"■^ ^

r

COMPOSITE DESIGN SYNTHESIS

BATTELLE COLUMBUS LABORATORIES

PREPARED FOR

OFFICE OF NAVAL RESEARCH

ADVANCED RESEARCH PROJECTS AGENCY

JUNE 1974

DISTRIBUTED BY:

AD-780 425

KHr National Technical Information Service U. S. DEPARTMENT OF COMMERCE

—————

• ■ M>i

^"T1 JW

Und ass i f ird SECUHITY CLASSIFICATION Of-' THIS PAC.t Clthiri DIIIII /irli-fi-.-fJ /^z) z?0 f^:

REPORT DOCUMENTATION PAGE I. REPORT NUMÜf-R 2. GOVT ACCLSSION NO

RF.AD INSTRUCTIONS BEFORE COMPLETING FORM

3 RECIPItNT'S C AT ALOG NUMBLH

4. TITLE fand Suhtlllr)

Composite Design Synthesis

7. AUTHORf.v>

Milton Vagins

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Bat'.ielle's Columbus Laboratories 505 King Avenue

Columbus, Ohio 43201

5 TYPE OF REPORT A PERIOD COVERED

Final Techni:al Report, 1 year

6 PERFORMING ORC. REPORT NUMbER

6. CONTRACT OR GRANT NUMBER^)

N0001/4-72-C-0195

II. CONTROLLING OFFICE NAME AND ADDRESS

Office of Naval Research 800 North Quincy Street Arlington, Virginia 22217

T« MONITORING AGENCY NAME * ADDRESSfi/ di»efpm from Conlrolllnt Ollice)

Advanced Research Projects Agency 1400 Wilson Boulevard Arlington, Virginia 22209

10. PROGRAM ELEMENT. PROJECT, TASK AREA & WORK UNIT NUMBERS

12. REPORT DATE

June 14, 1974 13. NUMBER OF PAGES

80 15. SECURITY CLASS, (ol Ihlt report)

Unclass ified

15«. DECLASSIF1CATION DOWNGRADING SCHEDULE

N/A 16. DISTRIBUTION ST AT EMEN T Co/ fh/> Rfpof(>

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abilract enltied In Block 10, II dlllerrnl Iron Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on re"ef«e «id« 11 neceefry mnd Identify by block number)

Composite Design Synthesis Rotating Disks Pressurized Thick-wall Cylinders

NATIONAL TECHNICAL INrORMATlON SERVICE

U 8 Department of Commerce Springfield VA 22151

20. ABSTRACT (Continue on reverie »id« (f noceseery end Identity by block number)

The research carried out deals with the problem of design synthesis in heterogeneous elasticity. Design synthesis is defined as the achievement of a des red design criterion, i.e., stress distribution, strength-to-weight ratio, etc., by preselecting a stress or displacement pattern in a stretched plate and then determining the variation of the elastic moduli that is required to permit the desired effects. This accomplishment requires the solution of the governing equations of %o

DD FORM

I JAN 73 1473 EDITION OF I NOV 65 IS OBSOLETE

I. Unclass ified

' \

SECURITY CLASSIFICATION OF THIS PAGE! (Wlim Pitta t'ntrrrft)

j -* —^

I .

I

I 1

I .

I

I 1

Best Available

Copy

■

^ ^m 7

Unclassified SECURITY CLASSIFICATION OF THIS PAGEfHTirti HnCa Entered)

elasticity, particularly the compatibility equation, in terms of preselected stress fields in the body of the plate for unknown material \. -operties which are spatial functions.

During this work, solutions to the moduli variation problem for annular disks have been achieved for two stress criteria: constant hoop stress, and constant in-plane shear stress. The disk may be rotating and have boundary traction. A computer program, D0M0V1, was developed to carry out these solutions. Attempts to solve the moduli variation problem for a hole in an infinite plate (two-dimensional), subject to certain stress distributions, were unsuccessful. However, great insight was gained into this problem for future work.

Basically, this work has shown that the concept of design synthesis, as defined here, is a workable discipline and,in the case of rotating annular disks and pressurized thick-wall cylinders, can be applied utilizing the present state-of-the-art fabrication technology, but that its application to complex problems requires additional work.

IL

Unclassified SFClimTY CLASMTIC ATIOM OF THIS PAGF'Whon IlKIn hnli-tr,lt

M^

■^r ^

!

FINAL TECHNICAL REPORT

COMPOSITE DESIGN SYNTHESIS

[ to

i

i

OFFICE OF NAVAL RESEARCH DEPARTMENT OF THE NAVY

June 1A, 1974

Milton Vagins

D D C prapmni?

UüisisEüins c

BATTELLE Columbus Laboratories

505 King Avenue Columbus, Ohio 43201

X r

X

Battelle is not engaged in research for advertising, sales promotion, or publicity purposes, and this report may not be produced in full or in part for such purposes.

III.

I>13f ■■ , ,

M^

^r ^M T

June 14, 1974

llBäireiie Columbus Laboratories 505 King Avenue Columbus, Ohio 43201 Telephone (614) 299-3151 Telex 24-5454

i

i:

I]

II fl II 0 n r r o 0 ö

Director Advanced Research Projects

Agency 1400 Wilson Boulevard Arlington, Virginia 22209

Attention Program Management DODAAD Code HX 1241

Dear Sir:

Form Approved Budget Bureau No. 22-R0293 Contract No. N00014-72-C0195

Enclosed is the Final Technical Report on the cc "-ract titled "Composite Design Synthesis".

Sincerely,

Milton Vagins Principal Investigator Applied Solid Mechanics Section

MV:Jlg

Enclosures (3)

iV

T "H-

Final Technical Report

on

Composite Design Tynthesis

14 June 1972

I I r i

Q

I) I

1

AR PA Order Number

1977

ProKram Cod« 1 Number

2N10

Contract Number

N00014-72. •C-0195

Scientific Officer

Director Structural Mechanics, Mathematical and Information Science Division

Office of Naval Research Department of the Navy

Name of Contractor

Batteile Memorial Institute Columbus Laboratories

505 King Avenue Columbus, Ohio 43201

Principal Investigator

Milton Vagins Phone (614) 299-3151 Ext. 1659

Effective Date of Contract

30 December 1971

Contract Expiration Date

28 February 1974

Amount of Contract

$39,400.00

Sponsored by Advanced Research Projects Agency

ARPA Order No. 1977 dated 18 October 1971

The views and conclusions contained in this document are those of the author and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the Advanced Research Projects Agency or the U. S. government.

v.

—

■^7 ^*

TABLE OF CONTENTS

i

II

11 li

IS I r

ii I

Pa&e

REPORT SUMMARY i

SYMBOLS ii

INTRODUCTION 1

GENERAL DISCUSSION 4

APPLICATIONS 12

Rotationally Symmetric Problems 12

Non-Syinmetric Problems 35

Limits and Other Considerations 42

Two-Diinenslonal Boundary Considerationr ^';

Material Considerations 48

SUMMARY 49

REFERENCES 50

APPENDIX A

INPUT DATA FOR PROGRAM D0M0V1 A-l

APPENDIX B

LISTING OF PROGRAM DOMOVi B-l

APPENDIX C

OUTPUT OF PROGRAM DOMOVI C-l

LIST OF FIGURES

Figure 1. Pressurized Annular Disk 13

Figure 2. Modulus Variation for Orthotropic Disk, with Pressur- ized I.D., with aa ■ Constant 18

Figure 3. Modulus Variation for Isotropie Disk with Pressurised I.D., with aQ = Constant 19

Ml

x

■ i * Ifr i^

^

LIST OF FIGURES (Continued)

Pag(

i i

II n ; i

il 11 (1 D H II II I

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12,

Figure 13

Figure 14

Modulus Variation for Orthotropic Disk with Presour- ized O.D., with 0o = Constant 21

0 Modulus Variation for Isotropie Disk with Pressur- ized O.D., with o ■ Constant 22

Rotating Annular Disk 23

Modulus Variation for Rotating Orthotropic Disk with no Edge Loads, with o - Constant 29

Modulus Variation for Rotating Orthotropic Disk, Ortho- tropic Ratio of 2.5 with no Edge Loads and a0 = Constant 30

Modulus Variation for Rotating Orthotropic Disk with no Edge -oads, and with Constant In-plane Shear Stress . . 31

Modulus Variation for Rotating Orthotropic Disk Ortho- tropic Ratio of 2.5 with no Edge Loads and with Constant In-plane Shear Stress 32

Modulus Variation for Orthotropic Rotating Disk Stressed on O.D. RPM - 2000; Density =0.1 lb/in.3; Stress on O.D. ■ 10,000 Psi; O.D. ■ 60.0 inches; I.D. ■ 6.0 inches, aQ ■ Constant

Mo ulus Variation for Orthotropic Rotating Disk Stressed on O.D.: RPM - 4000; Density - 0.1 lb/in.3; Stress on O.D. - 12,000 Psi; O.D. - 60.0 inches; I.D. ■ 6.0 inches, a« - Constant 34

Small Hola in Infinite Plate Subjected to Uniform, Uniaxial Tensile Stress Field 36

, Two-Dimensional Finite-Difference Model 40

M

VII.

y*

^

\

r 11 !:

II II II 0 II n ö

fi i

REPORT SUMMARY

This is the Final Technical Report describing the work done and

the accomplishments of the research effort for ONR-ARPA titled "Composite

Design Synthesis".

The research carried out deals with the problem of design syn-

thesis in heterogeneous elasticity. Design synthesis is defined as the

achievement of a desired design criterion, i.e., stress distribution,

strength-to-weight ratio, etc., by preselecting a stress or displacement

pattern in a stretched plate and then determining the variation of the

elastic moduli that is required to permit the desired effects. This accom-

plis'iment requires the solution of the governing equations of elasticity,

particularly the compatibility equation, in terms of preselected stress

fields in the body of the plate for unknown material properties which are

spatial functions.

During this work, solutions to the moduli variation problem for

annular disks have been achieved for two stress criteria; constant hoop

stress; and constant in-plane shear stress. The disk may be rotating and

have boundary traction. A computer program, D0M0V1, was developed to carry

out these solutions. Attempts to solve the moduli variation problem for

a hole in an infinite plate (two dimensional), subject to certain stress

distributions wire unsuccessful. However, great insight was gained into

this problem for future work.

Basically, this work has shown that the concept of design syn-

thesis, as defined here. Is a workable discipline and in the case of

rotating annular disks and pressurized thick--'all cylinders, can be

applied utilizing the present state-of-the-art fabrication technology,

but that its application to complex problems requires additional work.

VJIII

\

Tq—

SYMLOLS

Symbol Quantity

a,b inside and outside radius

a. material coefficients

il

[1

c dimensionless shear-modulus coefficient ! ='^7,)

A .B »G .C,.C„ constants of integration o' o' o' 1' 2

orthotropic ratio [-p) e

E. modulus of elasticity corresponding to the subscripted direction (i • r,9)

';|] i A f1(e),f2(9) functions of 9 only

G modulus of rigidity ( = G Q = 1 00

g acceleration of gravity

k radius ratio ( =b/a)

\ i-t k. orthotropic ratio ( =e)

k- Poisson's ratio in tangential direction ( svfl )

n cos 0

n sin 0

P internal pressure

q external pressure

r radius

T temperature difference function

u,v displacements

R,X,Y body forces

U specific strain energy

U total strain energy

•■■;• II

^

I n

(I i

I'

l ii

U Q 0 II 0 e

i

Symbol

or

0

Y

e

I

9

\

v

Y

U)

6

n

Quantity

coefficient of linear thermal expansion

exponent (2-k,/k«)

material density

strain component wich one or two subscripts

body force in tangential direction

angular position

exponent

Poisson's ratio in tangential direction ( - vQ ) ü r

exponent ( H - [(J , ^ ]) -1C1 ■ K^ ^>

X/J ) J\r\m

dimensionless position ratio (=r/b)

coordinate system offset angle

stress function

rotational velocity

variation factor

arbitrary function

*.

J 11

■^r wm mm^^i ^

K. - — —-

I

in In

(

n fi n

11

i.

INTRODUCTION

It has long been recognized that structural ele.nents composed of

composite materials, such as glass, boron, carbon, or other filaments, embedded

in a suitable matrix, such as epoxy or polyester, offer outstanding strength-

to-weight ratios. The potential of such materials is considered so great that

material scientists and enginaers believe that they will form the bulk of the

structural materials of the future. Though the application of such materials

has been a growing part of the state of the art for structural components,

particularly in the aerospace industry, thi translation of the concept of

fibrous composites into a primary load carrying structure has been and remains

a challenging process.

The present and growing u^e cf structural elements fabricated from

composite materials creates the need for the development of a rational analytic

design basis, which to a great extent is presently non-existent. This is not

to be construed as meaning that little or no rtcearch on composite materials

has been carried out. On the contrary, a large amount of literature has been

generated dealing with both the determination of the mechanical properties of

these materials and the analysis of specific structures fabricated from them.

In general, from the microscopic viewpoint, research on the mechani-

cal properties of composites has dealt with the determination of such properties

for materials having given component elements ordered in fixed spatial

relationships. The spatial relation in these cases might have a high degree

of symmetry, as in long or continuous filament composites, or a completely

random or homogeneously disordered array as usually employed in short carbon

or boron fiber composites. Such research has been directed towards the

creation of analytic or experimental methods of determining the mechanical

properties of composite structures in terms of the known properties of the

composites' components, characteristic of »"h^ approach are the works of

Sayers and Hanley [1] , Chen and Cheng [2], Hill [3], and Gaonkar [4], among

others.

In the analysis of structures composed of such composites, the

material has generally been treated as exhibiting gross, homogeneous.

.—- '

Numbers in brackets are references found at the end of this report.

' \ mm

j _-* U li ^

I

(I Isotropie or anisotropic mechanical properties. These gross properties, when

entered into the constitutive equations defining the material, have allowed

analyses of such st uctures through the classical methods of the theory of 1

elasticity, plates and shells, vibration, and others. Along these lines, the \

concept of the "unidirectior.il lamina" was introduced and utilized as the l

"fundamental unit of material" in design and analysis. This procedure I

employs test data obtained from a unidirectional lamina as the basis for the

design of laminated components and structures. Exemplary work done along

these lines has been carried out by Dong [5], Tsai [6,71, Tsai and Azzi [8],

Whitney [9], and Whitney and Leissa [10,11], also among others.

All of these analyses have dealt with materials and structures that

have predetermined mechanical and behavioral characteristics. That is, once

the geometric array and the constituents of the composite are prescribed,

then the mechanical properties of the material and the response characteristics

of a structure fabricated from such a material have been inherently established.

An;lysis will merely determine what these properties and response charac-

Ceristics are.

When dealing with composite materials, the analytical procedures

discussed above appear to be highly inefficient in many applications. The

designd. of fibrous composite structures is presented with numerous degrees

of freedom and an opportunity to exercise ingenuity totally unavailable to him

with conventional materials. Composites, whether filanentary, fibrous, or

sintered or fused metallics, are capable of being tailored to meet specific

requirements. When considering specific structural applications for such

materials, it would be logical to assume that a structure could be optimized,

depending, of course, upon the optimization criterion, by varying the mechani-

cal properties of the material throughout the structure. Further it would be

logical to bypass analysis completely and define this now nonhomogeneous

structure by some means of design synthesis. Admittedly, the creation of a

design synthesis procedure to adequately handle most problems in structural

design is quite difficult. However, the concept of design synthesis tc deter-

mine the variation of the mechanical properties of a material within a structure

so as to achieve a desired stress or deformation pattern ir. that stvueture is n

one capable of being developed.

1

I!

(1

(i \

—

.^L-

"^^ «^n

^

I

I i

The work described herein deals with the first pi.ase of an effort

to develop a design synthesis methodology for composites. It is directed

specifically to the problem of heterogeneous plane elasticity.

(1 i II II

l\ ' • M«

. "^v. 7

GENERAL DISCUSSION

Consider the following question:

Given a plane elastic body with known boundary tractions and/or

displacemtnts, can the mechanica) properties of the continuum be described

such that an "arbitrary" stress distribution within the body is met?

The term "arbitrary" is to be understood as defining a family of

stress distributions that are preselected but still conform to equilibrium

requirements and boundary conditions. To answer this question we start by

making the following two basic assumptions:

(1) the classical equations of linear elasticity are valid in this

application, and

(2) the mechanical properties of the continuum can be expressed as

s atial functions.

Following from these assumptions the well known governing relations of gener-

alized plane stress, given in rectangular coordinates are as follows:

Equilibrium Equations

äa äcr xy ox T by + X

da Ba ~L + r-2 + Y »= 0 ,

öx öy '

Strain-Displacement Equations

(v/

II

du öv x dx ' y dy

dv . du xy dx dy *

(2)

li

Compatibility Equation

^2 .2 .2 de de de

^ 2 ^ . 2 dxdy dy dx

(3)

■^ *■ • ■ L"

^

Assuming that the continuum exhibits orthotronic material proper-

ties and neglecting time and strain rate effects, Lhe constitutive equations

can be expressed as generalized Hooke's Law as

e X 'all a12 0 ' 'V V

e y

■ a21 a22 0 ay

+ < a;T

c 0 0 a66 4 .^

. o J (4)

f]

where a^ and a?2 are the coefficients of thermal expansion and T represents

the temperature difference distribution. The relations defined by Equations

(4) are valid when the axes of the material properties of the continuum are

coincident with the axes selected for the differential equations of the

problem. If the axes are not coincident, except for the z-axes, and the

other two axes of the material properties are rotated about the z-axis through

some angle, «(, in relation to the geometric axes, then more complicated

relations between the stresses, temperature and strains are developed.

Lekhnitskii [12] pr.sents these relationships in some detail. For th^ gener-

alized plane stress case in point the material coefficients are related to the

two axis system by

I

1

[!

II v

- —

•u •'u "'21

=

''22

■v

4 2 2 2 2 4 .2 2" m mn mn n 4Lin

m2„2 J* • 11,11 m n m n m n -4m n

m2n2 J* 4 2 2.22 n> n n m m n -4m n

2 2 2 2 4 ,22 mn tn n m 4m n

22 22 22 22 ,22 m n -ran -ran ran (m -n )

'11

'12

'21

l22

'66

(5)

where ra • cos rf, and n ■ sin i.

No generality will be lost by continuing with the constitutive

equation as given by Equations (4). Substituting Equations (4) into (3),

«- ^v *m im "

assuming that the a,,'s are spatially dependent and carrying out the required

differentiation yields:

{a12

^2 .2 .2 ,2 .2 0 C" _5si 4

&g« , Lfs a ill 66 dxdy all . 2 + an .2 a22 . + a.

dy

+ ^ (flfjl) + ^-J (o2T)} öy

öy

äx2

öx öx

{■ ä2a 11

9 + öy 2W

X

2 2 2 9 a12 ^ a21 ö a22

ay dx öx

2 2 ö a66 v , ^ all ÖCJx . , a +2 —: • t r 2 ^12 . day , 2

da21 OCT.

öxöy xy öy öy öy öy dx öx

öa2? öa^ aa66 *,„ **u ^xvl n

öx öx öx öy öy öx J (6)

Assume that the body forces have a potential, V, such that

I!

i; ii

öx

öy '

and choose Airy's stress function, Y, in the form

ay

(7)

.^4.V

y ax2 (8)

2 ö Y

CTxy " öxöy

^^

^

Making the appropriate substitutions into Equation (6) yields

( ox äx dy ay

Ä . ._ . . d2v . Ö2

+ (all + al2> ft + <a22 + ^l) 2 + H ^1T) + S ^2T>} äy dx dy öx J

I ^o a,, d a K ,i a^ + b2a ^ ^

öy~ dx' öy' N dy2 dx2 ' dx2

9 a,, ^2.,. ^2

dx ÖXÖ

2 2 2

f ' ti; + 71 (a22 + «ll) ' V + S (all + a12) * V 2

+ S; (2a12 - a66) S3Y

öx öy + fc <2a21 •w>

)3Y öxay2 ay ^3

JX % ax My^u^^'-^^^a^^-i}-0 (9)

D 0 I i i! I \

and, of course, the equilibrium equations are met exactly.

Notice that in Equations (4), (5), and (9), a12 has not been

equated to a21. Norm' lly, under the limit of small displacement theory

dealing with linearly elastic materials which are conservative, the material

coefficient matrix, defined in Equation (4), wou'J be symmetric and a.- would

equal a^. However, some recent work by Bert and ^uess [13], among others,

shows that there exists experimentally derived data which indicate that for

some types of composite materials exhibiting orthotropic properties the

material coefficient matrix is not symmetric and a is not equal to a. .

To limit the growing complexity of this work, the material coefficient matrix

will be taken as symmetric, at least for the initial phase of this effort.

Equation (9) can also be expressed in polar coordinates as

follows:

(Note: Here, the a.., are in polar coordinates and are

not equal to the a.., for rectangular coordinates.) lj 8

—

■ »-P T ■^ ^■p

^

I

i

I

!

II

(fa ä\ + 2 121 äi _ ^1 A . il 8! . au aS o« är r dr r "' r a«

i) . >> / . 2,.2 ar as'

,2.u ♦ 2ai2 4. ^

^ r^ -ae

2a

^]

+ fa ä-r + I 21 : allNi a- 4. fU a2 1 A, •! or ,2

r 09'

a,.. .2

1 dr Lr ar2 r2 arae2 " r3 ae2 ar "~^~J

09 Lr2 ar2ae r3 öe J ör L2 ^ + r dr2 + 2 äT + "f J

r Lr3ae2 r2*r *■> 09 I 4,fl3 + r3ä7ä? + -2ärJ

aa "a^

aa "a MIX dfi Ip a3Y 1 . ^66

r Lr3 ae2 ' r2 arae2j ^ 1 Ä i- &. * J^ a3Y 1

.r4 ae ■ r3 arae + j ZXH r ar ae-

ar2 Lr2ae2 r ör rJ ae2 U2 ar2 r ^ :r- V eft

2 <.2a 22 a2a

ar "-ar'

a2.

Lar2 9j ae2 '-r^fl2 sar^^J

arae [2 arae " 3 aeJ J " 0 (10)

———— ■

^: r*m ^mm

T?

with / being the stress function defined by

^r r 8r 2 ^rt2 + vr

r dB

'• ■3+T« (ID

CTre or Vr 09/ '

Equations (11) satisfy the equilibrLura equations formulated in polar coor-

dinates for plane stress which are

or r 99 r R = 0

!Zr9 + l^+2^i + 0=o

(12)

In order for Equations (11) to meet Equation (12) exactly the body functions

V and V must be defined as r 9

fl \ ' h (rVr) " R

99 U '

(13)

thus putting a rather narrow interpretation on the body forces.

In the classical approach to the problem of orthotropic plane

elasticity, the coefficients a., are either constants, as in the case of the

homogeneous condition, or as in some rare cases, are special functions of

position. In the first case all the terms within the second set of large

braces, 1 |, in Equations (9) and (10) are zero leaving the remainder of

these two equations in the form of the well-known, homogeneous, orthotropic.

■^7 ***M 7

10

I I I !

i !

I

compatibility equations in terms of the stress function Y. In the second

case all the partial derivatives of the a, 's which appear in these second

sets of braces are capable of being evaluated, resulting in extremely compli-

cated fourth order partial differential equations wirh variable coefficients.

Proceeding along classical lii.es, these equations must be solved for Y which

c^ncains arbitrary constants of integration. These constants are then deter-

mined by evaluating Y in terms of the stresses on the boundaries.

Suppose it is assumed that the material coefficients, the a./s. ij •

aie unknown but that the stress function Y is a fully defined function of the

spatial coordinates. That is, the stresses throughout the body as well as on

the boundary are known. In such a case. Equations (9) and (10) reduce to

second order partial differential equations with variable coefficients, in

Leras of the a 's. The solution of these equations and the resultant deter-

mination of the magnitude and distribution of the material properties throughout

the body is defined in this worV as design synthesis.

It is believed that Equations (9) and (10) have not been previously

published. Bert [14] derived an equation similar to (10) wherein he reduced

the unknown material property coefficients from four to one. His formulation

is as follows:

c r A .l^Y e_iVp_^I+ 2(c-v) A U,.4 r . 3 2 . 2 ^ 3 or 2 „ 2iQ2 or ar r ar r r or 89

. 2(c-v) B3Y + 2(c-v+e) ^Y + e_ A 1

r3 örö92 r4 M2 r4 Ö94 J

3 SJY + dSr ^1 2^Ö?I e dY 2(c-v)

dr L ,. 3 r .2 " 2 dr 2 ^2 or or r r ^rdb r

.2„ . : 2(c-vHe A 1 4, A P-^- £ dl &. äV]

r3 ö92j dr2 ^r2 " r ör " r2 d92J

LV .2 " r ör " 2 ZJ '^V or r oo

^S^^-^blK)-- 2 ^

09 ^ (14)

m^

■^r •^ i

ii

ji

ii

v,ith the thermal terms oiritted. Equation (14) Is equal to Equation (10)

with the following identities:

S = a

ii

I

!

i

(15)

22

e - a11/a22

c ^ a66/2a22

vH " a12/a22

with S being the dependent variable and e, c, and v being fixed ratios. It

is anticipated that most real composite materials will exhibit material

property characteristics as defined by Equations (15).

The solution of equations such as (9) and (10) where there are 4

or more independent a.,'s, even where these a..'s are assumed independent of

temperature, is quite difficult but certainly not impossible. The simpli-

fying assumption made in Equations (15) leading to the formulation of Equation

(14) reduces the problem to only one unknown parameter.

J ""«i imiM UTi

T: ^* ^m T 12

APPLICATIONS

Rotattonally Symmetric rroblems

Example 1: The pressurized annular disk, internal pressure: Consider a

pressurized annular disk as shown in Figure 1. In the case where the ring

is Isotropie and homogeneous, the stress distribution is as developed by

Lame (1852) and is given by (c.f., Ref. [I5l, p. 60),

a *- r (VO 1 p

k - 1 p

(16)

where p » r/b, and k ■ b/a. These relations lead to the following con-

clusions:

(1) lcJ>!cr | for all p and all k ö r

(2) (aQ) occurs at the inner boundary (p ^ 1/k), and thus

I (Vmax P(4^)>P \- - v (17)

From the stresses so generated, which for the homogeneous ring, are quite

Independent of the material properties, it is clear that the material is not

being used efficiently, particularly as the thickness ratio, k, increases.

For the homogeneous, orthotroplc annular disk Bienick et al. [16]

shows that the stresses are given by

-(f+1) , f-1 ar " cl c2r

79 * -(f+lK r f-1 -c.fr v +c0fr

(18)

1/2 where f ■ (aii/a22^ » an^ ci an^ c2 are determined from the boundary con-

ditions, in this case CT ■ -P at r ■ a and j ■ 0 at r - b. This work and

X ♦ \

\

J ^-^ ^^

^

13

I

II

I

ii

FIGURE i. PRESSURIZED ANNULAR DISK

-,„'

A^

_ ^T ■ v r_n—:—. . J *■ "N

I

ll! II

!

I Li '\.

T^-

14

vork done by Shaffer [17] show that as k gets large the limit of the stress

concentration approaches the orthotropy ratio, f, avid does so rapidly. Both

Bienlck and Shaffer deal with the nonhomogeneous case, but both assume a

special form of nonhomogeneity, mainly that

all " •ll ^

a„ - a90 r *22 *i22

a12 " a21 ' a12 r

where \ is ^al and a.., a22, and a^2 are constants. Both carry out optimi-

zation by varying \ and observing the results, but no attempt was made ^o

carry out design synthesis directly.

Consider now that the disk is composed of a heterogeneous, ortho-

tropic material where the material properties are undefined but are functions

of the radius of the disk. The equilibrium equation for this case reduces to

dT + ^T"1-0- (W)

The compatibility equation reduces to

lr ("e5 " er " 0 • (20)

and ehe stress strain relation for an orthotropic medium is

er " ^r + aI2CTe

ee " a12CTr + a22CTe

(21)

where a^^, aj«, and a22 are functions of r. Assuming a stress function

such that

■

■ '

^r -^ ^^^^i ^

^

15

'.-i' (22)

and substituting Equations (21) and (22) into Equation (20) and carrying out

the required differentiation results in

•22f,, + C,

22*i*22>f,4'<J*,l2-i2*ll>,r "0. r (23)

where the prime marks designate absolute derivatives with respect to r.

Equation (23) is a single differential equation with three independent

material coefficients as the operational parameters. When dealing with specific

materials they may be found to be completely independent or show some type of

defined relationship. As a first step we will assume that there does exist a

simple ratio relationship between them that is expressable as

a99 * a99 » a 99 B *' '22 22 22 22

a11 «^22, a'11 ■ V'lJ (24)

a12 " k2a22' a,12 ' k2al22

Making the appropriate substitutions in Equation (23) yields

fl [V +7^ Y] a'22 + [Y" +7Y, - -|Y] a22 - 0 . (25)

f

Thus if Y is known a22 is fully defined. Let us suppose that for effective

material utilization it is desired that a be constant throughout the

disk; i.e., a. " A0. Integrating the second of Equations (22) gives

Y - A r + B . o o (26)

Applying the boundary conditions that a (a) « - P and a (b) = 0 yields

J ^4 ■ It ^

^r 9m V 16

Pb S^T tp - n

P rl ,, (27)

k - 1

with p ■ r/b and k ■ b/a as before. It is interesting to compare Equations

(27) with Equations (16) and (17). From Equations (27) it can be seen that

if k is greater than 2, a becomes less than the pressure P, and a , which

equals P at p ■= 1/k, becomes the maximum normal stress in absolute value.

Thus for such a stress function and geometry there is no effective stress

concentration. This is, of course, never true for the homogeneous disk.

With the stress function now fully defined. Equation (25) becomes

[(l+k2) - ~|-] ,22 + [—r^Hr]* 22 (28)

which has the solution

II

0 r (i

ii ii

l22 •V»>MI/*a-v] (29)

where p ■ r/b

I - - kj/k,

k -k S L(l-k2)k2J •

and for the modified orthotropic condition as defined by Equations (21) and

(24)

a22 B a22 " EQ 9

aii h k, ■ c „ 1 a22 Er

(3o)

k .J» 2 "^ " ver ' k .

■ — ——

* -» ~^ rta^

^r^ . . jm ^&~*m Tf—

17

i i:

The inverse of Equation (29) or Ee(r) is shown plotted in Figure 2

for various k's while k- was held constant at 0.5. This figure shows the

strong dependency of the modulus distribution upon the orthotropy ratio k..

The ratio k» has a lesser effect as shown by Figure 3. Here the isotropic

case (k. ■ 1) is shown plotted for five values of k2.

Example 2. Pressurized annular disk, external pressure: Consider the

pressurized annular disk as shown in Figure 1 but with the pressure acting on

the OD rather than the ID as shown. If the stress criterion is retained; i.e.,

aft ■ constant, then the stress function becomes

k ■j [1 - pk]

9 . -a- [i . k] r k-1 p

(31)

I '• k-l

I

where q ■ external pressure

k •= b/a

p ■ r/b .

The compatibility equation (25) becomes

k„a (1-kJ k.a. [(i+k2) .-|-]a.22 + [-rL + -±-] a22 =0 (32)

Equation (32) is the same as Equation (28) except that a replaces b. The

solution of Equation (32) is

II l22 -V')M[ä*<l-vJ< (33)

where 0, §, and p are defined in Equation (29). Comparing Equations (33)

and (29) we see that they are not the same, differing by the quantity 1/k

■ ■ Mt riMM

-X — •^m ^^

18

i:

i'

!

!

I

1 !

i tl

1 .0

0.9

0.P

0. 7

0.6

tu V

0.««

0.3

0.?

0. 1

K, =Variable, E = Eg

K2=0.5

o. o l

o.o

K, = 0.01

o. ? o.«« o. f.

flÜOULUS V.S. BA01US

0. 8 1 .0

FIGURE 2. MODULUS VARIATION FOR ORTHOTROPIC DISK WITH PRESSURIZED I.D., WITH a- - Constant

0

■ ■ -^ MM

"^7 *9 w mm 7 19

II

r

r

I,

■

f (1 I

E

/ E n A

x

I .0

0. 1

o. e

0.7-

0 . 6

0 .5

0 .•*

0 . 3

0 . 2

0 . I

K, = 1.0 (Isotropie Cose) E = EQ= E r

Kg5 ü= Variable

o.o

K2=0.5

K2=0.4

K2=0.3

K2=0.2

K2 = 0.l

o. o o.? o.H 0.6

MODULUS V.S. RADIUS

O.P I . 0

FIGURE 3. MODULUS VARIATION FOR ISOTROPIC DISK WITH PRESSURIZED I.D., WITH ae - Constant

"^ - - i i ■>*

■^7 m T?

X

20

!

within the second factor. Thus the distribution of the material parameters

through the thickness of the disk must be different as compared to the

internally pressurized disk. This is shown in Figures 4 and 5. Also notice

that for the internally pressurized disk the hoop stress aQ tends to zero as

k tends to infinity while the limit on CTQ for the externally pressurized disk

is q, the pressur?. Thus for the externally pressurized case the material is

not being as effectively utilized and perhaps some other criterion might more suitably apply.

If the annulus has both Internal and external pressure and the same

stress criterion is applied, i.e., C7Q « constant, then

^ ^ kTT [p " qk] »

(34)

and the material property variation is given by

*•*.<*"&* aM&l (35)

I I I 0 II

Example 3: The rotating disk: Consider the rotating, uniform thickness

annular disk as shown in Figure 6. The equilibrium equation governing this case is

fc (rt7r) " CT9 + g ^ - 0 ' (36)

The compatibility equation in terms of strain and the stress-strain relations

for an orthotropic material are given by Equations (20) and (21), respectively.

Substituting Equation (21) into (20) results in

0-a,12CTr + a12CT'r + a,22<7e + a22CT,e + r [(a12-all>CTr + <a22-a12)(V' (37)

i^te

^N Pi l^-ü w w~- *.——

21

!

r

n

i i

i

o

ii

/ E n A

X

1 .0

0.9

o.e

0.7-

0 . 6

0.5-

0.««

0.3-

0.2 -

0. I -

0.0

0.0 0.1 0.6

nODULUS V.S. RADIUS

I . 0

FIGURE 4. MODULUS VARIATION FOR ORTHOTROPIC DISK WITH PRESSURIZED O.D., WITH 0o ■ Constant

«U

-

"T ^m ■ m •Pi 7 22

r

i

I

I

E

/ E n A X

I . 0

0.9-

o. e -

o . r

o . 6

0.5-

o. H

0 . 3

0 . 2

0. I -

0 . 0 0 . 0 0.2 OH 0.6

nnouius vs. RADIUS

0 . * I . 0

FIGURE 5. MODULUS VARIATION FOR ISOTROPIC DISK WITH PRESSURIZED 0. D., WITH aa - CONSTANT

n

"^7 —-^

0

in 1

/ n

i

i

23

■J-Jl ZÄ l' -

Boundary Stress

FIGURE 6. ROTATING ANNULAR DISK

11 J

)

\

^.^^L. ^^

II M ^7 . . 9m p—

^

I 24

Assume the stress function

i.

f + — CD r

(38)

g

i

It is possible that the material density, y, could be a function of tie

radius. At this stage such a condition introduces an unnecessary complica-

tion and will not be considered. This being the case the governing

compatibility equation becomes

r Y 2 2 / , , a22 a12\ n (39)

Again, for simplicity, the modified orthotropy relations as defined by

Equations (24) will be adapted making Equation (39) take the form

0 - a'22[r+^ Y+^ü2r2] + a22[r'+f-"f+ ^a)2r(3-k2)] (40)

I

The stress criterion will be the same as before, i.e., aö B

constant. Using this criterion and operating on the second of Equations (38)

together with the assumed boundary conditions that a (a) = a (b) ■ 0 yields

Yv2 / k \ r, 1 11 2 ^ ^ -r-3rfc)L1-^-^ + ^3-p +kJ (41)

II fi

Yv '9 "It (Ä) M

.J l l «I*

^ —■-^

25

I i

0 I

where p • r/b

k » b/a

v B ujb * tip velocity

u) » rotational velocity, radians

g •« acceleration of gravity.

Here we note that in the limits, when k -» 1

w2

which is the stress in a rotating thin ring, and when k

very small)

Vv2

(42)

-> ■ (i.e., when a

(43)

which is smaller than exists for the Isotropie, homogeneous case by the

ratio

i 1

i [ II 0 II

u

where aQ • hoop stress for heterogeneous case öl

a« ■ hoop stress for Isotropie, homogeneous case o

v » Poisson's ratio for Isotropie, homogeneous ease .

Substituting Equation (41) into Equation (40) and carrying out the

required differentiation yields

where

, fAr 4- Br 4- D ~| 22 " I 4 , „ 2 . „ J

Fr + Gt + Hr- a22 -0 (45)

(kj - 3k2)

B - (1 - k.) rbi^ lJ L b

3-,

^^

^ .— > « ^

26

D - k^abHb+a)

F - k K3 3

H - k2(ab)(b+a)

!

i

1° 1

1

II

Equation (45) is not readily solvable in closed form. Before proceeding

to the solution of Equation (45) by some numerical means, it is convenient

to digress at this point to discuss another stress criterion which may be

applied to the implementation of design synthesis as defined in this

work. This criterion is that through the body of the disk, the in-plane

shear stress, T, is to be a constant. For a body of revolution, acted

upon by symmetric loads,

or

t - (ae-ar)/2

T - oQ-o,. - Constant ■ C or o (46)

Applying this concition to Equations (38) and integrating leads to

1 2 2

^ - C £nr - ^-^- + Cn r o 2g 1 , 2 2

4" - C £nr + C - 4 Yu r + C, o o 2 g 1

r R MU »

requiring that on the boundaries;

a (b) - o r ' o

a (a) - o, rv i

leads to the following relations.

(47)

A_

•^m

^

27

i f i 1

!'

i

I

i

I

0 I 1

If

i 2 2

a - ±y = C Inv - Y^ r r r o 2g + C,

2 2 yi + y"> r

g

2 2 Co<tnr + l) -■i|i^+C1

£n {<<'o-0l' * »Sift2-2)}

£n - a

!o.J!,nb - a Zna I i o

2

2g (b2Jina - a2£nb) (48)

These relations, as well as those defined by Equation (41) (a. = constant)

were applied to Equation (40) and were solved numerically. A digital

computer program, Determination Of Modulus Variation 1 (D0M0V1), was

structured for the solution of these sets of equations. A finite dif-

ference method of solution as developed by Manson[l8] was used for the

calculation algorithm. This program, which is detailed in the Appendices

was used to solve several problems as follows.

• The modulus variation for an orthotropic disk with

a pressurized I.D. for o - constant (disk not rotating)

• The modulus variation for an Isotropie disk with

pressurized I.D. for.o - constant (disk not rotating)

• The modulus variation for an Isotropie disk with

pressurized O.D. for o - constant (disk not rotating)

• The modulus variation for a rotating orthotropic disk

with no edge loads for OL ■ constant o

• The modulus variation ior a rotating orthotropic disk

(Poisson's ratio variable) with no edge loads for

a = constant I

• The modulus variation for a rotating orthotropic disk

with no edge load for constant in-plane shear stress

• The modulus variation for a rotating orthotropic disk

(Poisson's ratio variable) with no edge loads for

constant in-plane shear stress

.-

V

A.

» ■ ■■ p

^

I 28

The modulus variation for a rotating orthotropic

disc with 60.0 inch O.D., 6.0-inch I.D., turning

at 2,000 RPM, stressed on the O.D. with an

uniform edge load of 10,000 psi for o = constant

The modulus variation for a rotating orthotropic

disk with 60.0-inch O.D. 6.0-inch I.D., turning

at 4,000 RPM, stressed on the O.D. with a uniform

edge load of 12,000 psi for o«" constant.

r

i:

f

The solutions for the first four problems were checked by use of the closed

form solutions. Equations (29) and (33) and are those shown plotted in

Figure 2, 3, A, and 5. The solution to the remaining problems are shown

plotted in Figures 7 through 12. It should be noted here that when

boundary tractions as wel_ as body loads are applied to a rotating disk,

the solution is specific as regards the magnitude of the edge loads and

the rotational velocity of the disk. HowevPt, where only one type of load

is imposed, then the solution is generalized and is independent of the

magnitude of the load. (Note: The stresses remain directly dependent

on the load.) This solution dependence upon load is shown very clearly

by comparing Figures 11 and 12. Here the modulus variation is shown to

change with the change of the boundary tractions and the rotational

velocity. All these rigures have been non-dimensionalized by the expediency

of plotting E/E and R/B where r 0 max

E Ea, the modulus of elasticity in the 9 direction. 6

E ■ the maximum E„ calculated in the body of the disk, max 6

R ■ the radius of the point in the disk at which the

modulus is being calculated.

B ■ the outer radius of the disk.

The computer output for the curves plotted in Figure 12 is also found here

in Appendix C. For this case, o. equals 28,45A psi throughout the disk.

For an Isotropie, homogenious, flat-angular disk of the same material

■

^cr

ii

E

/ E «

I 0

O.*»

0.8-

0. T

0.6-

0 .5

O.t

0. 3

0. 2

0. I -

0.0 0. 0 0.2 0.I 0.6 0.^

• /• nnouL us v.5. RAOI us

1 . o

I \

FIGURE 7. MODULUS VARIATION FOR ROTATING ORTHOTROPIC DISK WITH EDGE LOADS, WITH o. - CONSTANT

B

NO

—————— ,

• ■ Mlii

30

II II r r r i

fi ii i fi J

E

/ E n A

x

I .0

0. V

0 . 8

0. 7

0.6

0 .5

0.««

0 . 3

0. 2

0 . 1

0.0 0. 0

K2=0.4

K2=0.5

K, = 2.5.E = EÖ

K2 = ü = Variable

-L.

0.2 0 . H 0.6

nnnuLus 9.%. RAOIus

o e i . o

FIGURE 8. MODULUS VARIATION FOR ROTATING ORTHOTROPIC DISK, ORTHOTROPIC RATIO OF 2.5 WITH NO EDGE LOADS AND oe - CONSTANT

•

\. 1

11

Mb

1

1

i

0 i I J

I . 0

0 . «»

0 . 8

0.7-

0 . 6

0 .5

0 .t

0. 3

0 . ?

0 . I

0 .0 0.0 0 . ? 0 . M 0.6

R/B nonuLUS vs. BADIUS

o . e i . o

FIGURE 9. MODULUS VARIATION FOR ROTATING ORTHOTROPIC DISK WITH NO EDGE LOADS, AND WITH CONSTANT IN-PLANE

SHEAR STRESS

I

I

D i 0

i

E

/ E n A X

I . 0

0 .9

0.8

0 . 7

0 . 6

0 .5

O.t -

0 . 3

0. 2

0 . I

0.0 0.0

-L -L

K, =2.5 E=Ee

K2-u= Variable

i

0.2 0.1 0.6 R/B

«OOULUS V.S. RADIUS

0 . P I . 0

FIGURE 10. MODULUS VARIATION FOR ROTATING ORTHOTROPIC DISK ORTHOTROPIC RATIO OF 2.5 WITH NO EDGE LOADS AND WITH CONSTANT IN-PLANE SHEAR STRESS

\

■ ■ -^

I .0

I

I

I

I

0 . 2 0.1 0.6

nnouLus v.s. RADIUS

0 . 8 » .0

FIGURE 11. MODULUS VARIATION FOR ORTHOTROPIC ROTATING»DISK STRESSED ON O.D. RPM - 2000; DENSITY «0.1 LB/IN. ; STRESS ON O.D. - 10,000 PSI: O.D. - 60.0 INCHES: I.D. = 6.0 INCHES, 0o - CONSTANT o

^JT

I I !

1 1 I

i

!

I

E

I E

1 .0

0.9

0.8

0.7-

0.6

0.5

O.t

0. 3

0.2

0. I

0.0 0.0 0.2 0.1 0.6

MODULUS V.S. RADIUS

0 . 8 1 .0

FIGURE 12. MOPULUS VARIATION FOR ORTHOTROPIC ROTATING DISK STRESSED ON O.D.: RPM - A000; DENSITY = 0.1 LB/IN.J; STRESS ON O.D. - 12,000 PSI; 0. D. - 60.0 INCHES; I.D. - 6.0 INCHES, oa - CONSTANT 8

^

35

I

I

!

1

density ofl would equal

b -a

ae - 35,810 + 24,2A2 = 60,052 psi.

Thus, by the use of design synthesis, the maximum stress in such a disk

can be reduced by a factor of 2.11. Further, Figure 12 shows that this

can be achieved with a modulus variation through the disk of approximately

2.5 to 1 for a material that exhibits and orthotropic ratio k, of 2.5

(the similarity of the two ratios is coincidental and bears no significance).

Non-Svtnnetric Problems

Example 4: Small hole in an infinite plate. Figure 13 represents a small

hole in an infinite plate which is subjected to a uniform tensile stress, P,

in the x-direction. For the homogeneous, Isotropie condition, the stress

distribution around the hole is well known as given by Timoshenko [15] as

r r r

2 A. ae -li1*3^) -K1^) cos2e

r r (50)

P / 3a 2a \ ar9 ' " 2 I1 " "T + TV 8in 2e '

The maximum stress occurs at r » a, 9 • (TT/2,3TT/2) , and is

max 9 r-0,e™n/2

For the homogeneous, anisotropic >. dition work by Green and Zerna [19],

Hearmon [20], Savin [21], LeckhnisKii [12], and (as directly applied to

composites) by Gveszczuk [22], shows that the maximum stress at the hole is

usually greater than for the Isotropie case and can reach values as high as 9P,

■ 1 Jiw

I

I i i

I

i

I

0 !

FIGURE 13. SMALL HOLE IN INFINITE PLATE SUBJECTED TO UNIFORM, UNIAXIAL TENSILE STRESS FIELD

■■»■■»■■■——————

I 37

Referring to Figure 13, consider the portion of the plate within

a concentric circle of radius b, large in comparison with a. It can safely

be assumed that the stresses at radius b are effectively the same as in a

plate without the hole and can be given by

I

(a) «T P (1 + cos 2 9) r r«b *

(a.) « - 7 P sin 2 9 . rö r«b z

(51)

From Equation (50) it can be seen that

((7Ö) WT P (1 - cos 2 9).

It seems reasonable, then, to choose a stress criterion for the plate

rQ ■ function of 9 = j P (1 - cos 2 9). (52)

From the second of Equations (11), with V equal to zero the stress function

becomes

Y -^ Pr2 (1 - cos 2 9) * f1(9)r + f2(9) . (53)

where f1(9) and f2(9) are functions of 9 only. Applying the first and third

of Equations (11) to ¥ results in

a_ am (i dr Vr ae/ "-CT_ft-TPsin29--ir 1 df2(9)

'r9 2

1

2 d9

1 d2f1(9) . d2f2(9) .

(5A)

rt^ '^H^r «lP(l + cos29) ^^^-^-^-—^-^ t^ r do ard9 r d9

Applying the boundary conditions

I (•J - (arö) - 0 » r-=a rö r«a

yields

38

I i

f2(e) » - ^- cos 2 9

d fl(e) P hr— + f^O) - - f ll + 3 cos 29] .

de2 1

(55)

Choosing as a particular solution to the second of Equations (55)

)

i

yields

and results in

f1(9) " Cj^ + C2 cos 2 3

fl(e) - " f^ [1 ' co829l

Y - | {(r2 - 2ar) - (r - a )2 cos 2 9}

CTr -Kl-^l-cos 29+ (l-^)cos29]

P '9 2 ■ 1 ■ "ö . - - cos 2 9)

^--K1-^)81"29-

(56)

(57)

i

39

The stress function, y, as defined by the first of Equations (57) was

applied to Equation (10) with the body functions and temperature difference

taken as zero. In order to implement a solution, the following relation-

ships among the material coefficients i re assumed.

i

all " kla22

a22 " a22

I

I

a12 " k2a22

a66 ' k3a22 '

A finite-difference algorithm was structured to carry out the solution.

The method of solution applied was that usually referred to as the

"relaxation method" which is discussed in detail by Shaw[23], Hildebrand[25],

Allen[26], and Richtmyer[27].

Due to symmetry, only one-fourth of the plate was modeled. A

square mesh was used and the quarter plate separated into a square array

of 61 x 61 nodes. It was assumed that the circular hole has a radius of

1.5 Inches and the mesh distance, h, the distance between nodes, is 0.25

Inches. The width of the quarter plate model, thus became 15.0 inches.

This gave a b to a ratio (see Figure 13) of 10:1, thus minimizing the

far-field effects of the outer boundaries upon the stress around the hole.

This plate model is shown in Figure 14. Rectangular coordinates were

employed and the differential equation that was differenced was that shown

in Equation (9). Equations (57) were converted to rectangular coordinates

for input to this program. In the solution procedure, it is necessary

to assume the values of the modulus parameter a „ at all boundary nodes.

The solutions were then to proceed in an iterative manner until the

values for a.« were determined at each interior node in the model. No

success was attained by this method. The program never was able to

converge to a solution. In fact, a strongly divergent tendency was noted

(i.e., each iteration on the a^'s at each node point was markedly greater

^_JL_

AO

!

I i I i f

1

!

Outer Boundary Points

eg 531 6QI 58,1 Line of Symmetry

7,1*6,1 5.1 4.1 3,1 2,1

Inner Boundary Points

FIGURE 14. TWO-DIMENSIONAL FINITE DIFFERENCE MODEL

_J

41

on each iteration in the relaxation process than the iteration which it

succeeded, regardless of the boundary values assumed.) Extensive investi-

gations indicated that thera was no error in the solution process. It

appears that there must be some other governing relations which hav as

)■

(

I I I !

i

I

ji J

yet not been expressed.

As a result, no solutions to tne two-dimensional problem in

design synthesis have as yet been accomplished.

-

■ * -^ MM

42

Limits and Other Considerations

In the preceedlug discussion, it was pointed out that a solution-

for the two-dimensional problem could not be achieved. This raised ques-

tions as to whether all governing equations have been generated. In the

application of the one-dimensional program, D0M0V1, ir was found that

under certain conditions, the only mathematical solution developed for

the modulus variation in a disk would require that at some points in the

disk body, the value of the modulus must be negative. Though this is

perfectly acceptable from a mathematical viewpoint, it obviously cannot

be implemented from the viewpoint of real materials. To illustrate this,

consider the case of the rotating annular disk with boundary tractions

when subject to the stress criterion that the in-plane shear stresses

throughout the disk must be constant. From Equations (48)

2 2 a - C (Anr + 1) - ^^-^ + C. ti o o 2g 1

i

!

I

1.1 II

2 2 2 0 ■ e tar - Y o y + ci TO 2g 1

C - JL- {(o -a.) + *f-(b2-a2)} (48) 0 £n b I o i 2g

a

1 / 9 9^ C. - ^ ia.inh-a £na - **£- (b Jlna-a Ä,nb)?. 1 b \ i o 2g )

a

Take the condition where the disk is not rotating; i.e., u ■ 0, then o -o..

An — a a.Änb-o £na

C, - ^ r-* (59) 1 in^ a

o - C (Ä,nr+1) + C. 9o 1

o - C Anr + C. . (60) r o 1

• i M>i

^s TTH • ■ P

^

43

Substituting Equation (59) in (60) yields

T \(.a -a.)(inr+l) + a.£nb-a Una}

0 Ü f

\

or - r Uao-a )£nr + a J,nb-a £na> .

a

Now, further assume that a - 0, a - -P . Then, Equation (61) become

(61)

in — a

^ I" n- (62)

Note that for all values of r <^ b, a < -P in the algebraic sense.

However, if - > e, (2.71828), then at r - a. In - > 1 and aQ is negative. a a 6

Defining the displacement at r = a as

Va) -{m '"e-^r1) lr.

V.) -utr tt u •ta i*v *• «'I u (r-a) " E^iT^b [1 " ^a * (1-v)] (63)

Ä,n - a

From Equation (63), if In — > -j— then the term in the brackets is negative

and U or E(8) must be negative. If either is negative, then the system

must do negative work, which is not possible for real materials. This,

of course, can be overcome in a numerical sense by simply requiring that

only those solutions are acceptable which yield a material coefficient

matrix that is positive definite at each point in the body; i.e., the

following conditions must all be met:

M ' -L ^

-^-

A4

all^0

alla12 " a12 ^ 0

alla22a66 " a12a66 i 0

(64)

Thus, It is clear that the "arbitrariness" of any stress conditions selected

are restricted to more than conforming to the equilibrium equations and

the boundary conditions.

Two Dimensional Boundary Considerations

!

It is instructive to approach this problem from the viewpoint

of the calculus of variations. Consider the total strain energy in a

stretched plate uf unit thickness:

i;

i M

I

f

r fi

U - //. U dxdy , A O

where

U = strain density at a point

■■T-[oe +oe +T Y ]. 2 xx y y xy xy

Assume a Hookien material, neglecting time and temperature effects,

e ■ 3,-0 + a,-o x 11 x 12 y

e ■ a.-o + a-.o y 12 x 22 y

xy 66 xy

and a stress function

(65)

(66)

■ J -^

_ ' _^^ ^

45

2 o. ay2

3x2 (67)

[ r r

i

xy ax3y

Such Chat Equation (65) becomes

U - // i[-u($)2-u($X$) + a 22 (S) + a66 (|S7) J^^ .)"}

Following the method of Weins took [ 28 ], the extremization of (68) is

affected by forming the integral 1(6) by replacing f in the integral

of (68) by

4 - nx.y) + 6n(x,y) ,

(68)

(69)

Ma

I !

r !

I I I I

'■■•.

1 X J

\

where ^(x.y) is the actual extremizlng function and n(x,y) is an ar-

bitrary function that is twice continuously differentiable. Then, the

integral 1(6) is an extremum for 6 - 0, so that

I'CO) - 0 .

Writing f(y , y , y ) 0 xx yy' xy the integrand of Equation (68), and here the

subscripts refer to differentiation with respect to x and y. Then,

according to (68) and using (69) to compute

ay XX

3t

36 - n J2. 3f

xx* 36 JÜL

'yy* 36 xy

* I -^

— III!

M^M^Mi

(70)

(71)

i HUM

j m

Tl

A6

and I'(6) Is formed and 6 Is set to zero, resulting In

I,(0> " "A (#" \x + #" V + ^ V)dxdy = 0' (72) v xx yy xy v

according to Equation (70). Integrating by parts and employing Green's

theorem results In the transformation of Equation (72) to

//, \{im*im^m\ Mnt*kv)+^ (%)]"" 3f 1

- T n

dxdy

3f / y 3* 2 "x a* (

yy yy)

dx

, r >f 1 9f c) 'x a* 2 'y ay

{ xx ' xy

[» /»f \ . 1 3 / 9f \"| / .

r Noting that

0 . (73)

3f 324' ■'324'

xx 12(;y2)+a22l,3x2J

3f 34»

yy

3f 3*

xy

'11

l66

/32Y\. /As l3y

.2

^3x3y ' "Yxy

(74)

(Note: Confusing subscripts. Subscripts on strains do not refer to

differentiation.)

thus, the area Integral becomes

//. K 32e 32e

ax 3y

32Y

3x3yjj dxdy . (75)

■ ' Mfci ■*

^s ^

47

Since (73) must hold for arbitrary n, it must In particulrr hold for those

n for which n = nx = n «= 0 on the boundary C. For such n Equation (73)

reduces to the well-known compatibility equation as previously derived

(see Equation (3)) and the resulting (Equation (9)). For arbitrary n, n , n , x y

other than zero, the boundary relations must be derived from the remaining

line integral portion of Equation (73). The first part of the line inte-

gral becomes, along C on y - constant,

4 3all »2¥ 3^ 8y 2y2 ^V

+ Üü A + (a + I . j JL1. 3y . 2 + u12 + 2 a66; ! 2,

3x 3x 3y

iÜ66 2 3x )x3yj

I

r 32*. aM if sM 663x3yJ

The last two terms in Equ^ tion (76) are

- n. [Cx]-[\ky] • If these strains are not prescribed as zero, then along this portion of

the boundary n

fied is

n ■ 0, and the boundary equation which must be satis-

3a 11 3*4' a.^3* . 3a12 S2* ^ , 1 v 3 f

(76)

(77)

1 ^66 _£l 2 dx 3x3y 0. (78)

and a similar set of relations can be written for the second part of the

line integral. These equations have, at best, a very limited application

other than showing that other constraints do exist on the boundary.

Their limited application is due in great part to the choice of a rectan-

gular coordinate system. All attempts to date to express these relations

in terms of other coordinate systems, in particular those using generalized

normal and tangential components have been unsuccessful. More work must

be done along these lines.

111

■ ■ -*-"

•^m ^T

i r r !

I ■

I I 1

A8

Material Considerations

Finally, just a brief note on the possibility of achieving a

material with tallorable properties. It is sufficient here to say that

work by Adams and Tsai[29], Dimmock and Abrahams[30], Hewitt and Malherbe[3l]

Halpin[32], Halpin and Pagano[33], Kohn and Krumhansl[34j Tabaddor[35],

Fotlnich[36], Wang[37], and Fokin and Shermergor[38], among others, have

clearly established that material properties for composites of various

types can be established by a knowledge of the known properties of the

constituent materials, their orientation and their relative density.

J ■I ■■ WP w^-n-*

^s ——. jm

^

I I

I i I

I

I II

111

49

SUMMARY

Mathematical design synthesis has been shown to be possible In

certain specific applications. The selection of a design criterion In

the cases discussed, two dealing with stress distributions, and the

development of the material property distribution within a plane body

such that compatibility Is satisfied, appears to be a rational basis

of design for composite materials. Difficulty has been encountered In

the solution of two-dimensional problems employing this concept due to

the, as yet, undefined boundary restraint requirements which affects

the selection of the stress criterion to be met.

■ ■ -^

TT^ ^~

r I i -

I

I

I

I

1 fl

N

3u

REFERENCES

[I] Sayers, K. H., and Hanley, D. P., "Test and Analysis of Graphite- Fiber, Epoxy-Resin Composite", AlAA Paper No. 68-1036 (1968).

[2] Chen, C. H., and Cheng, S., "Mechanical Properties of Fiber Reinforced Composites", J. of Composite Materials, Vol. 1, p. 30 (1967).

[3] Hill, R. C, "Evaluation of Elastic Modulii of Bilaminate Filament Wound Composites", Experimental Mechanics, Vol. 8, No. 2, pp. 75-81 (Feb., 1968).

[4] Gaonkar, G. H., "Average Elastic Properties and Stress for Composite Continua". Collection of Technical Papers on Structures and Materials, AIAA/ASME 10th Structures, Structural Dynamics and Materials Conference, New Orleans. La., pp. 172-182 (Apr.il 14-16, 1969).

[5] Dong, S. B., et al., "Analysis of Structural Laminates", Air Force Report ARL-76 (Sept., 1961).

[6] Tsai, S. W., "Strength Characteristics of Composite Materials", NASA CR-224 (1965).

[7] Tsai, S. W., "Mechanics of Composite Materials^ Part I - Introduction, Part II - Theoretical Aspects", Air Force Materials Laboratory, AFML- TR-66-149, Part I (June, 1966), Part II (Nov., 1966).

[8] Tsai, S. W. and Azzi, V. D., "Strength of Laminated Composite Materials", AIAA Journal, Vol. 4, pp. 296-301 (Feb., 1966).

[9] Whitney, J. M., "The Effect of Transverse Shear Deformation on the Bending of Laminated Plates", J. of Composite*Materials, Vol. 3, pp. 534- 547 (July, 1969). *

[10] Whitney, J. M. and Leissa, A. W., "Analysis of Heterogeneous Anisotropie Plates", J. Applied Mechanics, Vol. 36, No. 2, pp. 261-266 (June, 1969).

[II] Whitney, J. M. and Leissa, A. W., "Analysis of Simply Supported Laminated Anisotropie Rectangular Plate", AIAA Journal, Vol. 8, No. 1, pp. 28-33 (Jan., 1970).

[12] Lekhnitskii, S. G., "Theory of Elasticity of an Anisotropie Elastic Body", Holden-Day, Inc., San Francisco, Calif., (1963).

[13] Bert, C. W. and Guess, T. R., "Mechanical Behavior of Carbon/Carbon Filamentary Composites", Composite Materials; Testing and Design (Second Conference). ASTM STP 497, American Society for Testing and Materials, pp. 89-106 (1972).

»

■ ■ -^ - — _^^_^^^——^^M^_^. _J

^T mmam W ^ -r—7-Tz _ •

11 [ [

!

Ifl

i 1 i !

I

51

[ 14] Bert, C. W., "Analysis of Nonhomogeneous Polar-Orthotropic Circular Plates of Varying Thickness", Bulletin No. 190, Engineering Experi- Ment Station, College of Engineering, The Ohio State University, Columbus, Ohio, Vol. 31, No. 2 (March, 1962).

[Li] Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, Second Edition, McGraw-Hill, New York (1951).

[16] Bieniek, M., et al. y "Nonhomogeneous Thick-Walled Cylinder Under Internal Pressure", ARS Journal, Vol. 32, pp. 1249-1255 (August, 1962),

[17] Shaffer, B. W., "Orthotropic Annular Disks in Plane Stress", J. of Applied Mechanics, Vol. 34, pp. 1027-1029 (Dec, 1967).

[18] Manson, S. S. , "Determination of Elastic Stresses in Gas-Turbine Disks", NACA Technical Report No. 871, Flight Propulsion Research Laboratory, National Advisory Committee for Aeronautics, Cleveland, Ohio, February 27, 1974.

[19] Green, A. E. and Zerna, W. , Theoretical Elasticity, Oxford at the Claredon Press (1954).

[20] Hearmon, R.F.S., Applied Anisotropie Elasticity, Oxford University, Press (1961). ^

[21] Savin, G. N. , Stress Concentration Atound Holes, Pergamon Press, New York, N. Y. (1961).

[22] Greszczuk, L. B. , "Stress Concentrations and Failure Criteria for Orthotropic Plates with Circular Openings", Composite Materials: Testing and Design (Second Conference), ASTM ST? 497, American Society for Testing and Materials, pp. 363-381 (1972).

[23] Shaw. F. S., "An Introduction to Relax-ition Methods", Dover Publications Inc., New York, 1953.

[24] Hildebrand, F. B., "Finite Difference Equations and Simulation", Prentice Hall, New York, 1968.

[25] Southwell, R. V., "Relaxation Methods in Theoretical Physics", Oxford at the Claredon Press, 1946.

[26] Allen, D. H. de G. , "Relaxation Methods", McGraw Hill, New York, 1954,

[27] Richtmyer, "Difference Methods for Initial-VAlue Problems", Inter- science Publishers, Inc., 1957.

[28] Weinstock, R. , "Calculus of Variations, With Applications to Physics and Engineering", McGraw Hill Book Company, New York, 1952.

•

^

52

I

r i

I I I r

II

[29] Adams and Tsai, "The Influence of Random Filament Packing on the Transverse Stiffness of Unidirectional composites", the Rand Corporation, RM-5608-PR(1968).

[30] Dimmock, J., and Abrahams, M., "Prediction of Composite Properties from Fibre and Matrix Properties". Composite, December, 1969, pp 87-93.

[31] Hewitt, R. L., and Malherbe, M. C., "An Approximation for the Longitudinal Shear Modulus of Continuous Fibre Composites", J. of Composite Materials, Vol 4 (April, 1970) pp 280-283.

[32] Halpin, J. C., "Stiffness and Expansion Estimates for Oriented Short Fiber Composites", J. of Composite Materials Vol. 3 (October, 1969) pp 732-734.

[33] Halpin, J. C, and Pagano, N. J. , "The L minate Approximation for Randomly Oriented Fibrous Composites", J of Composite Materials, Vol. 3 (October, 1969) pp 720-724.

[34] Kohn, W. , Krumhansl, J. S., Lee, E. H., "Varnational Methods for Dispersion Relations and Elastic Propertiets of Composite Materials", J. of Applied Mechanics, June, 1972, pp 327-336.

[35] Tabaddor, F., "A Note on Effective Moduli of Composite Materials", J. of Applied Mechanics, June, 1972, pp 598-560.

[36] Fotinich, 0. V., "Determining the Equivalent Modulus of Corded Rubber Test Pieces", Mechanics of Solids (Mekhanika Tverdogo Tela) Vol. 4, No. 3, 1969, pp 162-166.

[37] Wang. F. F. Y., "Calculation of the Elastic MoHnl1 of Binary Composites", Materials Scienre and Engineering, 7 (1971) pp 109-116.

[38] Fokin, A. G., and Shermergor, T. D. , "Calculation of Effective Elastic Moduli of Composite Materials with Multiphase Interactions Taken Into Consideration", Journal of Applied Mechanics and Technical Physics (Zhurmal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki) Vol. 10, No. 1 (1964), pp 51-57.

—

^^N . _j w

^

i I I

!

f 0 I I I I !

!

I

APPENDIX A-|

INPUT DATA FOR PROGRAM DOMOV1

Five cards constitute the input data for D0M0V1 for each

variation to be studied. These five cards represent three (3) data

sets. As many runs may be stacked behind the initial set as required,

The computer run will terminate when an End of File card is read.

Data Set 1 (Title Cards)

READ (5, 10) (ITIL(I, 1), I - 1, 24)

10 Format (8A10)

Any information can be placed on these three cards. It is

necessary to have three cards, although any can be left blank.

Data Set 2

READ (5, 20) RO, RI, RPM, DENS, N, IBOND, 1ST

20 Format (4F 15.0, 3115)

RO - Outer Radius, Inches

PI ■ Inner Radius, Inches

RPM ■ Revolution per minute of Disk

DENS - Material Density of Disk, lb/in.3

N ■ Finite Difference Increment and Printout Number

IBOND - Type of Boundary Condition on Inner Radius

0 * A stress condition

1 "■ A displacement condition

1ST - Type of Stress Criterion Chosen

1 - Constant theta stress

2 = Constant in-Plane shear stress

• i ■!>

'^^N ^m V

!

[

[ i

!

I

I I fl W I!

I

A-2

Data Set 3

If IBOND Equals 0,

READ (5, 40) SI, SO, ORTHO, PO, ETHETA

40 Format (2F 10.0, 3F 15.0)

SI - Radial Stress on Inner Radius

SO - Radial Stress on Outer Radius

ORTHO - Orthotropic Ratio, E /E0 r o PO - Poisson's Ratio

ETHETA - Modulus of Elasticity in Theta

Direction at the Inner Radius

If IBOND Equals 1

READ (5, 30) UI, SO, ORTHO, PO, ETHETA

30 Format (2F 10.0, 3F 15.0)

UI - Selected Radial Displacement of Inner Radius

i i i aJti ^fc_^_^_ M*

•^1 w Reproduced from best available copy.

APPENDIX B

LISTING OF PROGRAM DOMOVl

C

•')

PROGRftH 00V0V1 (INOUT.OUfUT.TAPFSsINPUT.TAPFftsOUTPUT)

OIMENSION RdMI.-TMruu-'KrOCCi:-)) ,/i22 (l D TJ ) , CC (ICC » , 00 (1C 2 I , t ... PHi(i::),PHio(i:a> ,PHiPP(ict:i,siGRiij:»,STGT(icci ,u(i:o)

_DIKZNSION 33(lJii) ,AA(1.21»££(1Ü3J c c

c c

COHION R,PHI,PHIP,OHIPP,SIGR,SIGT,U

_5-_PE.A0 (5,1: 10 FORMAT (BA IF(£Of,5)

15 RFAT (5,? 20 FORMAT iUF

IFdHOND-l -__Z5_PEAO (5,3.

3C FORMAT (2F C RO . C RI

X.. RPM . C OENS

.C __N C C C C C

.C C C C C C .C. c c c c c

_c c c c c c

IDOMO

1ST

UI SI SO ORTMO PO

.ETHE.T.A.

ROO EOC

PHI PHTP r>HIPP SIG«> SIGT U ETHET

)- (ITILU.l) UI 12.,15 1 RO,RI,RPH, 15.:,315) , ) 35,25,2^ ) UI,S0,r«9TH 1C.;,3F15..)

»0UTrP. PA = INflrR "»t sREV^LUTT sMATEojAL

-FINITE_C =TYPC Oc

0 = A ST 1 = A 01

sTYPI OF 1 = COMS

2. =. CONS sRAOTAL n »RAOTAL S sRAOTAL Z =0RTH0TO0 SPOIS«;OMS

. =.M00UL'JS. THt INHC

sRATIO OF SRATIP rm FOUND I*

=THE STRF . ..sTHE.FTRf;

»THp's^nr »RAOIAL t roipru^pf «THE PAOl «THE CALC

,l=UZkL

OENS,N.IOOMO,1ST

0,PO,ETHETA_

OIU';,INCHES OIUS, INCHES CMS 0Z0 MINUTE OF DISK

OENSITY OF niSK, LR./C IPFERCMCE. INCREMENT. NUM nnuNriARY CONDITION ON I RESS CONHITION . _ SpLACEMENT rONniTION ST^TSS CRITERION CHOSEN TANT THETA STRESS TANT IN-PLANE. SHEAR ST«» TSFLACEMENT OF TNNE« RA TPE3S ACTINC ON INNER P TOESS ACTTHF, ON OUTER 9 0IC RATIO, E-SU9-R/E-SU

PATIO IN THE THETA-R 0 Qr_-LASIIC T.l.Y_.IM_ J_H£„TM 0 RADIUS

TH". RADIUS OVER THE OU THE MODULUS OVER THE H TH" BOOY nfL THE_OISK

SS FUNCTION T P-RIVATTVr OF_THF_STR NO OERIVATIVE' CF'THE" ST TPESS .. R"MTIAL STRESS M 0ISoLACEMENT ULATEO riRCUMFFÄFNTIAL

mic INCH ?EF. »•'NER RADIUS

PRESENTLY

ESS HIUS AOTUS AOIUS 9-THETA IRECTION ETA DIR£CTIOM_AT.

TEP RADIUS AXIMUM MODULUS

rSS FUNCTION RESS FUNCTION

MODULUS AT FACH

OOM DOM DOM OOM OPH OOM. DOM DOM 00" DOM DOM DOM OOM OOM OOM 00^ OOM DOM OOM D0M

00" DOM DOM DOM. OOM DOM OOM 00M

DOM 00" PPM OP" OOM DOM OOM DO" 00" DO" OOM OOM DO" OOM OOM DO« OOM OP" OOM

1 ? j

(. 5

.. 6 7 3 o

10 11

.12 13 1« 15 16 17 .1! 1° 2'. II 22 2?

.2«. 25 26 27 29 2° 3: 31

33 3«. 30 3t 37 3? 39 <•:

«.2

^7

■ " -^

-^7 W

B-2

C CALCULftTnn POINT IN THE DISK OOH . _C .. A22 sl.C/FTHfT POM

GO TO us noM __35 f?FAn f5.£.Jl «JT^OfnoTHO^n^THFTi pnM

5: SI

UD FOo^ftT IZriQ.fitSriS,:) OOM 5?

OMEGAr(POM.OT,/,^..j» ppn 51.

G«?HO=<nENSI»(0MrGA»»2.:)/(3l6.i.) DOH 56 57

ETHET{l)=LTHrTA OOH RMlrPT OOM

5? 5°

OELi (f?0--?I)/(N-l» 00" 00 11, T = ?.»J pnM

5:

J=l-1 OOH PtTl=?<.)Wnri priM

62

T

5C CONTIVUE 00« 6!4

65

i: n

CALL ST-?-SS (N,iRA,r,^HrtSI,SO,UT,Ol,THO,PO,ETHETA,IPONO,IST,RI,RO» OOH _. DO 55 1=1.N . ........._. OOH

AA(I»=PMTP(I)MPn»PHT(in/(R(I))*{GRH0»(R(I)»»2.:n OOH -BB(I)=PHI0P(I)t?HIP<I»/R(I)-. (ORTHO^PHK 1))/(??(I)»*2.;i ♦(GRHO'Rd D0U

$)M3.:-O0)) 00" r,n(T) = mtT)/AAm nn-

66 67 6° 6 = 7: 71

55 CONTINUE OOH 00(1)=?.: nnn

72 7^

EEdlsC! OOH DO fir T=?.N noM

7<. 7t;

J=l-1 OOH 0D(I)=1. }*CC(l)»(n(I»-p|J))/(2.LI no-

76 77

EE(T» = I. '-cctj»•(--■(n-o«jn/2.: DOH . •Ctltl«fCCftl/QOIIIt«IAttfJII _._ .. . _ . DOM

ETHET(I)= l.i/A22(T» 00« ., _ feC CONTINU- nn"

7» 7°

i

EMAX=-,: noH 00 «i^ T=l fN pnH

1! 1'

EHAXrAMAXKETHTTd» ,EMAX) OOH 65 CONTINUE DOM

DO 7 C 1 = 1. N 00". ponm=Rm/Rn non

•»6

EocmsETMETm/pHftx noH U(T)=A??(T)»«>(T)»(SIGT r)*P0»STGPITH OOH

14

7C CONTINUE OOC. . P0=-1.?*D0 . DOH

MPITE (5,75» l|TKttttl«f«tUtM 00" . .. 75 FORMAT (1H1.8A1:/1H ,eAl.-'lH ,8Ai:///) _._. . .. . .__ ,. DO"

I«"(IPONO-1» i;»ti4«t*<l OOM 81 WRITE (5,95)P0,i'I.!:!:,H,5<)t^i,FTHrTA,.?O,OPTH0.0ENS.N . __ . DO"

91. 95

'—

- - 'I Jfc I

B-3

85 FORMAT (l;x,51Hr)ET^OHTV4TTO,l Of MODULUS Vft^TftTTON IN AN ANNULAR HIS SK.//,1X,1-HTNPUT 1ATA,/,iy,lCH»»»»»»»»»»,//,5X,15HOUTE9 RADIUS = , SFIC.I.,?/.ISHINNER "AOTUS = ,P1 J.««,2X,6HoPK s , Fl 0 , 2,//,5V , 3C HP AOI A

.-. »L STRESS, OUTFR PAOIUf. = , Fl t. ?. 2» ,3 CHRA PI AL STOfSS, INNE" RAOIUS t» ,Fi: .2,//,5X,25wrTH: TA AT INNER RAnjUS = , C0?:i5 . 7, 2X , 1 7w0nT SSONS

S RATIO = ,FlJ.9,2X,2:HQRTHOTPOPIC RATIO = ,F1C .«..//, 5X , iqH"ATE RI AL t DEHSITY = ,m.B,2X,25^NtJHPER OF RAOIAL "OINTS = ,13,///»

9C WRITE (6.95) - —•• - 95 FOPMAT (lX,ll"OUTt>UT PATA,/, 1X,11H»»»»»»»»»»»,//,7X,*)HRA0I"JS,PX,6U

SETHETA,'iXt''HSIr,1A-P,,<X,7H3IGMA-T,l',iX,«.HR/PO,lCX,«»HE/En,13X,7HU-SUn t_o,/,/x, ^M»»»»»», qy, fcH» • ••••,<»X, 7H»» •••••, 8X, 7^ ••••••»ICX.UH»»»»,

JllX««iM»»»».iliX.7-H»»».»l»S.»//J WPITf (6,1 JO) (R(I),rTMrTm , SI GR ( T ) , ST GT (I) , POT «I) ,E OC (I) , U (I) , 1 = 1

1C( FORMAT r>,,Fl..«»,2y,lPcl3.6,flX,F7.C,flX,F7.0,*;X,F«.5,6X,Pq.i«,6X,

tE12.6) GO TO IIS

.105 »"?ITE(6,liJ) PO,RI.Rn^,SO,UI,ETMETA,Pn,ORTHO,OENS,N 11C FORMAT (5Y,5^HO=:TrpMX'4ftTnN OF MOnuLUS VARIATION IN AN ANNULAR OIS

1:K,//,1X,1*HINDUT ^ATA./^X, ll:H»»»»»»*»»»,//,5X,15HOUTER R^OIUS = , SFlC.^X^SHIKNrTR PAHIIIS ■ , Fi 0 , (, , jy ,f MRP" = , Fl <; . 9,//, <; X, 3! HO AO IA

SL STRESS. OUTER RADIUS = , FIT.. 2 , 2X , 3?HRA?I AL DISPLACEMENT, INNE? R SAnilJS = ,Fi: .7,//,5X,?,;MTH-:TA AT INNER RADIUS = ,CPF15.7,2X, 17MDOI

tSSONS RATIO =_,F.i:.3 ,2X, ZIHDRTHOTOOPIC . RAT 10. =_, F13 . i.,//,^X, 1°HM AT tFPJAl DENSITY = ,Fi:.5,2X,26HNUMnER OF RAOIAL o0INTS = ,13,///)

.... GO TO 9: 115 CONTINUE

. .. GO TO 5 12P COWTINUE C^LL. EXLT -—

END

DOM 96 DOM 97 DOM qp

DO^ 99 DP« li? DOM 1 Jl OOM 102 OOM 133 DP** n«. OOM l!5 DOM i'.% DOM 1C7 DPM fin DOM 159 OOM 11C OOM 111 OOM 112 .DOM 113 POM liu DOM 115 DPy 116 OOM 117 POM 11.1 DOM 11° DOM i?: OOM 121 OOM 122 DOM 123 OOM 12<4 .DOM 125 DOM 12*

■/

L i

B-A

SUD90UTINE STRE^SCN.Pft.GRHO.SI.SO.UI.ORTHCPO.ETHETA.IBONO.IST.RI, tRO» ...

oiHEKsioM R(in»,rTHET(ic:i,coc(i:o),a22(ico),cc(ioc),no(i2oi, $ PHH^-i» .DHI0(1-:» ,PHIPP(K:»,SIG9C10C ),SIGT11C;.) ,u<103) -i 1 ROB f 1C : 11 LT.XL (2 «111 DIMENSION RR(IJC),Äa«iji),EEIIP:»

_ c c COMMON R.0HI,PHIP,PMipp,siGR,SIGT,U

c c c c c r c _c. c c c c c -C c

THI5 SU^RfluTIN- CAIC'JL4TES THE STPfSS FUNCTTOK' AND ITS FTPS! ANO SECÖ"n DERIVAT IVESFOP THE FOLLOWING CONDITIONS, IF lOnHD = ] ANO IS! = 1 WE HAVE A STRESS CONniTION 0^ THE INNEP RADIUS AND A STRESS CRITERION OF CONSTANT TH^TA

STRESS JJL1PON0-« 9.ANO 1ST -_2_HL.HAVE A.ST'ESS CONDITION ON THEL

INNER RAPTU'; AMI 1 STRESS CPIT^inN OF CONSTANT IN-PLANr

. -. SHEAR STRESS . . IF IOONO = I ANO IST = 1 HE HAVE A OISPLACc-MENT CONDITION ON THE IHHf». PACIUS WITH_A STRESS CONOTION OF CONSTANT _

THETA ST^rSS. T-HESE..ARE . ItiE_rtlLY._C0bL1III9-NS_u30C-PA.MyEQ

.

IFIinoMO-l»?^?,!.1

5 GO TO (1.,,20»,1ST 1C CONTINUE

C JB C C

.THIS FOR

SECTION OeL A CONSTANT

THE .SUBROUTINE: CONTAINS THE. CALCULATIONS CIRCUMFERENTIAL STRESS CHNDTION

i" - I"

D n

-fii-L

C C C C

*r. = c j.o/co-cm» f (SO»RO-«;I»PTI*((G?HP/3,U»»(RT"»3.r-pi»»-».j)n _ P? = J(!?O»RI)/('>O-PI))»(.(SI-SQ).-.(GPHO/2.0)»(RO»»2.C-RI»»2.:)) DO 15 I»l,N

PHI ( 11 =A J »R 11) -_( G 'HO/' . r ) »(R(I>»»3. ; | » PHI^dlsA.-CGRHO)» {R(I)»»?. j)

_ *>HIPP(I» = <-2.;)»(GPH0)'R(I) SIGR<n=PHI<I»/R(T)

. - SIGT(I)=oHIP(I) + GRM0»(R(I)»»2.a) 15 CONTINUE GC. TO >*■■_ 2L CONTINUE

THIS SECTION OF THF SUBROUTINE CONTAINS" THE CALCULATIONS .FOR A CONSTANT IN-^LAME SHEAR..STRESS.CRITERION

STR STR STR STR STP -SIR STR STR STR STR STR .SIR. STR STR STR STR STR STR STR STR STR STR STP 5TR. STR STo STR STR ST» STR STR STR ST? ST» STR STR_ STR STR STR STR STR 5TR STR STR STR STR STR

1 2 3 u 5

_ 6 7 8 9

IG 11

_12 13 l^ 15 16 17 ie 19 2C 21 22 23 2«« 25 26 27 29 29 j; 31 32 •» -i

7«, 35 3t 37 •»P

39 kZ kt 1*2 i»?

(,5 1,6 «.7

• - MI mm—am

B-5_

Cir(17c7^Ln'r<^ä)TM<;i«ALnr.(R0)^SO»AL"cG(9I»-(G«?H0/2.C ) M «RO " 2. 0» ^ «LOGtPI)- |»X»»2.DI*C»LPC(W)II1 1-1.".

■•'" C'=ll.3/aLnG(PA))MSn-ST*(GOH0/?.^MPn^2.ü-PI^^C))

PHippmsco/Rm-'.o'GRHO'Pm _STp 55

__ . .8TWfII«0HHl»/«tX1 STR 56 SIGT(T) = 0HI0(n*6oHnMR(I)"2.:) STR 57

_25 CONTINUE ' ' ST«? 5 <J GO TO *•! _STR-_59

JCCONUNUI — STR &: C THIS SECTION OF THE SUIRO'JTINE CONTAINS THE C»LCUL»TIO»« STP 61

- C FO^ A OlUuCE-MT CO-OITIOH AT THE INNER PAOIUS WITH A STR. S2 C STRESS CRITEPIOM OF CONSTANT THETA STRESS »JJ JJ

!'r,V^Ti--TAr/rR^rr; ♦on.-Po.Ro,i• ,,iT?ov;227T7^'p^(GrMoVA2FriTsTTR »« f(l.V3.:)»POMRO»'3.--RI^3..n) Tl? %j,

BC=S«»ROMG?MO/3. •)'(P'>,'3.])-Aj»PO STR ^^

DO 35 I = 1«N ... cTo 7- — nHi(T»=A-»R(n-»r,"MO/:.:»MR«n^3.:)*90 JTR ^ SHl»II»«\--.t5i».H0liW<X»t?l«iJ—. — — sT9" 72

PHippin = «-?.:>'«F,pMn)»p«Ti STO 73

SIGR(I)=owTn)/=!(T» - -— ST9 7u sicTm = pHi?m*GoH0M'(i»"2.")» STP 75

CONTINUE ST^ 7fc COMTfHUS STR _77 RETURN ~ ' STR 7« END

t

r I 1

1

• ■ ■-** ^m

■)

I : z

• 3

r

n

i •

t

Z

• 5 ■> m'. • SI • ü n — -J X a _i

■" •»>,

S £

i z

4i i 3

J 3 I

V J — I .".

3— ■»

t o

o

3 Jl

« •

i a

o 1^

j a

4 .r a Ü

H

n J

if a 3

fc I Reproduced from best available copy.

I I

• ••

~ lx

r ,—

I t I • • I ■ • • I II II II II II II II I I I I M'tt »J J -J -kl ^J Jj Ul J* J il »J J ii U J U Oi'AJ xJ iJ -J -J J li U il

r^~i,-.>-n,o-«Mrin o Jl oa— o \j o oi-«-«*iOo-«/» Of« fM/>-*03T>#M-n>»')-«-«-«"n 03-»> />-••» t — X> fl -^ MOMOOilU-t 3->i>l>'>>><M>>3'33'-a-\i.\J ll*!*!/» Ol ♦ en ■tlAJM.'XJM.MMc-«-• -•-< -•-• -« M.M.-\J.MM,M.\J.\J.M.\J/M

300100300313000 OOlOOOOOOlOOOOOO

»« ♦ m >-« 3 -. r^ rj" > m (D CD ö -vj c -D T) ^- » o D tr pg i/l o r^ •"•o>jini*-j>n* m\J-« i>'f-in-ni3>~>j--«i>ir»i-«' » O O \) X)

00330033000 OOOOiOOOOOOOOOOiO

oo>:or>-4>if>ir<*rt.M-«o9>3or-.o'.ni/i ♦ mvj-io^ar-o

3i>.o— oin»i'\j'-» j37>(0-~«o r» ♦•niM -•'■> »•> of-oi 33—•|\i.'»i»in-oi»io.»>i 3».-< Mimtin Of~-o~i*" 3-> -* \j.ni

M rvi -M-vi i M M . vi M IM rv» i \j .i -n n -n •n

O 3 3

-il ^ •* » i a 3 «I 4

• «• • 3 • X • •- • 3 • 3 •

\n n s> * t * «OS »MiM !M

• n

l-M •

» ♦ f~ O » <n * m > o B j> > » o a> ~- o j»i«-- » >> 3 -«

3 O

»• O • > •> i J ■ • T J» ■ n ■

3». Jl 3 M IT >■*• t .- M R

•n.-n n

3 3 3 313 3 3 3: 300033 3303 3 o 3 3 3 3

O O O O O D VAI Ol MM M

3^33 3 >

f **■»■■»*• t -T * ■* » •<• * * * * * ■000000:0000 00 M .M M M M M M.M r\JM MM

OOOOOO MM M M,MM

rvirviji— IM-M—^ «Otn.Mi'Cx »r-^^D » »- f"- "i o rvj1^ 3 r >• .3<n^n ~- ■y —1.1 * o Miv>i**JiiJ>o oo-«.».|^.^.r-oin 00

I

O O O 3 O O 3 3 3 3 3 3

O O O O O 3 3 3 .J 3 ■> ■>

J J J J J J iJ ij i.' Ü U J -J J

/>\l ->M-<-Jl->»>0 O JI-OjO'?' -T ~t c-n O » !> -J3>J>

o o 0000 3 3. > 3

il -.) -J jiü J

» >• T rf J> 11 \i u g ••» r »» J » r> ■»» ~

r^ o o 0 i> "»

3 313 3 3 3

A, > O » — » U 3 •» O J> -• *» 3 •»: » O II » 1 O l* ■"

■n », r ♦ » n

— J« X> »I J> Ui \l Jl Al — T ■"!!> W II U 1 ^.''l • * ■•— —— MMMI on|o* r »ion

3 91 9 3{ J 3

r> y\ a ■>- ji (\( T "»• * M n -u

3 3| 3 »I 3 X

I ! > O' * — O ul

« 1 O > — » 1 -O n • - »

■o 3 >- n v j> * J vi n u j M » ^ t t-* > - "> J II U J

nn ««.oo^"-*-—,00 o >> > o 3

n ••■

J ^k J

C^2

0a>303»30339330333>a 303303 93»93.33333 330333^^0333 33O3300

—

•J Jj ll Ll Jj Xl li ll *i Al Xl kJ Li Lj ll li Xi ij O jj Li Li Li li Li Li Li Li tt ~ Li Li Li li JlJillJljLJUUlJJlJillJOU it il U ij U ^llvl

#M3>I>^«l*"\l><>'\J"--<'#-flOn'ni3i>>'M1'VOO-^*/»i>'0>i»<^OOlOMO>3T»i-»-.-n~-3-«oo»J»'VJ#-» M

«-oi»'3 3-« \j »>. *•* . n oi»-'^oi>"3 3-« vi vj ni ». t n n OIOI^-^-OIOU»!^« 3-3 -»-»-»VJM vjiiT.n.-nr'rtifrjn n n in ■MVM n ■nim-ni-n.m-ni^inini-»ii'>ir>i#i*.#1 *•.♦.♦ i ».#,».*i*.».♦.».*,»1»..t,nn n n n n n n n'f> n-n n n n n n n n r> n\n

OO 300*0 0000 OOJOOJOOO^» -> *0 3303 O O O ^ 01003030030 0300 O CO OO 03 03 O O O

I I

Cjl!>cni)Df^-r^r^r^f^r-'D(T>—•'»»oonr^-viD^-viocr^ o-^in^-#—•XIXID'^CI'-ID^'NJ—4"^»^^ O9so^o4>in4 #o A M •■• «>0;iM.'0' ♦ 3 oi't^in-^e ♦t-»^-#-<»*i-<5>0',niOij(a ■*>-«> r-voi^M-^ 3i>>x>">-oio o ii 'n o o o ^ a* wOi^ r-- x) n > o o -» ^ t\j n r-i ^ ^ u> ü ü r- n c C7> oo—<rvjr\jii*inx>x>r-~jOD'J'o-<ivjn»^jior~i.^<_'—'»jn^jio»^ rv») roncim** ♦•♦ ■»■» ♦•# •» ♦ ♦ ♦•♦i* ■»••♦ -n in vn m in j) iftininxi J-, JI JI JI ^} i o A>O £ £ x) o « <i^>>.r-f~t>-r^(>- ^r-

03 o o o o o oo-ooooioooooo.ooooooio;ooooo|ooooooJooocoo 3 3 O 30 33 OO O O O

t/i'«''nM-«3>a>r-<>Oinin'»fnfM^d3>i)Fwx>xim«rnl'\i^o»3>t>.x>l/iin«rnrM-«o>3>r-j>i/)^«'rn(M Jo» -o r- o JI Ln infl^-U3> 30>^rvjm»JI«'f>-»J> 00-»'\J1*/1iJ"-DJ»00—<-y»1 » n O >- U'> 33-<\irn.»->n^)r«-»3>l30-<'\|T»*/1 r^vj -• > » j>iO'-■■o /» ».-ni v-- ■»■ » >.oi»-'oin »••>i\i.-« a 3-MO-^to: o* II\J.-«|3 3>*o—o'n »mM-^ a. ^ j>. o ^-Oin *. r» Oi^- o »>< 3 o—. yiitcnai-tn».^ >-<j.nirinDi^it)M >. »—• Mini fin-Oi^'O'X 3a-< vi.icj-ift.oi^.-o.>o- i-« M Titiin *»r> -ni-n-n* ♦!*• ».♦ ♦■»*i» »»jinnnjinjin-nnno 004>i« aaöO0Of«-~-i,--'>-,>-i«-'>.^>.f^-~|03 o o D o o

C 3 3,3 3 3 O

in n in j) n ft /» « » * » * » » O O O O O O JO ,VM .V M M \J M

.

O O O 3333333333 03 33 3)3 03

»■•ri-»!» * ♦ S\ J* S* St J\ T\ *■**■*** 9 * -f * * * t * * »>>

a s o o,o on o o a a on o o o

O o o 3i3 3

-I •. ., •! *| »1

j>ji/>inift/>j>ifty»j)n jv,n s> n st jy j\.n JI a s< rt si ■••**♦■»♦*♦ *.*• r * t * t

oo-oooooooooo

3 o JJ 3 3 3 3 3 3 O 3 3 3 3 3 313 3

«I« • »i« •••!• .

ftift/>x>jijiftj>j>jinxijiftftj) j)/> o o t ooonooooooooo oo \i >» V ."« -M \» \J .M,.M "a M M M \J.\J -y VtV

Ln -»

to

r

I i

!

i — » iT PO 3^ , *> — T "~ »Mi * n n -v n N» » - - ^ j

V3 -• •« -• M Si ■n n » » ■^ -^

o n 9 O •»■ «- O O O T>l> • ••••••••••■•■

vj ivj 1-^ ^ —4 »4 —4 -^ V ^J M V» <\l -M •M 'M M rM

i

•I u

t r f n n in M M VJ \J \J iM

^k

4

J

• i 333 333333331303 II »till »I If 11 •

Cf3

•<

I i n

i

r r

">"-•--•>» o-«> n o n -• * r> *>> > t D -«M -. O >0 "VJ O J' -• -«o »I* -» n (i o o o o n n t i o u-i > o n n ii oi» »» on /> /» «ft /»-ft »i»i it ft xi .ft n rt n ft ft/» ft/» n n /»•/»■ ^-«-^-^-^ ^-#. -^ ^ «^ «4 ^ --i —• ■*•< • • • • t ••■•• •• ••]•• • 0)3000 30)3 33300309

f\l */» ^»^'MiftDi-^ «-SMOIO A) O — M i'n Jl J< -^I .V C> —. ,\J » Xl -^ o CD.OXDOX'COi.O:?- t> 5- J- J. CJ> J-

300 0'33|30303a

«II»» -VI -• o o B OP>. Oi3> O 31 IIM-^ > 3 >. t> OI^O.J»' 3> 3—« »ID O O 7>l> >

^ « 1ft J» •#• M t»V # .ft o ~-Ol/> »•■•>. M- TlfiftOl- J> !>• > > J> J.

31 3 3. 3 3' 3 3 3 9|0 3 3 3

j>;y> xi .ft JI x» x»

O.D X> JO O O O TJ, M y \J M \| -.J

M • 3 m T -< JJ

—4'n xi * ^- x x J D JJ ^ VJ "5 il

Ifti »• T »• » » ")

3 O O O o O J 3 J ."> J

•J _) U ij jj u ü — ■» 3 .T O O 1 *> J ij —i ^i j) —i -(-" "J I» U M V

•>i T J> T l> Jl .J »j f> n o o ^ D nl -i n ■> •> n in

1 > 3 » » •

m it >lf^ XI — T -^ 9 M O ft >i »ir«. # • • * •' V ft

n o D o ■- ■» •«.

»•♦Ifi*!*. » X> ft XI ift X> XI

O O D SOX) •M M .M M M M

D J) T I-- O XI « T "Vi 3 s i)

t) o o o o o 3 > 3 3 3

Ü _/ U J iJ ii

■n u» o o .» .\

W U « "I ^ u o ^ > -. i — •» "O •> ♦ t »

3 3 > 3

(Nl r* O ■• — C x > ■n o J> ■" •* >■ o •> » o T> > "» I»

o o u o (V IM; \i »I M V

O 3

C J> o -• .3, 3 0.> 3- > > O

3 ■•«

»t

X) /1ft ♦ •I« o o o

it -«o

« -V» 3

t t

XI »k 3 r ft M

^.A ^ J

^

!

}

H

•J1

•> t

o

P-L -

1 3

I/» 3 ^^ 3- 4 ■X

•r

E pa n

«t

I

-4

X • • l» D » 31»

03 31-33900 0030 • • ■ ■ I • ■ ) I > I I

MX"-—•-•'x» n-o -«o o o M.»»-*!*!/»^. 3 »• T>1 I. ^ -^

0 o 1 I

.n *

/> 1. — •»

O O • . I •J J O \J o -•

0 o 1 I

-• o

■n n O 3 -« NJ MM

0 o 1 I U xl J1 o o ^. ■n n .-• \J ■ni». M M

e» o i i i

i > o -^ ■no * IO

MM

O O 3 I I I il li jj

O T O — omi

>■ M Or "-•>i3. M M ni

3 33 I III U JUJ M ^iro

M-^in

:

3 a

-i

3 4 X

a •

y. » r- ja j> j> * n rvj _ oo «h-ojtji «moi^ooisp»- äJ» 3-«IM -^ » J> O l«-!t> 3> 3 3-« Mi.-n « J1 « r» S > 3 3-. M 11* — -- ■»••^ .i . *, ■ - M j- ^ _r ~. \j . ■ I >I

3.». D '- o r» »i-n^Mi^ o -> » D ^ > o n

3 3 O

-^ -"■—-« — ■^l'< -^ M M M IMl \J M M M M

330 3J0|3333333 3>3 C»-3 3

ii\j -.; ao >.a — om oi^-a. >■ ■>■ 3-« \j. mt-i

♦ f* ■»(♦■*■*.<i* » • i

. •»•<*.*i*.* J> x» r» XI ^ J1

»■■».* .»-i*.» Jl X» .0 aT J> X>

-***-»■»♦.■»♦#»**■ X>OOir>OOOOOODXiOOXHOOOOOD MjM M M fM (M MM M M M, \* M M .M M MM M M M

A « I • « » X • 9 • -• • v •

4 • — • 4 • 3*

• ^« 3« X • *- • 3« 3«

. « * O » OS XI

ul 1i^ ^ o ^ ^ i>

'3

»Jl O 9

3 3 3 3 J .J ♦ ♦ ♦ ♦ .♦ ♦ •i ü J .J Ll J

O i 0 O » 1: I 1 3:'>» o-l r.j'xi >

» ^.11 l> W -J * > U .->-. T i) \i j o - o o

> »/> 3 Jl r* -* .M .M ■/> —. ,\j J. ♦ — "- 1 i M

M .-n -n -n -o i

3 3.33 3 3,3

•i m, «, * *■ •»■ ». »■♦ * X1X> X>X) X( JtlA ♦ ♦♦♦•• -»I* JO o o o o olo M \l M M >M MiM

i-i ~ — o JJ r ^ > -«ai fi ■»• _ j. ^ *xt."v,.oxi >r^ -»»«i. r*- 3 * !*•■ 3 n. jl r» ^ ,- en ^i^>

3 3 3 3 3 3

— — in > «I XI 3-- r ■awn • • •

» 3 3 3 3

'i t) Ü U ij » .1 | 5 -.

3 3, 3 3 3 31 3 3

^ J ,) J J J 0 0 > 3 -i Jl

— 1 .i .>J 3 * 3>- vl J J o !•( II ,'| _ j U||-«. (y » ■•) — u T _»» \J '»I \j .*"w *! J 4> W **» O"~''»o.')»|vi — :>o-«-/>

^ r- o 3 o 3 3 3 3 3 3

I

3 313 3 3 3

I» .O t — X Xl !\d j. o ♦I— X) n ', •} r~.f\ <\j

>» 1 >UO\J J-» -.^iJI >o.i ♦ * •^^no»'>'o^-<»

r v) J 3

^ ^ »^ 3 3 3 ♦ ♦ ♦ 1. Ü ij > / » 3 i) > — U Jl 1» « ^> — I O 3 O .1

3 J XI > u\ r ■Ul-T -lo n -i

3 33 3.3 3l ■>

■ A n> o o o t. -,» o

i I 1 - .^- X. %. >

■» -» \j n o J »J 3 •>. * -<

• • • • • »j • J > > > > J ^

• I M>i

i

J

IC-5

33 3333333303930330 33 3 9 3 3 3 3

I |

OO 3 3 3 3 3 3 3 3 333333333 »333333 • 3333333

O 3 9 3 3 33 3 3 3

—

3 3 3 3 3 3 3 3

"^•~0"-M>>M--»-nn ^—< O-n S-^r-M^-«/)-«« 3* OMO 3»1X>> M-»09 J> 3-«-« -«'3 ii »1« »i-* »jioi 3 •!<■»•-» /> T-iroii'Divj-« »IOIJ»"-» »IOI>>-< niOdj' ■«'»in »-•3 M ♦ioiO'-<v>in ^«x >-«-oin xii3.vii»--->i\i»"Oi»"-« »■I>I>I3-3 0-3-«-«-« -»M.XJM.M -niiii-ni*!».*!*.*!/» n n ittn

a o » — n M /> ♦ >.3 O * -Ni ^ ♦■-< n--J,>>-~'3».0«3 MA *■«-/>■«■ »lO-<-«*M •»-•1 a"-^ «»i\i.-«-« -«-• M on 3-« T/» r~'>.-. mn ^■ rtn «*.«.* «i »in n st n

0333330t030033|<303 33|303333|3 0OOa3i3^3303lO3'0O3^tO33000|r390303

9'M>'M3m-«mr)j>(M«dom'm«-M-«i«>4om»a>o-#r\i'«-*'>'0«r-'«3>4>ioa]',)or^«<3>'^>otn-«^'n<i>-«3>i<o.fl I M9i •»•3^'#..,\f3>io »■o>'3-<-oin^)i-«i#aMo;3ji'-««i-MX).» 3»-.#l-«3«ioi*i"v«3>r>-oij>i»*.ni'ni-ni'>i-n.-ni**ino

oe r-«*ioi\i''^j>JOi^oJi«n-»riM'«-<io3>^»j) f» r- »o -o ji ji x»,* <• ♦ncnf,»^n.,>jrw(\)rMrv»'\if\jr\irg-vj."v)i\)(\)f\jc\*fM i r-n>Dß4jo«-o.ojiinLninininJi>ninjiiA>n*-»j*».»a*»»^»»-^j- t t t ■* t ■* t t' * -t ■* ■$ ■*■ tut -f •» & -t ■*

33 33 3 3 > 3 3 3 O O < 03 030003333I090 000

..

3 - 9 0901030300(030300000000

«ri*r>"u-<oa>io^«inin-» •nrvi'-«o>«f"OX»xi* rVtM -*o(^a>r-«j»j>-»mi\»^'3>s)f«-<onnjt-»(n(M-«o3>Bf^vOinln nc ^.I)3»33l-<M',1»inOir-XI0>33-«Mr)»J1J)»-;XJ-I»39-«M"^*J>-0^-Mi3'33—(\)rn.»/«a^--0 5>t93-«.M'n-»/l «•■ni M-« »»^IOI^-'OO^-'OIM—• *■•• >••)«• •ifl »-'niNj.-« 3-3j><»^ioin*,''>'M-«i3-3>'Oi^oin »■i»>i\<-»>i3>io'-oi/»**' noi ~' oi J>I »»I-« vi.iinnoi-oi^. >. 3-«ivj.m*-inoi>-<0'>>>> a-< Mioinnoi—'Oi^'i-a-» v».»>i»in3i~--oi»>i-»3-« g ni»in-

■9 3 3 3 3, 3 3 3 3 3 3 3 Of 3 3. 3 31 3 3

#M ■»•♦•♦i*'* ♦-»(■••••»•I »••* *■■». »i»i» •» IAA />i>tninj)rJtJiJ>J> n jii ji n /i .n ji JI

•i 30 33 39 9309 3 3 9'33|93 3 9'33 3 3)3 30 3 33

•1

♦ ♦ »»■»■»•» tt o s o a o B •Vl M MM M' M M

♦ •*■♦ .»i*i« «••♦i*'••.♦.-••

»•■»■»■♦*■*!•»■»■•■»■»• -»I **♦•♦* »I ♦*♦ »*♦

•r 3 3 3 3 3 9

«•■■»i-«- «■<»• * *<*■*■*'*■ t\*'-*>*-t'*-T *♦■-»■*■♦•■»♦' n j\ j\ A n n\f* f) nr> rt /> x> ^ n r>/>J> nj>n/>rj>infl

D D D 4>o oio o o o a a a a a o o oo a a a o m o o D O O OI O O D X> S OS O SO O a O O O soaio mm MM MIM'M-M:Mf\I..MMMMM \i M M M M 'VI M.M (MMMMMi.MMM MMMMMM MMM MM MMMMM MM

r^.o ■> a — — Ml'nfnmr'i »* a a o > 3> r 7>

ma T vi j- ji \J 3 3'-^ r svjsji ^ JI —• \j I- D J J> -» vi -u n —< —• oi o j vi v» u 911 •> r — n n u J U'viO uo u ~- r u o a

•» »»tn.'n'mVrVIVI»"-" 3^>»"~4>J1|.».-'>(\I— 3J>'i(~-i>»'^.M3J' VöiO"!— •>B-~/>">

J -> J J3 J3J333

J J t, J J J J nu »> J a .»

j

ft n.t It A al; V 4

Jl^ — ^ il 4» V| J /■ X ij u n'» vi — »Ji n. T •> .1 .i »

J ! i «I— Bun 3r- 0> -" » ^ 3 M 1 > o n M »>•

iJ ^ " I ■>- » VI

I 3 -• -• -«..y« \4

>>i^ >•>« >l> J»i« J'^^y^^^^-j«'» a a a ».> mo t> D on'

u J J ^1 L' JJ

f» O .VI * VJ *l ■u ~ a jt u T\

— ~. o n t nl 9 lit t T r

3 3 3 3 >

■< o > o ■n il IVI JI I VJ » T

-> J J J -> -> 3 3 J 333

O o o u o oon n in n n

3 3 3 9 3 :9| 3 3 3 .> > 3

J J J iJ iJ i.1 J Ü U J u JUiJJOiJJOilJ ■J-iryO-gvivi-j'i JM- •ton ^T«)a »m-«!^-« VIJU>«- g ."n a -> r > .vi /■ o 4 3 g — •vj — ''»f» ui 11 •- ^vi J J v *i "i — »- »ixi * g vi "i ii J> "1 ~ — 11 — U VI 41 T * *< "O »• -T— >> ■*■/>•• —

>>j>>o —■«-'•«•aoonnl."»»»» t t >^^•^•»->1'>■^^■o■^■n•n•n•^

3->3^.3.J3 33^ >3 I I (

3 3 3 313 3 3

Tl \l T O * — j> a, .»■»»« > » -•.> o •> 3( 11 tli ■> •> o r.

3 3

— r o n «i »

t r

3 3

■J 3 * »

3 3

r- St MJl

3 3 3 3 3 3

ig > o « — x O 31 u > -. ^ > O 1 3 O

g iiiii > •> u »-

3 3 3 3 3 3

.r» eil 3 r~ .n M » •- j g (i o r> J 3 ■>• » — » ~- » vi n w

g MI i •» n n

I

» « t n n o n o o o o i»!-» »-'O o o o

J J J .i iJ Ui .» il O .J u iJ, .1 .J'iJ U ,1 J U — g-o — 3l»-»o — lolot f^-ua-» o-'>iJ>3|Mt»aji jviyi-«^ r vio-n > y » 't 'i ii j> 'i i ■■»• o o.■»• > -"> u v» gi — 3 Al -.uii T1 n "M •! "Vi vi gl g "i •> "»i T JijO

» »'n,^'0'n'ni',i'>"ni"i'>'ini'ii*» ill I'nininin-nniiinii'nnrn

3 3 3 3 3 3

i I ^ a •* — a n J i a >i— r

»> 3 •> son 3 •» O •••• •

3 3 3 3 3 .9 > "3 3 3 3 3

I

■»- »gno .'iti-^— r~- ^vinos'*»» g^~»-<»>nT30">j v~»—»a'^ * >gnu "^nn »-«»•«■ 3gfiu 3 »IIO

3 3 3 313 3

»>3> 1» >• 3 Jli^^-^^M'VI VI g "»'»^ -•-«-«-< g g .vi M M IM M .v< M .g. g ig M ig

» t t » in ni/> g .g g ig IM .g g

**~± ^

:-6

33 30

J, 3"n on > D 3\J •O O

3030030 3 330*903 333 3 0i03 3 330303 •••♦♦»♦♦• ••!••« • l iJ iJ J J ii jj J u .1lvki.UuJ.1J /> \JJI -u-oi^-n-, 3_ xi^; n

■^nt)----.on/> oonjloo o *-« t> n — -o -* o a T o > -«

90 -3 9 3 9 <a OOO 3 3 aoo

<?3J0 3OO(O3.oai3ÖO0l0

in? 2i|r,0'>5D'^'0'n'n*,'ft''- a^ •jru -• 1 > »..öl- oi n ».»>. \i-« V3

Xt» 3 O > > »

- 3 3, 3 3 3 3

U--11 »'/I Ol ^( OlT. > > > J> > > > > • •• ••J.« 3 3 3 3 3a 33

u I

trtiA -O /I J1 ^1 IO * * t » J -» * ox) .o o s o a ■>IM r^ VVJ M M O D O O O o

M (M (M .\i M (\

• • • • • • Ifl X> Jl «^ 3 TO

ifl* * •» » ♦ "1

* * ■# a o o M (Ml V

a • - *- •« l« J» o »

O ■*; rvj j JJ x) ■^^»'■«1 .■»» "M -v

a o ■t Vi

•H u •»1

a a * » 3 j

iJ O .1 iJ ii a i o - •» -. ^ \i j> » T "» .y ^ r 9 J W f'l Jl

XI -<( 3 m i>J o » iV 3

•M "MM

IT»

I

- »•» > v » O n M ■> >. »

• • • v v y v u ^ ^

\

ii ii t \j •o ■>

» ii <> -ni • • > 3

V "» A» ^ _• 1 — i)

n no ni D u o

3 3(3 3 3 3

(\i a«« *,. xj.n m

• »"l •> •» a' r» vj

IM >'U At >M \l > T»! 3

\4,\l M •)

I

^k

I

lO

.if

i j -« • r' n j

J a. U! 3 i

— n: « ■

J ■>

— ^ r 3

I

n

i • ; ■»

I '

12 ft iz

«:• ' ü

i 3 — i»

QLI»

« X

X iJ

Ü

at

*■•

a o a t- o r ►—

at o

I i

r

X • »I* 31*

a

0.

J « 3 ■a x u. o

IS - B

I- ir

^ 4

•3

3« u. • ►-•

3« 3«

I 330333033 33 3 O O. 3 O 33 33 33 • III 111 III I I I III II II II* il .J i iJ aJ »J J U iJlAl Ü »J J »J »J J U U UlU tl U li »■no-««» n'"-—•o.\j*(\i>iOO-n/i>i-ni

».■»^l>•^o*■n♦l-<o»•>.o^t)•^ M -n x) n ■» -n o M Ti-. \i o. oni^ »i o o^t. j n in n \j o /> iix ii-ni*-uiOi,-,>i-<'>iO.>'"ni>-'M'«--"ni3 0ioiO"s-3

•I • O 30' 3 3IO 0'3 33aia33<330l

>.-4

3 3

-• O

■ni/>

>3 3 > 3i 3 • I* .tji.lt 0>3

n>- »WO«'

3 O 33

»■if» -<i3^.'r~'.o^""tr»-« O'*'».^-j* ♦■^- ♦ ■ni*!.— n-r^ >i'M ojni

o o o -<030i03000033l0009 3000000

oox> r-4>i/)>n^m:iM^3i>3>r>-4>ini/>« mM-^oj« -»r- oj» a 3-<"M n>»j»o~-n>> oo-^rxji'o^jij»»- »•> o 3"<r\mi* 3.-». o .^-oin ♦•••»IVM 3> >o->->o.n'»'niM.-«i33^D.~'Oin -•-•-•—•—•-<-•-.-•—.-•TJ-Vi\jrvj1\ji\jAj\i\jvji\j-'1 •!.-»» i-o mi*»

i ♦'♦•* lOin x»

I •♦ ♦ ♦ i » o a

IM \i M OOOUDOOOOO o x> i o xi X) o o u o o o o o o i a

-VJ AJ iVI \l .M \J \j -vj rvt \J >M r\J M I\| i IM M vj \J M .M M -VJ -vj \j -vj

* ■•»

hi«

3 3 3

O 3 33 3'3 3333333 33 3333t33

♦ »•-/*♦*'♦♦ t * -t ■*

■»■..» i .»i »I-»••*• *•»'* 9-1«« *. IA Jl.>A 41 .or» r» .o .n .n x» .n .n

o o o o o o o.o

'tlf»>J>3t)X',.'\J'\J/>—(NJ> >-3>^J) r> r<'i C D>"-*ff>m >• T O J> ~ ill» — — T) JJ l\) ~- J-»>- jr»l|n^3v— o*«

;> 3 3 .> 3 .y > J J 3 _> J -^ D J ^ ^JJJJ3i_-i

UUJi.iLliJiUüaJiJ'üiJüjJaJl<JuJJ 'J »i üü.Uüü J03 '-.TJt'l^vJO»» >ln»)rr'> >:-«OTO»jn » .'i n I'I T» j> ji x> >» •» J j« > ">| -• o ■»> i-n » '•> o o * * j> n m # ^»«all^ ^ fl Ji ^ ^»M ^-^ •» M «f9 -$ v*4* ^>^l'•»^ j*-»^*tT

■jji^-IJiuxjujirl— 'i u -• •» <j u .» i» vj vj i'iiA» j ^- » n vj'*> ttO— Ot— Of—'^l *-0»)iJ-Ot— »-otnj —

OOO oooan.-i-i»j-»n-injvjyj\j

..* »> ir> 3 M I«

■» ^ r

■n i •»

> 3 J i J ■»

I

> > 3 3 3 3

MTV o-» — onnjr-j-ioj u -»mij'^ — r-*- » vj ri ti -«^ 30 >0|!1 J »».»-< U » »1 U'lv —t» - » VI 11 O

•» ♦ t r t ft n n o o o o ^. >. ^ ~ o o

3 *| > 3 > 3

jv .O * — X J1 J 1 O 7v -« » > O 1 J O l>

• a a 4| s a

m o *» in »M > a * J vj n X) »t *l g » ~ * «>■! • J vj it ij >' n

O J> > >v Jv J 3

• - Ji J

r

M

r I I

3 33 3 3 3

I »> ~- -y o » i 3 o -n -«-n

in >I-O.'»M.

3 - 3 3 • ♦ U 111« 3f^ . O J> . * n • ft \t-

4> O r

C-8 I I

• 3 30 ' 3 3 O ~*'*'*'*-*'*'*-q^,*^~*^^

3333333330 i i :

3 3 33'3 3^2^^^t'^r-r-ff-•-—-.--I- 33 33333 3

yj o rt

>iSIM ,3'«--n

iJ il J ■0*3 n > M — T ->

3 — VI

«~ o »M 3 » •>• J«i3

■^ o 3/1 t'Oi • l>l

^3 30030333330

• -^ -^, -* ^^ -y

• * "» »

■♦•-• >J t

M.»>| M, V

iJ J O tl ij J /> -• »^ > ■ o "o 'O I -• ^> ■ 3 .M . O -• ♦ \J f> /> > \j - 1,7-. on. noi «>•>•-«.-m >JiM M.'VKn'l

30 ♦ 1 • Jj.tt r- -. M 3 >i ♦ -• >

3333 333333 3 3 o 3 3 o 0.3

1

< ^ /> a> o, <o > X) >■— -n. o 3 ♦ >

« a» o a> a) tX)p^>^ « —i —• o O o o o '••••■•••

I O O O 3 O O J

3 3 3 3 3 3 3

> "0 M /I D ^ O.X>i 3\J. »■♦■

I CT> Lf) Oi «39 <0 4> y> o o s

> ö j» r»' »\j M > O i^ i —•

O 3 -3 C- o

3033I33O03033300

xJ iJ J

1l* ■/! JII^Mi >r^ M

J u u > /> n » D —• —< 3 o <o o

•>• J>I M.D /i — • 3 \|

ü \ü

a> o « r- mo » \j ^ IO »*> n a o o o

o-»- a

l:30|iaJ>'>3(3333 3 3 3313

» O 3 ->- O » \J o -- o ^ > >' \J o J » o *"•» >iM *lOi O O O r- p~ — • • • •

' O 30 3 3)3

o o a 3^ot3o<

'n-vjnj'vrvjfu.M,M,M>j_,X-,^Xt"zIi'il »•'>''>'^M MM-•-•-.-.0303

33 0,3003 30 30 OO O »OO 30 00 Ooi io O io io' O ^3*

1^1 "^ -o ■n ♦ * q • * • • • • -»3 3 3333

Jj n r» .n • » »• » a o o o ntM .>» -vi

* » VI .M

♦♦♦»**! ♦♦*j»y)j>

3 333)3333

>n| /> ■/» ^

so« "VIAI rviM

/».n ji, 'O j» n ♦ ■*■?!»♦,♦■ 0 0 0,000 >» ■>< .\« "M -Mi M

n .n ft jt i\ st\ n j\ 0000 • •••mm' 3 33 30 3, 3 3 3 3 3 3

- u t v * ji \i a 3-. » a ü

000 IM -VJ

o o a o o ö Al M \i -M \i M

i^^l^^^^^-^-^^^ 3 >| 3 3,3 «»|3 33 3.3 3

33333 333 3 333

■3 n f» .jr»; _ v,.0 -o, „ ,, c, ,,.'o in _ _ J •

mmm&Mmmm X) ! O O o o o U,^»»» S^^So »^'^Uo*!

M W M V x m

U Nl M S. 3«

M \l M -U MM M M^ M M M ■Vi M Al M M "M ■010

^iiH^Hni^fi^^^^ilsilllHlH^

I 9 > *

a; tr,

V,

0 3 0 9 0 0 ■» .» » 3 4 a ♦ ♦.♦..» -J u J J U u -- ) - g >> » ^ i/ ^ H M «g *• -'•' • K « r v. -' -1 u » ' » o 1 »I ♦

o ~ 00 n| n • ••••/•

- tl JV n JlJif)

o o 3 -> ♦ ♦

r •>

■O .1 il J

■m-f • • ■» >

00 3 o o o o o 33JJJJJJ ♦ • •;♦ •* ♦♦♦ UfuJ ^ U AJ J xJ u ^J y * u(r^ 'IO g** ^—Ol 3>.1X>J>XI i> v»l/J>|»J> vi -i /• r U U -. - T «II

000 J J J

o o 3 O

li Jj

1 • -» 4

c 1 J — t - V< II

--^^z-:-jia^z:if.i2a2iii»ii^i££i£i

\v

\

MO» V » -« > II >

# t--n-»> OTMM

^j 3 ^ , 3, , ,, ,

ü - > f t> «

—1~ "I -« üi» 1 M Ml J MM • I • • •

" »I *

!♦ ♦ ♦ ♦

O »j - J- o t < o o

J

•i ji 1 <• .■>

it j- Ji u J> u

J ^ M -• • • •43

I

O O O 0 J 3 3 3 ♦ ♦ ♦ ♦

n r n ; » j» i»- o o •> * — r — ft u

•*■ -^ o n

o o ^ 3

000

o 'ni-»-

« » ♦ ♦

o o a .

III U • "O t •« > O III o « •*

3 3' > 3, 3 a

*'•- ST) O > — f "» a o; n »■ " —I »

- * M M > ^ - ' VI • •

o o

1/1 vi n v » -• >i o

^1 j j MOO *> » — U 11 r-

— il — n M M

-1 J n .1 M Li

a » a >

« * -, "n u ^ O 1 ». •» Ü

o o a _> ♦ ♦ AI *i

VI 4J * Jl

a a

o n -1 a

•1 n vi n

» J-

n n ♦ ♦ xl O

» P- JLil ' MM

o n 1

.' a

3 M

u n o Hi

n />in o n n n nn aa^iijjaaj

ilxIklUtiJJUJ'iJ T-<r»—i"> >-»J3 •»iu'~m — ■») n^, j, — Jl "l TiV .j » i: —

0*1— J^iOt» ^*>

J a

■Ol 3

T O ~ J J » 3 1

"I W ^ X> fvj > •I il I» »1 •

* M II

VI VI J -4

M M, ■« M "M 'VJ

o »'— r n n 1 ii ^ _« « «^

on > o o \J ■I O 'i" — T "^ • • • • • • VI VI M 1 O •> vi (M vi 'vi. g M

• "^ -^ N» r^. N.

• • • • • • > 3 a 1 a 1

»M 11 U » - » -. • vi n u • • • •

VI(M|IM M

» -nlo * 3 •»

> I'o m m\ m

■\ iin vl "V fM

^k

(fc-9

»M M AJ V ■VJ .M fM

it O U I AJ ^J U xl Utl^JjJiJtJ^JiJ »M »«- 3-<D-«> 3 -- ■ -VJ n M >

*—'n o > »o t>i'nip- « c * > o o -« -*» 0033>>r^->JD - - fl> o>j ->^ _ f> j I,^,_ n 3 ». ") --J^c-vj.fi >— *0>'-«,mOit)l03 3 03-<-« •~s r c s > >IJ>.>I-« -i

03 ooooooooooqotso

-4OD»~ ^> in ■» ^ M —• o » j> 3j aji«- ^O >■> >.> 7.13« >i3> 3»,^ t> XI X)IO ^^-H oooo^oooooaooo oo ooooojOooooaoao

Ooooooooooooaooo

*">- O •> 3 3-N| M n * .•> o f-

s; j> o a > > >

43 3 3 3.

J*J» n n n n y» ♦ •»»*♦» a o o o -a o a ■VIM M M M «M M

•j» * 3 in » — c

-) u o r M J v j}

3 3 O O o 3

ft '.o .*> n .n 'ft ♦ r * * * « o o o o o o ■U MM M M M

M >H O o > o -4 3 3 Oi T> 3- > > O

3 O

*■•■»• ft J» ■«• -» o o M M

n t » r .r * m yt rtrn

r

lift n r» o ♦ • ♦

Af O ij ^ u *■!-• O .■> j

ig o — > u ■»1 n » II u f> ^. o r> T o o o p n

O ft 3 3 ♦ *

- n

i> »«• u — ■»» •»

o r>

^ * > * 3 3 3

.5 .ft ••> J "» J» — » ^ » M ft

*> \i n o »

3X1 Ol » -V) ■n M M|M M

ft ftl ft ft /» 1 3 3 3 3 3 4 ♦ ♦ !♦ ♦ il il J J tl U ft ft| > > ^ ft > 301 n » j

.ft j*.ri u -i j* V —i— • i ». o r o o o ft

r» o n o ~ >* > J M M M IM

> 3 - 3 3 3

S 3'^ o •» — » i^ 3 ■*>3ft »Oft Mi» o »'no»»—)»».!»

o o •3 M

3 3 ♦ ♦ U u _ M »ift '3 -ftil«-

p > ft ft

— D O O O K M MiiW g r\i [\i \j

I i

i

-fc. I t ■ i*