Thermal bending of sandwich panels under uniaxial loading


Material Information

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


Subjects / Keywords:
Sandwich construction   ( lcsh )
Buckling (Mechanics)   ( lcsh )
Engineering Mechanics thesis Ph. D
Dissertations, Academic -- Engineering Mechanics -- UF
bibliography   ( marcgt )
non-fiction   ( marcgt )


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

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 13919585
notis - ADA4823
sobekcm - AA00004947_00001
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Full Text






April, 1964


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.



ACKNOWLEDGMENTS ...................................... ii

LIST OF TABLES .. ... .......................................... v

LIST OF FIGURES .................... .. ............... vi

LIST OF SYMBOLS ...................................... vii

ABSTRACT ............ .... ........................... xii


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




Chapter Page


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


Table Page

1. Experimental Data ............................. 87

2. Theoretical Results .......................... 90

3. Comparison of Buckling Load Values ............ 104

4. Experimental Temperature Gradient ............ 110


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 = R. sin 1T sin ITI
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 = R, sin 7r sin 7rq
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 = Ro sin Tr sin 7T7 as a function of
edge loading ......................... ......... 105

10. Dimensions and data points of the sandwich
panel ....................................... 109


a -


b -




Ci -

F ej
CIj ,

D -

D -



E -



F -



Length of panel in x-directlon

Elements of the determinant 6j

Length of panel in y-direction

Quantity defined in equations (3.34)

B* Be B1j Fourier coefficients defined in equa-
tions (3.27)

Quantity defined in equations (3.34)

Function defined in equations (3.22)

Cj Fourier coefficients defined in equations (3.27)

Differential operator d/dg

Plate modulus

Determinant formed from 6j (see p. 63)

e,y,-ey Linear components of strain

Young's modulus

f/ Auxiliary functions

S Function defined in equations (3.23)

Fqj Fourier coefficients defined in equations (3.27)

ax/ y

g Auxiliary functions

Gx, Gy Orthotropic shear moduli of core

h, h" Distances to "neutral" surface

ho, h, Auxiliary functions

Hij Fourier coefficient

Hx, Hy, Hy Couples formed from resolution of membrane

H7 Thermal couple formed from resolution of thermal
membrane forces

J Function defined in equations (3.20)

ke Dimensionless edge loading parameter -oNy/Pe

Ki Fourier coefficient

m E' t/E"t"

M,, May, My Bending moments in faces

MT Thermal bending moment in faces

M,* My, My*'- Bending moments applied on edges of faces

Ng, Nxy, Ny Additional membrane forces relative to state
of initial stress

N7 Thermal membrane force

Nx,, Nxy, Ny Resultants of additional membrane forces

NT Resultant of thermal membrane forces

oNx, oNsy, oNy Initial membrane forces

oNx, oN, y* Ny4- Applied initial edge loads

.N<, oNy oNy Resultants of initial membrane forces


7Nx, Ney, 7NT Total membrane forces, defined in equations


Buckling load of a simply supported, infinitely
long sandwich strip with a rigid core

Sum of buckling loads of each face of a simply
supported, infinitely long sandwich strip with
a rigid core

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)

(t + t')

- |(i + t")

q -


re -

R -

S' -

S -

Si -



t ,

tx I

Pf -

Function defined in equations (3.21)

Sy, Sy Components of initial stress

t Face thicknesses

Core thickness
1- t' t"
1 +
w(t + -t + t-
t 2 2

tay, ty Components of stress relative to initial state
of stress

Change in temperature from a constant initial

T, Coefficients of the temperature distribution




Xc, Yc


u, v -

u, v -







U/ ,

Displacements of middle surface of upper face
relative to "neutral" surface, defined in equa-
tions (3.1)

Displacements of "neutral" surface, defined in
equations (3.1)

x- and y-components of displacement in core

', v', s" x- and y- components of displacement in the

Functions defined in equations (3.19)

SVI, Va Coefficients of displacement distribution
functions in the core

Virtual work of edge moments Mx,* Mxy, My

Virtual work of edge forces oN* oNy* oNy*

Virtual work of transverse loading q

Virtual work of edge shear forces Q,* Qy*

z-component of displacement

Dimensionless deflection parameter

Function defined in equations (3.19)

Strain energy of core

Strain energy of faces

Z Solutions to ordinary differential equations; see
equations (3.25)

, Z, Complementary solutions

, Zp Particular solutions

VM -

v -

v, -

va -

w -


Wr -

X, Y

CK -

oe -


Cx 6xy

C -

A; -

Coefficient of linear thermal expansion

Dimensionless parameter 2t/b

Aspect ratio of panel b/a


y Complete components of strain

z- s


Quantity in cubic equation

Coefficient of exponent in complementary solution;
see equations (3.33)


/- Poisson's ratio

p Dimensionless parameter 2

(x, Wy, wx Components of rotation

Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy



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


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.




Historical Background

Legend has it that the term "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 than 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


The original, and still most widespread, application

of sandwich construction is in air and space vehicles; its

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

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,

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
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,

Williams, Leggett, and Hopkins and Leggett and Hopkins5

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

thicknesses and elastic properties.7'8'9 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.10 Reissner developed differential

equations for the finite deflection of sandwich plates.11

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-

wich plate.12 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

in sandwich plate analysis.14 Goodier and Hsu considered
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.16 Recently, Yu, in a series of papers, has

included the effects of rotatory inertia and shear in the


Thermal stresses in sandwich plates were treated by
Bijlaard, who used a homogeneous plate approximation.21

Chang and Ebcioglu extended their approach to include the

effect of a constant temperature gradient,22 and Ebcioglu
effect of a constant temperature gradient, 2and Ebcioglu

later solved the problem for a temperature gradient which

varies arbitrarily across the area of the plate.23 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.24

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.

11. -1 4MJ












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 /I. 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' = TI/ + T/

T" = T//V + T2"

where T', TI," Tz', and T"/' are functions of x and y only;

=z s (1.2)


s = s' = -(1 + t') for the lower face

s = s"= l-( + 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

each face there act applied axial forces oNO oNxy*, and oN",

applied moments Mx* Mxy* 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 T. Ty, and Tz 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.



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.


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

on that of Timoshenko25 and Herrmann.26

The initial components of stress are denoted by Sx,

Sy, and Siy. 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:

oN, = tS,

,Ny = tSy (2.1)

ONxy = tSxy

Since the plate is initially in equilibrium, the fol-

lowing equilibrium equations may be written for each face:

c) ,Nx 6 oN, y
+ = 0
ax 6 y
C +oNxy l)oNy
+-- = 0
b x 6y

The boundary conditions for the initial state are

oNy*= oNy

ONxy = oNxy

on y = 0, y = b, and

oN%* = ONK
oNxy* = oNXY

on x = 0, x = a, where oNx*, oNY*, and oN 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

x e + (W2 + W (2.4a)

If Y ey + 4(CJ 2 + .zz) (2.4b)

6z ez + (y2 + ) (2.4c)
z =ez + y2


Y = ey -1 x y (2.4d)

6yz = eyz a.y 4Jz (2.4e)

,zx -- ezx Wgz Wx (2.4f)

where e., ey, e, exy, eyz, and ex are the linear strain

components and wx, wy, and w. are the rotation components.

Based on the Kirchoff hypothesis, the deflections in

the faces are given by

u = u __
v = 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 9 is given by equation (1.2).

The rotations coz are considered to be negligible for

plate deformations. Then the use of equations (2.5) in

equations (2.4) produces the strain-displacement relations:

= au + 1(_2w), (2.6a)
SC)x 2 c)x x-(2.6a)

Vy 1( W)Z (2.6b)


1u +v + w w 2t w(

2x x ~y +--25 ) (2.6c)
2 ay 6x ax by x(wy

C ( + l() (2.6d)
2 Sx 2 b

x = 0 (2o6e)

S= 0 (2.6f)

The additional components of stress tx, t)y, and ty

are assumed to be related to the additional strains by the

Duhamel-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 Ojz will be neglected in ac-

cordance with classical plate theory. Therefore, the fol-

lowing equations for plane stress27 will apply:

tX = E (x + -4EY EOT
1 (E ET

ty = E (CY + 1x) EoT
1 / 1 -

1- = E Ex
S1 +4 Y

where o is the coefficient of linear thermal expansion.

The following stress resultants are defined in the

faces. The membrane forces per unit length are

Nx= tid =
J- /.

+ /1v + (6w)
6 ^

Et [1 x
1 -/Z X

Ny = tyd/ = [-- Et|[b
J -V1 /" C) y

+ p b
-b x

(t/I r
tEt )u
J 2(1 +,/)[(y

+ w2
2 (y

+ )2
2 ay

+ "V+ aw 6w]
Sx ax 5yJ

where the thermal force per unit length is

=1 -E x

S EocT2t
Td =-/
1 -/

The bending moments per unit length are

th7 ttd D 2
M, = jtd = -ID

M = txD[ +W] -2WM

-ti/I xy2
My = j tyd = -D(l +/x)-
-, /-V .

- N7

+ ,5,1

- Nr



+ ,. by -

where the thermal moment Mr is

MTr = D(I +/ )T,

and the plate modulus is

D Et3
12(1 (1 )

The strain energy WF in the faces is given by the


Wr =1 it 6 + ty~y + 2txy Ery 2(xT(t, + ty)]dV

+ j (Sx, + Syty + 2Sxyxy)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

W = D (V w)2 2(1 /x) -w (- w)2
2 1 6x+ Ty2 yxdy +

2(1 + /1)xT, V2wdxdy +

S{(oNx + )[bu + l(w)2]
( +2 c) x 2+ 2x

(.N), + INY) av+ -1(2kw
2 lapy 2 by

(,Ny + !:Ny [av+ u bw BW
2 a ox by +ox ~yj

OTa (N, + Ny)}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 wz are neglected, the ortho-

tropic stress-strain-displacement relations are

f = 2 x= G (- + )
)z )x

= = Gy( +w)

where Gx and Gy are the orthotropic shear moduli.

It follows from equations (2.11) and from the assump-

tion that iz = 0 that the displacements in the core are

given by the relations

U = U, z + U,

V = V, z + V,


where the functions U (x,y), U,(x,y), V, (x,y), and V,(x,y)

are found from the conditions of continuity of displacements

at the interfaces between the core and the faces. In this


U, -- U .
S= u" +

U + +
Ut + U/
U2 2 +

V/ V
V V +

Vp +Vr
V2 = ++


core. The

t" + t" 4w
2t bx

t" t" w
4 ax

t + t" w
2t by

t t" Iw
4 ay

following stress resultants are defined in the

shear forces per unit length are

- F u" A
f !Fix dz = tGx( + t i)

th bx

Q,= Tdz = tG(V :' v" + )




A t t")
t = (t + + -) (2.14)
t 2 2

The strain energy W, of the core is of the form

WC = (Tx^

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 = A I 6 aw
We = J G[x(Ut + Tx + Gy (V2 + )y dxdy (2.16)
Do "

The Equilibrium Equations

According to the principle of virtual work,

SW/,' + SW," + SWo SVt SVN SV" SV' SV"-

SVY' SV"' 0 (2.17)

where the virtual work due to the applied loadings is as

follows: (1) from the transverse pressure

JVq = (q"
"0 "0

- q')Swdxdy

(2) from the edge forces on each face

SVN = (,N1*Su + Nxy, Sv)dy + (0Ny*Sv + 0Nxy*Su)idx

(3) from the edge moments on each face

*b r a
j ax y
v =- M," 8(-y-)dy My"s

J'o yo

3M"y* 10~ ks
+ f sy Swdy + 2 Swdx 2M,
0 0

(4) from the shear forces on each face

SVO = QJo,*Swdy + QY'Swdy
^o 0o

In forming the variations use is made of the formulas

rxa d y x -a
j F dxdy = JF(xy) dy
S~O x
J ob Jo 3C

Iy- dxdy =

*a y =
F(xy) dx
0 y-o

and of integration by parts. Equation (2.17) becomes

f[(D' + D")V7w + V(Mi + Mr) (rN/' + N') aw
,T / W (W +T "" 62w q/ W 1
2( N + TNtY ),xy (TNy '+ TNy)" + q' q
my ~ ~~~ axy()Ty1+V

- ([ (u au" +) + aILW 'by +
L x dx + x y oy
fw ix., +_ y y

+ bxxy a (/ + a ) Su'

ax ay t ax J
--- + -Ty- + G ( + 5 uw

a INI' (.v'--- + t X')V
) +x C) t C) Sj

c rTN _V/ _" ^ C) w It
Ix y t y J

by t ~ yl6vId

+ -x ( + M,+)8( ) + (-x + 2 --r + ,N +
fo [ lCa

+ t~x -$ )u+ ( -

+ (7N. oNXy*)Sv]dy +

tt 8w
by2 'Jl


N )wy
rN .,Y y

I0 "0

(-My + My)S (w) + my aM2 Y
( :-'y+ 2 "-'-+rNyT + TN' I_.w
7y dy Cx y 'f x
+ J Y N, *)+
+ ty a Qy')6w + (rNy y)u

+ (rNy -oNy')8v dx + 2(M.y M.y') Sw = 0 (2.18)
0 o

In expression (2.18) the total membrane forces are defined


7N. = oN. + N,

rNy = oNy + Ny (2.19)

7rN> = oNjy + 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 Su', Su", Sv', 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' + N ,")*w (N,'+ Ny)T and

(Nxy' + Ny,) are neglected. The terms in parentheses are

membrane stresses due to bending of the plate after initial

stress and are negligible.

(D' + D") Vw (oNx' + Nc") a 2 2+ -N ). W
,, 2,W 2 o(Mi + N[b~
awzM + ("- a ul au" -^Pw)
(,N + N+ )-M-- + M") t G ( + tt .
+ y I V -ax Sx. ax

+ y( y- + tt y)1 + q' q"= 0 (2.20a)
by C)y t y J

aNx' 3 U' U" A
'my + NW' U + t = 0 (2.20b)
+x -ay =x

3N," N,," bN b./ / 0'
Nx" +_ u" _w + t ) 0 (2.20c)
ax xy ( x

c)N y' +N T(v' v" ( 0w)
X T )y G t oy

D N. t ~" -VI -a

+NX+ N/ (v" V" + t T) = 0 (2.20e)

These five equations in five unknowns reduce to those de-
rived by Ebcioglu23 for the case of constant temperature

variation through the plate thickness and for small


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 Gz = Gy-, then it follows from equa-

tions (2.11) that xz. = 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-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


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.

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


u' =0 (2.21a)


Nxy = 0 (2.21b)


u" = 0 (2.22a)


N /y =0 (2.22b)


v' =0 (2.23a)


N = 0 (2.23b)


v' = 0 (2.24a)

N = 0


w= 0

(M + My) + 2 y-(M.' + M."') + (.N + Ny)aw
3y y yx o y

+ ^ay = Q;+'+ Q+ )(M.;' + M.;)


3 = 0

Mi + M" =

M*+ MY/f

(2) on the boundaries x = 0 and x = a


u'. =0


Nx'= 0


UH = 0


N= 0


V/ = 0











Nx = 0



V I = 0

N" =0




w= 0


b-(M/+, M+') + 2 -(M, + M,;') + (,N,'+ N+ )

+ tQ, = Q,*+ Q'I"+ -(M4' + M,')


_w= 0

Mx, + MI,= M:,+ M.*"

(3) for the corners (0,0), (a,0), (0,b), and (a,b)


w= 0





I Iu a
Mv C = Mxy 1(2.33b)
0 o 0 o0

The boundary conditions for a rectangular sandwich

panel simply supported along the edges perpendicular to the

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

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

v' = oc Tzdy (2.34a)

v" = T"dy (2.34b)
J 2

and conditions (2.21a) and (2.22a) are replaced by

u' = O'j T,'dx (2.35a)

u" = O" Tj"dx (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

oNy! 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' = Tdx (2.36a)

u" = T.'dx (2.36b)

NY = 0

N/'= 0

w= 0





My'+ M/'"= 0

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, for the case of simply

supported edges,

N,' = 0

N/y = 0







N,Y" = 0

w= 0

M/ + M."= 0

For the case of clamped edges

U, = 0



U 1 = 0

Ny' = 0 (2.38c)

Nxy"= 0 (2.38d)

w = 0 (2.38e)

)w = 0 (2.38f)

For the case of free edges

N,'= 0 (2.39a)

Nx"= 0 (2.39b)

Nxy = 0 (2.39c)

Ny = 0 (2.39d)

(ONx" + ON)Tx + x(Mx' + Mx') + 2 (Mx/ + Mx')

+ 6, = 0 (2.39e)

M y+ My"= 0 (2.39f)



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")

v = m (v" v")
u= 1(U" + mu')
1 + m

= --- (v"+ mv')


E" t"

It is possible to give a physical interpretation of

the quantities in (3.1). The variation of u 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 u. 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

where h and h" represent the distance from the neutral
h i A

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 ; 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.



Figure 2.

Variation of displacement
thickness of the sandwich

u through the

----- N."

N % H

---- ; -^ J H

______ N.'

Figure 3.

Resolution of membrane forces into a force and
couple system at the "neutral" surface


U" +t a
u 2 ax

u, + w

2 Sx

t' w
U 2 x





Substitution of the transformations (3.1) into the

equations for membrane force resultants (2.7) yields the

corresponding transformations for these quantities:

ONx = oNx'+ oNx"

Ny = oNy' + oN/'

I / If
oNly = oNy + oN^y

N, = Nx'+ N,"

Ny = Ny + Ny"

Nxy = Nxy' + N~y"

NT = N/'+ Nr"


H = t (N.' mN')
l+ m

Hy = (Ny' mN"')

Hy = tt (N,/ mN'y)
Hy=1+ m Y y

1 + m 7'

where t has been defined in equation (2.14).

The quantities NK and Hx may be interpreted as the

statically equivalent resolution of the membrane forces Nx'

and N",. which act in the middle planes of the faces, into a

force and couple system acting at the neutral surface (see

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

(D' + D")V'w RN- 2Ny-- ,N 2M

-t[Gx(1-+ mm u + W w) + G +( m + )w)
[ m Tx o )x- m Ty t y2


2m- )t 2 -+ (1 (1 +1
2m(1 1.4 [2 ox2 6y 2 axdyj

x+w 1 ) Hr
5,(+m u + t 2
mtx tt x

E/t/ ) +2 c +(1 c))
2m(l -/"A) 2 +y2 (- x2 1 3 + y

v1 + m) = 1 H
mt y tt y

bx^ My =

aI~*y a~
ax +-^-ay= 0





Similarly, the boundary conditions become as follows.

(1) On the boundaries y = 0 and y = b

Ny = 0

Nxy= 0

U = m (oc'T' o"T/)dx
1 + m 2

HW= 0
w = 0

My/+ My"= 0

(2) On the boundaries x = 0 and x = a, for the case of sim-

ply supported edges,

NR = 0

N"y = 0

Hx = 0

HXy = 0






w= 0







For the case

For the case


M'+ Hmy"= 0

of clamped edges

u = 0

Nuy = 0

u = 0

Hry = 0

w= 0

aw 0
-- = 0

of free edges

Nx = 0

N~y = 0

Hx = 0

Hxy = 0

My+ My"= 0

+ Mx') + 2 -y(M"/ + M"') +t0

OX 0
(O~' + ax"












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 oNx, oNy, and oNAy 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 oN*, oNy*, and oNxy* are

constant, then the initial membrane forces are constant.

Since oN..= oNxy*= 0, there results

oNx = oNxy = 0

oNy = oNv

and therefore

oNX = oNxy = 0

:Ny = yO

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

ou= 0

oNxy = 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 Nty 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,

the compatibility equation may be expressed as27

V,(N + y) + (1 -/i)VNT = 0

Now the classical methods for plane stress problems28 can

be used for the solution of the state of stress.

If Nx'= NY*= Ny* = 0 and Nr is a linear function of

x and y, where

N7 = 1(E'o'T2t' + E O'Tt")
1 -i/.'

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

units. The scheme used by Ebcioglu23 is followed. Dimen-

sionless constants, variables, and reference quantities are

defined as follows:




Pe= 7 E-- (h') (1 + m)
b2 1 -/AL

Pf (D' + D")

kie =


t tGy


1+ m%

= 2t

The temperature gradient is nondimensionalized by the intro-

duction of the quantity

l+ 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

assumption that the faces act as membranes. PF 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:

rI = a u C) a 2 v

2p ^i Trr1 <
A[2gA + (1 + (3d +AA )- 2v

m'oc/3 w, 2mbre 6R
2 -~ 1 9 7 ) 31a

r.2c2 /3 7(1 1A)a v + /'3(1 + -u2v

=___/ a)? r R (3.14b)
2/i TT'2 >i

m~ r w w 2] u ;gzm a=w
21T bG 4a- + 23 )w + 43w

+m m I (r ek,, 1) 2W qmb mbVM1
-/P VM= -=- (3.14c)
4 z 2GY 2GY

and the transformed boundary conditions, written in terms of
displacement components: (1) On the boundaries = 0 and
7 = 1


U = mb R(9)dt
+ 1( +<) JI

bv + 13,ua u = mb R

;-? at

w = 0

Ir az MT





(2) On the boundaries 4 = 0 and 1 = 1, for the case of sim-

ply supported edges,

S+ = MbR

u+ 1 -
a7q a


Af WT i6 s= "MT

For the case of clamped edges

u = 0

au+ /A- =0
377 at







w = 0 (3.17c)

w = 0 (3.17d)

For the case of free edges

au + = mbR (3.18a)

+ /3 0 (3.18b)

P f 3 w + z, -w (3.18c)

P# 3 [1 w 3 + W 1 + m
rr [f3i 3 + (2 mA ()tal tGM u

+b(wat.4) = A d
+ (t9tG = b d Mr (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-

iary functions29 the nonhomogeneous boundary conditions on

the sides 7 = 0 and )7 = 1 can be reduced to homogeneous

ones. Then it is possible to select trigonometric functions

of that satisfy these homogeneous boundary conditions.

These functions may be employed in the Levy method25 of

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.( ,) + [f, (,) fO(g)] + fO(V) (3.19a)

v 0 ~ag -1 df, (U)
v = V. ( 7.) + 7 (t) 1, f ]

(1 o [go(g) 4/ df() (3.19b)

w = w.O,(,) + ?(, 1)[()7 + l)h,( )
6 L

+ (2 )?)h(07) (3.19c)

where the auxiliary functions denoted by f, g and h repre-

sent the following nonhomogeneous boundary terms:

o mb R(, 0)dt
fo () =/3(l + u )

f W mb R(, l)ds

f1( 3(1 + 1-)J/ ,

g.(-) = mbR(t,0)

g, (1) = 1 mbR(9,l)

ho() = -M(,(0O)

h,() = -MT(t,1)

It may be readily verified that the desired reduction
is accomplished by means of substitutions (3.19). The equa-
tions governing the quantities u., vo, and wo are then es-
tablished by the substitution of relations (3.19) into equa-
tions (3.14) through (3.18). The following equations result:

re 2 zu 3 2u_ ) _v
[ 2/ 2 + (1 /U)a7U2 + ( + u) -

mgcx4 wo 2i3mbre )R
c + Jf (,77) (3.20a)
2/o t r 2 -4^ -

r) 2 v o 0v )zu, 2v
S ; -1 + 1 -art + (1 +] 2Vo

t'4m awo 2mbre R+ R
= + J)- + (,) (3.20b)

m [ o w +-wo] U0
Pf Av-bS L, -O + 2/,, 21#,, + '-w,j o3 u

w3tgm =Zw0 avo amt2(rek,6 1) 32wo

2qmb 2 v 2MT + J3 (9,2) (3.20c)
2GF 2e cae y

The boundary conditions on = 0 and = 1 become, for the
case of simply supported edges,

13au, + A,)-- -- mbR + S, ( (3.21a)

auO + 13 av S M ( 7, ) (3.21b)

w, = S3 (9,7) (3.21c)

13= -M2 + 2.W ) (3.21d)

For the case of clamped edges

u0 = C, (, 7?)


Wo= S (j)

Fo = cs (c,?)

For the case of free edges

Uo _v
au0 av
1132-- + 1.

mbR + S, (,?7)

uo dVo

P 2+ /1 0+ .3-s)

7 )- = -s + F7 )




P,/3 [ 2CJ+wo 3w + m(.-aw.
-7-2b [3C3 + (2 -/-) 1 m+ u bzotG,)
.)t C3722 m b C)

/3 dM7-
- 2 + F, (M,2)
b dt


In the above boundary conditions the functions JL (4,),

Sj (,??), Ci (9,-), and Fj (t,-), where i = 1, 2, 3, 4, repre-

sent the additional terms arising from the substitutions


The boundary conditions on ,7 = 0 and 7 = 1 are





uO = 0

V" + 131A a u =

Wa = 0

62W W0
--) =




It is seen that the boundary conditions (3.24) along

the sides 7 = 0 and '7 = 1 are satisfied by terms of the form

uoj = Xj (9) sin J7r?1

voj = Yj (E) cos j7T7

woj = Zj () sin J7Trj

Thus the following solutions are assumed:

U0 = 211

V= I

WO =

Xj (t) sin Jir7

Yj (9) cos j7rr7

Zj (t) sin j1rr




However, before attempting an infinite series approach,

certain mathematical questions must be examined.

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 7T, simply supported, with an

applied sinusoidal bending moment at one end is represented

by the following equation:30

a Y + a2 )Y = 0
at2 a'x

with the boundary conditions

y(0,t) = y(7,t) = 0

a2y(ot) = 0

a 2Y(Tt) = b sin 4t

If a solution is attempted by the method of finite

sine transforms, the result will be the infinite series

y = 2 a2 bn (1) (sin wt -W sin an2t) sin nx
17 a2 n' -W an-

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-

tinence may be noted. The first is as follows.31

Theorem 1. Let there be given a series fn(x)
whose terms are differentiable in the interval

J = [a,b] and which converges at least at one point
of J. If the series 2 f,'(x) deduced from it by
term-by-term differentiation converges uniformly in

J, then so does the given series. Furthermore, if

2: f,(x) = F(x) and I f.'(x) = p(x), then
n.0 A-0

F'(x) = o(x)

It is therefore necessary first to perform formally the

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 term-

by-term differentiation of Fourier series.32

Theorem 2. If f(x) is a continuous function of

period 2i 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


A corollary of this theorem applicable to the inter-
val (0,1) is given as follows.32

Theorem 2(a). If f(x) is continuous and absolutely

integrable on (0,1), then the Fourier cosine series

I a, cos nnx of f(x) on this interval may be dif-

ferentiated term by term, while the Fourier sine

series 72 bn sin nnrx 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.

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 Condition (3.24a) implies that
-- = 0 along ) = 0 and 7? = 1, and thus condition (3.24b)

may be replaced by

a- 0 (3.26)

The cosine series (3.25b) for vo can be differentiated once

with respect to ), 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 w, may be differentiated four times with re-

spect to n.

It is concluded that it is permissible to perform

all partial differentiations with respect to q 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

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 E.

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 5 x !! 1 is

f(x) = + aj cos jinx


aj = 2 f(x) cos jflxdx

(2) The Fourier sine expansion of a function f(x) defined

on 0 x 1 is

f(x) =

bj sin jnx


bj = 2 f sin J7Txdx

In this way the following relations may be written:

S-2 KR (K ) sin j7r?


() Kj () cos JIT;7 + 1 K,0 (6)
arj 2

q = 2 K3j (4) sin j7Tr

V2MT = T2 KjJ (W) sin jrrj

Furthermore, the remaining terms on the right-hand

sides of equations (3.20) are given by

H,1 (9) sin JT??

J, (MO ) =

J2(.i) = L


H2 (j ) cos j7T? + H2 (9)

H3j () sin j7T?

In a similar manner the nonhomogeneous right-hand

sides of the boundary conditions (3.21) through (3.23) on

S= 0 and = 1 are

mbR(4,7) + S, (4,?) =

S. = B (

S3 (91_) =

Bj (9) sin JIT?


Bej () sin jTTj


-Mr(,) + s4(,1) =

C, (,) = 1 Coj (k

B3j (9) sin jTri

) sin jrn?

C.9 (4) sin jvTr





c#(9,1) =




) COS j'Wq + -1 B,. (t)

-M7(-,) + F, (,) = 2 Fej (9) sin j-u

b dM( ) + F ((,t) = 7 Fy (4) sin jrr'



As a result of applying the procedure described above,

the following systems of ordinary differential equations re-

sult (D denotes the operator -):

r D2 [r, (1 + + 2g Xj r/( DY
7r 11 7r

mgt c/3 2/3nibre
2, DZj = 17 Kz + Hij
2/o 772


re3(l + A) j DX, + r32(- D' (2rj + 2) Y+
Ir 1 7 1

t2 XmjT 2mbrc
Z =-- KIj + H2j
21o J r" a


fPlimb> Pm/32j2 + 32f Zgm] DZ
-p3DXf + pjYj + 2 b m [V + -b o 4 ]

Pf mj lT72
+ [ 2bG, +

(nm ,(l rek,e )jZ2TZ]
4 zE

mb mb
=-- + -K KGj + Hjj
2Gy2GGy KK 3 -


r1,,(l -/)D' 1 Yo = mbr K2o + H20 (3.28d)
[2n' 7 7T 2

where j = 1, 2, ....

The boundary conditions (3.21) on = 0 and = 1

become, for the case of simply supported edges,

3d ,uJY (k) = 2Boj () (3.29a)

jnx, () + / d- = Ba (=) (3.29b)

Zj () = Bej (t) (3.29c)

d z(U) = Bg ( ) (3.29d)

dYo ( )
1 dY = B' ( (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:

Xj = Xcj + Xpj


Y = Ycj + Ypj (3.30b)

Zj = Zc] + Zpj (3.30c)

Yo = YOc + Yop (3.30d)

Since equations (3.28) are linear with constant co-

efficients, solutions of the following form are assumed:

X' = Aje j (3.31a)

Ycj = Bi exk (3.31b)

Zj = C.e j (3.31c)

Yoc = B0e o (3.31d)

Substitution of the above relations into the homo-

geneous form of equations (3.28) reveals that Aj satisfies

the following quartic equation:

2 2P e 2Pf
(ree2 ) biA ei- g(i + bir ) -rekt j eJ

+ j2ke + j2 re( 1) {[brz ( 2) -

+ j2k,e (1 0 (3.32)


e (= -) j


A0 = + re(l -/4)

If there are no repeated roots of equation (3.32),

the complementary solutions are given by

x = 2 Aj e" (3.33a)

YCj= 21 Bjie e (3.33b)

Zc' = 2 Cj1 e Aj (3.33c)

Yoc = B, eXo + B,, e-X (3.33d)

The constants Bji and Cji are related to the constants Aji

by the relations

BjiL = bj, Aj,

CjL = cji Aji


b irr r,/32 [2 1 +,1)] AI2- _7[r. (1--)j + 2]
4A^ reg/32(l-/ 1 Aj,- 2 ,[2g(rej + 1) j2re(l+)] J

2 1o 2r e 13' r .r ( 1 + /,A) j /3 j i b
2/o___ [2re i3aAjc" J___2g_ b__
CJ m,3 2' oyAj j 772 7( J) re-2j- biI

The constants Aji 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

Aji = (3.35)

where Aj is an eight-by-eight determinant

A1 = Ia,|I

whose elements are given as follows for the case of simply

supported edges:

a/L = (/3Aj -/,ujTbiL )ejL aLl = (/3Aj /#j;7bL )e-jl
a2L = (/3AA /AjTbjL ) a,, = (/3AjL ,.jffbjL )
ajL = (JTT +/3A;L bL )eJL, a.3,, = (Jrr + /3AjL bJL ) e-bi
aL = j rr + /3AjL bL a,^ = j + /AjL bjL

a.L = cjL eAJL a,L, = -cjL e'AL
a6 = cJL a&,,,? = -cjI.

a = AL 2c eAjL a,,L, = AjL cj. e-x
a, = Aj 2cjL a1,, = Aj CjL

where L = 1, 3, 5, 7. The Dj, are the determinants formed

by replacing the i-th column of Aj by the column vector

2B./ (1) 1g a 1 + /aj7Ypj (1)

dX,. (0)
2B0j (0) -/ d-- ( + 1jfYpi (0)

dYp, (i)
B,4 (1) jxpj (1) -1 dYj(1)

dYo; (0)
Bcj (0) jl7Xpj (0) /3 d-(

Be] (1) Zpj (1)
B'1 (0) Zp, (0)

B3j (1) ____ -1(-

B (0) d2Zpj (0)
B31 (0) -

dY0P (0)] + dY0, (1)
e- Bo (0) -B,(1)d /3 dE
Boy = YZ ]d
/3Ao(e- A eO )

e[B(0) dYo (0)] -B () + dY ()
B e, CO (0) d % B '%' ) d'
/AoA(e-" e )

For the case when the edges k = 0 and = 1 are

clamped, the elements of Aj are given by

aL = eJL a, ,,
e AA
aZ. i a2,L 1

a., = (j r + /3AjL bjL )eAJ aL/

aL = Jr +/3 A L bJL a%,,

a., = CJ e, a,,,,,

a. cLl. a=,Z,,

alL = AjL CJL e JL a,,,,

aL = Aji. CiL. a,,,

where L = 1, 3, 5, 7. The Djj are





Bj (0)







C9i (0)

= e-L

= 1

= (Jir + 3 AjL bjL )e-A

= jPT + /3 AjL bjL

= -cj e-Ais
= -CjL

= AjL c e'-JL


formed from the column

- xpj (1)

- Xpj (0)
dYp1 (1)

- jTTXpj (1) /31

- jItXpj (0) 1 dYpj (0)

- Z ./ (1)

- zpi (0)

dZpi (1)
d t

dZpi (0)

For the

there results

case when the edges t = 0 and 4 = 1 are free,

a,, = (/3 j, mjfbj, ) e AjL

a2L = 3AjL /uJlbjL

aJL = (jTT +/iAJL bij )eAjL

aL = j1T + /3AjLbjL

as, = (/32AjclCjL AJ7'i2CJ.L )eA'.

a6, = /3AjL CJL Uj 7C

a7L = {f!1 [32AjJ- (2 -)JZa7Aj] CJL

/3 -t- j.A. e J
b ItGcAjeJ

aPL = /3 xJI3- (2 1)j2.2JL CJL
arL 7- &b -IAL j

-1 z-t )
- -t tG )c" Aj.

= -(/3AJL /4JTbjL )e-'"

= -(3JL J7bj L )

= (j77 + /?Xj bJL )e-JL

= J1T + /3A.JL bJ,

= -(3'2AJLCj, ,.j2l72cjc )e-AL

= -(/3"ALcJLj -/,J1a rCi. )

-{ [32 -( ^J72
IPf ) [13 2 L- p)jzIkjLJ CiL
17 z-br I

1 + min
m t

b -(12t CjLjl e-
b I

1 + m -
m tGX

1+ mi

a, L .l

a.,, Z. -!


a /2 3 (P'f [ 1] _l4 71A+ m tG.
a L,L. = A (2 /7)j1 1 I JL m

2(t tG)cjLAjL

where L = 1, 3, 5, 7. The Djz are formed from the column


dX,- (1)
2Baj (1) /3 + ,hdjfrYe1 (1)
2Boj (1) /3 dXpi (0) f+ /.ijlrYpj (0)

ddXpj (i)

Be, (1) jrX1 (1) 13 dY ,j (1)

Bej (0) jTXpj (0) /3 dY (0)

7r Fej (1) 1 dZj (1) + 1 J) iZpj (1)
P" d +Z

P F, (0) dzf.1 + /jI77Z,,j (0)

3 Ad' ,,i (1)1^ \1''J
F,1 (1) P- / [ # /(1) (2 j.ITL dZ(1)1
7r2b I dd Z j
d- dZ (l
+ --- tGX p (1) + dZ tG (1)
m b d

Fyj (0) 1-- 3 [ djZe(0) (2 -/4)jZ" dZ(0)
772b [ dt3 d7

+ t m GX PJ (0) + -(- ,)dzZ j(0)
m b d

Therefore, from equations (3.19), (3.25), (3.30),

(3.31), and (3.35), the solutions for the displacement

components u, v, and w are

u(9.") = f, () + (1 I)f ()

+ 21 { DBi e Xj + X'.' (k) sin JlTr
F_ I
J -f :1 )~


[gj ( ,) / I df, () 1 (1- g- [ M df0 ()
2 d t 1 2 d I

+ LLi bjiD eA-f + YP, ()cos jTr
j e B,, +

+ B01e + Bi, e-A'1 + yoP m

w(.?) =


77 1 ( l) (? + l)h, () + (2- )ho(

+ m j

cji Ai eA + Z^ (e) sin jlrY[
7z \ Y z,


These equations completely determine the displacement


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 .

Unfortunately, because of the extreme complexity of the

terms in each infinite series, it is not possible to

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


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'(4) = fm) b R(9,l)dt (3.37a)
f' (= + )

g, () = g, (S) = mbR

h, () = h0() = 0




MT = 0

The particular solutions can be determined from equa-

tions (3.28) after the following relations are noted:

r,/3mb 2 _R
T,,) = ( /. + 1 +/a)-( ,1)

+ 2g 1)mb l J-R( Rl)d 1
13(1 + U)

T'(%,) = -re/3~ -/4 ( lmbl
(1 1u) (2- 1) mb(1 -2 R_

/ [
+ (2- 1)Inb (1

T(,) = /onmb(l + g 1 )R(+ ,)

and after the following series expansions are noted:

1 si 4 sin j
1 /7 ,...





1 4 ~ j- cos jr?7 (3.39b)
i. ,j...
1? 2 17'~____

bR(t',) 2 [' )R(4,)) sin j7T d sin j7rl (3.39c)

_R(_,__ 2 aR( ,',) (3.d9d)
R(t) 0 =2 j R(, cos jTT- dn cos JT (3.39d)
i'',3 0

where the series in the last two equations contain only

terms with an odd-numbered index because of the symmetry of

R(t,7). Thus from equations (3.38) and (3.39) there result

K,j = 2 L R sin jrTT d7 (3.40a)

6 R
K j = 2 cos jITJ? d?? (3.40b)

Kjj = q0 (3.40c)
K 1J = 0 (3.40d)

K'j= 0 (3.40d)


= 0

2 i +/f) 1)d ] j34f
4 r3 ___' R( i)

H, =--^ re- lmbl ...
H r. 2 + 1 +

+ mb(l + ((3

H2 = mb(l + g 1 R(4 1) + (3.40h)
nPL!L g ( 1+/^~ )R(,l

H80 = 0 (3.40i)

where j = 1 3, 5, ....

Then the particular solutions Xpj, Yj, and Zpy are
found from equations (3.28) after P. 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

Y0p = 0

The particular solutions for two specific temperature

distributions were found. For a constant distribution R = RO

these solutions are

Xpj = Ajt + B

Ywj = Cy

Zp- = Di



A = Bamb R
S r/3(il +/ )j [re(l -/a)J3 + 2g] R

Bj = 1 Aj

=. y41 ^ ^'3 J1

and = 2(rj + 1)
l= 2(rej2 + 1)

1j- 2 j7

= [ /d )R- -
j 1=2J2 mb(1 +/ J

8r, mbgRo
rr'[r e(l )j2 + 2g]

.j = pjrr

gy =

r(mZ(l rek,e )7'2j2

S= Tj\ 2G +[ 1 +/" J

8pmb g2R
~ 7j(l +P)[re(l -/L)j' + 2E]

Here j = 1, 3, 5, ....

For a double sinusoidal distribution of the form

R = R, sin nt sin rrj, the solutions for j = 1 are

Xpj = A1 cos 1T E

Ypj = Ej sin Tr + F; (3.42)

Zpj = Hj sin t + Ij


Aj = 9Sj pli

Aj = Gi P5 j

H1 = ,-L


S= R,

Pj is the determinant

V .j Vi

x /3j


i = 2r,/ 3 + r e(l 14) + 2

-X, = r,13(l +,)

rj = log1

S= re/3'2(1 -/u) + 2re + 2

mgt of/
= IT

.j= -4 (g -, rekve + 1)

and Pii is the determinant formed by replacing the i-th

column of Pi by the column vector



F = -A j I -3j 5

= A j 7. j /M.i


A = 4 mbqo
Trji 2Gy

For j = 3, 5, 7, ...

Aj = Ej = Hj = 0

The coefficients in (3.27) are determined from equations

(3.37) as

Boj (0) = B, (1) = {mb[+ (1 -a)]R(l,1)J

+ 2mb R(l,r) sin jTu d7 (3.'

4 cR( I1,1) ( (3.
B, (1) = -B,j (0) -;T7 mb 1) (3.,

Bej (1) = Bej (0) = 0 (3.,

B5i (1) = B,, (0) =0 (3.





Bo (1) = B.o (0) = 0


C.j (1) = Coj (0) = 0 (3.43f)

C,1 (1) = Cj (0) = 0 (3.43g)

Fej, (1) = F,1 (0) = 0 (3.43h)
m) tG b^ 1! 1
F3j (1) = -Fj (0) 4 (1 M R(tul) dJ (3.43 )
^ (l)-7^l3(l + 1-4 JO .~

where j = 1, 3, 5, ....

The assumption of a temperature distribution sym-

metric in the 9-direction makes possible a simplification

of the complementary solutions (3.33). The function E1(M)

is defined as

Ej () = ej(4' Y) + e-)(j

and is seen to be even with respect to the line = 1/2,
which bisects the plate; the function

j0, () = e kj( Y' ) e- ('

is seen to be odd with respect to F = 1/2. Each pair of

terms of the form

Aj eAj4 + Aj,., e- ki

that occurs in the complementary solutions (3.33) may be

written in terms of the newly defined functions as

1l(Aj, e' Ai + A,,,, e- Aj )Ej + l(Ai e'i Aj,., e-'1/ AJ )0

Under the present assumptions, the horizontal dis-

placement component u will be odd with respect to 9 = 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 E, and Oj becomes

U mb R(, l)d + 2 1{21 (DJ)2A e/AXJ
1 3(1 + ,a) 2 f A1 1e
J/j= f, 3

+ D,/ e-'Jk )Ej, + D.;, -, e ,

-Dj, e- ) iJ sin j17-

Since u must be odd and the functions Ejk and Oj, are lin-

early independent, the coefficients of Ejk must equal zero;

there result

Dz = -e Dj,

Dj, = -e'2 Dj3

Dj6 = -eAJs Djf

Therefore the solutions (3.36) become

u mb R(t,l)dt +
1+ s1 +s) J;

+ XpjI sin JT?

e j, [ e e*- 1 Dji
y.j.~J .. .I


v = "- J[ mb(l --) R(t,l)
2 1 +1( 1

D (3.44b)
bj- j ex + ej'j + Ypj cos J77(
^j j

j-- 7,... *

/ ''
w = DAj [eA1 + ei ] + Zpj sin j771 (3.44c)
i., 1~... .

Here the Ajj are the roots of the cubic equation

(re, ej ) (g-j + rek,, j'8O Zj'ke )

+ j're(g 1) ej- j'kie (1 ----) =


e = ( 3) -j

Further, Aj is the three-by-three determinant


Az = aKL (3.46)

where K,L = 1, 2, 3; the elements aK. are, for the case when

the edges = 0 and F = 1 are simply supported,

a,,L = (e + 1) (/3AjL uj7TbjL )

a2L = (ezjL 1) (jrr + 13AjL bjL )

a3L = (exJL + l)CJL

where L = 1, 2, 3. The Dyj are the determinants formed by

replacing the i-th column of Aj by the column vector

dXp, (1)
2B.j (1) /3 d- + /4jlTYJ (1)

Bej (1) jTTXP (1) A dYpj (1) (3.47)

-z>,, (1)

where the coefficients B.1 and Bej are given by expressions

(3.43a) and (3.43b).

For the case when the edges = 0 and E = 1 are

clamped, the elements of Ay are

a. = ej 1

a2, = (e j 1) (jTr + /3AiL bL )

aJL = (e JL + 1) cJL

and the Dji are formed from the column vector

-xpj (1)

dYp1 (1)
Bej (1) jnXe, (1) 1 a(

-zpj (1)

For the case when the edges = 0 and = 1 are free,

the elements of elj are

a,L = (e*JL + 1) (/3A, /,jnbJL )

a2,L = (e JL 1) (jiT + /3AjL bjL )

a., = (eAJL l)[1+ m 13m Aj-
[ m b

and the DjL are formed from the column vector

dX,., (1)
2B.j (1) / d-- + /ujrrY/p (1)

dYe, (1)
B,1 (1) jnX-j (1) -

T. ) J "^_ ni ^ dZp; (1)
1 + m 13 -t Ed.) d
F~j (1) M tG., Xp bi dt't,

The coefficients F9j are given by expression (3.43i). The

quantities b., 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 + aA2 + bA+ c = 0 (3.48)


A= A=j

it is seen from the definitions and physical meaning of the

coefficients that

a< 0

b > 0

c < 0 for ke < J
1 + rej2

c > 0 for k,e >
1 + rj

From Descartes' rule of signs it follows that the number of

positive real roots of the cubic equation will be either one

or three if ke < 1 I -r or zero or two if ke > + e-r.
1 + r, j 1 + rej
In general, then, as ke which represents the compressive

edge loading, increases past this value, complex values of

X are to be expected.

A further observation is that when oNy, = 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 oNyy 0:

mb( R(,I)d + 2[ej" (1 eFj)e
Z 1,3..

+ 2 A [eAjA e""("] + XPi sin jTv (3.49a)

mb (1 )R(_,_)
v = (2n l)- l )R(+1)

+ 21 [ -bj,[ ei + (1 )e

+ ej, [e' + e J

+ b [e + e "] + Ybj cos j+rr (3.49b)
(* 1

w = r c1,[Sex'' + (1 )e''- ]

+ gj, [e -. 4 + e "1 (1-]} Dj3
j A

+ c. [e'-" + e""1-1) -A- + Z Pj sin jTr (3.49c)
+L ci I? ZJ

Here the Ay are the roots of the equation and are easily

found to be

= ~j 17

7=7 /2g + rej2 ( -/4)
Aj = re,(l -F/)

and Aj represents the determinant

AL = IaI

whose elements are given by

a,3 = /31 + (1 + Aj,)ej'] /,j7[e., + (by, + ey,)e']

a,. = jnei' + /3 {-(Aj, ej, + b,/) + [(bj, + ei,)A\, + b,,] eJ'}

a3. = gj, + (cj, + g,)eA,

= (1 +

= (e AJL
= (eZ;

e j ) (/3AjL /j7rbjL )

- 1) (ji +/3bJAJL )

+ )cJL

where L = 1, 2. The Dji are the determinants formed by re-

placing the i-th column of Aj by the column vector (3.47).

The bji and cji are given by equations (3.34);

r. rjz(3 -) (1 ( ) + 2g
ej n j rej2 (1 + P) (1 g) 2g

g4o= -morJz reJ (3 -,1-) + g reJ(1 +1,) + 2

while Xpj, Ypj, and Zpj are the particular solutions of

equations (3.28).



Although there are many reports of experiments on

rectangular sandwich panels without thermal effects, for ex-

33 34
ample, those of Boiler33 and Kommers and Norris,34 the only

experiment involving thermal deformation known to the author
is the one performed by Ebcioglu et al.24

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.

/= 0.3

E" = E" = 30x106 psi (4.1)

G = 39,825 psi

Gy = 66,400 psi

a = b = 6 in.

and thus

kle = 0

P, = 1,853 lb/in

t= 1.0253

r, = 0.067214

0' = ac" = 7.34xl0-6 in/in OF (4.2)

q = 0

g= 0.6

m= 1

,4 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

R = (1.25 sin 77? + 1.373 sin rTy sin rrt)10 (4.3)

A measure of the error caused by representing the tempera-

ture distribution by equation (4.3) was calculated from the


E o aIa e(4
Error = ----- (4.4)
A Tmax F_ i

where the ai are the experimentally recorded values of the

temperature gradient, the c. 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.



Temperature Measured
Trial Gradient Error Deflection (in.)

1 R +0.005 0.013

2 1.1 R +0.023 0.0135

3 1.2 R +0.043 0.0195

4 1.8 R +0.065 0.021