On Thin Shallow Elastic Shells Over Polygonal Bases...ON THIN SHALLOW ELASTIC SHELLS OVER POLYGONAL BASES by DOUGLAS S. WALKINSHAW, B. ENG. ·A Thesis .Submitted to the Faculty of

ON THIN SHALLOW ELASTIC SHELLS OVER POLYGONAL BASES

ON THIN SHALLOW ELASTIC SHELLS OVER POLYGONAL BASES

by

DOUGLAS S. WALKINSHAW, B. ENG.

·A Thesis

.Submitted to the Faculty of Graduate Studies

in Partial Fulfilment of the Requirements

for the Degree

Master of Engineering

-McMaster University

October 1965

MASTER OF ENGINEERING (1965)

TITLE: On Thin Shallow Elastic Shells Over Polygonal Bases

AUTHOR: Douglas S. Walkinshaw, B.Eng. (McMaster University)

SUPERVISOR: Professor G. ~. Oravas

NUMBER OF PAGES: viii, 141

SCOPE AND CONTENTS:

This thesis proposes to demonstrate, by m~ans of

numerieal examples, the applicability of the approximate

solution for shallow,. spherical, calotte shells enclosing

polygonal bases for the purposes of practical design.

The theoretical solution is based on a collocation

procedure by means of which prescribed boundary conditions

are satisfied at discrete boundary points and is derived from

the general theory of MUSHTARI and VLASOV in which the trans-

verse shear deformation of the shell is neglected in com-

parison with its transverse bending and extensional surface

deformation.

ii

ACKNOWLEDGMENTS

.The writer wishes to express sincere gratitude to

his ~esearch supervisor, Dr. G. m. Oravas, for his in

valuable guidance throughout this investigation.

Grateful acknowledgment is due to Professor J. N.

Siddal for his assistance in the execution of the experi

mental analysis, and also to G. E. Riley and T. E. Michels

for their important contributions to this work.

iU

TABLE OF CONTENTS

CHAPTER

~ INTRODUCTION

II SHELL ENCLOSING HEXAGONAL BASE

III SHELL ENCLOSING RECTANGULAR BASE

IV SHELL ENCLOSING TRIANGULAR BASE

V SUMMARY

APPENDIX A

APPENDIX B

BIBLIOGRAPHY

.iv •

PAGE

i

5

72

84

95

98

111

140

u n

F

F ( o-) I F ( cr) I 2

F ( cr) I F ( cr) rr ee

F(cr) 1 F(cr) re er

F(cr) I F(a) rn en

M( a) I M( cr) I 2

M( cr) 1 M( cr) re er

M(cr) 1 M(a) rr ee

p n

NOMENCLATURE

Normal displacement

VLASOV's stress function

Stress resultants per unit length of parametric lines in middle surface of shell

Normal components of stress resultants in middle surface of shell

Tangential shear components of stress resultants in middle surface of shell

Components of transverse shear stress resultants to middle surface

Stress couples per unit length of parametric lines in middle surface of shell

Flexural components of stress couples

Torsional components of stress couples

Load intensity per unit area of middle surface of shell

Load intensity component per unit area normal to middle surface of shell

v

D

E

h

r

2- ..d._ d v =d-. d-ro ro

Flexural rigidity of shell

Modulus of elasticity of shell

/

CAUCHY-LAME elastic constants

POISSON's ratio

Shell's thickness

Radius of curvature of middle surface of spherical shell

Radial parametric coordinate of shell

Radius vector to arbitrary point in shell

Radius vector to middle surface of shell

Radius vector of arbitrary point in shell's surface relative to middle surface

EULER - LAPLACE differential operator

Parametric coordinates in middle surface of shell

Parametric coordinate normal to middle surface of shell

Infinitesimal line segment along parametric coordinate oc.

~

vi

e 1 e r e

e n

e s

n

J{x) n

Y(x) n

I(x) n

K(x) n

ber ( x) 1

n

ker {x) I

n.

bei(x) n

:Kei{x) n

i = /-1

Middle surface strain parallel to shell's boundary

Direct middle surface strain

Unit base vectors in shell's middle surface

Unit base vector normal to shell's middle surface

Unit vector in surface of shell tangent to shell's boundary

Unit vector in surface of shell normal to shell's boundary

Standardized n-th order BESSELL function of first kind of argument x

Standardized n-th order BESSELL function of second kind of argument x

Standardized n-th order modified BESSEL function of first kind of argument x

Standardized n-th order modified BESSEL function of second kind of argument x

n ... th order KELVIN functions of first kind .of argument X

n-th order KELVIN functions of second kind of argument X

Imaginary unit

v'ii

I I

ber (x) 1 bei (x} n n

II N

ber ( x) 1 be i ( x) · n n

First derivatives of n-th order KELVIN functions of first kind with respect to argument x

Second derivatives of n-th order KELVIN functions of first kind with respect to argument x

viii

CHAPTER I

INTRODUCTION

The purpose of this investigation was to show that the

•approximate theoretical solution for shallow, thin, calotte shells

of spherical middle surface, subjected to isothermal deformation

by uniform normal pressure, given initially by ORA VAS in 1957(1) *

and further studied by him in 1958 and later by RILEY in 1964,

will give reliable results for practical design.

TOLKE's Boundary Collocation Method forms the basis of

this solution whereby a rigorous satisfaction of the'prescribed

boundary conditions is collocated for a number of discrete points

located on the boundary of one of the shell's rotationally per

iodic segments. The solution obtained for the characteristic

segment of the shell is applicable to the entire shell since in

the Semi-Direct BERNOULLI's Method of solution of the partial

differential equations, the rotationally periodic symmetry of the

shell's comportment was anticipated in the choice of the direct

part of the functional form of the solution series.

RILEY's investigation raised certain fundamental

*References are given chronologically in the BIBLIOGRAPHY.

1

2

questions pertaining to the extent of the applicability of

the collocative boundary value problem to thin shells and

the general nature of its numerical accuracy.

It was subsequently established that the number and location

of the collocation points are not of such paramount signifi-

cance to the practical reliability of the results as was

. -initially believed to be the case. Instead, the reliability

of the results is largely dependent upon the degree of

accuracy employed in the numerical calculations. Therefore

McMaster University's I.B.M. 7040 computer was used to per-

form the extensive numerical computations required for the

requisite high degree of accuracy in the solution of the

polygonal spherical shell problem.

KELVIN functions, bern(x) and bein(x), constitute an

important part of the truncated series in the collocative

solution of the spherical calotte shell problem. The wide

range of the orders of magnitude of bern(x) and bein(x) over

their functional order n, produces widely ranging orders of

magnitude for the coefficients of the linear collocation

equations. A satisfactory solution of these equations as

well as subsequent computations required machine computation

by double precision techniques which employed 17 figure

accuracy.

Theoretical solutions by the collocation method were

obtained for spherical shells enclosing hexagonal, rectangular

and triangular bases. The distribution of normal displace

ments, stress resultants and stress_ couples .along radial

lines are graphically depicted for the characteristic segments

of the three calotte shells.

Comparison of the theoretical and experimental results

for the spherical shell enclosing an hexagonal base revealed

that the theoretical solution by the collocation method is of

acceptable accuracy in view of the unavoidable geometric im

perfections in the structure of the experimental shell and of

- the differences between the actual and theoretically conven

ient boundary conditions.

The results obtained for the spherical shell over a

rectangular base were compared with a solution by another

method first given by DIKOVICH in 1960. There are certain

differences in the results of the two solutions, some of

which, no doubt, were caused by the fact that the boundary

conditions of the two shells were not entirely identical.

It was observed that while the stress resultants of both

solutions were of comparable magnitudes, the normal dis

placement and the stress couples were of considerably

larger magnitudes for the solution by the collocation method.

The solution for the shell over a triangular base

has been given without comparison, as no other theoretical

solutions are known to exist for such shells.

4

CHAPTER II

SHELL ENCLOSING HEXAGONAL BASE

The experimental results given by RILEY in 1964 for

'j:he shallow spherical shell enclosing an hexagonal base were

used for the purpose of verifying the reliability of the

theoretical solution by the collocation method. Since this

method satisfies prescribed edge conditions only at.a set of

discrete points on the shell's boundary known as collocation

points, there should be some minimum number of points for which

the solution will become sufficiently accurate for practical

design. Logically, an increase in the number of collocation

points above this minimum number should only cause an insigni-

ficant increase in the accu:r;acy of the practical solution.

This section gives a thorough comparison of the sec-

tional resultants obtained experimentally and theoretically for

various number and distribution of collocation points (see FIG-

URE 7). The distribution of sectional resultants F~F), F~~),

M(9a) and M( o-) and normal displacement u are depicted graph-r er _ n

ically along radial· lines emanating from· the shell's apex (see

FIGURE 6).

s:

6

A vectorial representation of the sectional result-

ants is given in FIGURE 1. For a spherical shell,coordin-

ates 1 and 2 become the cylindrical coordinates r and e.

For this shallow polygonal spherical shell of 6-ply

· periodicity, the boundary conditions

F (a) = 0 (II-1) nn

fi(~) = 0 (II-2)

Un = 0 (II-3)

E ss = 0 .( II-4)

were used for all the collocation points except at the shell's

corners since the strain was certainly not zero at the corner

points of the shell structure. Therefore the boundary condi-

tion f = 0 has been omitted at that point. The boundary ss

conditions (II-1) to (II-4) appear as linear boundary equations

of the coefficients of the truncated series solution and are

given in APPENDIX A.

It was found that there were some incongruities be-

tween theoretical and experimental results especially near the

shell's boundary. The theoretical boundary conditions were

not rigorously satisfied since the shell structure exhibited

some constraint against normal displacement and some rotation of

the boundary. Better over-all consistency between the experi-

mental and theoretical results was obtained by introducing the

7

~bserved experimental average normal boundary stress resultant

and average boundary rotation

F(CT) = -240 lb./in. nn

c (~n) --o an- Q.OOOSS rad.

in the boundary equations ( II-1) and ( II-2) respectively.

These modified boundary equations are denoted by (II-1*) and

(II-2*) respectively. Part of the deviation between the

theoretical and experimental results, near the re-entrant

corners of the shell, was due to the stress concentration

brought about by the discontinuity of the boundary members

at the shell's corners.

A typical I.B.M. 7040 computer programme .. by which

the theoretical solution may be obtained is now given.

8'

Computer Programme

. ~.$IBFTC . .. . . ... . . .... . . C B0UNOARY CeLL0CATl0N F0R SECTI0NAL RESULTANTS .

00UGLE PRECISI0N FR(20,53l,FIC20,53l,UC20l,EC20l,BER0C20l,BEI0( 12C),BER!2U,14),8EIC20,14),8ERIC20,7l,BEII(20,7l,BERIIC20,7),

·------ .... 2 B E I I I ( 2 0 , 7 l , H C 2 0 l , 0 ( 2 0 ) , A A ( 2 0 ) , R R ( 2 7 l , D ( 2 7 l , 3 R C 2 0 l , Z l ( 1 0 l , Z ( 1 0 l , A C 2 7 , 2 7 l , 0 Q ( 1 0 l , Q ( 1 0 ) , C 0 ( 2 7 ) , C I ( 2 7 } , OI S ( 15 ) , . 4R0T(l5l,UN(7,20l,FRRC7,20l,F00(7,20),8MR0(7,20),8M0R(7,20),ARG(20) 5,FK,CC,Y,CK,G,ZZZC10l,SI0(10l

-~'C,CY,DlrX,T,PRESS,RAO,VJOO,W,BB,CJ~Z~,APl,R0,CKl,CK2,C2Kl,C2K2, 7C2K,FK1,FK2,F4K,F4K2,EF. . . .. . .. .. . . . ... .. .. . . ....... . . C I r-1 ENS I 0 N N I C l 0 8 l , W 0 R K ( 54 ) . L=7 L ........ LU=9+L . LC=<J+L

LL=L+l LO=L-1

·---- .. . . . ... R E A C 2 , U\ R G ( J l , J = 1 , L CJ READ 3,(AA(J),J=l,LCl REAC l,CBER0(J),J=l,LUl REAC l,(8tl0(J),J=l,LU)

"·-·-·-· .. REAC l, (uERI (_K, l l ,K=l,LC). READ l,(bEli(K,ll,K=l,LC) READ 5,(QQCil,I=l,7)

1 F0R~AT (1020.14) ..- ; .... ~ ... 2. F0R~A T ( lUY. 2 l

3 F0K~AT (107.0) 5 F0RMAT (7C7.l)

EF=l.D+7 I

i 1.

. PRES S =:- 2 0 • 0 0 RAD=64.00 V=.33DO CO= EF *.37500**3/(l2.DO*Cl.OO-(V**2)))

.. .... . ..... .. w ::: < 1 2 • 0 0 * ( 1. 0 0- < V * * 2 l ) l * * • 5 0 0 I < ( E F > . * C • 3 7 5 0 0 * * 2 l ) £lO=C ( 12.00*( l.OO-(V**2) l >**•500/(64.~'00*-•. 37500) l**•SDO API=3.1415926535900 ZB-=API/6.00

.... R 0 = 25 • 0 0 0 * 0 C 0 S ( Z.0 ). 00 773 l=l,LU

773 t<Cll=ARG(Il/l.lB 00 77'-t I=l,L IF (R( Il-R2J 371.,771,77.2

771 R< I l=R0 772 SIOCI)=(R(l)**2-R0**2)**0.500 774 C0NTINUE

, ....... 77 s 2 ~z 1 I~=~ I 6 ( 1 >lR e· 00 776 I=l,L

776 Z(IJ=OATAN(ZlZ(I))

-·--777 2~ d1~z i I}!~ lac.oo; APil PRINT l50,Z0 PRINT 150,R0 PRINT 6 1 (H(ll 1 l=l,LUl PRINT 7,(ZZ<I>,I=l,L)

6 F0RMAT (6El8.8/6El8.8/6El8.8/6El8.8/3El8.8)

f "'I

.]

···-·~ ....... 3 FZRI'AAJ (7D.l8 .• 10J ... . NN=5l Nl=I\N+l N2=1\N+2 NNl=l\N-1 ····-. NN2=N~~:_2 :\1\3=1\N-3 Nf\4=1\N-4 C=NN oe e I=l,T

8 Q(l)=(API/l80.00)*QQ(l) ce g J=l,LU

9

__ ...... U ( J l =-ARG ( J) /2. 00**0. 500 E ( J l =A R G ( J l I 2 • 0 0 * * 0 • 50 0 ........... ·······-···-~····"---·· .. ·• ···--·----~----···--··---·-·-·----·-·--··---···-----~---· ----9 CZNTINUE

CZ=l.DO CY=2.00 oe 10 J=l,LU X=2.00/(U(J)**2+E(J)**2l FR(J,ll=O.O

......... FI(J,1l=O.O FR(J,2l=AA(J) F I ( J , 2 l =0. 0 FR(J,3l=X*<C-DZl*U(J)*AA(J)

....... FI(J,3l=~X*(C-0Zl*E(J)*AA(J) F R < J, 4 l =X* (C-OY)* U ( J) * F R ( J, 3 l _;AA ( J l +X* <C-OY)* ETJ) * F I ( J, 3T ---··~--------FI(J,4l=-X*(C-DY)*E(J)*FK(J,3)+X*(C-OY)*U(J)*FI(J,3)

10 C~NTINUE C0 11 J=l,LU. CZ 11 I=l,NN3 Y=NN2-I T=(2.00/(U(J)**2+E(J)**2) )*Y FR(J,I+4l=T*U(Jl*FR(J,I+3l-fR(J,I+2l+T*E(J)*FI(J,I+3) FI(J,I+4l=T*U(Jl*FI(J,I+3l-FlCJ,I+2l-T*ECJl*FRCJ,I+3)

11 C0NTINUE 00 12 J=l,LU H ( J l = ( F R ( J ,N 1 l *BE R 0 ( J l + F I ( J , N 1 ) *BE I 0 ( J ) ) I ( BE R li: ( J ) * * 2 +BE I 0 ( J ) * * 2) 0(Jl=<-FR(J,Nll*BEI0(J)+FI(J,N1l*BER0(J))I(BER0(J}**2+BEI0(J)**2)-

l2 C0NTINUE 00 13 J=l,LU ce 13 I=l6,Nl,6 Lf'A=N2-I L N = ( ( L M- l l I 6 l + 1 BER(J,LNl=(Fl(J,Il*0(J)+H(J)*FR(J,Illi(H(J)**2+0(J)**2) BEl(J,LNl=l-FR(J,ll*lil(J)+H(J)*Fl(J,I))/(H(J)**2+0(J)**2) _, 13 C0NT INUE . ····· ..... ······. . . .. ... .... . ... . .. . .. ....... .... .. . 00 778 J=l,Lu OZ 778 I=l7,NN4,6 Lr-'=N2-I LN=LM/6+7 8cR(J,LNl=(Fl(J,I)*0(J)+H(J)*FR(J,I))I(H(J)**2+0(J)**2) BEI(J,LNl=(-FR(J,l)*0(Jl+H(J)*FI(J,Il)I(H(J)**2+0(J)**2)

-~ ... ..7.7.8 ... C0NT INUE 0 0 14 K =' l ' L. c ~-~ ,.,~ "'~'·'o~-_,A __ , __ ,_

0~ 14 M=l,6

10

I=t':+7 8=6•M BERl(K,M+1)=-(l.D0/2.00**•500)*{BER(K,Il+BEl(K,ll)-(B*BER(K,M+l))

liARG(K)

1 ~ ~ h6 ~ ~) M+ 1 l = C 1 •. 00/2 • 00 ** • 5DO) •J BER. ( ~, I) ~.BElliS.dJ >:-:_( --~~-*B .. f:JJ 1\_, M::t-J.tL __

14 C0NTINUE . . 00 15 K=1,LC 00 15 M=1,7 C C =A e S ( 6 * ( M- l ) ') .. -- ....... --- ....... - . ~-· ----- -----dERil(K,Ml=-CDZ/ARG(Kll*BERICK,~)+((CC/ARG(K))**DY)*BER(K,M)-BEI

1( K 'f'l l . BEIII(K,Ml=~(Ol/ARG(K)}~BEII(~,Ml+((CC/ARG(Kl)•*DYl•BEl(K,Ml+BER( l K, M ) . . .. . . · · ·· · ··· · · ·•·· ··--: ·----· · •·· ··--····· ···-··-········ ...... '".. •..•. .. ---··---------------

15 C2l\:TINUE PRINT 150,88

.. ______ . . P R. I NT 15 0 , W PRINT 150,00 PRINT 150,(8ER(l,l)) PRINT l50,CBEICl,l))

...... 150. F0Rfv'AT ( 1C20.12 l r-.; = 4 * L -1 · · · ·· ·· LLL=2*L+l LLLL=3•L+l L2=2•L L3=3•L LLO=L+3 Lll=L+2

... LL2=2•L+2 ... LL3=3*L+2 C T= 1. D-20 00 20 J=l,L I=J A ( J, ll = ( DZ I W l * (BH/RO l ) •BE I I C I, U * (DC0S { ZCI)) ··•2r•cT····-------------------

l + ( ( 8 B * *-2 ) I W l * B c I I I ( I , 1 ) * ( D S 1 N ( l ( I ) l * * 2 ) * C T A(J,2l=-CDZ1Wl•(BBIR(l))*BERI(I,ll•CDC0S(Z(I))**2)*CT

l+((Bb*•2liWl•BERII(l,ll•CDSINCZCill•*2)*CT A ( J , 3 l = 0 • G . . . . . . .. .. .. .. . . . .. .. . . ... ····-··· ......... -- ..... --------------······---·-··---·

20 CIZNTINUE DZ 21 J=LL,L2 I=J-L A ( J, l >=-tib•BER n I, ll * { DC0S (Z( n l f•CT·--· .... --~--------·----------------·----------------------~------

ACJ,2l=-BB*BE1Iti,ll•COC0SCZ(Illl*CT A(J,3l=O.C

_21 CCNTINUE 00 2 2 J =LLL, L3 .. I::J-2*L A(Jtll =BEI{(Itll•CT A ( J , 2 l = 8 E I ( I , 1 l . *C T A(Jdl=l.CO•CT ... -

22 C2NTINUE 02 23 J=LLLL,N I=J-34:·L A ( J , 1 l = ( ((D zn·n * ( B B I R C !} )• BE I IT I , U-V * ( ( B 8 * •2TlW )*BE I I IC I, 1 ) ) *

lCCSINCZCill**2l+(((bB**2)/Wl*BEIII(I,ll-(v/Wl•(BB/R(l))*BEll(l,l)l

11

2•(0C0S(Z(ll l**2ll*CT , A < J , 2 l = ( ( V * ( ( B B * * 2 l I W l *BE R I I ( r ,T l - B 8 I C W * R CT l l* BE R ( I ,Tl ) * ( 0 Sl N (l rr-, ll l**2l+( (V/Wl*(BBIR(Il}*BERICI,ll-C(BB**2)1Wl•BERIICI,ll) 2* ( OCt?,S ( l (Ill **2 l l •CT

A(J,3l=C.C "23 C0NTINUt:

ce 25 K=4,LL1 ce 2~ J=l,L M=K-3+1 l=J CK=FLZAT (6•(K-3)) CK1=CK-l.L:O CK2=CK-2.CO A(J,K)=( ( ( COZI~l•CBB/R(Il l•BEII CI ,Ml-CDZ/W")*( CCKIRCill**2)*BEICI,w

ll l*(UC~S(Z(Il l**2l+((BB**2l1Wl•BElll(I,~l*CCOSIN(Z(llll**2l)* 2(0C0SCCK•Z( I) l )+( (-CK/Wl*(BB/R( Il l•BEII (I ,M)+(CK/W)tt(OZ/(R(l)**2)) 3•bEI( I,M) l*CDSINC2.DO•ZCilll•(OSINCCK*Z(l))))*CT 25 CZNT lNUE " , ,. , . , " " " ,~ -- ···~,,. ·- ···--··········-·~--···--···

00 26 K=4,Lll 00 26 J=LL,L2 .t-t.=K-3+ 1 l=J-L CK=FL0AT (6*(K-3ll CK 1=CK-1. CO CK2=CK-2.CO A ( J , K l = ( - ( 8 B *BE R I CT, M ) * ( 0 C 0 S C Z C I l l l * C 0 C 0 S C C K* Z ri l ) l r...;·c C K lR ( I ) *··-~······

lHER(I,Ml•(DSINCZ(ll ll*DSINCCK•ZCill ll•CT 26 C0NTINUE

CZ 27 K=4,Lll oe 27 J=LLL,L3 fvl=K-3+1 I=J-2*L CK=FL0AT (6•(K-3)) CK1=CK-l.DO .... C.K2=CK-2.CO A(J,Kl=(tlERCI,Mll*CDC0S(CK*Z(IlllttCT

_27 CZNTINUE L0 28 K=4,lll 00 28 J=LLLL,N t;,=K-3+1 I=J-3•L CK=FL0AT C6*CK-3lr. CKl=CK-l.CO CK2=CK-2.CO A(J,K)=(( ((Ol/hl•CBB/R(Il l•BEIICI,Ml-COZ/Wl•CCCK/RCill**2)•BEICI,M

1 l- V * ( ( F1 B * * 2 ) I l·d * 8 E I I I C I , M l l * C CDS INC Z ( I l l > * * 2) + ( ( ( B B * * 2 l /W > *BE I I I { I 2,M)-(V/W)*(BB/K(IllttBEll(l,Ml+(V/W)•((CKIR(l))**2lttBEICI,Mll*(( 30C0S(Z(Ill l•*2ll•!OC0S(CK•Z!Ill)+(COZ+V)*(CK/Wl*CCBB/R(lll*BEIICI, 4M)-(0Z/R(Il**2l*BEICI,Mll*(OSINCOY*Z(I))))*(DSIN(CK*Z(l))))*CT 2 8 C 0 N T I N lJ f . . . .. . . .. . . . . .. ... . . . .. . .. . . . .. . . . . . . . .. .

. C0 29 K~LLO,Lll 012 29 J=l,L M=K-3-Lu+l I =J .. . CK=FL0AT C6tt(K-3-L0))

12

13

....... ~ ... 34 CZNT INUE ....... . 02 35 K=LL2,L3 00 35 J=LLL,L3 t4=K- 3-2 *L 0+ 1 I=J-2*L ..... - . C K = F L 0A T. ( 6 * ( K~3...;2* LO rr·------- ·-·· . -------~---- . ···----.----~----·-·---~--~----------··--·----CKl=CK-l.CO CK2=CK-2.CO C2K=CK/2.00 A ( J, K l = ( R ( l l **C2K l*CT*TRrlT**C2K l *OC0S ( CK*ZriTr··-~·---------------···-···-·-w

35 CZNTINUE OZ 36 K=LL2,L3 ue 36 J=LLLL,N M=.<.-3-2*LO+l ... I=J-3*L A(J,Kl=O.O

.. .36 C0NT I NUE C?, 37 K=LLLL,N 0~ 37 J=l,L t"=K-3-3*LC+l I=J C K = F L 0 AT ( 6 * ( K- 3-3 * L 0 ) r·· -CKl=CK-1.00 CK2=CK-2.DO C2K2=CK2/2.CO A ( J , K l = ( K. ( I ) * *C 2 K 2 )* C T * ( R ( I )* *C 2 K 2T*CK * CC K 1/ WT*T ( (( 0 S 1 N ( Z ( I ) n * *2 ) ..

l-((CC0S(Z(llll**2})*(0C0S(CK*Zllll)-(0SIN(OY•Z(l)))*(DSINtCK*Z(l)) 2))

37 CI2NTINUE . 00 38 K=LLLL ;·N ............. . Ck:: 38 J=LL,L2 ~=K-3-3*LO+l I=J-L A(J,Kl=O.G

38 C0NTINUf: C0 39 l<=LLLL,N 00 39 J=LLL,L3

. r~=K-3-3*LU+l ··-'··· I=J-2*L A(J,Kl=O.O

3.9 C0NTINUE Cl2 40 K=LLLL-,N Cfl 40 J=LLLL,N M=K-3-3*LG+l I=J-3*L

.. CK=FL2lAT (6*(K...;3;.:3*LO'lr CKl=CK-l.CO CK2=CK-2.CO C2K2=CK2/2.00

······ A(J,Kl=(H.( I l**C2K2litCT*CRCTl**C2K2T*C (DZ+VT/WT-*CK*CKl*{ (( CDCflS(Z(I' 1 l l l * * 2 l- ( ( D SIN ( Z ( I l l ) ** 2) ) * (DC 0 S ( CK * Z ( I ) ) ) + (0 SIN ( DY * Z (I ) ) l * (OS 1 N ( 2CK* Z (I) l) )

40 C0NTINUE C0 < l l = 1. C 13"~-----------------------··--· ...................... ~--------------------·-------~~----~--------·~·------ .. -·-------.. ·-----~· "-----------------· C0<2 l=l.D 13

. C0(3 l=l.C.l5 U~(4 )=1.0 14 C0(5 l=l.C 16 C0(6 l=l.O 19

... C£(7 l=l.O 23 C0(8 l=l.O 27 C0(CJ l=1.C 31 C0(10l=1.C 14 C0(ll)=l.O 16 C0(12)=l.D 19 C0(13l=l.O 23 C:Z(l4l=l.D 28

... C.e(15l=l.O 3L C0(16)=l.C 07 CZ(l7l=1.C-02 C0(18l=1.C-10 C0(lCJ)=l.lJ-l8 C0(20l=1.0..:..26 C0(2ll=l.C-34 CZ(22l=1.C C7 c e < 2 3 > = 1 • c-o 2.. ... . . . . . ............ _ c 0 < 2 4 > = 1. c- 11 C0(25l=l.0-19 C0(26l=l.0-27 c z < 2 1 > = 1 • o- 3 s. flRit\T 150,88 PRINT l50,W PHif\T 150,00 00 4 2 I= 1, N. 00 42 J=l,N A ( I , J ) =A ( I , J l *C 0 ( J)

42 C0NTINUE 00 ao I=l,L

80 CI(Il=DZ DZ 81 I=LL,L3

81 Cl(Il=l.0+5 .00 82 I=LLLLrN

82 CI(Il=OZ 00 43 J=1,N 00 43 I=l,N

43 A(l,J)=A(l,Jl*CI(l) PRif\T 150,(t3ER(1,1l) PRINT 150,(BEI(l,1l) .

52 F.CRMAT (7El8.8/7El8.8/7El8.8/6El8.8//)

14

....... 41. CALL 01'-liNVS (A,27,N,O.,IERR,NI,W0RK) P R I NT 15 0 , B jj . . . . . . . ... .. .. . .. .. . ... . ..•... ···········--·--·--· .. ·--·~····-·· ..... -- ··---·-·-·····-···--· .... - ... ·-·----·-·····-

PRINT 150,1--; PRlNT 150,00 PRif\T l50,(BER!l,lll PRINT 150, (BE I ( 1, 1 )' )-~ "~~~~--~~ .. ~-, C0 56 I=l ,N 00 56 J=l,N A(I,Jl=A (l,J)*CI(J)

56 CeNTINUE ...... . 00 45 J=l,L

45 C ( J l =.-PRES S*RAD/CY-240. DO 00 46 J=LL,L2 .. . . ··~-·······

46 0(J)=O.OOC55DO 013 47 J=LLL,L3

47 O(J)=-PRESS*(RAD **2)/(EF•0.37500) 00 ''8 J=LLLL,N . . ....... -- ...... . 48 O(J)=-(OZ-Vl*PRESS*RAO/DY

D0 49 I=l,N .49 RR(I>=O.O

00 50 I=l,N 00 50 J=1,N

50 RR(ll=RR(l)+A(l,J)•O(J) Of: 57 I=1,N ...... . RR(Il=RR(l)*C0(1)

57 C0NTINUE PRINT 5l,(RR(Il,I=l,N)

15

51 F0RMAT (5X,8El5.8//SX,8El5.8//5X,8El5.8//5X,3El5.8//) 0 0 16 3 I= 1, 1 . . . . . . . . ................... " . . ...... ···"······· .. . ... ······-·-··--···-·--···-·-·

00 163 J=1,l PRINT 158

158. ~0R~AT (//5X,9HDISP C0MP/) PRINT 150,(BtR(l,l)) .... PRINT 150,{Bcl(l,l)) PRINT 150,CT PRlf'\T l50,RR(l) PRU\T 150,RR(2) DIS(ll=PRESS*(RAD**2)/{EF*•37500) PRINT159,(0IS(l)) 0 I S ( 1 ) = R " ( 2 ) * B EI ( J , l J * C J PRINT159,(01S(l)) DIS(l)=RR(l)*BER(J,l)*CT PRINT159, (DIS( 1)) OIS(ll=RR(3)*CT PRINT159, CDIS(l)). 00 160 K=l,LO K6=K+l FK=6*K FKl=FK-1.00 FK2=FK-2.CO K3=3+K KL3=3+LO+K KLL3=3+2*LO+K KLLL3=3+3*LO+K F4K=FK/4.CO F4K2=FK2/4.DO

·a IS< K l =RR ( KL 3) *BE ICJ, K6) •CT*DC0S ( FK-*Q (IT) ······-···"' PRI~T l59,(0IS(K)) DIS(K)=~R(K3l*BER(J,K6)*CT*DC0S(FK*Q(l)) . PRINT 159,(01S(K)} 0 IS ( K l = < ( < ( RR ( KLL3) *R C~f>**F4Kl *R ( Jl**F4Kr•RfJT**F4KT*CTT•RTJ )**F4K~

l*CCZS ( FK*C (I)) PRINT 159,(0l$(K)}

159 F0RMAT. ( S .. ~, lEl~.8{J. ···~·'"·-···· 160 C0NTINUE

PRINT 161

•

_ 161 F0R~AT (//5X,8HR0T C0MP/l R0T(l)=PRESS*RAC*((R(J)**2)/4.00) PRINT 159,(R0T(Ill R0f( ll=CDZ/Wl*RR(l)*BEI(J,ll*CT PRII\T 159, (R0T( I l) . . " ... . .... ReTCll=-(OZ/Wl*RR(2)*BER(J,ll*CT PRINT l59,(R0T(l)) 00 162 K=l,LO K6=K+l FK=6*K F-Kl=FK-l.CO FK2=FK-2.CO K3=3+K KL3=3+LO+K KLL3=3+2*LO+K KLLL3=3+3*LO+K F4K=FK/4.00 F4K2=fK2/4.00 R0TCKl=(DZ/W)*(RRCK3l*BEI(J,K6ll*CT*DC0S(FK*Q(I)) PRINT 159,(R0T(K)) R0T(K)=-(OZ/~l*(RR(KL3l*8ER{J,K6ll*CT•DC0S(fK*Q(Ill

16

PRINT l59,(R0T(Kll . . .. RCT(K)=(OZ/Wl*((((RR(KLLL3l*R(J)**f4Kl*R(Jl**F4Kl*R(Jl**F4Kl*CT)

l*R(Jl**F4K*DCeSCFK•Q(Ill PRINT l59,(R0TlKl)

162 CZNTINUE 163 CZNTINUE

c~ 60 I=l,7 00 60 J=LL,LC . . UN(I,Jl=PRESS*(RAC**2)/(EF * • 3 7 5 0 0 )+ ( R R ( 2) * BE 1 C J , 1 r - _,

l+RR(1)*1JER(J,1l+RR(3) )*CT FRR(I,Jl=PRESS*RAD/DY+(RR(l)*(BB/(k*R(J)))*BEIICJ,ll l-KR(2)*(38/0~*R(J) l l*BERI (J,ll l*CT

F Z 0 ( I , J ) = PRE S S * R AD I 0 Y + ( R R ( 1 l * ( ( 8 B * * 2 ) I W ) *BE I I I( J , 1 ) - R R ( 2 }* ( (B 8 * * 2 ) 1/hl*BERII (J, 1 l )*CT B~R0CI,Jl=DD*(-RR(l)*((BB**2l•BERI1(J,l)+V*(BB/R(Jll*BERI(J,l))

l-RR(2)*( (bB**2l*BEI1I (J,ll+V*(BB/R(J) l•BEII(J,l) l l•CT HM0R(I,Jl=DO•(RR(l)*((BB/R(J))•8ERI(J,l)+V•(BB••2l•BERfllJ~l))+RR(

l2)•((b8/RCJ)l•BEII(J,l)+V*(BB**2)*8EIII{J,l)))*CT 60 C0NTINUE

PRINT 150,88 PRif';T 150,~-.J PRII\T 150,00 C0 61 1=1,7 CZ 61 J=LL,LC 00 61 K=l,LO K6=K+l FK=6*K FK1=FK-l.CO FK2=FK-2.CO K3=3+K KL3=3+LO+K KLL3=3+2•LO+K KLLL3=3+3*LO+K F4K=FK/4.DO

17

*.37500)+(RR(2)•BEI(J,l)

18

2*DC0S(FK*ZCil l . . · F R R C I , J l = F R R ( I , J ) + ( ( R R ( K 3 l * ( ( - 0 Z I W ) * ( ( F K /R ( J ) l * * 2 ) * B E I ( J , K 6 l - - .. ····~-·-· 1+(88/(V.:*:{(J) l l*BEII <J,K6) )+RR(KL3l•( CDZnoJl•{ CFK/R{Jl l••2l• 2BER(J,K6)-(BB/(W*R(J})l*BERI{J,K6)))*CT 3-FK*CFKl/w)*((( (KR(KLLL3l•RCJl**F4K2l•R{Jl••F4K2l•R(J)**F4K2l•CTl 4 * R ( J l * * F 4 K 2 l * 0 C 0 S ( F K * Z ( I l l . . . . . . . . . . . .. . . . --·· ..

F00CI,Jl=F00CI,Jl+( CRRCK3l*(((B8••2l/Wl•BEIIICJ,K6l l-RR(KL3l*((( ll3B**2)/Wl*BEKII (J,K6)) l•CT 2+FK*(FKl/W)*( ( ( CRRCKLLL3l*R(Jl••F4K2l*R(Jl••F4K2l•R(Jl**F4K2l•CTl 3*R(J)*•F4K2l•OC0S(FK*ZCil) . . . . . .. . .. - -.

BMR0 (I ,J)=BMR0( I ,Jl +00•( CRR(K3) •< (-BB••2 l*BERII (J,K6l+V•( (FK/R(Jl) 1 * *2 l * BER ( J, K 6 l- V * ( B B I R ( J l ) * B ER I ( J, K6) l +RR ( K L3 l * (- ( B B••2 l •BE I I I { J, 2K6l+V*( (FK/R(J) l*•Zl*BEI (J,K6l-V*(B8/R(J) >•BEII(J,K6l l l*CT 3-CCZ-Vl*FK*FKl*((((RR(KLL3)*R(Jl••F4K2)•RCJl••F4K2l*R(J)•*F4K2l 4•CTl*R{Jl**F4K2l*DCeSCFK•Z(Ill B~0R(I,Jl=B~0R(I,Jl+DC*((RR(K3)•(-((FK/R(J))••2l•BER(J,K6}+(BB/

lR(Jll*BEKI(J,K6l+V•CBB••2l•BERIICJ 1 K6ll+RR(KL3l•(-((FK/R(Jll**2l• 2BEICJ,K6l+(BB/R(Jll•BEII(J,K6l+V*(HB**2l• BEIIICJ,K6lll•CT -- -3-(0Z-Vl•FK*FKl*((((RR(KLLL3l•R(J)**f4K2)*R(Jl••F4K2l•R(Jl••F4K2)* 4 CTl*R(Jl**F4K2l*OC0S(FK•Z(Ill

_40~_C0NTINUE . . . . .... PRINT 85, CZZ(J),J=l,Ll PRit\T 63

63 FZR~AT (28X,6HN~RMAL,2X,l2HDISPLACEMENT//) PH I t\ T 6 2 , ( ( UN ( I , J l , I = 1 , 7 ) , ..1 =L L , . L_C ) PRINT 85,(UN (l,Il,I=l,L) PRII\T 64

64 F0R~AT (2bX,6HRACIAL,2X,6HSTRESS,2X,9HRESULTANT//) ~~I~i ~§: ~~k~RH:Y~!f~t:l~,J=LL,LCl · ······. .. ·0

" • ..... ~--·"·--········--·· .. ••

PRINT 65 65 F0R~AT (2bX,6HCIRCUM,2X,6HSTRESS,2X,9HRESULTANT//l

P R IN T 6 2, ( ( F 0 0 ( I , J l , I= 1 77 ) , J =L L , l C ) PRINT 85, (F00 (l,I),l=1rl) PRINT 66 ·

66 FZR~AT.J24X,6HCIRCUM,2X,6HTRANSV,2X,6HSTRESS,2X,6HC0UPLE//l PRINT b2,((BMR0(I,J),I=1,7},J=LL,LC) PRii\T 85,(BMR0(I,Il,I=l,Ll .. . ... ... .. . .... . .. PRII\T 67

67 F0~~AT (24X,6HRAOIAL,2X,6HTRANSV,2XJ6HSTRESS,2X,6HC0UPLE//) PRINT 62, ( ( BIV10R( I ,J) t 1=1,7) ,J=LL,LC P R I 1\ T 8 5 , ( B M 0 R ( I , I ) , I = 1 , L) ··········· . . ... .. ... .... . ...... - ···-·-···----

62 F2~~AT (5X,7El8.8) 85 F0~MAT {//5X,7El8.8)

STZP Et\D

19.

l. LEGEND AND OUTLINE

(a) COLLOCATION POINTS

L = 7

(b) SHELL PARAMETERS

Tr - API

shell periodicity - k = 6

shell base circle - 25 in. radius

central angle of characteristic segment - Z~ = API/6

shell radius - RAD = 64 in.

coefficient of elasticity, E - EF = 107 lb./in. 2

(aluminium) POISSON's ratio, JJ - V = 0.33

normal pressure, p - PRESS = -20 p.s.i. n

(c) KELVIN FUNCTIONS

constant a - AA(I}

function Fn' ~ [ Fn ( ~ r)] - FR ( I, J}

J ( l=;.{i\r)J - FI( I, J)

constant 0C. I - H(I) - ~(I)

KELVIN functions of the first kind

zero order ber (~r} - BER~(I) b ,0 (ll.r) - BEI~(I) el.

0

order n ber (:hr) - BER(I,J} b .n (~r) - BEI(I,J) el.

n

first derivative ber' (1\r} - BERI(I,J) n bei' (.1\r) - BEII(I,J) n

second derivative ber 11 (1\.r) n

BERII(I,J)

where

and

bei 11 (1\r) - BE III (I, J) n

I represents the argument Ar

J represents the order n ,

20

(~) SIMULTANEOUS EQUATIONS SATISFYING BOUNDARY EQUATIONS

These equations are set up in the form

A(I,J) x RR(J) = D(I) (II-5)

. where

I represents row subscripts

J represents column subscripts ,.

A(I,J) represents the coefficients q>,

RR(J) the I 2

represents constants Ao, Aol

I I 2. ' 2 Eo, A 6f'\l A6n I c6n I c6n

I

and D(I) represents the non-homogeneous constants

in the simultaneous boundary equations.

Each row of the array given by (II-5) represents a

boundary equation. Since each boundary equation must

be satisfied at each collocation point except at the

• corner (i.e. e = 30 ) 1 there will be

4L - 1 = N

equations containing N unknown coefficients RR(J).

J

21

For L=7 these coefficients are

RR(l) = A! RR(2) = A.,.

0

RR(3) = E' 0

RR(4} = A~ RR(5) = A,~ . . . RR(9) = A;6

RR(lO) = A! RR(ll) = A:-&

• • RR(l5) r. = A36

RR(l6) = c' " RR(l7) = c,'t. . . . RR( 21) = c~"

RR(22) = c! RR( 23) = c,~

0 . . ~ RR(27) = c,,

The first three constants Al o'

A2 o'

El o' are the

same for any number of collocation points, while the

remainder are divided into four consecutive sets of

order L-1 (in this case sets of six constants).

The coefficients~. are given by A(I,J) and form

an N x N matrix. This portion of the programme commences

/· I

after ISN (statement) 15 and ends at ISN 40.

The matrix A(I,J) is then inverted (ISN 41) and

once the constants D(I) have been given (ISN-45, 46, 47,

48), the unknown constants RR(I7 may be calculated from

22 _, RR(I) =A (I,J) x D(J)

(e) NORMAL DISPLACEMENT AND SECTIONAL RESULTANTS

u UN(I,J) n

F (a) rr

FRR( I, J)

F ( cr) F~~( I, J) ee

M( cr) BMR~(I,J) re

M( a) BM~R(I,J) er

where I represents radial lines defined by e = constant,

J represents arguments "r and the constants

D - DD w-w 'A - BB

2. REFINEMENTS FOR RETENTION OF ACCURACY AND OVERMASTERY OF COMPUTER LIMITATIONS

Double precision techniques were used throughout

the numerical calculations. Limitations on the amount of

machine memory storage available were overcome by comput-

ing and storing KELVIN functions of order kn only, where

n = o, 1 ••• , (L-1) and k represents the rotational period-

icity of the shell.

N.A.S.A. tables by LOWELL in 1959 gave zero order

KELVIN functions to 12-14 figure accuracy for arguments

containing only two decimal places. In order to insure

the accuracy of the zero order KELVIN functions employed

2J

in the programme, it was essential to use only those

arguments which were tabulated.by LOWELL. Consequently

once ~was computed, radii r were calculated from the

arguments Ar - ARG (I).

The radii r- R(I), which described the collo-

cation points, were then used to determine correspond-

ing angles e - ZZ(I) such that in this case

rcose = 25.0 cos 1'T /6

Unique constants a-AA(I) were used for each KELVIN

function calculated by the "Backward Recurrence Technique"

in order to prevent both computer underflow (numbers

-38 smaller than 10 ) and overflow (numbers greater than

10+38).

A constant CT = 1. x l0-20

was introduced in the

calculation of the coefficients ~, to prevent computer

overflow. It was not removed until the calculation of

the sectional resultants was completed.

The rows and columns of the matrix A(I,J) were

multiplied by constants CI(I) .and c¢(J) respectively,

in order to reduce the elements to comparable orders of

magnitude with a maximum range of approximately 10+ 4

This step was essential to produce accuracy in the

inversion calculation which employs the "GAUSS Pivotal

24

Technique". These constants were removed after the

inversion was completed.

The accuracy of the matrix inversion was checked

by multiplying the matrix by its inverse

i.e. AxA-l =I

where I is the unit diagonal matrix having off-diagonal

elements of zero magnitude. A "perfect" inversion using

double precision techniques produces off-diagonal

elements whose magnitudes are of the order l0-16 •

3. CHANGES IN PROGRAMME FOR DIFFERENT SPHERICAL SHELLS OVER POLYGONAL BASE.

The shell parameters given in section (1-a) must

be changed for each particular shallow shell.

In·addition, the following changes may be necessary with respect to:

(a) PERIODICITY

(i) New KELVIN functions must be calculated.

(ii) The constant CK must be changed.

(iii) The constant FK must be changed.

(b) BOUNDARY CONDITIONS

( i) The coefficients A~( I., J) must be changed and rearranged.

( ii) . The constants D('J) · must be changed and rearranged.

25

n·iscuss ion of Results

1'. CONSISTENCY

The theoretical results given for the spherical

shell enclosing an hexagonal base were obtained by em-

ploying boundary co~ditions (II-1) to (II-4). Often the

boundary conditions (II-1) and (II-2) were replaced by

the special boundary conditions (II-1*) and (II-2*) res-

pectively as noted on the graphs.

It can be seen from FIGURES 8 to 41 that the

number and location of the boundary collocation points

had a relatively minor effect on the theoretical results

with the exception of some irregularities close to the

0 shell's boundary especially along the radial line, 6=30,

to the corner point of the shell. Reasonable agreement

between the experimental and theoretical results was

obtained when boundary conditions (II-1*, II-2*, II-3,

II-4) were employed. The greatest number of discrepan-

cies occurred near the shell's boundary, particularly

• on the radial line 9=30, some evidently brought about

by the stress concentration due to discontinuity in the

boundary members. Better agreement could sometimes be

obtained by using slightly different boundary conditions,

more in concert with the shell's structure. However,

26

the majority of discrepancies appeared to be the result

of the physical shortcomings of the experimental shell.

Elimination of the horizontal normal boundary

constraining force by using boundary condition (II-1)

instead of (II-1*) caused increases in the normal dis-

placement and the radial and circumferential stress

couples, decreases in the radial stress resultant,

0 0 especially on the radial lines e = 0 and e = 10, and

some changes in the circumferential stress resultant

at the boundary only.

Confining the boundary to be fully constrained

against rqtation by imposing the boundary condition

(II-2) instead of (II-2*) caused some reduction in the

normal displacement as well as some changes in the

radial and circumferential stress resultants and stress

couples in the vicinity of the boundary as was to be

expected.

2. ACCURACY OF THE THEORETICAL SOLUTION

Logically the accuracy of the theoretical solution

by the collocation method should increase with increasing

numbers of boundary collocation points. In order to verify

this reasoning, normal displacements were calculated at

. 27

the shell's boundary for solutions using three and seven

collocation points both of which satisfied the boundary

equations (II-1*), (II-2*), (II-3) and (II-4) (see

. FIGURES 42 and 43).

S.ince all the computations were done to 17 figure

accuracy, boundary condition (II-3),(i.e. u =O),would be n

"perfectly" satisfied when the magnitude of the normal

-17 displacement was approximately of the order 10 inches

for the entire boundary of the shell.

FIGURES 42 and 43 reveal that an increase in the

number of boundary collocation points produces a better

average satisfaction of the boundary condition ( II-3).

The maximum deviation of the normal displacement at the

boundary from the theoretical boundary condition for

three collocation points was about -5 x 10-4 inches, while

for seven collocation points it was about 2 x 10-4 inches.

However, the scales of these graphs do not permit to show

that the minimum deviation of the normal displacement at

the boundary from the theoretical boundary condition for

-16 three collocation points was of the order of 10 inches

while for seven collocation points it was of the order 10-6

inches. This indicates that an increase in the number of

28

collocation points gives better satisfaction of the pres

scribed edge conditions over the complete shell boundary

as long as the resulting increase in numerical computations

does not reduce the number of significant figures in the

computer calculations below a "safe" level. For seven

collocation points, there appears to be only six figure

accuracy in the calculations, which would certainly con

stitute a minimum requirement. The accuracy of the

theoretical solution for this shell would probably not

be increased by using more than seven collocation points

when double precision (17 figure) accuracy is used for

all computations. It is possible, however, that in this

case the accuracy might even decrease.

--

29

VECTOR DIAGRAM of SHELL ELEMENT SHOWING SECTIONAL RESULTANTS

MCO'_,CII tdOI) a z z F2<a,01

2td01

2)

F(a) • F(a)i 't FCa)i +FCo)i I II I 12 2 Iii n

F~a> .. FJr>e1 t FJr>i2 t FJg->in

M(a) • M(a)i t M(a)i I II I 12J 2

MCa) • M(cr)'i t M (CJ) i 2 21 I 22 2

P • plil t Piz tpnin

FIGURE

EXPERIMENTAL SHELL ON EDGE ROLLERS

FIGURE 2

30

31

DETAIL OF SLIT CORNER SHOWING

CIRCUMFERENTIAL STRAIN GAUGES

FIGURE 3

32

SHELL ASSEMBLY WITH DIGITAL STRAIN INDICATOR,

PRINT-OUT RECORDER AND SWITCHING UNIT

FIGURE 4

-~- 1-, : I

' I ' I

HEAD 8 DIAPHRAGM REMOVED FROM THIS HALF

0

0

0

0

' 0 '~

THE ORE TICAL SHELL DIA.

33

-4._-7~+-~~---------------------+----------------------~-----.+- 50" l I I

I 1 ' I

... l - + ..J _ ..

'

SHELL

0

0

0

) /

0

0

NOTE :-TIE RODS DELETED FROM THIS VIEW

TEST ASSEMBLY

FIGURE 5

0

SECTION THROUGH TIE ROD

PLAN VIEW of SHELL on HEXAGONAL BASE

SHOWING LOCATION of RADIAL LINES for

which SECTIONAL RESULTANTS are

CALCULATED

I I

I I

\

/ I

\ \

/ /

0 ,..,o ..,.,-----....__ ,, 0

..... ..... ..... 0 0

e

,~ '0 0

" ,o '0~ 0

' 0 e-:. \

\

I I

..... _ ....

. -- ..... ---~ radius==25"

FIGURE 6

34

BOUNDARY COLLOCATION POINTS LOCATION of

for SHELL on HEXAGONAL BASE

0

=---=:.....::..~.;;;;...-..,J 9 =2·6

3 COLLOCATION POINTS

9:;:30°

9=20°

s:\~: 9=10°

.o=....-..=..=-=.;;;~= e= 2·6°


9=2cf 9=14° 6= 10°

.-..-..:=--o:..=...;.;;;_----.;~ e =2· 6°

~ COLLOCATION POINTS

6=30°

9=24° 9:20° e=l"?! 9= 14~ e= 10°

.... e::;;;:==.-=-==-=:::....=::....=.::!J e = 2 · 6°


FIGURE 1

3~

t. 04

t. 02

t. 01 1-z LIJ :I LIJ u ~ .....J a.. (/)

0

.....J ~

0

~ -·01 0:: c z

-. 02

-· 03

-. 04

0

THEORETICAL II u II n PLOT SHOWING


NOT ·. norm ol pres sure·-~ -- 2?,~ s.i three col. ~ ts. at 6=2· 01403(0

SPE IAL I OUND ~RY ONOI IONS:

·~ -~-.v ·I II •

' "' ·000 J>5 rad.

LEG NO:

rodi I line - 6= 0 ...

~- ~c;o t'.:=.E.::: e=: -e= 00

~

~i /.· . ~17:·· ..

~ (o;·

~·

~~: I/ •• .. · . .. ..

4 8 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX , ..

'm.;

FIGURE 8

36

.. ... .. ...

24

t '04

I · 03

t. 02

t · 0 I .... z LLI :1 LLI u <(

..J Q. (/)

a ..J <(

0

~ -·01 a:: c z

-. 02

-· 03

-. 04

0

PLOT SHOWING THEORETICAL II u II

n for SHELL on HEXAGONAL BASE

NOT I : norm I pre! sure ,1 ,:·20 .s.i.

0 0 _a 0 ·~ ~~v • I" - - .. ,_, .-.............

SPE IAL I OUND ~RY ONDI ION:

,.. ~g-1 - "4::At' IIJof I

LEG NO:

rodi I line - e= 0

0

~- v r----r----e=~ o• -1--- ---e= o• ··········· ......

~ ~~·

~~~··

~ ~~

_., ,_.

~d ~·


'tr. .;

FIGURE 9

37

--

?.··--;-rrt ... ..

24

t. 04

I · 03

t. 02

t. 01 1-z LLI ::E LLI u <(

....J ll. (/)

0

....J <(

0

~ -·01 a: c z

- 02

-· 03

-. 04

0

II u II

n PLOT SHOWING

for SHELL on

THEORETICAL

HEXAGONAL BASE

NOT : norm ~~ pre ~sure, P.: -2( ~~<>~::4· :zo0.3~ o five ~ol ot! at "= 2·E

SPE< IAL E OUND RY oN or• IONS: F( CT) =-240 lb./ in

~) = ·000 55 rod

LEG NO:

radi I line - e= c 0

.o tJ- JV -~-F~ 9=~ oo - -- -9 = ~ oo •• ...............

~~ A/ ... .~ ..

V-'L·/ _L~···

lL~/ .

..d ~·~· 'it' -


'm.;

FIGURE 10

38

----

.. ... ...... ..

24

t '04

I ·03

t. 02

t. 01 ... z LU ~ LU u <(

....J n. Vl c ....J <(

0

~ -· 01 Ct: c z

- 02

-· 03

0

PLOT SHOWING

for SHELL on

THEORETICAL II u II n HEXAGONAL BASE

NOT I al prct. 0 p.s.i. norm sure , Pn=-2

14°17. 20°3C0 .·,y eo I ItS. 0 'e= ~ ·6°10°

SPEC IAL I OUNC ~RY ~ONDI IONS: f'(,V) =-24 ) lb./ n.

£(~ =·0()( 55 ro ~·

LEG NO:

radi I line - e= 0 .. o- IOo ~-E~-e=. --- -e=~ oo ··········· .......

I~;~ ~ / .. . ··

1/1~/ ,~/ ..

:L_;v.~·· /., .. . &

/·· .. .. · -"""

~tt .(tJ

4 8 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX I • ' 'm.;

FIGURE II

39

... ····· ...

24

t '04

t . 03

t. 02

t. 01 .... z lAJ :1 lAJ u <(

-' ll.

~ c

-' <(

0

~ -·01 a: 0 z

-. 02

-· 03

-. 04

0

THEORETICAL II u II n PLOT SHOWING


NOT ~-~· norrr ol pre ssure r:tr·2C D ~.s.i. 0 .,o __

0 ~ . ;.... .,.o 0 0 ww •v v- • IP" •· - - -1-, ,_,

' ,- ,_ ,-SPE riAL ~OUNil ~RY roNOI ION:

~-~ •\CJft . ..,.., !"'"'"'' , ...

LEG NO:

rodi I line - e= 0

o- 'ao -- r--

e = • -1"--r--e = ~ oo .......... ......

J,

1~ ~~I •

J- I.: 1 •• ~ .. L, ... ·

~· ~"I/ /...··

,_,~{-:···

k? ·'· ......

~> ?, ---...· ~~~./ / .

4 8 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX tin.)

FIGURE 12

40

. "" .. .. .. .•·

24

t •04

I ·03

t. 02

t. 01 1-z UJ ~ UJ u <( _J Q.

Vl 0

_J <(

0

~- ·01 a: C· z

-. 02

-· 03

- '04

0

PLOT SHOWING THEORETICAL "u " n for SHELL on HEXAGONAL BASE

NOT : norm I pre sure, P. = -2C p.s.i. n 0 0 0 0 0 0 0

""¥"' .. u •. ,..,,,.. "' Q-t;; o,•v,• •''•'' ,, .... , .. •V

SPEC IAL OUND ~RY ONDIT ONS: .:'I"\ -?4t'l lh Jin nn

6(~) =·000 5 ro .

LEG NO:

rodi I line - e= < 0 .. ... -· 1~0 - -

9=~ - ----e =~ o• ...................

A if;/ / .. 1. •• LW /.- .

~ t~-•

""""--!Jo!lll"",.,. .....

41

~;/L.······ I •

/ .. ·· . . ··

4 8 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX , .. \ m.;

FIGURE 13

...

24

c ' .0

t800

t600

t 400

:::: t 200

tz <( ... ...J ::> 0 (/)

LIJ 0::

(/) (/) LIJ- 200 0:: .... (/)

...J <[

0-400 <[ 0::

-600

-800

0

42 PLOT SHOWING EXPERIMENTAL 11 F,~a) 11


NOT ~: norn al pre ssure • P. • 20 J), .i.

LEC:: END; 0

I '"II ~· ...... v-

e= 1 )0 ·- ,_,_ .;....-, ...

8=2 no .... -- ....-e =! p• ... ....... .......

, __ / '

/ \ I \ l )I'

I / r,

\ t' ' \

j / _,.. /l

' / : .. •'

. ~

. . /

. ..,.,. . ···~: "·""' .... l'r':,-:- •

... . .. ~-- ,. \. .. ~-- - . ...

/ ~ .. ')! " . .. ' / .. :' ..

"\ .. . ~/ .. ...

' .. .. ...

4

HORIZONTAL 8 DISTANCE

12

FROM

. .. . I 6 ····-z.e.··· 2 4

SHELL APEX (in.)

FIGURE 14

tBOO

t600

t 400

1- t 200 z ~ ..J ;:::) (/)

UJ a:: 0

(/) (/)

UJ a:: 1-(/)

-200 ..J <l 0 <l a::

-400

-·600

·800

0

PLOT SHOWING THEORETICAL II F (o-) II

rr for SHELL on HEXAGONAL BASE

NOTI: norm pi pre! sure, =-20 .s.i thret col. bts o· 9=2· 0 14°3 bo

SPE IAL BOUN ~ARY CONDI IONS:

rr{nu 1 --c:"tU I'"'" m.

S~) =·000 ~5 rad

LEGI NO:

radic I line ~ 9 = Po o- I"' ·- :...:.~f~ e= 00 -9= ~00 •• ..............

L A f

Yj ~/ /

/ / /

/ ,.,,/

" ~ _,, ~ ..... ,.,, ._,.,.. __

..... -:::.::-: ~- ,..!- ..... .... ---·----····· ········ ····· .. .. .. .. ... .. .. ..

4 8 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 15

43

-

~

24

t800

t600

t 400

1- t 200 z ;: ...1 ::> C/)

L&J a: 0

C/) C/)

L&J a: 1-C/)

-' ~ 0 <( a:

-200

-400

-600

-BOO

()

PLOT SHOWING THEORETICAL "F(cr)" rr


NOTE: normCII pres ure, ij =-20~ s.i. f1ve ol. pt . at =2·6, 10,14, ~o. 3C

SPEC IAL EOUND ARY CON[ IT ION

F ( <~r) 1=-240 lb./ i~. nn

LEGE NO:

radic I line e = o: o- v ·- :_-.:.~f...: e= 00 -e= 00 •• ..............

/ /}1 /, / I

/ v /'r I

/ .L'' / ,, 7

~ ,/ I ,,, I

-:::;,~ V-.- .,.,,-_// _ ..

:-::.:::.-~~ ~/

······· -... -- 1-_ ... .. .. .. ... .. .. .. ... ··.

4 e 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 16

44

. . . . . . . . ~--

-:-

:--. ~ ......

. . . 1

24

tBOO

t600

t 400

c:

1- t 200 z <[ 1-_, ::J (f)

UJ 0:: 0 (f) (f)

UJ 0:: 1-Vl

-200 _, ~ 0 <[ 0::

-400

-600

-800

0

PLOT SHOWING

for SHELL on

THEORETICAL II F(o-) II

rr HEXAGONAL BASE

NOTE nor me I pres ure, p =-20 .s.i

j) 20°2! rf iiY~ lt:ol o ts CJt' A=?· S0 10° 14

SPEC IAL BpUNDA RY c .>NDITI PNS: F(cr) =-24C lb./ir.

'(~) =·000 [55 roc .

LEGI NO:

rodic I line e = o_: ICI" v. ·- :..-.:.-f:: e= o• -e = po• .. ................

~·

~ /

~ ;' /

,/,; ,/ / L_ / :L" .L

~,-'' /, _,..,.,.,.- :.;J -· . :'It. 0::: • ~ ...... .... !"""----······ ... ·~. .. .. .. .. .. ...


FIGURE 17

45

I /

24

t800

t 600

t 400

1- t 200 z ct 1-..J ;::) (f)

1.&.1 a:: 0

(f) (f)

1.&.1 a:: I(/)

-200 ..J !! 0 ct a::

-400

-600

-BOO

0

PLOT SHOWING

for SHELL on

THEORETICAL 11 F(cr)" rr

HEXAGONAL BASE

NOTE: norm I pret sure , ~=-2C p.s.i.

·;.., .... ~,/) o .. 0

... - - - , . -· '--. ,--SPEC IAL I OUND ~RY ONDIT IONS: Ftal -240 lb./in.

Ill

b(~ =·OOC 55 ra~.

LEGI NO:

radic I line e = p• o- u ·- :..-.:.~f~ e= o• -e= o• .. ..............

AI .

// ~~

I

~ / /

V,, /' / I

./ ,, /

-~-~' .,~ ~,

~-

__ , ~~.:: _ ... ,..

••...... J-- ..... ~-------····· ... .. .. ... .. .... ... ... 4 8 12 16 20

HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 18

46

I

i I

24

t800

t 600

t 400

.... t 200 z <t .... ..J :::> (/)

IJJ a:: 0 (/) (/)

IJJ a:: .... (/)

-200 ..J <t 0 <t a::

-400

-GOO

-800

0

PLOT SHOWING

for SHELL on

THEORETICAL "F(cr)'' rr

HEXAGONAL BASE

NOTE: norm pi pre sure, pn=-2 b p.s.i

A A

seve col. pts. a e = • ·6,10, 14,17;~ 10,24,~ 0

SPEC IAL f OUND RY C ONDI, ION:

~(~) =·000 155 ra

LEGI NO:

rod ill line e = o: o- u ·-

·=-~-~f~.~ e= 00 -e= 00 ••

ll I

v •' J /

I

V·l /

/ l /

•

/ / I "/ I

v / / ~/ I ~

,.,,, ~' 1---:': -~-~--... -, L' t':~ ...... _, ................

~---:;" ······ i---~--' ... . .. .. .. .. .. ...


FIGURE 19

47

•

24

t800

t600

t 400 .....

l-t200 z ~ ..J ::1 en 1.1.1 a: 0

en en 1.1.1 a: l-en - 200 ..J g Q Cl a:

-400

-600

-800

0

PLOT SHOWING

for SHELL on

THEORETICAL "F(cr) 11

rr HEXAGONAL BASE

NOTE: norm I pre sure, P.. =-20 P-.5.i.

,..,P, ,....c 0 ... A. """' ...... • ....... . ·- -·. .•. - - - , ·-J , --, ·-·-SPEC IAL OUND ~RY ONDIT IONS~ ... ,. ............... n'n 5~) =·000 ~5 rod

LEG NO:

rodi I line e = p• o- IV ·- ,_, f-' 6= o• - -- -• e= ~0 •• ···············

Ld' /.t I

l6 , / v: I''" / I

../ ·'""' I

-~ _,,, ,' "' ~~-

i""~.~ ·- , ...... r"""--1"----.... ····· .... ... .. . . .. .. ... . 4 8 12 16 20


FIGURE 20

48

/ i ~

24

-. e ' .a

t80 0

t60 0

=:: t 40 --1-z C( 1-..J :l t20 en

"' D:

en ~ 0 D: len

..J c --20 1-z

"' a:: "' IL

0

0

.... , :E -40 :;) u D: -u

-60 0

-80 0

49 PLOT SHOWING EXPERIMENTAL "F

8Cg>"


NOT E: norr hG I Dl lessur, . D:: 20D. a.i. . . . . . LE< END:

. . rad al line - 9 = 00 .

9= 0 . ---- ---- /1 .... ,.o - -0 fl 9= ~- ......... r······· . . . . . :

: . . : . . . . _if i :' , I ~I j

(/_ ! It

~~ . ~ .: u ~~ ! .

_4 : IIJ

/~

~ ~-···· ........ ········ .IJ "/ -"'-: ~~

.., v· ~-.,. ,_ --- _, /I

0 4 8 12 I 6 20 24


FIGURE 21

t 800

t 600

.....

. E ' ~t400 ..... tz ~ ..J

~ t200 LIJ a:: (/)

(/)

LIJ a:: 0 t-en ..I ~

t-z- 200 LIJ a:: LIJ LL ~ ::.> u_ 400 a:: u

-600

-800

0

PLOT SHOWING THEORETICAL "F(a)'' ee


NOT : norm pi pre! sure , 1 ~=-20 .s.i. thre col. bts a 9=2· 6°14°3 bo

SPE IAL E OUND ~RY COND11 IONS: . . .,., _, . ., "' : nn .

&<l¥n =·OOC 55 rod . . LEG NO:

. : . /

radi I line 6= ~0 .

I . . ~= 10 ·---- --· . I .

00 -'--· . e =:: r-- .

I . 00 ••••••••••• .

9= . 1 . . . ~ .

! . • . I . . . . . .

.1 . . . I . . : I . . / . : ! 1_/_ . IJ

~' . .

/ . !j_ I.J.

.fi /f .: I l I

/ I

~r .: I .: I .

lj_ . . .

. : I I '.L 1_

.... /# (/ :1 I .· I' 1j ..

..... /' / .1_ .. /, I v ..

.:;; "' ,;'~ ........ " ::. , -4 8 12 16 20


FIGURE 22

50

-- --

I r--

24

t 800

t 600

--~ ......

~t400 ..... .... z ~ ..J

~ t200

"" a: f/)

f/)

"" a: 0 .... f/)

..J <(

.... z- 200 ~ a: ~ u. :I :::> u-400 a: u

-600

-800

0

PLOT SHOWING THEORETICAL "F (crf' ee for SHELL on HEXAGONAL BASE

NOT I : norm~l pre sure, ~=-:C ~0 lb. Vin. five col. p s. ot 9=2- p, 10,1 14· 20, 0

SPE ~IAL OUNO ~RY C ONDil ION: F(O') 1=-240 lb./ ir. nn

·--LEG I NO:

rodic I line e = 00 e= 10~ _,_, _ _ ,.

e =4 00 -'--- I e= 00 ••••••••••• I

/ A

51

_J_

i I ,--·- ·--

.. ;I \

• 'I \ . /~'

. ~ ~ . \ . .

I \ : . . I . . . : ~

.' I . . .· I . . . ~I

,r, . . . .: i--r-t--~ ---. / .J.

. . . . .'_A . . ----')-----

fill_ . . . ·i-t/ . .

.~· v -~~

..... ~, / /7 ,.y :I /

,// I

... ·· ll" 1 'I .. /~ /J ... ··v v ... :;, ,,

·······,·.: ~~ ...... ,-'/ ~~

4 8 12 16 20 24 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 23

t 800

t600

.....

. s ' ~t400 ..... .... z ~ ...1

~ t200 1.&.1 ~

(/)

(/)

1.&.1 ~ 0 .... (/)

..J

"' .... z- 200 1.&.1 ~ 1.&.1 LL :::E ::::::1

u-400 ~ u

-600

-BOO

0

PLOT SHOWING THEORETICAL .. F(cr)" ee


NOT :

~~rm ~~ pre ssure, ~=-2< f;).S.i. 0 20~3 b- .

ive eo·t 1 ,ts ot : 2.· s•to~t . . . IAL I SPE OUNO RY C ONOI IONS: . . .

F,.{H) =-24C .

lb./ il . .

s{)!W) =·OOC ~5 rod. . . . . . LEG NO:

. . . . o· .

rodi I line e = : 6 = 10' -- - --· .

o• -'---.

e =~ ~- j / e= o• ........... L . I . . . : I : . . .

_L . . . . . . I . . .

I . _L . i . .

f I . . ll . 4n . . . . .: _l

l I .. If : I I. . . . : I

~ I •· I l . I ~~

. : .

/ / ./ . .~ . : I

.: I . -'I .. .· / .. ·· / • i I .. ,

7 .... ,, /'

~-::;. .. ::;:~ ,;.:;/

f::_.., , ._.

4 e 12 16 20 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 24

52

. ~----

~,

_!

24

PLOT SHOWING THEORETICAL "F(cr)" ee for SHELL on HEXAGONAL BASE

NOTE

53

norm~l pre sure, ~=-2( P;!i·i. : tBOO~--~a~·~~··~"~t·-~n•··~·~~--~~,~~~~~~A~o~bro.~,~~o~~~n·o~----r---~~--+---4-~

~ 6 = 1 o~ --r _--+_-__ --+--- -+----+-!;-+----;--- ----t ---, 9 =· 0 0 ------ I .G 9= 0 •••••••••••••••• ; - t 4 00 1---+---+---+---=--l-.,;:___, __ +---+---+---+-------+----~---- ..... ~ ! ,~

... ; I \ ~ ----+r { --r-, ·-- t,· ---

_. : I \ ~ t 2 00 1---+---+---+---+-----l!---+---+----+----t-;'--1+---t--- --t --

: I \ I

~ : I l l --+--+----+--~=---:L-r-1---l- ---

en : I • I en 5 I i ~ 0 ~---··--+----+----t---+---+--·--1----+---!--~: ....,.~,___ - ·-· ! --~

~ / i _/~1~------ J.l ~ /1 ,- 1

~ _ 200 ~------- -----+--r---- ___ +--~- "a / - ! I ~ ll / I i I i -;:7 -i- -r '-1 ~ .. 400 r·.?'l~ - -+- -t- : l,

u .. I / I I

----t---t---+--_,--t--+----•• ~ •• ~I?~ -r- ---t-·· -- ---- :· ~~·

- 600 1-~t---+----+---+---+--t:.•l~' /'/ . ~----+--- ..... ... ~ I .. ··t/ IV \ ll' ••., I

·~:.:.:::.~·;.... .. ;'i --- --- __ J __ --~ I

- BOO 1---+----+---t---+-_,--t---+----t----r---t--·--t-·--+---1

I i ~--'----~:---__._-~-~----:-'-:---'----:L::--_..__--::-L:::----L---- . L. ! 0 4 8 12 16 20 t'4


FIGURE 2 5

t 800

t600

-.5 ~ ~t400 ... z ~ ..J

~t200 ILl a: en en ILl a: 0 ... en _, c( -... z- 200 11.1 a: 11.1 I&. s :::> u_ 400 ~ u

-600

-800

0

PLOT SHOWING THEORETICAL "F(g)• •• for SHELL on HEXAGONAL BASE

NOT : norm 101 pre [Iaure P.•-2 P_..P·_~;. :

... n ~"'- ... •

seve COl. pu. a e = o.:: jti 1 1U11"! ,lr,i!U1 "Cf ,;)U

f SPE IAL 1 OUNO ~RY 1-0NOI ION: I', i~l •·OOC !55 ra~.

. l . . .

= _:_

LEG NO: .

I . i

radi 1 line •• p• _._ e• 10' ·- --- --· . i . ·~ o•-~--~-o• ... .

I •• ········ ..... . . . I . . : I : ..1.. . i • • . • • I _;_

I 'I • ~''-.... : _j . i / ~

: • • ! I ' •: I L r 1 l .1 _j_

: I ll .~~~ . l 'I • :I

~l I lJ .: I lf ::I

• • il

.:~i l l I v

.... ,, •• .. , lj _/

•• ,, I I .. .. ,; ... , ' .... ! • ., ,1'~

~ _,, v -


FIGURE 26

54

~, _;,_

' 1

' ' ' __l

I I .

24

PLOT SHOWING THEORETICAL ,. F(cr)" ee


NOT! :

55

normpl pre sure , ~=-2< p. s. i. i t800~--~~~···~·~~·.n~L~~·~·~nt~·A~-~·?~-~~t.n~~~~~~u~

0

:~~~4~~~~-o~~~o---r--~=~~----r-, , I :

SPE IAL OUND~RY ONOI110NS: F(a) -24C lb./ir.

"" t600

i(~ =·OOC 55 ra~. . . . . . . 1 __ 11;

LEG NO: i i radic I line e :: ~no f I I

-c:· 1---+---+---+--x-.,......rl"----lf-:--+-c--:---+---t---+---,,:o-+'·---t----+-, ···-·,·. e = 10~ ·----- --· ' e =· o: -

1---r-- ./ I ~t4oor---+---;--~-9~=~0-··~·-··_--_ .. _--~------·~--~r--~~.-+---~--- : .... : ,l !

z : / -.\ ' ~ f----t--+---t---t---+--+-~---~--~-f---v· ~------- !· --

..J / I j\ -~ . ~t200t---+---+---+----+----lr---r---+----t----~:--, •\

IU • I I \ a::: o 1_ I . f-L .. :· en i I ' \ : ~ : I I \ a:: o /

1_, ,r;,+-t- · ·· ·

t- • ~ \ I

en ; / , ..J : j ----'1 ~- --+- -.... ct 1 I / I ! I : I I i t- • I ' z- 200 H-r1 ----t-·--·----·; - · · ~ .: I I I I

~ /// ~~-~ .

~~ 400 _;, 11(--+- ~~ ~ u .... ~' /jl t' !

_ 600 ••••• ~~:f r~ . ~~ -1--+---t---+------+--'-+--.. -:::+·, .~,.::., ~~~"~+· v· 1-1-~ ~ •

j:.:".::.:':.:: ::.- ,/......... ' - ..-- ·---t-··-· '

I - sao t---+--+----+---+-~-~r--+--+----t----r----r-----···--+--· !

I ' I

~--'---~--'--~-_...---:'":---'----'-:---'---::-'-:::----L. _____ L 0 4 8 12 16 2 0 r i;


FIGURE 27

t80

t60

t40

-.~ ' . ·= t20

I

l&J ..J 0 0.. ::l 0 u

en 20 cn-l&J 0:: l-en

..J <t -40 0 <t 0::

-60

-80

0

56 PLOT SHOWING EXPERIMENTAL ''M~g)''


NOTI :

n or11 ol p essur .• p_ = 20p_. •. i. n

LEG END: rodi I line 8=0

8 =I< o -- --- --A-~'

0

8=3 ~· .... ········ . ...

t l . ··. t

~ ~ I ' ...,, ~. : ' \. I

'

"' ~· ' / I \ \,~ ... I'~ II I

'..

~ /f / \ ····· ··. ' . ... ·· '

. ·. I .... ' ' ' · .. I

I ~' .

~/ .

~··. . ...... . -...... -~ . . . .. . .

' ' ... (. /

. . . ~ ~-... /

4 HORIZONTAL

8 12 16 20 24

DISTANCE

FIGURE FROM

28

SHELL APEX (in.)

tBO

t60

t40

c: ...... .='t 2 0 I, Q

UJ ...J CL 0 :::> 0 u

Vl (/)

UJ o::-2 0 ..... (/)

..J <(

'-l

;i-4.0

-60

-eo

!

I

0

PLOT SHOWING THEORETICAL "M (cr)" re


NOTI : norrr ol pre ~sure, p.=- 20 ~.s.,i

~o· lL I thre1 col ~h. o1 "e=2 ,s '14° I I

SPE IAL I OUND ~RY ONDI IONS~ I I I

'j,'fl ,.-,.,.v "''"' '"· i i s<flr> =·000 ~5 rae. I

~·-- -~--- c-! LEG NO: I radi ol line- 9= 00 I

__L

tJ .. ~uo • ~-~-r-~- i e" 0 -1--.-.---e " 30° • ............... J

I I

_L

i J .!-I I

I

1 I

'" ~ I I

~ _j _/_ .. ~ ~ ij_ I

I ··~ ,""''l!!.. ..I.'; I ...,' I : ··~ ~,. . . ··~, I / : . . . ...... _, .

··.. J • . . .... .. .. .. . . .... ....

4 8 12 16 20


FIGURE 29

57

r-·--r-·-

I I

j I l

r---~ !

·······~. ) ~·· I ···.t ------r-· I I

! : I

-~

---~~ -----

24

t80

t60

t40

..... c:

.......

. £t20 I. J:l

LIJ ...J Q. 0 ~ 0 u

(/)

(/)

LIJ Q::-2 0 1-Vl

-' ~

'-l

}4.0

-60

-80

0

NOT

PLOT SHOWING THEORETICAL ''M (cr)" re


: norm ~~ pre ssure, Pn=-z p p.s.i five col. ll s. at 8=2· ~~ 10, I ~,20

SPE IAL BOUN ~ARY CONC IT ION F(cr) =-24) lb. I in. nn

I I ' ·--- ----·- t-1 LEG NO:

rodi ~I line- e = 00 I / .. I e = ~?o

,_,_, ___ . •' e = 0 - --'-- : / e = 30° • ............... J

I ~l L

I ~l I I

I;.~ v j'

-~ 1/

{/

~ /J I

"iii~ ~ ~/ I ll ': , __ ,;..../ I

·~, .1 ,/ ·.• ..... .,.. .. ·. .. . .. ·· .. . · . .. \ . . . . . . . . . . . . . ---t-

. . . . . . . 4 8 12 16 20


FIGURE 30

08

I

l I

I '

i

li I I

--

I

+-

24

t80

t60

t40

.-.. c:

......

.~t20

'· ll

IJJ _J

a. o :::> 0 u

(/)

Vl IJJ Q:-20 ~ Vl

...J <t :"::)

:i-4.0

-60

-80

0

PLOT SHOWING THEORETICAL 11 M (cr)'' re


NOTI: I j I norm I pre sure, ~=-20~p .s.i.

rf 3.0° fiV@ ~ol. nt at =2·11=:0 n°14°? I

SPEC IAL E OUND RY CONDIT ONS: !' !f F(cr) -240 lb I in

Ill ~ I

s<m> =·000 55 rae. , 11 +----

LEG NO: i I

radi pi line- e = 00 I I Ll

tt = ~uo - - --r -- i I e = 0 - ~--------e = 30° • ............... i !

I II I

I! I ll I

L" } 1

~~ I 'i I

' I / I

~"' I I

I ·~

~ /;l I I

I

'"' ~~ ·"'' / ~., ~/

···· ... -.. .. -r ... ········T·· ··. ·· .• . .

I

I

I 1

\ r\ . . . . . . . .

4 8 12 16 20

HORIZONiAL DISTANCE FROM SHELL APE X On.)

FIGURE 31

59

I :

I ~-

f-

l-I

r~ j

t I ~-~ I i i ! I I n I '

I I--

H I

.. -

. 24

180

PLOT SHOWING THEORETICAL "M (a-)'' re


~ ~

NOTE: norm~l pre! sure '"'=-20r.s. i. 0 .,~·. :1 o """'-o ~~ ~•n lv' I' '- -~~~-· 0 ·--·

~

SPE( I AL OUND~RY ~ONOIIriONS J:'ltr\ = -?4( lh /' nn

s<fW = · OO< 55 ra

• I

. . . . . . . . . . . . . . .

60

I t60 ~--+---4--~---+----1---r---+---+-----+--·---...... LEG NO ' ; .! 1

1

1 J radipl line- e = 0°.. .L

~--+---4---~--~~~-----r---+---1----~-----e = I' o o - ---- ( -- I : ! • e =~o -1----- : I e = 30° • ........ ....... I : ·

14 o ~--+---4-----+----+----1r---r---+----+---~---------r~---r-l.

c::: ...... . ~t20 I.

~

UJ ...1 ll. 0 :::> 0 u

(/)

(/)

UJ a::-20 1-(/)

...J ct :':) ! ;r4.0

1-----

I : J :

I . I :

~ -~ . I I .J

~ / ! I j

I '"'~ J l-r-1..-r--------·~---i "" I : I : ' A' I : i : ; :

~-----t---1----1------ll------11---·~---l··~~ ,_ ,/ / .:1--r-1 ··~·~ ....... _.,. ~·lfl tt ..... · . ·, ---t-·ll ..

1----

••• •• t •• •• •••• l -1--t----+---+----+------+----lr------t-='·· ..... _. - . - --r -l

ll -6 0 t---t----ll------1----l----1---t---+---+---+--+--+--t-

l ~-+--4---~--+--~--+----+---+-----+--·- .. ----- -· ..... _ ......... ..

-80~--r---+----r--+--4----t---+---~---r---+-----1r---~

0 4 8 12 16 20 24

HORIZONTAL DISTANCE FROM SHELL APE X (in.)

FIGURE 32

tBO

PLOT SHOWING THEORETICAL "M (cr)" re


NOT : ' .

61

: I ~--~n~o~rm~~~ll~p~r,etsr.u~re~,7P~n=,.-~2riCI~~··~~i.~~o~~o~--~--~L_I:4---~-1 seve coL pts. a e=2· ,10,1~ ,I r,;w:~'+,;:)U ! 1

SPE<IAL IQUND~RY ONDI110N: .~ ! l_j a(~ =·OOC !55 roc. • 1 I I

t60 1-: --+-L--E-G4-N-D-:~--~--t----t----+--t---- -- f --,:-----Jj ·J rodi~l line- e = 0°.,

t----+----+-----+--.....--::e=:-+.1',_... o _0--+--~-------+-1_--,_-+-----+---1---- -,1---t--r------ - · 1

e =~oo ... ____ f 1 r e =30 • •••••••• ....... i 1

--+--+--r--+-~~~---~--- -I I

--t-+----+-----+----+---1----+----+-~/!-+-. f- - rj --+---+---+---+-----+--+--·--t----t-f-1--! -+-='----- --[ l I ! I ; \

-+--+----+---+--+-- --+---+----+1--Hll "+/-+ ~~- - ---' -~ I IJ l I :

!----+---~--~---+--~~--~--+---~---.+-~ I" I I ! I -+--+---+---+--~--r----+---+-, -...f-l!.' 4 --- ' t---+--+--+----+--+-1,_---J,' 'lliik---t----Ti· --4t t~ i i

~ v;·-,! ....... ·.il •• \.· - ~ I I I l-----~-;-f---j

~----+----+----+---+---+----+---\~~f, t_f~- -~ I • 1 • I

\. I ·4- F~ i ~-+---+--+---+--+-- /'•. I --7 I' ---r---- :T-'1

•· ••.....• •• I : I I r i : i

-6 0 r---~-+------+--+----+--+---+----t-1--- --r--- ----.. -1Tl I I : '

t-' -+---r--+--+---+-------1t---+--+---+ + - i· ~-l

-8 0 ~-----+----+----+---+----1--+---~--+----+-------r-- -_i

c ·-........

:'t20 1------ --· I

J:l

r----· UJ ....J

I 0.. 0 ~ I 0 u

1--(/)

(/)

UJ a::-2 0 1-

I (/)

I ....J 1---<l Q <t ~r4.o

t40

0 4 8 12 16 20 24

HORIZONTAL DISTANCE FROM SHELL APE X On.)

FIGURE 33

PLOT SHOWING THEORETICAL "M,<g>"


NOT : I •

62

~!~:~I !:.~ !~:•·,~·-:~ ·.·~~·trl'lta0 t7°~n·~a·~ln° f t80 ~--~&U.-~~~~~~~~~~~~~~~--hK--~--~ SPE IAL OUND~RY C ONDiliONS:

11

~(§) •-24 lb./i~.

~ •·000~5 roc. •

t60 ~--+---+~--.~--~~--r---+---~--~--~------~ .• ~--+-; LEG~ND: 1

radi~l lint- 8 • o: I ~ • tu 0 • ---- --- I e •zo -~---- I

t40 r---+---+---;--•-~·a_o• __ .r··_··_··-··+·-··-··_··+---+---4---~~~-+---+-1 I I

c ' ' I '20 ~--+---+---~--~--~---+---+--~--~~._HI~-+---+-4 g I

I,

s I ....

"' ~ 0 ~ I

8 ''"" j / ....• . . . . . . • • . . .

E ~ _} / 1 \ ~2 0 1----+---+--~--+---~---+--J~ '""'-· ;/ /'----+:~. ~-H-\--+--1 : ~.="- /~'-~· --+-~!----lt--1 .116 ~~;___ ~. "' .. .. . c ... . .. · :. ~4..0 •••••••••• •

~ • . . ~

.. • \

-60~--r---+--~---~~-r--~---+--~--~-----~--r-~~

. . . . . .

-eor---r---+----~--~--~---r---+--~--~----~--~--~:~ .

0 4 B 12 16 20 24

HORIZONTAL DISTANCE FROM SHELL APE X (in.)

FIGURE 3 4

t80

t60

-c 't40 c

'· .a ......

I&J t20 ..J Q. :l 0 u

en o en I&J 0: t-en

..J -20 <[

t-z I&J 0: I&J lL -40 2 :::> u 0: u

-60

-80

PLOT SHOWING

for SHELL on

NOT :

norm ~· pnr "11urc, Pn- -e. ~ Jlll.l.

LEC: END:

rad1~l lint - 9= pv 9= 00 ·-1-----.. 5= ~u ---e= ~00 .. ........

, ___ -- ~,_.. ... ---~ "'-

EXPERIMENTAL "M~~)"

HEXAGONAL BASE

. . . .... .... . . . . . ---· . . . __ , . . . ....... .... . . . ...

•• : .

..... . : : .. .. .

.... .. ~--- ... _,;;_..-- ... --t7 ~ .........

~ ' .....

---- ,.,.,,_ --- ' ----~ _1

'

63

0 4 8 12 16 2 0 24


FIGURE 35

t80

t60

c

' . 5 t40 I,

.Q

LU ...J ~

t20 ::> 0 u

VI (/)

LU 0::: 0 1-(/)

...J <t

1- -20 z LU 0::: LU ~

::f ::> -40 u Q: u

-60

-80

PLOT SHOWING THEORETICAL "M(a-)• er


NOTE : norm ~I pre sure, =-201 .s.i thre col. bts a n 9 = ~ f6°14.0 130°

SPE IAL ~OUND ~RY ~ONDI IONS:

,;r;' - .. ~ ,.,,, ... 6(~) = ·00( 55 rc d;

-· LEGE NO:

radic I line 9= )0

e = 1 ~0 -· - -- - -e = 0 -,--+-e = 00 ...................

64

--r----f-----

-L ""' ~ -- -- ·--t--

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

~ I-'' r:-.~ .... ' :"-~· ~ I

~~.~··· .. ~, \ \ ..

~~ It'" \._ \

I \-·.~-

"' .\l , ... \ '.\ \j\ \ \' \ ~\ ,, \ ~\

\\J \\ '~ ' : ----- ' : --r--' : . 'i n .

--r--- ·-. . .

4 8 12 16 20 ?4 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 36

PLOT SHOWING THEORETICAL "M(o-)• er


NOTE:

65

tSO r---+----r."~o_rm~l~p-re4-su_r_e~,~~~·=~-~2~P~-~s-~i.~~~---+---+~·~r---+-1 five col. p 15. at e = 2·6 10, 14, ~0,30 ~

c::

' .E '· .Q

LU ..J ~ :::> 0 u

C/1 (/)

LU

t60

t 40

t20

a:: 0 1-(/)

SPEC AL J3 OUNO_} RY ONOiliON : : . ~<g-> -240 b./in. :

+----- ---LEGE NO;

rodic I line e = ~o

. ! I I --- r:---- ' - r --

j /1 I

--~·-tr·---- t- -o I I

1---1------1----t----+--+·~-+---+-----+---+--;Jl ..... t ~c- ~ ~,--::ft -1·· f

t------+----+-----+---+--+---+"""""""'~=-t---~ .... J\. . ·----t---~, ~ -.~;.:.··· \ I 1

.... - .•.. I'' I I ~ ~~~····-- I I

~~~ ... ,..... \, ~ __ L t--- .,

= -20 \ \ -!--1-~ \ \1_----t---1 ~ ~~~~~~~,-~+-----:::> -40~--+--4----r---+--~r---~--+--4----+--~~-

~ '\ u \

~--+---·+--4---4----+--~r---r---+---1-·--~,.--~---r~

\ -60~-+--~-~----+-~~---~--+--4----+---+-~+--r---+-i

\ ~--+--~--~---+--~----r---+---~---r---T--\:~-----r-

1 I

-80~--+---~--~---+--~~--r---4'--~r---~---~-~~r---·+-1 I

0 4 8 12 16 20 24 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 3 7

t 80

t60

c ' .e t40 '· .Q

l&J ...J Q.

t20 ::)

0 u

(/'1 C/)

l&J 0 0: ....

C/)

...J ct 1- -20 z UJ 0: UJ u. ~ ::) -40 u ~ u

-60

-eo

0

PLOT SHOWING THEORETICAL "M (a-)• er


NOTE : norm I pres ure, ~ ~=-20 p.s. i 0 five ol. pt . ot ~= 2·ff. 10°14° 0 30

SPEC IAL E OUND ~RY ONOI IONS:

Fn<rr> -2401 b./ in.

S(~) =·000 ~5 ro

LEGE NO:

rodic I line e= Po e = 1 Po ~~ ~-~..;.. ~-~

e =· 0 --- ~--•• J. ••••••.•••••••. e =• 00

~, ~~---~

66

. . : . . . .

·=-. . . . -:-. . . . . . . . . . --:---. . . ~ /' -· ~- .... ~ .....

~ ~;r-~ :;. ~.:-; .. ... ··~.

~~ ' ... '\·. ' .

~ \ .: ,, .. ·v:·· .... I \

' \

\ \ \

·-\ \ 1 \ \ \ \ _l

\

' I I

r 4 8 12 16 20 ?4


FIGURE 38

t80

t60

c ...... .e t40 I

.Q

~ ..J a..

t20 ::> 0 u

C/'1 (/)

1.1.1 a:: 0 1-(/)

..J <(

1- -20 z 1.1.1 a:: 1.1.1 LL. :E ::> -40 u ~ u

-60

-80

0

PLOT SHOWING THEORETICAL "M(a-)• er for SHELL on HEXAGONAL BASE

NOTE; norma I press ure , ~ =-20p s.i.

P.,no'2, o "' - ? . ..::.0 'no .~~o,.,

J ' . > ' SPEC AL 8PUNDA RY CCNOIT ONS: r::J . •.. \ • ?A f\ L~

n'n - - ·-· 6<!¥f) =·000 ~5 roc.

LEGE NO:

radial line · e= )0

e = 1 00 __ .,._ -----

e = 0 -~--+--e = 00 ···················

-""'-

~ .... ----~-~~ ~ , ~-:.:-.:: -.....

~~~ ..

'\~ ~ .. ..

~ ....... ··. ·. ,~, ..A. r- .

~\\ I ·· ... · . . ~~'

. . . . . \ . . .. . \ . . . \ . . . ' . ~ . . . \ . . I . . . . . . . . . . . . . . . . . . . . . . .

. . . .

4 8 12 16 20

HORIZONTAL DISTANCE F~OM SHELL APEX (in.)

FIGURE 3 9

67

---

/

f--

c::

' .e .. .Q

kJ ...J 0.. ::::> 0 u

VI (/)

kJ 0:: .... (/)

...J <[

.... z kJ 0:: LU u.. ~ ::::> u Q;; u

PLOT SHOWING THEORETICAL "M (o-)• er


NOTE:

68

tBO r---~--~n_o_rm_l~''--p~re_sr.u_r_e~,~~=~-2_0~t1~.s~·~i.~~~~~~--~---+----~ seven col. ts. a e= 2~,10,1~ ,17,2C ,24,3<

t60

t40

t20

0

-20

-40

-60

SPEC AL I OUND RY < ONDI ION:

b(~) =·000! 5 rod

LEGEND:

radicl line e = )0

e =to· __ ,..._ ----e = 0° ----1--· e = 0° .. .! •••••••••••••••.

0. \

•• \ I . ' I

. 0 .

0

r---~--~---+----r---+---~---+----r---+---~\~0--~---:--

0

0 0 0

r---~---r---+----r---+---~---+----r---+----r~;--+--~~

0 0 0 0 0 0 •

r---~---r---+~---r---+----r---+----r-~+----r-T;-+----:r-0 • • 0

: 0

: . : .

0 . . . -sor---~---r---+----r---+----r---+----r---~---r--~'+---~:--. .

0 • . . : 0

0 4 8 12 16 20 "4 HORIZONTAL DISTANCE FROM SHELL APEX (in.)

FIGURE 40

t80

t60

..... c ..... . & t40 .. .&:1 --"' .J Q.

t20 :::> 0 u

(/t Cl)

"' 0 a: ... en

.J C[

... -20 z LLI a: LLI LL. ::E :::> -40 u g; u

-60

-eo

0

PLOT SHOWING THEORETICAL ·M~y-)•


NOTE: 'h=-~C norm I pre sure , ,.p.s.i.

·-·~ ,, .. _. -· ... .... ...... ..... , .... - , , .... , ·--· , ........

SPEC AL BOUNDl RY ONDIT IONS: e 1 _, .........

'"' lo

nt:,~.

5(~ •·OOC !SS ro .

LEG I ND:

radic 1 line 8• ).

. " . o· • ~--~- ,__.,.

e • o• - --· --· e • o• .. :-········ ......

-~ -~--~~ ~ ... -: .-,

~ r.~.::: .... ...

'\ ••• ~

.. •·. '"' """""""'

!""' - • . . .

i I I I . I

I .\ '\ ~ . , .... . . • . . . . ~ . ~ . . ~ ~

~ . ~ • \ . . . . . . . . \ ; . . . . .

4 8 12 16 20


FIGURE 41

69

. . . . . . . : • : • .

24

.... ..,. •o

)(

c .._

.... z l&.l ~ l&.l (.) <[ ...J 0.. (/)

0

...J <t :::E a:= 0 z

70 VARIATION of

11Un 11 on the BOUNDARY of SHELL on

HEXAGONAL BASE for THREE COLLOCATION POINTS

LEC: END:

t2·0 lnte media e poi ~ts Coli catior poin s

""""' v ' [It_

ti·O

0 L \ -\ L , v -1·0 1\.

~ I -2·0 I \

I \ -3·0 ~

1\ \ ~ \ J

-4·0

:\ L \... v -5·0

0 5 10 15 20 25 30

CIRCUMFERENTIAL ANGLE , e (degrees)

FIGURE 42

-'It I

0

:1(

c: -.... z l&J ::E 1&.1 u c:( ...1 Q. (/)

0

...1 c:(

::E 0: 0 z

71 VARIATION of II un" on the BOUNDARY of SHELL on

HEXAGONAL BASE for SEVEN COLLOCATION POINTS

1·6

LEG ND:

Inter rnediat ~ Doin ts Colle cation points

1·4

1·2

1·0

0·8

0·6

0·4

I

I 0·2

-0·1

_.._ l - ~ v ~ 0

0 5 I 0 15 20 25 30 CIRCUMFERENTIAL ANGLE , e (degrees)

FIGURE 4 3

CHAPTER III

SHELL ENCLOSING RECTANGULAR BASE

DIKOVICH in 1960 gave a solution for a rotational

parabolic shell enclosing a rectangular base and subjected

to a uniform normal pressure. This solution also employs

the fundamental shallow shell equations which were derived

by MUSHTARI in 1938 and VLASOV in 1949 and are giv~n'in

APPENDIX A as (A-1) and (A-2). However, DIKOVICH obtained

one fourth order differential equation in normal displace

ment from (A-1) and (A-2) which are fourth order differential

equations containing both the normal displacement and the

stress function. This equation was modified for a shell with

rotationally symmetric p~rabolic middle surface. BERNOULLI's

Semi-Direct solution was applied and normal displacement was

assumed to be given by a trigonometric cosine series. The

homogeneous solution for the no~mal displacement was coupled

with the particular solution and the result was simplified for

rectangular symmetry. This solution for the normal displace

ment was substituted in the MUSHTARI-VLASOV equations to give

a solution for the stress function which also was assumed to

be given by a trigonometric cosine series.

12

•

73

Consequently DIKOVICH's solution does not neglect the

transverse bending stiffness of the shell as do membrane sol-

utions given by a plethora of authors. However, her solution

does limit the combinations of boundary conditions which can

be applied to lateral sides of the shell, since both the nor-

mal displacement and the stress function are assumed to have

trigonometric cosine series solutions. For example, if the

normal displacement is assumed to vanish at the shell's bound-

ary, then the radial and circumferential stress couples which

contain only second derivatives of normal displ~cement, must

also vanish.

ORAVAS gave a similar solution in 1957 (2) by means of

WEBB complex dependent variable technique for rotationally

symmetric, parabolic shells which may be considered to be

shells of translation as a special case.

)

In this chapter a comparison of normal displacements

and sectional resultants, computed by DIKOVICH's solution and

by the collocation solution for the radial line e = 0° , is

made for a pair of shells of similar middle surface which en-

close identical rectangular bases and are subjected to the

same normal pressure.

The geometry of the shell for which DIKOVICH's solu-

tion applied is described by the parameters

where

~ :: fl :: .Z•O 6

a. :. 25·0 C03 ~ : 17·66 i 1'\..

2f represents the height of the apex above the shell's base, 5.085 in.,

2a represents the horizontal length of each of the shell's four boundaries,

and 6 represents the shell's thickness, 0.432 in.

74

The shell for which the collocation solution applied

also had boundaries of the same horizontal length,2a, whose

corners touched a base circle of the same diameter,50 inches:

it had the same shell thickness and its spherical middle sur-

face had a radius of 64 inches so that the apex was the same

distance,5.085 inches, above the shell's base.

Geometrically the only difference between the two

shells lay in the curvature of their middle surfaces. The

curvature of the parabolic shell was slightly flatter near

the apex and steeper near the boundaries than was the curva-

ture of the spherical shell.

The boundary conditions which the two solutions

satisfied were, however, not quite· identical.

' 75

DIKOVICH's solution satisfied the boundary conditions

u = 0 n

M~cg) = 0 (III-1)

F~f[) = 0

At a = o· these boundary conditions become

u = 0 n

F~¥) = F~~) = 0 (III-1*)

Mfg) = Mbf) = 0

The collocation solution satisfied the boundary

conditions :

un = 0

M( cr) = 0 ns ( III;-2)

F~g) = 0

tss = 0

at all seven collocation points except at the corner for

• a = 45 , where the strain was not assumed to vanish. At

e = o·these boundary conditions become

un = 0

F ( o-) = . F ( cr) = 0 rr ee (III-2*)

M~~) = 0

Ess = 0

76

Comparison of the boundary conditions (III-1*) and

(III-2*) shows that the collocation solution replaces the

. 0

boundary condition M~¥) = 0 at e = 0 , used in DIKOVICH's

solution, by fss = 0.

The normal displacements and sectional resultants

calculated by both solutions are graphically depicted in

FIGURES 46 to 50. It can be seen that while the radial and

circumferential stress resultants concur for both solutions,

the normal displacement and the radial and circumferential

stress couples have significantly larger values for the

collocation solution.

The dissimilarities between the normal displacement~

and the radial and circumferential stress couples as cal-

culated by the two methods are probably aggravated by the

incongruity of some of the boundary conditions.

However, the stress resultant boundary conditions

in (III-1*) and (III-2*) were identical and consequently

the radial and circumferential stress resultants ·of both

solutions are quite similar. This indicates that comparable

results can be obtained by either method of solution pro-

vided that their boundary conditions can be made completely

compatible.

77

PLAN VIEW of SHELL on RECTANGULAR

BASE SHOWING LOCATION of RADIAL LINES

for which SECTIONAL RESULTANTS ore

CALCULATED

- -/ ....... 0

/ ........ 0

' ~

/ '0 / \ I \

I \ I \ I \

I I

\ I \ I \ I

\ I \ I

' /

' /

' / ...... ........ /'

- ~rodius=25"

FIGURE 44

LOCATION of BOUNDARY COLLOCATION POINTS

for' SHELL on RECTANGULAR BASE

7 COLLOCATION

FIGURE 45

6= 45° 6=44° 6=4cf

e =35°

6=28°

0

6= 19

0 6=2

POINTS

78

0 0 o,-. II-

CD

a:: Oo ~lb ---Xo LIJ• a.!!l ~

.J•

.J LLio :1:· en=

:E 0 a: LL.o

Q, LIJ u z ~ Cl) -o o,._

.• .J ~ 1-z 0 NO -lb a: 0 ::1:

0 ..,

0 ..:..

PLOT SHOWING THEORETICAL • U0

•

for SHELL on RECTANGULAR BASE

~~ ' ~~ ' ' '~ "'I

' --, ."

' " \ ~ \ "' \ ~

' \ \

\ \ \ \ \ \ . i , I

~ I \ I .

• i .. 0 ... l I II) t I .2 ., J ..

I ~

CD I a N I u

r:b ; ·= . - I 8 ci,..:' I I

I • I 0 . ~ c I 0 '0

a! .. _g a _I_ ~

• • . . .g • L. • I • E ::t .. . ~ I "' • >o

as I 0 J: 0 c .. ! z - .2

1&1 ~ c ~ .~ • ~

0 ~ : ; 1&1 ::.:: 0 z c • -J 0 .. I

~ ~ 0 s: co

0 h 0 0 0 c .i 0

FIGURE 46

79

.. ,

HORIZONTAL DISTANCE FROM SHELL APEX (in.) FOR 9 = 0°

~-10 c ·-' • .Q -... -20

'"T1 z G') ~

1-c _. :0 ::> 1'11 en

LLt 30 a:-~ ...... en

en LLt 0: 1-en -40

0 1·0 3·0 5·0 7·0 9·0 11·0 13-0 15·0 17·0

NOl E: V" ~ norr al pr ~ssure ~ .. - I p.s.i. - - - .. ~if sev n col pts. t e= 1·7,l ,.7,28, 35, 4C ,44,45

~ .,.,

... ~ ,.· v

..,·· ~ ~ ~

.-!!J:.

~ ~

..,~ ~

. .. -- I,;-..... _ ... ---._ __ ----

J

.... "tJ 0 r ""' 0

-{ (I)

::t: (I) 1'11 ::t: r 0 r

~ 0 z :::J

G')

:0 -{ 1'11 ::t: n fT1 -{

0 l> :0 z fT1 G') -{ c -r 0 l> l> :0 r

...J ~

0 ~ 0:

-50

LEG END:

DIK ~VICH 's sc lution -- -- f--

= a:J .....

l> ""'-(I) ""'q fT1 - =

sol tion t y met hod o ' colic cotior

.. Q)

0

0

00 II •

CD!::

0 -. ·It) c-.... X LLI Q.O ct!2 ...1 ..J 1.&1 J: cno

2: 0 a: I.L.

LLI<? Ucn z <t ~ C/) -Oo

,.:. ...1

,cl ~ z 0 No crib 0 J:

0 !'>

0

PLOT SHOWING THEORETICAL 11 F(a)'' ee

for SHELL on RECTANGULAR BASE

~ ........

"' ci. -~ .. ... ::I

"' :: Q.

~ ~ 0 l) z c

0

It) 'It . .,... 'It o· 'It . ~ aJ. N r:-!!! r; II

CD -v

.,; -Q.

8 ~ > ., "'

.... ~

0 I

"""-.

"""

0 N

I

~

"' ' ~ ~ \ \ \

I \ I I I I

11 I ,/

I I I I

I

I I

0

'?

I I

I I

I I

I

I I .

0 'It

I

I

/

.. 0 z 11.1 .., 11.1 ..J

I

! T

I I I I

c: -~ -::I 0

"' ., :I: ~ ~ :!11::

2i

0 It)

I

81

c: 0

+= ·o

!! 0 u

0

"0 0 .s= ., E >o

c: 0 -0 lilt

( 'UI /'q I) J.N VJ. 10S 3 ij SS3 ijJ.S 1\fiJ. N ::Jl:J 3::1Wn:> ijl :>

FIGURE 48

0 0 0~ II -

<D

a: 0 LLo

- ~ c -X w<? 0..1() c:t-

~ 0 0::

"-o I&Jc;, 0 z <( t-C/)

-o a~

.J

~ z 0 NO -.;, 0:: 0 :a:

0 ,;,

<? -10

0 ..

PLOT SHOWING THEORET I CAL '1 M(cr) 11

re for SHELL on R E C TANG U L A R 8 A S E

~ ~

/ I

/ 11

' I I I

I I I I l

I I

6

I

/ II

t---.... ........

,' /

I /

I/

0

I

~ ...... ~ ,, ' I

I _L

/ If

/ /

~~ L

L v

v

0 N

I

~ .........

~ I '

) / v

v

10 . 'It 'It o· l~.

~-co .. .1':-·- co

"!c-..

- ..:.. I u II (p

tr • 0

OJ .. ~ ,;

= ~ ~ ~ Q.

u .. 0 c

6 ~ ; u

z c

0 til I

Ill

I

J I .~ I ~ ·= I o u

... c 0 0 ·- "0 .. 0 E .~:. Ill OJ

E .. • >-

0 n c 0 z ~ L&J ..

w :.::-- 0 _, Q "'

('UI/'U!-'ql) 31dnO~ SS::I~J.S 1'110'1~

FIGURE 49

82

0 .;,

I

'"T'I --(i)

c :::0 m

01 0

HORIZONTAL DISTANCE FROM SHELL APEX (in.) FOR 9= 0° 1·0 J·O 5·0 7·0 9·0 11·0 13·0 15·0 • 17·0 -

~

' . 4·0 ~ I

.D --Ill ..1 3·0 a ~

NOT~: norn~ol pr4ssure,(Prf-1 Pj.s;.i. ..1 .. ..1 .. _a~ sev~n col.l pts. crt-a: f7,1a-=-7,~,35,'J0,44,~5

X ~· LEfEND:

DIKPVICHI's soau tion 1---~---~ soldtion I>Y me~hod lof cdllocotlon --1---1-- /v

Lr ' 0 u en en 2·0 Ill 0: 1-en

A ;r

..1 ct 1·0

v , 1-z Ill 0: Ill b. 0 ~ ~

-- --+--+--

-- ..,_- -!- .---1-- -~--- -- - --~------·--.... --~----- '"""'~ ',

u 0: -u -r·o L---~--J----L---L--~----L---~--~--~--~----~--~--~--~--_.--~~~

- , 0 ., r

0 (I) -i :r: m (I)

r- :r: r- 0

~ 0 z :::J (i)

:::0 m -i 0 :r: -i rrl

)> 0

z :::0

(i) rrl

c -i

r 0 )> )>

:::0 r

OJ = )> 3: (I) c:t-rrl .,!!.

= CD

""

CHAPTER IV

SHELL ENCLOSING TRIANGULAR BASE

A solution by the collocation method is given for a

shallow, thin, calotte shell of spherical middle surface en-

closing a triangular base. 'Shells of this type have been

constructed in practice. The shell which was solved had

the same thickness, 0.375 inches, spherical middle surface

radius, 64 inches, and base circle radius, 25 inches, as the

shell enclosing an hexagonal base studied in CHAPTER II.

The boundary equations

E-ss = 0

M( a) = 0 ns (IV-1)

un = 0'

were·· satisfied at all the collocation points except at the

shell's corner where the normal boundary force·, ·F ~g), was

not assumed to be zero which is more in concert with the

actual boundary condition of such shells. These boundary

conditions would occur in practice if the shell's boundary

was supported on a very narrow boundary diaphragm.

84

as

FIGURES 53 to 57 depict the values of the normal dis-

~lacements and sectional resultants which were computed theo-

ret~cally along the radial lines shown in FIGURE 51 by employ-

ing the seven collocation 'points shown in FIGURE 52.,

The distributions of the normal displacements and the

sectional resultants were consistent with the results given

for the spherical shell enclosing an hexagonal base in FIGURES

12, 19, 26, 33 and 40 of CHAPTER II, even though the boundary

conditions satisfied by both shells were not identicale The

boundary condition M(a) = 0 used by the shell enclosing a ns

triangular base was replaced by the condition

s(~~") = 0.00055 radians

for the solution of the shell enclosing an hexagonal base.

Also the shell enclosing an hexagonal base assumed that it was

the strain rather than the normal force which did not vanish

at the corner collocation point. ~he magnitudes of the normal

displacements and the sectional resultants became similar for

both shells near their apex.

A solution for the shell enclosing a triangular base

was attempted ·for· ten boundary collocation points; however the

results were inconsistent. This attempt supported the con-

elusion drawn in CHAPTER II which maintains that an increase in

the number of boundary collocation points above seven, would

8G

likely decrease the number of significant figures in the in

creased number of requisite numerical computations below a

"safe" level. An indication of the increase in the number

of computations involved is given by the size of the N x N

matrix of the boundary equation coefficients, where N=4L-l

and L represents the number of collocation points employed.

For example, seven collocation points necessitate the solu

tion of a 27 x 27 matrix while ten collocation points nec

essitate the solution of a 39 x 39 matrix.

In CHAPTER II a comparison of the degree of satis

faction of the boundary condition un = 0 for solutions,

employing 'three and seven collocation points respectively,

revealed a decrease in the number of significant figures in

the computations ·from sixteen to six when' double precision

accuracy was used throughout. Logically then, the solution

which satisfied ten boundary collocation points would require

approximately 20 to 25 figure accuracy throughout its compu

tations in order to yield reliable numerical results, p~ovided

that the same precautions outlined in CHAPTER II and APPENDIX

B were taken.

Some questions were considered with regard to the re

liability of the theoretical solution in which four boundary

conditions must be satisfied at all the collocation points

except one point where only three boundary conditions must be I

satisfied.

A solution for the shell enclosing a triangular base

was attempted using the same seven boundary collocation

points but assuming that the strain rather than the normal

force did not vanish at the corner boundary point. The re-

sults, which were obtained by this solution, were almost

identical to the results given in FIGURES 53 to 57, with the

exception of minor variations near the boundary corner. This

indicated that the non-homogeneous boundary collocation point

does not affect the reliability of the solution for seven

boundary collocation points. Obviously the perturbing effect

of the non-homogeneous collocation point on the accuracy of

the results would be greater when the solution employed fewer

collocation points.

PLAN VIEW of SHELL on TRIANGULAR BASE

SHOWING LOCATION of RADIAL LINES for

which SECTIONAL RESULTANTS are

CALCULATED

/ /

I I

I I I

I

I

\ \

\ \

\

'

/

/

' '

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

' ' \

\

' \ \ I

~------+----~ e = 60° I I

FIGURE 51

88

89

LOCATION of BOUNDARY COLLOCATION POINTS

for SHELL on TRIANGULAR BASE

0

9=60

I 9=40° I /

I / 0

/ e=27 I / / 6=20°

'/ / ./ -::: _. v / / -- e= 11° //. -- -- 0 ---~-:::...---- e= 3


FIGURE 52

NOl nor

seve

LE

t·03 rae

t·02

-c - t·OI

1-z lLI 2 lLI 0 u oct .J Q. en 0

-·01 ..J <[

2 0: 0 z

-·02

-·03

0

PLOT SHOWING THEORETICAL II u II

n for SHELL on TRIANGULAR BASE

E: ~al pr usure R.,=-2 0 J>.S.i

0 0 0 0 0 0 col. ts. a 8=3 11,2o~c; 7,40;!5 14,60

GEND

iol lin e- 8= o• e= l~o. r--- --8= 40 ·--- --8• so• . ········ ........

V/ I'

I J

j / I I I

1

lfti I / I

/ / I I I ... l;/ /I .. ·· .. ·· / ~ . II' /, .. · ~ ... ··

~ ,,/ /

.~ . 1'1. ~:::-- ~- . -- . ·· ... ········ .. ..

········ .. ....... .. ....

90

. . .· . . . . .

... ··· .·· ..... . .

/

4

HORIZONTAL 8 12 I 6 20 24

DISTANCE FROM SHELL APEX (in.)

FIGURE 53

.

tl60 0

tl20 0

teo 0

-c

' .c t400

0

en -40 en ~ 0:: 1-C/)

_.

0

~ -80 0

0

~ 0::

-12 nn

-16 00

0

NOT

norr

seve

LE

rae

E:

nal pr

col. ~

~END:

ial lin

PLOT SHOWING THEORETICAL 11 F (a)" rr


ssure, P. =- 0 p.s .. ts. at 9=3 • u"2o,4: , . i40,5 • • i4, so"

~- e = 0

0 e = 2d' ·---- ~---e= 40° . -- 1-- -·

e= 60 ........ ........

91

... . . . . . . . . . . . . . . . . v~

/ ~· v 1--' v ,, ,, ~ I

v ~, ...... -""' .-'' !--"'' ~,,., ...

... . . ~

..... .....

~-""' -- / .. J'-.. / ----- -- .· .. .. .. .. ·· .. .. .. .. .. .. ... .. ... .. ....... ..... ········

4

HORIZONTAL

8 12 16

DISTANCE FROM SHELL

FIGURE 54

.

. . . . . . . . . . . . . . . . . . . . . . . . .

20

APEX

. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . .

24

(in.)

tl600

t1200

-c

.0 t800 ....

.... z 4: .... ..J t 40 ::) en 1.1.1 0:

en en o 1.1.1 0: .... en

..J

0

4: -40 'I. .... z 1.1.1 0: 1.1.1 u.. -800 :IE ::)

u 0: u

-120 0

,, -160

NOT

norr seve

LE

rae

0

E:

al pr

PLOT SHOWING THEORETICAL .. F(o)" ee


ssure pn =-2 ~ p.s.i. . col. p ~· at 19= 3,1 ,20,2 ,40,5 •,60 . . . . . . . END:

. . . . le- e = o• .

ial lin . . e= 20"'

~~:~:::.~ ... . . .

40° .

e= . . • t' .... -"'"'

J I

~:

'! J.l

ll 1,'

,/: J'J:'

.I .. ·ill /tl •• ~· I'J ..

<f .. ... .. ./, / .. .. i .. .. ,.;

~r ....

.,.. . . . ... _ ... .,.. - ·J ,l

---~--- -~' / ..... ........ /

92

4

HORIZONTAL

8 12 16 20 24

DISTANCE

FIGURE FROM

55 SHELL APEX (in.)

-c ' c

t 80

t60

t40

'· t 20 .0

ILl ..J 0.. ;::) 0 0 u

(/)

(/)

ILl -20 0::: 1-(/)

..J <t 0 -40 <t 0:::

-60

-80

0

NO

nor

seve

LE

E:

PLOT SHOWING THEORETICAL "M (a)'' re


mol prjessurt 'P. =- i!!Op.s. . col. 1 ts. ot e = 3: lv20~i 7°40° ~4°,60 . .

' ' ' ' . . . . . . GEND~ . . . .

93

.

. . 00 .

~iallin e . ra ~ - . . . .

e 20v . ----r---· . . 40°

. e . --,--- . . . . . e 160 . ················ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .. . .. .. . . . ... .. '

. ~ .....

.. .. . ' '

.. f . .. . .. I .

' l'.. .. ! .

' . ·· ... .

1\\ I . \

. . ... II I I .

\ \ ·· ... l ~ .

) . \. ~~

. l j .

~ . . . .

I~ \ . ~/I I . ...... / .

I . \ . . I .

\ 1\ ~)~ I . . . .

i\.\.. I • I

....... ,\ . I

\1

4 8 12

HORIZONTAL DISTANCE

FIGURE

. . I . . .

I . . . '..:.<

. . . . . . •·.

FROM

56

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ·. •'j .....

16 20 24

SHELL APEX (in.)

t80

t60

-c

' c t40 I

.a

"' ..J CL t20 ::J 0 0

en en LtJ 0 0:: t-en

..J <( -20 t-z "' 0:: liJ NOT lL

-40 ~ ::J

norr sevet

0 0::

u LE

-60 rae

-eo

0

I /

...

..

E:

~al pr

PLOT SHOWING THEORETICAL "M (cr)" er


v .....

/

/ /

v /~ -I' ' ~I \

I / \... ... -, ,/I

,..., ·,

·' ' \

v / /, • ·······•···· / ~·

...... .. . .. . / /

.. . . .. . v' .. . . .. . . ,... .. · •· ... ,... .. ..

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

········· . . . . . . . . . . . . . . . . . . . . . [essure ' P,.=- 20 p.~ .i. . .

col. 1= ts. 9=3 ,11,20 27,40 54,6( . a

~END:

ial lin - 9= 00

20° 9= Q- l.tt. "0 9= 60°

4

HORIZONTAL

. --- ---· --........ ········

8 12

DISTANCE

FIGURE

FROM

57

16

SHELL 20 APEX

.

94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

(in.)

CHAPTER V

SUMMARY

The intent of this thesis was realized when it was sub

stantiated that the approximate solution given by ORAVAS in 1957

for shallow, spherical, calotte shells enclosing polygonal base

does indeed yield reliable results for the purposes of practical

design.

The problems encountered in this collocative solution

were largely numerical in nature as the computations tended to

be very extensive and involved numbers which had widely varying

orders of magnitude. Consequently the use of McMaster's I.B.M.

7040 computer was essential in the practical execution of the

solution. Techniques had to be devised to overcome computer

limitations and to maintain the greatest possible number of sig-

. nificant figures in all the computations.

The most detailed attempt to verify the reliability of

the theoretical results was made in the comparison of the theor

etical and experimental results obtained for a spherical shell

enclosing.an hexagonal base. The results were as compatible as

could be expected since the experimental shell was naturally

subject to physical limitations both in its construction as well

95

96

as' its boundary conditions. The experimental shell structure I t

exhibited a considerable degree of unperiodicity in its defor-

mation and, therefore, the experimental results can serve merely

as an indication for the general nature of the structural be-

haviour of the shell.

Another attempt to verify the theoretical results was

made for a spherical shell enclosing a rectangular base through

a comparison with a solution given by DIKOVICH for a similar

shell. The normal stress resultants were in satisfactory agree-

rnent in the two solutions even though slightly different boundary

conditions were employed in the two methods. The normal displace-

ment and stress couples in the collocation solution were markedly

larger in magnitude.

A solution for a spherical shell enclo~ing a triangular

base was ·included in the investigation since shells of this type

have been constructed in practice.

The results for the three shells demonstrate that the

periodic polygonal boundary of a 2pherical shell introduces per-~

iodic perturbations emanating from its nonrotationally symmetric

boundary in the rotationally symmetrical solution. ·The extent of

the penetration of these perturbations towards the shell's apex,

where the rotationally symmetric solution associated with the

zero order terms of the truncated series solution dominates, de-

97

pends upon the degree by which the polygonal boundary deviates

from the circular boundary of the rotational spherical shell

enclosing the polygonal shell.

The consistency of the results for the three shells,

which enclosed bases of differing periodic symmetries, indicates

that the solution by the collocation method is consistent for all

thin calotte shells which satisfy the conditions of shallowness.

The discrete satisfaction of boundary conditions tends to accum

ulate larger magnitude errors near the corners of the polygonal

calotte shell and, therefore, it is to be expected that the col

locative solution deviates more from the actual solution in the

neighbourhood of the corners of the shell.

Since the numerical solution is very sensitive to the

degree of accuracy employed in the calculations, ·it is con

sidered to be good practice to solve any shell using two inde

pendent sets of boundary collocation points in order to verify

the consistency of the results. The accuracy employed in the

theoretical computations of this i:nvestigation permitted the

introduction of a maximum of seven boundary collocation points

for one of the rotationally periodic segments of the shell. It

is considered that this represents a sufficient number of bound

ary collocation points in order to provide reliable practical

solutions for shells with as little as triple periodicity.

APPENDlX A

THEORY OF SHALLOW SHELLS

A detailed theoretical solution by the collocation

~thod for a spherical calotte shell over a polygonal base I

was first given by ORAVAS in a paper of 1957 (1) which how-

ever contains a number of misprints. Consequently, it is

necessary to give only a brief outline of the method and the

correct relations used in the solution of the problem.

The stress resultant tensor

FCo) Fca) e e II I I

tF(a)!~ 21 2 I

and the stres~ couple tensor

MCcr) :. MCa) e e1 11 I

t MCcr) e e zt 2 1

t

t

J:"(a) e e 12 I Z

Fta) e e 22 2 2

+ M<cr) e e L~ I 2

t. MCa) e e 2i 2 2

+ Fco-> e e 1n 1 n

t FCal!! an z n

for shallow shells are related by the force and moment equil-

ibrium equations. The expressions for F(o) and F(cr) obtained tn 2n

from the moment equilibrium equations can be simplified using

the first two kinematic compatibility equations. An auxiliary

stress function F is introduced in order to satisfy the first

two force equilibrium equations identically. Then the third

force equilibrium equation becomes a fourth order differ-

eptial equation in u and F. For shallow shells of spherical n

98

VECTOR DIAGRAM SHOWING RELATION OF

BOUNDARY COORDINATES n,s TO SHELL

INTERIOR COORDINATES r, e

FIGURE 58

t M(cr) I ns

99

100

middle surface, this equation becomes

(A-1)

where

)) = POISSON's ratio,

va -..!L .~·- a a + I a I a a. - eLi! d.i; - ar• r a;: + r• iJe•

and R = radius of curvature of the middle surface.

The third kinematic compatibility equation becom~s

the second fourth order differential equation in u and F n

and can be written as

(A-2)

for shallow spherical shells.

These two fundamental fourth order differential

equations (A-1) and (A-2) were given by MUSHTARI and VLASOV.

MARGUERRE also gave similar shallow shell equations in 1939.

The solution of (A-1) and (A-2) under certain re-

' strictions can be reduced to the solution of a set of three

differential equations.

\1& [\:t-i1\ a] V. = 1r (A-3)

[\/a -i ~2

J v'v:. = \J1

V, = 0 (A-4)

. (A-5)

101

where

V =u +iwF=V+V+V ~ 0 I & I

w = ) I

~nd V , V and V represent three linearly independent o I Z

particular solutions.

Solution of (A-3), (A-4) and (A-5) yields the approx-

imate normal displacement u and stress function F for a . n

spherical shell of k-tuple symmetry. Since the shells which

were investigated possessed no inner boundaries, terms con

-n taining ker (x), kei (x) and r were omitted from the solu-

n n .

tion because of their singular nature .at the origin. Finally

the solution becomes:

l\ =~n~:l + A~ber; (ar) +A:bei.(a~)+£~ + t[.{b~n {ar)

+Lla bei (1-r) + c' rjcos (kne) 'l.n kn · ltn (A-6)

(A-7)

102

Boundary Conditions

At the outer boundary TOLKE's Boundary Collocation

·Procedure, introduced by TOLKE in 1934 and employed ·.:i.n·:.this . . .

work, restricts idealized boundary conditions to be satisfied

only at discrete boundary points instead of along the entire

length of the boundary. Five different boundary conditions

were utilized in this investigation in which three distinct

shallow calotte shells were studied.

These boundary conditions were given.by ORAVAS in

1957 and are listed below:

1. Stress resultants normal to the boundary vaniSh:

F( 0') = 0 nn

2. The boundary undergoes no rotation:

6 (au; = o rn . 3. The boundary undergoes no normal displacement:

u = 0 n

4. The boundary of the·shell is fully restrained and con-

sequently undergoes no linear strain:

E = o ss

.5. The tangential stress couple vector vanishes along the

boundary edge:

M( 0') = 0 ns

103

The boundary conditions 1 to 5 can be expressed in

terms of the normal displacement un and stress resultant

function F as :

', .

+C2. ~] = -P" R kw. s 2 (A- 8}

]= 0 '(A- 9}

]=~ Eh

.(A-10}

(A-ll}

J = 0 (A-12}

The coefficients in the equations (A-8} to (A-12} are:

Ill _ "),. • _ 2 _ 2 .. a 't"a -- wr ber ("r)cos e + .2..- ber (~F) sin e

. 0 w 0

104

+ ~ bei (-Ar)sin:z.e cos(kne) 2. II ~ kn

+ ~- Jin.pa ber (?\r) + 'lJ<n ber.' (?\r)} sin z e sin (kne) W kn -wrr kn . .

' <.p_ =-'A ber (~r) cos e 0

4{ = -[ ~ ber"" sin a J s\n(kne)- ~ ber;~ (?tr)cos e]cos~ne)

105

{1\ kn-1 kr\-1 "'tto = - [knr sine] sin(kn e) - [kn r co~ a] ~os(kne)

'·

41. = ber (?\r) 0

411 = bei c~ r) 0

'

tP.a - ber ('Ar) cos(kn e) kn

'P.~ - bei (?\r) cos(kn e) kn

4?s = rkn cos(kn e)

~ bei ("'r)- :U.. be"• (?tr) san e + -t; bei (?\r)-~ bei (~r) cos a [ I a, II Q . a - u . II

1 ~ :&

0 w 0 0 0 •

lll = [~ bei I ('l\F)- .J . .'~\2.bei (~f)- ~bet (?\rjl sin 2 e

'rae liB kn -\. r/ kn ""' kn !J · ..

I 2. . a, II

+r-~ bei ('l\r)+ .:L(J:S.D.) bei ('llF) + 2. bei (,._r)1 cos a a} cos(kne) 1.: ~ kn ., · pr. kn w kn :J

lOG

107

where f, e are the coordinates of collocation points.

Simplified expressions for the stress resultants

and stress couples can be obtained by utilizing stress-

strain, moment-curvature, strain-displacement and.. curva-

ture-displacement relations and by enforcing the condition

wherever it is e~pedient for simplification purposes:

. [Ca) rr- (A-13)

F(a) re

= F(a) er

and

108

M,.(:} =- D [;a~}.' + ~( t~ ~a~~ + + ~~1'.)]

M ( ) D [ 1 a•un. + _I .E..!!.a. + '\> (}au .... ] e~ = r• a a" t" C) .... ~ r a

(A-14)

M M .&il!!.[~ ( I .2Y:A)J !~> =- ~a;; = ' ar r ~ a

Substitution of the expressions for u from (A-6} andF n

from (A-7) in (A-13) and (A-14) yields the sectional resultants :

109

a fi kn-JJ +Cknt_t(kn-a)r . cos(kne)

ata 11 ( )~ 1 +/).- -~ bei (?tr)+'\> 2 bei. ('Ar)- .::t.?\bei c~r)1 ""1&,. kn . ,. kn t- kn ~

110·

2. )2 I a II

+.A- ~ (m bei (?.r)+ ~ bei ('1\r) +'));\ bei (~r)l 1'"\n ,. kn kn kn :.1

I ~-j 1 +C E-kn(kn-•) c•-v> r } cosc.kne>J kn . .

First derivatives of KELVIN functions of kn-th order

with respect to ~r can be expressed by

ber'c~r) - 1221 [ber('Ar) + bei(i\r)l

~n kn-\ kT\-1 ~

,21 [ber (1\r) - bei C/\r)] - rkn bei Ci\r)l

" kn-1 kn-1 L! ~ Second derivatives of KELVIN functions of kn-th order

with respect to ~r can be expressed by

ber"(Ar) =- -1 ~er'(Ar) 1\n ?.r kn

+ r~rber{"Ar) - bei ("Ar) kn kn

• '' { :'1 I b · '( ) pe1 t..r, • = - ~r e1 "Ar Kn kn

t .{¥r)2bei (-~r) -t- berC'Ar) •

kn kn

APPENDIX B

KELVIN FUNCTIONS

one of the differential equations arising in the

theoretical solution of shallow, elastic, spherical shells

subjected to isothermal deformation is the modified BESSEL

equation

where n·; 0,1,2, ••• •

where

The standard solution of this BESSEL equation is

t.B X. [iCz)1 n n J

J is the standardized n-th.order BESSEL function n

of the f?-rst kind,.

· Y is the standardized n-th order BESSEL function n

of the second kind

and A and B are complex constants. n n

See the book by FARRELL and ROSS of 1963.

This solution can also be written in the form

11.1

where

In is the standardized modified n-th order BESSEL

!unction of the first kind,

Kn is the standardized modified n-th order BESSEL

function of the second kind,

the complex constants

.n A . n-1 I - I n

.and '2 • -(nt-2) B - I , 11' n

112

See the McLACHLAN text of 1955.

Substitution for z =IT 11.r yields

where

ber (.\r) and bei (Ar) are KELVIN functions of n-th n n

order of the first kind,

ker (~r) and kei (~r) are KELVIN functions of n-th n n

order of the second kind,

the complex constants

E .-nc =A -i-'B n ::. ·t n n n

. n D 2 ,·-2 B Gn =I n = 1T h • and

The shells under investigation possess no inner

boundary, hence KELVIN functions of the second kind, ker (Ar) n

and kei ("r) which pertain to ·the ·inner boundary effect, , ~ n

are not pertinent. Therefore the solution germane to the

shell problems investigated becomes

113;·,.;

• (B-1)

The most facile method to procure KELVIN functions

•' of the first kind for any order n is to employ the Backward

u ' Recurrence Te~hnique devised by J.C.P. MILLER and outlined

by MICHELS in 1964.

Backward .. R~cur:r;ence Technique

KELVIN functions of the first kind f decrease n

rapidly in order of magnitude with their increasing order n.

Forward recurrence techniques result in a loss of one signi-

ficant figure in f when it is computed from f for each · ntl n

power of ten of the ratio : f n / fn+t Consequently it is .

exceedingly difficult to obtain accurate higher order KELVIN

functions of the first kind from ~'emputed lower order functions

by means of forward recurrence techniques.

MILLER devised a scheme of Backward Recurrence by

which.the number of significant figures in the calculated

functions would actually increase with each successive

application of the recurrence relation. The standardized

114

BESSEL functions J , Y as well as the standardized modified n n

BESSEL functions In' ~ obey the recurrence relation

F(z) -t F(z) n-tl n-1

£D.. F(z) Z n

(B-2) .

where Fcz)represents a series of 'functions of argument z n , •

The collocation solution is concerned only with the

modified BESSEL function I (z). Since both I (z) and F (z) n n n

satisfy equation (B-2), a linear relationship between the two

functions,

FCz) n

o< I (z) 1""1.

where « represents a complex constant, exists.

(B-3)

For any known value of the complex constant «,the

modified BESSEL function I (z) can be calculated for any n

particular order n for which the function F (z) has been comn

puted.

Since Fn(z) is a linear function of In(z), its magni-

tude also decreases with increasing order n. Hence, computing

Fn(z) by means of a backward recurrence relation starting at

some arbitrary order n = m, gives increasing accuracy for each

successive recurrence computation.

MICHELS states that for single precision computation

routine (eight figure accuracy) it is safe to start the back-

119

ward recurrence at some order n = m, such that the ratio of

the value of the BESSEL function at which the recurrence was

begun to the value of the BESSEL function of order h, where

h is the highest functional order reguired in the solution,

should satis~y

Ul! I (z) h

-S' <10

This ratio should be even smaller to obtain double precision

accuracy (17 figure accuracy) in I (z). n

Since F (z) is an arbitrary functional-series go~~ n

erned only by the recurrence relation (B-2), it can be deter-

mined numerically by assigning arbitrary values to two suc

cessive terms, Fn and Fn+l, such that I~ I> :I ~,. 1 1'. Starting

the backward recurrence at some order n = m - 1 and using

Fm(Z) = 0

Fmi,~) = a .. ,

where a is any arbitrary real constant, the recurrence re-

lation (B-2) yields :

FCz) - ~ (m-1) m-2 z

F ('z) a [ 4(rn~).jm-2) _ I] -m-3

FCz) [ s{m-•)(m-z)( m-3) -z(m-3) 2(W>-I)] - a z3 z z . m-""'

• • •

116

'The backward recurrence is repeated until the functional-

series F has been calculated for all orders n, ranging n

from n = 0 to n = m.

The constant o< ;nay be calculated from equation {B-3)."

Thus

F lz) = 0{ 1 ( z) n n

where

Since~ is not a function of nand ber {Ar), bei {Ar) are 0 0

tabulated to a high order of accuracy, it is expedient.to

set n = 0 :

Substituting

where

/( [f;(z)] - real part of F0

{z)

and J [~(z)] - imaginary part of F0 {z) ~

in the relation above yields

'or = [ ?( [f=;,Cz)} be~("Ar) t J [F;;(Z)] be i/Ar) (

ber;(.Ar) + bei!('Ar) )

+ ·i { ~ [fo(z~ ber4 (t.r) - ?-?[f;(z)]be.i 0 (?S")J •

be r!(t. r) t be.i ~("A r)

117

(B-3*)

Consequently the constant ~ can be determined by equating

the real and imaginary parts of the equation (B-3*),

as:

o<r =

tX· ': I

._·...-

~ [F;,( z) J be ro (?.r) t J [F:(z: )] bei,b.r)

[ ber;('Ar) + bei;tAr)]

~[F;,(z) ]ber0 b,r) - i'?[F.Cz)]bei/;.r)

[ber;("r) t bei;('xr) ]

(B-4)

(B-5)

Equating real and imaginary parts of equation ( B-3 *) yields :

o(i ~ [Fiz>J = ol i "'r ber ~"r) - o<.z bei C'Ar) I 1'\

o< r tR IFJz)] = Dl i o(r be r,.{r.r) + o.; bei,/Ar)

Rearranging yields:

b er (1\r) oe; J[F/z)] + c< r ~ [ fnCz) J -n (c<i + a;) (B-6)

bei,t~r) ar: J [~(z)] o( .I ~ [ fo(z:) J (B-7) -

( o( f d;) + Now ber (Ar), bei (Ar) can be calculated for any order n,

n n

for which Fn ( z ) is computed.

It should be noted that both ~ and F are functions n

of the constant ''a", so that "a" is eliminated in equations (B-6)

and (B-7) and has no influence whatsoever on the values of

118'

ber CAr) and bei (/\r). n n

Numerical Calculations of KELVIN Functions

An important purpose of this investigation was to

ascertain whether the numerical solution was consistent for

various numbers of boundary collocation points. KELVIN

functions, ber (x) and bei (x), were calculated accurately n n

from orders 0 to 36 for arguments ranging from 3.0 to 10.0

to give a maximum range of seven collocation points for the

characteristic boundary segment of the shell enclosing an

hexagonal base. Consequently, the shell of quadruple period-

icity enclosing a rectangular base had a maximum range of ten

collocation points and the shell of triple symmetry enclosing

a triangular base had a maximum range of twelve collocation

points.

Double precision techniques were used throughout the

KELVIN function calculations.

LOWEL~ • s Tables of 1959 gave zero order KELVIN functions

to 12·- 14 figure accuracy. The backward recurrence was begun

at the order 51 to insure the accuracy of higher order KELVIN

functions. For a typical boundary point of argument 9.0, the

ratio of berm(;.r) to berh(~r) was

ber (9·o) 49

11'9

It was considered that the accuracy employed in the

backward recurrence computations would permit ber (~r) and n

bei (~r), for n ranging from 0 to 36, to have the same 10 to n

l2 figure accuracy as the initial zero order functions from

which they were calculated.

It was essential to use a different constant "a" for

each functional-argument 1 A. r 1 in order to overcome the float-

ing point underflow (numbers less than 10-38 are set equal to

+38 zero) and overflow (numbers greater than 10 cause the ter-

mination of the computer calculations) limitations of the

-37 I.B.M. 7040 computer. The constant "a" ranged from 10 to

l0-15

(see FIGURE 59) for arguments ranging from 3.0 to 10.0.

Since values of higher order KELVIN functions were

-38 less than 10 for arguments smaller than 3.0 (see TABLES 1

and 2 and FIGURE 60), it was impossible to accurately cal-

culate sectional resultants close to the shell's apex.

The distributions of ber (Ar)l as functions of n

arguments ~r ranging from 0.5 to 10.0 1 are given in FIGURES 61

to 76 for orders n ranging from 0 to 36. It was observed that

ber (Ar) are slowly oscillating functions of ~r which decrease n

rapidly in magnitude for increasing order n. Comparison of

TABLES 1 and 2 indicate that the behaviour of ber (Ar) and n

bei (Ar) is quite similar. See also the tables by YOUNG and n

KIRK of 1964.

BER(x) - n . 120

!\;{·(, . !<JRDER 0 _3RDER 6 ZROER u 0ROER 1fl ~oWER 24 0ROER 30 klROf:R 36 0.50 0.99902346'·00 GO -0.302751 0 1370-0H -0.124413 833 10-15 0.747o612611D-29 o. ~7260249340-38 0.00000000000-38 O.OOOOUOOU000-38 0.60 o.997-n?1139D 00 -0.13017613060- 0 7 - o. 1 1 o ' J 4 ? 1 n 21l- 1 4 (). 2'l66J490610-27 0.45519945230-36 0.00000000000-38 O.OUOOOOOOOOJ-38 0.70 ().9'1624882840 01) -0.44678515250-07 -0.7U~442570U0-14 0.6i'5j'>2 HH'tU-26 0.18404lt3018D-34 0.00000000000-38 O.UCOOUOOOU00-38 o.so 0.99360113770 00 - 0 .13002404050-06 -0. 3:>022 '122!l3D-13 0 • '} u 3 tl':i 'J i' 2 2 u IJ- 2 'j 0.45365486140-33 0.00000000000-38 O.OuOOOOOOOOu-38 0.90 0.98')75135670 ~.;o -0.33360177180-00 -o. ttt343424131J-12 0.9~312220460-24 0.76626847390-32 O.OOOOOOOOOOD-3B 0.00000000000-38 1. 00 O.Ci8438178120 ull -(;. 7749374753 0 -06 -0. 50q598~ J70D-12 o • 1 a J9 6 1 4 2 6 1 c.- 2 3 0.96062426170-31 O.U0000000000-38 O.OOOOOOU0000-3R 1. 10 0.9771379732(' 00 ~u.16610157560-U~ -o. 1':i'J'J212 302tJ-11 0. ':>2 7403'174 7L'-22 0.':14616811190-30 O.OOOOOOOOOOLJ-38 0.00000000000-38 1.20 lJ. 96 7629155 Rl' 110 - 0 .333157171 60-05 -0.45427LJ76'•6 1l -11 o. 300:>'· J992 30-21 0.76362658H7D-29 0.00000000000-38 o.ooooooouooo-3o 1.30 0 .9554Z8746 HO ()() - 0 .63197054900-05 -0.1186!H195271l-10 0.148~&472630-20 0.5<?138106190-28 0.00000000000-38 0.00000000000-3H 1.40 0.94J0150'>670 GO -u.J1431740770-04 -0.2887706019D-10 o.6~5oH06214D-20 0.30872164100-27 0.00000000000-38 0.00000000000-38 1.5() l).92l072!83~f' 00 -0.1984~127120-04 -0.6607247j34D-1 0 0 .2606610352U-l<J o.1616809824n-26 -0.12215660840-37 o.oooouoooooo-3A 1. 60 0.891091!3Ht>O ll -=· -0.332565707~0-04 -0.14Si0256H20-09 0.94762067100-19 0.7608817475D-26 -0.96345122650-37 0.00000000000-36 1. 70 0.86997123700 ()() -o. 5 1.00l)26 '>8'•0-0'• -0.29652d87~~D-00 0.31fl~61':16tJ50-18 0.32596737410-25 -0.67042171090-36 0.0000000000~-38

1.80 0.83o7217942U co -0.8~280482710-04 - 0 . 'j Btl 55H745'JI'-O'J 0. '19 9 1 77 1 3 H 1 0-18 0.12849769340-24 -0.41753564360-35 0.00000000000-38 1. 90 o. 797524Jo70IJ 1)0 -o.13138294R6U-U ~ -0.1125573047~-08 0 . 29460295720-17 0.47031485000-24 -0.23554710270-34 o.ooooooooooo-38 2.00 u. 75173t.1B27LJ GO -0.19795361850-03 -0.20819423170-08 O.H2173734120-17 0.16104979100-23 -0.12159331070-33 o.oooooooooou-38 2.10 0.69868500140 00 -0. 29232156061)-03 -0.37366512940-0B 0.21H0127312D-16 0.51931552320-23 -0.57936530380-33 0.0000000000)-38 2.20 0.6376904571D 00 -o.423R7475460-0J -o.65256o01670-08 0.55272196480-16 0.15H5730798LJ-22 -0.25671059320-32 O.OOOOUOOOOOLJ-38 2.30 0.56804892610 ou -o.6044936245u-01 -(). 11115 '150270-07 0.13444997050-15 . 0.46074212660-22 -0.10646067820-31 0.00000000000-38 2.40 0.489047772lD 00 -0.849047080 50-03 -0.1H507964550-07 0.31490014720-15 0.12792372H00-21 -0.41556212570-31 -0.19040106320-38 2.50 0.39996841711) 00 -0.117~9557251J-02 -0.30176091100-07 0.71233'>82550-15 0.34065791810-21 -0 .15343742010-30 -0.82765271300-38 2.60 0.300092090JO oc -0.1607826o/50-02 -0.48257'>83 268-07 O.l56U558~4'JD-14 0.87293094960-21 -0.53823224950-30 -0.33961296800-37 2.70 0.188706304Ull 00 -0.~1721641340-Ul -0.7~803931H31J-07 0.33190067360-14 0.2151!7091710-20 -0.18006575490-29 -0.13211962080-36 2.80 0.6511210843tJ-01 -0.29021594600-02 -0.11711103120-06 0.68675800520-14 0.516514421!20-20 -0.5765161!6610-29 -0.48919334990-36

. 2.90 -0.7136782583D-01 -0.3837563R90U-02 -n.1781479325n-o6 o.13tl5170589n-13 0.11985269900-19 -0.17719491490-28 -0.17299351950-35 13.00 -0.221380249 60 00 -0.502~6464470-02 -0.26710766BOLJ-06 o.272tJ083463 0-1 3 0.27026571120-19 -0.52426570900-28 !o.58609472840-35 3. 10 - U. '3 H 55 3 1 t, ~ ')0 0 uo -0.652l-23Bo~00- 02 -0.39510776o~D-06 0 . 5254652H260-13 0.59336749930-19 -0.14969140070-27 -0.19077167700-34 3.20 -0.564 3 76'•3050 00 -0.8j9286696 8tJ-02 -0.57707263120-06 0.4912309467LJ-13 o.1270517311lJ-18 -0.41339791!740-27 -0.59809646190-34 3.30 -0.75&4070121[) no -0.1071397165LJ-01 -0.832834H9320-06 o.1833(J0':)6590-12 0.26572217000-18 -0.11065123330-26 -0.18102603960-33 3.40 -O.<J6t!0389953D 00 -O.l35742113tJLJ-01 -0.118849~1420-05 0.33299921830-12 0.543590441'190-18 -0.28758819360-26 -0.53007186650-33 3.50 -0 .119]5'J811\UlJ 00 -0.2603740712U-03 -0.15027690480-05 -0 .1'1712 2597640-12 0.92826606690-18 0.21238964270-25 -0.12599803560-32 3.60 -0.1435305~220 00 0.1236'1H0303iJ-02 -0.2001646320tJ-05 -0.16570190330-11 0.17187622460-17 0.55291125210-25 -0.3259791578)-32 3.70 -O.l6932599H40 00 0.30317994110-02 -0.26179220230-05 -0.2~879845600-11 0.30933841000-17 0.13612852160-24 -0.81188782090-32 3.80 -0.1967423273LJ 00 0.5090064200U-02 -0.33623573960-05 -0.5152411247LJ-11 0.54146157000-17 0.31905875810-24 -0.19406081320-31 3.90 -0 • 2 2 57 59 q 4 ,, 6 c 00 0 • 7 3'• 53 7141 1 0-0 2 -0.42405381260-05 -0.85396906920-11 0.92210274970-17 o. 71489357180-24 -0.45104948100-3 1 4.00 -0.2563416557[) 00 0. 969't959U4 3D-:02 -0.5 25027 4 55 7t>-O') -0.13644112320-10 0.15281118340-16 0.15352352590-23 -O.l007531681D-30 4. 10 -0.28843057320 01 -0.59377320160-01 -0.10~45734260-04 0.14025'J90750-10 0.4824564619['-16 -0.11478640120-23 -0.446550(2210-30 4.2() -0.32l<J479A321l 01 - 0 .71696684860-01 -1).14 54162 6720-04 0.226LJ515179D-10 0.85906369350-16 -0.24811993300~23 -0. 10625479540-29 4.30 -0.35679108630 01 -0.8615201297U-01 - 0.19180156520-04 0.36306469170-10 0.1508!l179150-15 -0.52667557600-23 -0.24770821!490-29 4.40 -0.3928306622 0 01 -0.10303'/1U76 LJ 00 -0.25123960980-04 0.57452957710-10 0.26155438650-15 -0.10987450820-22 -0.56630819t40-29 4.50 -0.42990865520 01 -0.1226835648 0 0 0 -u.:S269165708(j-04 0.89976180680-10 0.44776688680-15 -0.22545529540-22 -0.12707631~90-28 4.6C -0.4678356~370 01 -o.14•;'t4ll83 9U 00 -0.42267420500-04 0.13951658520-09 0.75742741700-15 -0.45535368840-22 -0.28011080690-28 4.70 -0.50638855870 01 -o. 1 7170lLHI7 W 00 -0.54311701000-04 0.21428 441010-09 0.12666251040-14 -0.905~5151070-22 -0.6069~744070-28 0 4.HO -0.5453076175[) 01 -0.2 0 lllt11 69 {, ()0 00 -0. 6'1 S7 3 10 lt'>SLI-04 0.32613199930-09 0.20949711860-14 -0.17760739820-21 -0.12939700100-27 4 • ' ) () -0.58429474420 01 -0 • 2_ ·.LJ I, J') jiJ 7 6 [) 00 -O. BH10 142035f'-04 0.49203417200-09 0.34286476990-14 -0.34341558530-21 -0.27155273450-27 0

'5.un -0.62300824791) 01 - 0 . 275!l1> 1 1'·J:l6 1J 0() -0.11126lR1501l-03 0 .73611 757640-09 0.55547753000-14 -0.65520490740-21 -0.5613615lJ6D-27 5.10 - IJ. 66l 0 6 5'3 3 57 D 01 -u.3206b r32 01 4 D ou - 0 .1 3 97~000490-03 0.10'J2473782D-OI:I 0. B 911818 7000--14·--,..0.12 34130998 0-20 -0.11437904$10-26 0 '>.2 0 -0.6980346.4030 _0 l - 0 .371 4 14 05'> '• 0 00 -0. 11460836160-0J 0.160U6468980-08 0.14164235430-13 -0.22960H40140-20 -0.22983026210-26 5.30 -0.73343634350 01 -0.42&6B36593ll 00 -0. 2 1704266590-0J 0.23511577260-08 0.22309643020-13 --0.42214315020-20 -~0.45567236~tD-Z6 5.40 -0.76673943510 01 -0.49309110630 00 -0.26843921710-03 o.J4Ll716444o-oa 0.348HL95'461J-1J -0.7673063RC8D-20 -0.89l6622586D-Z6 )(

5.50 -o. 79735964510 01 -o. 5652768'•200 00 -O.J303Rl87020~03 0.49163A898JO-OI:I · 0.53934J6 :..49D- 13 -0.13194183700-19 -0.172~0325~60-25 II 5.60 -0.82465759620 01 -0.64590391460 00 -0.40466844190-03 0.70373236150-08 0. tj 28 311.! 69')6[)-1'3 -0.24536496180-14 -0. 3292969~ 60-25 5.70 -0.84793722520 01 -o. 73565320260 00 -0.49332577120-03 0.100082220 30-07 0.12621761530-12 -0.43199677780-19 -0. 62173555720-25 0 5.80 -0.86644452630 01 -0.83521749310 00 -0.59862253280-03 0.1414455183['-07 0.190H-r:327421~-12 -0.753101~2560-19 -0. 1160A3tl6i3D-2<r 0 5.90 -0.87936667530 01 -0.94529428060 00 -0.72307861!560-03 0.19869827590-07 0.236')3)80121)-12 -0.13 00 4003880-18 -0. 2 14~11984~J-24

6. 00 -0.88583159660 01 -o. 10665771460 01 -0.86947010560-03 0.27749603Y10-0'I 0.42709<'.14340-12 -0.22247931570-1 b -0.3~1915l4b8D-2~ I

6.10 -0.8849080413 0 01 -0.11997455690 01 -0.10408273010-02 0. 385352383611-07 0.63221<J<JJ <J10 -l2 -o.37724412810-1e -0.7091657~980-2~ Q 6.20 -0~87560624740 01 -0.13454530280 0 1 -0.12404245J40-02 0.532199673RU-u7 0. 9296 2u'i9 5 70 - 12 -0.63416 2 49500-18 -O.l2707J7~01D-21 6.30 -0.85687925930 0 1 -0.15043132200 01 -0.14717597720-02 . 0.73110434170-07 0. 135BOtJ9 942D - 11 -0.10571646460-1{ -0.?2555322.ff5C-13 6.40 -o. 82762498730 01 -0.16768842 520 01 -0.1738519R 59D-02 0.99917523970-07 0.14715?3 371D-1l -0.174808153h0-17 -0.39669239~20-23 6.50 -0.78668909280 01 -0.18636506320 01 -0.20445292610-02 0.13587122310-06 0.284462 f'2 730-11 -0.2SG791891L0-17 -0.6914976o99u-2~ w 6.60 -0.73286878850 01 -0.20650028900 01 -0.23936781300-02 0.1A386524730-06 0.40794 ~ 6 ?400 -11 -0.46693850361)-17 -0.119501772j0-22 I-6 . 70 -0.66491764630 01 -0.22812146860 01 -0.27890248630-02 0.24763841500-06 0.58 177915020-11 -t1. 'f5'th 38348 OD-1 7 -0.20479257850-22 0 6.80 -0.58155151150 01 -0.25124L72090 01 -0.323666R733D-02 0.33200118290-06 o.824 '~13 o3 ;21J-11 -0.1~10872JOOn-16 -0.34~10863090-22 6 . 90 -0.48145562000 01 -0.27585707410 01 -0.37375859700-02 0.44311750900-06 0.11632.30945~-10 -0.1929422989~-16 -0. 58705099410-22 z 7.00 -0.36329302430 Ol . -0.30194332180 01 -0.42954229901)-02 0.58885419610-06 0 • 16 3 1 5 Ll8 57 8 0- 1 C -0.305360075?0-16 -0.98240 96 71 00-ZZ 7.10 -0.22571442800 01 -0.32945256500 01 -0.49122389760-02 0.77921227180-06 o. 22763.284 '100-10 -0.48010331340-16 -0.1 63 L 760067 Ll-2L 7.20 -0.67369537910 00 -0.35830942820 01 -0.558R98'J1920-02 0.1026H577160-05 0.315Hf30L00-10 -0.75002088740-16 -0.269062998'-JO-Zl 7.30 0.11307996530 01 -0.38840693740 01 -0.63251393170-02 o. 134 77720790-05 0.436'tl800 7:)U-10 - Q.11b4403446D-1') -0 .4405Z345 86D-21 7.40 o. 31694573120 01 -0.41960205330 01 -0.711R199906ll-02 0.17620472560-05 o. ~i9 9 o 229 30 6u- 1 o - 0 .17967895710-15 -0.71627749400-21 7.50 0.54549621840 01 -0.45171085170 01 -0.79631688590-02 0.229485300'10-05 0 . 820469300RJ-I.O -J.2756282679D-15 -0.11568238t10-2J 7.60 0.79993824940 01 -0.4H4503349JO 01 -0.88518684470-02 0.29776107560-05 0.11170251~411-o -, -0.42038720270-1? -0.18560942400-ZJ 7.70 0.10 81396541!0 02 -0.51769797620 01 -0.9772162024D-02 0.3849412B57D-0'> 0 • 15 13 7 ~ 2 0 'tiW - (J 9 -0.6375S5006 3D-1'> -0. 2 ~')90337170-20

7.80 0.13908911710 02 -0.55095569910 01 -0.10707034121)-01 0 .49:>87 330040-05 0.20422;,1R710- C"l -0.9617 2513440-1? -0.46879633430-20 7.90 0.17293127650 02 -0.5838731!0660 01 -0.11633516040-01 . 0.6J654o0737o-os 0. 274 30 15d930-IJ -1 -O.l442934?~HD-14 - 0.73818870150 -20 8 .oo 0.20973955610 02 -0.61597937150 01 -0.12521437480-01 0.81434613350-0~ o.3668232ti160-o9 -0.215368465~0-14 -0.11554'17G900-l9 H.10 0.24956880800 02 -0.64672241400 0 1 -0.13331983200-01 0. 10 j 8 3 3 116 21)- 0 4 0. 4884 5 Tl1 79[)- (' '.,l -u.319S2426G ~D-14 -0.179 8 15 02920-1~

8.20 0.29245214fl00 02 -0.6 75 46f: 79330 01 -0.14016031960-01 0.13195992680-04 0.64768fl62330-09 -0.47259267~)0-14 -0. 2 782388400J-l; 8.30 0.33839755430 02 -0. 701492tl6700 01 -0.14512253360-01 0.16716932000-04 0.855270>;8530-09 -0.694 9591G6SQ-14 -0.42814573 J?0 -1 9 o-40 0.38738422960 02 -0.72396996470 01 -0.14744936880-01 0.21110920~f; 0-04 0.112476S fn30-08 -0.101713 05~90-13 -u.a?S239620LD-1~ 8.50 0.4393587?750 02 -0.74196873251) 01 -0.14621525400-01 0.26 :i77 R83730-04 0. 14 1'i222 11 490-0R -0.14R17H 324 c D-1J -U.9974561B520-L 9 8.60 0.49423084980 02 -0.7~444341510 01 -0.1402 9824~90-01 0.33359662670-04 0.19219397~60-0ti -0.214898099 60-13 -0.15L05 0 1369J-16 8 .70 0.55186932100 02 -0.76022615710 01 -0.12834858030-01 0 • '• 1 7 '• 7 ~ 5 5 1 9 {)- 0 4 0. 24974'<4 7 20D-08 - 0 .310784 0472LJ-1~ -0.2 2 757711280-18 8.80 0.61209725220 02 -0.75801941430 01 -0.10875336850-01 0.52093588600-04 0. 32326036 31,[)-08 - 0 .44 60 757864 ll-1C) -0.34116219290-19 8. 'i 0 0.67468740A50 02 -0.74638858310 01 -0.79597120B~ll-02 0.648173122AD-04 0.416l<Jf9657D-08 -0.63~5 M7 76~ ~D -13 - 0 . 50C>93343171l-16 9.00 0 . 73935729860 02 -0.723754961!60 01 -0.38617774130-02 0.80422320090-04 0.535J341 B07D-Od -0.91 0 4042JLJ 08-1J -u.'!~5~618561~-l~ 9.10 O.U05764111')0 02 -0.68838LJ2j120 01 0.16842100350-02 0 .9 9~0fl73404D-04 0.6il4CJ525719D-08 -0.1Z9266d 99Un -12 -u.11l64L986~~-17 9.20 0. 873499?2670 02 -0.631!40546510 01 0. H988 92 26290-02 0. 1 ?27903026 LJ -03 0.87304 J33Ut)-08 -0.18 2H16R105U-12 - u .16q 19 138640- 17 9.30 0.?4208443360 02 -0.57175608540 01 0.18413R99590-01 0.1':)111359020-03 0. 11085 fl3900 - 07 -0.25754 650590-12 - U. 2 lt04021232 D-17 9.40 0.10 10963':1970 03 -0.43622 771340 01 0.30378057~51)-01 0.18547995660-03 0.14022841560- 0 7 -0.36144328470-12 -0.35039 H9G630-l7 9.50 0.107}5003190 03 -0.379438277HO 01 0.45364800460-01 0.22707064810-03 0 • 1 7 6 ., 0 6 9 1 7 9 D - C 7 -0.50536143570-12 -0.5 0d 45513~6D-i 7

9.60 O. 1146 ·l711420 03 -0.248<33~56390 01 o. 639297511.40,-01 0 . 2 7727534211)-03 0 . 2218249810[) - 0 1 -J.704 000 972 RU-12 -0 .7 3471151250-17 9.70 0.12125606630 03 -0.9169746R220 00 0.86709122210-01 0 . 3377 2450640-()j 0.27739404460-07 -0.977201 8427[)-12 - c .L o~ 7uo~9~40-lu

9.80 0.12753565150 03 0.94865760240 00 0.11442873750 00 0 • 4 1 0 3 2 59 5 u 41) - u 3 0 . '3455'• ! 15d l0-C7 -0.13516516j3 J -11 - 0 .1514551905 0 -16 9.90 0.13343446030 03 0.31390699070 01 0.14791370130 0 00 0 .4 9 730595600-Qj Q.421!74S!:!250D-u7 - J.1B63l3L024 0 -11 -U.21h107583U C-l6

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TABLE

BE I hx) 121 .I

AKG. ORDER 0 ~ROER t> 0RDER 11 l'lRDEI\ lfl 0RDER 24 ~ROb{ 30 e RDER 36 (>. so 0.624932183UD- 0 1 (J . 33907237510-06 -t) •• , ')112'•'• 1 '• 74{)-1 f\ -0.?2f2k82075D-~6 -0.00000000000-38 -0.00000000000-38 O.OOOOuOOU000-3& (). (J 0 0 .8997975041 0-01 0.10124267750-05 -0. ((,f\() <)t,1 ~tl6U-l 7 -0.60~11363040-2~ 0.16387250830-38 -0.000 00000000-38 o.oooouooouoJ -3 H 0.70 0. 1224<t893900 I) l) 0.25528046240-05 -0.66476344 ~60-16 -0.97023112530-24 0.90182428500-37 -0.00000000000-3 8 o.oooouooooo0-38 o.so 0 • 1 5 9 8 8 6 22 9 5 j) 00 0 .56875 ilB602D-05 -0.431072~171 0-15 -0.10733075750-22 o.29034j06440-35 -0.00000000000-38 0.000 000000 00-3 8 0.90 0 .2 0226936351.) 00 0 • 1 1 52 8 B ~ 4 :> 0-0 1t -0.22422310120-14 -0.894253H0070-22 0.6206'1100010-34 -0.00000000000-38 O.OOOUOOOOOOJ-3 E 1.00 0 • 2 4 y 5 6 6 0 '• 0 (11) 00 0.216892794~0-04 -(). 'IM0117 63 1U il -14 -0.59577646290-21 0.96065619340-33 -0. 00 000000000-38 o.ooooo oooo o~-38

1. 10 0-30 17312(,92D on 0.38413926251)-04 -'J. 1721 '10'}JP,7!)-13 -0.33123414400-20 0.11'•491911HU-31 -O.OOu uOO OO OOD-3A O.OOOOUOOOOUD -3H l. LO 0 • 3 5 8 7 0 '• '• l 9 ~I) ()(; l) • 6 4 l z 50 2 6 14t) -ll4 - .) . l2:Jil32'tl0'i tl-12 -U.15Hu0154H6D-19 O. 1W1Y t.9A0780-30 -0.00000000000-3~ O.U000000000D-38 1.30 0 .4 20405 96 5 6 0 Q,j 0 • 1 0 '• 58 l 53 8 l D- 0 3 --U.1 55B 71 6 71 JD -l2 -0.669ti 8 057~9D-l9 O.fJK1<'1765210-30 o.919H1192660-38 0.0000000000)-38 1.40 0 .4 86733933 60 00 t.J.1630 5 117501l-0j -0.10H 89~ J 36~0 -11 -0.254264~4060-18 0.60S1716~l1Jn-L ':I 0.849620Y0061J-37 o .ooooo ooooo~ -3H

1. so 0.557S600o23D 00 0 .24649425510-Ui -0. 2H606 735 R30 -11 -0 .llAO 1 '1903131\-18 0.363H4343610-l8 0.67314490460-36 -o.oooooo ooouo~38 1.60 0.63272567700 00 0.36275786880-03 -0.7060S46360ll-11 -0.2A121~ 9H 600-17 0.19482816880-27 0.46660551510-35 -o.00000000000-3 b 1.70 0.712037292'•0 00 0.521 H8552fU-03 -0.16476'18YS60-10 -O. R3 7J3891030-17 o. 942 30731970-27 0.28760300260-34 -0.0000000000)-38 1. 8 0 0.7952619548[\ 00 0.7J37912369U-03 -O.~b71H 0 1J620-10 -0.2342335?520-16 0.416477855311-26 0.15976129370-33 -0.00000000000-38 1. 90 o • s s 2 1 n 3 4 M , D 00 0 .1 0 135417440-02 -0.7826Sj2021D-10 -0.61':175245000-16 0.16'JU5724350-25 0.80885330060-33 -O.OOOOOOOOOOD-3 U 2.00 0.9f22Q16273D 00 0.13764995750-02 -0.1&0463009 10-0Y -0.155986~~090-15 - 0.64'<5419 7640-25 0.376!:l0875490-32 -0.0000000000)-38 2.10 0.106 53 8H 1610 01 0.1841 067U86D-02 -O.J17651421AD-09 -0.3752937ti33D-15 0.22916630310-24 0.16283687940-31 -0.00000000000-38 2.20 0. 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PLOT of BER (x} 24

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FIGURE 76

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139

~ \ '

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1934

1939

1949

1955

BIBLIOGRAPHY

Tolke, F. "Tiber Spannungszustande in dunnen Rechteckplatten". Ingenieur-Archiv, vol. 5, p. 187.

Marguerre, K. "Zur Theorie der gekrummten Platte grosser Forrnanderung", Proc. 5th International Congress of Applied Mechanics, Cambridge, Mass., 1938.

Vla s ov , V. z . "Obshchaya teoriya obolochek i ee prilozheniya v tekhnike", c;ostekhizdat, Moscow, 1949, p. 295.

McLachlan, N.W. "Bessel Functions For Engineers", Oxford at the. Clarendon Press, Second Edition.

1957 (1) oravas, G. ~.

1959'

"Stress and Strain in Thin Shallow .. Spherical Calotte Shells", International Association for .·--Bridge and Structural Engineers, zurich.

(2) oravas, G. l-IE • "Transverse Be~ding of Thin Shallow Shells of Translation", Osterreichisches Ingenieur-Archiv, vol. 11 - 4, p. 264, 1957.

Lowell, H. "Tables of Bessel-Kelvin Functions ber, bei, ker, kei and their Derivatives for Argument Range 0(0.01) 107.50", Technical Report R- 32, National Aeronautics and Space Administration, Washington, D.C.

140

1960

1963

1964

1964

141

Dikovich, V. V. "Pologie Pryamougol' nye v Plane Obolochki Vrashcheniya", Gos. · Izd. Lit. po Stroitel' stvu, Arkhitekture i Stroit Materialam, Moscow -Leningrad

Farrell, o.J. and Ross, B .. "Solved Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions", The MacMillan Co. , New York •

Michels, T. E. "The Backward Recurrence Method for Computing the Regular Bessel Function", Technical Note D - 2141, National Aeronautics and Space Administration, Washington, D.C.

Young, A. and Kirk, A. "Bessel Functions Part IV Kelvin Functions", R.S.M.T.C. vol. 10, University Press, Cambridge.

Riley, G. E. "Stress and Strain in a Shallow Elastic Calotte Shell", Thesis, McMaster University.

Documents

On Thin Shallow Elastic Shells Over Polygonal Bases...ON THIN SHALLOW ELASTIC SHELLS OVER POLYGONAL BASES by DOUGLAS S. WALKINSHAW, B. ENG. ·A Thesis .Submitted to the Faculty of