oticthree shear moduli: the in-plane shear modulus g12, and two transverse moduli g13 and g2 3....
TRANSCRIPT
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LR 29763 November 30, 198t
TRANSVERSE SHEAR STIFFNESS Of T300/5208 :h'lc· •
. ' GRAPHITE-EPOXY IN SIMPLE BENDING
By : F. E. Sandor
I LR 29763November 30,1981
TRANSVERSE SHEAR STIFFNESS OF T300/5208
GRAPHITE-EPOXY IN SIMPLE BENDING
By: P.E. Sandorff, Staff EngineerI Structures & Materials Laboratory
I Accession For*-iNTTS GRA&IDTIC TAB [IUnannounced LJ 2
Ava-ilability CodesjAvail and/or
Dist Special
waaLockheed-.California CompanyA Division of Lockheed CorporationBurbank, California 91520
DISTRIBUTION STATEMENT AApproved for public releasel
Distribution Unlimited-
LOCKHEED • CALIFORNIA COMPANYA DIVISION OF LOCKHEED AIRCRAFT CORPORATION
Ir -
I REPORT NO. LR 29763
DATE _November 30, 1981
MODEL ID
TITLE COPY NO. __
jTRANSVERSE SHEAR STIFFNESS OF T300/5208
L GRAPHITE-EPOXY IN SIMPLE BENDING
IREFERENCE 41-5707-3040
CONTRACT NUMBER(S)
i
PREPARED BY_P. E. Sandorff, Staff EngineerStructures Laboratory
I APPROVED BY \'V- 7 "iW. F. Dryden, Department Engineer
Structures & 4aterials Laboratory
APPROVED BY -
? Fairchild, Division Engineer
I. ggineering Laboratories
II
The information disclosed herein was originated by and is the property of the Lockheed Aircraft Corporation,
I and except for uses expressly granted to the United States Government, Lockheed reserves all patent,
proprietary, design, use, sole, manufacturing and reproduction rights thereto. Information contained in this
Ireport must not be used for sales promotion or advertising purposes.
REVISIONSREV. NO. DATE REV. 9 PAGES AFFECTED REMARKS
FORM 402-2
I LR 29763
!!
FOREWORD!
This report describes work accomplished during 1980 and 1981 in the program,
Experimental Mechanics. This program is funded by the Lockheed Independent
Research and Development Program, and administered by the Structures and
Materials Laboratory.
Acknowledgement is due K. N. Lauraitis and J. T. Ryder for their helpful
discussions, and to R. C. Young, F. L. Imera, R. LaForce and F. M. Pickel for
special care in specimen fabrication and testing.
I!
'WLockheed-CaMfOMA Company i
FOREWORD
ABSTRACT
INTRODUCTION
OBJECTIVE
APPROACH
SPECIMENS
EXPERIMENTAL PROCEDURE
Axial Tests
Flexure Tests
TEST RESULTS AND ANALYSIS
Axial Tests
Flexure Tests
TABLE OF CONTENTS
Approximate Method for Shear Calculations
Overall
CONCLUSIONS
REFERENCES
APPENDIX
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I ABSTRACT
I.Buckling and crippling of a structural element is governed by its elastic
Jproperties. Nevertheless, a complete definition of the elastic constants of agraphite/epoxy laminate, including the transverse shear moduli (important
under high compressive stress), has never been accomplished. For this pur-
pose, tension, compression, and bending tests were conducted on square-section
specimens cut 0-degrees and 90-degrees to the fiber axis from 64-ply unidirec-
tional T300/5208 laminate. Axial test specimens were instrumented with tee
gages to obtain data on extensioral moduli and Poisson's ratios. Bending
J deflections were measured with high p-ecision and analyzed using a rigorous
finite difference solution for stress distribution to derive the shear moduli
Ialong with the extensional moduli acting in bending. Redundant determinations
were obtained for the nine elastic constants of Hooke's law as generalized for
specially orthotropic materisl.
Extensional moduli obtained under axial load and in bending agreed with each
other to within one percent. A self-consistent set of properties was deter-
mined which were generally in accordance with published data, and which
f established the material as being transvers. y isotropie and conforming to the
linear theory of elasticity.. The value obtained for the shear modulus G12
(=G13) was 4500 MPa (0.65 Msi), some twenty percent lower than that generallyaccepted on the basis of 4450 tension test data. The value of the transverse
shear modulus G23 was 3650 MPa (0.53 Msi), in accordance with published re-
sults of ultrasonic tests.I
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II INTRODUCTION
In the design and analysis of metal aircraft structures which buckle under
compressive loads, and those which deform under transverse bending load. it is
seldom necessary to consider the effects of shearing deformation. The trans-
verse shear stiffness of graphite-epoxy composite laminate, however, is no
more than a fifth that of aluminum alloy, and the adverse effects of shearing
I defcrmation will require consideration in many practical applications of
advanced design. Reference 1, for example, indicates a reduction in com-
pressive buckling stress of fifty percent for stiffener elements designed for
high stress (width to thickness ratio of 10). An evaluation of the transverse
T shear modulus of this material to a reasonably high confidence level is there-fore necessary for applying it efficiently in aircraft construction.
IThe elastic properties of composite laminate are usually represented as
specially orthotropic. In this analytic model, unidirectional laminate has
J three shear moduli: the in-plane shear modulus G1 2 , and two transverse moduli
G13 and G2 3 . There are several direct, relatively simple test methods for
I evaluating G12, but very few for the transverse moduli. A selected biblio-
graphy is presented in References 2 through 15.
The most commonly used method for evaluating the in-plane shear modulus of
composite laminate is the + 450 tensile test (Reference 5). In this test, a
Utee or rosette strain gage provides bi-directional stress-strain data; two-
dimensional transformation of stresses and strains identifies shear stress vs.
shear strain in the plane of the laminate. While this test is concerned only
with response in the plane of the laminate, the material is usually assumed to
be transversely isotropic on the basis of its structure, in which case 013
=G 12. Investigations have demonstrated this method to be equivalent (probably
to the limitations of experimental control) to the cross-sandwich beam, thin
'SkLockheed-Califforna Comping"
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tube In torsion, ten-deg off axis, panel shear, and rail shear tests (Refer-
ences 3, 6, 8). Tests of T300/5208 graphite/epoxy of 0.62-0.67 fiber volume
fraction by the +J45 ° tension method have indicated G - 0.T4 - 0.80 Msi- 12 2
(References 9.18). This value is generally understood to be the initial slope
of the stress-strain relation. A possibly significant aspect of all the test
methods in this category is that they usually furnish decidedly non-linear
shear stress-strain curves (for example, see References 3, 6, 8, 9, and 16).
The complete set of material elastic constants can be evaluated from the speed
of acoustic wave propagation in ultrasonic tests of specimens of special
geometries. Such experiments indicate G12 and G13 to be about thirty percent
higher than obtained with the + 450 static tension test; Reference 7, for
example, gives 1.03 Mai for this property, with G23 0.527 Hal. Engineers
hesitate to use such values in design because a one-to-one relationship
between microscopic dynamic oscillations and macroscopic static deformation
has not been demonstrated for graphite-epoxy; indeed, even in isotropic
metals, the two techniques usually provide slightly different results.
Since one of the engineering uses of the shear modulus is to improve the
analytic prediction of buckling phenomena, a practical approach is to test
simple buckling specimens, determine the reductions in critical load due to
shear effects, and back-figure to derive applicable values of the shear
moduli. This method was used to analyze some 1800 column test data in
Reference 9. For T300/5208 laminate, this approach indicated room temperature
values of G13 = 0.30 Msi and G23 = 0.045 Hsi. Besides being unexpectedly low,
these values conflict with the assumption of transverse isotropy.
In the past few years the Air Force Materials Laboratory has developed a
direct test method for evaluating two of the shear moduli. This method
utilizes a half-inch diameter, six-inch long cyclinder cut 90 degrees to the
fiber direction from thick unidirectional laminate. Shear strain gages are
mounted on the surface at 0 degrees and 90 degrees to the fiber and the
cylinder is loaded in torsion (Reference X). Tests of T300/5208 of fiber
volume fraction 0.63 reported in Reference 0 3 ive values of 0.866 and 0.506
Msi for G12 and G2 3P respectively.
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I Deserving of mention for its simplicity and low cost, a method known as the
Iosipescu shear test has recently been applied to composite laminates
(Reference 15). This test uses a specimen only slightly larger and harder to
fabricate than that for the ASTM D 2344 short beam shear test, and it makes
material characterization possible for both shear stiffness and strength in
all three planes. The method must be confirmed by test results obtained under
loading conditions realistic for structural applications, since (as with most
methods) the geometry is unusual and the purity of the stress condition might
f be questioned.
A method that is appealing from a practical standpoint determines the shear
stiffness from simple flexure tests in accordance with accepted handbook-type
formulas for bending and shear deflection (References 10 and 14). Application
of this method in Reference 10 indicated G13 of 16-ply T300/5208 laminate to
be 0.40 Msi. This method suffers from the crudeness of the handbook approxi-
mation made for shear deflection, as well as from the difficulty of achieving
adequate experimental precision.
The basis for an accurate analysis of flexure test data was established, in
Reference 19, by the development of a computer-aided solution for the plane
stress representation of a simple 3-point beam of orthotropic material. This
solution provides a precise determination of the complete normal and shear
stress distribution, and so permits accurate prediction of the bending and the
shearing components of elastic deflection. This predictive calculation can be
applied to determine the values of the elastic moduli which best fit redundant
sets of deflection data obtained in flexure tests at different spans. The in-
vestigation described in this report was planned to make use of this approach,
in order to evaluate the elastic constants of graphite/epoxy laminate under
loading conditions representative of typical structural applications.
II
'-q&-LOCkheed3-California Company
LR 29763
OBJECTIVE
The express purpose of this investigation was to evaluate the transverseshearing stiffness of T300/5208 unidirectional laminate when subjected to
simple bending.
In the completion of this objective, answers to the following additional
questions were also sought:
o Which if any of the existing test methods for shear modulus
could be confirmed.
o What causes the disagreement among values obtained by different
test methods.
o Can the material be considered transversely isotropic or not.
o Is more fundamental study required on the nature of the material
and the elastic model used to represent its response.
",'Lockheed-CeinrU Co wi 4
ILR 29763I
APPROACH
IFour elastic moduli govern the plane stress or plane strain response of
orthotropic material. In the x-z plane these are the two extensional moduli
Exx and Ezz, the transverse shear modulus Gxz, and the Poisson's ratio Vxz.
1 The bending stiffness of a flexure specimen is a strong function of Exx, a
weak function of Gxz, and almost independent of Ezz and vxz" Therefore, two
flexure tests obtained on the same beam at different spans provide sufficient
information to evaluate Exx and Gxz to good precision. To do this, however,
it is necessary to relate observed deflections to the moduli through an
accurate representation of the stress distribution.
The two-dimensional stress distribution for an orthotropic beam can be solved
quickly and to any desired degree of accuracy by the computer-aided relaxation19
technique described in Reference ,. This is a rigorous finite difference
analysis which derives values of the stress function over a matrix of interior
points to suit a specified set of boundary conditions. Once the stress
function matrix is determined, stresses are readily computed, and the strains
and deflections follow upon selection of the moduli.
The investigation was planned to take advantage of conventional axial load
* tests to identify the extensional moduli and the Poisson's ratios of a sample
unidirectional laminate. Identical square section specimens cut at 0 degrees
j and 90 degrees to the fiber axis from 64-ply graphite-epoxy were used for all
tests. A number were tested under axial load; these were instrumented with
strain gages. Replicate specimens were tested in three-point bending, the
stiffness being determined by measurement of central deflection. Each bending
specimen provided data in two planes of loading under identical test
I conditions. The unreduced test data therefore furnished qualitative
information regarding transverse isotropy and the relative magnitudes of G13
and G2 3. The experimental approach is illustrated schematically in Figure 1.
-" Lockheed 5-COMM&Company
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0 1 2
2 3 2
3::00, I-mer 900, 2-Oir 900, 301ir
El1 .~ E33V23, V21, V3 l ~ l E3& P13 E2,P1(Plans strain? (Pane strss) (Pu. iirum) (Plane stru)
.3 4
22 1
900, 3-Dir goo, 1-DIr 00, 3-Dir 00, 2-Dir
E22, G23 E22.G 1 2 E11,613 Ell,6G12
Figure 1. -Test Plan Schemoatic
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I LR 29763
IDeflection measurements made in the flexure tests included deformation
occurring in the test fixture and Hertzian-like bearing deformation of the
surface of the beam specimen. These effects comprised an error that was
J significant for tests at very short span. To provide correctional data,
supplementary tests were conducted on specimens loaded as a block in bearing,
and on the test fixture with the specimen removed. The correction for bearing
deformation was derived from the block-bearing test data with the aid of two
additional computer-aided relaxation analyses, by which the Hertzian
deformation occurring in the block specimen was related to that occurring in
the beam.
Reduction of the flexure test data was achieved by applying the analysis for
beam deflection to each test span and condition, using a trial-and-correction
procedure to obtain values of the moduli which best fit the multiple test
points. The procedure is illustrated by the the flow chart of Figure 2.
7
.LR 29763
AXIAL FEUEFIXTUREBARNLOAD TESTS DEFLECTION DEFLECTICNTESTS TESTS TESTS
BEARIN RATE AT
OVERALL BFFL, RATEVS. SPAN H .ER.TZI ANALYSES
BRG, IDR.., RATESURF. TO SURF.
VS. BRG. LENGTH
1 ,LENGTH
FLXLAX' __________________
PRELIM..qILSSIANAVLYSES
, BRG, DEFt. RATE ]
SURF. TO MIDPLANE
ANALYSESh9
Figure 2 - Flow Chart, Test and Analysis Procedure
"rLockheed 8-&fr bCOPM
LR 29763
SPECIMENS
To furnish stock for test specimens, a 64-ply unidirectional panel was
fabricated of T300/5208 low-resin content (35 %) prepreg tape. The layup with
symmetrical bleeder plies was placed under vacuum in a press for the standard
cure cycle (45 minutes at 1329C followed by 2 hours at 1790C); no post-cure
was applied.
To remove resin-rich surface material, a grinding operation using water
coolant was performed on both surfaces of the panel. Square specimens of
ten-inch length were then cut 0 degrees and 90 degrees to the laminate axis.
Final grinding reduced all sections to a uniform 10.59 x 10.59 mm (0.269 x
0.269 inch). Section dimensions were measured to ;0.0001 inch at stations
spaced every inch along the length.
Photomicrography of two specimens after test showed a high degree of
homogeneity, although traces of the ply boundaries indicated irregular
distortions from the plane. Photomicrographs are presented in the Appendix.
Resin analysis performed per ASTM D3171 on samples taken from the finished
specimens indicated a laminate specific gravity of 1.61 and a fiber volume
fractic;, of 68.6 percent. The specimens were dried at 66 °C (150°F) in vacuo
for two months prior to test, a procedure estimated to reduce the moisture
content to less than 0.1 percent.
Three specimens of each orientation were used for axial tests to obtain
extensional properties at 0 degrees and 90 degrees to the fiber direction.
The problem of applying test load in the thickness (3) direction was met by
cutting short lengths from specimens tested under axial load and reassembling
the short sections by adhesive bonding to form 3-tier and 5-tier compression
specimens.
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LR 29763
Additionally, to obtain specimens for axial load in the 3-direction which
were geometrically similar to the others, cubical elements were cut from a
fourth 90-degree specimen, reoriented through 90 degrees, and bond-assembled
as indicated in Figure 1. These assembled specimens were intended for test
under axial tension, but bond failure prevented tension loading of useful
magnitude. One of these specimens, which incorporated seventeen cubicalelements, was salvaged for axial compression testing by squaring the ends.
Three additional specimens each, of 00 and 900 orientation, were selected for
flexure tests. When supported for beam tests the central load was always
applied at the specimen midstation, with excess specimen length extending
symmetrically beyond the beam supports. Maxiumum variation in measured
section dimensions over the test span for any of these specimens was less than
0.3 percent.
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IEXPERIMENTAL PROCEDUREI
j Axial Tests
Axial test specimens were instrumented with tee foil gages centrally mounted
on all four faces and tested in a universal testing machine. Gage output vs.
load data were automatically recorded and reduced with the Rye Canyon
j centralized digital data acquisition system, which has a demonstrated overall
accuracy better than 0.05% of selected range. Tension loading not exceeding
0.003 strain was first applied using Templin-type mechanical grips. The
specimens were then reduced to approximately 10 cm (4 inch) length, the endsI squared, and compression loads applied between flat, adjustable platens. In
compression tests of built-up specimens in the thickness (3) direction, the
element bearing the gages was always the central member of a stack of
identical and identically oriented elements, in order to minimize end effects.
j Representative stress-strain curves obtained in these tests are presented in
the Appendix.
Flexural Tests
Flexural tests were conducted in the special fixture shown in Figure 3. Load
was applied and reacted through hard steel surfaces of 3.175 mm (0.125 inch)
radius. The width of the loading nose matched that of the specimen.
Deflection was measured between the loading nose and the fixture base with aoclip gage (MTS 632.02B01). The Rye Canyon centralized digital data system was
used to acquire, record, and reduce load-deflee ion data. End-to-end
calibration of the deflection measurement and recording system against a
certified Templin extensometer calibrator indicated overall accuracy and
Ilinearity of + 1.3 micrometer (0.00005 inch) maximum deviation over a range of2.5 mm (0.10 inch).
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Tests were conducted at eight different spans on each specimen, providing
replicate data from three specimens for each of the four beam orientations.
Test span lengths had to be short enough so that shear deformation was large
compared with the expected error in measurement, yet not so short that the
I error caused by local bearing deformation became intolerable. Maximum applied
loading corresponded to a beam theory strain Mc/EI of about 0.004.
I Representative deflection vs. load curves obtained in these tests are
reproduced in the Appendix. Except for initial softness, which may be
attributed to bearing on asperities and possibly to slight accomodation in
torsion, load-deflection relations were quite linear.
I The measured load vs. deflection included load carried by the clip gage as
well as that on the test specimen. A test was conducted on the clip gage
j alone, measuring deflection with a dial gage, to evaluate this quantity. The
results, which are included in the Appendix, provide a correction which was
important for the longest test spans.
A supplementary test was performed to evaluate the error introduced into the
flexure test data by deformations which occurred in components of the test
fixture included between the gage points. For this test the loading nose was
brought into contact with the base of the fixture, and the load-deflection
characteristics recorded as before. The resulting curve is reproduced in the
Appendix. (These measurements provided only an approximation to the behavior
in the flexure test because some deformation occurred in the base under the3 loading nose instead of at the beam supports).
Tests were also conducted with load applied to representative specimens as in
the flexure tests but with the specimens resting directly on the flat base of
the test fixture, in order measure specimen bearing deformation. Deflection
vs. load curves obtained in these tests are included in the Appendix.
" ,Lockheed 13
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TEST RESULTS AND ANALYSIS
Axial Tests
Compliance data under axial stress were derived by manually evaluating the
slopes of the stress-strain curves furnished by the data processing system.
This is an operation that produces repeatable values within one half to one
percent if the slopes are truly linear. Linearity existed over the maJor
portion of the curve for all tests except those for tension in the fiber
direction, in which stiffness increased slightly with load. In these cases
secant data were obtained over several ranges.
Corrections for strain gage transverse sensitivity were applied after
evaluation of the slopes. Tabulations of the material compliance
coefficients, obtained by averaging data from back-to-back strain gages, are
recorded in the Appendix. Mean values and coefficients of variation are
presented in Table 1.
With only moderate exception, these data are self-consistent and fit the
usually assumed model which displays symmetric coupling effects (S12 2 S21,
$13 = S3 1' $23 = $32) and transverse isotropy (S22 = S33 S12 = S13). A set
of coefficients for this model, obtained by averaging all determinations
bearing on any particular value, is as follows (units of Ue/MPa (ue/psi)):
6.92 (0.048) - 2.2 (0.015) - 2.2 (0.015)
= -2.2 (0.015) 94 (0.65) - 46 (0.32)
i,j = 1,2,3 -2.2 (0.015) - 46 (0.32) 94 (0.65)
Corresponding values of the engineeering constants are listed in Table 2.
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TABLE 2 RESULTS OF AXIAL LOAD TESTS
Coef
n Value Var'n
%
Extensional Moduli:
Ell 14 21.0 Hsi 2.1
E22, E33 27 1.54 Msi 2.9
Poisson's Ratios:
v12, V13 14 [ 0.31 3.8
V2 3 , V32 13 0.49 7.3
v21' v3 1 (V 12E22/E11) 13 0.023 8.8
I
N5LWckheed 16
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Flexure Tests
Recorded test results were provided by computer in the form of load-deflection
curves, which were analyzed manually to obtain values of slope. The resultant
deflection rate data (pin./lb) and the corresponding range of loading over
which the behavior was essentially linear are listed in Table 3.
The correctional data also were reduced to terms of deflection rate. The
correction for clip gage load was incorporated by combining it with the
average deflection rate obtained from the three specimens tested under each of
the thirty-two test conditions. These results are included in Table 3.
Bearing deformation effects were non-linear in nature, and reduction of the
data produced the deflection rate vs. load relations presented in Figure 4. A
preliminary analysis of the flexure test data was required before the
correction for bearing deformation could be applied.
The finite difference relaxation technique of Reference 19, which was used in
analyzing the flexure test data, is based on the stress function relation for
orthotropic material:
a4 04+2 + K 24 -0
z ax4 ax22 X O 4inaxaz2 x z4
Here, for plane strain, 2( E- 2y/E )K -Y x y (2)
x E XX/2G XZ-v X -v ,Y v Z E YY/E (2)x x/2x xz y Vx~ y ]zi
E (1 - v 2- YEz ) (3)
K - K XX z y /E ZZ2 X E (1 - v 2
zz xy Eyy/EXX)
Plane stress is expressed by placing E =0.
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Li 29763
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S0 I I I I I I I0 40 80 120 160 200
j LOAD P(LB)
Figure 4 - Experimental Deflection Rates (Slopesof Recorded Load - Deflection Curves)
" ,Loclkheed 19-California COMOan
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The stress function solution was applied to treat the central portion of the
test specimen out to a distance 2h (twice the beam depth) from the point of
load application. Deflection of the remainder of the span was calculated by
beam theory. The net of point values was spaced h/1O in the spanwise (x)
direction and h/20 in the depth (z) direction. The model is shown
schematically in Figure 5. Preliminary studies showed that this analysis
provided the desired precision in deflection calculation.
Preliminary solutions of the stress function matrix were derived for each of
the flexure test conditions, using elastic constants determined in the axial
tests and assuming values of the shear moduli. The case of beam load in the
2-3 plane (obtained in the 90-degree beam with fibers oriented transverse to
the load) was analyzed as plane strain, the others as plane stress. Best-fit
values of E and G were then derived by placing
(dw test/dP) = a (dwb /dP) i + b(dw s/dP) (4)
Here dwb /dP and dw s/dP are deflection rates attributable to bending stress and
to shearing stress, respectively, calculated at the beam midplane; a and b
are error factors. Defining, for simplification,
dws /dP dw test IdP
x dWb/dP ;Y - db/dP (5)
Then
Yi = a + bxi (6)
and a and b may be derived by linear regression of y on x. If bending and
shear deformation are assumed to occur independently of each other, a and b
become correction factors for the moduli Exx and G xz. The error in this
asssumption becomes negligible if the correction is small, making iterative
improvement of the relaxation solution a practical measure.
IVLockheed-Ca/ft t/ Coman 20
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LR 29763
Correction of the flexure test data was then made for bearing deformations
occurring between the surface at which load was applied and the midplane of
the beam. Analysis was required to derive this correction quantity from
measurements taken surface-to-surface on a block which carried no bending
stress. General solutions for the two cases (viz., block in bearing supported
at the opposite surface, and section of a beam supported by moment and shear
at the ends) were obtained by extensions of the stress function relaxation
analysis described above which used closer grid spacing. For each of the
flexure test conditions, a set of specific solutions for deformation under the
two support conditions was derived for a range of values of the width over
which the central load was distributed. The block-bearing test data could
then be used to evaluate a load distribution parameter for each condition. By
assuming this parameter applied also to the beam-supported bearing problem,the applicable solution was identified and the bearing deformation occurring
between the loading nose and the beam midplane was evaluated.
This analytic procedure is considered only an approximation, which is
especially limited for short spans and high loads, but adequate as a
correction procedure. To facilitate the correction process, the analytic
solutions are arranged in nomographic fashion in Figure 6. Entry to the lower
set of curves with the block-bearing deflection rate, obtained from Figure 4
at the median of the load range of interest, lead via the upper curves to the
surface-to-midplane deflection rate in the flexure test. Data derived by this
method for the loading station, and (under half the beam load) for the support
station, were applied to correct the flexure data in Table 3. Also included
in Table 3 are corrections for deflection occurring in the fixture, taken from
Figure 4.
With the flexure test data refined, additional iterations of the solution for
Exx and Gxz were performed until best-fit values to the eight test points of
each set were obtained. Variation in the Poisson's ratios from the values
determined in the extensional tests was not investigated, other than to
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"0Lo-----
4* "dir
I 90w, I-dir
0 .01 .02 .03.0.0.0.7.8
Lengt under uniform bearing Wstre, Ip tin)
Figure 6. -Nomograph for Converting Block Bearing Deflection Data toSurface-to-Midplane Bearing Deflection of Beam
23
-io~Lockheed-Ca/?twT7A Con man
LR 29763
demonstrate the effect to be too small to utilize. The results, which provide
at least one and in some cases two determinations of the extensional and the
shear moduli, are presented in Table 4 and summarized in Table 5. Extensional
and shear moduli derived from beams of different fiber orientation are seen to
be in good agreement, and agreement of the two transverse shear moduli G12 and
G13 (from 00 and 900 beams, respectively) again confirms the material as being
transversely isotropic. Furthermore, tie transverse shear modulus G23 is in
almost exact agreement with the value Wxpected by transverse isotropy, namely,
G23 = E22 /2(1+ v 23) or .52 Msi.
The fit to the test data achieved by iterative refinement of the relaxation
solution is indicated in Figure 7. While this plot demonstrates that the
computation provides an exceptionally good representation of total beam
deflection, it is misleading with regard to the precision of the shear modulus
determination. This process is better characterized by the plot of final
values of the deflection rate ratios presented in Figure 8.
IWrLoCkheed-CVmNCHpy 24
--
--
--
-.-
:---
-->
..
.~~
...---
--.
,...
---.
~ ~ ~
-
n~
TABL
E 4
. A
NA
LY
SIS
OF
FLE
XURE
DAT
A
II T
est
Res
ul1
s
Rela
xati
on
S
olu
tio
ns
Best
Fi
t An
aly
si
s
Bea
m O
rien
t.
Dep
th
Wid
th
Sp
an
w'<
tst)
A
ssu
mp
tio
ns
·::'
b w
's
w'<
to
t>
X
< i
>
Y<
i>
0-
de
<J,3
-d
ir
0.2
69
5
0.2
69
1
1.0
20
8 1
.24
6
.81
8
.0
5.5
13
5
6.4
76
7 0
-deq
, 3
-d
i 1·
0.2
69
5
0.2
69
1
1.5
20
1
7
Pla
ne
Str
ess
6.0
0
10
.79
16
.8
1. 7
98
6
2.8
33
0
1"=0
.99
99
I O
-d
e<J,
3-d
ir
0.2
69
5
0.2
69
1
2.0
24
3
1 E
xx
=2
1.2
M
si
15.9
1
14
.80
30
.7
0.9
30
6
1. 9
48
7
a=
0.9
97
3 0
-de
<J,
3-
di
r 0
.26
95
0
.26
91
2
.52
0 50
.5
Ezz=
1.5
5
Msi
3
2.2
4
13
.74
51
.0
0.5
81
2
1 • '
56
65
b
=0
.99
65
O-
de<J
,3-
di
r 0
.26
95
0
.26
91
3
.02
0
78
.6
Gx
z=0
.65
Msi
57
.28
2
2 .•
71
80
.0
0.3
96
4 1
• 37
21
E1
1=
21
.26
O-
de
<J,3
-d
lr
0.2
69
5
0.
26
91
4.0
20
16
8 V
xz=
0.3
1
13
9.8
8
30
.69
1
70
.6
0.2
19
4
1 • 2
01
0 6
13
=0
.65
2
0-
de<
J,3
-di
r 0
.26
95
0
.26
91
5
.01
7
31
1
27
6.4
8
38
.64
31
5.
1 0
.13
98
1
.12
49
0
-de
<J,
3-
di
r 0
. 26
95
0
.26
91
6
.00
5
52
3 47
6.0
3
46
.44
52
2.5
0
.09
76
1
.09
87
O-d
e<J
,2-d
ir
0.:
:!6
91
0
.26
9j
1 .0
20
8
1 .2
4
.s.8
1 !3
.1
5.4
85
0 6
.4
43
8 O
-d
e<J
,2-
di
r 0
.26
91
0
.26
95
1
.52
0 1
8
Pla
ne
Str
es
s 6
.04
10
.79
16
.8
1. 7
85
2
2. 9
77
7
r=0
.99
91
O-d
e<J
,2-d
ir
0.2
691
0
.26
95
2
.02
4
31
Ex
x=
21
.1
Hsi
1
6.0
3
14
.80
3
0.7
0
.93
06
1. 9
48
7
a=
1.0
06
8 O
-d
e<J
,2-
dir
0
.26
91
0
.26
95
2
.52
0 5
0.5
E
zz=
1.5
5
Hs
i 3
2.4
6
18
.73
51
.2
0.5
77
1
1 .5
55
7
b=
1.0
01
0 0
-d
e<J
, 2-
d i 1
· 0
.26
91
0
.~695
3.0
20
' 78
.4
Gx
z=0
.65
H
si
57
.67
2
2.7
1 8
0.4
0
.39
37
1
. 35
94
E 1
1 =
2(:>
. 96
0-
de
9,
2-
d i 1
· 0
.26
91
0.2
69
5
'1.0
20
16
8
Vx
z=>
.3
1 1
40
.90
3
0.6
9 1
71
.6
0.2
17
8 1
.19
24
6
12
=0
.64
9
0-d
e<J
, 2-
d i 1
· 0
.2
69
1 0
.269
5
5.0
17
31
1 2
78
.42
3
8.6
4
31
7.
1 0
.1
31:1
8 1
.11
70
O
-d
eg
,2-d
ir
0.
26
91
0.2
69
5
6.0
05
5
25
4
79
.76
4
6.4
4
526
.2
0.0
96
8
1 .0
94
3
'.10-d
eg
,3
-d
ir
0.2
69
5
0.2
69
2
1.0
20
3
9.
0 P
lan
e
Strain
29
.30
9
.45
:59
.1
0.3
22
7
1 • 3
31
3
~
90
-de
<J,
3-
dir
0
.26
95
0
.26
92
1
.52
0 11
7.5
E
xx
=1
.55
M
$i
102
.49
1
4.4
5
11
6.9
0
.1
40
9
1.
14
64
r=
0.9
98
6
Ut
?O
-d
e<J
,3-
dir
0
.26
95
0
.26
92
2
.02
4
26
9
Ey
y:=
21.0
Hsi
246
.57
19
.4
7 2
66
.0
0.0
79
0
1.0
91
0
•=
1.0
15
1 9
0-
de
<J,3
-d
ir
0.2
69
5
0.2
69
2
2.5
20
5
11
E
zz=
1.5
5
Msi
4
80
.07
2
4.4
2
50
4.5
0
.05
09
1 .0
64
4 b
=0
.97
09
'JO-
de
<J, 3
-d
i r
0.2
69
5
0.2
69
2
3.0
20
8
71
Gx
z=
0.5
2
Hsi
8
30
.77
2
9.4
1 8
60
.2
0.0
35
4 1
.04
84
E
?2=
1 .5
27
9
0-d
e<J,3
-d
ir
0.2
69
5
0.2
69
2
4.0
20
2
02
0
Vx
z=0
.49
19
68
.82
39
.38
2
00
8.2
0
.02
00
1 .0
26
0
62
3=
0.5
36
9
0-
de
<J,3
-d
ir
0.2
69
5 0
.26
92
5
.01
7 3
97
0
Vx
y=
0.0
23
38
35
.11
4
9.3
3
38
84
.4
0.0
12
9
1. 0
35
2 'J
O-
de
<J,3
-d
ir
0.2
69
5
0.2
69
2
6.0
05
67
80
V
zy
=0
.02
3
65
85
.70
5
9.1
9 6
64
4.?
0
.00
90
1.0
29
5
90
-de
<J,
1-
d i 1
· 0
.26
82
0
.26
95
1
.02
0
39
.1
31
.39
7
.71
39
.1
0.2
45
6
1 .2
45
6
90
-d
e<J,
1-
d i r
0
.26
82
0
.26
95
1
.52
0 1
15
.3
Pla
ne S
tress
10
6.2
6
11
.72
1
18
.0
0.
11
03
1
• 08
51
r=
0.9
87
6
90
-d
eg
, 1-d
i 1·
0
.26
82
0
.26
95
2
.02
4
'267
E
x x
=L
5
5
Ms
i 2
53
.70
1
5.7
5 26
9.4
0
.06
21
1 .0
52
4
a=
0.9
99
0 'l
0-
de
<J,
1-
dir
0
. 26
82
0
.26
95
2
.52
0 5
10
Ezz
=21
.2
Hsi
49
2.1
6
19.
71
'51
1.9
0
.04
00
1
.03
62
b=
0.9
6G>5
9
0-
de9
, 1
-d
i l"
0.2
68
2 0
.26
95
3
.02
0 8
65
G
xz=
0.6
5
Ms
i 8
49
.1
4
23
.79
8
72
.8
0.0
27
9
1 • 0
18
7
E2
2=
1.5
5!;!
9
0-
de
<J,1
-d
ir
0.2
68
2
0.2
69
5
4.0
21)
20
30
V
xz
=0
.02
3
20
07
.98
3
1.6
8
20
39
.7
0.0
15
8
1 .0
110
612
=0
.67
7
'}0
-de
<J,
1-
d i r
0
.26
82
0.2
69
5
5.0
17
40
20
3
90
7.6
3
39
.63
39
47
.3
0.0
10
1
1. 0
28
8 9
0-
de
<J,1
-d
ir
0.2
68
2 0
.26
95
6.0
05
6
'(9
0
67
04
.71
47
.52
675
2.2
0
. 0
07
1 1
• 01
27
- Be a•
<:1 i
to;e
ns
ion
s
in
inch
es.
Def
lecti
on
rate
w
'=d
w/
dP
,pin
/lb
. w
'b=
ben
din
g co
Mp
on
en
t,
w's
=s
he
ar
co
mp
on
en
t.
Best
fit
an
alY
sis
b
Y
lin
ear
re<
Jre
ssio
n
of
Y o
n
X;
X=w
's/
w'b
; Y
=w
'tst
/w
'b;
Y=
a+bX
t"' ,.. N
\0
....,
0\ ""
"'L···
-.-·
LR 29763
TABLE 5. SUMMARY OF FLEXURE TEST RESULTS
(Extensional and Shear Moduli in Ms)
TestCondition 00, 3-dir 00, 2-dir 900, 3-dir 900, 1-dir
N 22 22 20 21 1Ell 21.26 20.96 - -
E22 - - 1.527 1.552
E33 ....
G12 - -- 0.677
G13 0.652 0649 - -
G23 - - 0.536
26
W LoCkh'"d-CaAfrbnf Copany
I - , . - . -.-.-. , - . . . -- Lit 29763
50 0 0 . ~..-
I3000 .- -V-
I ~2000 .
1000 3.. . . -
j 500 9f0 beams .-
I -,. .... ~ l - -dir ti~
I -300 - 3
. - 2
1 300 - / -AE 1 00 -a 3
3 9 0 beano3 3-dir .
1001
10~ - V50 - - , -- -I , - - ~ * 4 -T -
-30 4' 4.6.2 5 .i
Fiur 7. Coprio-o Computedin fr
I~~~~ dean Delfto test TestDat
27
lqwLockheed-Callfolnia Company
LR 29763
1.5 7
1.4 6
A/1.3 5 -
90-dig beams
1.2 4/
1.1 3
1.0 2 0
()ata from table 4Points represent experimenta dite450 lines represent perfect fit
0.9 1
0I I I0 1 2 3 4 5 6
0 0.1 0.2 0.3 0.4 0.5 0.6xi
Figure 8. - Deflection Rate Ratios Corresponding to BestFit Solutions for E and G
xx xz
28
-LoCkheed 2Cua~rori Conpany
I
LR 29763
I Approximate Method for Shear Calculations
jShear deflection occurring in beams is usually estimated by assigning a value
to the Timoshenko shear coefficient K in the relation,I
Wol-ff dx dx + 'fS dx (7)
Here the first term is the Bernoulli-Euler bending deflection found in
handbook formulas; it recognizes no restraint of section warping. The second
term is a "shear correction" which includes all other effects, presumed to
cause a shearing type of deflection. K is generally thought to be a section
property that varies only with section shape; various sources cite values
ranging from 2/3 to 1.0. Consideration of the shear stress destribution in
beams makes it clear that K may vary over the span (Reference 19).
Therefore, any value which is to be used outside of the integral as in (7)
must be specific not only for load and section but for span.
The results of the current study can be utilized to develop rational values of
K. Such data, obtained by introducing the appropriate values from Table 4
into (7) and solving for K, are presented in Figure 9. (For the 900 beams
loaded in the 3-direction, the value of E was taken as E 22/(1- 21 V12) in
J accordance with representation as plane strain.)
Reissner (Reference 20), and Nair and Reissner (Reference 21), working from
fundamental energy considerations, have established upper and lower bounds for
the shear deflection that occurs in a beam which does not deform locally at
the load and support stations. In the 90-degree beam loaded in the 1-
direction, the high fiber stiffness produces boundary conditions which are
closely represented by Reissner's model, and the resulting values of K, as
plotted in Figure 9, are seen to lie close to the prescribed bounds calculatedI" Lockheed
S-Califorw Commo 29
-~*--'-~ *'4~~- - '~~-- -
LR 29763.
1.0
0
0.9 ,
O-deg beams0.8
A
2.0
AA1.5
S.0 90-dug beams loaded
. in 3-direction
0.9
z 0.5
1.0
90-deg beams loadedin 1-direction
0.9o~ ... _____ ' - 0.8
Thor.upr8ndloe bouds (u 21)
I I I
04 8 12 16 20 24
Span to depth ratio L/h
Figure 9. - Variation of the Timoehenko Shear Coefficient
with Span and Direction of Load
30
"wLockIleed-Calbrmi. Coalp
I
~I
LR 29763
from Reference 21. When the load is transverse to the fiber, however, local
deformation at the loading station is significantly greater, and consequently
jK for very short spans is increased and may exceed 1.0.
These differences in the value of K are produced by variation in the way shear
is introduced into the beam, interacting with section constraint. Shear
stress distribution adjacent to the loading station for the 90-degree beam, L
1.02 inch, loaded in the 3- direction, is contrasted with that for load in
the 1- direction in Figure 10 (data from the relaxation solutions). The high
peak stress and distorted distribution associated with loading transverse to
the fibers tends to produce large section warping. The central section cannot
Iwarp, however, and adjustment to this condition reduces overall beam de-
flection. Thus, short beams with low E zz may appear to be stiffer in "shear"
I than those for which E zz is large.
Accuracy in the calculation of K is adversely affected by limitations in ac-
curacy of the current set of relaxation solutions. While these data have a
precision of better than 0.1 percent, they suffer a small error as a result of
the finite dimensions of the grid. Such errors have a greater effect on K
than on the shear modulus, because after Gxz has been evaluated, the relax-
j ation solution values enter once again in thewtotal term of Eq. (7). and in
this case without the benefit of multiple points. For cases important in
j design, further investigation is desirable to confirm the results and to
establish a broader basis for extrapolation to other materials.
IIII
" L ockheed 31-CAWO A COMPWy 3
k _ _ . . .. ... . . . . .. . . . ... . .. . . . . .. .. . ... .
LR 29763
+0.5
/ / / /
"\ / / /" 90-deg bum, L=1.02
\ "\ ; / loaded in 3-dir"\ i I (parallel to fibers)
0 0. 0.2 0. 0.4 I!\ \ ;.i*. i\ \ '
i ' \ - \ -\.' \ \ '
I \ \ \ , \"
" =0 0.1 0.2 0.3 0)4 0,8, \ \ \\
-0.5Central
loadstation /+0.5
/ / / */
.1 ; / ./ /
90-dig bum, L:1.02loaded in I -dir(transverse to fibers)
\~. if i ii ;; * i
'I1 ' !
I, \ \ \\
x0 0.1 0.2 0.3 0.4 0.8
*\ \. \ \ .-0.5-
I I * .. I I I I 1 I a i i a I I Ii
Shw srew -- I I- One ksl
Figure 10. - Calculated Shear Stress Distributions for
90-degreee Beams, L = 1.02-in
32
")Lockheed-CaWA0n7 CoiNpW.y
iI
LR 29763
I Overall
Where data obtained in the axial load tests and the flexure tests overlap,
only small differences are seen to exist, and these discrepancies probably
represent the accuracy of the test methods. By taking rounded values which
are consistent with each other and the indications of transverse isotropy, aS ndepen'dlently evaluat
complete set of the n elastic constantsAis obtained and presented in TableI 6.
.These data represent the overall results of replicate tests and various test
conditions. Scatter of test results with different test procedures is no more
jthan a few percent.
These results were derived by assuming linear, orthotropic elasticity, an
assumption justified by the structure of the material and the linearity
exhibited in the basic load-deflection data. The fact that the final results
provide a complete, self-consistent set of properties confirms this model and
appears to refute suggestions of non-linear elasticity relations found in the
literature (Reference 16 and 17, for example). However, the scope of the
current program with regard to stress range and condition is insufficient to
resolve this question. For the type of loading and the material tested,
simple c-thotropy provides a satisfactory model.IIII
w)'Lockheed-CaklO d Compan 33CAMOMM coff-
TABLE 6 OVERALL SUMMARY
Elastic Constants of T300/5208 Laminate As Tested
I PROPERTY VALUE
j(GPa) (Msi)
E1 1415. 21.0
E2 , E3 j 10.7 1.55
G 12 G 131.50 0.65
G3.70 0.52
v12 Iv13 03
v23 9v32 0.49
V2 1 O310.023
nockheeo-C*A~7hnb C1PWY 34
5 LR 29763
j CONCLUSIONS
A complete set of the nine elastic constants experimentally determined for a 64-
ply T300/5208 unidirectional laminate is presented in Table 6. These data are
based on multiple independent experimental determinations, with the overall
means of the test values modified slightly to obtain a consistent set.
The results agree with values cited in the literature, except in the case of thein-plane shear modulus G 12 The value obtained here (0.65Msi) is somewhat lower
than that expected from + 450 tension tests (0.80 Msi) and the Pagano torsion
test (0.87 Msi), and substantially lower than reported for ultrasonic tests
(1.03 Msi).
The results obtained confirm, to the limits of the experiment, that the elastic
stress-strain relations are linear and that the material is transverse.Lj iso-
tropic.
Calculations of the Timoshenko shear coefficient indicate a substantial varia-
tion with the orientation of the orthotropic axes of the beam and with the span.
In some cases the calculated values exceed by a considerable amount the recog-
nized bounds for a cantilever which has slightly different end conditions.
Boundary conditions are thus seen to have an unexpectedly strong influence on
the deformation attributable to shear. Such an effect could explain variations
in performance associated with different test procedures, and would increase the
5 problems of analysis and design.
I
I
W, Lockheed
LR 29763
REFERENCES
1. L. D. Fogg, M. Feng, and J. P. Pearson, "Design and Analysis Methods for
Composite Structures," in "Independent Research and Development Plan,
1980," Lockheed-California Company LR 29364, April, 1980, page 7-251.
2. L. B. Greszczuk, "Shear Modulus Determination of Isotropic and Composite
Materials," ASTh STP 460, pp. 140-149, 1969.
3. P. H. Pettit, "A Simplified Method of Determining the In-Plane Shear
Stress-Strain Response of Unidirectional Composites," ASTH STP 460, pp.
83-93, 1967.
4. J. M. Whitney and J. C. Halpin, "Analysis of Laminated Anisotropic Tubes
Under Combined Loading," J. Comp. Mat., Vol. 2, No. 3 (July 1968), pp.
360-367.
5. B. W. Rosen, "A Simple Procedure for Experimental Determination of the
Longitudinal Shear Modulus of Unidirectional Composites," J. Comp. Mat.,
Vol. 6 (1972), pp. 552-5541.
6. C. C. Chamis and J. H. Sinclair, "Ten-Deg Off-Axis Test for Shear
Properties in Fiber Composites," Experimental Mechanics, September, 1977,
pp. 339-346.
7. R. D. Kriz and W. W. Stinchcomb, "Elastic Moduli of Transversely Isotropic
Graphite Fibers and Their Composites," Experimental Mechanics, Vol. 19,
No. 2, February, 1979, pp 41-49.
8. G. Terry, "A Comparative Investigation of Some Methods Of"Jnidirectional,
In-plane Shear Characterization of Composite Materials," eCos0tes,
October, 1979, pp. 233-237.
I,4Lockheed 36-C Tforn Cmwoy
I
LR 29763
j 9. K. N. Lauraitis and P. E. Sandorff, "Effect of Environment on the
Compressive Strengths of Laminated Epoxy Matrix Composites,"
1 AFML-TR-79-4179, December, 1979.
i0. P. E. Sandorff, "Experimental Mechanics - 1978," Lockheed-California
Company LR 28938, December 14, 1979.
11. G. Stoeffler, "Determination of Torsion Strength and Shear Moduli of a
Multi-Layer Composite," J. Composite Materials, Vol. 14, April, 1980 pp.
95-110 April. 1980.
12. N. J. Pagano, "Shear Moduli of Orthotropic Composites," AFML-TR-79-164,
March, 1980.
13. M. Knight and N. J. Pagano, "The Determination of Interlaminar Moduli of
Graphite/Epoxy Composites," presented at Mechanics of Composites Review,
28-30 October 1981, Dayton, Ohio.
14. H. D. Wagner, S. Fischer, I. Roman, and G. Marom, "The Effect of Fiber
Content on the Simultaneous Determination of Young's and Shear Moduli of
Unidirectional Composites," Composites, October 1981, pp. 257-259.
15. D. E. Walrath and D. F. Adams, "The Iosipescu Shear Test as Applied to
Composite Materials," to be published in Experimental Mechanics, 1981.
j 16. H. T. Hahn and S. W. Tsai, "Non-Linear Elastic Behavior of Unidirectional
Composite Laminate," J. Composite Materials, Vol. 7, January, 1973, pP.
U 102-118.
-4LoLckh~e- CaI/fo# W ConpW&y 37
___________'________________________'______________-_-_____-_--__-____-_-_-.-_-_-
_....__ll I
LR 29763
17. J. G. Davis, Jr., "Compressive Strength of Fiber-Reinforced Composite
Materials," ASTM STP 580, American Society for Testing and Materlils,
1975, pp 364-377.
18. A. M. James, et al, "Advanced Manufacturing Development of a Composite
Empennage Component for L-1011 Aircraft," Quarterly Technical Report No.
7, NASA Contract NAS1-14000, 14 October, 1977 (Lockheed-California Company
LR 28325).
19. P. E. Sandorff, "Analysis of Saint-Venant Effects in Orthotropic Beams,"
Lockheed-California Company LR 29357, April 15, 1980.
20. E. Reissner, "Upper and Lower Bounds for Deflections of Laminated
Cantilever Beams Including the Effect of Transverse Shear Deformation," J.
Appl. Mech; Trans. ASME, Ser. E, Vol. 40, December, 1973, pp. 988-991.
21. S. Nair and E. Reissner, "Improved Upper and Lower Bounds for Deflections
of Orthotropic Cantilever Beams," International J. Solids and Structures,
Vol. 11, No. 9, September, 1975, pp. 960-971.
"qgLockheed-CAM7 Company 38
LR 29763
!APPENDIXI
PAGE
Photomicrographs of Sections of Test Specimens 40
Representative Extensional Stress Strain Curves
Tension in Fiber (1) Direction 41-42
Tension In-Plane Transverse to Fiber (2) Direction 43-46
Compression in Thickness (3) Direction 47-50
Representative Load vs. Deflection Curves in Flexure
00 Beam, Load in 1-2 Plane 51-530° Beam, Load in 1-3 Plane 54-56900 Beam, Load in 2-3 Plane 57-59
090 Beam, Load in 2-1 Plane 60-62
Representative Load-Deflection Curves in Block-Bearing 63-66
Load vs. Deflection Curve for Fixture Alone 67
Load-Deflection Characteristics of Clip Gage 68
Table A-1 Compliance Coefficients, Tension Tests in 1-Dir 69
Table A-2 Compliance Coefficients, Compression Tests in 1-Dir 70
Table A-3 Compliance coefficients, Tension Tests in 2-Dir 71
Table A-4 Compliance Coefficients, Compression Tests in 2-Dir 72
Table A-5 Compliance Coefficients, Compression Tests in 3-Dir 73
Representative Computer Print-outs of Relaxation Solutions 74-76I
39
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