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Differential analyzer solution for the stresses in a rotating bell-shaped shell
DIFFERENTIAL ANALYZER SOLUTION FOR THE STRESSES IN A ROTATING BELL-SHAPED SHELL BY J. L. MERIAM I
ABSTRACT In an earlier paper (1) 3 by the author the theory of axially symmetrical shells was extended to include the body force of rotation about the axis of generation, and the results were applied to the conical shell. In the present paper this theory is used to obtain the solution for a bell-shaped rotating shell whose middle surface is generated by revolving a circular arc about a tangent line. The defining relation is an ordinary linear differential equation of fourth order with variable coefficients. A differential analyzer solution was used and this involved the problem of handling boundary conditions which were divided between both extremities of the range of the independent variable. Comparisons of the results with those of the conical shell and with the membrane theory are made. NOMENCLATURE
T h e following nomenclature is used in the paper: rl h
= constant radius of meridian curvature, = shell thickness, = angle b e t w e e n normal to axis of rotation and normal to shell meridian, 0 = angle measured a b o u t axis of rotation in plane of rotation, p = mass per unit area of shell, ~o = angular velocity of shell, Q = shear force per unit length of shell section, N , = m e m b r a n e force along meridian per unit length of shell section, No = m e m b r a n e force t a n g e n t to a parallel circle per unit length of shell section ("hoop stress"), M~ = bending m o m e n t in plane of meridian per unit length of shell section, Mo = bending m o m e n t in plane perpendicular to meridian per unit length of shell section, R = b o d y force of rotation per unit area of shell, v = displacement c o m p o n e n t along meridian, w = displacement c o m p o n e n t normal to meridian, /" = radial displacement (normal to axis of rotation), e, = strain in direction of meridian, e0 = strain in direction t a n g e n t to a parallel circle, x0 = change in c u r v a t u r e in plane of meridian, xo = change in c u r v a t u r e in plane perpendicular to meridian,
1 Assistant Professor of Engineering Design, University of California, Berkeley, Calif. The boldface numbers in parentheses refer to the references appended to this paper. 115
116
J.L.
E D
MERIAM
[J. F. I.
= Poisson's ratio, = modulus of elasticity, and = flexural rigidity.
Dimensionless quantities :
O'
= O_/ po 2r12
N;
=
No'
= No/po~2rl 2
M/=
Ma/p, r 3
M o ' = Mo/po~2rl 3 ~' = ~/(rl/Eh) h' = h/rl
INTRODUCTION
Solutions for rotors which are symmetrical with respect to a plane of rotation and whose axial dimensions are small compared with the radial
0
FIG. 1. Dimensions of bell-shaped shell.
dimensions are well known. The solution for the rotating conical shell (1) was a departure from this condition of symmetry. T h e analysis of the bell-shaped shell reported in this paper represents a still further departure and approximates the curved profiles of m a n y of the high speed p u m p impellers. Analyses of ring- or torus-shaped shells under static loading are available in the papers of Wissler (2) and Stange (a). T h e shell of the present investigation differs in t h a t it is subject to a
Aug., ~95o.]
ROTATING BELL-SHAPED SHELL
II 7
body force due to rotation and that its surface is defined by a portion of a torus with zero hole. The defining equations for the bell-shaped shell can be obtained from the relations of the earlier paper (1) by appropriate changes in variables and symbols. These substitutions are listed in the Appendix. In the present article the essential equations are cited t o maintain the completeness of the presentation and to provide necessary reference. THEORY
The middle surface of the shell is shown in Fig. 1 and is generated by revolving the circular meridian arc o a of radius,r~ about the tangent a
d
~
!
i
:,.,.~ \ / ,4
+d
V../
0 FIG. 2. Shell element with forces and moments.
(Shell thickness not shown.)
line oo. The angles/31 and /32 define the inner and outer boundaries, respectively, and the thickness h is assumed small compared with rl but large enough to support appreciable bending stresses. Rotation about the axis oo with a constant angular velocity provides the centrifugal loading. An element of the shell with the forces and moments acting on it is shown in Fig. 2. Equating to zero the forces in a direction tangent to the arc oa gives d d-~ (No[1 -- cos/3]) --
No
sin/3 + 0(1 -- cos 13) + poa=rl ~ sin/3(1
-- cos/3)2 = 0.
(1)
Likewise, equilibrium of forces in the direction of the radius rl gives d d--~(O[1 - cos/3J) + No cos/3 - N~(1 - cos/3) - - pw~rl 2 cos/3(1
-- cos/3)= = O.
(2)
J.L.
~I8
MERIAM
[J. F. I.
A third equation is obtained by equating to zero the moments about the upper edge and d d-~ (M~[1 - cos/3]) - M0 sin/3 -- Qrl(1 - cos/3) = 0.
(3)
Equations 1, 2, and 3 completely specify the equilibrium of the shell element. In Fig. 3 the positive directions of the two displacements a
~ 7~" w* dw
0 Fit;. 3.
Components of displacement.
v and w are shown.
The strain in a direction tangent to the arc is
e~ =
and
--w
rl
that
in a
Eh [ l(dv ~-~-w
direction
is
Substituting these expressions
e0 = (w cos/3 -4- v sin/3)/r1(1 - cos/3). into the Hooke's Law equations gives N~ = 1 - ~2Lrl
circumferential
) +
~(wcos/3+vsin/3)]) rl(1 - c o s / 3 )
Eh
[ w cos/3 + vsin B L(dv )] No = 1 - p~---3 rl(1 - c o s / 3 ) +rl ~-~ - w .
(4)
The moments are related to the changes in curvature by the equations Ms = - D(xa + vx0) [ Mo -- D(xo + vXa), !
(5)
where
x~ = X0 =
l d( d~) rl' d~ v + -- - ,-
rl 2
ctn /3 (sec t3 -{- 1) ( v +dw
(6)
Aug., I95o.]
ROTATING BELL-SHAPED SHELL
II9
Equiations 1 through 6 completely define the problem. Their combination is somewhat lengthy (1) but results in the two equations L(U) -
~' U = V E h -at- T [ Y1
L ( V ) -{- L V = rl
(7) U D
where L( ) -
sec/3rl
tan/3d( ) ld2( ) d/32 -[- rl d/3
U = rl(sec/3rl
sec/3 + 1 rl
()
1) Q d-~
and 3+~ 2
ooo2rl2 sin 2fl(sec/3 - 1) 5.
Eliminating first V and then U from Eqs. 7 gives L L ( U ) + u ' U = L ( T ) + r-~ L L ( V) + ~4 V
T D
(8)
where Eh #4-- D
u2 rf
Either of Eqs. 8 m a y be used to solve the problem. The homogeneous or reduced equation is the same for each but the particular integral is different. Since the first of the two involves the component of force Q it will be used in favor of the second which is expressed in terms of the rotation V of a shell element. Carrying out the operations indicated in the first of Eqs. 8 and substituting the dimensionless shear Q' = Q/pw2r12 results in the defining differential equation d'Q' dSQ ' . d~Q ' dQ' d~3--T -t- j3 ~ - -t- y 2 - ~ - nt- j l - d ~ + joO'-- j
(9)
where j3 -- 2 ctn/3(sec/3 + 1) (3 sec 13 + 1) j2 = (sec/3 + 1)2{ctn 2/3(13 sec 2 [3 - 1) -t- csc 4/3(7 - 5 cos/3 - 2 cos 2/3) } j l = (sec/3 nu 1)2{csc3/3(28 sec 13 - 8 - 5 cos 13) + csc/3(12 sec fl + 4 + 2 cos/3) }
J.L.
12o
MERIAM
[J. F. I.
j0 = (sec/3 + 1)3{csc ~/3(6 sec B - 9 + cos/3 + 2 cos 213) + csc4/3(3[10 sec/3 - 11] + 2 sin 2/312 sec/3 - 1] + cos/3 + 2 cos 2/3) r12# 4
-- ctn 2/3(sec/3 + 1)} + j
= (3 + v)(1 + cos/3)2 c t n 3/3{sec 3/3(2 sec/3 + 1) +(2+v)
(sec/3 -- 1) 2
sec /3 -- (5 + v) }.
W h e n t h e r e are no m e m b r a n e or shear forces on t h e b o u n d a r y / 3 = /32, equilibrium of t h e shell requires t h a t N/
= - Q' tan/3.
(10)
F r o m either Eq. 1 or 2 a n d Eq. 10 is o b t a i n e d No' = (1 - cos/3)2 _ (tan/3 sec/3)Q' -
(sec/3 -
1) dQ'
d/3"
(11)
T h e b e n d i n g m o m e n t s are derived from Eqs. 5 a n d 6 using t h e s u b s t i t u tion V = -
"(v+d-~)/rlandthe'--
f i r s t o f Eq. 7 relating V a n d
U.
Expressing U in t e r m s of Q a n d r e d u c i n g to dimensionless form there results [ d3Q ' i. d2Q ' dQ' ] Ma' = - p|k3-y~.,L ap. + ,~2---~ + k~--~ + koQ' - k j (12) ttp where h t2 p
m
12(1 -- v2) (sec/3 -- 1) 2 k3 = k2 = tan/3 (sec/3 -- 1) (4 sec/3 + 1 + v) kl = (sec/3 -- 1){(2 sec ~ + 1)(3 sec2/3 + [1 + ~] sec/3 -- [1 -- v] + s e c / 3 ( 2 s e c 2 / 3 - 1) -- v} tan/3 sec/3 (sec/3 -- 1)(6 sec 2/3 + 2 sec/3 -- 1) k0 + tan/3(sec/3 + v)(sec/~[2 sec 2/3 -- 1] + sec 2/~ -- [1 + v]) k = (3 + v)(sec/3 -- 1)2(sec/3 + 1) cos2/3{2 + v + csc2/3 (sec/3 -- 1)}. Also
• [ d3Q ' d2Q ' dQ' ] Mo' = - Plm3-d~- + m2--d~ + m,-d~ + moO' - m
(13)
where m3
=
~(sec/3
-
1) 2
m2 = tan/3(sec/3 -- 1)(4v sec/3 @ 1 + v) m l = (sec/3 -- 1)[(2 sec/3 W 1)(sec/3 @ 1) +v(8sec 3/3+5sec 2/3-2sec131 - ~)} m0 = tan/3{2 sec ~/3 + sec 2/3 - sec/3 - 1 + ~ sec/3(8 sec 3/3 - 3 sec 2/3 - 4 sec/3 - v) } m = (3 + v)(sec/3 - 1){v(sec/3 - cos 2/3) @ sin 2/3}.
Aug., i95o.]
ROTATING B E L L - S H A P E D S H E L L
12I
The radial deflection of any element is ~ = eorl(1 - cos fl). Substituting the Hooke's Law relation gives ~=
rl(1 - cos/3) (No - vN~). Eh
Changing to dimensionless form, expressing No' and Nff in terms of Q', and using Eqs. 10 and 11 give r l / E h - cos/3(sec ~ - 1)2
d/3
foQ' + f
(14)
where f0 -- ctn/3(sec/~ + 1)(sec/3 - 1) f
= cos/3(1 -
cos/3).
The rotation V of an element is needed in the process of deriving the expressions for M f f and Mo'. From the first of Eqs. 7 this becomes
v -
Eh
(see/3 -
Q'2 + e l -dO' [[ dd/3 ~ + eoO' - e_]
(15)
where el = ctn ~(sec/3 -/- 1)(2 sec/3 -I- 1) e0 = (sec/3 + 1)[1 + v + see/3 + csC/3(sec/3 -- v)] 3+p e s i n 2/3. 2 In Eqs. 10 through 15 the membrane forces, the moments, the radial deflection, and the rotation are all expressed in terms of Q' and its derivatives. SOLUTION OF THE DIFFERENTIAL EQUATION
The homogeneous part of Eq. 8 can be factored to L ( U ) -4- i#2U = O.
Thus, solution of the homogeneous equation is reduced to the solution of these two second order equations which, although simpler than Eq. 9, are still awkward for series expression when expanded. Wissler (2) and Stange (3) developed solutions of their similar second order linear equations but the resulting functions necessitate extensive calculation. The particular integral involved in the present shell also adds considerable complexity to the form of a rigorous mathematical solution. The availability of the differential analyzer at the University of California at Los Angeles suggested the investigation of a machine solution for this type of problem. The resulting advantages and disadvantages of this method will be shown subsequently. In setting up the analyzer it was found that the complete fourth order form, Eq. 9, was more convenient for use than the factored form by reason of the conservation of certain elements of the available analyzer equipment.
I22
J . L . MERIAM
[I. F. I.
The j's of Eq. 9 were first calculated using u = 0.3 and plotted against ~. These functions were then supplied to the machine on input tables manually operated. Due to the rapid changes in j, jo, jl, j2, it was found advisable to supply these functions to the machine in the form of their logarithms. This greatly reduced the manual errors due to the difficulty in following rapidly changing curves. These logarithms were converted by means of one integrator in each case to the function. The j-plots and the schematic machine diagram are shown in Figs. 4 and 5, respectively. IZ
/ i/
,o
80
9 (9 tD 0 ,.J
'
J3
-..,..__
f~
50
~ J ~
40
---._._,__,_
Oa...z d
4 35
~
ao
I
40
45
50
55
60 65 t8, D E G R E E S .
70
75
80
85
FIG. 4. j-Coefficientsof definingdifferentialequation. Problems of divided boundary conditions present a difficulty when a machine or step-by-step solution is attempted. The difficulty lies in the fact that in order to start such solutions the initial values of the dependent variable and its n - 1 derivatives must be assigned at the starting point before a solution m a y proceed. Thus, in general, it is necessary to adopt a trial and error process selecting various arbitrary values for the initially unknown derivatives until a combination is obtained which will satisfy the boundary conditions at the second point at which boundary conditions are specified. In the case of the fourth order equation of this paper two boundary conditions exist at each of
Aug., I95o.]
ROTATING BELL-SHAPED SHELL
12 3
the two edges B = fll and/ff = /~2 of the shell. Hence if the solution is begun at B = B1 then two of the derivatives will be unknown. Fortunately a m e t h o d for handling this problem of two-point boundary values is available for ordinary linear differential equations (4). In the case of a fourth order equation with equally divided boundary conditions the m e t h o d consists of m a k i n g three trial solutions with three arbitrary b u t different combinations of the two unknown derivatives at the starting point• The other derivatives are determined at this point by the two known boundary conditions. Each of the three trial solutions proceeds until the point at which the remaining two conditions are J
-! i l
1-/
•
O=-[
d2
®
t
s
+
SYMBOLS : J~EGRATOR
@
.
+
t
® +
o
t
®
-
®]
g tNPUT
OUTPUT
RAT;0-1:;.
ADDER
Fro. 5. Schematic machine diagram for solution of differential equation.
known is reached. The two quantities which take on prescribed values at this second boundary are next calculated for each of the three solutions at this point. If these quantities are expressible as linear combi nations of the dependent variable and its n - 1 derivatives, then by linear interpolation and extrapolation it is possible to determine the proper initial combination of unknown derivatives and also the correct values of the dependent variable and its derivatives for all points within the interval of solution. To illustrate this m e t h o d consider the case where the shell rotates with free inner and outer boundaries. T h u s the four conditions to be
J.L.
I2 4
MERIAlvI
[J. F. I.
satisfied are Q' = MS = 0 for both/3 =/31 and/3 =/32. If the solution begins at/31, then, in addition to Q' = 0, Eq. 12 requires that ~a-~-t-k2
/3 q - ~ l - d ~ - k
e=~l = 0 .
(16)
The values of any two of the derivatives in Eq. 16 must be assumed and the third calculated so as to satisfy this algebraic equation. If dQ'/d/3 and d2Q'/d/3 ~ are the two arbitrarily selected derivatives, then the possible initial choices may be represented as points when plotted against (dQ'/d/3)~l and (d2Q'/d/32)~l. In accordance with the method cited in the foregoing paragraph three trial runs are made with the initial values represented by points 1, 2, 3 in Fig. 6. When/3 = /32 has been
b,
df3ZJp, FIG. 6. Schematic control plot for solution with free inner and outer edges.
reached, the values of Q' are noted and the values of Mo' are calculated. By reason of the established linearity (4) of the boundary parameters the point al where Q' = 0 can be located by linear interpolation from the values of Q' at points 1 and 2. Likewise, points a2 and a3 are located and the parameter line ( Q ' = 0)e2 can be drawn. In like manner, points bi, b2, b3 where Me' = 0 are located from the values of Me' at points 1, 2, 3, and the (Me' = 0)e~ parameter line is drawn. The intersection of these two parameters gives the solution point S. Point S determines the proper initial combination of derivatives which would have resulted in the correct solution. Although this solution may then be made using these values it is not necessary to do so. The correct values of Q' and its derivatives may be obtained for all values of 13 by direct linear interpolation and extrapolation from the corresponding values at points 1, 2, 3 to the point S. This is done, for example, by interpolating to point c from 2 and 3, and then interpolating to S from 1 and c.
Aug., I95O.]
ROTATING BELL-SHAPED SHELL
I2 5
In the case of a shell with clamped inner edge and free outer edge Eqs. 14 and 15 require that d ~ -t- foQ' - f = 0
,Q
and
0
d2Q ' dQ' . ~d~ -1- el - ~ - -}- eoQ'
for ¢t = El.
e =
1
(17)
Again it is necessary to assume two initial values such as 0.28
I\ 1 \ I \
o.2o 0.16 O.IZ.
~" ~ '... 65"I lez="~~°"t
\
'\
' ~" \
&.m, ~ \
".% o.o,
~e~:
o
~;~'-~
~--'T
II
-5
-4
-3
-P-.
(eQ'
-I
0
÷I
d p21~,=ss" FIG. 7. Control plot for Case A, free inner and outer edges, h' = 1/20.
Q' and dSQ'/d~ ~. The corresponding initial values of the first and second derivatives are obtained from Eq. 17 expressed at/~1. If the outer edge is free, then the relations discussed in the preceeding paragraph hold at f12. The control points are plotted against the initial values of Q' and
,d~Q'/d~~.
J.L.
126
MERIAM
[J. t:. I.
In the procedure described it is necessary that the distances between trial points be of the same order of magnitude as the distance from the solution point to any of the trial points. Although the error due to extrapolation of non-linear quantities is absent, still small errors in the -0.004
-0.008
7¢
B-2
~,,eo',~ /
-0.012
q
.
\
#,=35
\
-0.016
-I00
\
-gO
/
/
/
\
-0.020 -I10
,.~ ~7,B- 3
/
-80
-TO
-60
-50
-40
d#"/O,=mFIG. 8.
Control plot for Case B, clamped inner edge and free outer edge, h' = 1/20.
0.6
o.5
~
....
o ¢~=e¢
._
?4 °
x'~ ,
C-Z
0.4
"a,
0,=~. 0.3
'
O,Z
0:I
-14
I
\
\
C-a
-IZ
-I0
-8
-6
-4
c-3
-2
0
2
4
d#Zl~,=85 • FIG. 9.
Control plot for Case C, free inner and
outer edges,
h t = 1/10.
values of the trial solutions will limit the extent of accurate extrapolation. Thus a preliminary estimate of the region of solution is very helpful and will alleviate the necessity for repeating the trial process
Aug., I95o.]
ROTATING BELL-SHAPED SHELL
I2 7
with three additional points closer to the solution point. In the case of the present problem a sufficiently close estimate of the region of solution was made with the aid of the results for the rotating conical shell (1). 0.05
It' INNER EDGE I/ZO FREE I/ZO CLAMPED 1/10 FREE
A 8 tC
/
o.oz 0.01 Q', hl;
OUTER EDGE FREE - FREE FREE q/ ~ - - - '
~--...
f \ \
0
"-~--
\
/ / /i .
~'~,.
"QOI
\
// .~B//, /
,,c "
-0.02
35
40
50
45
55
.sA
60
65
70
75
80
13, DEGREES FIG. 10. Variation of shear Q' and membrane force NO'. !
-
;-
0.4
-I 0.3
d~
,
/
I// CASE A B C
.\\i
h' INNER EDGE i/co FREE ' |/ZO CLAMPED |/10 FREE
OUTER EDGE
-2
FREE
FREE FREE
dB deC~ dPz
O.i
I \
!
-0.1
i ~...-&oo AT as*
,40
45
50
55
60
65
70
8, DEGREES Fie. 11.
Variation
o f d Q ' / d ~ a n d ~P~Q'//d/~:.
75
80
J.L.
128
~/[ERIA~
[J. F. I.
In addition to obtaining a solution for a given value of/32 the data of three runs enable solutions to be made for various values of/32. Thus a family of solutions may be obtained and indicated by the locus of solution points S on the control plot of Fig. 6. Due to the rapidly increasing j-coefficients in Eq. 9 the machine became somewhat unstable as/3 approached 90 degrees. This limited the calculation of Mo' for angles up to 75 or 78 degrees and the solution point to 80 degrees. By plotting the locus of solution points for smaller angles it was possible to locate the 80 degree point with sufficient accuracy.
80 CASE
60
~
INNER EDGE OUTER EDGE FREE ~REE I~#.0 CLAMPED PRE£ VIO FREE FREE
I/zo
A "B C
40
d/3 ~ 20
-2o 35
,,;,,. ,tO
45
50 FIG. 12.
5,5 60 19, DEGREES
65
70
75
80
V a r i a t i o n of d~Q'/d¢ 8. RESULTS
Three separate cases were investigated and the following tabulation gives the conditions of each case. Case A B C
hp 1/20 1/20 1/10
Bt, deg. Inner Edge 35 Free; Q --- 0, MB = 0 35 Clamped; ~'---0, V = 0 35 Free; Q -- O, M O = 0
Outer Edge Free; Q = 0, M~ = 0 F r e e ; Q = 0, M a = 0 Free; Q = 0, M~ = 0
Trial Solution A - l , A-2, A - 3 B-1, B-2, B-3 C-I, C-2, C-3
The value of B1 = 35 deg. was chosen to represent a reasonably sized hole. The ratio of 1/20 for h' = h / r l was selected so that direct comparison could be made with the results of the rotating cones (1) which were calculated for a comparable thickness. The value of h' = 1/10 for Case C was investigated as an indication of the effect of thickness. The
Aug., r95o.]
ROTATING BELL-SHAPED SHELL
I2 9
control plots, corresponding to Fig. 6, for each of the three cases are
shown in Figs. 7, 8, 9. Solution points for each of a number of values of ~2 were obtained in the manner described with Fig. 6, and their locus gives a family of solutions each with the same boundary conditions. 08
0.6
0.4 i
No O.Z
i:
0
-0.2
as
40
4s
50
55
6o
6s
70
7s
ao
~, DEGREES FIG. 13. Variation of "hoop stress" No' and radial deflection ~'. h' INNER ED(~E OUTER ED6 F' I/~.0 FREE FREE FREE I/ZO CLAMPED I/IO FREE FREE; 1 M~ ~ ,
CASE A
BC
60!
% 40'
M;
.
.
.
.
.
.
\ •. 2 0
25
-- / " I ' "
4o
45
50
55
60
~
70
75
8O
~B, DEGREES FIG. 14. Variation of moments Me' and M0'. Only the data for/~2 = 80 deg. were calculated. By linear interpolation and extrapolation from the three trial points in each case to the • 80-deg. point the values of Q' and its derivatives were calculated. The values used in this calculation were read directly from the machine plotted curves for each of the trial solutions. The variations of Me',
J.L.
I3o
~/['ERIAM
[J. F. I.
Mg, No', No', ~" with/3 were next calculated from Eqs. 10 through 14. The resulting curves are shown in Figs. 10 through 14. DISCUSSION
If the bending strength and resistance to shear are neglected a simple membrane theory results. The relations may be obtained by inspection from Eqs. 11 and 14 and are No'
= =
(1 (1 -
cos
[
cos ) . J
(18)
These expressions are plotted in Fig. 13 and it is seen that this approximate theory gives a good prediction of the results for these two quantities. Restraining the inner edge with a rigid clamp as reported in Case B has a very small effect upon the forces, moments, and deflection as is clearly evident from Figs. 10, 13, and 14. The effect of restraint is pronounced only in the immediate region of the clamp. Comparison of the "hoop stress" for the bell-shaped shell with free inner edge with that for the conical shell with free inner edge (I) is of interest. This comparison may be made in two ways as shown in Fig. 15. In part A of the figure all shells have the same dimension rl and the same thickness h = rl/20. It is interesting to note that for a cone whose angle a is the average of B1 = 35 deg. and ~ = 80 deg. or 57.5 deg. the "hoop stress" at the outer rim is the same (5) as that for the bell. Comparison may also be made. on the basis of equal outside diameters. This is shown in Fig. 15(b). The maximum rim stresses are all nearly the same except for the flat disk) It should be noted that in this comparison the thicknesses of the shells are slightly different. The effect of increasing the shell thickness from h ' = 1/20 to h' = 1/10 is to reduce the maximum "hoop stress" No' and the outer rim deflection ~' by 26 per cent and increase the maximum bending moment M~' by 60 per cent. For small values of ~t where the radius of rotation is small, there is undoubtedly appreciable error introduced in the solution for a ratio of h/rl as great as 1/10. The assumption introduced in the expression for curvature requires that the thickness be small compared with the radius of curvature. Thus the possible error in using the theory for such a thick shell is recognized. Nevertheless, the general effect of a thicker shell is clearly indicated. The effect of thickness can be seen further from the logej0 plot in Fig. 4. Thus for > 75 deg. the differential equation does not depend on h'. The j-coefficients in Eq. 9 approach infinite values very rapidly as ~t approaches 90 deg. This condition presented a difficulty in the 3 It was shown in the discussion (3) of the conical shell investigation t h a t the transition of stress values for cones to those for the flat rotating disk occurs abruptly as a changes from 83 to 90 deg.
Aug., 195o.}
ROTATING BELL-SHAPED SHELl+
13][
analyzer operation for large values of /~. If additional input equipment had been available it might have been possible to multiply each term in the equation by some function which would have eliminated or reduced the large values. Nevertheless, no trouble was experienced up to 80 deg. except for the computation of M~' at this point where small 0.18
0.4.,?.
,---.,.
JI -,--
.
_U___ "
-e--=
i_
. . . . t _1 4 1 ~ = i/+ , h : l , / z o
I
A ; VALUES OF N e FOR ,SAME q 0.69 FOR
~ ~ ~ ~ ~ ~ q
N ; AT OUTER 63 _o.m o.~
'i
L..---
lJ
RIM ~s~ a s 8
=-,s-o.63
__e__Jri _ (,U
B ; VALUES OF N e FOR SAME OUTSIDE RADIUS FIG. 15. Comparison of "hoop stresses" between bell-shaped shell and conical shells.
differences between large quantities began to assume importance. Uncertainty as to the location of the 80-deg. solution point was removed when additional solution points for lesser values of fl~ were constructed and the distances between these points measured along their locus were plotted against their values in degrees. In Fig. 13 the slopes of the Ne' curves are very small at f12 = 80 deg.
132
J . L . MERIAM
[J. F. I.
and it m a y be inferred t h a t for a shell of/32 = 90 deg. there will be a negligibly small difference in m a x i m u m No'. The portion of such a shell between/32 = 80 deg. and/32 = 90 deg. m a y be approximated by a flat disk or by a steep cone (1) in both of which cases the "hoop stress" No' is decreasing with increased radial dimension. The original intent was to determine the solution point during the operating period of the analyzer and obtain the curves for the proper Q' and its derivatives directly from the machine plots. Before this was shown to be not feasible, a n u m b e r of additional trial solutions, not indicated in Figs. 7, 8, or 9, were made. With these solutions the expected linearity of Q' and its derivatives with respect to values of the initially assumed derivatives was verified experimentally. The over-all accuracy of the results is difficult to'establish, particularly since the j-coefficients were manually supplied to the analyzer. Several re-runs which were made resulted in curves which were visually identical. The best measure of accuracy was obtained from the departure of the linearity observed when more than two trial points were taken on any straight line on the control plots. In one case where four such points were investigated, the resulting solutions were plotted against the distances between the points for various values of/3 up to and including 80 deg., and the width of the bands which included the values for all four points were measured. These measurements showed t h a t the errors for Q', dQ'/dl3, d2Q'/d~ ~, and daQ'/d/3 a were usually less b u t not greater than 4-0.00015, 4-0.0015, 4-0.01, and 4-0.15, respectively. The machine plotted curves were read to 0.0001, 0.001, 0.01, and 0.1 for these same respective quantities, and in no instance did these readings deviate more than the last digit from numerous recorded counter readings which were more accurate than the plots. APPENDIX
The defining relations in this paper m a y be obtained from those in the investigation of conical shells (1) by the following substitutions: Replace ro r~ Q~ N~ Mr w e, X~
by by by by by by by by by
~r/2 rffl - cost/) - rl Q No Mo - w ea Xo
All other symbols remain the s~ime.
Aug., ~95o.]
ROTATING BELL-SHAPED SHELL
I33
Acknowledgment The author wishes to acknowledge the helpful suggestions made by Professor P. L. Morton and the cooperation of the staff of the differential analyzer at the University of California at Los Angeles. The accurate checking and calculations of Mr. A. L. Hale are also acknowledged. REFERENCES
(1) J. L. MERIAM, "Stresses and Displacements in a Rotating Conical Shell," Journal Appl. Mechanics, Trans. A. S. M. E., Vol. I0, No. 2, p. A-53 (1943). (2) H. WISSLER, "Festigkeitsberechnung von Ringflachenschalen," Doctoral Thesis, Zurich, 1917. (3) K. STANGE, "Der Spannungszustand einer Kreisringschale," Ingenieur Archly, Vol. 2, p. 47. (4) J. L. MERIAM, "Procedure for the Machine or Numerical Solution of Ordinary Linear Differential Equations for Two-Point Linear Boundary Values," Mathematical Tables and Other Aids to Computation, The National Research Council, III, No. 28, Oct., 1949, p. 532. (5) Discussion on Reference (1), Journal Appl. Mechanics, Trans. A. S. M. E., Vol. 10, No. 4, p. A 233-234 (t943).
ABSTRACT In an earlier paper (1) 3 by the author the theory of axially symmetrical shells was extended to include the body force of rotation about the axis of generation, and the results were applied to the conical shell. In the present paper this theory is used to obtain the solution for a bell-shaped rotating shell whose middle surface is generated by revolving a circular arc about a tangent line. The defining relation is an ordinary linear differential equation of fourth order with variable coefficients. A differential analyzer solution was used and this involved the problem of handling boundary conditions which were divided between both extremities of the range of the independent variable. Comparisons of the results with those of the conical shell and with the membrane theory are made. NOMENCLATURE
T h e following nomenclature is used in the paper: rl h
= constant radius of meridian curvature, = shell thickness, = angle b e t w e e n normal to axis of rotation and normal to shell meridian, 0 = angle measured a b o u t axis of rotation in plane of rotation, p = mass per unit area of shell, ~o = angular velocity of shell, Q = shear force per unit length of shell section, N , = m e m b r a n e force along meridian per unit length of shell section, No = m e m b r a n e force t a n g e n t to a parallel circle per unit length of shell section ("hoop stress"), M~ = bending m o m e n t in plane of meridian per unit length of shell section, Mo = bending m o m e n t in plane perpendicular to meridian per unit length of shell section, R = b o d y force of rotation per unit area of shell, v = displacement c o m p o n e n t along meridian, w = displacement c o m p o n e n t normal to meridian, /" = radial displacement (normal to axis of rotation), e, = strain in direction of meridian, e0 = strain in direction t a n g e n t to a parallel circle, x0 = change in c u r v a t u r e in plane of meridian, xo = change in c u r v a t u r e in plane perpendicular to meridian,
1 Assistant Professor of Engineering Design, University of California, Berkeley, Calif. The boldface numbers in parentheses refer to the references appended to this paper. 115
116
J.L.
E D
MERIAM
[J. F. I.
= Poisson's ratio, = modulus of elasticity, and = flexural rigidity.
Dimensionless quantities :
O'
= O_/ po 2r12
N;
=
No'
= No/po~2rl 2
M/=
Ma/p, r 3
M o ' = Mo/po~2rl 3 ~' = ~/(rl/Eh) h' = h/rl
INTRODUCTION
Solutions for rotors which are symmetrical with respect to a plane of rotation and whose axial dimensions are small compared with the radial
0
FIG. 1. Dimensions of bell-shaped shell.
dimensions are well known. The solution for the rotating conical shell (1) was a departure from this condition of symmetry. T h e analysis of the bell-shaped shell reported in this paper represents a still further departure and approximates the curved profiles of m a n y of the high speed p u m p impellers. Analyses of ring- or torus-shaped shells under static loading are available in the papers of Wissler (2) and Stange (a). T h e shell of the present investigation differs in t h a t it is subject to a
Aug., ~95o.]
ROTATING BELL-SHAPED SHELL
II 7
body force due to rotation and that its surface is defined by a portion of a torus with zero hole. The defining equations for the bell-shaped shell can be obtained from the relations of the earlier paper (1) by appropriate changes in variables and symbols. These substitutions are listed in the Appendix. In the present article the essential equations are cited t o maintain the completeness of the presentation and to provide necessary reference. THEORY
The middle surface of the shell is shown in Fig. 1 and is generated by revolving the circular meridian arc o a of radius,r~ about the tangent a
d
~
!
i
:,.,.~ \ / ,4
+d
V../
0 FIG. 2. Shell element with forces and moments.
(Shell thickness not shown.)
line oo. The angles/31 and /32 define the inner and outer boundaries, respectively, and the thickness h is assumed small compared with rl but large enough to support appreciable bending stresses. Rotation about the axis oo with a constant angular velocity provides the centrifugal loading. An element of the shell with the forces and moments acting on it is shown in Fig. 2. Equating to zero the forces in a direction tangent to the arc oa gives d d-~ (No[1 -- cos/3]) --
No
sin/3 + 0(1 -- cos 13) + poa=rl ~ sin/3(1
-- cos/3)2 = 0.
(1)
Likewise, equilibrium of forces in the direction of the radius rl gives d d--~(O[1 - cos/3J) + No cos/3 - N~(1 - cos/3) - - pw~rl 2 cos/3(1
-- cos/3)= = O.
(2)
J.L.
~I8
MERIAM
[J. F. I.
A third equation is obtained by equating to zero the moments about the upper edge and d d-~ (M~[1 - cos/3]) - M0 sin/3 -- Qrl(1 - cos/3) = 0.
(3)
Equations 1, 2, and 3 completely specify the equilibrium of the shell element. In Fig. 3 the positive directions of the two displacements a
~ 7~" w* dw
0 Fit;. 3.
Components of displacement.
v and w are shown.
The strain in a direction tangent to the arc is
e~ =
and
--w
rl
that
in a
Eh [ l(dv ~-~-w
direction
is
Substituting these expressions
e0 = (w cos/3 -4- v sin/3)/r1(1 - cos/3). into the Hooke's Law equations gives N~ = 1 - ~2Lrl
circumferential
) +
~(wcos/3+vsin/3)]) rl(1 - c o s / 3 )
Eh
[ w cos/3 + vsin B L(dv )] No = 1 - p~---3 rl(1 - c o s / 3 ) +rl ~-~ - w .
(4)
The moments are related to the changes in curvature by the equations Ms = - D(xa + vx0) [ Mo -- D(xo + vXa), !
(5)
where
x~ = X0 =
l d( d~) rl' d~ v + -- - ,-
rl 2
ctn /3 (sec t3 -{- 1) ( v +dw
(6)
Aug., I95o.]
ROTATING BELL-SHAPED SHELL
II9
Equiations 1 through 6 completely define the problem. Their combination is somewhat lengthy (1) but results in the two equations L(U) -
~' U = V E h -at- T [ Y1
L ( V ) -{- L V = rl
(7) U D
where L( ) -
sec/3rl
tan/3d( ) ld2( ) d/32 -[- rl d/3
U = rl(sec/3rl
sec/3 + 1 rl
()
1) Q d-~
and 3+~ 2
ooo2rl2 sin 2fl(sec/3 - 1) 5.
Eliminating first V and then U from Eqs. 7 gives L L ( U ) + u ' U = L ( T ) + r-~ L L ( V) + ~4 V
T D
(8)
where Eh #4-- D
u2 rf
Either of Eqs. 8 m a y be used to solve the problem. The homogeneous or reduced equation is the same for each but the particular integral is different. Since the first of the two involves the component of force Q it will be used in favor of the second which is expressed in terms of the rotation V of a shell element. Carrying out the operations indicated in the first of Eqs. 8 and substituting the dimensionless shear Q' = Q/pw2r12 results in the defining differential equation d'Q' dSQ ' . d~Q ' dQ' d~3--T -t- j3 ~ - -t- y 2 - ~ - nt- j l - d ~ + joO'-- j
(9)
where j3 -- 2 ctn/3(sec/3 + 1) (3 sec 13 + 1) j2 = (sec/3 + 1)2{ctn 2/3(13 sec 2 [3 - 1) -t- csc 4/3(7 - 5 cos/3 - 2 cos 2/3) } j l = (sec/3 nu 1)2{csc3/3(28 sec 13 - 8 - 5 cos 13) + csc/3(12 sec fl + 4 + 2 cos/3) }
J.L.
12o
MERIAM
[J. F. I.
j0 = (sec/3 + 1)3{csc ~/3(6 sec B - 9 + cos/3 + 2 cos 213) + csc4/3(3[10 sec/3 - 11] + 2 sin 2/312 sec/3 - 1] + cos/3 + 2 cos 2/3) r12# 4
-- ctn 2/3(sec/3 + 1)} + j
= (3 + v)(1 + cos/3)2 c t n 3/3{sec 3/3(2 sec/3 + 1) +(2+v)
(sec/3 -- 1) 2
sec /3 -- (5 + v) }.
W h e n t h e r e are no m e m b r a n e or shear forces on t h e b o u n d a r y / 3 = /32, equilibrium of t h e shell requires t h a t N/
= - Q' tan/3.
(10)
F r o m either Eq. 1 or 2 a n d Eq. 10 is o b t a i n e d No' = (1 - cos/3)2 _ (tan/3 sec/3)Q' -
(sec/3 -
1) dQ'
d/3"
(11)
T h e b e n d i n g m o m e n t s are derived from Eqs. 5 a n d 6 using t h e s u b s t i t u tion V = -
"(v+d-~)/rlandthe'--
f i r s t o f Eq. 7 relating V a n d
U.
Expressing U in t e r m s of Q a n d r e d u c i n g to dimensionless form there results [ d3Q ' i. d2Q ' dQ' ] Ma' = - p|k3-y~.,L ap. + ,~2---~ + k~--~ + koQ' - k j (12) ttp where h t2 p
m
12(1 -- v2) (sec/3 -- 1) 2 k3 = k2 = tan/3 (sec/3 -- 1) (4 sec/3 + 1 + v) kl = (sec/3 -- 1){(2 sec ~ + 1)(3 sec2/3 + [1 + ~] sec/3 -- [1 -- v] + s e c / 3 ( 2 s e c 2 / 3 - 1) -- v} tan/3 sec/3 (sec/3 -- 1)(6 sec 2/3 + 2 sec/3 -- 1) k0 + tan/3(sec/3 + v)(sec/~[2 sec 2/3 -- 1] + sec 2/~ -- [1 + v]) k = (3 + v)(sec/3 -- 1)2(sec/3 + 1) cos2/3{2 + v + csc2/3 (sec/3 -- 1)}. Also
• [ d3Q ' d2Q ' dQ' ] Mo' = - Plm3-d~- + m2--d~ + m,-d~ + moO' - m
(13)
where m3
=
~(sec/3
-
1) 2
m2 = tan/3(sec/3 -- 1)(4v sec/3 @ 1 + v) m l = (sec/3 -- 1)[(2 sec/3 W 1)(sec/3 @ 1) +v(8sec 3/3+5sec 2/3-2sec131 - ~)} m0 = tan/3{2 sec ~/3 + sec 2/3 - sec/3 - 1 + ~ sec/3(8 sec 3/3 - 3 sec 2/3 - 4 sec/3 - v) } m = (3 + v)(sec/3 - 1){v(sec/3 - cos 2/3) @ sin 2/3}.
Aug., i95o.]
ROTATING B E L L - S H A P E D S H E L L
12I
The radial deflection of any element is ~ = eorl(1 - cos fl). Substituting the Hooke's Law relation gives ~=
rl(1 - cos/3) (No - vN~). Eh
Changing to dimensionless form, expressing No' and Nff in terms of Q', and using Eqs. 10 and 11 give r l / E h - cos/3(sec ~ - 1)2
d/3
foQ' + f
(14)
where f0 -- ctn/3(sec/~ + 1)(sec/3 - 1) f
= cos/3(1 -
cos/3).
The rotation V of an element is needed in the process of deriving the expressions for M f f and Mo'. From the first of Eqs. 7 this becomes
v -
Eh
(see/3 -
Q'2 + e l -dO' [[ dd/3 ~ + eoO' - e_]
(15)
where el = ctn ~(sec/3 -/- 1)(2 sec/3 -I- 1) e0 = (sec/3 + 1)[1 + v + see/3 + csC/3(sec/3 -- v)] 3+p e s i n 2/3. 2 In Eqs. 10 through 15 the membrane forces, the moments, the radial deflection, and the rotation are all expressed in terms of Q' and its derivatives. SOLUTION OF THE DIFFERENTIAL EQUATION
The homogeneous part of Eq. 8 can be factored to L ( U ) -4- i#2U = O.
Thus, solution of the homogeneous equation is reduced to the solution of these two second order equations which, although simpler than Eq. 9, are still awkward for series expression when expanded. Wissler (2) and Stange (3) developed solutions of their similar second order linear equations but the resulting functions necessitate extensive calculation. The particular integral involved in the present shell also adds considerable complexity to the form of a rigorous mathematical solution. The availability of the differential analyzer at the University of California at Los Angeles suggested the investigation of a machine solution for this type of problem. The resulting advantages and disadvantages of this method will be shown subsequently. In setting up the analyzer it was found that the complete fourth order form, Eq. 9, was more convenient for use than the factored form by reason of the conservation of certain elements of the available analyzer equipment.
I22
J . L . MERIAM
[I. F. I.
The j's of Eq. 9 were first calculated using u = 0.3 and plotted against ~. These functions were then supplied to the machine on input tables manually operated. Due to the rapid changes in j, jo, jl, j2, it was found advisable to supply these functions to the machine in the form of their logarithms. This greatly reduced the manual errors due to the difficulty in following rapidly changing curves. These logarithms were converted by means of one integrator in each case to the function. The j-plots and the schematic machine diagram are shown in Figs. 4 and 5, respectively. IZ
/ i/
,o
80
9 (9 tD 0 ,.J
'
J3
-..,..__
f~
50
~ J ~
40
---._._,__,_
Oa...z d
4 35
~
ao
I
40
45
50
55
60 65 t8, D E G R E E S .
70
75
80
85
FIG. 4. j-Coefficientsof definingdifferentialequation. Problems of divided boundary conditions present a difficulty when a machine or step-by-step solution is attempted. The difficulty lies in the fact that in order to start such solutions the initial values of the dependent variable and its n - 1 derivatives must be assigned at the starting point before a solution m a y proceed. Thus, in general, it is necessary to adopt a trial and error process selecting various arbitrary values for the initially unknown derivatives until a combination is obtained which will satisfy the boundary conditions at the second point at which boundary conditions are specified. In the case of the fourth order equation of this paper two boundary conditions exist at each of
Aug., I95o.]
ROTATING BELL-SHAPED SHELL
12 3
the two edges B = fll and/ff = /~2 of the shell. Hence if the solution is begun at B = B1 then two of the derivatives will be unknown. Fortunately a m e t h o d for handling this problem of two-point boundary values is available for ordinary linear differential equations (4). In the case of a fourth order equation with equally divided boundary conditions the m e t h o d consists of m a k i n g three trial solutions with three arbitrary b u t different combinations of the two unknown derivatives at the starting point• The other derivatives are determined at this point by the two known boundary conditions. Each of the three trial solutions proceeds until the point at which the remaining two conditions are J
-! i l
1-/
•
O=-[
d2
®
t
s
+
SYMBOLS : J~EGRATOR
@
.
+
t
® +
o
t
®
-
®]
g tNPUT
OUTPUT
RAT;0-1:;.
ADDER
Fro. 5. Schematic machine diagram for solution of differential equation.
known is reached. The two quantities which take on prescribed values at this second boundary are next calculated for each of the three solutions at this point. If these quantities are expressible as linear combi nations of the dependent variable and its n - 1 derivatives, then by linear interpolation and extrapolation it is possible to determine the proper initial combination of unknown derivatives and also the correct values of the dependent variable and its derivatives for all points within the interval of solution. To illustrate this m e t h o d consider the case where the shell rotates with free inner and outer boundaries. T h u s the four conditions to be
J.L.
I2 4
MERIAlvI
[J. F. I.
satisfied are Q' = MS = 0 for both/3 =/31 and/3 =/32. If the solution begins at/31, then, in addition to Q' = 0, Eq. 12 requires that ~a-~-t-k2
/3 q - ~ l - d ~ - k
e=~l = 0 .
(16)
The values of any two of the derivatives in Eq. 16 must be assumed and the third calculated so as to satisfy this algebraic equation. If dQ'/d/3 and d2Q'/d/3 ~ are the two arbitrarily selected derivatives, then the possible initial choices may be represented as points when plotted against (dQ'/d/3)~l and (d2Q'/d/32)~l. In accordance with the method cited in the foregoing paragraph three trial runs are made with the initial values represented by points 1, 2, 3 in Fig. 6. When/3 = /32 has been
b,
df3ZJp, FIG. 6. Schematic control plot for solution with free inner and outer edges.
reached, the values of Q' are noted and the values of Mo' are calculated. By reason of the established linearity (4) of the boundary parameters the point al where Q' = 0 can be located by linear interpolation from the values of Q' at points 1 and 2. Likewise, points a2 and a3 are located and the parameter line ( Q ' = 0)e2 can be drawn. In like manner, points bi, b2, b3 where Me' = 0 are located from the values of Me' at points 1, 2, 3, and the (Me' = 0)e~ parameter line is drawn. The intersection of these two parameters gives the solution point S. Point S determines the proper initial combination of derivatives which would have resulted in the correct solution. Although this solution may then be made using these values it is not necessary to do so. The correct values of Q' and its derivatives may be obtained for all values of 13 by direct linear interpolation and extrapolation from the corresponding values at points 1, 2, 3 to the point S. This is done, for example, by interpolating to point c from 2 and 3, and then interpolating to S from 1 and c.
Aug., I95O.]
ROTATING BELL-SHAPED SHELL
I2 5
In the case of a shell with clamped inner edge and free outer edge Eqs. 14 and 15 require that d ~ -t- foQ' - f = 0
,Q
and
0
d2Q ' dQ' . ~d~ -1- el - ~ - -}- eoQ'
for ¢t = El.
e =
1
(17)
Again it is necessary to assume two initial values such as 0.28
I\ 1 \ I \
o.2o 0.16 O.IZ.
~" ~ '... 65"I lez="~~°"t
\
'\
' ~" \
&.m, ~ \
".% o.o,
~e~:
o
~;~'-~
~--'T
II
-5
-4
-3
-P-.
(eQ'
-I
0
÷I
d p21~,=ss" FIG. 7. Control plot for Case A, free inner and outer edges, h' = 1/20.
Q' and dSQ'/d~ ~. The corresponding initial values of the first and second derivatives are obtained from Eq. 17 expressed at/~1. If the outer edge is free, then the relations discussed in the preceeding paragraph hold at f12. The control points are plotted against the initial values of Q' and
,d~Q'/d~~.
J.L.
126
MERIAM
[J. t:. I.
In the procedure described it is necessary that the distances between trial points be of the same order of magnitude as the distance from the solution point to any of the trial points. Although the error due to extrapolation of non-linear quantities is absent, still small errors in the -0.004
-0.008
7¢
B-2
~,,eo',~ /
-0.012
q
.
\
#,=35
\
-0.016
-I00
\
-gO
/
/
/
\
-0.020 -I10
,.~ ~7,B- 3
/
-80
-TO
-60
-50
-40
d#"/O,=mFIG. 8.
Control plot for Case B, clamped inner edge and free outer edge, h' = 1/20.
0.6
o.5
~
....
o ¢~=e¢
._
?4 °
x'~ ,
C-Z
0.4
"a,
0,=~. 0.3
'
O,Z
0:I
-14
I
\
\
C-a
-IZ
-I0
-8
-6
-4
c-3
-2
0
2
4
d#Zl~,=85 • FIG. 9.
Control plot for Case C, free inner and
outer edges,
h t = 1/10.
values of the trial solutions will limit the extent of accurate extrapolation. Thus a preliminary estimate of the region of solution is very helpful and will alleviate the necessity for repeating the trial process
Aug., I95o.]
ROTATING BELL-SHAPED SHELL
I2 7
with three additional points closer to the solution point. In the case of the present problem a sufficiently close estimate of the region of solution was made with the aid of the results for the rotating conical shell (1). 0.05
It' INNER EDGE I/ZO FREE I/ZO CLAMPED 1/10 FREE
A 8 tC
/
o.oz 0.01 Q', hl;
OUTER EDGE FREE - FREE FREE q/ ~ - - - '
~--...
f \ \
0
"-~--
\
/ / /i .
~'~,.
"QOI
\
// .~B//, /
,,c "
-0.02
35
40
50
45
55
.sA
60
65
70
75
80
13, DEGREES FIG. 10. Variation of shear Q' and membrane force NO'. !
-
;-
0.4
-I 0.3
d~
,
/
I// CASE A B C
.\\i
h' INNER EDGE i/co FREE ' |/ZO CLAMPED |/10 FREE
OUTER EDGE
-2
FREE
FREE FREE
dB deC~ dPz
O.i
I \
!
-0.1
i ~...-&oo AT as*
,40
45
50
55
60
65
70
8, DEGREES Fie. 11.
Variation
o f d Q ' / d ~ a n d ~P~Q'//d/~:.
75
80
J.L.
128
~/[ERIA~
[J. F. I.
In addition to obtaining a solution for a given value of/32 the data of three runs enable solutions to be made for various values of/32. Thus a family of solutions may be obtained and indicated by the locus of solution points S on the control plot of Fig. 6. Due to the rapidly increasing j-coefficients in Eq. 9 the machine became somewhat unstable as/3 approached 90 degrees. This limited the calculation of Mo' for angles up to 75 or 78 degrees and the solution point to 80 degrees. By plotting the locus of solution points for smaller angles it was possible to locate the 80 degree point with sufficient accuracy.
80 CASE
60
~
INNER EDGE OUTER EDGE FREE ~REE I~#.0 CLAMPED PRE£ VIO FREE FREE
I/zo
A "B C
40
d/3 ~ 20
-2o 35
,,;,,. ,tO
45
50 FIG. 12.
5,5 60 19, DEGREES
65
70
75
80
V a r i a t i o n of d~Q'/d¢ 8. RESULTS
Three separate cases were investigated and the following tabulation gives the conditions of each case. Case A B C
hp 1/20 1/20 1/10
Bt, deg. Inner Edge 35 Free; Q --- 0, MB = 0 35 Clamped; ~'---0, V = 0 35 Free; Q -- O, M O = 0
Outer Edge Free; Q = 0, M~ = 0 F r e e ; Q = 0, M a = 0 Free; Q = 0, M~ = 0
Trial Solution A - l , A-2, A - 3 B-1, B-2, B-3 C-I, C-2, C-3
The value of B1 = 35 deg. was chosen to represent a reasonably sized hole. The ratio of 1/20 for h' = h / r l was selected so that direct comparison could be made with the results of the rotating cones (1) which were calculated for a comparable thickness. The value of h' = 1/10 for Case C was investigated as an indication of the effect of thickness. The
Aug., r95o.]
ROTATING BELL-SHAPED SHELL
I2 9
control plots, corresponding to Fig. 6, for each of the three cases are
shown in Figs. 7, 8, 9. Solution points for each of a number of values of ~2 were obtained in the manner described with Fig. 6, and their locus gives a family of solutions each with the same boundary conditions. 08
0.6
0.4 i
No O.Z
i:
0
-0.2
as
40
4s
50
55
6o
6s
70
7s
ao
~, DEGREES FIG. 13. Variation of "hoop stress" No' and radial deflection ~'. h' INNER ED(~E OUTER ED6 F' I/~.0 FREE FREE FREE I/ZO CLAMPED I/IO FREE FREE; 1 M~ ~ ,
CASE A
BC
60!
% 40'
M;
.
.
.
.
.
.
\ •. 2 0
25
-- / " I ' "
4o
45
50
55
60
~
70
75
8O
~B, DEGREES FIG. 14. Variation of moments Me' and M0'. Only the data for/~2 = 80 deg. were calculated. By linear interpolation and extrapolation from the three trial points in each case to the • 80-deg. point the values of Q' and its derivatives were calculated. The values used in this calculation were read directly from the machine plotted curves for each of the trial solutions. The variations of Me',
J.L.
I3o
~/['ERIAM
[J. F. I.
Mg, No', No', ~" with/3 were next calculated from Eqs. 10 through 14. The resulting curves are shown in Figs. 10 through 14. DISCUSSION
If the bending strength and resistance to shear are neglected a simple membrane theory results. The relations may be obtained by inspection from Eqs. 11 and 14 and are No'
= =
(1 (1 -
cos
[
cos ) . J
(18)
These expressions are plotted in Fig. 13 and it is seen that this approximate theory gives a good prediction of the results for these two quantities. Restraining the inner edge with a rigid clamp as reported in Case B has a very small effect upon the forces, moments, and deflection as is clearly evident from Figs. 10, 13, and 14. The effect of restraint is pronounced only in the immediate region of the clamp. Comparison of the "hoop stress" for the bell-shaped shell with free inner edge with that for the conical shell with free inner edge (I) is of interest. This comparison may be made in two ways as shown in Fig. 15. In part A of the figure all shells have the same dimension rl and the same thickness h = rl/20. It is interesting to note that for a cone whose angle a is the average of B1 = 35 deg. and ~ = 80 deg. or 57.5 deg. the "hoop stress" at the outer rim is the same (5) as that for the bell. Comparison may also be made. on the basis of equal outside diameters. This is shown in Fig. 15(b). The maximum rim stresses are all nearly the same except for the flat disk) It should be noted that in this comparison the thicknesses of the shells are slightly different. The effect of increasing the shell thickness from h ' = 1/20 to h' = 1/10 is to reduce the maximum "hoop stress" No' and the outer rim deflection ~' by 26 per cent and increase the maximum bending moment M~' by 60 per cent. For small values of ~t where the radius of rotation is small, there is undoubtedly appreciable error introduced in the solution for a ratio of h/rl as great as 1/10. The assumption introduced in the expression for curvature requires that the thickness be small compared with the radius of curvature. Thus the possible error in using the theory for such a thick shell is recognized. Nevertheless, the general effect of a thicker shell is clearly indicated. The effect of thickness can be seen further from the logej0 plot in Fig. 4. Thus for > 75 deg. the differential equation does not depend on h'. The j-coefficients in Eq. 9 approach infinite values very rapidly as ~t approaches 90 deg. This condition presented a difficulty in the 3 It was shown in the discussion (3) of the conical shell investigation t h a t the transition of stress values for cones to those for the flat rotating disk occurs abruptly as a changes from 83 to 90 deg.
Aug., 195o.}
ROTATING BELL-SHAPED SHELl+
13][
analyzer operation for large values of /~. If additional input equipment had been available it might have been possible to multiply each term in the equation by some function which would have eliminated or reduced the large values. Nevertheless, no trouble was experienced up to 80 deg. except for the computation of M~' at this point where small 0.18
0.4.,?.
,---.,.
JI -,--
.
_U___ "
-e--=
i_
. . . . t _1 4 1 ~ = i/+ , h : l , / z o
I
A ; VALUES OF N e FOR ,SAME q 0.69 FOR
~ ~ ~ ~ ~ ~ q
N ; AT OUTER 63 _o.m o.~
'i
L..---
lJ
RIM ~s~ a s 8
=-,s-o.63
__e__Jri _ (,U
B ; VALUES OF N e FOR SAME OUTSIDE RADIUS FIG. 15. Comparison of "hoop stresses" between bell-shaped shell and conical shells.
differences between large quantities began to assume importance. Uncertainty as to the location of the 80-deg. solution point was removed when additional solution points for lesser values of fl~ were constructed and the distances between these points measured along their locus were plotted against their values in degrees. In Fig. 13 the slopes of the Ne' curves are very small at f12 = 80 deg.
132
J . L . MERIAM
[J. F. I.
and it m a y be inferred t h a t for a shell of/32 = 90 deg. there will be a negligibly small difference in m a x i m u m No'. The portion of such a shell between/32 = 80 deg. and/32 = 90 deg. m a y be approximated by a flat disk or by a steep cone (1) in both of which cases the "hoop stress" No' is decreasing with increased radial dimension. The original intent was to determine the solution point during the operating period of the analyzer and obtain the curves for the proper Q' and its derivatives directly from the machine plots. Before this was shown to be not feasible, a n u m b e r of additional trial solutions, not indicated in Figs. 7, 8, or 9, were made. With these solutions the expected linearity of Q' and its derivatives with respect to values of the initially assumed derivatives was verified experimentally. The over-all accuracy of the results is difficult to'establish, particularly since the j-coefficients were manually supplied to the analyzer. Several re-runs which were made resulted in curves which were visually identical. The best measure of accuracy was obtained from the departure of the linearity observed when more than two trial points were taken on any straight line on the control plots. In one case where four such points were investigated, the resulting solutions were plotted against the distances between the points for various values of/3 up to and including 80 deg., and the width of the bands which included the values for all four points were measured. These measurements showed t h a t the errors for Q', dQ'/dl3, d2Q'/d~ ~, and daQ'/d/3 a were usually less b u t not greater than 4-0.00015, 4-0.0015, 4-0.01, and 4-0.15, respectively. The machine plotted curves were read to 0.0001, 0.001, 0.01, and 0.1 for these same respective quantities, and in no instance did these readings deviate more than the last digit from numerous recorded counter readings which were more accurate than the plots. APPENDIX
The defining relations in this paper m a y be obtained from those in the investigation of conical shells (1) by the following substitutions: Replace ro r~ Q~ N~ Mr w e, X~
by by by by by by by by by
~r/2 rffl - cost/) - rl Q No Mo - w ea Xo
All other symbols remain the s~ime.
Aug., ~95o.]
ROTATING BELL-SHAPED SHELL
I33
Acknowledgment The author wishes to acknowledge the helpful suggestions made by Professor P. L. Morton and the cooperation of the staff of the differential analyzer at the University of California at Los Angeles. The accurate checking and calculations of Mr. A. L. Hale are also acknowledged. REFERENCES
(1) J. L. MERIAM, "Stresses and Displacements in a Rotating Conical Shell," Journal Appl. Mechanics, Trans. A. S. M. E., Vol. I0, No. 2, p. A-53 (1943). (2) H. WISSLER, "Festigkeitsberechnung von Ringflachenschalen," Doctoral Thesis, Zurich, 1917. (3) K. STANGE, "Der Spannungszustand einer Kreisringschale," Ingenieur Archly, Vol. 2, p. 47. (4) J. L. MERIAM, "Procedure for the Machine or Numerical Solution of Ordinary Linear Differential Equations for Two-Point Linear Boundary Values," Mathematical Tables and Other Aids to Computation, The National Research Council, III, No. 28, Oct., 1949, p. 532. (5) Discussion on Reference (1), Journal Appl. Mechanics, Trans. A. S. M. E., Vol. 10, No. 4, p. A 233-234 (t943).