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UNIVERSITY OF ILLINOIS
BULLEETIN
Vol. XXXVII September 26, 1939 No. 5
ENGINEERING EXPERIMENT STATION
BULLETIN SERIES No. 316
THE EFFECT OF RANGE OF STRESS ON THE
TORSIONAL FATIGUE STRENGTH
OF STEEL
BY
JAMES 6. SMITH
PRICE: FORTY-FIVE CENTS
PUBLISHED BY THE UNIVERSITY OF ILLINOIS
URBANA
[Issued weekly. Entered as second-class matter December 11, 1912, at the post office at Urbana,
Illinois, under the Act of August 24, 1912. Acceptance for mailing at the special rate of postage provided
for in section 1103, Act of October 3, 1917, authorized July 31, 1918]
T HE Engineering Experiment Station was established by act
of the Board of Trustees of the University of Illinois on De-
cember 8, 1903. It is the purpose of the Station to conduct
investigations and make studies of importance to the engineering,
manufacturing, railway, mining, and other industrial interests of the
State.
The management of the Engineering Experiment Station is vested
in an Executive Staff composed of the Director and his Assistant, the
Heads of the several Departments in the College of Engineering, and
the Professor of Chemical Engineering. This Staff is responsible for
the establishment of general policies governing the work of the Station,
including the approval of material for publication. All members of
the teaching staff of the College are encouraged to engage in scientific
research, either directly or in cooperation with the Research Corps,
composed of full-time research assistants, research graduate assistants,
and special investigators.
To render the results of its scientific investigations available to
the public, the Engineering Experiment Station publishes and dis-
tributes a series of bulletins. Occasionally it publishes circulars of
timely interest, presenting information of importance, compiled from
various sources which may not readily be accessible to the clientele
of the Station, and reprints of articles appearing in the technical press
written by members of the staff and others.
The volume and number at the top of the front cover page are
merely arbitrary numbers and refer to the general publications of the
University. Above the title on the cover is given the number of the
Engineering Experiment Station bulletin, circular, or reprint which
should be used in referring to these publications.
For copies of publications or for other information address
THE ENGINEERING EXPERIMENT STATION,
UNIVERSITY OF ILLINOIS,
URBANA, ILLINOIS
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN SERIES No. 316
THE EFFECT OF RANGE OF STRESS ON
THE TORSIONAL FATIGUE
STRENGTH OF STEEL
BY
JAMES 0. SMITII
INSTRUCTOR IN THEORETICAL AND
APPLIED MECHANICS
PUBLISHED BY THE UNIVERSITY OF ILLINOIS
FPICE: FORTY-FIVE C(ENT
4000-6-39-16786 7'7 LL"O
CONTENTS
PAGE
I. INTRODUCTION . . . . . . . . . . . . . 5
1. Purpose of Investigation . . . . . . . . 5
2. Acknowledgments . . . . . . . . . . 6
II. MATERIALS AND SPECIMENS USED, MIETHOD OF TESTING,
AND TEST DATA . . . . . . . . . . . 6
3. Materials and Specimens Used . . . . . . 6
4. Method of Testing . . . . . . . . . . 8
5. Test Data . . . . . . . . . . . . 10
III. INTERPRETATION OF TEST DATA AND DISCUSSION OF
R,ESULTS OF TESTS . . . . . . . . . . . 15
6. Method of Plotting and Meaving of Terms or Symbols 15
7. Modified Goodman Diagram . . . . . . . 19
8. Mean Stress-Alternating Stress Diagram. . . . 23
9. Range Ratio-Endurance Limit Diagram . . . . 27
10. Comparison of Results Obtained by Various Methods
of Interpretation . . . . . . . . . . 29
IV. CONCLUSONS . . . . . . . . . . . . . 30
11. Summary . . . . . . . . . . . . . 30
12. Conclusions . . . . . . . . . . . . 34
LIST OF FIGURES
NO. PAGE
1. Details of Test Specimens . . . . . . . . . . . . . . 7
2. Photomicrographs of Steel Tested, Transverse Section .. . . . 9
3. Repeated Torsion Testing Machine . . . . . . . . . . . 10
4. Stress-Strain Curve for Torsional Static Test . . . . . . .. . 11
5. Photographs of Specimens After Testing Under Repeated Torsional Stress . 14
6. S-N Diagrams for Heat-Treated S.A.E. 3140 Steel for Various Ranges of
Torsional Shearing Stress . . . . . . . . . . . . . 16
7. S-N Diagrams for Hot-Rolled S.A.E. 3140 Steel for Various Ranges of
Torsional Shearing Stress . . . . . . . . . . . . . 17
8. Stress Symbols for Varying Range of Stress ... . . . . . . 19
9. Goodman Diagram Showing Effect of Range of Stress Upon Endurance
Lim it . . . . . . . . . . . . . . . . . . . 20
10. Modified Goodman Diagrams Showing Effect of Range of Torsional Shearing
Stress Upon Endurance Limit of S.A.E. 3140 Steel . . . . . . 23
11. Modified Goodman Diagrams Showing Effect of Range of Torsional Shearing
Stress Upon Endurance Limit of S.A.E. 3140 Steel . . . . . . 24
12. Mean Stress-Alternating Stress Diagrams Showing Effect of Range of
Torsional Shearing Stress Upon Endurance Limit of S.A.E. 3140 Steel 25
13. Range Ratio-Endurance Limit Diagrams Showing Effect of Range of
Torsional Shearing Stress Upon Endurance Limit of S.A.E. 3140 Steel 28
14. Range Ratio-Endurance Limit Diagram Showing Effect of Range of Stress
Upon Torsional Endurance Limit as Reported by Various Investigators 31
LIST OF TABLES
NO. PAGE
1. Chemical Analysis of Steel Tested . . . . . . . . . . .. . 6
2. Physical Properties of Steel Tested . . . . . . . . . . . 12
3. Results of Repeated Torsional Shearing Stress Tests . . . . . . . 18
4. Comparison of Torsional Fatigue Strength as Calculated by Various
Methods with Actual Test Values . . . . . . . . . . . 22
5. Stress Concentration Factors for Transverse Holes in Specimens Subjected to
Completely Reversed Cycles of Torsional Shearing Stress . . . . 33
6. Stress Concentration Factors for Fillets in Specimens Subjected to Com-
pletely Reversed Cycles of Torsional Shearing Stress . . . . . . 34
THE EFFECT OF RANGE OF STRESS ON
THE TORSIONAL FATIGUE STRENGTH OF STEEL
I. INTRODUCTION
1. Purpose of Investigation.-Various investigators have shown
that the maximum tensile or compressive stress to which a steel
member may be repeatedly subjected without causing fracture (the
endurance limit) depends upon the range of stress applied, that is,
upon the values of the maximum and minimum stresses of a repeated
cycle, but very few experimental data are available involving varying
ranges of torsional shearing stress. The research committee of the
American Society for Testing Materials reported in 1937 that
"There is a paucity of data available covering fatigue tests showing
the effect of range of stress on shearing endurance limit." The limited
data presented in the report of this research committee indicate that
for unnotched specimens of several steels the minimum range of
shearing stress required to cause a repeated stress (fatigue) fracture
is nearly constant. Therefore, as the minimum shearing stress is
increased, the maximum value in the range of stress that causes
rupture after the application of a large number of cycles of stress
(the endurance limit for the given range of stress) is also increased
by a nearly constant amount, provided that the shearing elastic
strength of the material is not exceeded. Little information, how-
ever, is available indicating the effect of range of stress on the shearing
endurance limit for specimens of steel containing a notch, hole, fillet,
keyway, or other type of "stress raiser."
Most of the data available on the torsional shearing fatigue
strength of steel concern the endurance limit for a range of stress in
which the stress alternates in a cycle from a maximum value to a
minimum value of equal magnitude but of opposite sign; this range
or cycle of stress will be referred to as a completely reversed range
or cycle of stress.
There are many torque-resisting steel members in service, such as
helical springs, crankshafts, cam-shafts, propeller shafts, etc., which
are subjected to repeated torsional shearing stresses such that the
stress in each cycle varies from a minimum stress to a maximum
stress of a different magnitude. Moreover, in many of these torque-
resisting members there are stress concentrations caused by fillets,
oil holes, keyways, or other discontinuities.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 1
CHEMICAL ANALYSIS OF STEEL TESTED
The purpose of the investigation herein reported was to determine
the effect of the range of stress upon the torsional shearing endurance
limit of S.A.E. 3140 steel for specimens free from stress concentra-
tion, and also for specimens containing stress concentrations caused
by a transverse hole drilled through the specimens. The steel was
tested in the hot-rolled condition and also in a heat-treated condition.
Thus the tests show the effect of range of torsional shearing stress
for four possible conditions of use of the material.
2. Acknowledgments.-The investigation was carried on as a part
of the work of the Engineering Experiment Station of which DEAN
M. L. ENGER is director and of the Department of Theoretical and
Applied Mechanics of which PROFESSOR F. B. SEELY is the head.
The author is indebted to PROFESSOR SEELY and PROFESSOR
T. J. DOLAN for many helpful suggestions, and for their interest and
encouragement in carrying out these tests.
Appreciation is expressed for the generous cooperation of PRO-
FESSOR H. F. MOORE and MR. N. J. ALLEMAN in making available
the equipment of the Fatigue of Metals Laboratory used in this
investigation.
A considerable number of the tests reported herein was made
under the direction of PROFESSOR T. J. DOLAN by MR. D. D. STREID
and MR. R. A. MACGREGOR in satisfying the requirement for thesis
work for the degree of Bachelor of Science in Mechanical Engineering.
II. MATERIALS AND SPECIMENS USED, METHOD OF TESTING,
AND TEST DATA
3. Materials and Specimens Used.-S.A.E. 3140 steel was chosen
for these tests because of its frequent use in service for parts sub-
jected to ranges of repeated stress covered by the investigation.
This steel is generally used in a heat-treated condition, but tests
were made on it in both the hot-rolled and a heat-treated condition.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
(2al-)nno/c/ead Spec/men for Tors/'ona/ Fa/-que Tests.
(b)-Notched 5Spec /ten for Torsiona/ Faftiue Tests.
(c) - Tesion Spec/men for Sta0l'c Tests
(d)- Torsion Spec/'en for Sta/,ic Tests
Dimesion$s "a"aý d "'o " var/ed' s//'IA'h/t (See Ttbl/e )
FIG. 1. DETAILS OF TEST SPECIMENS
The chemical composition of the steel as taken from the heat analysis
is given in Table 1.
The specimens used were cut from seven 7%-inch round bars
received from the manufacturer in the hot-rolled condition, having a
Brinell hardness averaging about 245. Some of the specimens were
given a heat-treatment consisting of heating to 1510 deg. F. for
twenty minutes, quenching in light oil at 85 deg. F., and tempering in
a salt (NaNO3-KNOa) bath for about half an hour at 920 deg. F.
The general details of the specimens tested are given in Fig. 1.
The type of specimen shown in Fig. 1(a), which had a solid cross-
section and was free from abrupt changes in cross-section, is referred
to as the unnotched specimen. The diameter d of the unnotched
specimen ranged from 0.25 inch to 0.32 inch depending on the
strength of the steel. A variation in diameter was necessary in order
to keep the load within the capacity of the testing machine. The
ILLINOIS ENGINEERING EXPERIMENT STATION
type of specimen shown in Fig. 1(b) with the transverse hole is
referred to as the notched specimen.* The diameter of the hole, a,
was made such that as nearly as possible the ratio of a to d was 0.1
(a/d = 0.10). The nominal diameter d of the notched specimens
varied from 0.32 inch to 0.40 inch according to the strength of the
steel (see Table 3 for further details of the specimens).
The specimens which were heat-treated were rough turned to
about 0.050 inch (diameter) oversize, heat-treated, and then finished.
The surface in the neighborhood of the critical section of each
specimen was finished circumferentially with No. 00 polishing paper,
but since in torsion a fatigue crack may start on a longitudinal,
transverse, or diagonal plane it is not possible, by polishing in a
particular direction, to minimize the effect of surface scratches to the
same extent as is possible by polishing bending specimens in the
longitudinal direction. For the specimens with a hole the finishing
operation was nearly completed before the hole was drilled and thus
the sharp edge of the hole was rounded only slightly in completing
the polishing.
The photomicrographs of Fig. 2 show the structure of the steel
in the conditions as tested. In the hot-rolled condition, as shown
in Fig. 2(a), the steel is composed of medium to large size grains of
pearlite and ferrite, while for the same steel in the heat-treated con-
dition Fig. 2(b) shows a very fine-grained troostitic-sorbitic structure,
with traces of ferrite at high magnification (x2000).
4. Method of Testing.---Three repeated-torsion testing machines
were used in making the tests for the data reported herein; each
machine had a capacity of 300 in. lb. twisting moment. Figure 3
shows one of these machinest which is arranged to produce a con-
trolled angular rotation of one end of the specimen through the
chuck A' by means of cam C and lever arm B. The torque thus pro-
duced is transferred through the specimen S to chuck A" and thence
to the calibrated bar D, the twist of which is proportional to the
torque, and is measured by the dial gage M. By adjusting the set
screws F, which act on a lever fastened to the left end of bar D, the
desired range of twisting moment, and therefore the range of stress,
may be obtained. These machines operate at 1400 r.p.m., and cut
off automatically when a specimen fractures.
The nominal shearing unit-stress, s, in all specimens was calcu-
lated by means of the torsion formula, s, = Tc/J, in which T is the
*In fatigue testing any discontinuity such as a v notch, fillet, keyway, transverse hole, etc.,
causing an abrupt change in the cross-section of the specimen is frequently referred to as a notch.
tSee University of Illinois Engineering Experiment Station Circular No. 23 for further description
of repeated torsion nmachine.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
(b)
FIG. 2. PHOTOMICROGRAPHS OF STEEL TESTED, TRANSVERSE SECTION
(a) S.A.E. 3140 steel, hot-rolled (x 400, etched 2 per cent nital)
(b) S.A.E. 3140 steel, quenched and tempered (x 2000, etched 2 per cent nital)
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 3. REPEATED TORSION TESTING MACHINE
twisting moment, J is the polar moment of inertia of the circular
section where the critical stress occurs in the specimen, c is the radius
of the critical cross-section and s, is the nominal shearing unit-stress.
In making computations for shearing unit-stress in the specimens
with a transverse hole no allowance was made for the hole.
An endurance limit for each of four ranges of stress was established
for the steel in each of four conditions, namely, (1) the hot-rolled
unnotched, (2) the hot-rolled notched, (3) the heat-treated unnotched
and (4) the heat-treated notched conditions. All endurance limits
except those for completely reversed ranges of stress were established
by keeping the minimum stress of the range constant and varying
the maximum stress. All endurance limits are based upon 10 million
cycles of stress, except where conditions made a higher number of
cycles necessary.
5. Test Data.-Figure 4 shows the lower portions of the static
torsion stress-strain curves for the steel as obtained from solid un-
notched specimens. It will be observed that the steel did not have a
I,
K
~
It
K
*.-, I~
K
K
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
620 -
Y/e/d Strenqth,.i .
/ S.A.E. 3140 51eel
S/ ,Spec/men d/a'. 050 in.
2 0 -- _ _--------------- -
2OY,'e/d Strenqth /based
r- 10." on 0.2% offse.~
0 I7-- - - -
0.005 0.0/0 0.0/5 0.020 0.025
Shearing Unit Strain in Inches per Inch
FIG. 4. STRESS-STRAIN CURVE FOR TORSIONAL STATIC TEST
pronounced yield point in either the hot-rolled or the heat-treated
condition. The results of static tension and torsion tests are shown
in Table 2. Drawings of specimens used in these tests are shown
in Figs. 1(c) and 1(d).
The static torsional yield strengths reported in Table 2 were
obtained from cylindrical specimens of solid cross-section rather than
of hollow cross-section because the solid form of specimen corre-
sponds more closely to members as used in service. Furthermore, it
will be noted that the values of yield strength reported in Table 2
are based on an arbitrary offset of 0.2 per cent. However, the yield
strength of the material used in these tests was obtained using hollow
specimens as well as solid specimens, and the ratio of the yield
strengths of hollow and solid specimens was found to be from 0.80 to
0.85. This ratio corresponds closely to that obtained by Seely and
Putnam* for six steels.
Two types of failure were observed in the repeated stress tests,
namely, a progressive fracture or fatigue failure, and failure by general
yielding before a fatigue crack started. The type of failure depended
*F. B. Seely and W. J. Putnam, Bulletin 115, Engineering Experiment Station, University
of Illinois, p. 41.
v
12 ILLINOIS ENGINEERING EXPERIMENT STATION
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RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
largely on the range of stress to which the specimens were subjected.
Photographs of specimens indicating these two types of failure are
shown in Fig. 5.
Figures 5(a) and 5(b) show typical fatigue fractures which occurred
in these repeated stress tests. In a fatigue failure of an unnotched
specimen (Fig. 5b) the fracture usually began as a longitudinal shear
failure, and then followed a helicoidal path approximately along the
plane of maximum tensile stress in the specimen. For specimens with
a transverse hole the fatigue fracture always was first detected along
the plane of maximum tensile stress and followed a helicoidal path.
These facts agree with results of tests by Southwell and Gough,* in
which they found that when no stress concentration was present a
shear failure occurred in low carbon steel specimens subjected to
repeated torsional stress, but when stress concentrations were
developed (due to flaws in the material, or to abrupt change in the
shape of the specimen) a helicoidal fracture resulted, with fracture
progressing approximately along planes of maximum tensile stress.
This helicoidal type of fracture is characteristic of a tensile failure
in members subjected to static torsional loads.
Figures 5(c) and 5(d) are from photographs of unnotched hot-
rolled and heat-treated specimens, respectively, which have with-
stood 10 million cycles of torsional shearing stress at or slightly below
the endurance limit. It is evident from what follows that these
specimens, although they have not developed a fatigue crack, may be
said to have failed because they are so badly distorted. This type
of failure will be referred to as general yielding. The hot-rolled
specimen, Fig. 5(c), was tested with a range of torsional shearing
stress of from +30 000 to a calculated stress of +95 000 lb. per sq. in.
(calculated by the formula s, = Tc/J), although the static yield
strength of this steel was only 56 000 lb. per sq. in. It will be noticed
that the specimen was permanently twisted through an angle of
about 20 degrees, as can be seen from the relative positions of the
flat surfaces on the ends of the specimen which were originally in the
same plane. At the upper end of the specimen the flat surface is
almost perpendicular to the plane of the paper. Figure 5(d) shows
an unnotched specimen of the heat-treated steel which was subjected
to 10 million cycles of torsional shearing stress from +40 000 to a
calculated stress of 122 000 lb. per sq. in., the yield strength of the
heat-treated steel being 110 000 lb. per sq. in. The permanent angle
of twist for this specimen was 11.5 degrees.
*R. V. Southwell and H. J. Gough, "On the Concentration of Stress in the Neighborhood of a
Small Spherical Flaw; and on the Propagation of Fatigue Fractures in Statistically Isotropic Materials,"
Philosophical Magazine, Vol. 1, Jan. 1926.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 5. PHOTOGRAPHS OF SPECIMENS AFTER TESTING UNDER
REPEATED TORSIONAL STRESS
(a) Fatigue failure of notched specimen
(b) Fatigue failure of unnotched specimen
(c) Failure by general yielding, hot-rolled unnotched specimen
(d) Failure by general yielding, heat-treated unnotched specimen
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
It is evident, therefore, that the endurance limit has little signifi-
cance as a criterion of failure for ranges of stress that permit the
member to fail by general yielding before the endurance limit is
reached. This fact will be emphasized further in the section on
interpretation of test data.
It is recognized that values for s, calculated by the formula
s, = Tc/J for values of torque above the static elastic strength of
the material are not the stresses which actually exist, but for means
of comparison this method of calculation seems permissible.
The S-N diagrams shown in Figs. 6 and 7 give the individual
results for each specimen tested under repeated torsional shearing
stress. The ordinates of the points of these diagrams are the nominal
maximum torsional shearing unit-stress as calculated by the formula
s, = Tc/J, and the abscissas represent the number of cycles of stress
required to fracture the specimen, except that plotted points with
arrows attached indicate that the specimen did not fracture. Table 3
gives a summary of the data taken from these curves. Columns 1
and 2 of Table 3 give information concerning the details of the
specimens used to establish each endurance limit. Columns 3 and 4
give the results of the repeated torsional shearing stress tests.
III. INTERPRETATION OF TEST DATA AND DISCUSSION
OF RESULTS OF TESTS
6. Method of Plotting and Meaning of Terms or Symbols.-For
convenience in plotting the data to show the effect of range of stress
upon the torsional shearing endurance limit, three methods of speci-
fying a range of stress have been chosen. These three methods of
specifying a range of stress and the method of plotting the data corre-
sponding to each method are described in the following paragraphs.
First, a range of stress is the magnitude of the change of stress in
passing from the minimum stress to the maximum stress of a cycle.
A range of stress is not specified completely, however, unless either
the maximum or minimum stress is given in addition to the magnitude
of the range. Thus in Fig. 8 a range of stress may be specified by
stating the maximum stress, smax, and the minimum stress, Smin, or
by stating the magnitude of the range, As, with either s ax or Smin.
This method of specifying a range of stress will be used in plotting
the data on a modified Goodman diagram which will be described
in detail in the next section (see Fig. 10 or 11).
Second, a range of stress may be thought of as being made up of a
steady stress Sm with an alternating stress so superimposed upon it,
ILLINOIS ENGINEERING EXPERIMENT STATION
*^/
\I'
1:1
N
I
K
-K
1)
L~~~~~~~ ______________
0 6 0 0 8
AN, Number of Cyc/es of Stress for Fra'cure
A range of stress is 1
deftermined by sx anad sa,,,i
CoRp/etee \ iF
R/versed F7\ ,\ I
SLy I
Curve
U Symwbo/s 0
FIG. 6. S-N DIAGRAMS FOR HEAT-TREATED S.A.E. 3140 STEEL FOR VARIOUS
RANGES OF TORSIONAL SHEARING STRESS
13
u[r
Range, ± 6$
t 4$
N
1:O
:1'. 2
/ I0/ /0- 1 /0-
N, Number of Cyc/es of Sress for Fracture
FIG. 7. S-N DIAGRAMS FOR HOT-ROLLED S.A.E. 3140 STEEL FOR VARIOnS
RANGES OF TORSIONAL SHEARING STRESS
so that s. is completely reversed around s_.,. For example, in Fig. 8 a
range of stress is specified by choosing s,, and s,. This method of
specifying a range of stress will be used to show the effect of range
of stress upon the endurance limit by plotting s,,, as abscissas and
sa as ordinates. This diagram will be referred to as the "mean stress-
alternating stress" diagram.
Third, a range of stress may be specified by stating the ratio of
the minimum stress, smin,, to the maximum stress, sm.ax. This ratio
of the minimum to the maximum stress is called the "range ratio," and
is denoted by r, which may be positive or negative; in addition, either
the maximum stress, sx, or the minimum stress, s,,in, must be given.
For example, in Fig. 8 the range of stress is specified by stating the
value of s .. and r. This third method of specifying a range of
stress is used to show the effect of range of stress upon the endurance
limit by plotting as abscissas the values of r and as ordinates the
values of the endurance limit, s ...x, in terms of the endurance limit,.
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 3
RESULTS OF REPEATED TORSIONAL SHEARING STRESS TESTS
Specimen Details Range of Stress
Minimum Diameter Diameter of Hole Minimum Stress Endurance Limit or
d a smi Maximum Stress
in. in. lb. per sq. in. lb. per sq. in.
(1) (2) (3) (4)
I. S.A.E. 3140 steel, hot-rolled, unnotched
0.320 No hole -44 000 +44 000
0.250 " " -32 000 +64 000
0.320 " " 0 +79 000
0.250 " +30 000 +95 000
II. S.A.E. 3140 steel, hot-rolled, notched
0.400 0.040 -22 000 +22 000
0.320 0.032 -10 000 +38 000
0.400 0.040 0 +39 000
0.320 0.032 +20 000 +53 000
III. S.A.E. 3140 steel, heat-treated, unnotched
0.280 No hole -56 000 + 56 000
0.250 " "-35 000 + 80 000
0.280 0 +116 000
0.250 +40 000 +122 000
IV. S.A.E. 3140 steel, heat-treated, notched
0.380 0.036 -30 000
0.320 0.032 -15 000
0.400 0.040 0
0.320 0.032 +25 000
0.320 0.032 +45 000
+30 000
+38 000
+45 000
+62 000
+83 000
s_1, for the same type of specimens for completely reversed cycles
of stress. This diagram will be referred to as the "range ratio-
endurance limit" diagram.
The symbols which are used in this report are illustrated by
Fig. 8 in part and defined as follows:
s,.a = The maximum shearing stress for a range of torsional
shearing stress. If this range of stress is the maximum
range that can be repeated an indefinitely large number
of times without causing fracture then s,,,ma becomes the
endurance limit of the material.
s,,in = The minimum shearing stress for any range of torsional
shearing stress. min may be positive or negative.
s- = The endurance limit for completely reversed torsional
shearing stress. This symbol applies to the results
obtained from either unnotched or notched specimens.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
FIG. 8. STRESS SYMBOLS FOR VARYING RANGE OF STRESS
s., s. = The mean or steady torsional shearing stress s, upon
which an alternating stress sa is superimposed to create
a range of stress.
Note: It will be observed that for completely reversed
cycles of stress, s,- = 0 and smax = - Smin = Sa.
r = The ratio of the minimum stress, smin, to the maximum
stress, smax, for a given range of stress; r may be positive
or negative and is called the range ratio.
s, = The torsional static yield strength obtained from tests of
solid cylindrical specimens based on 0.2 per cent offset.
Su = The torsional modulus of rupture as obtained from tests
of solid cylindrical specimens.
As = The magnitude of a range of stress; the algebraic difference
between s,..ax and s,nin, When the magnitude As is such
that smax is the endurance limit, As is called the magni-
tude of the endurance range of stress.
7. Modified Goodman Diagram.-A diagram such as Fig. 9 for
representing the effect of range of stress upon the endurance limit
was devised by Goodman* based chiefly on test results by W6hler.
In this diagram the minimum stress, s,min, of an endurance range of
stress is plotted as an ordinate to the line of zero stress with an
arbitrary abscissa such that one extremity of the ordinate lies on a
line DB making an angle of 45 degrees with the line of zero stress.
He found that, if the maximum stress, smax, of the endurance range
is plotted as an ordinate at the same abscissa, the upper ends of
the ordinates (values of s..ax) lie approximately on the line CB where.
*"Fatigue of Metals," H. J. Gough, D. Van Nostrand, New York, 1926, p. 67, 68.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 9. GOODMAN DIAGRAM SHOWING EFFECT OF RANGE Or
STRESS UPON ENDURANCE LIMIT
FC is 1%s, and OA = AE = 1's2. From similar triangles in Fig. 9
the following equation may be found
As
2 = 13 (s, - s,) (1)
This equation has been used extensively to represent range of stress
data. Equation (1) may also be written in the form
As 8
2 1',s.( - ) (2)
Further, it has been found by other investigators that a better agree-
ment with experimental results for steels may be obtained if Equa-
tion (2) is modified by replacing the factor as,, by the endurance
limit, s-_,* for the same type of specimen for completely reversed
cycles of stress. Equation (2) thus becomes
As Sm
- s 1 - - (3)
Equation (3) gives the magnitude of the endurance range, As, when
the mean stress sm, of the endurance range is chosen, but it is com-
monly more convenient to have an equation from which the en-
*Moore and Kommers, "Fatigue of Metals," McGraw Hill Co., 1927, pp. 185-186.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
durance limit, smax, may be calculated more directly. From Fig. 9
it is noted that
As
- smax - Sm
2
Hence Equation (3) may be written
Smax - .. +8- (1 +--) (4)
Equation (4) is represented by a straight line on the Goodman
diagram similar to line CB in Fig. 9 in which, however, s_1 is not
equal to 1sS, but is the actual endurance limit obtained from
fatigue tests, and hence the name "Modified Goodman Equation"*
is given to it, and diagrams plotted from Equation (4) are called
"Modified Goodman Diagrams." Figures 10(a) and 10(b) are
modified Goodman diagrams plotted from test data for the notched
steel specimens, that is, specimens with a transverse hole. The line
CB is fixed by two points C and B determined from Equation (4).
For point C, sm = 0, smax = s_1. For point B, Sm = s., smax = s8.
The test data are plotted in the same manner as on a Goodman
diagram. It may be noted that the straight line CB represented by
Equation (4) agrees satisfactorily with the test data. A comparison
of values of endurance limit calculated by the modified Goodman
equation with the test data is made in Table 4, columns (4) and (5).
Column (4) shows the computed values, and column (5) shows the
ratio of the computed values to the test results.
Figures 11(a) and 11(b) are modified Goodman diagrams plotted
with the data for the unnotched steel specimens. It is clear from
Fig. 11 that the modified Goodman diagram does not agree well
with the test data for the unnotched specimens. The test data show
that line CA drawn parallel to the line DB such that the magnitude
of the endurance range of stress, As, is constant agrees satisfactorily
with the test data, provided that the torsional static yield strength
represented by line AE is not exceeded. Based on the hypothesis
of a constant endurance range of stress within the yield strength of
the material, the usable endurance limit may be written as
Smax = Sm + S-1 < S, (5)
*Note* The "Modified Goodman Equation" used here is not the same as the "Modified Good-
man-Johnson Formula" used by Moore and Kommers.
ILLINOIS ENGINEERING EXPERIMENT STATION
0
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RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
K
K
K
K
a
-'I,
-K
a
U~)
K
0
K
FIG. 10. MODIFIED GOODMAN DIAGRAMS SHOWING EFFECT OF RANGE OF TORSIONAL
SHEARING STRESS UPON ENDURANCE LIMIT OF S.A.E. 3140 STEEL
A range of stress in which As is a constant provided that the maxi-
mum stress of the range is within the elastic strength, is in agreement
with the results already reported by various investigators for steel
free from notches and stress concentrations (see Fig. 14).
8. Mean Stress-Alternating Stress Diagram.-Soderberg* pro-
posed to represent range of stress data by a diagram in which the
mean stress sm of a cycle is expressed as a fraction of the torsional
static yield strength and is plotted as an abscissa, and the superim-
posed alternating stress of the cycle, Sa, is expressed as a fraction of
*Trans. A.S.M.E., Applied Mechanics Division, July-Sept. 1933, Vol. No. 3, A.P.M. 55-16.
ILLINOIS ENGINEERING EXPERIMENT STATION
/20
. 80
40
K
/'-
%810
K 40
0
i 8o
. -40
-60
t
*'K
I::
'o 40
It
I 0
-40
-60
FIG. 11. MODIFIED GOODMAN DIAGRAMS SHOWING EFFECT OF RANGE OF TORSIONAL
SHEARING STRESS UPON ENDURANCE LIMIT OF S.A.E. 3140 STEEL
the endurance limit for completely reversed cycles and is plotted as
an ordinate. Figure 12 represents the data of the tests plotted in
this manner. Thus in these diagrams the point A represents the
range of stress for which there is no mean stress, and therefore the
alternating stress s. equals s_1. The point B represents the range of
stress for which the mean stress s, is equal to s», the torsional static
yield strength as obtained from tests of solid cylindrical specimens,
and consequently no alternating stress is allowable. Now let a
straight line be drawn joining A to B, and let the point C on AB
represent an intermediate range of stress. Then the mean stress sm
is given by the abscissa OE and the corresponding alternating
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
mearn Stress, s,,,
FIG. 12. MEAN STRESS-ALTERNATING STRESS DIAGRAMS SHOWING EFFECT OF RANGE
OF TORSIONAL SHEARING STRESS UPON ENDURANCE LIMIT OF S.A.E. 3140 STEEL
ILLINOIS ENGINEERING EXPERIMENT STATION
stress Sa is given by the ordinate OF according to this straight line
representation.
The equation of line AB in Figs. 12(a) and 12(b) may be written
in the intercept form as
8sm Sa
-+ - = 1 (6)
Sy 8-1
If Equation (6) is rewritten as
sa = s- 1 - -- (7)
an expression for the alternating stress is obtained. As already
pointed out, it is frequently more desirable to have an equation
giving the endurance limit. If the mean stress sm is added to both
members of Equation (7), the following equation for the endurance
limit Smax is obtained:
Smax = Sa + S_ = S. + 8_s 1 - -- (8)
((8)
It will be noted that, if in Equation (8) su is replaced by s,, the result
is identical with the modified Goodman equation (4). In fact, if the
straight line CG is drawn in Figs. 10(a) and 10(b) it will be repre-
sented by Equation (8). Equation (8) will be referred to as the
"mean stress-alternating stress" equation. Column 6 of Table 4
shows a summary of values of the endurance limit for notched
specimens of steel as computed by Equation (8) for the mean stresses
(see column 2) sm of the endurance ranges established by the fatigue
tests. Column 7 of Table 4 shows the ratio of the endurance limits
computed by Equation (8) to the endurance limits established by the
fatigue tests. It will be noted that the average deviation of the com-
puted values of endurance limit is - 6.0 per cent and the greatest single
deviation is -20 per cent, both values being on the side of safety.
In the mean stress-alternating stress diagram, Fig. 12, it
will be noted that the straight line AB represented by Equation (8)
agrees more closely with the data for the notched steel specimens
than with those of the unnotched steel specimens. It was decided
that the data for the unnotched specimens is more closely repre-
sented by a line AD parallel to the mean stress axis drawn from
a point A where smax = sa = s_- and sm = 0 to a point D where
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
Smax = s _ + s1 s,. The line AD is represented by Equation (5),
which has already been seen to give values of endurance limit corre-
sponding to a range of stress with a constant magnitude As. Since
endurance limits in excess of the torsional static yield strength s,
permit the material to fail by general yielding, the line DB is drawn
in the diagram such that points on this line represent cycles of stress
in which the sum of the mean stress and the alternating stress is
equal to s,.
The mean stress-alternating stress diagram for representing
endurance range of stress data from fatigue tests of metals under
direct tensile and compressive stresses or under combined direct
stresses and flexural stresses was used by various investigators,
including Soderberg, Haigh* and Wahl.t
9. Range Ratio-Endurance Limit Diagram.-In a study of range
of stress data for specimens with no abrupt change of section of
several steels subjected to fatigue tests under stresses consisting of a
combination of direct tensile stress and flexural stress, Howellt used
a diagram in which he plotted values of the ratio r of Smin to Smax as
abscissas, and as ordinates values of the endurance limit, smax,
expressed in terms of s_1, the endurance limit for the range of com-
pletely reversed stress. Moore and KommersI used the same diagram
for a study of range of stress data from torsional shearing fatigue
tests of several steels in which they used specimens with no abrupt
change of section. The data from the tests reported herein are
plotted in this way in Figs. 13(a) and 13(b). Moore and Jasper§
and McAdam** found that for fatigue tests in torsional shear of
polished steel specimens with no abrupt change of section, the
magnitude of the range of stress was nearly constant. They derived
an equation on this hypothesis as follows:
Smax = As + smin = 2s 1 + Smin
whence,
2s-1
s - (9)
1-r
Equation (9) will be referred to as the constant range equation.
*Haigh, B. P., Journal, Society Chemical Industry (British), Jan. 11, 1929, p. 33.
tWahl, A. M., "Analysis of Effect of Wire Curvature on Allowable Stresses in Helical Springs,"
Preprint of paper presented to The American Society of Mechanical Engineers, Dec. 1938.
IBulletin No. 136, University of Illinois Engineering Experiment Station, p. 67-89.
¶Moore and Kommers, "Fatigue of Metals," McGraw-Hill, 1927, p. 185.
§Bulletin No. 142, University of Illinois Engineering Experiment Station, p. 72.
**Proceedings of American Society for Testing Materials, Vol. 24, pt. II, p. 574, 1924.
ILLINOIS ENGINEERING EXPERIMENT STATION
N
-'4
K
/.O -0.8 -0.6 -0.4 -O.Z 0 +0.2 +0.4 +0.6 +0.8 +/.0
Rangqe Ratio, r i = M/n/ir7um Stress of CYc/e s
FIG. 13. RANGE RATIO-ENDURANCE LIMIT DIAGRAMS SHOWING EFFECT OF RANGE
OF TORSIONAL SHEARING STRESS UPON ENDURANCE LIMIT OF S.A.E. 3140 STEEL
In Fig. 13(a), where data for the unnotched steel specimens are
plotted, the curve of the constant range Equation (9) is drawn.
The curve seems to fit satisfactorily the data which are within
the torsional static yield strength of the steel. A straight line
whose equation is
Sma - s (7r + 15)
smax = (10)
8
seems to fit all the data for the unnotched specimens. Since endur-
ance limits greater than the static yield strength of the material are
of little importance, it would seem that the constant range equation,
which agrees with data for unnotched polished steel specimens
within the torsional static yield strength, can be depended upon
¢9
%.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
without serious error where there is no stress concentration. The
straight line representation of the data seems unimportant unless
endurance limits above the yield strength of the steel are desired.
The data for the notched specimens, Fig. 13(b), are now con-
sidered. First, the constant range curve of Equation (9) is drawn
and is seen to give values of endurance limit which are too large and
hence unsafe. It was thought that a curve whose equation is of the
same general form as the constant range Equation (9) could be
used to represent these data. Accordingly, an empirical equation
was written as follows:
2.7s_1
smax = - (11)
1.7 - r
Equation (11) is plotted in Fig. 13(b) and seems to represent the
data satisfactorily for these fatigue tests of notched specimens of
steel. Further, since data for notched specimens of only two steels
are available, it was thought that a third and still more conservative
curve representing the data might be chosen. Accordingly, a curve
whose equation is of the same form as Equations (10) and (11) was
chosen, as follows:
3s 1
smax -- (12)
2-r
Equation (12) was also used by Moore and Kommers for representing
range of stress data for unnotched specimens of several steels sub-
jected to fatigue tests under a combination of direct tensile stress
and of flexural stresses. In Table 4, columns 8 to 11, inclusive, give
some comparisons of values of endurance limit computed by Equa-
tions (11) and (12) with the actual experimental values.
10. Comparison of Results Obtained by Various Methods of Inter-
pretation.-Three methods of interpreting the range of stress data
of this report have been used. By each method of interpretation an
equation for calculating the endurance limit was found. A study
of Table 4 shows that with these formulas a computed value of
endurance limit agreeing satisfactorily with the experimental value
may be expected. There are, however, certain other comparisons of
these interpretations which should be made.
First of all, by each method of interpretation there are certain
experimental constants for the steel necessary for the computation
of an endurance limit. In the modified Goodman equation, Equa-
ILLINOIS ENGINEERING EXPERIMENT STATION
tion (4), the experimental constants required are s_- and su, which
are the endurance limit for specimens with the same type of stress
concentration for completely reversed stress and the static torsional
modulus of rupture, respectively. In the mean stress-alternating
stress equation, Equation (8), the experimental constants necessary
are s-1 and s,, the torsional static yield strength as obtained from
tests of solid cylindrical specimens. In Equations (9), (10), (11)
and (12) where the range ratio r is used, only one experimental
constant is necessary, namely, s--_. It is conceivable that the avail-
able experimental constants might be a dominant factor in the choice
of a method of computation of the endurance limit. For any one
of the methods chosen it is necessary to have at least the experi-
mental value s-_ for the type of stress concentration to be used.
It is frequently desirable to compare range of stress data for two or
more materials. The adaptability of a method of interpreting range
of stress for the comparison of test data for various materials may
therefore be important. The mean stress-alternating stress and the
range ratio-endurance limit diagrams are both used for comparing
range of stress data for two or more materials. Figure 14 is a range
ratio-endurance limit diagram showing range of stress data of the
tests herein reported, and also data previously reported by various
investigators for several steels and cast irons. Although the prop-
erties of the various materials* varied widely, this diagram shows
that for small unnotched polished specimens of steel the constant
range of stress curve fits the data satisfactorily.
IV. CONCLUSIONS
11. Summary.-The formulas for computing the endurance limit
for any range of stress in torsional shear are divided into two classes
depending upon whether the steel has a stress concentration or not.
The symbols used in these formulas have been given in Section 6.
The first class of formulas consists of those which may be used
to compute the endurance limit for any range of stress in torsional
shear for steel members not having a stress concentration. These
formulas are
Smax = Sm + S_l1 (I)
2s_-
smax -= -- (II)
1 -r
*For the tests of cast iron see, Moore and Picco, "Fatigue Tests of High Strength Cast Iron,"
Trans. American Foundryman's Association, Vol. 42, 1934, p. 525.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
K
-'4
K
. Mi/nim, Stress of Cy'c/e s,,,
FIG. 14. RANGE RATIO-ENDURANCE LIMIT DIAGRAM SHOWING EFFECT OF RANGE OF
STRESS UPON TORSIONAL ENDURANCE LIMIT AS REPORTED
BY VARIOUS INVESTIGATORS
Formulas I and II are based on the hypothesis of a constant magni-
tude of range of stress, a hypothesis which has been supported by
data from several investigators for small polished steel specimens
with no abrupt change of section.
The second class of formulas consists of those which may be used
to compute the endurance limit for any range of stress in torsional
shear for steel members having a stress concentration caused by a
transverse hole.
These formulas are:
S(s - s\ (III) (modified Goodman
smax . sm + s-1 -S1 formula)
\ u
ILLINOIS ENGINEERING EXPERIMENT STATION
smax = s + s_1 _ 'm (IV) (mean stress-alternating
Smax - S. i- -- - l stress formula)
2.7s1-i
smax - (V)
1.7 - r (range ratio-endurance
3si limit formulas)
smax - 2 r (VI)
Formulas (III), (IV), (V), (VI) are all based on the results of
fatigue tests in torsional shear of small specimens of S.A.E. 3140 steel
in the hot-rolled and a heat-treated condition with a stress con-
centration caused by a small transverse hole. Formula (VI) is the
same formula as that developed by Moore and Kommers from
results of fatigue tests of polished unnotched steel specimens sub-
jected to combined flexural and tensile stresses.
In all formulas given for computing the endurance limit the
experimental constant s_1 is required. If a formula for computing
the endurance limit for a range of stress in a steel with a stress con-
centration caused by a transverse hole is to be used, then the value
of s-1 is required, and must be experimentally determined for speci-
mens with this stress concentration. However, should information
be available for the endurance limit for completely reversed cycles
of stress for specimens without a stress concentration, then a stress
concentration factor may be used to estimate s_i for the specimens
with the stress concentration. If these formulas are thought to be
satisfactory for use where the stress concentration may be of some
form other than that due to a transverse hole the same procedure as
before may be used to find s_i. For the purpose of finding s_1 for
various stress concentrations, some factors are given in Tables 5 and 6.
The values of static torsional yield strength, sy, to be used in
Formula (IV) should be obtained from tests of solid cylindrical
specimens based on an arbitrary value of offset of 0.2 per cent.
If the value of s, is obtained from a hollow specimen, or if a smaller
value than 0.2 per cent offset is used, the calculated value of the
endurance limit, smax, will be somewhat less than the endurance
limit that the material may be expected to have.
The specimens used in these tests varied in diameter from 0.25
inch to 0.40 inch, but no study of the effect of size was made since
that was not the purpose of the investigation. However, data
obtained by Mailander and Bauersfeld* for the effect of size show
that for specimens of chrome-nickel-tungsten steel with no stress
*R. Mailander and W. Bauersfeld, "Einfluss der Probengrdsse Und Probenform auf die Dreh-
Schwingungsfestigkeit Von Stahl," Techniche Mitteilungen Krupp, Vol. 2, Dec. 1934, Pages 143-152.
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
TABLE 5
STRESS CONCENTRATION FACTORS FOR TRANSVERSE HOLES IN SPECIMENS SUB-
JECTED TO COMPLETELY REVERSED CYCLES OF TORSIONAL SHEARING STRESS
Results obtained by Dolan*
S.A.E. 1020 steel, hot-rolled.. 0.400 0.04 0.10 1.31
0.400 0.07 0.175 1.43 20 100
0.400 0.10 0.250 1.62
Rail steel, hot-rolled......... 0.400 0.04 0.10 1.64 37 000
0.400 0.10 0.25 2.26
S.A.E. 3140 steel, hot-rolled. . 0.400 0.04 0.10 2.00 44 000
0.400 0.10 0.25 2.25
S.A.E. 3140 steel, quenched
and tempered............. 0.380 0.036 0.095 1.87 56 000
Results obtained by Armbrustert
N-steel................... . 0.394 0.059 0.15 1.39 22 800
V-steel..................... 0.394 0.059 0.15 1.31 24 900
E-steel..................... 0.394 0.059 0.15 1.66 42 000
Results obtained by Mailander and Bauersfeldj
Cr-Ni-W steel.............. 0.55 0.078 0.14 1.60 39 800
1.18 0.118 0.10 1.88 33 400
1.77 0.197 0.11 1.74 28 400
Results obtained from "Plaster Model" method*
Pottery plaster ............. . 2.0 0.125 0.063 1.86 Tested in
2.0 0.25 0.125 1.89 static
2.0 0.50 0.250 2.12 torsion
*Dolan, T. J., University of Illinois Engineering Experiment Station, Bulletin 293, Table 5,
p. 27, 1937.
fArmbruster, E., "Einfluss der Oberfllchenbeschaffenheit auf den Spannungsverlauf und die
Schwingungsfestigkeit," Ver. Deutsch. Ing. Verlag 1931.
tMailander, R. and Bauersfeld, W., "Einfluss der Probengrbsse und Probenform auf die Dreh-
Schwingungsfestigkeit von Stahl," Technische Mitteilungen Krupp, Vol. 2, Dec. 1934, pages 143-152.
Allowance was made in these tests for the area removed by the hole; all other values in this table were
computed on the basis of the gross cross-section of the specimen.
concentration the endurance limit for completely reversed cycles of
torsional shearing stress for specimens of diameters 0.55 inch,
1.18 inch and 1.77 inch decreased 16 per cent in passing from the
0.55 inch diameter to the 1.18 inch diameter, and decreased 29 per
cent in passing from the 0.55 inch diameter to the 1.77 inch diameter.
They also found that for specimens of the same steel with stress
concentration caused by a transverse hole the endurance limit for
completely reversed cycles of torsional shearing stress decreased
19 per cent in passing from a diameter of 0.55 inch to one of 1.18 inch,
and decreased 28 per cent in passing from a diameter of 0.55 inch to
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 6
STRESS CONCENTRATION FACTORS FOR FILLETS IN SPECIMENS SUBJECTED TO
COMPLETELY REVERSED CYCLES OF TORSIONAL SHEARING STRESS
Minimum Stress Con- Theoretical
Diameter Ratio Ratio centration Value
Material d D/d r/D Factor of
in. K K*
Results obtained by Dolant
S.A.E. 1020 steel, hot-rolled.. 0.375 2.0 0.0053 1.15 > 3.3
S.A.E. 3140 steel, hot-rolled.. 0.300 2.5 0.027 1.54 1.95
0.300 2.5 0.067 1.57 1.52
S.A.E. 3140 steel, quenched
and tempered............. 0.300 2.5 0.0087 1.51 3.00
Results obtained by Armbrustert
N-steel.................... 0.394 1.4 0.17 0.96 1.45
N-steel.................... 0.394 1.4 0.03 0.99 2.36
E-steel ..................... 0.394 1.4 0.17 1.11 1.45
E-steel..................... 0.394 1.4 0.03 1.66 2.36
Results obtained by Mailander and Bauersfeld¶
Cr-Ni-W steel. ............. 0.55 1.8 0.14 1.10 1.56
Cr-Ni-W steel.............. 1.18 1.8 0.10 1.17 1.70
Cr-Ni-W steel.............. 1.77 1.8 0.11 1.03 1.65
*As obtained by Jacobsen Trans A.S.M.E. Vol. 47, No. 1974, p. 619, using the "Electric Analogy"
method.
tDolan, T. J., University of Illinois Engineering Experiment Station, Bulletin 293, 1937.
tArmbruster, E., "Einfluss der Oberflichenbeschaffenheit auf den Spannungsverlauf und die
Schwingungsfestigkeit," Ver. Deutsch. Ing. Verlag 1931.
I¶Mailander, R. and Bauersfeld, W., "Einfluss der Probengrosse und Probenform auf die Dreh-
Schwingungsfestigkeit von Stahl," Technische Mitteilungen Krupp, Vol. 2, Dec. 1934, pages 143-152.
one of 1.77 inch. The effect of size should be kept in mind when use
is made of the results of these tests.
12. Conclusions.-The following conclusions may be drawn:
(1) For ranges of torsional shearing stress other than completely
reversed cycles of stress, steel is likely to cease to perform satisfac-
torily as a structural or machine member by either one of two types
of failure. First, for some ranges of stress it may fail by developing
large permanent deformation at the torsional static yield strength
before the endurance limit stress is reached. Second, for some ranges
of stress it may fail by progressive fracture (fatigue) at a nominal or
calculated stress below the torsional static yield strength. These
two types of failure are illustrated in Fig. 5. Therefore, endurance
limits for various ranges of stress have little significance if their values
exceed the static elastic strength of the steel.
(2) For small polished specimens free from stress concentrations,
the torsional shearing endurance limit of S.A.E. 3140 steel in both
RANGE OF STRESS ON FATIGUE STRENGTH OF STEEL
the hot-rolled (as received condition) and the quenched and tem-
pered condition follows the constant range relation; that is, the
magnitude of any endurance range of stress is constant and equal
to the magnitude of the endurance range of stress for completely
reversed cycles of stress, provided that the maximum stress in the
range does not exceed the torsional static yield strength of the steel.
In order to possess a high endurance limit for various ranges of
torsional shearing stress a steel which is to be used without stress
concentration must also possess a high static elastic strength in
addition to a high endurance limit for completely reversed cycles of
stress. Therefore, heat-treated steels are particularly desirable for
use for ranges of stress other than completely reversed cycles of
stress, since heat-treatment usually raises the static strength prop-
erties more than it does the fatigue strength properties. The facts
on which this conclusion is based are shown by the diagrams of
Figs. 11, 12 and 13(b) and by Formulas (I) and (II) of the summary.
(3) For small specimens with a stress concentration caused by a
small transverse hole the torsional shearing endurance limit of
S.A.E. 3140 steel in both the hot-rolled (as received) condition and
the heat-treated condition is not independent of the range of stress.
The magnitude of the endurance range of stress is decreased as the
maximum stress in the range increases, that is, the difference between
the maximum and minimum stresses of the endurance range de-
creases as the maximum stress is increased. The effect of the range
of stress may be shown satisfactorily by any one of three methods of
interpretation of the test data. The facts on which this conclusion
is based are shown by the diagrams of Figs. 10, 12 and 13(a) and by
Formulas (III-VI), inclusive, in the summary. These formulas and
diagrams indicate that steels with high usable torsional shearing
endurance limit for various ranges of stress must possess high elastic
strengths as well as relatively low notch sensitivity; this last require-
ment insures a relatively high endurance limit for completely reversed
cycles of torsional shearing stress.
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by Hardy Cross. 1936. Thirty-five cents.
Circular No. 26. Papers Presented at the First Annual Conference on Air
Conditioning, Held at the University of Illinois, May 4 and 5, 1936. Fifty cents.
Reprint No. 6. Electro-Organic Chemical Preparations, by S. Swann, Jr. 1936.
Thirty-five cents.
Reprint No. 7. Papers Presented at the Second Annual Short Course in Coal
Utilization, Held at the University of Illinois, June 11, 12, and 13, 1935. 1936.
None available.
Bulletin No. 287. The Biologic Digestion of Garbage with Sewage Sludge, by
Harold E. Babbitt, Benn J. Leland, and Fenner H. Whitley, Jr. 1936. One dollar.
Reprint No. 8. Second Progress Report of the Joint Investigation of Fissures
in Railroad Rails, by Herbert F. Moore. 1936. Fifteen cents.
Reprint No. 9. Correlation Between Metallography and Mechanical Testing,
by Herbert F. Moore. 1936. Twenty cents.
Circular No. 27. Papers Presented at the Twenty-third Annual Conference on
Highway Engineering, Held at the University of Illinois, Feb. 26-28, 1936. 1936.
Fifty cents.
Bulletin No. 288. An Investigation of Relative Stresses in Solid Spur Gears by
the Photoelastic Method, by Paul H. Black. 1936. Forty cents.
Bulletin No. 289. The Use of an Elbow in a Pipe Line for Determining the Rate
of Flow in the Pipe, by Wallace M. Lansford. 1936. Forty cents.
Bulletin No. 290. Investigation of Summer Cooling in the Warm-Air Heating
Research Residence, by Alonzo P. Kratz, Maurice K. Fahnestock, and Seichi Konzo.
1937. One dollar.
Bulletin No. 291. Flexural Vibrations of Piezoelectric Quartz Bars and Plates,
by J. Tykocinski Tykociner and Marion W. Woodruff. 1937. Forty cents.
Reprint No. 10. Heat Transfer in Evaporation and Condensation, by Max
Jakob. 1937. Thirty-five cents.
Circular No. 28. An Investigation of Student Study Lighting, by John 0.
Kraehenbuehl. 1937. Forty cents.
Circular No. 29. Problems in Building Illumination, by John 0. Kraehenbuehl.
1937. Thirty-five cents.
Bulletin No. 292. Tests of Steel Columns; Thin Cylindrical Shells; Laced
Channels; Angles, by Wilbur M. Wilson. 1937. Fifty cents.
Bulletin No. 293. The Combined Effect of Corrosion and Stress Concentration
at Holes and Fillets in Steel Specimens Subjected to Reversed Torsional Stresses,
by Thomas J. Dolan. 1937. Fifty cents.
Bulletin No. 294. Tests of Strength Properties of Chilled Car Wheels, by
Frank E. Richart, Rex L. Brown, and Paul G. Jones. 1937. Eighty-five cents.
Bulletin No. 295. Tests of Thin Hemispherical Shells Subjected to Internal
Hydrostatic Pressure, by Wilbur M. Wilson and Joseph Marin. 1937. Thirty cents.
Circular No. 30. Papers Presented at the Twenty-fourth Annual Conference on
Highway Engineering, Held at the University of Illinois, March 3-5, 1937. 1937.
None available.
Reprint No. 11. Third Progress Report of the Joint Investigation of Fissures
in Railroad Rails, by H. F. Moore. 1937. Fifteen cents.
Bulletin No. 296. Magnitude and Frequency of Floods on Illinois Streams, by
George W. Pickels. 1937. Seventy cents.
Bulletin No. 297. Ventilation Characteristics of Some Illinois Mines, by Cloyde
M. Smith. 1937. Seventy cents.
Bulletin No. 298. Resistance to Heat Checking of Chilled Iron Car Wheels,
and Strains Developed Under Long-Continued Application of Brake Shoes, by
Edward C. Schmidt and Herman J. Schrader. 1937. Fifty-five cents.
tCopies of the complete list of publications can be, obtained without charge by addressing the
Engineering Experiment Station, Urbana, Ill.
ILLINOIS ENGINEERING EXPERIMENT STATION
Bulletin No. 299. Solution of Electrical Networks by Successive Approxima-
tions, by Laurence L. Smith. 1937. Forty-five cents.
Circular No. 31. Papers Presented at the Short Course in Coal Utilization,
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Bulletin No. 300. Pressure Losses Resulting from Changes in Cross-Sectional
Area in Air Ducts, by Alonzo P. Kratz and Julian R. Fellows. 1938. Sixty-five cents.
Bulletin No. 301. The Friction of Railway Brake Shoes at High Speed and High
Pressure, by Herman J. Schrader. 1938. Sixty cents.
Bulletin No. 302. Fatigue Tests of Riveted Joints, by Wilbur M. Wilson and
Frank P. Thomas. 1938. One dollar.
Circular No. 32. Two Investigations on Transit Instruments, by William H.
Rayner. 1938. Twenty-five cents.
Circular No. 33. Papers Presented at the Twenty-fifth Annual Conference on
Highway Engineering, Held at the University of Illinois, March 2-4, 1938. 1938.
None available.
Bulletin No. 303. Solutions for Certain Rectangular Slabs Continuous Over
Flexible Supports, by Vernon P. Jensen. 1938. One dollar.
Bulletin No. 304. A Distribution Procedure for the Analysis of Slabs Continuous
Over Flexible Beams, by Nathan M. Newmark. 1938. One dollar.
Circular No. 34. The Chemical Engineering Unit Process-Oxidation, by
Donald B. Keyes. 1938. Fifty cents.
Circular No. 35. Factors Involved in Plate Efficiencies for Fractionating
Columns, by Donald B. Keyes. 1938. Twenty cents.
Bulletin No. 305. Summer Cooling in the Warm-Air Heating Research Resi-
dence with Cold Water, by Alonzo P. Kratz, Seichi Konzo, Maurice K. Fahnestock
and Edwin L. Broderick. 1938. Ninety cents.
Bulletin No. 306. Investigation of Creep and Fracture of Lead and Lead Alloys
for Cable Sheathing, by Herbert F. Moore, Bernard B. Betty, and Curtis W. Dollins.
1938. One dollar.
Reprint No. 12. Fourth Progress Report of the Joint Investigation of Fissures
in Railroad Rails, by H. F. Moore. 1938. None available.
Bulletin No. 307. An Investigation of Rigid Frame Bridges: Part I, Tests of
Reinforced Concrete Knee Frames and Bakelite Models, by Frank E. Richart,
Thomas J. Dolan, and Tilford A. Olson. 1938. Fifty cents.
Bulletin No. 308. An Investigation of Rigid Frame Bridges: Part II, Labora-
tory Tests of Reinforced Concrete Rigid Frame Bridges, by W. M. Wilson, R. W.
Kluge, and J. V. Coombe. 1938. Eighty-five cents.
Bulletin No. 309. The Effects of Errors or Variations in the Arbitrary Con-
stants of Simultaneous Equations, by George H. Dell. 1938. Sixty cents.
Bulletin No. 310. Fatigue Tests of Butt Welds in Structural Steel Plates, by
W. M. Wilson and A. B. Wilder. 1939. Sixty-five cents.
*Bulletin No. 311. The Surface Tensions of Molten Glass, by Cullen W.
Parmelee, Kenneth C. Lyon, and Cameron G. Harman. 1939. Fifty-five cents.
*Bulletin No. 312. An Investigation of Wrought Steel Railway Car Wheels:
Part I, Tests of Strength Properties of Wrought Steel Car Wheels, by Thomas J.
Dolan and Rex L. Brown. 1939. Seventy cents.
Circular No. 36. A Survey of Sulphur Dioxide Pollution in Chicago and
Vicinity, by Alamjit D. Singh. 1939. Forty cents.
*Circular No. 37. Papers Presented at the Second Conference on Air Condition-
ing, Held at the University of Illinois, March 8-9, 1939. 1939. Fifty cents.
*Circular No. 38. Papers Presented at the Twenty-sixth Annual Conference on
Highway Engineering, Held at the University of Illinois, March 1-3, 1939. 1939.
Fifty cents.
*Bulletin No. 313. Tests of Plaster-Model Slabs Subjected to Concentrated
Loads, by Nathan M. Newmark and Henry A. Lepper, Jr. 1939. Sixty cents.
*Bulletin No. 314. Tests of Reinforced Concrete Slabs Subjected to Concen-
trated Loads, by Frank E. Richart and Ralph W. Kluge. 1939. Eighty cents.
*Bulletin No. 315. Moments in Simple Span Bridge Slabs with Stiffened Edges,
by Vernon P. Jensen. 1939. One dollar.
*Bulletin No. 316. The Effect of Range of Stress on the Torsional Fatigue
Strength of Steel, by James 0. Smith. 1939. Forty-five cents.
*A limited number of copies of bulletins starred are available for free distribution.
UNIVERSITY OF ILLINOIS
Colleges and Schools at Urbana
COLLEGE OF LIBERAL ARTS AND SCIENCES.-General curriculum with majors in the hu-
manities and sciences; specialized curricula in chemistry and chemical engineering;
general courses preparatory to the study of law and journalism; pre-professional
training in medicine, dentistry, and pharmacy.
COLLEGE OF COMMERCE AND BUSINESS ADMINISTRATION.-Curricula in general business,
trade and civic secretarial service, banking and finance, insurance, accountancy,
transportation, commercial teaching, foreign commerce, industrial administration,
public utilities, and commerce and law.
COLLEGE OF ENGINEERING.-Curricula in agricultural engineering, ceramics, ceramic en-
gineering, chemical engineering, civil engineering, electrical engineering, engineer-
ing physics, general engineering, mechanical engineering, metallurgical engineering,
mining engineering, and railway engineering.
COLLEGE OF AGRICULTURE.-Curricula in agriculture, floriculture, general home econom-
ics, and nutrition and dietetics.
COLLEGE OF EDUCATION.-Curricula in education, agricultural education, home econom-
ics education, and industrial education. The University High School is the practice
school of the College of Education.
COLLEGE OF FINE AND APPLIED ARTS.-Curricula in architecture, art, landscape architec-
ture, and music.
COLLEGE OF LAW.-Professional curriculum in law.
SCHOOL OF JOURNALISM.-General and special curricula in journalism.
SCHOOL OF PHYSICAL EDUCATION.-Curricula in physical education for men and for
women.
LIBRARY ScHooL.-Curriculum in library science.
GRADUATE ScHOOL.-Advanced study and research.
Summer Session.-Courses for undergraduate and graduate students.
University Extension Division.-Courses taught by correspondence, extramural courses,
speech aids service, and visual aids service.
Colleges in Chicago
COLLEGE OF MEDICINE.-Professional curriculum in medicine.
COLLEGE OF DENTISTRY.-Professional curriculum in dentistry.
COLLEGE OF PHARMACY.-Professional curriculum in pharmacy.
University Experiment Stations, and Research and
Service Organizations at Urbana
AGRICULTURAL EXPERIMENT STATION BUREAU OF BUSINESS RESEARCH
ENGINEERING EXPERIMENT STATION BUREAU OF COMMUNITY PLANNING
EXTENSION SERVICE IN AGRICULTURE BUREAU OF EDUCATIONAL RESEARCH
AND HOME ECONOMICS BUREAU OF INSTITUTIONAL RESEARCH
RADIO STATION (WILL) UNIVERSITY OF ILLINOIS PRESS
State Scientific Surveys and Other Divisions at Urbana
STATE GEOLOGICAL SURVEY STATE DIAGNOSTIC LABORATORY (for
STATE NATURAL HISTORY SURVEY Animal Pathology)
STATE WATER SURVEY U. S. SOYBEAN PRODUCTS LABORATORY
For general catalog of the University, special circulars, and other information, address
THE REGISTRAR, UNIVERSITY OF ILLINOIS
URBANA, ILLINOIS