ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR
ELASTIC SEMI-INFINITE CIRCULAR CYLINDRICAL SHELLS
By E.
REISSNER AND
J. G.
SIMMONDS
1. Introduction. The present paper contains further developments for the
problem of obtaining asymptotic expansions of the solutions of boundary value
problems of thin circular cylindrical elastic shells. The principle of the method
has been described earlier [4, 7] in conjunction with a part of the problem which
may be designated as the slow-circumferential-variation problem.
In what follows the earlier work is completed by removing the restriction of
slow circumferential variation. In so doing, it is established that there are two
two distinct classes of problems. In one class a slow circumferential variation of
variables is associated with two distinct types of variations in the axial direction,
one of the interior type and one of the edge zone type. For these two types this
paper obtains the appropriate two characteristic axial lengths as a function of the
characteristic circumferential length. In the other class of problems, a rapid
circumferential variation of variables is associated with only one characteristic
axial length.
The present paper once more shows the necessity of basing the asymptotic
developments on the joint consideration of differential equations and boundary
conditions in a manner which is described in the body of the paper.
New results which are obtained, in addition to supplying the solution in that
part of the range of parameter values which was not considered in the earlier
work, include the following.
It is shown, through use of tracer constants in the differential equations, that
certain simplifications in the differential equations are always permissible while
certain others are permissible only in certain ranges of parameter values.
It is shown that the appropriate form of the asymptotic expansions is obtained
much more easily once the problem has been reduced to a single differential
equation (for the transverse displacement w, and of the eighth order), in comparison with an earlier derivation based on the direct use of the three simultaneous differential equations for the three midsurface displacement components
u, v, w.
It is shown, as found by the junior author [9], that certain transformations of
the boundary conditions result in an essential simplification of part of the work.
This point is further developed in a sequel to the present paper dealing with the
subject of influence coefficients for semi-infinite and infinite shells.
2. The basic equations. The static equations of equilibrium of a circular
cylindrical shell, free of surface loads, are taken in the following form
+ N ex = 0,
Q~ + Qe + No = 0,
N~
M~
+
N~o
+ Ne
- cqQo = 0
(2.1a, b)
a(Nox - N xo ) - crrMox = 0
(2.2a, b)
Mex - aQo = 0,
M~o
1
+ Me
- aQo = 0
(2.3a, b)
2
E. REISSNER AND J. G. SIMMONDS
Relevant geometric quantities, stress resultants, and stress couples are defined
in Fig. 1. In the above equations, primes denote differentiation with respect to
the nondimensional axial distance x = z/ a, and dots denote differentiation with
respect to (J. The quantities Cq and Cm are tracer constants which are equal to
one in the exact equilibrium equations. Their purpose is to allow us to assess
independently, at any point in our calculations, the effects of neglecting the
transverse shear force term in (2.1 b) or the twisting couple term in (2.2b).
By elimination of N s,. - N,.s, Q,. and Qs , the six equilibrium equations reduce
)!("
Qx
N8X
Nx
Q8
FIG. 1. Geometrical notation and stress conventions
to the following three:
N~ + ~ (N,.s + N s,.)" + ~: M;,. =
0
(2.4)
~ (N,.s + N s,.)' + N; - ~ (M~s + M;) - ~: M;,. =
0
(2.5)
+ (M,.s + M s,.)'· + M;" + aN s =
0
(2.6)
M:
In accordance with recent work [1], we take as stress-strain and strain-displacement relations the set
N,.
aM,. =
+ liES), Ns = G(ES + liE,.)
N,.s + N s,. = (1 - II)G"{
D(</J~ + II</J;),
aMs = D(</J; + II</J~)
=
G(E,.
aM,.s = (1 - II)D</J; ,
aMsx = (1 - II)D(</J; - cww)
(2.7a,b)
(2.8)
(2.9a,b)
(2.lOa,b)
3
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
aE x
,
aEe
U,
V· -
acPx = -w' acPe = - (w·
+ U·
(2.11a,b,c)
2aw = v' - u·
(2.12a,b,c)
a'Y = v'
W,
+ c.v),
2
11 ), E is Young's modulus, 11 is
a
where C = Ehj(l - l), D = Eh j12(1 1
Poisson's ratio, and h is the shell thickness.
In the above equations we have introduced the two additional tracer constants
COl and C. to assess the influence of the rotation about the normal, w, in (2.1Ob)
and the influence of the v-term in (2.12b). For the principle of virtual work to
be valid in its usual form, we must have that c. = Cq and COl = cm •
By use of this same principle, a proper set of boundary ~onditions for the
semi-infinite cylindrical shell acted upon by a set of self-equilibrating loads at
x = 0 may be obtained in the form
1
2"
[(N x - Nx)ou
+
(Sx - Sx)ov
+
(Rx - Rx)ow
1
- a- (M x
-
(2.13)
M",)ow'j.,=o d8
=
0
where a bar indicates a prescribed quantity and where
as., = ta(Nxe + N e.,) - cqMxe - tcmMe.,
and
aR., = M~ + (Mxe + Me.,r
(2.14)
(2.15)
3. Reduction to a single differential equation for w. By substituting (2.7) to
(2.10) into (2.4) to (2.6), using (2.11) and (2.12), and setting
2C
2
A4 = ~ = 12a »1
D
h2
the equilibrium equations are expressed as a set of three simultaneous differential
equations for u, v and w.
Lnu + L 12V + L 1aw = 0
(3.1)
L 21U
L a1u
+L +L w = 0
+ L + Laaw = 0
22V
(3.2)
2a
(3.3)
a2v
In this the L i ; are constant-coefficient differential operators of the form
+ .1\. -4L(~)
L '3.. -- L(O)
"'3
"'3
where
[L:;'I
~ [(
)"+H1-II)(
H1+II)(
-II(
)
),.
..
HI + 11)(
H1-II)(
)'
-(
),.
)"+(
)"
) ••
-II(
-(
(
(3.4)
)']
)"
)
(3.5)
Eqs. (2.10) correspond to Eqs. (21) of [1]. It can be shown that they are equivalent,
except for terms small of higher order, to the relations
1
M%8 = M o", =
HI -
v)D(",;
+ "'; -
c",w)
4
E. REISSNER AND J. G. SIMMONDS
and
r
HI -
[LWl
=
v)cm cw (
-Hl- v)cmcw(
) ••
-HI -v)cmcw(
HI -
)"
l-l(l - v)c",( )'"
v )(cm Cw
+c q cv (
[tel - v)c",
-l(l - v)cm (
)"
+ 4c.cq )( )"
[!(1 - v)em
)/ ••
+ co< )...
) ••
+ c.l( )". + C.( )...
l
)/"
+ c.l(
J
(3.6)
\74
Eqs. (3.1) to (3.3) are solved by first obtaining from (3.1) and (3.2) two equations for u and v in terms of w,
(L11L22 -
L 12L 21 )u
=
(L 12L 23 -
L13L22)W
(3.7)
(L11L22 -
L 12L 21 )v
=
(L13L21 -
L 11L 23 )W
(3.8)
and by introducing (3.7) and (3.8) into (3.3). This leads to an equation,
Lw = 0
(3.9)
where the operator L has the form
L
=
1 L,; 1
=
L(O)
+ },.-4L(1) + },,-SL(2) + },.-12L(3)
(3.10)
Detailed inspection indicates that every derivative appearing in L(2) or L(3)
also appears in either L(O) or L(l). Since the basic equations (2.7) to (2.12) do
not correctly account for terms of relative order},.-2, we are justified in neglecting
terms of relative order },.-4 or higher in (3.10) so that, effectively,
L
The operator
L(O)
=
L(O)
+ },.-4L(1)
(3.11)
follows from (3.5) as
L(O)
=
1 L~~) 1
=
HI -
v)(l -
v2 )(
(3.12)
)""
To calculate L (1), we write
L(l)
where the
[MWl
,,~.
L...J 1. ,1=1
(-l)i+iL\~)M\~)
"'3
(3.13)
1.1
MW are the minors of 1LW I. We have from (3.5)
= t(l - v)
[
()"
()/
v( )", - ( )' ••
(
)/.
( ).. + 2(1 +v)(
)"
-(2 +v)( )'" - ( ) •••
V(
-(2
)", - ()/..
]
)'" - ( )...
+ v)(
(3.14)
\74
Introduction of (3.11) to (3.14) into (3.9) gives as a differential equation for w,
...s
vw
+
+ (Cq + Cv )W ...... + CqCvW....
+ [(1 - v)(c + c + (2 + v)(cq + cv)1V' w""
+ (co + cq)w"···· + [(1 - v)cmc + (3 + v)cqcvlw""
(1 _
V
2)"1\4W ""
m
2
w)
w
(3.15)
=
0
An analogous consideration of (3.7) and (3.8) reduces these equations to the
following explicit form
_
vu-
....4
vw -w••
["
+1
~
( Cm
+ 11_vCq
+v )
n2
vW ••
J'
(3.16)
5
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
{(2 + v)w" + w·· - h[Cq 2
+ (cm + 1~ v Cq) w"··]}" (3.17)
It is important to note that the approximation (1 + }.-4)( 1) ;::::::; (1) must not
¢v
=
V W"
be made at too early a stage in the calculations. The reason for this is shown by
(3.12): in calculating L(O) = I L;J) I all terms cancel identically, except those
involving ( )"". Hence none of the terms in L(1) except a ( )""-term can be
compared with those of L(O), and to calculate the terms of L(1) correctly, we must
retain all terms of relative order }.-4 in (3.1) to (3.3). Specifically, if we were
to replace LW by zero, since it contains the derivatives of the same type as
L~~), then the significant term cqCvw.... would not appear in (3.15).
4. Introduction of scale factors. For the purpose of solving the differential
equations (3.15) to (3.17), we introduce, in addition to the given large parameter
}., two parameters u and r by setting
~
=
uX
'11
= rO
( 4.1)
The new independent variables ~ and '11 are to be such that differentiation with
respect to them does not change orders of magnitude.
The parameter r is to be considered as given, a/ r being a characteristic circumferentiallength associated with the nature of the circumferential variation
of the prescribed edge loads or displacements.
The parameter u is to be determined suitably in terms of A and r. Determination of u is to be effected in such a way as to obtain asymptotic expansions in
terms of some small parameter without any loss of orders of the relevant systems of differential equations.
We introduce (4.1) into the differential equation (3.15) for wand now use
primes and dots to indicate differentiation with respect to ~ and '11. In this way we
obtain
u\ )"
+ i(
)..]4W
+ (1
- /})u 4"),.4W""
+ (cq + cv)r6w•••••• + cqc r4w····
(4.2)
V
+ k 1u\4W""" +
k2u 2r 4w"····
+
k au 2r 2w""
0
where kl' k2, and ka are certain constants expressible in terms of Cm , COl, Cq
and c. which are of order of magnitude unity if the c's are of order of magnitude
unity.
We will show that for the class of problems to be considered here the terms
with k. in (4.2) are always of a negligibly small order and that consequently
the terms with Cm and COl are always negligible.
To determine the possible values of u, we take u and r in the form
u
AP ,
r
q
A
( 4.3)
where q is assumed to be given and p is to be determined. The guiding principle
in this determination is that as A ~ 00 a reasonable limiting form of equation
( 4.2) must result.
6
E. REISSNER AND J. G. SIMMONDS
It is convenient to consider two cases in the determination of p: 0 ~ q < 1,
and q ~ 1. For both cases there are three possibilities to be considered: p > q,
p = q, and p < q.
The case 0 ~ q < 1. For p > q, equation (4.2), upon division by u 8 and useof
( 4.3), can be written in the form
w""""
+
(1 - p2)'A4(1-p)W""
+ O('A2(q-p»
= 0
( 4.4)
In order that the first two terms in (4.4) balance, which is necessary if, in the
limit as 'A ~ 00, (4.4) is to admit solutions which decay as ~ ~ 00, we must set
1,
p
I.e.
u
'A
(4.5)
8
For this choice of u, equation (4.2) can, upon division by u , be written in the
form
(T/'A)2( r·J4w
(1 - p2)W''''
'A- 2[k1(T/'A)2W""··
= 0 (4.6)
( )" +
+
+
+ ... ]
Since (2.7) to (2.12) already contain errors of relative order 'A-2, it is consistent
to neglect the underlined terms in (4.6).
).
For p = q, t h e term ( 1 - p2) u 4'A4""
w
= ( 1 - p2) 'A4(p+l) w " " .III ( 4.2,
III the
limit as 'A ~ 00, dominates all others. Since the equation w"" = 0 admits no
decaying solutions, p = q is not a possibility for 0 ~ q < 1.
For p < q, equation (4.2) can, upon division by T8 and by use of (4.3), be
written as
[w····
+ (c q + c )'A-2qw·· + cqc 'A-4Qw)""··
+ (1 - p2)'A4(1+P-2 Q)W"" + O('A2(p-Q»
v
v
= 0
(4.7)
To insure that (4.7), in the limit as 'A ~ 00, will admit decaying solutions, it is
sufficient to balance the first and fourth terms in this equation, which requires
that we set
p = 2q -
1,
i.e.
u = T2/'A
For this choice of u, (4.2) can, upon division by
[( T/'A)2( )"
+(
(c q
r·]4w
+
+
8
T ,
(4.8)
be written as
(1 - i)w""
2
Cv )T- W······
+ C CvT-4W···· + 'A-2[k 2w"···· + ... J =
q
0
(4.9)
Since (2.7) to (2.12) already contain errors of relative order 'A- 2, it is consistent
to neglect the underlined terms in (4.9).
8
The case q ~ 1. For p > q equation (4.2), upon division by u and use of (4.3),
becomes identical with (4.4). However, the choice p = 1, which is necessary
if (4.4) in the limit as A ~ 00 is to admit decaying solutions, contradicts the
assumptions p > q. Thus p > q is not a possibility for q ~ 1.
For p = q, equation (4.2) can, upon division by T8 and use of (4.3), be written
in the form
[( )" + ( r·tw + (1
- p2)'A1-
w"" + O('A-2Q ) =
Q
0
( 4.10)
7
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
If q = 1, all terms appearing explicitly in (4.10) are of equal order of magnitude,
whereas if q > 1, the dominant terms in (4.10) are
[( )" + ( t°J'w
In either case, (4.10), in the limit as A ~ 00, admits decaying solutions. Thus
the choice p = q, i.e. (J' = T, is valid for all q ~ 1. For this choice of (J', equation
(4.2), upon division by T 8 , takes the form
[( )" + ( t·J'w + (1 - V2)(A/T)4W""
( 4.11)
2
4
T- [(C q + c.)w······ + k1w""··J + T- [ •• • J = 0
+
Since T -2 ~ A-2 for q ~ 1, the underlined terms in (4.11) are negligible.
For p < q, equation (4.2) can, upon division by T8 and use of (4.3), be written
as
w········ + O(A2(p-q) =
0
( 4.12)
In the limit as A ~ 00, the only decaying solutions which (4.12) admits are nonperiodic. Since we are concerned with circular cylindrical shells complete in the
8-direction, such non-periodic solutions are unacceptable. Therefore p < q is
not a possibility for q ~ 1.
All possible values of (J' for any given value of T have now been determined.
Our results may be summarized as follows.
If 0 ~ q < 1, i.e. if 1 ~ T < A, then either (J' = A, in which case (4.2) can
be simplified to
[( )" + (T/A)2( t·J'w + (1 - v2)w"" = 0
( 4.13)
or else
(J'
= T2/A, in which case (4.2) can be simplified to
[( T/A)2( )"
If q
~
+ ( t·J4W +
1, i.e. if T
~
2
(1 - v )w""
A, then (J'
.... + (Cq + Cv ) T-2W •••••• + CqCvw-
=
[( )" + ( rtw
and (4.2) can be simplified to
+ (1 - V2)(A/T)4W"" = 0
0
( 4.14)
T,
(4.15)
The preceding results show that when 1 ~ T < A, there occur two distinct
types of solutions. One is an edge-zone (or boundary layer) type solution associated with the (dimensional) axial length scale v'oJi = OCalA). The second,
called here the interior-zone solution, is associated with the axial length scale
2
(a/T2 )VOJh = O(aA/T ). As T increases, the interior-zone length scale decreases
from its value aVOJh for T = 1. For T = Ai, it is of the same order of magnitude
as the midsurface radius a, and when T = A, it coincides with the edge-zone
length scale v'oJi.
These results further show that as T increases beyond A, all solutions are of
the edge-zone type where now the length scale is a/ T. As the dominant terms in
(4.15) indicate, the behavior of the cylinder approaches that of a flat plate for
sufficiently rapid variation of the edge-loading.
8
E. REISSNER AND J. G. SIMMONDS
The values which T can assume for T > "A are however, not unbounded. An
upper limit is imposed by the condition that the circumferential length scale
of the applied loading, a/T, remain small compared to the thickness h of the
shell, i.e. that T « "A2. Otherwise the basic assumption of shell theory that the
transverse shearing and normal stresses are small compared to the in-plane
direct and bending stresses is violated.
Our determination of the various values of the scale factor (j has three important consequences.
First, (4.13) to (4.15) are all special cases of the equation
[u 2 (
)"
+
i( t·tw
+
(1 _
p2)(j4"A
w""
4
+ (c q + Cv)T6W•••••• + cqCv/w····
0
(4.16)
Equation (4.16), with Cq = c. = 1, was first obtained in 1938 by Donnell [2],
who derived it in order to correct the defects known to exist in the simpler equation
[(j2(
)"
+ i(
t·],w
+
(1 -
w"" = 0
4 4
p2)U "A
(4.17)
obtained by him five years earlier [3]. Thus our analysis re-confirms the validity
of the "extended" Donnell equation, at least for the class of problems considered
herein. 2
Second, comparing (4.16) to (4.2), we see that in (4.16) all the k-terms, being
in all cases at most of relative order "A-2, have been neglected. Since the tracer
constants Cm and c'" appear in these k-terms alone, we conclude that neglecting
the rotation-about-the-normal term in (2.lOb) and replacing the moment
equilibrium equation about the normal (2.2b) by the approximation N x6 = N6x
has no significant influence on the differential equation for w.
And third, note that if we set Cq = C. = 0, the extended Donnell equation
(4.16) reduces to the simplified Donnell equation (4.17). A comparison of the
two equations shows that the simplified Donnell equation contains errors of
relative order T -2. This, of course, agrees with the generally recognized fact,
pointed out by Donnell himself [3], that the simplified Donnell equation is valid
only if the circumferential variation of the normal displacement w is sufficiently
rapid.
6. Asymptotic solution of the differential equation for w. We now seek to
simplify the solution of the extended Donnell equation (4.16) by expressing w
as an asymptotic series in powers of some suitably chosen small parameter. We
shall call the leading term of such a series "a first approximation to w", and shall
refer to the differential equation this leading term must satisfy as "a first approximation equation for w".
2 Another recent, independent confirmation of the validity of the extended Donnell
equation has been given by Johnson [8] who has concluded from an asymptotic analysis
of the full three-dimensional elasticity equations that the solution of the extended Donnell
equation for a semi-infinite cylindrical shell under self-equilibrating edge loads provides
an uniformly valid first approximation to w for all values of the circumferential length scale
at afT in the range a < afT « h.
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
9
A single asymptotic series for w, valid for all values of T, is of no use since it
would lead us back to (4.16) as a first approximation equation. Nor are simpler
first approximation equations obtained by sf>litting the range of T into 1 ~ T < A
and T ~ A even though this was the natural subdivision to use for the determination of rT. This is because in the range 1 ~ T < A and for the choice rT = i/A,
(4.16) would again be obtained as a first approximation equation for w.
An investigation of the various ways of subdividing the range of 7 in order to
obtain a reasonable set of first approximation equations for w reveals that a
suitable choice is 1 ~ T < At and At ~ T« A2. This choice, briefly, is based on
the following reasons. First, there must be at least one subdivision point within
the range 1 ~ 7 ~ A, otherwise one obtains a first approximation equation for w
identical to the extended Donnell equation (4.16). Second, one subdivision point
within the full range 1 ~ 7 « A2 is sufficient since, so long as solutions for the
full range of T are required, no further subdivisions of the range of T will lead to
first approximation equations all of which are simpler than those one obtains
with just one subdivision point. And third, only for the choice of subdivision
point T8 = At do the errors in the first approximation equations for T ~ 78 and
T ~ T8 become of the same order of magnitude when one sets 7 = 78 •
Solutions of the differential equation for w for 1 ~ T ~ A\ as reference to the
discussion at the end of Section 4 indicates, are associated with "slow circumferential variations" in the sense that the circumferential length scale alT (i.e.
minimum distance in the 8 direction over which significant changes in w
occur) is large compared to the axial length scale Yah. Solutions of the differential equation for w for At < T« A2 are associated with "rapid circumferential
variations" in the sense that the circumferential length scale alT can become
equal to or less than the axial length scale Yah.
The appropriate simplified forms of (4.16) for 1 < T < A! and A! ~ T« A2 can
be immediately deduced from (4.13) to (4.15). For 1 ~ 7 < At we find, upon
neglecting all terms of relative order A-2, that when rT = A, the associated simplified equation is
[Willi + 4(TIA)2W"" + (1 - p2)W]"" = 0
(5.1)
and that when
[w····
rT
= T2lA, the associated simplified equation is
+ (c q + C.)T-2W" + CqC 7- W + 4(7/A)2w" ..r .... + (1- p2)W"" = 0
4
V
(5.2)
Eq. (5.1) can be immediately integrated four times, to yield
W""
+ (1 -
p2)W
+ 4(TIA)2W""
= Co(.,,)
+ Cl("')~ + C2(.,,)~2 + C3(.,,)~3
(5.3)
Since the arbitrary constants Cj (.,,) do not lead to solutions for w which decay
as ~ ~ 00, we set them equal to zero. Thus for 1 ~ T < Al, the full equation (4.16)
for w simplifies to either (5.2) or to
Willi
+
(1 - p2)W
+ 4(TIA)2W""
= 0
(5.4)
Eqs. (5.2) and (5.4) are each of the fourth order in ~ so that, altogether, the order
of the original equation for w is preserved.
We denote the solutions of (5.4) and (5.2), respectively, by WeWe(~e,.,,)
10
and
E. REISSNER AND J. G. SIMMONDS
WiWi(~i,
'1/), where
~e
2
~i = (r /A)X
= AX,
(5.5)
and where We and Wi are edge and interior-zone reference displacements chosen
so that (We, Wi) = 0(1). The magnitudes of We and Wi are to be determined from
the boundary conditions. Because of linearity, the complete solution for W
can be written
W = WeWe(~e, '1/)
+ WiW,(~"
'1/)
Both (5.2) and (5.4) contain the small parameter (r/A)2, which suggests
the use of asymptotic series for We and W, of the form
We = WeO
+ (r/A)2Wel +
Wi = WiO
+
(r/A)2W il
+ ...
(5.6,7)
Substituting these series for W. and Wi into (5.4) and (5.2) respectively, and
equating to zero the coefficients of successive powers of (r/A)2, we obtain the
following two sequences of differential equations for W eO , WeI, ... , and W,o,
W il
, ••.
W;~"
W;~"
+ (1
+
2
(1 - v )W.0 = 0
- V2)W'1
+ 4W;~"
(5.8)
= 0, etc.
(5.9)
and
+ (Cq + c. )r-2W" + cqCVr-4W j•••• + (1 v2)W"" = 0
[Wii" + (cq + c.)r- 2Wii + cqc r-4Wid ....
+ (1 - V2)W~~" + 4W~~""" = 0,
[W ••••
iO
iO
iO
-
(5.10)
iO
V
etc.
(5.11)
Since (5.2) and (5.4) already contain errors of relative order A- 2, only the
first two equations in these sequences are physically meaningful. Beyond this
if r = 0(1), the expansion parameter (r/A)2 in (5.6) and (5.7) reduces to
2
A- , in which case only (5.8) and (5.10) are physically meaningful.
For At ~ r « A2 , reference to (4.13) to (4.15) shows that the three possible
simplified equations for ware all special cases of the single equation
[(o/r)2( )"
)"·tw
(1 - v2)(uA/r2)4w""
(cq Cv )r-2W...... = 0 (5.12)
+(
+
+
+
Since we wish to obtain from (5.12) a single first approximation equation valid
2
for all r in the range Ai ~ r « A , we cannot assign to U anyone value. This
means that, depending on the value of r some pair of the three terms in (5.12)
(U/
( " /r 2)4W" "
r ) 2W" " " " , W•••••• ,u"
will dominate the remaining one (unless, of course, r = A, in which case the
three terms are all of the same order of magnitude).
In any event, the last term in (5.12) is always of relative order r -2 or smaller
compared to the dominant terms (whichever ones they may be). This suggests
the following expansion for w,
11
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
+ r-2W1(~' TJ) + ...
W = WO(~, TJ)
(5.13)
Since r ~ A\ at most, only the first two terms on the right hand side of (5.13)
have physical significance.
Substituting (5.13) into (5.12) and equating to zero the coefficients of successive powers of r -2, we obtain an infinite sequence of equations, the first two
of which read
[(o/r)\ )"
[(O/r)2( )"
+(
)"·tW1
+ (1
+(
)"·two
+
(1 - 7})(A/r)4w~'" = 0
+ (c + cv)wo·····
- 7})(A/r)4iv~'"
q
(5.14)
= 0 (5.15)
In the next two sections, we shall study the three first approximation equations (5.8), (5.10) and (5.14) together with the set of auxiliary equations for
u, v, the stress resultants, and the stress couples associated with each of these
first approximation equations.
Once methods for solving the sets of first approximation equations and satisfying the boundary conditions have been established, solution of the second approxition equations is a straightforward matter, and we shall not concern ourselves
here with this latter problem.
6. First approximation equations for slow circumferential variations (1 ~
Having, in the preceding section, obtained first approxinlation equations (5.8) and (5.10) for the edge and interior-zone contributions to W for the
range of 1 ~ r < At, we list in this section the associated first approximation
equations for U and v together with expressions for stress resultants and couples.
All edge- and interior-zone quantities are assumed to have asymptotic series
expansions of the same form as the right hand sides of (5.6) and (5.7).
It is convenient, for what follows, to introduce dimensionless displacements
U and V by setting
"C
< J..!).
U
=
v = (Ve, Vi)V
(u e , Ui)U,
where U and V are assumed to be 0(1) and (u e , Ui) and (V e , Vi) are reference
displacements to be related to the reference displacements We and Wi by means of
the auxiliary equations (3.16) and (3.17).
Edge-zone equations. The first approximation equation for W eO , given by
(5.8), is
W~~"
+
First approximation equations for
U
= ue[UeO
+
(r/A)2U e1
(1 - p2)WeO
U
+ ... ],
= 0
(6.1)
and v follow upon setting
v = Ve[VeO
+
(r/A)2Ve1
+
"'J,
inserting the above expansions into (3.16) and (3.17), introducing the change of
variables
x = A-1~e
()
r
-1
TJ
and equating to zero the coefficient of the lowest power of (r/A)2. In this way
12
E. REISSNER AND J. G. SIMMONDS
we obtain the two equations3
AUeU;~"
"WeW;~/,
A2VeV;~"
= (2 + ")TWeW;~·
(6.2a,b)
In order that U eO and VeO be, at most, 0(1), we set
Ue = A- We ,
1
2
Ve = (T/A )We
(6.3a,b)
With the aid of (6.1), (6.2) may be integrated to yield explicit solutions for
U eO and VeO in terms of WeO . Since the homogeneous solutions so obtained do
not decay as ~e ~ 00, they must be discarded. Thus we obtain for U eO and VeO
the equations
(1 - ,,2)UeO = _"W;~/,
(1 - ,,2)Veo = -(2
+ v)W;~·
(6.4a,b)
By use of (6.3), (6.4), and the equations of Section 2, the leading terms of the
asymptotic expansions of the edge-zone contributions to the stress resultants
and couples may be expressed explicitly in terms of WeO as follows:
+ ... j
aN~/C
=
(T/A)2We[W;~··
aN:/C
=
-we[(1 - ,,2)Weo
(6.5)
+ ... j
(6.6)
+ ... j
- (T/A)We[W;~/· + ... j
- A-lWe[W;~' + ... j
-A-2We[W;~ + ... j
-A-2We["W;~ + ... j
-(T/A 3 )we[(1 - ,,)W;o + ... j
a(N~6, N:x)/C = -(T/A)We[W;~/·
aSx"/C =
aR~/ C
=
M~/C
=
M:/C =
(M~6, M:x)/C =
(6.7)
(6.8)
(6.9)
(6.10)
(6.11)
(6.12)
Eqs. (6.1) and (6.3) to (6.12) agree, respectively, with Eqs. (5.16a), (5.1),
(5.16b) and (5.19) to (5.24) of [4j upon setting 8 = r- 11/ in these latter equations.
Interior-zone equations. The first approximation for W iO , given by (5.1O),is
[Wio··
(c q
Cv )T-2Wio
Cq Cv T- 4W;0j""··
(1 - ,,2)W~~" = 0 (6.13)
+
+
+
+
First approximation equations for u and v follows upon setting
u = u;[U iO
+ (T/'A)2U;r + ... ],
v =
+ (T/A)2V + ... ],
V;[ViO
il
inserting the above expansions into (3.16) and (3.17), introducing the change of
variables
x = (A/T2)~i,
8
T
-1
1/
and equating to zero the coefficient of the lowest power of (T/A)2. In this way
3 Here, and in what follows, we assume P = 0(1) » (r/A)2, otherwise certain of our formulas must be modified. In particular if p = (r/A)2 po , with Po = 0(1), then (3.16) shows
that (6.2a) must be replaced by AU,U:t = (r/A)2w,(poW:~ + W;;), and (6.3a) by
u, = (r'/A 3)W, .
13
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
we obtain
- X-lWi W~O·,
u,Uio··
TV"
Vio··
Wi
Wio·
(6.14a, b)
In order that U iO and ViO be, at most 0(1), we set
A-1Wi,
U,
Vi
T
-1
(6.15a, b)
W,
Eqs. (6.14) may be integrated with respect to TJ. Since the homogeneous solutions so obtained are not periodic in fJ, they are discarded and we have
010
- W~o,
Vio
WiO
(6.16a, b)
By use of (6.15), (6.16), and the equations of Section 2, the leading terms of
the asymptotic expansions of the interior zone contributions to the stress resultants and couples may be expressed explicitly in terms of W iO as follows:
a(N~)"·/C
aN~/C
= -(r/A)2wi [(1 -
a(N~e, N~S·/C
=
a(S~)"··/C
=
aR!/C
=
M~/C =
M~/C =
(M~8' M~x)"/C
+ ... ]
(r/A)3w;[(1 - v2)W~~' + ... ]
(T!A)3Wi [(1 - v2)W~~' + ... ]
- (r /A )wi[(2 - v)(Wio + W,o)' + ... ]
-(r /A )wi[V(Wio + r- W;o) + ... J
2 4
2
- ( T /A )W,[Wio + r- W,o + ... J
= (T/A)4Wi [(Wio
4
5
2
4
+
+ ... ]
V2)W~~
2
T- W,O)"·
T-
2
2
2
5
= -(1 - 1')( r /A )w;[(Wio
+ T- W,O)' + ... J
2
(6.17)
(6.18)
(6.19)
(6.20)
(6.21)
(6.22)
(6.23)
(6.24)
In (6.17), (6.19), (6.20) and (6.24), it has been necessary to write the left hand
sides in differentiated form in order that the right hand sides of these equations
be expressible in terms of WiO and its derivatives.
Eqs. (6.13) and (6.19) to (6.24) agree, respectively, with Eqs. (6.17), (6.1),
(6.15) and (6.21) to (6.28) of [4] upon setting 0 = r- 1TJ in these latter equations
and upon writing the equations for N! , S~ and M!8 in the form of (6.19), (6.20)
and (6.24).
7. Leading terms in combined edge- and interior-zone expansions and dominant stresses for slowly varying edge loads. For easy reference, we list below,
for 1 ~ r ~ At, the leading term expressions for the midsurface displacements,
the axial rotation, the stress resultants, and the stress couples. These results
follow from the equations of Section 6 upon combining the edge- and interiorzone first approximation contributions to the various quantities. (The expressions for v = 0 listed below have been taken from Appendix D of [9].)
W"-' We WeO
W'(x)
,
---a- "-' A [ We WeO
+ Wi W iO
(7.1)
+ (r)2
X Wi W' ]
iO
(7.2)
14
E. REISSNER AND J. G. SIMMONDS
U
.. (TJ ) ""' _!'" [ We {[V/(1
- V2)]W~~'··,
( /,")2 '..
1\
r
-
1\
W eO
,
V~
V
+-V2 (r)2
"•
V·(TJ) ""' -1 [ -2- We WeO
r
I-v
X
aN~(TJ) ""' (~y [ We W~~····
a~o ""'
-(1 - ,})We WeO
OJ + W..W' Jl
= 0
,0
+ Wi W iO ]
(7.4)
- (1 - v2)W; W~~]
+ (~y wi(Wio +
r-2W io )··
a[NxO(TJ),
NOx(TJ)]···
(r)[ - We W",
•••• + (r)2 W.. W~,,]
-'--'--'--.:..:....:....,-;;-.-----'---'-'--- ""' >;:
eO>;:
,0
~x ""' _~ [ We W;~ + (~y Wi {(~~~o2~iOr,-2 W iO ),
Mo,......,
C
_~
X2
[We {vW;~ ~.. V ~
(r/X) W eO , V =
M;o(TJ)CMox(TJ) ,......, -(1 - v)
:
Ot. + (~)2
wi(Wio +
of
X
(7.3)
(7.5)
(7.6)
(7.7)
~ ~}]
(7.8)
r-2WiO)]
(7.9)
(i2) [WeW~~· + (~yWi(Wio + r-2WiO)']
(7.10)
Of particular interest are the dominant direct and bending stresses N /h
and 6M/h 2that occur in the shell for the four values (X/r )2,1, (r/X)2 and (r/X)4
of the displacement ratio We/Wi which are found to occur in the stress boundary
value problem, the first and last ratios occurring only if V = O( i/X2). The
results are as follows
We/Wi = (X/r)2. The dominant stresses are, from (7.6) and (7.8)
u; ""'
U8D
E""'
- (We)
a W eO
-0(1 _
v (:e) W;~,
2
)
(7.11)
UOB ""' VUxB
(7.12)
We/Wi = 1. The dominant stresses are, from (7.6), (7.8) and (7.9)
7; ""' _(:e) WeO
UxB ""'
E
_ M(
-V
3 1
a
- V2) (we) W"
eO
,
(7.13)
UOB ""' VUxB
(7.14)
We/Wi = (r/X)2. The dominant stresses are, from (7.5), (7.6), (7.8) and (7.9)
U;~TJ) ,......, _(~y (:i) W~~
(7.15)
An alternative to giving a separate treatment of the first approximation equation for
O(-r/}..)2, is to retain certain higher order terms, e.g. the term (-r/}..)2w.W:; in (6.2a). But
such a procedure would violate the purpose of a first approximation theory.
4
" =
15
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
UeD
-"" -
E
We/Wi
(r)2
A
(Wi)
-
a
W eO
(7.16)
u; '"" -0(1 - p2)
(~y
(:) [W~~ +
_/<13 (1 -UeB '"" -v.:>
E
(r)2
-
(Wi)
[pWI! + Woo + r -2W 1
-
p2)
A
a
eO
p(Wio
+ r-2W io )]
iO
iO
(7.17)
(7.18)
= (r/A)4. The dominant stresses are, from (7.5) and (7.9)
(:i) W~~
er;; '"" -vI3CI _ p2) (~y (:i) (Wio + r-2W io )
u;~~) '"" _(~y
(7.19)
(7.20)
We observe that, to a first approximation, the dominant stresses are produced
exclusively by the edge-zone contribution to W if We/Wi = (A/r)2 or if We/Wi = 1,
by a combination of the edge- and interior-zone contributions to W if We/Wi =
( r /A) 2, and exclusively by the interior-zone contributions to W if We/Wi =
(r/A)4. Furthermore, the dominant direct and bending stresses are of equal order
of magnitude.
8. First approximation equations5 for rapid circumferential variations (:1.1 <
"= «:1.\ In section 5 we assumed, for the range At ;;;;; T «>.,2, an expansion for
W of the form
W(~,~)
and we obtained for
Donnell equation
Wo
= wo(~,~)
+ r-2w1(~'~) + ...
an equation that was identical in form to the simplified
[(er/rn )"
+(
)"otwo
+ (1 -
p2)(uA/r2)4w~'"
= 0
(8.1)
It is easy to see from (3.16) and (3.17) that, upon assuming asymptotic expansions for u and v similar to that for w, Uo and Vo will satisfy the equations
[(u/r)2( )"
+(
[(er/r)2( )"
+(
)""]2UO = (er/r2)[p(a/r)2w~ - WOO]'
)"O] 2vo = r- 1[(2 + p)(a/r)2w~ + woo]"
(8.2)
(8.3)
which also are identical in form to the simplified Donnell equations for u and
v, [3].
Rather than working with (8.1) and the auxiliary equation (8.2) and (8.3),
it is here simpler to dispense with the use of scale factors altogether and to
reduce the first approximation equations to the solution of a single fourth order
differential equation for a complex displacement-stress function '1'. According to
Novozhilov [6, p. 90], the method of reduction which follows was first given by
S. Feinburg in 1936.
5 A treatment of the first approximation equation for the special case v = 0 may be found
in appendix C of [9J.
16
E. REISSNER AND J. G. SIMMONDS
It was remarked at the end of Section 4 that use of the simplified Donnell
equations is consistent with setting all tracer constants equal to zero. This
implies that the two in-plane force equilibrium equations (2.1), may be replaced
by those for a plate and hence can be statisfied identically by introduction of the
Airy stress function F, as follows. (For simplicity we henceforth drop the subscript zero.)
N.,
FOo,
Ne
N.,8 = -F'O
F",
(8.4a,b,c)
With c. = 0, (2.9) and (2.10) read
a2M8 = -D(w' o + vw")
a2M., = -D(w" + PWOO ),
a2M.,8 = iM8., = - (1 - p)DW'O
(8.5a,b)
(8.6)
Substituting these expressions for the stress couples along with (8.4b) into
(2.6), we obtain
D'ilw - a3F" = 0
(8.7)
A second equation relating wand F follows from the compatibility condition
of the in-plane strains which, when expressed in terms of F and w, reads
a'fiF
+ Ehw"
= 0
(8.8)
By multiplying (8.8) by ia(AD)! and adding it to (8.7), we obtain the single
differential equation
v\Ir -
i2K~"
= 0
(8.9)
where
a-'/,
2
W=w.2K
Eh F
(8.10)
K2 = v3(1 - p2) a/h = !~}..2
(8.11)
and
The effective edge in-plane and transverse shear stress resultants, given by
(2.14) and (2.15), now assume the form
8.,
N
_F'o,
R., = -a-3D[w" + (2 - p)WOO]' (8.12,13)
x8
The displacements u and v can be related to wand F by using any two of the
three equations (2.11a, b, c). Using the last two, we have
OO = -w' - (a/Eh)[F" + (2 + p)rO]'
U
(8.14)
v' = w
+ (a/Eh)(F"
- prO)
(8.15)
9. Stress Boundary Conditions. At an edge x = 0, the contracted stress
boundary conditions, in accordance with (2.13), assume the form
N.,(O, fJ) = Nx(fJ),
8.,(0, fJ) = 8.,(fJ)
(9.1a,b)
17
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
R:.(O,O) = R,,(O),
M,,(O, 0) = M,,(O)
(9.1c,d)
where S" and R" are defined by (2.14) and (2.15), respectively. We now consider the specific form the left hand sides of Eqs. (9.1) assume when the applied
edge loads are either slowly varying (1 ~ r ~ At) or rapidly varying (Ai ~
r« A2 ).
Because the statement of stress boundary conditions for rapidly varying
edge loads is straightforward, we treat this case first.
From (S.4a) , (S.5a), (S.12) and (S.13), we have for rapidly varying edge
loads that, to a first approximation, the stress boundary conditions (9.1) read
r o = Nx(O), F'o = N"e(O)
(9.2a,b)
D[w"
+ (2 -
v)w'··] = -a3R,,(O) ,
D(w"
+ vw··)
= -a2M,,(O)
(9.2c,d)
In general, the satisfaction of Eqs. (9.2) requires the solution of four simultaneous
equations, and further reduction of Eqs. (9.2) is not possible without making use
of the specific form of the general solution of (S.9).
We now consider the problem of satisfying Eqs. (9.1) for slowly varying edge
loads. Recall that in Section 5, the problem of solving the original eighth order
differential equation for w, for 1 ~ r ~ At, was reduced to the problem of solving
two sequences of simpler equations such that the complete solution for w could
be written as the sum of an edge- and an interior-zone contribution,
w = We[WeO(~e, 7])
+
(r/A)2Wel(~e, 7])
+ ... J
+ W;[WiO(~;, 7]) + (r/A) Wil(~i' 7]) + ... J
2
(9.3)
°
Boundary conditions for the functions We .. and W,n will then be conditions of
decay as ~e and ~, tend to infinity, plus suitable conditions at the edge ~e = ~, =
of the shell to insure satisfaction of Eqs. (9.1).
When the stress resultants and couples are expressed as the sum of an edge- and
interior-zone contribution and the variable 7] = TO introduced6 Eqs. (9.1) assume
the form
N:(O, 7])
R!(O, 7])
+ N!(O, 7])
+ R;(O, 7])
= N,,(7]),
= R,,(7]),
S!(0,1I)
M!(O,7])
+ S!(O, 11)
+ M~(O, 7])
= 8,,(11)
(9.4a,b)
= M x (7/)
(9.4c,d)
Upon writing (9.4a, b) in differentiated form and using (6.5), (6.S) to (6.10),
(6.17) and (6.20) to (6.22), the stress boundary conditions can be expressed
entirely in terms of the edge- and interior-zone contributions to w as follows
+ ... J + Wi[- (1 - V2)W~~ + ... J = a(A!r)2N;·(7/)jC
We[-W;~/···o + ... J + (r/A)2wi[(1 - V2)W~~" + ... J = a(A/r)8;··(7/)/C
2
We[W;~' + ... J + (rjA)4w ,[(2 - v)(Wio + r- W il )' + ... J = -aRx/C
2
We[W:~ + ... J + (r/A)2w;[v(Wio + r- W;o) + ... J = -M"jC
We[W;~·"·
(9.5)
(9.6)
(9.7)
(9.S)
6 In certain of the equations which follow 7J is taken as the circumferential variable while
in others (J is used. Where ambiguities might arise, the argument has been displayed.
18
E. REISSNER AND J. G. SIMMONDS
Eqs. (9.5) to (9.8) agree, respectively, with Eqs. (9.5), (9.6), (9.3) and (9.4)
of [4], upon setting fJ = T -171 in the latter equations.
In [4], the system (9.5) to (9.8) was reduced to a set of two boundary conditions for WeO and two for Wio by considering separately the four cases where one
of the quantities N." S." R., or 11£., was prescribed and the other three were zero.
This was done by determining a ratio of the edge-zone to interior-zone reference
displacements, We/Wi, which would yield an unambiguous set of boundary conditions for WeO and W iO such that neither WeO nor W iO vanished identically. By
superposition, boundary conditions for any set of prescribed edge loads could
thus be determined.
It was found in [4] that for the M.,-case, the proper disposition of We/Wi was
to set We = Wi, whereas for the N." S., and R.,-cases, the corresponding results
were w. = },.-2Wi . It was further noted that for an exceptional case, given when
S., = R;( fJ), superposition of the S., and R.,-cases would not give the correct result
since this exceptional case would lead to the identical vanishing of W io • This
case was treated by setting S., = R;(fJ) + },.-2tl S., and taking w. = Wi. For v = 0,
an additional exceptional case can occur if aN",
M;'(fJ) = 0, as indicated by
(9.13a) of [4].
In what follows, we describe a method, obtained by the junior author in his
dissertation [9], of rewriting the stress boundary conditions (9.5) to (9.8) in such
a way that no exceptional cases can occur in the determination of the boundary
conditions for WeO and WiO. The idea is to introduce the two load combinations
+
P.,(x, fJ)
= N.,(x,
fJ)
+
a-1M;'(x, fJ)
(9.9)
T.,(x, fJ) = S.,(x, fJ) - R;(x, fJ)
(9.10)
and then to express the stress boundary conditions in terms of the new set of
edge loads (p." T"" R"" M.,). The reason for taking these combinations is associated with the form of the leading terms in the We-expansions in (9.5) to (9.8).
In order to write the boundary conditions in terms of P., and T., , we need expressions for the edge and interior zone contributions to P., and T., . Considering
the interior-zone contributions first, we have from (6.17) and (6.20) to (6.22)
+ ... ]
+ ... ]
a(P!)"'/Eh = -(T/},.)2Wi[W~~
(9.11)
a(T!)"'/Eh = (T/},.)3Wi[W~~'
(9.12)
The determination of first approximation expressions for P~ and T! requires
additional calculation, since (6.5) and (6.8) to (6.10) show that, to a first approximation, N~ = -a-\2(M~)" and S~ = T(R~)". While it is possible to obtain
the relevant results by calculating higher order terms in the equations for N~, etc.,
it was observed in [9] that it is possible to obtain first approximation expressions
for P~ and T~ directly from the reduced equilibrium equations (2.4) to (2.6).
In what follows we shall take c. = cq = 1. Eliminating Ne between (2.5) and
(2.6), we get
!a(N.,e
+ N e.,)' -
M~" -
(M.,e
-
+ Me.,)'"
,
(M.,e + tMe.,) -
(M;'
+ Me)"
= 0
(9.13)
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
19
which, by using (2.14) and (2.15), may be written
a(S", - R;)' = (Me'
+ Me)"
(9.14)
Eliminating N xe + N e", between (2.4) and (9.13), we obtain an equation for
N", + a-1M;', which we write in the form
(aN",
+ M;')"
= {[(M",e
+ Mex)' + Me]"' +
(M",e
+ Me",), + Me}'
(9.15)
Finally, to obtain expressions for the edge-zone contributions to T", = S'" - R;
and P", = N", + a- 1M;', we introduce into (9.14) and (9.15) the edge-zone
variables ~e = AX and 1/ = TO, and the edge-zone contributions to M: e , Mex and
Me given by (6.11) and (6.12). Upon integrating with respect to ~e and retaining
only the particular solutions, we arrive at the desired formulas
aP~/C
aT:/C
= (T/A)4We [(2 - v) (W: o + T-2WeO )"'
=
(T/A)8We [-v(W: o +
T- 2W eo )"
+ ... ]
+ ... ]
(9.16)
(9.17)
For the new set of edge loads (p x , '1'x , Rx , M "'), the two stress boundary conditions involving Rx and M"" (9.3) and (9.4), are as before, while the two conditions involving P x and '1'"" to be expressible in terms of We [WeD + ... ] and
w.[W,o + ... ], are taken in the form
[P~(O, 1/)]"'
[T:(O, 1/)]'"
+ [P!(O, 1/)]"'
+ [T!(O, 1/)]""
= P;'(1/)
(9.18)
= '1';"(1/)
(9.19)
Inserting (9.11), (9.12), (9.16) and (9.17) into (9.18) and (9.19) and dividing
the resulting expressions by certain constants, we obtain, along with (9.7) and
(9.8), the following set of new stress boundary conditions expressed in terms of
the edge- and interior-zone contributions to W,
we[(2 - v)(W: o + T- 2W eO )"'" + ... ]
+ (A/T)2w;[-(1 - v2)W:~ + ... J = a(A/T/P;'(1/)/C (9.20)
We[-v(W;o + T- 2W eO ),· .. • + ... ]
(9.21)
+ w;[(1 - l)W~~' + ... ] = a(A/T)8'1';"(1/)/C
2
2
We[W;~ + ... ] + (T/A)2W,[v(Wio + T- W iO ) + ... ] = -A M x(1/)/C (9.22)
2
We[W:~' + ... ] + (T/A)4Wi [(2 - v)(Wio + T- WiQ)' + ... ] = -aR",(1/)/C (9.23)
We note that when the boundary conditions are written in the above form,
~e-derivatives and WiD and its first three ~i-derivatives each
appear but once in any of Eqs. (9.20) to (9.23), e.g. WeD and W:~ appear in
(9.20) but nowhere else, W;o and W~~ appear in (9.21) and nowhere else, etc.
We note further that as we proceed from (9.20) to (9.23), the relative order of the
edge-zone contribution compared to the interior-zone contribution increases by
a factor of (A/T) 2 for each equation.
As noted earlier, the method of obtaining boundary conditions for WeD and
WiD used in [4] is based upon the decomposition of the given stress boundary
WeD and its first three
20
E. REISSNER AND J. G. SIMMONDS
value problem into the sum of four simpler problems. An analysis of this method,
when applied to (9.20) to (9.23), reveals that to apply the arguments of [4] we
now need decompose the given stress boundary value problem into the sum of
only two problems, as follows
(1) Set the right hand sides of (9.22) and (9.23) equal to zero, and solve for
W.o and WiO . Denote these solutions by W;~) and W~~).
(2) Set the right hand sides of (9.20) and (9.21) equal to zero, and solve for
WeO and WiO • Denote these solutions by W;~) and W~~).
Because of linearity, the solution of the original problem is then given by
(I)W O )
(2)W(2)
d W
w.W.o = w.
.0 + w.
.0 an Wi
iO -- WiO)W(1)
iO + Wi(2)W(2)
iO.
The ratio W,/Wi and the boundary condition for problems (1) and (2) are
determined by requiring that, in the limit as A ~ 00, four boundary conditions
are obtained altogether for W,O and W.O(which in all present cases separate into
two sets of conditions for W.o and W,O, respectively), such that the set of these
boundary conditions does not lead to the identical vanishing of W. O or W iO (unless
W. or W, = 0, identically).
For problem (1) the right hand sides of (9.22) and (9.23) are zero; hence
W~l) = (T/A)4WP) or W~l) = (T/A)2WP), otherwise either the edge or the interiorzone contribution would dominate both of these equations and we would obtain
a set of homogeneous boundary conditions. If W~l) = (T/A)4W~1" we get two
boundary conditions for W,O from (9.20) and (9.21) plus a third from (9.22).
But this is too many, so the only possibility is that
(T/A)2WP)
W~l)
(9.24)
For this ratio of reference displacements, we then obtain from (9.20) and (9.23)
the following boundary conditions for wW and W~~),
wP)W~~)II
=
-a(A/T)2P;o(TJ)/Eh, W~l)W~~)'" = a(A/T)3'l';oo(TJ)/Eh
_ (W O)••
-2W~1»)
W~l)'"
0
W (l)II
.0
II
,0
+
,0,
T
,0
(9.25,26)
(9.27,28)7
For problem (2), the right hand sides of (9.20) and (9.21) are zero; hence
W~2) = (A/ T)2W~2) or W~2) = W~2), otherwise either the edge- or interior-zone con-
tribution would dominate in both these equations, and we would obtain a set of
homogeneous boundary conditions. If W~2) = (A/T)2W~2), we get two boundary
conditions for W;~) from (9.22) and (9.23) plus a third from (9.21). But this is
too many, so the only possibility is that
W;2)
W~2)
(9.29)
For this ratio of reference displacements, we then obtain from (9.20) to (9.23)
the following boundary conditions for W;~) and W;~),
(2)W(2)fI
_ "\2M()/C ,W.(2)W(2)II'
-- - aR- '" ( TJ )/C (9.30,31)
W..o
1\
'" TJ
.0
7 If JI = o (T2/X2), it is necessary to set w;l) = (T/X)4w)i) and to replace (9.27) and (9.28)
by new equations to avoid having w;~) "" O. For the special case JI = 0, the equations which
replace (9.27) and (9.28) are given in Appendix C of [9].
21
ASYMPTOTIC SOLUTIONS FOR CYLINDRICAL SHELLS
w~~)"
(1 -
0,
l)wg)II'
V(w~~)oo
+
T-2W~~»'OOOO
(9.32,33)8
Eqs. (9.24) and (9.28) show that problem (1) is characterized by the fact that
the interior-zone contribution to the normal displacement dominates the edgezone contribution, and that the boundary conditions for the interior-zone solution
are given directly in terms of the prescribed edge loads whereas the boundary
conditions for the edge-zone solution must be determined subsequently in terms
of the interior-zone solution. Eqs. (9.29) to (9.33) show that problem (2) is
characterized by the fact that the edge and interior-zone contributions to the
normal displacement are of equal order of magnitude and that the boundary
conditions for the edge-zone solution are given directly in terms of the prescribed
edge loads whereas the boundary conditions for the interior-zone solution must be
determined subsequently in terms of the edge-zone solution.
Introduction of the load combinations P., and '1'., defined by (9.9) and (9.10)
may be related to virtual work considerations as follows. We consider the expression for the virtual work of the edge forces, based on the assumption that the
transverse shearing strain is zero,
oIT.
Replacing
=
a
1 (N., ou + S., ov + R", oW 211"
0
N., and S., in (9.34) by the expressions
N.,
= P",
- a-1M;O(O),
S., = '1'.,
a-1M., ow') dO
(9.34)
+ R;(O)
(9.35)
and integrating by parts all 0 derivatives, we obtain
/lIT. =
a
1 [P",
2.-
0
OU
+ '1'",ov
- aR., OEe
-
M., o('Y° -
E~)] de
(9.36)
We now consider the partially inextensional deformation defined by the two
conditions
'Y=E8=0
(9.37)
Eq. (9.36) shows that P., and '1'., are simply those combinations of edge loads
that can do work when the midsurface displacements are constrained so as to
satisfy (9.37). It is because of this that (9.25) and (9.26) coincide with a set of
contracted boundary conditions derived by the senior author by entirely different
considerations for the case of semi-inextensional deformation for which
'Y = EO = 0, [5].
REFERENCES
1. E. REISSNER, "On the Derivation of the Theory of Thin Elastic Shells", J. Math. and
Phys., 42, 263-277, 1963.
2. L. H. DONNELL, "A Discussion of Thin Shell Theory", Proe. Fifth Int. Congress for
Appl. Meeh., 66-70, 1938.
8 If J) = 0(r 2/>..2) it is necessary to set W~2) = (>"/r)2w~2>, and to replace (9.32) and (9.33)
by new equations to avoid haying W~~) "" 0. For the special case J) = 0, the equations which
replace (9.32) and (9.33) are given in Appendix C of [9].
22
E. REISSNER AND J. G. SIMMONDS
3. L. H. DONNELL, "Stability of Thin-Walled Tubes under Torsion", N.A.C.A. Report
No. 479, 1933.
4. E. REISSNER, "On Asymptotic Expansions for Circular Cylindrical Shells", J. Appl.
Mech., 31, 245-252, 1964.
5. E. REISSNER, "Variational Considerations for Elastic Beams and Shells", ProG. Am.
SOG. Civil Engineers, J. Eng. Mech. Div., 88 (EMI), 23-57, 1962.
6. V. V. NOVOZHILOV, The Theory of Thin Shells, Translated from the 1st Russian edition
by P. G. Lowe.
7. E. REISSNER, "On Some Problems in Shell Theory", ProG. First Symp. on Naval Struct.
Meeh. (1958), Pergamon Press, New York, N. Y. 1960, 74-114.
8. M. W. JOHNSON, JR., "A Boundary Layer Theory for Unsymmetric Deformations of
Circular Cylindrical Elastic Shells", J. of Math. and Phys., 42, 167-188, 1963.
9. J. G. SIMMONDS, "Asymptotic Solutions for Circular Cylindrical Thin Elastic Shells",
Ph.D. Thesis, M.LT., Oct. 1964.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
UNIVERSITY OF VIRGINIA
(Received May 15, 1965)