TENSILE PROPERTIES Of GLASS-FABRI
LAMINATES WITH LAMINATIONS
ORIENTED IN ANY WAY
LOAN COPT
PLEASE RETURN TO:
No. 1853
1
•
Wood Engineering Research
Forest Products Laboratory
Madison, Wisconsin 53705
November 19515
Information Reviewed and Reaffirmed 1960
This Report is One of a Series
Issued in Cooperation with the
ANC-17 PANEL ON PLASTICS FOR AIRCRAFT
of the Departments of the
AIR FORCE, NAVY, AND COMMERCE
1111111111
1iN1
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simell11111111 H1111illill i llai
UNIFED STATES DEPARTMENT OF AGRICULTURE
FOREST PRODUCTS LABORATORY
FOREST SERVICE
MADISON 5. WISCONSIN
°operation with the University of Wisconsin
TENSILE PROPtETIES OF GLASS-FABRIC LAMINATES
WITH LAMINATIONS ORIENTED IN ANY WAY•
By
E. C. O. ERICKSON, Engineer
and
C. B. NORRIS, Engineer
Forest Products Laboratory,? Forest Service
U. S. Department of Agriculture
Abstract
A mathematical analysis of the elastic properties of glass-fabric-base
laminates in the plane of the laminate is presented. This analysis provides a theoretical means of determining the mechanical properties at
any angle in the plane of the laminate, based on the orientation and
basic properties of the individual orthotropic laminations. Comparisons
between computed and experimental. verification values in tension are
presented for two laminates composed of the same fabric and for one laminate combining three different fabrics.
Introduction
More efficient means of utilizing present-day glass-fabric-base laminates in structural applications for aircraft are now available. Previous studies have indicated that strength properties may be varied by
varying the orientation of the laminations, or by combining laminations
-This progress report is one of a series (ANC-17, Item 54-1) prepared and
distributed by the Forest Products Laboratory under U. S. Navy Bureau
of Aeronautics Order No. NAer 01610 and U. S. Air Force No. DO(33-616)
53-200 Amend. A2(55-295). Results here reported are preliminary and
may be revised as additional data become available.
?Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
Report No. 1853
-1-
Agriculture-Madison
of differing mechanical properties within the laminate. The purpose of
this report is to present and experimentally verify a mathematical analysis of the elastic and strength properties of a laminate made of laminations oriented in any desired way by testing three laminates, one of
which is orthotropic and the other two aeolotropic.
Each of six test panels made by the Forest Products Laboratory for this
study were laminated from 38-inch squares of finish 1141 fabric. Three
of these panels were parallel laminated and were tested to provide data
upon which the computations could be based. The laminations in the other
three panels were oriented in a way that seemed likely to best illustrate
the usefulness of the analysis.
The load-deformation curves obtained by test presented's choice of elastic moduli. Four such choices were made for each specimen tested Whenever such choices were possible. The values obtained by these choices
from the control panels were substituted in the equations of the mathematical analysis to obtain computed values comparable to those obtained
from tests of the other panels.
Materials and Fabrication
Materials
Three types of glass fabrics were used in'this study, namely 181-114,
143-114, and 162-114. Type 181-114 is a satin-weave fabric of approximately equal strength in the warp and fill direction. Type 143-114 is a
unidirectional weave fabric with a warp-to-fill strength ratio of about
8 to 1. The 162-114 fabric has a warp-to-fill strength ratio of about 3
to 2. The laminating resin, identified as batch No. 884, was a hightemperature-setting, low-viscosity resin of the polyester (styrene-alkyd)
type, known as laminating resin 2.
2Finish 114 fabrics were approved for use in this study in order to make
use of materials already on hand. These fabrics will not cOnform to
the wet-condition requirements of glass fabrics now specified for
aircraft laminates, but will conform to the standard dry-condition
requirements of Specification MIL-P-8013. As such, they are considered satisfactory fbr these experimental tests made solely to verify
the mathematical analysis of laminates with random orientation oflaminations. Since the study involves only tests made in the dry
condition and comparisons of test results with computed results, the
validity of the conclusion is in no way affected by the use of fabrics
with this finish.'
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Fabrication
Six test panels, each approximately 1/8 inch by 36 by 36 inches,*were
made at essentially the same time, using a fresh batch of the same resin.
The fabric and resin were laid up between cellophane-covered cauls. Each
sheet of fabric was grasped at its four corners and carefully tensioned
to bring the fill threads at 90° to the warp weave before it was finally
positioned on the assembly caul. A paperboard template considerably
larger than the cauls was placed under the bottom caul before assembly.
Several groups of parallel grid lines had been scored upon the template,
parallel to the several angles of orientation desired in the test panels.
The template enabled two,assembly men to orient the warp direction of
each sheet as nearly parallel to the desired direction with respect to
the chosen axis of reference of each panel as it was possible to aline
them by eye, before lowering the sheet into position upon the previously
positioned adjacent sheet and freshly spread resin. A considerable effort was made during layup to get a good distribution of resin, thoroughly wet all threads, and work out the air entrapped between plies. Minor
flaws, such as surface wrinkles, were present in some panels. The specimens were so chosen, however, that there were no noticeable imperfections
in the critical sections.
Each panel was cured in a hot press for 20 minutes at 220° F., followed
by 70 minutes at 250° F. Curing pressures, resin content, and general'
information on each panel are presented in table 1.
Two of three panels (Nos. 4 and 5) designed for comparison with computed
properties were composed of the same fabric, whereas three different
fabrics were combined in panel No. 6. Panel No. 4 consisted of 12 plies
of 143-114 fabric, laid up with the warp direction of 6 parallel alternate laminations at an angle of 30° to the warp direction of 6 parallel
adjacent laminations. Thus the natural axis of the laminate (which was
chosen to be the axis of reference) fell midway between the natural axis
of each parallel group of six laminations.
Panel 5 consisted of 13 laminations of 143-114 fabric with the warp direction of three groups of laminations' parallel and oriented as follows:
The laminations were numbered consecutively from 1 to 13; laminations
Nos. 1, 2, 4, 5, 7, 9, 10, 12, and 13 were laid up with the warp direction parallel to the axis of reference; laminations Nos. 3 and 11 making
an angle of 110°; and laminations Nos. 6 and 8 making an angle of 140°
with the same axis. The angles were measured positively in a counterclockwise direction from the arbitrary axis of reference.
The 12 laminations of panel 6 included 3 of fabric 162-114, 4 of 143-114,
and 5 of 181-114. The warp direction of each fabric was parallel and
was oriented as follows: Laminations 1, 6, and 12 were of 162-114 fabric
Report No.
185 3
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and ware laid up with their warp direction making an angle of 0° to an
axis of reference; laminations 3, 5, 8, and 10 were of 143-114 fabric
making an angle of 10 0 ; and laminations 2, 4, 7, 9, and 11 were of 181114 fabric making an angle of 140° to the same axis. The angles were
measured positively in a counterclockwise direction.
Three panels for test determinations of basic properties of each fabric
were parallel laminated with the warp direction of all laminations parallel to the axis of reference. Panels Nos. 1 and 2 consisted of 12
laminations each of fabrics 143-114 and 181-114, respectively. Panel
No. 3 consisted of eight laminations of 1627114 fabric.
Test Specimens
The location and direction of test specimens with respect to the trimmed
edges, warp direction of laminations, and axis of reference of the panels are indicated in the cutting diagrams, figures 1, 2, and 3.
Nine tensile specimens were taken parallel, 9 perpendicular, and 9 at
45° to the warp direction from each of the three parallel-laminated control panels (fig. 1). The axis of reference was taken parallel to the
warp direction, so that these directions are referred to as 0 0 , 90°, and
45°. Six tensile specimens from among those having the 0° warp direction and 6 from among those having the 90° warp direction (numbered 1,
3, 4, 6, 7, and 9) were used for tensile tests, and the 3 remaining 0°
and 90° specimens (numbered 2, 5, and 8) were used for the determination
of Poisson's ratios. Three compression specimens were taken from the 0°
and 90° directions of each of the control panels (fig. 1).
Three tension specimens were taken from panels 4, 5, and 6 in each of
the several directions, as shown in figures 2 and 3.
Each tensile specimen was 16 inches long by 1-1/2 inches wide before it
was shaped. Each was reduced at the center of its length to a minimum
section 0.8 inch wide and 2-1/2 inches long, which was connected to the
maximum end sections by circular arcs of 20-inch radius, tangent to the
minimum center section. Tension specimens were cut to approximate size
on a bandsaw, and finished to the desired shape and curvature by use of
an emery wheel mounted on a shaper head.
Compression specimens 1 inch wide and 4 inches long were cut to size
with a 1/8-inch emery wheel rotated at 1,770 revolutions per minute in
the arbor of a variable-speed table saw. Square, smooth edges were obtained by this method of cutting.
Report No. 1853
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All
and
der
but
specimens were conditioned to approximately constant weight at 75° F.
50 percent relative humidity before test. Specimens were tested uncontrolled conditions of 75° F. and 64 percent relative humidity,
exposed to these conditions for as short a time as possible.
Test Methods
Tension Tests
All tensile-type specimens were held in Templin-type grips and tested in
a hydraulic testing machine (fig. 4). Load was applied at a head speed
of 0.035 inch per minute. The first tensile specimen of each testdirection group was tested directly to failure in one continuous operation. The remaining tension specimens in each group were first loaded
to a load almost equal to the secondary proportional limit load as determined from the load-deformation curve of the first specimen of the group.
Specimens were then unloaded back to the initial load point, after which
they were immediately reloaded under a continually increasing load until
failure occurred. Load deformation data were recorded at convenient increments of load during the initial loading, the unloading, and the reloading operations. This loading cycle was used on all tensile specimens unless otherwise noted.
Data for the determination of Poisson's ratios were obtained from
tensile-type specimens held in Templin grips. Each specimen was loaded
and unloaded 4 times in one continuous operation at a rate of 0.035 inch
per minute; The first 2 loading runs were carried up to a load of about
three-fourths of the secondary proportional limit load of the material,
as previously determined from the initial tensile tests, before unloading. The third and fourth loading runs were continued to the secondary
proportional limit load before unloading. Once testing was begun, the
specimens were not completely unloaded until after the fourth and final
unloading run. Each of these specimens was loaded and unloaded in a continuous operation without any intentional rest or recovery periods.
Deformation data for specimens from panels 1 to 4 inclusive were obtained
parallel to the applied load only. Strains were measured in the stress
direction across a 2-inch gage length with a pair of Marten's mirrors
reading to 0.00001 inch.
Strains for Poisson's ratio determinations from tensile-type specimens of
panels 1 to 3, inclusive, were measured with SR-4 metalectric strain
gages reading to 0.000001 inch. Gages of 1-inch and 1/2-inch gage length
were applied to each side of each specimen at the net section, the 1-inch
gages parallel and the 1/2-inch gages perpendicular to the direction of
stress.
Report No. 1853
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Strains of the experimental specimens from panels 5 and 6 were measured
with metalectric strain gages of the 45° Rosette-type gage (AR-7) having
three elements at 0°, 45 0 , and 90° to the first element. These gages
were applied in pairs directly opposite each other, one on each side of
the specimen, with the two 45° elements always in the same plane. No
particular difficulty was experienced in using these gages, except as fbllows: A few gages loosened before testing began and had to be reglued.
Approximately 1 specimen in each like group of 3 had the 45° elements
oriented at 90° to those of the 2 other specimens. This circumstance
necessitated the use of individual rather than average experimental values in comparing these 45° stress-strain ratios with computed values.
Compression Tests
Compression specimens were loaded on their squared ends and were restrained from buckling by an apparatus similar to that shown under Method
No. 1021.1 of Federal Specification L-P-406b for Plastics, Organic. Load
was applied ate head speed of 0.012 inch per minute, and strains were
measured across a 2-inch gage length with a pair of Marten's mirrors.
Method of Obtaining Experimental Data
Experimental load-deformation data obtained from each tensile test were
plotted at convenient scales, and curves Were drawn. Most of these
specimens exhibited the usual two straight-line sections common to
parallel-laminated tensile specimens at 0° and 90° to the warp direction
and normally referred to as initial and secondary. Only the 0° specimens of the unidirectional fabric 143-114 in panel 1 failed to develop a
definite second straight-line portion in all six test specimens. In all
other instances at 0° and 90°, a full complement of six initial values
of modulus of elasticity and stress at proportional limit, based on the
initial straight-line portion of the load-deformation curve, was obtained, together with six secondary values based on the second straightline portions.
Perhaps unique among tests of this kind was the procedure of recording
load-deformation data during each unloading, after the secondary proportional limit load was reached, and during the subsequent reloading to
failure. Each of these plots exhibited definite straight-line portions
and each such portion, beginning at the initial loading increment, was.
used to obtain the respective unloading and reloading values. Thus
there were 4 slopes recorded, 1 from each phase of the loading cycle.
These slopes are referred to as the initial, secondary, unloading, and
reloading slopes.
Report No. 1853
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Representative load-deformation curves for the 0 0 0 90°, and 45° tension
specimens from the 143-114 and 162-114 fabrics used in panels 1 and 3,
respectively, are shown in figures 5 through 10. The load-deformation
curve of the first specimen loaded directly to failure is shown, for
purposes of comparison, adjacent to a companion specimen whose initial
loading was interrupted shortly after a secondary proportional limit
load was reached. Secondary load-deformation phases were not obtainable
from the 45° specimens, so all of the 45° specimens were loaded directly
to failure. Load deformation curves of the 181-114 fabric in panel 2
were quite similar to the curves of the 162-114 fabric in panel 3, shown
in figures 8, 9, and 10.
A few representative load-deformation curves from tests of specimens
taken from panels 5 and 6 at various angles to the arbitrary axis of
reference are shown in figures 11 through 15. Load-deformation curves
obtained from each of the three elements of the Rosette-type strain gage
are shown in adjacent plots for ready comparison of the behavior characteristics ' of the 0° (vertical) element, the 90° (horizontal) element,
and the 45° element of the Rosette gages. The vertical element of the
Rosette gage was always parallel to the direction of applied stress.
In general, the 0° and 90° gage elements gave fairly consistent results
among similar specimens. Contrarywise, the strain data obtained from
the 45° element of the gages were very inconsistent. _A number of the
45° gages first registered tension and later compression during the same
loading phase. The 45° curves of specimens 150°-5-1 and 150°-5-30 shown
in figure 12, illustrate such a condition and make obvious the reason
why only the initial load-strain phase of the data could be used.
Similar load-deformation curves were plotted from data from the vertical
and horizontal gages of the 0° and 90° specimens tested for Poisson's
ratios. Only ratios of the slopes of the first 2 of 4 loading runs were
used in'calculating Poisson's ratios.
Presentation of Data
Experimental Data
Experimental values obtained from the tests are presented in tables 2
through 7.
Table 2 presents the results of tension tests with the stress applied
at an angle of 0°, 90°, and 45° to the direction of the warp of parallellaminated panels 1, 2, and 3. Maximum, minimum, and average values of
modulus of elasticity are given for each of the four phases of loadReport No. 1853
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deformation cycles, together with corresponding values of proportional
limit stress and ultimate strength. Values of the modulus of rigidity
-) computed by means of equaassociated with the 0° and 90° axes (G090
tions (14) of the Appendix and the experimental values in this table and
in table 4 are also given.
Table 3 presents the results of compression tests with stress applied in
the 0° and 90° directions with respect to the warp of the parallellaminated panels 1, 2, and 3. Individual and average values of modulus
of elasticity, stress at proportional limit, and ultimate strength are
given.
Table 4 presents individual and, average values of Poisson's ratios from
tensile-type tests with stress applied in the 0° and 90° directions with
respect to the warp of the laminations in panels 1, 2, and 3. As far as
possible, ratios were calculated for each of the 3 laminates in each of
the 4 phases 'of the loading cycle.
Table 5 presents the results of tensile tests of panel 5 with the stress
applied at various angles to an arbitrary axis of reference (fig. 3).
Values of ultimate strength and stress-strain ratios for individual
specimens are given. Values of stress-strain ratios in the 0°, 90 °, and
45° directions with-rrespect to the direction of applied stress are presented. Values are shown for as many of the 4 phases of the loading cycle as were possible from the data obtained.
Table 6 presents corresponding tensile test data from panel 6 in the
same form as presented in table 5.
Table 7 presents the results of tensile tests with stress applied at 0°,
90°, and 45° to the axis of reference of the orthotropic panel, No. 4
(fig. 2), . Presented also in this table are the corresponding theoretical values computed for comparison with the experimental values.
Computed Data
Computed values of strength and elastic moduli corresponding . to the experimental test values of panels 4, 5, and 6 are presented in tables 7
through 11. The computed values presented in these tables were obtained
in accordance with the mathematical analysis presented in the Appendix.
The elastic properties obtained from panels 1, 2, and 3 for each of the
4 phases of the loading cycle were substituted in the equations of the
Appendix to obtain corresponding computed values for the 4 phases of the
loading cycle for panels 4, 5, and 6. Using these values, 4 corresponding values for tensile strength were computed.
Report No. 1853
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The computed values of elastic moduli presented in table 7 for panel 4
were obtained in accordance with section 7 of the Appendix. Computed
strength values for this panel were determined by means of the equations
of section 6.
Tables 8 and 9 present the computed values of stress-strain ratios for
comparison with corresponding experimental values from specimens of panElls 5 and 6 respectively. Stress-strain ratios in the 0°, 90 0 0 and 45°
directions with respect to the lengths of the specimens cut at various
angles to the axis of reference are given for each of the 4 phases of
the loading cycle. Values of the ratios at both plus and minus 45° were
computed so that they could be compared with the particular experimental
values obtained. These stress-strain ratios were computed according to
section 5 of the Appendix.
Tables 10 and 11 present computed values of tensile strength for comparison with experimental strength values of specimens from panels 5 and 6.
These computed values were obtained in accordance with the equations
presented in section 6 of the Appendix. The elastic properties computed
from each of the 4 phases of the loading cycle were used; thus, 4 computed strength values were obtained. Section 6rof the Appendix yields a
strength value associated with each lamination 044he laminate. Thus,
the computed strength values for panel 5 arer,eachlan,average of 13 atiqh
values, and those for panel 6 are each an average of 12 such values.
These averages are weighted according to the relative thicknesses of the,
individual laminations.
Figures 16 through 21 show, with one exception, curves of computed
stress-strain ratios and corresponding experimental values for the four
phases of the loading cycles for specimens taken from panels 5 and 6.
In most of these figures, the computed values are connected by a smooth
curve.
The experimental values were taken from tables 5 and 6 and the computed
values from tables 8 and 9. Experimental values in the stress direction
(0°) and perpendicular to the stress direction (90°) are, in each case,
the average value for each group of 3 test specimens at the respective
test angles and loading-cycle phase. Experimental values at.45 0 are in
each case individual values, because the values measured at plus 45°
could not, of course, be averaged with those measured at minus 45°. The
plots of these values are divided into 2 halves. Stress-strain ratios
at plus 45° are plotted on the left half and those at minus 45° on the
right half.
Corresponding computed and experimental strength values for specimens
from panels 5 and 6. are presented in figure 22. Experimental values are
indicated by solid circles and computed values by open circles. A smooth
Report No. 1853
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line has been drawn through the plotted points representing computed values for the initial phase. Plotted points for each of the other loadingcycle phases are also shown clustered around each corresponding initial
value. The experimental strength values were taken from tables 5 and 6,
and the computed values from tables 10 and 11.
Analysis of Data
Inspection of table 7 and figures 16 through 22, which show the degree
of agreement between the theoretical and experimental values, leads to
the following observations.
The computed values of the stress-strain ratios for panel 4 agree very
well with the experimental values in all but the secondary phase of the
loading cycle, as shown in table 7. Thus if initial, unloading, or reloading values of elastic properties are used in the equations of the
mathematical analysis, reasonably good estimates of the related moduli
of elasticity of panel 4 are obtained.
Figure 16 shows similar data for panel 5 and figure 19 for panel 6.
Reasonable agreement between computed and experimental values is Indicated.
Figure 17 shows the stress-strain ratios associated with Poisson's ratios for panel 5. Good agreement between the computed and experimental
values is obtained only for the initial phase of the loading cycle. Only general agreement is obtained for the reloading phase and poor agreement for the other two phases.
Similar information is given for panel 6 in figure 20. Again good
agreement between the computed and experimental values is obtained only
for the'initial phase. Values for the secondary phase were not plotted
because they were too erratic. The computed values for the other two
phases give only rough, unconservative estimates of the experimental
values.
Figure 18 shows stress-strain ratios at plus and minus 45° for panel 5
and figure 21 those for panel 6. The computed curves for the secondary
phase of the loading cycle are quite different from the computed curves
for the other phases. The experimental values for this phase do not
agree well with the computed values. The agreement between the computed
and experimental values for the other phases seems reasonable when it
is remembered that individual rather than average values are plotted.
Figure 22 shows the tensile strength values for panels 5 and 6, respectively. Computations of tensile strength based on elastic properties
Report No. 1853
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obtained from each of the four phases lead to substantially the same results. The computed tensile strength values are slightly conservative.
Conclusions
1. The mathematical analysis of elastic properties is not applicable to
the secondary phase of the loading cycle.
2. The mathematical analysis for the elastic properties is in reasonable agreement with experimental values in the initial, unloading, .
and reloading phases. The best agreement was obtained in the initial phase.
3.
The Mathematical analysis provides an excellent estimate of tensile
strength regardless of the loading-cycle phase used in the determination of the elastic properties. The values obtained from the
secondary phase, however, are the least reliable.
Report No. 1853
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APPENDIX
Mathematical Analysis
The first four sections of this appendix are taken from the material in
Chapters 1, 2, 3, and 6 of "The Mathematical Theory of Elasticity," by
A. E. H. Love but are restricted to two dimensions. This restriction
greatly simplifies the mathematical expressions, so that they can be
written in algebraic forms without being unduly cumbersome.
The fifth section of this Appendix applies this theory to laminates made
of aeolotropic laminations and closes with the special case of orthotropic laminations.
The sixth section derives equations for the strength of laminates made
of orthotropic laminations -- more particularly, those laminations made
of fabric or veneer. This derivation is based on the previous sections
and on some previously reported work of the Forest Products Laboratory.
Sections 7 and 8 deal with the special cases of a laminate made of two
identical orthotropic laminations arbitrarily orientated with respect to
each other, and of a laminate made up of a number of different orthotropic laminations having their natural axes parallel to each other.
1. Transformation Equations for Stress and Strain
Consider a sheet of material subjected to uniformly distributed loads applied at its edges and acting in the plane of the sheet. The stresses
and strains in the sheet are referred to the orthogonal axes x and E or
to the orthogonal axes t and n, The x axis makes an angle 9 with the
axis, which is measured positively counterclockwise from the t axis to
related to the x) E
,
the x axis (fig.23). The stresses are f x ,
axes, andf,f 0 f related to the t l aaxes. Those with a.single.subscript are direct stresses acting in the direction of the axis indicated
by the subscript. Those with two subscripts are shear stresses associated
with the axes indicated by the subscripts. Similarly, the strains are ex,
eie ,e,e,ande.
_La
The two sets of stresses are related to each other by the equations:
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f = fx cos2 0 + f sin2 0 - 2f sift 9 cos 8
• Y
f = f sin2 0 + f cos 2 0 + 2fxy sin 9 cos 9
x
(1)
f = f sin 0 cos 0 - f sin 8 cos 0 + f (cos2 8 - sin2 0)
En
x
xy
or
f = f cos2 8+f
+ f sin2 8+2f
+ 2f sin 8 cos 9
x
tn
f = f sin 2 9 + f cos 2 8 - 2f sin 9 Cos 0
xy
Y
fxy = 4-f sin 8 cos 8 +
t
(2)
sin 8 cos 9 + f (cos 2 8 - sin2 0)
The two sets of strains are related to each other by the equations:
e = e cos 2 8 + e sin 2 9 - e sin 0 cos 9
x
y
xy
2
2
eTI = e sin 0 + e cos 9 + e sin 0 cos 8
x
y
xy
eEn= 2e xsin 0 cos 0 - 2e sin 9 . cos 0 + e (cos2 9 - sin2 6)
xy
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(3)
e x = e cop?
sig2 9 + e n sin 0 cos e
+ e •
e = e sin2e
Y
g
•
cos 2 9 e tn sin cos e
(4)
(cos 2 0 - sin 2 0)
e = -2 sin
et 9 cos 0 + 2e sin 0 cos 0 + e
xy
tn
Equations (1), (2), (3), and (4) are arrived at by geometrical considerations. Therefore, they are independent of the properties of the material
to which they are applied. Their derivations may be found in most any
standard text on the theory of elasticity.
2. The relations Between Stress
464.0.0tr4itn
According to the generalized Hooks' law, each strain is linearly related
to the three stresses, which results in 9 elastic properties for each
material. These 9 properties are not independent of each other because
of the existence of a strain-energy function. The most general relation
between strain and stress, in two dimensions, is given by:
f+r
f+rf
=r
12
n r 13 El
11 t
e
f f +r
f +r
23 tn
22 n
n = r12 t
e=r
tn
f+r
13 t
(5)
f+rf
23 n
33 tl
The inverse of this group of equations is:
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f =s e+s e+se
13 tl
12 1
11 t
g
f =s e+s e+se
n
12 t
22 n
23 gn
(61
f tl = 8 13 e t B 23 en 8 33 etn
These two sets of equations are related by:
s
- 2
9 22 33 B23
r
11
r 13
r
533
r - 61 3 82 3 - 6 12
g
s
8
12 23
- s
12
2
r = B 11 8 33 613
22
s
13 22
5
B 13 B 12 - B
r
23
2
33
=Ft
8
8
11 22 33
-8
11 23
8
(7)
2
ll B 23
-
eS
El
=
8
12 23 13
6111
- 8
5 22
512
2
8
12 33
6
2
s
13 22
The inverse of this group of equations is identical in form; that is:
s
11
2
r22
r
33
r23
-
R
) etc.
(8)
2
2
R =r r r -r r + 2r r r -r r
12 33
12 23 13
11 22 33
11 23
2
r, r
13 22
The meanings of some of the coefficients in equations (5) are made evident by letting two of the stresses be zero; thus:
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r = –
11 E ,
r
12
r = 1
22 E
E 3
3
n
- n. = - taL
E
t
r =
G
Where E and E are the moduli of elasticity in the Ej and Idirections,
G is the modulus of rigidity associated with the Ej and
a
axes and p
and p are Poisson's ratios of a contraction in the direction of the
second subscript to an elongation in the direction of the first subscript
and
due to a tensile stress in this last direction. The values of r
13
and r have to do with strain ratios involving shear strains.
23
3..
Transformation of the Stress-Strain Equations
may be transformed to refer to the lc, zaxes by use of
Equations
equations 1 and (4). The transformed equations are:
fxy
f +a
f +a
e =a
13
12 y
11 x
x
e
y
f y
f +a
f +a
23 x
22 y
12 x
(9)
= a
f + a f +a f
33 xy
13 x
23 y
e =a
xy
Which are related to equations (5) by:
Report No. 1853
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5
tu
0
H
.......
CD
cb
CO
-I-1
-1*
CI
4.-1
1
ID
xi
01
ID
k
k
+
+
cr.
.3)
CI
V
5
41
KN
K \
• r9
al
CD
Cb
02
CD
14
0
U
(Nlm
CD
••••n
8
4......
to
K\
H
Ctl
al
+
80
V
CD
i
r
r2
I
"Th.%
cb
W4
K114
I
CV 0
H
C.1
cb
cr.
N
CU
+3
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0
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g m
u
41 03
K \
CV
g
ClL
414
Ntn
to
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a)
N KN
+
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+
1
k
02
CD
tes
14" \
8
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0-1c%
K1
tO
►-1
k
K \
kal
+
.......
+
a)
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o
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k
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cd
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K\
K1
a)
to
0
C)
0
U
.--...
0-..„
1N
I
K1
1-1
k
......
+
k
K \
1
C1
+
01
r-I
C'
+
m
PI
4-1
01
CD
K\
k
8
0
I
to
a
.4
cb
CD
---:
K N
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ta
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a)
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03
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O
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0
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ci)
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C33
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4.71
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cb
alK
9
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CU
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CD
133
+
+
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K \
gi
14
0-sic%
a)
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alto
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co
K1
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uq
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+
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8
C)
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CO)
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NC \ 1
al
I
a)
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03
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C)
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It
N
k
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II
II
K\
H
N
acd
K\
UN
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CD
t‘L
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Fi
O
K
iz.
aL8
91
to
Cb
No
tO
0
C)
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k
.-4
\
+
+all
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m4.)
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....+
+
49
K \
ti
Nr-1
ri
. -1* .1
U3
1
+
(3)
cb
(X)
Fr\
o
0-1(N
K1
1.1
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49
to
K \
to
a)
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to
to
a)
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to
o
0-41c%
1
a
-$
nm
to
0
H
to
o)
a)
k
tO
1
C)
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al
._. .
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CV
14
k
.4
I
I
+
+
tr.
t
CI)
$4
....1
8
O
K1
Cti
k
11
14- \
^ N
8
1
C.)
t4
4-1
4-2
4-1
K\
.F4
al
......0
0
e-1
+
tp
.1
to
1
ul
4-1
03 N1
K \
k
1Cy
0CO
4-1
Cb
orn
to
i
CJ
tO
CD
.,9
0
,c1
4-)
0
+3
tO
K \
k
.41
al
IC1
'd
941
to
k
1
M
1-4
1
cm
P4
mg
In
teN
,1
05
DO
9-1
0
+
K1
K1
1.4
Cb
.0
0
11
\
add
45-4
I
These relations are obtained by letting the stresses associated with the
x, axes be zero except one; then, substituting this one stress in equations M I values of f , f and f are obtained. These values are substituted in equations–T5) to obtain the related strains. These strains
are then substituted in equations (4) (one equation at a time) so that
the ratios of the strains to the stresses associated with the x, axes
are obtained, and the values of a are obtained by reference to equations
(9).
3.
Equations (6) may be transformed in a similar manner by using equations
(1) and (4). The transformed equations are:
e +b
e y
e +b
f =b
13 x
12 y
11 x
x
e y
ey+ b
f =b
e +b
23 x
y
12 x
22
e y
e +b
f y= b
e +b
33 x
23 y
13 x
x
Which are related to equations
Report No. 1853
(6) by:
-19-
CD
K\
111
•9
to
CD
..--..
03
AL
cy
o
91
141
0 •
a)
CD
ta
CO
141
cD
to
to
e-
0.1
i
0
tb
a)
Na
4-1
rz4
Ckld
141
1-4
too
.1q
1+
+
K1
K
K1
M
. I
a)
to
A
Al
8
+
I,
u
1
ti)
P
CO
1-1
to
O
0
c;
14
141
to
r-1
to
II
11
M
.-I
P
al
.0
0
fal
o)
r4
•
•
•
r4
to
CD
to
01
a)
co
\
to
to
w
4
+
•
01
05
ICI
4
8
tr
tO
+
CO
tr%
tO
0
a)
in
cr•
O
PrI
0
IA
CD
4-1
M
to
1
1
81
•
a)
teN
CV
1-1
co
tt•
t41
CO
*
These relations are obtained by letting the strains associated with the
x, axes be zero except one, then substituting this one strain in equations (3), and so on.
4. Orthotropic Materials
All of the equations given up to this point are perfectly general, and apply to all materials. They indicate that the most general material has .
radial symmetry. If various additional symmetries are imposed, the coefficients of equations (5) assume particular values. If a material has an
axis of symmetry (a) it also has another axis of symmetry (2) at right
angles to a, because of the radial symmetry, and the material is called
orthotropic. It can be shown that, if equations (5) for such a material
are written for the a and axes, rather than for the arbitrary and
axes, the values of r and r are zero and equations (5) become:
1
e = Ea
a
-
1100 f
p
E
f
f
e =
13
E'
a
(13)
Ep
+
These equations may be transformed to refer to axes x and making angles
(measured positively counterclockwise , from the a axis to the x axis as
shown in figure23)with the a and E. axes by use of equations (9T, so that:
ex = all fx + a12 fy + a13 fxy
(14)
ey = al2 fx + a22 fy a23 fxy
f +
f +a
13 x
23 y
e y= a
x
Report No. 1853
f y
33 x
-22-
0 Ctl
4-I
to
1 I-11 C,S. 1
1 Hi C.T.1
-
The inverse of this group of equations (13) is obtained. by using equations
(6) and relations (8). They are:
E
fa
a
Ep
a e 4. c_c2
x ep
x a
E
+ x e
f - elx 00 ea
(16)
f = G0 e
af3
13 a43
= 1 - 11 •
p,
op Da
be transformed to refer to axes x and xby use of
equations (11). The transformed. equations are:
These equations may
f = b11 + b .e + b e
12 y
13 xy
x • 11 x
fy = b12
e
x
+ b22 e
y
+
b
(17)
e y
23 x
exy
e +b
e +b
=b
f
xy
13 x
23 y
33
Report No. 1853
-24-
•-1
1
1-7.—i
.371
tc, 8
to
O
u
•
C
s
M
4M
,9
'
at
I1
e
gt
N
•••••••rn
6
4
I
a)
0 lir
...?
aet
0,
i
+
e.
e
1—737-1
g
...:
6
c.)
el
fa
I
ti
al
•
4S
at..9
1
g
e.
ti
V
1
M6
4-)
8
rtl ti
9.
tb
t
g
.0.
+
e
6\
t
ir - I
8u
cp
44
.e.
42
44
tu
to
tess
a
N
l
.8
N
ni 0
u
4-I
at
4,
to
....wt.
O
0161
6
s
0
r=1
–1-
g
1
tS
Cfl
M
+
PA
I
ti
Wm.
wPI
L.L:_i
,4
It
pa
0
ctl.
J.
+
•
-1r
09-
6
i
8u
01
ti
rI1
I
-1
6
El
I
m
re\
+
2 Li-.
0
CO.
r-i I g
i-i I r4
r-41 g
II
II
II
II
A
,i3
/41
r--1
141
CU
8
P4
ct.r
r-i I g
.0
1..
4.9-1
-d
.....Li
L..rzi
II
ri
e
4-1
a
PI
1--1
I
A- ,to-1
c0
(1)
0 -1
FA
l_____J
:31
4) I
ci
+)
k
ID
k 0
M
re N
A
41)
Xi
P4
01)
r4
5. Laminates
Consider a laminate made up of n individual laminations. The properties
of each lamination, associated with certain orthogonal axes E l'. and n i are
assumed to be known. The individual laminations are orientated in the
laminate so that the axes E i make angles , 4 i with an arbitrarily chosen
with the values of * i measured positively counterclockwise from the
ti axis to the E axis, as shown in figure 23. Equations (5) and (6) hold
for each lamination, and it is assumed that a sufficient number of the
r's and Ws are known for each lamination so that they are all known because of relations (7) and' (8). Equations (11) can then be written for
each lamination thus:
axis
e+be
fE1 =b 111 e+b
131 En
E
121 n
ri
F
f
tni
b
e + b22i
121
=b
t
(1 9)
+ b_ e
23i En
e+be
e+b
231 n
331 tn
13i a
The values of the coefficients to each lamination are given by relations
are those asand
(12) by replacing ...0 by *i• The stresses f ^ i , f
ti,
a axes in the ith lamination.
1!
sociated with the
Because the laminations are cemented together in the unstrained condition,
the strains in equation (19) apply to all the laminations and to the laminate. The values of the stresses in the individual laminations vary from
lamination to lamination according to equations (19). The average stresses
in the laminate are given by:
i=h
1
ft =t– > t f ti
f =
1
i=n
i=n
t
f
1=1
1=1
f
ni
an
tif f i (20)
=
1=1
in which t is the thickness of the laminate and t 1 is the thickness of an
individual lamination.
Report No.
1853
-26-
Substituting equations (19) into equations (20):
+
t b
1 12i
t b
+
i 111 t
t b
1 131
t
e,
t b
+e
221 + tom
i 12i t
tb
(21)
231
e
f tn
tb
t 1 +
i 13i t
1
23i
+e
-
t
t
These equations are identical to equations
s 11 t
b 12i
b
tib111
(6)
b
1 331
with:
0 13
=–
b
131
(22)
b
=
22 t
i 22i
0
2
t b
251
533
t b
331
They can be put in the form of equations (5) by use of the relations (7).
These equations can then be transformed to any arbitrarily chosen 2,
axes, the x axis making an angle 9 with the axis (as shown in figure 23)
according to equations (9) and relations (10J. Thus the properties of the
laminate are known in any direction.
If the individual laminations are orthotropic, it is convenient to choose
and 1 parallel to the natural axes a and p i . The coefficients of
i
i
equations (19) are then given by relations (18) rather than the more general relations (12).
Report No.
1853
-27-
6. Strength of Laminates Made of Orthotropic Laminations
It is assumed that the elastic properties of the laminate associated with
the j, a axes have been computed (that is, values of r for use in equations (5) are known).
The laminate is subjected to the stresses f f and f (associated
x/
/
with the x l . z axes) which are held proportional to each other until failure takes place. The angle 0, is, measured positively counterclockwise
from the taxis to the x axis (fig.24). The strains associated with these
stresses are obtained from equations (9), using the values of 15 previously obtained $ in equations (10) to obtain values of a. These strains are
transformed to the t i) n i axes, the natural axes of the individual laminations, by means of equations (3) (fig. 24) 1 thus:
eai = e
e
Si
=e
x
x
i
/ + 0) - e sin (* + 0) cos (* + e)
+ 0) + e sin 2 k
cos2 ‘lif
1
y
xy
i
sin2 (* + 0) + e cos 2 (*i + 0) + e sin (*i + 0) cos (* i + 0)
Xy
(21)
eao i = 2e sin (* + 0) cos (* + 0) - le y sin (* + 0) cos (* + 0)
x
Fet2
+ e
+ 0)'- sin2 (* + 0)1
The stresses to which the individual lamination is subjected are found
from these strains by means of equations (16), thus:
f
od
al
1
E
e
4al
alpal
X
e0i
E
eoi
f = Pi °Pi e al -EL
01
X
f
api
= G
(25)
e
api api
Report No. 1853
-28-
Each individual lamination will fail when
2
al fai fpi _fti
Fai Fai F 2
2
F
2
f 2
a
Pi
In which Fai ,
F,
2
F
:4A
.
(26)
0#31
and
are the strengths of the individual lamina-
F021
tions associated with the stresses f f
al L ,
and f , each acting alone.
and F i have two values, one in tension and one
a1
in compression. If the associated stress is negative, the compressive
strength is used; if positive, the tensile strength is used. The strengths
have the same signs as the stresses, so that the ratios are always positive.
Each of the strengths F
Thus, values of f f , or f are.obtained at which the individual lami-
x_x.
nations will fail, one value for each lamination. The value of this stress
at which the laminate, as a whole, will fail is at least as great as the
least of its values associated with failure of the individual laminations,
and will lie between this least value and the average of the values associated with the individual laminations.
7.
A Laminate Made of Two Identical Orthotropic Laminations
Consider a laminate made of two identical orthotropic laminations placed
so that the angle between their natural axes a 1 and a2 is 20. Choose the
direction of the I axis so that it bisects this angle, thus 0 1 $ and
(fig. 25). In making the summations indicated by equations (22),
0 =
2
Norris, C. B., and McKinnon, P. F. Compression, Tension, and Shear Tests
on Yellowpoplar Plywood Panels of Sizes That Do Not Buckle With Tests
Made at Various Angles to the Face Grain. Forest Products Laboratory
Report No. 1328. 42 pp. ) Illus. 1946.
2Werren, Fred, and Norris, C. B. Directional Properties of Glass-FabricBase Plastic Laminate Panels of Sizes That Do Not Buckle. Forest
Products Laboratory Report No. 1803. 49 pp., Illus. 1949.
6
Com–Norris, Charles B. Strength of Orthotropic Materials Subjected to34
pp.,
bined Stresses. Forest Products Laboratory Report No.
Illus. 1955.
Report No. 1853
-29-
1816.
it will be noticed that in relations (18) a change in sign of • changes
the sign of b and b2 and does not change the sign of the other cceffiENE
cients. Thus
B12 =b 121
b 111
813 =
(27)
s
22
=
b 22i
s
s33 = b331
= 0
23
Equations (6) became
f f . = b lli e
:/3123 el
f =b
12i et
t
22i n
(28)
ftn = b331 etl
These equations show that the laminate is orthotropic and that the and
a axes are its natural axes. By comparison of equations (28) with equations (16):
E a
blli
E, g
b
= k
12i
-
1.15
b
b 22i= x
33i
G tn
(2 9)
Solving these equations for the elastic properties, or using relations (7):
b
E = b
f
111 b
111
2
121
221
b 121
=
is b
E1 = b
b 121
b
2
12i
22i b111
2
= 1
111
Report Ro. 1853
b
-30-
111
b
221
(30)
8. A. Laminate Made of a Number of Different Orthotropic
Laminations Either Cross or Parallel Laminated
A laminate is made of a number of orthotropic laminations placed with
their natural axes (al. ) parallel to each other. The natural axis (a) of
the laminate is parallel to those of the individual laminations. Equations (22) became:
t
al
s
i xi
12
= —
t
L41
µA
P
13
=0
(31)
= 1-
13 22
t
t
i
s 23
Xi
=
=0
33
t
t
i
060
i
Using relations (7):
)
E2
1
a
Eaa
ti
Xi
1
t
(
pai
x
t
E
ti
t
Eai
Ex
=
G
ofs
=
i
ti Eai
Xi
E
ai
t
(32)
ai
ti
1113a
Bai 2
t
Eai
ti
ix
13 a
E
ti G p
Gap i
s
Report No. 1853
-31-
1.-68
Table l.--Assembly
al properties of the
,glass-; fabric laminates. Resin 21 was used.
Panel: Fabric :Orientation: Total : Curing : Resin :Average:Specific: Barcol
No. :
: of warp : number :pressure:content: thick-: gravity:hardness
: direction :of lami:
of :ness of:
•
:nations
panel : panel :
P.s.i. :Percent:
1
: 143-114 : Parallel
12
:
8
•
2
: 181-114 :....do 12
:
30
8
:
30
:
Inch :
34.6 : 0.121 :
1.e0
•
•
•
:
36.1 :
.125 :
1.79
36.9 :
.117 :
1.76
6
•
•
•
:
61
•
:
65
:
61
6
•
3
4
5
6
: 162-114 :....do •
•
: 143-114 •
(2)....:
: 143-114 : (2)....:
.•
:Composite -1.: a...#(2) 12
13
12
:
:
..
:
9
10
:
36.6 :
•
•
:
•.
30
:
36.9 :
.
34.8:
.
.128
:
•
•
1.76
:
62
•
•
.137 :
1.79
:
61
1.80
.•
:
61
•.
.133 :
•.
:
-Resin 2 is a high-temperature-setting, low-viscosity, laminating resin of the
polyester (styrene-alkyd) type.
?Illustrated in figure 2.
2111ustrated in figure 3.
Panel 6 consisted of 4 laminations of 143-114, 3 laminations of 162-114, and
5 laminations of 181-114 fabric.
Report No. 1853
0
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Table 3.--Results of compression tests of parallel-laminated
glass-fabric panels 1,'2, and 3.
Panel:
No. :
Stress at
Fabric : Stress :Modulus of:
: direction) :elasticity:
:Proportional: Ultimate
:and specimen:
limit
number
: 1,000 p.s.i.:
143-114
0 ° -- 1-1
0°
1-2
0° -- 1-3
Average
4,559
4,506
4,570
4,545
1,477
1,612
11610
1,566
181-114 :
0° -- 2-1
0° -- 2-2
0 ° -- 2-3
Average
3,221
3,213
3,211
3,215
90° -- 2-1
90° -- 2-2
90° -- 2-3
3,104
3,141
3,030
3,092
Average
3
P.s.i.
:
P.s.i.
:
33,740
30,100
33,13o
32,320
:
:
:
:
54,460
53,46o
54,840
54,25o
..
9,850
:
•
.
8,940
:
22,340
22,040
.•
9,020
9,270
:
:
:
21,970
22,120
21,740
24,900
23,760
23,470
:
:
:
:
:
:.
:
:
:
41,000
43,420
40,240
41,550
.
•
.
•
:
:
.
.
•
•.
:
.
:
:
:
.
.•
14,900
13,980
17,120
15,330
.
.
.
.
162-114 :
3-1
3-2
o ° -- 3-3
o°
o°
Average
90° -- 3-1
90° -- 3-2
Average
3,021
3,031
2,946
2,999
2,225
2,228
2,192
2,215
340120
34,17o
33,460
33,920
•.
.
.•
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13,530
13,910
12,610
13,350
:
:
:
:
21,650
19,640
20,180
20,490
.
:
:
:
23,030
.
12,790
13,590
13,480
.
13,290
:
20,120
•
•
:
•
21,330
16,010
-Direction of applied stress with respect to the warp direction
of the laminations.
Report No. 1853
2
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Table 8.--Computed values of stress-strain ratios for tensile specimens of panel 5.
:
Phase of :
loading : cycle :
.
:
Stress-strain ratio value (1,000 p.s.i.)
Stress direction with respect to axis of reference -:
0°
30°
:
45° :
90°
60° :
:
120° : 135° : 150°
0° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
Initial
Secondary
Unloading
Reloading
:
:
:
:
3,511
3,342
3,56 0
3,524
:
:
:
:
1,829 : 1,403 : 1,296
2,717 : 2,147 : 1,643
1,822 : 1,330 : 1,138
1,321 : 1,149
1,794
:
:
:
:
1,672
1,315
1,296
1,346
:
:
:
:
2,022
1,996
1,919
1,926
:
:
:
:
2,120 : 2,464
2,833 : 3,568
2,290 : 2,775
2,233 : 2,679
90° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
Initial
Secondary
Unloading
Reloading
: -15,380 : -6,936 -4,633 : -4,714 : -15,380 : -6,936 : -4,633 : -4,714
9,833 : 7,883 : 10,904
9,833 : 7,883 : 10,904 : 159,944 :
: 159,944 :
: -24,914 : -20,428 : -9,241 : -7,494 : -24,914 : -20,428 : -9,241 : -7,494
-6,467
: -21,736 : -14,078 : -7,401 : -6,467 : -21,736 : -14,078 : -7,401
45° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
:
Initial
Secondary :
7,634 :
5,708 :
2,603 : 2,731 : 3,681 : Unloading :
6,646 :
Reloading :
6,762 :
2,302 :
2,225 : 2,027
2,253 : 2,110 : 2,474 : 3,339 : 2,396 2,036 :
5,986 :
6,168 : 11,257 : 74,278
2,861
7,245: 7,673 : 6,439
4,200 :
6,929 : 11,071 : 24,070
6,721: 11,125 : 29,275
4,488 :
-45° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
Initial
: 11,257 : 54,383 : 7,634 : 3,476
Secondary :
7,673 : 5,876 : 5,708 : 4,778
Unloading : 11,071 : 19,807 : 6,646 : 3,216
Reloading : 11,125 : 23,554 : 6,762: 3,20.9
•
Report No. 1853
•
:
:
:
:
•
2,731
2,396
2,027
2,110
:
5,311 : 5,986 : 5,548
: 2,152 : 2,861 : 4,615
3,050 : 4,200 : 5,396
:
3,340 : 4,488 : 5,419
:
Table 9.--22auted values of stress-strain ratios for tensile specimens of panel
value (1,000 p.s.i.)
Stress-strain ratio
Phase of :
loading :
cycle
Stress direction with respect to axis
0°
:
30°
4 5 °
:
60°
L.
:
of reference -120° :
90° :
135° :
150°
1,978
1,917
1,830
1,840
2,223
0° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
2,796 :
Secondary : 2,524':
Unloading : 2,798 : Reloading : 2,771 :
Initial
:
2,560
2,571
2,550
2,512
:
:
:
:
2,245
2,491
2,198
2,161
:
:
:
:
1,995 : 1,772 : 2,046 : 1,311 :
1,544 :
1,883
1,862 : 1,558 : 1,831
1,478
1,622
1,643
:
:
:
:
:
:
:
:
2,440
2,150
2,143
90° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
-7,759 : -7,762
:
- 7,762 :
-8,731 :
-18,587:
57,608:
33,764
-9,956
:
57,608:
33,764
9,956
:
-18,587
:
Secondary : -10,026
:
-10,303
:
-9,840
:
-9,674
:
-10,026
:
-10,303
Unloading : -9,e40 : -9,674 :
Initial
:
45°
Initial
:
9,757 :
Secondary : 6,026 : 9,204 :
Reloading : 9,112 : Unloading :
:
:
:
Reloading :
ANGLE BETWEEN
-9,690 :
: -9,583 : -9,690
:
STRAIN AND STRESS DIRECTIONS
5,552 :
6,378 :
5,177 :
4,670 :
3,956 :
4,056 :
5,056 :
4,023 :
4,270 : 4,670 : 2,628 : 3,956 :
3,445 : 4,056 : 3,487 : 4,023 :
:
:
:
:
7,098 :
7,703 :
9,611
6,566
6,796 :
9,200
9,123
4,149 :
2,219 :
4,240 :
2,443 :
3,219 :
3,307 :
3,337 :
3,441 :
5,453
5,815
4,937
4,963
6,737 :
-45° ANGLE BETWEEN STRAIN AND STRESS DIRECTIONS
7,098 : 11,252 :
9,757 :
7,239 : 4,240 :
5,604 :
Secondary : 7,703 :
10,463 :
Unloading : 6,796
Reloading : 6,737 : 10,451 :
6,026 :
9,204 :
7,251 : 2,443 :
6,961 : 3,337 :
Initial
:
9,112 :
6,794 : 3,441 :
•
Report No.
1853
4,608
3,482
3,856
3,940
Table 10.--Computed strength values for tensile specimens of panel 5.
Tensile strength
Angle of : Number of : Warp
loading :laminations: direction::
Loading-cycle phases -Initial : Secondary : Unloading : Reloading
P.s.i.
•.
0°
9
2
2
9
2
2
Total
45°
9
•
0°
.
42,927 :
41,388 :
42,122 :
:
:
:
6,097 :
2,080 .
3,857 :
47,325 •.
:
42,213
5,894
5,668
53,775
0°
140°
110°
:
:
4,053 :
53,077 :
.
13,098 :
5,696 :
6,152 :
53,970 :
1,817 :
6,354 .
2,806 •.
18,454 ..
11,683
2,922
11,996
2,596
Total
0°
9
2
140°
:
2
,010
110°
2
6o°
Total
90°
0°
9
140°
2
2
110°
Total
o°
120°
9
4,708
2
140°
2
110°
•
21,960
Total
135°
:
9
0°
140°
110°
2
2
Total
150°
9
2
2
Total
P.s.i.
140°
110°
0°
140°
110°
2
2
P.s.i.
:
:
,,
:
:
Total 30°
P.s.i.
.
.
2,101 :
17,016 :
.
•
•.
9,345 :
1,466 :
1,905 :
8,042 :
3,131 :
1,474 .
2,259
12,716 :
:
7,899 :
1,424 :
12,647 :
.
8,637 :
1,374 :
:
11,333 :
r
1,100 :
11,111 :
:
8,162 :
19,351 :
:
842 :
3,731 :
14,120 :
1,132 :
21,325 :
2,227
2
,
.
12,057 :
:
5,195 :
:
15,065 :
4,931
4,365
24,361
:
:
:
18,986
5,496 :
4,348 :
28,830 :
13,679
1,212
3,202
18,093
2,586
17,178
2,663
17,268
9,064
:
0°
140'
110°
9,294 :
.
:
:
:
:
9,157
2,036
2,008
13,331 •.
2,035
13,228
•
8,379
1,984 ..
1,820 •.
12,183 •.
.
10,470 :
2,120 •
2,461 :
15,051
.
14,419 :
3,528 :
5,195
8,302
1,853
1,900
12,055
9,860
2,188
2,699
14,747
13,860
3,829
5,210
23,142 :
22,899
11,901 :
2,366 :
8,155 :
15,1,38 ;
5,o56 :
4,596 :
22,422 :
.
14,025 :
12,090 :
24,790 :
.
17,583
6,79 3 :
15,167
5,126
4,524
24,817
3,428 :
:
28,496 :
.
29,543 :
:
17,886
6,486
4,167
28,539
-Panel 5 consisted of 13 laminations of 143-114 fabric with the warp direction of
9 laminations making an angle of 0° with the axis of reference, 2 an angle of
140°, and 2 an angle of 110°.
Report No. 1853
Table 11. Computed strength values for tensile specimens of panel 6.
Tensile strength
Angle of : Number of :• Warp
loading :laminations: direction=:
•
Loading-cycle phases --
: Initial : Secondary : Unloading : Reloading
P.s.i.
4
3
Total 30°
Total 45°
Total 6o°
Total 90°
Total 120°
Total 135°
Total 5
4
3
5
10°
•
o°
140°
10°
0°
140°
4
3
5
10°
0°
140°
4
3
5
10°
4
3
5
4
3
5
4
3
5
150° 4
3
5
Total o°
140°
14,871
o°
140°
10°
0°
140°
10°
o°
140°
:
:
P.s.i.
13,571 :
13,930
P.s.i.
:
:
8,901 :
11,086 :
16,598
41,338
14,769 :
34,756 :
10,454 :
16,675 :
40,70o :
12,167
8,497
14,979
35,643
4,908 :
9,746 :
9,492 :
8,514 :
16,056 :
34,316 :
9,869
•
17,964 :
•▪
32,364 :
3,417 :
8,784
7,278 :
13,088 :
29,150 •.
.•
6,260 :
6,632 :
12,060 :
24,952
10°
P.s.i.
:
.
4,287 :
5,804 :
11,344 :
21,435 :
4,634 :
6,209 :
10,782 :
21,625 :
.
5,7 82 :
6 , 573 :
10,763 :
23,118 :
10°
8,003 :
o°
140°
7,446 :
11,638 :
27,087 :
9,366 :
20,661 :
33,444 :
3,231 :
8,467
14,335 •
26,033
.
2,939 •
5,671 :
8,392
17,002 .
4,800 :
5,720 •
9,300 :
19,820 :
7,290 :
7,274 :
14,226 :
28,790 :
:
6,308 :
6,590 :
12,282 :
25,180 :
.
5,872 •
5,635 :
9,989 :
21,496 •.
6,006 •.
5,744 ••
10,168 :
21,918 :
.
6,239 :
6,188 :
10,425
16,491
40,846
10,249
8,355
15,923
34,527
7,630
7,119
14,067
28,816
6,365
6,510
12,222
25,097
5,490
5,709
10,100
21,299
5,729
5,822
10,211
21,762
6,273
:
:
:
:
11,135 :
23,562 :
23,502
3,352 :
9,134 :
18,435 :
30,921 :
7,016 :
7,251 :
12,670
26,937 •
7,360
7,195
12,419
26,974
3,475
6,820
12,784
23,079
6,198
11,031
-Panel 6 was of 12 laminations cons is ting of 4, 3, and 5 laminations of 143-114,
162-114, and 187-114 fabrics respe ctively, with the warp direction of each
fabric parallel and oriented with respect to an axis of reference as follows:
4 laminations of 143-114 at 10°, 3 laminations of 162-114 at 0°, and 5 laminations of 181-114 at 140°.
Report No. 1853
•
Figure 4. --Tensile test used in testing glass-fabric-laminate
specimens.
Z M 80078 F
SUBJECT LISTS OF PUBLICATIONS ISSUED BY THE
FOREST PRODUCTS LABORATORY
The following are obtainable free on request from the Director, Forest Products
Laboratory, Madison, Wisconsin 53705.
List of publications on
Box and Crate Construction
and Packaging Data
List of publications on
Mechanical Properties and
Structural Uses of Wood
and Wood Products
List of publications on
Chemistry of Wood and
Derived Products
List of publications on
Pulp and Paper
List of publications on
Fire Protection
List of publications on
Seasoning of Wood
List of publications on
Fungus and Insect Defects
in Forest Products
List of publications on
Structural Sandwich, Plastic
Laminates, and Wood-Base
Aircraft Components
List of publications on
Glue, Glued Products, and
Veneer
List of publications on
Thermal Properties of Wood
List of publications on
Growth, Structure, and
Identification of Wood
List of publications on
Logging, Milling, and Utilization of Timber Products
Partial list of publications for
Architects, Builders, Engineers,
and Retail Lumbermen
List of publications on
Wood Finishing
List of publications on
Wood Preservation
Partial list of publications for
Furniture Manufacturers,
Woodworkers, and Teachers
of Woodshop Practice
Note: Since Forest Products Laboratory publications are so varied in subject no single
list is issued. Instead a list is made for each Laboratory division. Twice a year,
December 31 and June 30, a list is made showing new reports for the previous
6 months. This is the only item sent regularly to the Laboratory's mailing list.
Anyone who has asked for and received the proper subject lists and who has had
his name placed on the mailing list can keep up to date on Forest Products Laboratory publications. Each subject list carries descriptions of all other subject
lists.