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Bending of spherical shells Gurel, Ahmet Okan 1957

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B E N D I N G OP S P H - E R I C ' A L S H E ' L L S by Ahmet Okan GUREL DIpl.Ing., Technical University of Istanbul, Turkey, 19f?lj. A thesis submitted i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of MASTER OP APPLIED SCIENCE in the Department of CIVIL ENGINEERING We accept t h i s thesis as conforming to the standard required from candidates f o r the degree of Master of Applied Science Members of the Department of C i v i l Engineering THE UNIVERSITY OP BRITISH COLUMBIA July 19^7 B E N D I N G O F S P H E R I C A L S H E L L S by Ahmet Okan GUREL This study of the bending of a spherical s h e l l of constant thickness consists of f i v e chapters. In the f i r s t chapter,general d i f f e r e n t i a l equations of the problem are derived and presented with the variables i n dimensionless form. The second chapter deals with the solution of the two second order d i f f e r e n t i a l equations i n terms of Bessel Functioned. The solution i s applicable for s h e l l s with a colatitude of about 25 degrees or l e s s . Expressions are given for the i n t e r n a l stresses and displacements i n terms of these values at the boundary. The t h i r d chapter gives a numerical solution of the problem. Solutions are found f o r three, cases of radius to thickness r a t i o , namely 30, 100 and 500, and for colatitudes from $ to 90 degrees. In the preparation of tables the Alwac III-E Electronic D i g i t a l Computer was used. The flow diagram and programme for the Alwac LLL--E are given. The fourth chapter compares the numerical method with one of Timoshenko's approximate solutions. The wave lengths, deflections, rotations and moments are calculated by two methods and compared. As the conclusion to these comparisons a graph i s given, which shows the approximate error i n the damped harmonic solution. In the l a s t chapter, a numerical example i s solved using tables from the d i g i t a l computer. The r a t i o of radius to thickness i s 500; radius being 125 feet and the colatitude 30 degrees.The edges are considered as being on r o l l e r s . In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representative. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia, Vancouver 8, Canada. Date August 19, 1957 ACKNOWLEDGMENT The author wishes -to express his indebtedness to the members of the Department of C i v i l Engineering at the University of B r i t i s h Columbia. It was a great experience and extensive pleasure to work under h i s supervisor, Professor Dr. R.F. Hooley, who spent many discussion hours on th i s thesis and gave the most valuable suggestions. This post-graduate study was made possible through the f i n a n c i a l support of an assistantship i n the Department of C i v i l Engineering. The author wishes also to make p a r t i c u l a r acknowledgment to the University of B r i t i s h Columbia through Professor J.F. Muir, Head of the Department of C i v i l Engineering. July 1 9 5 7 Vancouver, B r i t i s h Columbia CONTENTS PAGE CHAPTER I Introduction 1 General Equations of a S h e l l of Revolution and Introduction of the Dimensionless Forms . . . 3 CHAPTER II Application of Bessel Functions to the Solution of Spherical Shells 20 CHAPTER III Numerical Solution of the D i f f e r e n t i a l Equations . . . . 35> CHAPTER IV Comparison of the Numerical Method with an Approximate Solution 89 CHAPTER V Numerical Example 101 BIBLIOGRAPHY 106 CHAPTER I I N T R O D U C T I O N Since 1910 the inv e s t i g a t i o n of the bending of shells was the object of many papers written by curious i n v e s t i -gators. But the complication of the solutions of the necessary d i f f e r e n t i a l equations f o r t h i s i nvestigation created a great deal of d i f f i c u l t i e s . After almost hal f a century, today one who wishes to look Into a s h e l l solution s t i l l finds a laborious work, I m p o s s i b i l i t i e s or great number of assumptions behind his p a r t i c u l a r problem. The form of the s h e l l , d i s c o n t i n u i t y of surface and variable thickness cause d i f f e r e n t problems i n the solution of the d i f f e r e n t i a l equations. Even In the case of the spherical s h e l l of constant thickness the d i f f i c u l t y of the problem forced Investigators to make ce r t a i n assumptions. The theory developed to date Is limited to sp e c i a l cases. For instance, the approximate method f o r the solution of a spherical s h e l l , the so-called method of asymptotic 1 2 integration i s valuable f o r very t h i n s h e l l s . Another approximate investigation of the bending of a spherical s h e l l gives f a i r l y accurate values f o r non-flat spherical s h e l l s . The object of t h i s paper was to investigate an exact solution of bending of the spherical s h e l l s by a numerical method. To analyze the problem i t was necessary to e v a l -uate some boundary values by tabulated functions. For t h i s purpose the solution of d i f f e r e n t i a l equations was made by the application of Bessel Functions around the top of the s h e l l . The solution by Bessel Functions gives almost exact values f o r 0 ranging from 0° and 20°-2$°. 3 GENERAL EQUATIONS OP A SHELL OP REVOLUTION AND INTRODUCTION OP THE DIMENSIONLESS FORMS Pi g . I. 1. NOTATION 0 : Angle f o r p o s i t i o n of a meridian. 0 : Colatitude. P : External force per unit area, perpendicular to the surface. Y : External force per unit area, tangent to the meridian. ^0,NQ: Magnitude of normal forces per unit length. M*0,Me: Magnitude of bending moments per unit length. : Magnitude of shearing force per unit length. T : Magnitude of h o r i z o n t a l force per unit length. r 0 : Radius of h o r i z o n t a l c i r c l e at 0=0. r-j_ : Radius of curvature at 0=0. r*2 : Distance between s h e l l and axis of ro t a t i o n on the projection of radius of curvature. r l c : Radius of curvature at the top of the s h e l l . C0,GQ: Magnitude of stresses. A : Horizontal displacement, v : Tangential displacement, w : Radial displacement. 2. EQUATIONS OP EQUILIBRIUM Let us take an element on the s h e l l bordered by two adjacent meridians and two adjacent p a r a l l e l s , I.e., meridians defined by 0 and © + d8; and p a r a l l e l s defined by 0 and 0+ d0 , as shown i n P i g . 1.1. In writing the equations of equilibrium we can obtain three equations which r e l a t e the normal forces N"0,NQ, the shearing force Q^ , and the bending moments M^ , MQ to each other. The following equations represent these r e l a t i o n s , Equilibrium of forces p a r a l l e l to the meridians: d — ( l 0 r o ) - ^Qr± Cos0 - Q 0 r o = - r ^ 7 (1.1.1.) d0 r Equilibrium of forces perpendicular to the element: d N 0 r o + N e r l S l n 0 + — < r o ^ ) = - P r i ^ o (1.2.1.) r d 0 Equilibrium of moments of a l l the forces with respect to the tangent to the p a r a l l e l c i r c l e : d — ( I L r 0 ) - MQr! Cos0 - Q^r 0r 1= 0 (1.3.1.) d0 y r If there i s no external forces, i . e . , 7 = 0 , P = 0 the above formulae can be written as follows: > " ¥ l C o s ^ " V o = 0 (1.1.2.) d ^0 ro + ^ 9 r l S i n 0 + ~ ( y r o ) = 0 (1.2.2.) 6 F o r the non-loading case we can express another set of two equations i n s t e a d of the formulae (1.1.2.) and (1.2.2.). From F i g . 1.2., s i n c e there i s no l o a d , the sum of the v e r t i c a l p r o j e c t i o n s of i n t e r n a l f o r c e s should be equal to zero, (1.1.3.) By s u b s t i t u t i n g t h i s r e l a t i o n i n e q uation (1.1.2.) we can o b t a i n the f o l l o w i n g r e l a t i o n between N E and §0 : \ - -d0 < V2 } F i g . I . 2. (1.2.3.) To s o l v e f o r f i v e unknowns, i . e . , N ^ J I ^ M ^ MQ . Q ^ , the above three r e l a t i o n s are not s u f f i c i e n t . The problem becomes inde t e r m i n a t e and we have to c o n s i d e r the geometry of the d i s p l a c e d s h e l l . By t h i s way, we w i l l Introduce s i x more unknowns, I.e., two displacements w , v , two s t r a i n s , £ 0 , £ q , and two c u r v a t u r e s X^ , Xg. But elev e n d i f f e r e n t r e l a t i o n s among these e l e v e n unknowns can be e s t a b l i s h e d and problem can be s o l v e d . In the next a r t i c l e s these r e l a t i o n s w i l l be shown. 3 . GEOMETRY OP THE SHELL OP REVOLUTION Let us assume that In the o r i g i n a l p o s i t i o n , before deformations, the s h e l l has the form of AB P i g . 1 .3* a f t e r deformation A'B». Therefore AA» and B*B» are the displacements of A and B respectively. Each of these d i s -placements can be resolved into two components, i . e . , w i n the d i r e c t i o n of the normal of the element, v i n the d i r e c t i o n of the tangent to the meridian. These displacements are shown on the Pig. 1 . 3 . The r e l a t i o n s between strains and displace-ments can be established by geometry. The element AB w i l l deform into shape of A'B', ^ therefore the change In the length of AB i s : d0 d0 w d< (a) Pi g . 1 . 3 . dv The portion — dp Is the extension of the element because of v , and w d0 i s the shortening of the element because of w. The r a t i o between the change and the o r i g i n a l length of the element, r^ d 0 , w i l l give us the s t r a i n of the s h e l l i n the meridional d i r e c t i o n . 8 1 dv w €.= . - — (1.4.1.) The s t r a i n In the p a r a l l e l d i r e c t i o n i s the r a t i o between the increase of the circumference of the p a r a l l e l c i r c l e and the o r i g i n a l length of t h i s circumference. This r a t i o Is simply equal to the r a t i o between the Increase of radius and the o r i g i n a l radius. Since t h i s increase i s , v Cos 0 - w S i n 0 (b) and the o r i g i n a l length of radius i s j r Q = r 2 S i n 0 (c) the s t r a i n i n the p a r a l l e l d i r e c t i o n becomes: 1 e 0 = — ( v Cot 0 - w ) (l.£.l.) r~ The changes i n curvature of meridian and p a r a l l e l c i r c l e can be expressed i n terms of the v and w displacements The t o t a l r o t a t i o n of upper side of the element shown i n P i g . I . l . i s , v dw 1 — + (d) r-|_ d0 r^ For the lower side rotation i s , v dw 1 d v dw 1 — + — — + (— + — ) d0 (e) r-j_ d<zf r 1 d0 r^ _ d0 r± 9 The change of curvature of the meridian i s simply the rotation per unit length, or: I d v dw 1 ~ — ( — + — — ) (1.6.1.) v r-j_ d0 d0 r ^ On the other hand, the change of curvature of the p a r a l l e l c i r c l e i s the r a t i o between the projection p a r a l l e l to the meridian of rot a t i o n (d) and the o r i g i n a l length r 0 d 9 . Since the projection p a r a l l e l to the meridian i s , v dw 1 ( — + — — ) Cos0 d9 (f) r l d0 r i and substituting (c) Instead of r 0 , the change of curvature of p a r a l l e l c i r c l e i s obtained as follows: _ v dw 1 Cot0 X = ( — + — — ) — — (1 .7.1.) r x d0 r1 r 2 So f a r , we have seven equations. The other four are r e l a t i o n s between force components, stra i n s and changes of curvature. In the next a r t i c l e these l a s t four equations w i l l be given. 10 4. RELATIONS BETWEEN FORCE COMPONENTS AND GEOMETRY Hooke's Law w i l l give us r e l a t i o n s between normal forces and strains as follows: From these r e l a t i o n s N^ and ITQ can be drawn out as functions of s t r a i n s : Eh _ _ y = 1^2 ( 60 d . 8.1.) __ Eh _ N9 = ^2 ( £ e + (1.9.1.) On the other hand the r e l a t i o n s between moments and changes of curvatures are as follows: _ Eh M , = ( X M + V Xfi ) (1.10.1.) 0 12(l - i > 2 ) ^ 0 MQ = -Eh 3 ( Xg + V X0 ) (1.11.1.) I 2 ( l - y 2 ) If we substitute the values of €0> ^0* -^ 0> which were given previously, i n the above formulae, we can obtain 11 the following r e l a t i o n s between forces, moments and d i s -placements: N0= dv w P ( — ) + — ( v Cot0 - w ) (l-V^) r 1 d0 r-j_ r 2 (1.8.2.) Eh r 1 i ; dv w l?e= =- f — ( v Cot0 - w ) - — ( ) 1 (1.9.2.) W (1-V2) L v 0 ? ! d0 r x J -Eh"* r l d v l d w o v 1 dw -, M> r — — (- + — — ) + ~ C o t 0 ( — + — — ) J (1.10.2.) 12(1-1^^) r-j_ d0 r]_ r i d0 r 2 i " ! rx d0 .-3 -Eh v 1 dw Cot0 v d v 1 dw -, + ( — + — — - ) ] (1.11.2.) 1 2 ( l - i ; ) rx r i d0 r 2 rx d0 r-j^ rx d0 Now we are to solve the eleven unknowns from the above eleven equations, numbered from (1.1.1.) to (1.11.1.). To deal with t h i s problem i s rather laborious work. To reduce this to an easier approach we w i l l introduce two new unknowns and eliminate the rest of the unknowns In the following a r t i c l e . $. DIFFERENTIAL EQUATIONS OF THE PROBLEM By introducing Z and U* as the new unknowns we can reduce the problem to a solution of two d i f f e r e n t i a l equations. Z* i s the angle of rota t i o n of a tangent to a meridian and equal to (d) of the previous a r t i c l e . 12 1 d w Z = — ( v + — ) r, 60 (1) The other unknown Is as following: U = r 2 Q0 (j) Substituting these variables into equations (1.1.3.) (1.2.3.) (1.10.2.) (1.11.2.), and defining the f i r s t and second d e r i -vatives of any function, F as follows, dP P = 60 P = d 2P 60Z (k), (1) we obtain the following r e l a t i o n s : 1 ? 2 1 U Co t0 u ( 1 . 1 4 . ) ( 1 . 2 4 . ) EE" 3 1 2 ( l - i ^ ) Eh 3 MQ= -I2(l-v^) 1 i i> — Z + — Co t0 Z _ Co t0 V Z — — + — Z r 2 r x (1.10.3.) (1.11.3.) The f i r s t equation f o r U and Z can be r e a d i l y obtained from Eqs. (1.8.2.) and (1.9.2.) by forming v - w, r l f , r2 r 2 ( ~zr~ ) + *1 r2 Cot0 -r-L h U _2 r l _ = — E h Z" r2 r l r rT 2 h — 7-7 Got 0-V -V— Cot0ItT r 2 L r 2 h J (1.12.1. The second equation f o r u* and Z* i s obtained from the sub s t i t u t -ion of M0 and Me into formula (1.3.1.) by t h e i r equivalents shown as Eqs. (1.10.3.). and (1.11.3.) r-. r rp rp 3?p . . • r i r 3v>Cot0 . ? i ? - i z + ~ (~) * + ~ Cot0 + ; r - = L H Z - V h + - C o t ^ J Z r2 r l r l r l h r2 h r 2 9 - 2 _ 12(1-1^) r-, e - U 33 ^ (1.13.1.) E h r 2 A solution must now be obtained from the above two simultaneous second order d i f f e r e n t i a l equations f o r U and Z. With U" and Z" known as a function of 0 a l l the force components and d i s -placements are found from the preceding equations. Up to now a l l the variables used were with dimension. To maka the analysis of problem easier we w i l l introduce dimensionless forms to our vari a b l e s . In the next a r t i c l e t h i s aspect w i l l be discussed. 6. DIMENSIONLESS FORMS So f a r we used the bar ( - ) above a quantity to Indicate that i t has a dimension. For the dimensionless quantities we w i l l introduce the same symbol without the bar. The tables which are given i n the next few page? w i l l l i n k the two systems. The f i r s t table i s divided into two parts. The part (a) contains the variables which were defined a r b i t r a r i l y In dimensionless form. In order to get the dimensionless re l a t i o n s which are shown In Table 1.2., It was necessary to 11+ define the rest of the variables as they are shown i n Table I . l . b . The l a s t table which i s numbered as Table 1.3. gives a l l variables In terms of IT and 2 . Before we presented the dimensionless forms, the two simultaneous d i f f e r e n t i a l equations, Eqs. (1.12.1.) and (1.13.1.) were obtained with the dimensions. After having the given tables we are able to transform these equations Into the dimensionless forms. For the general case these equations are r 1 U + — (—)' + — Cotfi- — (-)JU " — [ — Cot2(Z( - V r 2 X"! r1 r± h r 2 r 2 h r-, P Z -V (-) Cot0 U = — (l-V^) ( l . l i | . l . ) h r 2 r i r r P r 2 3 r ? h -, . r n r h Z + — [ ( — ) ' + — Cot 0 + — — J Z - — [i> - 3 V Cot0 — r 2 • . r-j_ h r 2 h r r 2 U + — CotZ0 ] Z = - 12 — ( l . l S . l . ) r 2 r 2 h For the case of the spherical s h e l l of constant thickness these equations reduce to the following: P (1-P 2) U + Cot0 U - ( Cot^0 -V ) U = Z (1.16.1) h 12 Z + Cot0 Z - ( Cot 20 +i>> ) Z = U (1.17.1.) TABLE: I . l . a . Relations between with dimension and dimensionless quantities Dimensionless quantities General case Spherical s h e l l case h h = h ? l c h = h a ^0 ro= ? 0 ? l c ro= *o a ? 1 r l = ? 1 ? l c a = a a = 1 ? 2 r 2 = ? 2 ? l c a = a a = 1 V v = V ? l c V = V a w w = w ? l c w = w a €0 h ^ = h " e h ^0 h z z = Z h 2 Z = Z h 2 TABLE: I . l . b . Relations between with dimension and dimensionless quantities Dimensionless quantities General case Spherical s h e l l case N 0 N, 9 0 ^9 §0 o e • A -,>2 l - l > N 0 = N 0 — Er l c 1-V>2 Ne= N, M0= 9 * 7 l e 12 (1-V 2) M0 ~Z2~— h E r l c 12 (l-i>2) M 9 ^ w -2 h E r l c 1-^2 Q 0 — — E r l c fy" X 0 P l C h 2 x e= *e r l c h u* = u* E r l c -i;2 1-V °0 = °0 h E a e=a 9-A = I E ha-i^ 2) F l c r2 N0= ^ 1-y2 E a l-p2 E a 12(1-V2) M0= M0 _ ?  r hE a^ 12(1-V 2) 9~h7?~ Q0 = Q0—— E a X0 = X0a h 2 X 9 = X e a h' 1-V U = U E a^ <J0 = 00 o«i h E 1-V2 h °Q = G 9 A = A E h ( l - V 2 ) 17 TABLE: I. 2. Dimensionless Relations General case Spherical s h e l l case N0 = - Q0Cot0 N0= - Q^Cot^ N e= 1 d " _ T : ( Q 0 r 2 ) r]_ d0 N Q= d - — (00) d0 ^ T = N0CoS0 - Q0Sin0 T = N0Cos0 - Q0Sin0 d — (M 0r o)=r 1MQCos0+12 r Q d0 h dv £0 = — ( v ) *! <30 i*lQ0 d — (M Sln0)=MQCos0+l2Sin0Q0 d0 dv ^ d0 %- A ( v Cot0 - w) h ( v Cot0 - w ) X 0 = b.2 d v dw 1 (-+ ) r]_ d0 r ^ d0 r± X0= Q d dw h 2 — ( v + — ) d0 d0 b.2 d v dw 1 ( — + ) r l 0^ r l 0^ vi Cot0 V P d dw h — ( v + — ) Cot0 d0 d0 N0= N0= 60 +^e e N e= € e + y 6 0 - ( x 0 + i> x e) M^ = - ( X^ +i? x e) - ( x e + v x^) MQ= - ( Xe +\? X0) C0 = o e = M0 N f l + 9 e- 2 °0 = 0 Q = M0 0- 2 Me N 9 ± — 2 A = (N Q -J> Np() Sin0 A = ( N 0 -VSty) Sin0 Z = b.2 dw — ( v + — ) r ^ d0 Z at dw hd ( v + — ) d0 TJ = r 2 Q 0 U = Q0 18 TABLE: I. 3. Dimensionless forms In terms of U and Z General case Spherical s h e l l case N0 = u Cot0 r2 N0= - U Cot0 1 . u r l • - u T = u T = u r 2 Sin0 Sin0 V3 U r2 TJ 1 . r-, (Z + — V Z Cot0 ) r x r 2 - (Z + V Z Cotpf ) V 1 r - i ( — Z Cot0 +v Z) r l r2 M = - (Z Cot0 +^ Z ) TJ . 1 . r, Cot 0 - — (Z+ -^ZCot0) r 2 2v1 r 2 °0 = + 1 • -U Cot0 (Z Z Cot0) 2 U A 1 r, - - ± - (—Z Cot0+^ Z) r x 2 r x r 2 1 -U ± — ( Z Cot0 +V Z ) 2 A = . 1 U - Sin0( U — ->> — Cot0 ) r l r2 A = - Sin0 (U - VU Cot0 ) 19 As It can be noticed, these two equations Involve two parameters; l.e.,V and h. Poisson's Ratio \> i s constant f o r a given material and changes very l i t t l e from one material to another. In Chapter I I I V w i l l be given an average value of 0.20. On the other hand h which l s the r a t i o between the thickness and the radius of s h e l l can vary within large l i m i t s . For each value of h, the solution of the above y i e l d s d i f f e r e n t functions f o r U and Z. In the next two chapters t h i s problem w i l l be solved by two d i f f e r e n t methods. F i r s t l y by Bessel Functions, and secondly by a numerical method. CHAPTER II APPLICATION OP BESSEL FUNCTIONS TO THE SOLUTION OF SPHERICAL SHELLS Last chapter, a f t e r presenting the dimensionless symbols we were able to simplify two simultaneous d i f f e r -e n t i a l equations f o r a spherical s h e l l of constant thick-ness to the following: U + Cot0 U - ( Coty-\) ) U = Z (2.1.1.) h • 12 Z + Cot0 Z - ( Cot2p + i) ) Z = U (2.2.1.) h In t h i s chapter we w i l l show that f o r small 0 a solution to the above equations can be obtained In terms of Bessel functions. If the angle 0 i s small, say less than 10° or 20°, 1 then i t Is permissible to replace cot0 by — with f a i r 0 accuracy. •20 21 For example, 0 = 10° 1/0= 5.72 cot0 = £.67 D l f f . = 0.88$ 0 = 20° 1/0=2.86 cot0 = 2.7S " = k'00% 0 = 30° 1/0 = 1.87 cot0 = 1.73 " = 8.09$ This approximation to cot 0 w i l l convert our d i f f e r e n t i a l equations to the following: TJ TJ (l-v>2) TJ + + V TJ = Z (2.1.2.) 0 0 2 h Z Z 12 Z + 5 - V Z = TJ (2.2.2.) 0 0d h By introducing the notation L(F) 1 . 1 L(F) = F + — F F (a) 0 0 2 where F Is any function, we can rewrite our equations i n the following s i m p l i f i e d forms: ( l - i > 2 ) L(TJ) +VTJ= Z (2.1.3.) h 12 L(Z) -v> Z « TJ (2.2.3.) h From these two simultaneous d i f f e r e n t i a l equations of the second order we can obtain f o r each unknown, an equation of the fourth order by operating with L on each of the two equations and eliminating one unknown each time. At the end of t h i s operation we obtain LL(TJ) + t^k TJ = 0 (2.3.1.) LL (Z) + Z = 0 (2.I4..1.) 22 where 1 2 ( 1 - / ) (b) AB a f i r s t step, l e t us solve f o r Z from Eq. (2.2+. 1.). This equation can be written i n one of the following forms: L L L(Z) + "H2Z ] - i ^ 2 ^ L ( Z ) + i£,2Z L(Z) - iJj 2 z ] + lfe,2 [ L ( Z ) - i 4 2 Z n = 0 = 0 (2.1+.2.) (2 . i| . .3.) Therefore the. solutions of the following two d i f f e r e n t i a l equations L(Z) - i ^ 2 z = 0 (2.5 .1 .) 1(Z) + H 2 Z = 0 ( 2.5 . 2 . ) w i l l be the solutions of the fourth order d i f f e r e n t i a l equation (2.I4..I.). Let us name the solution of Eq. (2.5 .1 .) Z-j_ and the solution of Eq. ( 2.5 . 2 . ) Z 2 and rewrite the equations by opening the L operator. z l ^ 1 Z-, +* ( i 4 2 + -—) Z x = 0 1 0 0 2 'Z2 p 1 , Zo + — + ( u 2 ) z 2 = 0 0 02 ( 2.5 . 3 . ) (2.54.) The above equations are Bessel's D i f f e r e n t i a l equation of f i r s t order and complex argument. The well known general Bessel D i f f e r e n t i a l equation i s d y 1 dy P nd - + + ( k ^ -) y = 0 dx^ x dx x^ ( 2 . 6 . 1 . ) and i t s general solution i s as below, y = A* J (kx) + B* Y (kx) (2.7.1.) i n I n In t h i s solution J f i(kx) and Y n (kx) represent F i r s t and Second Kind Bessel Functions respectively. A and B^ are ar b i t r a r y constants. In this equation i f n = 1 our solutions are called - Bessel Functions of F i r s t Order - i . e . , y = A* J (kx) + B* Y (kx) (2.7.2.) Modified Bessel Functions of F i r s t and Second Kind are I-^kx) and K^(kx) respectively. The r e l a t i o n s between non-modified and modified functions are as follows: J 1 ( k x i ) = 1 I 1 (kx) (c) 2 Y 1 ( k x i ) » — i K x(kx) - 1^ (kx) (d) These modified Bessel Functions are applicable i n cases where the argument Is imaginary. By knowing th i s information we can obtain the solutions of our d i f f e r e n t i a l equations. In the next two pages the solutions f o r both equations are shown i n a tabulated form to make easy the comparison between them. After following these tables step by step two solutions can be found such as: i i z l = H1! ( ^ i 2 ) + B i K l ( ^ 1 ) (2.8.1.) Z 2 = A 2 I 1 ( 4J2T i~*) + B 2K 1( 0 i " * ) (2.9.1.) 21+ These solutions involve 1^ and Modified Bessel Functions. For the Imaginary Argument, these functions can be expressed in terms of Thomson Functions which are tabulated i n most of the books on Bessel Functions. The r e l a t i o n s tretween Thomson and Bessel Functions are I l n ( k x i 2 ) = ber n(kx) + 1 bei n(kx) (e) l " n K ^ k x i ^ ker n(kx) + i kei n(kx) (f) where ber, be i , ker, kei are called the Thomson Functions. Recurrence formulae e x i s t giving the n + 1 order Bessel functions In terms of the preceding orders. Since we need the f i r s t order, and the zero order Thomson functions are extensively tabulated f o r the small kx i n t e r v a l s , we w i l l obtain the f i r s t order d i r e c t l y from these tabulated zero orders by the following recurrence formulae: i d l 0 ( k x i * ) 1 d l n ( k x i ) i It ( k x i B ) = — j — = — — 2 = I ( k x i 3 ) (g) d(kxi*) i a d(kx) 1 i dK D(kxi^) 1 dK 0(kxi2) i K* (k x i 8 ) = - r- = — — ^ = - K ^ k x i 2 ) (h) 0 d(kxi2) i s d(kx) For the case of n = 0 which i s zero order, (e) and (f) are x I 0 ( k x i 3 ) = ber(kx) + i bei(kx) (i) K Q(kxi») * ker(kx) + i kei(kx) (j) It i s customary to omit the zero subscript i n zero order function. I f we l i k e to express 1-^  and i n terms of zero order Thomson functions, from (g) and (e) I 1 ( k x i a ) = i (ber'kx + i bei'kx) Equations f o r 2]_ Equations f o r Z 2 . . 1 . o p 1 , 2 1 + 0 Z 1 + ' 1 4 " 0 2 > Z l = ° 2 since i = -1 Therefore solutions are: Z l = A l J l ( ^ 1 ' + B l Y l ( ^ ± 2 ] > «• i % 1 Z 2 = A2 J l ( 4 ^ ± Z ) + B2 Y l ( ^ In terms of modified Bessel Functions: 3 J x (40 i 2 ) = J x ( 4 0 i 2 i ) = i I ^ U ^ i * ) ^ ( 4 0 I 3) = JX(40 i i ) = i I x (40 i 3 ) 3 Y x (40 i 2 ) = Y 1 ( 4 0 I s 1 ) Y 1 (40 I s ) = Y i ( c = 0 1 i ) 2 i i = — i K x (40 i 3 ) - 1-^40 i 3 ) 2 _ i - i = - 1 ^(£0 i 2 ) - I i 2 ) Equations f o r Z-j_ (Cont'd.) Equations f o r 2 2 (Cont'd.) therefore i f we substitute these values Z 1 = A * i 1^40 i S ) + l S ) _ i # 2 "a Z 2= A * l 1^(40 i 2 ) + B 2 - i ^ ( 4 0 ^ ) - B* I 1 (40 1*) l - B I (40 1 ) 2 1 Z = (A*l - B*)I (40 I s ) + — B * l K (40 i S ) 1 1 1 1 J T 1 1 * * , " i 2 * , v , - l x Z = (A i - B ) 1 ( 4 0 1 ) + — B 1 K (40 1 ) 2 2 2 1 J l Z = A I U0 i*) + B K (40 I*) 1 1 1 1 1 _! - i Z = A I (40 l 2 ) + B K (40 I 2) 2 2 1 2 1 where % ft A]_ = ( A x i - B 1 ) A = ( A i - B ) 2 2 2 2 * B = — B i 1 J T 1 2 * B = — B i 2 J T 2 ro d ber(kx) _ i where ber'kx = . In the above formula 1 s can be d(kx) taken Into the bracket. F i n a l l y the f i r s t order Bessel functions w i l l be expressed i n terms of zero order Thomson functions as follow: 2, ~h i I 1 ( k x i 2 ) = i ber'kx + i 3 b e i ' k x (k) On the other hand the expression f o r can be obtained by the very same way, from (f) and (h) -K^kxlh = i 2ker»kx + i ^ kei'kx (1) Therefore the f i r s t solution Z 1 can be written i n following form by substituting (k) and (1) into Eq. (2.8.1.) Z 1=A 1(i 2ber « 4 0 +i sbel ' 4 0 ) " B 1 ( i ker '40 +i 3kei « 4 0 ) 2.8.2.) For the second solution Z 2 we can follow the same system, -n - i i I n ( k x i ) = ber nkx - i b e i n k x (m) l n K n ( k x i 3 ) = ker nkx - i ke i n k x (n) by replacing n = 0, I ( k x i 3 ) = ber kx - I bei kx (o) o -2. K 0 ( k x i ) = ker kx - i k e i kx (p) d I (kxi ) A d I n ( k x i 2 ) = I 2" ( ber'kx - bei'kx ) O _ * a • °  _ i ~ 1 d(kxi" a) d(kx) I 1 ( k x i 2 ) = I s" ber'kx + i 2 b e i ' k x (q) d K 0 ( k x i ) _ i d K 0 ( k x i ) = i 2 ( k e r kx - i k e i kx) d ( k x i s ) d(kx) -K-jJkxi 8) = i " k e r kx + I k e i kx (r) 28 Therefore, by substituting (q) and (r) into Eq. (2.9.1.) the second solution Z 2 can be obtained. _ i _ i Z 2 =A 2(i«ber»40 + i % e i ' 4 0 ) - B 2(I2ker»40 + i \ e i ' 4 0 ) (2.9.2.). These two solutions together represent the complete system of Independent solutions of Eq. (2.i|. 1.) . By adding up Z-^ and Z 2 we can write the solution as i n following form: Z = C berH0 + C 2 bei«40 + ker»40 + kei'40 (2.10.1.) where = 1 + A 2 i s C 2 = A1 i * + A 2 i " * C 3 = - (B 1 i ~ * + B 2 ih i -I = - (B x I s + B 2 i ) a r b i t r a r y constants. The solution f o r IT i s easy when the Z^ and Z 2 solutions are known. For t h i s we w i l l use Eqs. (2.2.2.) with (2 .5-3.) and ( 2 . 5 . 1 | . ) . For the f i r s t solution, by rearranging Eq. (2 .5 .3.) we obtain Zn 1 p z + —± - — zi = i 4 ^ z. 1 0 0 2 1 S 1 By use of equation (2.2.2.) t h i s becomes; 2 1 2 '(14 - v> ) Z-, = U h 1 from which h 2 U , = — ( V - 1 4 ) Z-, (2.11.1.) 1 12 29 For the second solution, the same replacement can be done between Eqs. (2.2.2.) and (2.5.1^..). By t h i s way the r e l a t i o n between u*2 and Z 2 can be found : h 2 Up = — (V + I T ) Zp (2.12.1.) 12 . Substituting Z-|_ and Z 2 with t h e i r equivalents i n terms of Bessel Functions i n the above r e l a t i o n s we can obtain the f i r s t and second solutions f o r U: 17, = — (v>-i 4 * ) A. (I ber»40 + I sbei«40 ) - B , ( i ker'40 1 3 L 1 N -1+ i 2 k e i « 4 0 ) ] (2.11.2.) h 9 i _ i i n = —(v + i 4 ) A P(i 2ber«40 + I 2bei'40 )- B 2(i 2ker'£,0 12 -g-+ i kei'40 ) ] (2.12.2.) By adding two solutions together and combining the c o e f f i c i e n t s the general solution f o r U can be shown as follow: U ^— [ ( C ^ - C2-42) ber«40 + (Cg> + C 1 e 2 ) bei'£,0+ [Cj>- C^f)ker»£0 + ( C ^ + C 3 4 2 ) kei ' ^ , 0 ] (2.13.1.) The Thomson Functions b e r 1 , b e i ' , k e r 1 , k e i * are shown on F i g . II.1.a and F i g . I l . l . b . U and Z Involve zero order Thomson Functions. To evaluate U and Z we have to know the f i r s t derivatives of b e r 1 , b e i 1 , ker', and k e i ' . The following r e l a t i o n s are 3 0 1.0 o 1.0 (a) F i g . II.1. given f o r t h i s reason: dx ( x ber'kx ) = - kx bei kx — ( x bei'kx ) dx — ( x ker*kx ) dx kx ber kx = - kx k e i kx (s) — ( x kei'kx ) dx kx ker kx Prom the above r e l a t i o n s we can derive (t) d(ber'kx) ber'kx + x = - kxbeikx dx d(bei'kx) bei'kx + x = kx berkx dx d(ker'kx) ker'kx + x = - kx k e i kx dx d(kei'kx) kei 'kx + x = kx ker kx dx therefore, In our case substituting k = 4 , x « 0 d(ber»40 ) 1 = - t, b e i 40 - — ber'40 d0 - 0 1 d(bei»40 ) = 4ber40 bei'40 — : — 0 d 0 1 (u) d(ker'40 ) = - 4kei40 - — ker'40 0 d0 d(kel«40 ) = 4. ker 40 kei«40 0 d0 The functions ber bei ker kei are shown on P i g . II.2.t and Pig. II.2.b. Thomson functions are tabulated f o r the values of £0 between 0 and 10. For 40 bigger than 10 there are "Asymptotic Expansions" given i n reference (1+) . Since we know Z , Z , U, U i t i s easy to express M ^ , M Q , N0, N Q , T , Q0 i n terms of Thomson Functions by using the r e l a t i o n s given i n Ch. I. i . e . , 32 (a) (b) P i g . II.2. N 0 0 Ne -T = Q0 = - U 0 - ( Z + V Z — ) 0 1 - ( Z — +• ^ Z ) (2.H+.1.) (2.15.1.) (2.16.1.) (2.17.1.) (2.18.1.) (2.19.1.) In the above re l a t i o n s C o t 0 = — and S l n 0 = 0 were used. Up to th i s point the solutions were given i n a general form. The c o e f f i c i e n t s C 2 are to be evaluated f o r the d i f f e r e n t problems. It i s obvious that to obtain these we have to have four boundary conditions. Two of these conditions belong to the top of the s h e l l and the other two are given around the periphery. For instance, l e t us take the case of a s h e l l with a hole at the crown. The boundary periphery, would give us values of the four c o e f f i c i e n t s . As an other example a concentrated load placed at the crown would y i e l d four d i f f e r e n t c o e f f i c i e n t s . In our case since there Is neither a hole nor a concen-trated load at the top the evaluation of the c o e f f i c i e n t s w i l l be considerably easier than the above two examples. By the symmetry, there can be no r o t a t i o n at the top. There-fore Z = 0 at 0=0 w i l l be the f i r s t boundary condition. On the other hand at 0 = 0 the shearing force i s equal to zero. Since TJ = TJ i s zero also. From these conditions we can conclude that the two c o e f f i c i e n t s of ker 1 must be zero since ker* approaches i n f i n i t y at the o r i g i n . therefore both and c o e f f i c i e n t s should be zero. As a r e s u l t we can see that the solution f o r TJ and Z does not contain either ker» or kei» at a l l . F i n a l l y our functions are as shown below: conditions, the values of and T at the top and at the From , TJ = 0 From Z = 0 (2.10.2.) J(2.13.2.) 31+ The f i r s t derivatives of the above functions are 1 1 Z = - C 1 ( 4 bei40 + -ber'£,0 ) + C 2 U b e r 40 - ^ bei«40 ) ( 2 . 2 0 . 1 . ) h _ P 1 U = L~ ( C1V~C2^ > U bei40 + 0ber'£0 ) "In 2 +' (C 2>> + C 1 4 2 ) ( 4 ber40 --^bei'40 )] ( 2 . 2 1 . 1 . ) The c o e f f i c i e n t s C-^  and C 2 are found by considering the boundary conditions at the support at 0 = o. . I f and T^ are prescribed at 0 = « then the following equations w i l l be necessary: M^ = C 1 ( 4 b e i 40 + - y ber'40 ) + C £ ( — bei»40 - 4ber40 ) ( 2 . 1 8 . 2 . ) h T e 12 ( 2 . 1 6 . 2 . ) C 1(i>ber«40 + 4 2bei'£0 ) + C 2(Vbei«40 - 4,2ber»4,0 f Prom these C, and C0 become, 1 p 12ocT« l-V M (i> bei ' 4 c x - £,n>er'4<* ) + ( ——bei» £ o < - £,ber4,o< ) C 1 = h * A 2 l2<xT f t . 1-D M a ( \> ber '4cx + £, bei»fc,c* ) + ~ ( — ber» + 4bei$,o< ) c2= ti « . A where 1-1) 2 A a= ( ber'4« + 4bei4<x ) ( \) b e i 1 4<* - 4 ber« 4<* ) -1- V> p ( b e i ' 4 « -4, ber4o< ) ( ber ' 4 ^ + c4HDei'4cx ) CHAPTER III NUMERICAL SOLUTION OP THE DIFFERENTIAL EQUATIONS Last chapter the solution of two simultaneous d i f f e r e n -t i a l equations was made i n terms of the Bessel Functions. As we have seen already, t h i s solution i s applicable f o r small values of $ . To extend the solution from the Bessel Function zone to 90°, we w i l l use a numerical method. Let us rewrite our equations. TJ + TJ cot0 -(cot 20-V 2 ) U 2 2 Z + Z cot 0 ~(cot0 ) Z The solution of these equations shows an o s c i l l a t i n g and exponential behavior. This exponential b u i l d up Increases very f a s t and exceeds the working l i m i t s of d i g i t a l computers. Since we w i l l be using a d i g i t a l computer f o r our computations, It i s more convenient to define two new functions u and z as 35 1-^ 12 U (3.1.1.) (3.2.1.) 36 follows: -A 0 U = XT e (a) z = Z e where k I 3(1- i>2) h 2 (b) As It w i l l be shown i n Ch. IV, t h i s A appears i n the approx-imate solution. These u and z functions have a small exponential b u i l d up and an o s c i l l a t o r y behavior. If 7v were some more complicated function of h then u and z would have no exponential build up. It was not considered necessary to use t h i s though. Two d i f f e r e n t i a l equations f o r u and z are obtained by substituting these expressions into the o r i g i n a l equations f o r tJ and Z. . 2 1 _ ^ u + (2A+ cot 0 ) u + (A^+A cot0 - cot 0 +v>) u = z h (3.1.2.) • • * 2 2 ^"2 z + (2 A + cot 0 ) z + + A cot 0 - cot 0 -v>) z = u h (3.2.2.) For further rearrangement we w i l l present a new variable x instead of 0 . The r e l a t i o n between them i s A0 = x (c) If we d i f f e r e n t i a t e t h i s once, then, A d 0 = dx (d) AB a dot (P) represents derivative with respect to 0 . Then we w i l l l e t a prime (P») represent derivative with respect to x. If we replace 0by x i n Eqs. (3.1.2.) and (3.2.2.) and divide by A 2 we obtain: u" + (2 + C0t-^r- C O t y - COt -y- V A ) u« + ( 1 + — ^ ~ + - ) u = A A A £ 1- iT (3.14.) z" + (2 + cot COty COt 2^- V> z» + ( 1 + * ^ - - ) z = 4 A A A 2 >2 A<= l-v> 2 (3.2.1|.) To simp l i f y the above equations i n writing, l e t us define a new shorthand such that: u" + a u' + ( b + - 3 ) u = Cz A V z" + a z' + ( b — ) z = D u A where a = 2 + cot' 0 A b = 1 + cot 0 cot 20 A A' f l - V C a (3.1 .5 . ) (3.2 .5 . ) (e) D = 1+ L-V2 In t h i s chapter up to now we have been reforming our two simultaneous d i f f e r e n t i a l equations of second order. F i n a l l y we obtained the forms shown i n Eqs. (3.1 .5 . ) and (3.2 . 5 . ) 38 To solve these equations we w i l l replace the f i r s t and second derivatives by approximate expressions derived from f i n i t e difference equations. Rather than deal with the continuous function u we w i l l consider only the values u n _ i » u n > u n + l e * ce^era. evaluated at a f i n i t e number of points n-1, n, n+1 ... on the curve u = f ( x ) . With reference to F i g . I I I . l . i t i s seen that at point n the func-t i o n and i t s f i r s t derivative are given by u^ and u^ while at point n+1 by un+-j_ and Each point i s dx apart. To replace the f i r s t d e r i -vative by an approximate expres-sion we w i l l argue that the average of the slopes at points n and n+1 i s du / dx, or: F i g . I I I . l . u n + l + < un+l " u n dx (f) S i m i l a r l y we can replace the second derivative by arguing that the average of the second derivatives at point n, and n+1 i s the change i n the f i r s t derivatives / dx, or: u n+1 n + S i t i u - u n+1 n dx (g) 39 Since t h i s i s also true f o r z we have: < Z ' „ + l + 2 i > / 2 = ( z n + l " V / d * (h> < z " n + l + z n > / 2 " ( zi +l " Zn> / Two more equations are known i f we write our d i f f e r e n t i a l equations f o r point n+1. u" + a ' _ u » + (b , + 4r ) u ^ = C z n (j) n+1 n+1 n+1 n+1 A 2 n+1 n+1 z v> + a , z» + ( b ^ , - ) z = - D u (k) n+1 n+1 n+1 n+1 A2 n+1 n+1 If the function and i t s derivatives are known at point n, then these s i x equations (f to k) w i l l enable us to solve f o r s i x unknowns at point n+1. These six unknowns are u . <i U'.T u" z„, n z* and z" , . With the function known n+1 n+1 n+1 n+1 n+1 n+1 at point n+1 i t may be again evaluated at point n+2. These six equations are shown solved f o r the values at point n+1 below: C e n + 1 " ^ ( d x / 2 ) 2 ] +P k. "n+1 x u" g n+l LQ n + l - j^ d*/ 2) 2J ' - -n+1 (3.3.!.) n + 1 [ e 2 n + l " ^ d x / 2 ) ^ ] + H F z" C i = C- i ( d x / 2 ) + K + ( d x / 2 ) < > C l = 2n+l ( d x / 2 ) + ( z n + <d*/2) z» ) un+l = *n + ( d x / 2 ) C l + ^ d x/2) Zn+1 = Zn + ( d x / 2 ) z ^ + 1 + (dx/2) 2 ; (3.5.1.) (3.6.1.) (3.7.1.) (3.8.1.) The fact o r s and c o e f f i c i e n t s used i n t h i s solution are shown below: 'n+1 = 1 + a n + 1 (dx/2) + b _ , (dx/2) n+1 C (dx/2)' H = D (dx/2)' K = % + (dx/2) u£ L M N n - i z<> + (dx/2) z» u + (dx) u» + (dx/2) 2 u" n n n z n + (dx) Zj!, + (dx/2) 2 z^ C N - a K - (b + —p ) M n+1 n+1 A k n-1 -D M - a L - (b n+1 n+1 N (1) In the above solution the main problem arises in choosing the right i n t e r v a l dx. If we take a large Interval the error i n accuracy i s great. On the other hand very small i n t e r v a l s kl cause quite appreciable round off errors which make the results u nreliable. Of course, the range of the optimum i n t e r v a l s varies with d i f f e r e n t h r a t i o s . In the numerical analysis, f o r each h value this i n t e r v a l was investigated and then by using these optimum in t e r v a l s the r e s t of the problem was solved with the above equations. Once u z and t h e i r f i r s t and second derivatives are found, the normal forces, bending moments and deformations can be evaluated. In order to get these values the r e l a t i o n s should be known. The Table I I I . l . i s given f o r t h i s reason. It can be noticed that a f t e r f i n d i n g the quantities shown by the small l e t t e r i n g symbols we can e a s i l y obtain the quantities shown by the c a p i t a l l e t t e r s which are the products of the previous values times A e r , the magnification f a c t o r . It was shown that to f i n d the next point values we have to know the values f o r the previous point. At the very f i r s t step we need to know the values f o r u,z,u',z',u",z". These star t i n g values taken at 0 = 0 0 are calculated by Bessel Functions. To obtain a f a i r l y good r e s u l t these s t a r t i n g values are supposed to be as close as possible to the exact values. For the small 0 values the Bessel Solutions give us an almost exact answers therefore i f we f i n d the values f o r a small 0 , say f>° - 10° by Bessel functions we can use the numerical solutions f o r the rest of the problem. In Ch. I I , U,Z,tI,Z were found f o r small 0 . Since the r e l a t i o n (a) i s known between TJ's and u's, substituting these into Eqs. (2.10.2.) (2.13.2.) (2.20.1.) (2.21.1.) we obtain the 42 Table I I I . l . The r e l a t i o n s between the functions and non-exponential values Functions Non-exponential values N0= - U Cot 0 N0= - U 1 T = - U Sin 0 iy= -(Z + V Z Cotjtf ) M Q = - ( \> Z + Z Cot i0) A r = Sin-0 ( v> U Cot0- U ) Z = Z A0 y = A e y A0 T = Ae t A0 M^= A e A0 MQ= A e nig A = A e F Z = A e z u ^ n^= - — Cot 0 n Q = - (u + u» ) u u 1 A Sin 0 Cot 0 i t i v = - (z»+z) +Vz Cot 0_ mQ = - [ (z' +z) v> + z — - — r Cot 0 -, & = Sin 0[u(—^-v>-l) - u' * z z = — -A following: C (3.9.1.) 1= -e u = '1_ [i A0 h i ; [~be±£,0+ — b e r « ^ 0 ] + - ^ [ - b e r ^ — bei'^jZfj- z A A0 e >° A A0 (3.10.1.) •12e^ "12e 12e (3.11.1.) u»= -h ^ e 2 h •12e 12e l2eJ 4 1 — bei40 + — ber« 40 A A0 — ber'40 - — bei'40 ] - u ( 3 . 12 .1 . ) •A A0 On the other hand the Eqs. ( 3 . 1 . 5 . ) and ( 3 . 3 - 5 . ) can be rewritten by rearranging ao that the second derivatives should be separated at the l e f t of the equations. z " = - D u - a z » - ( b p) z A C z - a u« - ( b +^2) u u" = ( 3 . 1 . 6 . ) ( 3 . 2 . 6 . ) In the above equations the. c o e f f i c i e n t s C.^  and should be determined. As we Indicated In the Ch.II, U and Z equal zero at the top of a spherical s h e l l . But U' and Z' can have any values a t - a l l . However, each of these p o s s i b i l i t i e s can be expressed as a l i n e a r combination of the following two cases: Case I : at 0 = 0 u ! = K, a constant z' = 0* Case II : at 0 = 0 u' = 0 z' = K, a constant. namely, A. (Case I) + B. (Case II) = Actual case Therefore i f we obtain the solutions f o r the Case I and Case I I , i t i s a matter of f i n d i n g the constants A and B f o r any p a r t i c u l a r case. Case I. Since at 0 = 0 the Thomson Functions are zero except that ber (0) = 1. Therefore, z» = 0 w i l l give us C 2 = 0 12 K u» = K A C l = h 4 2 4 If we choose K as such that = 12 , the Eqs. from (3.9.1.) to (3.12.1.) w i l l be as follows: z = 12 b e r » 4 0 (3.9.2.) z» = -12 r 4 i — be i 40+ ber' 40 A A0 - z u = hV ber'40+ 4 2h b e i ' 40 u«= -hV (3.10.2.) (3.11.2.) t 1 — b e i 4 0 + — b e r » 4 0 A A0 - u -4 1 — ber 40 bei' 4 0 -A A0 (3.12.2.) The equations (3.1.6.) and (3.2.6.) w i l l remain as they are. Case I I . z' = K w i l l give us C 2 = K A u» = 0 If we choose K such that C^ = 12 the above equations w i l l appear as shown below. 12 V * = - 42 12 J ,4 ber' 40+ 12 bei« 40 (3.9.3.) z' = 4' be ! 4 0 + — ber' 40 -A *0 + 12 re i — ber 4 0 bei' 4 0 - z A A0 (3.10.3.) u = --^ h L 4< fh L e + e 2h + a2h 4 ber' 40 b e i 40+ A0 ber1 4 0 - u 45 ( 3 . 1 1 . 3 . ) ( 3 . 12 .3 . ) In these formulae 4 and A depend upon the parameter h, assuming i? , Poisson's Ratio i s constant. In the Table I I I . 2 . , 4 and A are tabulated f o r d i f f e r e n t h values, by taking V = 0 .2 and the st a r t i n g angle as 0O = 6 ° . The values of the Thomson Functions at t h i s angle are obtained from Ref.(4) and shown on Table III.3. . The star t i n g values f o r the Case I. and Case I I . are calculated and shown on the Table III.l).. f o r the same values of the parameter h. Tables I I I . 2 . and 4« were calculated by Alwac III-E E l e c t r o n i c D i g i t a l Computer. TABLE I I I . 2 . 0.2 0 Q = 6C 1/h A 4 A0 ^ 0 30 7.135243 10.090747 .747200 1.056699 40 8.239069 11.651796 .862792 1.220172 5o 9.211559 13.027106 .964631 1.364194 100 I3.O27HI 18.423115 1.364194 1.929262 i 5o 15.954888 22.563617 1.670790 2.362854 200 18.423117 26.054221 1.929262 2.728389 250 20.597671 29.129504 2.156981 3.050431 300 22.563618 31.909775 2.362854 3.341580 400 26.054223 36.846235 2.728389 3.858524 5oo 29.129506 41.195343 3.050431 4.313961 46 TABLE I I I . 3 . V = 0 .2 0 O = 6° i A **0 ber40 o bei<# 0 ber»e0o b e i ' ^ 0 30 1.0567 .9805 .2786 - .0737 .5249 40 1.2202 .9654 .3708 - .1133 .6031 50 1.3642 .9460 .4625 - .1582 .6698 100 1.9293 .7848 .9082 - .4434 .8953 i5o 2.3629 .5195 1.3206 - .8022 .9912 200 2.7284 . 1549 1.6835 -1 .2090 .9761 250 3.0504 - .3024 1.9813 -1.6420 .8528 300 3.3416 - .8437 2.1990 -2 .0830 .6208 400 3.8585 -2 .1353 2.3389 -2 .9160 - .1709 5oo 4.3140 -3.6176 2.0045 -3 .5630 -1 .3890 TABLE III. 1+. V = 0 .2 0 = 6 ° Case I u u» u" z z» z" Case II u u' u" z z' z" 30 1.781075 .539149 - 3.293490 - .884040 - 2.660035 - 3.239540 .250045 .752373 .916281 6.300536 1.916634 -11.631543 40 2.046420 .213008 - 3.127757 - 1.359600 - 3.356808 - 2.767788 .384553 .949449 .782851 7.239202 .762987 -II .050151 5o 2.272741 - .090621 - 2.999983 - 1.898400 - 3.370957 - 4.017557 .536949 .9534^1 1.136339 8.039838 - .312449 -10.597082 100 3.037860 - 1.500082 - 2.553856 - 5.320800 - 6.191232 1.785724 1.504949 1.751144 - .505073 10.746736 - 5.296298 - 9.031352 i5o 3.363172 - 2.884882 - 2.130806 - 9.626400 - 7.023405 6.389965 2.722755 1.986517 - 1.807346 11.898181 -10.194088 - 7.538544 TABLE III. 4 . (Continued) u u» u" Case I z z» z" Case II u u» u" z z» z" 200 3.311783 - 4.287341 - 1.558924 -14.508000 - 6.542474 11.168408 4 . 1031+81 1.850491 - 3.158901 11.717474 -15.154191 - 5.518204 250 2.893183 - 5.688075 - .756350 -19.704000 - 4.784883 15.914215 5.573126 1.353368 - 4.501204 10.23821+4 -20.108150 - 2.681536 300 2.105674 - 7.048501 .309267 -24.995999 - 1.743280 20.414091 7.069928 .493073 - 5.773962 7.454510 -24.919558 1.085492 400 - .581511 - 9.456595 3.288541 -34.992000 8.124331 27.990059 9.897225 - 2.297906 - 7.916779 - 2.045645 -33.436529 11.618575 500 - 4.715846 -11.103581 7.401575 -42.756000 22.754273 32.555154 12.093220 - 6.435879 - 9.207986 -16.662962 -39.262461 26.160855 In t h i s chapter, so f a r , we obtained the mathema-t i c a l r e l a t i o n s between the known values u n , z n, t h e i r f i r s t and second derivatives at the point (n) and the unknown values UJJ^.^, z n +]_» t h e i r f i r s t and second derivatives at the next point (n+1). To obtain the next point values from the above shown formulae we w i l l use the Alwac III-E D i g i t a l computer. Since the accuracy of a numerical method depends upon the Interval dx, we face the problem of Investigating the most acceptable i n t e r v a l . In the following pages three d i f f e r e n t cases were examined, i . e . , l / h = 30 , 100, 500. To i n v e s t i -gate the i n t e r v a l f o r each case the s t a r t i n g values of Case I were used and the corresponding values were typed out fo r every 2 0 ° - i . e . , 2 6 ° , 1+6°, 6 6 ° , 8 6 ° - . The curves shown on Pig. I I I . 2.a.b.c. are f o r the z values at d i f f e r e n t h»s and every 2 0 ° . Prom these f i g u r e s , It l s easy to see that the thicker s h e l l ( l / h = 30) gives the larger i n t e r v a l s ; the thinner s h e l l ( l / h = $00) requires the shorter i n t e r v a l s . The curves f o r the thick s h e l l go almost In h o r i z o n t a l Bhape and even f o r the larger Intervals the obtained values come close to the smaller i n t e r v a l r e s u l t s . But f o r the th i n s h e l l , say l / h = $00, the very same curves show a large curvature and get stable and h o r i z o n t a l f o r only the quite small i n t e r v a l s . For Instance, f o r $00 d0 = 0.05 degrees was taken as the closest Interval to the best, and 0.25 f o r 30 . The reason f o r t h i s i s that the thi n s h e l l has many more nodes than the thick s h e l l . It requires a smaller Interval to approximate these many waves. 5o After the inves t i g a t i o n of the i n t e r v a l s the stress resultants were calculated with these i n t e r v a l s f o r the d i f f e r e n t cases. The r e s u l t s are tabulated on the Tables III.5-6. and 7. The flow diagram of the Programme i s shown on pages 54 and 55 . This seven channel programme was performed on Working Channel I. On the other hand the Working Channels II and III were used as the scratch pads f o r temporary stor-age, and are shown on page 56. The words which are indicated by an asterisk contain several intermediate computation steps. The value M shown i n Channel I I defines the type out i n t e r v a l . That i s the stress resultants were typed out at an angular i n t e r v a l of M.d0. During the calculations B which places the binary point, was taken as 23 i n decimal system which means 17 i n the hexadecimal system. The Main Memory Channels, numbered from 98 to 9e were used f o r the storage of the programme. The entrance word i s the f i r s t word of Ch. 98. After the operations are done, the programme stops at the second word from the end of Ch. 9e . It i s very i n t e r e s t i n g to compare the wave lengths for the d i f f e r e n t h values, i . e . , 30 100, and 500. The thickest s h e l l gives a wave length more or less four times bigger than the thinnest s h e l l does. The intermediate s h e l l gives a wave length which i s between the other two. In the Pigs. III.3.4'5. the iry curves are shown for the d i f f e r e n t cases. ttrrr. PLOW DIAGRAM OP PROGRAMME Decimal Input: 0°,u n,u^,u£, z n , z n , z j , V,A, d0°,M Store the above Input values i n Working Chan-nel II as noted. Calculate the following values and store i n the indicated places In Ch. II or I I I . C, D, d0 r a d, dx, (^|) 2, P,H. PH, V (dx^2 A 2 ' Put M i n the storage word (i|.5) Add d0f° to and put 0 n + 1 i n (20) Calculate: ^ S i n 0 n + 1 C o x 0 n 4 . 1 , C o t 0 n + 1 ] n+1 A ' A 2 Calculate: a n + 1 , b n + 1 , K , L , (2) Calculate: M, N, e n + l 'S n + l ' k n + l (1) Main Mem. Ch : 98 Main Mem. Ch : 99 Main Mem. Ch : 9a Main Mem. Ch : 9b 1 Calculate: u" z" u' z» u z at 0 = 0. n+1 Main Mem, Ch : 9c 1 Transfer u u' u" z z' z" to the old places. Copy 0 present to (5f) Subtract 1 from M at (1+5) . I f , to ( 2 ) A / 0 r A = 0 •0 n e m0 m e y •8-Calculate: n . n^ m^ m~ q,y t z, 5 Main Mem. Ch : 9d Decimal Output: 0 rty mg n^ ng t q^ z.*S Check 90 - 0 p r e 8 e n t. I f , to ( 1 ) A > 0 A < 0 STOP Main Mem. Ch : 9e The following charts show the working channels which were used as the scratch pads f o r the programme. WORKING CHANNEL II fo l a t t e r 0 ° + 1 u n »n < z n *A z" n A df M U' , n+1 u" •, n+1 zn+l z" Zn+1 me «v n 9 t :z S C D F H PH dj#(radians) dx WORKING CHANNEL III ( ^ ) 2 -v> /dxx2 T T / 2 9 0 ° Unity M (transferred) 0 n + 1 ( r a d i a n s ) S l n^n+1 Cot0 , 'n+l C c t 0 n + 1 /A an+l b n+1 K L M N en+l gn+l o kn+l ( e n + l ^ ( T } } n+1 A2. ^ ( e2 J > ( % + H F n+1 X* -a Note: In the above tables a l l the values are based on B = 1? (Hexadecimal) decimal point l o c a t i o n . 57 The complete programme which i s to be stored i n the Main Memory drum i s as follows: 9804 START 571f2800 00000000 00000000 00000000 48601704 00000000 00000000 00000000 (9800) 8 7 i e 5 7 i b 00000000 41270727 00000000 5 b l 7 1 l 6 0 00000000 a30965la 00000000 482bl70c 00000000 81991100 00000000 790ff782 00000000 00000000 00170010 11060000 00000000 00800000 OOObOOOO 00000000 00000000 03170619 00400000 9904 3128a317 e713©bl7 a309493c 4943791b e b l f 2 8 0 0 c53ee728 4l27e509 a7034944 a3098705 a l l 7 c 5 3 f eb282800 8 l 9 a l l 0 0 5 b l b l l 6 0 e73fa307 a317eb28 00000000 a3174939 49403a00 c 5 4 l 4 l 3 b 00c90fda 2800411b 4 l 3 9 e 5 l 8 e73ca309 2d000000 a31ceb39 a309493b 493d7913 00100000 c53a4l29 4l3ae500 49427917 01800000 9a04 792a4945 al09c547 a309494a 4l26e506 79206129 28004146 7944a701 a3086l25 49203128 a3095b0d 6149494b 494e4l22 e742eb43 11600011 6744674a e5lOa309 c5462800 a30eeb47 494c0000 6 l 2 l 4 9 5 e a3098706 c5482800 5b3f4l23 4l25e500 5 b l c l l e 4 a317eb28 e5lba308 a3096l24 00000001 c549e749 6l22494d 8l9bll00 9b 04 495d43i+0 e723a309 4l4oe726 a3096l£d 49504H+0 e7irCa309 J4.95c4l3f 9c04 67l4.H4.955 7951611+1 49563000 e755a309 6l3dl+957 1+I55e753 a309l+95d 1+I56e751+ 9d0l+ l5lf571b 78311+827 1701+7920 l+95f79l+5 671+1+1+931+ 1918111c 8l9all0l+ l+121e7l+9 9e0l+ 1+9321+11+9 e727a309 671+1+3000 e72la309 67223000 e7l+7a309 1+9380000 87If5b If e7l+ba308 6ll+l+6l5c 1+95H+127 2800a317 eb282800 a317eb28 c5523000 6ll+c3128 a309l+95e i+13be751+ a3096l5d 3128a317 eb57o52d 1+I3ce753 a309655e 3128a317 a3092e00 1+9337921 6l222e00 1+931+1+121 2800a317 eb28c536 2800a317 ebl+7312e 79201160 57175bl3 791bf78l 78351160 1785790f f78l5717 5b0b791b f78l7839 e7l+fa309 l+95bl+H+b e7l+da309 l+95al+139 e750a309 675a675b 1+953791+c 67523OOO eb57c530 79306126 3000e73f a3086l25 l+92f792d 61233000 e73fa308 6122592c 1+9351+121+ 2800a317 eb28c537 7921+6125 l+95cl+12l+ e727all7 e7l+9a309 6l>c2e00 11601719 790ff78l 791+36720 ldleOOOO 8l9allOO 00000000 00000000 lbOOOOOO O STOP e750a309 1+9591+ll+b e7l+ea309 l+9581+13a e74fa309 63586759 1+951+7951 8 l 9 c l l 0 0 792f6l25 3000e73f a3086l2i+ l+92e792c 61223000 e73fa308 6l2U+92b 8l9dllOO l+93H+l5c e727a309 l+95bl+12i+ e7l+9a309 6l5b2e00 8l9ellOO 00060000 lbOOOOOO 00000000 00000000 03170682 5b6d0000 03170611 00040000 58000000 03170005 TABLE I I I . 5 . a . h = 1/30 Cas e: I 0 n0 n e )egree a t V Z a 8 5.210935 -1.83728J+ 2.602285 .255700 -1.8191+01+ -.228393 -1.671+068 -.18231+3 10 6.275797 -1.390701 3.139213 .21+11+93 -1.369573 -.31+6051 -1.087089 -.11+1206 12 6.899218 -1.021+379 3.1+661+03 .212980 -1.001991+ -.1+61991 -.51+2312 -.071088 U+ 7.067032 -.721506 3.58261+5 . 171+51+8 -.700071+ -.563387 -.037007 .021+920 16 6.8027H4. -.1+70505 3.5ol+568 .129689 -.1+52278 -. 61+01+05 .1+221+31+ .11+1371 18 6.153980 -.2631+1+5 3.258890 .0811+09 -.250551 -.6861+21+ .821+826 .270370 20 5.181+81+1+ -.091+813 2.877771+ . 0321+28 -.089095 -.697915 1.157591+ .1+02015 22 3.970350 .039397 2.39591+9 -.011+758 .036528 -.671+168 1.1+09300 .525196 21+ 2.5921+12 .11+2339 1.81+8732 -.057895 .130033 -.61691+3 1.571I+03 .628569 26 1.136021 .216760 1.270569 -.095021 . 191+822 -.530086 1.639378 .701575 28 -.311+516 .265285 .693808 -.121+51+1+ .231+233 -.1+19115 1.613306 .7351+08 30 -1.679281+ .290596 .11+761+3 -.U+5298 .251661+ -.290780 1.1+98016 .72381+1 32 -2.886329 .295518 -.31+2795 -.156601 .250611+ -.152619 1.302866 .663852 31+ -3.87I+88I+ .283039 -.757193 -.158273 .231+650 -.0121+95 1.01+1217 .555998 36 -1+. 598075 .256291 -1.080607 -.15061+1+ .20731+3 .12181+1 .729667 .1+01+513 TABLE I I I . 5 . a . (Continued) h = 1/30 Case: I 0 m e n9 egrees t y z* 5 38 -5.0249I+0 .218498 - 1 . 3 0 3 7 9 7 - . 1 3 4 5 2 1 .172179 .243183 . 3 8 7 1 0 9 .217127 1+0 -5 . 141629 .172901+ - 1 . 4 2 3 3 0 7 -.111141 . 1 3 2 4 5 2 .345238 . 0 3 3 6 6 5 . 0 0 4 6 1 1 1+2 -4 . 9 5 1 7 2 2 .122687 -1.1+41292 - . 0 8 2 0 9 3 .091174 .422953 - . 3 1 0 4 2 4 -.219916 1+1+ -4-475627 . 0 7 0 8 7 2 - 1 . 3 6 5 0 9 6 - . 0 4 9 2 3 2 . 0 5 0 9 8 1 .472755 - . 6 2 5 9 0 9 -.I+41876 1+6 - 3 . 7 4 9 1 0 9 .020245 - 1 . 2 0 6 6 0 1 - . 0 1 4 5 6 3 . 0 1 4 0 6 3 .492718 - . 8 9 5 5 6 7 -.64621+0 1+8 -2.821010 - . 0 2 6 7 2 3 - . 9 8 1 3 7 7 . 0 1 9 8 5 9 - . 0 1 7 8 8 1 . 4 8 2 6 2 5 -1.105114 - . 8 1 8 6 0 2 50 - 1 . 7 5 0 3 0 9 - . 0 6 7 9 4 8 - . 7 0 7 6 8 1 .052051 - . 0 4 3 6 7 6 .1+43952 -1.21+3939 - . 91+6221 52 - . 6 0 2 6 7 5 - . 1 0 1 7 8 2 - . 4 0 5 3 5 4 . 0 8 0 2 0 6 - . 0 6 2 6 6 4 . 3 7 9 7 4 2 - 1 . 3 0 5 6 3 4 - 1 . 0 1 8 9 7 7 54 .553298 - . 1 2 7 0 5 1 - . 0 9 4 6 8 7 . 102787 - . 0 7 4 6 7 9 .291+412 - 1 . 2 8 8 2 7 9 - 1 . 0 3 0 1 5 6 56 1 .649958 - . 1 4 3 0 7 1 .204709 . 118611 - . 0 8 0 0 0 4 .193479 -1.191+480 - . 9 7 7 0 0 3 58 2 . 6 2 4 3 5 5 - . li+9646 .471+944 . 126907 - . 0 7 9 3 0 0 . 0 8 3 2 3 0 - 1 . 0 3 1 1 5 6 - . 8 6 1 0 1 9 60 3 . 4 2 1 7 0 4 -.11+7052 .700770 .127351 - . 0 7 3 5 2 6 - . 0 2 9 6 4 2 - . 8 0 9 0 8 7 - . 6 8 7 9 5 5 62 3 . 9 9 8 3 3 2 - . 1 3 5 9 9 4 . 8 7 0 3 2 3 . 120076 - . 0 6 3 8 4 5 -.1381+23 - . 5 4 2 2 5 8 -.1+67511 61+ 4 .323986 - . 1 1 7 5 5 6 .975653 .105659 - . 0 5 1 5 3 3 - . 2 3 6 7 5 8 - . 2 4 7 0 2 3 - . 2 1 2 7 5 9 66 4 -383361 - . 0 9 3 1 3 2 1 .013019 . 0 8 5 0 8 1 - . 0 3 7 8 8 0 - . 3 1 9 0 0 1 . 0 5 8 8 4 1 . 0 6 0 6 7 5 68 4-176784 - . 0 6 4 3 5 0 . 9 8 2 9 5 2 . 0 5 9 6 6 4 - . 0 2 4 1 0 6 - . 3 8 0 5 3 2 .357166 .335628 TABLE I I I . 5 . a . (Continued) h = 1 / 3 0 Case: I 0 rae n 0 n e egrees t z* 8 7 0 3 . 7 2 0 0 0 3 - . 0 3 2 9 8 2 . 8 9 0 0 5 5 . 0 3 0 9 9 3 - . 0 1 1 2 8 1 - . 4 1 8 0 0 1 . 6 3 0 4 2 4 . 5 9 4 5 2 5 7 2 3 . 0 1 | 3 1 2 5 - . 0 0 0 8 6 3 . 7 4 2 5 9 8 . 0 0 0 8 2 1 - . 0 0 0 2 6 7 - . 4 2 9 5 0 8 . 8 6 2 7 2 3 . 8 2 0 5 4 9 7 4 2 . 1 8 8 7 5 0 . 0 3 0 2 0 3 . 5 5 1 9 0 4 - . 0 2 9 0 3 3 . 0 0 8 3 2 5 - . 4 1 4 6 9 1 1 . 0 4 0 6 9 7 . 9 9 8 7 8 2 7 6 1 . 2 0 9 4 1 7 . 0 5 8 5 2 8 . 3 3 1 5 7 7 - . 0 5 6 7 8 9 . 0 1 4 1 5 9 - . 3 7 4 7 3 1 1 . 1 5 4 2 4 9 1 . 1 1 7 2 1 5 7 8 . 1 6 4 5 1 7 . 0 8 2 6 1 4 . 0 9 6 6 2 4 - . 0 8 0 8 0 9 . 0 1 7 1 7 6 - . 3 1 2 2 7 3 1 . 1 9 7 1 0 7 1 . 1 6 7 5 8 7 8 0 - . 8 8 3 1 7 3 . 1 0 1 2 2 3 - . 1 3 7 4 8 9 - . 0 9 9 6 8 5 . 0 1 7 5 7 7 - . 2 3 1 2 5 7 1 . 1 6 7 1 6 2 1 . 1 4 5 9 6 8 82 - 1 . 8 7 1 1 1 3 . 1 1 3 4 3 1 - . 3 5 5 7 8 2 - . 1 1 2 3 2 7 . 0 1 5 7 8 7 - . 1 3 6 6 8 0 1 . 0 6 6 5 6 2 1 . 0 5 3 0 5 6 8 4 - 2 . 7 4 0 5 9 9 . 1 1 8 6 6 7 - . 5 4 4 6 5 8 - . 1 1 8 0 1 7 .012404 - . 0 3 4 3 0 8 . 9 0 1 5 7 4 . 8 9 4 1 6 8 8 6 - 3 . 4 4 0 1 2 3 . 1 1 6 7 3 5 - . 6 9 2 7 0 2 - . 1 1 6 4 5 0 . 0 0 8 1 4 3 . 0 6 9 6 7 3 . 6 8 2 2 0 7 . 6 7 8 9 2 0 8 8 - 3 . 9 2 8 3 1 5 . 1 0 7 8 1 6 ' - . 7 9 1 3 2 9 - , 1 0 7 7 5 0 . 0 0 3 7 6 3 . 1 6 9 0 0 6 . 4 2 1 6 2 7 . 4 2 0 6 1 8 9 0 - 4 . 1 7 6 2 9 7 . 0 9 2 4 6 2 - . 8 3 5 2 6 0 - . 0 9 2 4 6 2 . 0 0 0 0 0 0 . 2 5 7 7 3 3 . 1 3 5 3 9 8 . 1 3 5 3 9 8 62 TABLE I l l . S . b . h = 1/30 Case: II 0 Degree! 0 n0 ne 3 t "0 z* s 8 -7.22^722 -.1+61+166 -7.62651+9 .061+599 -.1+5961+9 .901+936 -1.38191+7 -.179535 10 -1+. 836535 -.563658 -5.623202 .097878 -.555095 .855166 -1.664046 -.269680 12 -2.652985 -.6281+92 -3.939680 .130671 -.611+759 .751+815 -1.8281+1+1+ -.351+591 11+ -.653630 -.658681+ -2.515382 .159350 -.639118 .619335 -1.871037 -.1+21721 16 1.11+7000 -.65711+6 -1.311+108 .181131+ -.631690 .1+61035 -1.797762 -.1+60707 18 2.711+861+ -.628283 -.315393 .191+150 -.597533 .290521 -1.621103 -.1+61+018 20 1+.009352 -.577159 .1+92239 .1971+00 -.51+2353 .117392 -1.358026 -.1+27373 22 1+. 992867 -.509023 1.116262 .190683 -.1+71958 -.01+9531 -1.028591+ -.31+9958 21+ 5.637518 -.1+29019 1.563628 .171+1+98 -.391929 -.202266 -.651+860 -.231+473 26 5.929381 -.31+2019 1.81+3029 .11+9931 -.3071+05 -.333869 -.259835 -.086953 28 5.870768 -.252505 1.966195 . 11851+1+ -.22291+8 -.1+38683 .133548 .083630 30 5.1+8081+1+ -.161+1+90 1.91+81+48 . 08221+5 -.11+21+53 -.512566 .503463 .265977 32 1+. 791+885 -.0811+60 1.808670 .01+3167 -.069082 -.553069 .830193 .447257 31+ 3.8621+11 -.006321 1.56881+2 .003531+ -.00521+0 -.559533 1.097031 .614038 TABLE I l l . ^ . b . (Continued) h = 1/30 Case: II 0 m e 0 Degrees t q z* s 36 2.744448 .058630 1.253271 -.034462 .047432 -.533086 1.291047 .753282 38 1.510130 .111721 .887595 -.068782 .088037 -.476559 1.403662 .853340 ko .232883 .151913 .497687 -.097648 .116372 -.394298 1.431000 .904868 42 -1.013592 .178783 .108509 -.119629 .132861 -.291907 1.373987 .901596 kk -2.159293 . 19214.90 -.256977 -.133715 .138466 -.175918 1.238206 .840893 46 -3.141638 .193735 -.578800 -.139362 .134580 -.053424 1.033493 .724071 48 -3.908732 .183688 -.840799 -.136507 .122911 .068317 .773321 .556421 50 -4.422000 .163918 -I.031238 -.125569 .105364 .182286 •473990 .346955 52 -4.658060 .136302 -I.143181 -.107407 .083916 .282079 .153680 .IO7876 54 -4.609739 .102930 -1.174610 -.083272 .060501 .362250 -.168595 -.146186 56 -4.286185 .066009 -1.128289 -.054724 .036912 •418594 -.474060 -.399134 58 -3.712077 .027759 -1.011374 -.023541 .014710 .448358 -.745221 -.634478 60 -2.925995 -.009681 -.834819 .008384 -.004840 .450370 -.966836 -.836466 62 -1.978030 -.044342 -.612582 .039152 -.020817 .425076 -1.126736 -.991173 TABLE I l l . ^ . b . (Continued) h = 1/30 Case: II 0 n 9 Degrees t y z* s 64 -.926790 -. 0714,505 -.360704 .066965 -.032661 .374491 -1.216476 -1.087490 66 .I6k026 -.098765 -.096301 .090227 -.040172 .302060 -1.231769 -1.117937 68 1.229318 -.116083 .I63466 .107630 -.043485 .212439 -1.172677 -1.079223 70 2.206297 -.125816 .402398 .118228 -.043032 .111219 -1.043571 -.972548 72 3.0380I4.6 -.127735 .606178 .121I4.83 -.039472 .004590 -.852832 -.803584 74 3.676709 -.122019 .763150 .117292 -.033633 -.101019 -.612346 -.582159 76 I4..086H4.8 -.109235 .864935 .105990 -.026426 -.199309 -.336791 -.321658 78 4.243915 -.090297 .906831 .088324 -.018774 -.284475 -.042779 -.038171 80 4.142444 -.066418 .887994 .065409 -.011533 -.351533 .252105 .250546 82 3.789386 -.039039 .811386 .038659 -.005433 -.396601 .530316 .526231 84 3.207088 -.009757 .683505 .009704 -.001020 -.417120 .775362 .771317 86 2.431243 .019754 .513905 -.019706 .001378 -.411988 .972738 .970093 88 1.508775 .047831 .314548 -.047802 .001669 -.381620 1.110762 1.109751 90 .495113 .072898 .099023 -.072898 .000000 -.327917 1.181236 I.181236 65 TABLE III. 6 .a. h = l/lOO Case: I 0 m0 n0 n 9 Degrees t y ** 8 8 13.0k l071 -I .182708 6.6951+67 .161+601 -1.171198 - .598359 .182697 .058026 10 II .230198 -.1141+361 6.058271 .077162 -.1+37610 - .700207 1.596820 .292483 12 7.5H+625 .056301+ 1+.603750 - .011706 .055075 - .686563 2.573510 .532772 lk 2.775996 .3611+91 2.7151+96 -.0871+53 .350753 - .561065 3.005121 .710033 16 -2.038789 .501+017 .767108 - .138926 .1+81+1+92 -.350921+ 2.862312 .762251 18 -6.059520 .516657 -.92181+9 - .159656 .1+91370 - .098171 2.210673 .652767 20 -8.621853 .1+31+765 -2.119093 -.11+8698 .1+08545 .149653 1.200274 .382572 22 -9.36I+172 .295206 -2.701062 - .110586 .273710 .348568 .036941 - .006668 21+ -8.273068 .13351+8 -2.659161 -.051+319 .122002 .465889 -1.056603 - .439683 26 -5.668062 - .019198 -2.089280 .0081+16 - .017255 .485532 -1.881576 - .823316 28 -2.1281+00 -.1391+01+ -1.166153 .0651+1+6 - .123086 .410121 -2.298293 -1.067426 30 1.623985 -.212668 - .106909 .106331+ - .184176 .259631 -2.2493I+4 -1.106251+ 32 1+.863163 -.231+373 .869792 . 121+199 - .198760 .066940 -1.766908 -.915253 31+ 6.993691+ -.208917 1.582065 .116825 - .173200 - .128807 -.963302 - .519301 TABLE III.6.a. (Continued) h = 1/100 Case: I 0 0 rae n0 ne Degrees t V z s 36 7.656351+ -.11+7797 1.911+117 .086873 -.119570 -.28971+1+ -.006981+ .009951 38 6.7861+03 -.066883 I.831967 .01+1178 -.052705 -.386319 .911155 .5671+53 1+0 1+. 611+913 .01661+9 1.383519 -.010702 .012751+ -.1+02537 1.611+61+5 1.036231+ 1+2 1.611+1+82 .087277 .683781). -.0581+00 .061+859 -.3381+81+ 1.971+481+ 1.312507 1+1+ -1.598986 .133509 -.111303 -.09271+3 .096038 -.209730 1.93221+1+ I.3289O6 1+6 -1+.388781+ . 11+91+02 -.837121 -.1071+71 .103783 -.01+3833 1.50911+3 1.070655 1+8 -6.2201+19 . 135021+ -1.3521+35 -. 10031+2 .09031+8 .125350 .799691 .580858 50 -6.761783 .095676 -1.565320 -.0731+1+6 .061628 .261+375 -.01+8521 -.01+6611 52 -5.9I+1950 .Ol+H+61 -1.1+1+81+08 -.032672 .025526 ..3I+667I+ -.861+725 -.6851+36 51+ -3.958298 -.016726 -I.OI+IO27 .013532 -.009832 .357526 -1.1+88656 -1.202757 56 -1.231701+ -.067392 -.1+381+08 .055871 -.037685 .296615 -1.8011+61+ -I.I+87233 58 1.680526 -.1011+50 .2291+75 .086031+ -.053760 .177753 -1.71+7991+ -1.1+73261+ 60 1+. 196398 -.113508 .821x91+7 .098300 -.056751+ .025859 -I.3I+6271 -1.156075 62 5.823633 -.102611 I.230H+I .090600 -.01+8173 -.128151 -.682611 -.591+203 67 TABLE III. 6 .a. (Continued) h = 1/100 Case: I 0 ra 9 n 9 Degrees t y z S 64 6.251+51+2 - .072156 1.369626 .061+853 - .031631 - .253550 .106494 .1011+02 66 5.1+23312 - .029027 1.223929 .026518 - .011806 - .325831 .861968 .789604 68 3 . 5 i 5 i 5 o . 0178I+1+ .831581 -.01651+5 .006681+ -.3311+32 1.433767 1.328126 70 .926157 .059225 .279651+ - .055653 .020256 - .270235 1.710284 1.603334 72 -1.817581 .087311 -.31501+3 - .083038 .026981 -.155400 1.639868 1.554475 74 -1+. 1651+1+3 .097120 - .830173 - .093357 .026770 - .010591 1.240336 1.167141 76 -5 .651392 .087327 -1.162551 -.081+733 .021126 .134832 .594750 .572984 78 -5.985371+ .0601+01 -1.21+81+33 - .059081 .012558 .251689 - .165603 -.I6kl+4l 80 -5 .109281 .022032 -1.0751+76 - .021698 .003826 .316761 - .887569 - .874838 82 -3.206766 -.020015 -.681+170 .019820 - .002786 .317353 -1.426620 -1.412184 81+ - .665302 -.0571+13 - .158662 .057099 - .006001 .253739 -1.675460 -1.665087 86 2.001788 - .082860 .39102k .08265c; - .005780 .139034 -1.585144 -1.580129 88 1+. 257826 -.0911+51 .851682 .091395 - .003191 - .003480 -1.174605 -1.173251 90 5.61+9567 - .081590 1.129911+ .081590 - .000000 - .145021 - .526737 - .526737 68 TABLE III . 6.b. h = 1/100 Case: I I 0 m e Degrees t V z~"" s 8 - .197601 -1.216052 -4.019519 .169242 -1.204218 .582658 -3.447728 - .446311 10 5.322942 -I .140515 - .425201 .198048 -I .123189 .273636 -2.951741 - .473556 12 9.128823 - .933999 2.009041 .194189 - .913589 - .040580 -1.942739 - .365929 14 10.869460 - .655968 3.361844 .158693 - .636483 - .308531 - .657874 - .128358 16 10.464796 - .360098 3.735997 .099256 - .346148 - .490764 .645885 .197112 18 8.170506 - .089856 3.301526 .027767 - .085458 - .564353 1.730982 .540184 20 4.542661 .123760 2.295648 - .042328 . I I6296 - .525905 2.415369 .818149 22 .335185 .263183 .997017 - .098590 ,2k4019 - .391392 2.599784 .955614 24 -3.639638 .323977 - .312656 - .131773 .295967 - .192595 2.280783 .903602 26 -6.657900 .313111 -I .389020 - .137329 .281566 .029183 1.546856 .653411 28 -8.210224 .247086 -2.058941 - .116000 .218164 .230904 .241654 30 -8.084782 .146869 -2.2k1562 - .073435 .127193 .375642 - .484771 - .255104 32 -6.393242 .035729 -1.953141 - .018933 .030300 .439031 -1.381570 - .735332 34 -3.536503 - .065151 -1.295379 .036432 - . o 5 4 o i 3 .413191 -1.967313 -1.094067 36 - .118262 - .139425 - .429939 .081952 - .112797 .307484 -2.1k2985 -1.246355 69 TABLE III.6.b. (Continued) h = 1/100 Case: II 0 m0 m 9 n0 Degrees t y 2 B 38 3.176157 -.1771+80 .1+55786 .109267 - .139856 . ll+601+O -1.891513 -l.U+7311 5 .712217 -.177126 1.185191 .113855 - .135687 -.037361+ -1.278157 -.801+139 1+2 7.021+818 -.H+3078 1.621+679 .095738 - .106327 - .206075 -.1+35379 - .277096 kk 6.901310 - .085395 1.705981 .059321 -.0611+28 -.32761+9 .1+61+51+7 .331236 1+6 5.1+11+187 - .017235 1.1+3501+3 .012398 - .011972 - .379916 1.21+3730 .896387 1+8 2.898553 .01+7708 .8861+92 -.0351+51+ .031923 -.351+912 1.753016 1.298000 50 - .120003 .097611+ .1851+23 -.071+776 . 06271+5 - .259981 1.899972 i.l+i+5850 52 -3.032218 .121+1+32 - .519500 -.098051+ .076608 - .115920 1.665315 1.300212 51+ -5.2651+81+ .121+996 -1.086173 -.101121+ .0731+71 . 01+71+22 1.101+881 .881980 56 -6.391+337 .101196 -1.1+06550 - .083895 .056588 .197181+ .337060 .270053 58 -6.220089 .059285 -1.1+26361 - .050276 .0311+16 .303969 -.1+81608 -.1+13755 60 -1+. 801+866 .0081+1+6 -1.153585 -.007311+ .001+223 .31+7515 -1.187765 -1.029366 62 -2.1+51+319 -.01+1052 -.651+1+1+6 .03621+7 -.019273 .3201+72 -1.61+3318 -1.1+1+7560 70 TABLE III . 6.b. (Continued) h = 1/100 Case: II 0 m0 y n 9 Degrees t y z 8 61+ .31+6777 -.079790 - .038144 .071715 - .034978 .229590 -1.762056 -1.577438 66 3.032781 - .100880 .566320 .092159 - .041032 .094137 -1.525738 -1.386334 68 5.069719 - .101105 1.036480 .093743 - .037874 - .058104 - .986660 - .907791 70 6.060002 - .081339 1.280641 .076434 - .027820 - .196447 - .256393 - .235702 72 5.819033 -. OI4.6216 1.255325 .043954 - .014281 - .293400 .516946 •494361 71+ I4..I4.09093 -.003116 •972675 .002996 - .000859 - .330056 1.178337 1.132855 76 2.12i|359 .039304 .496615 - .038136 .009508 - .299736 1.596554 1.547284 78 - .569562 .072779 - .071228 - .071188 .015131 - .209180 1.689893 i . 65ooo5 80 -3.129306 .090976 - .612813 - .089594 .015798 - .077086 1.441963 1.416945 82 -5.042058 .090643 - I . O I 7 8 1 6 - .089761 .012615 .069702 .904487 .893186 81+ -5.927106 .072164 -1.205760 - .071768 .007543 .201577 .186667 .184144 86 -5.610777 .0391+21 -1.141762 -.039325 .002750 .292078 - .566741 - .565909 88 -4.160131 - .000985 - .842859 .000984 - .000034 .323134 -1.204288 -1.203547 90 -1.868956 - .041018 -.373791 .041018 - .000000 .288637 -1.597939 -1.597939 71 TABLE III. 7 .a. h = 1/500 Case: I 0 m0 y n e )egrees t y 7 -6.525208 2.60681+9 6.098029 - .317695 2.5871+18 -.91+6856 15.7561+19 1.857157 8 -31.1+1+8891+ 2.755683 -1+. 1+13855 - .383516 2.728861+ -.271+629 12.162363 1.616715 9 -1+6.791+61+0 2.263577 -11.721+693 -.351+101 2.235709 .390312 6.103398 .881+832 10 -1+9.887868 l . k09638 - l k . 881+017 -.21+1+781 1.388223 .901185 - .877380 - .200568 11 -kO.973178 .1+5681+2 -13.900183 - .087169 .kkgkkS 1.155255 -7.123819 -1.3761+01 12 -22.9878k5 - .378152 -9.630182 .078622 - .369889 1.111+285 -11.2]|]|)|16 -2.3221+63 13 - .752762 - .9k8503 -3.516666 .213367 -.92kl92 .809509 -12.1+11+977 -2.751179 lk 20.2231+31 -1.1881+37 2.771313 .287509 -1.153136 .330716 -10.5391+30 - 2 . k'93 923 15 35.01+3821 -1.11031+8 7 .716131 .287379 -I.07251I+ -.1971+35 -6.236938 -1.558720 16 ' ko.525771 - .789069 10.267951 .2171+97 - .758502 - .646011 -.661+1+35 - . 11+1329 17 35.859836 -.337193 10.035653 .098586 -.3221+60 - .911995 1+.778519 1.1+15958 18 22.6881+92 .123239 7.318370 - .038083 .117207 -.91+111+0 8.78971+2 2.708933 19 1+.607351 .1+8701+2 2.980272 - .158565 .1+60507 -•738kl+0 10.1+6821+2 3.378138 20 1 -13.773610 .685381+ -1.791+011 -.231+1+15 .61+1+051 - . 36I+2I4.0 9.511718 3.20911+0 72 TABLE III. 7 .a. (Continued) h = 1/500 Case: I 0 n 9 Degrees t z 21 -27.972286 .695439 -5.800948 -.249223 .649248 .082566 6.264759 2.198553 22 -3ii.7030ii6 •539335 -8.104486 -.202038 .500062 .489829 1.611296 .566136 23 -32.620488 .273775 -8.242075 -.106972 .252011 .759622 -3.258037 -1.292709 2k -22.571906 -.026516 -6.306839 .010785 -.024224 .831304 -7.144121 -2.903802 25 -7.313947 -.287424 -2.893189 .121471 -.260495 .694797 -9.129733 -3.836370 26 9.221728 -.451968 1.074309 .198129 -.406226 .391221 -8.792606 -3.818806 27 22.923286 -.491355 4.582518 .223070 -.437800 .001136 -6.292161 -2.816828 28 30.506373 -.408694 6.780844 .191870 -.360856 -.376388 -2.312616 -1.051824 29 30.290483 -.235209 7.181317 .114032 -.205719 -.648553 2.119764 1.047628 30 22.562391 -.020460 5.761776 .010230 -.017719 -.751334 5.897504 2.950521 31 9.449568 .180727 2.951306 -.093081 .154913 -.664322 8.105461 4.158659 32 -5.652695 .321424 -.493767 -.170328 .272582 -.414479 8.240114 4-337702 33 -18.949366 .372867 -3.689031 -.203078 .312713 -.068217 6.322537 3.409434 3k -27.189675 .329729 -5.844985 -.184382 .273358 .285998 2.879987 1.579895 73 TABLE II I . 7 . a . (Continued) h = 1/500 Case: I 0 0 n e Jegrees t q0 5 35 -28.1+51+307 .209557 -6.1+5861+3 -.120197 .171659 .560006 -1.196280 -.70581+9 36 -22.596676 »0I+69I+3 -5.1+27997 - .027592 .037977 .687689 -1+. 879317 -2.8721+53 37 -11.237868 - .115861 -3.063891+ .069727 - .092531 .61+0773 -7.2630I+2 -1+.359867 38 2.673711+ -.2391+75 .000313 .11+71+35 - . 1 8 8 7 0 9 .1+31+91+0 ' -7.783I+H+ -1+.768707 39 15.6201+87 -.29651+0 2.975955 .186618 -.2301+51+ . 121+963 -6.350616 -3.967562 ko 21+. 3981+1+1+ -.2771+1+5 5 .119602 .178338 -.212535 - .209698 -3.3611+93 -2.1331+01 1+1 26.901+572 - .191508 5.91661+1+ .12561+1 -.HLI+533 -.1+85106 .1+081+71+ .28691+7 1+2 22.636573 - .063602 5 .203015 . 01+2558 - . 01+7265 - .633753 1+. 001+716 2 . 686001 1+3 12.786982 .072713 3.196696 -.01+9590 .053179 - .620996 6.533725 k.1+1+8732 1+1+ -.081+210 .183728 .1+33523 - .127628 .132163 -.1+53031+ 7.38371+7 5 .110816 1+5 -12.710000 .21+3696 -2.371+362 - .172319 .172319 - . 171+621+ 6 .371798 k.1+81169 1+6 -21.91+0791 .21+0605 -1+. 5201+83 - .173077 .167138 . 11+2735 3.781867 2 . 696399 1+7 -25.525890 .178371+ -5.I+80316 -.1301+51+ .121651 .1+1901+7 .286615 .191823 1+8 -22.659122 .075112 -5 .038253 - .055819 .050259 .585880 -3.22618k -2.1+01+990 TABLE III. 7 .a. (Continued) h = 1/500 Case: I 0 y n e Degrees t y z 5 49 -lk.l5698k -.0kl939 -3.331+795 .031652 -.027511+ .603222 -5.877979 -4.432011 5o -2.2306k9 - .Lk3k25 -.823936 .109869 - .092192 .1+69011 -7.018L43 -5.362080 5 i 10.085625 -.205096-1.81+6810 .159388 - .129070 .219085 -6.382836 -4.940331 52 19.702830 -.2l3k66 k.002306 .168213 -.1311+23 - .082315 -4.155415 -3.253796 53 2k.2k8633 - .168656 5.1091+11 .131+695 - .101500 - .358970 - .914670 - .714275 5k 22.65001k - .083799 1+.907951 .067791+ - .0k9255 -.51+1875 2.514745 2.042439 " 55 l 5 .38k601 .018663 3.1+7108k - .015288 .010705 - .586379 5.271271 4 .316215 56 k. 31+2233 .112803 1.181225 - .093518 .063078 -.1+83036 6.672062 5.520928 57 -7.66I4.IO3 .175672 -1.371020 -.11+7331 .095678 - .259532 6.381495 5.335920 58 -17.612051 .192802 -3.5381+11 - .163505 .102169 .026672 4.491134 3.791366 59 -23.027995 .161462 -k. 7801+08 -.1381+00 .083159 .303050 1.492194 1.264803 60 -22.599957 .O909I4.2 -k.797302 - .078758 .01+5471 .500327 -1.851512 -1.611331 61 -16.1+93962 - .000103 -3.601991+ .000090 - .000050 .569779 -4.697406 -1+. 108432 62 -6.297868 - .088511 -1.512357 .078151 - .041554 .495225 -6.335778 -5.586819 75 TABLE III. 7 .a. (Continued) h = 1/500 Case: I 0 m0 n0 n e legrees t y z* 63 5.389725 - .152435 .932786 .135820 - .069204 .296762 -6.366208 -5 .659998 6k 15.619525 - .17667k 3.112018 .158794 - .077448 .025388 -4 .795070 -4 .295855 65 21.833507 - .156238 4.478638 .1416OO - .066029 - .250051 -2.030438 -1.828232 66 22.502635 - .097310 4.697259 .088897 - .039579 - .460277 1.223471 1.124927 67 17.50186k -.015450 3.725700 .014222 - .006037 - .552953 4.145110 3.816703 68 8.130695 .068460 1.822270 - .063475 .025646 - .505667 6.002442 5.560609 69 -3 .223185 .133496 - .522536 - .124629 .047841 -.331338 6 . 3358k l 5.906081 70 -13.690464 .163848 -2.711989 - .153967 .056039 - .074711 5.071511 4.755127 71 -20.643355 .152631 -4 .194481 - .144315 .049692 .199089 2.537485 2.389842 72 -22.353539 .103411 -4 .602040 - .098349 .031955 .421036 - .621172 - .596847 73 -18.420628 .029129 -3.841315 -.02785.6 .008516 •535563 -3 .606113 -3 .450170 74 -9.865164 - .051262 -2.114644 .049277 - .014130 .514433 -5 .666974 -5 .444726 75 1.135263 -.117595 .133502 .113588 - .030436 .363683 -6.289523 -6.069331 76 11.798799 - .153502 2.330576 .148942 - .037135 .121928 -5 .323660 -5 .158316 76 TABLE III. 7 .a. (Continued) h = 1/500 Case:. I 0 0 n0 n e Degrees t y 8 77 I9.I4I4.IO6I - . I 5 0 k 2 l 3 .92 l3k7 .11+6565 - .033837 -.11+9502 -3.0191+56 -2.9351+73 78 22.H4.9255 - .109615 k.507818 .107220 - .022790 - .382085 .037382 . 01+1021+ 79 19.259770 -.OI4.I736 3.9k8k96 .0k0969 -.007961+ - .517357 3.071+013 3.019097 80 11.52030k •03593k 2.392351 - .035388 .00621+0 - .521579 5.325387 5.21+3252 81 .896993 .103 81+6 .239325 - .102567 .01621+5 -.391+127 6.22651+1 6.11+6671 82 -9.9238kk . U4.5068 -1.962165 -.11+3656 .020189 -.16751+2 5.551+01+2 5.1+95990 83 -18.2131+11 .H4.9l4.79 -3.65k560 -.11+0365 .018217 .100762 3.1+81253 3.1+51688 814, -21.886995 .116230 -k.1+12009 - .115593 .01211+9 •3k3005 .533772 .5281+31 85 -20.0270k8 .053895 -l+.0l+72k9 - .053690 .00k697 .1+98131 -2.51+3538 -2.531+791+ 86 - 1 3 . H I 8 3 0 - .021727 -2.657751+ .02167I+ - .001516 .527157 -k. 971+31+3 -k.961922 87 -2.892775 - .09158k - .59983k .0911+58 -.001+793 .1+2291+9 -6.11+6256 -6.136875 88 8.OI+8088 - . 1 3 8 l k l 1.602512 .138057 -.001+821 .211972 -5.761+760 -5.760281+ 89 16.9k9055 - . I k9750 3.390690 .11+9727 -.002611+ -.0521+19 -3.927065 -3.9259I+I+ 90 21.561+290 - .1235kk 1+. 312859 .12351+1+ - .000000 -.3031+1+0 -1.097392 -1.097392 77 TABLE I I I . 7 . D . h = 1/500 Case: II 0 0 m0 0 n e )egrees t y z* 7 57.53814,87 -2.197530 20.287947 .267811 -2.181150 -1.122998 2.281837 .331249 8 144.937471 -.558131 18.249131 .077677 - .552700 -1.355871 9.005629 1.268724 9 23.170692 .705707 12.222947 - .110397 .697019 -1.252028 13.096118 2.026873 10 -2.108630 1.467874 4.291216 - .254893 1.445574 - .865644 13.821304 2.349838 11 -24.859754 1.712475 -3.448523 - .326755 1.681012 - .308463 II .25276O 2.082975 12 -40.011156 1.515873 -9.256492 - .315167 1.482747 .277709 6.205393 1.228517 13 -44 .546905 1.017838 -12.045404 - .228963 .991751 .754174 .014563 - .041343 14 -38.082677 .386656 -11.530111 - .093541 .375171 1.016419 -5.795084 -1.420109 15 -22.817555 - .215761 -8.203913 .055843 - .208409 1.016085 -9.870207 -2.543808 16 -2.895027 -.662898 -3.153959 .182719 - .637219 .769120 -11.334974 -3 .089210 17 16.658156 -.882272 2.236493 .257951 - .843721 .348768 -9 .973947 -2.866761 18 31.153966 - .861425 6 .627955 .266195 - .819263 - . 134422 -6.253420 -1.881778 19 37.335371 - .641532 9.029604 .208862 - .606580 - .560439 -1.181839 - .345272 20 34.087667 - .301218 9.003289 .103023 - .283052 - .828697 3.952375 1.371152 78 TABLE III.7.b. (Continued) h = 1/500 Case: II 0 *V ne Degrees t V it z 5 21 22.614955 .065166 6.726664 - .023353 .060837 - .881156 7.899589 2.826597 22 6.058570 .36981+0 2.909257 - .138544 .342910 - .714428 9.746920 3.625565 23 -11.333017 .549876 -1.410846 - .214853 .506163 - .378383 9.125233 3.525954 2k -25.270095 .578085 -5.135816 - .235128 .528107 .037935 6.278676 2.510805 25 -32.460958 .465002 -7.376006 - .196518 .421435 .429300 I .984409 .803025 26 -31.375980 .252421 -7.653792 - .110654 .226874 .700401 -2.653673 -I . I83183 27 -22.561131 .000708 -5.998169 - .000321 .000631 .788672 -6.483809 -2.943642 28 -8.438686 - .226762 -2.912683 .106458 -.200219 .678453 -8.588461 -4.013234 29 7.341896 - .378372 .769883 .183438 -.330931 .403316 -8.501255 -4.089399 30 20.832986 - .425019 4.106162 .212509 - .368077 .036346 -6.307991 -3.117185 31 28.764449 - .364825 6.278k33 .187899 - .312716 -.328935 -2.610197 -1.312138 32 29.327286 - .221227 6.790484 .117232 -.187611 - .602105 1.636346 .887014 33 22.579137 - .035426 5.577188 .019295 - .029711 - .717974 5.365632 2.925566 3k . 10.381992 .144659 3.004302 - .080892 .119928 - .651957 7.666416 4.273589 79 TABLE III . 7.b. (Continued) h = 1/500 Case: II 0 m0 n 9 Degrees t z* 5 35 -4.101407 .276151 - .237471 - .158394 .226209 - .k25092 8.002850 4-564292 36 - 17.218819 .330917 -3.314651 - .194508 .267717 - .097715 6.337759 3.693766 37 -25.741515 .301152 -5-462172 - .181238 .240511 .246371 3.130445 1.854999 38 -27.652594 .199817 -6.170893 - .123019 .157458 .521161 - .787733 - .504364 39 -22.619458 .056163 -5.306046 - .035345 .043647 .659766 -4.426870 -2.791411 40 -12.040706 - .092272 -3.129567 .059312 - .070685 .630570 -6.886785 - 4 . k l 7 6 4 9 kl 1.335636 - .209141 - .223561 .137209 - .157840 •444321 -7.578193 -k.951027 42 14.120057 - .267889 2.663429 .179252 - .199080 . l 5 0 6 l 4 -6.362772 -k.230880 43 23.134800 - .257543 4.807307 .175644 - .188355 - .175182 -3.579032 -2.4l5200 44 26.198967 - .184461 5.688265 .128137 - .132690 - .451128 .050356 .053415 45 22.652559 - .069849 5.115343 .049391 - .049391 - .609199 3.604808 2.555967 46 13.494279 .056123 3.266174 - .040371 .038986 - .611954 6.197996 4-452853 47 1.105371 .162062 .634059 - .11^524 .110526 - .461325 7.197707 5.247898 48 -11.365411 .222987 -2.102377 - .165712 .149207 - .197487 6.379125 4.718434 80 TABLE III.7.b. (Continued) h = 1/500 Case: II 0 0 m 9 n-0 n 9 Degrees t y z* 8 49 -20.797920 .226070 -4.252854 - .170617 .148315 .111764 3.974536 2.977231 5o -24.877550 .173171 -5.288331 - .132656 .111312 .388337 .608660 .449206 51 -22.660383 .079736 -4.970116 - .061967 .050180 .563470 -2.862681 -2.232519 52 -I4 .789I87 - .029545 -3.403916 .023282 - .018190 .594743 -5.569163 -4.385690 53 -3.311331 - .127132 -1.006829 . .101532 - .076510 .476302 -6.843247 -5.453036 54 8.850804 - .189446 1.602893 • .153265 - .111354 .239817 -6.384119 -5.146840 55 18.640703 - .202469 3.764381 .165853 - .116131 - .053912 -4.328195 -3.526421 56 23.632882 - .164797 4.940599 .136623 - .092153 - .330521 -1.209738 - .987638 57 22.631455 -.087527 4-850995 .073407 - .047671 - .520831 2.177268 1.834005 58 15.954960 .008895 3.537770 - .007544 .004714 - .57^085 4.980496 4.222898 59 5.339936 .099998 1.350281 - .085715 .051503 - .489390 6.502999 5.565325 60 -6.508605 .163406 -1.147323 -.I4l5i4 .081703 - .278569 6.375892 5.507529 61 -16.603989 .184260 -3.320897 - .161158 .089331 .000185 4-647330 4.049017 62. -22.428258 .158640 -4.626631 - .140071 .074477 .276188 1.766411 1.546496 81 TABLE III . 7 . . D . (Continued) h = 1/500 Case: II 0 0 m0 n0 n 9 Degrees t * z S 63 -22.558153 . 09k20ij. -4.746483 - .083937 .042768 .480128 -1.532997 -1.373531 64 -17.01158k .007989 -3.665184 - .007181 .003502 .561415 -4.418568 -3.972009 65 -7.230521 - .078036 -I .670240 .070725 - .032979 .500689 -6 .168851 -5.584897 66 k.292330 - . l 4 2 5 0 6 .724083 .130186 - .057962 .314407 -6.353113 -5.793264 67 14.646785 - .169905 2.908814 .156398 - .066387 .050414 -4.936996 -4.532305 68 21.238055 - .154257 4-334610 .L4302k - .057786 - .224301 -2.288151 -2.110820 69 22 .k35 l71 - .100384 4-649383 .093717 - .035975 - .440552 .918848 .864535 70 17.973357 - .022488 3.784870 .021131 - .007691 - .544337 3.873785 3.641611 71 9.011372 .059555 1.970950 - .056311 .019389 - .510279 5.834944 5.513379 72 -2.168307 .125215 - .325170 - .119087 .038694 - .347817 6.314797 5.998366 73 -12.739175 .158401 -2.518893 - .151480 .046312 - .098612 5.200878 4.964764 74 -20.043449 .151367 -4.056616 - .145504 .041722 .174099 2.781946 2.666155 75 -22.258618 .106494 -4.555005 - .102865 .027562 .401510 - .326613 - .320808 76 -18.851029 .035542 -3.896241 - .034487 .008598 .526561 -3.338923 -3.241410 82 TABLE III . 7.b. (Continued) h = 1/500 Case: II 0 n0 n 9 Degrees t z S 77 -10.703903 -.01+3398 -2.255635 . oi+2286 -.009762 .518223 -5.I+968IO -5.35I+O23 78 .110830 -. 1101+84 - .055205 .108070 -.022971 .379169 -6.260161 -6.118865 79 10.858111 -.11+9069 2.11+1+571 .11+6331 -.028)|)j)| .11+1+969 -5.1+1+1797 -5.336229 80 18.829788 -.H+9801 3.787117 .11+7525 -.026013 - . 121+991+ -3 .253231 -3.198683 81 22.0251+26 - .112866 1+.1+60210 .1111+76 - .017656 -.362538 - .250177 -.21+3609 82 19.653125 -.01+7853 3.99911+6 .01+7368 - .006660 - .507863 2.808218 2.782207 83 12.325270 .02871I+ 2.526888 - .028500 .0031+99 -.521+573 5.150830 5.HI7I+I 8)4 1.900920 .097551 .1+211+28 - .097016 .010197 -.1+08766 6.188537 6.152607 85 -8.98I+723 .11+11+31 -1.780992 -.11+0893 .012327 - .189939 5.662039 5.638036 86 -17.581+91+2 . H+9I+66 -3 .522125 -.11+9102 .0101+26 .076505 3-706500 3.695390 87 -21.733008 .119792 -1+. 362866 -.119628 .006269 .323251+ .81691+7 .811+575 88 -20.386817 .059991 -1+. 093725 - .059955 .002091+ .1+88055 -2.276761 -2.275792 89 -13.890120 -.011+828 -2.786896 .011+826 - .000259 .529378 -1+. 793861+ -1+.793082 90 -3.881+951+ -.085826 -.776991 .085826 - .000000 .1+36866 -6.099302 -6.099302 CHAPTER IV COMPARISON OP THE NUMERICAL METHOD WITH AN APPROXIMATE SOLUTION The approximate solutions of the d i f f e r e n t i a l equations are given i n many published papers. In t h i s chapter we w i l l define Tlmoshenko's solution (Reference 1, p. k69) In terms of the dimensionless forms and compare the numerical r e s u l t s with t h i s approximate solution. The two d i f f e r e n t i a l equations of dimensionless form are as follows: U + Cot0 U - (Cot 20 - V) U = ^ — ^ Z (k.1.1.) h 2 12 Z + Cotj2f Z - (Cot 0 +\>) Z = U (4.2.1.) h For the large values of 0 and a very thin s h e l l , U and Z functions are damped out very f a s t from the edge of s h e l l to the crown. The functions U and Z and t h e i r f i r s t derivatives 89 90 are comparatively much smaller than the second derivatives besides f o r large 0 value of cot0 goes to zero. Therefore as an approximation we can neglect them i n the l e f t side of our equations and obtain the following r e l a t i o n s . l - v > 2 TJ = Z (k . 1 . 2 . ) 12 Z = U (k . 2 . 2 . ) h Eliminating Z from these equations, ' T J* + k 7^ U = 0 (k . 3 . 1 . ) can be obtained, where k 2 1 2 A 4 = 3 ( l - V ^ ) ( — ) (a) h The general solution of t h i s equation i s Atf A0 - A0 - A0 . U = C^e ^ Cos A0+C2e SlnAfi+G^e Cos A0+c^e FS±n A0 (k.k.l.) Since at the crown there i s no hole and the function should go to zero, C^ and C^ c o e f f i c i e n t s must be zero. Therefore the solution w i l l be reduced to TJ = E A ^ {c1 Cos A0 + C 2 Sin A0) (k.k . 2 . ) On the other hand the r e l a t i o n between U and u was given i n the l a s t chapter as being u = e" ^  U (b) therefore the solution f o r u w i l l be u - CO S A0 + C 2 Sin A0 (k.5.1.) Considering Eq. (k.2.2.) solution f o r z can be written. ( - C n S i n A 0 + C 5 CosA0) ( k . 6 . 1 . ) ' 1 - V 2 1 t L The r e l a t i o n s between u, z and the stress functions are given i n the Table I I I . l . We w i l l compare the two ways by taking the example which Is shown on P i g . IV.1 . At the 0= , the angle at the support, only the hor i z o n t a l force exists and has a value of unity. In the two methods the c o e f f i c i e n t s 0 ^ , 0 ^ and A, B can be determined by equating t = 1 and m.^ = 0 . These c o e f f i c i e n t s are calculated f o r the three h values, 3 0 , 100 and f>00. In each case the angles at the support are taken as 9 0 ° , 6 0 ° , 3 0 ° . Now having these c o e f f i c i e n t s we can compare our two methods. The comparison w i l l be made in three ways. For the f i r s t comparison & , ho r i z o n t a l displacement and z*, the angle of rotation were calculated f o r both the methods. These r e s u l t s are shown on the Table IV.1 . The tt curves f o r S and z versus cx are drawn on the F i g . IV.2.a. and b. Comparing the two methods on these f i g u r e s , we can ea s i l y conclude that the two methods give more or less the same re s u l t s f o r the sti f f n e s s e s at the small values of F i g . IV. 1. 92 TABLE: IV. 1. o< Approximate method Numerical method C l c 2 z" A B 5 Z 30 90 - 8 4 7 5 . 5 2 4 . 3 -3.55 l . k l 1 1 . 9 11+.3 - 3 . 5 5 80 2.31+ 9 . 6 k 13.8 - 3 4 7 1 1 . 5 2.14 1 3 . 7 - 3 4 7 60 3 . 3 0 - 8 . 0 0 lO.k - 3 . 0 2 - 6 . 3 1 - 7 - 3 9 1 0 . 5 -3.17 30 2 . 4 9 . 6 3 6.82 - 3 . 3 8 1+.17 1 .27 3 . 3 6 - 1 . 8 7 100 90 1 3 . 5 - 1 2 . 5 2 7 . 0 - 3 . 7 9 -1+.87 -4-7 26 .1 - 3 . 5 0 60 1+.7 - 1 5 . 2 1 9 4 - 3 . o 5 -9.1+3 - 8 . 2 5 1 9 4 - 3 . 1 1 30 - 2 . 3 3 - 8 . 8 0 6 . 3 5 - 1 . 7 3 - 5 4 6 - 1 . 1 0 6 . 3 0 - 1 . 8 3 5oo 90 3 2 . 7 - 2 5 . 1 5 8 . 2 -3.51+ - 1 . 6 7 -9.26 5 8 . 2 - 3 . 5 o 60 - 3 5 . 3 1+.71 1+3.3 - 3 . 0 k - 2 . 1 1 7 - 3 0 k 2 . 6 - 3 . 0 9 30 1 9 4 6 . k 8 4 4 -1.71+ 2.28 -2.1+7 4 4 - 1 . 8 0 91+ s h e l l thickness, but f o r the thick s h e l l s two methods d i f f e r quite appreciably at the small values of the support angle The second comparison may be based on the wave lengths f o r the two methods. The wave lengths from numerical method were measured from the Pigs. III.3 .1+.5. and found 2 5 ° , ll+°, 6° approximately, f o r l / h = 30, 100, $00 res-p e c t i v e l y . On the other hand, the wave lengths f o r the approximate method were determined by equating Eq. (l+.jp.l.) to zero. zuaue Length too zoo 300 wo 500 P i g . IV. 3. C1 Cox A0 + C Sin = 0 from this equation It can be concluded that tan A0 (c) 95 therefore A0 = TT Prom t h i s r e l a t i o n , the wave lengths f o r the d i f f e r e n t h TT values can be determined as the r a t i o — . The r e s u l t s are A 2 5 . 2 , 13.8 and 6 .2 degrees f o r ~ = 30 , 100, 500 r e s p e c t i v e l y . Prom the F i g . IV.3. the curve of the wave length vs. the values of h, can be observed. It i s obvious that the two methods give almost the same wave lengths f o r each case. As the th i r d comparison curves were drawn f o r the two ways. F i v . IV.k. These computations were based on t = 1 and m . = 0 at the support o< = 6 o ° . The tables which contain the values of ny's f o r the d i f f e r e n t points are given as Table: IV.2. From here, again i t can be e a s i l y seen that the values f o r the two methods are close to each other i n the case of the t h i n s h e l l . But f o r the thick s h e l l , the difference between the two ways i s out of the tolerance l i m i t s . At the peak points the percentage of error gets larger from the th i n shells to the thick ones. For instant, 1.6$ f o r the case of h and about 3 .0$ f o r h = 1 From the above three comparisons we can draw a general conclusion. The Pigs. Iv.k.a. and b. show that the percent-age of error i n approximate method gets larger f o r smaller support angle and thicker s h e l l . This conclusion i s shown i n the error curves of Pig. IV. $. Therefore i n the case of s h e l l which plots i n Zone - II the approximate method can be used with an error of not more than 6$ . As we concluded i n TABLE: IV.2.a. Appx. Numr. 0 Appx. Numr. Dgr * y m0 Dgr. m0 1 60 0.00 0.00 1 60 0.00 0.00 h = h = 35 .5 30 56 21.0 21 .3 100 58 36 .1 A = 7.135 A * 13.03 56 63.8 64.3 52 36.8 38 .2 cx = 60° 60° C ,=-3.98 48 43 .8 46.7 4.70 54 78.4 80.8 cl= 52 81.0 44 40 .4 44 .2 JL 77.2 C P= 9.67 c2=--15.2 50 64.8 ko 26 .8 30.7 -9 .43 60 .2 A =-6.31 A = 48 34.7 36 7.7 8.73 30.9 B =-7.39 B = - 8 .25 -4 .68 - 3 . 2 8 32 - 14 .9 46 Degr. Appx. m0 Numr. 1 60 0.00 0.00 h = 500 59 86.0 87.5 A = 29 .13 153. 58 151. c* - 60° 57 178. 181. C = -35 .3 1 56 160. 163. c2= 4 .71 55 104. 101. A = -2 .11 54 17.3 16.9 B = 7.30 -75.2 53 -71 .7 97 TABLE: IV.2.b. 0 Apprx. Numr. 0 Apprx. Nurar. gree - m . 0 Degree y y 1 30 0.00 0.00 h - 100 30 0.00 0.00 26 24.2 12.3 A = 13.03 28 20.3 20.6 22 43 .0 22.9 « = 30° 26 36.6 38 .2 18 51.4 29 .1 -2 .33 24 45 .3 49 -2 c l = 14 47 .0 28 .6 22 44 .9 50 .8 c 2= -8 .80 10 30.3 20 .0 20 35.0 42 .1 8 18 .1 12.5 A = -5 .46 18 18.3 24 .1 B = -1 .10 16 - 2 . 6 5 -0 .33 h = A = 30 7.135 o « = 30 C-p 2.14 C 2= 9.63 A = 4 - 1 7 B = 1.27 0 Degree Apprx. 0 Numr. m / 0 1 30 0.00 0.00 h = 500 29 50.0 50.8 A = 29 .13 ri 28 87.3 90.2 CX SS 30° 27 103. 108. c l = 19.4 98.3 JL 26 92.1 6.48 d 25 53.9 63 .4 A = 2.28 24 9.46 10.9 B = -2 .47 -46 .3 23 -41 .7 Ch. I I , the Bessel solution was accurate enough f o r a s h e l l which i s located on Zone - I. On the other hand numerical method can be applied over entire surface. That l s to say, numerical method i s good f o r any thickness and support angle 500 400 300 200 100 0 30° 60° 90 Pig. IV.5 . CHAPTER V NUMERICAL EXAMPLE For a numerical example we w i l l consider a spherical s h e l l of 125 feet radius and 3 0 degrees support angle as shown on F i g . V.I. The loading consists of the dead load and v e r t i c a l l i v e load, kO psf and 20 psf respectively. The r a t i o of the thickness to radius i s 3 i n c h . / l 2 £ (12) = l/^OO. The edge i s considered to be on r o l l e r s so that only a v e r t i c a l reaction i s provided. The results of membrane analysis can be obtained e a s i l y from the s t a t i c s and are shown as follows: w-LL 1- COS0 ^ = - a ( 2 + w. DL sin 0 (k . 1 . 1 . ) l0 Ne (4 .2 .1 . ) a a therefore N Q = - (P + a (4.2.2.) a 101 where w = l i v e load, 20 psf. of horizontal plan. I i i i w DL dead load, kO psf. of s h e l l . Substituting these values into the formulae, N^ and N Q can be W = 40 psf PI . •. cy found. At the periphery, °< = 30 . = -3930 l b / f t . N e = -2280 l b / f t . On the other hand the h o r i -zontal force, f can be evaluated as the h o r i z o n t a l projection of N^. Therefore, T* = N(X coscx = -3k00 l b / f t . Since there i s only the v e r t i c a l reaction around the periphery, then T and must be equal to zero at the sup ports. In order to have these conditions, T", caused by \ r~y V P i g . V . l . membrane forces, should be eliminated by an opposite force of the same magnitude. Knowing the boundary conditions at the <x - 30° as T = - T* O T n and err 3400 v-Memb. 3950 (1970 3liOO = 0 we can e a s i l y obtain the c o e f f i c i e n t s A and B f o r the num-e r i c a l method from the following r e l a t i o n s : Bend. A t 1 + B t 2 = t (k . 3 . 1 . ) J970 Real Siipf. condition A m^x + B m^2 = 0 (k.k.l.) where t ^ , t 2 , ny^ rry 2 are taken from the Tables I I I . 7 . a and b. P i g . V . 2 . In these r e l a t i o n s , t i s evaluated by means of the known r e l a t i o n s : i - >-2 T = T E a T = t Q A A therefore, t = T 1- V 2 E a e -A^A Since ¥ = - (-31+00) l b / f t . , t w i l l be, t = 0.21k0 — 1 — 10 6 E Substituting the values which are taken from Tables III.7.a, into Eqs. (1+.3.1.) (k.i+.l.), A (-0.0205) + B (-0.k25) + 0.2lk0 —K— 10° E A (22.56) + B (20.83) = 0 and solving A and B from the above equations, the following cam be obtained: 1 A = 0.1+86 -B = -0.527 10 6 E 1 10 6 E Having A and B the stress functions and deformations can be calculated e a s i l y . m0 = A m01 + B m02 nig = A I H Q^ + B iriQ2 10k 0 A n ^ 01 + B > n e = A n e l + B n 9 2 A n tyl + B qj2f2 z = A z ^  + B Z 2 5 = A S 1 + B § 2 A0 Multiplying the above values by (e A ), M^MQN^Q^ Z A can be obtained. To transform to the values with dimension, these dimensionless quantities should be multiplied by the following f a c t o r s : hE a 2 M = M = M 2 .71 E f o r moments. 12(l-i>2) a E N = N jr- = N 130.2 E f o r forces. 1- V 2 Z = Z — i - = Z 2 £ . 1 0 k — a ) A = A = A 6 . 5 l 1 0 4 h ( l - i > 2 ) P i n a l values which are sum of membrane and bending solutions are computed and shown on Pig.V . 3 . F i g . V . 3 R E F E R E N C E S 1. TIMOSHENKO: Theory of Plates and Sh e l l s . McGraw-Hill Book Co. Inc. 1940 2. McLACHLAN, N.W.: Bessel Functions for Engineers. Oxford at the Clarendon Press, 1934 3. JAHNKE, E. & EMDE, F.: Tables of Functions. Dover Publications, New York, 1945. k. FLUGGE, W. Four Place Tables of Transcendental Functions. McGraw-Hill Book Co. Inc. 1954 5. KARMAN & BIOT: Mathematical Methods i n Engineering. McGraw-Hill Book Co. Inc. 1 9 4 ° 6. MILNE, W.E.: Numerical Solutions of D i f f e r e n t i a l Equations. John Wiley & Sons Inc. New York, 1953 7. SCHELKTJNOFF, S.A.: Applied Mathematics f o r Engineers and S c i e n t i s t s . D. Van Nostrand Co.Inc. 1948 8. LOGISTICS RESEARCH INC.: Description of operations Alwac III-E E l e c t r o n i c D i g i t a l Computer Log i s t i c s Research Inc. C a l i f o r n i a . 9. BEEK, A.: Elementary Coding Manual Alwac III-E. L o g i s t i c s Research Inc. C a l i f o r n i a . 

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