AXISYMMETRIC BUCKLING OF ANNULAR

SANDWICH PANELS

By

AMELIO JOHN AMATO

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

1970

UNIVERSITY OF FLORIDA

3 1262 08552 2943

To my wife,

Carol

ACKNOWLEDGMENT S

The author wishes to express his appreciation to the members

of his supervisory committee: to Dr. I. K. Ebcioglu, chairman, who

originally suggested the topic and, through his constant guidance and

encouragement, made this work possible; to Drs. S. Y. Lu, V. H. Kurzweg,

M. A. Eisenberg, and E. H. Hadlock for their helpful discussions with

the author and many valuable suggestions.

The author also wishes to express his thanks to the NDEA

Title IV program for the financial support accorded to him.

Finally, the author wishes to thank his wife, Carol, without

whose patience, understanding and encouragement he would not have

been able to complete this task.

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . .

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

LIST OF FIGU ES . . . . . . . . . . . .

KEY TO SYMBOLS . . .

ABSTRACT . . . .

CHAPTER

I. INTI ODUCTION . . . . . . . . . . .

II. BASIC CONCEPTS . . . . . . . . . .

1. Description of

2. Displacements

Sandwich Panel and Notations

3. Strain-Di splacement Relations . . . . . .

4. Stress-Strain Relations . . . . . . .

5. Stress Resultants . . . . . . . .

III. DERIVATION OF EQUATIONS IN CARTESIAN COORDINATES . .

Total Potential . . . . . . . .

Theorem of 1Minimum Potential Energy . . .

Equilibrium Equations and Boundary Conriitions

Simplification of Equations . . . . .

IV. EQUILIBRIUM EQUATIONS AND BOUNDARY CONDITIONS

IN CYLINDRICAL COORDINATES . . . . . . . .

Application of Covariant Derivatives . .

Transformation of Equations . . . .

Axial Stress Distribution . . . .

Reduced Equilibrium Equations . . .

Boundary Conditions . . . . . .

Comparison with Other Theories . . .

. . . . . . . iii

vii

. . . viii

. . . . . . . . . . xi

6

S. . 8

TABLE OF CONTENTS (Continued)

CHAPTER Page

V. AXISYMIMETRIC DUCKLING OF ANNULAR SANDWICH PANELS . . 42

1. Uniform Axial Stress Distribution . . . ... 42

2. General Solution . . . . . . . . . 46

3. Successive Approximations . . . . . . 53

4. Numerical Results and Discussions . . . ... 60

5. Error Bound . . . . . . . . . . 66

6. Remarks on Convergence . . . . . . 68

VI. CONCLUSION . . . . . . . . ... . . 71

BIBLIOGRAPHY... . . . . . . . . . ... . 73

BIOGRAPHICAL SKETCH . . . . . . . . . . . 75

LIST OF TABLES

Table Page

1. Comparison of Equations Governing Stability of Single-

Layer Panels and Sandwich Panels . . . . .... . 40

2. Lowest Value of pi Satisfying Equation (86) . . ... 44

3. Approximate Values of (N /A) for 2 = 0 . . . . 67

LIST OF FIGURES

Element of Sandwich Panel

Annular Sandwich Panel

Minimum Critical Values of

Minimum Critical Values of

Radius of Convergence

(N /A)

0

(N /A)

for N = N.

0 1

for N. = 0

1

Page

. . 34

. . 63

. . 64

. . 68

Figure

1.

2.

3.

14.

5.

KEY TO SYMBOLS

V V 3

U11' w,

x, y, z

S z

tc' tf

t

'V

Uc' U

W 11 V

q' N' P.

C, 3, y,

i, j, k

eij' ep, eu3' e.33

1 .. 7, T 7 33

13 Ccp 0C3 33

G

c

Ef

D

A,

N

-a

= Displacements

= Displacement components

= Cartesian coordinates

= Coordinates, defined b3 equation (1)

= Thicknesses of core and faces, respectively

= Superscripts indicating the lower and upper

face quantities, respectively

(t 4 t )/t

c f c

- Total potential

= Strain energies

= Work performed by external forces and moments

= Indices taking on values x or y

= Indices taking on values x, y or z

= Components of strain

- Components of stress

= Shear modulus of the core

= Poisson's ratio of the face

- Young's modulus of the face

= Bending rigidity of the face

=L(1-'-)/291 5 6 + 6 6. +[(2v )/(0-910f

~f [I y I,' '1 i P fP yfu

= Kronccker daela

= Pre-bucklliig nxial forces per unit length

viii

q

3

MB

N

(n)

I

nq

Xq

n,.,q,s,t,m,p

Lm

X Ys

n n

r, 9

2

u

u

Ni, N

i o

n, b

i), E

F, G, H, I

= Lateral load per unit area

= Externally applied moment per unit length

= Transverse shear forces per unit length

= Bending moments per unit length

= Axial forces per unit length corresponding

to small deformations during buckling

= Bracketed index indicating physical compo-

nent of a tensor

= Chri stoffel symbol of the second kind

= General coordinate variable

= Indices taking on values 1, 2 and 3

-= Metric tensor

= Arbitrary tensors

= Polar coordinates

= Laplacian operator

= Pre-buckling lateral displacement of middle

plane of faces

= Lateral displacement of middle plane of

faces for axisymmetric equations

= Compressive axial forces per unit length

applied to inner and outer edges, respectively

= Inner and outer radii of annular panel,

respectively

= a/b

= Defined by equation (65)

dw

dr

= Defined by equation (72)

A, B = Defined by equation (73)

f -= r/b

N, R, Q = Defined by equation (94)

t = 11 1

Ak = Coefficients of power series

P = Buckling coefficient for uniform axial

stress distribution

C1 = Arbitrary constant of integration

Jl Y1 = Bessel functions of order one

p = Radius of convergence

Abstract of Dissertation Presented to the

Graduate Council of the University of Florida in Partial Fulfillment

of the Requirements for the Degree of Doctor of Philosophy

AXISYIIETRIC BUCKLING OF ANNULAR

SANDWICH PANELS

By

Amelio John Amato

June, 1970

Chairman: Dr. Ibrahim K. Ebcioglu

Major Department: Engineering Science and Mechanics

The buckling of an annular sandwich panel is investigated

using the theorem of minimum potential energy. Governing equations,

derived in cartesian coordinates, are transformed into cylindrical

coordinates by means of covariant differentiation. Considering the

faces to be membranes and assuming an axisymmetric buckling mode,

the equilibrium equations are uncoupled through the application of

an improved technique.

For a clamped outer edge and "slider" inner edge a series

solution is applied to the general problem of radially varying

in-plane stresses. Critical stresses are plotted versus ratio of

inner and outer radii. The fifth approximation is shown to yield

acceptable results.

CHAPTER I

INTRODUCTION

A sandwich panel is defined as a three-layer panel, consisting

of two thin outer layers of high-strength material between which a

thick layer of low average strength and density is sandwiched. The

two thin outer layers are called faces, and the intermediate layer

is the core of the panel [1].

Among the main advantages of sandwich construction are:

a high rigidity to weight ratio, good thermal and acoustical insula-

tion, and ease of mass production. Some examples of core materials

are bnl sa wood, cellulose acetate and synthetic rubber. HolAever, in

more recent times thin foils in the form of hexagonal cells perpendic-

ular to the faces have been employed. Depending upon the intended

application, faces may be constructed of aluminum alloys, high-streng-th

steel, etc.

T'he practical importance of sandwich construction came into

prominence with the advent of the aircraft and space industries.

With the need for lighter, stronger and more stable structural com--

ponents, great emphasis was placed upon the design and analysis of

workable sandwich pane-ls.

Numbers in brackets refer to the Bibliography,

The essential difference between the analysis of single-layer

panels and that of sandwich panels is that the shear deformation

associated with the core of a sandwich panel may not be neglected.

Moreover, initially plane sections no longer remain plane during bond-

ing, and the existing plate theories [2,3] require extensive modifi-

cations.

Numerous authors have contributed to the development of

mathematical theories describing the behavior of rectangular sandwich

panels. Two of the more noted of these are Hoff [4] and Eringen [5] who,

early in the 1950's illustrated a concise and straightforward approach

to the problem using the theorem of minimum potential energy. In 1960

Chang and Fbcioglu [6] introduced continuity conditions for displace-

ments across the interface of two adjacent layers. This modification

has been shown [7] to contribute appreciably to the accuracy of the

derived equations, while introducing no additional mathematical compli-

cations.

In comparison, circular sandwich panels have been the subject

of relatively little investigation. In 1949 Eric Reissner [8], neg-

lecting the bending rigidity of the faces, solved the problem of a

circular sandwich panel subjected to axisymmietric transverse loading.

Later Zaid [9] included the effects of the bending rigidity of the

face layers. Huang and Ebcioglu [10], using a technique similar to

that employed by Zaid, recently investigated the axisymmetric buckling

of a circular sandwich panel subjected to uniform axial compression.

In their final results, the faces were treated as membranes.

Prior to the advent of sandwich construction, the stability of

circular and annular single-layer panels were extensively investigated

[3]. Mcissner [11] analyzed the axisynmmetric buckling of a single-

layer annular panel subjected to uniform compression along the outer

boundary. In this work, the inner boundary was considered free and

the outer boundary simply supported or clamped. Olsson [121 extended

Meissner's analysis by considering the outer boundary clamped and the

inner boundary "slider." Such a condition could be approximated by

allowing a shaft or rigid cylinder to occupy the central hole (see

Figure 2).

Prompted by the foregoing sequence of investigations, the main

objectives of the present analysis are: (i) to parallel the sequence

of analysis present in the literature of single-layer panels by inves-

tigating the axisymmetric buckling of annular sandwich panels; (ii) to

achieve a more satisfactory formulation of the theory through the use

of continuity conditions [G1 ; (iii) to modify the uncoupling procedure

introduced by Zaid [91 and later employed by HIuang and Ebcioglu [10],

thereby sign Ficantly reducing the complexity of the uncoupled equilib-

rium equations; and (iv) to apply the boundary conditions employed by

Olsson [12] to the present work.

For the initial derivation, the set of basic assumptions

employed in the present analysis are:

(Al) The effect of transverse normal stress in each

layer is negligible.

(A2) The core undergoes shear deformation only.

(A3) Displacements in each layer are linear functions of

distance from the median plane of the layer.

(A4) The median plane of the core remains neutral during

small deflections of the panel.

(A5) Each layer is homogeneous and isotropic.

(A6) The core is attached to the faces securely.

(A7) Hooke's law is valid throughout.

(AS) Local buckling of the panel does not occur.

These assumptions are similar to those used by Hoff [4] and Chang and

Ebcioglu [13].

Equilibrium equations and boundary condiLions are derived,

using the above assumptions and the theorem of minimum potential energy.

These equations are then transformed into polar coordinates through the

application of tensor analysis. In order to solve our problem, the

resulting equations are then simplified further using the following

additional assumpti ons:

(Dl) Lcrnoulli-Navier hypothesis is valid for the faces.

(132) Faces are considered to be membranes.

(13) Arnular sandwich panel subjected to uniform axial

compression buckles axisymmetrically.

The first assumption is valid when the shear deformation of the

face layers is negligible. Kim [7] has shown this to be true for most

practical applications. The second assumption, which has been employed

by many authors [8,10], is discussed at length by Plantema [1]. Finally,

the assumption of axisymmetric buckling cannot be rigorously verified

without solving the more difficult problem of unsymmotric buckling.

However, such an assumption would seem reasonable since Olsson [12] has

proved its validity for the analogous problem: of a single-layer panel

with similar boundary conditions.

5

When comparing the present work with existing and subsequent

theories, the words panel and plate may be considered interchangeable.

Numerical computations are carried out by a desk calculator.

CHAPTER II

BASIC CONCEPTS

1. Description of Sandwich Panel

and Notations

An element of the sandwich panel to be considered in this wurk

is shown in Figure 1. For convenience, three coordinate systems are

defined having common x- and y-coordinates, and transverse coordinates

related as follows:

t + t

c f

z = z ---

(1)

t + t

a c f

z = z +-

where t is the thickness of the core and t is the thickness of the

face layers which are identical. 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 unprimed quantities

are used for the core as well as for general relations. Subscripts

c and f denote quantities related to the core and face, respectively.

The total thickness of the panel is assumed to be very small

in comparison with the lateral dimensions. Also, each layer is con-

sidered to be isotropic and homogeneous, with the properties of the

faces characterized by E and V and those of the core characterized

by Ge.

IFx

Sz

/7"

t '

Figure 1. Element of Sandwich Panel

Arbitrary transverse load q, axial forces N ,, and external

moments are the only externally applied forces or moments which

may contribute work to the system.

For the initial derivation and formulations, indicial notation

will be employed. Greek indices Q, $, y and p will take on values 1 or

2, while Latin indices, when employed, will take on values 1, 2 or 3.

Any repealed index denotes summation, while any number of indices

preceded by a comma indicates partial differentiation with respect to

the coordinate variable represented by those indices.

2. Displacements

In the following relations, which are valid for small displace-

ments, it will be assumed that within each layer of the sandwich panel

plane sections are preserved, although not necessarily perpendicular to

the deflected middle surface. This will constitute an approximation

since the presence of transverse shear suggests a non-linear variation

of the longitudinal displacements through the thickness. Also, while

the panel is geometrically symmetric with respect to the middle surface,

the displacer.int relations will not, in general, reflect this symmetry,

since unequal bending moments may be externally applied to each face

layer. Therefore, with the aid of equation (1), our displacements can

be written in the following form:

V (x,y,z') = u (x,y) + z'tg(x,y) V (x,y,z') = w(x,y)

V (x,y,z) = z (x,y) V 3(x,y,z) = w(x,y) (2)

V-(x,y,z") = n (x,y) + z" (x,y) V\(x,y,z") = w(x,y)

C1 ce Qa

where (V ,V3) is the displacement of a generic point in the sandwich

panel; u. represents the displacement in the xy-plane of a point lying

in the median plane of either face; .,1 represents the angle that the

normal to the median plane of each layer rotates when the planes are

deflected; and i% is the transverse deflection which is assumed to be

constant through the entire thickness.

In order tc ensure continuity at the interface of any two

adjacent layers, the following conditions must be imposed on equation (2):

t t

(3)

t t

a f I C

u + T t

Using (3) to eliminate and u11 from (2), our displacement

relations become:

v u u + z V

z /

V (2u tt) V (4)

Vt t C3 (4)

t

V = '+ z' + ( ) V= w

3. Strain-Displacement Relations

Linear strain-displacement relations are defined by [11]:

1

e -- (V 9+ V )

e (, + V5)

3 2 ,3 3,0

e33 \3,3

Using

relations for

(4) in conjunction with (5), the strain-displacement

each layer become;

e = u + z + u' z '']

^3 = 2 Q p B, + /

el 1rW + %I-

e(3

33

Ze = t' + 2u -t ]

o f0

e 3 = t[u + (tcf,

C

33 0

t

e -u ( ) + z

t

e" = o

33

4. Stress-Strain Rolations

The generalize H]looke's law for a homogeneous isotropic body

can be written in the following forn [15]:

SE

ij 1-V [Cij 1-2\ ij kk

where the Latin indices i, j and 1: take en the values 1, 2 and 3.

When the transverse normal stress, T33, is neglected, (7) may

be rewritten as [15]:

E v

T = -[e + 6 eCp ]

C3 = 3'r = 2G e --3 T= 3 (8

T33= 0

or,

F

Ep 1 2 Ac/,, e .

(9

E

7 =- -- e

o3 1+V 0'3

where,

wi th,

A (5 + 2 + )

C4y4 2 cp 1-i-w K' 0 Y

A = A- = A A

Or.Yk poyp. &piPY Yl-^

If the lateral,

(9) reduces to:

in-plane stresses, T , are also neglected,

E

a =- e = 2Go

c'3 l+v C3 l

Assumption (Al) implies that equations (9) may be used for

the face layers, while assumption (A2) suggests the use of equation (10)

for the core.

5. Stress Resultants

In order to reduce the three-dimensional elasticity problem to

a two-dimensional one, the following stress resultants are defined:

t

2

t

f

2

tf

2

t

2

Q = 'iT,3 dz

t

_f

2

t

2

dz' ; N S- T dz'

t

7

2

t

z dz' ; M T z dz

2

t

f

2

2

t

c

2

dz" % Q T a3' dz

t

2

where M, N, and Q denote internal moments, in-plane stress resultants,

and shear resultants, respectively.

Expressions for stress resultants in terms of displacements

arc obtained by substituting strain-displacement relations (6) into

equations (9) and (10), yielding relations for the faces and core,

respectively. hTien these relations are then substituted into equa-

tions (11), and the indicated integration performed, the following

expressions for stress resultants and moments are obtained:

M' I D A 4

= D A ,

f

N' D A u

tE t

t fE f A F- t f (12)

N", = A -u' + ( (' ) (12)

2 1-v2 2 L ,, 2 yv, YJ

tE

ft

Q [ + w, ]

& 2(1+L ) f ,

tEr

Q [(1 + 'w

a 2(l+vv) c ,a

Q = 2G [u' + i(tw tu A)]1

c c a 2 c ,y f a

where the bending rigidity of the faces, Df, is defined:

3

Et

12(1-v )

CHAPTER III

DERIVATION OF EQUATIONS IN CARTESIAN COORDINATES

1. Total Potential

The total potential consists of the strain energy stored in

the panel during a small deformation and the total work performed on

the panel by the external forces and moments during that deformation.

In general, the strain energy stored in the body is defined

[14]:

U J j 7 e. dv (13)

2 1J 1J

In particular, as a result of assumptions (Al) and (A2), the

strain energies associated with the faces and core are:

U 1' (IT' e', + 2T3 e' ) dv'

f 2 ? Do aS Cd3 c3

u =-' (T 0c + 23 e 3) dv"

f 2 a c 3 o3

where the integration is performed over the volumes of the individ-

unl layers.

Also, since the only externally applied forces or moments

which are allowed to do work are the transverse loading function, q,

the axial forces N F, and the external moments M the work performed

on the panel is defined by:

W =IV qS w da

a

W = 3N ,(VI + VI + w w ) da'

(15)

N 4 --a$ CY, y ,aY ,p

f = f (it N(' i VN + ) )n (i

where the integration is performed over the areas and boundaries of

the individual layers.

In the second and third integral expressions of (15), which

are analogous to those employed by Timoshenko [3] and Eringen [5], it

is assumed that the total axial load is evenly distributed between the

upj.er and lower faces.

Therefore, the total potential, which is defined as

r = U' + U + U, w W' W (16)

f c f q N N (16)

may be rewritten, using (14) and (15), in the following form:

1 Jjff (T e', + 2r' e' ) dv

2 fp 03 C 3 cY3

v

+ Jf (2T 3 e) dv + LJ (T&B e"# + 2T e"3) dv"

v V

qw da + :J f E (V' + Vt +w, w ,) da'

a a

+ f N (V" + v" + w w ) da"

4 J-op ,8 g$, ,a ,z

a

f (M3 + / 3 ) ng ds (17)

2. Theorem of Minimum Potential Energy

The theorem of minimum potential energy states that of all

displacements satisfying the given boundary conditions, those which

satisfy the equilibrium equations make the total potential energy an

absolute minimum [14]. Therefore, equilibrium equations and boundary

conditions for the sandwich panel are given by the variational

equate ion:

6 = 0 (18)

However, before applying this extremum principle to equation

(17), we note that:

6 (T e) = 6 1 A ey e

2E

1I /

16

= 2r 6 '

oa ci

and similarly,

6 (T e') = 2T3 6e' (20)

S(3 e3 = 23 a3 (21)

6 (T3 e) 2T- 6e (21)

03 ea3 U3 a Q3

6 (r" e",) = 2T" 6e" (22)

a3 a3 a3 a3

6 (T" e" ) = 2T" 6e", (23)

Therefore, using equations (19-23) in conjunction with

displacement relations (4) and strain-di splacement relations (6),

the variation o. equation (17) yields:

' p = T L,(511 + z'6' ) + ai (64' + ), dv'

Y L : Q, 'Y, a3 acy ,

S 2 6ll ( t-

i r2T a'3 L 2 + 6a t 6 a J

V c c

+ 61a, + z""r + t f

+ ,F6t" + 6w dv"

q 6w da + 6 N 2w 6w + t (6' 6 ) da

a a

[M 65' + 6 n ds = 0 (24)

In equation (241) the symmietry of the stress tensors and

strain tensors has been employed. Also, the displacements

associated with N are those occurring in the median planes of the

face layers.

Using relations (11) to reduce the volume integrals ill

equation (24), and applying the twordimensional divergence theorer,

of Gauss to the resulting area integrals, the final integral equation

is obtained.

As an example of the above procedure, consider the fi-'st teri-i

in the integrand of equation (24):

t /2

T a 6u1, dv'- T 6, dz' da

a -1 /2

= f Nj 6u' da

a

= ff[(u' bu') N' 6u'] da

= N' 6u' n ds -J N' 6u' da (25)

s a

After applying a similar procedure to the remaining terrs,

and collecting coefficients of similar virtual displacements, a

complete description of the sandwich panel in the equilibrium state

is obtained in integral forn:

i = f- q Q N w Q Q<, 6w

+ N'33 + 4N' 6u, + L -

c c

t t

f/ f Ij

U 2

+ 1' f Qu + N wq + "6

+ [N5 N/J 61, + tf N + m, + tf N -

+ N + M" 6 n ds 0 (26)

,T -0a

L 4 -oS 2 of f g -a pj S^ Q

3. Equilibrium Equations and

Boundary Conditions

If .we apply the fundamental lemma of the calculus of varia-

tions [15] to equation (26), we obtain the following equilibrium

equations in terms of stress resultants and moments:

N,, w - (Q + 0' + Q") = q

20

,NI + N" 0

C

t t

-- M +0 N" = 0

t af aS,, a 2 Q@,P

SN" + Q" = 0 (27)

2 at,P ap, a

and boundary conditions,

S + Q + Q + N w w ;n ds = 0

5 [ N' N" 5Su' n1 ds = 0

f N i + M' + N 6' n, ds = 0

ri o + M - N *16 nc s = 0 (28)

If we substitute the stress resultant-displacement relations

(12) into equations (27) and (28), and collect similar terms, we obtain

the following equilibrium equations in terms of displacements:

4G tE

c 3 ff

[u + (t w t )] + A [- u + t (

t a 2 c ,a 2(-v 2) Oypv y, p f YP

)] 0

-2t t fE

fcu- + ( t ) + 2 [f + w

tc ae 2 c ,a f a 2(1+V ) C

t2E

+ A u D A- [4D 3" ] = 0

2(1-v yi y4,P f Y ,p y ,P

-it2: tE

Aaf-- A + u + [ + w D A [ 3

2 uy y (31v-) L ] f 0 (28)

+ 4 P] =0 (29)

and boundary conditions,

2G [u' + (t w)] + w+ + f [2w

c 2 c ,a ,p 2(l+v ) ,a

+ '+ 5] w n ds = 0

tf A [4u tf(f ]46u na ds = 0

s 2(1-v ) 2 ykl y, y -Y y a

f

tf

j I 4 -t3

s {- 4 --^

tE t

f 2 A (4 ) t u

2(1-v) Syp L2 yd y-2 -f dJ

+ D A 4 -M i n ds = 0

t- A 1 (E' ) --t 2

2(1-v 2 L 2 y, Y, f -Y,

+ D A 4" B3 84 ds = 0

O yk y1p a~ c

Boundary conditions (30) state that either the quantities

contained within the large brackets must vanish along the boundary,

or the variation of the displacement must vanish along the boundary.

Compared with equations (27) and (28), equations (29) and

(30) constitute a formulation through which the buckling problem can

be more readily solved. This becomes immediately evident by consider-

ing the classical concept of instability which is associated with

displacement fields rather than stress fields.

4. Simplification of Equations

In order to simplify equations (29) and (30), we exclude the

possibility of externally applied edge moments by specifying:

II I 0 (31)

which reduces the number of dependent variables from seven to five

since (31) suggests:

=t; ((32)

Also, if we assume that plane sections remain perpendicular

to the deflected middle surface of each respective face assumptionn

Bl), we have,

S= w (33)

which further reduces the number of dependent variables to three.

A final assumption (132) which simplifies the problem by

reducing the order of the differential equations is [1]:

t >> tf (34)

which justifies neglecting the bending rigidity of the faces, D ,

compared with the bending rigidity of the panel as a whole.

Consequently: when equations (29) and (30) are simplified

through the application of (31-34), we obtain the following equi-

librium equations:

2

c c t pa ,oan -aS ,cx

C

2t E

2G [-u + w ] A u = 0 (35)

c t ,a 2 oQyp y, P

f

and boundary conditions,

f2t t G\ + t + N w } 6w n ds = 0

L c c L 2 ,cd -c-y ,+ 0!

s C

r {--1- A u 1 6u n ds = 0 (36)

sI 2 ,

f

Equations (35) and (36) can be rev.ritten in extended form as:

2 r2

- t G --u + w t- G 1F- u f w N

c c t x,x ,xx c c Lt y,y ,yy

-K w v. -N w =q

-xx xx -xy xy -yy ,yy

c Lt x ,x 2 f x, xx y, xy

f

+ ( l-t) (u + u )v] = 0

f x,xx x,yy

F2 tE

2G u v ) ( )

c t 1-v2 y,yy x,xy

+ (1-v ) (u + u y)

f y,xx yyy)J

0 (37)

and,

SY{2t tG + w +N v, +N w 6 n d = 0

cc c- 2 ,x -xx ,x -xy ,yJ x -

2tG Y w +N w +N w Sw dx = 0

J2 c Lc Lt 2 ,y -yy ,y -xy ,x y

{ I uf + V u

1-v2 L x,x f yu j J x xdy

y tE

f

)-V [u + u ] u n dy = O

f f [u + u ] 6u n dx = 0 (38)

1+v x,y y,x x y

t + t

wherc ---- f

C

and primus and double primes have been omitted without confusion as

a result of equation (31).

Equations (37), which are a special case of equations (29),

are identical with those derived by Chang and Ebcioglu [13] if thermo-

elastic effects are neglected.

It should be noted that when simplifi cation (33) is introduced

into equations (29) and (30), equations (35) and (36) are not immc-

diately produced. The last two of equations (29) must be returned to

the area integral of equation (2G) where, as a rsu lt of (33), they

become coefficients of -6w Consequently, a transformation proce-

durec similar to that illustrated in equation (25) yields additional

terms which contribute to clquatioets (35) pad (3G).

CHAPTER IV

EQUILIBRIUM EQUATIONS AND BOUNDARY CONDITIONS

IN CYLINDRICAL COORDINATES

I. Application of Covariant Derivatives

Let us consider a covariant derivative of an arbitrary

covariant vector, Xn, [16]:

aX

X = -". ( I X (39)

n"q X q nq- 2

In (39) y1 is a general coordinate variable, { ] is the

nq

Christoffel symbol of the second kind, and all indices take on values

1, 2, or 3. Furthermore, when a vertical line precedes any number of

indices, it indicates covariant differentiation with respect to the

coordinate variables represented by those indices.

In considering the second covariant derivative of an arbitrary

vector we must recall that in order to sum two indices we must have

one covariant and one contravariant index. Therefore, introducing the

metric tensor, g (39) implies:

X g X = g -1[- (' J IX (40)

n q q nq

And since the covariant derivative of an arbitrary second order

mixed tensor, Y is [1 6:

n

6ya

,,s n + s Mm m s

we have, from (40) and (41):

X gpt ps s gps { sq n

8x x

[ sq fi s my q n mq

q Im x ( [ nq

gs g 1Xq J } (42)

In cylindrical coordinates, the Christoffel symbols and

metric tensor take the following form [16]:

f 2 2 1 1 x 1

12 21 1 22

all other nI = O

11 33

11 33 = = g = 1

g, = (X1 2 ; 22 (112 (13)

Equations (42) and (43) imply:

X = X (44)

Xn pt ntp (4)

Therefore, with the aid of (-13) and (44), (39) and (42) may be

expanded to yield:

Xli 1

X2 1

212 2 -ix XI

BX

ax1

|112 2 X

SX

-1

1 2

S-

X2 1 X 1 2

aX X

ax3

x x1

311 ay

2

a "_

21

1 11 1 2

2X

l a(x )

2X?

2

x a111

(; )

2

2

21 222 22

aX )

2J12 1a 2ax

BX EX c

2 2

1 ax2

1 ax 1

1 ax 2

2X

(X12

ax

WI

2X

2

1 2

2 ax9

x ax

-. 2X a +

2 ax

8X

1 ax2

x1 ax

ax1 2X2

ax

a x ax

ax3

-1 2 ax2

2x

-3

X3J11 = a(x1 2

3 1 3

X322 2 2 X (46)

a(X ) ax

In order to make each equation of (45) and (46) dimensionally

homogeneous, we replace the right-hand members by their physical com-

ponents, X (, through the relationship [16]:

(n) = X (No summation) (47)

(n) n

where a bracketed indc:x indicates physical component.

Furthermore, to ensure dimensional compatibility among equa-

tions, the following relations are used to introduce physical compo-

nents to the left-hand members of equations (45) and (46):

11)J ufg" r g X p1

x( = i /-T g Xn (Nc. sumiwation) (48)

The left-hand members of the resulting equations may be

interpreted as cartesian components. Therefore, after replacing

1 2

X by r, X by e, X(1) by ur, X(2) by Ug, and X(3) by w in the right-

hand mc-mbors, and X() by u, X(2) by u and X(3) by w in the left-

hand members, transforn.iation equations relating csrtesian and polar

coordinates take the iollow'ing form:

U =U

Xy

ux x = r, r

1 1

U I = U + -Ur

y y r 6,9 r r

1 1

u = -- u - 11

xj y r r,9 I

UyJx ,r

Ix ,r

x xx 1r, 1rr

1 2 1 1

u --u + -u --u

"xyy 2 i,SS 2 U,5+r r,r 2 r

1 1 1 1

S = u - u +

Sxy 2r r, r 2 r, 0 r 0,r 2 0

r r

u ik =u

S-1- 9,69 + 2 + -I r

1 2 1 1

+ e 2 96 2 ur, + r Cr r 2 Ur

I r

y -xy r ,r 2 r rr 2 r

1 1

In equation (49), r and 6 are polar coordinates defined in the

usual manner, while ur, u. and w represent displacements in the direc-

tions associated with this new coordinate system.

2. Transformation of Equations

If we interpret the derivatives appearing in equations (37)

and (38) as covariant derivatives and thereby transform them into

polar coordinates using relations (40), we arrive at the following

equilibrium equations:

- t 2G { L (r u ) t r- u t 2

c L r r r r 9, J

1 1 1 1

N 2N (- N (- N + -w = q

-rr ,rr -r9 r ,rO 2 ,a -9W 2 ,9 r ,r

r r

2G 1 2j t W 1 2 1u + f u

CLt r 1-2 ur 2 r 2( 2 6,6

f

-+ (u- u -.u )= 0

2r r,r6 0 rr r

20 2 tfEf r 2 1

cL2 u + r tE (1-v )(V u- 2 u)

1 1 (3-v -

+ (liv )(-u +i- u ) +- u 0 (50)

r r

and boundary conditions,

S2t 2 + rr wr + + N,9 6w nr d6 0

c

O02tf E 1 1

-r2 Lur,r r ,0 r r2Jr r

ri--- -[ u - u + uJ } 6uL n dO = 0 (51)

1+f r r,e r e 9,J r r

2 2 1 2 1

where V + +

ar2 r ar 2 a82

Since only complete annular or circular regions are to be

considered, the use of continuous cyclical functions of 6 eliminates

the need to specify boundary conditions along a radial boundary.

In the present work, which investigates the buckling of annular

sandwich panels, the transverse loading function, q, is not considered.

Such a restriction does not limit the application of the obtained

results, since, for snall deflections, the transverse loading function

does not influence the buckling load [3].

Finally, as a result of uniform compression along the inner

and (or) outer boundaries, an axisymmetric buckling mode is assumed

(B3) to result from the lowest critical axial load. Such an assump-

tion has been shown by Olsson [12] to be valid for single-layer

annular panels.

The governing equations thus become:

tt G { _I2 (ru) + V2 1 d (rN ') = 0 (52)

c t L dr r dr -rr dr

2trE d211 1 du 2

2G u + t + = 0 (53)

c Lt drj 2 2 r dr 2_j

c 1-v dr r

2 1 d d

where V ( - r )

r dr dr

and,

24 G- C + t -' + N 6w n d = 0 (54)

r c 2 d + r -rr dr r

-c

S"er + v xi 8u a d 0 (55)

In equations (52-55) the subscript r has been omitted from

the displacement u, without confusion. Also, N has been eliminated

by considering Ihcepre-buchling equilibrium of the face layers:

dN N N

-yr -rr --95

+ 0 (5G)

dr r(

3. Axi.al Stress Distribution

For two-dincnsional axisyneiotric stress distribution in pol ar

coordinates, the governing equations are [17]:

2- -

du 1 du u

S+, O (57)

dr2 r dr 2

dr

2'1 t

Ir 2 +dr f r/

J f

2t E

N f f + v (59)

1-v2 r dr

where u is the pre-buckling lateral displacement.

The general solution of equation (57) is:

u = C r I C /'r (60)

where C. and C6 are arbitrary constants of integration. Substituting

(6) into (55) yields:

2tF E (1- )F

N C(lv1+) C-- (61)

For the case of an annular panel subjected to uniform com-

pression along the inner and outer boundaries (see Figure 2), boundary

conditions are:

N (a) = N.

Krr 1

(62)

N (b) = N

-rr o

where a and b are inner and outer radii, respectively, and N. and

N are inner and outer compressive forces per unit length, respectively.

Imposing boundary conditions (62) on equation (Gl), we obtain

the following axial stress distribution:

N = -- (63)

-rr 2

N --- F (61)

-zGe 2

Figure 2. Annular Sandwich Panel

where,

b 2(N -N ) N PN

1 1 1 5)

and,

aB (66)

b

Since N and N are considered to be much larger than

-rr -06

the forces produced by bending during buckling, the axial stress

distribution remains essentially unchanged during buckling.

Also, if a = = 0, or N. = N it is easily verified that

equations (63) and (64) reduce to:

N = N = N (67)

4. Reduced Equilibrium Equations

From Chapter IV, Section 2, the equilibrium equations

(reproduced here for convenience) are:

C2 71 d 2 1 d (5w)

-t G -- I- (ru) + L vw ( r N 0 (52)

c tc t Lr r r dr -rr dr)

2t E 2

2G r2 + d 2- -[du 1 du o (53)

c Lt dr_ 2 2 r dr 2

c ]-v I r

Zaid [9] and Hiuang and Ebcioglu [10] uncouple similar equa-

tions by operating on the second with the linear operator

1 cl

L(y) -r dr )

and eliminating the first large common bracket in both equations.

The resulting equation is then directly integrated to yield:

1 f 1 dw C l r+ C2 r +

t tE r -rr dr 2 2 2 rj

fc f

(68)

where C', C', and C' are arbitrary constants of integration.

1 2' 3

Equation (68) agrees with Huang and Ebcioglu's results if

N is defined by equation (67) instead of (63). It should be noted,

however, that the uncoupling procedure described above yields five

constants of integration, while only four boundary conditions are

available. This is a direct result of the uncoupling procedure which

initially increases the order of equation (53).

As a consequence of the above-mentioned complications and

other considerations which will be discussed later, a different

technique is employed in the present work to uncouple equations (52)

and (53).

Multiplying equation (52) by r and integrating directly without

the aid of equation (53), we obtain:

t N C

c -rr 1 1

u + + (69)

S 2G 2tG r

c c

where,

dw

Y, J- (70,

and C is an arbitrary constant of integration.

Comparing equation (69) with equation (68), the advantage of

simplicity becomes immediately evident. Also, it will be made clear

in the following sections that the present procedure facilitates the

application of boundary conditions, and suggests analogies between

the classical single-layer plate theory and the present analysis of

sandwich panels.

Substituting equation (69) into equation (53),with the aid of

(63) and (70), we obtain, after some simplification:

3

2 2 d 2 d 4 12 BCr

(r ,2G)r r 4 (r -3G) r L + (Hr Ir 4 3G)= (71)

2 dr F

dr

where,

D BE F+BD EF

G= ; ; I =- ; F = A (72)

2F- F F 2

t 2G G (1- 2)

A-c c c f

A = c B (73)

2 tftEc (7f

and D and E are defined by equation (65).

Equations (69) and (71) represent the uncoupIl d equilibrium

equations. Since (71) is a second order differential equation, two

constants of intcgvration are generated. Together with C, and the

constant introduced through the integration of (70), we have four

arbitrary constants of integration to be determined by boundary con-

ditions at the inner and outer edges of the annular panel.

5. Boundary Conditions

From Chapter IV, Section 2, the boundary conditions (reproduced

here for convenience) are:

u dwl dw

S2t G + + N 6w n dB = 0 (54)

c +t 2 dr1 -rr dr r

C

S e C11+. 6u n. d = (55)

Integral (54) requires the specification of either the trans-

verse deflection w, or the resultant shear stress along the inner and

outer edges. Similarly, integral (55) requires the specification of

either the relative lateral movement of one face with respect to the

other, or the moment produced by tensile stresses on one face and

compressive stresses on the other, along the inner and outer edges.

Since we are neglecting the bending rigidity of the faces, the large

bracket in integral (55) represents the total edge moment, while u

becomes analogous to the slope, y, used in the formulation of boundary

conditions for the classical theory of single-layer circular plates [3].

Specifically, in the present work, integrals (54) and (55)

are satisfied through the following choice of boundary conditions at

the inner and outer edges:

tt N

At r = a: u C F c -rr ] cp = 0 u = 0 (74)

c

At r = b: w = O; u = 0 (75)

Conditions (75) are analogous to boundary conditions termed

clamped or built-in in the classical theory, while conditions (74),

termed "slider" in the present work, have previously been employed

for stability analysis of sandwich columns [1] and single-layer

annular panels [12]. Physically, such a restriction could be approx-

imated by allowing a shaft or rigid cylinder to occupy the central

hole (see Figure 2).

As the inner radius, a, shrinks to zero, conditions (74)

become identical to boundary conditions present at the center of

a circular sandwich panel, without central hole, constrained along

the outer edge only. This limiting process provides a check for our

final results since the stability problem associated with such a

panel yields a relatively simple solution.

6. Comparison with Other Theories

Comparison of the present work with existing theories can be

facilitated through the use of Table 1. Referring to this table, an

analogy between single-layer theory and sandwich theory is observed.

Meissner's [11] equation, which can be solved by means of bessel

functions, is reduced to a homogeneous differential equation (1-=0)

if the shear resultant is made to vanish at any arbitrary radius.

This becomes immediately evident if we compare the boundary condition

which specifies zero shear with the second form of the equilibrium

equation. Consequently, since the shear at the center of a circular

panel must vanish because of symmetry, Timioshoeko's [3] bessel equa-

tion of ordei one is a ho:mogoncous differential equation.

CO

u

4-

-9

u

C,)

+

C-l

I )

41 '4> <-4'

a- N

+ +

411

Cl +

'Cfltf,-

H C 0 II

0+0

Hr Cl -H

N^ .

N' CM') I

-^

c! U

tiC

4

,I

H

S

0,

0

CO

44'

Sa

41 9

,- 01)-

O 1-1-S-

t-44

0

H

'C

c'-;

0

a

0

9-

40

O 0

U

N

+

SL-W

t- +

0

41

44

0

a

41

I

N9

n +

-r4

T-f -.4

O- *r-t H

000t

00CC

0 4 H

( ) 0

M~ o

004441

0C o o

01 1 "i-s

O

9-

r-.

Cl C

01

-- 0 0

o O

rH -.+- -

M 0 0o

0 0 0

9 N 44

O

0

N

I I

.-1

Cl

o\ 'i-

*Q In

+

9 19

+

-9-It

0 0

A 9 N

1aI C

oM 4

0

H

*-H' 1C)

^-~T o a

0I M .

.II w

4-'

414

a

i Cl 0

1+

<4

''.4-'

k I3 -

441H 41

+ 41

1-3 +

,-1 44

0 At

01 o

spi

HO 41

*H 1-' 04 '

*-< 0 ',41 *r1

PIc C; ^ .

*^ V IS 0

~ I _~__

____1___1 I ___II_ ____I_____

Analogously, one of two equilibrium equations describing

an annular region for a sandwich panel reduces to a homogeneous

equation (C1 = 0) if the shear resultant is specified to vanish at some

arbitrary radius. Such a simplification also provides a direct

correspondence between u and y (see equation (69)). In the case of

a circular sandwich panel, a homogeneous equation again results as

a consequence of the symmetry involved, and one of the two equilibrium

equations yields bessel functions of order one, as is the case in the

single layer theory.

The second equilibrium equation for a circular sandwich panel,

which is attributed to Huanig and Ebcioglu, is deduced from equation (69)

rather than the original form in which it appeared (see equation (68)).

Consequently, without employing the present uncoupling technique, the

above stated analogies would not be evident.

Since the present work deals with an annular sandwich panel,

it becomes obvious that bessel functions cannot be employed unless

N = N. (see Equations (65) and (67)). Solutions for this special case

Sare obtained in Chapter V

and the more general case are obtained in Chapter V.

CHAPTER V

AXISYMMETRIC BUCKLING OF ANNULAR SANDWICH PANELS

1. Uniform Axial Stress Distribution

If the pressure along the inner edge of the annular sandxtich

panel is equal to the pressure along the outer edge, or:

N. = N (76)

1 o

then, from (65), (67) and (72):

D = G = 0; I = 1 ; E = N

o

N RN (77)

N = -N F A ; H 0

-rr o 2 A-(N /2)

0

which reduces equations (71) and (69) to:

2 B 0C r

Sdr LA- 2T r A-(N /2)(7

dtr o o

c c

Applying the first of boundary conditions (74) to equa-

tion (79) imnredliately reduces (78) and (79) to:

2 r BN r -

r2 d d o j 0 0 (80)

r r 2 dr A-(N /2) (

dr

rit N -

u = Z-r[- p (81)

2tG

c

Since (80) is Bessel's equation of order one, a general

solution of equations (80) and (81) takes the following form [18]:

1 1

r N 2 BN 2

A1 A-o/2) r + B2 1 A-(N /2) r (82)

O 0

1 1

ft N BN 2 BN 2

u L2c 7- {A 1 A-(N /2) r+A2Y1 L (rO ) r (83)

0 ?0

c

where J, and Y1 are bessel functions of order one of the first and

second kind, respectively, and A, and A are arbitrary constants

of integrati on,

If we now impose the remainder of boundary conditions (74)

and (75) on equations (82) and (83), we have:

[o Al[(c) + A2 Y1(i()16 0

2tG

c

(84)

SA (L + A Y (p)]

2- 2G 2 1 0

c

where

1

B(N )c 2 a

u [(N ) /2 b b (85)

o cr

These equations can be satisfied by taking A = A = 0. Then the

deflection at each point of the panel is zero and we obtain the

trivial, undeflected form of equilibrium of the panel. The buckling

form of equilibrium of the panel becomes possible only if equations

(84) yield values for Al and A2 different from zero, which requires

that the determinant of the coefficients of these constants vanish.

Therefore, after multiplying by (4/t t) and considering the first

of equations (73), our critical condition becomes:

[0) 2] rJ1()Y ) J1 (V1)Y(1) = 0 (86)

Equation (86) closely resembles the critical condition obtained by

Olsson [121 for a single-layer panel constrained in a similar manner.

For a given value of p, the smallest corresponding value of

[L for which the second large bracket in equation (86) vanishes is

given in Table 2 below [12].

TABLE 2

LOWEST VALUE OF pV SATISFYING EQUATION (86)

$ 0.0000 0.0256 0.0526 0.0909 0.1000

4 3.832 3.840 3.860 3.924 3.942

p 0.1111 0.1250 0.1433 0.1667 0.2000

p 3.966 4.000 4.045 4.116 4.235

$ 0.2500 0.3333 0.3956 0.5000 0.5461

p 4.445 4.905 5.355 6.394 7.016

@ 0.6285 0.6897 0.7634 0.8333 1.000

1 8.523 10.175 13.312 18. 73 w

And, from the first of (85):

N (2

Acr b2 2 /2

It can be shown that (N /A) given by equation (87) is a monotonic

increasing function of p. Therefore, the lowest value of p results

in the lowest value of (N /A) Also, since b B is always positive,

o cr

equation (87) yields values of (N /A) which are greater than or

0 cr

equal to two; and we can therefore conclude that the lowest root of

equation (86) is always given by equation (87) in conjunction with

Table 2. (See Figure 3.)

The first of boundary conditions (75) has not been used

to obtain the above results since the buckling load is independent of

a transverse translation of the panel as a whole. In this respect our

analysis parallels the classical single-layer theory [3].

For a = 0, equation (87) becomes:

2

) = (3.832) 2 (88)

cr b B+(3.832) /2

Equation (88) agrees with Huang and Ebcioglu's [10] results if the

present notation is used. The validity of this limiting process was

discussed in Section 5 of Chapter IV.

As G. approaches a very large value, the first term in the

denominator of equation (87) becomes dominant, and we obtain:

2,

(N 1 (S9)

o cr 2

b

where the "effective bending rigidity," D', is defined:

D' = (90)

2(1-v )

f

Equation (89) in conjunction with Table 2 agrees with the

results obtained by Olsson [12] for a single-layer annular panel

subjected to uniform inner and outer axial pressures of equal intensity.

For a = = 0, equation (89) becomes:

) (3.832)2D

o cr 2 (91)

b

Equation (91) is identical to the buckling load obtained by

Timoshenl:o [3] for a circular single-layer panel clamped along the

outer edge.

This limiting process is intuitively expected, since,

for G = m, the only structural function of the core is to control

the distance between the face layers. A similar relationship exists

between the web and flanges of an I-beam.

2. General Solution

We now return to a general solution of equations (69) and

(71). Introducing the dimensionless variable, 7], through the trans-

formation:

= r/b

equation (71) becomes:

2 b2BC E

where,

2

N = ; R = b N ; Q = IN (94)

Sir.:ila;ily, equation (69) becomes:

ft N C

c F -rrl 1 1

u I Ij + (95)

2 2iLG 2btG

c c

Since we seek a series solution of equation (93), and choose

to expand our series about the point T[= 1, the following additional

transformation is introduced:

S= 1 (96)

The reasons for seeking a solution about the point ,1 1 (= = 0)

will be discussed in Sections 5 and 6 of this chapter.

With the aid of equation (96), and after some simplification,

equation (93) becomes:

2

[(N+1) + 2(2N+1)5 + (6N+)1)2 + 4N 3 + N(4 d- w+ [(N-3)

dr

+ 30(-1) + 3N72 + N 3] -+ [(R--Qi3) + (4R 2Q)

2 3 b1 C1(1+)

+ (6R-Q) + 41HE 4 R4 ]T = -- (97)

To obtain a complementary solution of equation (97) the

following infinite series is employed [18]:

cP = Z A k (98)

k=0

where the coefficients A, are functions of the elastic and geometric

properties of the panel and the critical buckling load. The radius

of convergence of series (98) can be shown (see Section 6, Chapter V)

to be of sufficient magnitude for our particular problem.

Substituting (98) into a homogeneous form of equation (97),

and collecting coefficients of common powers of , we have:

[2(N+1)A2 + (N-3)Al + (R-Q43)A ] + [6(N+1)A + 2(5N-1)A

+ (R-Q+3N)A + (4R-2Q)A ]o + [12(N+1)A + (27N+3)A3

1 o 4 3

4 (R-Q3]fN-l)A + (4R-2Q+3N)A + (6R-Q)A ] 2

2 1 o

+ [20(N+1)A5 1 4(13N+3)A + (45N+R-Q)A3

+ (14N+41R-2Q)A + (N+6R-Q)A + 4RA ] 3

2 1 o

4 . . . . = 0 (99)

In order that this series vanish for all values of in some region

surrounding E = 0, it is necessary and sufficient that the coefficients

of each power of vanish [18]. This produces the following relations

in which some coefficients have been eliminated through the accumulative

introduction of previously computed coefficients:

A (R-Q+3) A 3)A (100)

2 2(N+1) 2(N+1) 1

A [(S-)(R-Q+3) (4R-2Q)] A r(5N-)(N-3) ( R-Q+3N) A (101)

2 fL -^ -^2 6(N+6)( 31)l

A_ (6R-Q) (R-Q+3)(R-Q+18iN-1) (27N+3)(4R-2Q)

4 12(N+1) 24(N+1)2 72(N+1)2

(27N-i3)(5N-1)(R--Q+3)1 A (4R-2Q+3N) (N-3)(R-Q+18N-1)

72(N+1)3 L 12(N+1) 24(N+1)2

(273i3) (R-Q+3N) (27K+3) (5N-1) (N-3) A (102)

72(N+1)2 72(N+1) 1

[ (13N+3)(GR-Q) (13N+3) (R-Q+3) (-Q+18SN-1)

5 GO(N+1) 120(N+1)3

(13N+3) (27N+3) (4R-2Q) (13N+3) (27N+3)(5-1) (R-Q+3)

3 + --- -- 4

360(\+1) 360(N+1)

(45N+R-Q)(5N-1) (R-Q+3) (453iX+R-Q) (4R-2Q) (R-Q+3)(14N+4R-2Q)

120(N+1)' 120(N+1)2 40(N+1)

4R (13N+3) (411-2Q3) (13+3) (-3) (R-Q+18N-1)

20N+ o 60(N+1)2 120(N+1)

(13N+3) (27Nt3) (R-Q4 3N) (13N+3) (27N+3)(5N-1)(N-3)

360(;+1)3 360(N+1)

(45N+R-Q)(5N-1)(N-3) (45N+R-Q)(R-Q+3N)

120(N+1) 3 120(N+1)2

(N-3)(14N+4R-2Q) (N+61R-Q) A (103)

40(N+)2 20(N+1)J 1

40(3+1)

. . . . . . . . * *

A-

where A and A remain arbitrary.

o 1

Because of the complexity of the computations, and the

immediate requirements, no recursion formula is sought in the

present analysis.

With the aid of (100-103), our complementary solution becomes:

^ Ai -( (R-Q+3) 2 + [(5N-1)(R-Q+3) (4R-2Q) 3

c 2(N+ (1) G(N+1)

6(N+1)

[ (6R-Q) (R-Q+3) (R-Q+18N-1) (27N+3) (4R-2Q)

+ 12(+1) 2 + 2

2(N+1) 24(N+1)2 72(N+1)

(27N+3) (5-1) (R-Q+-3) 4 (13N+3) (6R-Q)

72(N+1) 60(N+1)

(13N+3)(R-Q+3)(R-Q+18N-1) (133+3)(27N+3)(4R-2Q)

320(N+1)3 360(N+l)3

(13N+3) (27N43) (5N-1) (R-Q+3) (45N+R-Q) (5X-1) (R-Q+3)

360(N+1)4 120(N+1)3

(45X+R-Q)(4R-2Q) (R-Q+3) (14N+34R-2Q) 4R 5

120(N+1)2 40(N)+1)2 2 0 N+ +

4A 1{ (-3) 2 + (5N-1)(N-3) (R-Q+3N)3) 3

1 2(N+1) +L -.2 -6(N+ 1) ]

F (4R-2Q-F3N) (N-3) (R-Q-18N-1) (27N+3) (Rf-Q+3N)

L- 12(N+1) + ^ 2 .2

24(N+1) 72(N+1)

(27N+3) (5N-1) (N-3)1 4 r~(13.N+3) (4R-2Q+3N)

72(N+1) 60(N+1)

(13N+3)(N-3)(R-Q+18N-1) (13N+3)(27N+3)(Rfl-Q+3N)

120(N+1)3 360(N+1)3

(13N+3) (27+3) (5N-1) (N-3) (45N+R-Q)(5N-1)(N-3)

360(N+1)4 120(N+1)3

(45N+R-Q)(R-Q+3N) (N-3)(14N+4R-2Q) (N+6R-Q)f 5

120(N-l) 40(N+1)1) J

+ . . . . .} (104)

Following a procedure similar to that employed in the previous

section, equations (95) and (104), together with a particular solution

of equation (97), are constrained according to boundary conditions

(74) and (75).

Imposing the first of boundary conditions (74) on equation

(95) eliminates the need to find a particular solution of equation

(97), since we have CI = 0. The subscript c can therefore be

eliminated from equation (98) and our solution becomes a complete

solution.

From equations (92) and (96), we note that at r = b, 1] = 1

and 0 = 0. Therefore, from equation (95), recalling that C = 0

and N (b) = N (see equation (62)), the second of boundary condi-

tions (75) becomes:

[() 2 cp (:=0) = 0 (105)

And since (N /A) A 2 in general, condition (105) imposed on equa-

tion (104) yields A = 0

Similarly, the second of boundary conditions (74), evaluated

at r = a ( =3-l), and expressed through equation (95) becomes:

-i) 2 p (=0-1)= 0 (106)

After substituting equation (104) into (106) with A = 0,

and noting that in general (N./A) 4 2, we arrive at the general

buckling criteria for an annular sandwich panel:

(N-3) (1) 2 F(5N-1)(N-3) (R-Q+3N)] ( 3

S2(N+1) G(N+l)2 6(N+l)

F- (4R-2Q+3N) (N-3)(R-Q+1SN-1) (27N+3)(R-Q+3N)

12 (NT+1) 24(N+1)2 72(N+l) 2

24(N+1)2 72(N+1)2

(27N+3)(5N-1)(N-3) (1)4 (13N+3) (4R-2Q+3N)

72(N+1) 1- 60 (N+1)

(13N+3) (N-3) (R-Q+1-1) (13N+3)(27N+3) (R-Q+3N)

120(N+1)3 360(N+1)3

(13N+3)(27N+3)(5N-1)(N-3) (45N+R-Q)(5N-1)(N-3)

360(N+1)4 120(N+1)3

(45N+R-Q)(R-Q+3N) (N-a3)(14N+4R-2Q) (N+6R-Q) 0( -1 5

420(+1) (1

120(N+1)2 40(N+1)2 20(N+)

+ . . . . = 0 (107)

As explained in the previous section, the undeflected form

of a panel in equilibrium, i.e., When Al = 0, is of little interest.

Therefore, approximate critical buckling loads of the panel under

discussion may be computed by considering a finite number of terms

in equation (307).

The first of boundary conditions (75), as stated earlier, is

not used in obtaining the above results, since the buckling load is

independent of a transverse translation of the panel as a whole,

3. Successive Approximations

From equations (94), (72) and (65), we have:

N 2A(1-2 (N 0 N (108)

2 (N -N.) 2 (N -N.)

0 1 0 1

2b2B (No-2 N)

B 0 i (109)

2 (N -N.)

0 1

2A(1-2 (No N 2

Q + 2b B (110)

2 (x -N.) 2 (N -N.)

01 O 1

We designate the first approximation to be that which

considers only the first term of the infinite series (107), and

consecutively add a term for each succeeding approximation.

Therefore, with the aid of (108-110), and after some simplification,

we have:

First Approximation

The first approximation yields no results since the loading

functions are not present in the first term of series (107).

Second Approximation

S) ) 2[(N ) (N.) c r

2 +2

(p1) I 2A(1-- ) o cr 1c -31 0 (111)

( N[(N ) -(N.) i] 2[(N ) -(N.) ]

L o cr 1 c o cr 1 Cr

Third Approximation

6r 2A(1-02)

L2 [(N ) -(N.) ]

L 0 cr i cr

[(N ) -2 (N.)

o cr 1 cr

2

2[(N ) -(N.) ]

S10OA(1-p )

ocr i cr

[(N ) 2 (N.) I

o cr icrY

Socr i cr

[( cr -2Ni cr +1 3 2A(1-2)

$2[(N )[ (-() N ) cr-(N.) ]

o cr i cr L o cr i cr

I 2 [(N ) -_ (N.) cr

I 2A(1-5 ) ocr -3cr-i)

2 o [(N) -(N.) ] 2[(N ) -(N.)]c

5[(N ) -52(N.) 1

2 ocr 2 cr l 2A(1-S) 2

S2[(N ) -(N.) c -2[(N ) -(N.) ]

o cr i JL o cr i cr

- 3] (1)2

2 [(N) -( 2(N.)

S2A(1- ) o cr i cr

2[(N ) -(N.) ] (2[(N) c-(N.) ]

S4A(1-2)

-21(N ) -(N.) c]

- 2b2B (-1)2 = 0

Fourth Approximat ion

2[(N ) -5 2(N.)

2 [(N ) -(N.)

0 cr 1 cri

52( oc^(icrl^

2b2B[(Nocr- 2(N )cr

2 [(N ) -(N.) 1

O Cr I C

(112)

2)

2A(1-5 )

72 2A _;2

72Ls N 0) -(N.) ]

ocr i cri

[(N ) - (N.) ]-

Scr- i cr

2 + 1

a cr a cr -

- 36 F2A

S2[(N ) -(N.) I

cO Ca 1Ci

[(N cr- 2(N )

2 [(N ) -(N.) 1

O ~ CT1CT

-3 2A(1-$2 )

- 3Ik cr-(

L2[(N ) -(N.)]

-cO Ca 1Cr

[(N ) -5 (N.) ]

o cr i cr

52[(N ) -(N.) ]

o cr a cr

-+1 (B-1) + (121 10A(1-3 2 )

LP [(N cr-(N)

5[(N ) 2 (N .) ]

o (cr 1 crl

[ 2A(1-82) S

2 [(N ) -(N.) I r

[(N ) -2(N.)

o cr 1 cr+

2b 2B[(N ) 2( )]

+ 2

2[(N ) -(N.)

ocr 1 cr

2[(N ) (N.) I

o cr 1 cr

[1 2 2A(1- -$ )

o cr(N 1 cr

[3( -(N.-),,

[(No cr- 2(Nicr

2

L(No~cr (Ni)cr]

r 2

4A(1-B ) )

L2[(N ) (Ni) cr

- 2b B (3-1) + 6

2 r 2

2A(1-3 )

2[(cN ) -(N.) ]

o cr i cr

[(N ) -$2 (N.) 1

o cr i cr

3

3 [(N ) -(N.) )

ocr icr

12 I 2A(i-32)

2[(N ) 2(N.) I

2A(1-g5

32[(N,) ,-(N.)"I

ocr-P i crI

S[(N ) -c(N.) c

o cr i cr

Sb2 [(No) cr- 2(N) '

cr cr

- 4b2j + [ 2A( _2)

o cr-32 cr

82 [(N ) -(N.)

o cr i cr

[ 34A(1-22 )

2 [(N ) -(N.)"I

Oc C1 1Cr

1 2A(1-2 )

112[(N ) -(N.) ]

[5 o cr i cr

17[(NO cr 2 (N )cr

S2[(N ) -(NV.) ]

O crT 1C

[(N ) -2 (N.) I

0 cr I cr-3

32[(N ) -(N.) r

o cr 1 cr

32 [(No )cr-(N)cr

2 27(1-02 [(N) -32 (Ni ]

- 2b B +1 + 2A (1-32) 2 c

(N) cr-(N) c1 2 [(N ) -- (N) ]

O C cr O Cr 1 cr

54A(1-5 2 )

S[(NO cr-(N cr]

27[(No cr- 2(Ncr

5V[(N )c-(N.)c

0 cr dcr'

4A(1- 2)

2 cr(N c

2 [(N ) ,-2(N .) ]

2( cr-(N )cr

27[(No c 2 (N. )cr

2[(No cr-(Ni cr

82[(N ) -(N.) ]

___ 2A(l-g2)

o cr 1 cr

Fifth Approximation

2b 2B(N o 2 (N.) ,]

o cr 1 cra

3][1OA10(1-2)

[(N cr-(N )cr

[(No cr- 2(N)cr

2 [(N ) cr(Ni c

o cr 'c

2b2B 54A(1-B 2

S 2[(N ) (N.) ]

5[(No cr 2( i cr

S[(N ) -(N.) I

ocr icr 1

- 3 (-1)3 = 0

(113)

360 2A(1-_2)

62[(N ) -(N.)cr-

o cr a ci

[(N) cr- (N icr

p [(N ) -(N.)cr

o cr 1 cr

- 180 2 A(1-2) 2) -

0 [ c i cr

[(N cr-B2 cr

0 cr cr

2A(1-B2)

[(No cr(N i)cr

[(N) -F2(N.)I

o cr 1 cr1

2- N cr cr

cr 2 cr

S22A(1-B2)

L [(No )cr- (N )cr

(S-1) + 60 2 2(-n )

2[(No cr- (Ni)c

2

+1

10A(.-2 )

2[(NO cr-(N )cr

[(N )cr- 2(N )cr

[(No ) r-(N )c,]

5[(No) cr,-(N)i cr

2 [(No )cr-(N )c

o-cri^cr

- 60 2 2A(1-9-F2 .

S[(No cr-(N i)cr

[(No cr (N cr 3 4A(1-2

0 -1 cr

B2[(N ) -(N.) c]r LB[(N ) -(N.) ]

0 ia1c o cr i cr

2[(N )cr- 2(Ni)c

2[(N ) -(N.)

Scr i cr

+ 1 '

2b2B[(N ) -2 (N.) ] ii

ocr icr 2 2

+- 2-2b B (8-1) -

82[(N ) -(N.) J

ocr i cr

2A(1-2 )

0 cr i cr

(2N( 2 ) (N.)

o cr r cr

r 2[(N ) (N.) I

o cr 1 cr

2 o

o ccr (N i cr

S[(oN ) -(N.)

o cr i crl

2A(1-$2 )

2[(No) cr-(N )cr

[( cr cr

2 [(No c) -(N )r

4b2B + 15 2c(12

S(N ) cr-(N.) 3

2A(1-2) )

[(N ) -(N.)cr

[(No cr- 2()cr

2(NO cr- i cr

2 17[(: )

3 (1-l o cl

2[(N ) --(N) c2[ (N)

-2(N.) ] 2b2B[(N ) cr- (N cr

cl ( r cr+ 2 cr

-(N.) ] [2(N ) -(N.)

2r i cr o cr i cr

[(N )cr (N c

82[(N ) -(N.)

o cr i cr

S541A(1-3 )

' 2[(N ) -(x cr.)

L o cr 1 cr

27[(N ) -c- (N.)

0ocr icr)

2 + cr- c

2 [(No )r-(N.)cr

S4A(1-32) _

S (No cr i cr

2[(N ) (N.) ]

2 0 r i cr

2[(N ) cr-(N .) r

2b2 B[(N )cr- 2(Ni )cr

2[(No cr -(Ni)cr

r

5 2A(1-S2)

2 [(N ) -(N.) ]

Scr i cr

[(No) c- (Ni)cr I

F 2 [(-- c.- ( +

0 Cc 1 Cr

54A(1-82)

F 2[(N ) -(N.) ]

O cr i cr

27[(N ) 2 (N) 2 5[(N (

o cr 1(Ni cr 3 OA(1-2) 5(o ccr cr

82[(N) cr-(N cr [( ) -(N. )c i [(N ) -el cr(N.

( c c cr i cr cr icr

- 2b 2B

IJ

2A(1-2 )

S2[(N ) -(N.)

0cr 1 cr

[(N ) -$ (N.)

o cr i cr

2 [(N ) -(N.) ]

ocr i cr

2A(1- 2)

[(N )cr-(N)cr

O Cr i Cr

13[(N ) -(N.) ]

o cr 1 cr

8b [(N ) 2-()2

0 cr 1 cr

Sb B[(N ) -a (N.)

B2[(N ) -(N.) 1

o cr 1 Cr

[(No cr-82(N )cr+

[(N ) -c (N.)

a cr 1 Cd

2 +

5 [(N ) ,-(N.i )

2A(1-82 )

[(N )cr -(Ni

o cr icr

'1 26A(1-32)

JL2[(N ) -(N.)cr

-(N -$ 2(N )crI

0cr r icir

2[(N ) (N.)

2] F2A(1-2

- 4bB 3 2

L ((N ) -(N.) I

SO r 2T Cr

[(N ) -5 (N.) J

o cr i cr,

a cr 1 Cr

. 1 26A(1-5 2)

L2[(N ) -(N.) ]

cr 1 cr

13[(N ) ,- 2(N.)]

2 [(N ) -(N.)

o Cr 1 er

S o cr 2i cr

17 [(N ) -(N.) ]

o cr i cr

182[(N ) --(N.) l]

oc icr

- [No cr- ci 34A(1-52

(N ) cr-(N.) IJ 2[(N ) -(N.) r

a cr i cr a cr i cr

2b2 B[(N ) c- 2(N.) ]

o cr i cr

- 2b2B 1

2A(1-2)

0 cr 1 cr

z3[(No)cr 2(Ni )c

2

O cr 1 C

+ 3 4IA(1-2 )

i 2L(N ) -(N.)

o cr 1 cr

[(No cr- 2(ai c + ^1

2 L(N,),,-(N i) cr ]

2[(N ) (N.) i]

aocr i cr

2 [(N ) -(N.)

cr cr

52[(No) -(N.)i c

2GA(1-52)

27[(N ) 2(N .)

o cr a cr

2b2 I[(N ) -P2 (N.) ]

o cr i cr

2[(N ) -(N.) I

Oc iT1Cr

+

3 ($-1)3

-i

2b2B 26A(1-82) 13[N) cr-8(N) cr 3

2( L c -(N. )" 2[(N) c (N cr )

S54A(1-2) 2

2 [(N ) -(N.) ]

ocr icr

5[(N ) -(N.

o cr 1 cr

02[(N ) -(N.)

O Cr 1 Cr

44[(N ) -3(N.)]

o cr 1 cr2

5[(N ) -2(N.)

o cr 1 cr

44[ (N) -32 ),

(N ) -2(N.) ]

ocr (Ni cr

------------+ 3

[(N ) -(N.) I

o cr 1 cr

r 2

2A(1-4 )

2[(N 0)cr-(N el I

[(No cr-P2 (N i c

2[(N -(N.) ]

a cr i cr

[(N ) (N.)

0 Cr 2 1 cr

2A(1-) 2 )

$2 [(N) -(N)

0 cr i cr

o cr2(N cr

2

ocr 1 cr

1OA(1- 2

L8 2 [(Ncr-(N cW

(N 2 (N.) I

Scr I crl

S[(N ) -(N.) I

o cr i cr

Sr 1 CT

1 8SA(1-2 )

2b B 21AI-

ocr icr

[(N ) -2(N.) ]

o cr 1 cr

2 [(N ) -(N.) ]

o cr 1 cr

1

- 31i

j

88A(-2 )

I2 [(N) -(N.) ]

o cr Icri

22

2b21 -B-r -_$2

[(N- ) ,-(N.) ]r

o cr 1c

2[(N ) -2(N.) I 2b2B[(N ) -82 (N.) ]

o cr -c c -icr 2 2b2

2[(N ) -(N ) cr] 2 [(N ) -(N.) II

2 -cr ) cr o cr crJ

2A(1-2) (No) cr- 2(N r1 2 2A(1-,2

2[(No ) ^c.) ] [(N ) -(N.) ] 2 [(N ) r-(N.)

cr ocr icr ocr 1cr

[ 2A(1-P 2 )

S [(N ) -( .) ]cl

1- o cr i cr

[(N ) -$ 2(N.) ]r 24 12[(N ) 2(Ni .)cr

ocr 1 c 24A(1-c) o cr a cr

-o3

$2[(N ) -(N.) ] L2[(N 0)-(N .] )2[(No) -(N.)"]

8b 2B[(N ) 2(N.) c] r 2

2 cr cr 2402 2A(1-0

+ o 4b Ba is8 2

[(No 0cr-(N )c 2[(No cr-(N )cr

[(W ) -(N.) ] 12b2B[(N ) -2(N.) ]-N)

cr 1 cr o cr i cr 2b2 (o-1)4 = 0

2 [(N ) cr-(N.) ] 2[(N)-( N ) .)

o cr cr o cr a cr

(114)

Computation of the sixth approximation is unnecessary since

it can be shown that the fifth approximation yields acceptable

results (see Section 5, Chapter V).

Equations (111-114), each being self-contained, represent the

approximate critical buckling criteria for an annular sandwich panel

constrained by boundary conditions (74) and (75) and subjected to

uniform radial compressive loads, N. and N along the inner and outer

edges, respectively.

4. Numerical Results and Discussions

Once the inner compressive load, N., is prescribed to be some

multiple of the outer compressive load, No, the approximate critical

conditions given in the previous section are completely defined by

three dimensionless parameters: (N /A) b B, and p. Thus we obtain

o cr

algebraic polynomials in (N /A) which increase in degree as the order

of approximation increases (ranging from first degree in the second

approximation to fourth degree in the fifth approximation). For

obvious reasons we consider only the lowest positive value of (N /A)

o cr

satisfying each polynomial.

Three possible loading conditions are analyzed in the present

work: (1) N = N., (2) N = 0, and (3) N. = 0. However, it should be

0 1 0 1

noted that the techniques employed in these examples are applicable

for any ratio of N to N..

O 1

N = N.

o 2

If the inner and outer axial compressive loads are equal, then

the exact solution is given by equation (87) in conjunction with

Table 2. Thus, the approximation techniques employed in the previous

section are unnecessary. For this case, Figure 3 shows the existing

relation between (N /A) and 3 for various values of b 2B.

o cr

N = 0

o

If the inner edge alone is subjected to axial compression,

then, since equations (111-114) yield no positive values of (N /A) ,

it can be concluded that buckling never occurs. This would seem

reasonable, since, from equations (63-65), such a reduction results

in a relatively large tensile N compared with a relatively small

-96

compressive N Analogously, for a rectangular single-layer panel

subjected to compression along opposite edges and tension along adja-

cent edges, Timoshenko [3] has shown that a large tensile load will

prevent a significantly smaller compressive load from causing insta-

bility. However, we must keep in mind that, while N and N can

-xxe -yy

be varied independently in a rectangular panel, such is not the case

for a circular or annular panel, since the following equilibrium

condition must be maintained:

dN N N

-rr -rr -68 (56)

-dr + r = 0 (56)

N. = 0

1

If only the outer edge is subjected to axial compression,

the second approximation, equation (111), can be solved explicitly

for (N /A)

o cr

(N 2(1+)(3-) (115)

cr 3+23 + 33

Since b B does not enter into relation (115), it is obvious

that further approximations must be considered.

Due to the complexity of the calculations involved, a graphical

solution is employed for the succeeding approximations. In Figure 4

approximations two through five are compared by plotting (N /A)

o cr

versus $ for various values of b B.

All quantities appearing in Figures 3 and 4 are dimensionless.

Buckling loads may be obtained in the appropriate dimensions by using

the first of relations (73).

Figure 3. Minimum Critical Values of (N /A) for N

o o

(1 /.

p= aA

Figure 4. Minimum Critical Values of (N /A) for N. = 0

0 1

The following conclusions may be drawn from Figures 3 and 4:

1. All critical values of (N /A) are less than or equal to two.

0

(Figures 3 and 4)

2

2. As the value of b B approaches infinity, the results approach

those obtained for a single-layer panel in equation (89).

(Figure 3)

3. For p = 0, the present theory coincides with Huang and Ebcioglu's

results (equation (88)) for a circular sandwich panel. (Figures 3

and 4)

4. The second approximation in Figure 4 is the exact solution for

b B = 0.

5. An annular sandwich panel subjected to axial compression along the

outer boundary becomes stronger if an equal compressive load is

also applied along the inner edge. (Figures 3 and 4)

6. In Figure 4, the third approximations yield more accurate results

than the fourth approximations. This peculiarity and the error

bound associated with the fifth approximations will be discussed

in the next section.

7. A dual response is apparent in Figure 4. As the hole increases

in relative size, the panel may become weaker or stronger depend-

ing on the value of b B and the range of P being considered.

Such a behavior is possible because both the shear and the bending

stiffness of a sandwich panel enter into the analysis. An annular

single-layer panel, which can be described by only two dimension-

less parameters, exhibits no such dual response.

2

- 8. If buckling loads for values of b B or ratios of N to N., not

O 1

considered in Figures 3 and 4, are required, equation (114) may

be used directly. However, when considering only values of p

greater than one-half, the third approximation, equation (112),

yields acceptable results. (Figure 4)

The above conclusions are valid only for the special case in

which boundary conditions (74) and (75) are applied.

5. Error Bound

Since we chose to expand the solution of equation (93) about

the point = 1, the speed of convergence of the series solution

obtained, equation (104), depends on the proximity of the entire

annular region to that point. Clearly, as the hole increases in size

(5 approaches one), the solution converges more rapidly. This fact is

evident from Figure 4, and also from equation (107). Indeed, it can

be concluded that the speed of convergence is the slowest when $ = 0.

From equations (111-114) the approximate critical buckling

parameter, (N /A) can be solved explicitly for the degenerate cas'

of 0 = 0. The results of this simplification are found in Table 3.

However, for this special case, the exact solution (reproduced

here for convenience) is available from Section 1 of this chapter:

2 o 14.684 (88)

cr b B + 7.342

TABLE 3

APPROXILATE VALUES OF (N /A)

o cr

FOR P = 0

Order ef (N /A)

o cr

Approximation

First (No results)

Second, equation (111) 2

12

Third, equation (112) 2 12

bB + 6

10

Fourth, equation (113) 2

b B + 5

Fifth, equation (114) 2 15

b B + 7.5

If we compare equation (88) with Table 3, it can be concluded

that the fifth approximation, for = 0, is within 2.2 per cent of the

exact solution, even for large values of b B. And, since the series

solution converges more rapidly for other values of 5, equation (114)

yields results that lie within 2.2 per cent of the exact solution for

all values oC b2 B and .

Following the same reasoning outlined above, it can be concluded

that the third approximation yields more accurate results than the

fourth approximation. This peculiarity can be easily verified by

expanding a series solution of equation (78) about the point r = b

( = 0). If this is done, the resulting approximations, as expected,

coincide exactly with those listed in Table 3.

6. Remarks on Convergence

The series solution of an ordinary differential equation

possesses a radius of convergence at least as great as the distance

from the point of expansion to the nearest singularity [19].

Equation (93) possesses two regular singularities, one at

= 0, and another at (NQl +1) = 0; and its solution was expanded about

the point I = 1. Therefore, it must be demonstrated that these

singularities do not inhibit the validity of our solution throughout

the entire annular region of the panel.

Referring to Figure 5, it becomes obvious that the singularity

at T = 0 does not restrict the required radius of convergence, regard-

less of the value of 3. It is therefore necessary only to show that

the singularity occurring at (NJ 2+1) = 0, lies outside the annular

region and its reflection illustrated in Figure 5, for all values of 3.

refleded i ncae ot&JfZLdh

o 1 2-p 2

Figure 5. Radius of Convergence

With the aid of equation (108), (NT 2+1) = 0 becomes:

2

S[2A(1-B) (N -2 N.)] +1 0 (116)

S2(N -N.) 1

O i

Therefore, the position of the singularity associated with equation

(116) depends on the value of the critical buckling load which becomes

known only after the solution is obtained.

For the particular case in which N. = 0, (11G) is satisfied

if

(N) 2(l- 2P (317)

A /cr 2 2

However, from equation (115) and Figure 4, it is apparent

that, for = 0,

( ) 2(1+g)(3-) (118)

3 + 2 + 332

for all values of h 2'.

In order that conditions (117) and (118) be satisfied simul-

2

taneously, Tl must satisfy the following inequality:

2 2(3-) (119)

[2(3-3) 2(1-3)(3+2+33 )]

2 2 '

Equation (119) constrains fl to be greater than (2-s) for

all values of ranging from 0 to 1. It can therefore be concluded

that the singularity associated with equation (117) lies outside the

annular region and its reflection illustrated in Figure 5. Similarly,

it can also be sho'n that, for No = 0, the singularity associated with

equation (116) lies outside this critical region.

Thus, the radius of convergence, p, of the series solution of

equation (93), expanded about the point T1 = 1, is:

P A 1 (120)

which, as illustrated in Figure 5, is large enough to encompass the

entire annular region of the panel.

Care must be taken, however, when imposing boundary conditions

other than (74) and (75) on the solution of equation (93). Critical

loads resulting from boundary conditions or ratios of N to N. not

O 1

considered in the present work may satisfy equation (116) within the

needed radius of convergence. It would then become necessary to either

expand the solution of equation (93) about some other point, or employ

the techniques associated with analytic continuation. Tne difficulties

encountered in the latter approach would be enormous.

CHAPTER VI

CONCLUSION

The present work investigates the buckling of annular sandwich

panels. Equilibrium equations and boundary conditions satisfying

continuity requirements were derived in cartesian coordinates, using

the theorem of minimum potential energy. These equations were then

transformed into polar coordinates through the application of tensor

analysis.

Axisymmetric buckling being assumed, and the bending rigidity

of the faces being neglected, the equilibrium equations were uncoupled

by using a modified technique. The governing equations were then

compared with existing theories for single-layer annular panels [11,121

and circular sandwich panels [10].

For the general problem of an annular sandwich panel subjected

to unequal inner and outer compressive loads, and constrained by

boundary conditions similar to those employed by Olsson [12], a power

series solution was obtained. This series was shown to possess a

radius of convergence of sufficient magnitude. Successive approxima-

tions Aore then computed, and a graphical solution was employed for

various ratios of outer to inner compressive loads. Results from the

fifth approximation, which were shown to be within 2.2 per cent of the

exact solution, were compared with those obtained from earlier theories

[10,11,12].

72

The present work represents the first attempt to analyze the

stability of annular sandwich panels. Further extensions of the

present theory may be carried out by including the effects of the

bending rigidity of the faces or considering boundary conditions

other than those employed here. Furthermore, continued efforts should

be directed toward obtaining a solution to the unsymnetric buckling

problem. In this way, the assumption of axisymmetric buckling could

be justified, and the problems associated with angular dependent

loading functions could be analyzed.

BIBLIOGRAPHY

I. F. J. Plantema, "Sandwich Construction," Wiley and Sons, New York,

1966.

2. S. Timoshenko and S. Woinowsky-Kricger, "Theory of Plates and

Shells," 2nd ed. McGraw-Hill, New York, 1959.

3. S. Timoshenko and J. Gero, "Theory of Elastic Stability,"

2nd ed., McGraw-Hill, New York, 1961.

I. N. J. Hoff, "Bending and Buckling of Rectangular Sandwich Plates,"

NACA TN 2225, 1950.

5. A. C. Eringen, "Bending and Buckling of Rectangular Sandwich

Plates," Proc. First U.S. Natl. Cong. Appl. AMech. 1951, pp. 381-

390.

6. C. C, Chang and I. K. Ebcioglu, "Elastic Instability of Rectangu-

lar Sandwich Panel of Orthotropic Core with Different Face Thick-

nesses and Materials," Transactions of the American Society of

Mechanical Engineers, J. App. Mech., Vol. 27, No. 3, September

1960, pp. 474-480.

7. S. J. Kim, "Symmetric and Antisymmetric Buckling of Sandwich

Panels," Doctoral Dissertation, Dept. Eng. Sci. and Mech.,

University of Florida, 1969.

8. E. Reissner, "Small Bending and Stretching of Sandwich-Type

Shells," NACA TN 1832, 1949,

9. M. Zaid, "Symmetrical Bending of Circular Sandwich Plates," Pro\c.

of the 2nd U.S. Natl. Cong. Appl. Mech., 1951, pp. 413--422.

10. J. C. Ihuang and I. K. Ebcioglu, "Circular Sandwich Plate under

Radial Comprcssion and Thermal Gradient," AIAA Journal, Vol. 3,

No. 6, June 1965, pp. 1146-1148.

11. E. Meissner, "Uber das Knicken krcisringfodrmiger Schihben,"

Schweiz. Bauztg., Vol. 101, 1933, pp. 87-89.

12. G. GOsson, "Knickung der Krei sringplatte vcn quadratisch

veranderlicher Stcifigkcit," Ingr.-Arch., Vol. 9, 1938, pp. 205-

214.

73

74

13. C. C. Chang and I. K. Ebcioglu, "Thermoclastic Dehavior of

a Simply Supported Sandwich Panel Under Large Temperature

Gradient and Edge Compiession," J. Aero. Sci., Vol. 28, No. 6,

June 1961, pp. 480-492.

14. I. S. Sokolnikoff, "Mathematical Theory of Elasticity," 2nd ed.,

McGraw-Hill, New York, 1956.

15. Y. C. Fung, "Foundations of Solid Mechanics," Prentice-Iall,

Englewood Cliffs, New Jersey, 1965.

16. A. J. McComnnell, "Applications of Tensor Analysis," Dover

Publications, Inc., New York, 1957.

17. S. Timoshenko, "Strength of Materials," Part II, D. van Nostrand,

Princeton, New Jersey, 1956.

18. F. B. Hildebrand, "Advanced Calculus for Applications," Prentice-

Hall, Englewood Cliffs, New Jersey, 1962.

19. L. R. Ford, "Differential Equations," 2nd ed. McGraw-Hill,

New York, 1955.

BIOGRAPHICAL SKETCH

Amelio John Amato was born in Newark, New Jersey, on

January 20, 1914. He was graduated from Seton Hall Preparatory

School in June, 1962. In June, 1966, he received the degree of

Bachelor of Science in Mechanical Engineering from Newark College

of Engineering (New Jersey).

In September of the same year he entered the Department

of Engineering Science and Mechanics at the University of Florida

as a National Defense and Education Act, Title IV Fellow. Here,

in August, 1967, he received the degree of Master of Science in

Engineering prior to pursuing the degree of Doctor of Philosophy.

This dissertation was prepared undrc the direction of the

chairman of the candidate's supervis-oi / croniuitt ec aniijd har' becin

approved by all members of that committee. It a.'.s stul..mii tdcd to h11e

Dean of the College of Engineering aind to the Crnduate Council, rind

was approved as partial fulfillment of the rCequireinieits for the degree

of Doctor of Philosophy.

June, 1970

Pean, Coll gcz of En gicieering

DC-Mn, Gr1.Ddu1itC Sc-1ool

Supervisory Committee:

/W-
^\A '\^ V:Y~~

k //w^