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1

THESIS

This is to certify that the

thesis entitled

AN EXPONENTIAL RELATIONSHIP BETWEEN STRESS IND STRAIN
WITH APPLICATION TO PLASTIC BENDII‘I}

presented by

NELSON D. WOLF

has been accepted towards fulfillment
of the requirements for

MASTER OFW degree inWCHANICS

Mofiw

Major professor

 

Date_M§LCL1_8._1255__

0-169

AN EXPONENTIAL RELATIONSHIP BETWEEN STRESS AND
STRAIN WITH APPLICATION TO PLASTIC BENDING

by
Nelson David lglf

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics
1955

THEE)!

q

 

ACKNOWLEDGMENT

The author wishes to eXpress his appreciation

to Dr. Charles 0. Harris for suggesting the problem,
as well as for his many helpful ideas throughout its

undertaking. It has been through his guidance and

supervision that this paper was possible.

0r{‘”%?
e g.)- 3%}: :HJ

 

AN EXPONENTIAL RELATIONSHIP BETWEEN STRESS AND
STRAIN WITH APPLICATION TO PLASTIC BENDING

by
Nelson David Wolf

AN ABSTRACT

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics
Year 1955

Approved _§45z4a~fl24 C), 7JWLhJAJL6

Nelson David Wolf

The objectives of this study were to find a suitable
mathematical relation between stress and strain; to find a
relation between stress and bending moment where the stress-
strain law is non-linear; and to check this relation by
experiments.

The properties desired in a stress-strain relation are
listed, and a formula between stress and strain was found.
From this relation, the bending moment for a given cross-
section was found as a function of the strain at the outer
surface. Using the stress-strain.curve, a plot of «%§ versus
the maximum.stress on the section was drawn.

From.this study it was concluded that the proposed
mathematical relation is suitable and can represent stress-
strain data for certain types of magnesium.and aluminum
alloys. It was also concluded that the stress at the outer
surface can be fairly'well’estMmated by the use of this rela-
tion.

TABLE OF CONTENTS

CHAPTER Page
Statement of the Problem 1

I. Mathematical Relationships Between Stress and
Strain with Application to Magnesium and
A1uminum.Alloys . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . .
Properties Desired in a Formula . . . . . . . .
Previously Proposed General Formulas . . . . .

The Proposed Formula 0 e e e e e e e e e e e o

a) U1 to a: r4 I4

Method of Application of the Formula . . . . .

Application to Magnesium and Aluminum Alloys . 11

II. Application of the Stress—Strain Formula to
Plastic Bending . . . . . . . . . . . . . . . . . 23
Introduction . . . . . . . . . . . . . . . . . 23
Bending Moment for a Rectangular Cross-Section. 25

Bending Moment for a Circular Cross-Section . . 27

III. The Bending Tests . . . . . . . . . . . . . . . 30
Introduction . . . . . . . . . . . . . . . . . 30
Description of EXperiment . . . . . . . . . . . 30
The Aluminum Alloy . . . . . . . . . . . . . . 32

TheMagnesiumAlloy............. 33

IV.

Conclusions

Bibliography .

TABLE OF CONTENTS (Cont.)

LIST OF FIGURES

FIGURE PAGE
1. 9
2. Tension and compression tests for Alcoa 2hS-Th Al

Square Section 13
3. Tension and compression tests for Alcoa 2hS-Th Al
Circular Section 1h
h. Tension and compression tests for Mg, Type.
ZKCOA-B by Dow Company Circular Section 15
S. 17
6. Tension curve 20
7. Compression curve 21
8. 2h
9. 2h
10. 26
ll. 26
12. 31
13. Stress Vs. %% for Rectangular Section ans-Tu Al 3h
1h. Stress Vs. %g for Circular Section 2hS-Th Al 35
15. Stress Vs.M2 for Circular Section ZKéoA-B Mg 37

STATEMENT OF THE PROBLEM

The objectives of the problem are: to find a suitable
mathematical relation between stress and strain; to find a
relation between stress and bending moment when the stress-

strain law is non-linear; and to check this relation by

suitable experiments.

CHAPTER I

MATHEMATICAL RELATIONSHIPS BETWEEN STRESS AND STRAIN
WITH APPLICATION TO MAGNESIUM AND ALUMINUM ALLOYS

Introduction

The problem of representing stress-strain data by an
analytical expression is not a new one. Many studies have
been made on the subject with no one answer being the re-
sult. Below the proportional limit of the material Hooke's
law seems to be universally accepted, but beyond this point

no one formula is in general use.

Properties Desired in a Formula

In seeking formulas to represent stress-strain data,
it is possible to state certain properties that a formula
should have. This statement makes it possible to eliminate
quickly certain types of formulas that otherwise would seem
adequate at the first glance. The following is a list of
such properties.1

1. The equation should be simple.

 

1 "Stress-Strain Formulas," W. R. Osgood, Journal of
Aero,Science, Vol. 13, Jan. l9u6, pp. hS-u .

2. It should be able to represent a large number of
engineering materials.

3. It should go through the origin.

h. There should be at least two parameters in the
formula, and they should be easy to calculate.

5. The lepe at the origin should be equal to the
modulus of elasticity.

6. The equation should be easy to integrate when
substituted in integrals which arise in develop-
ing the theory of plastic bending or column
analysis.

Each formula should have the properties listed.above
if it is to be useful in the general case.

Previously Proposed General Formulas

2

The following is a list of formulas taken from.the

literature. The notation used is:

C is unit strain,
.3

a- g .jEr ,
S is unit stress,

E is the slope of the stress-strain curve at the origin,
e is the base for the natural log, and

a, b, c, d, k, orq ,fl , I are parameters.

 

2 Ibid.

1.

2.

3.

LL.

5.

6.

Hooke's law, the universal stress-strain formula,
6:6- ,
usually written as,
S = Ber,
needs no comment.
In 1729 Bulffingeri suggested,
egkr",
and it has been used in spite of the fact that the
slope at the origin is not B.

Riccati proposed two formulas,

file
a- : he

9

“(a
/I A.)

and d- .(

9

but they have only one parameter each, and the
slope at the origin is not equal to E.

Gerstner's formula,

6": E+b62’
also has only one parameter and therefore cannot
be general.

In general, there is no objection to Ponoelet's

formula except that the parameters are hard to find.

He suggested,

6: 0"[l + 3(6‘r-lfl

In‘Hertheimfis formula,

7.

9.

10.

11.

1 2
i ago-4780’,
the slope at the origin is zero.

Hodgkinson's formula,

I 4'
d’sé +652+CQ +J6 ,

is no more than a power series in 6' ,
and the parameters are hard to find.

Cox suggested three formulae,

6

0/(I +l|f)
a a
6=r+5r+yr

3
rat +66 +ce3

The first has only one parameter and the other two
are limited to only a few materials.

Inbert's formula,

I ‘6'
6 = (x)(* -'):
has only one parameter.

Hartiz suggested two equations,

‘=(é)(e“‘-')

6' =[e [(I-ége"

but they both have only one parameter which is not
enough to represent accurately wide ranges of stress-
strain data.

In Schiile's formula,

12.

13.

15.

a“ .a6”+6e’,
the slope at the origin is not E.
Prager's formula,

6" =Qé +6fa04[(“a)/b]6,
is good for many materials with a wide range of
strain values, but the parameters are hard to find,
and in problems in plastic bending the integrals
do not integrate in closed form.

Holmquist and Nadai suggested the use of two formulas,

6:0"

€=6+k(r'°?)a 5'40;
where d; is the proportional limit stress.
The two form.a powerful combination but the para-
meters are hard to find in the second equation.

Osgood and Ramberg suggested,

€e6+kd'h,
wiich represents a wide range of materials and the
parameters are easy to find.
Rao and Leggett suggested,
G a 0' +/3(Ce:ln 16‘ -l)
but the parameters are hard to find for the best fit.

The Proposed Formula

After making an intensive study of the subject, Osgood

and Ramberg suggested their formula,

G ==¢l"+kd"a
finich is a special case of Holmquist's and Nadai's equation
with 0; equal to zero. They thought that this relationship
was a good approximation to the stress-strain data and possessed
the other properties desired in a formula.
The relationship suggested here is similar to that of
Osgood and Ramberg with an interchange in the variables. The

equation now takes the form,

{r'==£ + l:e"',
This is similar to the relationship suggested by Gerstner
except that there are two parameters, whereas his formula
replaces n by the numerical value of two.

The relationship does possess the properties discussed
previously. First, the equation is certainly simple. Second,
as is shown later, it can represent the stress-strain data of
certain types of aluminum.and.magnesium alloys, and can there-
fore represent curves of similar shape for other materials.

It best represents materials with a slight break in the lower
range of the stress-strain curve. Third, the curve goes
through the origin, and fourth, the parameters are easy to
calculate. Fifth, the slope at the origin is equal to E, and
sixth, the equation can be integrated in closed form.when
substituted in the integrals of plastic bending.

It can be seen that the relationship possesses the de-
sired properties, but the equation does have limitations. In

attempting to apply the equation to a number of different
actual stress-strain curves for magnesium and aluminum alloys,
as well as a number of hypothetical curves, the following
observations were made.

The first and biggest limitation found was the fact that
the equation holds good over only a portion of the entire
stress-strain curve. For the cases tried, and for the best
fit, it was found that the equation holds for values of
strain up to the yield strength, at 0.2 percent offset, of
the material in every case, and in some cases up to one and
one-half to two times the yield strength strain. Therefore
it can be seen that the equation cannot be used over the
entire stress-strain curve, that is up to rupture of the
material, and the point up to which it holds is best found
by applying the relation to the particular data.

The second limitation found was the fact that for the
cases tried, usually, a perfect fit could not be obtained.
The closeness of the fit depended upon the range to which the
equation was being applied, as well as the material itself.
The lower part of the mathematical curve is not a straight
line like that found in many materials, but when plotted and
drawn to a reasonable scale it cannot be distinguished from

it.

Method of Application of the Formula

The general equation can be written in the following
.4

form.
r: e + k6 n (1)
r—e =k‘n (2)
log (f-‘) = log k + n log 6 (3)

The parameters to be found are k and n . Plotting
values of (f- 6) versus G , obtained from actual tension and
compressive tests, on log-log paper should give a straight
line if the data is to fit this type of curve. If the result
is a straight line, or nearly so, the constant k is the
intercept, and n the actual slope of the line on logélog
paper.

Before going further, the question of algebraic signs
should be investigated. Let M be any point on the stress-
strain curve past the proportional limit OP . See Fig. l.
The coordinates of this point are (tfi , 81). Let E be
the slope of the straight part of the curve. PN is a con-

tinuation of the straight line OP and has slope E. From

the Fig.,
52.2%,
and by definition,
i=6; .

llll'llllllll

 

 

 

 

 

FIG.

10

Again from the Fig.,

e! > £2 a
or 6| > a: o
In general
6 a 0"
and (7-6) is a negative quantity if 6 = 5- is not con-
sidered.3

From equation (2) it is seen that k must be a negative
number. Therefore the absolute values of the quantities

must be considered in equation (3).
log|T-€|=loglk‘+nlog€ (h)

The general procedure in calculating k and n is
as follows.
1. Pick a number of points from the stress-strain
curve that is to be fitted.
2. For each point (6 , S), calculate 0" .
3. CalculatelE- 4“ .
1;. Plot ‘5' -6‘ versus 6 on log-log paper.
5. Draw a straight line through the points.
6. Find the actual slope of this line which is n .L"

 

3 For (- =0' , the curve is a straight line. This is very
often the case for small values of G . When 6 is very small,

the term k‘ is much smaller than 6 and when 0 ti the
curve the term 6 cannot be distinguished fromG Elke; .ng

h It may be found that the curve is not a perfect straight
line. In this_case a compromise must be made in drawing the
line and determining the slope n .

11

7. Substitute a typical point in equation (A) and
calculate k .S
8. Plot the curve just obtained and compare with

the original stress-strain data.

Application to Magnesium and Aluminum Alloys

The aluminum.alloy used was ordinary stock, Alcoa 2hS-Th.6

Six tensile specimens and six compression specimens were
tested. They were cut from two different bars, each twelve
feet long; one being of square cross-section one inch on a
side; the other of circular cross-section one inch in diameter.
One of each type of specimen was cut from each end, and one

of each type from the middle of each bar.7

The magnesium.alloy was a laboratory produced experimen»

tal material of a type designated ZKooA-B by the Dow Chemical
Company. The bar was 30 inches long and had a circular cross-

section one inch in diameter. One tension specimen, and one

compression specimen was cut from each end.

 

5 Actually k could be read directly from the graph,
but due to the physical size of the graph paper and numerical
size of numbersinvolved, it is best found in this manner.

For properties and heat treat see, Alcoa Structural
Handbook, Aluminum Company of America, 1950, pp. IE, 1?, 22.

The reason for this procedure will be seen in
Chapter III.

12

The tensile specimens were standard8 0.505 inches in
diameter. The compression specimens were three inches long
and one inch in diameter.

The stress-strain tests were performed with a 60,000
pound, universal testing machine, with automatic stress-
strain recorder.

The procedure followed was that normally used when
running standard tensile and compressive tests.

The tests results for the aluminum.alloy, square section,
are shown in Fig. 2, for the circular section, Fig. 3. The
tests results for the magnesium.alloy are shown in.Fig. h.

The following example shows the procedure used in deter-
mining the stress-strain equation for the alwminum.alloy.
Notice how steps one, two and three have been tabulated in
Table I for easier calculation. Fig. 5 shows steps four,
five and six, where the graph has been plotted, the line
drawn, and the slope obtained from.it. Step seven is to
pick a point from the graph and calculate the value of k .

Thus, for a point at which

6 = 0.007
\e- 41$ 0.0021
and n 5 3

16310 i.0021| = 10310 ‘k‘ + 3 loglo (.007)

 

8 For standard tensile specimens see ASTM Standards,
125 , Part I, pp. 506, 507.

f
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TABLE I
s 6' ar' [GFN-lfl.
38,500 .008 .0037 .0003
h0.500 .000 .0039 .0001.
h3.000 .0088 .0081 .0003
un,000 .005 .0082 .0008
u6,300 .005 .00u5 .0005
u6,000 .006 .0088 .0016
u8,000 .006 .0086 .0018
50,500 .006 .0009 .0011
87,250 .007 .0085 .0025
51,500 .007 .0050 .0020
L;8,000 .008 .00116 .0031.
51,500 .008 .0050 .0030
Sh.000 .008 .0052 .0028
08,500 .009 .0007 .0003
58,000 .009 .0052 .0038
89,000 .010 .0088 .0052
53,000 .010 .0051 .0009
56,000 .010. .005u .0086

 

18

7.3222 - 10 = 10glo [kl + 3(7.8851 - 10)
loglo IkI =3.7869

lkl = 6120
Jun-6100
0' = 6 - 610063

Step eight is carried out with the use of the tabulation
shown in Table II. The curve is plotted in Fig. 2 and the
results observed.

The equation fits well for values of strain up to 0.008
in./in. In Fig. 2, this is somewhat past the yield point
strain of 0.007 in ./in. in tension and 0.006 in./in. in
compression at 0.2 percent offset. The equation as found
here is an average curve between the tension and compressive
tests; the reason for this procedure will be eXplained later.
If so desired, a curve could be found that would fit either
the tension or compressive curve more closely. For the
closer approximations, see Figs. 6 and 7. Notice the equation
of the curve is given in each figure.

If the curves shown in Figs. 6 and 7 are compared with
the actual stress-strain data, Fig. 2, very close agreement
can be seen.

The stress-strain data, and the curve plotted for the
equation is shown in Fig. A for the magnesiwm alloy. The
fit is almost perfect as can be observed, and the equation

holds for values of strain up to 0.012 in./in. Beyond this

l9

 

 

 

TABLE 11
6 63 61003€3 0" 3

0.001 1 10'9 ' 6.1 x 10"6 0.00100 10,250
0.002 8 10'9 u8.8 x 10"6 0.00195 20,000
0.003 27 10'9 165. x 10'”6 0.00288 29.100
0.000 6h 10'9 390. 2x 10"6 0.00361 37.000
0.005 125 10’9 762. x 10'6 0.00428 83,500
0.006 216 10"9 1320. x 10"6 0.00u68 88,000
0.007 303 10’9 2090. x 1077'6 0-00091 50,200
0.008 512 10"9 3120. x 10"6 0.00888 50,000
0.009 729 10"9 into. x 10""6 0.00856 86,700

 

 

 

 

 

 

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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22

point the curve falls off rapidly as is the case for the
other equations. There is a slight break in the curve for

low values of E 3 this leads to a better fit as well as a

wider range of application.

23

CHAPTER II

APPLICATION OF THE STRESS-STRAIN FORMULA TO PLASTIC BENDING

Introduction

The flexure formula,

-1212.
3‘1

found in elementary books on strength of materials, uses
Hooke's law in its derivation. Hooke's law,

8 = E€?,
can be used where the stress-strain data can be fitted to
a straight line. For non-linear stress-strain curves, and
for stresses past the proportional limit, the flexure for-
mula cannot be used because Hooke's law does not apply.
Therefore, when these cases exist, some other relationship
between stress and bending moment is necessary.

In general, when a bar is subjected to bending, the
stress distribution curve may be thought of as a stress-strain
curve, if the stress axis of the stress-strain curve is assumed
to lie in the neutral surface of the beam. See Fig. 8.
Section A-A is considered to be any cross-section. The
stress-strain properties in tension and compression are

assumed to be the same.

25

In applications this may not be the case, so the average
of the stress-strain curves in tension and compression could
be used for the stress distribution in the cross-section.
This is the reason why the average curve was calculated for
the aluminum.alloy in Chapter I.

.Plane sections are assumed to remain plane before and
after bending, in the elastic range as well as the plastic
range. I

From this assumption and Fig. 9,

5 $20.39
B + d d d p:= a?

y=Re ' mm

6

 

Bending Moment for a Rectangular Cross-Section

From Fig. 10,

M = I’ SydA
Hz
and from.equation (l)
s '=Ep(e+ ken).
"Iz
M=2b1ES (6+k6n)ydy
c
From (S) dy = Rdé (c
, I
and M =2blE (€.+k€n+l)d(

26

 

 

 

 

 

 

 

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/< 1
9‘

 

 

 

FIG.

27

where 5‘ is the strain at the outer surface.

2
Using (5) M = bJEh __t_‘_._ +1.6.“ (6)
2 ~3 n + 2

The values of k and n , as found in Chapter I, can
be substituted in equation (6). This formula gives the rela-
tion between the bending moment and strain at the outer sur-
face for a rectangular section.

'I-IIIE can be calculated and plotted against the maximum
stress on the cross-section by using the stress-strain curve.
This graph gives a useful relation between a given bending

moment and maximum stress on a rectangular section for a

given material.

Bending Moment for a Circular Cross-Section

The same assumptions are made here as in the treatment
of the rectangular section. From Fig. 11,
M = " SydA
dA = Zxdy

P
M = h ’ Sxydy
0

Using equations (1) and (S), with the equation of the

circle gives,

5.
M = 1, R3 ES (52 + ken + 1) (6‘2 _€2)'lz d6

28

Integration by parts, a sufficient number of times gives,
60*:

3 ”-6—;
M = u R E ‘d‘
... l
1 n 2 If” (G: _ (1)];
Making the substitution, 6 aft, the integral becomes,
’ n+3
*
3/ a“ J
2. a “
( - '2
, l a )
From.a table of definite integrals,

=—(”’3B ”:4 J.)

where B denotes Beta function. This can be written in terms

(8)

 

 

 

of Gamma functions as follows,

 

(hi3
LP ~)P(z) )6:¢39
2, mil
”(‘7 )
Substituting this back in equation (8), using (5) and noting

P(f)=fig1ves,

M =7Tr3 E 0.250 6‘ + 2 k T‘ LII—1A)“ (9)
r (n + 2) r‘ (1.1.1.5)6‘

The values of k and n as found in Chapter I can be

substituted in equation' (9). This gives the relation between
the bending moment and strain at the outer surface for a cir-
cular cross-section, when the stress-strain data for the

material can be fitted to equation (1).

 

co
9 r(m) =I Jen-'15.x dx

0

29

By using the same procedure as before, a graph can be

drawn that will relate ‘ME to the maximum stress on the

I

section. This will be for a circular cross-section and a

given.material.

30

CHAPTER III

THE BENDING TESTS

Introduction

The bending theory developed in the previous chapter
can be checked by performing bending tests. These test re-
sults can be used to show how close the stress predicted
by the theory approaches the actual stress developed at the

outer surface, and also to illustrate the procedure in

1:42

I diagram.

plotting the stress versus

Description of Experiment

Fig. 12 shows a diagram of the test setup. The load
was applied by the use of dead weights. Two SR-h type elec-
tric strain gages were used. ‘One gage was located directly
at the highest point of the section, while the other was
located at the lowest point of the section. The strain at
the outer surface was taken as the average of the two readings.
This was done to account for the fact that the stress-strain
properties in tension and compression were not the same as
assumed in the theory.

The Brush amplifier, along with the Brush oscillograph,

was used to record the strain values.

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32

The procedure was as follows:
1. Load increments of 100 lbs. were used.
2. Both values of strain were recorded for each load.

With this data, ‘Eg can be calculated, and the value

I
of stress can be found from the stress-strain diagram corres-
ponding to the average value of strain. These two values are

then plotted.

The Aluminum.Alloy

The tensile and compressive specimens were cut from.the
same bars that were used for the bending tests. In order to
determine if the properties of the bar were the same through-
out, the specimens were cut out of the bar as described in
Chapter I. I

From the average stress-strain curve of the aluminum
alloy, Figs. 2 and 3,

k = -6100
n = 3.

For the square section,
bl = 1 in.
n = l in.
E 8 10.5 x 106 psi,

and from equation (6),
M = 5.25 x 106 6, (0.330 - 12505‘2). (10)

For the circular section,

33

r = 0050 in.
E = lO.u x 106 psi

and from equation (9).

6
M = 1.09 6, (0.250 - 76263) 10 (11)
Calculating £19 , and substituting values of strain for Q

in (10 and (11), and finding the corresponding stress value
from the average curve, Figs. 2 and 3, the calculated curves
in Figs. 13 and 1h were drawn. The actual test results are
also shown for both the square and circular sections.

For the rectangular section there is good agreement be-
tween the theory and test data up to a stress value of about
30,000 psi, and again from about h5,000 psi to 50,000 psi,
Between these points the stress values are low. One reason
for the low values may be the fact that the average stress-
strain curve, Fig. 2, is not as good an average in this range,
as in the other parts of the curve.

For the circular section good agreement exists over the

entire range of the curve.

The Magnesium.Alloy

The same procedure was applied to the magnesium.alloy
as was applied to the aluminum alloy. From Fig. A, and from
the data on the circular section,

n = 2.6
k = -h25

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36

I' 0.50 in.
E = 6.3 x 106 psi
and from.equation (9).

M = 2.1+? €¢(o.2So - 60 6‘1"”) 10"

Using the same prooedure as before, the results are
shown in Fig. 15. Better agreement was expected because the
equation relating stress to strain agrees very well with the
stress-strain data. The error was assumed to be due to ex-

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38

CHAPTER IV

CONCLUSIONS

From.this study it may be concluded:

I. The proposed mathematical relationship is suit-
able and can represent stress-strain data for certain types
of magnesium.and aluminum alloys.

2. A relation between stress and bending moment was
found when the stress-strain law is non-linear.

3. The maximum stress at the outer surface can be

fairly well estimated by the use of this relation.

39

BIBLIOGRAPHY

gigoa Structural Handbook, Aluminum Company of America, 1950.

Bronwell, Arthur. Advanced Mathematics in Physics and Engineer:
ing. McGraw-Hill Book Company, Inc., New York, Toronto,

Coonan, F. L. Principles of Physical Metallurgy. Harper and
Brothers, l9h3.

Cozzone, F. P. "Bending Strength in the Plastic Range," Journal
of the AergEScience, Vol. 10, May 19MB, Pp. 137-151.

Gill, S. S. "Bending Strength of Materials with Non-linear
Stress-Strain Curves," Aircraft Engineering, Vol. 19,
July 19h7, Pp. 212-216.

Hoffman, 0., and G. Sachs. Theory of Plasticity. McGraw-
Hill Book Company, Inc., New Xork, Toronto, London, 1953.

Lee, G. H. An Introduction to Experimental Stress Analysis.
John Wiley and Sons, Inc., 1950.

MathematicalATables from.Handbook of Chemistry and Physics,
Ninth Edition, Chemical Rubber Publishing Co., 19h3:

Osgood, W. R. "Stress-Strain Formulas," Journal of the Aero.
Science, Vol. 13, Jan. 19h6, Pp. u5-h8.

Ramberg, W., and W. R. Osgood. Description of Stress-Strain
Curves by Three Parameters. N.A.C.A. T.N. No. 902, 19MB.

Timoshenko, S. Strength of Materials, Second Edition, Part I.
D. Van Nostrand Company, Inc., 19h1.

Williams, E. J., and N. H. Kloot. "Stress-Strain Relationship,"
léustralian Journal of Applied Science, Vol. 3, March 1952,
Pp. 1-13.

Dwight, H. B. Table of Integrals and Other Mathematical Data,
The Macmillan Company, New York, 1951, Second edition.