NOTICETHIS DOCUMENT HAS BEEN REPRODUCED FROMMICROFICHE. ALTHOUGH IT IS RECOGNIZED THATCERTAIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASEDIN THE INTEREST OF MAKING AVAILABLE AS MUCH
INFORMATION AS POSSIBLE

College of Engineering
Virginia Polytec %ic Institute and State University

Blacksburg, VA 24061

March 1581

VPI-E- 81-4

Calculations of the Room-Temperature Shapes of
Unsymmetric Laminates

Michael W. Hyer1

Deportment of Engineering Science and Mechanics
NASA Cooperative Agreement NCCI-15 Interim Report No. 24
Prepared for: John G. Davis, Jr., HeadMaterials processing and Applications Branch
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665

lAssociate Professor of Engineering Science and Mechanics
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061

i^

Preface
The enclosed report is a copy of a paper submitted for publication

'i

t
3:

in the Journal of Composite Materials, The issuing of this report is
not intended to duplicate the Journal's publication effort. This reportis intended to accelerate the dissemenation of information resulting
from investigations of advances: composite materials.

Abstract

The cured shape of unsyiAmetric laminates do not always conform to
the predictions of classical lamination theory. Classical lamination
theory predicts the room-temperature shapes of all uosymmetric laminates
to be a Saddle. 'Experimental observations, however, indicate some
unsymmetrie laminates have cylindrical room-temperature shapes. In
addition, some unsymmotric laminates, exhibit two stable room-tempierature configurations, both cylindrical. This paper presents a theory
which explains these charo,;., teris tics. The theory is based on all, ezteasion of classical laminaxion theory which accounts for geometric nonlinearities. A Rayleigh-Utz approach to minimizing the total potential
energy is used to obcain quantitative information regarding the roomtemperature shapes of square T300/5208 [0 2 /90,2 1,x, and(04/904'T

graphite-

epoxy laminates. It is shown that, depending on the thickness of thelaminate and the length of the side of the square, the saddle shapeeonfiguration is actually unstable. For values of length and thickness
that render the saddle shape unstable, it is shown that two stablecylindrical shapes exist. The predictions of the theory are comparedwith existing experimental data.

PIntroduction
Most calculations used to predict the response of laminates tostatic, dynamic, or thermal loadings are based on what has become to be
known as classical lamination theory(l)r Ill, C3) .

This is a linear

theory and is based on thefollowing major assumptions;(l) the displacements are continuous throughout the laminate,(2)

the Kirchhoff hypothesis regarding undeformed normals is

assumed to be valid,(3)

the strain-displacement relationship is linear,(G) the material is linearly ele.atic, and(5) the through-the-thickness stresses are small in comparison to
the in-plane stresses.
The theory smears the individual lamina properties by integrating
the constitutive equations through the thickness of the laminate. As aresult of this integration, force and moment resultants are defined. In
addition, the well.-known A, D, and D matricies are defined. While thistheory is quite capable of accurately predicting static deflections,natural vibration frequencies and mode shapes, buckling loads and modeshapes, and thermal expansion coefficients of laminates, there arephysical situations for which the theory fails to predict the correct
answer. Two situations of note are:the inability of the theory to

predict the response of thicker laminates, and, its inability to explain

the behavior of laminates 'near edges. The former problem has been'studied by several investigators (ai), (5; and satisfactory corrections
to the theory have been obtained. The latter problem is now a classic
and has been studied by many individuals, at least for the case of the2straight free.-edge. A survey of the edge problem has been put forth in

There appears to be another situation for which classical lamination theory fails to give the correct answer. Specifically, it
appears the theory is unable to correctly predict the room-temperatureshapes of thin unsymmetric laminates. Byer [7] has documented the roomtemperature shapes of several families of unsymmetric laminates andfound that the room-temperature shapes of some thin unsymmetric laminntes are closely approximated by right circular cylinders. In addition, some thin laminates have two room-temperature cylindrical. shapes.
These results are in contrast to the predictions of the classical ,hevary. The classical theory predicts the room-temperature shapes of all

unsymmetric laminates to be a saddle with unique (single-.alu€d) curva=
ture characteristics, Specifically, Hyer found that 100 x 100 mm and150 x 150 mm [0 2 /90 2 ] T `x300/5208 graphite-epoxy laminates cured to
become cylindrical at room temperature. In addition, they exhibited asnap-through or oil-canning phenomenon. The cylindrical shape could besnapped into another cylindrical, shape which had the same characteristics as the first shape. However, the second cylinder was oriented
perpendicular to the first cylinder and the curvature of the second
cylinder was of opposite sign. Hyer showed that thicker (say, 8-layer)
100 x 100 tuna unsymmetric laminates conformed to the predictions of thetheory.
To explain the behavior of unsymmetric laminates, it was assumed
that it would be necessary to incorporate a nonlinear effect into classical lamination theory. The existence of two room-temperature shapesaw

3

i.e. the two cylinders) essentially ruled out a linear extension to thetheory since a linear extension would lead to the prediction of aunique shape, albiet perhaps not a saddle shape, Furthermore, since theout-of-plane deflections of the :;.asymmetric laminates were on the orderof many laminate thicknesses, geometric non, nearities were felt to be
an important effect. Thu gs, in an effort to explain the behavior of
unsymmetria laminaten, classical lamination theory was extended toinclude geometric nonlinearities through the strain--displacement relationship. This extension was applied to the analysis of the COn/90naT,
n-1,2,..6 family of laminates. This family was chosen for study because
this class of laminates always exhibits a snap-through phenomenon, thereis data available for different thickness C0 n /90n 1 T laminates, and due.
to some of the A id , 13W and Dij terms being zero with this family, thealgebra associated with this family of laminates is simpl,ier than thealgebra associated with other families. The analysis of the roomtemperature shapes of this family of laminates is the subject of thispaper. The paper traces the development of the analysis which success-fully predicts the existence of two room-temperature cylindrical shapes,
and compares the predictions with the available data.

"problem Formulation

Since a problem .formulation which includes geometric nonlinearities
would result in nonlinear governing equations, it was assumed from the
beginning that obtaining a closed-form exact solution for the unsymmetric laminate problem would be difficult and not really necessary.
The occurence of the cylindrical, shape is so prevalant with thin laminates that it was hypothesised that one is dealing with a fundamental

i4phenomenon rather than some higher order effect. Thus any good mpproximato theory would reveal the mechanics of the problem. The pro!nlem isIdealized as follows. A curd l&;nnate (and uncured prepreg) is flat at
the elevated curing temperature, fig. la. Ao the laminate cools, it isassumed it is free from any external mechanical forces which produce net
work. it is Assumed the out-of-plane. deflections develop only becauseof the differences in thermal expansion properties of the individual
lamina. This idealization ignores the effects of any mechanical constraints the autoclaving and vacuum bagging process may exert on thelaminate. Upon cooling, the laminate deforms into one of the shapes
given by figs. lb , c and d. Of these shapes, the one that actuallyoccurs is the one associated with a minimum of the total potential
energy, The shape scenario given by figs, lb, c and d includes thesaddle shape observed for the thicker laminates and predicted by theclL,:,6ical theory, and the two possible cylindrical shapt.s observed forthe thinner laminates. It is assumed that in attaining these shapes,
geometric nonlinearities are important.
Since it is assumed external tractions are not important during thecooling process, the total potential energy, including the effects ofthermal expansion, is given by [$),
W - fWdVol,

(1)

Vol
(2)

W 2 C ijkl eij ekl $ijeijAT,
where W equals the strain energy density.

The Cijkl are the elastic

constants of the material and the 0 j are coefficients related to theelastic conatants and the coefficients ofthermal expansion of the mat-erial. Both the elastic properties and t' ze thermal expansion coeffi-

;kx

'1

5
cients are assumed to be temperature-indepondant. The e ij are thestrains in the material and AT is the temperature change in the material
due to cooling from curing. in eq. (2), since the problem is a planestress formulation, i< and j assumi the values 1 and 2. These values are
not directly related to the principal material directions of the lamina,
but rather, 1 and 2 are associated with the x and y directions of thelaminates (see fig. 1). 'Thus the following relations apply:

ell=,
2

()

Ox

2
^
^ O ,.^^,
e4)
C
22^`

y

Q

ell = ^xa - z

!W
a2w

►

( )

with

2
0
PO ax + 2 (aX)

(6)

CO a
y° + ^ (te
a >,
2 and
y
y
Y

x

2 (ay
e + 8x+ (a-X)% Y)).

(7)

(s)

As usual., z-0 is the midplane of the .laminate. The quantity u° is
the laminate midplane displacement in the x-direction, v° is the laminate midplane displacement in the y-direction, and w is the out-of-plane
displacement of the midplane.
»

Equations (6) - ( g ) represent the principal departure from classical lamination theory and include the usual, approximations associatedwith thin-plate theory when employing the nonlinear geometric effects inGthe strain-displacement relations. These approximations assume theelongation and shearing strains and the squares of the rotations are thesame order of magnitude and this order is small compared to unity (9)(101.
For a laminate the Gijkl 's can be related to the g ij 's, the reduced

stiffiivsses, and the 0 i j 's can be related to the & i j ' s and dcx , ay , and%y , the laminate thermal expansion coefficients in the x-y coordinate
system. Expanding eq. 2 yields

I41 2 Quell + ^)12e11e22+

2a 11 e 12 2 Q22e22(9)(^n'x + a 12 o'y )e 11AT - (&12"x + ^22 e y)e 22 AT .
(axy-0 for this family of laminates.) The problem has now been reduced
to one of finding the deformation a°, v°, and w as functions e)f x and ywhich, through eqs. (2) - (9), minimize eq. (1). As previously mentioned, approximate solutions to u°, v°, and w are sought. In seeking
realistic approximate solutions, two basic assumptions are made. ri'rst,
it is assumed that even in attaining the cylindrical shape, the midplane
elongatl.on strains, e* and e o , do not vary much from the linear prediction (i.e. eX and sy independent of x and y). second, it is assumed
that to the order of the nonlinearity considered here, the midplane
shear strains are negligible i.e. C* - 0. Since for [0 n /9011 1 lami-nates classical, lamination theory predicts C'to be zero and PX andC O to be constant, these two assumptions could be lumped into one bysaying it is assumed that even in attaining the cylindrical shapes,
magnitude of the midplane strains do not vary much from the predictionsof the classical, theory. However, for rationalizing the choice of the

Zfunctional form of thu approximate solutions, the two issues are`eanarated.
it is assumed w(x,y) is of the formw (x ,Y) "2 (ax 2 * bY 2 )(10)a and b being constants. With this functional form for w, both theclassical lamination solution, a - -b, and alther of the two cylindricalshapes, fig. lc and d, can be approximated. For fig. lc the solution
can be a # 0, b - 0 while for fig, Id the solution can be a - 0, b 0 0.
Using the kinematic assumptions rcgording the midplane strains, cx,y, and gxy, the approximate zviuMions for u° and v° are given by
3V*

abxy 2(x, y ) - dy - 9b 6 .... ab 4

(12)c and d being constants. Using eqs. (ll) and (12) in eqns. (6) - (8)yieldsOx

c - a4Y_(13)

ad.,abxx

(14)

Y

4

e000.
XY

(15)

Note that if it were not required to have exy° be zero, the third term ineach of eqs. (ll) and (12) would not need to be included and then thesecond term in eqs. (13) and (14) would not appear. However, with the(0n/90n ] T fa- ly, it is felt in-plane shear strains are impossible andthis is the prime factor in choosing the functional form for u° and v°.
'It this point a, b, a and d are considered as generalized coordi-

8nates and are to be datermin`d. The problem of finding a minimum fortil ^,otal potential energy, W, becomes a problem of finding solutions to
the values of a, b, c, and d so that the first variation of W is zero,
1.e.
SSW -

(^a)da + (ab)6b * ()dc + ()cSd 0(16}This variation is done. with all the assumptions in they Introduction
with the exception of (3).

Calculations Associated with the Solution

The x-y-z coordinate system in fig. 1 is assumed to be situatedsuch that at the elevated curing temperature the laminate is defined by
the region-Lx /2 < x < Lx/2
(17)

-Ly /2 < y < Ly/2
-h/2 < z < h/2 .

With these limits on the spatial variables and with the various material
properties involved, eq. (1) takes the form

1'x /2W•

fL y /jh/2

W(a,b,c,dvQijsax)ay,AT,x,y,z)dxdydz. (18)fx--T^x /2 y--Ly /2 ZW-h/2

The involved, but stra:.ight-forward, process of substituting eqs. (10)and (13) - (15) into eqs. (3) - (5), substituting these results into eq.
(9), performing the spatial integrations in eq. (18), and finally taking
the first variation, eq. (16), leads to an equation of the form;N - f 1 (a,b,c,d)6a + f 2 (a,b,e,d)db +f 3 (a,b,c,d)6c # f 4 (a,b, c,d)cSd = 0 .

9Equation ( 19) immediately leads to four equations;Clcb + C 2 ab 2 + 2C3 ab - 8 ll c +f1 ( a , b, c, d)

D11a - C4 cb + 20 5 0 2 - C6db +(20)
D 12 -» C7 db + C8 ab2 + C9b 2 +(L 2,/48)N Tb + MT + (0 / 48) N b w 0
f2(a,b,c,d) - -C l ac + C

2

2
a b + 2C 3a2 0 4 4c + 2C5 a 2 b + D12-

C 6 da - C 7 da + C 8 a2b + 2C 9 ab -T 22 d -D22

f 3 (a,b,c,d) " A IZ a

+ (L2 /48)NTa + (Lx/ 48)NT
A

(21)

+ MT
w 0

Crab hlla + Al2 d - C 4 ab - NT - 0 ,

(22)

R0 .

23)

12 - C6 ab - B22
f4 (a ' b,c,d) A+ A 22 - C 7 ab - N^

The constants C1 - C 9 are defined in the Appendix and A ij , Bij and Dij
have the familiar definitions associated with laminates. The otherdefinitions used in eqs. (20) - (23) are,h/2
INX - AT(Q ax + Ql2 ay )dz '

(24)

-h/2
h /2
t
Ny . AT ! (Ql2 ax + Q2 2 ay)dz ,.. 11/2(25)

h/2
Mx AT 1 (4 11
-h/2

ax

+ a12'y)zdz ,

(26)

t

10

h/2

My • ATj0(27)

12 % + g22 0ty) zdz_h/2
These quantities are immediately recognizable as the effective in -plane
thermal loads, NT and NT, and the effective thermal moments, MT and My.
It should be noted that when Lx-Ly-0, the coefficients C 1 through C 9 are
all zero and eqs. (20) - (23) reduce to the equations of classical lamination theory.
Solution of Equations, Numerica a.

Results

Solutions to eqs. (20) - (23) were obtained by solving eqs. (22) and(23) for c and d in terms of a and b and substituting these relations
into eqs. (20) and (21). Thus eqs. (20) and (21) become coupled cubic
equations for the quantiCites a and b. These two resulting equations
have the characteristics of being able to be reduced to a single cuL,:cequation for either a or b. Such an equation would have either one or
three real roots. This reduction approach was not used, however, andeqs. (20) and (21), in terms of a and b, were solved numerically. Solutions were obtained for several laminates using elastic and thermal
expansion properties of T300/5208 graphite-epoxy. It was assumed thecuring temperature of T300/5208 is 177° C (350° F) and that the laminates are cooled to a room temperature of 21° C (70° F). The material
properties used in the calculations were:E1 = 181 GPa (26.2 x 10 6 psi)
E210.3 GPa (1.49 x 106 psi)
v12 = 0.28
G12 - 7.17 GPa (1.04 x 10 6 psi)

ll

ot L

-0,106 x

10-6/° C (-0.059 x 10- 6 /° )
E

a2

6

w, 25.6 x 1 0" /° a (14.2 x 10 -6 /° F)

The elastic properties were taken from [2l while the thermal expansion
coefficients were taken from (111. Solutions were obtained for square
laminates (Lx-LyRL) ranging from 0 to 150 no ic,t length on a side. Two
thicknesses were considered, [0 2 /90 2 1 T and (0 4 /900 T .

Experimental data

were available for some checking of these two thickness cases. Figure 2
shows the characteristics of the predicted room-temperature shapes ofthe (0 2 /90 2 ] T laminate and fig. 3 shows the characteristics for the
thicker laminate. Each figure illustrates the effect of the size of theLaminate, L, on the room temperature shapes.

Immediately obvious from the figures is the existence of threepossible room-temperature shapes of the laminate if the, lengths of thesides are greater than some critical. value. For both 1<i^. ,,aates, at L-0
the room- temperature shape is the saddle predicted by classical laminaLion theory, a m-b. Th is solution is denoted by point A on figs. 2 and3. As the sides of the laminate increase in length, say to L=25 mm forthe [0 2 /90 2 1Tlaminate of fig. 2, the shape is still predicted to be asaddle but one which is shallower than the one predicted by the classical theory. As the lengths of the sides increase, the saddle shape isstill predicted to exist but it gets shallower and shallower. At someacritical length, the solution bifurcates. For the thinner laminate thecritical length is 35 mm while for the thicker laminate the critical•length is 71 mm. The bifurcation point is denoted as B on the figures.
For lengths greater than the critical length, three zoom temperatureshapes, each represented by a different branch on the figures, can12possibly exist. These branches are denoted as BC, BD, and BE on the
figures. Branch HD represents a continuation of thesaddle shape(a- -b) but the other two branches represent a radical departure from asaddle shape. Branch BC represents a shape which has

a large curvaturein the x-direction and practically no curvature in they-direction, fig.
le. On the other hand, branch BE represents a shape which has a large
curvature in the y-direction and very little curvature in the x- direcLion, fig. 1d. The shapes associated with these latter branches can beconsidered cylindrical because as the laminate gets larger, i.e. L
increases, the one curvature asymptotically approaches zero while theother curvature asymptotically approaches a non-zero constant value. As
seen from the figures, the latter two branches have certain symmetry
characteristics. These symmetry characteristics are such that for agiven length, the values of a and b associated with branch BC are equal,
respectively, to the values of -b and -a associated with branch BE.
Figures 2 and 3 show both laminates exibi,t similar behavior. There
are differences however. The two main differences are that the curvatures for the thicker laminate are less than the curvatures for thethinner laminate, and, the critical length for the thicker laminate is
greater. Thus, compared to a (0 2 /90 2 1 T laminate, a [0 4 /90 4 I T can bemade larger before the trl., phe-shape phenomenon occurs.

Stability of the Predicted Shapes

With multiple solutions to a nonlinear problem, the question arises
as to the stability of the various solutions. If any of the solutions
do not represent a stable solution, those solutions will not be physically realizable. Equating the first variation of the total potential

13energy to zero yields equilibrium positions of the laminate which either
maximize or minimize the total energy. For stable equilibrium, thetotal potential energy must be mimimized. Thus, for stable equilibrium
the second variation of the total potential energy, 6 2 W, must be posStability theory (123 for this discretized system,

itive definite.

stability of the r+q,!, -01br um positions for the laminate is possible if
and only if the following matrix of coefficients is positive definite:

af Iafl

af1

afl

Da

ab

ac

ad

af 2

af2

af2

af2

Da

ab

8c

ad

af3

af3

af3

af3

as

D

ac

ad

af4

af4

ac

ad

af
as

4

af
ab

4

(2s)

For a given equilibrium solution at a given length, e.g. the saddle
solution of the triple-valued solution at L=100 mm, each element of thematrix is evaluated numerically by substituting in the values of a, b,
c, and d corresponding to that solution. If each of the principal
minors of the matrix are positive definite, the matrix is positive
definite and the solution corresponds to a stable equilibrium solution.
Otherwise the solution corresponds to an unstable equilibrium solution.
Using this scheme for the solutions shown in figs. 2 and 3, it was foundthat the p addle solutions corresponding to the single-valued solutions,segments AB in the figures, were stable. On the other hand, the saddle
solutions corresponding to the triple-valued solutions, segments BC in
the figures, were unstable. The other two branches of the triple-,m^BH'

14
valued solutions, segments BD and AP, represent stable solutions.
Physically this means that for square laminates of the [02/902]T

family, if the length of the sides of the laminate exceed 35 mm, thesaddle shape does not exist. Instead, two cylindrically shaped equilibrium configurations exist. If

the laminate is from the thicker[04 /900 T family, the length of a side must exceed 71 mm before thesaddle-shape equilibruum configuration disappears and dual cylindricalshapes appear. The ,fact that two stable cylindrical equilibrium solutions are predicted to exist is felt to be significant since it correlates well with the the reported snap-through phenomena associated withthese types of Laminates.

Experimental Results

Shown on fig. 2 are two data points. These points correspond tothe curvatures of cylindrical (0 2 /902 ), laminates as measured by Hyer.
Figure 3 shows one data point. This point corresponds to the curvatureof a [+45 4 /-454 ] T saddle-shaped T300/5208 graphite-epoxy laminate asmeasured by Pagano and Hahn (13). The comparison between magnitudes ofthe predicted and experimentally measured curvatures is fair. More
importantly, however, the character of the measured shapes, i.e. cylindrical or saddle, compares well with the predictions. For the 150 x 150
mm Laminate shown in fig. 2, only the major curvature of 11 laminates
were measured in the original work and the average curvatureand therange are shown here arbitrarily as a y-direction curvature, b. These
curvatures could have just as easily been called an x-direction curvature. In this case the experimental data would have been associatedwith the variable a. Also the major curvature from the one 100 x 100 mm

15

specimen is also arbitrarily associated with the y- direction curvature.
For the data point in fig. 3, the curvature of the specimen was nevermeasured directly. The out-of-plane deflection across the diagonals of
a 63.5 mm (2.50 In.) square laminate was measured and the curvature wascomputed from this measure. Again this single curvature measurement wasarbitrarily associated with the variable b.
It should be noted in fig. 2 that the x--direction curvature for the
100 x 100 mm laminate was measured to be slightly negative. The theory
predicts this curvature to be slightly positive. The reason for thedescrepancy is not clear. However, it is not felt to be due to measuring error in the experimental determination of curvature. This
points needs further investigation.

Discussion
Despite the lack of large amounts of quantitative experimental datato comp are with the complete range of the numerical predictions, theresults of the work reported here are quite encouraging. First, thetheoretical calculations predict the disappearance of the saddle shape,a phenomenon observed by many investigators. Second, the snap-throughor appearance of two stable equilbrium states is predicted, anothk+r
phenomenon observed by investigators. Finally, the transition fromstable single-valued saddle solutions to stable cylindrical solutions is•linked with a size effect. The figures show that both the thickness of
the laminate and the length of the side determine whether the cylindrical shape exists or whether the saddle shape exists. This investigator, as well as others, has felt a size effect exists in unsymmetriclaminates and the model put forth here lends some credence to that

lei

notion.

While the predictions presented here exhibit all the importantfeatures associated with unsy ►mnetrie laminates, several comments are inorder before closure. first, the solution presented here is a one-term

Galerkin, or Rayleigh-Ritz, solution. Thus the solution is, as with all
one-term Galerkin solutions, over-constrained. This deficiency can be
remedied by using more terms, and hence more generalized coordinates, inthe assumed functional forms for a°, v°, and w. Modifying the current
approach this way would probably change the numerical values associatedwith each solution branch of figs. 2 and 3 but not the main featuresof bifurcation and triple-solution. Second, the effects of moistureabsorption, viscoelastic relaxation, or any other mechanism that alters
the intrcrral stress state of the laminate is felt to be i,nnurtant forlaminat«s with lengths near the critical length. Alteration of theinternal stress state most likely influences the numerical value of thecritical length. Thus a laminate sized just above the critical length
could, with time, actually be sized just below the critical length due
to any of the above mentioned time-dependent effects. Related to thisis the fact that unsymmetric laminates sized near

their critical length

could exhibit "strange" behavior, requiring practically no force to snap
them from one shape to another. More than likely, these multiple shapes

would be some barely .stable combination of shallow cylinders And shallow
saddles.
Finally, the analysis presented here is based on symmetric curing(symmetric about the z=0 plane) of the laminate and, as noted above, thelack of any effects to alter the internal stress state over and above

aq,.

t

17

that due to temperature change during the cool-down from curing. The
result is that for lengths greater than the critical length, two similiar shapes are possible. Each of these shapes has the same possi-

bility of actually occurring. however, any external perturbation whichis unsymmetric with respect to the midplane will cause one or the otherof the two possible cylindrical shapes to be favored. Such a perturbation could be unsymmetric curing (due to unsymmetric cooling) , unsym-

metric moisture aboorbtiono or any other process over which our control
is limited. Thus in reality it may require, for example, more force tosnap the cylinder of fig. le to the cylinder of fig. Id than it does tomake the reverse snap.

I I' ll, -- ­­­­_,^__ -1

Appendix
C1 0

A11L2/48

C3 w B

C2 . A11 L4/1280

L2/48

C4 . Al21X/48

Cy

Al2L2L2/2304

C6 - Al2Ly/48

C

A22L2/48

C8 " A22L4/1280
0 9 = 822L2/48

The coefficients A 11 , A1V A22 , 11 11 and B22 have the familiar definitions associated with classical lamination theory.

Werances

1. Jones, R. M., Mechanics of Composite Mate, McGraw-Hill DoakCo., Now Yoa:k, 1975, Ch. 4.
2. Tsai, S. W. and Hahn, H. T., Introduction to Comosite'Materials,
1°80, Chs, 4,4Technomic Publishing Co., Inc,, Westport, CT 0,
0688

5 and 6.
3. Agarwal, B. D. and Broutman, L.. J., Anal sia3 and Performance ofFiberS2myosites, John Wiley and Sons, New York, 1980, Ch, S.
4. Whitney, J. M., "Shear Correction Factors for ^rthotropic Laminates
Under Static Load," J. Applied Mme., Trans. OME, vol. 40, Series
E, no, 1, 1.973, p. 302-304.
5. Whitney, J. M, and Pagano, N. J., "Shear Deformations in Heterogeneous Anisotropic Plates," J. A 2lied Mech., Trans. ASMF, vol..
37, Series E, no. 4, 1970, p, 1031-1036.
6. Salamon, N. J., "An Assessment of the Interlaminaar Stress Problem
in Laminated Composites," J. Composite materials, to appear,

I. slyer, M.

W., !!

"some Observations fair 6

Cured

T
.
r til+n p ^. ^hin TIT^_"M

metric Laminates," submitted to J Composite Materials.
8.

Fung, Y. C., roundations of ,Solid Mechanics, Prentice -Hall Inc.,

Englewood Cliffs, NJ, 1965, p. 354.
9.

Chia, C.-Y. Nonlinear Analysis . of Plates, McGraw-Hill, Inc., New
York, 1980, Ch. 1.

10. Novozhilov, V. V., Foundations of the Nonlinear Theory of Elasti-

hester,, NY 1953, p. 836
Roc
cif, Graxyl.ock s,
Pres
11. Bowles, D. E., Post, D., Herakovich, C. T. and Tenny, D. R.,
Thermal Expansion cif Composites Using Moire Interferometry,

Virginia Polytechnic Institute and State University, College ofEngineering Report, VPI-E-80--19, 1980.
12, Simttses, G. J., Ali Introduction to the Elastic Stability: oflewood Cliffs, NJ) 1976, P. 8Structures, Prentice-Hall, Inc., Ing
14.

13. Pagano, N. J. and Hahn, H. T., "Evaluation of Composite Curing
Stresses," Composite Materials: Testing and Design, 4th Conference,
ASTM STP 617, 1977, p, 317-329.

19

Fig. l Laminate Shapes: (a) at the elevated curing temperature, andat room-temperature. (b) a saddle shape, (c) a cylindricalshape, (d) another cylindrical shape.

20

i

2

v
.L^

D

12

vr.3

Fig. 2 Roam-temperaturc shapes of square (0,/90„],7300/5208 graphiteG
4 1

epoxy laminates

i

22

23

1cE
v
Ca
i

X
z
W

d
U

150

50

V

LENGTH OF SIDE, L, mm
0

50

0+

100

150
D

E
-2
^yw

X

B

^. -4
z
w-6Cr.A

4
Cr —8

U

E
0
I cylindrical
specimen, ref. 7

range of iI

cylindrical
specimens, ref, 7

I

Fig. 3 Room-Temperature shapes of square (04/9041T T300/5208 graphiteepoxy laminates

24

25

E 4
v
w

^ 3
0
x

z 2
W
i

U 0
0

50

100

150

LENGTH OF SIDE, L, mm

0
E
.a
-1

co

r
z -2
w

Q

3
-4

0

50

100

150

R ^_
k

,y

26

i

Prof. Donald F. Adams
Dept. Of Mechanical Engineering
University Of Wyoming
Lar n fe, WY 82070

Mr. David Bowles!fail. Stop 1088
NASA-Langley Research Center
Hampton, Va. 23665

Dr. N. R. Adsit
General Dy,nanics Convair

Dr. H. F. Brinson
ESN Dept.
Virginia. TechBlacksburg, VA. 24061

P.O. Box 00837
San Diego, CA. 92138

Winfield H. Arata, Jr.

4414 Countrrrood. Drive
Santa Maria, CA 93455

Mr. Er lie Brooks
Code 1844
DTr SR DC
Bethesda, HD 20084

Dr. Clifford J. Astill.
solid Mechanics Program
National science Foundation
1800 G St. N.Y.

Washington, D.C.
AVCO, Systems Division
Subsystems 6 Meth. Structures

201 Lowell Street
Wilmington, MA. 01887

Dr. J. A. Bailie

D81-12 Bldg. 154
Lockheed Missiles 6 Space Co,Inc1111 Lockheed Way

Dr. dicharl P. Card
hail Stop 190
NASA-Langley Research Center
Hampton, VA 23665
Dr. C. Chamis
NASA-Lewis Research Center
2100 Brook Park Rd.
Cleveland, Ohio 44135
Dr. Paul A. Cooper
Mail Stop 190
NASA-Langley Research Center
Hampton, Va. 23665

Sunnyvale, CA. 94088

Dr. Charles W. Bert., Director
School Of Aerospace, Mechanical
6 Nuclear Engineering

The University of Oklahoma

Dr. Frank Crossman
Lockheed Research Lab
Org. 52-41, Bldg. 204
3251 Hanover street
Palo Alto, CA. 94304

Norman, Oklahoma 73069

Dr. I. M. Daniel, Manager

Dr. C. M. Blackmon
NSNC, Code 921

TIT Research Institute
10 West, 35 Street.

Dahlgren, VA 22448

Chicago, IL. 60616

Mr. Richard Doitnott

Dr. John R. Davidson
Mail Code 180E
MD Structural Integrity branch
Langley Research Center
Hampton, YA. 23665

Mail Stop 190
I

Hasa-Langley Research Center
Hampton, VA. 23665

27

Dr. John G. Davis, Jr.
Mail Stop 188A
Langley Research Center
Hampton, VA.

23665

Mr. Jerry M. Deaton
flail Stop 188A
NASA -Langley Research Center
Hampton, VA. 23665
Mr. H. Benson Dexter!fail Stop 188A

NASA-Langley Research Center
Hampton, VA. 23665
Mr. 0. Earl Dhonau
Section 2-53400♦ought Corp.
P.O. Box 5907
Dallas, TX. 75222
Dr. M. F. Duggan
52-33/205/2
Lockheed Pala Alto Lab.

3251 Hanover St.
Palo Alto, Ca. 94304
Prof. John C. Duke, Jr.
ESM Dept.
Virginia. TechBlacksburg, ♦ A. 24061

Mr. Gary L. Farleynail stop 168A
NASA-Langley Research Center
Hampton, VA. 23665
Mr. Larry nogg
Lockheed- California
Dept. 7572, Bldg. 63, Plant Al
P.O. Box 551
Burbank, CA. 91520
Dr. R. L. Foye

USAMR DLSAUDLAS (207-5)
Moffet Field, CA. 94035
Dr. D. Frederick
ESM Dept.
Virginia TechBlacksburg, VA. 24061
Mr. Samuel P. Garho
McDonnell Aircraft Co.
Bldg. 34 0 Post 350
St. Louis, M0. 63166
Mr. Ramon Garica
Mail Stop 190
NASA-Langley Research Center
Hampton, VA. 23665
Prof. Jim Goree

Prof. George J. Dvorak
Civil Engineering
University of Utah
Salt Lake City, UT. 84112
Dr. Wolf Plberbail Stop 188E
NASA-Langley Research Center
Hampton, VA. 23665

Dept. of Mechanical Engr.
Clemson University
Clemson, S.C. 29631
Dr. Login B. Greszczuk
McDonnell Douglas Asir. Co.
5301 Bolas Avenue

Huntington Beach, CA. 92647
Dr. O. Hayden Griffin, Jr.

Mr. Dave Erb
Aero E Ocean Engr. Dept.

Virginia TechBlacksburg, VA. 24061

Bendix Advanced Technology Ctr.
9140 Old Annopolis Road
Columbia, MD 21045

1

{

28

hr. Glen C. Grimes
Dept. 3852/82

Mr. Edward A. Humphreys
Materials Science Corporation

Northrop Corp., Aircraft Div.
3901 West Broadway
Hawthorne, CA. 90250

Blue Bell Office Campus

Dr, H. T. Hahn
Washington University
St. Louis, ho. 63130

ESA Dept.

Dr. J. C. Halpin
Flight Dynamics Lab
Wright-Patterson AFB
Ohio 45433
Professor Z. Hash in
School of Engineering
Solid tech. Materials & Struc.
Tel Aviv University
Tel Aviv, Israel
Dr. N. A. Heller
ESt Dept.
Virginia TechBlacksburg, VA. 24061

Blue Bell, PA. 19422
Dr. Michael W. Hyer
Virginia TechBlacksburg, VA. 24061
Dr. Eric R. Johnson
BSH Dept.
Virginia TechBlacksburg, VA. 24061
Dr. N. J. Johnson
Hail Stop 226
NASA-Langley Research, Center
Hampton, VA. 23665
Dr. M. P. Kamat
ESA Dept.
Virginia TechBlacksburg, VA. 24061
Dr. Keith T. Kedward

Dr. E. G. Henneke
ESh Dept.
Virginia TechBlacksburg, VA. 24061

Prof. Carl T. Herakovich
Laboratoire de llecanigue
des Solides
Ecole Polytechnique
91128 Palaiseau ceder, FRANCE
Professor Phil Hodge
107 Aeronautical Engr. Bldg.
University of Minnesota.
Minneapolis, MN 55455

Dr. K. E. Hofer
ITT Resoa.rch Institute
10 West 35 Street

Chicago, Illinois 60616

1768 Granite Hills Dr.
E1 Cajon, CA. 92021
Mr. John M. Kennedy
Mail Stop 188E
NASA-Langley Research Center
Hampton, VA. 23665
Hr. Eric Klang
ESM Dept.
Virginia TechBlacksburg, VA. 24061
Mr. James P. Knauss
Northrop Corporation
3901 West Broadway
Dept. 3852/82
Hawthorne, CA. 90250
Dr. Ronald D. Kriz

Dept. Cos. NBS Bldg. 2
Boulder, CO. 80302

29

Dr. S. V. Kulkarni
1342 Lawrence Livermore Lab
P. 0, Box 808
Livermore, Ca. 94550
Dr. Trent R. Logan
Mgr. structures, Design, Dev.
Boeing Commercial Airplane Co.
P.O. Box 3707 - M.S. 3M-23
Seattle, NA. 98124
Dr. M. R. Lout han
Materials Engineering
Virginia TechBlacksburg, VA. 24061
Mr. Vic Marzio
General Electric Co.
P.O. Box 8555
Bldg. 100, RR. M4010

Philadelphia, PA. 19101
Dr. Martin M. Mikulas
Mail Stop 190
NASA.- Langley Research Center
Hampton, VA. 23665
Mr. J. Steve Mills
A3-220 13-3 McDonald Douglas
5301 Bolsa Avenue
Huntington Beach, CA 96247

Dr. D. H. Morris
ESM Dept.
Virginia TechBlacksburg, VA. 24061
Mr. Anya Nagarkar
Material Sciences Corp.
Blue Bell office Campus
Blue Bell, PA. 19422

NASA Scientific B Technical
Information Facility
P.O.. Box 8757
Baltimore/Washington Inter. Air.
Baltimore, MD. 21240

Mr. Michael Nemeth
ESM Dept.
Virginia TechBlacksburg, VA. 24061
Newman Library
Virginia Tech
Mr. David A. O'Brien
5902 Kinqsford Pl.
Bethesda, MD 20034
Dr. Donald W. uplinger
Army Materials 6 Mechanics
Research Center
Department of the Army
Watertown, MA. 02171

Dr. Nicholas J. Pagano
WPAFB/MBM
Wright Patterson AYB
Ohio 45433
Mr. Michael Parin
3M Co., 3M Center
Bldg. 230 -1F
St. Paul, MN. 55101
Dr. Nicholas Perrone, Director
Structural Mechanics Program
Department of the Navy
Office of Naval Research
Arlington, VA. 22217

Prof. T. H. H. Pian
Mass. lust. of Tech.
Dept. of Aero. E Astr.
Cambridge, MA. 02139

Mr. Marek-Jerzy Pi.ndera
ESM Dept.
Virginia TechBlacksburg, VA. 24061
Dr. R. Byrou Pipes
Dept. of Mech. E Aero. Engr.
107 Evans Hall
University of Delaware
Newark, DE. 19711

1
30

Prof. Robert Plunkett
Dept. hero S Eng. Mach.
Aero 107
University of !Minnesota

Minneapolis, MN. 55455

Dr. .1. Wayne sawyer!tail Stop 190
NASA-Langley Research Center
Hampton, VA. 23665
Dr. George P. Sendeckyj

Dr. K. L. Rei fs aide r
ESM Dept.
Virginia Tech

Structures Division
Air Force Plight Dynamics Lab.

Blacksburg, ♦ A. 24061

Wright-Patterson APB
Ohio 45433

Dr. Gary D. Wenieri.

Mr. Steven M. Serabian
28 Berkeley Drive
Chelmsford, MA. 01814!McDonnell Douglas Astro. Co-East
P.O. Box 516
Bldg. 106, Level 4, Post C -5
St. Louis, M0. 63166
Dr. Michael W. Renieri
McDonnell Aircraft Co.
Bldg. 34, Post 350
St. Louis, M0. 63166
Dr. Larry Roderick!fail Stop 1882
NASA-Langley Research Center
Hampton, VA. 23665
Dr. B. W. Rosen
Materials Science Corporation
Blue Beal Office Campus
Blue Bell, PA. 19422

Mr. John S Short, Jr.
ESM Dept.
Virginia TechBlacksburg, VA. 240b1
Mr. Mark J. Shuart
Mail stop 188
NASA,-Langley Research Center
Hampton, VA. 23665
Dr. J. R. Stafford
B.P. Goodrich
500 S. !lain St.
D/6145, B110-E
Akron, Ohio 44318
Dr. James H. Starnes, Jr.

Dr. R R. Rowlands
Dept. of Engineering Mechanics
University of Wisconsin
Madison, WI. 53706

Dr. Edmund P. Rybick i.
Mechanical Engineering Dept.
The Univ.. of Tulsa

Mail Stop 190

NASA - Langley Research Center
Hampton, VA. 23665

Prof. Tehuda Staysky
Gerard Swope Prof. of Mech.
Technion-Israel Inst. of Tech.
Technion City, Haifa, Israel

Tulsa, OK 74104
Mr. Harminder saluja
Boeing Vertol Company
Structural Tecknology

P.O. :Box 16858
Philadelphia, PA. 19142

Dr. W. W. Stinchcomb
ESM Dept.
Virginia TechBlacksburg, VA. 24061

31

Dr. Davrel R. Tenney
flail Code 188E
RD-Materials Research Branch

Dr. Edward Yu
Lawrence Livermore Lab.
University of Calitornia

Langley Research Center
Hampton, VA. 23665

BOX 808, L--338

Dr, S o No Taal,
Nonmetallic Materials Division
Air Force Materials Laboratory

Mr. Thomas A. Zeiler
school of ►ero. s Asro.

Livermore, CA. 94550

Wright-Patterson APB

Grissom Hall
Purdue Univ.

Ohio 45433

West Lafayette, IN. 47907

Dr. J. R. Vinson
Dept. of Rech. 6 Aero. Engr.
107 Evans Hall
University of Delaware
Newark, DE. 19711

Dr. Carl H. Zweben
General Electric Co.
Space Division
P.O. Box 8555
Philadelphia, PA. 13101

Mr. M. E. Naddoups
General Dynamic Corp.
Port north, TX 76101
p rof. A. S. Yang
Mechanical Engineering
Drexel University
Philadelphia, PA. 19104
Prof . S. S. Wang
Dept. Theoretical E Applied
Mechanics
University of Illinois
Urbana, TL. 61801
Dr. T. A. Weisshaar
School of Aero . 6 Astro.
331 Grissom Hall
Purdue Unit.
West Lafayette, IN. 47907
Dr. J. N o Whitney
Nonmetallic Materials Division
Air Force Materials Laboratory
Wright-Patterson APB
Ohio 45433
Dr. Ernest G o Wolff
The Aerospace Corp.
P.D. Box 9295
Los Angeles, CA. 90009