A STUDY OF STRESSES IN CAR AXLES UNDER
SERVICE CONDITIONS
I. INTRODUCTION
1. Introductory.A study of failure of car axles has been made by
the Engineering Experiment Station of the University of Illinois
under a co6perative agreement with the Utilities Research Commis
sion. To date two bulletins* have been published in which the
problem of fatigue cracks in car axles was specifically studied.
The present bulletin is a study of the stresses occurring in a car
axle under service conditions with special reference to the life of the
axle, the magnitude, duration, occurrence per mile, and cause of stress.
In view of the large amount of mathematical computation presented
it is suggested that a preliminary idea of the results of the investigation
may be gained by reading first Section 3 and then the conclusions,
Chapter IV.
2. Acknowledgments.This study has been supported by funds
contributed by the Utilities Research Commission, Inc., WILLIAM L.
ABBOTT, President. An Advisory Committee was appointed as
follows:
H. A. JOHNSON (Chairman), General Manager, Chicago
Rapid Transit Company
E. A. BRANDT, Asst. to VicePresident, Ice Department,
Middle West Utilities Company
A. H. DAUS, Supt. of Shops & Equipment, Chicago Rapid
Transit Company
G. E. TEBBETTS, Engineer of Structures, Chicago Rapid
Transit Company
R. N. WADE, Engineer of Maintenance of Way, Chicago
Rapid Transit Company
This committee has acted as an advisory committee for all the
work reported in this bulletin, and several meetings have been held
to consider the progress of the work. The tests described in this bul
letin were carried out on the Rapid Transit System in Chicago, Illi
nois. The whole study has been closely allied with the previous
investigations of fatigue cracks in car axles.
The field tests in Chicago were greatly accelerated by the co6p
eration of the Shops & Equipment Department, Chicago Rapid
*Univ. of Ill. Eng. Exp. Sta. Bul. 165, 1927, and Bul. 197, 1929.
7
ILLINOIS ENGINEERING EXPERIMENT STATION
Transit Company, A. H. DAUS, Superintendent; L. C. ALMY, Master
Mechanic of the Wilson Avenue Shops; and KENT WOOLRIDGE,
appointed by the superintendent to assist with the tests. Special
acknowledgment is made to MR. S. W. LYON, formerly Research
Assistant with this investigation, for the preliminary design of the
strainrecording apparatus used, and to MR. E. C. BAST, student
assistant with the investigation, and to MR. M. K. SHAFER, instru
ment builder. Acknowledgment is made to PROF. H. R. THOMAS of
the Department of Theoretical and Applied Mechanics and to
PROF. J. K. TUTHILL of the Department of Railway Engineering for
valuable suggestions. MR. W. L. SCHWALBE of the Department of
Theoretical and Applied Mechanics rendered valuable assistance in
connection with the mathematical analyses of stresses in car axles.
The investigation has been carried on as a part of the work of the
Engineering Experiment Station at the University of Illinois and has
been under the general administrative direction of DEAN M. S.
KETCHUM, director of the Engineering Experiment Station, and of
PROF. M. L. ENGER, head of the Department of Theoretical and
Applied Mechanics.
3. Summary of Work with Car Axle Steel.The following para
graphs summarize the conclusions reached in the earlier part of the
investigation concerning the fatigue strength of axle steel.
"By means of simple methods available for railroad shops, fatigue cracks in car
axle steel were detected before complete failure of the specimen occurred. If the
specimen was subjected to a stress but little above its endurance limit, the crack
could be detected before onehalf the "life" of the specimen under repeated stress
had passed. If the axle was subjected to a very high stress repeated many times,
the crack was not detected until failure was imminent.
"From the results of tests on two sizes of specimens, it appeared that the size of
the smallest crack which could be detected is about the same for small specimens as
for large specimens. Hence when the crack is first detectable in a small specimen a
larger proportion of the metal has been damaged than is the case when the crack is
first detectable in a large specimen. For a small specimen, then, there will remain a
shorter proportion of the "life" between the detection of a crack and final failure than
is the case for a larger specimen. This indicates that the chances of detecting a
fatigue crack before failure is imminent is better for fullsize axles than for small
specimens.
"Test specimens in which a fatigue crack had been detected, and had spread to
a definite length, showed a continuing spread of such a crack to failure, under sub
sequent cycles of stress having a magnitude of 64 per cent of the endurance limit of
the virgin steel; but such a crack in a test specimen did not spread further under
cycles of stress having a magnitude of 50 per cent of the endurance limit of the
virgin steel.
"For the axles tested the range of strength values of the steel was found to be as
follows (in pounds per square inch):
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
"Proportional Elastic Limit, from 40 800 to 53 500
"Ultimate Tensile Strength, from 91 700 to 105 100
"Endurance Limit for an indefinitely large number of reversals of Flexural
Stress (Tests of small 'Farmer' specimens) about 35 000."
4. Scope of Bulletin.This bulletin deals with the data and re
sults of field tests conducted in Chicago, Illinois, on the Chicago
Rapid Transit System. Bending in one or two axles of a test car was
recorded in a series of tests covering about two hundred miles of
elevated and surface track. A continuous record was obtained of the
bending of an axle of the test car. The stress, in pounds per square
inch, corresponding to this measured bending was computed by the
use of the ordinary formulas, and by the substitution of proper
constants for readings of the continuous record. The derivation of
these stress constants, together with an analytical discussion of the
effects of the action of frogs, crossings, and other parts of track on
the stress in the axle, is therefore a necessary and an integral part of
this bulletin, and is given in Chapter IV, and in Appendix B.
There are two chief divisions into which the data have been
divided: (1) those data relating to the number of occurrences per
mile of any given stress, and particularly to the number of occur
rences per mile of stresses higher than the normal stress, and (2) those
data which show the effects of guard rails, frogs, crossings, or other
parts of track on the magnitude of stress in the axle.
II. INSTRUMENTS AND RECORDING METHODS USED IN TESTS
5. Instrument to Measure Bending in Car Azles.In order to
measure the amount of bending in a car axle, it was actually necessary
to measure the changes in slope between two definite crosssections
of the axle under bending moment. It was decided to use a tele
scoping tube placed between the inside flanges of a pair of wheels.
This measures the change in inclination of the car wheels and there
fore gives the required change in slope of the axle.
Mr. S. W. Lyon, Engineer of Tests, Fatigue of Metals Laboratory,
until May 1930, was responsible for the design of the telescoping
tube. This instrument is shown in Figs. 1 and 2. It consists of a
telescoping tube T supported above the axle, and held with its ends
against the inner sides of the wheel flanges by means of a spring S.
It was afterward found necessary to install a spring on the other end
of the tube to overcome inertia effects. The whole tube T is free to
slide axially in the bearings B' and B", so that any change in the
length of the tube T will be a measure of the relative inward or
ILLINOIS ENGINEERING EXPERIMENT STATION
Fro. 1. INSTRUMENT FOR MEASURING CHANGE IN INCLINATION OF CAR WHEELS
FIG. 2. RECORDING INSTRUMENT OF FIG. 1, IN POSITION
outward motion of the flanges. The bearing of the ends of the tube T
on the flanges is through ball bearings L' and L". A gear box is an
integral part of the instrument. Its function and action are described
in detail in the following section.
6. Recording System Adopted.It was desired to obtain a contin
uous record of bending of the axle as the test car moved along the
track under test speeds or in regular traffic. The instrument de
scribed in Section 5 was installed between the inner sides of the flanges
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
Fra. 3. DIAGRAMMATIC SKETCH OF RECORDING SYSTEM
of a pair of wheels which had been turned smooth in a lathe. Figure 2
is a view of this instrument in position above the axle with the car
body removed from the truck. Figure 3 is a diagrammatic sketch
of the instrument and the recording system used in connection.
The sliding motion of the tube T causes a rack R to rotate a pinion P';
on the same shaft as P' is gear G'. Gear G' rotates pinion P", and on
the same shaft with P" is pointer N which, as it swings, passes near
points Q'. The pointer N carries a high voltage current from one
side of the secondary of a 3in. induction coil C. Each of the points
shown at Q' is one end of a wire and is connected to a point in the
series Q", the other end of the wire. Over the points at Q" a sheet of
paper is drawn in a direction perpendicular to the plane of the draw
ing. A brass bar A extends above the paper along the line of points
Q" and is connected to the other hightension terminal of spark coil C.
As the car travels along the track, the pointer N moves as the wheel
flanges are inclined by the bending of the axle, and a spark jumps
from pointer N to the nearest point on series Q', thence to the corre
sponding point on Q", then through the paper, recording a small hole
with a slightly scorched edge. In the actual apparatus there are 50
points in the series Q' and Q".
The 3in. induction coil may be seen in Fig. 4 on the bracket at
the right side of the recording table. The induction coil was equipped
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 4. VIEW OF RECORDING TABLE IN TEST CAR
with an adjustable mechanical circuit breaker, allowing the interval
of make and break, or frequency, of the primary current to be changed
at will. Usually the frequency used in the tests was between 50 and
60 per second. The voltage of the secondary probably was as high
as 60 000.
In addition to the primary circuit breaker adjustments it was
found necessary to vary the spark gap between the pointer N and
the wires leading to the brass bar on the recording table and also the
gap between the ends of the wires and the brass bar in order to obtain
a good perforation of the paper. Moisture in the air, damp paper,
and frequency of make and break affected these adjustments. The
coil used is a standard commercial product.
The recording table was built in the shop of the Department of
Theoretical and Applied Mechanics, and is similar to the tables used
in the test car of the Railway Electrical Department of the Univer
sity. The paper was driven by a constantspeed 32volt d.c. motor.
For the tests, the current was supplied by storage batteries carried
in the test car.
7. Accelerometers.The values of the stress constants used to
convert ordinates of the paper record to stress in the axle depended
upon the magnitude, location, and direction of the forces causing the
bending of the axle. In order to aid in determining the nature and
direction of these forces three accelerometers were designed. One
instrument recorded acceleration of the test car along the track,
one acceleration crosswise to the track, and the third vertical accel
eration. These accelerometers were used for qualitative but not
quantitative results.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
These instruments, shown in Fig. 4, consist of cubical steel blocks
attached to the ends of flat steel springs. The springs pass through
the center of the blocks. The oscillations of all three accelerometers
were recorded on the paper record by pens, and the magnitude of the
oscillation is roughly proportional to the acceleration. By means of
bent levers and supporting arms these pens recorded each oscillation
of the accelerometer at right angles to the direction of travel of the
paper.
8. Speedometer.The speedometer used in all tests was a Weston
Model 44, Type C Magneto driven from a split pulley, mounted on
one of the trailer axles of the test car, by means of a spring steel belt.
The indicating instrument was graduated in miles per hour.
9. Distance and Event Markers.Two small electromagnets were
built to operate two recording pens which were used to locate the
test car on the track.
A circuit breaker was attached to the split pulley which was used
to operate the speedometer. At every revolution of the trailer wheel,
the circuit breaker closed and sent a current through an electro
magnet which in turn moved a recording pen in a direction at right
angles to the paper movement. By this means a Vshaped mark was
made on the paper record for every 7.39 feet of movement of the
test car along the track.
The other electromagnet was used as an event marker recording
the location of frogs, switches, stations, etc. It was controlled by a
handoperated key. The recording points of all the recording instru
ments were in line at right angles to the direction of movement of
the paper so that records could be correlated.
10. DistortionofWheel Test Method.The change in inclination
of the car wheels was recorded by the method outlined in Section 6.
This change in inclination is a true measure of the increase or de
crease of the change in slope of the axle provided the wheel does not
move on the axle or the wheel itself does not change in shape due to
applied live loads. Tests were conducted in the shops of the Elevated
Railway to throw light upon the latter phase of the problem.
The apparatus used to determine the amount of distortion of the
car wheels due to loads applied on the inside flanges is shown in Fig. 5.
It consisted of two frames attached to the ends of the axle; on these
frames Ames dials reading to one thousandth of an inch were mounted.
The recording instrument described in Section 5, with an auxiliary
attachment described in Section 11, was used also. A hydraulic jack
ILBINOIS ENGINEERING EXPERIMENT STATION
FIG. 5. APPARATUS FOR MEASURING DISTORTION OF CAR WHEEL
FIG. 6. AUXILIARY STRAIGHTEDGE FOR MEASURING DEFLECTION OF CAR AXLE
was placed horizontally between the inside flanges of the wheels.
Pressures were then applied in varying increments. All Ames dials
were read at each change of load. The spread of the wheels outward
at the bottom was read to the nearest one hundredth of an inch.
A second series of tests was made in which it was possible to read
the outward movement of the bottom of the wheels, as well as the
inward movement at the top of the wheels, to one thousandth of an
inch. An auxiliary straightedge, shown in Fig. 6, was used in this
test to obtain a check on the bending of the axle under this type of
loading. The change in slope between two given points on the axle
as calculated from the observed readings of the autographic record,
agreed within 1.25 per cent with the change in slope between the
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
FIG. 7. AUXILIARY ATTACHMENT TO RECORDING INSTRUMENT
same points calculated from the readings of the dial gage on the
straightedge instrument. This agreement shows that the axle bends
elastically in accordance with the theory of flexure for loads applied
laterally to the wheels. The results indicated that while there was a
slight distortion of the bottom half of a wheel, no appreciable dis
tortion of the top half was detected. The recording instrument used
in all of the test runs measures the change in inclination of the top
halves of a pair of wheels on an axle.
11. Method of Checking Bending in Axles.The maximum read
ings of the recording instrument described in Section 5 were checked
by means of a simple instrument which consisted of two brackets.
One bracket, on which was mounted an Ames dial, was attached to
the telescoping end of the instrument, the other to the gear box, as
shown in Fig. 7. By this means a number of maximum readings of
inward and outward movements of the wheels were obtained to
compare with the wheel movements as recorded on the paper record.
12. Special Telescoping Tube for Studying Forced Vibrations in
Wheels.After a few test runs had been made in which the recording
instrument described in Section 5 was used, it became evident that
the paper record was not a true measure of the change in inclination
of the wheels. It appeared that forced vibrations of the wheel caused
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 8. SPECIAL TELESCOPING TUBE FOR STUDYING FORCED VIBRATIONS
IN THE WHEELS
the recording instrument to move inward a distance greater than the
inward movement of the wheel flange. The original light spring built
into the instrument was not heavy enough to overcome or prevent
this inertia effect imparted to the mass of the instrument. In order
to study how to eliminate inertia effects in this instrument a special
telescoping tube was built.
This special instrument is shown in Fig. 8. It consists of a steel
tube, one inch inside diameter, with a ball bearing contact end as in
the instrument described in Section 5. The other end of the tube
was fitted with a closelyfitting round brass rod with a ball bearing
contact end. A clamp retarded the sliding of the brass rod inside the
tube as the wheels changed inclination. The first record of the change
of wheel inclination was made using an Ames dial as described in
Section 11. It was found that regardless of the pressure exerted by
the clamp between the brass rod and the tube, the head of the instru
ment always moved farther than the flange of the wheel. This pre
liminary test was made with about a 100lb. spring pressure against
one wheel.
The clamp of the telescoping tube was removed and a very heavy
spring substituted. The light spring also was replaced by a very
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
FIG. 9. TEST CAR
heavy spring. The Ames dial was replaced by a piano wire enclosed
in a coil spring similar to flexible steel belting. The piano wire and
coil spring were enclosed in a copper tube. One end terminated on
top of the recording table inside the test car and operated a recording
pencil. Thus the change in inclination of the wheels was recorded
mechanically and directly on the paper record without magnification.
From the result of these special tests it was possible to choose such
springs for application to the axle bending apparatus used in the
regular tests that vibration effects on the instrument were reduced
to an inappreciable amount.
III. TEST PROGRAM AND METHOD OF PROCEDURE
13. Test Car.The car selected as a test car was a motor passen
ger car with a steel body, 48 ft. in length overall, 33 ft. 8 in. center to
center of trucks, weighing 76 800 lb. Cars 4419 and 4438 are shown
in Fig. 9. Both cars are of the same type and weight. Car 4419 was
used as the test car. The trailer wheels were 31 in. and the motor
driven wheels 34 in. in diameter. The axle under test was a motor
driven axle, 6 in. in diameter.
14. Preparation of Wheels and Axle.The immediate object of the
tests was to measure the change in inclination of the wheels due to the
change in slope of the axle. This change in inclination has been
termed "wheel movement" throughout this bulletin. It was impor
tant that the inside flanges of both wheels have smooth surfaces and
ILLINOIS ENGINEERING EXPERIMENT STATION
that the planes of these surfaces be as nearly parallel as possible before
the initial loading. These objects were effected by machining the
inside flanges in a large lathe, using successively finer cuts as the
metal was removed.
The majority of runs were made with the recording instrument
above No. 1 axle* and in contact with No. 1 and No. 2 wheels.
Auxiliary runs were made with the recording instrument above No. 2
axle and in contact with No. 3 and No. 4 wheels. Both pairs of
wheels were prepared in the same manner.
The dimensions of the wheels and axle were recorded, including
the position of motor and journal loads, the clearances between the
ends of the axle and the end plate in the journal box, and the clear
ances between the motor bearings and the hubs of the wheels.
15. Installation of Recording Instrument.The instrument de
scribed in Chapter II, Section 5, was installed above axle No. 1 by
means of brackets attached to the motor housing. Figure 2 shows the
instrument in position. Each end roller of the telescoping tube of the
recording instrument makes contact with the smooth surface of the
inner side of a wheel. The supports of the instrument are brass
bearings, one being a twopoint bearing, the other a singlepoint
bearing. The instrument is free to move in the bearings in an axial
direction.
16. Loading of Test Car and Calibration of Instrument.The
weight of the motor truck was estimated and the resulting motor and
journal loads were calculated. Due to the dead load of the truck
alone there was a small amount of bending in the axle. (See Table 3,
load No. 6.) The recording instrument was placed in position after
this bending had taken place.
The electric recording system was attached to the instrument and
a line of perforations in the paper was obtained which corresponded
to the position of the wheels for dead load of truck only.
The body of the test car was then set on the truck and another
line of perforations was obtained corresponding to the new position
of the wheels. The difference between the two lines of perforations
gave the effect of the dead load of the car body.
The ordinate of the paper record between these two lines of per
forations corresponded to the change in slope of the axle due to the
difference in loading for the two cases.
Other loadings were used, brake shoes being distributed over the
*No. 1 Axle is the motor axle nearest the end of the car; No. 2 Axle is next to No. 1 Axle and is
also a motor axle.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
TABLE 1
SCHEDULE OF LOADING OF TEST CAR
Date
91630...............
91630...............
91630...............
91630...............
91630...............
111831...............
111831..............
Load
Number
1
2
3
4
5
6
7
Total Load of
Car and Cargo
lb.
77 960
80 960
82 710
89 960
92 660
21 380*
92 660
Estimated Journal
Load Due to Motor
Truck Alone
lb.
3745
3745
3745
3745
3745
3745
3745
Estimated
Journal
Loadt
lb.
5122.5
5497.5
5716.25
6622.5
6960.0
3745.0
10705.0
*Weight of motor truck only.
tThese journal loads used to obtain change in slope of axle and wheel movement.
TABLE 2
CALIBRATION OF RECORDING INSTRUMENT
Date
91530 ....................
91530 ....................
91530 ....................
91530....................
91530 ....................
91530 ....................
91530 ....................
91530....................
91530....................
91530 ....................
91530....................
91630 ...................
91630....................
91630....................
91630 ...................
91630 ...................
91630 ...................
91630 ...................
91630 ....................
111930 ....................
111930 ....................
111930 ...................
111930 .. .............. .
111930 .. .................
111930 ...................
5 131....................
Axial
Movement of
Instrument
in.
0.01
0.015
0.020
0.025
0.030
0.035
0.045
0.050
0.060
0.075
0.095
0.01
0.04
0.08
0.136
0.258
0.370
0.500
0.628
0.091
0.136
0.258
0.371
0.499
0.632
0.022
Corresponding Move
ment on Paper
Record
in.
(3)
0.38
0.57
0.72
0.93
1.17
1.33
1.70
1.87
2.33
2.94
3.57
0.38
1.50
3.05
5.00
9.60
14.10
18.65
22.80
3.51
5.18
9.27
13.74
18.72
23.74
0.80
Average .............. ... ................... . ...............
Wheel Movement* of
Test Car per Inch of
Paper Record
in.
Col. (2)  Col. (3)
(4)
0.0263
0.0263
0.0278
0.0269
0.0256
0.0263
0.0265
0.0267
0.0258
0.0255
0.0266
0.0263
0.0267
0.0262
0.0272
0.0269
0.0262
0.0268
0.0275
0.0259
0.0263
0.0278
0.0270
0.0267
0.0266
0.0275
0.0266f
*Wheel movement signifies the change in inclination of the wheels due to the change in slope
of the axle.
tThis value of wheel movement per inch of paper record was used in determining all stress
constants.
floor of the car to give additional weight. Loads were increased, in
varying increments, until a total of about 15 000 lb. of weight was
distributed over the floor of the car. This is, approximately, the
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 3
RELATION BETWEEN CHANGE IN SLOPE OF AXLE AND WHEEL MOVEMENT
Comparison of calculated wheel movement and observed or measured wheel movement
Distance
from Center Calculated Observed
Calculated of Axle to Wheel Wheel
Date Load Change in Center of Movementt MovementS
Number Slope of Recording in.
Axle* Instrument Col. 3 X
in. Col. 4 in.
(1) (2) (3) (4) (5) (6)
91530......... 6 3238.8527 X 107 15.312 0.0050 ......
91530 ........ 5 14564.658 X 107 15.312 0.0223 ..
11431........ 6 3238.8527 X 107 15.375 0.0050 0.0045
11431........ 5 14564.658 X 10' 15.375 0.0224 0.024
111831........ 7 17803.5107 X 107 15.375 0.0274 0.028
*The changes in slope of the axle under test recorded in all tables of this bulletin correspond to
that part of the axle between section a and section b, see Fig. 12.
tWheel movement signifies the change in inclination of the wheels due to the change in slope of
the axle.
SMeasured with an Ames dial reading to one thousandth of an inch.
TABLE 4
COMPARISON OF CALCULATED AND OBSERVED PAPER RECORDS
Calculated
Change in
Slope of
Axle
(3)
3238.8527 X 107
14564.658 X 107
10719.4 X 107
11504.1 X 10'
11961.9 X 107
13858.3 X 107
14564.658 X 107
14564.658 X 107
14564.658 X 107
17803.5107 X 107
Distance
from
Center of
Axle to
Center of
Recording
Instrument
in.
(4)
15.312
15.312
15.312
15.312
15.312
15.312
15.875
15.375
16.17
15.375
Calculated
Wheel
Movement
in.
Col. 3 X
Col. 4
(5)
0.0050
0.0223
0.0164
0.0176
0.0183
0.0202
0.0231
0.0224
0.0234
0.0277
Wheel*
Movement Calculated
per inch of Paper
Paper Record
Record in.
by Cali Col. 5
bration Col. 6
in.
(6) (7)
0.0266 0.1864
0.0266 0.8384
0.0266 0.6170
0.0266 0.6622
0.0266 0.6885
0.0266 0.7977
0.0266 0.8692
0.0266 0.8418
0.0266 0.8854
0.0266 1.029
weight of one hundred passengers. Ordinates on the paper record
were measured for each increment of load.
Calibration of the recording instrument was made with a series of
thin steel blocks from 0.01 to 0.628 in. in thickness. The blocks were
placed between the wheel and one end of the instrument, thus giving
it an axial telescoping movement, and the corresponding ordinate of
Date
(1)
91530....
91530....
91630....
91630 ...
91630 ...
91630 ....
2 731....
11431....
5 631....
111830 ....
*See Table 2.
Observed
Paper
Record
in.
(8)
0.840.93
0.660.71
0.68
0.69
0.81
0.901.05
0.840.90
0.901.05
0.901.05

~
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
the paper record was measured. Results of this calibration showed the
amount of wheel movement corresponding to one inch of ordinate on
the paper record to be 0.0266 in. Table 2 contains these calibration
data. This calibration was made in September, 1930, and repeated
in November, 1930, and May, 1931.
The change in slope between section a and section b, Fig. 12,
the wheel movement, and the ordinate to the paper record were
calculated for each case of loading given in Table 1. These calculated
quantities were then compared with the observed or measured
quantities as shown in Tables 3 and 4.
17. Program of Tests.A trial trip of the test car was made in
order to study the operation of the instruments, the accelerometers,
the recording method, and the amount of bending in the axle. Fol
lowing this trip, a series of runs was made at Niles Center, northwest
Chicago, on the surface, in order to study the effects of speed,
curvature, frogs, switches, difference in clearances, and other features,
on the amount of bending in the axle. At Niles Center there was
available a stretch of yard track about half a mile long, free from
traffic, with one 19deg. curve without restraining rail, a number of
crossovers, frogs, and switches, and a considerable length of
rough track.
On a trip of the test car from the Loop to Jackson Park, an ab
normally high paper ordinate was recorded on one curve. A number
of subsequent runs also revealed high paper ordinates. It seemed
impossible that these ordinates were true measures of the wheel
movements or the bending in the axle. A study of the cause of these
apparently abnormal readings indicated that under certain conditions
of track and above certain speeds of car the wheel was forced to
vibrate. The frequency of this vibration at times equalled and
exceeded the natural frequency of the spring holding the instrument
against the wheels. When this occurred, the recording instrument
was given a series of sudden axial movements and at such a high
frequency that the spring of the recording instrument was unable to
overcome the inertia effects of the moving instrument. This action
was eliminated by replacing the instrument spring by a spring with a
higher frequency of vibration and by installing another spring on the
other end of the instrument. A detailed study of this subject is
presented in Appendix C.
A series of final runs was begun on February 8, 1931 and com
pleted May 15, 1931. About two hundred miles of track was included.
ILLINOIS ENGINEERING EXPERIMENT STATION
IV. MATHEMATICAL ANALYSES
18. General Case of Bending in Car Axles.The value of the fiber
stress at any section in an axle subject to bending may be found from
the bending moment and the section modulus. The section modulus
may be determined from measured diameters of the axle. The bend
ing moments due to static loading may be determined directly as it
is not difficult to estimate the magnitude and positions of loads and
reactions for this case. Bending moments due to live loads cannot
be determined directly as in the case of dead loads.
In order to study the problem of bending in car axles, and more
particularly to obtain the bending moments, it is necessary to know
the change in slope of the elastic curve between any two points on
the axle. The change in slope of the elastic curve between any two
points on an axle subject to bending is given by the area under the
M
 diagram between the two points in question.*
El
It was not possible to measure directly the change in slope of the
axle used in the field tests in Chicago. It was possible, however, to
measure the change in inclination of the car wheels due to the
change in slope of the axle.
The relation between the change in slope of an axle and the change
in inclination of the wheels may be derived in the following manner;
In Fig. 10, line AB represents the elastic curve. Angle 0 repre
sents the change in slope between points P and Q. Distance Y is
the intercept on the vertical at Q of the tangent at P.
Q M QM
Y = (QP  x) dx and similarly Z =  xdx as
p El p El
given in Bulletin 108 of the Engineering Experiment Station of the
University of Illinois.
In Fig. 11, let P and Q be two points on the elastic curve as in
Fig. 10, and located on the center line of the axle at points exactly
below the inner faces of the two wheels. The points P and Q are
located slightly inside the hubs of the wheels on account of the dish
of the wheels. It is assumed that the effect of the wheels on the
amount of bending of the axle may be neglected. Let aa and bb
represent the position of the two wheels before the bending of the
axle, and a'a' and b'b' the positions of the two wheels after bending
of the axle.
*An  diagram is one plotted with values of bending moment divided by product of modulus of
elasticity and moment of inertia of crosssection as ordinates and values of distance along the axle as
abscissas.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
I 7 'j
A B
^ ' "i "
M
FIG. 10. ELASTIC CURVE AND El DIA FIG. 11. CAR AXLE SUBJECTED TO
GRAM OF A BEAM SUBJECTED TO BENDING MOMENT
BENDING MOMENT
It is desired to find the relation between (aa' + bb') and the
change in slope between points Q and P on the elastic curve AB.
Angle DQP = angle a'Qa from similar triangles, and tangent
DP Z
angle DQP = = 
QP QP
fQ M
From Fig. 10, Z = I x dx.
a'a Z
From Fig. 11, tangent angle a'Qa =  =  and
aQ QP
aQ aQ fQ M
aa' = Z  =  x dx.
QP QP JP EI
bP
Similarly bb' = Y  and as bP = aQ,
QP
aQ aQ fQ M
bb' = Y  =   (QP  x) dx
QP QPJP El
aQ i M M
Therefore (aa' + bb') =   (QP  x) dx + x dx
QQP P EI jP El
and (aa' + bb') = aQ  dx.
P El
Hence (aa' + bb'), the total change in distance between the two
wheels, is equal to aQ, the distance from the center of the axle to
the points a and a' multiplied by the change in slope of the elastic
curve of the axle between the points P and Q. Throughout this
bulletin (aa' + bb') is called wheel movement.
19. Causes of Bending in Car Axles.Bending moments causing
flexure in car axles may be caused by (a) dead load of car and cargo,
ILLINOIS ENGINEERING EXPERIMENT STATION
(b) lateral reaction of a restraining rail of curved track against the
flange on the inner side of a wheel, (c) lateral reactions of an outer
running rail of curved track against the flange of a wheel, (d) lateral
reactions of crossings against the inner sides of both wheels, and (e) an
abrupt lateral change in the running rail, such as switch points.
Frogs and their guard rails may have effects on bending of axles
similar to (b), (d), and (e).
Due to the diversity of positions of the points of application, or
lines of action, of the lateral forces of the various cases mentioned in
the preceding paragraph, it was found convenient to divide the
mathematical analyses into seven cases.
20. Classification of Forces Causing Bending in Car Axles.The
field tests conducted in Chicago on the Elevated Railway yielded
several thousand feet of continuous record of bending in an axle.
There were about three hundred thousand ordinates of this record
to be converted into bending stress. In order to facilitate these
calculations, there have been employed seven socalled "cases" of
analyses, one case for dead loads and six cases for live loads.
The case involving dead loads only is discussed in Section 21.
The cases involving the action of one lateral force only, Cases 2
to 5 inclusive, represent the action and effect of a single lateral force
acting on one wheel.
The cause of the force, its magnitude, and line of action must be
determined by inspection of the autographic record. When classified
according to the case to which it belongs, the resulting external
bending moment, the fiber stress in the axle, the change in slope of
the axle, and other information may be obtained from the test record
with the aid of the mathematical analysis for that case.
The bending stress due to the lateral force of each of these four
cases has been calculated and combined with the stress due to dead
load of car and cargo. Appendix B shows in detail the steps by
which these stresses were obtained.
Not all records, however, may be classified under one of these
simple cases. There are many records involving more than one
lateral force. A general discussion of the forces involved is given in
Appendix A.
21. Bending in Car Axle Due to Dead Loads.The axle used in
most of the field tests in Chicago was a motordriven axle, No. 2490
of car No. 4419. The dimensions of this axle are shown in Fig. 12.
Three sets of loads and reactions are shown in Fig. 13. These
loadings are for (1) dead load of truck and motor, which is designated
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
8000
o
0.00004
0.00008
e000
4000
6000
4/i 2' 4/1 S
 63( 0 /.7s.0 a94 2/5'
4 / ;7/6 32.610 / /01 09" 79644
( l .404 i " 
/242O /2/90
2 3 L a456 c 7 8 9 1d // I/ /lb A'4id/7
AMome/7z VGir~rm
'S
No./ Axle and iDagrams
I/ 54
/Z6z I
No. 2 Ax/e
FIG. 12. DIAGRAMS AND DIMENSIONS OF NOS. 1 AND 2 AXLES
Case la, (2) dead load of the remainder of the car plus dead load of
cargo of brake shoes, designated Case Ib, and (3) total dead load of
car and cargo, designated Case Ic.
M M
Table 5 gives moments,  units, areas under the  diagram,
El Ei
and the change in slope between section a and section b for Cases la,
I, ~Stress Viagro'm
v/
 4 5" 44" . " 47, " a
f 01753.4/ 0.I/ZS' t
c  7.93 '"  8/" 7.
«  t W <57.9t/'  / w
I
·Fa
o
Z 3  a 56 7 8 /0/ /,?4? h? 14151 1
~I
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 13. DEAD LOADS ON AXLE NO. 1, CASE 1
Ib, and Ic, together with the resulting fiber stresses at seventeen
sections along the axle for Case Ic.
It is to be remembered that during all test runs the test car was
loaded with dead load as in Case Ic and at times was subject to live
loads from causes enumerated in Section 19.
Figure 12 also shows No. 2 axle. This axle was used in conjunction
with other recording instruments to study the position of the wheels
on the rails, the amount of bending in the axle, and the vibrations
in the wheels.
22. Bending in Car Axle Due to Live Loads.The application of a
lateral force against the inner side of the wheel opposite the geared
end of the axle has been designated Case 2.
A common example of a lateral force acting against the inner side
of a wheel is the reaction of a restraining rail of curved track. Such
reaction is the angular accelerating force and is resisted by friction
and lateral impact of the rail against the wheel.
If P1 is the only lateral reaction against the wheel, the unbalanced
force system acting on the axle and wheels may be considered as
made up of two sets of couples and a force P1, acting as shown in
Fig. 14. The assumption is made that the force P1 may be replaced
by a force and a couple acting as shown and that the opposing couple
composed of forces P2 has its line of action through the treads of the
wheels. This assumption is made for Cases 2 to 7, inclusive.
From Appendix B it is seen that the relation between P1 and P2 is
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
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ILLINOIS ENGINEERING EXPERIMENT STATION
Disfance Along the Ax/e
M
FIG. 14. LOADING, MOMENT, AND El DIAGRAMS, CASE 2
given by: 17.5 P1 = 57.827 P2 or P2 = 0.3026 P1. The moment at
any section of the axle due to the force P1 is
Mý = 17.5 P1  P2 x = 17.5 P1  0.3026 P1 x in. lb.
in which x is measured from the left reaction and with limits from
0 to 57.827 in. The moment is considered positive as it causes
compression in the top fibers.
M M
Table 12 contains moments,  units, areas under the
EI El
diagram, and change in slope of axle. Figure 14 also contains dia
M
grams for moments and  due to lateral force P1.
EI
The method of analysis for the remaining five cases involving live
loads is quite similar to the analysis of the case just treated (Case 2),
and is discussed in detail in Appendix A.
23. Bending Stresses on Sections Lying Outside of the Wheels.
Section 22 dealt with bending in an axle due to live loads for sections
which lie inside or between the two wheels, i.e., between sections a
and b, Fig. 12. Section 19 dealt with bending due to dead loads for
all sections of the axle, Fig. 12. Table 5 gives the stress for any
section. At sections located on the axle outside of the wheels the
bending stresses are as high as 6000 Ib. per sq. in. This stress is
neither increased nor decreased by lateral live forces, but it may be
increased or decreased somewhat by a change in vertical reactions of
the wheels on the rails due to the speed of the car on curved track,
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
or to superelevation of the outer rail, or both. It will be observed
that sections 2 and 15 have practically the same stress. When the
stress at section 2 increases, the stress at section 15 decreases by
practically the same amount, and vice versa. Under the most adverse
conditions a change in stress amounting to about 1125 lb. per sq. in.
may be encountered. The maximum bending stress outside the
wheel, therefore, lies between 6000 and 7125 lb. per sq. in. at all times.
The fibers of sections 2 or 15 are consequently subjected to about
70 000 000 complete reversals of a 6000 lb. per sq. in. stress in
100 000 miles.
24. Shearing Stresses in Car Axles Due to Vertical Dead Loads.
Shearing stresses accompanying bending stresses, i.e., due to vertical
dead loads and lateral live loads applied on the axle and wheels, vary
from zero intensity at the outside fiber of the axle to a maximum at
the neutral axis. Shearing stresses may conveniently be discussed
under the following heads:
The maximum unit shear for a circular crosssection may be ex
V
pressed as S, = 4/3, where V = total vertical shear and A =
A
crosssectional area. The maximum total vertical shear on any sec
tion of the axle between the two wheels was found to be about 1715
1715
lb., or S, = 4/3 = 83 lb. per sq. in. and is negligible. The
27.69
maximum total vertical shear on a section of the axle on the outside
of the wheel was found to be about 10 700 lb. giving a unit shear on
the outer fibers
10 700
S, = 4/3 = 1068 lb. per sq. in.
13.36
and this is not high.
25. Shearing Stresses in Car Axles Due to Lateral Live Loads.The
shearing stress due to a lateral live load Pi may be found from the
general equation
) 0ý 0 ry 49 0rzZ
+ ++X= 0
Ox Oy Oz
In the case of an axle r,x = 0, and the body force, X, the centrifugal
force due to the mass of the pair of wheels and axle, may be neglected.
9 ax a Ty
The equation then becomes  +  = 0, in which ax is the
ax Oy
unit bending stress in the axle at any section, and Tr, is the unit
shearing stress at the same section.
ILLINOIS ENGINEERING EXPERIMENT STATION
From Appendix A, the bending stress due to the live force P1 is
y
S= (17.5  0.3026 x) P1 Y; y is the distance from the neutral axis
I
to the fiber under consideration and I is the moment of inertia of the
circular crosssection about an axis through its own center.
Differentiating the foregoing expression for ao, the following
0 Uý Y T Y
relation is obtained: 0.3026 P ; hence = 0.3026 Pi
9x I ay I
0.1513 P1 y2
and by integrating, Tx, = + C
I
When y = ±d/2 = +2.969 in., x,, = 0, and therefore
0.1513 P1 2.9692
0=  + C
I
 1.3337 P1
I
The unit shearing stress is, therefore,
0.1513 P1 y2 1.3337 P1 P 3  1.3337
T, =  I 0.1513 y2  1.3337
From this equation, and observed values of P1, the maximum shearing
20 000 (1.3337)
unit stress due to live load only is 7 = = 437 lb.
61.03
per sq. in., and may be neglected. There are no shearing stresses in
the axle outside the wheels due to live load P1. Maximum shearing
stresses due to dead load and live load forces combined occur simul
taneously with maximum bending stresses, but as the maximum
shearing stresses are on fibers at the neutral axis of the axle, and
maximum bending stresses are on the outside fibers it follows that
these shearing stresses do not affect maximum bending stresses.
26. Shearing Stresses in Car Axles Due to Torsion.The axle
under test of Car 4419 was a motordriven axle. Shearing stresses
due to torsion may therefore be developed. The unit shearing stress
is dependent upon the twisting moment and the area of the cross
section of the axle. The twisting moment may be calculated from the
power supplied to the motor and the angular speed of the axle. The
angular speed may be either calculated or observed. Tests were made
by the Shops and Equipment Department of the Chicago Rapid
Transit Company using cars of the same type as Car 4419 in which
the power supplied to the motor was determined by the use of an
ammeter and voltmeter. Readings were taken every five seconds.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS 31
The initial power input to the motor was approximately 200
amperes of current at 600 volts. The current increased in a series of
predetermined steps with slight fluctuations up to a maximum of
approximately 300 amperes at full speed of the test car. The maxi
mum twisting moment on the axle occurs at very low speeds of the
car (when the car is just starting from rest) rather than at high speed
of the car. This is due to the rapid increase in the value of the
angular velocity of the axle with increase in speed, and to the com
paratively small increase in the amount of current supplied to the
motor as the speed of the car increases.
The twisting moment on the axle may be calculated from the
following relations:
P
T =  where
c)
T = twisting moment in in. lb., P is the power input to the motor in
in. lb. per sec., and is equal to
I X V X 6600
746
in which I is the current in amperes, and V is the voltage.
2r
S=  n radians per sec.
60
in which n is the number of revolutions of the axle per minute.
The maximum twisting moment developed in the axle probably
occurs at a low speed of the test car, say at 1 mile per hour. The
value of the twisting moment is, for a 34in. diameter wheel,
P I X V X 6600 200 X 600 X 6600
T =  and P =
w 746 746
= 105 000 in. lb. per sec.
in which I and V were determined by tests.
27r
w =  9.88 = 1.04 radians per sec.
60
105 000
Hence T =  = 101 000 in. lb.
1.04
The tractive effort per axle due to twisting moment developed by
the motor is
T
T.E. =  in which T.E. is the tractive effort per axle in pounds,
and D is the distance from the center of the axle to the rail in inches.
and D is the distance from the center of the axle to the rail in inches.
ILLINOIS ENGINEERING EXPERIMENT STATION
The tractive effort per axle due to a twisting moment of 101 000
in. lb. and a 34in. diameter wheel is
101 000
T.E. =  = 5930 lb.
17
It was observed in the tests that the wheels did not slip when the
calculated twisting moment was 101 000 in. lb. or the tractive effort
per axle was 5930 lb.
The maximum unit shearing stress, lb. per sq. in., in the axle due
Tc
to a twisting moment is S, =  in which T is the twisting moment
rd4
in in. lb., c is the radius of the axle in inches, J =  , and d is the
32
diameter of the axle in inches.
The maximum unit shearing stress due to a twisting moment of
101 000 in. lb. on a 6in. diameter axle is
101 000 X 3 X 32
S, =   = 2380 lb. per sq. in.
64
A shearing stress of this magnitude on the outer fibers is not
likely to occur very frequently. However, when a car is started from
rest, especially on a rusty or a wet and sanded rail, this stress may
be developed. Shearing stresses due to torsion occur simultaneously
with bending stresses due to dead load of car and cargo only. Com
bining the tensile stress due to bending and the shearing stress due
to torsion, we have,
S' = 1/2 S + 1/2 /S2 + 4S82
= 1/2 X 3668 + 1/2 \/36682 + 4 x 23802
= 1834 + 3008 = 4842 lb. per. sq. in.
There are no shearing stresses in the axle on the outside of the
wheels due to torsion.
V. DATA AND DISCUSSION OF DATA
27. Division of Data.The data contained in this bulletin were
obtained from two sources: (1) mathematical analyses, and (2) re
corded readings of instruments and autographic records. Tables 1 to
4, inclusive, contain data recorded from tests made in the Wilson
Avenue Shops of the Chicago Rapid Transit Company, Chicago.
They also contain the calculated quantities derived from mathemati
cal analyses. Thus data for comparison of calculated and observed
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
TABLE 6
AXLEBENDING STRESSES OBSERVED AT CROSSINGS, FROGS,
SWITCHES, AND CURVED TRACK
Item
Crossings. ...........
Frogs. .............
Switches ..... ........
Curved Track........
Range of
Axlebending
Stress Remarks
lb. per sq. in.
3650 tension*10 250 compression
3650 tension17 400 tension
3650 tension9300 compression
3650 tension15 000 tension
3650 tension19 850 tension
3650 tension16 600 tension
Stress caused by too wide spacing of
guard rails.
Stress due to sudden change of di
rection of wheel flange.
Stress caused by too wide spacing of
guard rails.
Stress due to sudden change of
direction of wheel flange.
Stress due to sudden change of
direction of wheel flange.
Stress probably due in part to change
by "jumps" in direction of
wheel flange.
*The minimum value of 3650 lb. per sq. in, is the dead load stress.
quantities were obtained. The agreement is good. The data of
Tables 6 and 7 were obtained by the substitution of observed
ordinates of the autographic record in mathematically derived equa
tions for fiber stress. The choice of the proper case of analysis
depended upon several factors as discussed in Chapter IV.
The data have been divided into two chief divisions for purposes
of discussion: (1) those data which show the effects of restraining
rails, frogs, crossings or other parts of track on the magnitude of
bending stress in the axle, and (2) those data relating to the number
of occurrences per mile of any given bending stress and particularly
to the number of occurrences per mile of bending stress higher than
the normal bending stress.
28. Effects of Crossings, Frogs, and Other Parts of Track on Bending
Stresses in Car Axles.Table 6 summarizes the maximum ranges of
stress observed at section 5 of the test axle for crossings, frogs,
switches, and curved track.
In connection with Table 6 it may be noted that on the record
sheet, stress caused by too wide spacing of guard rails was distin
guished from stress due to sudden change of direction of wheel flange
by the direction of bending of the axle.
29. Occurrence of Bending Stress in Car Axles.Table 7 contains
data relating to the number of occurrences of bending stress in the
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 7
NUMBER OF OCCURRENCES OF BENDING STRESS IN CAR AXLE IN 100 000 MILES OF
SERVICE, AT SECTIONS 5, 10, AND 13
Based on tests covering 177 miles of track
Average duration of each stress is 0.51 revolution of the axle
Number of Number of Stresses
Per Cent of Fiber Stress Occurrences of Equal to or Greater
Nominal* lb. per sq. in. Stress in Than the Minimum in
Stress 100 000 Miles Column 2
Section 5
500550..................... 1850020000 2 800 2 800
450500...................... 1650018500 5 100 7 900
400450..................... 1450016500 87 500 95 400
350400 ..................... 1300014500 440 000 535 400
300350 ..................... 1100013000 821 000 1 356 400
250300..................... 900011000 3 361 000 4 717 400
200250 ..................... 7500 9000 8 228 000 12 945 400
150200..................... 5500 7500 15 122 000 28 067 400
100150..................... 3650 5500 33 013 000 61 080 400
50100..................... 1825 3650 43 492 000 104 572 400
0 50..................... 0 1825 12 052 000 116 626 400
Section 10
450500 ..................... 1650018500 600 600
400450 ..................... 1450016500 11 900 12 500
350400 ..................... 1300014500 141 000 153 500
300350 ..................... 1100013000 259 000 412 500
250300 ..................... 900011000 617 000 1 029 500
200250 ..................... 7500 9000 981 000 2 010 500
150200 ..................... 5500 7500 2 338 000 4 348 500
100150..................... 3650 5500 42 151 000 46 499 500
50100..................... 1825 3650 68 350 000 114 849 500
0 50..................... 0 1825 1 775 000 116 624 500
Section 13
400450 ..................... . 1450016500 600 600
350400 ..................... 1300014500 2 300 2 900
300350 ..................... 1100013000 151 000 153 900
250300 ..................... 900011000 330 000 483 900
200250 ..................... . 7500 9000 990 000 1 473 900
150200...................... 5500 7500 2 150 000 3 623 900
100150................... 3650 5500 1 975 000 5 598 900
50100... .................. 1825 3650 108 592 000 114 190 900
0 50..................... 0 1825 2 434 000 116 624 900
*By nominal stress is meant the bending stress in the axle resulting from dead load of car and
cargo when the car is standing still on straight and level track.
test axle of car 4419 obtained from runs on about 177 miles of track,
elevated and surface, of the Chicago Rapid Transit System.
30. Tentative Proposal for Formula for Lateral Forces Against a
Car Wheel.Sections 18 to 22, Chapter IV, and Appendix A are
devoted to a discussion of the bending of car axles under dead and
live loads. Appendix B is devoted to methods of determining stress
constants and fiber stresses in car axles in terms of ordinates to auto
graphic records obtained from runs made by the test car.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
A. M. Wellington in "The Economic Theory of Railway Loca
tion" treats very ably the subject of the positions of the wheels of a
car truck on the rails of a curve, without restraining rail. See para
graphs 294 and 298 of his work. Autographic records of tests reported
in this bulletin may be regarded as graphic representations of the
statements made by Mr. Wellington. In his book the slipping of
wheels and rotation of truck on the rails are treated in paragraphs
300, 301, and 302. Paragraphs 311 to 313, inclusive, contain a dis
cussion of the reaction of the upper rail against the flange of the
leading wheel due to the turning of the truck. This reaction or
horizontal thrust arises within the truck alone, and according to
paragraph 312, the magnitude of this force is equal to the force
necessary to slide three wheels, or is equal to the load carried by three
wheels multiplied by the coefficient of sliding friction. In the tenta
tively proposed formula for lateral forces against a car wheel,
Equation (1), the factor f is the lateral reaction discussed by Mr.
Wellington in paragraph 312, and by tests at Niles Center, Chicago,
the magnitude of f for bright dry rails was estimated to be about
8000 lb. or the equivalent of the load carried by three wheels, 40 900
lb., multiplied by the coefficient of sliding friction equal to 0.196.*
Paragraphs 322 and following contain a discussion of forces originat
ing outside of the truck itself. Centrifugal and centripetal forces and
forces due to superelevation are considered and the conclusion is
drawn that "the centrifugal and centripetal forces have but a trifling
effect on curve resistance" (see paragraph 329). This is due to the
fact that trains usually run at speeds near that corresponding to the
superelevation.
The ordinate to the autographic record is a fairly accurate measure
of the magnitude of the total net lateral force acting on the leading
pair of wheels, chiefly on the upper wheel, which causes bending in
the axle, irrespective of the origin or cause of any individual contrib
uting force. It is proposed, tentatively, to use a general equation of
the following form as giving total lateral force developed at the
leading pair of wheels, which causes bending in the axle of the test
car as it moves around a curve, applicable to curves with or without
a restraining rail.
Pi=f + F +It (1)
in which Pi is the total lateral force in pounds, developed at the
leading pair of wheels of the test car, which causes bending in the
*Wellington suggests 0.25 as a value to be assumed for the coefficient of sliding friction.
tIn this section the symbol I should be carefully distinguished from the symbol I for moment of
inertia and the symbol I for current used in other sections.
ILLINOIS ENGINEERING EXPERIMENT STATION
axle and is due to lateral slip, centrifugal force, and a factor I. f is
the lateral force, in pounds, developed by the turning of the car as
the wheels of the leading truck slip laterally or longitudinally, or
both, and was determined by tests in the following manner: When
the test car moved around a curve at such a speed that the centrif
ugal force was exactly balanced by the superelevation of the outer
rail, the ordinate to the autographic record (exclusive of slight fluc
tuations in the record) was considered to measure the lateral force f.
This lateral force is chiefly friction due to lateral slip of the wheels of
the leading truck on the rails. There is included under f only that
part of the total friction of the truck which is exerted on the leading
pair of wheels and recorded by the axlebending instrument.
F = F,  F2 (2)
where Fi is the lateral reaction of the wheel against the upper running
rail due to centrifugal force and F2 is the lateral reaction against the
lower rail taken as the horizontal component of the load due to the
superelevation of the upper rail.
M V2 e
F1 =  and F2 W (3)
R G
in which W is taken as the weight in pounds carried by the front axle
of the car, G is the gage of the track taken as 57.8 in. for the test car,
V is the speed of the car in feet per sec., R is the radius of curvature
of the track in feet, and e is the superelevation of the upper rail in
W
inches, M = , g is taken as 32.2 ft. per sec. per sec., and I is a
g
factor determined by tests. A car does not usually move smoothly
around a curve but moves in a series of "jumps". Hence there are
some irregularities or fluctuations in the autographic records of the
test runs. The fluctuating part of the ordinate to the record has been
taken as I. The factor I may then be due to a change in the lateral
reaction of the upper rail against the leading wheel due chiefly to
roughness or to a change in vertical reactions or perhaps to a com
bination of both. The observed values of I are not high as compared
with the total lateral thrust P1. Until further detailed study may be
made it is proposed to use a table of values of I found by tests on a
19deg. curve at Niles Center, Chicago.
From a series of test runs made on a 19deg. curve, with 12/ in. of
superelevation of outer rail, located in the yards of the Chicago
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS 37
TABLE 8
DETERMINATION OF VALUES OF f AND I FROM TEST RUNS
Run
(1)
35c. ...............
35d ................
35ab ................
35a4c................
34c. ...............
34f.................
34h................
36ait...............
36bit...............
Speed
mi. per f,
hr. I lb.*
(2) (3)
10 7 560
10 7 560
10 7 990
I
lb.*
(4)
590
590
590
10 8 670 590
10 9 220 590
10 8 540 1 025
10 9 220 1 025
10 3 075 1 025
10 13 000 1 080
Total
Lateral
Force Calcu
Pi lated
Col. 3 + FI
Col. 4
lb. lb.
(5) (6)
8 150 548
8 150 548
8 500 548
9 260 548
9 770 548
10 045 548
10 735 548
5 100 548
14 080 548
*For a speed of 10 m.p.h. and for this particular curve the lateral force due to centrifugal force is
balanced by the lateral force due to superelevation. Hence as may be seen from Equation (1) f is
determined directly from the ordinate to the autographic record, taking no account of fluctuations.
The magnitude of these fluctuations is a measure of I. Fig. 15 illustrates these determinations.
tWet rails, 12.8 miles per hour.
SRusty rails.
¶JNote that for this table I denotes a force in pounds not a moment of inertia in in.4
Rapid Transit Company at Niles Center, the following data were
obtained:
At a speed of 10 miles per hour on this 19deg. curve, the centrif
ugal force was exactly balanced by the force due to the supereleva
tion of the outer rail. At speeds higher than 10 miles per hour the
centrifugal force becomes greater than the force due to the super
elevation of the outer rail. This difference is shown in column 6 of
Table 9. Table 9 contains data of several runs of the test car made
at speeds of 15, 20, and 25 miles per hour. Column 3 is the value,
in pounds, of the ordinate to the smooth part of the record as ex
plained in the footnotes of Table 8. The values of f, column 4, are
taken from the runs at 10 miles per hour. Column 5 gives the com
puted values of Fi  F2, the difference between the values given in
column 3 and column 4. Column 7 shows the difference between the
computed and the calculated values of Fi  F2.
The values of I were determined from the autographic records of
tests made at Niles Center, Chicago, on a 19deg. curve. The value
of the irregular or fluctuating part of the record is taken as I. The
values of I given in Table 10 were obtained from several runs of the
test car at speeds of 10, 15, 20, and 25 miles per hour. Figure 15
shows the fluctuations in a typical record.
A study of the test values of f given in Table 8 shows that the,
Calcu
lated
FiF2
lb.
(7)
0
0
0
0
0
0
0
0
0
38 ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 9
COMPARISON OF COMPUTED AND CALCULATED VALUES OF (F1  F2)
Computed values were obtained from ordinates to autographic records of test runs; calculated values
were derived from Equation (3)
Run
(1)
35ci............
35d. .............
35bi.............
35ai .............
34c. ............
34f ..............
34h...............
36ait ............
36bit. ............
35c. .............
35di .............
35b. .............
35a. .............
34c...............
34f ..............
34h...............
36a. .............
36bl .............
36bi. .............
35dc. .............
35bi. . . . . . . . . . . . . .
35ai. .............
34c..............
34f . . ............
34h...............
36a ..............
36b¶i .............
Speed
mi. per
hr.
(2)
15
15
15
15
15
15
15
15
16
20
20
20
20
20
20
20
22
25
25
25
Value of*
Smooth
Ordinate
lb.
(3)
7 560
8 650
10 100
f
lb.
(4)
7 560
7 560
7 990
10 100 8 670
9 760 9 220
o 5in 8 540n
9 560 9 220 340
3 075
14 060
9 740
11 170
11 170
11 080
10 600
9 860
3 075
15 030
8 660
9 740
12 280
12 280
25
25 10 600
25 10 600
23 6 150
27 13 330
3 075 0
13 000 1060
7 n 5 100nn
7 560
7 990
8 670
9 220
8 540
9 220
3 075
13 000
7 560
7 560
7 990
8 670
8 540
9 220
3 075
*See Fig. 15.
tWet rails.
$Rusty rails.
$Rusty rails but becoming less rusty with each successive run of test car.
pressure due to lateral slip is approximately 8000 lb. for dry bright
rails, 3000 lb. for wet rails, and 13 000 lb. for rusty rails.* The data
of runs made at higher speeds, Table 9, show a fairly good agreement
between computed and calculated values of the net centrifugal force
(F1  F2), considering f to be constant. Hence f, the lateral force
due to slip of the wheels as the car turns, may be considered approxi
mately a constant for the 19deg. curve at Niles Center. The value
of f in Equation (1) may be taken, tentatively, as 8000 lb. for dry
bright rails, 3000 lb. for wet rails, and 13 000 lb. for rusty rails.
The calculated values of F or (F1  F2) of Equation (2) show a
fairly good agreement with computed values determined from test
*This variation of resistance to slip is a matter of considerable uncertainty.
Computed
Value of
FýF2
lb.
Col. 3 
Col. 4
(5)
0
1090
2110
1430
540
1020
Calcu
lated
FiF2
lb.
(6)
586
586
586
586
586
586
586
586
758
1545
1545
1545
1545
1545
1545
1545
1545
2006
2770
2770
2770
2770
2770
2770
2260
3356
Differ
ence
lb.
(7)
 586
+ 504
+1524
+ 844
 46
+ 434
 246
 586
+ 302
 455
+ 635
+1535
+1955
+ 215
+ 515
 905
+1545
+ 24
1670
 590
+1520
+ 840
 710
1390
+ 815
2180
3180
3500
1860
2060
640
0
2030
1100
2180
4290
3610
2060
1380
3075
on  C8
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
TABLE 10
VALUES OF I* FOR SEVERAL SPEEDS OF TEST CAR
On 19deg. curve at Niles Center
Speed Speed Speed Speed
Run mi. per I mi. per I mi. per I mi. per I
hr. lb. hr. lb. hr. lb. hr. lb.
35c ......... 10 590 15 590 20 1080 25 2160
35dl......... 10 590 15 590 20 1080 25 2750
35b ......... 10 590 15 1080 20 2160 25 2750
35al......... 10 590 15 1080 20 2160 25 2750
34c ......... 10 590 15 1025 20 2050 25 ....
34f.......... 10 1025 15 1560 20 2050 25 2050
34h.......... 10 1025 15 1370 20 1710 25 2050
36al......... 10 1025 15 1080 20 1500 25 2050
36b ......... 10 1080 15 1370 20 1080 25 1585
From the data given above an average value of I was estimated to be:
Speed I
miles per hour lb.
10 590
15 1080
20 2160
25 2750
*Note that in this table I denotes a force in pounds not a moment of inertia in in.4
Speead o A/les per Hoar
FIG. 15. AUTOGRAPHIC RECORDS, 19DEG. CURVE, NILES CENTER, CHICAGO
runs, see Table 9. Hence F may be calculated on the basis of the
total weight carried by the leading axle of the car.
Equation (1) becomes, for dry bright rails,
Pi = 8000 + F + I (4)
F is to be calculated and I to be taken from Table 10.
Table 11, gives a comparison of values of total lateral force as
determined from the records of test runs and as computed from
Equation (1) with values of f as given in Table 8 and values of I as
given in Table 10.
Speed /0 Mlles per Hour
[
i*m~
ILLINOIS ENGINEERING EXPERIMENT STATION
COMPARISON OF TOTAL
TABLE 11
CALCULATED LATERAL FORCE AND TOTAL LATERAL
FORCE FROM TEST RUNS
Run
35ci ......................
35d. .....................
35a. .....................
35b. .....................
34c.......................
34f.......................
34h......................
36al .....................
36b . ......................
35ci .....................
35d . ......................
35a ......................
35b. .....................
34c.......................
34f.......................
34h......................
36a. .....................
36b ......................
35c. .....................
35d ......................
35al. .....................
35b. .....................
34c.......................
34f.......................
34h......................
36a. .....................
36b. ....................
35c. .....................
35d ......................
35a. .....................
35b . ......................
34c ......................
34f ......................
34h......................
36a. .....................
36bi ......................
Speed
mi. per
hr.
10
10
10
10
10
10
10
10
10
15
15
15
15
15
15
15
15
15
20
20
20
20
20
20
20
20
20
25
25
25
25
25
25
25
25
25
Total Lateral
Force P1
from Records
lb.
8 150
8 150
8 500
9 260
9 770
10 045
10 735
5 100
14 080
8 150
9 240
11 180
11 180
10 785
11 120
10 930
4 155
15 430
9 730
10 820
13 330
13 330
13 130
12 650
11 570
4 575
16 110
10 820
12 490
15 030
15 030
12 650
12 650
8 200
14 915
There is a fairly good agreement between these two values, the
maximum difference being 2700 lb. and the average difference about
1150 lb.; and the values from the record are sometimes smaller than
those given by the equation.
The preceding paragraphs contain discussions of the factors
entering into the proposed formula, Equation (1), on the basis of
forces acting chiefly against the leading wheel on the upper rail of
curves without restraining rails. It is believed that Equation (1) is
equally applicable to curves with restraining rails. In this case the
lateral forces act chiefly against the wheel on the lower rail instead of
the wheel on the upper rail, and the direction of the bending of the
axle is opposite in sense to that occurring with curves without
restraining rails.
Total Lateral
Force P1
from Equation (1)
lb.
8 590
8 590
8 590
8 590
8 590
8 590
8 590
3 590
13 590
9 680
9 680
9 680
9 680
9 680
9 680
9 680
4 680
14 680
11 710
11 710
11 710
11 710
11 710
11 710
11 710
6 710
16 946
13 520
13 520
13 520
13 520
13 520
13 520
7 460
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
31. Customary Method of Design of Car Axles.In 1896 a Com
mittee of the Master Car Builders' Association, now called the Ameri
can Railroad Association, made a report in which is given a method
of calculation to determine the proper diameters of an axle.* This
committee made a search of the previous methods used and derived
the following formulas:
Wb Hh Hhx
"M  (x  b) + + H h +
2 m 1
/W H h
S h tan a
\2 m
M
f = 
0.0982d3
Where W = total vertical load per axle, car and cargo plus 26 per
cent which was found by test to be the oscillating load.
H = total of all horizontal forces the maximum of which
could not be more than 0.4W otherwise car would
tip over.
b = distance center of rail to point on journal where concen
trated load is assumed to act.
h = height of center of gravity of complete car, except
wheels and axles, above top of rail.
m = distance between centers of rails.
x = distance between point on journal where concentrated
load is assumed to act and point on axle under con
sideration.
hi = height of center of gravity of complete car, except
wheels and axles, above center of axle.
I = length of axle between points on axle where concen
trated load is assumed to act.
h2 = height of center of axle above top of rail.
tan a = tangent of angle which wheel tread makes to horizontal.
f = fibre stress.
M = total bending moment.
d = diameter of axle at point about which moments are
taken."
This equation is attributed to F. Reuleaux, M. C. B. A., Proceed
ings, 1896, Volume 30, pages 150153.
The term H, however, was defined as "H = horizontal force
caused by curves, switches, and wind pressure."
*Master Car Builders' Association Proceedings, 1896, Vol. 30, pp. 149172.
ILLINOIS ENGINEERING EXPERIMENT STATION
The committee on axle design also considered W6hler's equation
for total bending moment on an axle, derived chiefly from tests con
ducted on the Prussian railways in which the term H was definitely
evaluated for conditions existing at that time. The three components
of H according to the Wihler equation were: (1) friction between
rails and wheels, (2) lateral pressure against the flange of a wheel as
the car moved around curves or through switches, and (3) wind
pressure against the car.
The committee did not attempt to say how far the data obtained
by W6hler were applicable to American railways. It decided upon
the use of an arbitrary value for H. The value of H was not expected
to exceed the value of the force necessary to overturn the car. The distance
from top of rail to center of gravity of some cars of that period (1896)
was 72 in. The gage of the track was considered as 59 in., hence the
force necessary to cause overturning of the car was 0.4 W, where W
is the total weight of car and cargo in pounds. From this information
the committee decided to consider the value of H against one axle
as 0.4 times the load carried by the axle.
However, in incorporating the value of H in the general equation
of F. Reuleaux, it became 0.4 W, where W was the weight carried by
the axle plus 26 per cent. Hence car axles were designed for dead
loads they were to carry plus a force equal to 1.26 times the force
necessary to overturn the car.
Suppose it is desired to find the size of axle for a car of the type of
car No. 4419 with a cargo of 15 000 lb. The following values may be
substituted in the general equation:
W = weight carried by front axle plus 26 per cent.
= 24 600 X.1.26 = 31 000 lb. (exclusive of wheels and
axle; the weight 24 600 lb. includes the weight of the
motor).
All the motor weight does not appear as journal loads, for part
of the weight is carried directly on the axle inside of the wheels
H = 0.4 W = 0.4 X 31 000 = 12 400 lb.
b = 7.188 in.
hi = 31 in.
h = 48 in.
1 = 73.5 in.
m = 59.125 in.
h2 = 17 in.
x = 11.937 in.
tan a = 0.05
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
31 000 X 7.188 12 400 X 48
M = (11.937  7.188) +
2 59.125
12 400 X 31 X 11.937
735 + 12 400 X 17 +
73.5
(31 000 12 400 X 48
(31  12400 ) 17 X 0.05 = 341 720 in. lb.
2 59.125
The customary method of design also specifies that for motor
driven axles (as in this case) consideration must be given to the
moment exerted on the axle when the axle is being driven, according
to the following method:
( W \
"M = + h (x  b)
where M, W, h2, x, and b are as given above and d = distance from
center of axle to motor nose suspension.
A = coefficient of traction = 0.2
W1 = that part of the weight of the motor carried by the axle."
Substituting in the preceding equation the values for a car of the
type of car No. 4419, the bending moment is
1 r(31 000 X 0.2 X 17) 3200 11.937 7.188
M =    32001 11.937 7.188
2 L(30 + 0.2 X 17)
= 214 in. lb., and is negligible.
Suppose the allowable working stress of axle steel to be 16 000 lb.
per sq. in., the diameter of axle required for a bending moment of
341 720 in. lb. is, from the relation given,
M
F=
0.0982d3
341 720
d3 =
0.0982 X 16 000
d = 6 in.
It may be pointed out that the statement regarding the maximum
value of H, page 42, needs careful consideration. At the time of the
adoption of the M. C. B. formula (1896), the center of gravity of
some cars was located 72 inches above the top of the rail. The cal
culated force necessary to overturn a car was consequently 0.4 W
against the flange of each wheel on one side of the car, or, of course,
ILLINOIS ENGINEERING EXPERIMENT STATION
1.60 W acting through the center of gravity of the car. Some of the
cars of today have much lower centers of gravity.
The car used in the tests reported in this bulletin, when loaded
with 15 000 lb. of cargo, had its center of gravity approximately
48 inches above the top of the rail at the motor end. The calculated
overturning force is 16 560 lb. or 0.67 W, where W is the weight
(exclusive of wheels and axle) carried by the motor axle. The tests
of this bulletin have shown that it is possible for the flange of the
leading wheel to be subjected to a horizontal force greater than
0.67 W without the car tipping over. This means that a force greater
than 0.67 W may be developed against the flange of the leading wheel
before a force of 0.67 W can be developed against the flange of each
of the four wheels on one side. Horizontal forces as large as 0.81 W
have been observed. It is thus seen that the assumptions of the
customary method of design for car axles are not always assumptions
on the safe side.
VI. RESULTS OF TESTS AND CONCLUSIONS
32. Number of Cycles of Stress at Critical Sections per 100 000
Miles Service.The allowable mileage between examinations of axles
for cracks evidently depends on the magnitude and the frequency of
the axlebending stresses developed. An examination of the stresses
at different crosssections of the axle (see Table 5, Fig. 12, and
Appendix B) showed that section 5 (Fig. 12) and section 10 were the
ones receiving the highest stress under most of the occasions when
high bending moment was set up in the axle, and that section 5 was
the more severely stressed of the two in most cases. Thus, from the
records covering runs over 177 miles of track and some 207 000 stress
determinations, Table 7 has been compiled, showing the number of
times various stresses may be expected to occur in 100 000 miles of
service at sections 5, 10, and 13. Of course, this table is statistical
in its nature, the actual number of high stresses in another series of
runs over the track might be either greater or less than those observed
in the tests already made. However, it is believed that the figures
given in Table 7 form a fairly satisfactory basis for estimating the
frequency of occurrence of axlebending stresses of any given magni
tude, for the axle of a car in service over the whole system of the
Chicago Rapid Transit Company.
33. Number of Cycles of Stress for any Point on Circumference at
Critical Section.The number of times any given "fiber" of a given
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
section of car axle is subjected to a cycle of reversed stress evidently
depends not only upon the total number of stresses observed at that
section in a given distance, say one mile, but also upon the duration
of the individual stresses. If a stress lasts but a small fraction of a
revolution, only a few fibers on the surface of the axle may be sub
jected to the stress, and complete reversal does not take place for all
of them; on the other hand (e.g. on curves), a stress may continue for
several revolutions of the axle causing several cycles of reversed
bending stress in every fiber of the axle. In Table 7 the average
duration of a stress has been computed and is stated to be about
onehalf a revolution. It seems that a safe method of estimating the
number of cycles of any given stress for any fiber of the axle would
be to consider the number of cycles for any given mileage the same as
the number of times the axle is subjected to the given stress. This is,
of course, equivalent to considering the duration of each stress one
complete revolution of the axle, and this is assuming rather more
severe conditions than indicated by the average duration observed.
It is further to be noted that for the very highest stresses the duration
was usually for a comparatively short part of a revolution.
34. Fatigue Strength and Rapidity of Spread of Fatigue Cracks in
Axle Steel.Bulletins 165 and 197 of the Engineering Experiment
Station of the University of Illinois give data as to the fatigue strength
of specimens of steel cut from car axles. From results of tests of
specimens with a diameter of 1 in. and with a 14in. radius fillet at
the critical section, the average endurance limit of the uncracked steel
tested averaged about 26 000 lb. per sq. in. However, for specimens
in which a fatigue crack had been started by cycles of stress above
26 000 the endurance limit was reduced to some value between 50
and 65 per cent of the value for uncracked steel, say an average
value of 14 400 lb. per sq. in.
In the field tests the highest axlebending stress obtained was just
under 20 000 lb. per sq. in., and it seems that under any ordinary
conditions of service an axle may be expected to remain uncracked.
However, if by any accident a crack is started, the axle will occasion
ally be subjected to stresses which will cause that crack to spread,
and if the axle is left too long without inspection there may occur a
complete axle failure. In the tests recorded in Bulletin 165 of speci
mens of axle steel after fatigue cracks had been started, no specimen
with such a crack failed under less than 1 000 000 cycles of com
pletely reversed stress, although the stress was above 20 000 lb. per
sq. in. in some such tests. The number of cycles of stress above the
ILLINOIS ENGINEERING EXPERIMENT STATION
fatigue limit which will cause failure is notoriously uncertain, even
in laboratory tests. However, it seems reasonable to suppose that
under the stress ranges indicated by the results of the field tests a
cracked axle might be counted on safely to withstand 200 000 cycles
of stress before failure.
Examining Table 7, which shows the estimated number of stresses
of various magnitudes for a car used over the whole Chicago Rapid
Transit System, it is seen that, for stresses equal to or greater than
14 400 lb. per sq. in.the assumed limiting stress for the spread of a
fatigue crack in axle steelis 95 000. It would appear, therefore,
that inspection for cracks after each 100 000 miles of travel was safe
practice for a car so used,probably inspection after each 200 000
miles of travel would be fairly safe.
35. Allowable Working Stress and Actual Stress Developed.As
already discussed, on the basis of the customary method of design of an
axle to resist the stress due to bending moment, the allowable working
stress seems to have been at least 16 000 lb. per sq. in. This value
may be compared with the yield point value for axle steel, about
45 000 lb. per sq. in.; the fatigue limit for uncracked axle steel,
about 26 000 lb. per sq. in.; the stress necessary to spread a fatigue
crack once it is started, about 14 400 lb. per sq. in.; and the highest
observed axlebending stress in the tests, just under 20 000 lb. per
sq. in.
36. Conclusions.The following is a summary of the conclusions
drawn from this investigation.
(1) In the tests herein reported high bending stresses in the test
axle of the test car were found at crossings, frogs with guard rails,
switch points, and curves without restraining rails.
(2) At crossings two causes of high axlebending stress were found
to be: (a) Too wide spacing of guard rails, which causes pressure
against the inside of the wheel flanges, and (b) narrow gage at frog
and the tendency of slight changes in alignment of the track to change
the direction of the wheel rather suddenly, with consequent lateral
forces and axlebending stress. These two types of stress can be
distinguished by the fact that they tend to bend the axle in opposite
directions. From wide spacing of guard rails in crossings, axle
bending stresses as high as 10 250 lb. per sq. in. were noted; from the
second cause stresses as high as 17 400 lb. per sq. in. were noted.
(3) The two causes of stress noted in the preceding paragraph
also acted to cause axlebending stresses at frogs. Due to too widely
spaced guard rails, axlebending stresses as high as 9300 lb. per sq. in.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
were noted at frogs; due to sudden change of direction of wheel,
stresses as high as 19 050 lb. per sq. in. were noted.
(4) High axlebending stresses at switch points seem to be due to
the sudden change of direction as the wheel flange hits the point of
the switch. At switch points axlebending stresses as high as 19 850
lb. per sq. in. were observed.
(5) On curved track without restraining rail, axlebending stresses
as high as 17 400 lb. per sq. in. were observed. These stresses could
not be accounted for by centrifugal force alone. It was observed as
the truck of the test car passed along a curve that the amount of
bending of the axle fluctuated, as if the car truck adjusted itself to
the curve by "jumps," with appreciable impact at each jump. The
fact that with wet rails and, consequently, reduced friction the axle
bending stresses were reduced on curved track is in harmony with
this explanation.
(6) On curved track properly spaced restraining rails were found
effective in reducing axlebending stress. The effectiveness seems to
depend mainly on the space between restraining rail and running rail.
On some curves with restraining rails so spaced that the inner face of
the flange of the wheel on the "lower" rail bore against the restraining
rail the axlebending stresses were found to be not much greater than
the dead load stresses.
(7) From test results with different adjustments for lateral play
of axles it was found that, for the test car and test axle, a lateral
play* of 1/4 to 3/8 inch gave distinctly less axlebending stress than
a lateral play of either 1/16 inch or 3/4 inch.
(8) A statistical study of all the axlebending stresses observed in
the tests reported (some 207 000 in number, covering over 177 miles
of track) shows no axlebending stress higher than 20 000 lb. per sq.
in. A few higher stresses were observed in yard tracks.
These stresses are reversedbending stresses and for such stresses
the endurance limit of axle steel may be taken at 26 000 lb. per sq. in.
for uncracked steel, and at 14 400 lb. per sq. in. for axle steel in which
a fatigue crack has been started.t
(9) As seen from Table 7, the probable number of cycles of axle
bending stress greater than 14 400 lb. per sq. in. on the most severely
stressed section of the test axle is 95 000 per 100 000 miles service,
roughly one per mile. In no case did laboratory specimens of steel
in which fatigue cracks had been developed fail under less than
1 000 000 cycles of reversed bending stress, even when that stress was
*The term "lateral play" signifies the total clearance between the truck frame and the axle.
tSee Univ. of Ill. Eng. Exp. Sta. Buls. 165 and 197.
ILLINOIS ENGINEERING EXPERIMENT STATION
as high as 20 000 lb. per sq. in., a stress as high as any observed in
axles during the whole course of the field tests.
Hence, for a car which is to be used over the entire system of the
Chicago Rapid Transit Company, inspection of axles for cracks after
every 100 000 miles of service would seem to be an effective safeguard
against fracture of axle, even if accidental overloads started a fatigue
crack near the beginning of a 100 000mile period of service.
(10) It may, however, be pointed out that if a car should be put
in service on track which has an unusually large number of locations
causing high stresses, the axles might be subjected to more than
1 000 000 cycles of axlebending stress above 14 400 lb. per sq. in. in
100 000 miles of service.
(11) The allowable working stress used as a basis for axle design
is 16 000 lb. per sq. in. This figure may be compared with a maximum
observed service stress of 20 000 lb. per sq. in., which occurs very
rarely indeed; with observed stresses above 14 400 lb. per sq. in.,
which may be regarded as a fatigue limit for axles in which a fatigue
crack has been started, and which occur rarely; and with lower
stresses, which occur frequently.
APPENDIX A
MATHEMATICAL ANALYSES FOR BENDING IN CAR AXLES
DUE TO LIVE LOADS
1. General Statement.Chapter IV of this bulletin dealt with the
relationship between fiber stress in axles, change in slope of axles, and
change in inclination.of car wheels. It also enumerated some causes
of bending in car axles and classified forces causing bending into
seven "Cases." The first case of bending is that due to dead load,
see Section 21, Chapter IV. Each of the next five cases of bending,
due to live loads, is treated as if there were only one live load, acting
as shown in each case, causing bending in the axle. Case 7 deals with
more than one live load. The mathematical treatment of each of
these cases is precise. The decision as to how nearly service condi
tions approach one or another of the outlined cases is one of judg
ment. It is of interest to discuss briefly the probable number of
lateral forces involved as a car moves along the track, before proceed
ing with the detailed analysis of each case.
2. Effect of Change in Vertical Reactions on Cases 2 to 7.The
change in bending stress in an axle due to changes in vertical reactions
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
of the wheels on the rails from superelevation of outer rail and from
centrifugal force is not treated as a part of Cases 2 to 7 developed in
Chapter IV and this appendix. The estimated change in the vertical
reaction on the upper rail is usually equal in magnitude to that in
the vertical reaction on the lower rail, one increasing, the other
decreasing. Consequently, the change in slope of the axle from one
wheel to the other is approximately zero. Hence the autographic
records are not influenced by changes in vertical reactions.
Lateral forces, or horizontal components, accompanying changes
in vertical reactions do affect the ordinates to the autographic records
and are considered in each of the Cases 2 to 7.
3. Determination of Number of Lateral Forces Involved.Inspec
tion of the autographic record of bending of the axle will disclose the
speed of the car, the direction of curvature, the presence of restraining
rail and whether or not it is in action, the direction of acceleration of
the car, and the resulting change in inclination of the wheels due to
lateral forces acting on the wheels. The track conditions, such as
gage, superelevation of the outer rail, and radius of curvature, are
obtained from the plan of the track layout.
Change in slope of the axle due to lateral forces encountered on
straight track, at frogs, at switches, and on curved track with neither
restraining rails nor superelevation of outer rail, may be classified
directly as belonging to one of Cases 2, 3, 4, or 5. Case 2 was treated
in Chapter IV. Cases 3, 4, and 5 are treated in succeeding pages.
Curved track, with superelevation of the outer rail, whether with
or without a restraining rail, requires careful consideration. A sum
mation of the possible forces, chiefly lateral forces, for each case
follows:
For curves with no restraining rails there are two conditions,
which may be designated (a) and (b).
(a) When a car travels around a curve with no restraining rail,
and moves at a speed lower than that which corresponds to the super
elevation of the outer rail, the forces, chiefly lateral forces, which act
on the wheels may be:
(1) A lateral reaction of the lower running rail against the tread
of the wheel mainly by friction. This lateral force is a part of the
centripetal force and is very small at low speeds, and for the stress
calculations it may be neglected.
(2) A lateral reaction of the lower running rail against the tread
or flange of the wheel due to the horizontal component of the weight
of the car corresponding to the superelevation of the outer rail.
ILLINOIS ENGINEERING EXPERIMENT STATION
This force acts in a direction opposite to that of (1).
(3) A change in vertical reactions due chiefly to superelevation,
increasing on lower rail, decreasing on upper rail.
(4) A lateral force, Pi. P1 is the angular accelerating force which
is resisted by friction and impact of the rail on the wheel. It acts
laterally against the flange of the wheel on the upper rail.
(b) When a car travels around a curve with no restraining rail and
moves at a speed higher than that which corresponds to the super
elevation of the outer rail, the forces, chiefly lateral forces, which act
on the wheels may be:
(1) A lateral reaction of the lower running rail against the tread
of the wheel, mainly by friction. This lateral force is a part of the
centripetal force and is probably very small at the instant of lateral
slip of the wheels on the rails, and for the stress calculations it may
be neglected.
(2) A lateral reaction as in (a)(2) above.
(3) A change in vertical reactions due to superelevation and
speed.
(4) A lateral force, P1. P1 is the angular accelerating force which
is resisted by friction and impact of the rail on the wheel. It acts
laterally against the flange of the wheel on the upper rail.
The lateral force P1 in either case is the angular accelerating force.
It must be large enough to overcome friction of the wheels on the
rails. It must be equal to or greater than MV2/r or Mrw2, (the
centrifugal force). It may be much larger than Mrw2. It probably
is, at times, much larger than necessary to overcome friction. It
usually includes lateral impact of rail against wheel. This subject
was discussed in Section 30.
The distribution of the horizontal component of the weight of the
car due to superelevation of the outer rail is indeterminate. That
part of the horizontal component appearing on the lower rail,
(a)(2), causes bending of the axle in the same direction as does the
lateral force P1 in (4), and the horizontal component, if any, appear
ing on the upper rail causes bending in the opposite direction from
that of P1 in (4). Hence the proportion of horizontal component of
the weight of the car due to superelevation appearing as a force in
(a)(2) may be of importance in some cases. As the speed of the car
approaches that which corresponds to the superelevation, the force
of (a)(2) approaches zero and the force P1 in (4) becomes somewhat
larger. An approximate value of bending stress may be obtained
by use of Case 7.
It may reasonably be assumed that the angular accelerating force
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
P1 is the only live lateral force that need be considered as causing
bending in the axle when the speed is above that corresponding to
the superelevation of the outer rail.
Therefore, for curves without restraining rails, with the speed of
the car below that for which the superelevation is designed, Case 7
may be applied for approximate solution, and for speeds greater than
that for which the superelevation was designed Cases 4 or 5 may
be applied.
When a car travels around a curve having a restraining rail and
moves at a speed lower than that which corresponds to the super
elevation of the outer rail the lateral forces acting on the wheels and
axle are identical with those for a curve without restraining rail.
These lateral forces are classified in the preceding paragraphs.
Case 7 may be used.
When a car travels around a curve having a restraining rail and
moves at a speed higher than that which corresponds to the super
elevation of the outer rail, the lateral forces acting upon the wheels
may be:
(1) Under usual conditions the track gage on curves is such that
the flange of the upper wheel does not bear against the rail, but
friction exists between wheel tread and rail. There may be, therefore,
a lateral reaction of the upper rail against the tread of the wheel
approaching, in magnitude, at the instant preceding shifting of the
truck of the car, a maximum value of MV2/r, the centripetal force,
but never exceeding the value of friction between wheel and rail;
but immediately after shifting of the truck, this force is very small.
(2) A lateral force P1, the angular accelerating force, which is
resisted by friction and lateral impact of restraining rail and wheel.
It may, therefore, be concluded that for curves with restraining
rails, the car moving at any speed, lateral forces may act on both
wheels, and Case 7 may be used, or P1 may approximately represent
the lateral forces. Either Case 2 or Case 3 then applies.
4. Bending in Car Axles Due to Live Loads, Cases 8 to 7.The
mathematical analysis for each case of live load has been continued
from Chapter IV. The next case for analysis, the application of a
lateral force against the inner side of the wheel on the geared end of
the axle, has been designated Case 3.
A common example of a lateral force acting against the inner side
of a wheel is the reaction of a restraining rail of curved track. Such
reaction is the angular accelerating force, and is resisted by friction
and lateral impact of rail against wheel.
ILLINOIS ENGINEERING EXPERIMENT STATION
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STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
If P1 is the only lateral reaction against the wheel, the balanced
force system acting on the axle and wheels may be considered as
made up of two sets of couples and a force P1 acting as shown in
Fig. 16, Case 3.
From Appendix B it is seen that the relation between P1 and P2
is given by: 17.5 P1 = 57.827 P2 P2 = 0.3026 P1
The moment at any section of the axle due to the force P1 is
M, = 0.3026 Pi x
in which x is measured from the left reaction toward the right and
with values from 0 to 57.827 in. The moment is positive, causing
compression in the top fibers of the axle.
M
Figure 16, Case 3 presents diagrams of moments and  due to
El
M
lateral force P1. Table 12 presents moments, units, areas under
El
M
the  diagram, and change in slope of axle.
El
The application of a lateral force against the tread or flange of
the wheel opposite the geared end of the axle has been designated
Case 4.
This lateral force may be due to the reaction of the upper running
rail of curved track against the tread or flange of the wheel. This
lateral force is the angular accelerating force and is resisted by friction
and lateral impact of rail against wheel.
If P1 is the only lateral reaction against the tread or flange of the
wheel, the unbalanced force system acting on the axle and wheels
may be considered as made up of two sets of couples and a force P1
acting as shown in Fig. 16, Case 4.
From Appendix B the following values are obtained:
17.5 P1 = 57.827 P2, or P2 = 0.3026 P1 and
M, = 17.5 Pi + 0.3026 P1 x
in which x is measured from the left reaction toward the right and
with values from 0 to 57.827 in. This moment is negative and causes
tension in the top fibers of the axle.
M
Figure 16, Case 4 presents diagrams of moments and  due to
El
lateral force Pi.
M M
Table 12 presents moments,  units, areas under  diagram,
El El
and change in slope of axle.
ILLINOIS ENGINEERING EXPERIMENT STATION
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STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
The application of a lateral force against the tread or flange of the
wheel on the geared end of the axle has been designated Case 5.
This lateral force may be due to the reaction of the upper running
rail of curved track against the tread or flange of the wheel. This
lateral force is the angular accelerating force and is resisted by fric
tion and lateral impact of rail against wheel:
If P1 is the only lateral reaction against the tread or the flange of
the wheel, the unbalanced force system may be considered as made
up of two sets of couples and a force P1 acting as shown in Fig. 16,
Case 5.
From Appendix B the following numerical values are obtained:
17.5 P1 = 57.827 P2, or P2 = 0.3026 P1 and
M, = P2 x
in which x is measured from the left reaction toward the right and
with values from 0 to 57.827 in. The moment is negative and causes
tension in the top fibers of the axle.
M
Figure 16, Case 5 contains diagrams of moments and  due to
El
M
lateral force P1. Table 12 contains moments,  units, areas under
El
M
 diagram, and change in slope of axle.
El
A lateral force applied to the inner side of each wheel has been
designated Case 6.
The lateral reactions of rails against the inner sides of both wheels
may result from the guard rails of crossings and also from the wing
and guard rails of a guarded frog.
If P1 and Pi' are the lateral reactions of the rails against the inner
sides of the wheels, the unbalanced force system may be considered
as made up of four sets of couples and two forces, P1 and Pi', as
shown in Fig. 17, Case 6a.
From Appendix B the following numerical values are obtained:
17.5 Pi' = 57.827 P2 or P2 = 0.3026 P1' and
17.5 P1 = 57.827 Pa or P3 = 0.3026 P1
For the special case of P1 = Pi', P2 = Pa, the unbalanced force
system acting on the wheels and axle may be considered as made up
of two sets of couples and two forces, P1 and Pi', as shown in Fig. 16,
Case 6b. The bending moment at any section x located between the
left and right reactions is
M, = 17.5 P1,
a positive moment causing compression in the top fibers of the axle.
55 .
ILLINOIS ENGINEERING EXPERIMENT STATION
M M
Table 12 presents moments, units, areas under the  dia
EI El
gram, and change in slope of axle. Figure 16, Case 6b presents dia
M
grams of moments and  due to lateral forces P1 and Pi' when
El
P1 = Pi'.
The action of a lateral force against the inner side of one wheel
and the simultaneous action of a lateral force against the tread or
flange of the outer wheel has been designated Case 7.
In most cases of curved track with restraining rail, the restraining
rail reacts against the inner side of the wheel on the lower rail, pro
vided the restraining rail is spaced sufficiently close to the lower
running rail. This lateral reaction is the force necessary to give
angular acceleration to the car, and is resisted by friction of the
wheels on the rails and lateral impact of rail and wheel. In magnitude
this force P1 must be sufficiently large to overcome friction, and
usually is much larger.
There may exist a lateral reaction of the upper running rail
against the tread of the wheel as already explained.
There are two parts of Case 7. Case 7a is for action of restraining
rail against the inner side of the wheel at the geared end of the axle,
or for curvature to the left with No. 1 axle at the front end of the car.
Case 7b is for action of restraining rail against the inner side of the
wheel opposite the geared end of the axle, or for curvature to the
right with No. 1 axle at the front end of the car.
The pair of wheels whose change in inclination is being measured
is the leading pair of wheels of the leading truck as the test car
moves around a curve.
Case 7a and Case 7b.The unbalanced force system acting on the
wheels and axle may be considered as made up of couples and forces
as shown in Fig. 18.
Case 7a.Replacing F1 by a force and couple we have
17.5 Fi = 57.827 Fi', or Fi' = 0.3026 F1
and, similarly,
17.5 P1 = 57.827 Pi', or Pi' = 0.3026 P1
Then the bending moment at any section x, where x is measured
from the left reaction toward the right, and has values from 0 to
57.827 in., is
M, = 17.5 F1 + 0.3026 F1 x + 0.3026 Pi x
Case 7b.By similar procedure,
M, = +17.5 P1  0.3026 FI x  0.3026 Pi x
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
APPENDIX B
DETERMINATION OF STRESS CONSTANTS
1. General Statement.Chapter IV and Appendix A outlined the
several cases of bending in car axles. The mathematical analyses are
divided into seven cases, depending upon the type and position of the
force or forces causing bending of the axle. The moment equation
in terms of the unknown force or forces is given for each case.
In order to determine the stress in the axle corresponding to any
specific ordinate of the paper record, the value of the unknown force
must be determined.
An examination of the autographic record shows the speed of the
car, the direction of acceleration of the car, the direction of curvature
of the track, and the length of the ordinate to the record. Additional
information, such as superelevation of outer rail, degree of curve, and
presence or absence of restraining rails, is obtained from a plan of the
track layout. From a study of this information, the number of
forces acting and their lines of action may be determined and may be
classified according to case.
2. Determination of Stress Constants.The term "stress constant"
is here defined as that factor by which the ordinate to the autographic
record must be multiplied in order to obtain the maximum fiber stress
in the axle due to the bending moment caused by one or more lateral
live forces.
The maximum unit fiber stress due to both dead and live loads is
found by adding, algebraically, the unit stress due to dead load and
the unit stress due to live load.
As explained in Appendix A, changes in magnitude of the vertical
reactions do not affect the analyses of Cases 2 to 7. They do not
affect the determination of stress constants for these cases. Hence
the total fiber stress obtained by methods given in Appendix A and
Appendix B does not include any correction for changes in vertical
reactions due to superelevation or centrifugal force.
Conversion of paper ordinates into fiber stress, or vice versa, in
the axle may be accomplished in the following manner:
Case 1 deals with dead loads of car and cargo only. The fiber
stresses were calculated directly. The loading is shown in Fig. 13
and the dimensions of the axle in Fig. 12. The moments and stresses
at seventeen sections were calculated directly from the known load
M
ing. The area under the  diagram gives the change in slope of the
El
ILLINOIS ENGINEERING EXPERIMENT STATION
axle. For all cases of bending, the change in slope of the axle has been
taken between sections a and b, Fig. 12. The change in slope 0
multiplied by the distance from the center of the axle to the line of
action of the recording instrument gives the amount of wheel move
ment. The wheel movement divided by 0.0266* gives the paper
ordinate, Ord. The conversion of ordinates of the paper record into
moments and stresses is made in the reverse order.
Case 2 is the case of a single lateral live load force applied to the
inner side of the wheel opposite the geared end of the axle, Fig. 14.
The line of action of this force is estimated to be 17.5 in. from the
center of the axle. This force Pi may be replaced by a force and
couple as shown in Fig. 14. The assumption is made that the
reactions of the couple may be represented by another couple acting
on the wheels, and with a distance of 57.827 in. between the forces of
17.5
the couple. The value of P2 is then  P1 = 0.3026 P1. The
57.827
bending moment at any section x measured from the left reaction,
and with values from 0 to 57.827 in., is
Mx = 17.5 P1  0.3026 P1 x in. lb. (5)
The change in slope between sections a and b, due to P1, is the area
M
under the  diagram between points a and b and is equal to
El
24 066 X 1011 P1 (See Table 12, Case 2.)
The distance from the center of the axle to the line of action of the
recording instrument had various values, but for the final runs it was
16.17 in. This value will be used here. The wheel movement is then
24 066 X 1011 P1 X 16.17 = 389 156 X 1011 Pi (6)
389 156 Pi
and the paper ordinate, Ord. = 06 and
1011 X 0.0266
0.0266 X 10"
Pi =  Ord., or Pi = 6835 Ord. (7)
389 156
Substituting in Equation (5) we obtain the moment due to live load Pi
Mý = (17.5  0.3026 x) 6835 Ord. (8)
Stress due to Case 2 is usually in addition to stress due to Case Ic
and we are interested in the total stress due to both dead and live
load. A determination of the stress constant for any section, or any
value of x and the location of the section at which the combined'stress
is a maximum, may be made in the following manner:
By inspection, section 5, is the section having the highest stress
*One inch ordinate to the paper record corresponds to 0.0266 inch change in inclination of the
wheels, see Table 2.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
for dead load. Therefore, when the paper ordinate is zero, the maxi
mum fiber stress in the axle is at section 5 and equals 3668 lb. per
sq. in., Table 5. There are three sections of the axle which have
higher fiber stresses than other sections, i.e., sections 5, 10, and 13.
It is necessary to determine the moment and stress at these three
sections for various values of paper ordinate in order to find the
maximum fiber stress.
The general equation for moment due to P1 at any section was
given in Equation (8) as M, = (17.5  0.3026 x) 6835 Ord. The
moment equations then are
for section 5, x = 4.218 in., M5 = 110 888 Ord.
for section 10, x = 47.593 in., M10 = 21 169 Ord.
for section 13, x = 53.968 in., M1, = 7 982 Ord.
The fiber stresses at each section due to the force PI, only, are
S5 = 110 888 X 0.04865 Ord. = 5395 Ord., Ib. per sq. in. com
pression in top fibers
Sio = 21 169 X 0.04865 Ord. = 1030 Ord., lb. per sq. in. com
pression in top fibers
Si = 7 982 X 0.03764 Ord. = 300 Ord., lb. per sq. in. com
pression in top fibers
The stress constant for this case may be either 5395, 1030, or 300.
The stress constant to be used depends upon which of the three sec
tions has the highest stress due to combined dead and live loads.
This may be readily determined by solving the following equations
for various values of the paper, ordinate. The interval of measure
ment or the distance between punctures on the autographic record is
0.15 in. It is convenient, therefore, to vary the paper ordinate values
by 0.15 in. in the following equations:
Ss = 3668  5395 Ord. (9)
So1 = 3404  1030 Ord. (10)
Si1 = 2990  300 Ord. (11)
Example: Suppose the paper record ordinate is 0.15 in. Find
the maximum fiber stress in the axle.
From Equation (9), Ss = 3668  5395 X 0.15 = 2859 lb. per sq. in.
tension in top fibers
Equation (10), So1 = 3404  1030 X 0.15 = 3250 lb. per sq. in.
tension in top fibers
Equation (11), S13 = 2990  300 X 0.15 = 2945 lb. per sq. in.
tension in top fibers
The maximum fiber stress then is 3250 lb. per sq. in. tension in the
top fibers at section 10. The stress constant is 1030.
ILLINOIS ENGINEERING EXPERIMENT STATION
Example: Suppose the paper record ordinate is 0.90 in. Find
the maximum fiber stress. As before
Ss = 3668  5395 X 0.90 = 1186 lb. per sq. in. compression in
top fibers
Slo = 3404  1030 X 0.90 = 2478 lb. per sq. in. tension in top fibers
S13 = 2990  300 X 0.90 = 2720 lb. per sq. in. tension in top fibers
The maximum fiber stress then is 2720 lb. per sq. in. tension in the
top fibers at section 13. The stress constant is 300.
It will be found for this case, Case 2, that for paper ordinate
equal to 1.20 in. or more, the fiber stress will be greatest at section 5
and the stress constant will be 5395.
Case 3 is the case of a single lateral live force applied to the inner
side of the wheel on the geared end of the axle (see Fig. 16).
This case is similar to Case 2. The single force P1 may be replaced
by a couple and a force as shown in Fig. 16. P2 = 0.3026 P1 and the
moment at any section x of the axle due to P1, x measured from the
left reaction toward the right, and with values from 0 to 57.827 in., is
MX = 0.3026 Pi x (12)
The change in slope 0 = 22 687 X 1011 P1 and the wheel movement
is 22 687 X 10" X 16.17 P1 = 368 860 P1 X 1011
368 860 P1 0.0266 X 1011
Ord. = and P1 =  Ord.,
0.0266 X 1011 368 860
P1 = 7211 Ord. (13)
From Equation (12),
M, = 0.3026 X 7211 X Ord. x = 2182 Ord. x (14)
The moment equations then are
for section 5, x = 4.218 in.,
Ms = 4.218 X 2182 Ord. = 9204 Ord.
for section 10, x = 47.593 in.,
Mio = 47.593 X 2182 Ord. = 103 848 Ord.
for section 13, x = 53.968 in.,
M13 = 53.968 X 2182 Ord. = 117 758 Ord.
The fiber stresses due to the force P1 at each section are
S5 = 9 204 X 0.04865 Ord. = 448 Ord., lb. per sq. in.,
compression in top fibers
Sio = 103 848 X 0.04865 Ord. = 5052 Ord., lb. per sq. in.,
compression in top fibers
S13 = 117 259 X 0.03764 Ord. = 4433 Ord., lb. per sq. in.,
compression in top fibers
The stress constant for this case may be either 448, 5052, or 4433.
The stress constant to be used depends upon which of the three sec
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
tions has the highest stress due to combined dead and live loads.
This may be readily determined by solving the following equations
for various values of the paper ordinate. The interval of measure
ment or the distance between punctures on the autographic record
is 0.15 in. It is convenient therefore to vary the paper ordinate values
by 0.15 in. in the following equations:
S5 = 3668  448 Ord. (15)
S10 = 3404  5052 Ord. (16)
Sis = 2990  4433 Ord. (17)
It will be found that when paper ordinates from 0.15 to 1.20 in.,
inclusive, are encountered, maximum fiber stresses due to bending
occur at section 5, and for ordinates of 1.35 in. and above, the maxi
mum fiber stress occurs at section 10. Maximum bending stress does
not occur at section 13 for any value of paper ordinate.
The stress constants are then 448 for ordinates of 0.15 to 1.20 in.,
and 5052 for ordinates of 1.35 in. and above.
Case 4 is the case of a single live lateral force applied to the tread
or flange of the wheel opposite the geared end of the axle, see Fig. 16.
As in Cases 2 and 3, the line of action of this force is assumed to be
17.5 in. from the center of the axle. "This force may be replaced by a
couple and force as shown in Fig. 16. The reactions of this couple
may be represented by a couple acting on the wheels as shown. The
value of P2 = 0.3026 P1
The change in slope 0 = 24 066 X 1011 P1
The wheel movement = 24 066 X 1011 Pi X 16.17 = 389 156 X 1011P1
0.0266 X 1011
and P1 =   Ord. = 6835 Ord. (7)
389 156
Substituting in Equation (5), M, = (17.5  0.3026 x) 6835 Ord.
By inspection, the maximum deadload stress is found at section 5.
The force P1, applied as shown in Fig. 16, causes increased tension in
the top fibers of the axle. Section 5 remains the section of highest
bending stress for all values of P1.
The moment at section 5 due to P1 is (x = 4.218 in.) Ms =
110 888 Ord. and the fiber stress at section 5, due to P1, is S5 =
110 888 X 0.04865 Ord. = 5395 Ord.
The stress constant for all values of P1 or of paper ordinates is
then 5395.
The maximum fiber stress due to both dead load and live load is
then S = 3668 + 5395 Ord. (18)
Case 5 is the case of a single live lateral force applied to the tread
or flange of the wheel at the geared end of the axle, see Fig. 16.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIa. 17. SYSTEMS OF FORCES ACTING ON WHEELS AND AXLE, CASE 6a
This case is similar to Case 3. The direction of application of P1,
however, is opposite to that of Case 3.
P2 = 0.3026 P1 and M, = 0.3026 Pi x (12)
0 = 22 687 X 1011 P1
Wheel movement = 368 860 P1 X 1011 and P1 = 7211 Ord. (13)
Therefore, M, = 0.3026 X x X 7211 X Ord. = 2182 X x X Ord. (14)
By inspection, section 10 always has the maximum bending mo
ment. Equation (10) then becomes M,0 = 2182 X 47.593 Ord. =
103 848 Ord. and the stress at section 10 due to P1, only, is Slo =
103 848 X 0.04865 Ord. = 5052 Ord.
The stress constant for all values of P1, or for all values of the
ordinates to the autographic record, is 5052. The total bending
stress due to both dead and live load is
Sio = 3404 + 5052 Ord., tension in top fibers. (19)
Case 6 is the case of a lateral live force applied to the inner side
of both wheels simultaneously, Figs. 16 and 17.
The general case of P1 7 PI' is indeterminate. The special case of
P1 = Pi' may be solved for moments, change in slope, and fiber stress.
When P1 = Pi', P2 = P3 and the bending moment at any section,
x, due to P1 and Pi', x measured from the left reaction toward the
right and with values from 0 to 57.827 in., is M, = 17.5 P1 (20)
The change in slope of the axle between sections a and b, is
0 = 46 754 Pi X 1011 (Table 12)
Wheel movement = 46 754 P1 X 1011 X 16.17 = 756 017 P1 X 1011
756 017 Pi X 1011 Pi
The paper ordinate is, then, Ord. = 7  and
0.0266 3518
P1 = 3518 Ord. (21)
Substituting this value of P1 in Equation (20), we.obtain
M. = 17.5 X 3518 X Ord. = 61 565 Ord. (22)
By inspection of Table 5 it may be seen that there are several
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
'Case 7a, Case 7b
FIG. 18. SYSTEMS OF FORCEs ACTING ON WHEELS AND AXLE, CASE 7
sections of the axle having the same crosssectional area. Conse
quently, the stresses due to P1 and Pi', only, are of the same magni
tude at several sections. Section 5, however, has the largest fiber
stress due to dead load. Therefore, for small values of P1, the stress
will remain a maximum at this section. When the value of Pi be
comes large, section d will have the larger stress.
The stress for either section will be 61 565 X 0.04865 X Ord. =
3057 Ord. and the stress constant is 3057.
By calculation, it is found that the maximum fiber stress is located
at section 5 for all values of paper ordinate from 0.15 to 1.20 in.,
inclusive, and at section d for all ordinates from 1.35 in. and above.
The total fiber stresses due to dead and live loads are
S5 = 3668  3057 Ord. (23)
Sd = 3023  3057 Ord. (24)
Case 7 is the case of a lateral live force applied to the inner side of
one wheel and a lateral force applied to the tread or flange of the
other wheel, both forces acting in the same direction, that is both
tending to incline their respective wheels in the same direction,
Fig. 18.
Such forces might be found in a reaction of a restraining rail
against the inner side of one wheel and a simultaneous reaction of an
upper running rail against the tread or flange of the other wheel.
There are two parts'of this case, 7a and 7b. Case 7a is for curva
ture of track to the left, the pair of wheels shown in Fig. 18 moving
toward the reader and with the elevated rail on the left. Case 7b is
for curvature of track to the right.
In Case 7a or 7b there are two unknown forces, F1 and P1. From
the test records the net change in inclination of the wheels is recorded.
If FI were equal to zero, the bending of the axle would be as for the
simple Case 3. But F1 may not have a value of zero. The maximum
value it could have is assumed to be the full centrifugal force due to
ILLINOIS ENGINEERING EXPERIMENT STATION
onehalf the weight of the front end of the car. In order to have a
force Pi, there must be contact between the wheel and the restraining
rail. Consequently part, if not all, of the centrifugal force of the car
will react on the wheel on the lower rail, thus being divided between
both wheels.
On the assumption that F1 is not zero, but may be as great as the
centrifugal force, F1 = 0.289 V2 D in which V is the speed of the car
in miles per hour and D is the degree of curvature, the following
relations will give a solution for the bending stress:
The net change in slope of the axle due to F1 and P1 will be,
6c = 22 688 Pi X 1011  24 066 F1 X 1011 (25)
and the ordinate to the paper record will be
68 X 16.17
Ord. = = 607.9 0, (26)
0.0266
For any specific case, the ordinate may be obtained from the
autographic record. Then 0c may be found from Equation (26).
The value of F1, depending upon the degree of curvature and speed
of car, equal to 0.289 V2 D, may be substituted in Equation (25),
and the value of P1 determined.* The bending stress due to P1 and
F1 may then be found from the following equations:
M, =  17.5 F1 + 0.3026 (P1 + F1) x, x is any section measured
from the left reaction and having values from 0 to 57.827 in.
and S = 0.04865 [17.5 Fi + 0.3026 (Pi + F1) x] (27)
when x = 4.218 in. (sec. 5) or x = 43.597 in. (section 10).
Example: Given curve to left, D = 67 deg., V = 15 m.p.h.,
paper ordinate 2.10 in.; find the bending stress.
F1 = 0.289 V2D = 43571b. Net change in slope, from Equation (26),
2.10
is c, =  = 0.003454
607.9
Substituting in Equation (25),
0.003454 = 22 688 Pi X 1011  24 066 X 4357 X 1011
= 22 688 Pi X 1011  0.00105
Pi = 20 000 lb.
Substituting this value of P1 in Equation (27), for x = 47.593 in.
(section 10)
S = 0.04865 [(17.5 X 4357) + (0.3026 X 24 357 X 47.593)]
= 0.04865 X 276 280 = 13 440 lb. per sq. in., compression.
S due to dead load = 3 400 lb. per sq. in., tension.
Total bending stress = 10 000 lb. per sq. in., compression.
*See Section 36, paragraph 5, and Table 11 for discussion of lateral forces greater than the
centrifugal force at a curve.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
This is the actual stress when the total lateral force is divided between
P1 and F1. Assuming F1 is zero (Case 3), the stress would be 7200 lb.
per sq. in. compression in top fibers. The difference between the
stress for Case 3 and that for Case 7a in this particular example is 2800
Ib. per sq. in.
APPENDIX C
FORCED VIBRATION OF CAR WHEELS AND INERTIA EFFECTS IN
RECORDING INSTRUMENT
1. General Statement.In one of the early runs made by the test
car some very large movements of the recording instrument were
observed. Similar movements were observed in subsequent runs on
a few other sharp curves. Usually no abnormally large movement
occurred except at high speeds of the test car. In all but one case of
an abnormally large movement of the instrument, the test car was
moving around track having a restraining rail and curved to the left,
and with the pair of wheels between which the recording instrument
was installed at the rear end of the car. Abnormally large move
ments of the instrument were always in the same direction, i.e., they
caused a shortening of the recording instrument. This shortening
indicated an inward movement of the top flanges of the wheels
whereas the prevailing movement of the top flange of the wheel on
the upper running rail was outward, and the movement of the top
flange of the wheel on the lower running rail was usually very small
except at the moment of a sudden inertia movement in the instru
ment. Apparently the abnormally high movements of the instru
ment were caused by forced vibrations of a wheel. The inertia of the
instrument apparently caused a movement of the instrument far in
excess of the movement of the wheel flange itself. The frequency of
the light spring originally incorporated in the instrument, Fig. 2,
presumably was lower than the frequency of vibration of the wheel
so that it could not overcome or prevent the effect of the inertia of
the moving instrument.
2. Study of Inertia Effects.In order to study how to eliminate
inertia effects in the instrument, a special telescoping tube was built
with a light spring on one end and a mechanical clamp on the other.
This tube is described fully in Section 12. It was placed in position
above the test axle, temporarily replacing the original recording
instrument. The spring on one end of this tube gave about 100 lb.
pressure against one wheel. The mechanical clamp when tightened
ILLINOIS ENGINEERING EXPERIMENT STATION
resisted several hundred pounds pressure. ITest runs were made.
It was found that this tube shortened in a manner similar to the orig
inal instrument, but to a greater degree. From these data it was
evident that springs holding the ends of the instrument in contact
with the wheels should exert high pressures against the wheels and
have high frequencies of vibration.
The telescoping tube was modified by replacing the clamp and
light spring by two springs, one at each end, which required large
loads for small deflections and consequently had high frequencies of
vibration. A mechanical arrangement, described in Section 12,
recorded the change in inclination, or movement, of the wheels
directly on paper without magnification. There were no noticeable
inertia effects. From the result of these special tests it was possible
to choose such springs for application to the axle bending apparatus
used in the regular tests that inertia effects due to vibration were
reduced to an inappreciable amount.
3. Position of Wheels of Truck on Rails.A study of the position
of the four wheels of a car truck on the rails of curves having restrain
ing rails was made in order to obtain information regarding the source
and frequency of forced vibrations of the wheels. The data for this
study were obtained from test runs made with the recording instru
ment installed first above one axle of a truck and then above the
second axle. Figure 19 shows the positions of the wheels. All
four wheels are on the running rails, and in addition the inside flange
of wheel No. 3 makes contact with the restraining rail, the tread and
perhaps the fillet between tread and flange of wheel No. 2 makes
contact with the upper running rail. This is the general case. In
order to traverse the curved track, the truck must from time to time
change its position by rotating in a plane parallel to the track so that
the axle of the leading pair of wheels (Nos. 3 and 4) becomes more
nearly radial to the curve. When the pair of wheels, Nos. 3 and 4,
shifts so as to become nearly radial to the curve, the other pair of
wheels of the truck, Nos. 1 and 2, is forced to change its position.
It may be assumed that the inside flange of wheel No. 1, which does
not ordinarily make contact with the restraining rail, is brought into
contact with the restraining rail. Perhaps this action occurs as often
as the truck changes position. This contact causes an inward move
ment of the top of the flange of wheel No. 1. All autographic records
of test runs show small fluctuations in the length of the ordinates to
the curves, see Fig. 20. Such fluctuations are caused by changes in
the positions of the wheels and may be considered as occurring as
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
fa) Pos/ifon of Truck on Ra/i/s
L,/. I' /Vc/ IVA  fIa //o / /,VV II.
Curve No. 2  Ra'd//s 90 1 /50 ft
Shade'd reas in b, c, ~ad a indica/e those whee/s wh/ch re
ceive /laera/ reaction fro77 the restra'iing rai/ andd the c/pper r~n
n/ing rar/
FIG. 19. POSITION OF WHEELS OF CAR ON CURVED TRACK
often as the truck, or pair of wheels, changes position. These fluc
tuations usually represent very small movements of the flanges of a
wheel. The rate of fluctuation in the record appears to be a function
(,crve /vo. '1rLaawS eCIFF
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 20. TYPICAL AUTOGRAPHIC RECORDSHARP CURVE WITH RESTRAINING RAIL
of the speed of the test car and the length of track traversed between
two successive changes in position of the truck.
4. Rapidity of Change in Position of a Car Truck on Four Sharp
Curves.Inertia effects were observed in the recording instrument
during early test runs on a few sharp curves. Four of these curves
had radii ranging from 90 to 300 feet. The average length of track
traversed by the truck before changing positions, based upon fluc
tuations of the record as explained in the foregoing, is very nearly
constant for the four curves under consideration. It appears that the
car truck changes position on these four curves about once every four
feet. Table 13 contains data showing frequency of change in position
of a car truck on the rails, compiled from several of the later runs of
the test car when there were no appreciable inertia effects present in
the instrument. The number of changes of position per minute
depends upon the speed of the car. At ten miles per hour, the number
of changes of position of the truck would be 220 per minute; and if
wheel No. 1 is brought into contact with the restraining rail at each
change of position, the rate of forced vibrations of the wheel would be
220 per minute when the car moves at 10 miles per hour, or 308 per
minute at 14 miles per hour. It may also be reasonably assumed that
with each change in position of the truck the part of the flange
of No. 2 wheel which has been in contact with the upper running rail
moves away from it momentarily, thus allowing a small inward move
ment of the top part of the wheel. As the pair of wheels, Nos. 1 and
2, shifts as a unit, it is probable that this change in position of No. 2
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS 69
TABLE 13
FREQUENCY OF CHANGE IN POSITION OF A CAR TRUCK ON CURVED TRACK*
Curve
No.
1 ...................
1....................
1 ...................
2 ....................
2 ...................
2....................
3 ...................
3 ......... .. .........
3....................
4....................
1 ...................
2....................
3....................
4....................
Run Speed
No. mi. per hr.
36a2 11.5
40b2 12.2
35cz 10.8
36a2 10.5
36as 10.0
35c2 11.5
40a2 10.5
36as 11.5
35c2 9.6
36a2 10.5
35c2 10.3
35c2 11.6
35c2 10.8
35c2 11.0
Summary
Curve No. 1. .......................... leading 5.59
trailing 4.09
Curve No. 2.......................... leading 4.12
trailing 4.04
Curve No. 3. .......................... leading 3.83
trailing 3.78
Curve No. 4........................... leading 3.60
trailing 3.96
*Average length of track traversed by truck before changing its position on rails based upon
the fluctuation of the ordinates of the autographic records.
fNo. 1 axle is in a leading position when it is at the front end of the test car. It is trailing when
it is at the rear end of the test car.
wheel takes place at about the same time that No. 1 wheel comes in
contact with the restraining rail. The test records show that this
change in position of No. 2 wheel occurs at approximately the same
rate per minute. It is then to be expected that the movements of the
two wheels will occur in such a manner that they will be additive in
frequency as far as the vibration effects upon the recording instru
ment are concerned. The combined movements or vibrations in this
case might be expected to be as high as 440 per minute at a car speed
of 10 miles per hour. It is to be supposed that the condition of the
rails would affect the rate of vibration, but the expected number of
vibrations for the usual case might be from 220 to 440 per minute at
a speed of 10 miles per hour.
5. Calculation of Frequency of Vibration of Springs Used on
Recording Instrument.The early test runs were made with only one
spring on the recording instrument, see Fig. 2. The instrument was
placed between the car wheels, as shown, with an initial compression
of the spring. Under this arrangement, the cycle of vibration of the
spring due to forced vibration or movement of the wheel will be only
Length of Track
ft.
5.90
5.86
5.00.
3.70
4.33
4.32
4.10
3.59
3.79
2.98
4.23
3.33
4.96
3.96
Position of
No. 1 Axlet
leading
leading
leading
leading
leading
leading
leading
leading
leading
leading
trailing
trailing
trailing
trailing
ILLINOIS ENGINEERING EXPERIMENT STATION
a part of a complete cycle of simple harmonic motion. Due to the
presence of the car wheel the instrument can vibrate in only one
direction from a position of zero spring pressure. A position of zero
spring pressure is the position the spring would take if there were no
wheel present. This removes onehalf cycle of vibration. The initial
compression of the spring also removes much of the range of ampli
tude of the half cycle of harmonic motion. The remaining part of a
cycle of vibration will then have an amplitude equal to any additional
compression of the spring due to an inward movement of the wheel.
The time required for the execution of onehalf cycle of this oscillation
may be assumed to be the difference between the period of onefourth
of a complete cycle of simple harmonic motion for the system and the
time required for the system to travel from a position of zero spring
pressure to its position of initial spring pressure.
The time of contact of the instrument and the car wheel was
calculated, and found to be about one and onehalf per cent of the
total period of vibration of the instrument at the maximum amplitude
shown in Fig. 21. This is so small that it was neglected in the
calculations given in the following paragraphs.
The period of onefourth cycle of simple harmonic vibration is
S Im*
T, =   (28)
2 c
where T, is the period of vibration, m is the mass of the vibrating
parts, and c is the spring constant or the force required to compress
the spring one inch. In this equation T, is independent of the
amplitude of vibration.
The period of vibration corresponding to any amplitude equal to
or less than the maximum may be found by integrating the equation
v fx dx
v v = a dx, substituting  for v and integrating again, from
J,, Jo dt
1 /k
which we get t = are sin  X (29)
c
in which t is the period of vibration, k = , c and m are as defined
m
in Equation (28), and Vo is the velocity the mass m would attain if it
were allowed to travel to the position of zero spring pressure.
The period of oscillation of the spring in the early tests is, conse
quently, 2(T,  t).
*S. Timoshenko, "Vibration Problems in Engineering," p. 2.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
0/0
0.08
0.06
004
0.02
!/o
O.O8
0.06
0.06
0.04
aoe
0
0./0
008
0. 06
0.04
0.02
0
70 /5
90 2000 2500 300 3500 4000 4500
Sma// end of /nsfrume/nt
Sas used /' f7Aia/ te7 s runs_
600 /. int/fi/ pressure on 3.0/b spr/r'g
 Wefh of v/raf/in oparts 2. 85/ 
h
0 4000 8000 /10 /6000 000 24000 Z000 32000 36000
(c)
L L'rge end of ,nstrumrent
as used i/n f'?a/ test runs
600/b an'd /00/b. i/nita/ spr/ng pressure
on 350 and 495/1 sprTins
SWe/ihf of vi/ra/in g p4ar/s /8.7/b._
\== 0nn O
_ _ _ _ _ s , _ _ _ _ _ _
0 5
(.
c:Z
0o /1
l I/VU WU JVVV 'V foV O v 00 UU( Ouvv .vvv
Frequency: V/brations per M/U/fe
FIGa. 21. FREQUENCY OF VIBRATION OF SPRINGS USED IN TESTS
j
I I I I I I
Largqe end l' or/'y/'a'l i's/rumen/
I/O /b. fi'///i/ pressure on /65/I spr/'ny
We/y/h of /bra/g par/s /8. ? /
0
ILLINOIS ENGINEERING EXPERIMENT STATION
By substituting the proper values of m, c, and x in Equations (28)
and (29), the curves shown in Fig. 21 were obtained.
Later runs of the test car were made in which two springs were
used on the recording instrument, one on each end. The arrange
ment of springs, bearings, and brackets of the recording instrument
was such that one end of the instrument received the benefit of the
combined pressures of both springs, whereas the other end of the
instrument received the pressure of one spring only. A modification
of Equation (29) was necessary when considering this combination of
springs.* This modification is treated in detail in Sample Calcula
tions, (2), and the data are plotted in Fig. 21.
Sample Calculations.(1) Find the frequency of vibration of a
single spring, as shown in Fig. 21, when the weight of the vibrating
parts is 18.7 lb., for a wheel movement of 0.10 in.
18.7 lb. sec.2
m = = 0.5807
32.2 ft.
c = 165 lb. per in.
c 165
m 0.5807
Initial compression of spring = 110 lb.
110
Initial deflection of spring =  = 0.667 in. = X
165
The maximum deflection of the spring is then
0.10 + 0.667 = 0.767 in.
From Equation (28) T, = time of onefourth cycle of simple
w m 7
harmonic motion =  0.0035197 = 0.0931956 seconds.
1 V/ k XX
From Equation (29), t = arc sin
Vk Vo
Now the potential energy in the spring at the end of its travel,
where velocity is zero, is equal to 2 m k X2. This is equal to the
kinetic energy of the spring if allowed to travel to the position of
zero spring pressure, at which point the velocity is a maximum.
Hence 12 m k X2m.. = 12 m vo2 or vo = /k ma,., hence Equation (29)
1 . Vk X
becomes t =  arc sin
k Vk X Xm...
X is the initial deflection of the spring.
*This problem is treated by S. Timoshenko, "Vibration Problems in Engineering," pp. 14.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
Xm.a. is the initial deflection plus the wheel movement and
0.6667
t = 0.05933 arc sin
0.7667
0.6667
sin =  = 0.86957
0.7667
0 = 60 deg. 24.533 min. = 1.05433 radians.
Then t = 0.05933 X 1.05433 = 0.0625534 seconds.
The period of actual vibration is then
2(T,  t) = 2(0.0931956  0.0625534) = 0.0612844 seconds.
60
The frequency per minute is then  = 979. (See Fig. 21).
0.0612844
(2) Find the frequency of vibration of a system of two springs
when the weight of the vibrating parts is 18.7 lb. Since the two
springs had different constants and different initial deflections, the
acceleration is a different function of the displacement X from that
in the case of only one spring. It is necessary therefore to integrate
again the equation v dv  a dx = 0 to find the period correspond
ing to any given amplitude.
In order to get the acceleration a the forces acting on the heavy
end of the instrument must be considered.
Constant of the large spring = 350 lb. per in.
Constant of the small spring = 495 lb. per in.
Initial pressure of large spring = 600 lb.
Initial pressure of small spring = 100 lb.
600
Initial deflection of large spring = = 1.714 in.
350
100
Initial deflection of small spring =  = 0.202 in.
495
This means that the large spring was compressed (1.714  0.202)
= 1.512 in. before the small one came into action. The forces exerted
to accelerate the mass of the instrument then are
(a) by the large spring 350 X and
(b) by the small spring 495 (X  1.512)
The accelerations per inch deflection X are
32.2
by the large spring 350 X  X = 602.673 X
18.7
32.2
by the small spring 495 X  (X  1.512)
18.7
= 852.353 (X  1.512)
ILLINOIS ENGINEERING EXPERIMENT STATION
The acceleration of the heavy end of the instrument is the sum of
these two accelerations or
a = 602.673 X 852.353 X +1288.76
Now, v dv = a dx = 1455 X dx + 1288.76 dx
and v dv =  1455 X dx + 1288.76 dx
The constant of integration for this equation may be determined from
the equation that the loss in kinetic energy is equal to the gain in
potential energy when set up in algebraic form and solved, and for
this case is equal to 1948.5.
Hence, v2 = (vo2 + 1948.5) + 2577.5 X  (1455) X2
dx
v =  = (vo2 + 1948.5) + 2577.5 X (1455) X2
dt
dx
odt = o V(o2 + 1948.5) + 2577.5 X  (1455) X2
S1 2577.5  2910 X
t = are sin
/1455 /5820 vo2  4 696 661
We now have a general expression for the time required for the instru
ment to move from a position of zero spring pressure to any distance
X from it.
By equating the potential energy at Xma.. to the kinetic energy at
the position of zero spring pressure, as was done in the case of the
single spring, we can find Vo. Let us assume the wheel movement to
be 0.10 in. and thus X.. = (1.714 + 0.10) = 1.814 in.
The potential energy stored in the springs at this amount of
compression is
350 X2 495 (X  1.512)2 845
E =  + =  X2  748.44 X + 565.82
2 2 2
= 1390.28  1357.67 + 565.82 = 598.43
E, = Ek = 2m vo2 = 598.43.
1196.86 X 32.2
vo2 = = 2060.9 ft.2 per sec.2
18.7
By substituting this value of vo2 and X = 1.814 in the preceding
general equation we get the time required for onefourth cycle of
what corresponds to the simple harronic motion in the previous case
when only one spring was acting. As a matter of fact we shall pres
ently see that in this case it is the same as for simple harmonic
motion.
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
t = 0.02621 arc sin 2701
\2701/
sin 0 = 1
0 = 90 deg. = 1.5708 radians.
t, = 0.02621 X 1.5708 = 0.0411707 seconds.
Now the time required to go from the position of zero pressure a
distance X = 1.714 in. is
2410.24
t = 0.02621 arc sin
2701.4
sin 0 = 0.89222
0 = 63.154 deg. = 1.10224 radians
t = 0.02621 X 1.10224 = 0.0288897 seconds.
(t,  t) is the time required for the instrument to travel one half
of its actual cycle. The period is, therefore,
2(t,  t) = 2(0.0411707  0.0288897) = 0.024562 seconds
and the frequency per minute is then 2443 (see Fig. 21).
6. Spring Characteristics, Observed and Calculated Vibration Fre
quencies for Early Tests.The spring originally built into the record
ing instrument required 165 lb. compression for each inch of deflection.
This spring was usually compressed 110 lb. for test runs. In the early
runs whenever inertia effects were observed in the instrument, this
spring was always at the end of the instrument in contact with No. 1
wheel, see Fig. 19. The weight of the parts of this end of the instru
ment was 2.85 Ib., while that of the balance of the instrument was
18.7 lb. Thus the original spring opposed the movement of either or
both ends of the recording instrument. As already explained, the
greatest inward movement of the instrument would be expected to
come from wheel No. 1 when it comes in contact with the restraining
rail. The mass of the vibrating parts, the spring constant, the initial
compression of the spring, and the amount of inward movement of
the wheels determine the frequency of vibration of the spring. In
Fig. 21 wheel movement was plotted against frequency of spring
vibration for the initial spring pressure generally used in the early
tests. The calculated frequency of the spring under this initial
pressure was 1350 per minute for an inward wheel movement of
0.05 in. The weight of the vibrating parts was about 18.7 lb. There
was probably some friction between the instrument and the bearings.
This friction was probably negligible for a strong spring but may have
had the effect of greatly reducing the frequency of vibration of the
original light spring. Thus the frequency of 1350 per minute expect
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 14
NUMBER OF FORCED VIBRATIONS IN CAR WHEELS
Estimated
Number of Observed
Curve Speed of Forced Movements of
No. Test Car Vibrations Instrument
mi. per hr. in Wheels per minute
per minute (average)
1............................. 4 86172 114
6 129258 282
8 172344 366
10 215430 501
12* 258516 404
14 302604 498
2........... ................. 4 88176 80
6* 132264 420
8 176352 304
10 220440 353
12 264528 349
14 308618 494
3............................ 34 93186 324
6* 140280 300
8 183366 327
10 233466 398
12 280560 306
4.............. ................ 4 98196 128
6 147294 294
8 196392 265
10 244488 376
12 293586 300
14* 342684 417
*Inertia effects were observed at this and all higher speeds.
ed of the original light spring may have been greatly reduced by a
few pounds of friction.
Table 14 contains data relating to four curves, and shows the
expected number of forced vibrations of the wheels, estimated on the
basis of the rapidity of change in position of the truck and the
number of movements of the instrument on the basis of the fluc
tuations observed on the autographic records.
It will be observed that inertia effects in the instrument were ob
served in curves Nos. 1, 2, and 4 when the observed frequency of
vibration of the wheels and the estimated frequency of the spring of
the instrument was about 400 per minute. These are simple curves
of 90 or 100 ft. radius. Curve 3 is a reversed, compound curve with
radii of 300, 150, and 90 ft. Inertia effects were present in this curve
at an observed frequency of from 140 to 280 at a speed of 6 miles
per hour. At higher speeds, the number of observed movements of
the instrument was somewhat lower than was estimated. It is to be
remembered that the observed movements of the instrument per
minute is an average number for the whole time the test car was
completely in the curve. The number of vibrations immediately
STRESSES IN CAR AXLES UNDER SERVICE CONDITIONS
2 DAO PExCeIS R UN 35a, DE UR E N ERIC
be ccu ArtelydAteWmeine
ce /.C .  . . .. . ....... elimi ate i e t
fe w m e the rsu o  Runo /h . te o  No.)Z
3.... e/. 3/ /34 b0300
S3 T;;  W3 =3 0  3000
L54 1c 6  00 000
60a I I IZ)/,' 15
1.6I Cir ur to P/'b
/0 /0 14 /6 /8 26O 0z 24 e6 2E
wileoerd o// the in0trun
preceding the occurrence of inertia effects in the instrument cannot
be accurately determined. pre sure s 6. (ty s
7. Selection of Suitable Springs to Eliminate Inertia Effects from
Recording Instrument. The selection of springs to eliminate inertia
effects was made from the results of trial runs of the test car using
different combinations of spring pressures. The original light spring
was replaced by a heavy spring of high frequency of vibration and
requiring about 495 lb. to compress it one inch. A similar spring
was installed on the other end of the instrument. Runs with the
test car were made around many curves, among which were the four
curves mentioned in the pre cedinSg paragraph. Spring pressures
holding the ends of the recording instrument against the wheels were
always high. Any friction between instrument and bearings was very
low in comparison with the pressure exerted by the springs. The
very high readings on the autographic record were immediately
eliminated. Tests were continued, using different springs and various
spring pressures, to determine whether or not there were still some
slight inertia effects present. Figure 21 shows the computed fre
quency of vibration of the systems of springs used in these tests.
Figure 22 is a graph of the data of several special runs made by
the test car on a 19deg. curve at Niles Center, Northwest Chicago,
in which the spring pressures against the wheels were varied. Average
in which the spring pressures against the wheels were varied. Average
ILLINOIS ENGINEERING EXPERIMENT STATION
ordinates to the autographic record were obtained by determining the
area under the curve by the use of an integraph. These average
ordinates are plotted in the figure for various speeds. The curvature
of the track was to the left for all runs except 35 a,. The data from
run 34 show that about 600 lb. spring pressure against the wheel that
is receiving the largest amount of vibration or movement, in this case
wheel No. 1, is sufficient to prevent inertia effects in the instrument
due to that wheel. The data also show that an increase in spring
pressure against No. 2 wheel, above 200 lb., decreases the magnitude
of the ordinates. Run 35al was made with the curvature of the track
to the right, thus causing wheel No. 2 to receive the largest amount
of vibration or movement. This run indicates that a high spring
pressure against wheel No. 2 was also necessary. Other runs were
also made with spring pressures higher than in runs 34 and 35.
Maximum ordinates are plotted against speed in Fig. 22. These data
show somewhat lower ordinates for spring pressures of 700 and 600 lb.
than for spring pressures of 600 and 300 lb. Average ordinates for
these runs and maximum ordinates for several other curves having
restraining rails indicate that about 600 lb. spring pressure against
each wheel was necessary and sufficient to prevent inertia effects in
the recording instrument. Spring pressures above 700 lb. did not
decrease the ordinates to the autographic record, but did cause the
rollers of the ends of the recording instrument to cut into the steel of
the wheel flanges. It was decided to use 700 and 600 lb. of spring
pressure against the wheels for test runs over the whole railway system.
Results presented in previous chapters of this bulletin were based
upon test runs in which 100 lb. of compression was used on one spring
and 600 lb. on the other spring, as shown in Fig. 21. Due to the
arrangement of instrument, springs, and brackets, one car wheel
received the combined compression of both springs, 700 lb., whereas
the other wheel received the compression of one spring only, 600 lb.