Discontinuity stresses at the junction of a pressurised spherical shell and a cylinder

CoA Note No.80 lECHNIS'- ^--^- "'-^^

VLIEC

ï'.anaodïtmat 10 - DEli f

THE COLLEGE OF AERONAUTICS

GRANFIELD

DISCONTINUITY STRESSES AT THE JUNCTION OF A

PRESSURISED SPHERICAL SHELL AND A CYLINDER

by

NOTE NO. 80. J a n u a r y , A95&' T H E C O I, L E G E O P A E R O N A U T I C S C R A N P I E L U A n o t e on t h e d i s c o n t i n u i t y s t r e s s e s a t t h e j u n c t i o n of a p r e s s u r i s e d s p h e r i c a l s h e l l and a c y l i n d e r b y -D.S.Houghton, M . S c , ( E n g ) . , A . F . R . A e . S . A.M.I.Mech.E. and A.S.L.Chan, 13.So., M.Sc. ( E n g ) . , D . I . C . , A . P . R . A ü . S . SUM',tARY

An analysis has been made of the forces and moments occurring at the junction of a pressurised spherical shell with an intersecting cylinder. The additional effects of having a temperature gradient along the

length of the cylinder and the effect of a jointing ring have been considered.

corn^ws

Pa^e

1, Introduction

2. Oase 1 — Cylinder joining on the cutside

• of a sphere. , 1 3. Case 2 - Cylinder extending into the sphere. 5

4. Case 3 - Cylinder joining on the outside of a sphere, with a temperature change in the sphere and a temperature

gradient along the oylindero 7 5. Case 4 - Cylinder extending into the sphere,

v/ith a temperature change in both the sphere and the portion of

cylinder outside, 13 6. Case 5 ** Effect of a heavy ring at the joint. 15

1. Introduction

This note is a study of the discontinuity stresses which arise when a press^arised spherical shell is constrained by a hollow cylinder whose axis passes through the centre of the sphere, (Pig. 1). The wall of the sphere is considered to be thin - less tlian l/lOth of the radius.

When a hollow thin v/alled sphere is under pressure, its shell is in a state of pure membrame stress v/hich is uniform throughout, and is in ooniplete equilibrium with the apjjlied pressure. There is a uniform expansion of the wall,

The presence of the cylinder will prevent free e3cpansion of the sphere at and near the joint, inducing discontinuity stresses both in the sphere and in the cylinder. Since there is no fe:ctra external force applied, the force system in the sphere thus prodaced must be self balancing, without any resultant force whatever. The same is also true with the cylinder. Therefore, by St. Venant's Principle, the effect of the constraint must be entirely local, and the discontinuity stresses diminish at small distances away from the structural discontinuity.

In the f«"llowing analysis, five cases have been considered. (See Contents). In all the cases the joint between the cylinder and the sphere is assumed to be rigid. That is, the angle between the tangent to the sphere and the wall of the cylinder remains unchanged after deformation,

It is further assumed that the sphere and the cylinder, and later on the jointing ring, are of the same material thus having the same physical constants. It will only be a very sirnple matter to extend the analysis to allow for the use of different materials. The physical properties are also ar-sumed to be imaltered through the range of

temperature concerned. 2. Case 1.

Cylinder joining on the outside of a. sphere

Vflien a sphere of radius R and skin thickness t is subjected to a pressure p^the radius is increased by the amount

-2-p cjii..

where

n

is the Poisson's Ratio

and E is the Young's Modulus of ElasLicitj'-.

Now, if a cylinder of radius r is joined en to the sphere,

the expansion at the joint normal to the axit; of the cylinder, had

the cylinder offered no resistance, would be equal to (see Pig,2).

5 = (AR) sina,

P P '

and since sin a = r A ,

6 = prR /. \

f . \

However, the presence of the cylinder will restrict the

movement, and the deformation at the joint will finally take the

shape as shown in Pig.2.

If the sphere is considered to be cut along the joint and

free from the cylinder, (Pig. 5 ) , there will be a distributed moment

m, and a distributed load q normal to. the axis of the cylinder, around

the edge of the cut on each of the three free bodies.

The force system on each free body is therefore self-equilibrating.

There can be no load in the direction of the axis of the cylinder

(aue to the restraint of the joint) because there is no extra

externally applied load to balance it- Hence the forces must die

away hyperbolically ai the distance from the joint is increased, and

the effect of the disturbance is therefore purely local.

The relation between the deformation and the forces

described above are now examined.

(a) Por the cylinder (subscript o ) .

The deflection 6 and rotation Ö at the end of the

cylinder due to moment m and force q (for sign convention see

Pig.3) are given in Ref.1 (pp. 392-393) as

3

-and e = - — - ( ^ m + q ),, (2b) 2i3 D ° where and 3(1-^'!) " 2 . 2 ' E t ^ o ' 1 2 ( 1 - / ) . wl.ere

(3)

t = thickness of the cylinder.

(b) Por the portion of spherical shell outside the

circujXiference of the cylinder (subscript 1).

Ref.1 (pp.4-70-471) gives the deflection and rotation

at the edge of the shell as

8 _ ^^ËHL^ZLiii n -

^!^n(2J±l ^

, (Z^)

1 - Et ^ Et 1 '

^^'

, n 4"^^ 2^^sin(7r-a)

,,^.

and ^ = M t "^1 it ^1 ' ^^^-^

X'^= 3(1-M') (|)2 , (5)

(c) Similarly for the portion of spherical shell inside

the circum.ference of the cylinder (subscript 2 ) .

Q 2 I. 2^Rsin~oc 2 \ s i n a /^ \ §2 = Et ^^2 + - E T - ^'2 > ^^'"^^ ;, fi 4^^^ 2"^^sin'^ /^. X ^ ^ ®2 = M t ^ 2 ^ ~ ^ t — ^12 • ^^^^ Now, assuming t h a t t n e j o i n t i s r i g i d , t h e n t h e r o t a t i o n s i n a l l t h e t h r e e p o r t i o n s must be i d e n t i c a l , vAiich g i v e s , 0 - 6 _ b . o - 1 ~ 2 * Hence: l^h'' 2k s i n g 1 e^y. „ \ rs (-i\

I 2 3 and 2 \ s i n a / \ 4'^ / \ r^ /ON Po.r c c n t i n u - i t y t h e d e f l e c t i o n of t h e two p o r t i o n s of t h e s p h e r e must a l s o be t h e same, t h a t i s 6 = 6 ' ^ 2

Hence

2 2

2\Rsiii

a / \ 2X

s:ina / \ ^

/n\

Prom Pig„2, the deflection of the sphere and that of the

cylinder must add up to the deflection 6 of Eq.1. Alternatively

locking at it another way, the total inward deflection of the sphere,

which is equal to 6 - 6 , must be the same as that of the cylinder,

which is 6 .

o

Hence 5^ - 6 = 5 .

1 o p

o \ü

and 27\sin a 2>.3ina 1 //^ \

— E T — '^^ - -Ë™- "^^1 - ~ 3^ (^o + %^

= f f ( i - . ) . (10)

And f i n a l l y , for e^;ailibrium at the j o i n t ,

and m + m + n^ = 0. (l2)

The 6 unknown quantities q , q^^, q^, m , m , and m^

can now be found by solving the 6 equations 7 to 12 simultaneously.

The solution of the equations are:

M M, M„ 1

5 -•(röiere a n d

N= a \ l + ^ ) + V^a^-bd (1+1^),

2 M = 8y9a d. e , M = a ( a ^ + hp\di - 4 9 a d ) e = - '^^ - 1 Q "p p.T ° > M, ,= - a ( a + kPiid. + 4vS'ad)( Q^= - 2 c ( a ^ + 4 3 ^ d ) e , 2 2 Q = c ( a + 2j/9 "db - ^ a d ) e M a ^ 2 2c ' Q c

- Jo - -,

2 •4a o ' Q,= c ( a + 4[?%d + ^ a d ) e = - ^o ^o V— + -j— M^ 2 4a o ; *

x"^

a b c d

3 Ü V ) .

" r ^ t 2 0 = 3(1-M^) 2X2. - R ' 2 2-Ar - p^ 4X3 - R '

~-p\ 5

j ( | ) ' y -2 >

t

e = ^ ( l - u ) , (Note t h a t bc = 2,?). \

ï

•^ )

j

(13a) ( 1 4 ) 3 . Case 2 . C y l i n d e r e x t e n d i n g i n t o t h e s p h e r e . I f t h e c y l i r i d e r i s e x t e n d e d i n t o t h e s p h e r e ( P i g ^ ) , and p r o v i d i n g t h a t t h e l e n g t h i n s i d e i s s u f f i c i e n t l y l o n g , t h e n i t can b e t r e a t e d i n t h e same manner as d i s c u s s e d above. The c y l i n d e r c a n a l s o be c o n s i d e r e d a s c u t a l o n g t h e j o i n t i n t o two p o r t i o n s . Por t h e p o r t i o n of c y l i n d e r i n s i d e t h e s p h e r e ( s u b s c r i p t 3 ) , t h e r e l a t i o n

6

-TECI-LNISCHE HOG£SCMO<^' VUEGTUiG50UWKUKi.-.t Konaalstraat 10 - nri.i f

V.

between the deflection and rotation at the edge of the cut in terras cf the distributed force and moment, with the sign convention the same as shown in Pig.3. can be written as

and 2/3^ ( - / ^ 3 + ^^)f (2/9m^ - q ^ ) . (I5a) (I5h)

Por continuity of the cylinder, 5 = 5^ and 0 = 6, .

o 3 o 3 Hence from equations 2 and 15,

/9(m^ + m^) + (q^ - q ^ = 0, and ?v3(m^ - m,) + (q_^ + q^) = 0.

T,'ith the addition oi' m, and q, at the joint the equations 3 3

of equilibrium 11 and 12 become:

qo + 0, + q2 + «13 = 0 »

and m + m, + m„ + rn,

o 1 2 3

0

The ar-guments from r-tiich equations 7,8,9 and 10 are

derived still apply to this case. Hence using these four equations, together with equations 16, 17, 13 and 19, the eight unlmown forces and moments at the joint can be obtained.

The solution to the set of equations is:

where ce

"% = ^ = ^ = -^3 = T '

ae m^

=

-m^

= y ,

ce o 3 2/5'5' ' 2 dc 2 dc $ = b c - a + — = ^ + 2 ~

1

1-The coefficients a, b , c, d, e, and/? are the same as given in equation 14. (16) (17) (18) (•<9) (20)

~7-4. Case 3»

Cylinder joining on the outside of a sphere with a temperature change in the sphere and a temperature gradient in the cylinder.

C'^nsider the geometry to be the same as that of Case 1, and let the initial temperature of the Tirhole system be T . The sphere is then heated (or cooled) to a temperature T . By conduction, the temperature at the end of the cylinder joining on the sphere is also T , v.hilst at the far end the temperature remains at T .

4a. It is now convenient to detach the cylinder from the sphere at the joint (Pig.5) and consider the effect of the temperature on each item,

(a) Por the sphere.

Let the change of temperature T - T = AT.

The increase in radius of the sphere due to cnis temperature change is

(AR)^ = y R, AT ,

where y = coefficient of expansion.

The component of expansion at the joint normal to the axis of the cylinder is

S. = ( A R ) ^ sina = yr. AT, (2l) (b) For the cylinder

Por convenience, the temperature gradient along the length of the cylinder is here assumed to be linear^ In fact it may be taken to be any other function without affecting the following reasoning,

The change of tenperature at any point x from the far end of the cylinder is then given by

A T ( x ) . ^

-Vi-^^'^'J

•

Using the same argument as given in Ref.1 (pp, 2^23-425), we can now assess the deformation of the isolated cylinder due to this change of temperature.

-8-Ccnsider a srra.ll ring element dx separated from the I'est of the cylinder. The expansi.on due to the change of temperature AT(JC)

is equal to Vr, Aï(x). It produces no stress in the ring.

Imagine now an external pressure p' which is applied on the ring element to restore it to its original diameter. The contraction due to p' must be equal to the expansion due to A T ( X ) ,

o

Hence p'r*^ = Vr. A T ( X ) , Et

from which p' = ^ ^ AT(x).

The pressure p' produces a hoop stress in the ring,

In fact, there is no actual applied pressure on the cylinder, Therefore a pressure -p' must be apjilied to the now un-distorted

cylinder. This latter pressure produces longitudinal bending stress along the cylinder axis as well as hoop stress. The hoop stresses of the two pressure systems of course cancel cash other, leaving the longitudinal bending stress produced by pressure -p' on the straight cylinder as the sole equivalent of the temperature gradient effect, This stress should be included in the final stress system in the cylinder.

The stresses produced by -p' can be obtained if the deflection curve along t.he cylinder is known, and the skin deflection w (positive inwards towards the axis) is given by the differential equation (Ref,1, p,392, eq.230. See also p.424, eq,h.)

— ^ + 4/9 w = - ^ . dx

()5and D are t h e same a s given i n e q . 3 ) .

We a r e here i n t e r e s t e d not so m.iK5h i n the bending s t r e s s as the d e f l e c t i o n and r o t a t i o n a t the j o i n i n g end ( t h a t i s v/hen x =•& ) of the detached c y l i n d e r due t o the temperature g r a d i e n t . These can be obtained e a s i l y by solving t h e above equation Vvlth t h e appropriate end c o n d i t i o n s and s u b s t i t u t i n g i n t o t h e s o l u t i o n the end value.

9

-After s u b s t i t u t i n g for - p ' , the equation above becomes

4 , i ^ 4 „ , . ^ AT{x).

The general s o l u t i o n of t h i s equation, assuming T(x) t o be l i n e a r i s :

w = c cosl^xcog5x + c cosh/^xsir^x + o-sinh^xcos/5x

+ c, sinlV^xsin^x ~ c ' x , (22) 4

where o ' a —ï— • T~ = "y~ » AT ,

The unknown constants c. .,. c, will depend on the

condition at the far end of the cylinder (x=o) where the temperature is kept constant at T .

(i) If the cylinder is free at both ends 2 3 then d w _ , d w „

— ^ = C and — ^ = 0 dx dx when X = 0 or X = •&.

These conditions give

°1 = °2 = °3 = °4 = °' and eq, 22 becomes simply

w = -yr, AT. ;^ . (22a) This infers a linear deflection with a rate proportional

to AT,

The deflection and rotation at the ;^cint (x a^) are given by:

w^ = -yr. AT, - (23a)

and e^ = ( g ) = - y , AT. | . (25b)

Note that for this case, -w^ = 6 (see eq, 2l), i.e;-both the cylinder and sphere are displaced by the same amount in the

-10-direction normal to the axis of the cylinder. It can also be seen that there is no longitudinal bending stress anyv.liere on the cylinder due to this temperature gradient.

(ii) If the far end is fixed, then the boundary conditions are dw at x = 0: w = 0 and -r- = 0, dx ' • . 2 and a t x = - & : d w ^ j , 3 o —=p = 0 and d w = 0. dx , 3 dx follows:

These c«»nditions give the values of the constants as

C | = 0 , cosh P^ c„ = c COS P^ 4. cosh P^ 2r. COS -/9e c , = c COS /5^ + c o s h ^^ sin/9.foos/:^.e - sinh/Sfcosh/g'g-a n d Cj = c p „ > "•• cos pi + cosh I3i.

where e = 7^ = - ^ , AT , (24)

/5 (3&

P u t t i n g these c o n s t a n t s into e q , 2 2 , t h e deflection a n d rotation at t h e joint x =•& a r e given by:

•.,' , - cpsh/9esir#6 + s±nbj3l^osj3l' s «'or:„^

w ^ a «o(/3-e '^ ^—-—2""^ ) ' (25a;

cos /9-6 + cosh /?•&

.. 6^ , (^) . ... (oosh^e - cos^^)^ (25b)

x=-£ cos z?^ + cosh /36

4B, The above calculation for the deflections and rotations of the sphere ajid cylinder due to temperatiire variation assumes that the

joint is detached and that no moment or force exist at the joint. The fact that the cylinder and sphere are actually attached will induce forces and moments on the edges of the cylinder and the two portions of sphere

-11-at the joint. The magnitudes of these forces are such th-11-at the deformations of all the parts at the joint are consistent. The treatment of the problem is then exactly the same as that of Case 1,

The relations given in eqs, 2, 4 and 6 still hold, and if the joint is assumed rigid, the deflection and rotation of the tv/o portions cf sphere cut at the joint must be the same. Hence eqs. 8 and 9, which give 6 = 6 and 6 = Ö must still apply.

If a pressure p is also applied in the sphere the sum of the deflection of the sphere in the direction normal to the axis of the cylinder is ö + 6, as given in eqs, 1 and 21. Adding to this

P t

the deflection of the cylinder (see Pig,5), the total 'gap' developed over the detached joint is (6 + 6 + w ^ ) .

Therefore, the right hand side of eq, 10 iiistead of being 6 , now becomes (6 + 8 + -57^) and the equation reads

6 - 5 = 5 + b + Tffr, ,

1 o p ^ t ^

T h i s means t h a t t h e t o t a l inwai'd d e f l e c t i o n of t h e s p h e r e 6 - 6 ~ ^+ Kiust be t h e same as t h a t o-^ t h e c y l i n d e r " + w^,

2 2

„ 27\Rsin a 2X s i n a 1 / o >

^^^°^

Et

"h

-

- Ë t — "^1 - ^ (^% -^ ^o^

Similarly, since we assume that the joint is rigid, the rotation of the sphere 6 must be equal to the total rotation of the cylinder 6 + 6, (see also Pig,6).

Hence 6^ - 9 = 6^ ,

1 o •t^ '

or 4X^ 2x2sin(7r-a) i . N ^

mt "^1 Ët — ^ ^ -717 (^^ -^

%^ = h '

2/3 D (27) w^ and 6^ in eqs, 26 and 27 are given from either

eq. 23 or eq.25, according to the condition at the far end of the cylinder.

-12-The conditions of equilibrium at the joint as given by eq.11 and 12 are still true. Therefore, we have new a set of six equations:

r < and c(m2 - m^) -I- a(q^ + qg) = 0, a(m^ + m2) + •^(qg - q^) = 0, q^ + q^ + q2 = 0, "^o •*• '"l •*• "^2 "" *^' bq. - am - d(/9m -j- q ) = e + e' L cm^ - aq^ -^d(a5m^ + q^) = f.

where a, b , c, d, and /9 are the same as given in eq,14,

e ' ^ + ( y r , A T + w ^ E t , 1

f =

e^. Et.

J

(8) (9) (11) (12) (26) (27) (28) I t can be seen t h a t t h e f i r s t f o u r e q u a t i o n s a r e i d e n t i c a l w i t h t h o s e u s e d i n Case ^ , Eq. 26 comes from e q , l O , b u t w i t h e . r e p l ö s é d

b y e -»- e ' ; and e q , 2 7 d i f f e r s from e q , 7 o n l y i n t h e r i g h t hand s i d e where i t i s now e q u a l t o f i n s t e a d of zero„ YiTien t h e r e i s no t h e r m a l

e f f e c t , e ' = Oand f = 0, t h i s s e t of e q u a t i o n s i s t h e n r e d u c e d i d e n t i c a l l y t o t h a t of Case 1, The s o l u t i o n of t h e e q u a t i o n s i s ; m^ M +M ' m = o o o N M,+M,'

m^ = r 1

N M2+M2' ~ N (25) 0 Q ' ° N ' ^ N ' ^ N J

where N, M , M , M Q , Q., Qp, are given in eq.l3a with the important difference that e is replaced by e + e' in the expressions. The other quantities M ' ... Q ' are:

-13-M ' = -2b(a + 2cd)f,

M ' = b(a^ + 2od - 2Aid)f = - V - ^ ,

^ " 2 4a ^o *

M • = b(a + 2cd + 2/9ad)f = - 2 2 _ __ n

^

2 ^ 4a ^ ' ,

Q^' = 8/9a^df,

-^N

2

2b o

^ • = ri 4^a^d + a(a^ + 2cd)~| f = - ^

„• = f- 49a^d - a(a^ -H 2cd) 1 f ^ - ^ + L M^'

> (29a)

^

5. Case 4,

Cylinder extending into the sphere, with a temperature

change in both the sphere and the portion ^f cylinder

inside it, and a temperature gradient in tne portion

of cylinder outside.

The geometry of this case i3 the same as that of Case 2,

and the same thermal changes as that of Case 3» The change of

temperature from T to T , applied tilso to the portion of cylinder

inside the sphere. The arguments that lead to equations 8, 9, 18

and 19 for Case 2 are still valid. By a similar consideration as

that employed in Case 3 from which yields eqs, 26 and 27 (which still

hold true), we obtain the relations between the deflection and

rotation of the two portions of cylinders at the joint.

w^

The s e t of e q u a t i o n s f o r t h i s c a s e it> t h e r e f o r e f c(ra2 - m^) + a (q^ + q2) = 0 a(m^ + mg) + b ( q 2 - q^) = 0 q^ + q^ + qg + q3 = 0 m + m^ + m^ + m, - 0 0 1 2 3 1 bq^ - am^ - d^Sm^ •(• q^) = e 4- e ' (31) (8) (9) (18) (19) (26)

cm

-14-.^ - aq^ - /?d(2^m^ + ^o^ = ^ ^^"^^ ^(m^ + m^) +(q^ - q^) = -^'/d (30a)

L 35(m^ -

m^)

+

(q^

+

q^)

=

- /Pd.

(31a)

where e' and f are given in eq.28 and the other coefficients in eq,14.

The left hand side of the equations is of course identical with that of Case 2, but e' and f are newly introduced into the right hand side as a result of the temperature change,

The solutions of the equations are:

m = o m^ = ^ 2 =

"3 =

^10 = ^ = ^ = ^3 = cu a u 2* au - 2 * cu

4/^

cu cu 2* cu 2* ^ c u ~ 2f bv ' 2 ^ -b v 2^i » b v ' 2?i ' f 4/5^ d bv • f J9bv 20 av 2^ ' av 2<^ ' /9bv ^ 2 9 ^ + e ' 2d * e ' 2d ' • " N

J

(32) where u = 2e + e' - /2/9 V ^ /9e' - f VI; 2 d c * = a s- 2-2 2-2 2-2, ji = be - a" + /? bd = a + /!? od ^ (32a)

-15-6, Case 5«

Effect of a heavy ring at the joint

The presence of a ring can be dealt with in exactly the same manner as the preceding cases.

Isolating the ring from the rest of the structure and letting the force and moment on it be q. and m. , inducing radial deflection 6, and rotation Ö, (see Pig,7).

The relation between 5, and q, is easily seen to be 4 4

6, = r^

where A = area of ring cross section, which is assumed constant,

Assuming that q. is applied on the centroid cf the ring and therefore inducing no rotation, the relation between 6, and m, is

4 4 ft 2 ^ 4 = ^ ' 2 \ ' E(I - I ^) X xy ' I y

where I^ = moment of inertia of the ring cross-section about an axis in the plane of the ring centroid, I = moment of inertia of the ring cross-section

about an axis perpendicular to the plane of the ring centroid,

I = product of inertia referring to these tvro axes.

xy ^ & It is now simply a matter of equating these displacements to the rest of the structure at the joint, two extfa equations, most conveniently

S, = ^ . 4 o M

> (when there is no thermal effect).. and Ö , = 6 "^

4 o

a r e derived. Also, extra terms q, and m, must be added t o t h e

4 4

-16-variables q. and m. are added to the original equations, but the two ex±ra equations will enable the new set of equations to be solved.

If the temperature of the ring is also changea from T to T., then the radius of the ring, when isolated, is changed by an amount yr.A T. i.e., there is no relative movement between the ring and the wall of the sphere due to the temperature effect. The deflection and rotation can more conveniently be equated with those of the sphere to give

4 1 p ' and 6, = 6^ ,

4- 1

which together v/ith the original equations will enable all the unknowns to be found.

REFEKBNCE

1, Timoshenko Theory of Plates and Shells. McG-raw Hill Book Co. 1940.

FIG. I CYLINDER JOINING ON TO A SPHERE

PORTION OF SPHERE OUTSIDE T H E CYLINDER (SUBSCRIPT 1)

*m,^e

Q—^..

_*%

PORTION OF SPHERE INSIDE THE CYLINDER

(SUBSCRIPT .2)

FIG. 2 DEFORMATION AT JOINT WHEN THE SPHERE IS UNDER PRESSURE.

FIG. 3 SIGN CONVENTION OF FORCES AND DISPLACEMENTS FIG 4 CYLINDER EXTENDING INTO THE SPHERE. AT THE JOINT.

SPHERE AND CYLINDER AT INITIAL TEMPERATURE T Q

SPHERE

TEMPERATURE T |

TEMPERATURE T o

FIG. 5 EFFECT OF TEMPERATURE ON DETACHED SPHERE AND CYLINDER.

FINAL POSITION OF SPHERE AND CYLINDER

ORIGINAL POSITION OF I SPHERE AND CYLINDER

e, 4 ( - ö o ) : « f POSITION OF CYLINDER

WHEN THE JOINT IS CONSIDERED DETACHED.

FIG. 6 RELATION BETWEEN ROTATIONS OF THE WALLS OF SPHERE AND CYLINDER.

•v*

^ ( - ^ ^ - <

FIG. 7 FORCE AND MOMENT ON A RING AT THE JOINT.