QA1Gf7ESS1ON, TENSION ANV, SHEAR TUTS ----LOW
PLYWOOD PANELS Cr it
irtiAT DV NOT MCKEE Willi TIMIS MAIM
Att VARIOUS ANCLIES IC THE FACE GRAIN
"IL vtdi
INFORMATION
AND REAFFI'
1962
LOAN COPY
Please return to:
Wood Engineering Research
Forest Products Laboratory
Madison,Wisconsin 53705
This Veport is Ctio of Scrio
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Ne. 1328
STATDEPARTMENTOF AGRICU
FallEST SERVICE
IS LABORATORY
_sons Wisconsin
die n University of Wisconsin
COMPRESSION, TENSION, AND SHEAR TESTS ON
YELLOW-POPLAR PLYWOOD PANELS OF SIZES
THAT DO NOT BUCKLE WITH TESTS MADE AT
VARIOUS ANGLES TO THE FACE GRAIN!
By
C. B. NORRIS, Engineer
and
P. F. McKINNON, Engineer
2
Forest Products Laboratory,— Forest Service
U. S. Department of Agriculture
Summary and Conclusions
This report is a study of the elastic properties and the strengths, at
proportional limit and ultimate, of plywood subjected to stresses acting in the plane of the plywood and at various angles to the grain. Investigation of the theoretical relationships is based on tests made on
yellow-poplar plywood, specimens so proportioned that they would not
buckle or fail because of elastic instability.
Incorporated in this report are the theoretical development of equations
for calculating the elastic moduli and the strength values of plywood
subjected to tensile, compressive, or shear stresses acting in the
1
— This is one of a series of progress reports prepared by the Forest
Products Laboratory relating to the use of wood in aircraft. Results here reported are preliminary and may be revised as additional data become available. Original report published in 1946.
2
— Maintained at Madison, Wis. , in cooperation with the University of
Wisconsin.
Report No. 1328
-1-
Agriculture-Madison
plane of the plywood and at various angles to the lace grain. Graphs
showing the comparison between the experimental data and theoretical
curves are included.
Supplements A, B, and C, which were prepared previously, deal with
compression, tension, and shear, respectively. They consist of descriptions of the testing techniques used, the data obtained, and such
analyses as do not require consideration of the interrelations among
the results from the different kinds of tests. The purpose of this report is to correlate the data presented in the supplements by analyses
derived from the mathematical theory of elasticity and from a second
power interaction formula for ultimate strength.
A review of the analysis and the various graphs indicates that as a
whole the agreement between theory and experiment is good.
The experimental data used in verifying the theory developed were from
tests on yellow-poplar plywood. It is assumed, however, that these
equations are applicable to plywood of other species.
-rom the analysis, the following conclusions were drawn:
(1) The tension tests of plywood at various angles to the face grain yield
moduli of elasticity which agree with the theory of elasticity.
(2) The compression tests yield moduli of elasticity which, in general,
lie between the maximum and minimum values predicted by the theory
of elasticity. The test specimens that were narrowest and, therefore ,
were subjected to the least restraint, yielded the least values of modu lus of elasticity; and the widest specimens, the greatest.
(3) It is difficult to determine accurate values of modulus of rigidity experimentally by means of a shear test in which direct measurements of
stress and strain are employed; the revised apparatus described in,
supplement C, however, yields values that are in reasonable agreement with those obtained by use of the theory of elasticity.
(4) Accurate values of the modulus of rigidity for the one case of face
grain parallel and perpendicular to the direction of the shear stresses
can be determined by means of three tension tests upon matched material; one parallel, one perpendicular, and one at 45° to the direction of
the face grain.
Report No. 1328
-Z-
(5) The tension and compression tests at various angles to the face
grain yield proportional limits which agree reasonably well with theory.
The effect of restraint of the specimens in the compression tests upon
the value of the proportional limits obtained is small.
(6) The proportional limits obtained in the shear tests with the revised
apparatus described in supplement C agree reasonably well with theory.
(7) The ultimate tensile strengths at angles to the grain, as obtained
from test, agree closely with those obtained from the second power interaction formula.
(8) The compressive strengths obtained from test lie, in general, between the maximum and minimum values predicted by the second power
interaction formula, In general, the specimens that were narrowest,
and, therefore, subjected to the least restraint, yielded the lowest
values; and the widest specimens yielded the highest values.
(9) The shear strengths obtained by use of the revised apparatus agree
well with those predicted by the second power interaction formulp.
(10) Substantially the same values of shear strength at an angle of 0°
were obtained by use of either the original or revised apparatus described in supplement C.
Introduction
3 issued by the Laboratory dealing with the behavior
In various reports—
of plywood, extensive use has been made of the mathematical theory of
elasticity as applied to orthotropic materials, that is, materials having
three mutually perpendicular planes of elastic symmetry. Although
this theory has been experimentally verified in connection with the elastic stability of plywood plates loaded in various ways and the deflection
of plywood plates loaded normally to their surfaces, it has not previously
3
—Forest Products Laboratory Reports Nos. 1300, 1301, 1304, 1312, and
1316 with Supplements A to I, inclusive.
Report No. 1328
-3-
been directly verified by tests in which forces are applied in the plane
of the plywood under conditions such that the plywood does not buckle.
A partial verification has been obtained for the case of tensile stresses
4
by Lowell M. Moses.— The fundamentals of the theory have been presented in two Laboratory reports.-5 The purpose of this report is, in
part, to verify this theory by the results of tests in which compressive,
tensile, and shear stresses are applied in the plane of the plywood.
In the design of aircraft parts involving the use of plywood in which the
stress is applied in the plane of the plywood and at an angle to the face
grain, it has been assumed that the proportional limit of the plywood
is reached when any one of the components of the stress, resolved in
the direction of the natural axes of the plywood, reaches the proportional limit. It is the purpose of this report, in part, to verify this
assumption.
The strength of plywood when subjected to stresses in its plane and at
an angle to the direction of the face grain is also needed in the design
of aircraft parts. The Forest Products Laboratory suggested the use
of an interaction formula of the second power for this purpose.. The
stresses are resolved into components in the directions of the natural
axes of the plywood. The condition for failure is then given by the
formula:
t aa
2
k a!
2f
4-
+( ‘ab
1
(1)
ab
4
—11 An Investigation of the Elastic Behavior of Spruce Plywood Subjected
to a Single Tensile Stress," by Lowell M. Moses. A thesis submitted to the Massachusetts Institute of Technology (1932).
—Forest Products Laboratory Reports Nos. 1317 and 1503.
6
—"ANC Handbook on the Design of Wood Aircraft Structures," (July
1942). Also ANC Bulletin No. 18 "Design of Wood Aircraft Struc-
tures," (1944).
Report No. 1328
-4-
in which:
t
t
t
= direct stress in the direction of the face grain.
aa
bb
= direct stress at right angles to the direction of the face
grain.
ab
= shear stress parallel and perpendicular to the direction
of the face grain.
F a = strength under a direct stress acting in the direction of
the face grain.
F
b
= strength under a direct stress acting perpendicular to
the direction of the face grain.
Fab = strength under a shear stress acting parallel and perpendicular to the direction of the face grain.
It is the purpose of this report, in part, to verify this interaction formula. An approximate theoretical determination of this formula is
given in Forest Products Laboratory Report No. 1816.—
Source of Material and Data
This report is a summarization of three separate studies reported in
the supplements A, B, and C, and no data are included that can be found
in the supplements. In the supplements will be found such information
as source of material, matching and marking, test methods and procedures, and other related information.
•
7
--Forest Products Laboratory Report No. 1816, "Strength of Orthotropic Materials Subjected to Combined Stresses."
Report No. 1328
-5-
Tables and Figures
Tables- 21 through 24 contain values from tension and compression
tests which were made upon plywood matched to that used in the shear
tests in which the revised shear apparatus, described in supplement C,
was used. The data from the corresponding shear tests are reported
in tables 19 and 20 of supplement C. In tables 21 through 24 columns
1 and 2 give the panel and stick numbers. Columns 3 and 4 give the
moisture content based on weight when ovendry and the specific gravity
at test, respectively. Proportional limit stresses are listed in column
5, moduli of elasticity in column 6, and ultimate strengths in column 7.
Figure 54 is a graphical representation of the stresses developed in a
plywood prism subject to external forces applied at an angle to the face
grain and in the plane of the plywood.
Figures 55 to 60, inclusive, contain graphs of the relation of the elastic moduli to the angle of application for tension, compression, and
shear. Similar graphs of proportional limit stresses for 'tension, compression, and shear are shown in figures 61 to 66, inclusive. Figures
67 to 72, inclusive, show graphs of the ultimate strengths in tension,
compression, and shear. Figure 73 contains graphs of the modulus of
rigidity, proportional limit in shear, and the shearing strength from
tests in which the revised shear apparatus was used.
In all these graphs the angles between the face grain and applied stress
are plotted as abscissas and the stresses or elastic moduli as ordinates.
Theoretical
General Elastic Equations
Figure 54 represents a small square CDEF drawn upon a piece of plywood of thickness h. The length of each side of the square is 2m. The
X and Y axes are chosen parallel to the sides of the square as shown.
:1--3-The tables and figures of this report are numbered consecutively with
those of the previous reports of this series, 1328-A, 1328-B, and
1328-C.
Report No. 1328
-6-
Another square GHIJ is drawn within CDEF and with its sides parallel
and perpendicular to the direction of the face grain (A) of the plywood.
The length of each side of this square is 2n.
When direct stresses t and t and shear stress t are applied to
xx
xy
YY
"V
the edges of the square CDEF it is deformed to the rhomboid CDEF.
The heavy arrows show the forces which are assumed to be distributed
uniformly over the edges of the square. The inner square GHIJ is de. The stresses which cause this deforformed to the rhomboid G'
as shown by the heavy arrows.
and t
t
mation are t
ab
aa
' bb'
for square CDEF and e ,
and e
, e
aa
xy
yy'
xx
for square GHIJ and are obtained from the displacements
The resulting strains are e
e
and e
bb '
ab
of the center points of the sides of the squares.
The angle between the X-axis and the A-axis is denoted by 0 as shown.
The relationships between the stresses and the angle 0 as given by
equations (35) of Forest Products Laboratory Report No. 15032 are:
t aa =t xx cos 2 0 + t
sin 2 0 + 2t xy sin 0 cos 0
yy
t
t
sin 0 cos 0
= t
sin 2 0 + t
cos 2 0 - 2f
xy
bb xx
yy
ab
= t
xx
sin 0 cos 0 + t
YY
(2)
sin 0 cos 0
+ t xy (cos 2 0 - sin 2 0)
The relationships between the strains and the angle 0 as given by equations (37), (38), and (39) of the same report are:
Forest Products Laboratory Report No. 1'503, "Stress-Strain Relations in Wood and Plywood Considered as Orthotropic Materials.“
Report No. 1328
-7-
= e
aa
e
e
= e
bb
ab
cos 2 + e
xx
sin 2 0 + e
xx
= - 2e
xx
+ e
xy
sin 2 0 + e
YY
YY
cos 2 - e
sin 0 cos 0 + 2e
YY
xy
xy
sin 0 cos 0
sin 0 cos 0
(3)
sin 0 cos 0
(cos 2 0 - sin 2 0)
The relationships between the stresses and the strains are given by
equations (36) as:
0"
e
1,
ba
=---t
---t
aa E aa E
a b bb
ab
e bb =-----tt
= E
a
e =
ab
1
+—t
E tbb
b
(4)
P-ab ab
Equations (4)can also be written:
t
aa
=
Ea
(e
+e
CT
X aa
bb ba )
E
t
t
bb
=
X.
(e
bb
+e
(5)
aa ab
= e
Fl
ab ab ab
Report No. 1328
-8
In which
X = 1 - cr
cr
ab
ba
E = modulus of elasticity of the plywood in the direction of
a
the face grain.
E - modulus of elasticity of the plywood in the direction
b
perpendicular to that of the face grain.
ab = modulus of rigidity of the plywood associated with a
shearing strain in the AB plane.
a- ab = Poisson' s ratio of the contraction of the plywood parallel to OB (fig. 54) to the extension parallel to OA associated with a tension parallel to OA.
0-
ba
is similarly defined.
The Interaction Formula for
Maximum Stress
The interaction formula -- equation (1) -- can be generalized by substituting in it the values of the stresses as given by formulas (2). The resulting equation is:
Fa
2 xx
cos 2 0 + t
1
F
2
xx
YY
sin 2 0 + 2t
sin2 0 + t
YY
xy
sin U cos 0)2
cos 2 0 - 2t xy sin 0 cos 0)2
b
1
F 2
ab
(6)
- t
xx
sin 0 cos 0 + t
-2
+ t xy (cos 2 0 - sin 2 0)
Report No. 1328
-9-
=1
yy
sin 0 cos 0
Modulus of Elasticity at an Angle to the Face Grain
Tension
=0
= t
If txx is the only stress applied, t
= 0. By putting t
= t
yy xy
yy xy
in equations
in equations (2) and the resulting values oft
and t
t
ab
as
' bb '
and e
(4), e , e
are obtained in terms of t and 0. When these
aa
ab
xx
bb'
—
values of strain are substituted in equations (3) and the values of e
YY
and e
eliminated
by
use
of
equations
(3)
the
following
equation
results:
xy
t
xx = El'
1
cos40 +
x Ea
xx
T
2 ab sin
g
Ea
E
11
sin4 0+
p.ab
b
cos t 0
(7)
This equation yields the modulus of elasticity (E x ) in the direction X at
angle 0 to the direction of the face grain. It should be noted that when
0 = 0, E = E , and when 0 = 90° E = Eb.
—
x
a
x
The values of E and E used in computing values of E for various
a
b
x
values of 0, were averages for 0 = 0 and 0 = 90° as taken from tables
10-15, inclusive, in supplement B.
Poisson' s ratio,
was computed using equation (8) (Report No. 1317)
crab'
which applies to plywood made of rotary-cut veneers and is:
(Tab
Report No. 1328
E L °TL
E
b
(A)
-10-
In which
EL = modulus of elasticity of the wood, of which the plywood
is made, in the direction of the grain.
TL
= Poisson s ratio for the wood; the ratio, of the contrac
tion in the grain direction to the elongation in the tangential direction due to a tensile stress in the grain
direction.
-From the definitions of E and E it follows that10
a
(B)
E +E =E +E
Ea bL T
= modulus of elasticity .in the tangential direction of the
T
wood of which the plywood is made. Using this relationship and the fact
E
= 0. 036, the value of E L can be obtained and
that for yellow-poplar
in which E
substituted in equation (A), which gives
E
(rab =
cr.
a
b
+1
TL
1. 036
It is evident from the preceding equations that the value of cr ab will be
different for each construction of plywood.
Values for p.ab for each construction were obtained by substituting for
Ex in equation (7) the experimental value from tension tests at 45 0 and
These values of p.ab were used in subsequent computaab
tions for the moduli of elasticity in compression, modulus of rigidity,
solving for
10
-Forest Products Laboratory Report.No. 1503, "Stress-Strain Relations in Wood and Plywood Considered as Orthotropic Materials,"
equations (53) and (55).
Report No. 1328
-11-
and proportional limit strength, instead of those obtained from the shear
tests because the method of shear test used was in question. Values obtained in this way are accurate because, at an angle of 45°, the term of
equation (7) which contains [Lab is large with respect to the other terms,
that is, Ea and Eb could be somewhat in error and yet not have a significant effect on the value of p. ab . This matter will be discussed subsequently in greater detail.
Figures 55 and 56 are graphs comparing the theoretical curves with the
experimental data; the agreement between the two is good. Experimental values for the intermediate angles are given in tables 10 to 15, supplement B, inclusive. The experimental values for each angle plotted
in figures 55 and 56, for a particular construction, is the average of all
tests made at that angle.
It may readily be seen that the theoretical curves and the experimental
data must agree when 0 = 0°, 45°, or 90°. The differences between the
two at the intermediate angles are small.
Compression
If the specimen were unrestrained, equation (7) could be used for computing from the results of tests the modulus of elasticity in compression
at various angles. Compression specimens, however, are usually made
much shorter than tensile specimens to avoid elastic instability, and
their central sections may not be entirely free from restraint imposed
by the heads of the testing machine. If the plywood is restrained in this
manner, t and t do not equal zero and equation (7) does not apply
xy
YY
exactly. If complete restraint in the Y direction is assumed, e
=e
yy xy
= 0. By substituting these values of strain in equations (3) and the resulting values of e , es and e
in equations (5), t , t , and t
aa
ab
bb
aa bb
ab
are obtained in terms of e and 0. These can then be substituted in
xx
equations (2) and t YY and t eliminated by combining these three equaxy
tions. The following equation results:
Report No. 1328
-12-
t
e
xx
XX
cos 4 0 + E sin4 0 + (4X Fl ab
=XE
b
x = Ea
) sin g 0 cos h 0
+ 2E oba
a
(8)
This equation yields a value of the modulus of elasticity (E) in the direction X at angle 0 to the direction of the face grain under the assumption that the plywood is completely restrained except for deformations
E
a
in the direction of X. It should be noted that when 0 = 0, Ex
= k
E
b
Similarly, when 0 = 90°, E = - For plywood, X is so nearly equal
X. •
x
to unity that it can be taken as such.
The values of E and E used in computing E from equations (7) and
x
b
a
to be substituted in these
and cr
(8) and for computing values of cr
ba
ab
equations were the averages from tables 2 through 7, of supplement A.
The value of E a for a particular construction, is the average value of
is the average value of all tests
all tests when 0 = 0°. Similarly ,
Eb
for 0 = 90°. The value of a b as computed for the tension modulus also
applies to the compression modulus. Poisson' s ratio, crab' was comwas
puted according to equations (A) and (B). Poisson's ratio of cr
ba
computed according to equation (8) of Report No. 1317, which is:
E
L TL
E
(C)
a
The relationship
E +E =E +E
T
a
b
L
Report No. 1328
-13-
was used as in the determination of cr
ab
and the following equation ob-
tained for yellow-poplar plywood:
E
or
a
b+1
Ea
a-
1.036
Figures 57 and 58 contain graphs of the theoretical curves and the experimental data. The data are given in tables 2 through 7, inclusive,
of supplement A. As the legends on the graphs indicate, each point
representing experimental data for a particular angle is the average of
values from all tests at that angle for specimens having a certain width.
Taken as a whole, the agreement between the theoretical and experimental data is reasonably good. Values from equation (7) for the case of no
restraint are the least that are to be expected in test, while those from
equation (8), the case of complete restraint, are the greatest. Values
between these two should be obtained from tests, if the specimens are
neither completely free nor completely restrained.
In general, the points plotted from data taken from the narrowest (2inch) specimens are lowest and those from the widest (6-inch) specimens
highest, which agrees with the concept that the wider the specimen the
greater would be the restraint (all specimens were approximately of the
same length). Both theoretical curves were made to agree exactly at
= 0° and 0 = 90° with the average of the data from tests at these angles.
Because the modulus of rigidity, 1.1.
as computed from the tension
ab'
tests at 45° was used in the determination of the theoretical curves, however, the curves do not necessarily have to agree exactly with the average of tests at 45°.
It is evident that at angles other than 0° and 90° the two values of modulus of elasticity given by equations (7) and (8) can be very different and
that the intermediate angles are more conservatively represented by
equation (7).
Report ij . 1328
-14-
Modulus of Rigidity at an Angle to the Face Grain
Two values for the modulus of rigidity at an angle to the face grain can
be obtained by methods similar to those used in the case of modulus of
elasticity in compression, one for the case of no restraint and one for
= 0, (no restraint)
= t
the case of complete restraint. When t xx yy
ex xy
1
1
cos 2 20 +
xy Flab
Also, when e = e =
xx yy
crab sin 2 20
1
+ 2
+
E
a EbE a
1
complete restraint)
t
(E + E
os
cos 2 20 +
xyc
p.
b
4K a
e
xy =
xy
- 2E a Tba ) sin2 20
Each of these equations yields p. = p. when 0 = 0° or 90°, but differ
xy a
at other values of 0.
The tests reported in supplements A and B yielded slightly different
values of E and E in tension than in compression. It was considered
a
b
proper to use in equations (9) and (10) the value in each case associated
with the stress which was obtained.
and E
a
b
change as the values of 0 change from negative to positive. When 0 is
negative, Eaand E b
are the moduli of elasticity in compression parallel
Therefore, in these equations, the meanings of the symbols E
and tension perpendicular to face grain, respectively. When 0 is positive, Ea and Eb are the moduli of elasticity in tension parallel and compression perpendicular to face grain, respectively.
Report No. 1328
-15-
The values of E
used in plotting the theoretical curves
Eb, and p.
ab
a'
representing equations (9) and (10) were obtained as described for the
were comcases of tension and compression. Values of cr and oab
ba
puted according to equations (A) and (C). The experimental values
were taken from table 18 in supplement C. Each point represents the
average value of all tests of a particular construction at the angle
shown.
Graphs of the theoretical curves and the experimental data are shown
in figures 59 and 60. It is evident that the correlation is not good.
Actually, the method employed in the shear tests represented by the
plotted points in figures 59 and 60 was not such that close correlation
with theory could be expected.11
- The specimen was not free to deform as it chose as assumed in the derivation of equation (9) nor was
it restrained as assumed in the derivation of equation (10).. The apparatus was revised as described in supplement C and check tests
were then made. The results of these check tests in shear are given
in tables 19 and 20 of supplement C and of tension and compression
tests on matched specimens in tables 21 through 24 herein. The theoretical curves and the data obtained are plotted in the bottom section
of figure 73. The experimental values check the theoretical curves
reasonably well. In general, they lie between the graphs of equations
(9) and (10) and closer to the first than the second, thus indicating that
the revised method of test is much better than the original as a means
for determining modulus of rigidity. It is evident that the values obtained by the original method are much too high. Because of this,
moduli of rigidity derived from tension tests at 45° were used instead
of those from the original shear test method in calculations of E x from
equations (7) and (8). Also, the moduli of rigidity obtained by means of
11
--The experimental difficulties encountered in the determination of the
modulus of rigidity by direct measurements of the shearing stresses
and strains are avoided by the plate method described in Forest
Products Laboratory Report No. 1301, "Method of Measuring the
Shearing Moduli in Wood." The moduli obtained by this method are
equal to those obtained by the direct measurement of the stresses
and strains if all of the plies of the plywood tested are of the same
species. The plate method, however, will not yield proper values
at angles other than 0° except for a few special constructions of
plywood.
Report No. 1328
-16-
the revised apparatus were unsuited to this use because the tests were
on plywood that was not matched to that used in the other tests.
The values given by equation (9) are minimums, but experimental procedures and methods of computation may yield values higher than those
from equation (10) because restraints in addition to those assumed in
the derivation of this equation may be present. Shear tests are difficult
to perform accurately, and measured values of the modulus of rigidity
can readily be greater than those given by equation (10).
Proportional Limit of Direct Stress
at an Angle to the Face Grain
It is assumed that the stress t will reach the proportional limit p x
xx
reaches its proportional limit p a' when t bb reaches p b' or
when t
aa
equations for the
There are, therefore, three
reaches p
when t
ab •
ab
proportional limit stress p . Of these the applicable one is that which
x
gives the least value.
•••••••n• nnnnnnn•••
0 in equations
= t
is the only stress applied, putting t yy xy
xx
(2) gives the three equations for proportional limit stress directly:
When t
P
px
a
cos t0
p
b
.2
sin 0
px
p
=
Report No. 1328
Pab
sin 0 cos 0
-17-
Equations (11) assume that there is no restraint by the testing machine
or otherwise to give rise to stress components t and t . If, however,
xy
YY
there is complete restraint so that e = e
= 0, equations (3) give e ,
yy
xy
as
e , and e
in terms of e
and 0, and substitution of these in equations
bb
ab
xx
(5) gives t , t , and t
by
in the same terms. Then by replacing e
as bb
ab
xx
t
XX
in which E is the modulus of elasticity computed by inserting exE '
x
x
perimentally determined values of E and E in equation (8), and replaca
ing t by the p appropriate to each case the following three equations
xx
result:
p XE
a
x
px
E cos 0 + o-
XE
px
E (sin
b
x
2
ba
sin g 0)
x
+ T
ab
(12)
cos t0)
P ab E x
2 p.ab sin 0 cos 0
Values intermediate to those represented by equations (11) and those
from equations (12) may be expected from compression tests because
neither complete freedom nor complete restraint is likely to be attained.
The correlation of the values of modulus of elasticity with equation (7),
as illustrated by figures 55 and 56 and discussed earlier, indicates that
in these tests there was not significant restraint and that consequently
proportional limits experimentally determined from tension tests may
be expected to conform to equations (11).
Positive direct stresses are tensile, and negative direct stresses are
compressive. The tensile stress t aa reaches the tensile proportional
Report No. 1328
-18-
limit p and the compressive stress -t reaches the compressive proaa
a
portional limit -p . Also these two proportional limits are not numeria
cally equal. The same is true of t and t . However, the proportional
xx
bb
limit in shear, p
is numerically equal to -p . It is necessary,
ab
ab'
therefore, to determine whether the tensile or the compressive proportional. limits should be used in each of equations (11) and equations (12).
This can be determined by the rule that if p is positive the right hand
x
member of each equation of (11) and (12) must also be positive, and if
px is negative the right hand members must also be negative.
Tension
Figures 61 and 62 are graphs showing average values of proportional
limit stress for the various angles as taken from tables 10 to 15 of supplement B. Shown also are curves established by substituting in the
first and second of equations (11) the average values for 0° and 90° and
in the third equation one-half the average value at 45° (because p
ab
= 1 /2 px at this angle).
It may be noted that agreement at the intermediate angles represented
by the tests (15°, 30°, 60°, and 75°) is good. Tests were not made for
angles near 0° and 90° (such as 7-1/2° and 82-1/2°) where changes
from one to another of equations (11) occur.
Compression
In figures 63 and 64 agreement of experimental values with theory is
illustrated by a comparison of graphs of equations (12) as well as equations (11) with plotted experimental data.
The values p , E , p , and . E to be substituted in these equations were
a a
taken from tables 2 through 7, supplement A, and, for a particular construction, they are the averages for all tests at the corresponding angles
of 0° and 90°. The value of p used is the same numerical value as
ab
Report No. 1328
-19-
that used in the case of tension. Values of E
were computed according
x
to equation (8), and 1.1.
according to equation (7). cr
have the
and cr
ab
ab
ba
same values used in equations (7) and (8) and obtained by use of equations (A) and (C).
As the legends in figures 63 and 64 indicate, each plotted point represents the average value from all tests at the indicated angle on specimens of the particular width.
The agreement between theoretical and experimental data is fair. The
data indicate that the influence of restraint is not as great in the case of
the proportional limit as it is for the modulus of elasticity.
The equations and the graphs show that when 0 = 0°, p = p
and when
x
a
= 90°, p = p
At the intermediate angles, there is a considerable
x
b
difference between values obtained from the two sets of equations (11
and 12).
Proportional Limit of Shear Stress
at an Angle to the Face Grain
It is assumed that the stress t will reach the proportional limit p
xy
xy
when one of the stresses t , t or t
reaches its proportional limit
as bb ab
p , por p and, therefore, there are three equations for the proa b
ab
portional limit in shear just as for the direct stresses. The one giving
the least value is the one to use. In the case of no lateral restraint
t = t = 0, and the three equations can be obtained directly from
xx yy
equations (2). They are:
Report No. 1328
-20-
Pa
P xy =2 sin 0 cos 0
-P b
P xy 2 sin 0 cos 0
P
(13)
-p ab
xy.
2
cos 2 - sin
0
-
In the case of complete lateral restraint e = e = 0, and a method
xx yy
similar to that used for the direct stresses can be used. In this method
t
xy
the expression - is substituted for e . The value of p. used in this
xy
xy
P-xy
substitution is obtained from equation (10). The equations obtained are:
P axy
P xy = E (1 - j ) sin 0 cos 0
a
ba
xy
p xy
E
Pb Xilxy
-o- ) sin 0 cos 0
ab
(14)
P ab Pxy
cos 2 0 - sin 2 0)
ab
I-k(
Equations (13) give minimum values of the proportional limit in shear,
but the values given by equations (14) need not be maximum values because of the difficulty in performing accurate shear tests.
As for the direct stresses, the proper proportional limits must be substituted in equations (13) and (14). The same rule holds that if p is
xy
Report No. 1328
-21-
positive, the right hand members must also be positive and vice versa.
The sign for p must be chosen correctly. This can be done by referxy
ring to figure 1 in which the shear stress t is shown positive. The
xy
symbols p a and pb have different meanings when they are positive than
they have when they are negative. When they are positive, tension
values should be used and when they are negative, compression values.
In equations (14) values for p, p , and p
are the same as in (13).
a b
ab
Also in (14), when p is a compression, values of E were taken from
a
a
the compression tests, and when p is a tension, values of E were
a
a
taken from the tension tests. A similar procedure was used in the case
of p and E . Values of oand o- were computed according to equaab
ba
tions (A) and (C). Proportional limit values for p and p in compresa
sion and tension were taken from supplements A and B, respectively.
Values of E and E were taken from the same supplements.
a
b
Figures 65 and 66 contain graphs of the theoretical curves and of experimental data taken from table 18 in supplement C. The agreement
between the two is reasonably good for all angles except -45°. The low
values at this angle can be attributed to the apparatus used. A similar
graph in figure 73 shows a much better correlation between the theoretical and experimental data. In this figure, the experimental data,
which are given in supplement C, are from tests in which the revised
apparatus was used. In comparing the two sets of graphs, it is evident
that at 0° the agreement between experimental and theory is good, no
matter which apparatus is used; but, to obtain a good evaluation at all
angles, the revised apparatus should be used.
Maximum Direct Stress at an Angle to the Face Grain
If t
is the only stress applied so that t = t
= 0, equation (6) yields
xx
yy xy
the maximum direct stress, that is, the maximum value of t = F .
xx
Report No. 1328
-22-
This is true of a long narrow test specimen which is free to change dimension in the Y direction. The equation reduces to:
4
4
2 0
2
cos 0 sin 0 sin
cos 0
2 F 2
2
F F
F2
x
a
b
ab
1
(15)
There are two values each of F and F one for tension and one for
a
b'
compression. If F is tensile, the tensile values of F
and F
are used.
x
a
b
If F is compressive, the compressive values are used.
x
Equation (15) gives the maximum direct stress if the test specimen is
not restrained in any way. If restraint does exist, then t and t are
xy
YY
not equal to zero and greater strengths than those given by the equation
will be obtained. The greatest possible strength predicted by equation
(6) can be obtained by equating to zero its partial derivatives with respect to t and with respect to t and solving these equations for t
yy
xy
yy
and t . These values are substituted in equation (6), and it is solved
xy
for the value of twhich is then the predicted maximum value of F .
xx'
x
The resulting equation is:
F
x
2
2
4
4
2 . 2
2
= F cos 0 + F sin 0 + 4 F
sin 0 cos 0 (16)
a
ab
As in equation (15) if F is tensile, tensile values of F and F are used,
xa
and if F is compressive, compressive values are used. It is interestx
ing that equations (15) and (16) are very similar to equations (7) and (8).
Tension
In evaluating F in tension according to equation (15), values of F and
a
Fb were taken from supplement B, tables 10 to 15, inclusive. F, the
a
Report No. 1328
-23-
maximum tensile stress parallel to face grain, is the average value of
all tests of a particular construction for 0 = 0° Similarly, Fthe
b'
maximum stress perpendicular to face grain, is the average value of
all tests for 0 = 90°. The term F is a shear term taken from shear
ab
tests at 0° and was taken from supplement C, table 18. As is pointed
out in supplement C, the shear strength which is obtained when 0 = 0°
with the original apparatus is about equal to that obtained with the revised apparatus. Thus, the values of F ab obtained by use of the original apparatus are taken as accurate and used in the determinations of
the theoretical curves.
Graphs showing the comparison between the theoretical and experimental data are presented in figures 67 and 68. Each point on a graph is
the average value for all tests of that construction for the particular
angle. The source of data is the same as that for F and F , supplea
b
ment B.
Of course, the theoretical and experimental data exactly coincide at 0°
and 90°. The correlation between the two at the intermediate angles is
good. What variations do exist can possibly be attributed to variations
in material.
Compression
In evaluating F
x in compression, both equations (15) and (16) were used,
since the conditions of restraint in the tests are not known. Values of
F and F in compression were taken from tables 2 through 7, supplea
b
ment A. The maximum stress parallel to face grain, F, is the average
a
value of all tests for a particular construction when 0 = 0°. Similarly,
F the maximum stress perpendicular to face grain, is the average of
b'
all tests when 0 = 90°. The term F retains the same values used for
ab
the tension case and was obtained from the shear tests of supplement C.
Figures 69 and 70 contain graphs showing the comparison between the
theoretical and experimental data. The latter were taken from the
same tables as F and F b, supplement A. As the legends on the graphs
a
Report No. 1328 -24-
indicate, each point representing experimental data for a particular
angle is the average value of all tests for one width of specimen at that
angle.
Inspection of the graphs reveals that for angles from 0° to 45° the theoretical and experimental data are in good agreement. Although, in general, the test results lie between the two curves with the narrow specimens nearest the bottom curve and the widest specimens nearest the top
curve, as would be expected, for angles from 45° to 90° this is not true.
The agreement for angles from 45° to 90° is not quite so good; the test
values seem, in general, to run a little higher than the theoretical
curves.
Maximum Shear Stress at Any Angle to the Face Grain
= 0 and equation (14) yields
= t
xx yy
the maximum shear stress, that is, the maximum value of t = F .
xy
xy
The equation reduces to:
If t
X y
is the only stress applied, t 1
r
.2
+ 1
2 20+1
sin 2 0
c os
2
2
2
F
F
L. a
J
xy Fab b
1
(1 7)
If the shear stress F xy induces a tensile stress in the direction of the
face grain, the tensile value of F and the compression value of F
b
a
should be used. If the stress in the direction of the face grain is compressive, then the compressive value of F and the tensile value of F
a
should be used.
Equation (17) gives the maximum shear stress if the test specimen is
are
not restrained in any way. If restraint does exist, then t and t
xx
YY
not equal to zero and greater shear strengths can be obtained. The
greatest possible shear strength predicted by equation (6) can be obtained in a manner similar to that employed in the case of direct stress.
and t . The following
The differentiations are taken with respect
xy
totyy
equation results:
Report No. 1328
-25-
F
2
xy
= F
2
ab
2
2
cos 2 0 + 1
+ F 2 sin 2 0
4 a
(18)
The meaning of the symbols F a and F b in this equation are determined
in an identical manner to that used for equation (17).
The similarity of equations (17) and (18) to equations (9) and (10) is
interesting.
Figures 71 and 72 contain graphs of the theoretical curves and the experimental data. The tension values of F and F
and the compression
a
b
values of F and F used in the determination of the theoretical curve
a
are the same as those substituted in equations (15) and (16), respectively,
and were obtained from supplements A and B. The experimental data
were taken from table 18, supplement C. It is evidenced by these graphs
that the agreement between the two is reasonably good. When 0 is -45°,
however, the experimental data fall well below the theoretical curves.
This variation is due primarily to the type of apparatus used for these
tests (see supplement C).
Data and theoretical curves from the tests, previously referred to, in
which the revised apparatus was used, are plotted in figure 73. In this
case, good agreement was obtained, indicating that the revised apparatus is much the better.
When 0 = 0° (equations 17 and 18) both the theoretical curves and experimental data must coincide exactly. Furthermore, the graphs of shearing strength in figures 71 and 73 for five-ply plywood show a good agreement for shear strength at 0° as obtained from tests with the two different sets of apparatus.
Report No. 1328 -26-
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Figure 54.--Forces and deformations in a plywood prism under external
forces acting in the plane of the plywood at an angle to the face grain.
'?O4 F
Figure 55.--Theoretical and experimental values of modulus of elasticity
in tension. Experimental data from tests on plywood made of 1/16-inch
Figure 56.--Theoretical and experimental values of modulus of elasticity
in tension. Experimental data from tests on plywood made of 1/32-inch
veneers.
7c,86 F
Figure 57.--Theoretical and experimental values of modulus of elasticity
in compression. Experimental data from tests on plywood made of 1/16inch veneers.
67087 F