General nonlinear theory of sandwich shells.

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General nonlinear theory of sandwich shells.
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viii, 85 leaves : ill. ; 28 cm.
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Huang, Ju-Chin, 1934-
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Sandwich construction   ( lcsh )
Shells (Engineering)   ( lcsh )
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non-fiction   ( marcgt )

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Thesis--University of Florida.
Bibliography:
Bibliography: leaves 83-84.
Statement of Responsibility:
Ju-Chin Huang.
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Manuscript copy.
General Note:
Vita.

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University of Florida
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Full Text








GENERAL NONLINEAR THEORY

OF SANDWICH SHELLS





















By
JU-CHIN HUANG


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
December, 1965














ACKNOWLEDGMENTS


The author wishes to acknowledge Dr. I. K. Ebcioglu, chairman

of his Supervisory Committee, for his invaluable suggestions and

guidance throughout the entire period of this study. He wishes to

express his sincere appreciation to Dr. W. A. Nash, chairman,

Department of Engineering Science and Mechanics, for his arrange-

ment of the financial support as well as his constant encouragement.

He also thanks Dr. S. Y. Lu, Dr. J. Siekmann, Department of

Engineering Science and Mechanics, and Dr. R. G. Blake, Department

of Mathematics, for their serving on his Supervisory Committee

and their valuable comments and advice.

Finally, the author is very much indebted to Dr. L. M. Habip,

Department of Engineering Science and Mechanics, for his generous

discussion and suggestions and also for his reading the manuscript.

The author would also like to acknowledge the financial support

of NSF Grant No. GP515, a Graduate School Fellowship, and a College

of Engineering Fellowship from the University of Florida.















TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS . . .

LIST OF SYMBOLS . . .

ABSTRACT . . . .


CHAPTER


I. INTRODUCTION . . .

II. PRELIMINARIES . . .


2.1 Outline of Tensor Analysis .
2.2 Concepts from Three-dimensional Theory of
Elasticity . .
2.3 Surface Geometry . .
2.4 The Relationship Between Space and Surface
Quantities in Normal Coordinates .
2.5 Modified Hellinger-Reissner Variational
Principle . . .

III. GENERAL NONLINEAR THEORY OF SANDWICH SHELLS. .


3.1
3.2
3.3
3.4
3.5


5
. 5

. 10
. 14

. 18

. 20


. 26


Change of Reference Surface .
Equations of Motion . .
Strain-displacement Relations .
Constitutive Equations . .
Boundary Conditions .. .


IV. SPECIAL APPROXIMATIONS . .

4.1 General Nonlinear Membrane Theory of Sandwich
Shells . . .
4.2 Partially Nonlinear Theory of Sandwich Shells
4.3 Analogy to Donnell-Mushtari-Vlasov
Approximation . . .
4.4 Partially Nonlinear Membrane Theory of Sandwich
Shells. . . .
4.5 Sandwich Shells with a Weak Core .
4.6 Linearization of General Equations of Motion.


. viii














TABLE OF CONTENTS (Continued)


V. COMPARISONS AND CONCLUSION . ... 68

BIBLIOGRAPHY ......................... 83

BIOGRAPHICAL SKETCH ...................... 85













LIST OF SYMBOLS


.o






A

.b

b b


Ci


c'
.c
C








f


F
r 4


components of acceleration vector

base vectors in a Euclidean Space of

normal coordinates

first fundamental form of shell middle surface

area of shell middle surface

components of body force per unit mass

second fundamental form of shell middle surface

isothermal stiffnesses

effective external couple resultants measured

per unit area of the shell middle surface

edge curve of shell middle surface

elastic coefficients

acceleration resultants defined in (106)

prescribed external edge forces on facings

base vector in cartesian coordinates

elastic coefficients

acceleration forces

body forces

effective external edge moments defined in

(106)










3I


2hk ,2 2.







7n


n,.

wn-",


n n"
*j o/1 YZ,


pfi


r, r; R







5

5

" -, ,
t t /t


base vectors in a Euclidean Space

metric tensors

thickness of core and facings, respectively

volume integral defined by (93a)-(93c)

surface integral defined by (93d)

prescribed edge moments

acceleration moment resultants

body moment resultants

moment resultants due to S and

5"/ respectively

components of normalvector in E-3 space

stress resultants due to / and

"S respectively

effective external loads measured per unit

area of shell middle surface

stress resultants due to S S and

respectively

position vectors

arc length of a curve on the shell middle

surface

area of shell surface

prescribed edge forces

stress tensor

moment resultants due to "S and

$^ respectively













I j



L vi ,
'U, j- 4 V



/ V4 V4'







Zi
c.r
CXt





Z L







I



fe /


-I Q

0)/i,


temperature field

thermal stress and couple resultants

components of displacement vectors

components of displacement vectors defined

by (100)

component of displacement vectors in E-3

space

cartesian coordinates

thermal coefficients

strain tensor

Christoffel symbols

variation symbol

Kronecker delta

convected coordinates

strain measures defined by (97)

strain energy function

density

expressions defined by (65) and (66)

respectively

inverse of AA defined in (69)

component of normal vector to normal surface

strain energy per unit area of shell middle

surface


vii














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

GENERAL NONLINEAR THEORY OF SANDWICH SHELLS

by

Ju-Chin Huang

December, 1965

Chairman: Dr. I. K. Ebcioglu

Major Department: Engineering Science and Mechanics


A general nonlinear theory of sandwich shells to the full

extent of the nonlinear strain-displacement relations

j = ( V 1 -*+ iVrl )
has been obtained by means of the modified Hellinger-Reissner

variational principle of three-dimensional elasticity. The

fundamental equations are in tensor notation and in terms of the

undeformed state. By this same technique and by making certain

simplifying assumptions in the strain-displacement relations, par-

tially nonlinear theories for sandwich shells are also obtained.

These approximations are based on the assumption of small strain

but large deflections, thin facings, and soft cores, respectively.

These intermediate theories are more suitable for application.

The intermediate and the linearized form of the various sets

of equations derived herein coincide with known theories.


viii














CHAPTER I


INTRODUCTION


A sandwich structure is formed by two thin facings of a

strong material between which a thick layer of very light-weight

and comparatively weak material is sandwiched. The advantage of

this kind on construction is the large moment of inertia of the

section provided by spacing far apart the load-carrying facings.

Accordingly, the applications of sandwich structures in various

areas, especially by the aerospace industry, have increased in

recent years. As a result, more research work concerning this

kind of structure has become desirable. A series of extensive

bibliographies (1), (2)* and comprehensive reviews on the analysis

of sandwich structures by Habip (3), () have appeared recently.

In the early work of Reissner (5), (), (7) and Wang (8),

the simplest model consisting of two facings acting as membranes

and a core resisting transverse shear and normal stresses has

been employed for deriving the governing equations of sandwich

structures. In recent years, it appears that changes in technol-

ogy and a concern for the optimization of structural elements



*Underlined numbers in parentheses refer to the Bibliography
at the end of this dissertation.










subjected to thermal as well as mechanical loads have brought about

several studies of a new type of sandwich construction with strong

cores. The theory then takes the flexural rigidity as well as

transverse normal deformation of the core into account while

including, as usual, the flexural rigidities of the upper and lower

facings about their own middle surfaces. Grigoliuk and Chulkov (9)

have presented a paper on this subject for the case of small de-

flection theory of sandwich shells. They consider the core as a

three-dimensional body and assume that the displacements can be

expressed approximately as a linear function of the transverse

coordinate. Wu (10) generalized this to a large-deflection theory

of orthotropic sandwich shallow shells. In the latest works of

Ebcioglu (11), (12), the nonlinear field equations for the sandwich

plates and shells have been obtained by means of the Hamilton

principle. For plates, the nonlinear strain-displacement relations

are used to the full extent, but partially nonlinear strain-displace-

ment relations are used to the full extent, but partially nonlinear

strain-displacement relations only are employed for shells. The

present dissertation is inspired by Habip's works (13), (14). The

fundamental equations of the theory of plates and shells have been

obtained by Habip from the three-dimensional theory of nonlinear

elasticity by integration across the thickness of the undeformed

plate and by means of a modified version of the Hellinger-Reissner

variational theorem of three-dimensional continuum dynamics,

respectively, for the case when the "shifted" components of dis-

placement can be assumed to vary linearly through the thickness of










the shell.

The present study is an attempt at obtaining a general non-

linear theory of sandwich shells to the full extent of the general

nonlinear strain-displacement relations (39) in tensor notation and

in terms of the undeformed state by means of the modified Hellinger-

Reissner variational principle of three-dimensional elasticity.

The method employed here is similar to that used by Habip but the

order of variation and "shifting" is slightly different. In this

work, we introduce the geodesic normal coordinate system into the

variational equation before we carry out the variation. In the

general theory, the results are identical regardless of the order

of variation and "shifting" but in partially nonlinear theories the

results come out quite different. We shall discuss this in more

detail in Chapter V.

In Chapter III, the "exact" fundamental equations, in the

sense of using the complete general nonlinear strain-displacements

relations, are given. No assumptions about the state of deformation

have been made except that the displacements vary linearly through

the thickness. But these "exact" equations are, for practical

purposes, too complicated; so we introduce some simplifications

in Chapter IV. These approximations are based on the assumption

of small strain but large deflections, thin facings, and a soft

core, respectively. By making these simplifications we arrive at

several approximate theories suitable for applications. When these

theories are compared to some known results, they do agree exactly.










The present work, as in (11), (12), takes into account the

effects of transverse shear and normal stresses as well as rotatory

inertia, with different material densities and material constants

in each layer of the sandwich shell. In addition, each of the

three layers is of different thickness, and no a priori limitations

are imposed upon the displacement functions until Chapter IV. It

is also assumed that the facings and the core are anisotropic,

having elastic symmetry with respect to the middle plane of the

layers. The effect of steady thermal gradients is also included

in the stress-strain relations.














CHAPTER II


PRELIMINARIES


In order to formulate a geometrically nonlinear theory of

sandwich shells, some fundamental concepts from tensor analysis,

surface geometry, and elasticity will be used repeatedly in this

study and are reproduced here for convenient reference. More

detailed treatment of these subjects may be obtained in references

Q5), (6), and (U).


2.1 Outline of Tensor Analysis


A. Convected coordinates and base vectors. Let Z be a

set of fixed rectangular cartesian coordinates. The e; are

unit base vectors, and the position of a point in space with

coordinates EZ can be defined by the position vector r

where


r = z"ei (1)


The range of Latin indices is 1, 2, 3. Repeated indices are to be

summed over their range.

The differential of the position vector in (1) is


d = dz= d (2)
rz J









The length of a line element is defined as

J1= d ir r =e **j 4zJ Szjd'd~W = '(3)


where <;j denotes the Kronecker symbol.

Let us now introduce a general convected coordinate system

9 defined by the coordinate transformation

8"= Cz' ZE) (4)


or

Z' = Z'e ( ee, Q ) (5)


provided that the Jacobian of (4) does not vanish.

If r is regarded as a function of e and using

a comma followed by a subscript to denote partial differentiation

with respect to ,


dr= r, de = J e, (6)

where

,--Z
L (7)


are the base vectors of the convected coordinates.

By the chain rule of partial differentiation, the differential

cd e can be expressed as


de = (8)









Substitution of this expression into Equation (6) yields

d =(9)


which, by comparison with (2), leads to

Sje (lOa)


J a = (10b)

From the definition of the law of covariant transformation we know

that & are covariant base vectors for the e coordinate

system.

By using (3), the length of a line element is


dd = d F = g 'de = g dj dej (11)

where


j = (9 9j (12)

is called the covariant metric tensor for the coordinate

system. From Equations (3), (11), and with the help of Equation

(8),


3m 9 on n (13)


B. Reciprocal base vectors. There is no distinction between

covariant and contravariant components of the base vectors e

in a Cartesian coordinate system. Thus we can write e = e ,









and define contravariant base vectors in the e coordinate

system by the law of contravariant transformation (17), namely,

0e" -
= ~ e (14)


In view of Equation (10b),


S== j, (15)

so that, in general, the covariant and contravariant base vectors

in the 6 coordinate system are orthogonal and reciprocal to

each other. By analogy with Equation (12), the contravariant

metric tensor is defined by


-" (16)

The following useful formulae are derived from Equations

(14), (10), and (15).


= (17a)




9 9j = (17c)

Thus, the metric tensor can be used to raise or lower indices.

The volume element for the convected coordinate system Q is

given by


d = e'de e0e (18)









where


9= 9j (19)


C. Derivative of a vector. The covariant base vectors

9L are expressed in terms of the position vector r by
-=b
Equation (7). Since r is assumed to be a continuous function

of 1 it follows that


g.,j = 9jL (20)

Differentiating Equation (10a), and using (10b), we obtain

_z = ain!M 9 r
9LM -^eM J 3 er

Introducing the definitions of Christoffel symbols of the first

and second kind, respectively,




and

wi = 9 r jr (21)

the derivative of base vectors (20) becomes


FZJ- &J g(22)

Now for any vector V which is expressed in terms of

its components by


V -V 9 V O (23)









the derivative can be written as

""J = V 9 i +V, s (24)
(, = V < ),j = V',j> S+ V Sij (24)

By using Equation (22),

V, V 9. (25),

where

v'lj = v,+ r v (26)

is the covariant derivative of the vector Vm

Similarly,

V', = Vmj 3 m (27)

where

VIj= Vi,j (j V, (28)

is the covariant derivative of the vector V1

2.2 Concepts from Three-dimensional Theory of Elasticity

A. Strain tensor. Let the undeformed body be described
by a general right-handed convected coordinate system. 0
If the position vector of a point in the undeformed body is
denoted by

r = (\, d ) (29)

then when this body is subjected to load, it will deform into
a new configuration

R = R(o ,e e o) (30)
which is the position vector of a point in the deformed body.









The 0 are material coordinates which are associated with

corresponding points in the deformed and undeformed body.

In the undeformed body, the length of a line element is

given by

cd I = 9 de'dej (31)


In the deformed body, the length of the same line element becomes


dS'= Gij eadej (32)

where rLj is the metric tensor in the deformed state. The

difference in lengths of these line elements is a measure of the

deformation experienced by the body as it moves from the undeformed

to the deformed position. This deformation is described by the strain

tensor defined by


d S'-d = 2 .j c ej (33)

from which it follows that


=-ij- )j (34)

This shows that i"j is a symmetric tensor.

The strain tensor can be expressed in terms of the displace-
-->
ment vector V by noting that


R = r + V (35)

By using the formulae given in the previous section, the base






12

vectors and metric tensor for the convected coordinates e may

be defined in both the undeformed and the deformed body. Thus,


i= CrL Gc R = RL

3ij Ji j CTJ i 4-; 6 3
(36)



f= ft r r G

In view of Equations (27) and (35), the base vectors C4 and

metric tensor zrj become

6 =( + +v ) ?I (37)


Gj= (L +VI)(S> V/Ij) (38)

Then, substituting these into Equation (34),


j -= ( Vlj + V4l + Vr Vrj ) (39)


B. Stress tensor. The state of stress is here defined in
-4P
terms of the three-dimensional stress vector, of per unit area

of the undeformed body and associated with a surface in the deformed

body whose unit normal in its undeformed position is ,n namely

(18),

.=s (40)


where












By using Equation (37), expression (40) becomes


5 = s (Ir).nf ^ = t j.n, .- (41)


The 5" and tj are contravariant stress tensors measured per

unit area of the undeformed body and referred to base vectors in

the deformed and undeformed body, respectively.


C. Stress-strain relations. The conventional Duhamel-Neumann

stress-strain relations for an isotropic material can be generalized

for an anisotropic material as follows (13).


Y E= F 5' s"" + c~j T (42)


The inverse of (42) is

S '= C#j rs ( Yrs- CXr5T) (43)


The coefficients C.. Cjrs are the elastic coefficients of

the medium. They depend on the metric tensors and physical

properties of the undeformed body. The oC;j are the coefficients

of thermal expansion. All of these coefficients are also functions

of 0 through the steady temperature field T and satisfy the

following symmetry conditions

Epjn E Ejnm E I
G". j S = Sj P

C,.ij r s C irS = srS CrrSiJ


OC Ij = cxjb









For a medium having elastic symmetry with respect to the surface

3 s= const., all coefficients containing the index 3 either once

or three times vanish (15), i.e.,


CG r-= Ca = 0 (44)

Equations (43) then reduce to (14)

5"= C "P' ( oC,, T) + c'( ,-ocaT) (45a)

5s z c'3^ ( 3 c T) (45b)

= C3 '(,-o,,T) C (3 ,,-3r3T) (45c)



2.3 Surface Geometry


A shell is a body occupying the space between two surfaces

(called faces) a small distance apart. The coordinate system is so

chosen that the surface defined by e)= 0 lies midway between the

faces. It is called the middle surface. When 93 is measured

along a line perpendicular to the middle surface, the coordinate

system is called normal (17). In a normal coordinate system the
-P
position vector r for the undeformed body becomes


r. r( ', r -e a (46)

The vector r* locates points on the middle surface, and 0

and e are general curvilinear coordinates on this surface. The

vector a, is the unit normal to the middle surface, here directed










outward on a surface of constant positive Gaussian curvature.

The covariant base vectors of the middle surface are


= -- (47)


Here and in what follows, we use the convention that Greek indices

take on the values 1, 2. The metric tensor of the middle surface is


The contravariant components are defined by


a" a= &


The contravariant base vectors are defined by


y = a' t'p


The area of an element on the middle surface is


Ah= 6J de"


where

a=-a,6 = a,,a, a (a,,)4


(48)





(49)




(50)





(51)


(52)


The coefficients of the second quadratic form are defined by


(53)


he me cp on, t= a3o ar.,,a

The mixed components of the second fundamental form are

b b(


(54)










and the contravariant components are


A cA b 1 (55)

The Christoffel symbols for the middle surface can be obtained

by evaluating Equation (22) at e = We have

Y ay+ "-t+ (56)


where an asterisk denotes quantities on the middle surface.

Equation (56) may be rewritten as


|i= A -3 (57)


Here the double vertical line stands for covariant differentiation

based on the metric tensor lp

From Equations (53) and (21), when evaluated at 0 o


Is C L, ,,= ba., (58)

and hence


LM= lip a'. (59)

Multiplying Equation (53) by CL we obtain


3,/ =-(60)

or

L = 4 =- (61)
111M a~=b~









which is Weingarten's formula.

From Equations (59) and (60)


b'16Y b
which are the Mainardi-Codazzi relations.

The covariant base vectors of the space (46) can now be

written as (16)

9<= ,+ 6 3~,-= f( a ,(63a)


3= (63b)

and the metric tensor of the space (46) becomes


=/ p (64a)

,3= 0 (64b)

9 =- (64c)


where


S= S; 0' 6 (65)

From Equations (64) and (65),


/4= / == (3/ ) (66)

Raising the indices o0 in Equation (63a) leads to

(67)









or

E= g= Y is (68)

where

S1 (69)

is the inverse of A (16).


2.4 The Relationship Between Space and Surface Quantities in
Normal Coordinates


Differentiating Equations (63) with respect to 6

= /a .+ (70)


From Equation (21),

(71)
P- I Y + 3*

Substituting Equation (56) into (70), and combining with (71),

r r.9+ 3, a S ) + s / (72)

Eliminating and 53 from Equations (63), Equation (72) yields

<( a^t r: .#r: ( -0 ) -- 0 (73)

where

Sr,/ (74)

Equating the coefficients of a; and a, to zero, respectively,

r -P (' (75)

3
r^ = P. (76)






19

These are the relations between space and surface Christoffel
symbols. Equations (75) and (76) have been derived by Naghdi
(16) but in a much more lengthy and indirect manner.
From Equations (23),



S v lt + v', (77)


Referring V to the space (46) with base vectors t, ,

It -, ft ri -*) V033
V = V~a V, a = V A.*- a (78)

Comparing Equations (77) and (78), with Equations (63) in mind,


V, = / V, v = f') v" ,
V.= .V' V (9



V, = V-= v- = v

Equations (79) are the relations between space and surface vectors
(16). V is a surface invariant. The /4, and its inverse act
as "shifters" in the space of normal coordinates.
From Equation (28),


= v -rF.4v- fr',,V3 (80)

which, with the help of Equations (65), (76), and (79), reduces to


V.|, = /< ( Vy, b6 V" ) (81)









Similarly,


V01= C); (v,(-6 v") (82)

For the other components of Vklj and V j we obtain, in a
similar manner, the following expressions (16)




V31,= v,. vV,= v., (83)

V =I3 V313 = V3,3 = V,= V*j,

Equations (81), (82), and (83) are the relations between the
space and surface derivatives of a vector.

2.5 Modified Hellinger-Reissner Variational Principle

For an elastic body (18)

sJ = l ) (84)

where 4 is the strain energy function, measured per unit volume
of the undeformed body, and satisfying the following relation

= S' ,- (85)

where & stands for variation.
The modified Hellinger-Reissner variational theorem can be
stated as follows: The state of stress and displacement which
satisfied the differential equations of motion and the strain-








displacement relations (39) in the interior of the body, and the

conditions of prescribed stress on the surface of the body, is
determined by the variational equation (14)


g [f ( (+
OT ( i ^ s ) y T ^


+Vrvrl)J), f. f.n- "1Vivt S = o,
v J.,S

in which the components of the strain tensor Yj stress tensor

5" and displacement V4 are allowed to vary independently.
The S. and .b denote respectively the density and the com-
ponents of body force per unit mass of the undeformed body, OL'
the components of the acceleration vector, oZ the volume of the
undeformed body, .S its total boundary, where only the stress,

t is prescribed, and d4 and JS represent the corresponding
elements of volume and area, respectively.

By Equations (18), (51), and (66) the element of volume is


d -c= .- dede'de~


36)


= d+4deo


(87)


For the face boundary, the element of area is


(88)


and, for the edge boundary (16),


.n, dS = / .., / d e


(89)








where o)/. are the components of the unit vector normal to the
intersection of the undeformed middle surface and the edge
boundary, and d is the line element of that intersection.
The variational Equation (86) can be written as



+ V V, i) + 2 ( Vl *V 4v + Vs .l, V, + v, ) +,

+ ,, v,, v dVis v,,,V ,)] n) +j>.(.0
*r
.ad)v, + S.C.b'-.) ) v ] + (fc.. 'v, .
.O J
.-oj^ve)dS } = o
I. (90)

By using (81), (82), and (83) we can formulate Equation
(90) in terms of the "shifted" components of displacement V;
and V as follows.

[S '---,) _. -f jL v;

/,(v ..v + ( v' -b v ( V -b ) ,

+ ( v +b v+ )(vb v )] +x S I +




.I& b* +
3, 4V. '.. J (9 1),




V, -.a +N ) V, -=C +.
5 ., Jr.v .C + .,, v 3 o (91)








We now carry out the indicated variation in (91), using
Green's transformation and Equation (87), and also dividing the
resulting volume integral into three parts, to obtain.

6 + I. SI, -J = o ,(92)

where

nI = [[^/i ^ sv v:) s^v'



s '.,s(veSb ) V 4(s"- v ],,





(Vs' (v,,+ *v r ;)+. f(,. v;,)]l/ +

+[-Cs" 'c *va, vl .a), s"<, v:,,)>,, +






= f{ r^- il V;^-b ;) -t(Vbw)
.,v
"pE fI -" a") SV&4 #( (93a)







Sv v )( v ,-bsv ,) .. v. +b. V),

lvf;>jr* \f SS+ + +'-ctr.. ,

-, + v; + < v% + b V3;) r v, + C V1,0. +










+ v v, t (= +
+ V $,S,, + V, V,, /j4 (93b)









u 'svy,] v' + -, ( g)
[ ", i'1 s J 3o (93c)-





V"













Here ,C represents the intersection of the unreformed reference
nr s*'/~s /v' t V"' +:


+ s J ,14 V;'j 3 +^ ^















the undeformed reference surface.
Since the displacements, strains, and stresses are regarded
as arbitrary functions and allowed to vary independently, we can
set I I and J equal to zero, respectively, and

thus obtain the four sets of fundamental equations. In what follows,
thus obtain the four sets of fundamental equations. In what follows,







25


we shall apply this principle to the three-layered sandwich shell

in order to obtain the corresponding equations of motion, strain-

displacement relations, constitutive equations, and boundary

conditions.













CHAPTER III


GENERAL NONLINEAR THEORY OF SANDWICH SHELLS


3.1 Change of Reference Surface


Let z2 2h 2h be the thickness of the upper facing,

core, and lower facing, respectively. From now on, the single

prime, bar, and double prime are employed to designate the quantities

referring to the upper facing, core, and lower facing, respectively.

We choose our right-handed general convected coordinates G' as

a set of geodesic normal coordinate system situated on the middle

surface of the core. It is found advantageous to choose similar

coordinate systems, e and in the middle surfaces of the

upper and lower facings, respectively. Then, by the definition of

geodesic normal coordinates, we have the following relations


9' = '9' = '' + '-.'e" (94)


where


-=- ^h Z= -c' )

3-'h 3h "e' h ((95)


Also let Vi V V be the components of the dis-

placement vectors in the upper facing, core, and lower facing,








respectively. By using Equations (79) the "shifted" components
of V V; and V; take the following form.




v, =(96)

V.. 1v v;
'v. v; 'v, {v)

In general /V V' ,and V are functions of 0 ,
and could be expressed as an infinite power series in terms of the
thickness coordinate. But, for the sake of simplicity, we shall
assume that they vary linearly as follows.

V:=, "- (- ) (97a)

V--= -+ = (97b)

VI = Ut. (4 ) (97c)

Substituting Equations (92) and (95) into (97), and con-
sidering the continuity conditions of the displacements at the
interfaces, we obtain

U, = U, -- + (98a)

S s- a, -h' --hf(98b)

Elimination of ILi and IA. from Equations (97), by using Equations
(98), yields










V'=


V=


' I.- e9 t


V -.- 0 e',


* == i; -< (^- )



'7iF LLfVT L
a= Tl -1( .^


Equations (99), imply the following strain distribution for

the upper facing


-:Y.,X= *(3 + 3'"'
'y,=- 'x.+^xe:,r


(101a)


(I01b)


(101c)


=y3 -Y.I


Similar distributions for the core and lower facing corresponding

to Equations (99b) and (99c) can be obtained by replacing the

prime by a bar and a double prime, respectively.


3.2 Equations of Motion


We now write the volume integral CI, of (93a) for each

layer, consider Equations (99) (101), and integrate with respect


where


(99a)


(99b)


(99c)


(lOOa)


(lOOb)


(l00c)








to a through the thickness of the undeformed composite shell,
to obtain

sI,= 'I,.- + T (102)

where

a 'I, = { (S,-bV.) +'m( -b,-,)




+ *b ) + '4 "qO( )] + 'f) f +






+ ^ l' )] + b '?"( f+ +rt SL )-
r 9



+'m, c' /^ 'St S^'lls.+ I'^^ )
+ +'h -b r) 4 C + + i s-b+ 't)



+'q 4"J 'f f } + l'r3 +

-+ ( +b, )],j^:'m S6.4.

b76 7 ir + 'k 9,'r 4 >]. 1 Lis
+ ^. ;) ^b-+ ot-I ,]

*'C()+ t> + ('t"-'n">)( (1,) 03
4 'C' I 5 d A (103)








Expressions for 'I, and Sl, can be obtained by replacing
the prime by a bar and a double prime, respectively.
The various stress, moment, body force, and acceleration
resultants appearing in (103) are defined as follows.


F, f de




'M"= / '/4 .. O' d .
*-h




m = ," f 'S\de
'= 'P4 'g' Ve


F_ = sr P. +,d I-
L



m= \fSA ,Lr '.
fr-I

't.'i




'rn= [# "a' o'de\
'/4 -ISM

S-' h







--'h


2-s







a.\-,',


(104)


^'"16z'


p = +I'sr/ + +e '(1.,+, ,I) +









+ S ( -T1 ) e' = 'I 'k
'G = Iq e t 3 I + eO'' V1 )] +

C { e' V"F ,

'eS= 4"s^, ) + 4 0s' VCF, +




Substituting Equations (100) into (103), and setting the
volume integral SI, equal to zero, for arbitrary and independent
variations of the twelve unknowns SLu; St and ,
we obtain the following twelve equations of motion




n &os + -bj j i ') ? .r n I&
+ +


Kq 1' -1 S

+(''- J Ib, $)" '+ q rT


( f + I J(Oa +


0J .7 (105a)

Tui=+ [^








+ 4',, ) ( "m ) (1 ) (, + e ,,
+ )( ) "- 'A #vf,

+Y + l n 'r ) +

4 [ h'.A~'n '., ] f b"-,) -m ,b;

(m I, W )- 5 (r'" I ", .) + ( 1. )
\+

-, ; ) -(tA'Y ,-b5 ) +Y4- /. )^1+

+t b* p = O > (105b)


S ;: [1 i9. nl -w ] c)( S M' -('a,) -Ci 'E+| *( -
-_rn")] [ ri r+I"( '1)] (f l-{>i.) +

-4 h( i' ;*) (- b+ ',) Ij (*- '"



(WYi* +;*<'n,

+V j)( ,-p + >i) + 1 C P ( rr +

+br ) S <*m' Ih b X) ,', p .* 6, ) +

qr'( .)-4'tlb*,)]h) bl +( X -

"n" b- (L,) +
6 r
C- r) o = 0 (105c)

)] -'44C n I) + i b' ~;) + jl- I T'4
YI ) c~u~* b6h~~.p'~~4-








,)E ,,) (9[rbo+) (' I .f-( ,,) (" ) ~ JI ))

m J+kO^ "n^-V] (E;a i G u-b, ) *s
+ h('*fr )]

()


(',' ) c( ) + c' -e = (od)











..Q i ( 'tt' 'U. ) -' C' ,
S+ : [ C',- i ir,) [ wl (, ,( -









_- ) -] C d '; (10e)- -




Stf1 ^<'wi'^-. hn-) ) + s 6C(fs -
I+r +3 ('fl n 4,- '.6





% y )] b, u, k + 0 A (5


6 1



-'r)f) -+ s, -allb+r c Ft' -










- ^)[i, (,; ( .)] -b'y I b-
- [ .s l ,- ~ 6 b ( ) -
- ,)] +('t"I.-'")( g r1 '- ( = o (105f

.+ i,: b) ., i ( I,' <'') -












+ (tOM-
- 6; < ') ] + ( I' ) 1,4 s

- ( + )] + 1 T^ 1' 4,, {( "'^




*;i S ) [ H 'A i )- %) (,,) + b. "-



- b', :-J- l). 6 = (105g)



-r b<( i- )] + (c'+ ( +
+ b^ 9 ) + '"q I << '* ) [ + ('lt



- 1, b ( I' c I ) m' ( 'tm l r+

- tc ('+1)] + q*" bj y -i -

- 1',,-) b I b ': g > +









+"('L, '1)( I ,) .- = o


Qp I -f *

n" = 'in'" o'f "t *

S _= r *( '





= F 'p+ -r ,


C = Fl + (F 'F'- ) ('P.f +:*)

'f = 'M'- 'F' 2,'f' ,


9'= 'M 'F ; 2"


r "= _- ,;fi
-I Lh A~
ac 4Af-f)


where


(105h)


(106)









3.3 Strain-displacement Relations

Writing (93b) for each layer, considering Equations (99) -
(101), and integrating across the thickness of the composite
shell, we obtain

SI,= s I, + I (107)

where



+( ),+f- +^)C -byv,) + ( -Sb )( +
AA



+ (.-t .l't, ( -kL ( 1) (15) -+









+ M +







4~, 1,^),,],J 2.-. -L( .+ ) CVi.










+b + ) 1Y* ) (4" I-

+u; j'1 J } Zt +

34.)' 3 S" A (108)

Expressions for I1 and 1 are similar to Equation (108).
Setting 1I equal to zero, the vanishing of the coefficients
of the arbitrary and independent variations of the stress and couple
resultants in (108) leads to the strain-displacement relations. By
using (100), we obtain

'X= a= t IL/+ 4^ cit-^-) S*1 -6 6y





(,3 -~ C 3- Z)] h b. ,-1 4 -





+ ( ^.+ 6 +) -


(109a)
4 cb~fi;.]













+ hO +-'I4g) bpi, 4 1 b6C,-4)] -




+ U -hI~,~(i .i^
J! E. a !. ( >)] < h,, f, '

I [, ^(T -L b) 6 T b c -
ir- S- +







-) (109b)






+ >ib > < 7 ) (109c)



s)J ( ,- [ I t L -6
S a)~ r C b4 (3)- c)'; >


(109d)

,Y., = i 't p ( L
(109e)

I 'q (+, (109f)









The strain-displacement relations for the core and lower facing

can be obtained by replacing the prime by a bar and a double
prime, and letting "* and h '- respectively.

3.4 Constitutive Equations

By analogy with Equations (45), the stress-strain relations

for the upper facing can be written as

'S = C ( 'Y, -, 'T) + C( Y, ,T) (11a)

5"- 2 ('Y o T (110b)

5"= 'c'C '( ,,-o',T) + C ((,- 'oT,,T) (110c)


Similar relations hold for the core and lower facing, respectively.

Writing the volume integral I 3 of (93c) for each layer,

considering (99) (101), and integrating across the thickness,

we obtain


S= 'VI+ I,+ 'I, (111)

where





I I

*,7 3
1 ,?-, )J X +[
.^ )]^ ^ ['r-^e)].









( I-7-t-a )]^ i <(112a)

and

= / de (112b)

is the strain energy function per unit area of the undeformed
reference surface oA

The strain energy function, for the elastic anisotropic

medium subjected to a prescribed steady temperature field, T is

now assumed in the following form (14).

S 'S j ( CoK T ) (113)

Substituting Equations (110) and (101) into (113) and then (112),

and setting the Equation (111) equal to zero, the following

constitutive equations are obtained


n 0 '" ,, "" 'B + *+

0T -(114a)

^= ',By^ ^ "^^ "^ : "


~"~( (114b)




J' A (114c)

= 2 (. B'' ., ,, ," (114d)

S== 2 ( ,I ,1 (114e)










,+33 ,
11 ."ti + O" 2 6 I',,

+,-


-'k


=-f


W.4(u .I (0)" d ,


..$3 (8 1de3


(M 0, 1#,, 3, 4)




(= o0, 1,2., )


r/1
'C (0') d dO3




I, = 3333 30"" 3
=f 'C'
n



."r ~f.t-^e'


are defined as isothermal stiffnesses, and


1 'T ( ~i- o + t'ot, )) (e')" J


3 r
/'T + o )'

-s .


( nr o, 1)


Z'*1


(116c)


where


(114f)


(115a)


(115b)


( nt o, a )


(115c)


(115d)


.' > (116a)





(116b)








as thermal stress and couple resultants per unit length of co-
ordinate curves on the undeformed reference surface.
Similar expressions can also be obtained for the core and
lower facing, respectively.

3.5 Boundary Conditions

The surface integral J of (93d) can be split into two
parts J, for the edge boundary surfaces, and SJ, for the
upper and lower boundary surfaces, so that

J J J, (117)

where


f f {r-r s"(/4. V'iL-b'. )-

"gj r s V:,
r s' v ; v; + u t" ,.
So' (,v,, ) ] j V, f a V Ja

(118a)


ce 3 V,$


-b.v: )-r Sv', 3] v +

+ ["At -r Is"( 1,,10 b v+

s"( V,,)] v; 4Ac
S(118b)








In a similar manner, we write S J, of (118a) for each layer,
substitute Equation (89) into (118a), and integrate across the
thickness of the composite shell, to obtain

S= *T. s' (119)

where







'i~{'k- .r L; 'I v R] -bk)+ 4-'.



+ bJ j) + b'c^, ) +



Here

e= (121a


{r z t 3 (121b)


Expression for J, and J, can be obtained by replacing
the prime by a bar and a double prime, respectively.
Substituting Equations (100) into (120), setting Equation
(119) equal to zero, and for arbitrary and independent variations








of the displacement components S :i 4 i S i' T
we obtain
jA A A&




+-n)J ('n ] b ) + ('m* )4

+ ( -1 ', ) + (0 ] V ^ ( P i-

3- ,) + V^ (122a)

eV% ,V3 ,i
S = 'S 5 S

) n ( a,' +ib .) ;(n -r +

+- '

-= w ( +f t ) 4 'f+} (122b)





+ [^ 1 \" ^ )J (> ]< ) -


"A n h ( 'nll+ 6 ,,) ")1- ( -


6 t,) V < Y^A- "'t) > (122c)

& 3 &' I*3 *^i3
L, = (S -'s )









= k [hCI+'n^ h ^M-) +
+^^ I Ih ('n ^C- ( g 'b;f) +










i 1 ( ',- ,) + (',*- iE' ) -



ae^ i'- vs"




= +1
4 -) ( 1 ( ,)(- 1 (- > 12f-)
A ,



"e = H- 'S

., {(c'. I- ^ )., [ ,^. E I,.,-( .,; -


+- f) ( ,^ 'i- ), t '6( 4, y)2+
+i [+'9i C +
+ ~ b (122g)








e= Ls


6- c )Js1 ) (^.+ ji ) +
+ ( ,*+y ,) e .(122h)


B. Boundary conditions at the faces. Evaluating the
integrand of Equation (118b) at the upper and lower faces,
respectively, and using relations (100), we obtain

P- ) +-i






S+ ) A (123)
-'y~)~ sl) F P+ ) ; c^ -





P-')j. dJ = (123)

where


f,= .,i f tt ,(124)

Setting Equation (123) equal to zero,


= (125)










Here the signs of the prescribed stresses are considered positive

when their direction coincides with the positive directions of

the normal coordinates.

Up to now all equations are "exact" in the sense that the

complete general nonlinear strain-displacement relations have

been used. No assumptions about the state of deformation have been

made except, of course, that the displacements are linear functions

of 9 The "exact" problem of composite shells is governed

by the equations of motion (105), the stress-displacement relations

(109), the constitutive equations (114), and the boundary conditions

(122) and (125). These equations-represent twelve equations for

twelve quantities Li I 1 and, thus form a

.complete system of equations for the problem. In this form, however,

they are, for most practical purposes, too complicated and approxi-

mations must be introduced.












CHAPTER IV

SPECIAL APPROXIMATIONS

4.1 General Nonlinear Membrane Theory of Sandwich Shells

Equations (105) can be simplified considerably for sandwich

shells with very thin facings of identical thickness, h For

this case

t= t= == =o 's= 's= o ,

and the last four equations corresponding to the variations S ,

&f S and a1 are dropped. The results follow.






^+ b^ i, ) C [MI-4 Cn -ir)]' ( (, ) *
-m --


+ V +9- f = 0 .(126a)







4fr'j) b.r'c'91)1I1 ai"{E- b') +"+"i)









+[ +'(.-' n -.n,)] < i -+ ,) -
+ I-Or-- -- i"

6+6rOler +0-f O (126b)






m 0 (126c)






+ +"n' ) ]<,
+ i ; (,',"') ] ( + + ) -'
+ n l ) --ls

9 ( ", url b u, t, ( t n )t

+C (126c)






S[ V I <'m5; (k "n">] ( le b? ,) 1i'



( [^ rcm'n>] (br, -b ( r.

+ ( )+( t+ (l-"") ( +' s)+ C'- m]= (126d)









These are six equations corresponding to the six displacement
functions i and T .
The related boundary conditions can be derived from (122)
and (125) as follows














r h .'r L" n( ,,s 6 u + bb m )+- ('- ) (127d),
S- (127e)

C= .[l: (',l/1S.7h(C)] (S 1 ^ -Y'J ^) .4+

A further simplification of (126) and (127) is possible for








the case when 0 so that one more equation, corresponding
4 6('rJ= -[ )])b(',frb; ) + **r )
(127c)






and

(127e)

A further simplification of (126) and (127) is possible for
the case when i = 0 so that one more equation, corresponding
to the variation S 3 should be suppressed in (105). The
equations of motion then are









Su,3 ; [ n ( u b ) + [r 't( -
-'#N )] t I b + }I -

n"\ (5,,,*6.2) [ cl' ( n -"1- )] k L' +
f = (128a)






4c(f'l-anu1)] rh *Fts qI-f}. *

3 5
4 P = 0 I (128b)






x i g' + [ C"r<' -"n")] (+Yt

+ + ( ) + r 'n" '"n">] @ 9 b -


C F = 0
cc (gnA x









(128c)

and the boundary conditions are
tS A
-= ,MI 4 (2-b9a)) r
-'n^ )]> 1 m1*i > (129a)










S n +fC)(
-Vl)] b;' + 4, (129b)




+ +t145)J jI j .i)bJ ij (129c)


and

p, = (- (229d)


where the definitions of P P. are also simplified accordingly.

4.2 Partially Nonlinear Theory of Sandwich Shells

The modified Hellinger-Reissner variational theorem of
Chapter II is now used in the formulation of a simplified non-
linear theory of sandwich shells on the basis of the following
partially nonlinear version of the strain-displacement relations (39)

j = Vi lj + Vjli + v1I VVl ) .(130)

These relations had been used by Ebcioglu in (12) and (23) for

deriving nonlinear theories of plates and shells, respectively.

Using (130), the variational equation (86) becomes


fJ S a'), -f j 5imnlj VY v'-V Vlj )0(c +1


+ J 0..-., J. = o (131)
ft .?








With the help of Equations (83) (83), we can shift the components
of displacement into the reference surface, so that





oI J t 3
S4.., ( -.b^V v 1,e + V )
4CS)rr 5" + <+ 6 VE + ( V y, j +




4 .( 4b- ) V* dt- + +v n t Vt +

4(.flt J%.nzt )V J = (132


)


Executing the

s1,


variation and using Green's transformation, we obtain

+ SI, ~ 3 + EJ = 0 (133)


where


SI, = { <[/ s'tep [ s't( v b v2) *s ,.




V-;,,,JB; +s- r; <

+ V ], [r 5 '"( 4})
+ I.)], [ +.( 1-,0 -V bo A +




+ V 4V V...1, ) (V;,b V ) J +


(134a)









{ I [r" v',, v + v,
bv: ) V,,] 5+ -4[zVS
4 V, V ] S" 1 Ad3 4 (134b)


7= [-( + )] br ,A (134c)


= J j"w {E r ^S"] VJ; V, i+

; -

v{, 5r r.) sn v sV L .r
s5' v<,, b; )-p S('
+V,)] V: V} (134d)

In a manner similar to that of the previous chapter we obtain
the field equations of a partially nonlinear theory of composite
shells as follows.
The equations of motion are

+&( M (nx-) l fw ( 0 ) +
+C n ( na14n t) ) +

+C(.,^ 1.0 -('n) +



'0 (135a)












-- bA ) ('-p* b ',) 4

+ 0(0 + )

3 +


S f 0 (135b)

Ur ": [~m '-r <'nr_*I )]- [ii' rm*-'m" ,,(nJ k

;[ { I ('n<1n ) ( rn [ 6+ E ) 9 [n F+ l\n*

S", b )](i' t)- h C r' n ") fl-

+ T;(^ %n") ( In ^) + [ '( 't,= -14 (*

SW)] 1 b C V o (135c)



n iA







W c no [ m^--m )] } ^f E W.A.+

+ E ))+( t ') ( + z) m = O (135d)









S [('m" ')-J 'i;n ) ( -l.() ] (' 1-

--h ")j +( I -
*-t 1>]j-ii < >; < (RIp b(
60- = 0 ,0 (135e)





) -" -1 'I';(
U ;5}|" )(*-i ") b11 J ] -
.11,,p + [t' '
S6 +( -^h, ) + 6 ] +
S .-n") + ) 0 (135f)






w e: .e 1'v 1* 4. () b 1] n 1 "
+ 'n) + 4- 6Y ) i-



= o (135g)

s l',: [( '^ [ 1 ; n c <. -; ) + j Us y ^ ^.






+ (t.l- "") ( = o0 (135h)

where


b'C ) ( t'e ]>'- (136a)









CtI
+ (I "t',) (1 'e ', (136b)


and the definition for and 'M are based on Equation (104).
Similar definitions for the core and lower facing can be obtained
by replacing the prime by a bar and a double prime, respectively.
The strain-displacement relations for the upper facing are

/ = i [ ) ( V bt- ^ )

+ "( -,,, b ) ., + ,)J (137a)

Xy',t i .(-b/4) ) (rrl*,- br."l',) -

+ (<,,. + b^Kj) ) (,. .b;,. ) C ("f,,. +

b ^ ,V ) ( ,,^ b .) (137b)

^ +I [(:'Pr^)J (137c)

,,= C" ( ,, b6) ( ,) + Y (137d)

,, i ("*) 3 (137e)

C2 (137f)

Similar relations for the core and lower facing can also be
obtained.
The general constitutive equations for 'nf" r' '
etc. are similar to (114), keeping in mind the simpler functional








relationship, expressed by (137), between the components of
strain and displacement.
Finally, the boundary conditions for this case are to pre-
scribe
/V L / i f t It 6
f.= P-- ,.

and
S= .'(n m ) (138a)



+(inf-r_ h; nC) ( 'b; i^) (<, i; n( ^J 4

)1 i~ *"^^, + 4, > (138b)


er x= Y t'n s/'n P 7] [ 4 10 '- ) 0] ( 1 3 8 c )




+ r. ) +e(nj (j ) -fe I Cm a (8d



= 'm -'n ) *- ( ') (138e)





+ 1) 't(^, )- 'iP( ) +( (138f)


S= .^[(h" 1;r n ^) ( ^ 'e ) 6 ] (138g)








"e= (V I '* )[ %, -; ( ,- ) .; ,-
-T,(=- )J +("' -i; ) (", -
+b; J.) #F( ,) ('^.) (138h)

4.3 Analogy to Donnell-Mushtari-Vlasov Approximation

Simplification of the above equations is possible under the
assumptions discussed, for example, by Mushtari and Galimov ()
and Novozhilov (20). This consists in neglecting the terms
containing the product of b1 V, in the expression for Y1 with
the following results for the strains
= [ (V ; b V v (139a)


"Y(. =i [ ,, ., V, ,] (139b)

Y,)= I-- [ 3. + V V (139c)

The variational equation (86), compatible with the nonlinear
strain-displacement relations (139) becomes

il V3: ) +



1. +3 +-

V. V"5 f f.tx + (f[ (.b"-,(.) b + 7.. l)V. d t'+
OT

SC( t t + t ." (14
x V, d5J 0 (140)








S I, +S I 91, + J = o


where


s J(/t -1A 4 v;,




3,, 11 +
+ f. (.b -.a)}SV AJ ,

Y.,-r c/.(V-, v) 6(v.. -b; ,) +
OT




3 + C V, ,sV ) Vil +
+ ,o v .t -7 S" ) { v,[ -
4 PS v,,. v,)] v -











By the procedure used in Chapter III, we obtain the
1 A V ,J} S)" d+ VJ 1 4

+ /4.4t r S + Y3/[ 1 -
Y # V^ $(,. v<;,,) J V; ] ^^ ^



1- 1A s )" 5s'( )] v } J 4

By the procedure used in Chapter III, we obtain the
corresponding nonlinear equations of motion as follows.


(d ; (nb- Jrr. )|,, -' + = 0


(141)


(142a)







(142b)

(142c)








(142d)


(143a)









(GIw o [ + + ('"*- "" ) f,, r (,' o 'n, +

+ ( IwV) Y, + 4. +9- +

+(In ," b, = 0 (143b)
(143b)
S [ +h'nl. h"- I] [C .I(n%"'-k )] 6 }1, *
J -t S S
+(-$~)i.,-9 C -m = o
S(143c)
= t[ ('n'-b)l, ,] ,, + V ('n "nP), +
+ ("'m-,+; '-#-) ; (omC") nA +


j L -(lg -+
n 7m 0 (143d)
Eor-i- m

= 0 (143e)



.- ', i ',) [( w )

-(+F' -w"li' ) 3 -3 [,,,.) c -

t (tI 'n") I+I) '+) = 0 (143f)

f(: ; : [ ".I .. '') ( P -,{)

+ d = 0 (143g)



+ m ") *, *' h +









+IW1-) (Y^ W^)SC ^ ]. -RU -

-T( -- )] *('tj-n"') (<*'r,) +
+* = O .

The corresponding boundary conditions are to prescribe

Pj 'P __ ,


= A( A ') )
V = jn^ <,,; [ + 1('d n ')J> z,,





s= + Li[ s ;n, -[+ (#. ;-h9 ,
' = .y; {[ t' S 4.h [ -V [ O I"( j I) +



+nt F t,] ,f +T ( ^- hn s t 1( E +



S= C 'm i'n) (.k I-4.)
0 4-_ ;fI s ,s j










.= 3 [(Wm* k n i) + ( hnAC) J


(143h)


(144a)




(144b)

(144c)





(144d)

(144e)



(144f)

(144g)









= {(4ut1#I+;
+ 4( E i ) "I + t( ) +

+ c" (144h)

where


,, = [i I(^-l1.))+,J.h,) ]

and 'f ', are defined in Equations (136).

4.4 Partially Nonlinear Membrane Theory of Sandwich Shells

When the facings are so thin that they are effectively
membranes, we set

-= -- = = -,- = O

and

5 -- 5 = O ,

in Equations (143) and (144). Accordingly, we may suppress the
equations corresponding to the variations S I', ,
and IT, The resulting equations can be simplified considerably
for sandwich shells. The equations of motion are

S: (n- m b,. )L T J f P- ft = 0 (145a)




+= + ( (145b)
+ p f = o (145b)











F[
q + C +1 = 0







T-h e(dge '-)ry,) C'i- m = 0


The edge boundary conditions are


ey



s'V



f3
I =


( n+ rn' 1 )

ntFcuf +

+ 5 '( ,)}


+nP] ) ,g + tn )


4.5 Sandwich Shells with a Weak Core


Assuming that the components of stress

are of negligible importance, they may be set

the components of transverse shear and normal

5 only are retained. The equations of

boundary conditions (146) then become


5 in the core

equal to zero, and

stress, 5 and

motion (145) and


t : ( n'- rr )ikf d = 0


(145c)


(145d)


(146a)




(146b)


(146c)




(146d)


I (147a)









no A 3,0 + "'1 (l + o

+ ) b f o 0 (147b)


s9: Ick' n"-""')-n ( ',-* -)6 ]1- q -5+ o (147c)



[C (iC$) 1 ] -r',.^

+(t"L,- "3) ('+,+) C = (147d)
'+ M (147d)


and

S= ^ ( 'pnp- r bnj) (148a)

= .,{ ^ ,,, C^ i; ('n' En( ^Y, +

+i ,)} (148b)


= k 4 [t I [ ^-"n") 'r- ) (148c)




(t -C+ h)} (148d)

respectively, where the definitions of )11 i ,
and are also simplified accordingly.

4.6 Linearization of the General Equations of Motion

Dropping all nonlinear terms in.the equations of motion






66

(105) and in the boundary conditions (122), we obtain the
following equations of motion

(: ( ^-- r ) (149a)

Sz:, qlIW + (n"'- .a) ) 0. f f 0 (149b)



('9"- '"q"6 b) ( a T = o
(149c)



-$c. I -

0 (149d)




-' lis = o (149e)

: c' -'. ),l'. = 0 (149f)

',: [, ^) -*'-'". +'m-"') b~ ](. 1 -

+ = 0 (149g)

: E 1 1 .^ 0 (149h)

and boundary conditions
v = .A- ) (1
S rip- MASJA) -(150a)









~" > (150b)



= --( -y)]J (150d)

= ., A- n ) (<' 'n ), ] (150e)



=e [( "n J -( "n )) bJ (150g)

(150h)













CHAPTER V

COMPARISONS AND CONCLUSION


The nonlinear fundamental equations of a theory of sandwich

shells in terms of the undeformed state have been obtained from

three-dimensional continuum mechanics with the help of the modified

Hellinger-Reissner variational theorem. These include strain-

displacement relations, equations of motion, boundary conditions,

and constitutive equations for elastic, anisotropic, composite

shells subjected to large displacement gradients under the influence

of mechanical and thermal loads. Several approximate systems of

equations which may be suitable for application to cases in which

the displacements and rotations are restricted in magnitude have

also been obtained.

For a sandwich plate, /A is simply the Kronecker symbol,

since then b6 = 0 Equations (105) become

+"P : hw)) < ( + '2 ) (

+ (w I-'r") A, (+ ( \ j) JW +


+'lot" +1JIOCI f= 0 (151a)

:i { In" a,,p A [ <,-."'n)] +


+ (-) ff, j.i. f= (% +bq
+ '"1 lot +, }!^3l, 0 (151b)














I^) i; ( "t- ^ (*1d-


- s ) q ( + 4. i l.















{(d h' ) [ 5


4 ( -5. ya)4 -
4/1


) C '- +s 0 ,


S+ [ t' +






u U,, +(th -

3
m = o ,


- + ( 1;-;t )] ;


a1)] = 0 '
+ ( l, -'n")T -'f [- (+ "-



+ J



- %.<,) + = 0 (1


151e)


L51f)


(151c)













(151d)


/'sl :














+ SOL -<) ^ + -:

+ 0 o (151g)




SC IL "r) Li ),,) -' ;f [ I( -13
"M + ( ) +




+ + + q A-r +-^ -


+ 4 d = 0 (151h)


It is easy to see that our results can be reduced to those
obtained by Ebcioglu (11), and shown by Ebcioglu to contain the
earlier works of Eringen (21) and Yu (22).
For homogeneous shells, we simply drop the upper and lower
facings of the composite shell and retain only the core layer.
Then, all terms including quantities denoted by single and double
primes and the last four equations of Equations (105) should be
suppressed. We have





+ ) + I" + ) ) +

+ P f = (152a)










+ i + Ii I -
)1' + 6 U ) + i b -


+Th" 'lio- b", ) -+ {' + +


+ -f (152b)



.,l 6. F


b^+ no )(tt-n -J

d ) C W, = 0 (152c)



4 0( a' Z15- ) +c (Ft -

b,; b^ c, z i) +

B ") + (> L)3 + W= 0 (152d)


This is in agreement with the results of a nonlinear theory of
elastic shells obtained by Habip (13), (14). In the absence of
curvature effects, these also reduce to those of Habip (13) for
plates in the reference state.
In the partially nonlinear case, our Equations (135) agree
with the results of Sanders (24), given for the small strain

approximation, if we drop the effects of the facings and adopt
Kirchhoff's hypothesis for the resulting homogeneous shell.








By setting ,=, = = \ = o and noting that

wi -s _n b-r h n n
Equations (143) lead to

r: -' f= (153a)

: ( ip + u}+ + + +- AW = (153b)


+ C = 0 (153c
: C o (153c)



S o (153d)



S0 = O (153e)

These are identical with Ebcioglu's equations (12).
For comparison with the well-known works of Reissner (6),
(7), we transform Equations (147) into their conventional form,
in terms of physical components, as shown in the Appendix. In
order to show that these previous results are contained in (147)
we use the following notation.

9, = X -= 9 -aT,7,= =1 R,=R,= o



Q,= Q = M,,=M ,



.,- M .,- M W,,- M..= M P,= =C. = o ,
p. t







73


and also omit the effects of thermal gradients and acceleration

resultants. Equations (A8) become


+ = o


+ -N 0


Q,


4Q, Oi N T w +


(154a)


(154b)


(154c)


(154d)


+Y
1 )


M. '" -+ M,,

+- o ,
1 y

a2 3 ) +
O( M. + =0, + -


- ,( (+ e) + ==
ri


(154e)


(154f)


These are the quations given by Reissner (6), (7).

Completely linearizing Equations (A8) and neglecting the

acceleration resultants, we obtain


S-N,,T ^


- X(T.. ,) Ni-l( .N.,)


-- N,(-.. ( P,) == o
9 O, R,


2
-4-
a e


(155a)


-4.
9 @9


" 4~0 -


(155b)











J c4 QG,) 4 c (a, )- rr


.) P = 0 (155c)


(---. (. ,, e. "

'- -.. (1,- .) = )o (155d)




S(1a. M11)M (



5 -,-( )--= 0 (155f)
th R, R.

These are the equations obtained by Reissner (5) for the small

bending and stretching of sandwich-type shells.
The above comparisons reveal that the modified Hellinger-

Reissner theorem is particularly suitable for the consistent
derivation of intermediate theories. However, during the course

of this derivation, we found that the order of the variation and
"shifting" is important. If we execute the indicated variation

in Equation (86) before shifting the components of displacement,
i.e., before we employ the normal coordinates in the three-
dimensional body, some inconsistency will appear between strain-
displacement relations and the equations of motion, forcing us to

drop certain terms in the strain-displacement relations in order
to make them compatible with the equations of motion (14). This







75


arises only in partially nonlinear theories. For the general

non-linear case, the order of variation and "shifting" is im-

material.













APPENDIX


For the purpose of comparison, we now proceed to record

the physical components of the equations of motion (147) for the

nonlinear membrane theory of sandwich shells with a weak core and

identical facings, i.e., k = h = t The material of the

composite shell is isotropic, and the coordinates are in lines of

curvature. The body forces are neglected.

Following Ebcioglu's procedure (12), assuming the membrane

and transverse shear stresses to be uniform across the thickness

of the facings and the core, respectively, and with the transverse

normal stress given as

= ) (Al)
Z th

we obtain the stress and couple resultants in terms of physical

components as follows





'( = +, -I, = NU ) -


,,) r ,) *4 1 N,,


+, 2t
fntls) 'S










R2
1,, = ',, (, + ) e, =

ii* zt
'8<*=,>= 3/ (eR =
fla)= k) O


'n (,,) = n (,)


= N, =N


r-0

.Ii -it ( -%) (I- )
-PIZi-zt


-1R
J-i-2


h ,) = N ,


S"n (a) = N


"n,.) = '5<,., (,* ) d o,
J-l-2t

t
^ .)= 's..).--ile
J-li-0t


= N,,

=4 t


, = .'(.,I = 4 = N,,


--r'-I
., ., d 8, = Q


= 5 ,) d= a0



-2


t4,, S3,,) de = o



-h


N k( )

R,
'N-.t+ t
+ .


(A2)


= N..0- )













2;
-4




where

'N, = 'S( zt 'N, = 'S.., 2 t ,

'N,,= S(1.,) t = 'N, = 'S, 2 t


Sf (( A3)
'N,, 'Sc,, z2t Ni t = $6S 2 t
'N,.= 'S(,.) zt = =, (A3)= ,

-= T (n .)



and Z3,, represents the value of the effective transverse

normal stress.

From Equations (A2), we can deduce the following definitions


n(,,) = n ,,) =

;+t t /
=( I. ) )N,, +(l N, = N




= (1" t -




n(,,,)= 'n*,,, n,,) = N,, N,, = ,










n ,,, =- 'n ,) Q*-+ .) =


+ ( N + ) N, N,






= (l, -- '-, ) (l-),N., = N.,
R, R.

," r(,, ) V N ,i tI) N,1, N,,

-+







S,,_ ,, =


,, M,, t



kC ,-^.. + -.0.










-l r R, N .

h ('v, .1,,Z)) N- N) = = M H



-Nil









In the same way, the following effective external force and

moment intensities are obtained



Rk t h +t g 1ot
+ ( ) ) c> +
R~r P




f l'* (A5)
C +t *t)('- ,.(, =






,- t (- )._o,J = C, -2
-(I)( '- Re.,t


-where +O) and P-ci are the prescribed loading at the upper
and lower facings, respectively, and


r^-^It)(. X. lL .-OX ) r. t (A6)

If we let

U = a,, ) -= -(A

V 7= a(3) -(A7)


where e represents the effective transverse normal strain for
the composite shell, the equations of motion (147) then become

,(^(.1) -.a-) 4 +
ae, ~eNot-






81


G p c P, -p ) > (A8a)

(Ja- N) 2 ( ,-) + MN, N,,


p0, Jz e, 7 o ,
+. ( + P. ) = > (A8b)

.-(, [ a ) N, ) + ( fe


+- ) ) +

Sl ( ,, ,, ,, + 0, ,

+ ( Q) .+-. Q,), +e 4 e[

+ ( -+ e -a, 2- 2-(A8c)

.. ,,) r )










,, -) I I,
S- (.- )= 0 (A8d)


R ,7
- ( I ) M, w- (^* ( M. ) -




SM,, arr la "



+N )z+ (> Ne e ] 0e(
*4 K8 8 a





R ,, u e ) (A8f)
R. /b a J" e,






82


where .1 T < <3 represent the physical components

of the inertial forces and moment resultants.














BIBLIOGRAPHY


1. Graziano, E. E., "All-metal Sandwich Structures Except Honey-
comb Core: An Annotated Bibliography," Rept. 3-77-62-5/
SB-62-8, Lockheed Aircr. Corp., Sunnyvale, Calif., ASTIA
AD 275280, 1962.

2. Foss, J. I., "For the Space Age, a Bibliography of Sandwich
Plates and Shells," Rept. AM-42883, Douglas Aircr. Co.,
Santa Monica, Calif., 1962.

3. Habip, L. M., "A Review of Recent Russian Work on Sandwich
Structures," Int. J. Mech. Sci., vol. 6, pp. 483-7, 1964.

4. Habip, L. M., "A Review of Modern Developments in the Analysis
of Sandwich Structures," Appl. Mech. Rev., vol. 18, 1965,
pp. 93-98.

5. Reissner, E., "Small Bending and Stretching of Sandwich-type
Shells," NACA TN 1832, 1949.

6. Reissner, E., "Finite Deflection of Sandwich Plates," J.
Aero. Sci., vol. 15, 1948, pp. 435-440.

7. Reissner, E., "Errata-finite Deflection of Sandwich Plates,"
J. Aero. Sci., vol. 17, 1950.

8. Wang, C. T., "Principle and Application of CompleAentary
Energy Method for Thin Homogeneous and Sandwich Plates and
Shells," NACA TN 2620, 1952.

9. Grigoliuk, E. I., and Chulkov, p. p., "Theory of Sandwich
Shells with Strong Core," Soviet Phys. Dokl., vol. 8, 1963,
pp. 310-312.

10. Wu, N. C., "A Large Deflection Theory of Orthotropic Sandwich
Shallow Shells," Ph.D. Dissertation, University of Florida,
April, 1965.

11. Ebcioglu, I. K., "On the Theory of Sandwich Panels in the
Reference State," Int. J. Engng Sci., vol. 2, pp. 549-564,
1965.










12. Ebcioglu, I. K., "On the Nonlinear Theory of Sandwich Shells,"
Final Rept. for Contract No. NA58-5255, NASA, George C.
Marshall Space Flight Center, Huntsville, Ala., 1965.

13. Habip, L. M., "Theory of Plates and Shells in the Reference
State," Ph.D. Dissertation, University of Florida, August,
1964.

14. Habip, L. M., "Theory of Elastic Shells in the Reference State,"
Ing.-Archiv, vol. 34, 1965, in press.

15. Green, A. E., and Zerna, W., Theoretical Elasticity, Oxford
University Press, 1960.

16. Naghdi, P. M., "Foundations of Elastic Shell Theory," Progress
in Solid Mechanics IV, North-Holland, 1963.

17. Synge, J. L., and Schild, A., Tensor Calculus, University of
Toronto Press, 1949.

18. Green, A. E., and Adkins, J. E., Large Elastic Deformations
and Nonlinear Continuum Mechanics, Oxford, University Press,
1960.

19. Mushtari, Kh. M., and Galimov, K. Z., Nonlinear Theory of
Thin Elastic Shells, Israel Program for Scientific Trans-
lations, 1961.

20. Novozhilov, V. V., Foundations of the Nonlinear Theory of
Elasticity, Graylock Press, 1953, pp. 186-198.

21. Eringen, A. C., "Bending and Buckling of Rectangular Sand-
wich Plates," Proc. 1st U. S. Nat. Congr. Appl. Mech.,
ASME, pp. 381-3, 1952.

22. Yu, Y. Y., "A New Theory of Sandwich Plates, General Case,"
AFOSR TN-59-1163, Polytechnic Inst. of Brooklyn, 1959.

23. Ebcioglu, I. K., "A Large-deflection Theory of Anisotropic
Plates," Ing.-Archive vol. 33, Oct. 1964, pp. 396-403.

24. Sanders, J. L., Jr., "Nonlinear Theories for Thin Shells,"
Quart, Appl. Math., vol. 21, 1963, pp. 21-36.














BIOGRAPHICAL SKETCH


Ju-Chin Huang was born on July 2, 1934, at Canton, China.

In August, 1953, he was graduated from Chee Yung High School, Cholon,

South Viet Nam. In July, 1958, he received the degree of Bachelor

of Science in Civil Engineering from Cheng Kung University, Tainan,

Taiwan, Republic of China. From August, 1958, until December,

1961, he served as a teaching assistant in his Alma Mater.

Since February, 1962, he has been enrolled in the Graduate

School of the University of Florida, has worked as a research

assistant in the Department of Engineering Mechanics, and was

awarded a Graduate School Fellowship, which he has continued to

hold until the present.

He received the degree of Master of Science in Engineering

in April, 1963, before pursuing his work toward the degree of

Doctor of Philosophy.

Ju-Chin Huang is married to the former Pi-chen Kuan and

is the father of two children. He is a member of the Phi Kappa

Phi Honorary Society.














This dissertation was prepared under the direction of the

chairman of the candidate's 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 fulfillment of the requirements for

the degree of Doctor of Philosophy.



December 18, 1965




Dean, College of Engineering






Dean, Graduate School


Supervisory Committee:



Cha n

ft. M ^^-































UNIVERSITY OF FLORIDA
II 16 0 5 39II II1III III II 11
3 1262 08553 9897


4