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DESCRIPTION OF STRESS-STRAIN CURVES BY THREE PARAMETERS
By Walter Ramberg and William R, 0sgood
SUMMARY
A simple formula is suggested for describing thestress-strain curve in terms of three parameters: namely,
Young_s modulus and two secant yield strengths. Dimension-less charts are derived from this formula for determining
the stress-strain curve, the tangent modulus, and the
reduced modulus of a material for.which these three param-
eters are given. Comoarison with the tensile and compres-sive data on aluminum-alloy, stainless-steel, and carbon-
steel sheet in NACA Technical Note No. 840 indicates that
the formula is adequate for most of these materials. The
formula does not describe the behavior of alclad sheet,
which shows a marked change in slope at low stress. It
seems probable that more than three parameters will be
necessary to represent such stress-strain, curves adequately.
INTRODUCTION
An assembly of the tensile and compressive stress-strain .curves for sheet materials characteristic of air-
craft construction is being obtained at the National
Bureau of Standards as the principal objective of a re-
search project for the National Advisory Committee forAeronautics. Stress-strain, stress-devlation, secant-
modulus, tangent-madulus, and reduced-modulus curves have
° been presented in reference 1. for various grades of sheet
materials of aluminum alloy, carbon steel, and chromium-
nickel steel. A second objecti_e of the same research
project is a search for yield parameter,a that give a betterdescription of the stress-strain curve than those in use
at present.
The conventional descripti-on of the s.tress-strain
curve of metals by the two parameters, Young_s modulus and
yield strength, iS .inadequate for the efficient design ofmembers unless the material follows Hooke's law up to. a
"_/• /.k
- p._
r
_ i _
2 NACA Technical Note No. 902
yiel_ point at which it yields indefinitely under constant
stress. This special behavior is approached, for example,
by certain steels (fig. l) and by certain low-strength
magnesium alloys, but it is not characteristic of many
high-strength alloys for aircraft.
Examination of the stress-strain curves for aluminum-
alloy sheet and chromium-nickel-steel sheet given in ref-
erence 1 shows, particularly for the compressive stress-
strain curves (figs. 2 and 3), a gradual transition from
the elastic straight line for low loads toward the horizon-
tal line characterizing plastic behavior. The type of
transition varies widely. Hence there is no hope of reduc-
ing all stress-strain curves to a single typ_ of curve by
uniform stretching, or affine transformation of coordinates.
This rules out the possibility, which exists for affinely
related stress-strain curves (reference 2), of complete
description in terms of only two parameters, Youngts mod-
ulus and secant yield strength. A minimum of three param-
eters will be required to describe the changes in shape
for different materials.
Several proposals have been made for describing the
stress-strain curve in terms of three or more parameters°
Donnell (reference 5) suggests as two yield parameters
the stresses s I, s_, at which the slope of the stress-strain curve is equal to 3/4 E and 1/4 E, where _ is
Young_s modulus. The stress-strain curve is then derived
from these two parameters on the assumption that the
slope varies linearly with the stress. This procedure
gives a good descriotion of many tensile stress-strain
curves of aluminum alloys, but it does not seem adequate
for the highly curved tangent-modulus curves found for
the compressive stress-strain properties in reference l,
from which figure 4 is taken. Furthermore there are
practical difficulties in determining the stresses corre-sponding to a tangent modulus of 8/4 E and 1/4 E
quickly from the stress-strain curve.
Esser and Ahrend (reference 4) noticed that the
stress-strain curves for many materials may be approxi-
mated by two straight lines when they are clotted • on log-
log paper. They orooosed to define yield strength as
the stress at the intersection of these two lines.
Description of the stress-strain curve above the yield
strength _ould be obtained from the slope of the upper
straight line. The proposal is doubtless an advance over
the description by an offset yield strength. It has the
.... • ................................ . • . . ................
..<
- - c
• i_̧.,i
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NACA Technical Note No. 902 3
disadvantage, however, of requiring the plotting of suf-
ficient stress-strain data on log-log paper to determine
a straight line through the points. Furthermore it gives
no informationabout the shape of the important transitionregion near the intersection of the two straight lines.
An analytical expression for the stress-strain curvewhich is suited for theoretical studies of plastic buck-
ling was proposed by Nadai in 1939 (reference 5). The
expression is
e = s- s'< SpE
e = ..... S > s-
ey- Sp
-\
[/
(i)
where
e strain
s stream
ey
Sp
n
strain corresponding to yield strength
proportional limit
constan_
Sy
If the iogarithmof both sides is taken in equation (_),
it can be seen that equation (1) approaches Esser and
Ahrendls two straight lines as asymptotes for low and for
high stress, respectively. The description of the transi-
tion region is obtained by increasing the number of param-eters from three to four.
In the study of plastic bending, the second author
found an analytical expression containing three parameters
that appeared to be well adapted for representing stress-strain curves. Further examination of the exoression in
the light of the data given in reference 1 confirmed thisview.
I
4 NACA Technical Note No. 902
:• !ii •ii
ANALYTI CAL EXPRE SSI ON
Stress-Strain Curve
The proposed analytical expression is
E •
where K and n are constants.
becomes the same as equation (2) if
(2)
Nadaits expression (1)
, • r._
Sp= 0
ey _ = K
(3)
that is, if the proportional limit is taken as zero, and
the requirement is dropped that ey is the strain corre-
spond__ng to a yield stress Sy.
The expression (2) may be written in dimensionless
form in terms of the follo_ving variables (reference 6):
S I
S 1
J
(4)
where s I is the secant yield strength, equal to the
ordinate of the intersection with the stress-strain curve
of a line through the origin having a slope equal to m I E, < i(fig. 5),• m I being a chosen constant 0 < ml .
Since m I is fixed, the transformation (equation (_))
reduces all affinely related stress-strain curves to a
single curve.
The abscissa of the intersection for a stress-strain
curve described by equation (2) is
NACA Technical Note No. 90_
sle i - + KmE E
1
Inserting equation (4) in equation (2) gives
Sl )n-I n
From equation (5)n-i
I mlK V - m I
Inserting equation (7) in equation (6) gives
5
(5)
(6)
(7)
i -- m I n= a + a (8)m I
f
- ,. "-
. - .
",L
Affinely related stress-strain curves that may be described
by equation (8) are characterized by having the same value
of n. Figure 6 shows a family of curves for a number of
different values of n, and m I = 0.7.
Stress-Deviation Curve
The stress-deviation curve is obtained by plotting
stress against difference between measured strain and
elastic strain corresponding to Hooke's law. For the
stress-strain curve given by equation (2) the deviation
is given by
d = e- s_ = K(S (9)E \ _,J
or
log d = log K + nlog s_ (I0)E
d
that is, a log-log plot of deviation against stress would
be a straight line. The deviation may be written in dimen-
sionless form as
6 - Sd (ll)S
1
1
6 NACA Technical Note No. 90,2
From equations (9)., (4), and (8)
or
- n
m 1
1 - m 1log 8 = log + nlog G (13)
m I
The family of straight lines corresoonding to various
values of n and to m I = 0.7 is shown in figure 7.
Tangent Modulus
The tangent modulus at a given stress is defined as
the slope of the tangent to the stress-strain curve at
that stress. The reciprocal of the tangent modulus is
from equation (2):
! = __ = 1 + nK shnE, d. { - {] (.14)
This may be written in dimensionl'ess form by making use
of equations (7) and (4):
- k
..,.. • .- •k
n(1 - m_ ) n-lE = 1 + c (15)E I ml
Figure 8 shows the tangent-modulus ratio E'/E plotted
against stress ratio _, with m I = 0.7.
Reduced Modulus for Rectangular Section
The reduction in buckling stress when the stress
exceeds the proportional limit is frequently estimated
(reference 7, pp. 159 and 274) by replacing Young's mod-
ulus E by a reduced modulus E r. Thus in the case of
columns the actual buckling stress sr would be esti- _mated as
_ Ersr - -- s e (16)
E
• i_ _ _
NACA Technical Note No. 902
wh,ere_ se is the buckling stress comouted from elasticL
theory and, for columns of rectangular crGss section,
from reference 8,
Er = _EE' (17")2
(4_ + Jgr)
Dividing equation (17) by E gives
mr = 4E '/E (18)
Figure 9 shows the reduced-mo'dulus ratio Er/E plotted
against stress ratio a for different values of n and
with m I = 0.7.
Equation (16) may be solved for Er/E as follows:
Let (_e = Se Sr--; (_r --S 1 Sl
(19)
so that equation (16) becomes
Or Er
(_e E
(20)
Thus, the desired value of Er/E is the ordinate of the
intersection of the straight line Er/E = O/(_ e with the
reduced modulus curve, equation (18), for the material
(n) in question. Straight lines w.ith the given slope
1/(_ e may be drawn conveniently by connecting the origin
with the proper point on the circular curve in figure 9.
DERIVATION OF EMPIRICAL CONSTANTS FROM STRESS-STRAIN CURVE
The adequacy of equation (2) was tested by plotting
on log-log paper the stress-deviation curves for the sheet
materials given in reference 1. The points should lie on
a straight line according to equation (10) if equation (2)
r i
r • !
3
i i_? _; .... [
r
8 NACA Technical Note No 902
is an accurate description of the stress-strain curve°
From the slope and intercept of such a straight line theconstants K and n can be de_ermined for a best fit.
Straight lines were obtained for all the compressive
stress-deviation curves and for all but four of the ten-
sile curves for stresses greater tlian the stress at which
the secant modulus was equal to 90 percent of Young's
modulus. The exceptions had stress-strain curves which
had a gradual change of slope throughout their entire
length. Thim indicates that any value of m I less than
0.90 would give an approximate fit to compressive stress-
Strain curves and to most tensile stress-strain curves
at stresses above that corresponding to m I = 0°90°
It appeared desirable to choose the value of m I
such that the secant _'eld_ strength s I would approxi-
mate the widely used yield strength So. s for 0.2--oercent
offset. In other words, s I should be chosen to satisfy
appr o ximat ely :
so.oo2 = e- _! (21)
E
where (see fig. 5)
S 1e - (21a)
miE
inserting equation (21a) in equation (21) and solving for
1/m I gives
1 0.002 0.002--= I + - 1 + _-- (21b)
ml sl/E So.alE
Examination of tables III and IV of reference 1 gave val-
of so.e/E for the aluminum alloys and the chromium-ues
nickel steels which ranged from 0.00258 to 0.00675; the
1025 carbon steel in reference 1 was not included because
of its relatively low value of this ratio. The average
value was
S
= 0.00486 (21c)E
Substituting this average _n equation (Ylb) and solving
for m I , gives
" . • • " , L • .
.. ..: .j:!
NACA Technical Note No " 902
m I = 0.709 (21d)
It was decided, therefore, to use for ml the value
m I = 0.7 (22)
for determining the secant yield strength sI.
W_nen E is known and sI has been determined, it
is still necessary to know the shape parameter n in
order to establish the shape of the stress-strain curveaccording to equations (8) and (4).
The shape parameter n is conveniently derived by
the use of a second secant yield strength s2, corre-
sponding to a second secant modulus rosE, as follows.In analogy to equations (5) and (7),
•i
. k
es -. - + KmsE E
._-= z +z(,,.m 2
Solving both equations (7) and (23) for K gives
1
,; C<-,)so that
_i__ 1
_h- lm I
Solving for n gives
(23)
(24)
(25)
(26)
m_ I- m Ilog
mI 1-m sn = 1 + .......
log s-klsm
(27)
l0 NACA Technical Note No. 902
The value of m s
was chosen asfor the second secant yield strength
m s = 0.85 (2,8)
since this value lies midway between 0.7 and 1.0 and
since it is on the safe side of the limiting value
m = 0.90 up to which equation (2) is an adequate descrip-
tion of most of the stress-strain curves in reference 1.
Substituting equations (22) and (28) in equation (27)gives:
17
log-_ 0.3853
n = 1 + = 1 + (29)8 1 S 1
log -- log --S 8 lO S8
A plot of this relation on log-log paper is given in fig-ure 10.
COMPARISON _ITH EXPERIMENTAL STRESS- STRAIN
AND TANGENT-MODULUS CURVES
The order of approximation w.ith which equation (2)describes the stress-strain curve for materials with val-
ues of n from 3.08 to c_ is brought out in figures ll
to 13, in which the data were taken at random from refer-
ence 1. The approximation appears to be adequate for most
practical purposes. In these figures E, el, and ss
were obtained from the stress-strain data, and n was com-
puted from equation (27). Equation (8) then gave the rela-.
tion between c and o, and the relation between e and
s was obtained by multiplying ¢ by sl/E and a by s I.A better fit would have been obtained if n and K had
been determined from a plot of the data, as exolained in the
first paragraph of the preceding section (p. 7); but the
procedure used is simpler and probably adequate in mostcases.
A much more severe test of the adequacy of equation (2),
than the comparison with the stress-strain curve, is a com-
parison with the tangent modulus - that is, the slope of the
stress-strain curve. Such a comparison seems to be advis-
able since the tangent modulus must be computed for evaluat-
ing the reduced modulus in compression. (See equation (17).)
• i
NACA Technical Note No. 902 ll
The tangent moduli in compression of reference 1
are plotted on a dimensionless basis in figures 14 to 21
together with computed moduli as given by figure 8. The
value of n from equation (29) for each material is given
in figures 14 to 21. The computed moduli are shown for
integral values of n and for n = 2.5. To appreciate
the closeness of fit, therefore, it is necessary to inter-
polate between the curves by using the particular value
of n applying to the plotted data. Except for the
curves with a very sharp knee (n > 10) the experimental
values of tangent modulus for stresses below the secant
yield strength s I differ less than _0.07 E from the
values corresponding to equation (15). In the case of the
values with the sharp knee (fig. 21) the maximum differ-
ence was considerably greater. These differences do not
detract seriously from the usefulness of equation (2),
however, since the region in which the agreement is not
good comprises a limited stress range. Consequently, in
this range the difference between the experimental and
the computed values of O corresponding to a given value
of tangent modulus are small.
The comparison was confined to the materials in ref-
erence l, which did not include alclad aluminum alloys.
In the alclad aluminum alloys a change in slope at low
stress is observed which corresponds to the yielding of
the aluminum coating. It seems probable that inclusion
of this effect will require the addition of at least one
more parameter to the three contained in equation (2).
EXAMPLE FOR APPLICATION 0F THREE-PARAMETER METHOD
/
Computations based on elastic theory give a value of
S
se = 87 × l0 pounds per inch e
for the critical compressive stress of a given specimen.
The material of the specimen has the compressive stress-
strain curve shown in figuA'e 22. It is desired to deter-
mine the stress:
E rS r -- Se
E
which is an estimate for the critical stress after taking
account of the olastic yielding of the material.
12 NACA Technical Note No. 902
, From figure 22 are obtained the two secant yield
strengths|
s_ = 43.0 X 103 pounds per inch s
ss = 38.0 x i03 poundsper inch s
so that
S 1- I. 132
S_
From figure i0 this corresponds to a shaPe parameter
n= 8.15
Entering figure 9 with this value of n and with th_ratio
I
gives
s_1- _ 43.0 _ 0.494
se 87.0
_E _ 0.473E
so that the corrected critical stress is
3 3" 2
s r = 0.473 × 87 × lO = 41.2 X lO pounds per inch
National Bureau of Standards,
Washington, D. C., April 8, 1943.
•j
?
L •
NACA Technical Note No. 902 13
REFERENCE S
i. Aitchison, C. S., and Miller, James A.: Tensile and
Pack Compressive Tests of Some Sheets of Aluminum
Alloy, 1025 Carbon Steel, and Chremium-Nickel Steel.
T.N. No. 840, NACA, 1942.
2. 0sgood, W. R. : A Rational Definition of Yield Strength.
A.S.M.E. Jour. App. Mech., vol. 7, no. 2, June 1940,
pp. A61-A62.
3. Donnell, L. H.: Suggested New Definitions for Propor-
tional Limit and Yield Point. Mech. Engineering,
vol. 60, no. ll, Nov. 1938, pp. 837-38.
_. Esser, Hans, and Ahrend, H.: Kann die 0.2% Grenze
durch eine ubereinkommenfreie Dehngrenze ersetztwerden. Arch. f. Eisenhu_tenw., vol. 13, no. 10,
1939-40, pp. 425-428.
5. Holmquist, J. L., and N_dai, A.: A Theoretical and
Exoerimental Approach to the Problem of Collapse
of •Deep-Well Casing. Paper presented at 20th Annual
Meeting, Am. Petroleum Inst., Chicago, Nov. 1939.
6. 0sgood, W. R.: Column Curves and Stress-Strain Diagrams.
Nat. Bur. of Standards Jour. Res., vol. 9, Oct. 1932,
pp. 571-82.
7o Timoshenko, S.: Theory of Elastic Stability. McGraw-
Hill Book Co., Inc., New York, N. Y., 1936.
8. von Karman, Th.: Untersuchungen uber Knickfestigkeit,
_._iitteilungen uber Forschungsarbeiten. Ver. deutsch.
Ing., Heft 81, 1910.
r•
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• L
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NACA Technical Note No. 90230
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Figure 2.- Sfr-ess-st-roin our-ves withblunt knee, aluminum-alloy
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NACATechnical Note No. 902
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Figure 4.- Non-linear variation of tangent modulus with
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56
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NACA Technical Note No. 90_ Fig. I0
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NACA Technical Note No. 902
200
150
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O,
I00
80
_b
uf
2O
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Figs. 1,12I I I I I I I I I
Figure II.- Experimental and computed .,,_..,-,<'_"stress-strain curves_ --_----
Cr-Ni steel, fUlll-hord, thicknessl_J _" /--11
0.020 inch. X../r" _ l
I ' ' ' /X.; _ ' ¢' / l
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NACA Technical Nofe No. 902
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Figure t3.- Experimenfol end compufed sTress-sfroin curves, 1025corbon-sfeel, fhickness=O.054".
rl
o 292
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Mu far / e I , /nch
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Figs. 13,14
0 .20 .40 .80 .80 LO0 lQO /,40G=S/8_
Figure 14.- Experimenfal ond compufed fongenf moduliin compression,2<n<3.
< ,.. ' ,.• LI I
L
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NACA Technical Note No. 902
LO0 _ _ ,,3.10X
_a'_ _ 3.08 "
.80 _\
.60
-E-
.40
Figs. 15,16I I I I I I I I I
Mo fer / o I Inch
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.... 314 ......" " full " " '_ "....... .024 ".... I/4 " " .020 trans.
_0 X
o_ _ o
.20 o t0
CDOD
I
0
0
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0 .20 .40 ._0 .80 /.00 /.20 /.40d=S/S,
Figure t5.- Experimenfa] and comoufed fomc)enf moduli in
compPessiom, 3<n<4.
0498 Cr-/vi steel ful/-hord 0.0275 long.1.00 ---_= A 410 ...... 1/2 " .0 0 tr- s.
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t_._.ge .......... .o2.+.80 -,-,_ ,.
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0 .20 .40 .80 .80 /.00 /20 140
Figure 16.- ExpePimenfal and compufed fongen¢ moduli incompnessioP,, 4<n<5.
,. . =*.. #
: L L :..,. ;i
-...
Technical Note No. 902
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.80
.60 .
in compression 6<n<7
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n Mc/fe/-iol Incho 6.72 Or--N/Sfeel full-ho/-d fh. 0.02
Figs. 17, 18
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71 Mafer/'al
a Z82 Al-ol/oy 24S-To 7.60 "
Inch
fh. 0.08/ /on_7-" .032 "
I I
Figure 18.-Experimental andco.mputed tangent modu
in compresslon. 7<n<8. ..Irl= 7
0 20 dO 60 .80d=s/sl
zOO 120
. .-<. , r:
i
NACA Technical Note No. 902
.80 _
n AAofer/'d/
E' o 8.59 AI-olloy 24S-RTth. long.7_- + 8.32 "x 8.01
8.20
._0
.2O
IFigure 19-Experimenta end com-
puted tangent moduli ncompression. _<_<9
0
ZOO I
.80
6O
E'-E-
4O
.20 4O 60 .80#=s/s,
ox 9.07
9.88+ 9.O7
Mafer/al Inch
A/-o//oy/TS-T fh.0.032 long...... 24S- T ...... --.......... 084 " ".... 243-RT ......
.20 --Figure 20-Experimenfo and com-
puted tangent moduli ncompression. 9<n</O
0 •20 .40 6o .8od--s/s,
Figs. I9,20
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00
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A
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180 Z __lu /# s# /# ii 1_
_90" 0__Iii ii #i II ii
I _.0 " ,, ........ 01__1 ,4--........... P_ZI
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l_90" . ...... l<gZI <_<_g'O' ,, _.Z-..q'/7_ . ,, 6'_/ 0/_.co" ,........ I0_1 [] - Og
_0" ,, i -SZ I/o//o-/V 6__I x
0_I ,, _ _" 0_ < +P_,cO" . I _ ,-,.<.,,_O_Ol O_C< o_I_0"0 "L/¢ I-S/�/o//o-IV _01 vqOUl
0_I<U "UOlSSaad
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-!,.Jadx_ -18 e.Jn6!g
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NACA Technical Note No. 902 Fig. 22
- •i!
i r.
• J .
5O
qO
_,30
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0
O
O
,_ 20 L
l0
0
I
I
I
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• 85 Ee
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.OO2 •004 •006
Strain
Figure 22.- Compressive stress-strain curve.
.OO8
