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Thermal bending of sandwich panels under uniaxial loading

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Title:
Thermal bending of sandwich panels under uniaxial loading
Creator:
Bernstein, Edward Linde, 1938-
Publication Date:
Language:
English
Physical Description:
xiii, 115 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Bending ( jstor )
Boundary conditions ( jstor )
Buckling ( jstor )
Experimentation ( jstor )
Infinite series ( jstor )
Sandwiches ( jstor )
Sine function ( jstor )
Structural deflection ( jstor )
Temperature distribution ( jstor )
Temperature gradients ( jstor )
Buckling (Mechanics) ( lcsh )
Dissertations, Academic -- Engineering Mechanics -- UF
Engineering Mechanics thesis Ph. D
Sandwich construction ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 111-114.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
022655263 ( ALEPH )
13919585 ( OCLC )
ADA4823 ( NOTIS )
AA00004947_00001 ( sobekcm )

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Full Text


THERMAL BENDING OF SANDWICH PANELS
UNDER UNIAXIAL LOADING
By
EDWARD LINDE BERNSTEIN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
April, 1964


ACKNOWLEDGMENTS
The author wishes to express his appreciation to the
members of his supervisory committee: to Dr. I. K. Ebcioglu,
chairman, for the advice and encouragement which he tendered
throughout the course of this investigation? to Dr. W. A.
Nash, whose many kindnesses have been of great assistance;
and to Drs. A. Jahanshahi, J. Siekmann, and R. W. Blake for
their helpful discussions with the author.
The author wishes words were adequate to express his
appreciation to the only member of his special committee--
his wife, Eileen. She has served as secretary, editor, ad
visor, and most importantly, inspiration to success.
11


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES V
LIST OF FIGURES vi
LIST OF SYMBOLS vii
ABSTRACT xii
Chapter
I.INTRODUCTION 1
Historical Background 1
Definition of the Problem 5
II.DERIVATION OF THE EQUATIONS 9
The Strain Energy Expression 9
The Equilibrium Equations 18
The Boundary Conditions 24
III.SOLUTION OF THE EQUATIONS 32
Transformation into an Uncoupled System 32
Solution of the Membrane Force Equations .... 39
Solution of the Second Set of Equations 41
Convergence of Series Solutions 51
Development and Solution of the Ordinary
Differential Equations 55
Solution for a Simplified Case 68
IV.COMPARISON WITH EXPERIMENTAL RESULTS 85
iii


TABLE OF CONTENTSContinued
Chapter Page
V. COMPUTATION AND DISCUSSION OF THE SOLUTION .... 92
Computation of the Solution for Various
Parameters of the Panel 92
Discussion of the Results 96
VI. SUMMARY 108
APPENDIX 109
LIST OF REFERENCES Ill
BIOGRAPHICAL SKETCH 115
iv


LIST OF TABLES
Table Page
1. Experimental Data 87
2. Theoretical Results 90
3. Comparison of Buckling Load Values 104
4. Experimental Temperature Gradient 110
v


LIST OF FIGURES
Figure Page
1. Dimensions of the rectangular sandwich panel .. 6
2. Variation of displacement u through the
thickness of the sandwich panel 34
3. Resolution of membrane forces into a force and
couple system at the "neutral" surface 34
4. Comparison of theoretical and experimental
values of the deflection at the center
of the plate 89
5. Deflection at the center of a plate under a
temperature gradient R = R0 sin rr sin mi
as a function of edge loading 97
6. Deflection at the center of a plate under a
constant temperature gradient as a
function of edge loading 98
7. Deflection at the center of a plate under a
temperature gradient R = R0 sin rr| sin 7
as a function of edge loading 99
8. Deflection at the center of a plate under a
constant temperature gradient as a
function of edge loading 100
9. Comparison of the deflection at the center
of a plate under a temperature gradient
R = Rc sin 7r| sin 7r?¡ as a function of
edge loading 105
10.Dimensions and data points of the sandwich
panel 109
vi


LIST OF SYMBOLS
a -
Length of panel in x-direction
a> "
Elements of the determinant Aj
b -
Length of panel in y-direction
b*y -
Quantity defined in equations (3.34)
Bj Bcj Bej By, Fourier coefficients defined in equa-
tions (3.27)
CJ£ ~
Quantity defined in equations (3.34)
C i -
Function defined in equations (3.22)
Caj C9J
- Fourier coefficients defined in equations (3.27)
D -
Differential operator d/d5
D -
Plate modulus
DJi -
Determinant formed from A,- (see p. 63)
e, exy ey Linear components of strain
E -
Young 1s modulus
fas f, ~
Auxiliary functions
F ~
Function defined in equations (3.23)
Fe> Fy
- Fourier coefficients defined in equations (3.27)
g -
G/Gy
90> g/ -
Auxiliary functions
vii


Gy -
Orthotropic shear moduli of core
h', h" -
Distances to "neutral" surface
h, -
Auxiliary functions
H*y -
Fourier coefficient
Hjf Hy,
Hy Couples formed from resolution of membrane
forces
Hr "
Thermal couple formed from resolution of thermal
membrane forces
Jc ~
Function defined in equations (3.20)
k/e "
Dimensionless edge loading parameter -0NyyPe
Kv -
Fourier coefficient
m -
E't'/E^t"
Mjf M *y ,
My Bending moments in faces
Mr -
Thermal bending moment in faces
M/, M
My*- Bending moments applied on edges of faces
H* NXy ,
Ny Additional membrane forces relative to state
of initial stress
Nr -
Thermal membrane force
Nx xy ,
Ny Resultants of additional membrane forces
T -
Resultant of thermal membrane forces
0NX, oNxy, 0Ny Initial membrane forces
oN* 0Nxy* 0Ny* Applied initial edge loads
N(, 0Ny <,Ny Resultants of initial membrane forces
viii


7 N *, TN,y rNv Total membrane forces, defined in equations
(2.18)
Pe Buckling load of a simply supported, infinitely
long sandwich strip with a rigid core
Pf Sum of buckling loads of each face of a simply
supported, infinitely long sandwich strip with
a rigid core
q -
Q X t Qy
Qx* Qy*"
re "
R -
s
s
// _
s -
Distributed transverse load acting on faces
Shear force resultants in core
Vertical shear forces applied on edges of faces
Dimensionless parameter
Dimensionless temperature gradient parameter de
fined in equation (3.13)
Art + t->
- fit + t">
Function defined in equations (3.21)
Sx/ S*y, Sy Components of initial stress
t', t" Face thicknesses
t -
Core thickness
t -
t*, t*y, ty Components of stress relative to initial state
of stress
T Change in temperature from a constant initial
temperature
T,, Tz Coefficients of the temperature distribution
function
ix


U, V -
Displacements of middle surface of upper face
relative to "neutral" surface, defined in equa
tions (3.1)
/W A/
U, V -
Displacements of "neutral" surface, defined in
equations (3.1)
, v -
x- and y-components of displacement in core
A / A //
U u ,
v', v" x- and y- components of displacement in the
faces
U, V0 -
Functions defined in equations (3.19)
U,, U2 V, V2 Coefficients of displacement distribution
functions in the core
v* -
Virtual work of edge moments Mx* Mx/, My*
V* -
Virtual work of edge forces eNx* 0N*y* <,N/
v? -
Virtual work of transverse loading q
Va -
Virtual work of edge shear forces Qx* Qy*
w -
z-component of displacement
A
W -
Dimensionless deflection parameter
Wo -
Function defined in equations (3.19)
Wc -
Strain energy of core
w, -
Strain energy of faces
X, Y, Z Solutions to ordinary differential equations; see
equations (3.25)
Xc Ye ,
Zc Complementary solutions
XP, Y,,
Zp Particular solutions
x


o -
O -
ft -
A;
*,
5 -
n ~
ej
\-
A,
M ~
i -
P -
cj*
Coefficient of linear thermal expansion
Dimensionless parameter 2t/b
Aspect ratio of panel b/a
Determinant
xy £y Complete components of strain
z s
y/b
Quantity in cubic equation
Coefficient of exponent in complementary solution;
see equations (3.33)
Poisson's ratio
x/a
1 m A
Dimensionless parameter ^t
u>y, cjt Components of rotation
xi


Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
THERMAL BENDING OF SANDWICH PANELS
UNDER UNIAXIAL LOADING
By
Edward Linde Bernstein
April, 1964
Chairman: Dr. I. K. Ebcioglu
Major Department: Engineering Science and Mechanics
The deflections of a rectangular sandwich panel sub
ject to a thermal gradient, transverse load, and combined
edge loading were calculated. Two opposite edges of the
panel contained rigid inserts and were simply supported.
The other two edges were simply supported, clamped, or free.
Small deflection theory was assumed, as well as an ortho
tropic weak core. The effects of bending in the faces were
also included. The five coupled, linear partial differen
tial equations governing the deflections were transformed
into two sets of equations, one of which is the well-known
equation of plane stress. The remaining set was simplified
to neglect all edge loading except a uniaxial force and was
xii


solved by the use of Levy-type infinite series, which reduce
the equations to ordinary differential equations with con
stant coefficients. The convergence of these series was
investigated. A standard solution of these equations was
obtained, and the resulting expressions for the displace
ments were written in closed form.
Values obtained from the solutions were compared with
experimental values reported in the literature, and agree
ment was found to be good, although shortcomings of the ex
perimental values are mentioned.
The variation of the deflections and the buckling
loads for panels with varying elastic properties, dimen
sions, and edge loading were calculated on the University
of Florida IBM 709 computer, and these data are presented
in graphs. The values of the buckling loads were found to
be somewhat less than those reported in a less exact analy
sis. It was noted that, in general, a reversal of deflec
tions took place and that buckling modes were characterized
by a large number of half-waves.
xiii


CHAPTER I
INTRODUCTION
Historical Background
Legend has it that the terra "sandwich" arose in 1792
when the Fourth Earl of Sandwich devised a convenient way to
have his lunch without interfering with his daily card game.
Since that time the term sandwich has come to mean more than
merely a way to eat. The term applies to any object con
structed like a sandwich, that is, composed of three layers,
the middle one being of a different material tha$ the outer
layers. Sandwich construction is a type of composition that
has recently begun to achieve wide application. The con
struction is characterized by two thin outer layers known as
the faces and a thicker inner layer known as the core. The
faces are made of a stiff dense material and resist bending
and membrane stresses, while the core is made of a light ma
terial offering resistance mainly to transverse shear and
compression.
The original, and still most widespread, application
of sandwich construction is in air and space vehicles; its
1


2
high strength-to-weight ratio is well suited for this pur
pose. One of the earliest applications was for the skin of
the World War II DeHaviland Mosquito bomber, in which birch-
wood faces and a solid balsa core were used.'*' At the pres
ent time the faces are frequently made of steel or aluminum
with a core made of either an expanded cellular material or
a corrugated metallic sheet.
Sandwich construction is also found in curtain wall
2
panels for office buildings; with a foam plastic core it is
used in refrigerator construction; it is also suitable as a
hull material for boats when a balsa core is used.
Because of this wide use of sandwich construction,
especially in panel form, the elastic analysis of a sandwich
plate is a timely and important problem. The nature of the
core necessitates the consideration of several effects not
present in the usual analyses of thin plates. The strength
of corrugated cores will obviously depend on the direction
of the corrugations. Cores made of expanded materials are
constructed of thin strips of material which run in one di
rection. The properties of either type of core are thus
orthotropic. Because of the relative thickness of the core,
the effect of shearing deflection is important. Since cores
contain air spaces, and thus act as thermal insulators,


3
sandwich construction used in an air vehicle will be subject
to a temperature difference between the two faces. Thermal
stresses arise from this temperature gradient.
The earliest study of a sandwich plate was performed
3
by Gough, Elam, and DeBruyne; they considered the problem
of a thin sheet supported by a continuous elastic foundation
in order to study "wrinkling," or instability of the faces
characterized by a large number of buckling waves. Later,
4 5
Williams, Leggett, and Hopkins and Leggett and Hopkins
examined the Euler type of buckling of sandwich plates as a
whole; they accounted for shearing deformations in the core
by using the "tilting method," as termed by March,6 in which
a line originally normal to the middle surface tilts during
deformation through an arbitrary angle. Use of an approxi
mate strain energy method determined the buckling load.
March and others extended this method to include the effects
of orthotropic cores and orthotropic faces of differing
7 8 9
thicknesses and elastic properties. ' Libove and Batdorf
developed a method of analysis in which the sandwich plate
is treated as an orthotropic plate with the effect of shear
ing deformation included.'*'6 Reissner developed differential
equations for the finite deflection of sandwich plates.'*'1


4
Hoff used the assumption of a weak core which resists
only vertical shear and the principle of virtual displace
ments to derive exact equations for the bending of a sand-
12
wich plate. He was able to solve the problem for a simply
supported plate by assuming that rigid inserts were set into
the core edges. In this way the boundary conditions were
altered to permit a solution.
In 1951 Eringen used the principle of virtual work to
derive extremely general equations which account for the ef
fects of compression and flexure of the core.^ In 1952
Gerard, by an order of magnitude analysis, examined the
limits of validity of the simplifications commonly assumed
14
in sandwich plate analysis. Goodier and Hsu considered
15
nonsinusoidal buckling modes. In 1959 Chang and Ebcioglu
extended Hoff's theory to include an orthotropic core and
dissimilar faces and found the buckling load for a simply
supported plate.10 Recently, Yu, in a series of papers, has
included the effects of rotatory inertia and shear in the
faces.17'18'19'20
Thermal stresses in sandwich plates were treated by
21
Bijlaard, who used a homogeneous plate approximation.
Chang and Ebcioglu extended their approach to include the
22
effect of a constant temperature gradient, and Ebcioglu


5
later solved the problem for a temperature gradient which
23
varies arbitrarily across the area of the plate. Utiliz
ing Hoff's approximation for the boundary conditions, he
presented a solution for the bending of a plate simply sup
ported on all edges. Ebcioglu et al. also give experimental
results for the deflection due to a thermal gradient.2^
Definition of the Problem
It is the aim of the present analysis (1) to extend
Ebcioglu's work in order to develop equations which include
the effects of bending of the faces, a linear temperature
distribution through the thickness, and various boundary
conditions, (2) to solve the equations for the exact bound
ary conditions rather than making Hoff's approximations, and
(3) to present graphs illustrating the effects of various
parameters of the plate on the deflection and buckling load.
A Cartesian coordinate system is defined for a rec
tangular sandwich plate as shown in Figure 1. The middle
plane of the core lies in the xy-plane, with two adjacent
edges lying along the x- and y-axes; the z-axis points down
ward. In the notational system employed here a single prime
denotes quantities related to the lower face, a double prime
those related to the upper face, and a superposed bar those
related to the core.


Figure 1.
Dimensions of the rectangular sandwich panel


7
The plate has lateral dimensions a and b along the
x- and y-directions respectively, and the thicknesses of the
upper face, core, and lower face are t", t, and t' respec
tively. The faces may differ in Young's modulus E, but have
the same Poisson ratio /j.. The core is assumed to be homo
geneous and orthotropic.
The temperature change T of each face is defined as
the change from some constant initial reference temperature
and is arbitrarily distributed in the x- and y-directions;
the variation in the z-direction through the thickness of
each face is given by the following equations:
T' = T/S + T/
pff = rp H ^ ip U
(1.1)
where T/, T/', Tz', and T2" are functions of x and y only;
? = z s (1.2)
where
s = s' = (t + t' ) for the lower face
s = s/y = -i(t + t") for the upper face.
Arbitrary transverse pressures q' and q" act normal
to the outer surface of each face, while on the edges of


8
each face there act applied axial forces 0NX', 0NXy*, and 0Ny*
applied moments Mx* Mx/* and My? and applied vertical shear
forces Q* and Qy*, where each of these quantities is defined
per unit length of the boundary. The subscripts follow the
usual convention for stress notation. These forces are uni
formly distributed in the z-direction on each face and do
not act on the core, but on any edge where external loading
is applied a rigid insert is set into the core for its pro
tection. A more complete description of the insert is given
in the section on boundary conditions.
The Kirchoff hypothesis is assumed to hold for the
deflections of the faces. The temperature is considered to
be in steady-state equilibrium and the effect of deformation
on temperature is neglected. Account is taken of the fact
that elastic properties vary with temperature by permitting
each face to have a different value of E. The "weak core"
assumption is made, that is, the core is considered to be
so weak in its resistance to bending and axial stresses that
only the antiplane stresses Tz% tzy and fzz are present.
The core is considered incompressible in the z-direction.
It is assumed that neither local buckling of the faces nor
bond failure between the faces and the core occurs.


CHAPTER II
DERIVATION OF THE EQUATIONS
The Strain Energy Expression
A macroscopic approach has been chosen for the deri
vation of the differential equations governing the five in
dependent components of displacement. It is felt that this
approach, based on the theorem of virtual work, more clearly
shows the assumptions involved in formulating a mathematical
model which adheres to physical reality yet allows a solu
tion; this approach also permits a more straightforward
derivation of the boundary conditions than would the alter
nate approach based on the microscopic equilibrium of an
element of the plate.
The theorem of virtual work states that if an elastic
body is in equilibrium, the variation of the strain energy
equals the virtual work of external forces acting on the
body. The expression for strain energy for each layer of
the sandwich plate is written and the variation taken by
applying the variational calculus.
9


10
The derivation of the strain energy expressions for
the faces proceeds from the consideration that they are ini
tially in a state of equilibrium under the action of large
membrane stresses. From this initial state the faces under
go additional deformations. The approach used will be based
he p
on that of Timoshenko and Herrmann.
The initial components of stress are denoted by S*,
Sy, and SXy They are assumed not to vary through the thick
ness of each face, and to be of an order of magnitude higher
than the stresses produced during the additional deformation.
The following initial stress resultants are defined:
oNx = tS*
oNy = tSy (2.1)
tS*y
Since the plate is initially in equilibrium, the fol
lowing equilibrium equations may be written for each face:
^ o^xy
x + d y
o N y
x + by
(2.2)
0


11
The boundary conditions for the initial state are
oNy*= 0Ny
(2.3a)
on y = 0, y = b, and
(2.3b)
on x = 0, x = a, where N,*, 0Ny* and 0Nxy* are the initially
applied loads on the edges of each face.
The additional components of strain are assumed small
compared to unity; it is further assumed that the elongation
and shear terms are of the same order and the rotation terms
are of a smaller order. However, since products of initial
forces and rotation terms will occur, these latter terms
will be retained. Then the components of strain relative to
the initial state are
(2.4a)
(2.4b)
(2.4c)


12
xy ~ e*y ~ x ^ y (2.4d)
Lyz 6yi ~ ~ Ct-Jy xJz (2.4s)
ZX ~ ex ~ <^z *^x (2.4f)
where e,, ey, e2, exy e/z_ and eiX are the linear strain
components and u>x, uty, and u> are the rotation components.
Based on the Kirchoff hypothesis, the deflections in
the faces are given by
u
A
V =
(2.5)
where u and v represent the displacement components due to
the stretching of the middle surface in each face from the
initial equilibrium state, and ? is given by equation (1.2).
The rotations col are considered to be negligible for
plate deformations. Then the use of equations (2.5) in
equations (2.4) produces the strain-displacement relations:
du 1 /dwv2 w
x 2 x c)x2
(2.6a)
4
d v 1 ,dwd
y 2{c)y
(2.6b)


13
f ~ A + x + Ah! Ah! 2Z w
*y 2 by bx bx by bxby
(2.6c)
er =
A(|)' + 1(W
2 dx 2 dy
(2.6d)
= 0
(26e)
yi
~ 0
(2.6f)
The additional components of stress t*, ty, and txy
are assumed to be related to the additional strains by the
Duhame1-Neuman equations. The error introduced by this
assumption is of the order of the ratio of the initial
stress to the elastic modulus of the face.
26
Furthermore,
the transverse normal stress cordance with classical plate theory. Therefore, the fol-
27
lowing equations for plane stress will apply:
tx = ^ -(£* + yuEy) -
1 /O 1 -
E*T
/**
t y =
1 -X
(Ey + mEx) -
ET
1 -
t x y
1 + yU
where

14
The following stress resultants are defined in the
faces. The membrane forces per unit length are
*./z
N, =
t,d? =
Et
la +/|v + &if + (|w,*
1 //6iz[_c)x by 2 dx 2 by
~ N,
Nv =
%
t/d? =
Et
/_
V*
1 ^
by ox 2 by 2~ bx
- N
T
(2.7)
N,
rVa
tx/ d? =
Et
bu bv
bw bw
2(1 +/0
by bx
bx by
where the thermal force per unit length is
* t/i
N =
E (X
^ 1 -/<
Td? =
J.
*/z
ET21
1 /U.
The bending moments per unit length are
M, =
?t,d?= -D
J.
t/l
b1 vi + ^ bw
>x2
by 2
- M.
M, =
?tyd? = -D
J.
Vi
bl w b1w
by 2 ^ bx1
- M
(2.8)
M,y =
*/*
St*yd? = -D ( 1 yu)
J_
bzw
bxby


15
where the thermal moment Mr is
Mr = D (1 + M) T,
and the plate modulus is
D =
Et
12(1 M)
expression
w = 1
The strain energy VIF in the faces is given by the
27
t X + ty£y + 2ty *y 2T(tX + ty)
dV
(S,, + Sy£y + 2SXy£xy)dV
(2.9)
and the integration is extended over the volume of the body
in the initial state.
The stress resultants (2.1), (2.7), and (2.8), to
gether with the strains (2.7) and the temperature distribu
tion (1.1), are substituted into expression (2.9). After
integration through the thickness, there results
Wr =
o J
(V2 w)2 2(1 -/x)
bl w bl
w o w ^ 2w
xA dy2 bxdy
- 2(1 + /x) T, V2w| dxdy +


16
_
+
i ( 0NX + Ny )
du + l^W'2
[ 2
c)x 2 x
0 J
0
+
( 0Ny
+
+
1(|W)>
2 iy
+ (oNv
dv + du dw _dw
dx c>y dx dy
- oct* (Nx + Ny ) j dxdy (2.10)
The first integral in (2.10) represents the strain energy of
bending, while the second represents strain energy of the
stretching of the middle surface.
Since a "weak" core that is inextensible in the z-
direction is assumed, only the shear stresses Tzx and Tzy
exist. If the rotation terms u>z are neglected, the ortho
tropic stress-strain-displacement relations are
T2X = 2G*£2X = G*(-| + |^)
a z ex
(2.11)
fZy = 2GylZy = G y ( + ^)
' az oy
where Gx and Gy are the orthotropic shear moduli.
It follows from equations (2.11) and from the assump
tion that z = 0 that the displacements in the core are
given by the relations


17
u = U, z + U,
v = V, z + Vj
(2.12)
where the functions U, (x,y)f U2(x,y), V, (x,y), and V2(x,y)
are found from the conditions of continuity of displacements
at the interfaces between the core and the faces. In this
way
U
;
U2
V,
V2
- u" + t/ + t" aw
t 21 bx
u/ + u" + t' t" aw
2 4 dx
v/ v" + t/ + t" aw
t 21 by
v/ + v" + t' t" aw
2 4 ay
The following stress resultants are defined in the
core. The shear forces per unit length are
Q*
ri% dz =
tG* (iL=-
-*/i
u
+
(2.13)
Qy =
V*
J-*h
TZy dz
ay


18
where
t
(2.14)
The strain energy Wt of the core is of the form
wc
(^zx ^L y ^ly ) dV
J v
(2.15)
The expressions (2.12) are substituted into the
orthotropic stress-strain-displacement relations (2.11) and
then into the strain energy term (2.15). After integration
through the core thickness, there results
wc = -r
Gx (U,
dx
Gy(V,
V .
dxdy
(2.16)
The Equilibrium Equations
According to the principle of virtual work,
SW/ + 8W," + SWC iVj SV/ SV/' SV^' sv" -
8 V,,' 8V where the virtual work due to the applied loadings is as
follows: (1) from the transverse pressure


19
=
r b
'o
(q" q')8wdxdy
(2)from the edge forces on each face
SV* =
(0NX* Su + 0Nxy* Sv)dy +
(0Ny*Sv + oNxy'SuJdx
(3)from the edge moments on each face
sv* =
Mx* S(|£)dy
My 6(~) dX
a>y
dJVL *
jry
c)y
<$wdy +
wdx 2M>
Sw
(4)from the shear forces on each face
r b
sva =
Qx*8wdy +
Qy* 5wdy
In forming the variations use is made of the formulas
b r a
rb
0 ^ 0
c>F
dx
dxdy =
F (x, y)
dy
K 0


20
> (+ Q
O Jo
Sf ,
dxdy
ey
F (x,y)
r = *>
dx
and of integration by parts. Equation (2.17) becomes
o Jo >
(D' + D")Vvw + V2(Mr' + M') (rN/ + rN")
dxa
- 2(tn.; + ,N./')£^ t + q' q"
M5* &'
du
dx
f drN/
brNx/
l dx
dy
drN,"
d r^xy
dx '
dy
' d rN*y ,
dyNy
dx
dy
drNXy"
b rNy'
+ tt -£-?) + Gy (
dx2
dv' dv" .
dy by by2 .
Svr
~ Gx (
+ G, (
b x by
+ + Gy (
u'
-
u"
t
U7
u"
t
V7
V"
t
V'
v"
+ 1 SU'
+
- Gy ( I"* + t ^)j
6u'
6v'
+ i Sv
dxdy
b r
(-M, + M*)5(~) + (~ + 2 ^1L + r^TT. + TX*y ~
bx dx by dx 1 by
A_ SMXy*
+ tQ* Q*) Sw + (rNx 0n/)5u
dy
+ (TNxy 0Nxy*)6v dy +


21
(-My + My*)S(|^) + (|^ + 2 + rNy|^ + rN.y|^
oy dy dx 3y dx
dMxym
+ tQy
dx
- Qy*)6w + (rNxy gN + (TN y 0Ny*)8V
dx + 2(Mxy My*)
Sw = 0 (2.18)
In expression (2.18) the total membrane forces are defined
by
tNx = 0NX + N*
yNy = 0Ny + Ny (2.19)
rNxy = aNxy + Ny
The integrals extend over the body in the initial state, and
virtual displacements are taken as relative to that state.
Since the virtual displacements 5u', u", 8v', Sv",
and Sw are arbitrary within the plate, their coefficients
in equation (2.18) must vanish. In this manner the follow
ing equations of equilibrium result. In the first of these
equations the terms (N/ + Nx")-^~, (Ny' + N y ") and
(Nxy' + Ny") are neglected. The terms in parentheses are
membrane stresses due to bending of the plate after initial
stress and are negligible.


22
(D' + D")Vvw (0N/+ oNx'O-^-T 2(0NX/ +
ox' 7 oxay
UV+ T + v' G.+ tt |i^)
OX OX Ox'
+ Gv(Sv'
_ dv" T dw%
dy c>y by1
+ q' q" = 0
(2.20a)
N/ + a>Nx/
dx by
G
+
0
(2.20b)
N/' b Nxy"
dx c>y
G*
+
dw
3
0
(2.20c)
an,/ ay
dx by
0
(2.20d)
dN*/' SN/'
dx dy
Gy (~ v" +
0
(2.20e)
These five equations in five unknowns reduce to those de-
23
rived by Ebcioglu for the case of constant temperature
variation through the plate thickness and for small
deflections.


23
Equations (2.20) can be seen to contain the equations
of homogeneous thin plate theory. If the core is assumed to
be rigid, i.e., if Gxx = Gyx. -=, then it follows from equa
tions (2.11) that xi = yz = 0 and there is no contribution
from the core to the total strain energy. The coefficients
of G and Gy in equations (2.20) disappear and, except for
the thermal terms, the resulting equations are the classical
ones of Saint-Venant, describing two plates constrained to
have a common vertical deflection and subjected to lateral
and edge loads. These plates correspond in their dimensions
and properties to the faces of the sandwich panel.
Saint-Venants equations also result if the thickness
t of the core is allowed to go to zero, since in this case
the integral (2.15) then degenerates to zero. In either
case the effect of the core is only to constrain the two
faces to have a common vertical deflection. The core has
no stress-resisting properties of its own, so the fact that
it retains its thickness for the first case is of no
importance.
It is noted that a formal application of these limit
ing processes to equations (2.20) is meaningless and pro
duces incorrect results, since considerations preceding the
derivation of this equation are involved.


24
The Boundary Conditions
The boundary conditions arise from consideration of
the line integrals occurring in the variational equation
(2.18)o These line integrals may be made to vanish by set
ting either the variations or their coefficients equal to
zero. The boundary conditions are linearized in the same
manner as equation (2.20a). After use of equations (2.3),
they are found to be: (1) on the boundaries y = 0 and y = b
either
u' = 0 (2.21a)
or
N*/ = 0 (2.21b)
either
u" = 0 (2.22a)
or
N*y = 0
either
v' = 0
(2.22b)
(2.23a)
or
N / = 0
either
v" = 0
(2.23b)
(2.24a)
or


25
Ny" = O
either
w = O
(2.24b)
(2.25a)
or
(My' + My) + 2 ( M x y/ + M x y") + (,N/+ 0Ny)^
dy ox oy
+ tQy = 0;'+ Q*"+ ^(M,*' + M;")
(2.25b)
either
= 0 (2.26a)
dy
or
M/+ My' = M y*'+ My*" (2.26b)
(2) on the boundaries x = 0 and x = a
either
u' = 0
or
N/ = 0
either
u" = 0
or
Nx" = 0
either
v7 = 0
(2.27a)
(2.27b)
(2.28a)
(2.28b)
(2.29a)
or


26
N / = O
either
v" = O
or
N = 0
either
w = 0
(2.29b)
(2.30a)
(2.30b)
(2.31a)
or
y-(M/+ M") + 2 (M y/ + Mxy") + (.N/ +
ox oy ox
+ tQx = Q*'+ Q*"+ ^-(M>y*' + M,;')
Sy
(2.31b)
either
^ w
>x
0
or
(2.32a)
Mx/ + Mx" = M**'+ M/" (2.32b)
(3) for the corners (0,0), (a,0), (0,b), and (a,b)
either
w = 0
(2.33a)
or
O
a
= Mxy*
0
0
0
(2.33b)
The boundary conditions for a rectangular sandwich
panel simply supported along the edges perpendicular to the


27
y-axis and variously supported along the other two edges,
containing rigid inserts set into the edges of the core that
are perpendicular to the y-axis, may be selected from the
preceding sets of boundary conditions.
A simply supported edge may be supported either on
knife-edges or by pins of negligible cross-sectional dimen
sions, about which the panel may freely rotate. If the
latter case is considered for a sandwich panel, the pins
may be imagined as running through both faces, parallel and
arbitrarily close to the edges. They restrain the boundary
from displacement perpendicular to itself, but permit paral
lel motion. The restraint in the perpendicular direction
will cause a normal stress at the boundary.
If, on the other hand, the knife-edge type of simple
support is envisioned, the edge of each face is considered
to be in contact with a wedge-shaped support along the line
forming the vertex of the wedge. In this case the edge is
free to translate in directions both parallel and perpen
dicular to itself.
Finally, the effect of a thin rigid insert is con
sidered. The insert is envisioned as being set into the
core so that it is flush with the edges of the upper and
lower faces and welded to these faces at its top and bottom


28
edges. Its effect is to prevent the edges from having dis
placements in the direction parallel to themselves. If the
insert is perfectly insulated, so that thermal effects cause
no expansion of it, boundary conditions (2.21a) and (2.22a)
or (2.27a) and (2.28a) will apply. If, however, thermal
expansion of the insert is allowed, the boundary conditions
must allow for some displacement of the edges parallel to
themselves. In general, vertical displacements of the edges
would also occur, but it is assumed that the supports pre
vent such displacements. The extreme case occurs when the
insert offers no resistance to the free thermal expansion of
the edges. Then boundary conditions (2.29a) and (2.30a) are
replaced by
r y
v' = oc' T/dy
(2.34a)
v" = or
T"dy
(2.34b)
Va
and conditions (2.21a) and (2.22a) are replaced by
u' = oc' T2'dx
a/i
(2.35a)


29
u" = O'
T/'dx
Ja/
(2.35b)
In general, the displacement along the edge will be some
where between zero and these values, depending on the co
efficient of thermal expansion of the material of the insert.
The vertical deflections at the center of the plate were
calculated for both zero deflection and the extreme values
in equations (2.35); only a slight difference resulted.
The problem to be examined here is a plate containing
a rigid insert along the two edges parallel to the x-axis,
which are simply supported on knife-edges. The two edges
parallel to the y-axis may be simply supported by knife-
edges, clamped, or free. Furthermore, the only external
edge loading applied to the face edges is the initial load
0Ny* which is assumed to be constant along the edge. No
transverse loading acts on the lower face (q' = 0).
The boundary conditions to be used are as follows.
(1) On the boundaries y = 0 and y = b
u
/
= or
T/dx
*/i
(2.36a)
u
or
T/dx
Jo.
'/i
H _
(2.36b)


30
N/ = 0
Ny" = 0
w = 0
My' + My" =
(2.36c)
(2.36d)
(2.36e)
(2.36f)
where the right-hand sides of equations (2.35) should be set
equal to zero if the inserts do not expand when heated.
(2) On the edges x = 0 and x = a,
supported edges,
N/ = 0
N/'= 0
N= 0
Ny" = 0
w = 0
M/ + M*" = 0
for the case of simply
(2.37a)
(2.37b)
(2.37c)
(2.37d)
(2.37e)
(2.37f)
For the case of clamped edges
u' = 0 (2.38a)
u" = 0 (2.38b)


31
Ny' = 0
(2.38c)
Ny" = 0
(2.38d)
w = 0
(2.38e)
*2. = 0
3 x
(2.38f)
For the case of free edges
N*7 = 0
(2.39a)
Nx"= 0
(2.39b)
N*y' = 0
(2.39c)
Nxy" = 0
(2.39d)
(,N/+ 0NX")|^ + ^(M*' + M*") + 2 + M*P
+
f+>
Oi
K
II
o
(2.39e)
My' + My" = 0
(2.39f)
0
(2.39f)


CHAPTER III
SOLUTION OF THE EQUATIONS
Transformation into an Uncoupled System
The solution of equations (2.20) is difficult because
they are coupled in the four dependent variables u', u", v',
and v". By a transformation of variables two sets of equa
tions more amenable to solution may be developed. One set
resulting from this transformation is composed of two equa
tions in two variables. These equations are then the equa
tions of plane elasticity including the effects of a body
force. The second set of equations may be solved by a
separation of variables.
To effect this transformation, new displacement com
ponents are defined as follows:
u =
m
(u'

u")
1
+
m
V =
m
(v,
V")
1
+
m
II
13
1
(u"
+
mu')
1
+
m
V =
1
(v"
+
mv')
1
+
m
*
32


33
where
It is possible to give a physical interpretation of
the quantities in (3.1). The variation of and the varia
tion of v through the plate thickness are given by straight
lines with discontinuities of the slope. Figure 2 illus
trates this variation for . If, however, it is imagined
that these variations are given by lines of constant slope,
then for pure bending a neutral surface exists, which is
defined by the quantities
/\
h' = -
1 + m
h" = tt
1 + m
where h/ and h" represent the distance from the neutral
surface to the middle planes of the lower and upper faces
respectively. Then u and v represent the displacements of
the neutral surface, that is, since the neutral surface was
defined for pure bending, u and v represent the displace
ments due purely to membrane effects. Furthermore, u and
v represent the displacements of the middle surface of the
upper face relative to the neutral surface of the plate.


34
t
Figure 2. Variation of displacement through the
thickness of the sandwich panel
//
jr
;
H,
Figure 3. Resolution of membrane forces into a force and
couple system at the "neutral" surface


35
Substitution of the transformations (3.1) into the
equations for membrane force resultants (2.7) yields the
corresponding transformations for these quantities:
0NX = eNx' + aN/'
0y = 0Ny' + sN/'
0Xy = 0NXy' + o Nx y "
Nx = Nx' + Nx"
Ny = Ny' + Ny"
Nxy = Nxy' + Nxy
Nr = Nr' + Nr"
Hx
Hy
H
*y
H
/V
tt
1 +
m
/S
tt
1 +
m
A
tt
1 +
m
A
tt
1 +
m
(Nj/ mN;')
(Ny' mN")
(Nx/ mNxy")
(N/ mN/')
where t has been defined in equation (2.14).
The quantities Nx and H may be interpreted as the
statically equivalent resolution of the membrane forces Nx'
and Nx"( which act in the middle planes of the faces, into a
force and couple system acting at the neutral surface (see


36
Fig. 3). The interpretation of the other quantities follows
in a similar manner.
Substitution of the transformations (3.1) into the
equations of equilibrium (2.20) yields
(D7 + D")VVW 0N,-2-^ 2 0N,
~ d2w o cf bzw
)x
- yjrr +
y dxdy Y by
- t
- ,1 + m u :
Gx (
+ tt ~) + Gy (
ox ox m ay
- ,1 + m dv
m
+ tt
t d
w,
by
- q
//
o
(3.2a)
E t'
2m(l yu) L dx
by
d2v
' dxby
- 5u + t H) = 1 SHr
mt
dx tt d:
(3.2b)
E't'
, 2 (i + u
2m(l /2) L dy dx5 ~ dxdyj
- Gy(i4-£5 v + t &?) = 1 dHr
mt
dy tt dy
(3.2c)
d Nx dNxy
+ r = 0
dx dy
(3.3a)
dNxy dNy
dx + dy
= 0
(3.3b)


37
Similarly, the boundary conditions become as follows.
(1) On the boundaries y = 0 and y = b
Ny = 0
N,y = 0
(3.4a)
(3.4b)
r *
u =
m
1 + m
(ot' T a" T/0 dx
/z
(3.5a)
Hy = 0
(3.5b)
w = 0
(3.5c)
M/+ My//= 0
(3.5d)
(2) On the boundaries x = 0 and x = a, for the case of sim
ply supported edges,
Nx = 0
(3.6a)
Ny = 0
Hx = 0
Hxy = 0
w = 0
(3.6b)
(3.7a)
(3.7b)
(3.7c)


38
M/+ My" = O ( 3.7 d)
For the case of clamped edges
u = 0
(3.8a)
N*y = 0
(3.8b)
u = 0
(3.9a)
X
*
II
o
(3.9b)
w = 0
(3.9c)
= 0
ax
(3.9d)
For the case of free edges
* = 0
Ny = 0
H* = 0 (3.10a)
Hxy = 0
My' + My"= 0
j(M/ + M") + 2 ^p(M*/ + Mxy") + tQx
+ (0N/ + 0N,")^ = 0
(3.10b)
(3.10c)
(3.lOd)


39
Solution of the Membrane Force Equations
The membrane forces in the initial state of stress
are governed by equations (1.2) and (1.3). In general, a
complete solution for the initial state of stress requires
the further specification of the compatibility condition.
However, since the forces 0N*, 0Ny, and 0NXy have been as
sumed to be an order of magnitude higher than the linear
additional stresses, the strain-displacement relations will
include nonlinear terms. As a result, the compatibility
equation becomes nonlinear and, in general, a solution of
the state of stress is not possible.
If the initial edge forces 0N**, 0Ny* and 0N*y* are
constant, then the initial membrane forces are constant.
Since 0N** = 0Nxy* = 0, there results
0N* = 0Nxy = 0
0Ny = oNy
(3-11)
and therefore
0NX = 0NXy = 0
0y = i/
(3.12)


40
In the case when the edges of the plate x = 0 and
x = a are clamped, the second set of boundary conditions
(1.3) is replaced by the conditions
u = 0
0 NX y. = 0
The solution for the membrane forces requires the specifi
cation of the compatibility condition. If it is assumed
that the plate is clamped after the initial stress is ap
plied, then equations (3.12) will also apply to this case.
The membrane forces Nx, Ny, and N*y arising from the
additional stresses satisfy equations (3.3) and boundary
conditions (3.4) and (3.6). These equations are identical
to those just discussed for the initial stresses. However,
since these forces are within the realm of linear elasticity,
27
the compatibility equation may be expressed as
Vv(Nx + y) + (1 -/<)V2r = 0
28
Now the classical methods for plane stress problems can
be used for the solution of the state of stress.
If Nx*= Ny*= Nxy* 0 and Nr is a linear function of
x and y, where


41
7 = -r^ (E'oc'T.'t' + E"."T,"t")
1 -
then the equilibrium equations, compatibility equation, and
boundary conditions are satisfied by
* = Ny = Nxy = 0
and thus
Solution of the Second Set of Equations
Once the membrane forces have been found as indicated
in the preceding section, the next step is to solve the set
of three partial differential equations (3.2), which are
subject to the boundary conditions (3.5) and (3.7), (3.9),
or (3.10).
First the system is transformed into dimensionless
23
units. The scheme used by Ebcioglu is followed. Dimen
sionless constants, variables, and reference quantities are
defined as follows:
\ = £
a


42
pe
n e't' ^ _x
= b* i 1 (1 + m)
pf
= pr(D' + D")
k;e
o /
Pe
r
Pe
tZtGy
g =
Gx
Gy
a =
b
a
p =
1 + m £
2
cn. =
_2t
b
The temperature gradient is nondimensionalized by the intro
duction of the quantity
R =
1..+ ? (oc/ T/ oc"t") (3.13)
1 + m
The quantity Pe has been defined as a reference quan
tity with dimensions of force per unit length; it represents
the buckling load of a simply supported, infinitely long
sandwich strip with a rigid core, calculated under the


43
assumption that the faces act as membranes. represents
the sum of the buckling loads of each face of the strip.
Substitution of these quantities into the appropriate
equations and boundary conditions yields the transformed
equilibrium equations:
re
TT1
l 32U
+
(1
-A)
32u
*1*
+ /3{ 1 + jul)
bz y
2gu
mgta/3 ^ = 2/3mbre ^
2p 31 rr1 (3.14a)
r
TT
V
w
+ /32(1 -/Of|i + /3(1 + /<>
d2u
dldri
- 2v
= 2mbre
2/0 dr¡ TT2 d*[
(3.14b)
m
27r bGy L
d*
w
+
2/32
w
'w
aq/3
a/321 zgm
4
32w
3v mt2(rek,( 1) 92W qmb mb
/>_ + -- = = = V nT
' 4 dr¡ 2Gy 2Gy 7
(3.14c)
and the transformed boundary conditions, written in terms of
displacement components: (1) On the boundaries r¡ = 0 and
* 1


44
u
mb
/HI +yu)
ri
R(5)d£
J'/z
(3.15a)
by
d7l
+
/3 m
' M b%
= mbR
(3.15b)
w = 0
(3.15c)
P* 3*w
-m7
(3.I5d)
(2) On the boundaries 5 = 0 and | = 1, for the case of sim
ply supported edges,
^fr+ -*R
(3.16a)
(3.16b)
w = 0
(3.16c)
w
= -M.
(3.16d)
For the case of clamped edges
u = 0
(3.17a)
du dv
+
o
(3.17b)


45
w = O (3.17c)
Ir = O (3.17d)
o 5
For the case of free edges
+
/u = mbR
(3.18a)
3u
dri
+
/3 =
' w
T
+ /x
a>?l)
-Mr
(3.18b)
(3.18c)
P*/9
rr2h
+
d\3
(2 -yU)
w
c>5 4
1 + m ?
+ tG,u
m
£*'*>!?
(3.18d)
It is seen that the system of equations (3.14) is
linear and nonhomogeneous and is accompanied by nonhomo-
geneous boundary conditions. By the substitution of auxil-
29
iary functions the nonhomogeneous boundary conditions on
the sides r¡ = 0 and >7 = 1 can be reduced to homogeneous
ones. Then it is possible to select trigonometric functions
of 77 that satisfy these homogeneous boundary conditions.
o c
These functions may be employed in the Levy method^ of


46
separation of variables so that the partial differential
equations are reduced to ordinary ones. It is necessary to
expand the nonhomogeneous parts of the boundary conditions
and equations in a series of these trigonometric functions;
furthermore, the final solution is expressed in such series,
and the convergence of these series is investigated in order
to validate their differentiation.
In order to reduce the boundary conditions (3.15) to
homogeneous form, the following substitutions are made:
u = u (4,j) + v [f, (|) f0(5)] + f0U) (3.19a)
v = vo (S>?) + n*
1 df,(5)
g, (4) 2^-JT~
- (i y¡r
1 df0(5)
god) -
(3.19b)
W = VJg (5,^)
(n + Dh, (4)
+ (2
>?) h o (4)
TT1
(3.19c)
where the auxiliary functions denoted by f, g and h repre
sent the following nonhomogeneous boundary terms:


47
fo (i)
mb
/3(1 + /<)
- i
R(4,0)d4
J'/i
f, (4)
mb
/?(! + /)
R(4,l)d4
g<, (4) = ^ mbR(4,0)
g, (4 ) = -^ mbR( 5,1)
h0(4) = -MT(4,0)
h, (4) = -Mt(4,1)
It may be readily verified that the desired reduction
is accomplished by means of substitutions (3.19). The equa
tions governing the quantities u0/ v0, and we are then es
tablished by the substitution of relations (3.19) into equa
tions (3.14) through (3.18). The following equations result
£,
n
,2
dzu.
d2v
2/3i^+
- 2guc
mgt2(X/d dw0 2/3mbre
2/o d 4
TT 1 d4
+ J, (4,77)
(3.20a)


48
La
_ 2
d 2v,
d2v
2 ---j- + /32(1 /-<)- £- + /3(1 + /w)
dr¡
c>u 0
?
- 2v,
t2am dw0 2mbre
2p <$ri 772 dr¡
+ J U,7¡)
(3.20b)
m
'f 2TTlhGy
A'ypr + 2/3'
di dqz drj'
3u
- />g/3
4
a/32tgm d2wc ~v0 %*e./e
4 2>4 2 70 d*?
dv0 rtm.2 (rek/e 1) d2wc
d^2
qmb mb 2
= v Mr + Jj (4 ,r¡)
2Gy 2Gy
(3.20c)
The boundary conditions on £ = 0 and 4=1 become, for the
case of simply supported edges,
ft
du0
^4
+
mbR + S, (4 r¡)
(3.21a)
du0 dv0
dr¡ + ^ d4
s2 (4,
(3.21b)
w0 = Sj(4,>?)
(3.21c)
32w0
34*
-Mr + s¥ (4 rj)
(3.21d)
For the case of clamped edges
uo = C,(4,^)
(3.22a)


49
,* + /* 31 **>
(3.22b)
W0 = Sj (5,77)
(3.22c)
dl Cv
(3.22d)
For the case of free edges
du0 dv0
/3 + / ^ ^ = mbR + S, (§,77)
(3.23a)
du0 3v
~d^+ put = s (l,7?)
(3.23b)
7r o | 0 7^
(3.23c)
P,/3
7Tb
/3
c>Jw,
+ [2 yU)
d3w0
<55 d7jz
1 + m is
tG* u
m
b o5
/3 dM7-
- b-dT +
(3.23d)
In the above boundary conditions the functions J (5,^),
st (?>?)/ C U.tj), and Ft- (4,77), where i = 1, 2, 3, 4, repre
sent the additional terms arising from the substitutions
(3.19).
The boundary conditions on r¡ = 0 and r¡ = 1 are
u
o
0
(3.24a)


50
d v 3u.
+ = 0
(3.24b)
wQ = 0
(3.24c)
d2wfl
dri
f = o
(3.24d)
It is seen that the boundary conditions (3.24) along
the sides r¡ = 0 and r¡ = 1 are satisfied by terms of the form
u0j = Xj (?) sin j7T7?
v0j = Y, (5) cos j7T>7
woy = Z^ (?) sin jTrr¡
Thus the following solutions are assumed:
u0
oc
X, (4 ) sin
a =;
(3.25a)
v0
oo
1
Y, (?) cos jrr^
(3.25b)
w0 =
z, (?) sin jTr?7
(3.25c)
However, before attempting an infinite series approach,
certain mathematical questions must be examined.


51
Convergence of Series Solutions
Once solutions of the form of (3.25) have been pro
posed, it is desirable to develop some assurance that the
series can satisfy the required equations, in other words,
that term-by-term differentiation of the series will produce
a series converging to a proper derivative.
Such considerations must not be thought a mere mathe
matical superfluity. Quantities of physical interest occur
ring in the deformation of a plate involve second and third
derivatives of the displacements; if the displacements are
represented as infinite series, it may happen that term-by-
term differentiation does not provide series that converge
to their physical counterparts. For example, the deflection
y of a vibrating beam of length n, simply supported, with an
applied sinusoidal bending moment at one end is represented
by the following equation: ^
with the boundary conditions
y (0, t) = y(77, t) = 0
0


52
If a solution is attempted by the method of finite
sine transforms, the result will be the infinite series
sin an2t)sin nx
z
which, upon term-by-term differentiation, will be found to
satisfy neither the differential equation nor the nonhomo-
geneous boundary condition; nor will the internal bending
moments calculated in this way converge to their true values.
In order to be assured that term-by-term differentia
tion of a convergent series, particularly a Fourier series,
will lead to another convergent series, two theorems of per-
31
tinence may be noted. The first is as follows.
Theorem 1. Let there be given a series f(x)
whose terms are differentiable in the interval
and which converges at least at one point
oo
of J. If the series
deduced from it by
term-by-term differentiation converges uniformly in
J, then so does the given series. Furthermore, if
oo
f(x) = F(x) and
F' (x) = It is therefore necessary first to perform formally the


53
term-by-term differentiation and then to investigate the
uniform convergence of the result in order to determine the
validity of the operation.
The second theorem relates specifically to the terra-
32
by-term differentiation of Fourier series.
Theorem 2. If f(x) is a continuous function of
period 2n with an absolutely integrable derivative,
then the Fourier series of f' (x) can be obtained
from the Fourier series of f(x) by term-by-term
differentiation.
A corollary of this theorem applicable to the inter-
32
val (0,1) is given as follows.
Theorem 2(a). If f(x) is continuous and absolutely
integrable on (0,1), then the Fourier cosine series
Y' a cos nnx of f(x) on this interval may be dif
ferentiated term by term, while the Fourier sine
series ^ b sin nttx can be differentiated term by
term if f(0) = f(l) = 0.
It is seen that the Fourier cosine series is necessarily
always continuous everywhere; if the conditions in the corol
lary are met, the sine series is also continuous everywhere.


54
Theorem 2 applies to functions of one variable only.
However, it may be extended to series of the form of equa
tions (3.25) if the variable £ is treated as a parameter.
It is now possible to proceed with the justification
of the assumed solutions. From Theorem 2(a) and boundary
condition (3.24a) the series for u0 may be twice differen
tiated with respect to rj. Condition (3.24a) implies that
c>ue
as
= 0 along = 0 and r¡ = 1, and thus condition (3.24b)
may be replaced by
avo
an
= o
(3.26)
The cosine series (3.25b) for v0 can be differentiated once
with respect to r\, and condition (3.26) implies the permis
sibility of a second differentiation. Finally, conditions
(3.24c) and (3.24d), together with Theorem 2(a), imply that
the series for w0 may be differentiated four times with re
spect to y\.
It is concluded that it is permissible to perform
all partial differentiations with respect to n required to
satisfy the differential equations and boundary conditions.
The validity of taking derivatives with respect to £,
according to Theorem 1, can be established only after the
actual solution is obtained and the differentiations in


55
question performed. Some further remarks will be made when
this stage in the analysis is reached.
Development and Solution of the Ordinary
Differential Equations
The next step in the solution involves the develop
ment of ordinary differential equations in the variable 5 .
These relations arise from equating coefficients of trigo
nometric terms after substituting the assumed solutions
(3.25) into equations (3.20). In this process the nonhomo-
geneous terms of equations (3.20) are expanded in Fourier
sine or cosine series on the interval (0,1) according to
the following formulas.
(1) The Fourier cosine expansion of a function f(x) defined
on 0 x ^ 1 is
where
f (x)
oo
J=1
COS j77X
r I
= 2
f(x) COS ji7xdx
(2) The Fourier sine expansion of a function f(x) defined
on 0 x 1 is


56
where
f (x) = j bj sin jT7x
j = t
r1
bj = 2
f (x) sin ]Trxdx
In this way the following relations may be written
R
*1
K,j U) sin j7r^
dR
K2j (K) cos jrrr¡ + k20 (5)
q
sin
V*Mr =
sin jrr^j
Furthermore, the remaining terms on the right-hand
sides of equations (3.20) are given by
J, (5.>7)
H,j () sin j7T?2


57
OO
J2(,T) = H2j (4) cos jrrn + Hz0 (5)
.¡-i
Jj U
oo
= £
Hj. (?) sin jTr?7
In a similar manner the nonhomogeneous right-hand
sides of the boundary conditions (3.21) through (3.23) on
4 = 0 and § = 1 are
mbR (4,>?) + S ,U,t?) = Baj (4) sin j n >¡ (3.27a)
-1
oo
S2U,7j) = ^ <5) cos jr^ + Bc<} (4) (3.27b)
j--1
S. U^) = BCJ (4) sin jtt>7
M y (4, ) + S ^ (4, *7) ^ j j
j = '
c, (4, t) = cOJ (4) sin
oo
c*(4,>?) = ^ CS7 (4) sin j7r^
1 = 1
(3.27c)
(4) sin jTrri (3.27d)
(3.27e)
(3.27f)


58
PO
-M + Fj (?,j) = Fej (5) sin j TT>i (3.27g)
y* >
fi dMrU.n)
b d £
I
J = i
+ Fv(5,7,)= ) F,, (4) sin jfru (3.27h)
As a result of applying the procedure described above,
the following systems of ordinary differential equations re
sult (D denotes the operator -%):
d4
2r_ /3
7t
D2
- [re (1 /u) j1 + 2gj
X, -
re/3(l + /y)j
77
DY;
mg t2a/3 2/3mbr
DZ; = rL K,, + H,
2/o
77* 7 7
(3.28a)
re /?(1 + /O j
TT
DXj +
re /32(1 yu) ,
D2 (2rc j + 2)
77
t3a mj 77
2/o
2mbre
zy = K*j + H*j
(3.28b)
P, m/3v
+ + { D* -
Pfm/32j'2 bGv + 4
D
P# mj 'n2 o(mt2(l re k/e ) jztt 2
2bGy + 4
K,; + K + H
20
Vy
2GV
'Vy Jy
(3.28c)


59
re
2nl
ft\ 1
mbre i
+ iH^
(3.28d)
where j = 1, 2, ....
The boundary conditions (3.21) on 4 = 0 and 4 = 1
become, for the case of simply supported edges,
ft /jY; (l) = 2B0;' (4)
d l
(3.29a)
jTTX, (4) + ft ^7- = BC; (4)
d4
(3.29b)
Z; (4) = Bej (4)
(3.29c)
*%<*> E (4)
d4 (4)
(3.29d)
dY0(4)
ft d4 co (4)
(3.29e)
where j = 1, 2, ....
The boundary conditions for the clamped and free
edges similarly follow.
The complementary and particular parts of the solu
tions to equations (3.28) are denoted by the subscripts c
and p respectively as follows:
X, = Xcj
+ xPJ
(3.30a)


60
Y; = Yc j + Y pj (3.30b)
Zj = Zcy + "Lpj (3.30c)
Y0 = Yoc + Yop ( 3.30d)
Since equations (3.28) are linear with constant co
efficients, solutions of the following form are assumed:
Xcj
= AjeXj^
(3.31a)
Yc,
= Bjeij5
(3.31b)
Zc ,
= Cjex'*
(3.31c)
Y0c
= B0ex*
(3.31d)
Substitution of the above relations into the homo
geneous form of equations (3.28) reveals that Xj satisfies
the following quartic equation:
(re, )
1 -/x
2Pj
T7H , g (1 + ~ i ) j rt^ie j j
bG y t OL ~ ~
bGyret + gj^k
ie
+ j 2re (g 1)
2P,
T~(r~ 1) 1
bGy ret oc 1 yu
,
+ 11 -
= 0
(3.32)
where


61
ej - J
Further,
TT
A = TT
/? / re (1 -yu)
If there are no repeated roots of equation (3.32),
the complementary solutions are given by
g
c = /
(3.33a)
(3.33b)
(3.33c)
Y
OC
B,
X o 5
+ B,
-x01
(3.33d)
The constants Bj and CJL are related to the constants Aj
by the relations
Bji b JL A j
Cj = c Jt A ji
where


62
j rr
re/32 [2 g(l + /<)]aj¡* tt2
re(l-//)j + 2g]
/3*jc
reg/3\l rr1 [2g(rc
j" + 1) j 2 re(1 t/')'
2/>
Cjc =
m/3gt2a AJ(
2re/92A)
v'
0.34)
re( 1 +/J) j/3 Ay,
(1 /-<) j2 re 2g by,
The constants A_,, are determined by substituting the
complete solutions (3.30) into boundary conditions (3.29);
eight simultaneous algebraic equations result, and their
solution may be written in the form
where Aj is an eight-by-eight determinant
Aj = | a*, |
whose elements are given as follows for the case of simply
supported edges:
a/
=
{/ikJL MiTThJL )eAjt
ai,L*/
-
(/3Ay /^j ^by_ ) e
a2L
=
i/3*jL ~ )
a2,L*l
=
~{/3Xjl )
a3L
=
(jrr + ftkJL hjL ) eXjL
aj,L*l
=
(jrr + byL ) e"^
a
=
]n + /3Xjl bJL
aH,Ltl
=
jrr + (3Xjl hjL
=
CJL ejL
aS)L-'l
=
-cJL e~^JL
=
CjL
at>,L*l
=
~CjL
=
A ^
*JL CJL e
a7
=
~XjlcJl e"Aji
a?i
=
Ai CJL
a %,L<1
=
V 2
*JL CJL


63
where L = 1, 3, 5, 7. The DJt are the determinants formed
by replacing the i-th column of Ay by the column vector
2Bqj (1) ft
2BOJ (0) -ft
dXp, (1)
dS
dXpj (0)
d
+ /iYPj (1)
+ /jttYPj (0)
Be,
(1)
- j 77 X P y (1)
" ft
Be,
(0)
- j 77 X pj (0)
~ ft
Bey
(1)
- Zpy (1)
B (0)
- Zp, (0)
BSJ
(1)
d2Zpy (1)
dS*
B 5,
(0)
dlZPj (0)
d4
dY PJ (1)
di
dYp, (0)
d%
Also,
B
Bco (0) ft
dY,
- B.(
n J
(1) + /3
dY0P (1)
d^
or
/3A0(e' ex )
B,
Bco (0) ft
dYOP (0)
dl
- Bco (1) + ft
dYop (1)
d£
ft^o(eAo e>0 )
For the case when the edges £ = 0 and £ = 1 are
clamped, the elements of A; are given by


64
au
ss
A;,
e JL
a Ij L + t
=
e ~XjL
aZL
=
i
1
a3i
=
(j TT + ft XjL bJL ) e
XjL a
aj ,L*I
=
(j TT + ft KjL hJL )e'JL
a m
=
jrr + /3Ah bJL
a*,L'1
=
j77 + fi AJL
bJL
aSL
=
cJL e '*
a5,L'l
BB
-c JL e ~XjL
a<,L
=
CJL
=
~CJL
au
SB
^JL CJL e
*JL
ai)Lti
=
A c 0
*JL '-JL e
atL
=
*JL CJL
at,L + r
=
^JL CJL
where
L = 1,
3, 5,
7.
The Dj are
formed from
the column
vector
cj
(1)
- XPj (1)
Cqj
(0)
- XpJ (0)
Be j
(1)
- jTTXpj (1) -
ft
dY pj (1)
dt
B< j
(0)
- jITXpj (0) -
ft
dYpj (0)
d4
Be j
(1)
- ZPJ (1)
Be,
(0)
- Zpj (0)
c,y
(1)
dZ pj (1)
dK
C3J
(0)
dZ pj (0)
dt
For the case when the edges 1=0 and 1=1 are free,
there results


'Y-
lrXirO( x0574)^
UI
-z ui + I
ir~
7 r.
ir-
T-
Z) 77 -/l ^7^
' Vti/J ¡r'aj
Jt/- lr3ilrYtSZ) ~
1Ti-e
tjr/- ^o/'Y'/)-
=
/7Vb
7rq 7/v6/ + ixf
=
/*7fAc
.9 ( 7rq irYi/ + /if)
=
( 7"q//fr/ irY(/)-
s
/'7*7^
( 7'q//(>/ lry {/) -
=
"7V*
V'oCD^jS. -
*0}
ui
7 7.
ui + x
Yru7C(rS- z) r ir\ z £/
q,//
^'d
Ify 9
Vr=COit3)2 -
UI
'01
i ui + x
77.
7A
7 r.
7 f-
If
=
7 T3
rf7/_ 7\?£/
S3
79p
Or/- 'r0;rYii/)
=
7ie
7rq 7r\e/ + /if
=
7*p
( 7rq irYC/ + z/f)
=
7fp
7rqutr/ 7rY£/
=
7/p
7rquCr/ 7ry (/)

7/p
S9


66
Pf/3
[/)V (2 /*) J1^
772b
Vi
1 + m
CjL ~ tG,
m
where L = 1, 3, 5, 7. The DJt are formed from the column
vector
dX.j (1)
2Baj (1) -ft tL + MjrrYpj (1)
2Ba. (0) /3
ax
axpj (o)
ax
+ /ujnYpj (0)
Bcj (1) jnxpj (1) ft
Be j (0) j TTXpj (0) ft
aYpj (i)
ax
aYpj (o)
ax
,2 a2Zpj (i)
77 Fej (1) ft 2 j TT1Zpj (1)
d4
- Fc; (0) /3* d -Z ^-Q)- + /ij 2nz ZPj (0)
(1) ~
7T b
d|
_ (2 )jVdz(l)
' d£3 d£
+ 1- + m tG tX pj (1) + ^(ttG, )dZ^(1)
m b dS
(0) -
Pf/3
TT^b
/3:
.a^Zpj (o)
d£
x-/ , j2 dZ(0)
f-1 (2
dl
+ m tGxXPj (0) + ^(ttGjd--^--(--
m b d X
Therefore, from equations (3.19), (3.25), (3.30),
(3.31), and (3.35), the solutions for the displacement


67
components u, v, and w are
u(§,t) = ^f, (4) + (1 *j)f(4)
+
77- eA>,{ + X,, (4)[ sin jrr^
(3.36a)
j-.t c--l
v(|,>?) = 7¡l
9, (5) -
fiM df, (V
2 d4
- (1 y¡r
9U) ~
fiM df0U)
2 d?
= 1 C--t
*>J eA-? + YPJ (4)
^j
cos jrr^
B0/ eAo + B e-Aot + Y0p (4)
(3.36b)
w (£,>?) =
/7
6P,
>7 (>7 ~ 1)
(? + l)h, (4) + (2 rj)h0 (4)
J 1 t
Dji
'ji
i */ ^
+ z
PJ
(4)
sin j?T>7
(3.36c)
These equations completely determine the displacement
components.
Now that the solutions have been completely specified,
further remarks can be made about the validity of forming
partial derivatives of these quantities with respect to 4>
Unfortunately, because of the extreme complexity of the
terms in each infinite series, it is not possible to


68
actually examine the uniform convergence of the derivatives.
All that can be said is that actual evaluation of the series
for w at the center of the plate indicated that the terms
converged. Uniform convergence was not shown.
Solution for a Simplified Case
In the preceding part of this analysis the solution
of a very general case of loading on a sandwich plate has
been presented. The remainder of the analysis will utilize
certain restrictions which aid in the tractability of the
problem, while still representing a broad category of
results.
The following assumptions are now made:
1. Bending of the faces is neglected (Pf = 0) ;
2. The temperature is constant through the thickness
of each face;
3. The temperature distribution and the edge loading
are symmetric with respect to perpendicular bi
sectors of the edges of the plate;
4. The transverse loading q" is constant.
As a result of these restrictions, the following re
lations hold:
f, (5) = f0 U)
mb
r (
RU.DdS
'h
/Hi + m)
J
(3.37a)


69
9/ U) = 9o (£) = n>R
h, (?) = h0 (?) = 0
Mr = 0
c
c
c
The particular solutions can be determined from
tions (3.28) after the following relations are noted:
T, (?, rj) = -
re/3mb. 2 .SR^ .
a + i +/<)>rr(5'i)
7r 1 + // o?
r ?
+ 2g
mb
/3(1 + /<)
R(?i l)d?
'/a
c
t2 (,*?) = -
^(1-^(2,- l)mb,l _^_,0
+ ( 2 1)
mb (1 - ^ ) R(£, 1)
1 + /<
c
Tj (?, >?) = /O
mb (1 + g ) R (?, 1)
1 + yU 1 + /U
c
and after the following series expansions are noted:
PO
1 =" H isin }nn
7 =
c
. 37b)
.37c)
. 37d)
equa-
. 38a)
. 38b)
. 38c)
. 39a)


70
_4_
77 a
oo
...
j rrri
(3.39b)
bR(S,n)
OO
~
^0
dR(K,n)
sin ]v r\ d>i sin jtttj
(3.39c)
R(£,/?)
dr¡
2L
<3 R (£ rL)
b-q
COS jTTrj dq cos JiT^
(3.39d)
where the series in the last two equations contain only
terms with an odd-numbered index because of the symmetry of
R{%,r¡). Thus from equations (3.38) and (3.39) there result
f 1
K
n
f? sin j nr, dn
K = 2
R
bn
cos jrrr} dr¡
K" nj
K*/ = 0
K
20
(3.40a)
(3.40b)
(3.40c)
(3.40d)
0
(3.40e)


71
H.
4
TT
re/3
TT
mb (
1 + /J-
i xdR(5,l)
- r + 1
- 2g
mb
/3( 1 +/0
I
RU,l)dS
'/z
1
J J
( 3.40f)
H
2./
" £ V 1,*1 -
4*
mb (1 -
M
1 + //
)R(C-1)
(3.40g)
hjj ~ n P
mb(1 + g
1 + /2
)RU,1)
1 +/x
1
j
(3.40h)
h20 = 0
(3.401)
where j = 1, 3, 5, ....
Then the particular solutions XPJ YPJ and Zpy are
found from equations (3.28) after Pf has been set equal to
zero and expressions (3.40) have been substituted for the
quantities on the right-hand sides of these equations.
Furthermore, it is found that for any symmetrical tempera
ture distribution
op
0


72
The particular solutions for two specific temperature
distributions were found. For a constant distribution R = R
these solutions are
= A; 4 + By
Y Pj = C j (3.41)
za/ = Dy
where
8mbg R
77/3(1 H-yuJj [re(l -yu)j2 + 2g]
C;
ay Sj /3j &j
Dy
y yy y
aj ?y ftj
and
; = 2 (re j 2 + 1)
fij =
tzo(mj7T
2/>


73
v1 j
mb (1 -
1 +/<
)R
8re mbgRc
^[ri(l ~^)j2 + 2g]
Sj = p]TT
5/ =
ofmt^d rek,e )n-ijz
X, =
mbq0
= + p
2Gy ^
mb (1 + ) Re
1 +
8/ombgR
77j(l +/u)[re(l pi) y + 2gj
Here j=l, 3, 5, ....
For a double sinusoidal distribution of the form
R = R, sin 7T ^ sin Trr¡, the solutions for j = 1 are
A pj
Y,,
Zpj
where
Aj
Ej
= A j COS TT £,
- Ej sin tt\ + Fj
= Hj sin + Ij
(3.42)


74
Here
/ = -
2mbre
77
R
P, is the
determinant
*y
ftj
0
fi
Uj
where
Vy = 2rt/3x + re (1 //) + 2g
7C; = re/3(l + /*)
rj = p g /3
^ = re/3a(l /u) + 2re + 2
mgtV/i
p, = L 77
' 2 P
77ai2m
H = 4 (/3 9 rekre + D
and Py is the determinant formed by replacing the i-th
column of Py by the column vector
ft
1
0


75
Further,
= -A,
a,
- /3j Sj
where
a,
= A,
X>j /3j Sj
A,
4 mbqc
7T j 2Gy
For j = 3, 5, 7, ...
A, = Ej = H, = 0
The coefficients in (3.27) are determined from equations
(3.37) as
Bay (0)
Bay (1)
4
77
mb
A
(1 m)
R(l,l)
1
j
+ 2mb
R(1, r\) sin jTT >7 d>7
BCy (1)
-BCy (0)
9R(1,1)
(1
1 + //
Bej (1) = Bey (0) = 0
BSJ (1) = (0) = 0
Bco (1) = Bco (0) = 0
(3.43a)
(3.43b)
(3.43c)
(3.43d)
(3.43e)


76
Caj
(1)
=
(0) = 0
( 3.43f)
(1)
= c9J
(0) = 0
(3.43g)
F*J
(1)
= F;
(0) = 0
(3.43h)
(1)
= -Fs, (0) =
4(1 + m) tGb
r J
R(?,l)d?
0
1
(3.431)
F3J
n/3(l + m) j
j
where j = 1, 3, 5, ....
The assumption of a temperature distribution sym
metric in the ^-direction makes possible a simplification
of the complementary solutions (3.33). The function Ej (X)
is defined as
Ej (?)
a,
e J
+
- X. (5 '/*)
e J
and is seen to be even with respect to the line % = 1/2,
which bisects the plate; the function
0, (?)
e
A j U-'/i)
e
X J ($ Vi )
is seen to be odd with respect to ? = 1/2. Each pair of
terms of the form
VI
+ A
. *;
that occurs in the complementary solutions (3.33) may be
I


rH| CM
77
written in terms of the newly defined functions as
(A
+ A
'/2 j
)E;
i(A-
- A
- '/j A,
)0
Under the present assumptions, the horizontal dis
placement component u will be odd with respect to \ = 1/2,
while the horizontal displacement component v and the verti
cal displacement component w will be even. The expression
(3.36a) for u in terms of the functions Ey and 0y becomes
mb
u =
/Hi + /0
r 5 M
r *
f 3 r
R(5,l>d4 + ) <
£ 2
L. L
A. 1
Dy,
Ik 1
'/i x.
D
y, 2 k
V,
A,
)E,* + (
Dy',z/k r
'/J Xy,
- h x j
) 9/a
sin j77 yj
Since u must be odd and the functions EJk and 0Jk are lin
early independent, the coefficients of E must equal zero;
Dyy = -ekjl Dy,
Djy = -e^1 Dyj
Dy* = -eAji Dj j
there result


78
Therefore the solutions (3.36) become
u =
mb
/3(1 +/u)
~ 5 oo
j
R(5,Dd4 +
t
r a,, t Aj.d-i)
e J e
4*
c = 7
Djj
A;
+ X
Pj
sin jrrr¡
(3.44a)
v
mb (1
M
1 +yu
)R(M)]
+
j=l,S. <'
A,, *
X.| 0 ?)1
N
e J
+ e J
+ Y pj

J
(3.44b)
COS j 7T 77
W
jV
D
JL
+ e
(i-V
+ Z
/>./
sin jTrr¡ (3.44c)
Here the are the roots of the cubic equation
(rGJ i' ) (9G/ + rk/e J Zqj ~ gJik/e )
U.
+ j J re (g 1)
/- j2k/e (1 ~~)
J. /A
= 0
(3.45)
where
6/
(
2
j
2
Further, Aj is the three-by-three determinant


79
Ay = I aKL I (3.46)
where K,L = 1, 2, 3; the elements aKL are, for the case when
the edges £ = 0 and ^ = 1 are simply supported,
a/ = (eXjL + 1) ^jnhJL )
a2L = (eXjL 1) (jrr + fthJL hJL )
a3L = (eXjL + l)cA
where L = 1, 2, 3. The Dyt are the determinants formed by
replacing the i-th column of Aj by the column vector
dXPJ (1)
2Baj (1) /3 f- + /ujnY'j (1)
Bcj (1) jrrXp. (1) ft
-Zpj (1)
dYPj (1)
d|
(3.47)
where the coefficients Baj and BCJ are given by expressions
(3.43a) and (3.43b).
For the case when the edges \ = 0 and ^ = 1 are
clamped, the elements of A¡ are
a
tL
= e
- 1
a
ZL
(e
- 1) (jrr + fthjL bJL )


80
aJt = (eXjl + D CJL
and the Dyi are formed from the column vector
-Xpj (1)
Bcj (1) ]TtXPj (1) /3
-Z?j (i)
dYPj (1)
For the case when the edges £ = 0 and \ = 1 are free,
the elements of A j are
aIL = (eXjL + 1) (/3Ay /*}rrbjL )
a2L = (eXjL 1) (jTT + /3jl hJL )
aJL = [eXjL 1)
l + m^- /3 ^ a
tG, £(tatG, )cjL j
m b
and the Djl are formed from the column vector
dXPj (1)
2Baj (1) /? -TZ + M^YPS (1)
Bc; (1) jttXp/- (1) /3
dYPj (1)
d l,
1 + m /3 .a2-- dZPj (1)
F3j 1) tG, XPj (1) Ht'tG.)
m b d X


81
The coefficients F$J are given by expression (3.43i). The
quantities bJt and c_,, are given by expressions (3.34)
An observation can be made on the roots of the char
acteristic equation (3.45). If the equation is expanded
into a cubic equation of standard form
A3 + aA1 + bA+ c = 0 (3.48)
where
A = A/
it is seen from the definitions and physical meaning of the
coefficients that
a < 0
b > 0
c < 0 for k,e < r
1 + rej3
j 2
c > 0 for k,e > ~¡-
' 1 + rej
From Descartes' rule of signs it follows that the number of
positive real roots of the cubic equation will be either one
j1 J2
or three if k/e < rr or zero or two if k/e >
l+rej l+rej
In general, then, as k/e which represents the compressive
edge loading, increases past this value, complex values of
are to be expected.


82
A further observation is that when 0Nvy = 0, i.e.,
no compressive edge loading is present, a double root for Aj
occurs; in this case the solutions (3.44) approach the fol
lowing limits as 0Nyy -* 0:
u
mb
/3(1 + ju)
* I
R(£# 1) d£, +
J'/z
(1
5)e
*i
\j, 0-0
+ X
pj
sin jrrri
(3.49a)
v = (2r! 1)^(1 )R(£,1)
1 2 1 + /<
o
Dyi
. A'
*>,7
t Aj f 5 / i r \ A ,i (* U
5 e + (1 5 ) e "
jd,i
.
ejt e + e
V, (-<>'
Y" hJi
O
*
<
1
A, '
1
cos jrrri
(3.49b)
oo
f
r
Cj,
Ke'J' + (1 ^)eA/,(,'iJ
i
* /.j...


83
+ 9 it
+ e
('->
Dj j
A,
z
r c
p ^
Dh z
/1 c-,
t -;
0
+ e j
Aj + Z"
sin j7T>]
(3.49c)
Here the Ayl are the roots of the equation and are easily
found to be
*#,
/3
'Si
77 /2g + re j 2 (1 -/*)
ft
reg(l -/*)
and A; represents the determinant
I aK
whose elements are given by
a/j = ft
1 + {1 + Xjt)e
- /ujn
e + (bj, + e)e'
= j"e Jl + /3
~(xjiej/ + hjft +
(bj, + + by,]
+ (Cj, + g)eA-
a/ = (1 + eXjL ) (/3AJL /U)nbJL )
a2L = (eXjL 1) (jrr +/3bjL*jL )
a
3L
(e
1) c
JL


84
where L = 1, 2. The Dj are the determinants formed by re
placing the i-th column of Aj by the column vector (3.47).
The hJC and cJt are given by equations (3.34);
. A re J 2 (3 ~ M) (1 ~ §) + 2g
Bji nj rej2 (1 + /j) (1 g) 2g
4/0/3
tjt
mt loc tt 2 j 2
re j2 (3 /Of) +
r e j 2 (1 + a) + 2
while Ypj and ZPJ are the particular solutions of
equations (3.28).


CHAPTER IV
COMPARISON WITH EXPERIMENTAL RESULTS
Although there are many reports of experiments on
rectangular sandwich panels without thermal effects, for ex-
33 34
ample, those of Boiler and Kommers and Norris, the only
experiment involving thermal deformation known to the author
24
is the one performed by Ebcioglu et al.
In this experiment a rectangular sandwich panel was
simply supported on all four edges and a temperature gradi
ent was applied by simultaneously heating and cooling oppo
site faces of the panel. No compressive load was applied
and the deflection was measured at the midpoint of the panel.
The properties of the panel were given as follows:
t' = t" = 0.010 in.
t = 0.395 in.
/u = 0.3
E' = E" = 30xl06 psi (4.1)
G* = 39,825 psi
Gy = 66,400 psi
a = b = 6 in.
85


86
and thus
Ke = 0
Pe = 1,853 lb/in
t = 1.0253
re = 0.067214
o' = a" = 7.34xl0~6 in/in F (4.2)
qQ = 0
g = 0.6
m = 1
/3 = 1
From Ebcioglu et al. and the original data sheets of
the experiment, it was found that for the four trials made
in the experiment the distribution of the temperature gradi
ent R, defined by equation (3.13), could be approximated by
multiples of the function
-4
R = (1.25 sin 77>7 + 1.373 sin nrj sin rrl)10 (4.3)
A measure of the error caused by representing the tempera
ture distribution by equation (4.3) was calculated from the
formula
L la et |
A Tmax H i
Error
(4.4)


Full Text
H0026



THERMAL BENDING OF SANDWICH PANELS
UNDER UNIAXIAL LOADING
By
EDWARD LINDE BERNSTEIN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
April, 1964

ACKNOWLEDGMENTS
The author wishes to express his appreciation to the
members of his supervisory committee: to Dr. I. K. Ebcioglu,
chairman, for the advice and encouragement which he tendered
throughout the course of this investigation? to Dr. W. A.
Nash, whose many kindnesses have been of great assistance;
and to Drs. A. Jahanshahi, J. Siekmann, and R. W. Blake for
their helpful discussions with the author.
The author wishes words were adequate to express his
appreciation to the only member of his special committee--
his wife, Eileen. She has served as secretary, editor, ad¬
visor, and most importantly, inspiration to success.
11

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ii
LIST OF TABLES V
LIST OF FIGURES vi
LIST OF SYMBOLS vii
ABSTRACT xii
Chapter
I.INTRODUCTION 1
Historical Background 1
Definition of the Problem 5
II.DERIVATION OF THE EQUATIONS 9
The Strain Energy Expression 9
The Equilibrium Equations 18
The Boundary Conditions 24
III.SOLUTION OF THE EQUATIONS 32
Transformation into an Uncoupled System 32
Solution of the Membrane Force Equations .... 39
Solution of the Second Set of Equations 41
Convergence of Series Solutions 51
Development and Solution of the Ordinary
Differential Equations 55
Solution for a Simplified Case 68
IV.COMPARISON WITH EXPERIMENTAL RESULTS 85
iii

TABLE OF CONTENTS—Continued
Chapter Page
V. COMPUTATION AND DISCUSSION OF THE SOLUTION .... 92
Computation of the Solution for Various
Parameters of the Panel 92
Discussion of the Results 96
VI. SUMMARY 108
APPENDIX 109
LIST OF REFERENCES Ill
BIOGRAPHICAL SKETCH 115
iv

LIST OF TABLES
Table Page
1. Experimental Data 87
2. Theoretical Results 90
3. Comparison of Buckling Load Values 104
4. Experimental Temperature Gradient 110
v

LIST OF FIGURES
Figure Page
1. Dimensions of the rectangular sandwich panel .. 6
2. Variation of displacement u through the
thickness of the sandwich panel 34
3. Resolution of membrane forces into a force and
couple system at the "neutral" surface 34
4. Comparison of theoretical and experimental
values of the deflection at the center
of the plate 89
5. Deflection at the center of a plate under a
temperature gradient R = R0 sin rrí sin tt>i
as a function of edge loading 97
6. Deflection at the center of a plate under a
constant temperature gradient as a
function of edge loading 98
7. Deflection at the center of a plate under a
temperature gradient R = R0 sin rr| sin 7T>j
as a function of edge loading 99
8. Deflection at the center of a plate under a
constant temperature gradient as a
function of edge loading 100
9. Comparison of the deflection at the center
of a plate under a temperature gradient
R = Rc sin ttZ sin 7r?i as a function of
edge loading 105
10.Dimensions and data points of the sandwich
panel 109
vi

LIST OF SYMBOLS
a -
Length of panel in x-direction
a*y -
Elements of the determinant A,
b -
Length of panel in y-direction
b*y -
Quantity defined in equations (3.34)
BOJ , Bcj , Bej , B9J - Fourier coefficients defined in equa-
tions (3.27)
CJ£ “
Quantity defined in equations (3.34)
C<- -
Function defined in equations (3.22)
C°j ' C9J
- Fourier coefficients defined in equations (3.27)
D -
Differential operator d/d5
D -
Plate modulus
Dji -
Determinant formed from A; (see p. 63)
e„, exy , ey - Linear components of strain
E -
Young 1s modulus
f, ~
Auxiliary functions
F¿ ~
Function defined in equations (3.23)
Fey « F»y
- Fourier coefficients defined in equations (3.27)
g -
G*/Gy
9o> 9, -
Auxiliary functions
vii

Gy -
Orthotropic shear moduli of core
h', h" ~
Distances to "neutral" surface
h, -
Auxiliary functions
H*y "
Fourier coefficient
Hy / d y y ,
Hy - Couples formed from resolution of membrane
forces
Hr "
Thermal couple formed from resolution of thermal
membrane forces
Jc ~
Function defined in equations (3.20)
k/e "
Dimensionless edge loading parameter -0NyyPe
Kv -
Fourier coefficient
m -
My , M yy ,
My - Bending moments in faces
Mr -
Thermal bending moment in faces
My‘, My y*
My*- Bending moments applied on edges of faces
My , Nyy ,
Ny - Additional membrane forces relative to state
of initial stress
Nr -
Thermal membrane force
ÑX , Ñyy ,
Ny - Resultants of additional membrane forces
ÑT -
Resultant of thermal membrane forces
0N*, oNxy, 0Ny - Initial membrane forces
»NX* oNxy* 0Ny*— Applied initial edge loads
oN„, 0N,y , <,Ny - Resultants of initial membrane forces
viii

7 N *, TN,y , rNv “ Total membrane forces, defined in equations
(2.18)
Pe - Buckling load of a simply supported, infinitely
long sandwich strip with a rigid core
Pf - Sum of buckling loads of each face of a simply
supported, infinitely long sandwich strip with
a rigid core
q -
Q X I Qy “
Qx* Qy*"
r e "
R -
s
s
// _
s¿ -
Distributed transverse load acting on faces
Shear force resultants in core
Vertical shear forces applied on edges of faces
Dimensionless parameter
Dimensionless temperature gradient parameter de¬
fined in equation (3.13)
Art + t->
- fit + t">
Function defined in equations (3.21)
Sxi S*y, Sy - Components of initial stress
t', t" - Face thicknesses
t -
Core thickness
t -
t*, t*y, ty - Components of stress relative to initial state
of stress
T - Change in temperature from a constant initial
temperature
T,, Tz - Coefficients of the temperature distribution
function
ix

u, V -
Displacements of middle surface of upper face
relative to "neutral" surface, defined in equa¬
tions (3.1)
A# A/
U, V -
Displacements of "neutral" surface, defined in
equations (3.1)
Ú, v -
x- and y-components of displacement in core
A / A //
u , u ,
v', v" - x- and y- components of displacement in the
faces
U„, V0 -
Functions defined in equations (3.19)
U,, U2 , V, , Vz - Coefficients of displacement distribution
functions in the core
v* -
Virtual work of edge moments Mx* M,/, My*
v* -
Virtual work of edge forces eNx* oN^y*, <,N/
V? -
Virtual work of transverse loading q
Va -
Virtual work of edge shear forces Qx* Qy*
w -
z-component of displacement
A
W -
Dimensionless deflection parameter
Wo -
Function defined in equations (3.19)
Wc -
Strain energy of core
w, -
Strain energy of faces
X, Y, Z - Solutions to ordinary differential equations; see
equations (3.25)
Xc , Ye ,
Zc - Complementary solutions
XP, Y,,
Zp - Particular solutions
x

a -
« -
ft ~
¿*,
? -
n ~
ej
A;
A,
/*“
4 -
/» “
U>XI
Coefficient of linear thermal expansion
Dimensionless parameter 2t/b
Aspect ratio of panel b/a
Determinant
yy , £y - Complete components of strain
z - s
y/b
Quantity in cubic equation
Coefficient of exponent in complementary solution;
see equations (3.33)
Poisson's ratio
x/a
1 m A
Dimensionless parameter ^t
coy, cjt - Components of rotation
xi

Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
THERMAL BENDING OF SANDWICH PANELS
UNDER UNIAXIAL LOADING
By
Edward Linde Bernstein
April, 1964
Chairman: Dr. I. K. Ebcioglu
Major Department: Engineering Science and Mechanics
The deflections of a rectangular sandwich panel sub¬
ject to a thermal gradient, transverse load, and combined
edge loading were calculated. Two opposite edges of the
panel contained rigid inserts and were simply supported.
The other two edges were simply supported, clamped, or free.
Small deflection theory was assumed, as well as an ortho¬
tropic weak core. The effects of bending in the faces were
also included. The five coupled, linear partial differen¬
tial equations governing the deflections were transformed
into two sets of equations, one of which is the well-known
equation of plane stress. The remaining set was simplified
to neglect all edge loading except a uniaxial force and was
xii

solved by the use of Levy-type infinite series, which reduce
the equations to ordinary differential equations with con¬
stant coefficients. The convergence of these series was
investigated. A standard solution of these equations was
obtained, and the resulting expressions for the displace¬
ments were written in closed form.
Values obtained from the solutions were compared with
experimental values reported in the literature, and agree¬
ment was found to be good, although shortcomings of the ex¬
perimental values are mentioned.
The variation of the deflections and the buckling
loads for panels with varying elastic properties, dimen¬
sions, and edge loading were calculated on the University
of Florida IBM 709 computer, and these data are presented
in graphs. The values of the buckling loads were found to
be somewhat less than those reported in a less exact analy¬
sis. It was noted that, in general, a reversal of deflec¬
tions took place and that buckling modes were characterized
by a large number of half-waves.
xiii

CHAPTER I
INTRODUCTION
Historical Background
Legend has it that the terra "sandwich" arose in 1792
when the Fourth Earl of Sandwich devised a convenient way to
have his lunch without interfering with his daily card game.
Since that time the term sandwich has come to mean more than
merely a way to eat. The term applies to any object con¬
structed like a sandwich, that is, composed of three layers,
the middle one being of a different material tha$ the outer
layers. Sandwich construction is a type of composition that
has recently begun to achieve wide application. The con¬
struction is characterized by two thin outer layers known as
the faces and a thicker inner layer known as the core. The
faces are made of a stiff dense material and resist bending
and membrane stresses, while the core is made of a light ma¬
terial offering resistance mainly to transverse shear and
compression.
The original, and still most widespread, application
of sandwich construction is in air and space vehicles; its
1

2
high strength-to-weight ratio is well suited for this pur¬
pose. One of the earliest applications was for the skin of
the World War II DeHaviland Mosquito bomber, in which birch-
wood faces and a solid balsa core were used.'*' At the pres¬
ent time the faces are frequently made of steel or aluminum
with a core made of either an expanded cellular material or
a corrugated metallic sheet.
Sandwich construction is also found in curtain wall
2
panels for office buildings; with a foam plastic core it is
used in refrigerator construction; it is also suitable as a
hull material for boats when a balsa core is used.
Because of this wide use of sandwich construction,
especially in panel form, the elastic analysis of a sandwich
plate is a timely and important problem. The nature of the
core necessitates the consideration of several effects not
present in the usual analyses of thin plates. The strength
of corrugated cores will obviously depend on the direction
of the corrugations. Cores made of expanded materials are
constructed of thin strips of material which run in one di¬
rection. The properties of either type of core are thus
orthotropic. Because of the relative thickness of the core,
the effect of shearing deflection is important. Since cores
contain air spaces, and thus act as thermal insulators,

3
sandwich construction used in an air vehicle will be subject
to a temperature difference between the two faces. Thermal
stresses arise from this temperature gradient.
The earliest study of a sandwich plate was performed
3
by Gough, Elam, and DeBruyne; they considered the problem
of a thin sheet supported by a continuous elastic foundation
in order to study "wrinkling," or instability of the faces
characterized by a large number of buckling waves. Later,
4 5
Williams, Leggett, and Hopkins and Leggett and Hopkins
examined the Euler type of buckling of sandwich plates as a
whole; they accounted for shearing deformations in the core
by using the "tilting method," as termed by March,6 in which
a line originally normal to the middle surface tilts during
deformation through an arbitrary angle. Use of an approxi¬
mate strain energy method determined the buckling load.
March and others extended this method to include the effects
of orthotropic cores and orthotropic faces of differing
7 8 9
thicknesses and elastic properties. ' ' Libove and Batdorf
developed a method of analysis in which the sandwich plate
is treated as an orthotropic plate with the effect of shear¬
ing deformation included.'*'6 Reissner developed differential
equations for the finite deflection of sandwich plates.'*'1

4
Hoff used the assumption of a weak core which resists
only vertical shear and the principle of virtual displace¬
ments to derive exact equations for the bending of a sand-
12
wich plate. He was able to solve the problem for a simply
supported plate by assuming that rigid inserts were set into
the core edges. In this way the boundary conditions were
altered to permit a solution.
In 1951 Eringen used the principle of virtual work to
derive extremely general equations which account for the ef¬
fects of compression and flexure of the core.^ In 1952
Gerard, by an order of magnitude analysis, examined the
limits of validity of the simplifications commonly assumed
14
in sandwich plate analysis. Goodier and Hsu considered
15
nonsinusoidal buckling modes. In 1959 Chang and Ebcioglu
extended Hoff's theory to include an orthotropic core and
dissimilar faces and found the buckling load for a simply
supported plate.10 Recently, Yu, in a series of papers, has
included the effects of rotatory inertia and shear in the
faces.17'18'19'20
Thermal stresses in sandwich plates were treated by
21
Bijlaard, who used a homogeneous plate approximation.
Chang and Ebcioglu extended their approach to include the
22
effect of a constant temperature gradient, and Ebcioglu

5
later solved the problem for a temperature gradient which
23
varies arbitrarily across the area of the plate. Utiliz¬
ing Hoff's approximation for the boundary conditions, he
presented a solution for the bending of a plate simply sup¬
ported on all edges. Ebcioglu et al. also give experimental
results for the deflection due to a thermal gradient.2^
Definition of the Problem
It is the aim of the present analysis (1) to extend
Ebcioglu's work in order to develop equations which include
the effects of bending of the faces, a linear temperature
distribution through the thickness, and various boundary
conditions, (2) to solve the equations for the exact bound¬
ary conditions rather than making Hoff's approximations, and
(3) to present graphs illustrating the effects of various
parameters of the plate on the deflection and buckling load.
A Cartesian coordinate system is defined for a rec¬
tangular sandwich plate as shown in Figure 1. The middle
plane of the core lies in the xy-plane, with two adjacent
edges lying along the x- and y-axes; the z-axis points down¬
ward. In the notational system employed here a single prime
denotes quantities related to the lower face, a double prime
those related to the upper face, and a superposed bar those
related to the core.

Figure 1.
Dimensions of the rectangular sandwich panel

7
The plate has lateral dimensions a and b along the
x- and y-directions respectively, and the thicknesses of the
upper face, core, and lower face are t", t, and t' respec¬
tively. The faces may differ in Young's modulus E, but have
the same Poisson ratio The core is assumed to be homo¬
geneous and orthotropic.
The temperature change T of each face is defined as
the change from some constant initial reference temperature
and is arbitrarily distributed in the x- and y-directions;
the variation in the z-direction through the thickness of
each face is given by the following equations:
T' = T/S + T/
ip= fji 11 g p 11
(1.1)
where T/, Tf", Tz', and T2" are functions of x and y only;
S = z - s (1.2)
where
s = s' = (t + t' ) for the lower face
s = s" = - -i(t + t") for the upper face.
Arbitrary transverse pressures q' and q" act normal
to the outer surface of each face, while on the edges of

8
each face there act applied axial forces 0NX', oNXy*, and 0Ny*
applied moments Mx* Mx/* and My? and applied vertical shear
forces Q* and Qy*, where each of these quantities is defined
per unit length of the boundary. The subscripts follow the
usual convention for stress notation. These forces are uni¬
formly distributed in the z-direction on each face and do
not act on the core, but on any edge where external loading
is applied a rigid insert is set into the core for its pro¬
tection. A more complete description of the insert is given
in the section on boundary conditions.
The Kirchoff hypothesis is assumed to hold for the
deflections of the faces. The temperature is considered to
be in steady-state equilibrium and the effect of deformation
on temperature is neglected. Account is taken of the fact
that elastic properties vary with temperature by permitting
each face to have a different value of E. The "weak core"
assumption is made, that is, the core is considered to be
so weak in its resistance to bending and axial stresses that
only the antiplane stresses Tz% , tzy , and Tzz are present.
The core is considered incompressible in the z-direction.
It is assumed that neither local buckling of the faces nor
bond failure between the faces and the core occurs.

CHAPTER II
DERIVATION OF THE EQUATIONS
The Strain Energy Expression
A macroscopic approach has been chosen for the deri¬
vation of the differential equations governing the five in¬
dependent components of displacement. It is felt that this
approach, based on the theorem of virtual work, more clearly
shows the assumptions involved in formulating a mathematical
model which adheres to physical reality yet allows a solu¬
tion; this approach also permits a more straightforward
derivation of the boundary conditions than would the alter¬
nate approach based on the microscopic equilibrium of an
element of the plate.
The theorem of virtual work states that if an elastic
body is in equilibrium, the variation of the strain energy
equals the virtual work of external forces acting on the
body. The expression for strain energy for each layer of
the sandwich plate is written and the variation taken by
applying the variational calculus.
9

10
The derivation of the strain energy expressions for
the faces proceeds from the consideration that they are ini¬
tially in a state of equilibrium under the action of large
membrane stresses. From this initial state the faces under¬
go additional deformations. The approach used will be based
he p
on that of Timoshenko and Herrmann.
The initial components of stress are denoted by S*,
Sy, and SXy . They are assumed not to vary through the thick¬
ness of each face, and to be of an order of magnitude higher
than the stresses produced during the additional deformation.
The following initial stress resultants are defined:
oNx = tS*
0Ny = tSy (2.1)
— tSxy
Since the plate is initially in equilibrium, the fol¬
lowing equilibrium equations may be written for each face:
áx + d y
^ ()o^y
ó x + by
(2.2)
0

11
The boundary conditions for the initial state are
oNy*= 0Ny
(2.3a)
on y = 0, y = b, and
(2.3b)
on x = 0, x = a, where aN/, aNy* and 0Nxy* are the initially
applied loads on the edges of each face.
The additional components of strain are assumed small
compared to unity; it is further assumed that the elongation
and shear terms are of the same order and the rotation terms
are of a smaller order. However, since products of initial
forces and rotation terms will occur, these latter terms
will be retained. Then the components of strain relative to
the initial state are
(2.4a)
(2.4b)
(2.4c)

12
(2.4d)
CJ y CaJ z
(2.4e)
1
2
(2.4f)
where e„, ey, ez, exy , e/z , and eiX are the linear strain
components and u>x, u>y, and are the rotation components.
Based on the Kirchoff hypothesis, the deflections in
the faces are given by
(2.5)
v
where u and v represent the displacement components due to
the stretching of the middle surface in each face from the
initial equilibrium state, and ? is given by equation (1.2).
The rotations u>z are considered to be negligible for
plate deformations. Then the use of equations (2.5) in
equations (2.4) produces the strain-displacement relations:
(2.6a)
(2.6b)

13
f ~ A (Aü + AY + dw dw _ 2 / w
*y 2 dy ix dx dy dxdy
(2.6c)
er =
A(|ü)' + i,W
2 dx 2 dy
(2.6d)
= 0
(2„6e)
-yz
= 0
(2.6f)
The additional components of stress t*, ty, and txy
are assumed to be related to the additional strains by the
Duhame1-Neuman equations. The error introduced by this
assumption is of the order of the ratio of the initial
stress to the elastic modulus of the face.
26
Furthermore,
the transverse normal stress crzi will be neglected in ac¬
cordance with classical plate theory. Therefore, the fol-
27
lowing equations for plane stress will apply:
t* = —^ -(£* + yClEy) ~
1 - yOÍ 1 ~
E«T
/**
t y —
1 -X
(£y + ¿¿Eje) ~
E«T
1 -
t X y —
1 + yU
where
14
The following stress resultants are defined in the
faces. The membrane forces per unit length are
â– *4.
N, =
J.
t,d? =
Et
*4
+ /, |v + I(|w^ + ¿ (|w(*
1 - //6iz[_c)x c>y 2 bx 2 by
- N,
Nv =
%
t/d? =
Et
«/_
*/z
1 -
by ^ c)x
+ h^)‘ * 4<^)2
2 by 2r bx
- N
r
(2.7)
N,
Í v2
tx/ d? =
* --t/~
Et
bu , bv
bw bw
2(1 +/0
by bx
bx by
where the thermal force per unit length is
* t/i
N„ =
E (X
^ 1 -/*
Td? =
J.
*/z
E«T21
1 - /U.
The bending moments per unit length are
•Vz
M, =
?t,d?= -D
J.
t/l
b2vi + ^ bXw
bx2
by 2
- M.
M, =
*4
?tyd? = -D
J.
Vz
bl w ib2 w
by1 ^ bx1
- M
(2.8)
M„y =
*/*
St*yd? = -D ( 1 - yu)
J_
Vz.
bz w
bxby

15
where the thermal moment Mr is
Mr = aD(l + M) T,
and the plate modulus is
D =
Et
12(1 - M¿)
expression
w = 1
The strain energy VIF in the faces is given by the
27
t„ ¿X + ty£y + 2t„y - 2«T(tX + ty)
dV
(SXd, + Sy£y + 2SXy£xy)dV
(2.9)
and the integration is extended over the volume of the body
in the initial state.
The stress resultants (2.1), (2.7), and (2.8), to¬
gether with the strains (2.7) and the temperature distribu¬
tion (1.1), are substituted into expression (2.9). After
integration through the thickness, there results
Wr = —
o J
(V2w)2 - 2(1 -/x)
bl w bl
w o w _ ^ á¿w
bx* by1 bxby
- 2(1 + yu) aTy V2wi dxdy +

16
,
_
+
J ( oN* + -¿N, )
du + l^W'2
1 2
c)x 2 ¿x
J
0 J
V
0
+ (0Ny + -ÍNy)
c>v
C>y
+
1(|W)‘
2 dy
+ (oNv
dv + du dw _dw
dx c>y dx c)y
- oct2 (Nx + Ny ) j dxdy (2.10)
The first integral in (2.10) represents the strain energy of
bending, while the second represents strain energy of the
stretching of the middle surface.
Since a "weak" core that is inextensible in the z-
direction is assumed, only the shear stresses Tzx and Tzy
exist. If the rotation terms u>z are neglected, the ortho¬
tropic stress-strain-displacement relations are
T2X = 2G*£2X = G*(-|ü + |^)
a z ox
(2.11)
fZy = 2GylZy = G y (“—“ + ^)
' 1 oz oy
where G* and Gy are the orthotropic shear moduli.
It follows from equations (2.11) and from the assump¬
tion that ¿z = 0 that the displacements in the core are
given by the relations

17
u = U, z + Uj
v = V, z + Vj
(2.12)
where the functions U, (x,y), U2(x,y), V, (x,y), and V2(x,y)
are found from the conditions of continuity of displacements
at the interfaces between the core and the faces. In this
way
U
/
U2
V/
V2
- u" + t/ + t" aw
t 21 bx
u/ + u~ + t' - t" aw
2 4 dx
v/ - v" + t/ + t" aw
t 21 by
v' + v" + t/ - t" aw
2 4 ay
The following stress resultants are defined in the
core. The shear forces per unit length are
Q*
-*/z
— -- n ' - U "
T** dz = tG„ (——= +
(2.13)
Qy =
Vz
Tz y dz
J- Í
*/l
ay

18
where
t
(2.14)
The strain energy Wt of the core is of the form
wc
(^z* ^L y £ zy ) dv
J V
(2.15)
The expressions (2.12) are substituted into the
orthotropic stress-strain-displacement relations (2.11) and
then into the strain energy term (2.15). After integration
through the core thickness, there results
Wc
Gx (U,2 +
dx
+ Gy (V,2 +
9w/
V
dxdy
(2.16)
The Equilibrium Equations
According to the principle of virtual work,
SW/ + $W," + SWC - JVj - SV/ - SV/' - sv„' - sv„" -
8 V,,' - 8V where the virtual work due to the applied loadings is as
follows: (1) from the transverse pressure

19
=
rb
'o *o
(q" - q')8wdxdy
(2)from the edge forces on each face
=
(0NX* Su + 0Nxy* Sv)dy +
(0Ny*Sv + oNxy'SuJdx
(3)from the edge moments on each face
sv„ =
Mx* S(|£)dy
My * 6(~) dX
a>y
3m,v*
xy
<$wdy +
— 6wdx - 2M„>
Sw
(4)from the shear forces on each face
r b
sva =
Qx*8wdy +
Qy* 5wdy
In forming the variations use is made of the formulas
b r °
rb
0 ^ 0
c>F
dx
dxdy =
F (x, y)
dy
X â–  o

20
é> (+ Q
O Jo
0f , ,
— dxdy
ey
F (x,y)
y = *>
dx
and of integration by parts. Equation (2.17) becomes
o >â– 
(D' + D")Vvw + 72(M/ + My) - (rN/ + yNjr,/)^lw
dxa
- 2 M5'&'
du
dx
f brK/
bT N,/
l dx
dy
‘ OyN,"
0 ^Nyy
dx '
Oy
' OrN,/
OyNy
dx
dy
OrNXy"
b rNy'
+ tt -2-S) + Gy (
dx2
bv' _ bv" . . d2Wx~
by by by2.
Svr
~ Gx (
+ G, (
b x by
+ ~C + Gy (
u'
-
u"
t
u'
u"
t
â–  v'
V"
t
V'
v"
+ t |^)t Su'
+
- Gy p-r^- + t |=)j
5u'
Sv'
+ * t?> Sv"
dxdy
(-M, + Mx*)5(|^) + (~ + 2 ■^1L + r^TT. + TX*y —
Ox dx by dx 1 by
A_ SMXy*
+ tQ* Q*)Sw + (rNx - 0n/)5u
Oy
+ (rNxy - „Nx/)Sv dy +

21
dw, ,dMy - »M-y
(-My + My*) 6 (~^) + (- + 2
dy 3y dx
+ rN,
dw
rKyay + rF,,y d
dw
x
dM*y*
+ tQy -
dx
- Qy*)6w + (rNxy - 0N„y*)Su
+ (rNy - 0Ny*)$v
dx + 2(Mxy - Mxy*)
Sw = 0 (2.18)
In expression (2.18) the total membrane forces are defined
by
tNx = 0NX + N*
TNy = 0Ny + Ny (2.19)
yN„y = flNXy + N„y
The integrals extend over the body in the initial state, and
virtual displacements are taken as relative to that state.
Since the virtual displacements 5u', ¿u", 8v', Sv",
and Sw are arbitrary within the plate, their coefficients
in equation (2.18) must vanish. In this manner the follow¬
ing equations of equilibrium result. In the first of these
equations the terms (N/ + Nx ") (Ny' + N y ") , and
(Nxy/ + N„y") are neglected. The terms in parentheses are
membrane stresses due to bending of the plate after initial
stress and are negligible.

22
(D' + D")Vvw - (0N/+ 0n;')4-T " 2(0Nx/ + 4N,;')^f
ox' 7 oxay
(.N/ + T + v' g. OX OX ÍX¿
+ Gy(Sv'
_ dv" tí: ^Wv
dy dy by1
+ q' - q" = 0
(2.20a)
¿N/ + 3N,/
bx by
G*
+
0
(2.20b)
SNx/'
dx c>y
G*
+
dw
a
0
(2.20c)
dN,/ dNy'
dx dy
0
(2.20d)
aN*/ bn/'
dx ay
Gy (—- v-'- +
0
(2.20e)
These five equations in five unknowns reduce to those de-
23
rived by Ebcioglu for the case of constant temperature
variation through the plate thickness and for small
deflections.

23
Equations (2.20) can be seen to contain the equations
of homogeneous thin plate theory. If the core is assumed to
be rigid, i.e., if Gxx = Gyx. -»=*>, then it follows from equa¬
tions (2.11) that ¿Xi = from the core to the total strain energy. The coefficients
of G„ and Gy in equations (2.20) disappear and, except for
the thermal terms, the resulting equations are the classical
ones of Saint-Venant, describing two plates constrained to
have a common vertical deflection and subjected to lateral
and edge loads. These plates correspond in their dimensions
and properties to the faces of the sandwich panel.
Saint-Venant“s equations also result if the thickness
t of the core is allowed to go to zero, since in this case
the integral (2.15) then degenerates to zero. In either
case the effect of the core is only to constrain the two
faces to have a common vertical deflection. The core has
no stress-resisting properties of its own, so the fact that
it retains its thickness for the first case is of no
importance.
It is noted that a formal application of these limit¬
ing processes to equations (2.20) is meaningless and pro¬
duces incorrect results, since considerations preceding the
derivation of this equation are involved.

24
The Boundary Conditions
The boundary conditions arise from consideration of
the line integrals occurring in the variational equation
(2.18)o These line integrals may be made to vanish by set¬
ting either the variations or their coefficients equal to
zero. The boundary conditions are linearized in the same
manner as equation (2.20a). After use of equations (2.3),
they are found to be: (1) on the boundaries y = 0 and y = b
either
u' = 0 (2.21a)
or
N*/ = 0 (2.21b)
either
u" = 0 (2.22a)
or
N„/ = 0
either
v' = 0
(2.22b)
(2.23a)
or
N / = 0
either
v" = 0
(2.23b)
(2.24a)
or

25
Ny" = O
either
w = O
(2.24b)
(2.25a)
or
^-(M/+ M") + 2 r— (M x y/ + Mx") + (0Ny' + 0Ny)-^
dy ox ' oy
+ tQy = 0;'+ Q*"+ ^(M,*' + M«;")
(2.25b)
either
= 0 (2.26a)
dy
or
M /+ My = My*'+ My*" (2.26b)
(2) on the boundaries x = 0 and x = a
either
u' = 0
or
N/ = 0
either
u" = 0
or
Nx" = 0
either
v' = 0
(2.27a)
(2.27b)
(2.28a)
(2.28b)
(2.29a)
or

26
N«/ = O
either
v" = O
or
N " = 0
either
w = 0
(2.29b)
(2.30a)
(2.30b)
(2.31a)
or
y-(M/+ M") + 2 (M „ y/ + Mxy") + (oN/+
ox oy ox
+ tQx = Q*'+ Q*"+ ^-(M,y*' + M,;')
Sy
(2.31b)
either
^ w
a>x
o
or
(2.32a)
Mx/ + Mx"= M/'+ M**" (2.32b)
(3) for the corners (0,0), (a,0), (0,b), and (a,b)
either
w = 0
(2.33a)
or
*
a
= M,/
0
0
0
(2.33b)
The boundary conditions for a rectangular sandwich
panel simply supported along the edges perpendicular to the

27
y-axis and variously supported along the other two edges,
containing rigid inserts set into the edges of the core that
are perpendicular to the y-axis, may be selected from the
preceding sets of boundary conditions.
A simply supported edge may be supported either on
knife-edges or by pins of negligible cross-sectional dimen¬
sions, about which the panel may freely rotate. If the
latter case is considered for a sandwich panel, the pins
may be imagined as running through both faces, parallel and
arbitrarily close to the edges. They restrain the boundary
from displacement perpendicular to itself, but permit paral¬
lel motion. The restraint in the perpendicular direction
will cause a normal stress at the boundary.
If, on the other hand, the knife-edge type of simple
support is envisioned, the edge of each face is considered
to be in contact with a wedge-shaped support along the line
forming the vertex of the wedge. In this case the edge is
free to translate in directions both parallel and perpen¬
dicular to itself.
Finally, the effect of a thin rigid insert is con¬
sidered. The insert is envisioned as being set into the
core so that it is flush with the edges of the upper and
lower faces and welded to these faces at its top and bottom

28
edges. Its effect is to prevent the edges from having dis¬
placements in the direction parallel to themselves. If the
insert is perfectly insulated, so that thermal effects cause
no expansion of it, boundary conditions (2.21a) and (2.22a)
or (2.27a) and (2.28a) will apply. If, however, thermal
expansion of the insert is allowed, the boundary conditions
must allow for some displacement of the edges parallel to
themselves. In general, vertical displacements of the edges
would also occur, but it is assumed that the supports pre¬
vent such displacements. The extreme case occurs when the
insert offers no resistance to the free thermal expansion of
the edges. Then boundary conditions (2.29a) and (2.30a) are
replaced by
r y
v' = oc' T/dy
(2.34a)
v" = or
T"dy
(2.34b)
Va
and conditions (2.21a) and (2.22a) are replaced by
= or
T2'dx
%
u'
(2.35a)

29
u" = Oí'
T/dx
Ja/
(2.35b)
In general, the displacement along the edge will be some¬
where between zero and these values, depending on the co¬
efficient of thermal expansion of the material of the insert.
The vertical deflections at the center of the plate were
calculated for both zero deflection and the extreme values
in equations (2.35); only a slight difference resulted.
The problem to be examined here is a plate containing
a rigid insert along the two edges parallel to the x-axis,
which are simply supported on knife-edges. The two edges
parallel to the y-axis may be simply supported by knife-
edges, clamped, or free. Furthermore, the only external
edge loading applied to the face edges is the initial load
0Ny* which is assumed to be constant along the edge. No
transverse loading acts on the lower face (q' = 0).
The boundary conditions to be used are as follows.
(1) On the boundaries y = 0 and y = b
u
/
= a
T/dx
(2.36a)
u
cc'
T/dx
Jo.
’/z
u _
(2.36b)

30
Ny/ = 0
Ny" = 0
w = 0
My' + My" =
(2.36c)
(2.36d)
(2.36e)
(2.36f)
where the right-hand sides of equations (2.35) should be set
equal to zero if the inserts do not expand when heated.
(2) On the edges x = 0 and x = a,
supported edges,
N/ = 0
N/'= 0
N= 0
N„y" = 0
w = 0
Mx' + M/'= 0
for the case of simply
(2.37a)
(2.37b)
(2.37c)
(2.37d)
(2.37e)
(2.37 f)
For the case of clamped edges
u' = 0 (2.38a)
u" = 0 (2.38b)

31
N,y' = 0
(2.38c)
Nxy" = 0
(2.38d)
w = 0
(2.38e)
= 0
3 x
(2.38f)
For the case of free edges
N*' = 0
(2.39a)
Nx"= 0
(2.39b)
N*y' = 0
(2.39c)
Nxy" = 0
(2.39d)
(«N/+ 0n;o|^ + ¿(m; + mx") + 2 ^(mx; + m„")
+
rt>
Oi
K
II
o
(2.39e)
M/ + My" = 0
(2.39f)
0
(2.39f)

CHAPTER III
SOLUTION OF THE EQUATIONS
Transformation into an Uncoupled System
The solution of equations (2.20) is difficult because
they are coupled in the four dependent variables u', u", v',
and v". By a transformation of variables two sets of equa¬
tions more amenable to solution may be developed. One set
resulting from this transformation is composed of two equa¬
tions in two variables. These equations are then the equa¬
tions of plane elasticity including the effects of a body
force. The second set of equations may be solved by a
separation of variables.
To effect this transformation, new displacement com¬
ponents are defined as follows:
u =
m
— (u7
—
u ")
1
+
m
V =
m
(v7
V")
1
+
m
II
13
1
(u"
+
mu')
1
+
m
V =
1
— (v7/
+
mv')
1
+
m

33
where
It is possible to give a physical interpretation of
the quantities in (3.1). The variation of ü and the varia¬
tion of v through the plate thickness are given by straight
lines with discontinuities of the slope. Figure 2 illus¬
trates this variation for ü. If, however, it is imagined
that these variations are given by lines of constant slope,
then for pure bending a neutral surface exists, which is
defined by the quantities
— /\
h' = -&S—
1 + m
h" = ——— tt
1 + m
where h/ and h" represent the distance from the neutral
surface to the middle planes of the lower and upper faces
respectively. Then u and v represent the displacements of
the neutral surface, that is, since the neutral surface was
defined for pure bending, u and v represent the displace¬
ments due purely to membrane effects. Furthermore, u and
v represent the displacements of the middle surface of the
upper face relative to the neutral surface of the plate.

34
t
t'
Figure 2. Variation of displacement ü through the
thickness of the sandwich panel
//
jr
;
H,
Figure 3. Resolution of membrane forces into a force and
couple system at the "neutral" surface

35
Substitution of the transformations (3.1) into the
equations for membrane force resultants (2.7) yields the
corresponding transformations for these quantities:
0NX = eNx' + 0NX"
0Ñy = 0Ny' + 0Ny"
0ÑXy = 0NXy' + o Nx y "
Nx = Nx' + Nx"
Ny = Ny' + Ny"
Nxy = Nxy' + Nxy
Nr = Nr' + Nr"
H,
Hy
l*y
H r
— /v
tt
1 +
m
— /S
tt
1 +
m
— A
tt
1 +
m
— A
tt
1 +
m
(N/ - mNx")
(Ny' - mNy")
(Nx/ - mNxy")
(N/ - mN")
where t has been defined in equation (2.14).
The quantities Nx and Hx may be interpreted as the
statically equivalent resolution of the membrane forces Nx'
and Nx'' which act in the middle planes of the faces, into a
force and couple system acting at the neutral surface (see

36
Fig. 3). The interpretation of the other quantities follows
in a similar manner.
Substitution of the transformations (3.1) into the
equations of equilibrium (2.20) yields
(D7 + D")Vyw - 0N,-2-^ - 20N,
~ d2w o cf bzw
)x
- + vjmt
ydxdy ° yby
- t
- ,1 + m du , c)2
Gx (
+ tt ^) + Gy (
ox ox m dy
- ,1 + m dv
m
+ tt
T d¿
W,
by â– 
- q
//
o
(3.2a)
E t'
. , 2 + (1 + (1 + /u)s .
2m(1 - yu ) l dx2 by2 dxdy
d2v
. 5x(i4JS u + t 4s) = 1 SHr
mt
dx tt d:
(3.2b)
E't'
2m(l - /¿2) L dy2 ' dx2 ~ dxdyj
- Gy(i4-£5 v + t |S) = 1 dHr
mt
dy tt dy
(3.2c)
d Nx dNxy
+ —r—— = 0
dx dy
(3.3a)
dNxy dNy
dx + dy
= 0
(3.3b)

37
Similarly, the boundary conditions become as follows.
(1) On the boundaries y = 0 and y = b
Ny = 0
N,y = 0
(3.4a)
(3.4b)
r *
u =
m
1 + m
(ot' T¿ - ot" T") dx
a/z
(3.5a)
Hy = 0
(3.5b)
w = 0
(3.5c)
M/+ My//= 0
(3.5d)
(2) On the boundaries x = 0 and x = a, for the case of sim¬
ply supported edges,
Nx = 0
(3.6a)
N„y = 0
Hx = 0
Hxy = 0
w = 0
(3.6b)
(3.7a)
(3.7b)
(3.7c)

38
M/+ My" = O ( 3.7 d)
For the case of clamped edges
u = 0
(3.8a)
N„y = 0
(3.8b)
u = 0
(3.9a)
HXy = 0
(3.9b)
w = 0
(3.9c)
= 0
ax
(3.9d)
For the case of free edges
Ñ* = 0
N„y = 0
H* = 0 (3.10a)
Hxy = 0
My' + My"= 0
-^(M/ + Mg) + 2 ^p(M*/ + Mxy") + tQx
+ (0N/ + 0NX")|^ = 0
(3.10b)
(3.10c)
(3.lOd)

39
Solution of the Membrane Force Equations
The membrane forces in the initial state of stress
are governed by equations (1.2) and (1.3). In general, a
complete solution for the initial state of stress requires
the further specification of the compatibility condition.
However, since the forces 0N*, 0Ny, and 0Nxy have been as¬
sumed to be an order of magnitude higher than the linear
additional stresses, the strain-displacement relations will
include nonlinear terms. As a result, the compatibility
equation becomes nonlinear and, in general, a solution of
the state of stress is not possible.
If the initial edge forces 0N**, cNy* and 0N*y* are
constant, then the initial membrane forces are constant.
Since 0N** = aN*y* = 0, there results
0N* = 0Nxy = 0
0Ny = oNy”
(3-11)
and therefore
0NX = 0Nxy = 0
0Ny = 0Ny*
(3.12)

40
In the case when the edges of the plate x = 0 and
x = a are clamped, the second set of boundary conditions
(1.3) is replaced by the conditions
„u = 0
O NX y. = 0
The solution for the membrane forces requires the specifi¬
cation of the compatibility condition. If it is assumed
that the plate is clamped after the initial stress is ap¬
plied, then equations (3.12) will also apply to this case.
The membrane forces Nx, Ny, and N*y arising from the
additional stresses satisfy equations (3.3) and boundary
conditions (3.4) and (3.6). These equations are identical
to those just discussed for the initial stresses. However,
since these forces are within the realm of linear elasticity,
27
the compatibility equation may be expressed as
Vv(Nx + Ñy) + (1 -/<)V2Ñr = 0
28
Now the classical methods for plane stress problems can
be used for the solution of the state of stress.
If Nx*= Ny*= Nxy* — 0 and Nr is a linear function of
x and y, where

41
Ñ7 = ——— (E'oc'T.V + E"a"T2"t")
' 1 - /X
then the equilibrium equations, compatibility equation, and
boundary conditions are satisfied by
Ñ, = Ñy = Nxy = 0
and thus
Solution of the Second Set of Equations
Once the membrane forces have been found as indicated
in the preceding section, the next step is to solve the set
of three partial differential equations (3.2), which are
subject to the boundary conditions (3.5) and (3.7), (3.9),
or (3.10).
First the system is transformed into dimensionless
23
units. The scheme used by Ebcioglu is followed. Dimen¬
sionless constants, variables, and reference quantities are
defined as follows:
\ = ^
a

42
pe
n¿ e't' ^
= b* i 1 (1 + m)
pf
= £t(D' + D")
k;e
o Ñ/
Pe
r«
Pe
t*tG y
g =
G*
Gy
a =
b
a
p =
1 + m £
2
a s
2t
b
The temperature gradient is nondimensionalized by the intro¬
duction of the quantity
R =
1---+ ^ (oc7T/ - oc"t") (3.13)
1 + m
The quantity Pe has been defined as a reference quan¬
tity with dimensions of force per unit length; it represents
the buckling load of a simply supported, infinitely long
sandwich strip with a rigid core, calculated under the

43
assumption that the faces act as membranes. represents
the sum of the buckling loads of each face of the strip.
Substitution of these quantities into the appropriate
equations and boundary conditions yields the transformed
equilibrium equations:
re
TTl
l 32U
+
(1
-A)
32u
+ /?( 1 + jUL)
bz y
2gu
mgtl(K/3 ^ = 2/3mbre ^
2p 31 rr1
(3.14a)
r
TT
v
32
u
v
+ + /3<1
- 2v
t¿cXm _ 2mbre ¿R
2/0 dr¡ TTÍ 3>j
(3.14b)
m
27r bGy L dfi
d*
w
+
2/32
w
’w
35*3;
3^
pq(.3
a/3*t zgm
4
á2w
_3v «mt2(rek,( - 1) 92W qmb mb
p-— + -—r = —=— - —— 7 Mr
' hr\ 4 dr¡¿ 2Gy 2Gy T
(3.14c)
and the transformed boundary conditions, written in terms of
displacement components: (1) On the boundaries >¡ = 0 and
>7=1

44
u
mb
/3( 1 +yu)
ri
R(5)d£
(3.15a)
/3 m —
' M b%
= mbR
(3.15b)
w = 0
(3.15c)
P* 3*w
-m7
(3.I5d)
(2) On the boundaries 5 = 0 and £ = 1, for the case of sim¬
ply supported edges,
¿if + '“tí-”4’*
(3.16a)
(3.16b)
w = 0
(3.16c)
d2w
= -M.
(3.16d)
For the case of clamped edges
u = 0
(3.17a)
du „ dv
+
o
(3.17b)

45
w = O (3.17c)
Ir = O (3.17d)
o 5
For the case of free edges
+
/u = mbR
(3.18a)
3u
dri
+
/3— =
w
T
+ yU
a>?z
-Mr
(3.18b)
(3.18c)
P*/9
rr2h
+
^ dl3
(2 -yU)
w
c>5 4
1 + m ■? —
+ tG,u
m
(3.18d)
It is seen that the system of equations (3.14) is
linear and nonhomogeneous and is accompanied by nonhomo-
geneous boundary conditions. By the substitution of auxil-
29
iary functions the nonhomogeneous boundary conditions on
the sides r¡ = 0 and >7 = 1 can be reduced to homogeneous
ones. Then it is possible to select trigonometric functions
of 77 that satisfy these homogeneous boundary conditions.
o r
These functions may be employed in the Levy method^ of

46
separation of variables so that the partial differential
equations are reduced to ordinary ones. It is necessary to
expand the nonhomogeneous parts of the boundary conditions
and equations in a series of these trigonometric functions;
furthermore, the final solution is expressed in such series,
and the convergence of these series is investigated in order
to validate their differentiation.
In order to reduce the boundary conditions (3.15) to
homogeneous form, the following substitutions are made:
u = u„ (4 , T) + >? [f, (I) - f0(l)] + f0(4) (3.19a)
v = vo (S»>?) + n*
1. df'($)
g, (4) - 2^-JT~
- (i - y¡r
1 df„(S)
god) -
(3.19b)
W = VJg (5,^)
(n + i)h,(4)
+ (2
>?) h o (4)
TT¿
pf
(3.19c)
where the auxiliary functions denoted by f, g and h repre¬
sent the following nonhomogeneous boundary terms:

47
fo (i)
mb
/3(1 + /<)
- i
R(4,0)d4
f, (4)
mb
/?( 1 + /¿)
R(4,l)d4
J'h
g0 (4) = j; mbR(4,0)
g, (4 ) = -^ mbR( 5,1)
h0(4) = -MT(4,0)
h, (4) = -Mt(4,1)
It may be readily verified that the desired reduction
is accomplished by means of substitutions (3.19). The equa¬
tions governing the quantities u0/ v0, and we are then es¬
tablished by the substitution of relations (3.19) into equa¬
tions (3.14) through (3.18). The following equations result
£,
n
,2
dzu,
d2v„
- 2guc
mgt 1a/3 dw0 2/3mbre
2/j ^ 4
TT 1 d4
+ J, (4,77)
(3.20a)

48
_ 2
3 2v,
c>2v„
2 -~-j- + /32(1 - l~ + /3(1 +/“)
€)>?
32u0
?
- 2v,
t2am dw0 2mbre
2p <$ri 772 dr¡
+ j¿ (4,>?)
(3.20b)
m
'f 2TTlhGy
a*w%
Pipr + 2A
di dr¡z drj'
3u
- />g/3
¿4
a/3212gm d2wc ~v0 v-e-,e
4 H 2 70 d*?
dv0 oani.2 (rek/e - 1) d2wc
d^2
qmb mb 2
= v Mt + Jj (4 ,7?)
2Gy 2Gy
(3.20c)
The boundary conditions on 4 = 0 and 4=1 become, for the
case of simply supported edges,
ft
du0
34
+
mbR + S, (4 , r¡)
(3.21a)
du0 dv0
dr¡ + ^ d4
s2 (4,
(3.21b)
w0 = Sj(4,>?)
(3.21c)
32w0
-Mr + s¥ (4 , rj)
(3.21d)
For the case of clamped edges
u0 = C,(4,^)
(3.22a)

49
»,* + fi aC ’ s<«•*>
(3.22b)
wc = Sj (5,>7)
(3.22c)
d| “ C* ^
(3.22d)
For the case of free edges
du0 dv0
/3 + /á ^ ^ = mbR + S, (§,^)
(3.23a)
du0 dv„
~d^+ put = s¿ (l,7?)
(3.23b)
--«r+ r.lf,,)
TT c)| ar¡
(3.23c)
P,/3
7T¿b
/?
+ [2 - yU)
a3w0
<55 d7jz
1 + m 4.=
tG,u0
m
b o5
/3 cLM-r
- - b-dT + p»(*'’)
(3.23d)
In the above boundary conditions the functions J¿ (5,^),
st (?»>?)/ C< U.tj), and F¿ (4,77) , where i = 1, 2, 3, 4, repre¬
sent the additional terms arising from the substitutions
(3.19).
The boundary conditions on r¡ = 0 and r¡ = 1 are
u
o
0
(3.24a)

50
dv„ 3u.
3Í + ^W = °
(3.24b)
w0 = 0
(3.24c)
d2 w
dri
T = 0
(3.24d)
It is seen that the boundary conditions (3.24) along
the sides r¡ = 0 and r¡ = 1 are satisfied by terms of the form
u0j = Xj {%) sin j7T7?
Voj = Yj (5) COS j7T>7
woy = Zj (?) sin j7T7]
Thus the following solutions are assumed:
u0
oc
X, (4 ) sin
a =;
(3.25a)
v0
oo
1
Y, (4) cos jrr^
(3.25b)
w0 =
Z; (4) sin jTr?7
(3.25c)
However, before attempting an infinite series approach,
certain mathematical questions must be examined.

51
Convergence of Series Solutions
Once solutions of the form of (3.25) have been pro¬
posed, it is desirable to develop some assurance that the
series can satisfy the required equations, in other words,
that term-by-term differentiation of the series will produce
a series converging to a proper derivative.
Such considerations must not be thought a mere mathe¬
matical superfluity. Quantities of physical interest occur¬
ring in the deformation of a plate involve second and third
derivatives of the displacements; if the displacements are
represented as infinite series, it may happen that term-by-
term differentiation does not provide series that converge
to their physical counterparts. For example, the deflection
y of a vibrating beam of length n, simply supported, with an
applied sinusoidal bending moment at one end is represented
by the following equation: ^
with the boundary conditions
y (0, t) = y(77, t) = 0
0

52
If a solution is attempted by the method of finite
sine transforms, the result will be the infinite series
sin an2t)sin nx
z
which, upon term-by-term differentiation, will be found to
satisfy neither the differential equation nor the nonhomo-
geneous boundary condition; nor will the internal bending
moments calculated in this way converge to their true values.
In order to be assured that term-by-term differentia¬
tion of a convergent series, particularly a Fourier series,
will lead to another convergent series, two theorems of per-
31
tinence may be noted. The first is as follows.
Theorem 1. Let there be given a series f„(x)
whose terms are differentiable in the interval
and which converges at least at one point
of J. If the series
deduced from it by
term-by-term differentiation converges uniformly in
J, then so does the given series. Furthermore, if
f„(x) = F(x) and
F' (x) = It is therefore necessary first to perform formally the

53
term-by-term differentiation and then to investigate the
uniform convergence of the result in order to determine the
validity of the operation.
The second theorem relates specifically to the terra-
32
by-term differentiation of Fourier series.
Theorem 2. If f(x) is a continuous function of
period 2n with an absolutely integrable derivative,
then the Fourier series of f' (x) can be obtained
from the Fourier series of f(x) by term-by-term
differentiation.
A corollary of this theorem applicable to the inter-
32
val (0,1) is given as follows.
Theorem 2(a). If f(x) is continuous and absolutely
integrable on (0,1), then the Fourier cosine series
Y' a„ cos nnx of f(x) on this interval may be dif¬
ferentiated term by term, while the Fourier sine
series ^ b„ sin nttx can be differentiated term by
term if f(0) = f(l) = 0.
It is seen that the Fourier cosine series is necessarily
always continuous everywhere; if the conditions in the corol¬
lary are met, the sine series is also continuous everywhere.

54
Theorem 2 applies to functions of one variable only.
However, it may be extended to series of the form of equa¬
tions (3.25) if the variable £ is treated as a parameter.
It is now possible to proceed with the justification
of the assumed solutions. From Theorem 2(a) and boundary
condition (3.24a) the series for u0 may be twice differen¬
tiated with respect to rj. Condition (3.24a) implies that
c>ue
as
= 0 along = 0 and r¡ = 1, and thus condition (3.24b)
may be replaced by
avo
an
= o
(3.26)
The cosine series (3.25b) for v0 can be differentiated once
with respect to r\, and condition (3.26) implies the permis¬
sibility of a second differentiation. Finally, conditions
(3.24c) and (3.24d), together with Theorem 2(a), imply that
the series for w0 may be differentiated four times with re¬
spect to y\.
It is concluded that it is permissible to perform
all partial differentiations with respect to n required to
satisfy the differential equations and boundary conditions.
The validity of taking derivatives with respect to £,
according to Theorem 1, can be established only after the
actual solution is obtained and the differentiations in

55
question performed. Some further remarks will be made when
this stage in the analysis is reached.
Development and Solution of the Ordinary
Differential Equations
The next step in the solution involves the develop¬
ment of ordinary differential equations in the variable 5 .
These relations arise from equating coefficients of trigo¬
nometric terms after substituting the assumed solutions
(3.25) into equations (3.20). In this process the nonhomo-
geneous terms of equations (3.20) are expanded in Fourier
sine or cosine series on the interval (0,1) according to
the following formulas.
(1) The Fourier cosine expansion of a function f(x) defined
on 0 í x í 1 is
where
f (x)
oo
J=t
COS j77X
r I
= 2
fix) COS ji7xdx
(2) The Fourier sine expansion of a function f(x) defined
on 0 ^ x ^ 1 is

56
where
f (x) = j bj sin jr7x
j = t
r i
hj = 2
f(x) sin j^xdx
In this way the following relations may be written
áR
a?
K,j U) sin j7r^
áR
KZJ (K) cos jrrr¡ + K20 (5)
q
sin
V*Mr =
sin jrr^j
Furthermore, the remaining terms on the right-hand
sides of equations (3.20) are given by
J, (5.>7)
H,j (€) sin j7T?2

57
OO
J2(5,T) = a2j (4) cos jrrri + hz0 (4)
Jj U
oo
= £
Hj, (4) sin jTr?7
In a similar manner the nonhomogeneous right-hand
sides of the boundary conditions (3.21) through (3.23) on
4 = 0 and § = 1 are
mbR (4,>?) + S ,U,t) = BOJ (?) sin j n >¡ (3.27a)
-1
oo
S2(4,tj) = ^ Bc> COS + 2 Bco (*} (3.27b)
j-1
S. u^) = BCJ (4) sin jtt>7
— M 7- (4 , t7) + S^(4,tj) — ^ j
j='
c, (4,7?) = cOJ (4) sin jrrTj
00
Cv(|,»?) = ^ Cs, (^) sin j7r^
¡=1
(3.27c)
(4) sin jTrri (3.27d)
(3.27e)
(3.27f)

58
PO
-MriS.ri) + FJ(5,»?) = Fej (?) sin jtt^ (3.27g)
y* >
fi dMrU.n)
b d?
I
J • i
+ F* (*.*!>“ ) F*y (4) sin jrrxi (3.27h)
As a result of applying the procedure described above,
the following systems of ordinary differential equations re¬
sult (D denotes the operator -^r) :
d?
2r_ /3
7t â– 
D2
- [re (1 - /u) j2 + 2gj
X, -
re/3(l + /y)j
77
DY;
mgt^a/J 2/3mbr«
DZ; = —- K,; + H;
2/o
TT 1 LJ L->
(3.28a)
re/3{l + /u) j
TT
DXj +
re /3 2( 1 — jj.)
— D2 - (2rc j 2 + 2)
V
t2a mj 77
2/o
2mbre
ZJ = ~i Kzj + Hyj
(3.28b)
P, m/3v
-^/SDX, + + < D* -
Pfm/32j'2 bGv + 4
D
P^mjv77'2 oant2(l - re k/e ) jz7T 2 i
2bGv + 4
K,; + K„ + H
20
vy
2GV
lVy ' “Jy
(3.28c)

59
—h/rti ~^°2 ~ 1
2tt
mbre
Y „ = — K
TT
20 + "2H20
(3.28d)
where j = 1, 2, ....
The boundary conditions (3.21) on 4 = 0 and 4 = 1
become, for the case of simply supported edges,
ft ~ MWi (K) = 2Boy (4)
d §
(3.29a)
jTTX, (4) + ft = B.y {\)
d4
(3.29b)
Zj (4) = Bej (4)
(3.29c)
d2Zy (4)
d4
= B9y (4)
(3.29d)
„ dY*<*)
ft d4 " ®co
(3.29e)
where j = 1, 2, ....
The boundary conditions for the clamped and free
edges similarly follow.
The complementary and particular parts of the solu¬
tions to equations (3.28) are denoted by the subscripts c
and p respectively as follows:
Xy = XCj
+ Xpy
(3.30a)

60
Yj = Yc j + YPj (3.30b)
Zj = Z cj + Z pj (3.30c)
Y0 = Yoc + Yop (3.30d)
Since equations (3.28) are linear with constant co¬
efficients, solutions of the following form are assumed:
Xcj
= A^e*'*
(3.31a)
Yc,
= Bj eXji
(3.31b)
Zc ,
= CjeÁJi
(3.31c)
Y0c
= B0ex°*
(3.31d)
Substitution of the above relations into the homo¬
geneous form of equations (3.28) reveals that Xj satisfies
the following quartic equation:
(r e 6j - â– â– â–  )
1 -yU
2Pj
T'H i~2 - g (1 + — £ i ) ©j - j ©j
bG y t a ~ .u ~
bGyret a
+ §j2k
re
+ j2re (g - l)
2P,
T~ (t ~ 1) - 1
bGy ret of 1 -yU
©,
= 0
(3.32)
where

61
ej - - J
Further,
n
A„ = ± TT
ft / re ( 1 -yu)
If there are no repeated roots of equation (3.32),
the complementary solutions are given by
g
c = 1
(3.33a)
(3.33b)
(3.33c)
Y
OC
Br
X o ^
+ B,
-x„£
(3.33d)
The constants BJL and CJL are related to the constants A
by the relations
Bji — bJL Ají
C ji = c jL A j í
where

62
j rr
re/32 [2 - g(l + /<)]aj¡* - tt1
re(l-//)j¿+ 2g]
/3*jc
reg/3\l rr1 [2g(rc
j" + 1) - j 2 re (1 + /*)"
2/>
c^ =
m/3gt2a ÁJ(
2re /3iÁJ¿i
v'
0.34)
re(l +/J) j/3 AJt
(1 - ,«) j2 re - 2g bJ¿ •
The constants AJt are determined by substituting the
complete solutions (3.30) into boundary conditions (3.29);
eight simultaneous algebraic equations result, and their
solution may be written in the form
where Ay is an eight-by-eight determinant
Aj = | a*, |
whose elements are given as follows for the case of simply
supported edges:
a/¿
=
(/3ájl - MinTDjL )eAjt
ai,L*r
-
(/3Ay¿ — /^j^by^ ) e
a^
=
(/3ÁJL - MlTTbjL )
a2,L*l
=
-(/?Ay¿ - /*jnbJl )
aj/.
s
(jTT + /3ájl hJL ) eXjL
=
(jrr + /3\jl bJL ) eX^
a^
=
]n + /3Xjl bJL
=
jv + /3ájl b jL
a^i
=
CJL eÁjL
aS}L* t
=
-Cyy e'^JL
=
C Jl
at>,L*l
=
-CjL
ar¿
=
A ^
*JL CJL e
a7,¿*1
=
-Ay/Cy¿ e“Aji
a*¿
=
^JL CJL
a %,L<1
=r
V 2_
*JL CJL

63
where L = 1, 3, 5, 7. The D,, are the determinants formed
by replacing the i-th column of Ay by the column vector
2Bqj (1) - ft
dXp, (1)
d5
2Boy (0) -ft
dXpJ (0)
d 4
+ /¿i”YPj (1)
+ /¿inYpj (0)
Be,
(1)
- j 77 X P y (1)
~ ft
Be,
(0)
- jnXpj (0)
~ ft
Bey
(1)
- Zpy (1)
B (0)
- zPJ (0)
BSJ
(1)
d2Zpy (1)
dC
B sj
(0)
dlZPj (0)
dil
dY PJ (1)
di
dYp, (0)
d 4
Also,
B
Bco (0) - ft
dY,
^1 - B.,
14 J
(1) + /3
dY„„ (1)
d4
or
/3A0(e'Á° - eÁ° )
B,
Bco (0) - ft
dY0p (0)
d4
- Bco (1) + ft
dY0p (1)
d4
ft^o(e‘Ao - e>0 )
For the case when the edges 4=0 and 4=1 are
clamped, the elements of A; are given by

64
au
ss
A j,
e JL
atj L* t
=
e ~XjL
aZL
=
i
a2,¿'/
=
1
a3i
=
(j TT + ft XjL bJL ) e
kjL a
aj,¿»/
=
(jrr + /3 áJl bJL )e'ÁJL
a
=
jrr +/3 A jL bjL
=
j77 + fi AJL
b7i
=
A
CjL e JL
a5,L'l
=
-cJL e ~XjL
=
CJL
a¿,¿-/
=
~CJL
au
s
^JL CJL e
*JL
at}Ltl
=
A c 0
*JL '-JL e
au
=
*JL CJL
at,L+r
=
where
L = 1,
3, 5,
7.
The Dj¿ are
formed from
the column
vector
c°j
(1)
- XPj (1)
Cqj
(0)
- XpJ (0)
Be j
(1)
- jTTXpj (1) -
ft
dYpy (1)
dC
Be,
(0)
- jITXpj (0) -
ft
dYpj (0)
d4
Be j
(1)
- ZPJ (1)
Be,
(0)
- Zpj (0)
c,y
(1)
dZ pj (1)
dK
(0)
dZpj (0)
For the case when the edges 1=0 and 1=1 are free,
there results

'Y-
7rXirO(
UI
-z ui + I
ir~
7 A
t r-
yr-
Z) _ 7r -/] ^7-“
' Vti/J ¡r>aj
Jt/- lr3tlrY'SZ) -
/'77b
1*7'»^
tjr/- 7'Wri/)-
=
7rq 7/ye/ + ixf
=
/*7'*c
.9( 7rq 7/yV + ut)
=
"7'^
( 7rqnCr/ - irYc/)~
=
/'7'7^
( - vYf/) -
=
"7''P
7V'o< "0*^)5. -
*0}
ui
it.
-- ui + x
Xtu^{r/ - Z) -r7ryze/
qr-u
£/*a
Ify 9
7Vr=COir3)2 -
UI
'01
- - UI + T
ir.
7 A
7/".
7 r-
'•"'.-Jíí)
=
7iü
rf7/ - 7rY? £/
=
79p
Or/- 7Wr£/)
=
7 *X!
7'q lry e/ + uz
s
7Ap
( 7rq iry¿/ + uZ)
=
7fp
lrquZr/ - lry £/
=
7rqufr/ - lry g/)
—
7,e
S9

66
pf/3 r , ,
A,/ - (2 - /6<) j277zA
1 + m
cJL - tG,
m
- ^Vi&,)oJLKJL
where L = 1, 3, 5, 7. The DJt are formed from the column
vector
dX-, (1)
2B0j (1) -ft ^ + yujrrYpj (1)
2B0. (0) - /3
d{
dXP7 (0)
dl
+ /ujnYpj (0)
Bc, (1) - j TtX pj (1) - ft
Be j (0) - j TTXpj (0) - ft
dYpj (1)
dl
dYpj (0)
d|
a dl2tPj (1)
77 Fej (1) - ftx +/Uj‘7rIZP; (1)
d|'
- Fc; (0) - ft1 d + /¿i*rr*Zpj (0)
F9j (1) -
l±A
tt
d|
/3¿ ^Z^-3(--} - (2 - /u) jíTTí
7 d|3 dl
+ 1- + m tG ,X pj (1) + ^(t¿tG, )dZ^(1)
m b dl
F9j (0) -
h£
7T2b
A
d3Zpj (0)
d|
x-/ , , j2 , dZ(0)
f-1 - (2
dl
+ ■" —-m tGxX p j (0) + ^(t¿tGjd--^--(-°-
m b d I
Therefore, from equations (3.19), (3.25), (3.30),
(3.31), and (3.35), the solutions for the displacement

67
components u, v, and w are
u(§,t) = r¡£, (4) + (1 - >j)f0U)
+
+
x„>
sin
j rrr[
(3.36a)
g, (5)
/V df, (4)
2 d4
g0 U)
Au df0 (4)
2 d?
*>><â– 
£!¿i_
+ YPJ (4)
COS j7T^
B0/ eAo€ + B„ e-Aot + Yop (4)
(3.36b)
w(4,>?) =
77 ‘
6P,
n(n - i)
(»? + l)h, (4) + (2 - rj)h0 (4)
J • 1 ¿'I
'ji
i */« ^
+ Z
PJ
(4)
sin jTrrj
(3.36c)
These equations completely determine the displacement
components.
Now that the solutions have been completely specified,
further remarks can be made about the validity of forming
partial derivatives of these quantities with respect to 4.
Unfortunately, because of the extreme complexity of the
terms in each infinite series, it is not possible to

68
actually examine the uniform convergence of the derivatives.
All that can be said is that actual evaluation of the series
for w at the center of the plate indicated that the terms
converged. Uniform convergence was not shown.
Solution for a Simplified Case
In the preceding part of this analysis the solution
of a very general case of loading on a sandwich plate has
been presented. The remainder of the analysis will utilize
certain restrictions which aid in the tractability of the
problem, while still representing a broad category of
results.
The following assumptions are now made:
1. Bending of the faces is neglected (Pf = 0) ;
2. The temperature is constant through the thickness
of each face;
3. The temperature distribution and the edge loading
are symmetric with respect to perpendicular bi¬
sectors of the edges of the plate;
4. The transverse loading q" is constant.
As a result of these restrictions, the following re¬
lations hold:
f, (5) = f0U)
mb
r(
RU.DdS
'/i
/Hi + m)
J
(3.37a)

69
9/ U) = 9o U) = n±>R
h, (?) = h0 (?) = 0
c
c
Mr = 0
c
The particular solutions can be determined from
tions (3.28) after the following relations are noted:
T, (?, rj) = -
re/3mb. 2 . .3R,t . .
— m + i +/<)>rr(5'i)
TT ¿ 1 + // d?
r ?
+ 2g
mb
/3(1 + M)
R(?i l)d?
’/i
C
T2 U,>7) = -
^(1-^(2,- iwu - ^->0
+ ( 2 - 1)
mb (1 - - ^ - ) R(£, 1)
1 +/A
c
Tj (?, >?) = /O
mb (1 + g i— ——) R (?, 1)
1 + yU 1 + fU
c
and after the following series expansions are noted:
PO
1 =" TL isin 3n>i
j=â– >,*â– â– â– 
. 37b)
.37c)
. 37d)
equa-
.38a)
. 38b)
. 38c)
. 39a)

70
_4_
77*
oo
...
j rrn
(3.39b)
bMt.n)
oo
J-f,*
~ Í
^0
dR(K,n)
sin ]v r\ d>7 sin jmj
(3.39c)
*R(S,t?)
drj
2 I
<3 R (£ , rL)
d-q
cos jnr¡ dq cos jfrq
(3.39d)
where the series in the last two equations contain only
terms with an odd-numbered index because of the symmetry of
R{%,r¡). Thus from equations (3.38) and (3.39) there result
f 1
K
tj
f? sin irrr, dr,
K „ = 2
Sr
bn
cos j7r^ dq
K- “ iti
K,; = 0
K
20
(3.40a)
(3.40b)
(3.40c)
(3.40d)
0
(3.40e)

71
H.
4
TT
re/3
TT â– 
mb (
1 + /J-
i , ,dR(5,l)
~ r + 1 +-^)-sP—
- 2g
mb
/3( 1 +/0
Í
R(?,l)d?
'h
1
J J
( 3.40f)
H
u
- - £ ¿V - D-Xl -
4*
mb (1 -
1 + //
•) R(£, 1)
(3.40g)
hjj ~ n P
mb(1 + g
1 + /U
-~—)RU,D
1 + /<
1
j
(3.40h)
H20 = 0
(3.401)
where j = 1, 3, 5, ....
Then the particular solutions XPJ , YPJ , and Zpy are
found from equations (3.28) after has been set equal to
zero and expressions (3.40) have been substituted for the
quantities on the right-hand sides of these equations.
Furthermore, it is found that for any symmetrical tempera¬
ture distribution
â– op
0

72
The particular solutions for two specific temperature
distributions were found. For a constant distribution R = R„
these solutions are
= A; 4 + By
Y pJ = Cj (3.41)
~ Dy
where
8mbg R
77/3(1 + /u) j [re(l ~yU)j2 + 2§]
C,
ftj
- ftj &j
Dy
Mj - *7
aj Zj - ftj
and
ocj = 2 (re j 2 + 1)
ftj =
tzo(mj7T
V

73
TT1 j
mb (1 -
1 +/<
)R«
8re mbgRc
^[ri(l -^)j2 + 2g]
6j = pjrr
5/ =
ofmt^d - rek,e )tt2jz
Xj = —
mbq0
—— + p
2Gy ^
mb (1 + —) Re
1 +
8/ombg¿R
77j(l +/u)[re(l - pi) jJ + 2gj
Here j=l, 3, 5, ....
For a double sinusoidal distribution of the form
R = R, sin tt % sin ttt¡, the solutions for j = 1 are
A pj
Vrj
Zpj
where
Aj
Ej
= A j COS TT £,
= Ej sin + Fj
= Hj sin tt% + Ij
(3.42)

74
Here
®/ = -
2mbre
rr
R»
P, is the
determinant
Vj
Vj
ft i
0
ft
Uj
where
V; = 2re ft1 + re (1 - //) + 2g
7C; = re/3(l + /*)
rj = ft g /3
^ = re/32(l - /u) + 2re + 2
rngt^or/?
p, = L 77
' 2 P
77ai2m
uj = —4—(/? g ~ rek,e + 1)
and P«y is the determinant formed by replacing the i-th
column of Py by the column vector
ft
1
0

75
Further,
= -A,
/3j
a.
S; - /3j Sj
where
a,
= A,
OC,
- /3j Sj
A,
4 mbqc
77 j 2Gy
For j = 3, 5, 7, ...
Ay = Ej = Hy = 0
The coefficients in (3.27) are determined from equations
(3.37) as
Bay (0)
Bay (1)
4
77
mb
A
(1 - /¿)
R(l,l)
1
j
+ 2mb
R(1,77) sin jTT d>7
BCy (1)
-BCy (0)
9R(1,1)
(1
M
1 + JU
Bej (1) = Bey (0) = 0
Bsj (1) = (0) = 0
Bco (1) = Bco (0) = 0
(3.43a)
(3.43b)
(3.43c)
(3.43d)
(3.43e)

76
Caj
(1)
= cmJ
(0) = 0
(3.43f)
(1)
= cgj
(0) = 0
(3.43g)
(1)
= F.y
(0) = 0
(3.43h)
(1)
= -V,j (0) =
4(1 + m) tG„b
n 1
R(?,l)d?
0
1
(3.431)
F3J
n/3(l + m) j
j
where j = 1, 3, 5, ....
The assumption of a temperature distribution sym¬
metric in the ^-direction makes possible a simplification
of the complementary solutions (3.33). The function Ej (X)
is defined as
Ej (?)
a,
e J
+
- X; (5- M
e J
and is seen to be even with respect to the line % = 1/2,
which bisects the plate; the function
Oj (?)
e
A j U-’/i)
e
A j ( S ' Vi )
is seen to be odd with respect to ? = 1/2. Each pair of
terms of the form
* jit
VI
+ A
- XJ i
'J ,1*1
that occurs in the complementary solutions (3.33) may be
I

rH| CM
77
written in terms of the newly defined functions as
(A,,
+ A
2 * j
)E;
i V,"'
- x,
)0
Under the present assumptions, the horizontal dis¬
placement component u will be odd with respect to 5 = 1/2,
while the horizontal displacement component v and the verti¬
cal displacement component w will be even. The expression
(3.36a) for u in terms of the functions Ej and <3, becomes
mb
u =
/?(1 + ju)
r l
r *
f J r
R(£,l)d£ + ^ -
£ 2
J'/z
L—. ¿ L
/<• t
2k- 1
'/i X,
D
J, 2 k
V,
A,
>E,* + (
D/, ih. 1
'/« xy.
- h x j
)0y,
sin j77 yj
Since u must be odd and the functions Ey>t and Oy* are lin¬
early independent, the coefficients of E• must equal zero;
Dy2 = -eX" Dy,
Dyv = -eÁjl Dj3
Dn = -eX'J Djj
there result

78
Therefore the solutions (3.36) become
u
mb
/3(1 +/0
R(?,l)d4 +
'h
oo r j
Aj.Ci-i)
Dy>
A;
+ X
Pj
sin
(3.44a)
v
mb (1
1 +yu
)R(M)]
+
I'I
Áj¿
X.| 0 - 5 J1
N
e J
+ e J
+ Y pj
â– 
J
(3.44b)
COS j 7T 77
W
D
JL
+ e
0-U
+ Z
sin jT7?7 (3.44c)
Here the are the roots of the cubic equation
(r«eA “ i _ ' ) (9®/ + r«k/< ji0y - gJik/e )
-L a
+ j 3 re (g - 1)
e/- J‘k- 11 - irh)
= 0
(3.45)
where
©7
(â– 
2
j
2
Further, A, is the three-by-three determinant

79
Ay = I aKL I (3.46)
where K,L = 1, 2, 3; the elements aKL are, for the case when
the edges £ = 0 and £, = 1 are simply supported,
a/¿ = (eXjL + 1) (/3Ayt - /UjirbjL )
a2L = (eXjL - 1) (jrr + fthJL hJL )
a3L = (e XjL + l)cA
where L = 1, 2, 3. The D,t are the determinants formed by
replacing the i-th column of Aj by the column vector
dXPJ (1)
2Baj (1) - /3 f- + /u}nY,j (1)
B.y (1) - jrrXp. (1) - ft
~1PJ (1)
dYPj (1)
d|
(3.47)
where the coefficients B0J and BCJ are given by expressions
(3.43a) and (3.43b).
For the case when the edges \ = 0 and ^ = 1 are
clamped, the elements of Ay are
a
tL
= e
- 1
a
2 L
(e‘
- 1) (jrr + fthjL hJL )

80
aJL = (e Jl + 1) cJL
and the D/i are formed from the column vector
-XPj (1)
Bcj (1) - jnx^ (1) - ft
-Kpj (1)
dY PJ (1)
d %
For the case when the edges £ = 0 and \ = 1 are free,
the elements of A j are
aIL = {eXjL + 1) {ftkJL - /ujrrbJL )
a2L = (eXjL - 1) (j7T + /3ájl bJL )
a3L = (e>lJL ~ D
1 + m ^ /3
tG„ - — (t tG« ) cy¿ AjL
m b
and the Djl are formed from the column vector
dXPj (1)
2Baj (1) - ft + /UjrrYPj (1)
d£
Bcy (1) " j nXPj (1) - ft
dY,y (1)
d l,
l + m^_ /3 . a2-- . dZ^; (1)
F3j 1) tG, Xpj (1) - Mt'tGj —
m b d X

81
The coefficients FSJ are given by expression (3.43i). The
quantities bj¡ and c7; are given by expressions (3.34)
An observation can be made on the roots of the char¬
acteristic equation (3.45). If the equation is expanded
into a cubic equation of standard form
A3 + aA1 + bA+ c = 0 (3.48)
where
A = A/
it is seen from the definitions and physical meaning of the
coefficients that
a < 0
b > 0
c < 0 for k,e < —r
1 + reja
j *
c > 0 for k,e > 77-
' 1 + rej
From Descartes' rule of signs it follows that the number of
positive real roots of the cubic equation will be either one
j1 J2
or three if k„ < tt or zero or two if ~k/e > — •
/e 1 + re j ,e 1 + re j
In general, then, as k,c , which represents the compressive
edge loading, increases past this value, complex values of
\j are to be expected.

82
A further observation is that when 0Nyy = 0, i.e.,
no compressive edge loading is present, a double root for Aj
occurs; in this case the solutions (3.44) approach the fol¬
lowing limits as 0Nyy -* 0:
u
mb
/3(1 + /u)
~ X
R(£# 1) d£, +
J'/i
(1
5)e
^,('•5)1
id
Kj, ('-()
+ X
PJ
sin jnrt
(3.49a)
v = (2r? - 1)^(1 - —— )R(£,1)
1 2 1 + /<
o©
D ,j
. A'
*>,7
t ^j f 5 / i r \ Ai . (* ‘ 5)
5 e *" + (1 - 5 ) e "
j.r,t
.
ejt e " + e
V, (»•«>'
Y" hJ‘
O
-
1
A, '
«r
cos jrrri
(3.49b)
oo
f
r
Cj,
Sex" + (1 -
—i
* /.j...

83
+ 9 it
+ e
Dj j
A,
z
r c
p ^
D- , z
/ 1
t -;
e
+ e j
Aj + Z"
sin j7T^
(3.49c)
Here the ÁJ¿ are the roots of the equation and are easily
found to be
n
ft
'Si
77 /2g + re j 2 (1 -/*)
ft
reg(l -/*)
and A; represents the determinant
Ay — I aK¿
whose elements are given by
a/j = ft
1 + {1 + Xjt)e
- yujn
e„ + (bJt + e„)e'
a2j = J' + ft
~(Áy/ej/ + hjft +
(bj, + ejft^ji + by/] eA"
= 9^1 + (cjt + g,/)eA-
a/¿ = (1 + eXjL ) {/3XJl - /UjnbyL )
a2L = (eXjL - 1) [jrr +/3hJLXJL )
a
3L
(e‘
1) c
JL

84
where L = 1, 2. The Dj¿ are the determinants formed by re¬
placing the i-th column of Aj by the column vector (3.47).
The b,t and cJt are given by equations (3.34);
. A re J 2 (3 ~ M) (1 ~ §) + 2g
Bjl nj rej2 (1 + /j) (1 - g) - 2g
4/0/3
tji
mt l re j2 (3 - yu) +
r e j 2 (1 + a) + 2
while , Ypj , and ZPJ are the particular solutions of
equations (3.28).

CHAPTER IV
COMPARISON WITH EXPERIMENTAL RESULTS
Although there are many reports of experiments on
rectangular sandwich panels without thermal effects, for ex-
33 34
ample, those of Boiler and Kommers and Norris, the only
experiment involving thermal deformation known to the author
24
is the one performed by Ebcioglu et al.
In this experiment a rectangular sandwich panel was
simply supported on all four edges and a temperature gradi¬
ent was applied by simultaneously heating and cooling oppo¬
site faces of the panel. No compressive load was applied
and the deflection was measured at the midpoint of the panel.
The properties of the panel were given as follows:
t' = t" = 0.010 in.
t = 0.395 in.
/u = 0.3
E' = E" = 30xl06 psi (4.1)
G* = 39,825 psi
Gy = 66,400 psi
a = b = 6 in.
85

86
and thus
Ke = 0
Pe = 1,853 lb/in
t = 1.0253
re = 0.067214
oí' = a" = 7.34xl0~6 in/in °F (4.2)
qQ = 0
g = 0.6
m = 1
fl = 1
From Ebcioglu et al. and the original data sheets of
the experiment, it was found that for the four trials made
in the experiment the distribution of the temperature gradi¬
ent R, defined by equation (3.13), could be approximated by
multiples of the function
-4
R = (1.25 sin nyj + 1.373 sin m-j sin rr£)10 (4.3)
A measure of the error caused by representing the tempera¬
ture distribution by equation (4.3) was calculated from the
formula
L la¿ - et |
A Tmax H i
Error
(4.4)

87
where the a¿ are the experimentally recorded values of the
temperature gradient, the cL are the values of the gradient
calculated at the same points from formula (4.3), and ATmax
is the maximum difference among the experimentally recorded
values of the temperature gradient.
The following table gives the temperature gradient
distribution, experimentally recorded values of the deflec¬
tion at the center of the plate, and measure of the error as
given by equation (4.4) for each trial.
TABLE 1
EXPERIMENTAL DATA
Trial
Temperature
Gradient
Error
Measured
Deflection (in.)
1
R
+0.005
-0.056
0.013
2
1.1 R
+0.023
-0.089
0.0135
3
1.2 R
+0.043
-0.079
0.0195
4
1.8 R
+0.065
-0.080
0.021

88
Since no edge loading was applied, the analytical
solution to this problem is given by equation (3.49c). The
particular solutions for the temperature distribution R as
given in equation (4.3) are
X„ = -(0.262 cos n £)10-3
Yp = -(0.477 + 0.262 sin rr£) 10-3 (4.5)
Zp = (9.011 + 5.066 sin v \)10"3
for j = 1, and
XPJ = 0
YPj = 0
for j = 3, 5, ....
Substitution of the values given in (4.1) and (4.2)
and the particular solutions specified above into equation
(3.49c) produces the theoretical results given in Table 2
for the four trials reported in the experiment. The com¬
parison between the theoretical and experimental results is
illustrated in Figure 4.

Deflection in inches
89
Temperature gradient R
Comparison of theoretical and experimental
values of the deflection at the
center of the plate
Figure 4.

90
TABLE 2
THEORETICAL RESULTS
Trial
Theoretically
Calculated
Deflection (in.)
%
Error
1
0.01335
0.0
2
0.01468
8.9
3
0.01602
17.9
4
0.02403
14.3
The agreement with the experiment is seen to be ex¬
cellent for trial 1, good for trial 2, and poor for the last
two trials. Nevertheless, the results, while somewhat in¬
conclusive, do tend to indicate satisfactory agreement with
the theory, since, as Table 1 shows, the assumed temperature
distribution most closely fits the actual distribution in
trial 1. Furthermore, the experimental data, as presented
in the appendix, show that for all four cases there is some
doubt whether the experimental temperature distribution was
actually symmetrical in the y-direction; since data were
taken at only two points in the half of the plate necessary
to establish this fact, no definite conclusion can be

91
reached. For trials 2, 3, and 4 the data show a definite
lack of symmetry in the 4-direction. It is felt, therefore,
that the data of trial 1 are the most significant, and the
large errors for the other three trials are not alarming.
The need for further experimental evidence is indicated.

CHAPTER V
COMPUTATION AND DISCUSSION OF THE SOLUTION
Computation of the Solution for Various
Parameters of the Panel
In order to investigate the dependence of the deflec¬
tion and buckling of the panel on its properties and dimen¬
sions, equation (3.44c) was evaluated on the University of
Florida IBM 709 computer for a variety of cases. It was
decided to limit consideration to panels with faces of equal
thickness, which are simply supported on all four edges and
are not subject to a transverse load; in this way the number
of variables was reduced to manageable size, while the
ability was retained to vary such quantities as the ratio of
face thickness to core thickness, aspect ratio of the plate,
shear and Young's moduli, and the temperature distribution
and edge compression.
The properties and dimensions of sandwich panels used
in practice are as varied as their applications. Reference
to industrial and design journals produced values of the di¬
mensions ranging from t' = t" = 0.006 in. and t = 0.44 in.,
92

93
as used in the wing of a supersonic bomber,35 to t' = t" =
0.064 in. and t = 18 in., as used in curtain wall construc¬
tion. Honeycomb core shear moduli varied from G* = 9,600 psi
and Gy = 21,600 psi to G* = 145,000 psi and Gy = 81,000 psi.3^
The usual facing materials were steel, with a Young's modu¬
lus E = 30x10^ psi, and aluminum, with E = 10x10^ psi.
A useful range of variation of the dimensionless pa¬
rameter re, given by the relation
r _ rr1 E't' t_
c 2b1 (1 - yu)Gy
can be chosen from the above information as
0.01 < re < 2.00
The values re = 0.017 and g = 0.6 given in the experiments
24
reported by Ebcioglu were chosen as typical values. A
typical value of the aspect ratio /3 was selected as 2.0.
The nondimensionalization of the deflection w was
accomplished by the definition of w as
w - w
R0b/2ta
where R0 is the maximum value over the surface of the plate
of the temperature gradient function R given in equation

94
(3.13). The quantity w was calculated as a function of the
applied edge loading parameter k,e . The calculations were
performed for the two temperature distributions
R = R„ = constant
R = R0 sin tt t, sin n
for varying re and for varying ft.
The effect on the deflection of the sandwich panel
due to the degree of lengthwise thermal expansion of the in¬
sert was also examined. Boundary condition (3.15a) was de¬
rived under the assumption that the thermal expansion of the
edges = 0 and >7=1 was not inhibited by the insert. At
the opposite extreme is the assumption that no thermal ex¬
pansion at all is allowed. The deflection was calculated
for both extremes for varying re.
A computer program, written in the FORTRAN symbolic
language, was designed in order to accomplish the tasks de¬
lineated above. This program instructed the computer to
evaluate equation (3.44c), with provisions for varying re,
/3 , g, /j, and k,e . An option was included so that those
terms in the quantities Dy{- which arise from the right-hand
side of boundary condition (3.15a) could be set equal to

95
zero. Furthermore, either of the particular solutions (3.41)
or (3.42) could be substituted into the equation, depending
on the temperature distribution.
The cubic equation (3.45) was solved by iteration of
Newton's method. Three forms of equation (3.44c) were used
in the program: one as given; one with real and imaginary
parts written for the case when two of the roots Aj of equa¬
tion (3.48) are complex conjugates; and one with real and
imaginary parts for the case when one of the roots Aj is
negative and A, =is thus imaginary.
The infinite series in the solution was truncated at
N terms either when the series was monotonically decreasing
and the N-th term of the series was a given percentage of
the first term or when N equaled a given maximum number.
A few limitations were imposed on the calculations.
When /3 was less than 0.75, the roots of the cubic equation
were so large that terms of the type eXjl involved in calcu¬
lation of the quantity A; exceeded the maximum magnitude
allowable in FORTRAN. For the same reason, no more than
fourteen terms of the infinite series could be calculated.
About fifteen minutes of actual running time on the
IBM 709 was required for all of the results obtained.

96
Discussion of the Results
The results of the calculations discussed in the pre¬
ceding section are shown in Figures 5 through 8. Each of
these graphs shows the dimensionless deflection parameter w,
evaluated at the center of the plate, as a function of the
edge loading parameter k,e . In Figure 5 the surface of the
plate is subject to a double sinusoidal temperature gradient
distribution given by
R = R0 sin Trt, sin n
and the effect of varying the parameter re is shown. In
Figure 6 the same information is presented for a constant
temperature distribution. Figures 7 and 8 illustrate the
effect of a varying aspect ratio ft on a plate subject to the
two temperature gradients.
A basic difference between the solution for a double
sinusoidal temperature gradient distribution and for a con¬
stant distribution (when there is no transverse load q) is
that for the former case the infinite series of the solution
(3.44c) reduces to its first term. This fact is seen from
the particular solutions (3.42), which are zero for j > 1.
From Figures 5 and 6 it is seen that the deflection
increases throughout most of the loading process as the

Figure 5. Deflection at the center of a plate under a temperature gradient
R = Ro sin sin irq as a function of edge loading

Figure 6. Deflection at the center of a plate under a constant temperature
gradient as a function of edge loading
010

Figure 7. Deflection at the center of a plate under a temperature gradient
R = R0 sin tt% sin 77*7 as a function of edge loading

k^e
Figure 8. Deflection at the center of a plate under a constant temperature
gradient as a function of edge loading
100

101
parameter re increases, while the buckling load decreases.
A marked difference between the two graphs is observed. The
deflection monotonically increases until it becomes unbounded
and positive at the buckling load in the case of a double
sinusoidal temperature gradient distribution; when the dis¬
tribution is constant, the deflection curve starts to in¬
crease for lower values of kie , but then turns downward and
decreases until the deflection becomes negatively unbounded
at a lower buckling load than for the corresponding case in
Figure 5. This effect is known as "reversal of deflections."
The difference in the response of the panel to the
two temperature distributions is due to the presence of all
the terms in the infinite series for the second case, as
mentioned above. Buckling occurs at the lowest value of k/e
so that the denominator Aj of a term in the series approaches
zero. The lowest such value of k,£ need not cause the de¬
nominator in the first term of the series to vanish; the
buckling load may be determined by any term in the series,
depending on the properties of the plate. This phenomenon
25
occurs in homogeneous thin plate theory also. In order to
describe this effect, the term "n-th mode buckling" is in¬
troduced, meaning that the critical load occurs when the
denominator of the series term with index j = n vanishes.

102
From another point of view, if the external forces acting on
the plate were due only to the compressive edge loading and
no temperature gradient were applied, then the problem would
become the associated eigenvalue problem of the one studied
here; in this case the critical load would cause the plate
to buckle into the n-th mode shape, or into n half-waves in
the £-direction. When n > 1, the effect is denoted as
"higher mode buckling." The buckling load is always of the
higher mode type when /3 = 2.0, g = 0.6, and re varies from
0.01 through 2.00, as seen in Figures 5 and 6.
In the evaluation of equation (3.44c) on the computer
it was found invariably that each term in the series alter¬
nated in sign. This result is not unexpected because of the
presence in the solution of the factor sin jTrrj, where >7 = 1/2
at the center of the plate; however, the sign of the remain¬
ing part of the solution could not be determined beforehand
for the general case. It is interesting to observe in Fig¬
ure 6 that whenever higher mode buckling occurred, the mode
involved had an index of j = 3, 7, 11, . .., so that the term
had a negative sign and thus the deflection always became
negatively unbounded.
The deflection at the center of the plate varies
similarly when re is held constant and /3 is varied. As /3

103
increases, the deflection for a given value of k/e decreases
and the buckling load increases, as is shown in Figures 7
and 8. From Figure 8 it is seen that the values of the pa¬
rameters chosen permit first-mode buckling for a range of
values of /3. Once again the heuristic result is noted that
higher mode buckling was characterized by a negative value
of deflection.
One of the results of the present analysis that
should be useful is the calculation of more accurate values
of the buckling load. When specific values of the proper¬
ties and dimensions are known, the buckling load can be
found by evaluation on the computer. Unfortunately, the
expression (3.46) for Aj is so cumbersome that an analytical
solution for the buckling load is forbiddingly difficult.
However, the deflections and buckling loads were found to
23
agree closely with Ebcioglu"s results. His expression for
the buckling load is
k
(5.1)
+ re /3i2+g(l-/¿) +ji2g+(l-/¿) + 2g

104
A comparison of the deflection calculated from Eb-
cioglu's results with the deflection calculated from the
present results is shown in Figure 9. Table 3 shows the
comparison of the buckling loads. The error is never great¬
er than 10 per cent. Therefore, it is felt that using equa¬
tion (5.1) for approximate calculations of the buckling load
and its behavior as a function of re, ft, and j is justified.
TABLE 3
COMPARISON OF BUCKLING LOAD VALUES
k,e at Buckling
re
fi
Ebcioglu's
Results
Present
Results
0.025
2.0
13.5
12.9
0.050
2.0
10.6
10.1
0.100
2.0
7.41
7.1
0.250
2.0
3.85
3.9
0.500
2.0
2.00
2.0
1.000
2.0
1.00
1.0
2.000
2.0
0.50
0.5
0.017
0.75
2.36
2.4
0.017
1.00
3.83
3.6
0.017
2.00
14.8
14.1
0.017
3.00
25.6
23.8
0.017
5.00
43.5
42.7
0.017
10.00
58.9
59.4

105
Figure 9. Comparison of the deflection at the center of a
plate under a temperature gradient
R = R0 sin 77-5 sin 77-77 as a function
of edge loading

106
The buckling mode is determined by the equation
dk,e
dj
0
(5.2)
where k/e is given by expression (5.1). This equation lo¬
cates the value of j at which the smallest critical value of
k/e occurs. Since j must be an integer, this equation may
only produce an approximation of the exact value of j.
It is interesting to note that cases exist where
there are no real roots of equation (5.2). When (5.2) is
written in full, a quartic equation results. If, for exam¬
ple, /3 = 2.0, g = 0.6, and re > 0.495, all coefficients of
this equation have like signs, and it follows from Descartes"
rule of signs that there are no positive values of j satis¬
fying this equation. Therefore, the function k/e is mono¬
tonic for j > 0, and a simple check at j = 0 and j -» <*> veri¬
fies that k/e monotonically decreases to its asymptotic value
It follows that the critical value of k,e is determined by
the mode corresponding to j ■* «*>. As a result, the sign of
the deflection as k,e approaches the critical load becomes
indeterminate.

107
The explanation of this effect is that as the number
of buckling waves increases, the bending stresses in the
faces will dominate the membrane stresses. However, terms
due to bending effects were neglected in the derivation of
equations (3.44c) and (5.1). Finite roots should result if
these terms are retained.
As a result of the preceding discussion, it is neces¬
sary to make a qualifying remark about the deflection curve
in Figure 6 for re = 0.5, 1.0, and 2.0. According to this
discussion, the portions of the curves near the buckling
load should be indeterminate. The values shown were actu¬
ally calculated from an infinite series truncated at the
term j = 23.

CHAPTER VI
SUMMARY
The equations governing the deflections of a sandwich
panel subject to a general system of loads and a thermal
gradient were developed and solved for the special case of
uniaxial loading. Application of a Levy-type solution en¬
abled the problem to be solved without making simplifying
assumptions about the boundary conditions. The solution for
the vertical deflection was evaluated on a digital computer.
It was seen that when bending stresses in the faces
are neglected, the response of the panel is determined by
three nondimensional parameters. The buckling loads were
seen to increase as the aspect ratio increased and to de¬
crease as the parameter re increased. Reversal of deflection
generally occurred. It was noted that buckling is charac¬
terized by a large number of half-waves for panels with
properties and dimensions commonly used in practice. An
unusual effect of an indeterminate buckling mode was noted
and accounted for by the fact that the bending stresses
should be included for high buckling modes.
108

APPENDIX
The dimensions of the sandwich panels used by Ebcioglu
24
et al. and the eight points on the faces at which the tem¬
perature was measured are shown in Figure 10.
0
a
.5"
o o o
12 3
2
.5"
o o o
3
- 6
4 5 6
1
.0"
o
7
1
.5"
o
_i
8
0
5"(1
["*—1.5" -*)^1.0,,4‘1.0"^^-1.5"-^
<
-0.5
â– < 6" >
Figure 10. Dimensions and data points
of the sandwich panel
109

110
The temperature gradients that were measured are
given in Table 4.
TABLE 4
EXPERIMENTAL TEMPERATURE GRADIENT
Data
Point
Temperature
Gradient (°F)
Trial 1
Trial 2
Trial 3
Trial 4
Number
1
12
12
16
18
2
16
19
21
33
3
14
19
19
30
4
35
37
39
58
5
53
56
60
97
6
53
59
63
97
7
48
51
58
80
8
26
23
21
30

LIST OF REFERENCES
1. E. W. Kuenzi, "Paper Honeycomb as a Core for Structural
Sandwich Construction," Symposium on Structural Sand¬
wich Construction, ASTM Special Technical Publication
No. 118, 1951, p. 70.
2. A. Marshall, "Honeycomb-Sandwich Structures," Machine
Design, Vol. 30, No. 10, May 15, 1958, p. 126.
3. C. S. Gough, C. F. Elam, and N. A. DeBruyne, "Stabili¬
zation of a Thin Sheet by a Continuous Supporting Medi¬
um, " J. Roy. Aero. Soc., Vol. 44, 1940, p. 12.
4. D. Williams, D. M. A. Leggett, and H. G. Hopkins, "Flat
Sandwich Panels Under Compressive Edge Loads," Report
No. A.D. 3174, British Royal Aircraft Establishment,
June, 1941.
5. D. M. A. Leggett and H. G. Hopkins, "Sandwich Panels
and Cylinders Under Compressive Edge Loads," Report No.
S.M.E. 3203, British Royal Aircraft Establishment, 1942
6. H. W. March, "Sandwich Construction in the Elastic
Range," Symposium on Structural Sandwich Construction,
ASTM Special Technical Publication No. 118, 1951, p. 32
7. H. W. March and C. B. Smith, "Various Types of Edge
Conditions—Buckling Loads of Flat Sandwich Panels in
Compression," Forest Products Laboratory Report No.
1528, 1945.
8. H. W. March, "Effects of Shear Deformation in the Core
of a Flat Rectangular Sandwich Panel," Forest Products
Laboratory Report No. 1583, 1948.
9. H. W. March and W. S. Ericksen, "Compressive Buckling
of Sandwich Panels Having Faces of Unequal Thickness,"
Forest Products Laboratory Report No. 1583-B, 1950.
Ill

112
10. C. Libove and S. B. Batdorf, "A General Small Deflec¬
tion Theory for Flat Sandwich Plates," NACA TN No. 1526,
1948.
11. E. Reissner, "Finite Deflections of Sandwich Plates,"
J. Aero. Sci., Vol. 15, No. 7, 1948, p. 435.
12. N. J. Hoff, "Bending and Buckling of Rectangular Sand¬
wich Plates," NACA TN 2225, 1950.
13. A. C. Eringen, "Bending and Buckling of Rectangular
Sandwich Plates," Proc. First U.S. Natl. Cong. Appl.
Mech., 1951, p. 381.
14. G. Gerard, "Linear Bending Theory of Isotropic Sandwich
Plates by an Order-of-Magnitude Analysis," J. Appl.
Mech.. Trans. ASME, Vol. 74, 1952, p. A13.
15. J. N. Goodier and C. S. Hsu, "Non-sinusoidal Buckling
Modes of Sandwich Plates," J. Aero. Sci., Vol. 21,
No. 8, 1954, p. 525.
16. C. C. Chang and I. K. Ebcioglu, "Elastic Instability of
Rectangular Sandwich Panel of Orthotropic Core With
Different Face Thicknesses and Materials," J. Appl.
Mech.. Vol. 27, No. 3, 1960, p. 477.
17. Y. Y. Yu, "A New Theory of Elastic Sandwich Plates—
One-Dimensional Case," J. Appl. Mech., Vol. 26, 1959,
p. 415.
18. Y. Y. Yu, "Flexural Vibration of Elastic Sandwich
Plates," J. Aero. Sci., Vol. 27, No. 4, 1960, p. 277.
19. Y. Y. Yu, "Nonlinear Flexural Vibrations of Sandwich
Plates," NASA TN No. 12, 1962.
20. Y. Y. Yu, "Damping of Flexural Vibrations of Sandwich
Plates," J. Aero. Sci., Vol. 29, No. 7, 1962, p. 790.
P. P. Bijlaard, "Thermal Stresses and Deflections in
Rectangular Sandwich Plates," J. Aero. Sci., Vol. 26,
No. 4, 1959, p. 210.
21.

113
22. C. C. Chang and I. K. Ebcioglu, "Thermoelastic Behavior
of a Simply Supported Sandwich Panel Under Large Tem¬
perature Gradient and Edge Compression," J. Aero. Sci..
Vol. 28, No. 6, 1961, p. 480.
23. I. K. Ebcioglu, "Thermoelastic Equations for a Sandwich
Panel Under Arbitrary Temperature Distribution, Trans¬
verse Load, and Edge Compression," Proc. Fourth U.S.
Natl. Cong. Appl. Mech., Vol. 1, 1962, p. 537.
24. I. K. Ebcioglu, J. A. Roberson, J. R. Schmermund, and
K. D. Shimmin, "Bending of Sandwich Panels Under Ther¬
mal and Mechanical Loads," To be published in Proc.
Eighth Midwestern Conf. of Solid Mech., 1963.
25. S. Timoshenko and S. Woinowsky-Krieger, Theory of
Plates and Shells, 2d ed., McGraw-Hill, New York, 1959.
26. G. Herrmann and A. Armenakas, "Vibrations and Stability
of Plates Under Initial Stress," Proc. ASCE, J. Engr.
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27. B. A. Boley and J. H. Weiner, Theory of Thermal Stress,
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28. N. I. Muskhelishvili, Some Basic Problems of the Mathe¬
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29. R. V. Churchill, Fourier Series and Boundary Value
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30. S. Timoshenko, Vibration Problems in Engineering, 3d
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in Compression," Forest Products Laboratory Report No.
1525-E, 1948.

114
34. W. J. Kommers and C. B. Norris, "Effects of Shear De¬
formation in the Core of a Flat Rectangular Sandwich
Panel," Forest Products Laboratory Report No. 1583-A,
1948.
35. J. C. Joanides, S. C. Mellin, and L. M. Lackman,
"Honeycomb Excels for Mach 3 Wing," SAE J., Vol. 69,
No. 3, 1961, p. 62.
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Vol. 30, No. 356, 1958, p. 294.

BIOGRAPHICAL SKETCH
Edward Linde Bernstein was born in Newark, New Jersey,
on February 9, 1938. He spent the early part of his life in
the East and later moved to Chicago, where he received his
secondary education. He entered Purdue University in the
fall of 1955 and received a Bachelor of Science in the field
of Engineering Sciences in June, 1959. He then enrolled at
Cornell University in the Engineering Physics curriculum; he
received a Master of Science in this area in September, 1961,
having held a Cornell Graduate Fellowship and a teaching as-
sistantship. Immediately following the granting of this de¬
gree, he entered the University of Florida to pursue studies
leading to the degree of Doctor of Philosophy. He has held
the positions of Assistant in Research, NSF Cooperative Fel¬
low, and NASA Pre-Doctoral Trainee. He is currently a Re¬
search Associate.
115

This dissertation was prepared under the direction of
< the chairman of the candidate^ supervisory committee and
has been approved by all members of that committee. It was
submitted to the Dean of the College of Engineering and to
the Graduate Council, and was approved as partial fulfill¬
ment of the requirements for the degree of Doctor of
Philosophy.
April 18, 1964
Dean, College of Engineering
Dean, Graduate School
Supervisory Committee:

UNIVERSITY OF FLORIDA
3 1262 08556 7344



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