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Stress
Strain
tension) and shear stresses acting
upon this plane can be described in
terms of two principal stresses and
the angle between the direction of
stress normal to the plane and the
direction 2 of one of the principal
stresses.L1.
The stresses are described by the
following well known equations:
f - f cos 2 0
u
f e
s
e
f
.
:extension) of this line and its rota:tion with respect to a principal axis
:of strain, both caused by the appli:cation of the stress, can be described
:in terms of two principal strains and
:the angle between the line and the direc2
:tion of one af the principal strains.—
The strains are described by the
:following well known equations:
. 2
+ e s= so')
e cos
u
fe
(f )
sinOcose
u
e
= angle of normal of plane to
s,J)
= - (e
3
u
e
(1)
v
) sinCnosD
angle of line to principal axis
(u) of strain measured positive
counter clockwise.
principal axis (u) of stress
measured positive counter
clockwise,.
e = component of stress normal to
= longitudinal strain of line.
plane.
fs
e
= component of stress parallel to :eso = angle of rotation of line with
plane (shear stress).
respect to a principal axis of
strain.
fu = principal stress.
: e u = principal strain.
f v = principal stress. :e v = principal strain.
Both sets of equations are purely geometrical, and apply no matter what
kind of material is considered and no matter what the elastic constants of the
material are.
The rotational strain (e
s q) is half of the conventional shear strain.
It has positive or negative values, depending upon the direction of rotation.
It is used, rather than the conventional shear strain, to make the two sets of
equations identical in form and to facilitate the use of the strain circles.
2
--The descriptions are a little more general than the case of plane stress,
but are applied only to the case of plane stress in this report.
*"Strength of Materials," Timoshenko. Ft.1,p.50, also Zivilingenieur,1882,p.113.
Note that 6 is an angle related to stress and that a) is an angle related to
strain. In some cases both angles Till appear upon a single diagram so that
the distinction is necessary.
Mimeo. No. 1317 -2-
Mohr represented the above relations graphically, as is shown in
figure 1. Such graphs .are called Mohr's stress circles or Mohr's strain
circles, depending upon whether the particular diagram represents stress or
strain.
•
-,.n••••n•••n••n•
Strain
Stress
The unit extension of the line,
e
The stress normal to the plane, lotted
hori-:previously
described, is plotted horireviously described, is p
:zontally
positive
to the right and the
ontally positi w to the right, and
:rotation
of
the
line
is plotted posi
the shear stress upan the plane is
:tive
downward
if
a
counter-clockwise
lotted vertically. Positive shear
:rotation is considered positive,
stresses are plotted downward if a counter-clockwise shear is considered :Points upon the circumferen ce of the
ositive, Points upon the circumfer- ;circle (fig. 1, B) give the relations
:between the strains indicated by the
ence of the circle( fig. 1, A), give
the relations between the stresses :above equations. A simple geometric
indicated by the above equations. A :analysis will demonstrate this factb
simple geometric analysis will
demonstrate this fact.
z
p
p
There are a number of useful geometrical relationship s in the strress
and strain circles which apply to either circle. The most important of these
have to do with the strains (or stresses) in two mutually perpendicular directions (1 and 2) and to the strains (or stresses) in a direction (3) at 45
degrees. These relationships involve the radius (r) of the strain (or stress)
circle and the distance (c) of its center from the origin. Both distances
be obtained by the use of the similar triangles shown in figure 2, which
can
is a strain circle. The relationshi p s are:
Stress
C = 1/2 (f 1f2)
f sl =
Strain
+ 1/2 (( e l +02'
(2)
(2) !e si = e 3c
-C
o
C)
2
il dmeo. No. 1317
2
/(e 1
+ (e sl )
2
Stress and Strain Circles for Wood
In an isotropic material. (such as steel or aluminum) the principal
axes of strain are always parallel to the principal axes of stress, so that fl)
is always equal to e.
These angles are not shown as equal in figure 1, and
they are not necessarily equal - for an orthotrOpid material like wood.
Again, in an isotropic material the principal strains are related to
the principal stresses in a way which is independent of the direction of the
principal stresses. In an orthotropic material this is not true; however,
simple relationships exist between the stresses and strains if the stresses
act in the directions of the natural axes of the material, such as the grain
or longitudinal (L) direction, tangential (T) direction, and radial (R)
direction of wood. (See fip--) . 1, C.) Since this report is concerned only
with the two dimensional case, the results will be given in terms of the (L)
anct the (T) directions, but can he used for any other two directions by
making the necessary substitutions.
If the stresses are resolved into the (L) and (T) directions, the
relations between th3 stresses and the strains in those directions are
given by the follo*ing equations:
f
f
e L =
, 7 'TL
z
f
em
=
J_
E
f
e
T fL
T
L
(3 )
L m?
SL
SL
Vhere
EL = modulus of elasticity in the (L) direction.
ET = modulus of elasticity in the (T) direction.
modulus of rigidity in the (LT) plane.
' LT
= Poisson's ratio of the contraction in the (L) direction, due
to tension in the (T) direction, to the extension in the
(T) direction.
1/LT
J, , Jmeo.
,T2
Poisson's ratio of the contraction in the (T) direction, due
to tension in the (L)) direction, to the extension in the
(L) direction.
No.
1317
-4-
Solving equations (3) for the stresses, we have:
'L
.
L
f
E
T
f
eTPTL
SL
e T + e L LI
T 1 - LT TL
(4)
2G e
LT SL
(The modulus 'of rigidity is multiplied by 2 because the rotational-strain
) is only one-half the conventional shear .strain.)
(e
SL
Determination of Strains from Stresses
lathematical equations have been developed for orthotropic materials,
but these equations are complicated. The use of the stress and strain
circles provides a geometric method of calculation in place of these equations and gives a better picture of the resulcs obtained. The following is
an example of the method.
Figure 3 shows a block of wood laid out at an angle to the grain direction. The block is so cut that its face, which lies in the plane of the
flat page, is parallel to the L, T plane of the wood; that is, i is part
of a flat-sawn board. The complete block shown in figure 3 is located at a
distance from the edges of the board, and the ends and edges of the board are
loaded in a manner which will cause the stresses at the edges of the block
which are indicated in figure j.
These stresses are:
fi = 80 pounds per square inch tension. f 260 pounds per square inch tension.
f Sl = 40 pounds per square inch shear.
f
82-
-7-- 40 . pounds per square inch shear.
Of course, fS2 will always equal -f 51 because of equilibrium conditions.
The stress circle can bc drawn from these values:
1/2 (12 1 + f2 ) = 70
0,
R - \/(fl
Yimeo. Foe 1317
) + (f
)
S1
2
_5_
41.25
The values of the r,iven stresses allow the determination of the direction in , Ihich those stresses act with respect to the stress circle (fic,. 4,
A). The circle is drawn to the proper radius (R). The origin is located
the distanc e (J) to the left (right if C is negative) of the center of the
circle. The stress (fs1) is positive and therefore down and because it
causes counter-clockwise rotation. A point (1) upon the circle is found
which satisfies the conditions f l = 80 and f 31 = 40. The line (H,1) is
drawn in the direction of the stress (f ). (See fig. 1, A,) The grain
1
direction makes an angle of (a) to line (H,1) and therefore the grain
direction can be located upon the stress circle. This direction is given
'by the line (H,L),
This stress circle completely determines the stress distribution,
and the stresses in any direction can be found by ,1 -;he use of simple geometry
and trigonometry. The stresses in the directions of the natural axes of the
-rood can be found as follows or by any other geometric method:
f
Tr?
=
S1
; _TTT
1
0
+ R
-7
'
e = 142°
'1
= tr-)
-L
180°
2t
+ a
168° 36
-
'
= 11° 2 4 '
The stress circle is redrawn in figure 4, B omitting some of the
lines which are no longer needed.
From similar triangles HLB and HLF (fig. 4, B):
,
L - 2R sin (180° —00 coo U_60
f, = 2k cos l. (
f T = 2C -
•
-
= 15.98
+ C = 108,05
I
(from equations(2))
31.95
and
f
01 =
f
- 3L = -15'98
Thus, all the stresses upon the inner block of figure 3 Nhich are
parallel to th natural axes of the wood have been determined. They can be
substituted in equations (3) to determine the strains parallel to the natural
axes. From these strains the strain circle can be drawn. It may be noted
that the stress &rc2e need not be drawn accurately to scale,as it serves only
to determine the method of calculaton. The same is true of the strain circle.
Ydmeo. No. 1:317 -6-
If the wood is spruce, we have
E = 0,036 E L
G
= 0.037 E
LT
L
= 0.0194
O.539
'LT
If these values are substituted in equations (3) together with the
stresses obtained above, the following, values of the strains result,
91.2
eL
L
= 828.8
T
e
SL
=
216
E
L
These strains are given in terms of the modulus of elasticity in the
(E ) and therefore this constant need not be substidirection of the
tuted in the equations until the problem is solved, thus avoiding the use of
exceedingly small numbers in the calculations.
From equations (2) we have:
c = 1/2 (e L + em) =
r =
2
(e Lc)
460
2
427.5
(eSL) =
The strain circle can now be drawn with the radius (r) as shown in
_A- An axis is drawn through the circle and the distance (c) laid
fir,-ure
negative) of the center of the circle. The
off to the left (right if (c)
value of (eSL) is positive, so that ib is laid off downward. A point (L) is
216
0 1.2
found upon the circle -which satisfies the conditions e Land en = -°- E
.
L
Line A, L is dra-wn which gives the direction of the grain vdth respect to
the strain circle. acferrinE , to figure 5, A:
•
e SL
LB =
= -368
tan %L =
c+e
AB r
L
:)T = 107 12
The strain circle completely determines the strain distribution, and
the strains in any direction oan be determined by the use of simple geometry
and trigonometry. It is evident from a comparison of figure 4, A, with
figure 5, A, that the principal axes of strain are not parallel to the
principal axes of stress. The major principal axis of stress can be drawn
in figure 5, A, by drawing a line through point (A) at angle (
BL ) to the
grain direction, as shown.
To illustrate the method, the strains in the directions of the
original stresses shown in figure 3 will be determined. The direction of
stress (f 1 ) acts at angle (a) to the grain direction; thus, the direction
of this stress can be shown by line A,1 in figure 5, B (which is a reproduction of figure
A, -with some lines omitted).
m 78 0 38
D
From similar triangles AlF and An
es 1 = -2r cos (15 1 vin C=
165.2
L
r
e l = 2r cos 2 17 1 + c
2c - e
e1
65.73
E
L
= 0,9,27
1
E
(from equations (2))
L
and
e
S2
=
e
S1
1 )542
EL
The strain 'can also be deterMined in any other direction. In particular, the strain (e 3 ) in a direction at 45° to (I) and (2) ad in the order
(1,3,2) counter clockwise can be found by U4e of equations (2) thus:
e 3 = 0 31 + c
2:4,4
E
L
Determination of Stresses from Strains
In testing procedure, it is often desired to measure strains and to
compute the stresses from the measurements. Linear strain 8 are more con
veniently measured than rotational strains. With records of such strains in
three different directions, the strain circle can be drawn and the stress
circle derived. The work is greatly simplified if two of the directions are
mutually perpendicular and the third is at angles of 45° to the others. The
work for wood is further simplified - if the mutually perpendicular directions
are those parallel and at right angles to the grain; the more general case
will, however, be illustrated.
Mimeo. No. 1317
_8_
Fio.ure 6 shows the directions in which the strains are measured
relative to the grain direction upon a face of a plane sawed spruce board.
The plane of the paper of figure 6 is therefore parallel to the L,T plane
of the spruce. The strains obtained in the previous example will be used
here. We have:
e
1
e2
=
65.73
=
8514.27
E
L
EL
From equations (2):
4Co
1/2 (e l + 6 2) =
e
r
1 6 c-,..2
=)
= e, -
S1
427.5
(eS1)2
V(e lc)2
EL
The strain circle can now- be drawn as shown in figure 5, B. Line
A, 1 gives the direction of (e ) relative to the strain circle.
1
-S1
tan C = Di =
i AD
C
1
+
r
=4,975
- 7 0 ° 3'
The direction of the grain can be drawn upon the strain circle at •
angle (a) to A, 1 as shown, and
(±)
2
= (1)
+
a = 105' 12'
From similar triangles ALP and BLF (fig. 5, A):
- 3L
=
c
2r cos 2I
e
=
L
e,
2c -
-L = 828'8
L
Mimeo. No. 1317
216
E
L
-2r cos n sin
L
-9-
r
=91.2
E
L
(from equations (2))
These strains are parallel to the natural axes of the wood and can be
substituted in equations (4) to obtain the corresponding stresses. The followinc7, valuer, are obtained using the elastic constants for spruce previously
given. "1f L = 1.08.05
31-95.
f T
f SL =
Usin,7 equations (2) we have:
C
1/2
+ fT)
= \j“'
70
0) + f
1
S1
= 41.25
The stress circle can now be drawn (fig. 11-, B), The grain direction
is given by the line H,L.
f
FL
HE"
tan OL
SL
fL C R
= -0. 2017
= 1b8° 56'
L
The stress in any direction can now be determined. For example, the
stress ,-.;s in the directions of the original strains shown in figure 6 will be
determined .(fig. 4, A).
The line 11,1 can 1-)e drawn at angle (a) to che line H,L. This line
gives the direction of the strain (e, ) relative to the stress circle.
1
-
From similar triangles Elf and 111K
eL - a = 142' 2'
we have:
21,-), cos (180-
Si
= 2R cos (180 -•
f
2
2 1"
'
)sin (180-
1
) +
-1
= /1-0
R = 80
f = 60
1
Lit will he noted that the valu-e-, .of E cancels out. If the original strains
L
had not been expressed in terms of E L ; this constant would appear in the
values " of stres . t as a multir)lier.
Mimeo. No. 1317
-10-
and
f
= -f. = LO
S1
These are the stresses shown in figure
3.
Stress and Strain Circles for Plywood
Plywood can be considered as an orthotropic material provided the
grain direction of each ply is perpendicular to that of the adjacent plies
and the surfaces of the plywood are not bent or warped by the application
of the loads. The assumption is slightly in error, as will be pointed out
later, but the error is exceedingly small. The strains in plywood can be
obtained from the stresses, or the stresses from the strains, in exactly the
salrie manner as described for wood if the apparent:1 elastic properties of the
plywood . are used instead of the elastic properties of wood and the direction
of the face grain of the plywood is used in p lace of the grain direction of
the wood.
It remains only to derive the apparent elastic properties of plywood
in terms of the elastic properties of wood. Figure 7 shows a small unit
square of plywood located at a distance from the edge of the plywood panel.
The grain directions of the various plies are parallel both to the edges of
the small square and those of the panel. A compressive force is applied to
the upper edges of the panel, causing the small square to become deformed,
as shown by the dotted lines in figure 7. The top edge of the small square
will exert an average compressive stress such as to oppose the load, but the
compressive stress in any one ply need not equal the compressive stress in
another ply. . The compressive stresses in the various plies are indicated in
figure 7. The plies are all glued together, so that the compressive strain
is the same for all plies as shown.
The small square will expand sidewise due to the apparent Poisson's
ratio of the plywood. Since the plies are glued together, this expansion
will be the same for all plies. Each ply, however, would expand according
to its own Poisson's ratio if it were allowed to do so. As it is restrained
by the glue, the stresses sho-yn at the left of figure 7 are induced by shear
stresses across the glue lines near the edges of the panels. These shear
stresses, however, are limited to parts of the panel that are close to its
edges, and hence do not appear in the small square.
The relations between the deformations and the stresses for a single
ply (the ith ply) are:
-"The word "apparent" is applied to a summation average value of each property.
Its meaning is made clear in the subsequent discussion.
Mimeo. No. 1317
-11-
ai
f.
i.
11,- •
rY),_
bi
f.
ai
f.
'ja
f
e =
b
E.
abi
ai
Ebi
where:
. -
stress in the ith ply in direction (a).
=
stress in the ith ply in direction (b)
al
' bi
E a = modulus of elasticity of the ith ply in direction (a)
direction (b)
ci = modulus of elasticity of the ith ply in
= Poisson's ratio of the ith ply ratio of expansion in the (b)
direction, due to a compression in the (a) direction, to the
contraction in the (a) direction.
= Poisson's ratio of the
ply: ratio of expansion in the (a)
direction, due to a compression in the (b) direc lAon, to the
contraction in the (h) direction.
These equations are identical with the first two of equations (3).
They can be solved for the stresses, and the following, equations are obtained:
f
f
ai
bi
=
E .
e a
+ e,
o
Da a..
bi
The force upon the top eage of any one ply is obtained by multiplying
the stress by the area of the ply, and, since the small square is a unit
square, the area of the bop edge of a sil]Kle ply is equal to its thickness.
The sum of these forces is equal to the total force, or the average stress
multiplied by the total thickness. The following equation results:
e
pt =
r
a
1 -
abl bat.
The apparent modulus of elasticity of the plywood is the average
stress divided by the deformation, or
b ur
1 +
E
Mimeo No. 1317
a
=
7.1 tf• E
bai
•
al-71--- abilbai
-12-
but
eb =
1-Lab
a
= the apparent Poisson's ratio of the plywood.
ij' ab
Therefore
a
(5)
1 -
t'abir-bai
the apparent modulus of elasticity of the plywood in the
(a) direction
t = the thickness of the plywood
= the apparent Poisson's ratio of the plywood: ratio of expansion in the (b) direction, due to a compression in the (a)
direction, to the contraction in the (a) direction.
Similarly:
=1
t E
1
-
i bi
Eb
abi
it is evident that each of the numerators and denominators in the
is only slightly less than unity, and by very similar
and E
equation for L
b
a
amounts. The following equations ar e , therefore, good approximations:
E
1
t
5
E =
b
T
approx.
t•E .
a_ al
(6)
a-oprox.
t E
i bi
The summations of the forces upon the left edge of the sma l l square
must be equal to zero, since no load was applied to the plywood in this
direction. It follows that:
b
t -v., • --
1) 1 1 -
ea
abi
4
= 0
ai
Separating the terms in the numerator:
e.
t. E
1 b i
n•• nn•••n •
ab i lo a i
Mimeo. No. 1317
t E
bi k
a bi
e
a — 1 _
-13-
Dividing both sides of the equation by e a and remembering that
eb
Pab
th
-Piab
1
-
tE
bi
+
.
abi
birabi
0
Sabi ^bai
t.Ebi pomi
1
bai
(7)
tihbi
1 - i4bi
Similarly:
t i Eal. tbal.
1-
t'abi
t.E
ai
Pba
1-
Pbai
The denominators in both the numerator and denominator are all slightly
less than Unity, and all by about the same amounts. Therefore, the following
equation is a good approximtion:
t i obi
ba abi
7 t. E .
bi
d ab
•
or
=
tL b
and
iLba
11D i
Ic]
f" a b i
from equations (6))
i -ai bai
The accurate value of (La ) can be obtained by substituting the
accurate value of 4±r ,00 ) in the equation for (E a ), thus:
Mimeo. No. 1317
(8)
i bi P'abi
1 1 - ha"
1
a
t
Ebi •
2"
-t-T;'
1
ab
•
-
(9)
abi
The appro x imate equ a tions are, however, sufficiently accurate for practical
use in connection with plywood.
In setting up these equations it has been assumed that the small square
of plywood shown in figure 7 is located at a distance from the edge of the
larger sheet. If the small square is located at the left edge of the sheet,
the stresses shown acting upon the left edge of the square are each equal to
zero and the argument used in the derivation of the equations breaks down.
A complicated distribution of stress and strain exists in this case. Very
near to the edge, the value of (L a ) approaches more nearly the value which
would be obtained if the various plies were not glued together. This is t
approximate value given by equation (6), which has been justified by test._
It remains to determine the apparent modulus of rigidity of plywood.
Imagine a small square of plywood similar to that shown in figure 7, except
that the direct stresses are replaced by shear stresses. The shear strain
is constant throughout the small square because the various plies are glued
together. The shear stress in the lth ply is:
f
si
=
2e
Sabi
S1
It is evident at once that the average shear stress is
2e
3i
L.
s
Gabi
and that the apparent modulus of rigidity of the plywood is
fs
G ab
2es,
t
b. G
.
a.bi
(Twice the rotational strain is used in these equations because . the rotational
strain is half of the conventional shear strain.)
Forest Products Laboratory Restricted Mimeograph No. 1315, "Tentative method
of calculating the strength and modulus of elasticity of plywood in compression."
himeo. No.
1317
-15-
Examples
It is quite possible to solve typical problems by means of the stress
circles, obtaining, a general formula for a specific case. Two examples are
given for plywood, one with tension at 45' to the direction of the face
grain and the other with shear at 49' to the direction of the face grain.
Plywood in Tension at 45' to the Direction of the Face Grain
FiFure
shows the specific case considered. From equations (2)
1
f1 for f 2 = 0
2
=
1
f
2 1
R=
for f
s1
=0
The stress circle can nolif ha drawn as shown in figure 9. The circle
passes through the origin, since C = R. It is evident that:
fa
1
'1
fb
f
sa
1
=— f
2 1
It may be recalled here that f a and f sb act parallel to the face grain and
f sa acts at riht angles to these two,
The strains in these directions
tions (3):
f
f,
a
0a
I-43a 1;0
ob =
e
fb
can be obtained by the use of equa-
-
-
\
."13
La
1
f a ,
ba)
(1
1
—
2
(1
f l
-
Ea'
f sa
1
1
f 9r,
sa
1 "ab
2S ab2
= The strain circle can nova be drawn, and is shown in figure 10.
_Numerical values for drawing it are not available because a general case is
being dealt with. The radius of the circle is chosen at random.
31imeo, No. 1317
-16-
L'_ax-,ell's theorem shov,is:
hpa = Pab
E
a
b
a
It follows from equations (11) that if E>.
E b , e a <e b , and figure 10 was
drawn with this fact in mind. It should be noted that the principal axis
of strain is not parallel to the direction of the tonsil() stress.
The first of equations (2) gives:
( 1
11
,
.11
=
ea +
2
1
+
-
'b
Lb
P' ab
a
The second of equations (2) gives:
a
ksalo +
1
)
7---Tab
a
'b
111--
1 4,( 1
'1 E esa
el = '13
This relation could have been obtained directly from the geometry of
figure 10.
The modulus of elasticity at Li-° to the direction of the face grain
is defined as the ratio of a tensile (or compressive) stress in this direction
to the strain in this direction which the stress produces, hence
f
L 450
1
1
1
Ea
ab
1
ba
Gab
"n'b
However, there will be associated with the tensile strain a shear strain
(e s1 ). It is found from the gemoetry of figure 10 that:
e
sl
e _ c
4 1 E
b
E
a
'.-2J
a
The use of :!.ia.xwell's relation reduces this equation to:
e
sl
l, ( 1
1
IF - 1 EtEa
The special case in -Jvhich E t = E a is interesting. In this case
- 0, and the principal axis of strain is parallel to the direction of
l
s
the stress. Then;
e
Mimeo. No. 1317
-17-
a
+
20ao
Plywood in Shear at 45' to the Direction of the Face Grain
Figure 11 shows the specific case considered. From equations (
w.e have:
C = 0, because f and f 2 - 0.
.1
It follows that the center of the stress circle is at the origin, and
R = f si , because C and fl0,
The stress circle can now he drawn and is shown in figure 12,
The figure shows that the principal axis of stress is parallel to the grain
direction. It also shows that;
sl
fa
f = f
sl
b
fs
a=0
The resulting strains can be found by substituting these values in
equations (3):
f,
a
1
77--)
b
E a E b Pl a = sl
kb
ra
Eh E a ab
e
f
sl
( 1
E,
+
/lab
ha
sa , 0
ab
As :In the previous specific cse e a ( (3 13 (numerically) if
Ea .> Et . The strain circle can be drawn, as before, usinE, an arbitrary
value for
ra(lius. Again using; equations (2):
1
—
lame°. No. 1317
a + -b
-
2 sl
-18-
la b a
1
7 -
T-
a
Using Yaxwell's relation, this reduces to:
1
=
1
( 1
f
The value of 0
The resulting strain circle is shown in figure 13. is negative (fsl is negative), therefore the center of the circle lies to
the left of the origin. Also the direction of e a , and therefore the
direction of the face grain, is parallel to the direction of the principal
axis because esa = 0.
It is evident from the geometry of the strain circle that:
C
esl
Caa
Both e 51 and C are negative and e a is positive.
f sl
+ 7 f sl
-b
/ 1
1
f sl
e sl =
-b
T7--
'a
1
aB
Ea - 7a
/1
1
'ba\
(1
0 51
.6b
1
E aEb
The modulus of rigidity at 45: to the grain direction (G4 5
0)
is
defined as the shear divided By the shear deformation, hence:
f
.)
Le
•
sl
1
=
(-4
45°
•
and
=
B
ID
Pjab
E
a
+ lib a
1
a
E
b
Associcted with the shear strain is a direct strain e l .
The
strain circle shows:
1
/1
1
e lC=—f
2 s l EtLa
Again it is of interest to examine the special case in which E a = Et.
In 'this case e l = 0 and the center of the strain circle coincides with the
origin. The formula for the modulus of rigidity at 45° becomes (using
Naxwell's relation):
E
a
1
G
2 1
45°
It is interesting to note that this is exactly the relation obtained
for isotropic materials.
Mimeo. No. 1317
-19-