' the tapered collet may be forced into or
removed from the tapered chuck B. without the necessity
of using a hammer to tighten or to loosen the specimen.
At the outer end of the specimen is a gimbal G carrying the
ball bearing B" , and from this gimbal are hung weights

TESTING MACHINES AND SPECIMENS

95

W. The weights are attached to the gimbal through a
spiral spring P which minimizes any tendency to surge.

The machine is driven by the spiral gear / and the drive
gear D. A counter is provided to record the number of
cycles of stress for each test. These machines are used in
McAdam's laboratory in blocks of four machines driven
by one motor.

In 1892 Sondericker^ used a modification of the Wohler
rotating-beam machine in which the specimen was a
rotating beam supported at the ends and loaded with two

Fig. 26. — Diagram of rotating

beam-type testing machine.
Farmer.)

{Sondericker,

symmetrical loads. This type of machine has the advan-
tage that between the symmetrical loads the bending moment
is constant and the shear is zero; the action is pure
flexure. Figure 26 shows a machine of this type following
closely a design used by Farmer.^ This machine has been
widely used in recent fatigue tests of metals. The specimen
A is driven through a leather flexible coupling H by the
drive pulley K. The specimen is mounted in ball bearings
B, C, D, and E. A tapered collet fastens the specimen in
each bearing as is shown for bearing D. Gimbals attached

1 "Some Repeated-stress Tests," Tech. Quart., April, 1892.
' Proc. A?n. Soc. Testing Materials, vol. 19, Pt. II, p. 709, 1919.

96

THE FATIGUE OF METALS

"^S

to the two central bearings
C and D carry an equalizer
bar M, and from the middle
of this equalizer bar are sus-
pended weights W, hanging
from a short spiral spring
not shown in Fig. 26. The
specimen drives a counter N,
t and when the specimen
^ breaks, the counter auto-
es^ matically stops. In addition
g, there is provided a device
. (not shown) for throwing
I open the motor switch when
% a specimen breaks.
w) For tests of the stronger
I metals the type of machine
« shown in Fig. 26 serves very
^ well indeed, but for tests of
§ small specimens of weak
M metals, such as pure alumi-

1 num, straight bearings have
S been found to give smoother
° running than ball bearings.,

2 Figure 27 shows such a test-
■2 ing machine designed by R.

I R. Moore and used in the
^ McCook Field (Dayton)
^ laboratories of the U. S.
Army Air Service. The ma-
chine is similar in its general
action to the machine shown
in Fig. 26, but in place of
ball bearings straight ring-
oiling bearings are used. It
is possible to make the action
of this machine so smooth

TESTING MACHINES AND SPECIMENS

97

Fig. 28. — Holders for short specimen in rotating-beam-type testing machine.

iOno.)

Fig. 29. — Rotating-beam-type testing machine. (R, R. Moore.)

98 THE FATIGUE OF METALS

that the vibration of the specimen is only a few ten-thou-
sandths of an inch.

Figure 28 shows a device for using short specimens in a
Sondericker-type machine. This special device was used
by Prof. Ono of Kyushu Imperial University, Japan. ^
It has been applied to the type of machine shown in Fig. 26.
Figure 29 is from a photograph of a very recent model of
rotating-beam testing machine designed by R. R. Moore
and combining the straight bearings shown in Fig. 27
with the chucks for using short specimens shown in Fig. 28.

In the Sondericker type of machine the computed unit
stress at the critical section of the specimen is given by the
flexure formula

■^-2773

in which

S is the unit stress in the extreme fibers of the specimen,
in pounds per square inch,

W is the total weight in pounds (hung at W in Fig. 26),
a is the distance in inches along the specimen from
center of bearing E, Fig. 26, to center of bearing D (or
from center of bearing B to center of bearing C),

I/c is the ''section modulus" of the specimen in inches^,
for a specimen of circular cross-section I/c = 0.0982
d^, when d is the diameter in inches of the specimen at
the critical section.

The rotating-beam type of repeated-stress testing machine
(including the rotating-cantilever type) is inexpensive,
is practically independent of speed effect, and can readily
be run up to speeds of 2,000 r. p. m. ; also the value of
bending moment set up can be computed accurately. It
is the most generally useful type of repeated-stress testing
machine.

2. Rotating-spring-type Machines. — In some repeated-
stress tests it is desirable to have the specimen stationary
so that it can be examined during the test. Figure 30

1 Mem. Coll. Eng., Kyushu Imp. Univ., vol. 2, No. 2, 1921.

TESTING MACHINES AND SPECIMENS

99

shows in diagram such a machine, designed by H. F. Moore,
for tests under cycles of reversed flexure. One end of the
specimen S is held rigid in the vise V, and the other end,
which runs in the bearing B, is rotated in a small circle.
Side wise pressure, which can be adjusted by means of a
screw, is brought on the bearing -S by a calibrated indicator
spring I. The compression of the spring, and hence the
load and the bending moment, on the specimen, is measured
by means of a strain gage spanning the gage holes GG shown
near the ends of the spring. The rotating spring is carried

Fig. 30. — Diagram of reversed-flexure rotating spring-type testing machine.

{H. F. Moore.)

in the cross-head C. Sidewise motion of the bearing B
is prevented by placing the bearing in a slot, and excessive
displacement of the bearing after the specimen breaks is
prevented by the rod R. The cross-head C is driven by a
shaft H, a pulley P, and a motor not shown. The number
of revolutions of the shaft is measured by the counter K
which is driven by a worm on the drive shaft.

When the specimen breaks, the broken end of the speci-
men hits a screw Q and kicks out the lever L. This releases
the spring W which then opens the motor switch D, thus
stopping the machine.

100

THE FATIGUE OF METALS

This machine sets up cycles of completely reversed
flexural stress and can be run at a speed of 1,800 r. p. m.
As in all spring-type machines, it is necessary to read the
load-indicating device occasionally, especially during the
earlier part of a test, and to adjust the load if found
necessary.

3. Spring-type Machines. — Figure 31 shows a type of
machine in which a specimen is subjected to cycles of
flexural stress which are set up by the reciprocating action
of a crank and connecting rod and whose magnitude is
measured by the compression of calibrated springs. Power
is furnished by a motor M (or from a
line shaft). A crank C with adjustable
throw is attached to a connecting rod
R which bends the specimen S back
and forth. The motion of the speci-
men is resisted by springs G acting
through a bent lever A. The magni-
tude of bending moment applied to the
specimen may be varied by changing
the throw of the crank and is measured
by the compression of the springs G.
The magnitude of compression of
springs is measured by the throw of the,
arm 7, to the end of which is attached
a pencil which records the throw on paper wrapped round the
drum D, which is rotated by a worm-and- wheel drive from the
main shaft of the machine. There is thus recorded on the
paper a diagram whose width is a measure of magnitude of
bending moment at the critical section of the specimen
and whose length is a measure of the number of cycles of
stress applied. The number of cycles is also indicated by
a counter K. The stress in the specimen at the critical
section is computed by the usual flexure formula (see p. 4).
This type of machine may be used to produce either cycles
of reversed flexural stress or, by varying the relative initial
pressure on each of the springs G, to produce cycles of
flexural stress with varying ratios of minimum stress to

m)mimimiimmiwnimjMih

Fig. 31. — Diagram of
repeated-flexure spring-
type testing machine.
( Upton-Lewis.)

TESTING MACHINES AND SPECIMENS

101

maximum stress. The Upton-Lewis machine is the com-
monest example of this type of machine. It is made in two
styles, one which runs at about 250 r. p. m., and the other
which may be run at 1,000 r. p. m. if the springs used have a
sufficiently short period of natural vibration.

Figure 32 shows a repeated-stress testing machine of the
spring type designed for making flexure tests of thin, flat
specimens. This machine is similar in principle to the
machine described in the foregoing paragraph. The
specimen N is fastened at one end to the calibrated flat
spring M, and the other end of the specimen is vibrated
back and forth by the connecting rod K, which is operated

Fig. 32. — Repeated-flexure spring-type testing machine for thin flat specimens.

(H. F. Moore.)

by the variable-throw crank D. If the throw of the crank
is increased, the bending moment on the specimen is
increased, and the deflection of Q, a mirror attached to
the calibrated spring, is also increased, causing motion
of a beam of light reflected from the lamp L to the screen S.
There is provided an automatic trip which is operated
by dropping of connecting rod K when the specimen
breaks, and which opens the motor switch, thus stopping
the machine. A counter is also provided which records
the number of revolutions. This machine, developed by
H. F. Moore of the University of Illinois, operates at
1,300 r. p. m. and has proved especiaUy useful in testing

102 . THE FATIGUE OF METALS

specimens from thin material and from locations very
close to the sm^face of metal.

4. Inertia-type and Magnetic-type Machines. — The pos-
sibiHty of application of inertia stresses or of cyclical mag-
netic pulls to the construction of the repeated-stress testing
machines for flexural stress is obvious. Machines of
these types have been built ^ and used, but the convenience
and simplicity of the rotating-beam type and the spring

Fig. 33. — Diagram of reversed-torsion rotating-specimen-type testing machine.

{H. F. Moore.)

type have made these the common types of repeated-stress
testing machine for flexure tests.

Repeated-stress Testing Machines for Cycles of Tor-
sion. 1. Rotating-specimen-type Machines. — The same
general types of device for measuring the twisting moment
(and resulting shearing stress) in specimens subjected to
repeated torsion are used as in repeated-stress testing
machines for cycles of flexure. Figure 33 shows a machine

1 GouGH, "The Fatigue of Metals," pp. 32-34.

TESTING MACHINES AND SPECIMENS 103

analogous in its action to the rotating-beam type of
machine.

The specimen S is attached at one end to the rotating
head H, and at the other end to a rotating flexible beam B.
At the end of this beam a load P is applied by means of
weights or a spring balance through the ball bearing R.
When the axis of the specimen is horizontal and the speci-
men is on the left-hand side of the shaft (as shown in Fig,
33), the twisting moment on the specimen is counter-
clockwise; when the shaft of the machine has rotated 90
deg. from the position shown, there is no twisting moment
on the specimen; when the shaft of the machine has rotated
180 deg., the twisting moment is clockwise. There is a
complete reversal of torsional (shearing) stress during
a rotation. Knowing the length of the flexible beam B,
the pull P at the end of the beam, and the moment set up
in the specimen by the weight of the beam, the torsional
moment and the nominal shearing stress in the specimen
can be computed. The beam B is made flexible, especially
in one direction, to minimize vibration, and a dash pot D
is also of service in this respect. The machine is operated
at a speed of 1,000 r. p. m.

2. Spring-type Repeated-torsion Machines. — Figure 34
shows in diagram the Olsen-Foster machine for cycles of
torsional stress. The specimen S is held at one end keyed
in the vibrating arm A, and at the other it is keyed into the
pivoted lever L. The arm A is vibrated by means of the
adjustable pin which in turn is driven back and forth by
sliding block B in the intermediate vibrating arm M.
M is slotted and is driven by the crank pin Q which in turn
is driven by the shaft and the drive pulley D (or a motor
may be directly connected to the machine). By varying
the position of the pin R up or down the arm A, the throw of
the arm may be varied. The pivoted lever arm L bears
against two calibrated springs P' and P", and the compres-
sion of these springs measures the twisting moment trans-
mitted by the specimen S. This compression is indicated
and recorded by means of the arm H and the lever J

104

THE FATIGUE OF METALS

which carries a pencil at its outer end and records on the
drum T a diagram whose width is a measure of the twisting
moment on the specimen. The drum T is driven by the
worm and wheel W which, in turn, is driven by a belt from
the drive shaft of the machine. A counter K is also
driven from the axle of the drum T. This machine is
similar in its general action to the Upton-Lewis machine
described on page 100, and the two machines are manu-
factured by the same firm. The Olsen-Foster machine

Section a+ Y-Y

Fig. 34. — Diagram of repeated-torsion spring-type testing machine. {Olsen-
Foster.)

usually is equipped with an attachment for making flexure
tests. For torsion tests it can be run at speeds up to about
300 r. p. m.

3. Inertia-type Machines. — Fatigue-testing machines for
cycles of torsional stress which use the inertia of a flywheel
for producing the stress have been developed by Stromeyer^

1 Stromeyer, C. E., "The Determination of Fatigue Limits under Alter-
nating Stress Conditions," Proc. Roy. Soc, vol. 90, p. 411, 1914; McAdam,
D. J., Jr., "A High-speed Alternating Torsion Testing Machine," Proc. Am.
Soc. Testing Materials, vol. 20, Pt. II, p. 366, 1920.

TESTING MACHINES AND SPECIMENS

105

and McAdam. Figure 35 shows in diagram the machine
developed by McAdam. The specimen S is keyed at one
end to a chuck in a shaft which is turned back and forth by
the vibrating arm A, actuated by the connecting rod K and
the variable-throw crank C. The drive pulley D is made
with a heavy rim to give a flywheel effect. If the machine
is equipped with direct motor drive, a flywheel is placed
on the drive shaft. The right-hand end of the specimen
S is keyed in a chuck which is a part of a shaft attached
to the flywheel /, which is shown made up of several sepa-

FiG. 35. — Diagram of reversed-torsion inertia-type testing machine. (McAdam.)

rate disks. To the shaft of flywheel J is attached a mirror
which reflects a beam of light from the lamp L to the scale
Q. When the machine is running, the oscillation of fly-
wheel J causes a band of light to appear on the scale Q.
The width of this band of light is a measure of the ''throw"
of the flywheel J; and knowing the speed of the machine,
the throw of the crank C, and the dimensions of the rods K
and A, the maximum angular acceleration of the flywheel
J can be computed, and from this the maximum and
minimum values of twisting moment on the specimen during
a cycle of stress. This machine has been used at a speed of
2,100 r. p. m. As in all inertia-type machines it is very

106

THE FATIGUE OF METALS

necessary that the speed control be very close, since the
stress developed varies as the square of the speed.

Repeated-stress machines for torsion tests using alter-
nating-current magnets (analogous to the Haigh machine
for tension-compression tests) and machines using the
action of centrifugal force have not been developed, but
the application of either of these agencies could easily be
made.

Repeated-stress Testing Machines for Tests under
Combined Stresses. — Repeated-stress testing machines

Fig. 36. — Diagram of testing machine for combined reversed flexure and steady
tension. {H. F. Moore.)

have been developed for tests under cycles of reversed
flexure combined with steady tension, and for tests under
cycles of combined flexure and torsion. Figure 36 shows a
machine developed at the University of Illinois for tests
under cycles of reversed flexure combined with steady ten-
sion. One end of the specimen S is held rigid in the vise V,
and the other end, which runs in the bearing B, is rotated
in a small circle. Sidewise pressure, which can be adjusted
by means of a screw, is brought on the bearing B by a
cahbrated indicator spring I. The compression of the
spring and hence the bending moment on the critical section
of the specimen are measured by means of a strain gage

TESTING MACHINES AND SPECIMENS

107

spanning the holes GG. The rotating spring is carried
by the cross-head C. The bearing B is placed in a radial
slot in the cross-head. The method of driving the machine
is evident from the figure. The steady-tension load on the
specimen is set up by the action of the spiral spring Q,
which is fitted at each end with a pair of crossed knife-
edges £" and E", so as to cause uniform tension in the
specimen and at the same time not to interfere with the

Fig. 37.-

-Diagram of testing machine for combined reversed flexure and reversed
torsion. {Stanton and Batson.)

flexure of the specimen. There is a shoulder on the specimen
at F which supports one end of the spring apparatus; the
spring is compressed by tightening the nut L. As the
specimen is bent, the fibers are subjected to a maximum
stress, which is the sum of the direct tensile stress and the
tensile stress due to bending, and then to a minimum stress
which is the difference of the direct tensile stress and the
compressive stress due to bending.

108 THE FATIGUE OF METALS

Figure 37 shows in diagram a machine used by Stanton
and Batson^ for tests under cycles of combined flexure and
torsion. The machine is a combination of the Wohler
rotating-beam principle and the principle of the rotating-
specimen torsion machine shown in Fig. 33. As a matter
of fact, the machine shown in Fig. 33 was developed from
the consideration of the work of Stanton and Batson.

In Fig. 37 S is the specimen which is located with its
axis along a radial diameter of the rotating jaw C. One
end of the specimen is rigidly fastened to the rotating jaw C.
The arm B is attached to the free end of the specimen S,
and its axis coincides with the axis of the drive shaft Y.
The weight W is attached to the free end of the arm B,
and as the jaw C rotates, the specimen is subjected to cycles
of reversed bending moment varying from + Wd to — Wd,
and to cycles of torsional moment varying from -\-Wh to
— Wh. Evidently the deformation of the specimen S
and arm B will cause the axis of B to get somewhat out of
line with the axis of Y as the machine is operated, but by
proper proportioning of parts and by the use of a dash
pot attached to the rod carrying the weights W, the vibra-
tion resulting from this misalignment can be kept within
workable limits. Stanton and Batson used a speed of
2,000 r. p. m. for this machine.

Ono^ has used a rotating-beam machine for tests under
cycles of reversed flexure combined with steady torsion,
the steady torsion being set up by using the specimen
as a drive shaft for transmitting power to an electrical
absorption dynamometer.

Constant-deformation Fatigue -testing Machines. —
Figure 38 shows a type of machine which has been used
by Arnold and others for quick shop tests of resistance to
repeated violent stress. No mechanism for measuring
load or moment on the specimen is provided, but for each
cycle the specimen is given a definite deformation — in

1 Brit. Assoc. Repts., p. 288, 1916.

2 "Fatigue of Steel under Combined Bending and Torsion," Mem. Coll.
Eng., Kyushu Imp., Univ., vol. 2, No. 2, 1921.

TESTING MACHINES AND SPECIMENS

109

the machine shown in Fig. 38 a definite range of deflection.
Specimens of various materials are subjected to this
arbitrary deflection, and the value of the material is judged
by the number of cycles withstood before fracture. Usually
the deflection is such that the specimen is stressed well
beyond the yield point of the metal.

It has so far proved impossible to correlate the results of
tests in which length of endurance under definite cycles of
deformation is regarded as the index of value of a metal
with the results of tests in which the stress corresponding
to indefinitely long endurance is regarded as the index of
value. It is sometimes found that the use of short-time

Counier.^So

.'Vafr/ab/e-
Throw Crank

V////7////////////////////////////////.
Fig. 38. — Diagram of constant-deformation repeated bending testing machine.

constant-deflection tests will arrange metals in a quite
different order of merit from that found by the use of long-
time tests at various stresses to determine the limiting
stress for indefinitely long ^'life." The short- time tests
seem to be more a measure of ductility or of toughness of
metal than of its ability to resist millions of cycles of a given
working stress.

Repeated-impact Machines. — Testing machines have
been used in which specimens have been subjected to flexural
action produced by means of repeated blows of a swinging
pendulum or of a falling weight. Usually there is a definite
amount of energy in each blow, but it is, in general, impos-
sible to translate this energy into terms of stress in the
specimen, so that direct correlation of test results on stress-

110 THE FATIGUE OF METALS

measuring fatigue-testing machines with test results on
repeated-impact machines is, in general, impossible.
Repeated-impact machines using some arbitrary amount of
energy per blow and using the length of ''life" under
repeated blows have been used for acceptance tests of metal
for automobile parts, because such tests were supposed to
simulate service conditions. For such tests there must
be used a specimen of a certain arbitrary size and shape,
usually a notched specimen.

Perhaps the most widely used repeated-impact machine
is one designed by Stanton.^ In this machine the specimen
is a simple beam in flexure. It is notched and is struck at
the middle by the head of a small trip hammer which is
driven by a motor. Between blows, the specimen is
rotated 180 deg. around its axis, thus being subjected to
reversed flexure. Various other investigators have devel-
oped machines utilizing this general idea.

Another form of repeated-impact flexure machine was
designed by Gustafsson.^ In this machine the specimen is
a vertical cantilever held in a vise and struck by a pair
of swinging pendulum hammers, first on one side and then
on the other.

In repeated-impact tests, if the energy per blow is rela-
tively high, the test results will arrange metals in an order
of merit similar to that given by single-blow impact tests
(Izod or Charpy tests). If the energy per blow is small,
the order of merit will be similar to that given by repeated-
stress tests, say on a rotating-beam machine. This has
been shown by tests by Stanton, McAdam, and Lessells.^
Repeated-impact tests which approach the conditions of
ordinary repeated-stress tests are very time consuming
indeed, since it is usually not feasible to use a speed of more

^ "Repeated Impact Testing Machine," Engineering {London), July 13,
p. 33, 1906.

2 Roos, J. 0., "Some Static and Dynamic Endurance Tests," Proc.
Intern. Assoc. Testing Materials, Paper V2b, 1912.

3 Stanton and Bairstow, Proc. Brit. Inst. Mech. Eng., November, p. 889,
1908; McAdam, Proc. Am. Soc. Testing Materials, vol. 23, Pt. II, p. 56, 1923;
Lessells, Proc. Am. Soc. Testing Materials, vol. 24, Pt. II, p. 603, 1924.

TESTING MACHINES AND SPECIMENS 111

than 100 blows per minute, whereas the repeated-stress
test may be run at speeds up to 2,000 cycles per minute.

The interpretation of the results of repeated-impact tests
is a matter of no small difficulty. No correlation with
stress values for the material is possible. Moreover, the
use of the number of blows to cause failure as an index
of merit of the material means that results will show a
great deal of "scatter," since a slight variation in the
energy per blow makes a very considerable difference in
the number of blows a specimen can withstand.

Specimens for Repeated-stress Tests. — The values taken
for maximum stress in repeated-stress specimens are those
given by the ordinary formulas of mechanics of materials.

For the specimens ordinarily used these formulas are

P
For tension-compression specimens, S = -r;

For flexure specimens, S = -^■,

For torsion specimens (circular), Ss = -j-

In the above formulas

S = the maximum unit stress in pounds per square inch,

Ss = the maximum shearing unit stress in pounds per
square inch,

P = the axial load, in pounds, on a tension-compres-
sion specimen,

M = the bending moment, in inch-pounds, at the critical
section of the specimen at which the maximum unit stress
is S,

T = the twisting moment, in inch-pounds, at the critical
section of the specimen at which the maximum shearing
unit stress is Ss.

I = the moment of inertia, in inches^, of the area of the
critical cross-section of the specimen (flexure),

J = the polar moment of inertial, in inches'^, of the area
of the critical cross-section of the specimen (torsion),

J = 0.098 d^ for a circle of diameter d,

c = the distance, in inches, from neutral axis to outer
fiber of flexure specimen,

r = radius of circular specimen in inches.

112 THE FATIGUE OF METALS

The foregoing formulas hold only within the proportional
elastic limit of the material (for ductile metals they hold
practically up to the yield point) and give values of unit
stresses for a certain definite section of any given specimen.
If specimens have sharp fillets, notches, screw threads, or
other abrupt changes of form, maximum stresses may
exist which are higher than those given by the nominal
formula for the specimen. In the case of a test specimen
for a "static" test, in which the load is increased gradu-
ally from zero to a maximum, the localized high stress at
a notch or a sharp fillet has no marked effect on the ultimate
tensile strength of a ductile metal on account of a general
readjustment of stress distribution after the stress at such
a point passes the yield point of the metal. In static-test
specimens of brittle materials and in fatigue-test specimens
of all materials, localized high stresses are of importance.
In a fatigue specimen such localized high stress may be the
source of a spreading crack which will cause final failure
under repeated stress. For fatigue-test specimens, then,
it is especially important that the specimen and the grip-
ping devices be so designed that there are set up no appre-
ciable localized stresses whose magnitude cannot be
calculated.

Tension-compression Fatigue Specimens. — It is assumed
by some engineers that fatigue tests using cycles of alternate
axial tension and axial compression give more reliable
results than do tests under cycles of fiexural stress, because
in tension-compression tests the stress can be computed
even beyond the yield point of the metal. Such engineers
usually neglect the fact that it is not at all easy to design
specimens for repeated axial stress so that the load shall be a
purely axial one without any flexure. In static tension-
test specimens truly axial loading can be secured by the
use of a long specimen and by using special spherical-seated
grips. Figure 39 shows such a form of gripping device.
This form of gripping device might be used in a repeated-
tension test in which the load varied from minimum to
maximum tension, but evidently could not be used in

TESTING MACHINES AND SPECIMENS

113

tests in which compressive stress was apphed to the speci-
men. It is evident that if an additional bearing were
used so that the device shown in Fig. 39 could be used to
apply compression, the freedom of adjustment of the grip-
ping device would be hampered. Moreover, a specimen
to be subjected to cycles of alternate axial tension and
compression cannot be long; else there is danger of buckling.

- ^ .\— Hardened
' ^feel Plug

^ .-'•• Machined af
>^^ the Same

Seffing

Fig. 39. — Shackles for insuring axial load on tension test specimen.

{Robert-

Figure 40 shows the specimen used by Haigh in his axial-
stress machine (see Fig. 22). To secure the necessary
combination of rigidity of grip and true axial load, he uses
a threaded-ended specimen and depends on careful machin-
ing and adjustment of the parts of the machine. The
change in cross-section from the enlarged threaded ends
to the middle is made gradually by means of a long taper.
Great care must be taken to avoid a ''stress-raising" notch

114

THE FATIGUE OF METALS

at the junction of tapered portion and straight portion of
the specimen.

Figure 41 shows a form of tension-compression specimen
used with marked success by Irwin. ^ The ends of the
specimen are shouldered, and the reduction in cross-section
from shoulder to midlength is made by turning down the

«i(V|

4-1

T

"■foO.30''

"7 — "^

^Radius c^^Juflch'on
of Taper and Parallel

20Thds per inch
Fig. 40. — Specimen for repeated axial stress. {Haigh.)

specimen with a tool swung on the arc of a circle. Using
this specimen with carefully adjusted grips in a Haigh
machine, Irwin found substantial agreement between the
fatigue strength of several kinds of metal as determined by
a tension-compression test and the fatigue strengths of
the same kinds of metal as determined by reversed-flexure
tests.

Fig. 41. — Specimen for repeated axial stress. (Irwin.)

Specimens for Repeated-stress Tests in Flexure.

Rotating-beam Specimens.— li in a rotating-beam specimen
for two symmetrical loads (see Fig. 26) no reduction of
cross-section is made to localize the fracture, the localized
stress under the bearings carrying the load introduces
an uncertainty as to the actual value of maximum

^Proc. Am. Soc. Testing Materials, vol. 25, Pt. II, p. 53, 1925.

TESTING MACHINES AND SPECIMENS

115

stress. Hence, it is customary to reduce a portion of a
specimen between the middle bearings so that it will
be certain that the maximum stress occurs there. Figure
42(a) shows a specimen in which there is a reduced central
portion consisting of a straight portion connected to the
larger end portions by fillets. The flexure formula would
neglect the localized stress at the junction of straight por-
tion and fillet, and recent work by Timoshenko and Dietz^
has shown that for the dimensions given in Fig. 42(a)
the stress at the junction of straight portion and fillet would

h"^':

■13"

X

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(a)

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0.27% 0.30" . 0.400"

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No.14-20 Righi-Hand
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0.^7 io 030 ATaperl"perfoo+

No. 14-20
Leff-Hand Thread

-9.85 Rad
(C)
Fig. 42. — Rotating-beam specimens

be theoretically about 1.35 times the stress at the middle
of the straight portion.

Figure 42(6) shows a form of rotating-beam specimen in
which the reduced portion is formed by a lathe tool swung
on the arc of a circle of radius much larger than the diameter
of the specimen. The effect of stress concentration for
this specimen is negligible. This form of specimen has
the disadvantage that the maximum stress occurs only at
one section and not along a length of the specimen. With
the dimensions shown, however, the variation of maximum
stress over the middle 0.1 in. is only 1 per cent.

^ "Stress Concentrations Produced by Holes and Fillets," Trans. Am.
Soc. Mech. Eng., vol. 47, p. 199, 1925.

116

THE FATIGUE OF METALS

It is sometimes inconvenient to use specimens as long as
those shown in Fig. 42(a) and (6); Fig. 42(c) shows a short
rotating-beam specimen designed to be used with holders
such as those shown in Fig. 28.

Figure 43(a) shows a rotating-cantilever specimen as
used by some experimenters. This form of specimen has the
disadvantage of stress concentration at the fillet coming
at the point of maximum stress, and also the disadvantage
of having only one cross-section under maximum bend-

^

ft

^

(a)

asis'

i_

4.27

tz iA y k--- -7-^ >j

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0.500

-0.468"

0.812
i

- ^J Taper 0. 25' per fooi-

kV

■lO'

(C)

Fig. 43. — ^Rotating-cantilever-beam specimens.

ing moment. Figure 43(6) shows a similar specimen
except that it is drilled hollow so that the highly stressed
outer fibers have very little backing of inner understressed
fibers. In such a specimen the finish of the surface of the
hole is of great importance. Figure 43(c) shows a rotating-
cantilever specimen used by McAdam.^ This specimen is
tapered for a considerable portion of its length so that at
the large end it is certain that the stress at the fillet is not
the maximum stress in the specimen, and so that for about
\y2 in. of length the maximum stress is very nearly constant.

1 Froc. Am. Soc. Testing Materials, vol. 23, Pt. II, p. 68, 1923.

TESTING MACHINES AND SPECIMENS

117

Figure 44 shows a form of flexure specimen used at the
University of IlHnois for tests under a combination of
reversed flexure and steady tension (see Fig. 36). This
same form of specimen (with screw thread omitted) has
been used for fatigue tests at elevated temperatures, in
which case it is an advantage to have the maximum stress
located within a short range of length of specimen so that

L

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m

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Fig. 44. — Specimen for combined reversed flexure and steady tension.

it may be certain that temperature is measured at the point
of maximum stress.

It may be noted here that specimens of the general form
of Fig. 42(5) have been used by Gillett and Mack^ in the
Upton-Lewis machine (see Fig. 31).

Figure 45 shows a specimen used at the University of
Illinois for tests of thin sheet metal under reversed flexure

'^i^f>— /

^M

^\

W/

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^r^r///

0»|V>

Ui'i<

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Spec/wens are
0.05" TMc/f

Fig. 45. — Specimen for fatigue tests
of thin sheet metal.

Fig. 46.

-Specimen for fatigue test in
torsion.

(see Fig. 32). Values for fatigue strength given by this
specimen are usually lower than values given by rotating-
beam specimens of the same metal. This is probably due
to stress concentration at the fillets. For comparative

1 " Molybdenum, Cerium, and Related Alloy Steels," Am. Chem. Soc,
Monograph, p. 259

118 THE FATIGUE OF METALS

results, however, the specimen shown in Fig. 45 ib
satisfactory.

Specimens for Repeated -stress Tests in Torsion. —
The general design of repeated-stress specimens for torsion
tests is similar in general character to the problem of
repeated-stress specimens for flexure tests. Figure 46
shows a form of specimen quite commonly used for fatigue
tests in torsion. The specimen is fastened into a chuck
by a key which bears on the flattened portions of the
tapered ends. This use of a flattened portion rather than
the use of a keyway sunk in the specimen tends to lessen
stress concentration at the grips.

Surface Finish for Repeated-stress Specimens. — A rough
surface finish at the critical section of a fatigue specimen may
reduce the fatigue strength by as much as 15 or 20 per cent;
it is highly essential that a fatigue specimen be highly
polished near its critical section. A good shop polish using
No. 00 emery cloth is ordinarily sufficient.^ For tension-
compression and flexure specimens, circumferential scratches
do more damage than longitudinal scratches, and where
feasible it is slightly preferable to polish the specimens by
rubbing with emery cloth in an axial direction, although
this is usually rather inconvenient. For torsion speci-
mens circumferential scratches do less harm than longi-
tudinal, and polishing should be done by rubbing in a
circumferential direction.

It is important to be sure that the polishing of specimens
removes all tool marks or deep scratches near the critical
section.

1 Experiments at the University of Illinois indicated that rouge polishing
added fatigue strength only 1 per cent above that shown for specimens
polished with No. 00 emery cloth. See Univ. Illinois Eng. Exp. Sta., Bull.
124, p. 108, 1921.

CHAPTER VI
CHARACTERISTIC RESULTS FOR FATIGUE TESTS

The S-N Diagram. — For determining the fatigue strength
of metals from the results of repeated-stress tests — whether
tension tests, flexure tests, torsion tests, tests under cycles
of reversed stress, or tests under cycles of unidirectional
stress of varying intensity — it is convenient to use diagrams
in which values of computed unit stress are plotted as ordi-
nates and values of number of cycles of stress for fracture
are plotted as abscissae. Such diagrams are called stress-
cycle diagrams by some experimenters and S-N diagrams
by other experimenters {S for unit stress, N for number of
cycles). The term S-N diagram will be used in this book.

Three methods of plotting S-N diagrams have been used :
(1) plotting the values of both S and A^ to ordinary Carte-
sian coordinates, (2) plotting values of S to Cartesian
coordinates and values of N to logarithmic coordinates
(semilogarithmic plotting), and (3) plotting values of both
S and N to logarithmic coordinates (logarithmic plotting).
Figure 47 shows S-N diagrams for a number of steels plotted
to both Cartesian and semilogarithmic coordinates. For
all wrought ferrous metals tested, and for most non-ferrous
metals the S-N diagrams become horizontal, as nearly as
can be determined, for values of N ranging from 1,000,000
to 50,000,000. This seems to indicate a fatigue limit or an
endurance limit, a unit stress below which the metal will
withstand an indefinitely large number of cycles of stress
without failure.

In investigations carried on in the United States, it has
been customary to use either logarithmic or semilogarithmic
plotting. The reason for this is twofold: (1) The use of a
logarithmic scale for values of number of cycles makes it
possible to plot on the same diagram both small and large

119

120

THE FATIGUE OF METALS

values of N with the same percentage of accuracy; and (2)
in a Cartesian diagram there is the danger that the general
tendency towards curvature of the S-N diagram will
lead the investigator to assume that an endurance limit
has been reached at a comparatively low value of N when
such is not the case. Figure 48, in which the data of tests
on unannealed hot-rolled monel metal are plotted to Car-
tesian coordinates (upper), and to logarithmic coordinates

110,000'

100,000 {

90,000

80,000

^70,000

J" 60,000
in"
± 50.000

«P

3 40,000

7.0.000

20,000

io;ooo

0

I

6MI

20 Car

bon,Hea

fi-TrecH

■ed

'*,

[j.SN

'j'ckelSl

eel,Hec

ii Trea

W

n

" "

p-

1

\SM, 0.37Carbon,Hecif Treoifed

^

""

1
»• o Shel, 037Carbon, Artnea/ed

S

^ \ \ \ \

%_Sfeel, 0.02Carbon, as Rolled

I

0 2 4-63

Millions of Cycle'; of 5fr«ss for Failure
(a)

10 10

10^ 10^ lO'' lO''

Number of Cycles for Failure
(b)

Fig. 47. — S-N diagrams — Cartesian and semilogarithmic.

(lower), illustrates this last-named point. The Cartesian
diagram might lead the experimenter to report an endurance
limit of 33,000 lb. per square inch, or if experiments had
been carried only to values of N of 50,000,000, an endurance
limit of 39,000 lb. per square inch might have been reported.
The semilogarithmic and the logarithmic diagrams indicate
that a well-marked endurance limit has not yet been
determined.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 121

In general, there does not seem to be any marked dif-
ference between the semilogarithmic and the logarithmic
method of plotting as regards the determination of values
for endurance limit. The criterion for reporting an endur-
ance limit for a metal is that the S-N diagram shall become

/OtP 200 30O 400

Cc/c/es for Rupfure, f/VJ. /n M////ons

60000

50000

^ 40000
S 30000

o^

n^^

^

Se/Tjj

1

L^

Coora'/na/'is's

^J

^^^-a^

i 2,L

o o

^■■^JS

rTljb*"*^

8 ?^g-

^"■'^

>^ 60000
"^ SO 000

I

5^ 40000

30000

/O'

/o

Ct/c/es for /?upfure, fA/J

/<p'

Cyc/es for Rupft/rej fA/J

Fig. 48. — S-N diagram for special lot of monel metal hot-rolled without anneal-
ing— Cartesian, semilogarithmic, and logarithmic coordinates.

horizontal, or shall approach a horizontal line as an asymp-
tote. Logarithmic S-N diagrams seem to show a ^'knee"
(where the diagram approaches a horizontal line) more
frequently than do semilogarithmic S-N diagrams. The
choice between semi-logarithmic coordinates and logarith-
mic coordinates does not seem to be a matter of very deep

122

THE FATIGUE OF METALS

significance. The authors, however, do recommend that
either semilogarithmic or logarithmic plotting be used for
S-N diagrams.

Typical S-N Diagrams for Various Metals. — Figures 49
to 52 give typical S-N diagrams for a number of metals.

S 40,000

30,000

Number of Cycles ■for Frac+ure

Number of Cycles for Frac+ure

Fig. 49. — S-N diagrams for plain carbon steels. Upper, quenched; lower, not

quenched.
Numbers on diagrams refer to numbers of steels in Tables 2B and ZB.

In a general way three kinds of S-N diagrams are shown:
(1) diagrams such as those for the wrought ferrous metals.
Figs. 49 to 50, and for certain non-ferrous metals {e.g.,
No. 1, Fig. 52, light non-ferrous metals) with a well-marked

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 123

horizontal portion; (2) diagrams, such as that shown for
No. 3, Fig. 52 (lower part), and No. 9, Fig. 51, in which
there is shown a distinct tendency for the diagram to become
horizontal, and in which the diagram still has an appreci-
able downward slope at the greatest value of N observed;

20,000

I08

105 10^ 10''

Number of Cycles •for Frac+ure

Fig. 50. — <S-7V diagrams for alloy steels. Upper, quenched ; lower, not quenched.
Numbers on diagrams refer to numbers of steels in Tables 45 and 5B.

and (3) diagrams, such as that shown for No. 2, Fig. 52
(lower), in which a straight-line relation (for logarithmic
plotting) seems to hold between S and N even when tests
are carried to several hundred milhons of cycles of stress.
It may be noted that S-N diagrams of this third type have

124

THE FATIGUE OF METALS

not been found for any of the ferrous metals, and are unusual
for non-ferrous metals.^

Evidence for the Existence of an Endurance Limit. — The
endurance limit is evidently a very significant physical
property for any metal to be used in structural or machine
parts which in service are to be subjected to cycles of
repeated stress. It seems fitting to examine the evidence
for the existence of this limit. The results of long-time

70,000 r
60,000 ■

40,000

30,000

20,000

15,000

10,000

Number of Cycles -for Frac+ure
Fig. 51. — S-N diagrams for cast steels and cast irons.
Numbers on diagrams refer to numbers of metals in Table 6B.

tests furnish three items of evidence for the existence of an
endurance limit for wrought ferrous metals and for most
non-ferrous metals.

1 McAdam does not consider this third type a real type of S-N diagram.
He holds that when such diagrams are obtained, it is because of corrosion-
fatigue or some other influence not due to the nature of the metal. Prob-
ably if data were available for extending these "third-type" diagrams to a
still greater number of cycles of stress, these "straight-line" diagrams would
be found to bend to approach a horizontal asymptote. In any event, such
diagrams are sometimes, though rarely, met with in making fatigue tests,
especially if the tests are not carried to a very great number of cycles of
stress.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 125

1. For high values of N, the S-N diagrams, even when
plotted to logarithmic coordinates, become horizontal,
at least as nearly horizontal as can be determined by ordi-
nary plotting. For all wrought ferrous metals tested, the

c 40,0"00

S" 30,000

-t 20,000c-.

10,000

10^ 10^ iO""

Number of Cycles -for Frac+ure

10'= I 10'' I0« 10-

Number of Cycles for Frac+ure

Fig. 52. — S-N diagrams for non-ferrous metals. Upper, heavy metals; lower,

light metals.
Numbers on diagrams refer to numbers of metals in Tables 7B and 8J5.

horizontal part of the diagram is developed for values of N
less than 10,000,000.

2. Specimens tested to millions of cycles of stress at or
near the endurance limit, when retested under cycles of
higher stress, have uniformly shown some gain in fatigue
strength. Below the endurance limit the application of

126 THE FATIGUE OF METALS

repeated stress seems actually to improve the metal, rather
than to injure it.

3. For wrought ferrous metals (and for some non-ferrous
metals) at stresses near the endurance limit there can be
noted a distinct rise in temperature. As is pointed out
elsewhere, this is probably an indication of slip rather than
of incipient fatigue failure, but for wrought ferrous metals
slip usually precedes the formation of a fatigue crack. In a
stress-temperature diagram there is near the endurance
limit a fairly well-marked ''knee"; below this "knee" the
rise of temperature is very slight indeed, and the absence
of continuing slip below the endurance limit as determined
by long-time tests seems an indirect piece of evidence in
favor of the existence of an endurance limit.

McAdam has given careful study to the form of the
S-N diagram, giving particular attention to that part of
the diagram corresponding to high computed stress in the
specimen. He finds that under high computed stresses,
especially for specimens not artificially cooled, there tends
to be a reversal of curvature in the diagram, as indicated in
the diagrams for steel No. 15, Fig. 49, and steel No. 12, Fig.
50. Of course, under such high computed stresses it is
frequently the case that inelastic conditions prevail and the
computed-stress value is purely nominal. Inelastic action,
however, would tend to cover up such a reversal of curva-
ture, and its persistence in diagrams is good evidence of
the existence of such a tendency. Most of the diagrams
in this book have not been carried to high enough stress
values to show this tendency plainly.

It cannot, of course, be asserted dogmatically that for
any metal there has been determined a limiting unit stress
below which it is certain that the metal can withstand an
infinite number of cycles of stress, but the authors believe
that the data of long-time tests do show that for ferrous
metals and for most (probably for all) non-ferrous metals
the assumption of the existence of an endurance limit seems
reasonably safe, and that such a limit in all probabihty
exists.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 127

Number of Cycles of Stress Necessary to Develop Endur-
ance Limit. — In Figs. 49, 50, 51, and 52 are shown typical
S-N diagrams for various ferrous and non-ferrous metals.
They are plotted to logarithmic coordinates. From an
examination of these diagrams and of other S-N diagrams,
the following lengths of rotating-beam test have been found
necessary to determine directly and accurately the endur-
ance limit of a metal :

1. For wrought ferrous metals, from 500,000 cycles
for very hard steel to 5,000,000 cycles for soft structural
steel.

2. For cast steel and cast iron, not less than 10,000,000
cycles.

3. For non-ferrous metals the range of cycles necessary
is very large, varying from less than 1,000,000 cycles for
certain magnesium alloys to several hundred million cycles
for some unusual lots of duralumin, and copper-nickel
alloys. Usually 50,000,000 cycles are sufficient.

For certain lots of monel metal and duralumin, 500,000,000
cycles of stress failed to develop a well-marked endurance
limit. Such results, however, are unusual and were
obtained on unannealed, hot-rolled metal.

The test data plotted in Figs. 49 to 52 inclusive are from
tests under cycles of reversed flexure, and from the data
available it seems that under cycles of direct axial stress
(tension-compression) the endurance limit is, in general,
developed with a smaller number of cycles of stress.

In all metals tested it is found possible to make a close
estimate of the endurance limit from tests run to not more
than 10,000,000 cycles of stress. Even if the S-N diagram
has not become horizontal, the curvature is usually suffi-
cient to enable a close estimate to be made of the location
of the horizontal line which the S-N diagram approaches,^
or if the S-N diagram seems to be a straight line (a rare

^ For a systematic method of estimating the location of this horizontal
line by extrapolation, see McAdam, "Stress-cycle Relationships and
Corrosion Fatigue of Metals," Proc. Am. Soc, Testing Materials, Pt. II,
p. 224, 1926,

128

THE FATIGUE OF METALS

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CHARACTERISTIC RESULTS FOR FATIGUE TESTS 129

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CHARACTERISTIC RESULTS FOR FATIGUE TESTS 131

case), that line can be extended to cover the probable
''life" required.

Table 3 A. — Chemical Composition op Plain Steels, Not Quenched

B

Material

o o

6 ^

1^

a

c g
o

bO »
03 a>

3

2 «
n «

3 a

1

0.02 per cent C, as rolled (hot) Armco iron
0 . 023 per cent C, as rolled (hot) ingot iron

0. 13 per cent C, as rolled hot

0. 13 per cent C, annealed at 1350°F

0.02

0.023

0.132

0.13

0.14

0.18

0.21

0.27

0.30

0.32

0.32

0.37

0.42

0.4S

0.49

0.52

0.56

0.77

0.80

0.93
1.20

0.02
0.005
0.028
0.17

0.06
0.08

0.03
0.037
0.300
0.56

0.37
0.82
1.06
0.52
0.31

0.58
0.60
0.60
0.46
0.56
0.55
0.55
0.51

0.38
0.25

0.005
0.002
0.028
0.008

0.013
0.060

0.010

0.032
0.010
0.010
0.017
0.037
0.023
0.037
0.029

0.017
0.021

0 042

9,

0 031

3

0 017

4

0 047

5

6

0. 18 per cent C, as rolled (hot)

0.21 per cent C, as rolled (hot)

0.27 per cent C, as rolled (hot)

0.30 per cent C, annealed at 1200°F

0.32 per cent C, as rolled (hot)

0 039

7
8

0.017

0.206

0.080

q

0.17

0 051

in

11

0.16
0.19
0.19
0.12
0.24
0.08
0.18
0.12

0.03
0.19

1?

0.37 per cent C, normalized at 1495°F. . ,

0.42 per cent C, annealed at 1560°F

0.48 per cent C, annealed at 1350°F

0.49 per cent C, normalized at 1700°F. . .
0.52 per cent C, normalized at 1550°F.. .

0 . 56 per cent C, annealed at 1470°F

0.77 per cent C, annealed at 1350°F

0.80 per cent C, annealed at 1470°F

0.93 per cent C, normalized at 1600°F.,

annealed at 1450°F

1.20 per cent C, normalized at 1460°F.. .

0 035

13

0 038

14

0 038

15

0 029

16

0 029

17

0 035

IS

0 047

IP

0 036

20

0 045

1?1

0 021

Values of Endurance Limit under Cycles of Reversed
Flexure. — The commonest fatigue failure in machine or
structural parts is a failure under repeated-flexure action,
and the most convenient fatigue test to make is a fatigue
test under reversed flexure. Accordingly there are given
in Tables 2B, SB, 45, 5B, QB, 7B, and SB values of fatigue
limit in reversed flexure. The values of fatigue limit under
cycles of axial stress, of fatigue limit under cycles of shear-
ing stress (torsion), and of fatigue limit under cycles of
stress not completely reversed are discussed in subsequent
paragraphs or in a subsequent chapter. In connection
with Tables 2B to 85 inclusive, reference should be made
to Tables 2A to 8A inclusive. In Tables 2B to SB inclu-
sive, in cases where the endurance limit was not clearly
defined, a limit was determined by extending the S-N
diagrams to a value of N of 1,000,000,000 cycles.

132

THE FATIGUE OF METALS

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CHARACTERISTIC RESULTS FOR FATIGUE TESTS 133

In using the values given in Tables 2B-8B inclusive it
should be remembered that they have been obtained from
tests of small specimens in which the heat treatment given
penetrates the specimen very thoroughly. Values as high
as these could not be expected for machine parts made of
metal of the same nominal composition and under shop
conditions, given the same nominal heat treatment as the
specimens giving the values in the tables. This is especi-
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ing these test results to actual design problems.

Fatigue Strength under Cycles of Axial Stress (Tension
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cycles of alternating tension and compression have proved
decidedly difficult to carry out. There are two reasons for
this difficulty. First, a slight deviation from true axial
loading in a specimen causes serious flexural stresses.
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opportunity for adjustment to true axial stress that appreci-
able eccentricity of unknown amount is very easily
introduced. Second, it is very difficult to design tension-
compression specimens so as to avoid high localized stress at
shoulders. In static tension tests of ductile metals, this
high localized stress has very little effect on the tensile
strength, although with brittle materials it has a very
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are usually less satisfactory than flexural tests. Under
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cracks even in ductile metals, and the actual maximum
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indeterminate and appreciably higher than the nominal
stress obtained by dividing load by minimum area of
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Various series of axial-stress tests have been made, and
the reported ratio of endurance limit obtained to that
obtained from reversed-flexure tests of the same material
has ranged from 0.7 to 1.0. A recent series of tests under

134

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reversed axial stress has been reported by Paul L. Irwin. ^
These tests were made on a Haigh testing machine in the
laboratories of the Westinghouse Electric and Manufac-
turing Company. Extreme care was taken to insure
axial load during the test and the specimen used was very
carefully designed to avoid localized high stress. Irwin has
tested in this way several kinds of steel and a few non-
ferrous metals. Some of the metals tested had an endur-
ance limit below the proportional elastic limit, and some
had an endurance limit above the proportional elastic
limit. He has found that endurance limits under axial
stress had practically the same value as endurance limits
in reversed flexure, obtained on a rotating-beam testing
machine.

It seems desirable to point out that in machine parts and
structural members subjected to repeated axial stress,
localized stress at shoulders, screw threads, etc. is of very
great importance, and is not usually computed or even
estimated; moreover, very few machine or structural
parts are subjected to pure axial atress without any flexural
action. It is doubtful whether structural or machine parts
which are subjected to reversed axial stress in engineering
practice would be likely (because of the accompanying
indeterminate bending stresses) to develop a nominal
fatigue strength higher than about 70 per cent of the endur-
ance limit given by tests of rotating-beam specimens. In
the case of parts with U. S. Standard threads, it is doubtful
whether the computed endurance limit at the root of
thread (load divided by area at root of thread) would be
higher than 25 per cent of the endurance limit determined
by tests of carefully designed specimens under reversed
flexure.

Fatigue Strength under Cycles of Shearing Stress. — In
determining fatigue strength under cycles of shearing stress,
the endurance limit under cycles of repeated torsion is taken
as the index of fatigue strength. Test data for repeated-

^ Irwin, "Fatigue of Metals by Direct Stress," Proc. Am. Soc. Testing
Materials, vol. 25, Pt. II, p. 53, 1925, and vol. 26, Pt. II, p. 218, 1926.

142

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CHARACTERISTIC RESULTS FOR FATIGUE TESTS 147

Table 9. — Fatigue Strength under Reversed Shearing Stress
AND Reversed Flexural Stress

Material

Endurance limit,
pounds per
square inch

Reversed
torsion
(shear)

Reversed
flexure

Ratio
endurance

limit in

torsion to

endurance

limit in

flexure

Investigator

Plain carbon steels:

Armco, annealed

0.24 carbon, as rolled

0.37 carbon, normalized

sorbitic

0.38 carbon, oil quench annealed. .

normalized

oil quench, drawn 1250°F

oil quench, drawn 1000°F

oil quench, drawn 800°F

0.49 carbon, normalized

sorbitic

0.52 carbon, normalized

sorbitic.

0.81 carbon, annealed

0.93 carbon, annealed

sorbitic

troostitic

1.20 carbon, normalized

sorbitic

Average for plain carbon steels
Alloy steels:

3.35 nickel, annealed

oil quench, drawn 900°F

oil quench, drawn 950°F

oil quench, drawn 1150°F

3.50 nickel, annealed

oil quench, drawn 1100°F

oil quench, drawn 1200°F

3.60 nickel, annealed

oil quench, drawn 900°F

water quench, drawn 900°F. . . .

water quench, drawn 1100°F. . .
Chrome-nickel, annealed

oil quench, drawn 700°F

oil quench, drawn 1200°F

Average for alloy steels
Non-ferrous metals:

Nickel, cold-rolled

Monel, hot-rolled

Monel, cold-rolled

Constantin-hot-rolled

Copper-nickel-zinc alloy, hot-rolled
Average for non-ferrous metals

12,700
14 , 000
16,000
32 , 500
17,000
17,500
21 , 500
16,500
20,500
20,000
28 , 000
22 , 000
31,500
19,000
16,300
29 , 000
52 , 000
24 , 500
48,000

28,000
35,000
45,000
37,500
29 , 000
36,000
35,500
22 , 500
38 , 000
47,000
46 , 500
25 , 000
38 , 000
33 , 000

17,400
18,600
19,000
15,800
13,000

26,000
25,500
33 , 000
57,000
30 , 000
32 , 000
33 , 500
33 , 500
33,500
33 , 000
48,000
42 , 000
55 , 000
31,500
30 , 500
56,000
98 , 000
50,000
92,000

49 , 500
54 , 500
63 , 500
54 , 000
54 , 000
64 , 000
63 , 000
51,000
66 , 500
74 , 000
72 , 000
49 , 000
68 , 000
66 , 000

32,400
35,300
38,000
34 , 500
21,900

0.49
0.55
0.49
0.57
0.57
0.55
0.64
0.48
0.61
0.60
0.58
0.52
0.57
0.60
0.53
0.52
0.53
0.49
0.52
0.55

0.56
0.64
0.71
0.69
0.54
0.56
0.56
0.44
0.57
0.63
0.64
0.51
0.56
0.50
0.58

0.54
0.52
0.50
0.46
0.59
0.52

Moore and Jasper

McAdam

Moore and Jasper

McAdam

Moore and Jasper

Moore and Jasper

McAdam

Moore and Jasper

Moore and Jasper

McAdam

Moore and Jasper
McAdam

Moore and Jasper

McAdam
McAdam
McAdam
McAdam
McAdam

torsion tests are fewer than data for reversed-flexure tests.
Table 9 gives the results of fatigue tests under cycles of
reversed torsion. The tests quoted in Table 9 were made,

148 THE FATIGUE OF METALS

some at the U. S. Naval Engineering Experiment Station
and some at the University of Ilhnois. Other tests have
been made at the British National Physical Laboratory,
and Gough in his book, ''The Fatigue of Metals, " quotes the
results of 49 series of fatigue tests which gave an average
value for ratio of endurance limit for reversed shearing
stress to endurance limit for reversed fiexural stress of
0.56. An examination of Table 9 shows values of the above-
named ratio ranging from 0.44 to 0.71 with an average of
0.55. Most of the values found lie between 0.49 and 0.60.

The endurance limit of metals in shear may then be
regarded as having a value of about 55 per cent of the endur-
ance limit in tension-compression. The general result
of fatigue tests thus adds weight to other existing test data,
which tends to show that the maximum-shear theory of the
failure of metals is a safe approximation for ductile metals,
but is not an exact statement of fact.^

Fatigue strength under cycles of torsion not completely
reversed is discussed in Chap. VII.

Accelerated Tests for Fatigue Strength. — Fatigue tests
to give data for determining endurance limit from an S-N
diagram are very time consuming, and various attempts
have been made to devise accelerated fatigue tests.

An accelerated fatigue test which is frequently proposed,
consists of comparative tests between specimens of different
metals using a standard computed unit stress (or a standard
deformation of specimen) for the series, and taking the
length of ''life" under this standard unit stress (or deforma-
tion) as an index of fatigue strength. In the opinion of
the authors this test is a very unsatisfactory one. In
the first place, a very slight accidental variation in stress or
deformation makes a large change in the "life" of a speci-
men, and the results of such tests show a great deal of
"scatter," so much as to render doubtful the quantitative
value of the results. In the second place, if a number of
metals are tested by this accelerated method and are then

^ TiMOSHENKO and Lessells, "Applied Elasticity," Chap. XVII,
Westinghouse Technical Night School Press.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 149

arranged in order of fatigue strength, the order of arrange-
ment will depend on the severity of the standard stress used.
A violent short test under high stress tends to emphasize
the effect of ductility, while as the stress is lowered, the
effect of strength is emphasized. The authors cannot
recommend the use of this accelerated test.

An accelerated fatigue test which seems to have a limited
field of usefulness is the rise-of-temperature test. In
1 855 Lord Kelvin^ called attention to the fact that a material
subjected to elastic stress is cooled under tensile stress and
heated under compressive stress, but that inelastic stress
causes heat for either tension or compression. In 1913
C. E. Stromeyer devised and used an apparatus for deter-
mining fatigue limit by means of the heat generated under
repeated stress. He used an inertia-type torsion testing
machine in which a stream of water flowed over the speci-
men, and delicate mercury thermometers measured the
temperature rise in the water. He did not, however, check
his fatigue limits by means of long-time tests to destruc-
tion. In 1921 Putman and Harsch developed an apparatus
for measuring the rise of temperature under repeated stress,
using a delicate thermocouple to indicate rise of tempera-
ture. They found a good correlation between the endur-
ance limit determined in this way and the endurance limit
given by long-time tests, studying some 20 wrought ferrous
metals. Gough developed a similar test at almost the
same time.^ The rise-of-temperature tests seems to give
fairly reliable results for many wrought ferrous metals.
It has not given uniformly reliable results for non-ferrous
metals, especially for cold-worked non-ferrous metals.

Bauschinger in his classical work on the fatigue of metals
always emphasized the idea of an elastic limit which was
gradually acquired by a metal under repeated cycles of
stress, which might be either higher or lower than the

^ Quart. Math. Jour., 1855.

2 References for further study of the rise-of-temperature test are : Unw.
Illinois Eng. Exp. Station, Bull. 124, 1921; Stromeyer, C. E., Mem. Chief
Engineer, Manchester, England, Steam Users' Assoc, 1913; Gough, H. J.,
"The Fatigue of Metals," Chap. X.

150 THE FATIGUE OF METALS

elastic limit of the metal in its primitive state, and which he
regarded as the fatigue limit of the metal. Gough^ devel-
oped this general idea into an accelerated test for fatigue
strength. In his apparatus the stretch of a tension speci-
men or the deflection of a flexure specimen was measured
while a repeated-stress test was in progress. On a graph
plotted with computed unit stress as ordinates and ''run-
ning" stretch or deflection as abscissae, the endurance Umit
was located at the point of deviation of the graph from a
straight line. Lessells also has used this accelerated test.
This ''running-deflection" test seems to be of the same
order of reliability as the rise-of-temperature test. It does
not give altogether trustworthy results for non-ferrous
metals. It seems doubtful whether either the rise-of-
temperature test or the running-deflection test would
distinguish between effects due to sudden temporary slip
(heat bursts) and effects due to the beginning of fatigue
cracks.

In the opinion of the authors both the rise-of-temperature
test and the running-deflection test give indications of the
beginning of serious slip in the metal, and frequently,
though not necessarily, they indicate the beginning of a
fatigue crack, which seems for many metals to occur under
about the same conditions which cause slip.

An interesting accelerated fatigue test has been used by
McAdam. For the test an inertia type of testing machine
is used, and the tendency for stress to increase as deforma-
tion increases (see p. 89) serves to cause the rapid spread
of a fatigue crack. If the stress is below the endurance
limit (with a number of cycles not sufficient to cause appreci-
able strengthening of the metal), the relation of stress (as
shown by the amplitude of oscillation of flywheel, Fig. 35,
p. 105) to strain (as shown by nominal amplitude of oscilla-
tion given by crank pin radius) remains constant. At the
endurance limit this ratio does not remain constant during a

1 GouGH, H. J., "Improvements in Methods of Fatigue Testing," The
Engineer, {London), p. 159, Aug. 12, 1921; also "The Fatigue of Metals,"
Chap. X.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 151

run, but as the incipient fatigue cracks spread, the oscilla-
tion of the flywheel increases. The actual spread of the
fatigue crack is accelerated, once it is started. McAdam
has used this test for determining endurance limit under
cycles of reversed torsion. Data are lacking on which to
base an opinion of its general reliability, but it seems to be
a promising test.^

It is the opinion of the writers of this book that for
cases where it is not feasible to determine the endurance
limit of a metal by a long-time test, the best accelerated
test to use is a series of short-time tests to fracture, using
varying values of stress, and estimating the location of the
horizontal line which the ^S-A^diagram presumably approaches
(see p. 127, especially reference in footnote). Long-time
tests running to several million cycles of stress, however,
should always be made when at all feasible.

Effect of Rapidity of Application of Cycles of Stress. —
Various experimenters have run fatigue tests at different
speeds in attempts to determine whether or not the value
of the endurance limit is affected by the rapidity of applica-
tion of cycles of stress. ^ In general, it seems that for speci-
mens of the size usually used in fatigue testing, which are
free from high localized stress, the value of endurance
limit is very little affected by variation of the speed of
testing over a range from 200 cycles per minute to 5,000
cycles per minute. Recent tests in England by Jenkin
indicate that, at speeds of 20,000 cycles of stress per minute
and higher, the endurance limit is higher than for speeds
up to 5,000 cycles per minute. It should again be noted
that the studies of effect of rapidity of application of cycles
of stress have all been made on specimens of steel reasonably
free from flaws and inclusions and free from high localized
stress.

^ For details of this test, see McAdam, "Accelerated Fatigue Tests and
Some Endurance Properties of Metals," Proc. Am. Soc. Test. Materials, vol.
24, Pt. II, p. 454, 1924.

2 For references to this subject see Univ. Illinois Eng. Exp. Sta. Bull.
124, p. 27, 1921 and Bull. 136, p. 58, 1923.

152 THE FATIGUE OF METALS

Effect of Cold Working of Metals on Fatigue Strength. —

The outstanding effect of cold work on metals is to raise
the elastic strength in the direction of the rolling. Com-
mercial cold-drawing and cold-rolling processes markedly
increase the elastic strength of steel, and have a still greater
effect in increasing the elastic strength of non-ferrous metals.
The effect of these processes is to increase the ultimate ten-
sile strength of metals noticeably, but not to so great a
degree as the elastic strength is increased.

Commerical cold-drawing or cold-rolling of steel seems
to increase the fatigue strength to about the same degree
as the ultimate tensile strength is increased. With the
non-ferrous metals tested a variety of results has been
obtained for the effect of cold-drawing on fatigue strength.
Tests of cold-drawn brass and copper rods in which there
had been brought about a reduction in area of 55 per cent
in a single pass of the cold-drawing process showed no
increase or only a slight increase in fatigue strength over
the strength of the same metal hot rolled. Tests of nickel
and of other non-ferrous metals subjected to a somewhat
less drastic reduction than that mentioned above showed
appreciable increases in fatigue strength over the same
metal annealed. Heating cold-drawn non-ferrous metals to
a temperature well below the critical range distinctly
improves their fatigue strength.^

Cold working of metal seems to exert two opposing effects:
(1) It elongates the crystalline grains in the direction of the
cold-drawing or the cold-rolling and seems to reorient crys-
talline grains into more favorable positions for resisting
slip and fracture; and (2) it tends to start new minute
fractures, or at least to set up severe internal stresses in
the metal so that fractures are likely to be started by a
small additional applied stress. For some degree of severity
of cold-rolling or cold-drawing, there is a maximum of net
benefit, and for more severe cold working, the damage

1 See Univ. Illinois Eng. Exp. Sta., Bull. 124, p. 104, 1921 and Bull. 152,
p. 56, 1925; also McAdam, D. J., Jr., article in Trans. Am. Soc. Steel Treating,
December, 1925.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 153

done increases more rapidly than does the benefit. This
picture is in harmony with the fact often observed by metal
workers that overdrawn metal is weakened. It seems
probable that the internal strains set up play an important
part in weakening metal, in view of the improvement in
fatigue strength noted after slight heating of cold-worked
non-ferrous metals. In this connection the prevention of
''season cracking" in brass by slight heating is of interest.

It is also of interest to note that polished soft steel cold
worked by direct tension in a testing machine and not
afterwards re-polished has its endurance limit lowered/
whereas cold-stretched soft steel polished after stretching
has its endurance limit raised. In commercial cold-drawing
and cold-rolling there is exerted a heavy lateral pressure on
the steel as it is reduced in cross-section, and a smooth sur-
face is produced. Possibly this explains the difference in
results for commercial cold-drawn and cold-stretched steel.
A further study of the internal stresses set up in cold working
might throw further light on this subject.

Cold-drawing and cold-rolling may, then, be regarded as
possible means of increasing the fatigue strength of steel,
but their usefulness is limited by their tendency to set up
severe internal strains in steel and, possibly, to cause
minute cracks.

The Effect of Heat Treatment on Endurance Limit. —
From what has been said regarding the correlation between
endurance limit and ultimate strength, it is obvious that
heat treatment of steels may greatly influence the magni-
tude of the endurance limit. Figure 53, taken from the
results of the Illinois investigation on a 0.93 and a 1.20
per cent carbon steel, illustrates what may be expected.
In the case of the 0.93 per cent carbon steel, the material
in the relatively soft pearlitic condition had an endurance
limit of 30,500 lb. per square inch. This was increased
84 per cent when the steel was given a sorbitic structure,
and 221 per cent when it was given a troostitic structure,

1 Univ. Illinois Eng. Exp. Sta., Bull. 136, p. 60, 1923 and Bull. 124, p. 104,
1921.

154

THE FATIGUE OF METALS

by heat treatment. The curves for the 1 .20 per cent carbon
steel show an increase in endurance limit from 50,000 to
92,000 per square inch, or 84 per cent.

140,000
120.000

20,000

' 10*' lO"'

Number o-f Cycles -for Frac4-urc

10''

120,000
S'lOO.OOO

40.000

10

105

10& lO"'

Number o-f Cycles -for Frac+ure

Fig. 53. — Effect of heat treatment on endurance limit.

The effect on endurance limit of various drawing tempera-
tures is discussed in Chap. VIII.

Fatigue Strength of Steel at Elevated Temperatures.—

Repeated-stress tests of steel at high temperatures have
been made by Lea at Birmingham. University, and by Moore

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 155

and Jasper at the University of Illinois.^ The testing
machine used in the tests at lUinois was of the type shown
in Fig. 30 (p. 99) with the addition of an electric furnace
to heat the specimen. A thermocouple was attached to
the specimen at the necked-down part, at the point of
maximum stress, and this thermocouple was attached to
a recording-controlling potentiometer, which maintained
and recorded any desired temperature up to 1800°F.
The fatigue tests were tests in reversed flexure, run at a
speed of 1,500 r. p. m.

In connection with these fatigue tests static tests under
high temperatures were also made. Some static tests
were made in the ordinary manner, and other tests^ were
run, holding each increment of load until no increase of
stretch (creep) could be observed after several minutes
under load. These tests were called "prolonged and
retarded" tests. Figures 54, 55, and 56 show graphically
the results of tests at Illinois. It will be noted that, in
general, the endurance limit does not fall off so rapidly
under high temperature as does the tensile strength. It
will be further noted that for some tests the endurance
limit approaches in value the ultimate static strength
given by a prolonged and retarded test. At temperatures
so high that these two values become equal, fatigue failure
ceases to be a matter of interest to the machine designer.
In general, at elevated temperatures the ratio of fatigue
strength to static ultimate strength is higher than it is
for ordinary room temperatures. For the steels tested
there does not seem to be much reduction of fatigue
strength below 800°F., except for the Cyclops metal and for
the 1.02 per cent carbon steel. Both these steels were heat-
treated steels and the effect of fairly low temperatures

^Lea, F. C, "The Effect of Low and High Temperatures on Materials,"
Proc. Brit. Inst. Mech. Eng., Dec. 5, 1924, and Univ. Illinois Eng. Exp. Sta.,
Bull. 152, 1925.

2 French, at the U. S. Bureau of Standards, has made much more pro-
longed static tests at high temperatures than these "prolonged and retarded
tests." He has found that the ultimate tensile strength is still further
reduced by further prolonging the time of test.

156

THE FATIGUE OF METALS

/ 20 000

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Fig. 54. — Effect of temperature on mechanical properties of metals.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 157

200000

200 400 600 800 /OOO /200

Tesf/ng Tempero'fc/re /n c/egf. F.

200000

200 400 600 800 .1000 J 2 00

Tesf/ng Tempera' /c/re //? c/eg. /T

Fig. 55. — Effect of temperature on mechanical properties of metals.

158

THE FATIGUE OF METALS

would be to weaken the steels by '^drawing," so that it is
natural to find their endurance limits falling off with any
increase of temperature above ordinary room conditions.
Under high temperatures the ductility of steel is increased,
and it seems reasonable to suppose that the ability of the
steel to withstand plastic action without starting fatigue
cracks is also increased.

A few fatigue tests on monel metal and on cast iron under
elevated temperatures give indications that for these metals
the fatigue strength is proportionately less reduced by high
temperatures than is the case for rolled steel.

a-/ 60 000

200 400 600 300 1000 /200

Tesf/ng^ Tempero'fc/re /n c/eg^. /T

Fig. 56. — Effect of temperature on mechanical properties of metals.

Corrosion of steel and spreading fatigue cracks are
mutually accelerative destructive factors. This action is
discussed further in Chap. VIII on the effect of '^ stress-
raisers" in metals.

The Effect of Case-carburizing on the Endurance Limit.
Moore and Jasper^ report the results of tests made by Muller
on a 0.21 per cent steel and on Armco iron, after these
materials had been carburized. The 0.21 per cent carbon
steel was carburized in a gas-fired furnace at 1750°F. for 3 hr.,
producing a case thickness of 0.06 in. The Armco iron was
carburized in an electric furnace at from 1650 to 1675°F.,

1 Univ. Illinois Eng. Exp. Sta., Bull. 152, p. 63, 1925.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 159

for periods of S^i, 2^^, and 2 hr., respectively. The depths
of case resulting were 0.075, 0.025, and 0.015 in., respectively.

Table 10. — Heat Treatments Used with Specimens of Case-carburized

Steel

Steel

Heat

num-

Designation

treat-

Description

ber

ment

52 .

0.20 carbon steel. .

A

Heat to 1600°F., hold 15 min., quench in
oil, reheat to 1450°F., hold 15 min.,
quench in oil, reheat to 1200°F., hold
30 min., cool in air

B

Heat to 1600°F., hold 15 min., quench
in water, reheat to 1450°F., hold 15
min., quench in water, reheat to
1200°F., hold 30 min., cool in air

C

Heat to 1600°F., hold 15 min., quench in
oil, reheat to 1450°F., hold 15 min.,
quench in water, reheat to 1200°F.,
hold 30 min., cool in air

D

Heat to 1600°F., hold 15 min., quench in
water, reheat to 1200°F., hold 30 min.,

0.02 carbon steel

cool in air

9

(Armco)

E

Heat to 1600°F., hold 15 min., quench in
oil, reheat to 1450°F., hold 15 min.,
quench in oil

F

Heat to 1600°F., hold 15 min., quench in
oil, reheat to 1450°F., hold 15 min.,
quench in oil, reheat to 1200°F., hold
30 min., cool in air

G

Heat to 1450°F., hold 15 min., quench in
oil

H

After carburizing allow steel to cool in
furnace

Table 10 shows the heat treatments to which the speci-
mens were subjected after they had been carburized;
Table 11 shows the results of the fatigue tests. Figures
57 and 58 show the S-N diagrams obtained from these tests.

Table 11 shows that the outside shell of high-carbon
steel which is produced in the case-carburizing process is
very effective in increasing the endurance limit, reaching
a maximum of 162 per cent for Armco iron, heat treatment
E. Treatments A, G, and E were particularly effective,

160

THE FATIGUE OF METALS

while treatment H, which was practically an anneal, was
apparently not at all effective.

Table 11. — Results of Fatigue Tests of Case-carburized Steel

Specimens
All fatigue tests were made on a rotating-beam testing machine

Designation

Case-carburizing treatment

Endurance
limit,
pounds

Increase of

Steel

Depth of case

Heat

endurance
limit over

num-

that of

ber

Inches

Percent-
age of

treat-
ment

per

square

inch

untreated
steel,

diameter

per cent

52

0.20 carbon steel

0

0

as
received

33 , 000

0

0.06

20

C

45 , 000

36

0.06

20

B

48, 000

45

0.06

20

A

55,000

67

9

0.02 carbon steel (Armco). . .

0

0

as
received

26,000

0

0.015

5.0

F

37,000

42

0.015

5.0

E

44 , 000

69

0.025

8.3

H

27 , 000

4

0.025

8.3

G

56 , 000

115

0.025

8.3

E

57,000

• 120

0.075

25.0

F

50,000

92

0.075

25.0

E

68,000

162

Figure 59 shows the relation between depth of case and
endurance limit for Armco iron. The curves indicate
that there is evidently a limit to the depth of case which is
effective in increasing the endurance limit.

The results of tension tests on case-carburized specimens
showed that, as might be expected, carburizing is less
effective in increasing the strength of tension members,
which have approximately uniform stress over the cross-
section, than it is for flexure members, in which only the
outer shell carries the high stresses.

Correlation of Fatigue Strength with Other Physical
Properties.^ — It is not to be expected that there will be found
any precise correlation between fatigue strength and any
one other physical property of a metal. Probably elastic
strength, ultimate strength, and ductility all have an effect

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 161

on the fatigue strength. Fatigue failure, however, seems
to be a progressive tearing apart or shearing apart of

9000^

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70000
60000

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3 for case-carburized Armco iron (0.02 carbon steel) .

metal, and it is not surprising that there seems to be
closer correlation between fatigue strength and ultimate

162

THE FATIGUE OF METALS

tensile strength than there is between fatigue strength and
any other one physical property. Figure 60 shows a
correlation diagram between values of endurance limit and

10^ 10^ lO''

Number o-f Cycles for Frac+ure

Fig. 58. — S-N diagrams for case-carburized steel (0.21 carbon).

ultimate tensile strength for the metals listed in Tables 2B
to 8B inclusive. For ferrous metals the comparatively

/s

^

^

Het
eafrr

'/

/

^

Tr

7e/?f

£

/

p

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ase

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Ar/

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2S

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Dep/h of Case
In Fer Ce/7/ of D/ame/^er of Spec//?? en

Fig. 59. — Effect of depth of case (Armco iron).

30

narrow ''scatter" of plotted points indicates a good degree
of correlation, and as a rough approximation for obtaining
the fatigue strength of a sound wrought ferrous metal for

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 163

Ul+ima+e TensilcS+rengfh.lb.persq.in-Wrought Ferrous Me+als

120,000

100,000

.80,000

&0,000

40,000

20,000

Cois+ Ferrous Me+als Non - Ferrous Me+a!s

UHima-kTensil6 5+reng+h,lb.persq.'m.
Fig. 60.' — Correlation diagram, endurance limit and ultimate tensile strength.

Brine!! Number- Wrought Ferrous Mekils
,0 100 200 300 400 500

14VJ,UUU

/

120,000

/

/

A

"i- 100,000

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y^

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i 60,000

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H

\

0

/'

^

U^i

r

/

0 100 200 0 10,0 200

Cas+ Ferrous Me+als Non-Ferrou'= Me+oiis

Brinell Number

Fig. 61. — Correlation diagram, endurance limit and Brinell number.

164 THE FATIGUE OF METALS

which fatigue-test data are not available, the following
formula may be used up to an ultimate tensile strength of
about 200,000 lb. per square inch:

F.L. = 0.50 T.S.,

in which F.L. = estimated endurance limit of the wrought
ferrous metal under cycles of reversed flexure (pounds per
square inch),

T.S. = ultimate tensile strength of the metal (pounds per
square inch) .

For cast ferrous metals the data are few, but such data

as are available indicate that the following formula may be

used to give a rough estimate for the endurance limit of

cast steel free from blowholes and abnormal inclusions:

F.L. = O.4OT.1S.,

in which the symbols are the same as in the equation for
wrought ferrous metals.

For non-ferrous metals, as might be expected, there is a
wider range of ratio of endurance limit to ultimate tensile
strength, the value of the ratio varying from 0.25 to 0.50.
No general equation for non-ferrous metals can be given at
the present time.

The Brinell number for wrought ferrous metals usually is
proportional to the ultimate tensile strength. Figure 61
shows, as might be expected, a fairly good correlation
between Brinell number and endurance limit for wrought
ferrous metals. For purposes of estimation of fatigue
strength of sound wrought ferrous metals, the following
formula may be used:

F.L. = 250 BHN,

in which F.L. denotes the estimated endurance limit for
cycles of reversed flexure (pounds per square inch),
BHN denotes the Brinell number.

For cast ferrous metals the data are too few, and for
non-ferrous metals there is too much "scatter" of results
to justify giving a formula correlating estimated endurance
limit with Brinell number.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 165

Figure 62 gives a correlation diagram for endurance
limit and proportional elastic limit for the metals listed in
Tables 2B to 8B inclusive. Here the correlation is dis-
tinctly poorer than that shown for wrought ferrous metals
in Figs. 60 and 61. Since slip, which seems to be associated
with elastic-limit phenomena, occurs before the start of
fatigue fracture for most, if not all, metals, it would seem

140.000'

120,000

•-. 100,000

^ 80,000

■J

% 60,000

s
o

u

[5 40,000
20,000
0

Propor+ionalElas-ficLimi+.lb.persq.in.

C> C3 C> O ^ 000

g §. o § S- S- S- S.Wrouqh+ Ferrous Me4als
^ S § S £j :g jg

0

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0

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0

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r

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g o S

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S <=> c>

S o <=>_

C3~ o" C2

Casf S+tel Non-Fcrrous Me+als

Proportional Elas+ic Li"m'i+,Ib.persq.in
Fig. 62. — Correlation diagram, endurance limit and proportional elastic limit.

that the elastic strength of a metal is to be regarded as a
factor in fatigue strength, but a minor one.

In Fig. 63 are plotted values of endurance Umit against
values of reduction of area of fractured tensile specimen — a
value generally accepted as an index of ductility. No
correlation is shown in this figure for wrought ferrous metals,
cast steel, or for non-ferrous metals. Probably ductility
is a major factor in determining resistance to a single impact
(Charpy or Izod value) but a very minor factor in deter-

166

THE FATIGUE OF METALS

mining endurance limit. Ductility is, however, a valuable
property of metals, and possibly it is a major factor in
determining the effect on fatigue strength of regions of
localized high stress.

Charpy impact values were not available for many of the
metals hsted in Tables 2B to SB inclusive. In Fig. 64(a)

80

c20
-I 10

io

50
40
30
20
10
0

-t-

fo

<

5 0

9

0

-Ni

1 1 1 1

■)N-FERROUS METALS

0^

0

0

0

0 n

0 t

0

^-D-

0

>

•

•

•

CAST STEEL

•

P

<?

i*o^

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0
00

0

°=h

°%

0

P "

°0

^

8°°

0

0

u 0
0

0

0 r^

0 0

0

u
0

0

°WR0U6HT FERROUS METALS

1 1 1 1 1 1 1

" 9

0

0

0

°o

0

0

>

0

-. — — S £i! ^

Endurance Li'm'i+Jb.persq.in.

Fig. 63. — Correlation diagram, endurance limit and reduction of area.

is shown a diagram for some 40 steels tested at the Uni-
versity of Illinois, in which endurance limit is plotted
against Charpy impact value. No correlation can be
detected from a study of the diagram. Fig. 64(6) shows
a diagram for the same steels in which endurance limit
is plotted against the results of repeated-impact tests

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 167

/00 000

^ ;| 80000
^\

'^ ^ 60000
\ % 40000

\\

•s

0
0

0
0

o

p

0

0

o

0 <
0 o

0

3
0

0

b

Po

0

o

0

c

o°o

o

0

0

0

oo

o

0

0

0

o

20000

" 0 /O 20 30 40 JO 60

C/7c/rpc/ S/ng'/&>-B/oi^/ /mpac/- Tesf /n /}f-/fe

Fig. 64(a). — ^Correlation diagram, endurance limit and Charpy value.

^60 000

o

o

o

o

"

o

8cc

CO

oc

k

o

o

o

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o

n°°

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.^- 100000
is -^80 000

n\
K\

>^ ^40000

iS 0

^ 0 200 400 600 300 /OOO /200 /400 /600

/V umber of Doujb/e B/oivs-/^epeo/ec/ //7?p(7C/- Mach/ne

Fig. 64(6). — Correlation diagram, endurance limit and results of repeated-
impact test.

168

THE FATIGUE OF METALS

on a special double-hammer machine. No correlation is
shown.

"Scatter" of Fatigue-test Data and Its Significance.' —

In fatigue tests of some metals the values determined lie
very close to the line representing the S-N diagram, while
in tests of other metals the S-N diagram is a line drawn
through the estimated middle of a rather wide zone con-
taining the plotted test results. Figure 65 shows two
sets of data illustrating the above-mentioned distinction.
In any laboratory in which the testing machines are kept
in careful adjustment, and careful test methods are used,

100,000

80,000

60,000

40,000

Number of Cycles, -for Frac+ure
Fig. 65. — "Scatter" of data of fatigue tests.

marked '^scatter" of fatigue-test data for a metal may be
interpreted as indicating non-uniformity of structure of
the metal — dirty metal, metal with minute cracks in it, or
badly segregated metal. If, for example, the metal con-
tains many small particles of dirt, the strength of any test
specimen depends to no small degree on the chance loca-
tion of a piece of dirt near the critical section of the speci-
men. If a large number of specimens of dirty steel are
tested, it may be expected that some would show normal
strength and others low strength, depending on the chance

1 So far as the writers have been able to ascertain, the term "scatter"
as appUed to the irregularity shown by plotted test data was coined by Prof.
G B. Upton of Cornell University.

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 169

location of pieces of dirt with reference to areas of high
stress.

A quantitative measure of '^scatter" would be the aver-
age deviation of a test result from the line representing

Number of Blows +o Fracture
Fig. 66. — Results of impact-endurance tests — Specimen A. {Baaed on McAdam
in Proc. Am. Soc. Test. Materials.)

mean results. The determination of this average devia-
tion would involve a rather tedious use of the method of
least squares. A quahtative idea of the amount of ' ' scatter' '

170 THE FATIGUE OF METALS

may be obtained from the area covered by the plotted
points, as indicated in Fig. 65.

Impact-endurance Tests. — Machine parts are sometimes
subjected to repeated impacts, and it is of interest to
inquire in what way the effect of repeated impacts is related
to the result obtained from an ordinary endurance test,
and also how it is related to the case of a specimen ruptured
by a single blow.

McAdam^ has made tests on various carbon and alloy
steels, determining the ordinary rotating-beam endurance
limit, an impact-endurance limit, the energy of rupture for
a single blow in a modified Charpy machine, and the energy
of rupture in slow bending.

Two types of specimen were used in the impact-endur-
ance machine: type A, having a diameter of 0.5 in., diam-
eter at bottom of notch 0.4 in., notch sides parallel,
radius at bottom, Ke in.; type B, diameter 0.75 inch,
diameter at bottom of notch 0.6 inch, angle between sides
of notch 60 deg., radius at bottom of notch, >^ mm. The
specimens were supported at the ends in the impact-endur-
ance machine, as a simple beam, and subjected to reversed
impacts by the drop of a hammer.

The purpose of the impact-endurance test was to deter-
mine the relation between the energy of blow and the num-
ber of blows necessary to cause fracture. By having
various hammers and using various heights of drop, the
number of blows necessary to cause fracture could be
varied from about 500 to many millions.

In Fig. 66 are shown the results obtained with specimens
of type A . On the left of the diagram are the results obtained
from the slow-bending tests and from the single-blow test
on a modified Charpy machine. It will be noted that the
curves obtained are similar to the ones obtained in a rotat-
ing-beam test, in this case the horizontal asymptote being
reached in the neighborhood of 10,000,000 impacts.

1 "Endurance Properties of Steel: Their Relation to Other Physical
Properties and to Chemical Composition," Proc. Amer. Soc. Testing Materials
vol. 23, p. 56, 1923,

CHARACTERISTIC RESULTS FOR FATIGUE TESTS 171

In Table 12 are shown the rotating-beam endurance
limits for 100,000,000 cycles and the impact-endurance

Table 12. — Impact — Endurance Properties of Carbon Steels and op

Alloy Steels

Reported by D. J. McAdam, Jr., from the U. S. Naval Engineering

Experiment Station, Annapolis, Md.

Steel

Charpyi
value,
foot-
pounds

Impact-
endurance
limit, foot-
pounds

Rotating-
beam

endurance
limit, 2 pounds
per square inch

Ratio of
rotating-beam
endurance limit
to impact-
endurance
limit

1-2 chrome-molybdenum „ .

I-l chrome-molybdenum

H-2 nickel-molybdenum

H-1 nickel-molybdenum

J-2 chrome-vanadium

M-2 silico-manganese

6i.5
92.6
68.0
46.3
45.6
56.4
93.2

0.100
0.075
0.100
0.085
0.110
0.098
0.048

67,500
47,000
57,000
46,500
67,000
62,000
30,000

670,000
630,000
570,000
550,000
610,000
630,000
620 000

' A modified Charpy test was used. Values given have relative significance.
2 Rotating-cantilever type of testing machine used.

1000

100

2-1.0

0.1

0.01

=:

=

=1

... ;

^

^4.r

-V,

-_.— . T„,J.

—

'^/7 and Low CarbonSfeel '
ahrlalS-l Hi'ofh Carbon

t^ Broken A Unbroken '-.
^rial 0-f Low Carbon
jecimenTypeB o Broken ■
jfside Diameter OJS" \
'ameferctiBoiiomofNokhO.6,
igleofNofcfieO" \
idi'us al-BoHom ofNoicfi/ifinm:

H

\

r M

ir

^ =

=

= z

->. ^- -

.. :,,^j:.

c

■»?^

S*

i-^

0
Dj

ir

^

T

^=

= z

--'■■■ — /?y

"v

oi.

f-,

^

H

\

^.

If

i:z=

=

•^^<s-

y^

5= = -

. -... ...

= ;

"'T

^

\

r^^e^

ffii^

■ n

^

=

= =

W^

^:;

t

—

—

n

100

Number o-f Blows ■io Fradure

Fig. 67. — Results of impact-endurance tests — Specimen B.
in Proc. Am. Soc. Test. Materials.)

(Based on McAdam

limits for 10,000,000 cycles. As the last column in the
table shows, the ratio of the two endurance limits is fairly
constant.

172 THE FATIGUE OF METALS

McAdam says:

Evidently, therefore, the ordinate at the extreme left of the impact-
endurance graph depends on the impact properties, and the ordinates
at the extreme right depend on the endurance properties of the metal.
Between these two extremes the influence of the impact properties
decreases and that of the endurance properties increases with increase
in the number of blows necessary to cause fracture.

Figure 67 shows the results obtained on type B specimen
on a high- and low-carbon steel. These tests were made
because the type A specimen had a larger radius at the
bottom of the notch than the standard impact specimen.
At the left of the diagram in Fig. 67 the ordinates are
approximately proportional to the Charpy values, and
at the extreme right the ordinates are approximately pro-
portional to the rotating-beam endurance limits for 10,000,-
000 cycles. The conclusions drawn from Fig. 66, therefore,
apply also to this diagram.

CHAPTER VII

THE EFFECT OF RANGE OF STRESS ON FATIGUE
STRENGTH

Range of Stress Defined. — While the most common
laboratory tests of fatigue of metals are those in which
the stress is completely reversed, yet in structural and
machine parts there are numerous cases in which stress
fluctuates between minimum and maximum limits which in
some cases are the same in sign and in others opposite in
sign. "Range of stress" is the algebraic difference between
stress in the cycles to which a test specimen or a machine
or structural part is subjected. This range of stress may
be conveniently indicated by a numerical value which may
be called the ''range ratio," and which indicates the alge-
braic ratio of minimum stress to maximum stress during a
cycle of stress. Thus a range ratio of —1,0 indicates that
the ratio of minimum stress to maximum stress is — 1.0, or in
other words the stress is completely reversed during a cycle;
a range ratio of 0 indicates that during a cycle the stress
varies from 0 to maximum; a range of ratio of —0.25 indi-
cates that during a cycle the stress varies from a maximum
to a value of 25 per cent of the maximum but in the opposite
direction.

If /Sniax denotes the maximum unit stress during a cycle,
;Smin denotes the minimum unit stress during a cycle,
R denotes the range of stress during a cycle,
r denotes the range ratio for the cycle {r is never

numerically greater than 1),
S^^ denotes the mean stress during a cycle,

then R = S,^ — S,^, remembering that S^^^ and S,^ may
be opposite in sign,

173

174 THE FATIGUE OF METALS

T — S,^/S^^, remembering that r may be either + or — ,
-S^av = mSn,^ + S^), again remembering that S^^ and
S,^ may be opposite in sign,

'Jmax ^^ ^B.v \ 7y i^ and Omin = Oav n ^

Evidently the endurance limit for a metal will have
different values for different ranges of stress. As indicated
on page 23 the endurance limit for cycles of stress varying
from zero to a maximum (range ratio 0) will be greater
than the endurance limit for completely reversed stress
(range ratio —1). For cycles of stress in which the stress
varies between two limits of the same sign (range ratio +),
the endurance limit is still higher. The developing of
formulas to express the effect of range of stress on endur-
ance limit has occupied the attention of a number of
investigators, and some of the more important formulas
will be discussed in this chapter.

Experimental data available for the study of the effect of
range of stress are much fewer than could be wished. This
is largely because of the fact that for every pair of points on
a range diagram, that is, for every value of range ratio for a
metal, it is necessary to make a number of tests sufficient
to give data for an S-N diagram from which to determine,
the endurance limit for that particular range of stress.
It is obvious that to get the data for determining a com-
plete stress-range diagram requires a long time and many
tests.

Gerber's Formula. — One of the early students of Wohler's
work was Gerber, who in 1872 proposed the following
formula for effect of range of stress on endurance limit. ^

^... = I + ^SJ - nS^R,

in which S^^ is the endurance limit for any given range
of stress,
R is the range of stress during a cycle,

1 "Relation between the Superior and the Inferior Stresses of a Cycle of
Limiting Stress," Zeit. Bayerischen Arch. Ing.-Vereins, 1874.

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 175

Su is the static ultimate tensile strength of the

metal,
n is an experimentally determined constant,

usually between 1.33 and 2.00.

Gerber's formula with a proper value for n fits test
results fairly well, but, as pointed out later by Barr,^ a
formula in terms of range ratio is a more convenient for-
mula to use than one in terms of range of stress.

Formulas of Launhardt and Weyrauch. — Two early
studies of effect of range of stress were those of Launhardt
and of Weyrauch, who derived formulas from considerations
based on the elastic action of metals and on the formulas
of mechanics of materials, checked by a few experimental
results. In 1873 Launhardt^ suggested the following for-
mula for cases in which the stress was not reversed (range
ratio from 0 to +1.0)

fSmax = ^0 + l^ySu — So)

in which. So is the endurance limit when r = 0, and the
other symbols have the same values as on pages 173 and
174; r is always + for Launhardt's formula.

In 1876 Weyrauch^ suggested the following formula for
cases when the stress was wholly or partly reversed (r is
negative) :

Sjnnx = So ~r TySo — S^i)

in which S-i is the endurance limit for complete reversal of
stress (r = — 1) and the remaining notation is the same as
in Launhardt's formula.

Figure 68 shows their two formulas plotted in a diagram.
It is inconvenient to have to determine both So and *S_i for a
metal, and a consideration of the combined Launhardt-
Weyrauch diagram led J. B. Johnson to suggest a combined
straight-line diagram which is equivalent to the Goodman
diagram discussed in a succeeding paragraph.

1 Kimball and Bare, "Elements of Machine Design," p. 86, New York.

2 Zeit. des Arch. Ing.-Vereins, Hanover, 1873.

3 Proc. Brit. Inst. Civil Eng., vol. 63, p. 275, 1880-1881.

176

THE FATIGUE OF METALS

+ 60,000

+ 50,000

+ 40,000
n-
I. + 30,000

^ +2o,ooa
<s

i + lO.OOO

-10.000

-zo.ooo

Ulh'mafe Tensile S-frengfh,Su--' ^^^

Fig. 68. — Launhardt-Weyrauch diagram for range of

E Ulhmcrte Tensile Sirengf^h Sg „

-0.33

Fig. 69. — Goodman diagram for range of gtress.

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 177

The Goodman Diagram. — Figure 69 is a stress-range
diagram proposed by Goodman^ and covering the field
of both the Launhardt and the Weyrauch formulas. The
ordinate of the line EB represents the static ultimate ten-
sile strength of the metal. The minimum stresses are
plotted along a 45-deg. line DOB. The horizontal line
through 0 is the line of zero stress, tensile stress being
plotted above the line and compressive stress below it.
According to the ^'dynamic theory" of suddenly applied
loads, the minimum or dead-load stress plus twice the live-
load stress equals the static ultimate strength; and the
maximum applied stress should fall on a line CAB, such that
the point A is five-tenths of the ultimate static strength.
Goodman plotted endurance limits obtained by various
investigators after the material had been subjected to
over 4,000,000 cycles of stress, and found that these experi-
mental points fell fairly well on the straight line CAB.

As the diagram shows, when the minimum stress is zero,
then the maximum stress for indefinite endurance should
be five-tenths of the ultimate static strength. When the
stress is completely reversed at CD, then the plus and minus
stresses should be one-third of the ultimate stress. Pre-
sumably a diagram similar to the one shown in Fig. 69 would
hold when the stress above the zero line is compression and
that below it is tension. Experimental data, however, are
very meager for these combinations of compressive stress.

According to the Goodman diagram the range of stress
(algebraic difference between maximum and minimum)
is greater for reversed stresses, and decreases as the maximum
stress is increased above one-third of the ultimate strength,
being actually zero when the maximum stress coincides
with the ultimate. In other words, as the maximum stress
is increased, the minimum stress must also be increased
algebraically in order that the material may be stressed
indefinitely without fracture.

^ "Mechanics Applied to Engineering," 9th ed., p. 634. See also Fidler,
"Practical Treatise on Bridge Construction."

^

178 THE FATIGUE OF METALS

Criticism of the Goodman Diagram. — The statement has
been made that the Goodman diagram does not always
give values which are accurate, and that sometimes it gives
values which are not safe.^ The Goodman diagram and
the Johnson-Goodman formula (see p. 179) imply that the
endurance limit of a metal under cycles of completely
reversed stress is one-third of the static ultimate tensile
strength. An examination of Fig. 60 (p. 163) shows that
for ferrous metals, both wrought and cast, the endurance
limit is higher than this value, for non-ferrous metals this
ratio of one-third averages results fairly well, but several
values of endurance limit fall below it. So for completely
reversed stress the Goodman diagram and the Johnson-
Goodman formula seem to give results which are on the
safe side for ferrous metals, but which are not on the safe
side for a considerable number of non-ferrous metals.
The writers of this book have examined with especial care
all the recorded results which they could find for metals
tested under varying ranges of stress. The list included
results of experiments by Wohler, Bauschinger, Haigh,
Smith, Smith and Wedgwood, and Moore and Jasper.
The discussion will be limited to the two important cases
in which the stress is completely reversed, and in which the
minimum stress is zero. The authors find that in only
two out of twenty-four series of tests were both endurance
limits respectively less than 0.33 and 0.50 of the ultimate
tensile strength as called for by the Goodman diagram.
For one case one endurance limit was 0.25 and the other
0.47 of the ultimate, while for the second case the values
were 0.29 and 0.493, respectively. In one other series of
tests the endurance limit for minimum-stress zero was less
than 0.50 of the ultimate, and this series gave a value of
0.44. For this last case no information was given for com-
pletely reversed stresses. In five cases out of the twenty-
four series of tests the endurance limit for completely
reversed stress was less than one-third of the ultimate
tensile strength, but in all these cases the endurance limit

1 GouGH, "Fatigue of Metals," p. 72.

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 179

for miniinuni-stress zero (r = 0) was equal to or greater
than one-half the ultimate tensile strength. These five
cases gave values for ratio of endurance limit for completely-
reversed, stress to ultimate tensile strength of 0.270, 0.314,
0.299, 0.302, and 0.263, respectively. The results cited in
this paragraph were from Wohler's tests and from Bausch-
inger's tests, and four of the five results cited in the sentence
before this were from Bauschinger's experiments on axial
stress, for which it is admittedly difficult to avoid the set-
ting up of unknown bending stresses.

Another series of experiments which has been cited ^ as an
argument against the Goodman diagram, is a series by
J. H. Smith^ on plain carbon and nickel-alloy steels. In
these experiments the endurance limits obtained by
Smith's ''yield" experiments, were checked by actual
repeated-stress tests. Most of these tests, however, were
carried out to less than 1,000,000 cycles, so that the endur-
ance limits would in general be too high. Gough calculated
the value of the range of stress when the minimum stress is
zero, and compared these values with the experimental
values. He found by this method that five cases out of
nineteen gave computed values which were greater than the
experimental values, the worst case being 7 per cent in
error. "**'

The Johnson-Goodman Formula. — Working independ-
ently of Goodman, J. B. Johnson ''straightened out" the
Launhardt-Weyrauch diagram (Fig. 62) and obtained a
diagram like the Goodman diagram.^ Johnson devel-
oped a formula which could be used in place of the dia-
gram, and this formula was afterwards simplified by Barr.^
The formula is

^^.. 1 - 0.5r
in which the notation is that given on page 173. It is

1 Gough, "Fatigue of Metals," p. 78.

2 Jour. Brit. Iron and Steel Inst., No. 2, p. 246, 1910.

3 Johnson, "Materials of Construction," 5th ed., p. 781.
^Sibley Jour. Eng., December, 1901.

180

THE FATIGUE OF METALS

-^60000

Fig. 70. — Comparison of experimental results with Goodman's diagram for 3.5
per cent nickel-steel (A and E). {Bull. 136, Univ. of III. Eng. Expt. Sta.)

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 181

again called to mind that r, the range ratio, is positive
if the stress Umits of a cycle are both tension or both

120000

100000

-60 OOO

Fig. 71. — Comparison of experimental results -nath Goodman diagram for 3.5
per cent nickel-steel (C and D). {Bull. 136, Vniv. of III. Eng. Expt. Sta.)

compression, but is negative if one limit is tension and the
other compression.

182

THE FATIGUE OF METALS

The Goodman diagram not only implies that jS_i:
^u = yi, but also that Sq: aS_i = 1.5 (see p. 175 for notation) .
The writers of this book wish to examine this second impli-

/ 00 000
.^ 80000

^ 60000

\

\40000
rs 20000
0

I

(P^-20000

^-40000
100000

■^ eoooo

^ 6000O

\

\40000

aoooo
o

I

<?)-20000

^ -'^0000

Stee/ A/o /c><7, 0.53 Corbon, A/or/vc/Z/zec:^

-60000

Fig. 72. — Comparison of erperimental results with Goodman diagram for 0.53
per cent carbon steel. {Bull. 136, Univ. of III. Eng. Expt. Sta.)

cation of the Goodman diagram in the light of the available
test data for fatigue tests with varying range of stress.
Examining the test data referred to in the foregoing para-
graphs, it is seen that, taking first the results for tests made

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 183

by applying repeated stress to destruction (omitting for
the moment the tests by Smith and Wedgwood), only two
series show a ratio of So'.S-i less than 1.5. One of these is
a series of tests of nickel steel by Moore and Jasper which
shows a ratio of 1.47,^ and the other is a series of tests by
J, H. Smith in which the value of the ratio is 1.44, but in
which So was 85 per cent of the ultimate tensile strength,
probably above the elastic limit of the material, so that
failure was elastic failure rather than fatigue failure.

Figures 70, 71, and 72 show results obtained at the
University of Illinois^ on a number of different steels tested
under various ranges of stress. It will be noted that in a
number of these steels the maximum unit stress was above
the proportional elastic limit of the material, and in no case
was the maximum unit stress less than that given by the
Goodman diagram. In view of the fact, however, that
Bairstow found that when the mean stress is tension, the
specimen suffers a permanent extension, it is probable that
it would be unwise in any case to use a stress range in
which the maximum stress exceeded the elastic limit.
For the same reason, when the Goodman diagram is used,
it seems desirable not to use a maximum stress which exceeds
the elastic limit of the material. It must always be remem-
bered that elastic failure is usually as important a consid-
eration as is fatigue failure.^

^ Moore and Jasper report some other tests with values of ratio So'- S-i
less than 1.50, but in all cases the stress at So was above the proportional
elastic limit of the metal, and failure was probably elastic failure rather than
fatigue failure.

2 Univ. Illinois Eng. Exp. Sta., Bull. 136, pp. 67-69, 1923.

3 As noted on p. 188, results for repeated torsion tests indicate that there
is only a slight difference in range of stress for various range ratios for cycles
of shearing (torsional) stress. McAdam is of the opinion that Figs. 70, 71,
and 72, showing results for flexure tests, support the idea that for cycles of
tensile-compressive stress the endurance range is practically constant within
the elastic range. In the figures named, he suggests that the line for maxi-
mum stress may be drawn parallel to the line for minimum stress up to
values slightly above the proportional elastic limit, and that beyond that
portion the maximum-stress line should be drawn horizontal (see footnote,
p. 190).

184

THE FATIGUE OF METALS

Considering now the experiments by J. H. Smith in which
endurance Hmit was determined by ''yield" method, and
to which reference is made on page 179, the writers of this
book would like to present a comparison of experimental
values and computed values, the computed values of aSq
being obtained by multiplying the experimental values
of iS_i by the factor 1.5. Since there is not for all metals a
constant ratio S^i'.Su, this method tends to correct errors
due to individual characteristics by reducing or increasing
computed values of So when the value of S^i happens to
be below or above normal. This computation gives the
values for Smith's results shown in Table 13.

Table 13. — Experimental and Computed Values of So the Endurance

Limit for Cycles of Stress Varying from Zero to a Maximum

Based on tests made by J. H. Smith

Experimental

Computed value

Error based on

Series

value of »So,

of *So,

experimental

number

tons per

tons per

value,

square inch

square inch

per cent

1

18.5

17.8

- 4

2

19.4

18.9

- 3

3

19.2

21.6

+ 13

4

18.6

19.7

+ 6

5

20.6

20.3

- 1

6

21.2

20.6

- 3

7

21.7

21.7

0

8

22.4

22.2

- 1

9

23.6

23.2

- 2

10

19.3

19.3

0

11

25.6

25.5

0

12

22.3

21.4

- 4

13

27.1

26.6

- 2

14

23.2

20.3

-13

15

21.2

21.5

+ 1

16

23.2

21.8

- 6

17

27.4

28.4

+ 4

18

28.0

29.4

+ 5

19

22.0

25.7

+ 17

As already explained, the experimental method used by
Smith would tend to give values of *S_i which are too large.

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 185

Since the computed values of Sq are 1.5 times S^i, these
values would also tend to be too large. Table 13 shows
six computed values which are higher than the experi-
mental, and only two of these are seriously in error. It is
interesting to note that the average error in the table is
+0.4 per cent.

It is the opinion of the writers of this book that the
criticism of the Goodman diagram and of the Johnson-Good-
man formula is justified in so far as it is a criticism of the
assumption that for all metals the ratio S-i'.Sy, has the
value }i. It is believed that the impUcation that the ratio
So'.S-i has a value of 1.5 for all metals is a reasonably safe
assumption so far as available test data show.

Modified Johnson-Goodman Formula. — The writers of
this book wish to suggest a formula for effect of range of
stress, a formula which has the general form of the Johnson-
Goodman formula, but which is not based on any assumed
ratio of S-i:Su, but rather on an experimentally determined
value of /S_i for each metal. The value of 1.5 for the ratio
So'.S-i is retained, and the proposed formula is

1.5o_i S^^^ 3

>S„„, = 3 r^^ or

1 - 0.5r " >S_i 2 - r

in which the notation is that given on page 173.^ If

o

r = 0, -^r^= 1.5, the Goodman ratio.

In Fig. 73 this equation is plotted as a dotted line, and
there are also plotted test results from the University of
Ilhnois and from the work of Haigh at the Greenwich
Royal Naval Academy. This formula seems to fit the
experimental data available fairly well, and where it differs
from such data, it usually gives results which are on the
side of safety. It is to be noted that in the modified
Johnson-Goodman formula *S_i is an experimentally deter-

^ This modified Johnson-Goodman formula was used by Moore, Kommers,
and Jasper as the basis of the curve shown in Fig. 21 of the paper on " Fatigue
of Metals" presented before the American Society for Testing Materials in
1922, and the formula is given in Moore's "Textbook of the Materials of
Engineering," 3d ed., p. 52,

186

THE FATIGUE OF METALS

mined endurance limit — an endurance limit which may be
determined by a series of rotating-beam tests. It should
be further noted that neither the modified Johnson-Good-
man nor any other formula justifies the consideration of
unit stresses which are high enough to cause static failure —
either elastic failure or rupture.

+06

ri r-, j_- Minimum 5+ress dunnq a Cucle

Hanqe Ratio t- rr j-^ — ~F i"

•^ Maximum Stress, during a („ycla

Fig. 73. — Diagrams for modified Johnson-Goodman formula, Howell formula,
and strain-energy relation for effect of range of stress.

The Howell "Straight -line" Formula. — HowelP has sug-
gested an empirical ''straight-line" formula based on test
results obtained at the University of Illinois. The Howell
formula is

r+ 3

in which the notation is that given on pages 173 and 175.
The graph of the Howell formula has been plotted in Fig.
73, and it is seen to agree fairly well with test data for
values of r from —1.0 to 0. For this range of values of r
the modified Johnson-Goodman formula gives slightly

1 Univ. Illinois Eng. Exp. Sta., Bull. 136, p. 89, 1923.

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 187

''safer" values than does the Howell formula. For values
of r above 0, the Howell formula is the more conservative.

The Strain-energy Hypothesis.^ — In 1919 Haigh^ sug-
gested that the strain energy absorbed within the elastic
Hmit might be of more general application as a criterion of
failure than the hjrpotheses of Saint Venant, Rankine, or
Guest.

In 1923 Jasper^ applied this method to the case of repeated
stresses and suggested that the limiting energy per unit
volume per cycle of stress might be found to be the same
for the cases of reversed stresses and those not reversed.
For the case in which the maximum stress is of opposite
sign to the minimum stress, he derived the formula

Sf 2

For the case in which the maximum and minimum stresses
are of the same sign, he derived the formula

S^ 2_

S^i " 1 - r'
in which the notation is that given on page 173.

In Fig, 73 the graph of the strain-energy relation is plotted.
The modified Johnson-Goodman formula, the Howell
''straight-line" formula, and the strain-energy relation all
fit the experimental data fairly well for values of r from
— 1.0 to 0. In this range the modified Johnson-Goodman
formula is slightly more conservative than either of the
others. Beyond this range, that is, for positive values of
r, the strain-energy relation gives values of ^Sjjjax lower than
the other two formulas, and also lower than test results.
Test data, however, are very few for tests with positive
values of r, and the value of endurance limit in most cases
exceeds the elastic limit, with the result that for structural
and machine parts subjected to cycles of stress varying
between a maximum and a minimum of the same sign, the
danger of elastic failure is usually greater than the danger
of fatigue failure.

1 Brit. Assoc. Repts., p. 486, 1919.

2 Phil. Mag., p. 609, 1923.

188 THE FATIGUE OF METALS

The writers of this book beheve that in the present state
of knowledge of fatigue of metals the modified Johnson-
Goodman formula is a safe and a convenient formula to
use in the design of structural and fatigue parts subjected
to cycles of tensile-compressive stress. This formula
should be regarded, however, as a tentative empirical
formula which may be modified or superseded as the result
of further experimental investigation.

Range of Stress in Torsion. — Moore and Jasper^ have
made some torsion tests on six series of steels in various
conditions of heat treatment, including 0.49 per cent car-
bon, 1.20 per cent carbon, and 3.5 per cent nickel-alloy
steels. The ratio of the endurance limit for various values
of r to the endurance limit for complete reversal varied
from 1.08 to 2.

The cases in which the stress was completely reversed
and in which the minimum stress was zero are of particular
interest here. The ratio So:S-i varied from 1.85 to 2.
According to the Goodman diagram this ratio should be
1.5.

It is desirable to translate these results in terms of range
of stress. According to Goodman's diagram the range of
stress for the case in which the minimum stress is zero
should be 0.75 of the range for complete reversal. The
above results show that the range of stress, for minimum-
stress zero, was either the same as for completely reversed
stress or in the worst case 0.93 of that range. This indi-
cates, therefore, that for the case of torsion the range of
stress is much more nearly constant than it is for the case
of bending stress and axial stress.

Similar results have been obtained by McAdam^ at the
U. S. Naval Engineering Experiment Station at Annapolis
in eight series of tests including plain carbon steels and
alloy steels. He found that in the worst case the differ-
ence in range of stress was 10 per cent, and averaged about
5 per cent.

1 Univ. Illinois Eng. Exp. Sta., Bull. 142, p. 72, 1924.

2 Proc. Am, Soc. Testing Materials, vol. 24, Pt. II, p. 574, 1924.

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 189

It is interesting to note that McAdam found in his experi-
ments with stress of the same sign that there seemed to be a
maximum stress beyond which the upper hmit of stress
could not be moved without a corresponding increase in the
minimum stress, and therefore a decrease in the range of
stress. He found this "endurance yield point" to be in

3.0r

E 2.0

1.0

1 1

o Resul-fs ai Annapo
(McAdam)

„s

(M

su/is a-tlllihoj

's

/

■>ore an

■:ijasp

er)

/

J/

X

x/ °

<y _.iV/ fhfn ihi's range are /O -hs-t

J i resulis : Sfrom Annapolis

/

na OTromjiiinc

/s

^

A

4^'

3*'

/

' o

//3

A

y<

■■ -.Si

on

-^

:^1

1.0 -0.8 -O.fc -0.4- -0.7 0 +0.2 +0.4 +0.6

n o i- Minimum S'^ress during a Cucle

Ranqc Ratio r"= r: — ■■ ft: j — -^ — ~~'-r

3 Maximum i+ress during a Cycle

+0.8

Fig. 74. — Diagrams for modified Johnson-Goodman formtila, strain-energy
relation, and constant-range relation for effect of range of stress in torsion.
(Shear.)

the neighborhood of the elastic limit and yield point.
This fact seems to indicate that in torsion tests, also, the
upper Umit of any range of stress should not exceed the
elastic limit of the material.

Figure 74 shows the results of the tests at Annapolis
and the tests at Illinois.^ The graphs indicate that for
cycles of shearing stress the constant-range relation fits

^ Omitting the cases for which the stress was above the elastic hmit of the
metal.

190 THE FATIGUE OF METALS

experimental results better than the strain-energy relation,
or than the modified Johnson-Goodman formula.

From these results it may be concluded that for cycles of
torsional (shearing) stress the assumption of a constant-
range relation involves no serious error, at least for stresses
below the proportional elastic limit of the metal. It is
also evident, as pointed out by McAdam, that for torsion
stresses a steel of high elastic ratio as well as high tensile
strength is desirable for machinery parts to resist fatigue.

On the basis of the constant range hypothesis the follow-
ing formula may be used :

2^_i

*Jmax — 2o_i + S^i^ or Oj^ax —

1 - r

Here *S_i denotes the endurance limit under cycles of com-
pletely reversed stress, and S^^ and r are minus if the
stress is wholly or partly reversed. When >S_i is known,
S>^^ may be calculated if aS^„ or r (the range ratio) is
known. ^

Formulas Involving Number of Cycles. — Before it was
definitely established that metals had an endurance limit,
a number of formulas were developed which attempted to
show the relation between the maximum stress at which a
material would fail and the corresponding number of cycles
for rupture. One developed by Moore and Seely^ may be
cited as an example:

o _ ^

Here B was a constant depending upon the kind of material,
r the ratio of minimum to maximum stress, and N the num-
ber of cycles of stress necessary to produce rupture. The
formula was based upon the assumption that the S-N
diagram when plotted on log paper was an inclined straight

1 McAdam holds that this formula, involving a constant range relation,
may be used for cycles of flexural stress or of compression-tension, with the
limitation that *Smax must never be considered to be higher than the elastic
limit of the metal.

^ Proc. Amer. Soc. Testing Materials, vol. 15, p. 438, 1915, and vol. 16,
p. 470, 1916. (The 1916 paper corrects a numerical error in the 1915 paper.)

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 191

line which extended to any value of A^, however large.
That assumption is now known to be wrong, and the formu-
las have no value for stresses below the endurance limit.

It does not seem at the present time that there is any
great need for a formula involving the number of cycles for
rupture. Even when the number of cycles to which a
structural or machine part is to be subjected is definitely
known (which is rarely), it does not seem probable that a
stress could be chosen with the degree of precision which
is here contemplated. It would seem that if the endurance
limit for a particular material is known, there is available
all that is necessary for the designer. At this point of
departure nothing can take the place of engineering judg-
ment in determining in a particular case what factor of
safety is to be allowed for such contingencies as unexpected
loads, danger to life in case of failure of a part, and the many
other factors which particular conditions bring up for
consideration.

If at any time a general diagram should be desirable
showing the numbers of cycles of stress which can be
withstood under various conditions, the authors wish to
suggest one which may prove useful as a basis for a rough
estimation. The form of the diagram was suggested by a
diagram published by Stromeyer,^ but the authors wish to
apply it to a modified Goodman diagram, based on an
experimentally determined S-N diagram for any given
metal under cycles of reversed stress and the modified
Johnson-Goodman formula. Figure 75 shows such an endur-
ance diagram for 1.02 per cent carbon steel, oil quenched
from 1450°F. The S-N diagram for reversed stress showed
the endurance limit developed at 500,000 cycles, a '4ife" of
100,000 cycles for a stress range of ± 115,000 lb. per square
inch, a "life" of 10,000 cycles for a stress range of + 130,000
lb. per square inch, and a ''life" of 1,000 cycles for a stress
range of + 150,000 lb. per square inch. To construct the
diagram, locate the pairs of points AC, DE, FG, and HK
from the S-N diagram for reversed stress for the metal.

1 Proc. S. Wales Inst. Eng., 1922.

192

THE FATIGUE OF METALS

Locate S at a height corresponding to the ultimate tensile
strength of the metal, and at any convenient distance to
the right of line KOH. Draw the straight lines CB, EB,
GB, and KB. Draw the curved lines AB by the use of the
modified Johnson-Goodman formula, making QF = 1.5 OA.
Draw the curved lines DB, FB, and HB, making the vertical

, Endurance, 1 000 Cycles
! ^Endurance, 10, 000 Cycles

IndemiielyLongll^^"^"''"^''' mOOOCydes
^Endurance / / \ rUlj-jmaieTensile Sirengf-h

+ 200,000

23

-100.000

Fig. 75. — Diagram for estimating length of endurance under repeated stress
(1.02 per cent carbon steel, oil quenched).

spacing from the line AB the same for any given abscissa
as the corresponding spacings for the lines C-B, EB^ GB,
and KB.

In Fig. 75 the Hues APB and CQB determine the stress
ranges for indefinitely long endurance; the lines D 75 and
EB determine the stress ranges for an endurance of 100,000
cycles; the lines FTB and GB determine the stress ranges
for an endurance of 10,000 cycles; and the lines HRB and

EFFECT OF RANGE OF STRESS ON FATIGUE STRENGTH 193

KB determine the stress ranges for endurance of 1,000
cycles.

As an example of the use of such a diagram, suppose
that a machine part made of this steel will be satisfactory
if it lasts for 10,000 cycles of stress, and that a maximum
stress is to be 175,000 lb. per square inch; it is desired to
determine the range of stress to which the part may be
subjected. From the line FTB, where it crosses the ordi-
nate for 175,000 lb. per square inch at M, drop a vertical
line MN to the line GB. The intersection of MN and GB
is found at —25,000 lb. per square inch, so that the range
of stress will be from 25,000 lb. per square inch in one
direction to 175,000 lb. per square inch in the other.

This method is, of course, a rough graphical method
rather than a careful analytic method, but in view of the
great variation found in length of endurance for any given
stress, such a rough graphical method is believed to be as
precise a method as the circumstances justify.

The Effect of Steady Torsion on the Range of Stress in
Reversed Flexure. — Shafts transmitting power are fre-
quently subjected to a combination of cycles of reversed
flexure together with a constant twisting stress. Recently
Lea^ has made fatigue tests on specimens of three kinds of
steel subjected to varying combinations of reversed flexure
and steady torsion. The three steels tested were a chrome-
nickel steel, a 0.14 per cent carbon steel, and a 0.32 per
cent carbon steel.

Lea found that so long as the shearing stress due to the
steady torsion was below a critical value, no marked effect
on the endurance limit was noticeable. Above this
limiting value the endurance limit was markedly lowered.
His tests are not sufficient in number to determine definitely
this limiting value of shearing stress, but it seems to be
nearly equal to the endurance limit for reversed flexure.
For example, in his tests of 0.14 per cent carbon steel, the

1 Oxford Meeting, Brit. Assoc. Advancement Sci., 1926; also Engineering
(London), Aug. 20, 1926. In discussion, Ono reported results confirming
Lea's results, in a general way.

194 THE FATIGUE OF METALS

endurance limit for reversed flexure with no steady shearing
stress was about 37,000 lb. per square inch. For reversed
flexure together with a steady shearing stress of 33,800 lb.
per square inch the endurance limit was about 39,200 lb.
per square inch (an actual slight increase) , and for reversed
flexure combined with a steady shearing stress of 43,500
lb. per square inch the endurance limit was about 32,500, a
distinct decrease.

CHAPTER VIII

"STRESS RAISERS" AND THEIR EFFECT ON FATIGUE
STRENGTH— STRESS AND CORROSION

Effect of Internal Flaws. — Gillett and Mack^ coined the
term "stress raisers" to denote internal flaws, abrupt
changes in cross-sections, and other factors which tend to
cause local increase in stress not taken into account by the
ordinary formulas of mechanics.

The work of Griffith in showing the effect of scratches in
glass in producing high local stresses has been mentioned
in Chap. IV. He showed that the computed local stresses in
glass were at least ten times as high as the ordinary ultimate
strength. He then carried his argument a step further by
proving that it was possible to have stresses in glass of this
high order of magnitude. He did this with very thin fibers
of glass. He believes that the weakness of ordinary solids
is due to discontinuities and flaws whose ruling dimensions
are large compared with molecular distances.

If these extremely minute flaws assumed by Griffith
really exist, they are of a smaller order of magnitude than
the inclusions, dirt, minute cracks, blow holes, etc., which
can be detected in unsound steel either by the unaided eye
or by the microscope. These minute fiaws, if they exist,
must be very generally and very uniformly distributed
throughout the mass of a piece of metal, since the test
strength of sound metal is found to be uniform and reliable.
It is again noted that instead of assuming these minute
flaws, it is possible to visualize a picture of the mechanics
of fatigue failure either on the hypothesis that in metal
there are high internal stresses which make possible the
start of cracks with slight additional imposed stress, or on

1 Proc. Amer. Soc. Testing Materials, vol. 24, Pt. II, p. 476, 1924.

195

196 THE FATIGUE OF METALS

the hypothesis that minute surface irregularities are the
starting points of fatigue cracks.

Consideration will now be given to the effect of such
minute flaws as can be seen in steel, either by the unaided
eye or through the microscope.

Gillett and Mack have done a considerable amount of
work in studying the effect of non-metallic inclusions and
other inhomogeneities. One series of tests which they
carried out was with steels containing cerium.^ These
steels were always dirty, that is, full of non-metallic inclu-
sions, and gave, on the average, lower results for fatigue
strength than would be expected from the tensile-test
results and the usual relation of endurance limit to tensile
strength. The results on these steels also showed a wider
''scatter" of results than the other steels.

While they found that the greater amount of evidence
indicated that clean steels gave better results than dirty
steels, yet often the opposite appeared to be the case.
They came to the conclusion that it is practically impossi-
ble to polish the surface of a fractured specimen and show
the actual starting point of failure. They are of the opin-
ion that polishing removes some of the material and thus
destroys the evidence, so that metallographic examination
merely shows the condition more or less remote from the
actual point where failure began.
They say :

Examination of successive surfaces showed that the distribution of
non-metallic inclusions is so extremely non-uniform that unless tedious
study of many surfaces indicates that the specimen is uniformly very
clean or very dirty, it is quite impossible to say that the steel was clean
or dirty at the actual point of fracture.

Gillett and Mack further point out that because the
volume of metal subjected to the maximum unit stress in
an ordinary fatigue test is very small, it is quite possible
for the element of chance to play an important part in
determining whether this small volume is clean or dirty.

1 "Molybdenum, Cerium, and Related Alloy Steels," Chap. VIII, espe-
cially p. 158, The Chemical Catalog Company, New York.

"STRESS RAISERS" AND THEIR EFFECT 197

They point out that an inclusion which is some distance
away from the point of maximum stress need not necessarily
cause failure, and that a flaw or longitudinal scratch which
lies parallel to the direction of stress does not markedly
increase the local stress. They examined some specimens
of normalized 0.52 per cent carbon steel which had been
received from H. F. Moore. This material had an endur-
ance limit of 42,000 lb. per square inch. A specimen was
run at 40,400 lb. per square inch for 100,000,000 cycles
without failure. This specimen had a large inclusion,
lying in the longitudinal direction, with its tip 0.01 in.
below the surface and about 0.1 in. away from the point of
maximum stress. Evidently this flaw did not cause a
local stress greater than the endurance limit. Another
specimen of the same steel failed after 3,500,000 cycles at a
unit stress of 40,400 lb. per square inch. This specimen, under
examination, showed some deep circumferential scratches
and a finish decidedly poorer than the unbroken specimen.

Another pair of Moore's specimens of a 3.5 per cent
nickel steel seemed to indicate that inclusions caused failure
in one case, and circumferential scratches caused failure in
the second case.

A pair of specimens of molybdenum-nickel steel sent to
Gillett and Mack by McAdam seemed to indicate that one
specimen failed at a low stress and after comparatively few
cycles because of inclusions and poorer surface finish than a
second specimen which ran at a higher stress for 250,000,000
cycles before failure.

Gillett and Mack conclude that inclusions appear to act
as local stress raisers, and that when they are so shaped, so
oriented, and so placed with respect to the direction of
stress application as to produce a local stress higher than
the endurance limit, they may start fatigue failure even
though the nominal computed stress is below the endurance
limit of the material.

Moore and Jasper^ report having made some tests on
''dirty" steel. They also report erratic results, some

1 Univ. Illinois, Eng. Exp. Sta. Bull. 142, p. 65, 1924.

198 THE FATIGUE OF METALS

specimens giving fatigue results as high as those for clean
steel, while other specimens gave low results.

McAdam^ found that for crank-shaft and propeller-shaft
material the endurance limits were usually lower for speci-
mens taken in a transverse direction than in a longitudinal
direction, although the tensile results did not differ greatly.
He thinks this is probably due to the unfavorable orienta-
tion of inclusions in the transverse specimens.

Lea- speaks of examining bolts taken from couplings and
connecting rods that had broken in service. These bolts
revealed no weakness in tensile tests, but microscopic, and
even naked-eye examination, often revealed slag inclusions
or planes of separation, at which cracks undoubtedly
started which led to failure. It is his opinion that it is a
mistake to use wrought-iron bolts in such cases.

In a discussion of materials used in aircraft construction
Aitchison^ says that one of the most potent groups of imper-
fections in metals is the one including slag, non-metalUc
inclusions, and the like. He points out that the effect of
these imperfections on ductility and toughness is much the
same as their effect on fatigue strength.

Effect of Abrupt Changes in Cross-section. — The effect
of external cracks, scratches, notches, and other abrupt
changes in cross-section is attested by the results of a
number of different experimenters. The results of Moore
and Kommers^ may be cited as an example of the effect of
abrupt changes in cross-section. Figure 76 shows the five
different kinds of specimens which they used. Figure 77
shows the results of the rotating-beam tests, the upper part
of the figure giving results on a heat-treated 0.49 per cent
carbon steel in the sorbitic condition and the lower part on
Armco iron. The endurance limit for the specimen with
the 9.85-in. radius was about 49,000 lb. per square inch.
With a ^'^-in. radius the endurance limit was reduced 8 per

1 Proc. Amer. Soc. Testing Materials, vol. 23, Pt. II, p. 100, 1923.
^ Proc. Inst. Civil Eng., 1923; Engineering {London), vol. 115, p. 253.

3 Engineering {London), p. 90, Jan. 18, 1924.

4 Univ, Illinois Eng. Exp. Sta., Bull. 124, p. 20, 1921.

"STRESS RAISERS" AND THEIR EFFECT 199

cent, with square shoulders it was reduced 51 per cent, and
with a V-notch it was reduced 60 per cent. With the
Armco iron the percentage of reduction was not quite so
great.

Stanton and Bairstow^ found that specimens with Whit-
worth screw threads, and also those with square shoulders
plus a small fillet, suffered a reduction in endurance strength

l0.275"d/am.

X ^

^9.85"rad/us ^0.40"c//am.

0.275"c//am.

j:

^ /"rad/us ^0.40"d/am.
,0.27S"c//am.

^ M r^Tj^h fC \ y

i

^ n7c//us ^0.40"<:f/a/r?.
0:27S"d/am.

^2^ ^0.40"d/a/v.

0.27£"d/an?.

i

^ 0.40"d/a

^ao'l/nofcfy ^ 0.40 d/am.

Fig. 76. — Specimens for study of effect of shape on endurance limit. (Bull. 124,
Univ. of III. Eng. Expt. Sta.)

of about 30 per cent for hard steel, for soft steel, and
for wrought iron, while specimens with square shoulders
suffered a reduction of about 50 per cent for hard steels
and from 25 to 45 per cent for mild steels and wrought iron.
Eden, Rose, and Cunningham- found that a sharp V-notch
reduced the endurance strength of bright-drawn mild

1 Proc. Brit. Inst. Civil Eng., vol. 4, p. 78, 1905-1906.

2 Proc. Brit. Inst. Mech. Eng., vols. 3 and 4, p. 839, 1911.

200

THE FATIGUE OF METALS

steel about 25 per cent. Square shoulders reduced the
strength of both hard and soft steels by 40 per cent, while
keyways at flange couplings reduced the strength of steel

10,000

0.49 Per Ceni- Carbon Sieel; Waier Quenched-^ Drawn ail200 F.

10^

io5 10^ iC

Number of Cycles for Frcxc+ure

IC" 105 10^

Number of Cycles for Fraciure

Fig. 77. — S-N diagrams for specimens of different shapes. (Bull. 124, Univ. of

III. Eng. Expt. Sta.)

by 50 per cent, and the strength of wrought iron by 23
per cent. Wohler^ found in some tests on axle steel

^Engineering {London), vol. 11, 1871.

"STRESS RAISERS" AND THEIR EFFECT 201

stressed from zero to a maximum in repeated tension, that
specimens with square shoulders as compared with speci-
mens having well-rounded shoulders, had their strength
reduced about 37 per cent. On rotating-beam specimens
of wrought iron the reduction of strength due to square
shoulders ranged from 11 to 22 per cent.

R. R. Moore^ has fOund that a single circumferential
groove around a rotating-beam specimen reduces the
fatigue strength much more than does a length of thread
cut with the same tool as the groove. His results were
confirmed by H. F. Moore. From this it is judged that a
thread with nuts taking up most of its length would weaken
a rod of metal in fatigue more than would the thread alone,
and that the fatigue strength of the rod might be somewhat
increased by cutting a longer thread on it.

All these results indicate the importance of avoiding abrupt
changes of section in members of machines which are to be
subjected to repeated stresses. Whenever a change of
section is necessary, generous fillets should be provided
at all shoulders.

Effect of Surface Finish. — Moore and Kommers^ studied
also the effect of surface finish on endurance strength. They
used five degrees of smoothness: (1) a high polish in which
after using Nos. 0 and 00 emery cloth, the specimens were
polished with emery papers Nos. 1, 0, and 000, and finally
with rouge and broadcloth, a microscope with a magnifica-
tion of 100 diameters being used to make sure that all
scratches were removed; (2) their standard finish, using
Nos. 0 and 00 emery cloth; (3) a ground finish, obtained
with a grinding wheel; (4) a smooth-turned finish using a
lathe tool; and (5) a rough-turned finish, using a lathe tool.
These tests were made on a heat-treated 0.49 per cent
carbon steel in the sorbitic condition, and a few tests also
on Armco iron.

Figure 78 shows the results of these tests. For the 0.49
per cent carbon steel the rough-turned specimens, which

1 Proc. Am. Soc. Testing Materials, vol. 26, Pt. II, p. 255, 1926.

2 Univ. Illinois Eng. Exp. Sta., Bull. 124, p. 108, 1921.

202

THE FATIGUE OF METALS

were the weakest, had their endurance Umit reduced about
18 per cent below that of the rouge finished. For the
Armco iron the turned specimens had their endurance limit
reduced from 8 per cent to 12 per cent compared with speci-
mens of standard finish.

Eden, Rose, and Cunningham^ found that polished speci-
mens of mild steel which had their surfaces scratched with

10,000
. 60,000

t 50,000
X" 40,000

% 30,000

D-
E
o

20,000,

0.49 Per Ceni Carbon she/; Waier Quenched; Drawn a-h / 200 F.

V)'^

\<fi 10^ 10''

Number of Cycles for Froic+ure

70,000

10

I05 10&

Number of Cycles for Fracl'ure

10^

Fig. 78. — S-N diagrams showing effect of surface finish on endurance limit.
{Bull. 124, Univ. of III. Eng. Expt. Sta.)

an ordinary sewing needle suffered an appreciable reduc-
tion in fatigue strength. Specimens of Bessemer steel with
a turned surface showed a fatigue strength about 18 per
cent lower than specimens of the same material which
had been turned and polished. Sondericker^ found that a
rotating-beam specimen of soft steel with a groove 0.003 in.

^Proc. Brit. Inst. Mech. Eng., vols. 3 and 4, p. 839, 1911.
2 Tech. Quart. {Boston), March, 1899,

"STRESS RAISERS" AND THEIR EFFECT 203

deep, cut with a diamond point, had its fatigue strength
reduced by 40 per cent. In some tests in which annealed,
cold-rolled steel was stressed in reversed bending beyond
the yield point, Kommers^ found that specimens which
had been turned in a lathe and specimens which had been
turned and then filed had their life reduced 30 per cent and
18 per cent, respectively, as compared with specimens which
had been turned, filed, and polished.

Table 14. — Effect on Fatigue Strength of Various Workshop

Finishes
Results obtained by W. Norman Thomas of the staff of the British Aero-
nautical Research Committee

Maximum Reduction
in Fatigue Strength
from Polished Surface,
Finish of Surface Per Cent^

Turned 12

Coarse file 18 to 20

Bastard file 14

Smooth file 7.5

Coarse emery (No. 3) 6

No. 1 emery 4

No. O or FF emery 2 to 3

Fine carborundum 2 to 3

Fine ground finish 4

Accidental scratches (maximum found) 16

1 These values were estimated by determining the ratio for the various scratches of
depth of scratch to radius of curvature at the bottom of the scratch. Values of depth and
radius were determined by making gelatine casts of the surface of the metal, slicing the casts
with a microtome, and then magnifying the outline of the slice by means of a projection
apparatus. From these values it was possible to make an estimate of the probable effect
of scratches on fatigue strength on the basis of fatigue tests on specimens scored with
72-deg. V-grooves with various ratios of depth to radius of curvature at the bottom of
the scratch.

W. Norman Thomas^ of the staff of the British Aeronau-
tical Research Committee has made an extensive study
of the effect of scratches and grooves resulting from various
workshop processes. The materials used were tool steel,
a 0.33 per cent carbon steel, a 0.13 per cent carbon steel,
aluminum, and cast iron. Table 14 gives values which are
a rough indication of the maximum effect on fatigue strength

^ Proc. Intern. Assoc. Testing Materials, art. V4a, 1912.
2 Engineering (London), p. 449, Oct. 12, 1923.

204 THE FATIGUE OF METALS

to be expected from various finishes. Attention is also
called to Fig. 16 (p. 76).

In order to determine whether the effect of the size of
scratches would be appreciable, Thomas made some
additional tests in which the depth of the scratches ranged
from 0.0051 to 0.0448 in. instead of the maximum value of
0.00244 in. in the previous experiments. These grooves
showed reductions in strength varying from 32 to 55 per
cent, considerably larger than the reduction shown for the
smaller scratches.

These results by various investigators all indicate that
the surface finish of a machine member subjected to fatigue
may have an appreciable effect on the fatigue strength.
A poor surface finish may lower the fatigue strength of a
metal by as much as 15 to 20 per cent. The results of Moore
and Kommers indicate that fine grinding would probably
be a satisfactory commercial finish.

Effect of Internal Stress. — The presence of internal stress
in a metal will be such as to increase the resultant maxi-
mum stress above the computed stress, when the applied
stress is of the same kind as the internal stress. The
result will be an apparent reduction of the endurance hmit
as computed on the basis of the applied load. Experiments
on quenched and tempered steels seem to indicate that the
quenching operation introduces internal stresses which may
be relieved to a considerable extent by heating, even below
the critical temperature for the metal.

Table 15 gives some results obtained by Aitchison^ on a
0.30 per cent carbon steel containing 0.56 per cent molyb-
denum, 4.30 per cent nickel, and 1.44 per cent chromium,
air hardened from 1480°F.

These results show that the drawing temperature of
390°F. decreased the ultimate strength but actually
increased the endurance limit. As the drawing tempera-
ture was increased, the ratio of endurance limit to ultimate
strength did not change greatly, but the endurance hmit
decreased with decrease of ultimate strength. There is

1 "Engineering Steels," p. 209, 1921.

'STRESS RAISERS" AND THEIR EFFECT

205

evidently a particular drawing temperature which relieves
the internal stress considerably and produces the greatest
absolute value of endurance limit.

Table 15. — Effect of "Draw" on Static and Fatigue Properties

Drawing
tempera-
ture,
degrees
Fahrenheit

Propor-
tional
elastic
limit,

pounds

per square

inch

Yield

point,

pounds

per square

inch

Ultimate

tensile

strength,

pounds

per square

inch

Elonga-
tion in
2 in.,
per cent

Reduc-
tion of
area,
per cent

Endurance

limit,

pounds

per square

inch

Ratio of
endurance
limit to
ultimate
tensile
strength

Values quoted from Aitchison for a nickel-chromium air-hardened steel

None

45,000

176,000

244,000

11

36.5

102,000

0.418

390

81,000

173,000

227,000

12.5

41.5

115,000

0.507

750

119,000

179,000

220,000

10

36

106,000

0.482

930

105,000

159,000

185,000

15

46.5

93,000

0.502

1,110

91,500

141,000

157,000

17.5

55

79,500

0.506

Values quoted from Moore and Jasper for a 0.49 carbon oil-quenched steel

None
600
800

72,000

80,900

126,500

12.5

52

65,000

73,300

80,800

126,800

11.5

52

68,000

75,800

78,800

121,800

11.5

51

64,000

0.513
0.536
0.526

Whyte^ found that the endurance limit for a nickel-
chromium steel rose for drawing temperatures up to 750°F.
and then started to fall.

Moore and Jasper^ report the results shown in the second
part of Table 15 for a 0.49 per cent carbon steel quenched
in oil from 1450°F. Here again the drawing temperature
of 600°F. had the effect of relieving the internal stresses and
actually increasing the endurance limit.

Gillett and Mack^ report that a 0.31 per cent nickel-
chromium-molybdenum steel, quenched in oil at 1500°F.
and drawn at a temperature of 980°F. had an endurance
limit of 100,000 lb. per square inch. This endurance limit
was increased to 112,000 lb. per square inch by raising the
specimens to a temperature of 1110°F. and cooling slowly.
Another similar steel with 0.44 percent carbon was quenched

^ Proc. (British) Inst. Automotive Eng., vol. 15, p. 512, 1921.

2 Univ. Illinois Eng. Exp. Sta., Bull. 136, 1923.

3 Proc. Amer. Soc. Testing Materials, vol. 24, Pt. II, p. 476, 1924.

206 THE FATIGUE OF METALS

at 1460°F. in oil and drawn at 980°F. This steel had its
endurance limit increased from 87,000 to 123,000 lb. per
square inch by 2)4 hours more of heating at 980°F. This
additional heating produced no decrease in hardness.
Several other steels are cited to show the same beneficial
effect on endurance limit of more prolonged drawing.

Moore and Kommers^ report a result on a 0.24 per cent
carbon chrome-nickel steel which illustrates the effects
of double heat treatment. The steel was first quenched
in oil at 1525°F., reheated to 700°F., and again quenched
in oil. This treatment gave an ultimate strength of 138,700
and an endurance limit of 68,000 lb. per square inch.
Specimens of this steel were also quenched in oil at 1525°F.,
but then reheated to 1450°F., quenched in oil, and next
reheated to 1200°F., held for 1 hour, and quenched in water.
This treatment gave an ultimate strength of 114,200 and
an endurance limit of 67,000 lb. per square inch. In
other words, the second method gave a much decreased
ultimate strength with an endurance limit almost equal to
that of the first method. Undoubtedly, the internal
stresses set up by the first method were much greater than
by the second method.

These results indicate that the influence of quenching
temperature, drawing temperature, and especially length
of draw, are very important; and that it appears to be
possible to reduce greatly the internal stresses and thereby
increase the endurance limit by proper procedure in heat
treatment. Very often this is accompanied by increase in
ductility and no appreciable decrease in ultimate strength.

Discrepancies between Experiment and Theory. — The
mathematical investigations of Suyehiro,^ Inglis,^ and
Griffith'* and the static experiments of Coker and Scoble^
and others have shown that the effect of holes, scratches,

1 Univ. Illinois Eng. Exp. Sta., Bull. 124, 1921.

2 Engineering {London), p. 280, Sept. 1, 1911.

3 Trans. Brit. Inst. Naval Arch., Pt. I, p. 219, 1913.

* Brit. Advisory Comm. Aero. Rept. and Mem., No. 12757, December, 1916.
s Trans. Brit. Inst. Naval Arch., Pt. I, p. 207, 1913, and other papers.

"STRESS RAISERS" AND THEIR EFFECT 207

and discontinuities in general is to produce high local stress.
On the other hand, experiments in fatigue have shown that
the endurance limit of specimens which have been provided
with holes and scratches does not show the reduction in
value which would be predicted by the mathematical
theory of elasticity or by the static experiments.

An interesting series of experiments relating to this
question are those of Thomas^ (see also p. 203). His results
showed that the increase of stress due to scratches, provided
the variation in depth is small, depends approximately
on the ratio d/p, in which d is the depth of the scratch and p
is the radius of curvature of its extremity.

Now Inglis^ has shown by an elaborate mathematical
analysis that, for the case of a flat plate notched at one edge,
the stress at the bottom of the notch is approximately

Si — S\

(1+2 J)
\ \p/

in which Si = the unit stress at the bottom of notch,
S = the mean stress in the plate,
d = the depth of the notch,
p = the radius of curvature at the extremity of
the notch.

The assumption is made that the elastic limit of the material
is not exceeded.

Furthermore, A. A. Griffith has shown by means of his
soap-film^ apparatus that the stresses in a shaft due to a
twisting moment are greater at the bottom of a V-shaped
groove than at the surface of a similar unscratched shaft,
according to the values shown in Table 16. Griffith also
showed by mathematical analysis^ that when a shaft was

1 Engineering (London), p. 449, Oct. 12, 1923.

2 Trans. Brit. Inst. Naval Arch., Pt. I, p. 219, 1913.

3 This ingenious device is described in Engineering (London), p. 546, Dec.
21, 1917, and in the Proc. Brit. Inst. Mech. Eng., October-December, p. 755,
1917.

4 "The Effect of Surface Scratches on the Strength of Shafts and Other
Members," Brit. Advisory Comm. Repts. and Mem., No. 1275T, December,
1918.

208

THE FATIGUE OF METALS

Table 16. — Theoretical Stress Concentrations at the Bottom of
Longitudinal V-grooves in Shafts under Torsion^

Angle of

Ratio of computed maximum stress at bottom of groove
to the surface stress in an unscratched shaft

V notch,
degrees

Values of -

li

1 3

5 9

0
60
90

120

1.85
1.84
1.81
1.66

2.01
2.00
1.95
1.75

2.66
2.54
2.40
1.95

3.23
3.06
2.64
2.06

4.54
3.99
3.12
2.13

1 The values in this table were obtained by the use of the Griffith and Taylor soap-film
method for determining theoretical stress.
d = depth of groove in inches.
p = radius of curvature at bottom of groove in inches.

subjected to a bending (as contrasted with twisting)
moment, the ratio of the increased maximum tensile stress
to the original maximum tensile stress could be obtained by
multiplying the values in Table 16 by the factor

1 +2^

1 +

d

This is true for grooves perpendicular to the direction of
stress (that is, circumferential). The ratio was less for
grooves in other directions.

Returning to the consideration of Thomas' experimental
study of the structural damage done by stress concentra-
tions at V-grooves, it is to be noted that he judged the
amount of this structural damage by the results of fatigue
tests, using a 0.33 per cent carbon steel. Two different
grooves were used, one made by a lathe tool with an angle
of about 72 deg. and the other made by a diamond — a
shallow groove having an angle of about 120 deg. The
size and shape of the grooves were determined accurately
by making gelatine casts, slicing these casts with a micro-
tome, and then magnifying the slices with a projection

"STRESS RAISERS" AND THEIR EFFECT

209

apparatus. The fatigue tests were made on a rotating-
beam testing machine (see also Fig. 16).

If Si denotes the intensified stress at the root of the
groove, S denotes the nominal stress in the bar, as computed
by the common flexure formula, and if the ratio of increase
of intensified stress to nominal stress be assumed to be
proportional to s/d,^ there results

♦St o

s

= c

d Si- S

-, or c = — ^ —

in which c is a constant to be determined from the results of
the tests.

Table 17. — Comparison of Theoretical Stress Concentrations at
THE Roots of V-grooves with the Effective Stress Concentra-
tions AS Shown by Fatigue Tests
Results obtained by W. Norman Thomas of the staff of the British Aero-
nautical Research Committee

I

II
Theoretical

III

IV

Values

semi-

Soap- film

Results of

Item

of

elliptical

experiments.

fatigue tests,

d/p

groove.

72-deg. V,

72-deg. V

Inglis

Table 16

(small)

formula

1. Values of C

2.0

1.75

0 15

2. Nominal.stress at fracture, tons per

square inch

1

6.1

6.1

15.8"

4

3.7

4.0

14.0"

7

2.9

3.2

13. 0"

3. Approximate decrease in strength,

per cent

1

66.7

66.7

13.5

4

80.0

78.0

23.5

7

84.0

82.5

29.0

4. Ratio of maximum stress to nomi-

1

3 0

3 0

1 16

4

5.0

4.5

1.31

7

6.3

5.7

1.42

" Endurance limits.

Table 17 shows some of the theoretical and experimental
results obtained by Thomas. It was found that an

1 This assumption implies the use of the Inglis formula (p. 207) rather than
the Griffith formula (p. 208). For the range covered, however, the values
given by the two formulas differ only slighth\

210 THE FATIGUE OF METALS

unscratched specimen had an endurance limit (nominal stress
at fracture) of 18.3 tons per square inch, and it was assumed
that at the endurance limit for a scratched specimen the
unit stress was also 18.3 tons per square inch. When c = 2
and d/p = 1, then from the Inglis formula (p. 207) there
results

S

S

= 2V1 or Si = SS.

The theoretical ratio of maximum stress to nominal stress
is therefore 3, and this value is recorded in column II of
Table 17 over against item 4 for the value d/p = 1. One-
third of 18.3 tons per square inch is 6.1 tons per square
inch, and this value is recorded in column II over against
item 2 from d/p = 1. The values in column III of Table 17
are obtained from Table 16, using a similar procedure.
Column IV gives the experimental results for endurance
limit over against item 2, and 18.3 tons per square inch
divided by these values gives the values in column IV over
against item 4.

It is evident from the table that, theoretically, the ratio
of maximum stress to nominal stress varied from 3 to 6.3,
while, experimentally, the ratio varied from 1.16 to 1.42.
While the theoretical decrease in strength varied from 67 to
84 per cent, the experimental values varied from 14 to 29
per cent. This makes plain the fact that the effect of
grooves and scratches is not so serious as the mathematical
theory of elasticity would indicate. This conclusion is borne
out by the results of a number of other investigators and of
tests on stress concentration at small holes as well as at
the root of grooves.

Moore and Jasper^ investigated the effect of small holes,
0.055 in. in diameter, on the endurance limit of a number
of different metals. They used both round and flat speci-
mens in reversed bending. Their results are given in
Table 18.

1 Univ. Illinois Eng. Exp. Sia., Bull. 152, p. 25, 1925.

"STRESS RAISERS" AND THEIR EFFECT

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212 THE FATIGUE OF METALS

The seventh column in the table gives the ratio of apparent endurance
limit given in the fifth column for flat specimens with holes to that given
in the second column for rotating-beam specimens without holes.
According to the theory of elasticity, this ratio would be 0.333. The
test results, however, give values ranging from 0.567 to 0.818, indicating
that under fatigue loading a small discontinuity is not nearly so serious
in its weakening effect as is indicated by the formulas of the theory of
elasticity. It is to be noted that the weakening effect of a small hole is,
in general, more marked in the alloy steels tested than in the carbon
steels.

The eighth column in Table 18 shows that the ratio between apparent
endurance limit for flat specimens with yi inch fillets and that for
rotating-beam specimens without holes varies from 0.677 to 0.970.
Professor Coker has shown that under static load within the elastic
range this ratio should be 0.696.^ The test results show a value
approximately equal to this for only two materials, both heat-treated
alloy steels.

Timoshenko and Dietz^ have made an experimental
study and a theoretical investigation of the stress con-
centration around holes and fillets, and its effect on fatigue
strength. They find that stress concentrations lower
endurance Hmit less than the amount indicated by the
mathematical theory of elasticity, and that specimens
of chrome-nickel steel were more damaged by stress con-
centration than specimens of carbon steel.

Wilson and Haigh^ found that perforated thin plates under
repeated axial stress did not fail under such low stresses as
might be predicted by the mathematical theory of elasticity.

R. R. Moore^ found that for six non-ferrous metals and
one steel the reduction in endurance limit caused by a
notch ranged from 25 to 45 per cent for the non-ferrous
metals, and 58 per cent for the steel. The mathematical
computation of stress indicated that the endurance Umit
would be reduced 78 per cent.

All these results reinforce the conclusion that the destruc-
tive effect of stress concentration cannot be neglected in

1 Brit. Assoc. Advancement Set., Rept., 1924.

2 Trans. Am. Soc. Mech. Eng., vol. 47, p. 199, 1925.
^ Brii. Assoc. Advancement Sci. Repts., p. 368, 1923.

* Proc. Amer. Soc. Testing Materials, vol. 24, Pt. II, p. 547, 1924; and vol.
26, Pt. II, 1926.

"STRESS RAISERS" AND THEIR EFFECT 213

parts subjected to repeated stress, but it is not so great as
that indicated by the mathematical theory of elasticity,
and the investigations of Moore and Jasper, of Timoshenko
and Dietz, and of R. R. Moore also indicate that stress
concentration produces different degrees of damage for
different metals. Tempered alloy steels seem to be more
damaged by stress concentration than other metals studied.

The explanation for these results offered on the basis of
Griffith's theory is that metals have in them minute cracks
and flaws, so that when experimental scratches are so small
as to be comparable with the cracks already existing, their
effect will be small, and that the theoretical reduction in
strength will be approached only when the grooves are large.

Another explanation for the observed results is based on
the fact that experiments seem to show that redistribution
of stress by slipping seems possible even within the endur-
ance limit of a material. The tests of Gough and Hanson^
in which they found slip lines in Armco iron below the
endurance limit seem to bear out this statement. The
results of Moore and Kommers^ on Armco iron, in which
the yield point was below the endurance limit, of Hankins^
on nickel, and of R. R. Moore^ on seven different non-
ferrous metals, in which the proportional elastic limit was
below the endurance limit, all indicate that slip and hence
cold working can occur below the endurance limit of a
material. The results of repeated stresses would, there-
fore, be in the nature of repeated cold work, which not only
permits redistribution of stress but strengthens the material
against fatigue.

Attention is called to the discussion in Chap. IV (p. 75)
of this discrepancy between the theory of elasticity and
results of fatigue tests of metals.

Effect of Corrosion on Fatigue Strength. 1. Corrosion of
Unstressed Metal.' — Corrosion roughens the surface of metal

1 Proc. Roy. Soc, vol. 104A, p. 538, 1923.

2 Univ. Illinois Eng. Exp. Sta., Bull. 124, p. 98, 1921.

3 Brit. Advisory Comm. Aero., Repts., vol. 2, p. 414, 1922-1923.

* Proc. Amer. Soc. Testing Materials, vol. 24, Pt. II, p. 547, 1924.

214 THE FATIGUE OF METALS

and causes many minute pits and grooves which act as
"stress raisers." The tests of McAdam and R. R. Moore^
on the fatigue strength of steel, corroded previous to testing
for fatigue strength, show a reduction of fatigue strength
of steel varying from 1 to 12 per cent. Corrosion of
unstressed metal seems to have a purely mechanical effect
and is comparable, as to injury caused to a poor surface
finish.

2, Effect of Simultaneous Corrosion and Repeated Stress. —
In 1917 Haigh^ reported fatigue results of some brasses in
contact with strong corrosive agents. He observed that
when corrosion was simultaneous with the stress application
the stress cycle graph under some conditions was slightly
lowered. When the corrosion was prior to the stress appli-
cation, the stress-cycle graph was not lowered. He made
no tests on steel.

In 1926 McAdam^ reported test results for a large num-
ber of ferrous metals subjected to the simultaneous action
of a stream of fresh water and of cycles of reversed flexural
stress. He found that for constant corrosion intensity
there is a definite fatigue limit and that this limit is usually
below (sometimes much below) the ordinary endurance
limit. To this phenomenon he gave the name "corrosion-
fatigue," and to the fatigue hmit obtained under such
conditions he gave the name "corrosion-fatigue limit."
Using specimens like that shown in Fig. 43(c) he found
the following reductions of fatigue strength of specimen
subjected to corrosion-fatigue as compared with specimens
subjected to reversed flexure alone.

1 Proc. Am. Soc. Testing Materials, vol. 26, Pt. II, 1926.

2 Jour. (British) Inst. Metals, Sept., 1917; see also Engineering (London),
Sept. 21, p. 315, 1917.

3 Proc. Am. Soc. Testing. Materials, vol. 26, Part 2, p. 224, 1926. Trans.
Am. Soc. for Steel Treating, 1926.

"STRESS RAISERS" AND THEIR EFFECT 215

Per

Cent

3 . 5 per cent nickel steel:

quenched and drawn 64

annealed 41

0 . 49 per cent carbon steel :

quenched and drawn 62

annealed 32

0.36 per cent carbon steel:

quenched and drawn 63

annealed 26

0 . 24 per cent carbon steel :

annealed 41

0.11 per cent carbon steel :

annealed 36

Ingot iron (average value) 23

High chromium-nickel steel (average vakie) 16

Stainless iron (average value) 29

Corrosion-fatigue tests with salt water as the corroding
agent showed markedly greater reductions of fatigue
strength than those listed above.

The corrosion-fatigue limit is surprisingly little affected
by heat-treatment or chemical composition, except as such
heat-treatment or composition affects corrosion-resistance.
For ''stainless" (high chromium) and other corrosion-
resistant steels the corrosion-fatigue limit is higher than for
carbon steels and other alloy steels.^

When corrosion and repeated stress act together there is,
in addition to ordinary mechanical stress, an action which,
following a suggestion by McAdam, may be called ''chemical
stress." By means of micrographs of the surface, McAdam
has shown that fatigue cracks start from spots corroded so
slightly that the corrosion can scarcely be detected except
by examination with a microscope. Once corrosion and
stress together start a fatigue crack, it apparently spreads
much as does an ordinary fatigue crack.

^ An interesting question, as yet unanswered, is, "What connection, if any,
exists between the corrosion-fatigue results reported by McAdam and the
'caustic embrittlement' of boiler plate under combined steady stress and
corrosion?" This is discussed by Parr and Straub. See Proc. Am. Soc.
Testing Materials, vol. 26, Pt. II, 1926.

216 THE FATIGUE OF METALS

An interesting problem in corrosion fatigue requiring
experimental study is the comparative effects of corrosion
fatigue on small specimens and on large pieces.

Corrosion of unstressed steel seems to be a rather minor
factor in reducing its fatigue strength. Corrosion and
fatigue acting simultaneously seem to constitute a factor of
major importance, one which must be given careful con-
sideration by the machine designer and the structural
engineer.

Significance of Ductility. — Moore and Kommers^ have
pointed out that it is unlikely that ductility, as represented
by the percentage of elongation and the percentage of reduc-
tion of area, will have much direct influence on the fatigue
strength. Ductility is based upon the action of a bar as a
whole, and in ductile materials is dependent upon the final
necking down after the ultimate has been reached, while in
fatigue failures there is no necking down. Furthermore,
fatigue failures are extremely localized and involve only a
small portion of the bar. A study of fatigue results shows
no correlation between ductility and endurance limit (see
Fig. 63, p. 166). The authors mentioned above cite the
case of a 0.93 per cent carbon troostitic steel which had low
elongation and reduction of area, but a high endurance
limit and a high ratio of endurance hmit to proportional
elastic limit.

Ductility, therefore, does not influence the endurance
limit directly, but the authors wish to emphasize the fact
that ductility is for other reasons one of the most valuable
properties of metals. Its influence on toughness is partic-
ularly important, toughness being defined as the quality of a
material which permits it to absorb large amounts of energy
without shattering failure. This quality is dependent on
the two factors of strength and ductility.

A bar of brittle material which has local concentration of
stress, due perhaps to an abrupt change in cross-section,
would be almost sure to fail when subjected to a shock.
If the material is ductile and tough, however, these quali-

1 Univ. Illinois Eng. Exp. Sta., Bull. 124, 1921.

"STRESS RAISERS" AND THEIR EFFECT 217

ties will permit permanent deformation to take place
without actual failure. Such permanent deformation at a
point of high local stress produces redistribution of stress
and thus relieves the situation at the local point.

Moore and Kommers^ have shown that a stress which is a
considerable percentage above the endurance limit may be
applied from 1,000 to 5,000 times without greatly influenc-
ing the fatigue strength under subsequent application of
lower stresses. Their results on a heat-treated 0.49 per
cent carbon steel in the sorbitic condition showed that a
stress 10 and 20 per cent above the endurance limit applied
5,000 times, a 29 per cent overstress applied 1,000 times, and
a 38 per cent overstress applied 100 times, did not appreci-
ably reduce the endurance limit as subsequently deter-
mined. However, an overstress of 35 per cent applied 1,000
times reduced the endurance limit 4 per cent, while an
overstress of 29 per cent applied 5,000 times reduced the
endurance limit about 11 per cent.

A heat-treated 1.20 per cent carbon steel in the sorbitic
condition whose original endurance limit was 50,000 lb. per
sq. in. was subjected to 20 per cent overstress for 5,000 and
10,000 cycles, respectively. The endurance was reduced
12 per cent and 14 per cent, respectively. Comparing the
result of 20 per cent overstress applied 5,000 times in the case
of the 0.49 per cent carbon steel, whose Brinell hardness was
197, and the 1.20 per cent carbon steel, whose Brinell
hardness was 369, it is seen that the harder steel was much
more influenced by the overstress than the softer steel. It
should be noted here that the absolute value of the over-
stress was practically the same in the two cases, because the
endurance limits did not differ greatly.

Moore and Jasper^ report some results on the effect of
overstrain of a different type. They appUed a heavy axial
tensile load twenty times, producing stresses greater than
the original endurance limit by various percentages. They
found that the endurance limit was not affected appreci-

1 Univ. Illinois Eng. Exp. Sta., Bull. 124, p. 112, 1921.

2 Univ. Illinois Eng. Exp. Sta., Bull. 136, p. 60, 1923.

218 THE FATIGUE OF METALS

ably until the maximum stress applied approached the
static proportional elastic limit, which was about 41 per
cent above the original endurance limit. For stresses near
or above the proportional hmit the endurance Hmit was
decreased from 18 to 22 per cent. The specimens were
not pohshed after being overstressed. There appeared to
be Httle difference in the results whether the overstressed
specimens were tested immediately, were immersed in
boiling water before testing, or rested three months before
testing.

Moore and Jasper^ did some further work on the effect
of overstress in reducing the endurance limit. The results
are shown in Table 19. In these tests the amount of axial
overstress ranged from 15 to 80 per cent and in all but one
case was applied twenty times. It will be noted that in all
cases except one the effect of overstress was to reduce the
subsequent endurance limit below the original value by
amounts ranging from 3 to 23 per cent.

The tests on the annealed specimens of 0.49 per cent car-
bon steel require some explanation. The annealed A
specimens were annealed at 1500°F. and then polished;
and they gave an endurance limit of 32,000 lb. per square
inch. The B specimens were annealed as were the A speci-
mens, then given an overstress of 80 per cent applied twenty
times, and repolished, with the result that the endurance
limit was 31,000 lb. per square inch. The C specimens
were annealed as were the A specimens and polished,
reannealed and repolished, giving an endurance limit of 33,000
lb. per square inch. The D specimens were annealed as
were the A specimens, then given an overstress of 80 per
cent applied twenty times, reannealed, and repolished,
giving an endurance limit of 30,000 lb. per square inch.
The E specimens were annealed as were the A specimens,
then given an overstress of 40 per cent apphed twenty
times, reannealed and repolished, giving an endurance
limit of 29,300 lb. per square inch.

1 Univ. Illinois Eng. Exp. Sta., Bull. 142, p. 32.

"STRESS RAISERS" AND THEIR EFFECT

219

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220 THE FATIGUE OF METALS

For specimens A and B the overstress reduced the endur-
ance limit 3.1 per cent. For specimens C and D the over-
stress reduced the endurance hmit 6.2 per cent, even though
the specimens were annealed and polished after the over-
strain. For specimens C and E the overstress reduced the
endurance limit 8.4 per cent, even though the specimens
were annealed and polished after the overstrain. The
beneficial effect of reannealing and repoHshing on endur-
ance limit seemed negligible.

The 0.49 per cent carbon steel in the annealed condition
is evidently better able to withstand overstress than in
the sorbitic condition. The sorbitic steel had its endur-
ance limit reduced 22.9 per cent by 70 per cent overstress,
while the annealed steel had its endurance limit reduced
less than 10 per cent by 80 per cent overstress. Apparently
the inherent ability of the metal for "healing" scars due to
slip, is diminished by any heat treatment which raises the
strength and lowers the ductility.

One of the valuable characteristics of materials which are
ductile and tough comes into play both under static and
repeated stress, this characteristic being the one which
permits the material to deform under an unexpected, high
stress. Such permanent deformation without actual fail-
ure gives a warning of impending failure which a brittle
material cannot give.

The Effect of Understressing. — Moore and Jasper^
investigated the effect of subjecting specimens to reversed
bending stresses at or near the endurance limit. They
retested 118 specimens which had received at least 10,000,-
000 and in some cases 100,000,000 cycles of stress without
failure, the specimens being subjected to stresses which were
increased by small increments until the specimens failed.
In some cases, the unit stress at fracture was 25 per cent
above the original endurance limit. Figure 79 shows the
results obtained on some of the ferrous metals which had
originally been subjected to 100,000,000 cycles of stress
without failure. The figure shows two different groups of

1 Univ. Illinois Eng. Exp. Sta., Bull. 142, p. 27, 1924,

'STRESS RAISERS" AND THEIR EFFECT

221

metals; those in the upper group show a very marked
increase in fatigue strength, while those in the lower group
show comparatively little increase in strength. It is evi-
dent, therefore, that all metals are not equally susceptible to
increase in strength due to understressing. Those metals
which have had their strength materially increased by heat

70,000

feO.OOO
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Hi '

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. 252 Per CeniCoirbon, Normalized

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Number of Cycles for Rup4ure,(N)

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5 80,000
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Trooshhc

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Chrome -NicUel, Oil Quenched^ Dravjn ai 1200 °F

10^ lO'^ 10^

Number of Cycles for Rup+ure (N)

I09

Fig. 79. — Effect of " understressiag " on endurance limit. {Bull. 142, Univ. of
III. Eng. Expt. Sta.)

treatment apparently do not have their strength increased
much by understressing.

Bauschinger's conclusion given as No. 12 in Chap. II

says:

Repeated stresses between zero and an upper limit in tension which
coincides with or lies slightly above the elastic limit will increase the
elastic limit, and the more so the greater number of repetitions, but not
above a certain limiting value.

222 THE FATIGUE OF METALS

The experiments of Moore and Jasper indicate that for
reversed stresses an analogous increase of endurance Hmit
may be expected in steels susceptible to cold work when
the stresses are at or slightly below the original endurance
limit. ^

It is of course well known that certain kinds of cold work
increase the static and fatigue strength of metals, and it is
conceived that repeated stressing tends to produce
repeated cold work over minute areas. There is evidently
the possibility of a localized rearrangement of particles
which were disadvantageously placed initially; thus con-
siderable strengthening is produced at critical locations.

Another series of tests on understressing was carried out
by Moore and Jasper on a 1.20 and a 0.49 per cent carbon
steel, both in the sorbitic condition, and also on the 0.49
per cent carbon steel in the annealed condition. In these
tests the specimens were first subjected to a small number
of cycles of stress above the original endurance limit, then
subjected to 10,000,000 or more cycles at or near the origi-
nal endurance limit, and finally tested for endurance limit
in the usual way. Table 20 shows the results of these tests.

As pointed out previously, the overstressing would tend
to reduce the endurance limit while the understressing
would tend to increase it. Table 20 shows that in every
case there is some restoring action due to the understress-
ing. In some cases the effect of overstressing seems to be
entirely overcome by the effect of understressing.

Tests were also made to determine the effect on static
properties of reversed axial stresses. Specimens were
subjected to 10,000,000 cycles of reversed axial stress at or
near the endurance limit, and subsequently tested in
static tension. Table 21 shows the results of these tests.
All the steels except one show an increase in static ultimate
strength due to the understressing, and most of the steels

^ Recent tests by N. P. Inglis at the University of Illinois indicate that
the fatigue strength of cast iron may be materially increased by under-
stressing. This interesting result on a brittle metal indicates the need of
further study of understressing.

'STRESS RAISERS" AND THEIR EFFECT

223

Table 20. — Summary op Results for Steel Subjected to Stress above

THE Original Endurance Limit but Not Tested to Failure; Then

Subjected to 10,000,000 or More Cycles of Stress at ok

NEAR THE ORIGINAL ENDURANCE LiMIT OF THE MeTAL

Results obtained by Moore and Jasper in the Joint Investigation of the Fatigue of Metals
at the University of Illinois.

In considering the values given in this table three endurance limits must be kept in mind:
(1) the original endurance limit of the metal, (2) the endurance limit of the metal after over-
stress, and (3) the endurance limit of the metal after overstress (not carried to failure) fol-
lowed by cycles of stress at or near the endurance limit. Endurance limit (3) is greater
than endurance limit (2), as is shown in this table, but endurance limit (3) is, however, less
than endurance limit (1).

Specimen

Total cycles
of stress

Increase over endurance limit
after overstress (2) of endur-
ance limit after subsequent
understress (3), per cent

Greater
than

Less
than

Amount of overstress applied

Excess over

Number

original

of

endurance,

cycles

limit

per cent

Steel No. 1, 1.20 carbon, sorbitic

IFOC

46,536,000

5.0

7.5

5,000

20

1F52G

42,615,900

18.8

21.4

5,000

20

1F39F

43,065,800

4.1

6.5

5,000

20

1F39D

80 , 686 , 600

13.5

16.1

5,000

20

1F39A

63,974,800

20.7

22.6

10,000

20

1F26B

66,153,200

19.9

24.4

10,000

20

Steel No. 10, 0.49 carbon, sorbitic

10B26D

13,594,600

0

1.0

100

38

10B169B

11,395,600

0

1.0

100

38

10C13B

36,453,200

12.7

15.8

100

38

10D104D

22,157,400

0.7

3.3

1,000

35

10G143A

10,646,200

0.8

2.9

1,000

29

10F26A

38,656,400

15.1

18.6

5,000

29

10F13C

142,769,300

23.2

26.8

5,000

29

10C104C

11,496,500

0

2.1

5,000

20

10B65D

26,447,900

6.9

10.1

5,000

10

10G26D

13,443,600

2.5

4.8

10K143B

116,649,100

6.0

7.8

20 axial

15"

10K143X

31,045,100

2.0

4.0

20 axial

IS''

10N13C

138,780,600

18.1

21.3

20 axial

15-:

10K52B

104,506,800

0

2.5

20 axial

30«

10K156N

37,416,200

7.4

9.8

20 axial

306

10K69A

115,870,600

2.1

4.7

20 axial

30<=

10K25A

123,751,900

6.4

8.5

20 axial

30=

10L39C

106,154,200

4.4

6.8

20 axi 1

SC

lONOC

166,415,100

41.5

45.2

20 axial

40''

10K78B

78,951,800

26.2

28.8

20 axial

40*

10K156B

36,717,900

17.3

19.5

20 axial

40*

10L182C

128,534,700

10.0

12.1

20 axial

SO"

10K78

102,378,900

36.0

39.0

20 axial

50*

10K156

57,604,300

22.4

24.8

20 axial

50!-

224

THE FATIGUE OF METALS

Table 20. — {Continued)

Specimen

Total cycles
of stress

Increase over endurance limit
after overstress (2) of endur-
ance limit after subsequent
understress (3), per cent

Greater
than

Less
than

Amount of overstress applied

Number

of

cycles

Excess over

original

endurance

limit

per cent

Steel No. 10, 0.49 carbon, sorbitic

lOKOC

103,670,300

0

1.0

20 axial

50=

10L65A

122,138,600

4.0

6.3

20 axial

50<^

10K91B

123,804,400

15.6

19.0

20 axial

50''

10M143A

138,875,300

17.1

20.2

1 axial

60=

10M143B

86,184,100

20.2

23.2

1 axial

60"

10M127B

86,089,400

13.9

15.9

1 axial

60«

10M78B

52,016,900

17.1

20.2

1 axial

60"

10M65B

40,580,700

0

1.9

10 axial

60"

10M130A

81,138,500

17.0

20.0

10 axial

60"

10M169

131,945,300

11.7

15.1

20 axial

60"

10K91A

121,695,900

7.1

10.2

20 axial

OO"*

lOLOA

139,603,700

14.6

17.1

20 axial

mi

10K169B

136,771,700

18.1

21.6

20 axial

70"

10N13B

140,016,200

14.6

17.2

20 axial

70-

10K117A

127,660,900

9.7

12.4

20 axial

70=

10K13B

108,764,000

0

23.8

20 axial

70=

lONOA

77,084,000

15.1

18.1

20 axial

70=

10K104C

120,925,900

6.7

9.0

20 axial

70i

10N52A

125,299,000

9.5

11.7

20 axial

70=

10N52D

147,039,500

21.2

24.0

20 axial

70=

Steel No. 10, 0.49 carbon, annealed

10V117C

107,500,100

10.3

14.0

20 axial

80=

10V117A

112,388,800

9.7

13.2

20 axial

80=

10V78B

137,581,300

16.1

20.6

20 axial

80=

10U117D

121,017,300

20.1

23.6

20 axial

40=

° Specimen tested immediately after overstress.

*> Specimen rested from 3 to 15 days after overstress.

= Specimen boiled in water for 1 hr., cooled and tested immediately.

<* Specimen rested 3 months after overstress before testing.

= Specimen polished after overstress and tested immediately.

Note: Specimen 10G26D, 0.49 carbon, sorbitic was subjected to 2,000,000 cycles of stress
10 per cent below the original endurance limit of the metal, and its endurance limit after
these cycles of understress was raised more than 2.5 per cent and less than 4.8 per cent.

show a decrease in percentage of reduction in area. It will
be noted that this effect of increase of ultimate strength
and decrease of reduction area is precisely the effect which
static cold work produces, and indicates, therefore, that the
understressing is in the nature of repeated cold work.

"STRESS RAISERS" AND THEIR EFFECT

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CHAPTER IX
FATIGUE FAILURE UNDER SERVICE CONDITIONS

Laboratory and Service Conditions. — In a laboratory test
to determine the endurance limit of a metal, specimens are
subjected to cycles of known stress. It is obviously a
prime essential in laboratory work that tests be performed
under known conditions. It then becomes very necessary
that the machine designer keep in mind the fact that in
machine parts and structural members the range of stress
developed is not constant, and the magnitude of the stresses
is not known with any high degree of accuracy. Freight-
car axles are subjected to occasional high stresses of very
uncertain magnitude, due to flat wheels, lateral flange
pressure at frogs, and bad rail joints. There are very
few stress-carrying joints in boiler plates which are not
subjected to occasional very high localized stress at the
edges of rivet holes. The steering knuckle of a motor car
is subjected to frequent repetitions of rather violent stress,
due to shocks transmitted from rough pavement. These
stresses are, however, quite impossible of computation.

The endurance limit determined by laboratory tests is a
very useful index of the strength of a metal under cycles of
normal stress. It is a value which should be known by the
designer before he designs machine parts to be made of the
metal, but it is by no means the only physical property
to be considered. No one physical property of a metal
is sufficient to enable a designer to design a machine part or
a structural member so that it will be both safe and
economical.

It has been previously noted (see p. 165) that there seems
to be little correlation between endurance limit and any
index of elastic strength, such as the yield point or the
elastic limit (however determined). Some metals show

226

FATIGUE FAILURE UNDER SERVICE CONDITIONS 227

endurance limits under reversed stress above the elastic
limit of the metal as received. If, however, parts in
service were subjected to widely distributed"^ stresses above
the elastic limit, failure would be likely to occur, not a
fatigue failure, but a failure by plastic distortion of sufficient
magnitude to impair the working of the machine or the
integrity of form of the structure.

Effect of Occasional Overstress.- — As noted in the fore-
going paragraph, one factor affecting the serviceability of
metal in machine parts is the effect of occasional overload —
load which causes stresses above the yield point or above
the endurance limit of the metal. It is to be noted that such
stresses change the metal. A stress beyond the yield point
improves the elastic strength after a period of rest, and a
stress beyond the endurance limit starts minute fatigue
cracks in the metal. Both experience and laboratory
tests indicate that once a fatigue crack is started it will
spread under subsequent cycles of stress somewhat below
the original endurance limit of the metal. How much
below the original endurance limit the subsequent stress
must be to be incapable of spreading the existing crack is a
matter of uncertainty.^

Metals seem to vary markedly in their ability to with-
stand occasional overstress without developing disastrous
fatigue cracks. In general, ductile metals are superior to
brittle metals in this respect, but ductility is not the only
factor involved. Certain ductile alloy steels seem to be
highly susceptible to damage by occasional overstress.
Occasional overstress in machine parts is frequently applied
very rapidly, and the ability of the part to absorb the

1 The term "widely distributed stress" is used to exclude the action of
localized stress. The stresses computed by the ordinary (Rankine) for-
mulas of mechanics of materials may be regarded as widely distributed
stresses.

2 The statement that overstress lowers the stress at which fatigue cracks
will continue to spread seems to be contradicted by the raised fatigue
strength observed in cold-drawn and cold-rolled iron and steel. It should
be noted, however, that drawing through dies, or passing between rolls,
causes lateral compression during cold working, and leaves a very smooth
surface. Cold working by simple overstress tends to roughen the surface.

228 THE FATIGUE OF METALS

energy of the cycles of overstress is in some cases as impor-
tant as the abiUty to withstand high unit stress.

The ''life" of a machine part or of a structural member
may be considered as made up of two parts: (1) cycles of
normal working stress, which cover (say) 99 per cent of the
''life" of the part, and (2) cycles of abnormally high stress,
which cover (say) 1 per cent of the "life." To insure
satisfactory service under (1), it is necessary that the work-
ing stresses shall be well below both the endurance limit
and the yield point (if any exists) of the metal. To make
probable the satisfactory service under (2), it is necessary
that the metal shall be tough, so that the damage done by
the occasional periods of high stress will not start disastrous
fatigue cracks.

Warnings of Impending Fatigue Failure. — For parts
made of ductile metals fatigue failures are likely to be more
disastrous than are dead-load failures in machine and struc-
tural parts. Dead-load (static) failures of ductile metal are
usually failures by plastic yielding unless the member is
long enough to collapse by buckling; such failures usually
occur without causing serious injury to the structure as a
whole. For example, under an accidental overload, a
steel crane-hook may be badly distorted without causing
it to let go its load.

For all kinds of loads on brittle materials and for repeated
loads on ductile metals, failure, if it occurs, usually means
a shattering fracture without much warning. In some
cases, however, careful systematic inspection will show
signs of approaching fatigue failure. For example, in wire
ropes bent around sheave wheels, approaching failure may
usually be detected by the snapping of individual wires
at the surface of the rope.

In car axles it is frequently possible to detect incipient
fatigue failures by careful periodic inspection.^ Experi-

1 An effective method of detecting incipient cracks is as follows : Oil the
surface of the axle over the portion where cracks are expected to develop.
Wipe off the oil on the surface. Then coat the surface with a wash of
whiting and wood alcohol. This soon dries and then, if the axle is rotated
and struck smartly with a hammer, the oil which has penetrated into any

FATIGUE FAILURE UNDER SERVICE CONDITIONS 229

ments in the Fatigue of Metals Laboratory of the Univer-
sity of lUinois indicate that in axle steel such cracks can

Fig. 80. — Fatigue crack in car-axle steel.
The specimen had been coated with whiting for the oil-whiting test and some of the
whiting had worked into the crack.

be detected at about 50 per cent of the 'life" of the axle
for stresses slightly above the endurance limit. Figure 80

Fig. 81. — Fatigue crack in steam turbine disc.

shows a crack in axle steel at an early stage of development.
For higher stresses the chance of detection is, of course,

little cracks will be forced out, discoloring the whiting coating. By this
means, cracks invisible to the unaided eye may be located.

230 THE FATIGUE OF METALS

less, but in general, the higher the stress the more rarely
is it developed, and the fewer the number of cycles at any
one period of such high stressing. Fatigue cracks have
been detected in some steam-turbine disks before the cracks
had spread to failure (see Fig. 81), although in other cases
disastrous failures have occurred before any cracks were
detected. In machine parts the commonest form of fatigue
failure seems to be one in which a crack or cracks are started
during short and infrequent periods of overstress and then
spread slowly under normal loads or occasional periods of
loads slightly above normal. For such failures the chance
of detecting cracks before a disaster occurs is fairly high
if periodical inspections can be ma\ie.

Probably examination under a high-power microscope
would detect a fatigue crack at an earlier stage than the
whiting-and-oil method outhned in the footnote on page 228,
but the search over any considerable area of metal for a
microscopic crack would involve so much labor in polishing
surfaces and in traversing them with the microscope that
the method would rarely be feasible in practice.

Another method which has been used with success in
detecting small cracks in iron and steel pieces is that of H. S.
Rawdon of the U. S. Bureau of Standards.^ In this method
the piece to be examined is polished, magnetized, and
covered with a wash of finely divided iron (''iron mud"
from cast-iron lapping disks) suspended in kerosene. The
gathering of iron particles shows the location of cracks.

Typical Fatigue Fractures. — Whether a failure in a
machine part is due to the progressive spreading of a
fatigue crack or to some other cause can frequently be told
by an examination of the fracture. Figure 82 shows three
fatigue fractures which are typical of shafting failures under
reversed flexure, such as occur in car axles. These partic-
ular failures were failures of laboratory specimens loaded as
rotating-cantilever beams. In each failure two parts can
be clearly distinguished: (1) a relatively smooth surface
which marks the spread of the fatigue crack and which has

1 U. S. Bur. Standards, Tech. Paper, 156.

FATIGUE FAILURE UNDER SERVICE CONDITIONS 231

232

THE FATIGUE OF METALS

been battered smooth by the repeated opening and closing
of the crack, and (2) a rough ''crystaUine" surface which
represents the final sudden failure of the small area of
sound metal not reached by the fatigue crack when final
failure occurred.

Figure 82(a) shows a fatigue crack which started all
around the circumference and spread very evenly inward,
leaving the remaining sound metal almost circular in shape.
On the surface of the fatigue crack will be noted a number of
radial lines, marking the edges of axial ''steps" in the

.- Rough Sur-facej Final Sudden Failure,

^=3=^

ZSii;

Smooih Surface wiih Ripple Marks
Progressive Failure

Failure s-hriedaiHole

Fig. 83.

-Diagram of typical fatigue failure caused by axial vibration in a thin
disc.

surface and bearing some resemblance to ripple marks left
by flowing water on sand or clay. Such ''ripple marks"
are frequently found on the surface of fatigue cracks.

Figure 82(5) shows a fracture in which fatigue cracks
started at opposite sides of the shaft and spread toward a
diametral line, leaving the remaining area of sound metal
in the approximate shape of an elongated ellipse. Figure
82(c) shows a fracture in which a fatigue crack started at
one side only of the shaft and spread inward, leaving the
remaining sound metal in the shape of a segment of a circle.

Figure 83 is a sketch showing a typical fracture due to
axial vibration in a thin steel disk rotating at a high speed.

FATIGUE FAILURE UNDER SERVICE CONDITIONS 233

The axial vibration caused reversals of radial stress in the
metal and the high speed of rotation set up steady tensile
radial stresses. The fatigue fracture started at the inter-
section of a deep tool mark and a hole and spread over the
length ah. At that stage the remaining sound metal ac, bd
was so reduced in area that the steady centrifugal force
caused a sudden tensile failure. The portion of the frac-
ture ab showed a smooth surface with "ripple marks" Uke
the surface shown in the outer ring of Fig. 82(a). The
parts ac and bd of the fracture and the upper edge of the

Faiigus Fraciure in Torsion
Transverse Shear

CO)

Fahgue Frac'hure in Torsion
Transverse and LongHudmal Shear

,- Fracfures-fari-edoii
Corner ofKeywoiy

Fai'iofue Fraciure in Torsion
Tension on indhned Sec-fion

(b)

Fig. 84. — Typical fatigue fractures in torsion.

center portion showed a rough ''crystalline" surface like
the surface of the fractured core shown in Fig. 82(a).

Figure 84 shows typical fatigue fractures under repeti-
tions of torsional stress. Figure 84 (a) is a sketch showing
the development of longitudinal and also circumferential
shearing fractures. It should be borne in mind that the
longitudinal shearing unit stress in a shaft subjected to
torsion is as great as is the transverse shearing unit stress.
Figure 84(6) shows a failure not by shearing under torsion
but by tension along an inclined plane. On a plane making
45 deg. with the axis of a shaft the extreme tensile unit
stress is as great as is the extreme shearing unit stress on a

234 THE FATIGUE OF METALS

section at right angles to the axis or the extreme shearing
unit stress parallel to the axis of the shaft. Such ''spiral"
fractures under torsion as that shown in Fig. 84(6) are
characteristic of rather hard, brittle metals, though under
repeated stress such fractures sometimes occur in fairly-
ductile shafting steel. Figure 85 is from a photograph of a
shaft which failed in service under cycles of torsional stress.

Fracfure si anted af
corner of key way

Fig. 85. — Fatigue fracture of shaft in service under repeated torsion.

The peculiar star-shaped fracture indicates a series of inclined
tensile-stress failures. The failure started at a corner of
the keyway, which is a point of high stress concentration.
Figure 86 shows a fatigue fracture of a bolt at the root
of the screw thread under repeated axial loading. The
rather irregular distribution of smooth fatigue-crack sur-
faces and rough final-failure surfaces is in marked contrast
with the regular distribution of those surfaces shown in

FATIGUE FAILURE UNDER SERVICE CONDITIONS 235

Fig. 82 for specimens which failed under cycles of flexural
stress.

Typical Fatigue Failures in Service. 1. Structural Mem-
bers in Bridges and Buildings. — Fatigue failures are very
rare in structural members of bridges and buildings. Such
members are, for the most part, subjected in service to

Fig, 86. — Fatigue fracture of bolt under repeated axial tension,

rather narrow ranges of stress, and very few such members
are subjected to reversals of stress. Certain members in
lift bridges are, however, subjected to partial stress reversal
in service. Most structural members in bridges have rivet
holes in them and at the edges of rivet holes there are high
stress concentrations; probably under normal load the

236 THE FATIGUE OF METALS

localized unit stress at the edges of rivet holes is frequently
as high as the yield point of the metal. In a few cases in
practice fatigue cracks have developed at the edges of rivet
holes and have spread into the member. In all cases with
which the writers are familiar, such fatigue cracks were
detected before a disastrous failure had occurred and the
parts affected were replaced or patched.

2. Boiler Plates. — Fatigue cracks sometimes develop in
boiler plates usually extending from one rivet hole toward
the next. Both localized stress and corrosion effects are
most marked at the edges of rivet holes, and corrosion and
localized stress are mutually accelerative. Usually before
a disastrous failure occurs, a crack can be detected by the
leakage of steam or water through it; but there have been
cases in which the combined effect of corrosion and stress
caused a sudden tearing of plate through a large number of
rivet holes at once, and a disastrous explosion followed,
although there had been no leakage detected.

3. Car Axles. — Fatigue cracks sometimes develop in
railway car axles. Such cracks practically always occur
near fillets, where the localized stress is higher than the
value computed by the ordinary formulas of mechanics of
materials. Probably fatigue cracks begin under the occa-
sional high stresses to which all car axles are occasionally
subjected, high stresses caused by flat wheels, wheel flange
pressure against ''tight" frogs, bad joints in rails, etc.
Once started, such cracks will spread under stresses lower
than those necessary to cause the first fatigue cracks.
Some railroads scrap axles after a certain mileage, while
some street railroads take out axles after a certain mileage
(usually about 100,000 miles), take off the wheels, and make
a careful search for fatigue cracks. If no cracks are found,
the axles are put back into service; if cracks are found the
axles are scrapped. In detecting fatigue cracks in axles the
process described in the footnote on page 228 is used. In
view of the comparatively rare occurrence of periods of
overload and rough service, the method of periodic inspec-
tion of axles for cracks seems to be a fairly effective pre-

FATIGUE FAILURE UNDER SERVICE CONDITIONS 237

caution against disastrous fatigue failures in service.
Axles cannot be inspected for cracks while in service, and
when fatigue failures of axles occur, they occur suddenly,
frequently with disastrous results.

4. Automobile Axles. — Automobile driving axles are not
infrequently subjected to severe repeated stress, both tor-
sional and bending. There is no opportunity for careful
inspection of axles in service, and when fatigue failure occurs,
there is no warning. Fatigue failure generally starts at
the edge of a keyway, at a deep tool mark, or at a rough
spot on the surface of the axle. In most cars the breaking
of an axle does not usually cause a wreck, and a broken
axle can easily be replaced.

5. Automobile Steering Knuckles. — Steering knuckles are
subjected to occasional sudden, severe loads, usually not
reversed. Fatigue cracks cannot well be detected in service,
and a failure is frequently the cause of a serious accident.
The only precaution available against fatigue failure seems
to lie in the choice of the metal for the knuckle and in the
careful design to minimize localized stress in the knuckle.

6. Bolts and Studs. — Bolts and studs have high stress
concentrations at the roots of the threads, stress concen-
trations reaching probable values as high as four times the
average stress on the section at the root of a thread.
There is very little chance for inspection, and when fatigue
failure occurs there is no warning.

Bolts and studs are frequently subjected to load very
rapidly applied — shocks and blows, for example. Under
rapidly applied load, an important criterion of strength is
-^the ability of the bolt or stud to absorb energy without
fracturing. This ability is somewhat different from the
ability to carry load without fracture, and both ductility
and strength contribute to the ability to resist energy load-
ing. When it is feasible, the reduction of area of the shank
of a bolt to a size slightly smaller than the section at the
root of the threads (as shown in Fig. 87(6)) increases the
energy-absorbing capacity of the bolt. In Fig. 87(a)
the energy of a sudden load will be absorbed almost entirely

238

THE FATIGUE OF METALS

in the very short sections of metal at the roots of the threads,
and this metal cannot stretch sufficiently to absorb the
energy of a heavy shock without fracture or, at least, the
starting of a crack. In Fig. 87(&) the reduced shank will
stretch appreciably, as well as the metal at the root of the
threads. As stretch is one factor in energy absorption,
the stress will be less for a bolt such as that shown in Fig.
87(6) than for a bolt like that shown in Fig. 87(a). Many
years ago John Sweet stopped the frequent failures which
occurred in the connecting-rod bolts of the ''straight-
line" steam engine by changing the design of the bolts
from that shown in Fig. 87(a) to that shown in Fig. 87(6).

(a) (&)

Fig. 87. — Two designs for bolt to resist energy loading in tension.

7. Springs. — Springs are usually designed to give large
elastic deformations and to absorb energy. The capacity
for elastic-energy absorption varies as the square of the
stress, and hence the prime requisite for the material is a
high elastic strength. Springs are sometimes fitted with
stops to prevent overstress, and are usually made of hard,
brittle steel. The failure of a spring rarely causes a
serious accident. Fatigue failures are not uncommon and
occur without warning; usually the fatigue crack is started
by a period of unusually high stress, and the crack spreads
gradually with final failure often occurring under normal
load. Careful lubrication of leaf springs increases the
deformation under any given load, and reduces stress con-
centration due to wear at bearing points.

FATIGUE FAILURE UNDER SERVICE CONDITIONS 239

8. Railroad Rails. — Railroad rails are subjected to partial
reversal of high stress in service. Normally, the head of the
rail is worn out by traffic ; this factor necessitates the use of
rather hard steel not very high in ductility. The passage of
loaded wheels over the rail cold rolls the steel in the head.
This one-sided cold rolling in all probability sets up heavy
stresses in the interior of the rail head. In exceptional
cases the combination of high stress due to heavy wheel
loads and high internal stress due to cold rolling of the

Fig.

-Fatigue fracture of rail, started at a transverse fissure.

surface of the rail start a '^shattered zone" or a ''transverse
fissure/' apparently from a focal point in the interior of the
rail head. Figure 88 shows a rail fractured by a progressive
failure starting from a transverse fissure ABC, which
apparently started at a focal, minute, area 0.

The whole subject of transverse fissures in rails is a very
fertile field for debate; metallurgists and engineers are
divided in opinion as to whether abnormal rolling-mill
conditions which produce poor steel or severe service condi-
tions should bear the chief blame for their existence.
Transverse fissures usually develop much more frequently

240

THE FATIGUE OF METALS

in heats of steel from certain rolling mills and seem to
develop from some defect which is initially in the rail and
which acts as the nucleus of a fatigue failure under the
high stresses set up by heavy wheel loads in service. As
noted above as a matter of experience, transverse fissure
failures in rails are not at all common. When they do
occur, they occur without warning.

9. Rotating Disks. — Thin rotating disks have critical
speeds at which severe axial vibration (''fluttering") is
likely to occur, with consequent cycles of reversed flexural
stress in a radial direction. Under such conditions a fatigue
crack may be started. Such disks may be subjected to
many thousand severe vibrations before there occurs a

(a) (b)

Fig. 89. — Wire rope bent round sheaves.

chance for inspection; hence there may occur a disastrous
fatigue failure before a crack is detected. Figure 83 shows
such a failure and Fig. 81 shows a disk in which a crack
was detected before it had spread to failure. Most of such
failures start at a deep tool mark, a hole, or other point of
high localized stress.

The available means of safeguarding disks against this
fatigue failure are: (1) careful surface finish, (2) avoiding of
holes at regions of possible high stress, and (3) the designing
of the disk so that the running speed of the machine will not
approach closely the critical speed causing axial vibration
in the disk.

10. Wire Ropes Bent around Sheaves. — The wires in
wire ropes running over sheaves are subjected to cycles of
flexural stress of a magnitude depending on the size of the
individual wires and the diameter of the sheaves. Figure

FATIGUE FAILURE UNDER SERVICE CONDITIONS 241

89(a) shows a rope bent around sheaves in which there
would be repetition of stress but not reversal. Figure
89(6) shows a rope bent around sheaves in which there
would be reversal of stress. Usually, before complete
fatigue of the rope occurs, individual wires snap, and as the
wires on the outside of the rope are subjected to wear as
well as flexural and tensile stress, the outside wires usually
snap first. Hence the failure can be detected before it
causes a serious accident. For wire ropes inspection
should be frequent. The rope should be replaced when
wires begin to break.

Summary of General Principles of Design of Members
Subjected to Repeated Stress. — Members subjected to
repeated stress should be designed so that the normal work-
ing stress will be well below the endurance limit of the
metal for the range of stress imposed in service. In
addition to this precaution the designer should so shape
a machine part as to reduce localized stress to a minimum,
and he should calculate or estimate the magnitude of
localized stress when feasible. This means that he must
avoid as far as possible sharp corners and notches in the
outline of the parts, must provide generous fillets at
shoulders, must avoid holes in regions of high stress (or
must allow for a stress concentration of about twice the
nominal computed stress if holes cannot be avoided),
and should avoid using screw threads to transmit repeated
stress, as far as is feasible.

In choosing metal for such parts, the designer must
consider not only its strength, but also its ability to with-
stand occasional overstress without starting a progressive
fatigue crack. ^ The designer must consider how serious
would be the results of fatigue fracture of any part and use
lower stresses, and should devise safeguards to minimize
damage in the cases of a member whose failure would cause
a disaster, and in the case of a member which cannot be
frequently or readily inspected for incipient fatigue cracks.

^ Usually this means choosing as ductile a metal as is consistent with the
necessary strength.

242

THE FATIGUE OF METALS

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FATIGUE FAILURE UNDER SERVICE CONDITIONS 243

Estimated Nixmber of Cycles of Stress for Various
Machine and Structural Parts. — Table 22 gives values of
the estimated number of cycles of stress which must be
withstood in the normal ''lifetime" of various machine
parts and structural members. The values given are to be
regarded as rough estimates, giving the ''order of magni-
tude" of the number of cycles of stress rather than anything
even approaching a precise number. The values given
for probable range of stress are also to be regarded as esti-
mates rather than precise values.

CHAPTER X
FATIGUE OF WOOD

Fatigue Failure of "Wood in. Service. — Structural and
machine parts subjected to repeated stress are rarely made
of wood. Bridge timber, electric-wire poles, and floor
beams in mill buildings are occasionally subjected to
"vibration" which is equivalent to a reversed flexural
stress superimposed on a steady stress, so that the net
range of stress is narrow. On account of its light weight,
wood has been widely used in airplane construction, and
in this service it is subject to a considerable amount of
repeated stress. Fatigue test data for wood are very
scarce, and only a few tentative conclusions can be drawn
as to the fatigue strength of wood.

Vibration Tests. — It will be of interest to study some of
the results of repeated-stress tests which have been obtained
at the Forest Products Laboratory at Madison, Wis. These
results have not been published hitherto.

Some of the first tests which were made were suggested
by the fact that certain members of airplanes were subject
to vibration. It can be shown^ that the frequency of
vibration of a simple beam is given by the formula

in which F = frequency of complete vibrations per second,
E = the modulus of elasticity in static bending, in

pounds per square inch,
I = moment of inertia of the cross-section, in

inches'^,
W = weight of the stick between centers, in pounds,
I = length between centers, in inches.

1 See MoRLEY, "Strength of Materials," p. 406.

244

FATIGUE OF WOOD 245

For the case of a vibrating-cantilever beam the formula is

EI
F = 11.04^^.

The experiments consisted in determining the experi-
mental constants for simple and cantilever beams, which are
given as 30.8 and 11,04 in the above formulas.

The apparatus consisted of a vibrating beam having a
brass stylus which traced a record on a rotating drum. An
electric tuning fork traced a record on the same drum, thus
making it possible to obtain the frequency. Since the
other quantities in the formula could also be determined, it
was possible to compute the constant. This constant
was found to have an average value of 31.2 for simple
beams, and 10.5 for cantilever beams. There was com-
paratively httle variation in the constant for the various
cross-sections and lengths of beam used.

The dimensions of the beams used in the experiments
varied from }'2 to }^ by 35 in. long, to ^i by ^i by 70
in. long for the simple beams, and from 3^^ by J4 by 24 in.
long to 1 by 2 by 42 in. long for the cantilever beams.
The species tested were red gum, yellow birch, yellow pine,
white pine, hard maple, black walnut, Douglas fir, and
Sitka spruce, a total of 128 specimens being tested.

Damping of Vibrations. — Some experiments were also
made on the damping of vibrations in wooden beams.
Since the specimens were fastened to a heavy concrete
column, it was thought that the energy loss due to vibra-
tions imparted to the column must have been very small.
The energy loss due to air friction was also found to be
small, so that most of the energy loss must be largely due to
mechanical hysteresis. The tests indicated that there was
a difference in damping of vibrations in the various species
of wood, and that this damping was independent of the
modulus of elasticity of the wood.

Effect of Vibration on Strength and Stiffness. — The next
series of tests was made to determine the effect of vibra-
tions on the strength and stiffness of relatively long Sitka

246 THE FATIGUE OF METALS

spruce specimens. The dimensions used were % by 2 by

52 in. long, and ^Ke by 2 by 78 in. long. About one-half

of the tests were carried out on matched pairs of test pieces,

one specimen being vibrated and one not vibrated, a total

of 44 specimens being tested.

The number of vibrations used in the tests was 900 per

EI
minute. Then using the formula F = 31.2^^^, and

knowing E, I, and the weight per cubic foot of the beam,
and having adopted a length for the beam, it was possible
to compute the depth of cross-section required. Assuming
a value of 7,200 lb. per square inch for the elastic limit for
air-dry Sitka spruce, it was possible to calculate the ampli-
tude of vibration for the specimen from the ordinary deflec-
tion formula, in order to make sure that the elastic limit of
the material would not be exceeded. The amplitude of all
test pieces was then kept at about one-half of that computed
from the elastic-limit stress.

Specimens were vibrated and then tested statically, the
time of vibration varying between 15 min. and 96 hr.,
representing 13,500 and 5,184,000 cycles of stress, respec-
tively. The modulus of elasticity was not greatly affected
by the vibration, although, in general, there was a reduction
in modulus due to the vibration which varied from 1.5 to
10.5 per cent. This reduction seemed to be as great after
1 hr. of vibration as after 16 hr. While the conclusion was
drawn that the change in modulus might perhaps be due to
changes in moisture content and temperature of the speci-
men due to vibration, yet the results obtained on concrete
(see Chap. XI) would lead one to suppose that such reduc-
tion of modulus of elasticity due to repeated stressing might
well be expected.

It maybe noted here that weight and moisture determina-
tions were made during the tests, and in most cases practi-
cally no difference in weight could be determined, indi-
cating that, in general, variations in moisture content were
negligible.

FATIGUE OF WOOD 247

The effect of vibration on the elastic hmit and modulus
of rupture could not be detected from the results, the values
being about the same for the specimens which had been
vibrated and those which had not been vibrated.

Fatigue Tests of Wood. — Another series of tests which
was carried out at the Forest Products Laboratory con-
sisted in making rotating-beam tests on wooden specimens.
Specimens 2 in. square were gripped in a lathe chuck, and
the projecting portion was then turned down to a diameter
of % in. A generous fillet joined this portion to the fixed
end of the specimen. At the free end of the specimen a
brass ferrule was attached, and through this the specimen
was loaded by means of a lignum-vitae roller. Forty-
five specimens each of kiln-dried Sitka spruce, kiln-dried
Douglas fir, and green southern white oak were tested in
fatigue, and five specimens of each species were tested in
static bending. Some air-dried specimens of Douglas
fir were also tested. The static bending tests were made
on specimens held and turned in the lathe and loaded
in a manner exactly like the fatigue specimens. Half
the tests in static bending were made with the plane
tangent to the annual rings in a vertical position, and
half with the plane tangent to the annual rings in a hori-
zontal position, the load in all cases being applied ver-
tically. The speed used in the fatigue tests was 2,880
cycles per minute.

Table 23 shows the results obtained from the tests.

Figure 90 shows the S-N diagrams plotted from the
results of these tests.

The fatigue tests were not carried out to a sufficient num-
ber of cycles to make the determination of an endurance
limit certain, but the indications are that the endurance
limit of wood can be determined at a much smaller number
of cycles than is the case with metals. In this respect
wood resembles concrete (see Chap. XI). In all cases tests
were carried out at least to 300,000 cycles. The curves
show that when the applied unit stress is one-third of the

248

THE FATIGUE OF METALS

Table 23. — Results of Static Tests and of Fatigue Tests of Wood

Test results obtained at the U. S. Forest Products Laboratory,

Madison Wis.

Kind of wood

Moisture
content,
per cent

Static
modulus
of rup-
ture,
pounds
per square
inch

Estimated
endurance

limit
(rotating-

beam

test),

pounds

per square

inch

Ratio of

endurance

limit to

modulus

of rupture

Sitka spruce, kiln dried

13.8

above
fiber
satura-
tion
point
14.3
23.8

12,100
10,600

15,000
12,800

3,200
3,200

4,000
3,900

0.27

Southern white oak, green

Douglas fir, kiln dried

0.30
0.27

Douglas fir

0.31

6000
5000

4000
3000

eooo

5000
i-4000

PL.

^' 3000
J 7000

2 6000
I 5000

I" 4000

^ 3500

6000

5000

4000

3500. ■ -^-^ j^—r-

o g g o^Denofes g

§ g S^ Specimen^

- 2 g /(fo-f Broken E

Number of Cycles for Fracture

Fig. 90. — S-N diagrams for fatigue tests of wood. {U. S. Forest Products Lab.)

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FATIGUE OF WOOD 249

static modulus of rupture^ or a little larger, failure may be
expected to take place quite rapidly. The indications for
endurance limit are in the neighborhood of 0.25 of the static
modulus of rupture. It is of interest to note that the kiln-
dried Sitka spruce and Douglas fir show a lower ratio of
endurance limit to modulus of rupture than the other two
woods which had a much greater percentage of moisture.

Tests Made at the National Physical Laboratory. ^ —
Stanton reports some tests made on spruce wood which was
to be used for airplane wing spars. This material had an
ultimate tensile strength of 6,800 lb. per square inch,
and the fatigue specimens were tested in rotating bending.
The specimens were selected so as to have similar distribu-
tion and thickness of the annual rings.

When stresses of +2,510 lb. per square inch were applied,
the specimen began to show cracks at 6,000,000 cycles
and failed after 16,860,000 cycles. With stresses of
+ 1,970 lb. per square inch cracks developed at 16,250,000
cycles and failure set in after 16,860,000 cycles. With
stresses of + 1,625 lb. per square inch, no cracks were visible
even after 125,700,000 cycles. Evidently, therefore, the
endurance limit of this material lay between 1,600 and 1,970
lb. per square inch, and Stanton concluded that + 1,875 lb.
per square inch would be below the endurance limit. The
endurance limit was therefore about 25 per cent of the
ultimate tensile strength given above.

Repeated-impact Tests.^ — The Forest Products Labora-
tory also made some tests on the effect of repeated impacts
on Douglas fir specimens. The machine used for these
tests dropped a heavy hammer (about 500 lb.) repeatedly
through a distance of 0.02 in. This action produced
a stress in the specimen which was a little greater than the
elastic limit in static bending. After specimens had

^ The modulus of rupture is a value obtained by dividing bending moment
at fracture by the value J fc for a flexure specimen. Modulus of rupture is
measured in pounds per square inch and serves as a comparative measure
of static fiexural strength.

2 Engineering {London), p. 605, June 23, 1916.

250 THE FATIGUE OF METALS

been subjected to this repeated-impact test, in some cases
to as many as 8,000 impacts, they were tested in static
bending. When these results were compared with results
on similar specimens which had not been subjected to
impact, it was found that the repeated impact had produced
no significant change in the properties of the wood.

CHAPTER XI
FATIGUE OF CEMENT AND CONCRETE

Fatigue of Concrete in Service. — Concrete in service is
most commonly subject to steady loading. Reinforced-
concrete bridges and concrete arches are subjected to
loads varying from dead load to dead load plus live load,
cycles of stress not involving reversal. Concrete highway
slabs are subjected to repeated load varying from practi-
cally zero to a maximum. The significant endurance hmit
for concrete is the endurance limit for cycles of stress vary-
ing from zero to a maximum.

Limitations of Experimental Study of the Fatigue of
Concrete. — Fatigue tests of concrete must cover only a
short time or must be made on concrete several months old;
else the results will be affected by the natural gain in
strength with age. Concrete test specimens must be of
considerable size; else their strength is determined largely
by the strength and location of a few large pieces of gravel
or stone. Large-size specimens require testing machines of
high capacity, of much higher loading capacity than the
testing machines used for fatigue tests of metals. The
machines used have been either single-lever machines or
ordinary "static " testing machines designed to be automat-
ically operated between definite limits of load. In either
case the speed of testing was slow, and judged by its
performance under static load tests, the strength of con-
crete is markedly affected by the speed of testing. As a
result of these conditions, the test data available for deter-
mining the fatigue strength of concrete are very meager,
and long-time test data are almost entirely lacking. Most
of the tests involve cycles of stress varying from a small
compressive stress to a large compressive stress. Some
estimates of the fatigue strength of concrete are made in this

251

252

THE FATIGUE OF METALS

chapter, and these estimates are based largely on extra-
polation of the available test data, assuming a general
similarity of behavior under test between concrete and
metals.

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\

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Fig. 91.-

Numberof Repe+ilions, Producing Failure ""

(a) Upper (b) Lower

-S-N diagrams for neat cement cubes under repeated compression.
{Basdeon Van Ornum in Trans. A. S. C. E.)

Fatigue Tests of Cement. — Van Ornum^ reports some
fatigue tests at Washington University, St. Louis, made on
92 2- by 2-in. cubes of neat cement. The static ultimate
compressive strength was determined, and then various

1 Trans, Amer, Sac. Civil Eng., p. 443, 1903.

FATIGUE OF CEMENT AND CONCRETE 253

specimens were subjected to cycles of stress with a range
from practically zero to a maximum. The maximum unit
stress varied from 55 to 95 per cent of the ultimate compres-
sive strength. The tests were made on blocks 4 weeks old.
Figure 91(a) shows the results which were obtained.
While the tests were not carried out to a sufficient number
of cycles to determine the endurance limit of the material
accurately, yet the indications are that the endurance limit
for cycles of stress ranging from zero to a maximum,
(r = 0) would probably be found to be about 50 per cent
of the ultimate static strength. Figure 91(6) shows Van
Ornum's diagram to a logarithmic scale.

The similarity of this curve to the ^S-A^" diagram for
metals is obvious, and suggests that cement subjected to
repeated stresses behaves in a manner which is similar to
the behavior of metals under similar conditions. Further-
more, the indications are that the material has an endur-
ance limit which has a relation to the ultimate static
strength of the material.

Fatigue Tests of Concrete.^ — In the investigation men-
tioned above, similar tests were also carried out on 18
concrete cubes 7 by 7 in. in cross-section, subjected to a
range of stress from practically zero to a maximum. The
results indicated that concrete also obeyed the same general
law shown in Fig. 91, breaking under repeated stresses
which were much less than the ultimate static strength.

Van Ornum,3--4n"a later research, made tests on both
concrete compression blocks and reinforced-concrete beams
under cycles of stress ranging from a small value to a maxi-
mum. Crushed limestone was used in a 1 : 3 : 5 mix, and the
loads were apphed by means of an oil-pressure piston at
the rate of from 2 to 4 per minute.

The tests on the compression blocks were made at ages
of 1 month and 1 year. The number of repetitions before
rupture varied from 1 to 83,000, and the maximum stresses
used in the fatigue tests were various percentages of the
average compressive strength as determined from static

1 Trans. Amer. Soc. Civil Eng., p. 294, 1907.

254

THE FATIGUE OF METALS

tests of similar blocks. Three specimens loaded at 55 per
cent of the ultimate static strength withstood over 40,000
cycles without failure. The present authors have plotted
these results to logarithmic coordinates in Fig. 92(a).
There is some evidence of a break in the curve for the cylin-
ders aged one year and tested at 55 per cent of the ultimate
static strength, indicating an endurance limit at this value
of stress.

The tests of the reinforced-concrete beams were made at
ages of 1 month, 6 months, and 1 year. The number of

CD

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:? 90

lo E

£ E ?"

100

90
80
70

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100

1000

10000

1

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Number of Cycles forFroic+ure

— a.

o Beams Acred 0

neMonihX

^

^\

-^

"^,

>fe

0

100

000

10

000

100

DOO

Fig. 92.

Number of Cycles for Fracf ure
(a) Upper (&) Lower

S-N diagrams for concrete cylinders and beams under repeated stress.
(Based on Van Ornum in Trans. A. S. C. E.)

repetitions before rupture varied from 1 to 14,000, the
stresses being again various percentages of the load causing
failure in static tests. The results are plotted in Fig. 92 (6)
to a logarithmic scale. There is no evidence of a break in
the curve, and presumably the endurance limit of the
material had not been determined.

The failure of the beams began with tension cracks, then
usually (but not always) diagonal tension cracks, and finally
a compressive failure at the top of the beam near the loading
point. These indications began with minute amounts
which gradually increased until failure occurred.

FATIGUE OF CEMENT AND CONCRETE

255

It was found that in the large majority of cases the grad-
ual and progressive destruction of the bond between the
steel and the concrete had an important influence upon the
results.

Van Ornum concluded that the stress-cycle curve became
horizontal at about 50 per cent of the static ultimate
strength. The present authors feel that the tests were not
carried out to a sufficient number of cycles to make this
conclusion certain. Evidence which will be mentioned
later, however, indicates that the material can adjust itself
to a certain cycle of stress, and presumably withstand this
stress indefinitely.

3,4,000,000^

'■$ 3,000,000

E 2,000,000

J 1,000,000
•f n

)

100

Number of Repe+I+ions
200 300 400

500

600

700

k

^

^

Fig. 93.

Defledfons: each li'ne represenHng 0. 00025 inch.

-Elastic behavior of concrete cylinders under repeated compression.
(Based on Van Ornum in Trans. A. S. C. E.)

Elastic and Inelastic Behavior of Concrete under Repeti-
tion.— The above research by Van Ornum included some
very interesting observations on the elastic behavior of
the compression cylinders subjected to repeated stresses.
Compressometers with gage lengths of 8 in. were used to
measure the compressive strains. It was found that
the stress-deformation curve for the first loading was a
straight fine for the lower stresses, becoming convex
upward for the higher stresses, as shown in Fig. 93. For
the second loading the stress-deformation curve was
practically a straight line up to the maximum stress applied,

256 THE FATIGUE OF METALS

this straight line being parallel to the straight portion of
the curve for the first loading. It was found that this first
stage under repeated loading might continue for a consider-
able number of repetitions. The second stage of the test
was characterized by a gradual decrease in slope of the
straight line. This stage did not continue long for cases
in which the number of cycles required for rupture was small.
The third stage began relatively near the failure point, and
was characterized by increase of deformation, which pro-
duced a curve which was concave upward. Finally, the
fourth stage added to this curve a portion which was con-
vex upward for the higher stresses, the deformations con-
tinuing to increase until failure occurred. These various
types of deformation curves are shown in Fig. 93 (c/.
Fig. 101).

The modulus of elasticity of the blocks was computed,
based on the maximum stress and the corresponding defor-
mation. The curve for modulus showed an increase for
the second loading, a decrease during the next few applica-
tions to its original value, and then a gradual straight-line
decrease during the greater portion of the test, terminating
in a downward curve as the failure point was approached.
This is shown in the upper part of Fig, 93.

A few specimens were also observed when subjected to
maximum unit stresses which were too small to cause final
failure. In these tests, as shown in Fig. 94, the first two
stages of the test were the same as before, but the third and
fourth were absent. For the case of the test data shown
graphically in Fig. 94, the specimen was subjected to 30,000
cycles at a stress of 1,000 lb. per square inch without failure.
The modulus curve decreased to a constant value equal to
about two-thirds of its original value, after about 12,000
cycles. The stress-deformation curves, after the modulus
of elasticity had become constant, were parallel straight
lines.

Another phenomenon w^hich was observed during these
tests was that a permanent set occurred in all specimens
during the first few loadings. For specimens which finally

FATIGUE OF CEMENT AND CONCRETE

257

failed, the permanent set became comparatively small dur-
ing the second stage,^ but again increased during the third
and fourth stages. For specimens which presumably

Number of Repe+i+ions
10,000 15,000 20,000 25.000

30.000 35.000

Dsflecfions:each Imerepresenfing 0. 0002Sinch

Fig. 94o — Elastic behavior of concrete under repeated compression below endur-
ance limit. {Based on Van Ornum in Trans. A. S. C. E.)

would have withstood an indefinitely large number of
cycles, the evidence of permanent set rapidly disappeared.^

Fig. 95.

1000

4000

eooo 3000

Number of Repe+i+ions
-Elastic behavior of reinforced concrete beam under repeated flexure.
{Based on Van Ornum in Trans. A. S. C. E.)

Beams under Repeated Loading. — ^^Observations similar
to the above were also made on reinforced-concrete beams,

1 In the opinion of the authors of this book this statement appHes only
after the adjustment to a certain cycle of stress is completed.

258 THE FATIGUE OF METALS

the deflections of the beam being read when the maximum
repetition load was on and also when it was off. These
results are shown in Fig. 95, which shows the total deflection
curve and the permanent-set curve. The shaded area
represents the range of deflections throughout the test.

The figure indicates five stages during the progress of
the test. The first stage represents initial adjustment of
the beam to the applied stress, accompanied by the develop-
ment of small tension cracks in the beam. The second
stage is characterized by a very slight downward trend in
the curve, indicating fairly stable conditions. The third
stage shows a rather rapid downward slope in the curve,
accompanied by the breaking of the bond between the steel
and concrete, and the enlargement of tension and diagonal
cracks. The fourth stage is again nearly horizontal on the
curve, indicating fairly stable conditions. The fifth stage
shows a rapid downward trend in the curve, accompanied
by the failure of the beam in compression and ending in
complete failure.

It was found that when a beam was subjected to stresses
which presumably could have been withstood indefinitely,
only the first two stages shown in Fig. 95 were developed.
The curves became horizontal, and the vertical distance
between the two curves remained constant with increase
in the number of cycles.

These results are especially interesting in indicating
that concrete, under stresses which will not produce final
failure, apparently acts just as metals do under similar
conditions. Concrete, like metals, evidently has the power
to adjust itself to cycles of stress, when the maximum
imit stress is within a certain limit. These experiments
show that after such adjustment the behavior of the
material is elastic, and that the modulus of elasticity
reaches and maintains a constant value. The similarity in
behavior between concrete and the metals is obvious, and
tends to reinforce the conclusion that the endurance limit
of a material is a definite physical property.

FATIGUE OF CEMENT AND CONCRETE 259

Tests of Bond between Concrete and Steel. — Van Ornum
also made tests on the effect of fatigue on the bond between
steel and concrete. He used beams 4 by 4 in. in cross-
section, and 15 in. long, with a /-i-in. square plain steel bar
placed with its center 1 inch from the tension side of the
beam. The specimen was clamped on a fairly rigid frame.
A machine was devised which had two metal struts, having
a reciprocating motion. On the front end of these struts
was a metal cross-head which was faced with 1}^ in. of oak.
By means of this device, the specimen was subjected to
blows at the rate of 150 per minute, thus being subjected to
impact, bending, and vibration. The tests were made
when the concrete was 1 month old.

To obtain comparative results, 18 specimens had the
steel rods pulled from the concrete in static tests without
preliminary fatigue treatment. Under these conditions
the average bond strength was 150 lb. per square inch, and
the frictional resistance after the bond was broken was
100 lb. per square inch.

Thirty specimens were subjected to fatigue treatment,
the average number of blows being 50,000. After this
treatment the steel was pulled from the concrete and
developed an initial average bond strength of 125 lb. per
square inch, and a subsequent average frictional resistance
of 90 lb. per square inch. The bond strength had therefore
been reduced 17 per cent, and the frictional resistance 10
per cent, due to the repeated blows.

Withey^ also made some tests on the effect of fatigue on
the bond between steel and concrete in reinforced-concrete
beams. In these tests the steel was imbedded at the ends of
the beam and exposed near the middle of the beam. This
made it possible to attach extensometers to the exposed
rods to determine the deformation in the steel and the load
on the rods. The beams rested on end supports and had
the load applied at two symmetrical points on the top of
the beam. The repeated loads were applied at one of the
supports by means of a cylinder and piston, at the rate of

1 Univ. Wisconsin, Bull. 321, 1909.

260 THE FATIGUE OF METALS

100 repetitions per minute. The concrete for the repeated-
stress tests was a 1.2. 2 A A mixture, having crushed lime-
stone for the coarse aggregate. The tests were made on
concrete 1 month old.

From static tests the conclusion was drawn that the
maximum bond strength for plain rods less than % in. in
diameter would be about 250 lb. per square inch and for
rods of larger size about 200 lb. per square inch. The static
bond for rusted rods was considerably greater than for
plain rods, and was about twice as great for corrugated rods
as for plain rods.

Under repeated loading the results showed that about
50 to 60 per cent of the static ultimate bond stress could be
repeated many times on plain rods without failure. Rusted
rods showed better bond strength than plain rods. The
tests on corrugated rods indicated that about 60 to 70 per
cent of the static bond stress could be repeated many times
without failure. The number of repetitions used in these
tests varied from 1,000 to 104,000 for the different beams,
and in quite a few cases the tests were not carried out to
destruction. No attempt was made to determine the bond
stress which could be withstood indefinitely without
failure.

Beam Tests by Berry. — Berry^ made some tests at the
University of Pennsylvania on reinforced-concrete beams
8 by 11 in. in cross-section, using a span of 12 ft. A mix of
1:2:4 was used, the age of the beams when tested was 4
weeks, and the rate of load application was 30 per minute.
Compression cylinders showed that the concrete had an
ultimate strength of 1,630 lb. per square inch at the age of
6 weeks.

Four beams were subjected to repeated stresses, and
three of these had duplicates which were subjected to an
ordinary static test. The tensile stresses in the steel varied
from 14,300 to 18,300 lb. per square inch for the different
beams, and the compressive stresses in the concrete varied
from 628 to 940 lb. per square inch. After receiving from

1 Proc. Amer. Soc. Tesiing Materials, vol. 8, p. 454, 1908.

FATIGUE OF CEMENT AND CONCRETE 261

200,000 to 1,100,000 cycles of stress, the beams were sub-
jected to increasing stresses until failure occurred. The
results showed that the maximum deflection at failure and
the ultimate load for the two beams of each set were much
the same. The results also indicated that the hundreds of
thousands of cycles of stress which were applied did not
have any marked effect on the static strength and the
deflection at static failure of the beams tested.

In the fatigue tests the deflection of the beam increased
with the number of repetitions. The elastic deflection
for any constant load remained nearly constant, but the
permanent set increased. At least one-third of the set
present after from 300,000 to 1,000,000 repetitions occurred
during the first 10,000 cycles, and a very considerable part
of the set occurred during the first few cycles. Berry says:

While it is evident that the rate of increase in the set is relatively very-
large for the first few applications of the load, there is nothing to indicate
that for a working load the set would cease to become greater.

On- the basis of the tests performed by other investigators
it seems clear that if the maximum stress applied in fatigue
is sufficiently low, the permanent set will cease to increase
and the beam apparently will withstand the stresses
indefinitely.

Berry drew the following conclusions from his tests:

1. That the ultimate static strength of a reinforced-concrete beam is
not materially affected by 1,000,000 repetitions of high working stresses.

2. That the maximum deflection is not affected.

3. That hair-line cracks become visible for such loads at intervals
of 6 to 8 in., and grow deeper as the number of repetitions is increased;
but that for 1,000,000 repetitions no crack extended beyond the neutral
axis.

4. That the bond between the steel and the concrete is .not appreci-
ably affected, as shown by the difficulty with which the steel was
removed in breaking up the beams.

5. That the position of the neutral axis is not changed by repeti-
tions of load.

6. That the greater part of the set in the deformation in the plane of
the steel occurs in the first few thousand applications of the load.

262 THE FATIGUE OF METALS

7. That the set in the deformation on the compressive side of the beam
is also relatively large for the first few thousand repetitions, and that it
increases with the stress applied and the number of repetitions.

Compression Tests by Williams. — Williams^ made some
repeated-stress tests on concrete cylinders, using a 1:2:4
mix, and testing at the age of 28 days and 6 weeks. The
number of repetitions applied was small (less than 75),
and his results indicated that the permanent set increased
somewhat with increase of repetitions, and that the modulus
of elasticity increased also. The modulus of elasticity
increased only slightly, and since the concrete was only 6
weeks old or less, it is probable that the increase in strength
with age affected the results. The present authors do not
consider this evidence sufficient to controvert the findings
of Van Ornum that the modulus of elasticity decreases with
increase in number of repetitions. It will be recalled that
this decrease to some constant value occurred even when
the stress was low enough so that failure by fatigue did not
occur.

Beam Tests by Bureau of Standards. — Slater^ and his
associates at the Bureau of Standards carried out some
tests which were different from any of those mentioned
hitherto, in that they were tests of double-reinforced-con-
crete beams subjected to reversed stresses. This work was
done in order to get information which would be of use in
the construction of concrete ships. Five beams 6 by 8 in.
in cross-section were tested on a span of 8 ft. One of the
beams had an I-shaped cross-section and web reinforce-
ment in addition to the longitudinal rods. Four beams
were made from a l:%:l}i mix, and the fifth was made
from a l-.^/i-.l}^ mix. The strength of the compression
cylinders made from the same concrete varied between 4,200
and 6,200 lb. per square inch. The beams were from 2 to
5 months old when the tests were begun, and the rate of
application of load was 17 cycles per minute. The load

1 Proc. Amer. Soc. Testing Materials, vol. 20, Pt. II, p. 235, 1920.

2 U. S. Bur. Standards, Tech. Paper 182, 1920.

FATIGUE OF CEMENT AND CONCRETE 263

was applied and released by lowering and raising weights
acting at the ends of levers.

Beam 5A1 had a measured maximum stress in the steel
which varied between 5,400 compression to 21,600 lb. per
square inch tension. The maximum compressive stress in
the concrete was 1,565 lb. per square inch. This beam
failed after 709,041 cycles of stress by rupture of the steel
in fatigue. During the first 1,000 cycles, the width of the
tension cracks increased to about 0.02 in., the steel deforma-
tion changed slightly, and the beam deflections changed
markedly. After the first 1,000 cycles the crack widths,
deformations, and deflections remained practically constant
for about half the ''life" of the beam, and finally the deflec-
tion downward and the crack widths increased gradually
until failure occurred.

" Beam 5J51 had a measured maximum stress in the steel
which varied between zero and 22,800 lb. per square inch
tension, and a maximum compressive stress in the concrete
of 1,210 lb. per square inch. The beam failed after 59,377
cycles by rupture of the steel in fatigue. During the first
300 cycles the deflections increased rapidly, and during
the first 7,000 cycles the crack widths did the same. After
that, the deflections and crack widths remained practically
constant until failure was imminent.

Beam 5 CI had a measured maximum stress in the steel
which varied between zero and 11,000 lb. per square inch
tension, and a maximum compressive stress in the concrete
of 1,425 lb. per square inch. These stresses represented
the working stresses used in concrete ship design. The
beam did not fail, and the test was discontinued after
2,008,000 cycles. The downward deflection, downward
permanent set, and crack width at the bottom (where the
steel stress was a maximum) increased gradually for about
400,000 cycles, and then remained practically constant.
When the test was discontinued, the beam showed no indi-
cations of approaching failure.

Beam 5F1 had the I-shaped section and the web rein-
forcement, and was designed to give a computed unit stress

264 THE FATIGUE OF METALS

of 175 lb. per square inch in the web. The measured maxi-
mum stress in the steel varied between 7,000 compressive
and 11,700 lb. per square inch tensile unit stress, while the
computed stress in the concrete was 1,641 lb. per square inch.
The beam failed after 544,448 cycles by rupture of the
steel in fatigue. During the early stages of the test the
deflections increased rapidly with a gradual increase of
crack widths. At 3,600 cycles a horizontal crack appeared
at the top edge of the web. As the test continued, small
diagonal cracks developed downward from the horizontal
crack toward the center of the web. From 3,600 cycles
until about three fourths of the life of the specimen,
deflections and crack wddths were practically constant, but
unit deformations varied considerably.

Beam 5L1 had a measured maximum stress in the steel
varying between 4,000 compressive and 18,000 lb. tensile
unit stress, and a maximum compressive stress in the
concrete of 2,080 lb. per square inch. The beam failed
after 451,821 cycles by rupture of the steel in fatigue.
It had been designed to develop a bond stress of 161 lb. per
square inch. During the first 70,000 cycles there was a
gradual increase in deflections, crack widths, and permanent
set. The downward permanent set, deflection, and bottom
crack widths increased gradually throughout the test. It
is interesting to note that up to about 7,000 cycles the
sUp of the steel bars was less than 0.001 in., that is, less
than the amount which tests of bond between steel and
concrete have shown to be a criterion of safe condition.
Due to repeated stress, however, this slip had been finally
increased to 0.06 in., and failure seemed imminent due to
this cause, when the steel failed in tension.

The ultimate tensile strength of the steel used in these
tests varied between 53,000 and 57,000 lb. per square inch.
None of the four beams which failed by fatigue of the steel
developed an endurance limit which might have been
expected, considering the ultimate static strength and the
maximum repeated stress in the steel. Three of the four
beams had steel which was high in phosphorus, and two

FATIGUE OF CEMENT AND CONCRETE 265

out of the four failed where gage holes had been drilled in
the steel. While these factors may have contributed to low
fatigue strength, it would seem that another cause had a
larger influence. This was the fact that in all four beams
large cracks extended entirely cross the section of the beam
at the places where the tension failure of the steel occurred.
It seems likely that a sharp local bending of the steel may
have been induced by the presence of these large cracks.
In any case these tests indicate that when steel is subjected
to large stresses in reinforced-concrete beams, developing
large cracks in the concrete, then the fatigue strength of the
steel will not be so great as is normally to be expected.

Beam Tests by Clemmer.^ — Investigation of concrete-
pavement failures in Illinois suggested that the failures were
due to the effect of repeated stresses, and since the standard
design in Illinois was based on the theory that the corners
were the weakest part of the slab, and acted as cantilever
beams, it was decided to devise a fatigue apparatus to
produce similar conditions. Furthermore, since most of
the loads were applied to roadways by means of rubber tires,
it was decided to stress the specimens in the same way. This
work was carried out by the Illinois Division of Highways.

The apparatus for these tests was arranged so that con-
crete beams radiated from a central support like the spokes
from the hub of a wheel, the beams being rigidly held at
the center and being free at the outer end. Concrete blocks
were placed between the ends of each pair of beams thus
completing a circular track upon which the loading device
could travel. The track was made as level as possible to
prevent impact.

The loading device was constructed by using the rear
wheels and axle of a Ford automobile, the span between
the wheels being increased to 7 ft. By means of a vertical
housing attached to the differential casing, and a horizontal
pulley attached to this vertical housing, the wheels could
be driven in a circle on the track which, had been prepared.
Weight boxes placed on the horizontal housing near the

1 Proc. Amer. Soc. Testing Materials, vol. 22, Pt. II, p. 408, 1922.

266 THE FATIGUE OF METALS

wheels permitted various loads to be applied to the speci-
mens. The rate of apphcation of load was 40 per minute.

The first series of tests consisted of 15 plain concrete beams,
6 in. square and 36 in. long, of a 1 :2:3H iiiix. Two of the
beams were tested under static load to determine their
modulus of rupture, and the fatigue tests were then carried
out on 7 beams using a stress equal to 50 per cent of the
modulus of rupture. After 1,130,976 appHcations of load
no failures had resulted.

The same beams were then subjected to 60 per cent of
their modulus of rupture (as determined after the beams
had failed). When a beam failed, a new beam was put in
its place. Under the stress 7 beams failed under applica-
tions ranging from 16,782 to 199,836 in number. The
average age of the specimens which failed was 174 days.

Some of the original beams which had been subjected to
1,130,976 cycles at 50 per cent of the modulus of rupture,
and to 409,655 cycles at 61 per cent, were now stressed to
70 per cent of the modulus of rupture. Under this stress
all the remaining specimens failed.

It was noted that beams which had resisted 1,000,000
cycles of stress, presumably less than the endurance limit,
resisted more applications than new specimens which had
not been so stressed. Furthermore, tests showed that the
moduli of rupture of understressed specimens were some-
what higher than those of beams which had not been sub-
jected to repeated stress. This strengthening effect of
understressing is a well-established phenomenon in the fatigue
testing of metals, and this evidence of a similar phenom-
enon in concrete is of interest.

In a second series of tests a 1:3:5 mix was used. When
these beams were subjected to 70 per cent of their modulus
of rupture, all but one failed at 1,525 cycles or less. This
single beam was then broken statically, giving a modulus of
rupture of 772 lb. per square inch. The unstressed end of
the beam gave a modulus of rupture of 808 lb. per square
inch, so that presumably the beam would have failed under
more applications.

FATIGUE OF CEMENT AND CONCRETE 267

A third series of beams was made with a 1 : 3 mortar,
these also being subjected to 70 per cent of their modulus of
rupture. These beams all failed at 5,280 cycles or less.

A fourth series of tests was made to determine the rela-
tion between the increase in strength of concrete with age
and the decrease in strength due to fatigue. Three sets of
beams were made of 1 :2:3>^ concrete, each set consisting of
15 beams. In all cases the beams of this series were placed
in the machine at the age of 30 days.

The beams of the first set, under a stress of 62 per cent
of the modulus of rupture, all failed at 7,000 cycles or less.
The beams of the second set, under a stress of 51 per cent
of the modulus of rupture, all failed at 33,000 cycles or less.

The beams of the third set were stressed to 48 per cent of
the modulus of rupture, and 2,000,000 cycles applied with-
out a single failure. A stress of 50 per cent of the modulus
of rupture, at the attained age of 90 days, was then applied
512,000 times without a single failure. A stress of 51 per
cent of the modulus of rupture at the then attained age of
120 days was then apphed 441,000 times without a single
failure. A stress of 54 per cent of the modulus of rupture
was then applied (at the age of 129 days), and all the beams
failed after 700,000 cycles or less.

The deflections and the permanent sets of the beams were
measured with a strain gage and with an Ames dial. The
curves indicated that there was a deviation from the straight-
line relation of load and deformation or of load and deflec-
tion at about 50 per cent of the modulus of rupture.

Figure 96 shows typical curves of deflections and recovery
during the progress of a fatigue test. These curves indi-
cate that as the fatigue test proceeds, the deflections increase
and the recovery decreases. The curves are so drawn that
the ordinates between the two curves indicate permanent
set, and it is apparent that the permanent set increases with
the number of applications. It should, however, be pointed
out that this was the case for beams which were stressed
above the endurance limit, so that they finally failed.
It will be recalled that Van Ornum's tests show that when

268

THE FATIGUE OF METALS

the stresses are below the endurance Umit, the permanent
set would not continue to increase.

Clemmer drew the following conclusions:

1. Concrete beams will fail under a number of repetitions of loads
which produce stress equal to or greater than a certain percentage of that
required to cause transverse failure when tested under one application
of load.

2. Loads which produce stress less than a certain percentage of the
modulus of rupture as determined in the testing machine will not cause
failure on repetitions, but rather, as indicated in the first and fourth
series, the strength of the specimens would actually be increased by this
condition of load if the load is near (but below) this critical percentage.

3. For cycles of stress ranging from zero to a maximum (r = 0),
the critical percentage or limit of endurance of the concrete specimens

o.on

0.018
g 0.019

1 0.020
'§ 0.021
'% 0.022

2 0.023
0.024
0.025

■

^

Recove

'y

■

■^

^

fe^

\

-'^o.

^y

-^

^

^

*v

si

3rof

re

\

\

Brok

?-^

n0.019
- 0.018. E
-0.017 f
■0.016 I
0.015'^
■ 0.014-

0 12 3 4

5 6 7 a Q 10 II 12 13
Ti'me, mlnules

Fig. 96. — Deflection and recovery during test of concrete beam under repeated
flexure. (Based on Clemmer in Proc. Am. Soc. Test. Materials.)

was between 51 and 54 per cent of the modulus of rupture as determined
from one apphcation of load.

4. For the same percentage of ultimate strength, a considerably less
number of applications of load is required to cause failure in the 1:3:5
mix specimens than in the 1:2:3^ mix specimens.

5. Stresses below the limit of endurance do not cause permanent
deformation in the specimen.

6. For stresses beyond the limit of endurance, the number of repeti-
tions of load required to produce failure decreases with increase of
percentage of stress.

In connection with this same research of the IlUnois
Division of Highways, Older ^ reported that test beams
subjected to a unit stress of about 50 per cent of the modu-

1 Trans. Am&r. Soc, Civil Eng., p. 1180, 1924.

FATIGUE OF CEMENT AND CONCRETE 269

lus of rupture had withstood 5,000,000 repetitions of stress
without failure.

Concrete slabs under actual traffic conditions were
studied. A corner break on a slab 4 in. thick, under a
3,500-lb. wheel load, indicated that at the corners the
slab was subjected to stresses equal to or exceeding the
critical value of 50 per cent of the modulus of rupture.
An increase of wheel load of 1,000 lb. would therefore
increase the stresses to about 78 per cent of the modulus of
rupture, under which conditions rapid failure might be
expected to follow. This actually proved to be the case,
since many corners were broken and the progressive
destruction was quite rapid. This same phenomenon had
been noted on Illinois highways in service, which had
withstood normal traffic for a number of years, and then
began to give way under increased highway loadings.

It may be of interest to note here that the Illinois high-
way tests have led to the following formula for computing
the depth of the slab :

/3F

in which W = the maximum wheel load, in pounds,

S> = modulus of rupture of the concrete, in

pounds per square inch,
d = depth of the concrete slab, in inches.

When S is the modulus of rupture, W would represent the
breaking load. It is recommended that for design pur-
poses S be taken equal to 50 per cent or less of the modulus
of rupture of the concrete. The various other recom-
mendations on slab design may be found in the original
paper.

Compression Tests by Probst. — Some interesting experi-
ments by Probst^ were made on concrete compression
specimens 7 by 7 cm. in cross-section and 28 cm. long.
Two specimens were stressed with a maximum stress of
1,848 lb. per square inch, but one had a minimum stress of

1 Festshrift zur Hundertjahr feier Tech. Hochs., Karlsruhe, 1925.

270 THE FATIGUE OF METALS

114 and the other 1,422 lb. per square inch. The maximum
stress was about 70 per cent of the static ultimate com-
pressive strength. The first specimen withstood 341,000
cycles before failure, and the second 1,500,000 cycles with-
out failure.

These specimens were also measured for deformation, a
Martens ' mirror apparatus being used on a gage length of
20 cm. Unit deformations could be read directly to
0.00001 and by estimation to 0.000001 cm. per centimeter.

In one case a specimen had a static ultimate strength of
2,104 lb. per square inch, and was stressed with a minimum
stress of 114 and a maximum stress of 789 lb. per
square inch, the maximum stress being, therefore, about
38 per cent of the ultimate static strength. The loadings
were repeated at the rate of 60 per minute. With increase
in number of cycles the elastic deformations increased at
first faster than the permanent sets, but later this was
reversed. The elastic deformations finally reached a
constant value and somewhat later the permanent sets also.
(It will be understood that the elastic deformation plus the
permanent set equals the total deformation at any stress.)
After 453,000 cycles the test was discontinued because both
the elastic and permanent deformations had reached a
stable condition. The modulus of elasticity had decreased
during the progress of the test, from an initial value of
3,590,000 lb. per square inch to a value of 2,810,000 lb. per
square inch at the end of 453,000 cycles.

After this test the specimen was subjected to increasing
stresses, beginning at 341 lb. per square inch, and the same
stress was repeated to determine whether stable conditions
of deformation had been reached. Up to a stress 1,000 lb.
per square inch, or 50 per cent of the ultimate static strength
determined after failure, there was no permanent deforma-
tion, and the modulus of elasticity had become constant.
At a stress of 1,068 lb. per square inch a unit permanent set
of 0.000005 was obtained, the elastic deformations were
stable after seven applications, but the permanent sets were
not. At a stress of 1,166 lb. per square inch and 10 repeti-

FATIGUE OF CEMENT AND CONCRETE

271

tions, neither the elastic nor the permanent deformations
were stable, and the permanent deformation had reached a
unit value of 0.000014, This condition existed also at
1,220 lb. per square inch with a unit permanent deformation
of 0.000022, and failure took place at 2,030 lb. per square
inch. The increase of permanent deformation had been
accompanied by decrease of modulus of elasticity. The
specimen had been made elastic by repeated stresses, even
beyond the original maximum value of 789 lb. per square
inch. This increase of elasticity by means of repeated
stresses is similar to that found by Bauschinger for metals.
Virgin specimens were then tested for elastic behavior,
and the average of three results is given in Table 24.

T.iBLE 24. — Tests of Concrete Compression Specimens for Elastic

Properties

Stress,
pounds

per
square

inch

Elastic Unit

Permanent

Modulus of

Number of
cycles re-

deforma-
tions, inch
per inch

unit deforma-
tion, inch
per inch

elasticity,
pounds per
square inch

quired for

attainment

of stable

Remarks

conditions

341

0.000081

0.000003

4,220,000

6

489

0.000120

0.000006

4 , 070 , 000

10

628

0.000158

0.00011

3 , 980 , 000

10

Stable for elastic but
not for permanent
deformation

■ 789

0.000222

0.000020

3,570,000

10

Not stable

940

0.000295

0.000031

3,180,000

10

Not stable

1,010

0.000332

0.000039

3 , 040 , 000

10

Not stable

1,068

0.000365

0.000047

2,920,000

10

Not stable

1,166

0.000428

0.000057

2,720,000

10

Not stable

2,104

Broke

1 Specimens not stressed previous to tests. Each value is the average of three test results.
Results obtained by Probst at the Karlsruhe Technical High School.

These results show that at a unit stress of 789 lb. per
square inch the sum of the elastic and permanent deforma-
tions is greater for the specimen subjected to repeated
stress than for the virgin specimens, the former being
0.000339 and the latter 0.000242. With increase of stress
the deformations of the virgin specimens surpassed the
deformations of the fatigue specimen, and that more rapidly
for the elastic than for the permanent deformations. By

272 THE FATIGUE OF METALS

permanent deformations for the fatigued specimen is meant
the permanent deformation which occurred after stable
conditions had been estabhshed by 453,000 cycles of stress
at 789 lb. per square inch.

When the logarithm of unit stress was plotted against the
logarithm of unit deformation, for elastic deformations,
straight lines resulted both for the fatigued and for the
virgin specimens. When the logarithm of unit deforma-
tion was plotted against the logarithm of cycles, the curve
showed a sharp break and became parallel to the cycles
axis, in a manner similar to the break in the S-N curve for
ferrous metals.

These results reinforce the conclusions arrived at by Van
Ornum. At a unit stress of 789 lb. per square inch, for
instance, the fatigued specimen reached a constant value
of modulus of elasticity which was about 79 per cent of
the value at the same stress for the virgin specimens.
This ratio undoubtedly would have been nearer the two-
thirds value found by Van Ornum if the maximum stress in
the fatigue test had been 50 per cent of the static ultimate
instead of only 38 per cent. These results indicate, as did
Van Ornum's, that concrete can adjust itself to a condition
of repeated stress, and presumably can withstand the
stress indefinitely if it does not exceed a certain maximum
value.

Beam Tests at Purdue University. — Hatt^ has reported
tests made at Purdue University on beams 4 by 4 in. in
cross-section, 30 in. long, and fabricated with a 1:2 mortar.
In order to minimize the effect of increase in strength due to
age, the tests were made on specimens which were over 6
months old. The specimens were subjected to bending
stresses, and Berry strain gages on each side of the speci-
mens measured the tensile and the compressive deforma-
tions over a gage length of 10 in. The fatigue testing
machine applied the load in such a way as to produce
reversals of stress on the two sides of the beam, which

1 Proc. Highway Research Board, Nat. Research Council, p. 47, 1925; also
Purdue Univ. Bull. 24, p. 46, 1925.

FATIGUE OF CEMENT AND CONCRETE

273

was placed in a vertical position. The tests to determine
the static properties were made in the same machine in
which the fatigue tests were carried out. In the fatigue
tests the rate of application of load was 10 per minute.

It was found that no definite endurance limit could be
determined for mortar at early ages, because the increase in
strength due to age might produce a greater effect than
the fatigue action. For beams only 28 days old the endur-
ance limit under cycles of reversed stress was found to be
as low as 40 per cent of the static breaking load.

_g 0.00035

'in
ifi

I" 0.00030

s

•I 0.00025
\ 0.00020
1^ 0.00015

<s
E

)^ o.oooio

Afark 160

Mix 1:2
' Age: l9monfhs

Beam: 4'x4 "cross sech'on

Speed; lOapplicafions of one stress per ml n'u-fe
- Load:^ S5 per cenf siaiic break incj load j

Loading: Aliernafe resfand-Fafigue loadmcf-fordG-AourperiocIs

Reversals of S+ress ^

Fig. 97. — Progressive deformation for concrete beam under reversed flexure.
{Based on Halt in Bull. 24, Purdue Univ.)

Figure 97 indicates the action of a beam in fatigue in
those cases in which final failure occurred. The deforma-
tion increased with increase of reversals of stress until
final failure occurred. Rupture occurred first on the
outer fibers where the deformation was a maximum, after
which failure was progressive toward the center of the
beam, sometimes requiring a considerable additional
number of reversals to make failure complete.

Figure 97 also indicates the recovery that occurred during
periods of rest, in this case periods of 96 hr. This decrease
of unit deformation after a period of rest was also observed
after 16 and 42 hr. It will be noted that this stiffening
effect due to rest is only temporary, and that the value

274

THE FATIGUE OF MET ALB

of deformation, existing before rest, is again attained after a
certain number of stress reversals. This stiffening phenom-
enon was noted at stresses above and also below the endur-
ance limit of the material.

In one case it became necessary to stop the machine for
5 weeks, and the specimens all showed a recovery in defor-

80

70

60

50

-40
30

— I 1 1 \

CARTES/AN COORDfMATES

Mix i-2\ r

Age&-l2rnon-f-hs ^

Cross Secfion4x4inch?s , ^ _

Speed: lOapplica-Hons of one siress per minufe

Loading : Faiigue dhoursperday rest period o verm^nr

J I) Contlnw^d-h 353, 000 reversals ofs^ess

(2)
(3J
(4)

1581,000
235,300 >•
355,700
U2I, 000 "
l,ni(>00 .. V
(a) Fafigue Loading 24hours perday
Specimen nofbrpkerj I

(b)

§■ i §

Reversals of Stress

Ti'me in Daus

100
"g 90

q. o

-^•^80
c c
?'-i 70

1-

60

50

LOGAklTHMIC COORDiNATFS

hv

c»

Speamet
NoiBroki

'rt

J

^

J^,

o

0

^^>

Number of Reversals o-f Stress

Fig. 98. — S-N diagram for concrete beam under reversed flexure.
Hatt in Bull. 24, Purdue Univ.)

{Based on

mation. One specimen which had been stressed to 55 per
cent of the breaking strength, and which was believed to
have been on the point of failure, showed the greatest
recovery. Under further stressing it gave a curve of
shghtly increasing and then decreasing deformation, indi-

FATIGUE OF CEMENT AND CONCRETE 275

eating apparently complete recovery from the previous
overstressing. Since these beams were 5 months old, it
was thought that the increase of strength due to increase
of age was of minor importance.

Figure 98 shows a typical S-N diagram. Figure 98(a)
is plotted to Cartesian coordinates, and Fig. 98(6) to
logarithmic coordinates. It should be noted that if the
stress is below the endurance limit, the specimen will not
fail, but if it is stressed only slightly above the endurance
limit, a comparatively few reversals of stress will cause
failure. It seems probable, from these tests and various
others which have been described in this chapter, that the
endurance limit of mortar and concrete can be determined
at a much smaller number of cycles than is necessary in
the case of metals.

It will be recalled that the tests of Van Ornum on com-
pression specimens and those of Clemmer on beams were
carried out between a minimum stress near zero and a
maximum stress of the same sign. They reported some
endurance limits lying between 50 and 55 per cent of the
static breaking strength. The Purdue tests employed
completely reversed stresses from tension to compression,
and the reported endurance limit hes between 50 and 55
per cent of the static breaking strength. While the mate-
rials experimented on are not strictly comparable, yet the
indication is that for these materials the range of stress
for completely reversed stress may be approximately twice
as great as for the case when the range lies between zero
and an upper limit.

The Purdue results showed that when a beam was being
subjected to a stress which would ultimately cause failure,
then the plastic set at zero load might be fairly large. For
stresses which could be endured a long time without fail-
ure, the plastic set was small and, after a period of rest,
was practically zero. The magnitude of this plastic set
seemed to be dependent upon the age of the mortar, being
inversely proportional to the age.

276 THE FATIGUE OF METALS

Another phenomenon which was observed was the streng-
thening effect of repeated stresses which were less than the
endurance Hmit. Understressing strengthened the beams
so that a later stress greater than the original endurance
limit could be withstood without failure. This effect,
therefore, seems to be a characteristic one for all materials
which have been subjected to fatigue stresses.

Hatt states the following conclusions drawn from the
Purdue tests:

1. (a) For 28-day tests: No definite endurance limit between 40
and 60 per cent of that static load required to break the beam under
a single application can be assigned to mortar of this age.

(6) For 4-month test: The load at the endurance limit is approxi-
mately 50 to 55 per cent of the static breaking load.

(c) For tests over 6 months : The load at the endurance limit is 54 to
55 per cent of the static breaking load.

2. The endurance limit does not seem to differ materially for beams
under continuous fatigue loading from that for beams under fatigue
loading with short rest periods.

3. The number of reversals of stress necessary to cause failure decrease
in a proportion to the respective increase of the percentage of static
load above the endurance limit.

4. Stresses above the endurance limit cause continual progressive
deformation.

5. Stresses below the endurance limit may cause progressive deforma-
tion for short periods with a tendency to become constant or to decrease
with continued loading.

6. The endurance limit may be raised by repeatedly stressing below
55 per cent of the static breaking load.

7. The amount of recovery in deformation seems to depend somewhat
upon the length of rest period.

8. Plastic set in fatigue is more pronounced in mortar of early age.
A sufficient rest period may reduce the plastic set to zero.

Tests by Mehmel. — The investigation of Probst (see
p. 269) was supplemented by important investigations by
Mehmel.^ He used compression specimens 7 by 7 cm. in
cross-section and 28 cm. long subjected to stresses from a
lower limit near zero to an upper limit. The concrete mix
was a 1 :6 gravel mix, and the water-cement ratio was 0.63.

1 Mitt. Inst. Beton Eisenbeton an der Tech. Hochs., Karlsruhe, 1926.

FATIGUE OF CEMENT AND CONCRETE 277

Careful measurements of deformation were made on a 20-
cm. gage length with Martens' mirror apparatus. The
repeated-stress testing was done at a rate of 60 cycles per
minute, and the average age of the specimens was about 1
year.

Tests Which Did Not Cause Failure. — A certain specimen
was loaded in compression with a minimum stress of 114
and a maximum of 704 lb. per square inch, the upper limit
being 29.5 per cent of the ultimate static strength deter-
mined from similar specimens. At various intervals, read-
ings of deformation Were taken so that stress-deformation
and deformation-cycles graphs could be drawn.

The original stress-deformation curve was convex upward,
and even after 10 cycles of stress the deformation had been
increased. The deformation-cycles graph with increase of
cycles became an inclined straight line, and the increase of
deformation was then so slow that many cycles of stress
were necessary to make the increase apparent. This was
true for the elastic as well as the permanent deformations.
The elastic deformation, corresponding to the maximum
stress, was 0.000158 at 10 cycles, increased to 0.000178 at
150,000 cycles, and then remained constant up to 610,000
cycles. The permanent set was 0.00001 at 10 cycles, in-
creased to 0.000086 at 400,000 cycles, and then remained
constant up to 610,000 cycles. The elastic deformation had
therefore increased 12,6 per cent, and the permanent set 760
per cent. The ratio of permanent set to elastic deforma-
tion was 0.0633 at 10 cycles, and 0.477 at 400,000 cycles.

The repeated loadings were discontinued after about
610,000 cycles, and the specimen was subjected to a static
test. The stable condition which had been reached by
the deformations was indicated by the value of modulus of
elasticity which was constant not only up to the maximum
repeated stress of 704 lb. per square inch, but even up to
803 lb. per square inch. At 917 lb. per square inch a
permanent set was developed, and at the same time the
modulus of elasticity decreased. Above the value of 704
lb. per square inch the stress-deformation curve was steeper

278

THE FATIGUE OF METALS

than for similar specimens which had not been subjected to
repeated stress. The specimen had evidently been strength-
ened by the repeated stressing even for stresses higher than
that used in the fatigue test. Continuing the static test to
destruction, it was found that the ultimate strength was
practically the same as for specimens which had not been
subjected to repeated stresses.

Tw^o other specimens were subjected to repeated com-
pression with the same minimum stress of 114 lb. per square
inch as before, but with maximum stresses of 790 and 1,138
lb. per square inch, respectively; which stresses were 37.5
and 47 per cent, respectively, of the ultimate static strength.

For the specimen stressed to an upper limit of 790 lb. per
square inch, stable conditions were reached at 169,000

0.0003

•5 § 0.0002
-5 ^

IXJ Q>

^ 0,0001

I 10 I02 10^ lO'^

Number of Cycles of S+ress

-Progressive elastic deformation of concrete specimen under repeated
compression. (Mehmel, at Karlsruhe.)

10-

10*

Fig. 99.

cycles for the elastic deformations, and at 260,000 cycles
for the permanent deformations. The ratio of permanent
to elastic deformation changed from 0.0128 to 0.208 in
going from 10 to 260,000 cycles, and the elastic deforma-
tion increased about 20 per cent.

For the specimen stressed to an upper limit of 1,138 lb.
per square inch, stable conditions were reached at 200,000
cycles for the elastic deformations, and at 400,000 cycles
for the permanent deformations. The ratio of permanent
to elastic deformation changed from 0.10 to 0.667 in going
from 10 to 200,000 cycles, and the elastic deformation
increased 24 per cent.

Figure 99 shows the logarithmic plot for these three
specimens, using unit elastic deformation as ordinates

FATIGUE OF CEMENT AND CONCRETE

279

and number of cycles as abscissae. Figure 100 shows both
the elastic deformations and the permanent deformations
plotted against number of cycles, using Cartesian coordi-
nates. In both curves it is evident that when stable
conditions are reached, the deformation line approaches a
horizontal line as asymptote.

For the three specimens above mentioned, which were
stressed to 29.5, 37.5, and 47 per cent, respectively, of the
static ultimate strength, the elastic deformation increased
during the period of repeated stressing 13, 20, and 24 per
cent, respectively, starting with a base of 10 cycles.

0.0004

200 300 400

Number of Thousands of Cijcles of S+ress

Fig. 100. — Progressive elastic and permanent deformation of a concrete specimen
under repeated compression. {Mehmel.)

A study of the relation between unit stress and unit
deformation under repeated stressing, showed that the
modulus of elasticity is not merely a function of unit stress,
but is dependent on the method of applying the stress, the
number of times the stress is applied, the pi'evious history
of stressing, and other factors. When a specimen has been
repeatedly stressed many times, it is possible for it to become
elastic within the limits of stress to which it has been
subjected, and within these limits Hooke's law is valid.
This condition may be reached for all stresses from zero up
to the endurance limit.

In this connection the authors of this book wish to point
out the decrease in modulus of elasticity which the repeated

280 THE FATIGUE OF METALS

stressing produced. Two specimens which were stressed
in a static test to 29.5 per cent of the ultimate strength,
had a secant modulus of elasticity based on elastic deforma-
tion, which varied from about 4,840,000 to 4,420,000 lb.
per square inch, showing that the stress-deformation curve
was not a straight line. A similar specimen after many
cycles of repeated stress had a modulus of 3,960,000 lb. per
square inch, a value about 85 per cent of the above.

Similarly, two specimens subjected to 47 per cent of the
ultimate in a static test showed a modulus of elasticity
varying from 4,730,000 to 4,280,000 lb. per square inch. A
similar specimen after many cycles of repeated stress had a
modulus of elasticity of 3,480,000 lb. per square inch, a
value of 77 per cent of the above.

This last specimen reached a stable condition for defor-
mations after 400,000 cycles. After 450,000 cycles had
been applied, the specimen rested for 36 hours, and during
this time the permanent set diminished from 0.000022 to
0.0000179; but the elastic deformation was not influenced.
At 600,000 cycles the previous stable condition had again
almost been reached. Evidently, therefore, the change in
permanent deformation was temporary; but it seems ques-
tionable whether there is ever established a condition of
stability in the strict sense of the word.

Tests Which Caused Failure. — In order to study the
behavior of concrete under repeated stress which would
finally cause failure, a specimen in Mehmel's tests was
stressed from 114 to 1,990 lb. per square inch, or about
80 per cent of the static ultimate strength. The first
period of stressing consisted of 1,470 cycles. The first
few loadings gave a stress-deformation curve convex
upward, but even after 20 cycles this had become concave
upward; and as the number of cycles increased, this con-
cavity upward increased, the curve being steeper at the
higher values of stress. This effect is shown in Fig. 101
in which the stress-deformation curves are drawn after
various numbers of cycles of stress had been applied.
In other words, for the first loading the increment of defor-

FATIGUE OF CEMENT AND CONCRETE

281

mation for a small increment of stress was greater at the
higher stresses than at the lower, but after repeated loading
this condition was reversed and the increment of deforma-
tion was greater for the lower stresses than for the higher.
During this period of stressing the elastic deformation
increased about 31 per cent, and the permanent deformation
increased about 244 per cent.

During the second period of stressing of this specimen the
maximum stress was decreased to 1,138 lb. per square inch,
a reduction of 43 per cent. The shape of the stress-deforma-
tion curve remained the same as in the first period of stress-
ing, and after about 800,000 cycles a condition of stability

Fig. 101.-

Ohs subdivision

=0.0001 unii -deforrnaiion

-Stress-strain diagrams for concrete specimen after cycles of com-
pression. (Mehmel.)

was reached by the elastic deformation but not by the
permanent deformation. A previous specimen had been
stressed to the same maximum stress as in this case, and
it seemed that the stress was below the endurance limit ;
but in the present case the first period of stressing to
80 per cent of the ultimate static strength had evidently so
weakened the specimen that even 1,500,000 cycles of lower
stress were not sufficient to produce stable conditions in
both the permanent and the elastic deformations.

After the second period of stressing of about 1,490,000
cycles, the maximum stress was again increased to 1,990
lb. per square inch. The character of the stress-deforma-
tion curve remained the same, but both the elastic and
permanent deformations increased. The third period of
stressing consisted of 1,460 cycles.

282 THE FATIGUE OF METALS

In the fourth period of stressing, the maximum stress
was again decreased to 1,138 lb. per square inch. After
86,000 cycles, occurred a period of rest of 12 days, during
which time the total permanent deformation decreased
about 14 per cent. The specimen was so near failure, that
this rest period did not affect results to any extent.

During the fifth period of stressing the maximum unit
stress was again increased to 1,990 lb. per square inch.
After only a few loadings the deformations increased rapidly
and in 588 cycles after the rest period the specimen failed.
Failure was preceded by a soft crackling noise which gave
sufficient warning so that the mirror apparatus could be
removed before complete destruction occurred.

A specimen was next stressed to a maximum of 1,848 lb.
per square inch, or about 70 per cent of the ultimate static
strength. The action in this case was much like that in
the previous specimen, similar stress-deformation curves
being determined which were concave upward. The test
showed very clearly that for an increment of stress at the
lower limit of the stress cycle the elastic deformation
increased from the beginning to the end of the test. For the
same increment of stress at the upper limit of the stress
cycle the elastic deformation decreased shghtly at first
with increase of cycles, and then increased very slightly
during the test, and slightly more just before failure.
For the purposes of this computation Mehmel employed a
range of stress from zero to 426 and from 1,422 to 1,848 lb.
per square inch. The specimen finally failed after about
341,000 cycles.

The character of the fracture of concrete failing under
repeated stress was found to be similar to that observed
in static tests.

The next test was one in which the maxium and min-
imum stresses were high but in which the range of stress was
low. The maximum stress was 1,848 and the minimum
1,422 lb. per square inch, thus stressing the specimen from
54 to 71 per cent of the ultimate static strength. The
deformation measurements soon showed that failure need

FATIGUE OF CEMENT AND CONCRETE 283

not be feared, and, indeed, the specimen withstood 1,500,000
cycles without failure. The specimen was then tested to
failure under a static load, and it was found that the repeated
stresses had reduced the static ultimate strength about 18
per cent. The curves showed that the permanent deforma-
tion had not reached a stable condition at the end of the
fatigue test, and it seems probable (to the writers of this
book) that failure would have occurred under continued
cycles.

In order to determine the ratio of endurance limit to
static ultimate strength, various other tests were carried
out beside those already mentioned. Two specimens were
tested with a maximum stress of about 60 per cent of the
static ultimate. After withstanding 1,500,000 cycles with-
out failure, these specimens were broken under static load,
and it was found that their ultimate static strength had
been reduced by 8.3 and 10.3 per cent, respectively.

Mehmel concluded that as long as the maximum unit
stress remained within a certain limit, it was possible for
the deformations to attain a condition of stability. When
stability was attained and when, furthermore, the ultimate
static strength had not been decreased by the repeated
stresses, it was concluded that the maximum stress lay
below the endurance limit. Under these conditions the
specimen subjected to repeated stresses showed a linear
relation between unit stress and unit deformation up to
the applied maximum stress and even higher.

When the maximum unit stress was increased, the stress-
deformation curve which at first was convex upward,
became a straight line, and next became concave upward.
Using a constant increment of stress of about 300 or 400
lb. per square inch, it was found that at the maximum stress
the total elastic deformation corresponding to this stress
decreased with increased cycles, while for the same incre-
ment of stress at the minimum stress of the stress cycle the
elastic deformation increased, so that the ratio e„^/e„i„
became less than unity. Up to rupture this ratio became
smaller for increased number of cycles. Figure 102 shows

284

THE FATIGUE OF METALS

the variation in this ratio for a certain specimen which failed
under fatigue.^

Bauschinger and Bach had previously found in tests of
various natural rocks that the stress-deformation curve
was concave upward, either from the origin or after a
certain unit stress was reached. Bach subjected speci-
mens of marble to a small number of repetitions of loading,
and found that after the fifth loading the deformations for a
constant increment of stress were larger for the lower
stresses and smaller for the upper stresses than they had
been for the first loading.

Since the aggregate for concrete has the same elastic
properties that natural rocks have, and since neat cement

1.5

1.0

E £
^ Ic

0.5

M

^-4i.

^

■o

-— .

100 200 300

Number of Thousomols of Cycles of S+ress

Fig. 102. — Progressive variation of the ratio emax/emin. (Mehmel.)

is known to give a stress-deformation curve which is convex
upward, Mehmel concluded that concrete gets its elastic
character from the cement content. Under repeated
stresses, however, when these are sufficiently high, the
elastic properties of the aggregate come into play.

The cement, therefore, is the first of the component
ingredients of concrete which becomes fatigued ; it takes on
less and less of the elastic stress energy, and becomes
plastic and acts merely as a binder, so that the curve of
elastic deformations is determined mostly or entirely by
the aggregate.

1 In this figure the Cmax refers to the elastic deformation corresponding to
the increment of stress from 1,422 to 1,848 lb. per square inch, and the Cmin
refers to the elastic deformation corresponding to the same increment of
stress from 0 to 426 lb. per square inch.

FATIGUE OF CEMENT AND CONCRETE 285

Mehmel concludes, therefore, that the shape of the defor-
mation diagram is a criterion Of the fatigue of concrete.
A weakening of the material shows itself by the reversal
of the stress-deformation curve from convex upward to
concave upward, and the weakening is greater according
to the amount of this reversal. This may be expressed by
saying that a weakening of the concrete has occurred when
the ratio of e^^/e,^ (meaning the same as on p. 284) becomes
smaller than unity, and is greater according as this ratio
becomes smaller. The ratio e^^Je,^ = 1 may be looked upon
as the critical value of the effect of repeated stresses on
concrete. Concrete will fail under repeated stresses when
the cement is so affected that it loses its ability to store up
stress energy. The tests show that concrete can probably
withstand a certain small number of repeated stresses
above the endurance limit without becoming fatigued,
provided that the ratio of e^„/e^;„ remains equal to unity.

Mehmel concluded from his tests that the determination
of the fatigue of concrete is measured much better by the
progress of the curve of elastic deformations than by the
curve of permanent deformations. He concluded also that
the endurance limit of concrete subjected to stresses from
zero to a maximum value lay between 47 and 60 per cent of
the ultimate static compressive strength.

Some measurements were made of the increase of tem-
perature of concrete specimens subjected to repeated
stresses. These measurements were made with thermo-
couples, and the test showed that the temperature rise with
increasing cycles of stress gave a curve similar to the curve
of permanent deformation as determined with the mirror
apparatus. Mehmel concludes that the thermal measure-
ments may prove useful for the determination of deforma-
tion changes shortly before the failure of a specimen, when
the mirror apparatus would have to be removed.

Summary of Conclusions. — The following is a summary
of the conclusions which Mehmel arrived at from the results
of his investigations:

286 THE FATIGUE OF METALS

1. The actions in a stress-deformation diagram of con-
crete, which at first are not reversible, may become revers-
ible through repeated loadings, but only so long as a
certain critical stress is not exceeded.

2. After repeated stressing the total deformation, the
elastic deformation, and the permanent deformation all
increase. The increase in the permanent set is considerably
greater than the increase in elastic deformation, so that
the ratio e-p/ee (in which ep = permanent and e^ = elastic
deformation) increases.

3. A stable condition is reached first for the elastic and
considerably later for the permanent deformation.

4. The attainment of stability is earlier or later depend-
ing on the absolute and relative increase in the permanent
and elastic deformation, and the increase in the quotient
Cp/ee is greater the higher the upper limit of stress.

5. The conception of stability is limited in the case of
concrete, a completely stable condition being apparently
not attainable. Elastic after-working occurred even after
100,000 cycles had not changed the deformations. The
stress-deformation curve was subject to change within a
given strip covering the stress-deformation line, but this
change affected only the permanent deformation.

6. A comparatively small number of cycles have the
effect of changing the original curved stress-deformation
curve to a straight line. The further cycles necesssary
to produce a stable condition do not influence the form but
only the position of the stress-elastic-deformation curve.
This curve, with increase of cycles, undergoes a turning
about the origin of coordinates.

7. The repeated cycles (within the critical stress) produce
a strengthening effect on the material above the upper limit
of the applied stress. The effect is only local, for the ulti-
mate static strength is not changed thereby. The elastic
properties of the concrete are changed, in a manner analo-
gous to steel under repeated stresses or under cold working,
so that the material is more favorably situated to withstand
loads ''gradually" applied. In relation to impact loading

FATIGUE OF CEMENT AND CONCRETE 287

two opposing influences may be noted: a decrease in the
capacity for plastic deformation, and a decrease in the
modulus of elasticity.

8. When the applied stress exceeds a certain critical value,
the specimen becomes fatigued sooner or later. The
cement is the first ingredient of the concrete to become
fatigued. The stress-elastic-deformation curve becomes
concave upward, so that the ratio e^^Je^^ (meaning the same
as on p. 284) becomes less than unity. A stress-deforma-
tion curve to which this criterion applies is called the fatigue
curve (see Fig. 101) . The degree to which concrete has been
fatigued is measured by the smallness of the ratio e^,^/e^;„.
A condition of stability is not reached, and the deformations
increase up to the time of failure. The increase in the elas-
tic deformations is such that the ratio e^^/e^-,^ decreases and
the weakening of the material proceeds in the lower stress
increments.

9. The character of the curve between stress and elastic
deformation is the criterion of fatigue, and not the per-
manent deformation.

10. All typical results of repeated stressing occur at a
slower rate according to the intensity of the unit stress.
The intensity is dependent on the upper and lower limits of
stress and on the rapidity of application of cycles, in such
manner that the intensity increases with an increase in the
upper limit of stress and with an increase of the time inter-
val per cycle. (The latter may not be true for extremely
rapid applications of stress.)

11. The endurance limit for stresses with the lower limit
zero, determined by this investigation, is identical with the
critical stress mentioned above. It lies between 47 and
60 per cent of the ultimate static compressive strength.
The factor of safety under repeated stresses is therefore
between one-half and three-fifths of the factor of safety
for steady loads.

12. It is possible to measure the thermal effects which
result from stress-deformation phenomena in concrete.

288 THE FATIGUE OF METALS

This method seems to be adapted to the study of the behav-
ior of the specimen shortly before failure.

Summary on Fatigue of Concrete. — The results on the
fatigue of concrete which have been reviewed in this chapter
indicate that concrete under repeated stresses is very similar
in its behavior to metals under the same conditions of stress.

While some of the conclusions which follow . must be
looked upon as tentative until verified by further experi-
ments, yet the indications are such that the following sum-
mary of the important phenomena may be regarded as
fairly well established:

1. Concrete will fail under repeated loads at unit stresses
which are much less than the ultimate static strength.

2. When the unit stress to which concrete is subjected
in fatigue is decreased, the number of cycles for rupture is
increased.

3. For concrete, as for metals, as the maximum limiting
stress in a cycle is increased, the minimum stress must be
increased algebraically if failure is not to occur. No
formulas for the effect of range of stress have been devel-
oped for concrete.

4. While tests of many millions of cycles of stress have
not been carried out on concrete, yet the indications are
that its endurance limit for cycles of stress ranging from
zero to a maximum (r = 0), is about 50 to 55 per cent of
the static ultimate strength, both for compression cylinders
and for beams.

5. Even when the cycle of stress (from zero to a maxi-
mum) is less than the endurance limit, a permanent set
occurs during the first few loadings. If, however, the
cycle of stress is such that the permanent set reaches and
maintains a constant value, then the indications are that
failure will not occur.

6. For stresses below the endurance limit concrete seems
to be able to adjust itself to the imposed cycles of stress, the
stress-deformation curve becomes a straight line, and the
stress can be withstood indefinitely.

FATIGUE OF CEMENT AND CONCRETE 289

7. In the above process of adjustment to a cycle of stress
the modulus of elasticity also reaches and maintains a
constant value.

8. Stresses above the endurance limit cause progressive
deformation and final failure.

9. Periods of rest seem to have only a temporary effect
on the recovery from deformation, and do not seem to
change the endurance limit.

10. Stressing concrete below the endurance limit increases
its strength, just as is the case with metals.

11. In order that the effect of increase of strength with
age shall not seriously affect the factors being investigated
in fatigue tests of concrete, it is necessary that tests be
made on concrete which has an age of 6 months or greater.

APPENDIX A

BIBLIOGRAPHY

In the preparation of this bibhography the authors
acknowledge their indebtedness to the bibhography
reported in 1913 by Mason in the Reports of the British
Association for the Advancement of Science, and to that by
Mailander reported in Stahl und Eisen of May 22, 1924.

This bibhography has been made inclusive rather than
selective, although the authors realize that, in all prob-
ability, they have overlooked many important contri-
butions. The reader is advised to pay especial attention
to the dates of the articles enumerated, if he wishes to look
up the results of the more modern investigations.

This bibliography is divided into three sections: Metals,
Wood, and Concrete.

FATIGUE OF METALS

AiTCHisoN, L., "Valve Failures and Valve Steels in Internal Combustion
Engines," Proc. Brit. Inst. Auto. Eng., vol. 14, 1919.

, "Engineering Steels, " Mac don ald and Evans, Chap. IV, 1921.

, ' 'Discussion on Automobile Steels, " Proc. Brit. Inst. Auto. Eng.,

p. 495, 1921.

, "The Low Apparent Elastic Limit of Quenched or Work-
hardened Steels," Brit. Iron Steel Inst., Carnegie Scholarship Mem.,
vol. 12, p. 113, 1923.

, "Materials in Aircraft Construction," Engineering {London),

vol. 117, p. 89, 1924.

and Jamison, "Effect of Grain Size on the Fatigue Strength

of Steel," Jour. Brit. Iron Steel Inst., vol. Ill, p. 351, 1925; also
Engineering {London), p. 585, May 8, 1925.

Albert, C. D., "Factors of Safety and Allowable Stress," Am. Machin-
ist, vol. 57, p. 54, 1922.

Andrews, T., "Microscopic Internal Flaws Inducing Fracture in Steel,"
Engineering {London), July, 1896.

Antisell, F. L., "Relation of Physical and Chemical Properties of
Copper," Trans. Am. Inst. Mining Met. Eng., vol. 64, p. 432, 1921.

290

BIBLIOGRAPHY 291

Arnold, J. 0., "Dangerous Crystallization of Mild Steel and Wrought

Iron," Proc. Brit. Inst. Civil Eng., 154, Supplement, 1903.
, "Fracture of Structural Steel under Alternating Stress,"

Brit. Assoc. Rept., Sec. G, 1904, p. 688; also Science Abstracts,

1929&, 27956, 1904; The Engineer (London), p. 227, Sept. 2, 1904.
, "Factors of Safety in Marine Engineering," Trans. Brit. Inst.

Naval Arch., vol. 50, p. 260, 1908; also Engineering (London),

p. 565, Apr. 24, 1908.
, "The Mysteries of Metals," Engineering (London), Pt. I,

p. 170, 1909.
, "Ghost Lines in Steel Forgings," Proc. Brit. Inst. Mech. Eng.,

p. 653, 1915; also Engineering (London), Nov. 26, 1915.
and A. A. Read, "The Chemical and Mechanical Relations of

Iron, Tungsten, and Carbon; and of Iron, Nickel, and Carbon,"

Engineering (London), Mar. 27, 1914; also Proc. Brit. hist. Mech.

Eng., 1914.
and , "The Chemical and Mechanical Relations of Iron,

Cobalt, and Carbon," Engineering (London), Mar. 26, 1915.
and "The Chemical and Mechanical Relations of Iron,

Molybdenum, and Carbon," Engineering (London), Nov. 26, 1915.
Archbutt, see Rosenhain.
Archer, R. S., see Zay Jeffries.
Bailey, R. W., "Ductile Materials under Variable Shear Stress,"

Engineering (London), p. 81, July 27, 1917.
Bain, Edgar C, and Zay Jeffries, "Mixed Orientation Developed in

Crystals of Ductile Materials by Plastic Deformation," Chem.

Met. Eng., p. 775, Oct. 26, 1921.
Bairstow, L., "The Elastic Limits of Iron and Steel under Cyclical

Variations of Stress," Phil. Trans. Roy. Soc, vol. 210A, p. 35, 1910.

, "The Fatigue of Metals," Beama, vol. 11, p. 817, 1922.

and A. J. S. Pippard, "The Determination of Torsional Stresses

in a Shaft of Any Cross-section," Proc. Brit. Inst. Civil Eng., vol.

214, Pt. II, p. 291, 1921-1922.
Baker, Sir Benjamin, "Some Notes on the Working Stress of Iron and

Steel," Trans. Am. Soc. Mech. Eng., vol. 8, p. 157, 1886. (Work

summarized in Unwin's "Testing of Materials of Construction.")
Baker, F., "Report of Tests of Metals," abstract in Jour. Brit. Iron

Steel Inst. Pt. II, p. 768, 1905.
Basquin, 0. H., "The Exponential Law of Endurance Tests," Proc.

Am. Soc. Testing Materials, vol. 10, p. 625, 1910.
Batson, R. G., and J. H. Hyde, "Mechanical Testing," vol. 1, Chaps.

14-17, inclusive, 1922.
Bauschinger, J., "Die Veranderung der Elasticitatsgrenze und des

Elasticitatsmoduls verschiedener Metalle," Mitt. Mech.-Tech.

Lab. Kgl. Tech. Hochs., Miinchen, Heft 13; see also Dingier s poly-

292 THE FATIGUE OF METALS

tech. Jour., Bd. 224, and Civilingenieur, 1881; also Unwin's "Test-
ing of Materials."
-, "tjber die Veranderung der Elasticitatsgrenze und Festigkeit

des Eisens," Mitt. Mech.-Tech. Lab. Kgl. Tech. Hochs. Miinchen,
1886; see also Unwin's "Testing of Materials."

Beare, T. Hudson, "The Fatigue Limit and the Proportionality Limit
of Monel Metal," abstract in Engineering {London), p. 88, July
20, 1923.

Beckmann, H., "Die Lorenz'sche Theorie tiber die Flieskurven Fester
Korper," Machinenbau, Bd. 1, p. 578, 1921-1922.

Beilby, G. T., "The Hard and Soft States in Metals," Jour. Brit. Inst.
Metals, Pt. II, p. 5, 1911.

Bengough, G. D., "A Study of the Properties of Alloys at High Tem-
peratures," Jour. Brit. hist. Metals, Pt. I, p. 123, 1912.

Berger, Karl, "Elasticity of Cast Iron Subjected to Repeated Tensile
and Compressive Strains," abstract in Proc. Brit. Inst. Civil Eng.,
vol. 136, p. 370, 1898-1899.

Berliner, S., "Behaviour of Cast Iron under Slowly Alternating
Stress," Ann. phys., vol. 20, June 3, 1906; Science Abstracts, 1528,
1906.

Blount, B., W. G. Kirkaldy, and H. R. Sankey, "Tensile, Impact-
tensile, and Repeated Bending Tests of Steel," Proc. Brit. Inst.
Mech. Eng., Pt. II, p. 715, 1910; Science Abstracts, 300, 1911.

BouDOUARD, 0., "Tests on Metals by Study of the Damping of Vibra-
tions," Compt. rend., vol. 150, Mar. 14, 1910; Science Abstracts, 645,
1910.

, "Essai des Metaux par I'etude de I'amortissement des Mouve-

ments Vibratoires," Bidl. soc. encour. ind. nat., Pt. II, p. 545, 1910.

, "Tests of Metals by the Abatement of Vibrating Movements,"

Compt. rend., vol. 152, Jan. 3, 1911; also Science Abstracts, 295, 1911.

, "Fatigue des Metaux," Rev. metal., p. 70, 1913; also Stahl u.

Eisen, p. 1757, 1912.
-, "Breakdown Tests of Metals," Proc. Intern. Assoc. Testing

Materials, 1912, Congress Paper V3.
BouASSE, H., and Berthier, "Decay of Oscillations, Ann. chem. phys.,

Feb. 10, 1907; Science Abstracts, 710, 1907.
and L. Carriere, "Decay of Oscillations," Science Abstracts,

1225, 1908.
Brown, Boveri, et Cie., "Machine fiir Dauerversuche mit Turbinen-

schaufeln," Rev. metal, p. 259, 1921.
Burr, W. H., "The Elasticity and Resistance of Materials of Engineer-
ing," p. 795, New York, 1915.
Burrows, C. W., " Some Applications of Magnetic Analysis to the Study

of Steel Products," Proc. Am. Soc. Testing Materials, vol. 17, Pt. II,

p. 88; also U. S. Bur. Standards, Sci. Paper 272.

BIBLIOGRAPHY 293

Capp, J. A., and T. R. Lawson, "Thermoelectric Indication of Strain

as a Testing Method," Proc. Intern. Assoc. Testing Materials,

Congress Payer IX, 1912.
Carpenter, H. C. H., and C. A. Edwards, "Properties of the Alloys of

Aluminum and Copper," Proc. Brit. Inst. Mech. Eng., Alloys

Research Comm., Eighth Rept., p. 57, 1907.
Charpy, G., and J. Dura,nd, "A Reason for Rail Failures," Genie civil,

Oct. 18, 1919; also Iron Age, p. 331; Jan. 29, 1920.
CoKER, E. G., "Endurance of Steel Bars Subjected to Repetitions of

Tensional Stress," Proc. Brit. Inst. Civil Eng., vol. 135, p. 294,

1898-1899.
, "Effect of Low Temperature on Overstrained Iron and Steel,"

Phys. Rev., Aug. 15, 1902; also Science Abstracts, 227, 1903.
, "On the Measurement of Stress by Thermal Methods," Trans.

Roy. Soc. Edinburgh, vol. 41, p. 229, 1904.
, British Assoc. Rept., Sec. G, 1910; also Engineering (London),

p. 412, Sept. 16, 1910.
, "The Optical Determination of Stress," Phil. Mag., vol. 20,

p. 740, October, 1910.

, "Photo-elasticity," Engineering {London), p. 1, Jan. 6, 1911.

, "The Determination by Photo-elastic Methods of the Distribu-
tion of Stress in Plates," Trans. Brit. Inst. Naval Arch., 1911;

also Engineering (London), p. 514, Apr. 21, 1911.
, "Optical Determination of Stress," Engineering (London),

p. 325, Mar. 8, 1912.
, "Stress Distribution in Materials," Engineering (London),

p. 261, Aug. 21, 1914; also British Assoc. Repts., Sec. G, p. 490,

1914.
, "The Effect of Holes, Cracks, and Other Discontinuities in Ships

Plating," Trans. Brit. Inst. Naval Arch., Mar. 25, 1920; also
Engineering (London), June 18, 1920.

— and McKergow, "The Relation of Thermal Changes to Tension
and Compression Stress: Experiments on Impulsive Stress,"
Trans. Roy. Soc. Canada, vol. 10, 1924.

and W. A. Scoble, "The Distribution of Stress Due to a Rivet

in a Plate," Trans. Brit. Inst. Naval Arch., Pt. I, p. 207, Mar. 14,
191S; Engineering (London), p. 439, Mar. 28, 1913.

Corse, W. M., and G. F. Comstock, "Aluminum Bronze, Some Recent
Tests," Proc. Am. Soc. Testing Materials, vol. 16, Pt. II, p. 118,
1916.

CzocHRALSKi, J., "On the Principles of the Phenomena of Strain-
hardening," Zeit. Metallkunde, p. 7, January, 1923.

, "Displacement Theory and X-ray Investigation," Zeit. Metall-
kunde, p. 60, March-May, 1923.

294 THE FATIGUE OF METALS

Dalby, W. E., "Researches on the Elastic Properties and the Plastic
Extension of Metals," Phil. Trans. Roy. Soc, vol. 221A, p. 11, 1920.

, "Further Researches on the Strength of Materials," Proc. Roy.

Soc, vol. 103A, p. 8, 1922.

Desch, C. H., "Chemical Influences in the Failure of Metals under
Stress," Trans. Faraday Soc, p. 17, April, 1921.

■ — , "Brittleness and Fatigue in Metals," Trans. Inst. Eng. Ship-
builders of Scotla7id, vol. 65, p. 585, 1922.

Drago, E., "Influence of Oscillatory Discharge on the Decay of Tor-
sional Oscillations," Science Abstracts, 1423A, 1911, and 2A, 1912.

Dudley, C. B., "Alternate Bending Stresses," Iron and Steel Mag.,
p. 134, February, 1904.

Eden, E. M., "Endurance of Metal under Alternating Stress and Effect
of Rate of Alternation," Proc Phil. Soc. Univ. Durham {England),
3, 5, 1910; Science Abstracts, 1384, 1910.

Eden, Rose, and Cunningham, "The Endurance of Metals," Proc.
Brit. Inst. Mech. Eng., Pts. Ill and IV, p. 839, 1911.

Eloy, F., " Influence des Chocs Repetes a la Compression sur les Aciers,"
Rev. ind. minerale, Pt. II, p. 603, 1921.

Engineer {London), The, Staff articles:

1. "Shock and Fatigue," Pt. I, p. 451, 1920.

2. "Machine for Fatigue Tests," Pt. I, p. 550, 1921.

3. "The Haigh Alternating-stress Machine," Pt. II, p. 116, 1921.

4. "Failures of Locomotive Cranks and Axles," p. 1, Jan. 5, 1923.
Engineering {London), Staff articles:

1. "Working Stresses," Pt. I, p. 653, 1908.

2. "Impact Testing Machine," Pt. I, p. 572, 1910.

3. "Stresses and Strains," Pt. II, p. 669, 1910.

4. "Testing of Materials," Pt. I, p. 121, 1912.

5. "British Exhibition," Pt. IV, p. 153, 1919.

6. "Spring Scragging Machine," Pt. II, p. 242, 1920.

7. "Failure by Fatigue," Pt. I, p. 525, 1922.

8. "The Elastic Limit," Pt. I, p. 715, 1922.

9. "Stress and Strength," Nov. 30, 1923.

Ensslin, M., "Briiche an Gekropften Kurbeln und Vorbeugungsmasz-

nahmen," Machinenbau, Bd. II, p. 107, 1922.
Ercolini, G., "Effect of Deformation on Torsional Couple Exerted

by Twisted Wire," Science Abstracts, 1807, 1906.
, "Recent Experiments in Elasticity," Science Abstracts, 965,

1909.
EwiNG, Sir J. A., "On Hysteresis in the Relation of Strain to Stress,"

Brit. Assoc Rept., Sec. G, p. 502, 1899.
, "Effect of Strain on the Crystalline Structure of Metals,"

Brit. Assoc. Rept., Sec. G, p. 657, 1906.

BIBLIOGRAPHY 295

— and J. C. W. Humfrey, "Fracture of Metals under Repeated
Alternations of Stress," Phil. Trans. Roy. Soc, vol. 200A, p. 241,
1903.

— and W. RosENHAiN, "Experiments in Micro-metallurgy — Effects
of Strain," Proc. Roy. Soc, vol., 65, p. 85, 1899.

and , "Crystalline Structure of Metals," Phil. Trans.

Roy. Soc, vol. 193A, p. 353, 1900.
Fairbairn, Sir W., "The Effect of Impact, Vibratory Action, and

Changes of Load on Wrought-iron Girders," Phil. Trans. Roy.

Soc, vol. 154, p. 311, 1864.
Faraday Society, The, "The Symposium on Failure of Metals under

Internal and Prolonged Stress," annual publication for 1921.
Farmer, F. M., "A Fatigue Testing Machine," Proc Am. Soc Testing

Materials, vol. 19, Pt. II, p. 709, 1919; also Am. Machinist, Pt. II,

p. 271, 1919.
Fea, L., "Mechanical Tests of Special Steels for Naval Construction,"

Proc Intern. Assoc Testing Materials, 1912, Congress, Art. Ii.
FiLON, J., and E. G. Coker, "Experimental Determination of the

Distribution of Stress and Strain in Solids," Brit. Assoc. Rept.,

Sec. G, p. 201, 1914.
FiNLEY, W. H., "Case of Failure of Iron from Fatigue," Ejig. Neivs, 55,

p. 487, 1906; also Scie7ice Abstracts, 1200, 1906.
FoPPL, A., " Dauerversuche von Bauschinger 1886 bis 1893." Mitt.

■ Mech.-Tech. Lab. Kgl. Tech. Hochs., Munchen, Heft 25, 1897.
■ , "Dauerversuche mit Eingekerbten Staben," Mitt. Mech.-Tech.

Lab. Kgl. Hochs., Munchen, Heft 31, 1909; also Stahl u. Eisen,

p. 409, March, 1909.
■ , "Schwingungs Beanspruchung und Rissbildung in besondere

von Konstructions Stahlen," Oest. Ing. Arch. Verein, February,

1923.
-, " Versuchsanordung zur Bestimmung der Schwingungsfestigkeit

von Materialen," Machinenbau, Bd. 2, p. 1002, 1923.
Foster, F., "Repetition of Stress," Mech. Eng., Nov. 22, 1902; also

Science Abstracts, 866, 1903.
Fraichet, L., "Essai Magnetique des Aciers a la Traction, Limites

Elastiques," Rev. metal, vol. 20, p. 560, 1923.
Franke, W. J., "An Apparatus for Delicate Flexure Tests," Proc. Am.

Soc Testing Materials, vol. 20, Pt. II, p. 372, 1920.
Freminville, Ch., "Influence des Vibrations dans les Phenomenes

de Fragilite," Rev. metal, p. 109, 1906.
• — , "Caracteres des Vibrations Accompagnant de Choc," Rev.

metal, p. 833, 1907.
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Genie civil, Oct. 22 and Nov. 19, 1910.

296 THE FATIGUE OF METALS

, "Alternating Stresses," Compt. rend., vol. 168, p. 348, 1919;

also Jour. Brit. hist. Metals, vol. 21, p. 469, 1919.
Fulton, A. R., "Experiments on the Effect of Alternations of Tensile

Stress at Low Frequencies on the Elastic Properties of Mild Steel,"

Brit. Assoc. Repts., Sec. G, p. 484, 1919; also Engineering (London),

p. 65, Jan. 9, 1920.
Gardner, J. C, "Effects Caused by the Reversal of Stresses in Steel,"

Jour. Brit. Iron Steel hist., Pt. I, p. 481, 1905; also Science Abstracts,

1804, 1905.
Gerber, W., "Relation between the Superior and Inferior Stresses of a

Cycle of Limiting Stress," Zeit. Bayerischen Arch. Ing.-Vereins,

1874; also Unwin's "Elements of Machine Design," vol. 1, Chap.

II.
Gibson, W. A., "Fatigue and Impact Tests of Aluminum Alloys," Proc.

A7n. Soc. Testing Materials, vol. 20, Pt. II, p. 115, 1920; also Iron

Age, Pt. II, p. 96, 1920.
GiESEN, W., "The Special Steels in Theory and Practice," Jour. Brit.

Iron Steel Inst., Carnegie Scholarship Mem., Pt. I, p. 1, 1909.
Gilchrist, J., "Wohler's Theories on Material under Repeated Stress,"

The Engineer (London), vol. 90, p. 203, 1900.
GiLLETT, H. W., "Endurance Tests of Molybdenum Steels," abstract in

Am. Machinist, p. 760, Nov. 22, 1923.

, "Dirty Steel," abstract in Iron Age, p. 1330, Nov. 15, 1923.

and E. L. Mack, "Notes on Some Endurance Tests of Metals,"

Proc. Am. Soc. Testing Materials, vol. 24, Pt. II, p. 476, 1924.
and , "Molybdenum, Cerium, and Related Alloy Steels,"

The Chemical Catalog Company, New York.
GouGH, H. J., "Improvements in Methods of Fatigue Testing," The

Engineer (London), p. 159, Aug. 12, 1921.
, "Elastic Limits of Copper under Cyclic Stress Variation,"

Engineering (London), p. 291, Sept. 8, 1922.
, "Some Experiments on the Fatigue of Materials under Alter-
nating Torsion," Brit. Advisory Comm. Aero., Repts. and Mem. 743,

vol. 2, p. 471, 1921-1922.
, "The Effect of Keyways on the Strength and Stiffness of

Shafts," Brit. Aero. Research Comm., Repts. and Mem. 843.

, "The Fatigue of Metals," London, 1924.

and D. Hanson, "The Behaviour of Metals Subjected to

Repeated Stress," Proc. Roy. Soc. 104A, p. 538, July, 1923.
and , "The Behaviour of Single Crystals of Aluminum

under Static and Repeated Stresses," Phil. Trans. Roy. Soc, vol.

226A, p. 1, 1926.
and H. J. Tapsell, "Some Comparative Fatigue Tests," British

Assoc. Rept, Sec. G, 1924; Engineering (London), p. 286, Aug. 29,

1924.

BIBLIOGRAPHY 297

Grammel, R., "Neurere Versuche tiber Elastische Hysterese," Zeit.
Ver. Deut. Ing., vol. 58, p. 1600, Nov. 28, 1914; also Jour. Am. Soc.
Mech. Eng., February, 1915.

Greaves, R. H., "Recovery of Elasticity by Iron and Steel after Over-
strain," R. D. Rept. 54, Research Dept., Woolwich Arsenal, 1922.

Green, C. W., "The Mechanism of the Failure of Steel upon and after
Hardening," Trans. Faraday Soc, p. 139, April, 1921.

Grenet, L., "Calcul du Travail de Choc, etc.," Rev. metal., p. 835, 1909.

Griffith, A. A., "Phenomena of Rupture and Flow in Solids," Phil.
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, "The Impressed Conditions of Fatigue Tests," British Assoc.

Rept, Sec. G, p. 325, 1924.
-, "Stress Concentrations in Theory and Practice," Brit. Assoc

Rept, Sec. G, 1921.
Grimaldi, G. and G. Accolla, "Influence of Oscillatory Discharge

and Magnetization on Elastic Hysteresis," Science Abstracts, 927 A,

1905, and 376, 1910.
Grimme, J., "Merkwiirdige Brucherscheinungen bei Eisenstaben,"

Zeit Ver. Deut Ing., p. 603, 1921, and p. 148, 1922.
Guest, J. J., "Strength of Materials under Combined Stress," Phil.

Mag., p. 69, July, 1900.
■ and Lea, "Torsional Hysteresis of Mild Steel," Proc. Roy. Soc,

vol. 93A, p. 313, 1916-1917.
GuiLLET, L., "Intervention de I'Amortissement dans I'Essai des Fers,"

Rev. metal., p. 885, 1909, also Stahl u. Eisen, p. 956, June, 1909.
■ , "Nouvelles Experiences de Chocs Repetes," Rev. metal., p. 755,

1921.
, "Quelques Essais aux Chocs Repetes," Rev. metal., p. 96, 1921.

Gulliver, G. H., "Internal Friction in Loaded Materials," Intern.

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1912; 224, 1910; 1189, 1910; Arch. Sciences, 26, pp. 136 and 263,

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Hahnemann, Hecht, and Wilckens, "Eine Neue Materialprufunngs-

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1915; also Engineering {London), Pt. II, p. 379, 1915.
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1917; also Jour. Brit Inst Metals, p. 55, Sept. 19, 1917.
— , "The Strain Energy Function and the Elastic Limit," Brit

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298 THE FATIGUE OF METALS

, "Elastic and Fatigue Fracture in Metals," abstract in Metal

Ind. (London), vol. 21, p. 466, 1922.
, "Thermodynamic Theory of Mechanical Fatigue and Hysteresis

in Metals," Rept. Comm. on Stress Distribution, Brit. Assoc, Sec. G,

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, "The Endurance of Metals under Alternating Stress," Jour.

West Scot. Iron Steel Inst., vol. 23, 1915-1916.
, "Slag Inclusions in Relation to Fatigue," Trans. Faraday Soc,

p. 153-157; 1924.
-, and A. Beale, "The Influence of Circular Holes on the Fatigue

Strength of Hard Steel Plates," Brit. Assoc. Rept., Sec. G, p. 326,

1924.
Hankins, G. a., "Alternating Stress Tests of Aluminum Alloys," Brit.

Advisory Comm. Aero., 12, 1917.
• , "Properties of Nickel in Fatigue," Brit. Aero. Research Comm.,

Repts. and Mem., 789, November, 1921.
-, "Commercially Pure Nickel as a Standard Material for Fatigue

Investigations," Brit. Advisory Comm. Aero., vol. 2, p. 414, 1922-

1923.
Hanson, D., " Intercrystalline Fracture in Steel," Trans. Faraday Soc,

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p. 197, 1923.
Harsch, J. W., "Heat Treatment and Strength of Steel under Repeated

Stress," Forging and Heat Treating, vol. 9, p. 57, 1923.
Hatfield, W. H., "Mechanical Properties of Steel," Engineering

{London), p. 615, May 9, 1919.
, "The Most Suitable Steel for Automobile Parts," Proc Brit.

Inst. Auto. Eng., vol. 14, 1920.
■ — , "Steel from the Standpoint of Marine Engineering," Jour.

West Scot. Iron Steel Inst, vol. 28, p. 52, 1921.
— ■ — — , "Further Notes on Automobile Steels," Proc. Brit. Inst. Auto.

Eng., p. 465, 1921.
-, "The Mechanism of Failure of Metals from Internal Stress,"

Trans. Faraday Soc, p. 36, April, 1921.
Heathecore, H. L., and C. G. Whinfrey, "Tearing Tests of Metals,"

Chem. Met. Eng., vol. 27, p. 310, 1922; also Stahl u. Eisen, p. 890,

1923.
Heaton, T. T., "Electric Welding," Engineering {London), Pt. I, p. 153,

1919.
Heyn, E., "Kerbwirkung und irhre Bedeutung fiir den Konstrukteur,"

Zeit. Ver. Dent. Ing., p. 383, March, 1914.
, "Einige Fragen aus dem Gebiete der Metallforschung," Metall.

u. Erz., Bd. 15, pp. 411, 436, 1918.

BIBLIOGRAPHY 299

HoNNEGER, E., "Das Verbulten Meclianisch Beanspruchter Metalle,"
Eisenbau, March, 1922, and following issues.

HoPKiNSON, B., "Effects of Momentary Stresses in Metals," Proc. Roy.
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300 THE FATIGUE OF METALS

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BIBLIOGRAPHY 301

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302 THE FATIGUE OF METALS

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304 THE FATIGUE OF METALS

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BIBLIOGRAPHY 305

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306 THE FATIGUE OF METALS

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308 THE FATIGUE OF METALS

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, "Versuch iiber die Spannungsverteilung in Gelochten Zug-

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, "Versuche tiber die Spannungsverteilung in Gekerbten Zug-

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, "Die Praktische Nutzandwendung der Prlifung des Eisens

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, "Die Festigkeit von Schweiszeisen gegen Stossbeanspruchung,"

Stahl u. Eisen, p. 1207, July, 1914.
-, "Kerbwirkung bei Dauerschlagbeanspruchung," Zeit. Ver.

Deut. Ing., p. 701, May, 1914; also Stahl u. Eisen, p. 1744, Novem-
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Primrose, H. S. and J. S. G., "Some Useful Testing Machines,"

Engineering {London), Pt. II, p. 387, 1918.
Prichard, H. S., "The Effects of Straining Structural Steel and

Wrought Iron," Trans. Am. Soc. Civil Eng., vol. 80, p. 1429, 1916.
, "Overstrain and Fatigue Failure of Steel," Eng. News-Record,

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Rasch, E., "Method for Determining Elastic and Critical Stresses in

Materials by Means of Thermo-electric Measurements," Proc.

Intern. Assoc. Testing Materials, 1909 Congress, Art. VII.
Rawdon, H. S., "The Presence of Internal Fractures in Steel Rails

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also Rept. 85 to Rail Committee, American Railroad Association,

and Trans. Faraday Soc, April, 1921.

BIBLIOGRAPHY 309

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310 THE FATIGUE OF METALS

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312 THE FATIGUE OF METALS

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BIBLIOGRAPHY 313

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314 THE FATIGUE OF METALS

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BIBLIOGRAPHY 315

Weyrauch, J., "On the Calculations of Dimensions as Depending on the

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Whyte, S., Discussion, Proc. Brit. Inst. Auto. Eng., vol. 15, p. 502, 1921.
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1920.
Wilson, J. S., and B. P. Haigh, "The Influence of Rivet Holes in Steel

Structures," Eyigineering {London), p. 309, Sept. 8, 1922; also

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861, 1923; also Stahl u. Eisen, p. 1109, 1923.

FATIGUE OF WOOD

Stanton, T. E., "Resistance of Wood to Stress Reversals," Engineering
{London), p. 605, June 23, 1916.

FATIGUE OF CONCRETE

Berry, H. C, "Some Tests of Reinforced Concrete Beams Under Oft-
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Clemmer, H. F., "Fatigue of Concrete," Proc. A??i. Soc. Testing Mate-
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Genie ciM, Staff article, "Machine a Essayer les Poutres en Beton Arme
a la Flexion Alternative," Pt. II, p. 446, 1909.

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Druckbeanspruchungen auf Druckelastizitat und Druckfestigkeit
von Beton," Mitt. Inst. Beton Eisenbeton an der Tech. Hochs.,
Karlsruhe, 1926.

National Physical Laboratory, "Reinforced Concrete Research,"
Concrete and Constr. Eng., p. 517, July, 1913; also Batson and
Hyde, "Mechanical Testing," vol. II, p. 231, 1922.

316 THE FATIGUE OF METALS

Older, C, "Highway Research in IlUnois," Trans. Am. Soc. Civil
Eng., p. 1180, 1924.

Probst, E., " Untersuchungen liber den Einflusz wiederholter Belast-
ungen auf Elastizitat und Festigkeit von Beton und Eisenbeton,"
Zeit. Tech. Hochs., Karlsruhe, 1925.

Slater, W. A., G. A. Smith, and H. P. Mueller, "Effect of Repeated
Reversals of Stress on Double-reinforced Concrete Beams," U. S.
Bur. Standards, Tech. Paper, 182, 1920.

Van Ornum, J. L., "The Fatigue of Cement Products," Trans. Am. Soc.
Civil Eng., p. 443, 1903.

, "The Fatigue of Concrete," Trans. Am. Soc. Civil Eng., p. 294,

1907.

Williams, G. M., "Some Determinations of the Stress-deformation
Relations for Concretes Under Repeated and Continuous Load-
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Withe Y, M. 0., "Tests of Bond between Concrete and Steel in Rein-
forced Concrete Beams," Univ. Wisconsin, Bidl. 321, 1909.

AUTHOR INDEX

Aeronautical Research Comm. (Brit-
ish), 203
Air Service (U. S.), see Moore, R. R.
Aitchison, 198, 204
Albert, 10
Archer, 33

Fidler, 177

Forest Products Lab. (U. S.), 244,

247
French, 50, 155

G

Gerber, 174
Gilchrist, 24
B Gillett, 117, 136, 195, 197, 205

Goodman, 24, 177
Bairstow, 20, 24, 35, 41, 48, 51, 53, Gough, 26, 30, 38, 44, 45, 54, 146,

54, 183, 199
Baker, 15
Barr, 175, 179
Basquin, 25
Batson, 86, 107

Bauschinger, 15, 58, 149, 221, 271
Beilby, 32, 74
Bengough, 34
Berry, 260
Budgen, 49, 57

C

Clemmer, 265
Coker, 8, 206, 212
Cunningham, 199, 202

Deitz, 115, 212

D

E

Eden, 199, 202
Engineering Foundation, 26
Ewing, 18, 23, 27, 30, 35, 64

F

Fairbairn, 11
Farmer, 95

148, 149, 150, 179, 213
Greenwich, see Haigh.
Griffith, A. A., 8, 71, 75, 195, 206,

207, 213
Guest, 37, 187
Gustafsson, 110

H

Haigh, 24, 89, 113, 132, 141, 185,

186, 212, 214
Hankins, 55, 213
Hanson, 26, 30, 38, 44, 45, 54,

213
Harsch, 28, 149
Hatt, 272, 273
Hopkinson, 24, 35, 89
Howard, 24
Howell, 186
Humfrey, 18, 23, 30, 64

Illinois Div'n of Highways, 265
Illinois, Univ. of, see Jasper, Kom-

mers, Moore, H. F.
Inglis, C. E., 206, 207
Inglis, N. P., 222
Irwin, 114, 141

317

318

THE FATIGUE OF METALS

James and Galton, 11

Jasper, 26, 31, 57, 84, 91, 130, 132,
136, 138, 144, 147, 155, 158,
183, 185, 187, 188, 197, 205, 210,
217, 220

Jeffries, 33

Jenkin, 151

Joffe, 77

Johnson, J. B., 24, 175, 179

K

Morley, 244
Muir, 31, 46
MuUer, 68, 158

N

National Physical Lab. (British), see

Bairstow, Gough, Hanson,

Rosenhain, Stanton.
National Research Council (U. S.),

26
Naval Eng'g. Exp. Sta. (U. S.), see

McAdam.

Kapp, 24, 89

Kelvin, 24, 149

Ivimball, 175

Kommers, 26, 30, 38, 54, 58, 130,
132, 136, 185, 198, 201, 203,
204, 206, 213, 216, 217

Launhardt, 24, 175

Lea, 37, 49, 50, 57, 132, 154, 193, 198

Lessens, 31, 110, 130, 148, 150

Lewis, 100

Lucas 68 (insert), 69, 73

M

McAdam, 26, 31, 83, 94, 105, 110,
116, 124, 126, 127, 130, 132,
136, 138, 144, 146, 147, 150,
152, 170, 183, 188, 190, 197,
198, 214

McCook Aviation Field, see Moore,
R. R.

Mack, 117, 136, 195, 197, 205

Mason, 47, 56

Mehmel, 286

Moore, H. F., 26, 30, 31, 38, 54, 56,
58, 99, 101, 130, 132, 136, 138,
140, 144, 147, 154, 158, 183,
190, 197, 198, 201,
206, 210, 213, 216,

O

185, 188,
204, 205,
217, 220
Moore, R. R.,

26, 31, 96, 98, 138,

144, 146, 201, 212, 213, 214

Older, 268

Ono, 98, 108, 193

Parr, 215

Pennsylvania, Univ. of, see Berry

Probst, 269

Purdue Univ., see Hatt.

Putnam, 149

R

Rankine, 3, 187
Rawdon, 230
Reynolds, 24
Rogers, 132
Rose, 199, 202
Rosenhain, 18, 27, 32, 34
Rowett, 36

Royal Naval Acad. (British) see,
Haigh.

S

Saint Venant, 3, 187

Scoble, 206

Seely, 190

Slater, 262

Smith, C. A. M., 24

Smith, J. H., 24, 39, 40, 48, 87, 183,

184
Sondericker, 95, 202

AUTHOR INDEX

319

Spangenberg, 15

Stanton, 24, 86, 107, 110, 119, 249

Straub, 215

Stromeyer, 104, 149, 1.91

Suyehiro, 206

Sweet, 238

Taylor, 8

Thomas, H. R., 67
Thomas, W. N., 76, 203, 207
Timoshenko, 115, 148, 212

U

Upton, 100, 168

Van Ornum, 252, 253, 255, 259, 262,
267, 272

W

Washington Univ., see Van Ornum.
Wedgewood, 40, 48
Westinghouse, 141
Weyrauch, 24, 175
Whyte, 205

Williams (concrete), 262
Williams (with Hopkinson), 35
Wisconsin, Univ. of, see Withey.
Withey, 259

Wohler frontispiece, 12, 14, 23, 85,
93, 95, 200

SUBJECT INDEX

Accelerated tests, 148
Amorphous cement, 34

effect on cracks, 73

metal, 32
Annealing, effect of, 218
Assumptions of elastic theory, 5
Atomic bond, 63
Automobile parts, fatigue failures,

237
Axial loading, 3
Axle steel, cracks in, 229
Axles, fatigue failiires, 15, 236

B

Bauschinger range, cyclic state, 42

and fatigue range, 44
Bauschinger's laws, 15, 58
Berry's conclusions on concrete, 261
Boiler plates, fatigue failures, 236
Bolts, absorption of energy, 238

fatigue failures, 237
Brinell number and endurance limit,
164

C

Case-carburizing and endurance

limit, 158
Cast iron, fatigue of, 11, 140
Cement, fatigue tests, 252
Chain, fatigue of, 11
Charpy tests and impact-endurance

tests, 172
Clemmer's conclusions on concrete,

268
Cohesion, 70

Cold drawing, see cold work.
Cold rolling, see cold work.

Cold work, benefit and injury, 152,
213

effect on endurance limit, 152

effect of mild heating, 152

effect of surface finish, 153, 227

ferrous and non-ferrous metals,
152

slip-interference theory, 33
Concrete, deflection and set of
beams, 261, 267

elastic and inelastic action, 255

failure of slab, 269

fatigue of bond strength, 259

fatigue in service, 251

fatigue tests, 253, 255

fatigue tests of beams, 253, 255,
260, 262, 265, 266, 272

fatigue tests in compression, 269,
276

Illinois formula for road slab, 269

strains and sets in compression,
270, 277

understressing, 266
Constant-range formula, 183, 190
Corrosion, and simultaneous stress,
214

-fatigue, 214

-fatigue of heat-treated steels, 215

-fatigue of stainless steel, 215

of unstressed metal, 213
Cracks, 64

and amorphous metal, 73

in concrete, 254, 265

detection of 69, 228, 230

formation of, 18, 198, 227

formation under strain hardening,
45

in reinforced concrete beams, 265

and slip bands, 23

spread of, 75

stress-concentration at, 72

321

322

THE FATIGUE OF METALS

Creep, 44

effect of understressing on, 51

and failure, 45

and fatigue failure, 51

at high temperatures, 49

and rebonding of atoms, 51

and slip, 50

stress, limiting. 45, 51

time element, 47

viscosity and adhesion, 51
Cross-section, abrupt changes in, 198
"Crystallization " of metals, 10
Cyclic state, 41

D

Deformation under repeated stress,

22
Design of machine parts, 241
"Dirty" steel, 168, 196
Discs, fatigue failures, 240
Ductility, 216

and fatigue strength, 165, 216

and overstress, 227

and toughness, 216

E

Elastic after-working, 45
Elastic deformation, 62
Elastic failure, 60, 183, 227
Elastic hysteresis, 54
Elastic limit, 9, 53

change under repeated stress, 15

determined by set, 53

effect of cold work, 152

"natural" 17, 149

proportional, 2, 53
Elasticity, mathematical theory of, 3

assumptions, 5

brittle and ductile metals, 61

discrepancies between theory and
test results, 210

limitations, 6, 61

and repeated stress, 61

"statistical" truth of, 6, 61
Elasticity, recovery of, 31
Endurance, length of , 190, 191, 192

of machine and structural parts,
243

Endurance limit, 9, 80, 119
alloy steels, 136, 138
axial stress, 133, 141
carbon steel, 130, 132
case-carburized steel, 160
cast iron, 140
cast steel, 14, 140
cement, 253

cold work, effect of, 152
concrete, 255, 268, 272, 276
correlation with other physical

properties, 160
corrosion-fatigue, 214
cycles of stress to develop, 127
ductility, effect of, 216
effect of annealing on, 218
effect of corrosion, 213
effect of simultaneous corrosion

and stress, 214
and elastic limit, 31, 53, 58, 226
evidence for, 124
fillets, effect of, 212
finish, effect of surface, 201
heat treatment, effect of, 153
high temperature, effect of, 57,

159
holes, effect of, 210
internal stress, effect of, 204
large and small pieces, 133
non-ferrous metals, heavy, 144
non-ferrous metals, light, 146
overstress, effect of, 217
reversed flexure, 131
scratches, grooves, notches, effect

of, 199, 203
screw threads, 141, 201, 237
shearing stress, 147, 148
shoulders, effect of, 199
significance, 226
tension-compression, 133, 141
torsion, 147, 148
and ultimate strength, 58, 163,

164
understressing, effect of, 49, 220
wood, 247
wrought iron, 14
Endurance ratio, 83
Exponential formula for repeated
stress, 25

SUBJECT INDEX

323

Factor of safety for fatigue, 133
FaUure, buckling, 228

dead-load, 228

elastic, 60, 183, 227

fatigue, see fatigue failure.

plastic yielding, 228

in service, 235

static, 228

static and fatigue compared, 9, 10,
64
Fatigue cracks, see cracks.
Fatigue failure, amorphous cement
theory, 34

automobile axles, 237

boiler plates, 236

bolts and studs, 237

concrete, 251

discs, rotating, 240

and hysteresis, 35

rails, 239

railway car axles, 236

reinforced concrete, 259, 261

results of, 241

springs, 238

steering knuckles, 237

structural members, 235

typical features of, 230

warnings of, 228
• wire rope, 240

wood, 244
Fatigue limit, see endurance limit.
Fatigue of metals defined, 10
Fatigue strength, see endurance

limit.
Fatigue tests, see tests.
Ferrite, 29
Fillets, 212
Flaws, sub-microscopic, 195, see also

"dirty" steel.
Flexure, common formula, 4

in tension members, 141

unsymmetrical cross-section, 4
Fracture, 64, 65

of bolts, 237

effect of temperature, 34

mechanism of progressive, 77

progressive, 60, 64, 65

Fracture and slip, 64, 150
sudden, 64
torsional, 233
typical fatigue, 230

G

Gerber's formula, 174
Girder, fatigue of, 12
Goodman diagram, 177, 178
Griffith theory, 71, 195, 213

H

Hardening by strain, 32
"Healing" of overstrained metal, 56
Heat bursts, 55

Heat treatment, effect on corrosion-
fatigue, 215

effect on endurance limit, 153

large and small pieces, 133

mild, effect on internal stress, 56,
153
Holes, effect of, 210
Homogeniety, 5
Hooke's law, 2
Howell formula, 186
Hysteresis, elastic, 54

below endurance limit, 38

and fatigue failure, 35, 38, 44

loop, 20, 40

mechanical, 35

mild heating, effect of, 37

overstrain, effect of, 37

rest, effect of, 37

temporary effects, 55

torsional, 37

Illinois highway formula for road

slab, 269
Impact, repeated 109, 166, 170, 249
stresses, 9
test results and endurance limit,

166
and toughness, 216
Impact-endurance tests, 170
Internal flaw theory, 71, 195, 213

324

THE FATIGUE OF METALS

Internal stress, 204
Isotropy, 5

Johnson-Goodman formula, 179
modified, 185

Launhardt's formula, 175
Length of endurance, 190, 192, 243
Loading and unloading, 46, 48
Localized stress, 7, 61, 141

M

Machine parts, length of endurance
required, 243
range of stress, 243

Mathematical theory of elasticity,
see elasticitj^

Mechanical hysteresis, see hystere-
sis.

Microscope, use of, 6

N

"Natural" elastic limit, 17, 149

O

Overstress, elastic properties, effect
on, 31
endurance hmit, effect on, 217
occasional, effect of, 227
relief by understressing, 222

Pearlite, 29
Pohsh, effect of, 201
Progressive fracture, 10, 60, 64
Proportional elastic limit, 2, 53

Rails, fatigue failures, 239
transverse fissures, 239

Range of stress, 10, 173

constant-range formula, 183, 190

Gerber formula, 174

Goodman diagram, 177

Howell formula, 186

Johnson-Goodman formula, 179

Launhardt-Weyrauch formulas,
175

in machine parts, 243

modified Johnson-Goodman for-
mula, 185

range ratio, 173

shearing stress, 188

steady torsion and reversed flex-
ure, 193

strain-energy relation (Jasper),
187

test data on, 187

torsion, 188
Recovery, by mild heating, 55

by repeated stress, 55

by rest, 55

temporary, 55
Reinforced concrete, fatigue tests,

253, 259, 262
Repeated-stress testing machines,

see testing machines.
Resilience, modulus of, 9
Rest, effect on deformation, 273

effect on hysteresis, 37, 42

recovery due to, 55

S

"Scatter" of test data, 168
Scratches, 198, 203, 207
Screw threads, 141, 201, 237
Set, permanent at low stress, 39
Shear, direct, 4
Shearing stress, see stress.
Shock, see impact.
Short-time tests, see tests, acceler-
ated.
Shoulders, effect of, 199
Shp, 63

and creep, 50

dui'ing loading and unloading, 46

and fracture, 64, 150

and redistribution of stress, 45

SUBJECT INDEX

325

Slip, under repeated stress, 29, 45

under static stress, 27

and strengthening of atomic
bonds, 64

of unfavorably placed grains, 45
Slip bands, 18, 27

and cracks, 29

and fatigue failure, 31
Slip lines, see slip bands.
Slip-interferance theory, 33
S-N diagram, 119

alloy steels, 123

carbon steels, 122

Cartesian, 119

cast iron, 124

cast steel, 124

cement, 253

concrete, 274

extrapolation, 127

"knee" of, 121

logarithmic, 119

non-ferrous metals, 125

semilogarithmic, 119

shape of, 126

typical, 119

wood, 247
Space lattice, 63

Specimens, abrupt changes of form,
112

brittle and ductile metals, 112

for fatigue tests. 111, 112

for flexure tests, 114

gripping devices, 112

localized stress in, 112

rotating cantilever beam, 116

rotating-beam, 114

surface finish, 118

tension-compression, 112

tension-flexure, 117

thin sheet metal, 117

torsion, 118
Speed of testing, 47, 151
Springs, fatigue failures, 238
Stainless steels, corrosion-fatigue of,

215
Strain, 1

adjustment to, 49
Strain hardening, 32

amorphous metal, 33, 45

Strain hardening, and fatigue cracks,
45

by repeated stress, 49

Rosenhain's theory, 32

slip-interference theory, 33
Strain-energy hypothesis (Haigh),

187
Strength, internal flaw hypothesis,
71

surface irregularity hypothesis, 76

theoretical and actual, 70
Stress, 1

alternating, 9

average, 39

and corrosion, 214

elastic, 63

internal, relieved by mild heating,
56, 153

limiting creep, 49

localized, 61, 112, 115, 141

localized, significance, 7

mean, 173

nominal and actual, 62

occasional high, 226

"raisers," 72, 75, 80, 195

range of, see range of stress.

range ratio, 173

reversed, 9

in service, 226

shearing, 4, 5, 141, 147, 148

significant, 7

steady, 39
Stress-concentration, 72, 207, 212

at fillets, 212

at grooves, 209, 212

at holes, 210

at keyways, 234

at scratches, 203, 207

at screw threads, 201, 237

theoretical and effective, 75, 80,
206
Stress-cycle diagram, see S-N dia-
gram.
Stress-intensification, see stress-con-
centration.
Stress-strain loops, 40
Structural members, fatigue failure,
235

326

THE FATIGUE OF METALS

Structural members, length of
endurance required, 243
range of stress, 243
Surface finish, 118, 201

Temperature, and fatigue strength,
57, 154, 155

and long-time static strength, 155

a,nd short-time static strength,
155
Tensile strength and endurance

hmit, 162
Tension members, flexure in, 141
Testing machines, alternating-cur-
rent magnet type, 89, 102

axial load, 84

centrifugal force type, 87

combined stress, 106

constant deformation, 108

elements of, 84

for high temperatures, 155

inertia type, 86, 102, 104

Olsen-Foster, 103

repeated impact, 109

reversed flexure, 92

rotating cantilever beam type,
93

rotating-beam type, 92, 95

rotating-specimen type, 91, 102

rotating-spring type, 98

for short flexure specimens, 98

spring type, 84, 100, 103

tension-flexure, 106

torsion, 102

types of, 83

Upton-Lewis, 100
Tests, accelerated, 109, 148, 150

discrepancies between theory and,
210

high temperature, 155

repeated impact, 110, 170, 249

rise of temperature, 149

running deflection, 150

speed, effect of, 151

static, at high temperatures, 155

tension, flexure in, 141

vibration, of wood, 244

Torsion, common formula, 5
fatigue fractures in, 233
range of stress, 188
steady and reversed flexure, 193

Toughness, 216

Twinned crystals, 29

U

Understressing, 220

and cold work, 222

concrete, 266

vs. overstressing, 222

static strength, effect on, 222
Unit strain, 1
Unit stress, 1

Vibration, damping of in wood, 245
strength and stiffness of wood,
effect on, 245

W

Weyrauch's formula, 175

Whiting method for detecting cracks.

228
Wire rope, fatigue failure, 240
Wohler's laws of fatigue, 13
Wood, endurance limit, 247

fatigue failure in service, 244

fatigue tests, 247

repeated-impact tests, 249

vibration, effect of on stiffness and
strength, 245
Wrought iron, fatigue of, 11, 14

Y

Yield, plastic, 45

range, 39, 44

static and fatigue, 51

stress, 39

time, effect of, 52
Yield point, 9

change under repeated stress, 15

and endurance limit, 40

MANUAL OF

ENDURANCE OF METALS
UNDER REPEATED STRESS

A Convenient Little Book for
Practical Use in

DESIGNING, INSPECTING, TESTING

Compiled by H. F. MOORE, D. So.

Research Professor of Engineering Materials, In Charge,
Investigatio7i of Fatigue of Metals, University of Illinois

AVITH THE OOOPBRATION OF

J. A. CAPP

Chief of Testing Laboratory
General Electric Company, Schenectady, New York

ALFRED V. DE FOREST

Research Engineer
American Chain Company, Bridgeport, Connecticut

H. C. DICKINSON

Chief, Heat and Power Division
U. S. Bureau of Standards, Washington, D. C.

F. P. GILLIGAN

Secretary-Treasurer
The Henry Souther Engineering Company, Hartford, Connecticut

ZAY JEFFRIES

Consulting Metalhirgist
Aluminum Company of America, Cleveland, Ohio

D. J. McADAM, JR.

Superintendent, Metallurgical Division
U. S. Naval Research Laboratory, Bellevue, Anacostia, D. C.

CHARLES A. MeCUNE

Director of Research
American Chain Company, Bridgeport, Connecticut

R. R. MOORE

Chief, Physical Testing Branch, War Department Air Service
Engineering Division, IMcCook Field, Dayton, Ohio

F. E. SCHMITT

Associate Editor
"Engineering News-Record," New York

Price $1.00 post-paid. Bound in cloth, 5 x 7}^ inches.

ENGINEERING FOUNDATION

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