AD—A056 295 NAVAL POSTGRADUATE SCHOOL MONTEREY CALIF

- - - -

F/S 12/1A COMPUTER SUBROUTINE FOR STRESS ANALYSIS OF ROTATING. NEATED D—ETC(U)MAY fl .1 E MOCK. R E MOWN

UNCLASS IFIED ,es—or7 e oSa P4.

ENDDATE

flLNt D

8-78000

I

S

_ _ _

L ~~~ 11 2.2

I I. ’ *

I ~Ii~L8

11111’ ~ HIII~ . HhII~MICROCOPY RESOL UTION TEST CI-I~~ 1

NM~ONM RUREAU M SrAt~ J4R OS

—~~~~~~ ~~~~~~~~~~~~~~~~ -~~~—-.‘

LEV~LI~NPS—69—78—012

NAVAL POSTGRADUA TE SC HOO LMontere y, CaIifor n~a

D D Cj i~ _ _ _ _ _ _ _

~~L6Ur~~~D

A COMPUTER SUBROUTINE FOR STRESS ANALYSIS OFROTATING, HEATED DISKS

by

John E. Brock

Robert E. BrownI

May 1978

Approved for public release; distribution unlimited.

Prepared for:Chi ef of Naval ResearchArl i ngton , Vi rginia 22217

78 07 10 019L~.. - ~~~~ ~~~~~~~

~~

.

--

~~ I

NAVAL FOSTGRADUATE SCI-[)OLMonterey, California

Rear Admiral Tyler Dedman J. R. Borsting •Superintendent Provost

A COMPUTER SUBRCJJTINE FOR STRESS ANALYSISOF ROTATING, HEATED DISKS

This report gives listing and instructions for using adigital computer subroutine for finding stres s distri-bution in a thin rotating disk with nonuniform heating;the probl en is axisynlnetric . An iterat ive method is used .Theoretical back ground is given.

‘

~7~~~~es~~~~~~~~ c!cai~~~~in±

Approved by:

( E ~~~t _ _ _ _ _ _ _ _

Allen E. Fuhs , Chairman W. M. Tolles

Mechanical Engineering Department Dean of Research , Acting

NPS— 69—78— 012May 1978

- - ~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - __________

-~~~~~~‘~~~~ - -

UNCI ASS IFTEDs~ eo~~ rv CLASSIFICATION OF THIS PAGE (W1..~i Dat a Ent.r.d) __________________________________

- f t -~ / REPORT DOCUMENTATION PAGE BEFORE ‘ORMI

~~~~p~~~~MJ~~aER GOVT ACCESSION P40. 3. R!CIPIENT S CATA LOG NUMBER

-‘ i~~

S_ 69_ 78 .øl2L.L~

) 4~~ TT’T~Lt (ond 5s~~tftt.) ,~~~~~~~~~.—.-_~——-.-—-~~~ —-.~~ 5. TYPE OF REPORT & .cmo o COvEREDj - -- —

4 COMPUTER ~JBROUTINE FOR STRESS ~NALYSIS OF ~1~O?ATING, H~ATED DISKS )

— -~~~ 5 J I. PERFORMING ORG. REPORT NUMBER

- .

.

.

- , - - - - . - - - - - . - --. - -

~\AuTHOR(a) S. CONTRACT OR GRANT NUMSKR(I)

~jo~n E./~~~~~~\LR0be~t E.fBrownj

_________________________L..PERRORMtP1O OR4ANIZ.~ TIQN NAME MAD 4DD~U3 10. PROGRAM ELEMENT~ PROJ ECT . TASIC

Frotessor Jonn t~. 1~rocic ~~oae ~ i1~c) A REA & WORK UNI T NUMB ERS

Department of Mechanical Engineering 61152N , RR000-01 -0lNaval Postgraduate School N0001 478 WR80023

II. CONTROLLING OFFICE NAME AND ADDRESS (

Naval Postgraduate School ~~~~~~~~~~ ~~!~

‘~‘ 78 -

Monterey, Cal i fornia 93940 13. ~TJM ,~~ #~~ AGES

14. MONITORING AGENCY NAME 4 ADDRESS(i t dlft.rsnt from Controlling WIle.) ¶5. SE URITY CLASS. (of thu repeat)

Chief of Naval Research -

Arlington, Vi rgInia 2221 7 - - IIS.. DECLAS ir ~~~~~~’::~~~~ RA DING

SCHEDULE

IS. DISTRIBUTION STATEMENT (of hi. Report)

Approved for public release; distribution unlimited .

IT. OISTRISUTION STATEMENT (of the ab.t,act entered In BIeck~~ , II dlIf. r.nt trou t Rep ort)

J IIIIIT,IS. SUPPL EMENTARY NOTES

II. KEY WORDS (Continu. on ,.r ra. aid. it n.c.. . y end Identify by block ni~ ib.t)

Stress analysisRotating disks, Heated DisksAxisyninetric Elasticity, Axisyninetric Disks, Elastic Disks

20. A~~~TRACT (C.nlinu. on rov r.• aid. If nec~e e y end ld.ntify by block nomb.t)

This report gives listing and instructions for using a digital computersubroutine for finding stress distribution in a thin rotating disk withnonuniform heating; the problem is axisynimetric. An iterative method isused. Theoretical background is given.

DD ~~~~~~~~

1473 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ UN~~~JIF1Et) 1 ~ç~

RITY CL I CATION OF THIS PAGE (Blisn Data tnt.rsd J

~r ( -~~~~~:

_._ ___~_

_ _ ~ _ ;_.~~ - - —-~~~~~--—~-— - -~~ --. -~,-~ --~ —~~-- -. —.—-- --- - - -

~~~~~~~~~ ~~~~T2=~~~— --

~-— -

~~~~~~~-----

~~~~~~~~~~

TABLE OF CONTENTS

Introduction 1

Fundamental Analysis 2

Power Law of Thickness Variation 3

Exponential Law of Thickness Variation 4

General Law of Thickness Variation 5

Computer Implementation 8

References 9

Appendix A 10

Appendix B 13

Appendix C 16

Initial Distribution List 20

~~~~ lIu*ii R I~PW kdlsiSN WI $INN tj

0MTIFICAIII ____..

4 j ...—.. — ..—...—

IL. .—. .. _~~~~~~

_

•muIswN/*AgIu~y NIS

11.~ LUlL U~/~ ~~~~

•

—

~- . . . ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ -

r ~~~~~~~~~ ~~~ —‘ —- —--

~~~---

~~ -

~~~~~~ -

~~~~~~~~~~~

A Computer Subroutine for StressAnalysis of Rotat ing, Heated Disks

byJohn E. Brock arid Robert E. Brown

Introduction

Although, as is irxlicated by the title hereof , the principal

purpose of this mono~~aph is to present tested arid proved digital com-

puter software for the analysis of stress in a spinning axisymetric

disk having a radially variable thermal strain field , the opport unity

is also taken of developing the theory and cresenting some analytic

solutions .

The method developed herein for computer analysis of disks

having a general law of thickness variation was suggested by the al—

goritirn contained in reference 2 arid it appears to have advantages

over such procedures as that of M. Donath , reference 3, which has been

widely circulated in a book by S. Ti.moshenko, reference 5.

In what follows we 1z~rnediately obtain a second order linear

differential equat ion with dependent variable u , the radial deformation,

and r , the radius . Analytic treatment is given for two part icular laws

of thickness variation. For the general case of thickness variation ,

the equat ion is recast as a second order linear differential equat ion

in which the dependent variable is radial stress, °r However, for

runerica]. treatment an alternate form is rr~ re useful and direct arid

this forms the basis of the digital computer software which is given

- .-- —-.--.~ — ~~

— - — —a-- ~~_-~-- —‘ — — ~~~~~~~~~ ‘~~~— ~~~~~~ g ~~~~~ — - .~. I__~_ ~~~. ~~~~~~~~~~~~~~~~ ~~~- ~~~~~~~

aid illustrated in the appendices hereto.

Fundamental Analysis

We presume that the disk is thin enough arid that the thick-

ness varies slowly enough with respect to radius that we are just —

if ied in neglecting all stress components excepting only the radial

stress a arid the circumferential stress cy~. Material properties

E , Young ’s modulus, aid v , Poisson’s ratio , are presumed to be in-

deed constant . The thermal strain field , aT, arid the density y

nay be specified functions of radius.

Two types of problem are considered :

1. Annular disk, 0 < a ~ r ~ b , with Cr (s) aid ar (b)

being specified.

2. Solid disk, 0 ~ r ~ b, with Cr(b) being specified.

We also use the symbols t = t(r) for disk thickness arid ~ for

angular velocity. Other simplifying notation will be introduced

later on.

Consideration of radial equilibrium leads without difficulty

to the equation

1 d(rtC)~~ = C — yw2r2 (1)~ 0

The thern~e1astic constitutive equations are

Eco = a o _ v C rr + ECLT; Ecr ar_ v a O + F ~T (2a ,b)

where the strain components are

= u/r ; tr du/dr (3a ,b)

Strain compatability leads to the equation

r ~F.(&a/r ) _( 1+v)(a~~ar ) (

~)

These equations r~ay easily be combined into the differential equation— 2 —

~~IiIillIL - — ~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~ ~ ..h —~~~~~. ~~~~~~~~~~~~~~~~ —~

_____ -..- ~- ~~~~~~

~~~~~~~~~~~~~ i~~~~. ~~T - -

u” + u ’/r — u/r2 + (t’/t)Uu ’ + vu/r — ( l+v)aT]* • (l+’u)(aT) ’ — (l—v 2 )yc~

2r/E (5)

where primes denote differentiation with respect to r.

Two particular laws of thic1~ ess variat ion permit sm i le

analytic treatment.

Power Law of Thickness Variation

Ift = t0(r/r0)1~ (6)

so that

(t ’/t) = n/r (7)

then equation 5 becomes

u” + (l+vn)u ’/r + (vn—l)u/r 2 = 8’ + n8/r — Ia’ (8)

where

8 = (l+v )aT , k = (l—v 2)y~2/E (9 , 10)

Equation 8 nay be rewritten as

~—[r1~~ ~~~r(m

~~~/2 u)] = r(2+fl~~~2 ~~ + n~,’r —Icr) (11)

where

m = ±v’(n 2 —4 n+14) = ±/E(n—2)2+’I(l— )n] (12)

arid either the positive or the ne~~tive sign nay be used Equat ion 11

nay be proved simply by performing the indicated operations arid com-

paring with equation 8.

The quantity on the right in equation 11 is well defined so

that the solution of the differential equation may be obtained simoly

by integration, nult iplication by rm~~, arid another integration. Two

constants of inte~ ’ation are introduced . For the solid disk (case 2)

— 3 —

-- ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~ -

~~~~~~--

~~ ~~~~~ -~~~~~~~~~~~~~~~_ - -~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~—-~~~~~-~~~—- -

—~~~~ ~~~~~~~~~~~~ :.~~ ~~~ 2J~~- --

~-

~~~~~~~~~~~~~~~~~~ _______

u(0) = 0 gives one condition arid the second canes from satisfying

the given value of ar(b)• For the annular disk (case 1) satisfying

the given condit ions ar (a) and Cr (b) permits evaluat ing the con-

stants.

An exa -e w ithn= —0. 142 s given in Appeidix C. Note

that the case n = 0 corresponds to a disk of uniform thickness.

Exponential Law of Thickness Variation

Ift = t0 exp ( —mr2 ) ( 13)

where m is a constant of appropriate dlmensionality, then

(t ’/ t) = — 2rm (1’I )

and equation 5 becomes

u” + ( h r — 2rm)u’ — (L/r 2 + 2vm) = —2rm8 + 8’ — Ic’ (15 )

If additionally w~ assume that B’ = 0 (which nakes the thermal

strain field constant — a triviality) the solution is simply

u = r(8+k/an)/(].+v) (16)

In this case we can easily find

Cr = C0 = yc~2/�n (17 )

which is independent of r. Thus , if an allowable norma]. stress CAis specified arid if blade or bucket loading at r = b is w (pounds,

say ) per unit circumference , then a disk having thickness

t = (w/a~ ) exp [(b 2 — r 2)( yo~2/2aA ) ] (18)

will be such that Cr If the failure criterion is the

naxinun shearing stress criterion (Tresca ’ a condition) , it is clear

that this disk is optinal in the sense of having least volume arid

- ___________ — ~~~ L.~~~ -~.-~~-- —j-

~~~~~~~ -~ - -- - —- - _________

:—- —- - --

~~~~~~_~r~~~~~~ - - -

~~~~~~~~~~~--- -

~

--

~~ —

thus having least weight. If the failure criterion were that of

von Mises , a slightly lighter disk would suffice.

General Law of Thickness Variation

Equations 3a and 2a give

= rEcT + r(a0_vclr ) (19)

arid by differentiation

Eu ’ = FaT + rE(aT) ’ + (Ce—var ) + r(a0 ~var ) (20)

~~om 3b and 2b we also have

Elit = E C L T + Cr _ V C O (21)

Subtracting 21 from 20 arid rearranging gives

E(aT) ’ + (l+v)(a 0~ar)/r + a0 ’ — va ’ = 0 (22 )

Equation 1 nay be rewritten as

rar ’ + 1’~Fa + ~‘o~2r2 (23)

where we have written

v t’/t (214)

for convenience. Differentiat ing 23 we ~et

a

~ r’ + rCr” + VCr

+ I’VCr ’ + PV ’Cr + 2y~2r (25 )

arid substituting 23 arid 25 into 22 gives

rar” + (3l~1~~~r ’ + [(2+v)v4rV ’)a + (3+v)yw2r + E(ctT) ’ = 0 (26)

This is a single differential equation with dependent - iar-

table a arid can be dealt with by standard mmerical methods . The

conditions for evaluating the constants of intepyation have been

mentioned earlier .

However, the preceding procedure is not particularly sat Is—

factory . For one thing, the solid disk case for which r can become

— 5 —

- —~~ — —n -- — — ~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - - - - - -

- - - ‘~#- -

~—,

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~-

~~~~~~~~~~-~~~~~

- . - , - _ .~~.

zero encounters numerical difficulties unless special treatment is

employed to evade them. More important ly , however, if T is given

by a graph or a numerical table , determination of’ (ST) ’ may involve

numerical difficulties which, ultimately, are due to our having per-

formed the differentiation to arrive at equation 20. Accordingly,

an alternate procedure which adheres nx re closely to the fundamental

mechanics of the problem is described in what follows.

We consider the annular disk first arid we represent the Un—

known stress difference C0~Cr in the form

= (A+Br~)r (27 )

where A arid B are unlax wn constants arid r~ is an unIa~ wn func tion

normalized so that

n(a) = 0, r i(b) = 1 (28a ,b)

Initially we make an assumption for r~ taking a linear var-

iation in the absence of better information. By the use of some of

the preceding equations we will be able to construct an improved

form for ~ arid will iterate until there is satisfactory convergence.

Letz = Dilr, w FaT, B = yo~ (29 ,30 ,31)

noting that w arid S have different meanings here than when they

were used earlier • We can recast equation 1 as

d( tar )/dr = (a0-~~ )t/r - yw~rt = t (A+Bn—yc~2r) (32)

Before performing the indicated integration we introduce two

convenient notational devices , viz .

p... = J ... th’~ *... = ~~‘~~r=b (33,314)

0Thus, from equation 32 we have

— 6 —

- - ,- — — ---~~~~~~—~~-~~~~~~~ ——-~~~~~~~ - ~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ -~~~~~ -- —-

r : - -~~~~~~~~~~

----~—--

~~-

~~~~~~~~~~~~~~~~~~~ :::-:-

____

tCr = (tCr)a + Apt + Bpr~t — psrt (35)

F~ran equation 14 arid equation 27 we have

dzldr = —(l+v)(A+B~ ) ( 36)

so that

z = (Z)a

— ( l+v) [A(r—a ) + Bpn ] (37)

Equation 2a is

= w + (Ce_C

r ) + (l_v )ar = w + (A+~ i) r + (l_V)Cr (38)

so that

(z) = (w) + Pa + (l_V)( Cr

)

(z)b = (w)b + (A+B)b + ( l_V)( Cr )b (140)

Evaluating 38 at r b and usIng 39 and 140 gives one equation

involving the unknowns A and B. Evaluating 35 at r = b gives a

second • These equat ions can be arranged as

[ *(Pj ) *(pflt) [Al = [*(pBrt ) + (tCr )b — (ta )

[ ( 2+v)(b— a ) b+(l+v)*(pn ) LB.1 L ) a_ )

b+ _ V

r)a_~~r)h~

(‘41)

so that erie can easily solve for A arid 3. Then Cr Is obtained f’rori

35, z is obtained from 37 arid 39, arid (Ce_Cr) and a0 are obtained

from 38. Then a new function n is calculated from

(142)

Using the new ~ the process Is iterated, convergence being

n~nitored by examination of the sequence of values of A and B that

are calculated . When convergence is satisfactory, the desired

functions Cr arid a0 are at hand .

The sItuation is simpler for the solid disk, case 2. (ar)ais not given but conditions of continuity require that (a0..ar )/r

vanish at r 0. Thus A is zero and B can be obtained fi’om

— 7 —

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~~~~~~~~~~~~~~~~~~~~ —~~ — ~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ -~ - -

~~~~~~~~~~~ - -

(w)0 — (w~ + c [l— (t )0/ (t ) ](ta ) + c*(p5r t)B b b r b ( 43)= b ÷ (l+v)*(p~ ) + c*(pr lt )

wherec = (l—v)/(t)0 (144)

Otherwise the procedure is as for case 1.

- Computer Implementation

The theory embodied in equat ions 27 through ~4 14 az-d the

associated procedure has been progra~mied for digital computer

using the FOR1’RAN language. An initial prograninlng based directly

on the preceding equat~.ons arid written in January 1978 by the

junior author hereof has been supplanted by a newer prograr~rdng

which is somewhat more compact due to the employment therein of

ancillary subroutines developed for use in another problem (the

lateral buckling of elast ic beams) on which we ar~ working. This

program, actually a subroutine called FtODISK is listed in Appen-

dix A hereof. This listing itself contains corrrrient s which ade-

quately explain the construction of a MAIN Program which supplies

necessary input information and which invokes RODISK. Appendix

B lists the ancillary subroutines . In each case coninents indicate

the purpose arid employment of the subroutine . These may prove

us.~fu1 in constructing the MAIN program. For this reason , the

ancillary subroutines DUPV arid PRIS! are given even through they

are rot called by 1~DDISK.

In the canputer implementation, the various funation8 of

r which appear in the theory are represented by vectors the -le—

ments of which are function values at equally spaced values of r.

— 8 —

I~~~~~~~ - - - -

~

- - - --- -“ ~~— - —

~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - --- -

~~~~~~~~~~ —- --

~-~-~~~~~

---—

The ancillary subroutines manipulate these vectors. All of these

are obvious except possibly fl’IW which performs an integration by

use of Milne’s formulas, cf. reference 14,

In the subroutine T~ DISK there is a slight departure from

the theory as given herein. As a first step , all quantities arid

functions were “dedimerisionalized” but otherwise the method is

just as described above. Somewhat finer details of what was

actually done may be found in reference 1.

Appendix C contains some examples arid remarks concerning

them.

References

1. Brook, J. E., Stress analysis of a~cisynvnetric rotating disks,

submitted for publicat ion .

2. Brook , J. E., A method for ana lyzing axiaymetric p lates with

complicating conditions, J. Appl. Mech., Vol. 29, larch

1962 , pp. 1 — 6.

3. Donath , M ., Die Berechnung rotierender Saheiben und Rinqe,

Berlin , 1912.

14• Milne, W. E., Numerical Calulua , Princeton Univ. Press , 19149,

p. 119.

5. Tinoshenko, S. Strength of Materials, Part II, 3r~ ed., D.

van Nostrand Co., Inc., 1956, pp. 223 — 228.

— 9 —

- - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

~~~~ rs B~s~’ ~ ThLtTY P1~a.CflCA~~~1F~D.M DDPY 7W~N,.I5HE.D I’Q I)I)C ... _..,. .—

Appendix A

Listing of subroutine RODISK

(The listing on this page , page 10, is of the coments whichprovide instructions for the use of F~JDISK. Comands appearon the following two pages.)

S U B R O U T I N E R O D I S K . JOHN E. BRO CK. , 1 MA Y 1978THIS IS A SUBROUTIN E FOR DET E RMIN ING RADIAL AND C IRCUN —F E F E N T I A L STRESS ES IN AN A X I SVMM ET PI C THIN E L A S T I C D I S K ,R O T A T I N G A~ A NGULAR VELOC i TY CME GA ( R A D I A N S PER SEC CND)AB OUT ITS AXIS OF SYMM ETRY AND HAVING AN A X ISYMMETR ICDISTRI BUTION CF THERMAL STRAIN . TWO TYPES OF PRCBL FM MAYBE TREATED :

TY PE 1. fNNUL4R DISK OF IN SIDE R A D I U S A R A D ANDOUTSIDE R A D IU S BRAD. THE RA DIAL STPESS IS SRA AT THEINNE R RA DI US AND SR8 AT THE OUTER RAD IUS. THE IN S IDERAD IUS MUST FE GREATER T~~AN ZERO.TYPE 2. SOL ID DISK OF OUTSIDE RADIUS BRAD AND WITHRACIAL STRESS SRB AT THE OUTSIDE RADIUS.

THE USER MUST PROVIDE 4 M A I N PROGRAM WHICH CALLS SUBROUTINEPOD ISK AFTE R IT HAS SUPPLIED THE FOLLOWING INFORMATI ON.U ) N, INTEGER. (p4— 1 ) IS THE NUMBER. OF EQUAL SUBDIVISIO N S

INTO WHICH THE AN NULAR RA DIUS (BRAD M INUS AP .AD ) ISD I V I D E D FOP. COMPUTATIONAL PURPOSES. ~HE PRESEN TDIME NS IONING CA N ACCOMMO DATE N NOT LARGER THAN 101.

(2 ) BRA D( 3 ) A RA D (NOT NECESSARY FOR PROBLEMS CF TYPE 2.)(4 ) SRB(5 ) SR.A (NOT NECESSAR Y FOR PROB LEMS OF TYPE 2.)(6 ) TEEB EE , DISK THICKNESS A l OUTSIDE RADIUS(7) POTS, POISSON’S RATIO.(8) KP(1) = 1,2. INTEGER TO DENOTE PRCB IEM CF TYPE 1,2.(9,) KP(2), INT EGER TO PROVIDE FOP SKIPPING W H ILE PRINTING

OUTPUT . FOR EXAMPLE, IF N=1O1 AND K P ( 2 ) = 5 , ONLYEV ER Y FIFTH SET OF VA LUES WILL BE PRINTED : 1ST , 6TH ,... , 96 TH , 101ST.

(10) K P ( 3 ) , INTEGER .SP EC I FY IN G THE NUMBER OF ITERAT I CNSTO BE PERFOR MED. USUALLY K P ( 3 ) = 1 3 IS SUFFICIEN T FORENG INEERING A CCURACY.

(11) KP(4). I~ K P(4 ) =0 ONLY FINA L A N S W E R S W I L L ~E PRINTED.IF K P( 4 ) = l A SEQUENCE OF ITE PA NT V~ LUES WILL BEPRINTED, INDICATING DE GREE OF CONVERGENCE.

(12 ) VECTORS X (i ,J), 1=1 ,2 ,3; J=1,2,...,N .VECTOR X (1 ,J) CONTAI N S V A LUES O~ TI- E RATIO (LOCAL

THICKNESS r’F O!SK)/(TEEBEE ) COM PU TED A T EQUALLY SPACEDRADII STARTING A T THE INSIDE AND ENDING AT THE OUTSIDE.

VEC TOR x (2 ,J1 CONTAINS VALUES OF (GAMMA ) (OMEGA—S QUA RED )TIMES (BPA O—S QARED) DIVIDED BY (SRB). FOR MOST PROBLEMSGAMMA DOES NOT VARY WITH RAD IUS AND THIS QUANTITY IS ACON STANT.

VECTOR X (3 ,J ) CONTA INS VALUES OF (F) (A LPH A ) (TEE )/(SRBJWHERE (E) IS YOUNG’S MODULUS , (ALPHA ) IS THE COEFFICIENT OFLINEAR THERMAL EXPA N SION, AND (TEE) IS TEMP ERATURE CHANGE.

THE MAI N PROGRAM MUST CONTAIN THE 5T .~T E M E N T S :IMPLICIT REAL*8 (A—H ,O—Z)INTEGER KP (4)COMMON X ,NCOM M ON /ONE /ARLD , BRAD SRA ,SRB TEEBE F POI S KP

FOLLOWING SUBROU fI NE RODI SK THE~~E AR~ S E V E R A LANCILLARY SUBROUTINES W HI CH PERFORM VA R IOUS OPERAT IONSON THE VE CTO PS X (I ,J ). THE FUNC T ION O~ EACH IS OBVIOUSFRO M THE LISTING . THEY MAY BE USED IN THE USER’S MAI NP R O G R A M . TWO CF THESE ANCILLA R Y SUBROUTIN E S WHICH AREA V A I L A B L E i~; THIS PACKAGE BL T W HICH AR ! NOT CALLED BYSUB ROUTINE POC ISK AS~ DUPV W HICH DU PLICATES A VEC TOR ANDPR IV WHICH PRINT S A V ECTOR.

— 10 —

-s _ _ _ —~ ~~~ -~~~~~

— —

- -

- - -~~~~

-

~~

‘~~Th PAGE I-S B1~ST Q1JAI.~If l ~~A~~1C~31~PROM COPY PlJB~US~iED TO DDC _~~~~ —

SUBROU TINE ROD ISKIMPLICI T P.EAL*8 (A—H ,0 Z)REA L *8 X (20 ,101 )INT !G~ R K P ( 4 )

~~~ ~~~E ,AsAD ,BRAD ,SRA ,SRB ,TEEBEE ,P01S, KPONE= 1 .O +3POIS 3 .D— l

Cs !E ) AP AD ZER O

RFiC A RAD/ BRADEN M=N— 1W R ITE (6 ,2 ) KP( 1, )

2 FC R M A T ( / / , 1 - ~X , ’ RO DISK PROBL EM OF TYPE •,t i, ’.’,/i)DC 5 I=1,NE IM =I— 1Y=E IM/ENMX ( 4 , I ) = R H O + ( O N E — R H O ) * YX (5,I)=Y

5 X (6,I )=YI TER=1 -ETA=X (1,1)/X (l,N)I F( K P( l h E Q . 2 ) GO TO 100

C THE PRO BLEM IS OF TYPE ONE : AN N ULAR DISKSRA T=S RA / S~ BC 1= (2.D+O+POI S ) *(ON~ —R F1O )CALL INTV ( 1,7)C 2 = X ( 7 , N )C 5 = X ( 3 , t ) — x ( 3 , N ) — ( - 3 N E — P O t S ) * ( O ~1E ~ SR A T )C A L L M U L V ( 1 , 2 , 8 )CALL MU L V ( 8 ,4 , 9 )CALL INTV (9 ,10)C 6 = X ( I C , N ) + ( O N E — E T A * S R A T ) / ( O N E — R H O )

20 C A LL INTV (6 ,11)C3~ ONE + (ONE_ RHO )* (ONE+POIS)*x (11 ,N )CALL MULV (1,6,12)C ALL INTV( I.2,13)C 4 X ( 13 ,NJD C 1 *C4—C2- ~ C3A = ( ’ 5$C4— C6 * C3 J /0B= (C1 *C6_C2*C5 ) /0IF (K PL 4 ) .E Q .1) W SITE (6,7) ITER,A , S

7 F O R M A T ( 5 X , I 1 O , 1 P2 E 2 0 . 5 )CALL MU LS ( 7 , 14 ,A )CALL MULS( ].3,15,B)CALL ADD V (14,15,15)CALL SUBV (15,l),16)S=CN E— RHOCALL MU L S ( 16 , 16 ,S )S = E T A * S R A TCALL A O D S ( 16 , 1 6 , S )CALL D IV V ( 1 6 , 1 . ,2 0 )l.j=A * RHO + X ( 3 , I ) - i ( O NE —PC j S ) *SRATCALL MULS( 1I ,] .1,~~)CALL MU L S ( 5 , 1 6 , A )CA LL A D DV( 11 ,16 , 16)S = — ( O N E—RHO )~~(O NE+PO I 5)CA LL M U L S ( 16 , 1 6 , S )CA LL A D D S( 16 , 1 9 ,Z 0 )S=POI S—ONECALL MULS(23,I8,S)CALL ADOV(18,19,].8)C A L L SUBV (18,3,19)IT E~ = ITER+1IF (ITER.GT.KP (3)’ GO TO 200CALL D IV V ( 1 9 , 4 , I9CA LL S ’ J BS ( 17, 17 A )CALL D IV S ( I 7 ,6 , b .~GO TO 20

— 11 —

_________________ -—

~

~~~~—0~~ ~~~~~~~ ~~~. ~~~~~~~ _______________________ _______

~~~~~~~~~~~~~~ -~~~--- - ----—--—w~ -P---- - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

-~~~~

IBIS PAG! IS BEST QUALITY PELCTIC.LJLZ7~OM CO.tY I 2~Is1~~ TO ~~Q

~~~~~~~~~~~~~~~~

100 C l = PCI S—0 NC2=ETAC 5 = X 1 3 , 1 ) — X ( 3 , N ) + P O I S — O N ECALL MU LV (I,2 ,7)CALL M U L V ( 7 , 4 , 8 )CALL I NT V ( 8 , 9 )C6 =JN E+X ( 9 , N)

105 CALL INTV(6 , 10 )C3= O NE+(ONE+PO IS )*X ( 1O,N)CA LL M U L V ( t ,6 , 1 t )CALL INTV( 11 ,12 )C4 X ( I2,N)D= C 1*C4— C2 *C3S RAT = (C5*C4_ C6* C3)/ DB = ( C1* C6— C2* C5 ) /DI F ( K P ( 4 ) . E Q . 1 ) ~S IT F ( 6 , 7 ) h ER, S R 4 T , BCA LL M U LS ( 12 , 13 ,B )CALL SLJBV (13 ,9 ,L’.1S =E TA ~ SRATCALL ADDSU4 ,20,S)CALL O IV V ( 2 3 , 1 ,2 0 )S=— 3* (CNE +POI SICALL MU LS( IO ,15 ,S )S= P OI S—ON ECALL M U L S I2O ,1 8,S)S = X ( 3 , 1 ) + ( O N E — P O I S ) * S RA TCALL A D D S ( 1 8 , 1 9 , S )CALL SUSV (1 9,3,19)S —B * (ONE +POIS)CA LL M U L S ( 1 O , 1 7 , S )CALL A D D V ( 1 9 , 17 , 1 9 )I TE R = ITER +1I F ( I T E R . G T . K P ( 3 ) ) GO TO 200C A L L MULS (4 , 16,B )X ( 16 , 1)=ONECA LL D IV V ( 19 , 16 , 6 )GO TO 105

200 CALL M U L S ( 4 , 7 , B R A D )CALL MULS (20 ,8 ,SRB )CALL MULS (1 9,9,SRB)CALL ADDV (8,9,9)S SR8/BRAD **2CAL L l’ULS ( 2 , 1) ,S )CALL M ULS (3 , 11 ,$ RB )C A LL MULS (1 ,15 ,TEEBEE )W R I T E (6,204 )

204 F O R M A T ( / / / f 1W R I TE (ó,205 )

2)5 FCRM A T (24X ‘RADIUS’ I1X, ’THICKNESS ’ ,6X ‘GAMMA.O ME C-A .SQ ’,8X,1 ’ E E .ALPH A .+ EE’ , BX, ’~~IGMA . Q A D I A L ’ ,7 X , ’S~ GMA. C IRC UMF’ )K SKI P=K P (2 )DC 210 I= 1 ,N ,KSKI P

I/ K S K IPW R I T E ( 6 , 2 1 1 ) J , X ( 7 , J ) , X ( 1 5 , I ) , X ( l ) , I ) , X ( 1 1 , I ) , X ( E , I ) , X ( 9 , I J

210 CONTINUE211 F Q R M A T ( h 1 O , 1 P 6 E 20.5)

R E TURNEND

— 12 —

- - - - - ________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - --- - -—~~~~~~ -

PluS PAGE IS B~ST QU1I~IT! P ICABLIFBOM COPY FUR1~ISH~ED 1’O DDC ~~~~~~ --

~ppen1ix B

Listing of ancillary subrout ines

SUBROLT INE ADDV (N1 ,N2 ,N3 )C ANCILLAR Y SUBROUTINEC ADC TWO VECTCR S TERM 3Y TERMC X (N 3 ,I) X (N1,I)4X (N2 ,I)

REA L*8 X (2),1-)1),SCOMMO N X ,N00 1 I=I ,N

1 X t N 3, I)=X (N 1 ,I)+X(N2 , I)R E T U R NEND

SU~ ROUTIN E SUBV (N1,N2 ,N 3 )C A NC I LLAR Y S UPFOU T I1~rC SU B TRA CT TWO VECT O RS T ERM 5y T R M

- -

C X ( N 3 , I )=X( ! 11, I ) — X ( N 2 , I )REA L ~~8 X ( 2 ) , 3 . O I ) , SC OMMON X ,NDO 1 I= ] , N

L X ( N3~~I ) = X ( N l , i I — X ( N 2 , I )END

SUBROU TI NE MULV (Nl ,M 2 ,N3 )C ANCILLAR Y SUBROUT INEC MLLT IPLY TWO V E CTO RS T E RM BY T E R MC X (N 3 ,I) X(N1,I)* X (N2 ,II H

REAL* 8 x (a3,t01 )~ SCO M M ON X,NDC 3. I=1 ,N

I X ( N 3 , I ) = X ( N 1 , I ) * X ( N 2 , I )RETURNEND

SUB ROUTI NE D IVV ( N1 , N2 , N3 )C ANCILLAR Y SUBR OUT INEC D IVIDE TWO V ECTORS TERM BY TERMC X (N3 ,I)=X (N l ,I3/X(N 2 ,~~

)REAL *E X (2O ,10I),S

• C tJMM ON X ,NDC I h=l ,N

I X ( N 3 , I ) X ( N 1 , I ) / X ( N 2 , I )RETURNEND

SUBROUTINE ADD S ( P J t , N2 , S )C AN CI LLARY SUBROU T I N EC AD D A SC4LAR TO EACH T ERM OF ~ VEC TORC X (N2, I)=X (NI ,I)+S

R!AL*8 X (20,101),SC O MMO N X , NDO I I=I,N

I X (N2 ,I )=X(N1.h )+SR ET UR’4END

- 13-

— ~~~~~ _______ ____

— - - ___________ -_~~~~~-_ _

~~~_

~~~~~~~~~~~~~~~ ~~~~_ . - -

~1~~IS PAGE IS BEST QUALITY PRACfl(!~&ftTLZ

QO~PY FLI~ USkisD TO J~DC

SUBROU TINE SUBS ( Nt ,N2 ,S )C A NCILLARY SUBROU T INEC SUBTRAC T A SCALAR FR OM EACH TER M CF 4 VEC TCRC XU%2,I1= X(Nt,I)— S

REA L* E X (20,101),SCOMMON X ,NDC 1 I=1,N

3. X (N2 ,L $=X (Nl ,I)-SRE TUR NEND

SUBROUTINE MULS (Nt,N2 ,S)C A N C I L L A R Y SUBROU TINEC MULTIPLY EAC H TERM OF A VECTOR BY A SCA LARC X (N 2 ,I)=X(N I ,I)*S

REA L*8 X (20,3.O1),SCOM M ON X,NDO 1 1 I,N

3. X (N2 ,I)=X (N 1,I)*SRETUR NENDSU BRO U TINE DIVS (NI,N2, S)

C ANCILLAR Y SUBROUTINEC D I V I D E E ACH TER M OF A VECTOR BY A SCALARC X (N 2 ,I)=X(N 1,I )/S

REAL* 8 X (20,1OI) ,SCOMMO N X ,NDC 1. I= 1 ,N

1 X (N2 ,I )=X (N 1 ,h )/SRETURNENDSUBRCUTINE DU PV (NI ,N2 )

C A N C I L L A R Y S U B R O U T I N EC DU PLICATE 4 VEC TORC X (N 2 ,I )=X (N l ,I)

REA L* ~ X (20 , lO lhSCOMMON X ,N00 1. I=1,N

1 X ( M 2 ,I )= X( M 1,f lRETURNENDSUBROUTINE INTV ( N3.,N2 )

C ANCILLA R Y SUBROUTINEC INTEGRAT E A VECTOR USING M ILNE’S METHOD

RE AL*8 X(20,I0t ) ,EN ,R,ADD ,N IN3 ,N TNQ,F IVO ,THT ’)COMMON X ,NEN= N— 1EN= 1 .O+O/E~l

NI N3 R*9.D+ONTNO=R *1 . 90+1F IVO R*5.0+ OTHT-J=R* 1 . 30+3.X (N2,1)=0.D +OX (N2 ,2)=N INO *X (NI ,1)+NTN O*x(Nl ,2)— FIvc *x(Nt ,3)+p *x (N1 ,4)N M 3 = N — 300 1 ( 1,NM 3KP1 K+1KP2 %+2KP 3aK+3

A D D~ T H T O * ( X ( N l , K P 1 ) + X ( N 1 , K p 2 ) ) — R * ( x ( N I , K ) + x ( N 1 , K p 3 ) )1 X (N2 ,KP2)ZX(N 2 ,KP 1)+A ClJ

X ( N 2 , N ) = X ( N 2 , N _ 1 ) + N I N O * X ( N j , N ) + N T N O * X ( N 3 . , N _ I ) _ F I V C * X ( N I , N _ 2 )L+R*X (NI,N—3 )

R E T U R NEN C

— lI~ —

— - —~ - ----i-- - -~~~-— - —‘—~~ -~~~• ~~~~~~~ ~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~~ -~- —~ • -

LZ

-

U.— LU..3

UXU 0 LU-J z

> -~ LUc~~~~ — 0 (~uZZ - IL

,_• LI) —

3 u-~ a~I-I

~~ ~dZ~~ . —~ .- Il)

~~ ~~i— )- 0 — —I— >~Zz-J U- •. IUI ~— W~~ U) >~ ~~~~~~~~ — 3 I—

0 — ~0flZ~- > U~ ~J ~-P )—U IL) ~~~ ..J Z

U0W~~~S4 L) LU ~)- W .-i>LLH— — U. C~__j > ~ z ‘-4f.- ww zw —

I I—LU I)- ~-Z~~ ‘-‘ U~t I— ~~— —.-,~~~—I—~~~— 2 ~ 3 2

LU — wZ’-”—4’-4’-4 — 0

0—a~~~~ cr —‘ —(D Q~Q- 0.0-0. LU C’~J0. I —

— z • —Jrn. t I~~

s.-. —3 ~ x 3

)Z I/) U)V)I#W)V) —-‘-4 ~J-J-LJ-J-J tt~ IX IX IX IX

i—i$—- ~~~~~~~~~~~~ — . 0 0 0 0.~~~ ~~~~~~~~~~~~~~ ~~c i— u— I— ~

—

Il) QQQ’~QQ c’Jrn4Ln .-(\j ~ ) L) U i..ZIXa. IUU ..WWWW —ft U lit ~J I LU Ui

OUOOLJO ZQ. > > ~~~~ >>:~~~,— .-‘ 1--1 1— 1-- l-- I--- ——4 — — —— —.-.V)I)O ZZZZZZ - 4 —~~ U-..-i W W W W W W t ~~0Cj Q0O .LA )X Z ( Z* —))(

• ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~ — ‘ 0 0 0 0c( 0 ~~~~~~~~~~~ — -rn -rn —rn -rn

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~-. — ‘- — — — -— —IXIXIXIXIXgXO-4(’Jrnttn •~Q~0 t’j ’~..~ rn’.. ~~~‘.. u’~~.

~ • ~ ~ ~ •.~ ~~~~ %. ~- ‘s

~~C~~~Q~~ QlI 0- ~O—~~~~~~~~~~~~~~~ O- -4~~ L) ‘-‘W ~~~~~~ .30C.)0L~~UJWW WI tJW~~. I—Z’~~ - — 1-- - 1.- ~~l—c z ~~ i ozzzZzz .. . . . . w w ~~ ou-<ow ~*c u <o

a ~~~~~~~~~~~~~~~~~~~~~~~--- -—-....— ~~~~~~~~~~~~~~~~~ ,~~~~~~

,~~~~~~

,~~~~~

IUD JU..WWWU.,LL.U..U- U-IL 0WIXU0i~~00~~~OC? IXC)0Z4/) ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

U.. U. IL IL. U.. U. -~~~ o~o ’J.-~ rn— ,~~.-. ~n -.‘-‘ ‘-“ — c’.j rn ~

I_)L) UUUUUL.)

— 15 —

______ -— -~ ~~~~~~~~~~~~~~ — - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - ~~~~~~~~~~~~ ~~~~~~~~~~~~~~~ ~~~~~~~~~~~ -~- -~

~ppendix C

Examples

The listing on the next page is of a MAIN pro~~am which

calls RODISK twice to solve two different problems. On sri I3~’1

360/67 canpile time (for the MAIN, RODISK, and the ancillary

subroutines ) was 12 s, link time was 2 s, and execution time

for both broblems was 1.5 s. In both problems we used N =

and iterated 10 times.

The first problem is of type 2 (solid disk) with b = 10,

t = IJ(1.6+ .008r2 ) , v 0.3, y~~ = 120, ar (b) = 1~IOOO , and

EaT = 25105 + 1300 lo~~(t) — r2(233+16t ) Units are inches and

pounds. The problem was made up from the exact solut ion

a = 9000 + 50r2 ; a = a + r2(120+16t)

The results, shown on page 18, show evaluations for the

stress cQ1~ponents which are correct within 0.03 psi even thougl-i

convergence was complete to only about L~ digits as indicated by

the sequence of values above the final tabulation; these are

values of ar (0)/ar (b) and of B.

The second problem is of type 1 with a 2.57 , b = 5.15,

ar (a) 18205, ar (b) 22000 , t — 0.1~493 r~~~2, T = 60 — l.6r2 ,

v 0.3, and Ea — 19~L3. The units are inches and pounds.

Since the thic~~ess variation is a power law, the theoretical

solution given in the body hereof nay be used to obtain the

theoretically exact solution •

a —113.95r2 + 15832r1’ — 1170 ~~~~~~~

122.80r2 + l3801r~ + 114310~~

— 16 —

~

._flt~~’ - ” _tS.~~..A_ .___ ~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~ - —~

— - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~ — -~~~~r .-—--- _— __ -- .~~

___ _ ._ - _-- -

THIS FAG! IS BEST QUMiIT! p&t-CtXCJ.~~~_._... 0~PY F B~N1SIi~D TO DDQ .~~~

—

C J. E. B~’OCK 1 M AY 1 576C THIS IS A MA IN PROG RAM TJ T~~~T MY SU3R O UTINE RO DI S K

iM PLICIT REA L *8 (A— H,O—Z~R E4 L*8 X (2 . ) , l ) 1)INTEGER KP (4)COMMON X ,NCCM .40N /ON~~IARAO ,BRAD,SRA,SR ~~,TEEBEE ,P 3r S,KP

C THE FIRST PROB LEM IS POP. A SOLID D ISKKP ( 1~ 2KP( 2 ) =4KP( 3) ~l0i (P(4) =1BRAD 1.D+].SR~ =1.4D+4N 4 1ENM=N -lDO 20 I=1,NEIM=I —1R E I M *BRAD /EN MTHICK =1 .D +0/ ( 1.6D+O+8.D—3 ’~R~ *2)X (2 ) ,I)=THICKX (2 ,I )=l20.D +3~ BR4D **2/SRBW 2 . 51050+4+1. 3O+ 3~ DL PG (THICK )— 1S**2) *(2 • 330+2+ 1.60-I-1*THICK )X (3 ,t )=W / SRB

20 C O N T I N U ES = X ( 2 C , N lTEESEE =SCE LL C IV S (2 0 , 1 ,S )Ct ~LL RODI SK

C THE SECO ND PROBLEM IS FOR AN ANNULAR DISKTãEBEE=1. 49 3D_ 1*5.15D+0*~ (—4 .2 0—1 )ENM=N -1KP ( l):1.4 RAD = 2. 570+ 03R~~O=5.I5D+ .)SRA= 1.82050+4SRB 2 .20+4BETA=5. )2473471C— 1DO 30 1 1,NE I M = I— 1Y= EIM/E NMXC 5, I)=YX( 6 ,I)=YR zA RA O + (BR A D_ AS A O )*YX ( 4 t 1 ) = R I B RA OT H IL K=1 .4 930_ 1* R** (_4 .20_ i )XC 1,1) = HICK /TEE~B EEX ( 2 , I ) = B E T AX (3 .1 ) = ( 6 . D + L — L .6D+O* P**2 )* 1 .94 3 D+ 2fS Rc ~30 CONT INUEC6LL ROD ISKST O PEND

— 1 7 —

- - - - ~~ - ~~~~~~~~~~~~~~~~~•~~~~~---~~~~ -- --

-~ - - - - - - - - - --- ~~- ~~-.-~~ - - ~~~~~~~~~

--~~~~- ~~

- -_

~-~~~

-~~~~ Z’)00 -~ O0 )O00o T0~~ O0 ~~Oo~~ 0

~~~P’ ~, o r)ocL r~)Jl~ J r~~~~l Q. 11’I “ n-O-t -o ~~~~~~~~~~~~~~~ “~7 .—~)~ J~ -4 O ’4C ’~.4 ~ 04 . - p ~~~— - ~~~~ pI’1J~~~~ ~~~~~ j~ —. O4~~c’1~’I 0 , 0

— f . _ ... ,o._o-? .p r— ,.Imo •S~ ~a- - 4 1~ ~~~~~~U~t ( ~~4 .0 4 ~~~~~~~~~ ‘fl.n .p~A.A -0

•I~> ~c~~ - F . . . .• . • i .~~~~~.00. 0. P44’.IeI (~~~~~~~~ SPJ I~J ~.Jf4. 1I’1N

In

4P~ ..O1’ mm~~Im~~~~ -t -?.P.~~

~*..t-4OC.)I 30

.U- U’II’~~ IA0IAO •~ 34. . 0 Or—~~ ,o0

.-.U3U’ ~~~~~~~~~~~~~~~~~~~~~In

us-,.,? -~~ .t.11.t-rmøs—4uj 0000O0’~~00’e0 W0000~~0O’)O00I- I p.•00I~~C.3O0CJ00CL0 •QC30 )Ci~~3QQ0L-30

40.00 ,0.op.P~InmmIAI.t.t ~~m~~~~’fl~~~’J-t ’ I0.0”? ~~~~~~~~~ U~,0(~IUI

4 .. r m(~J0. 4A c.sr— .o~~ .. t.O—.-Q—..OOcfltstJ’L’4.t ..w .0sn-r.r m

UI UI

Øp~~~~~~~~~~~~~~~~I’~JI%JrJI%4

~~0000O0 )0000 In0000~~000o00400 00000000

t,J0 ’L00(~0~~00~~0 t”~~~~~~~~ 0’ 0’U’O’ 0’0’~~o’~ oocto ooooo0O(~ 0OO0’)0000 0 ,0,0.0.0,0,0

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~• . • • .

4 4

~~c , ’ oo0o000 ~~00000001~ Oh u s h I S I S t i l l l i i i Ic~o oooooo0

It~-? OtfI ’.tt ’ oI\S-n ’ 1’I0.0 —I~~~ a - ~ ——.— -.—~—‘—.— .- -o— m m o 0 ~~ .0—P. 1C.JLMN

InOrI000000000 o.’o(~Lo~~ .t.t ~~ p—r-. In00~~I0000~~0OOt~ h i I I I I I I I I I I I~~ ’ I I I I I I I I I I

r ~-~~-c-ocor ’oo zc c.~~ot- ooooo c~j -~~...e00~~C~~~e ~~~ c-c r~~~~~~~ c~oo. . . . . • • ~~O’ IA ,(.I.*’O~~~ 0U” ’P • • •

‘)‘ .1’ LP Q~0.Q’ t..-0U~~~~~~~~ A,00O34,0 .0— —..— —~—.~~~— i~~~ hC. N~~~~-~—fII.OI I I S I I I I I I

~- ~~0P..t.-. p. O’OmO-P. ’ .LcO .OvI• . .s . • .* ~~~~• •

.0.~~CU~iA Ifl IAIfl *t.t.f

. 4 4 1 1 . . I .4(~I C~ ’~~~~00’~~0OO(J —~ ‘~ 00I)0O0O’~’Ou h t h l h I l t I ~~l ’ ’ ~~~~w C~~~~ 0oC.~0000 UI 00Q0~~ 00000. ~~~~~~~~~~ Id’mmU~ 0. , 0 , 0 . truI~1”t,- ~~~~~~~~~ 0~~~t,0In In ‘)00~~0000~~.-’ ). 0P.~~ InOOII0000’~~004~Ip ~ ~~000000 I— —I.t r4~-4~~0’ ~~0O. s000C,” OOf)

ILS. . ~~~~~~~~~~~~~~~~~~~U- t~~~~~~~~~~~~~~ 4~~ 0 00 000LJO (~~ U~G~ 000000C L)0000 . . . 4 ru0-%00~~~C~~’ h 0 0 . • . . . . • . . . 4(

~ O~~0000~~i0O~~~ .~~0C)h 4J00(J ’D ~~mr.44%l4~Jc’Ie.4c~a ~~ 0C~I~~C.IO0(_)L IQ0~~

0 0~~.0.r4’4.~Ui ~~0 0 0~)0~~0 W ~~~~~ P0,0-’t-ø~b0~In-~ 0.-I00~~0000~tO _I •.. .• . .• . .• . S

CI o m.sIn.or—So._ U0. 0.

In —~~~~.rv~~~r-su~e in— — .4

ao ‘I

50.0.4 .4

— 18 —

L~~~~~ -a.

-— — ~~ _ ~~~~~~ ~~~~~~~~~~ -~~~.- ~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~ - -~

“~~~~~ “~~~~~ ——-—- --,—~~--~~ - -—,. - - - ~~~~~~~ ‘~~~~~~~~“~~~ — ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ r’~ - ------~ _~,,,~~_ . —

where p .29171 aix! q —1.87171. The comput er results a~~ee

with five digit accuracy even though the sequence of iterants

(A ,B) shows only four digit convergence.

The same two problems were also worked with N = 101 aix! ~iterations • The execution tiit~ went from 1.5 s to 3.0 s. The

naxiiTun change in any stress value was 0.3 psi . From these aix!

other problems it nay be concluded that there are no difficulties

of accuracy , computer storage, or execut ion time .

Acknowledainent

Appreciat ion 1:8 expre ssed for parti al support by the

Naval Postgraduate Schoo l Research Founda tion.

— 19 —

— — - —~~--~~~~ - ~~~~~~~~~~~~~~~ ~—.~—- --.-—~~—- ~~~~~~ ~~~~~~~~~~~~~~ — ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~

— — - -—-.-— ---------~-.‘—--

~-,-

~—

~~~~~;-—_ -— -- - -

~ ---~~~---~ -—~~-~ --- - ~

—-- — ~~~~~~~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ - ,.

INITIAL DISTRIBI.rrION LI~r

1. Defense t~ cumentation Center 2Cameron StationAlexandria, VA 22314

2. Library 2Naval Postgraduate SchoolMonterey, CA 939140

3. Dean of Research, Code 012 1Naval Postgraduate SchoolMonterey, CA 93940

‘4. ChairTran 1Department of Mechanical EngineeringNaval Postgraduate SchoolMonterey , CA 93940

5. Professor J. E. Brock , Code 69Bc 15Naval Post~ ’aduate SchoolMonterey , CA 939140

6. LCDR Robert E. Brown, IJ~ J 2Naval Amphibious SchcslCoronadoSan Diego, CA 92155

7• ~~ Charles Miller 1NAVSEA Code 0333.Naval Ship Systems Conu~rx!Washington, DC 20362

8. Mr. Richard Carleton 1NAVSEC Code 6146Naval Ship Engineering CenterWashington, DC 20362

-2 0 -

- ~~~~~~~~~~~~~~~~~~~ . .~ ~~~~~~~~~~~~~~~~~~~~~~~~~~ -