ISBN
 978-3-642-8
e rial is concerne<
e of t ransla t ion,
reprintlng, re-use of i l lus t ra t ions , bro
adcast lng,
eans , and s torag
e In data ba
o r other
payable
nt
o
book
type, such
as shell roofs or pressure vessels, and consider all the minor
details
of stress calculations and even the design. On the other hand, he might
stress the mathematical side of the subject to such an extent that he
virtually writes a book on differential
equations
the
mechanical subject. The present book has been kept away from these ex
tremes. At first
mathematics
reader
has
been
in a theoretical subject such as
this
one, it is, of course, not possible
to get very far with the multiplication table and elementary trigonom
etry alone. The mathematical prerequisites vary widely in different
parts of the book, depending
on the subject. In some parts ordinary
differential
equations
In other
sections ordinary
equations,
variables will
encountered. However, the author wishes to assure
his readers nowhere in this book an
tool been used just for the sake of displaying it. No matter which mathe
matical tool
been used,
it had to be used to solve the problem at hand.
When
preparing
before the
question how to react to the spreading use of computers. Many a good
book of recent vintage
is, filled with
advice for the writing of computer programs. In the present book, the
challenge of the time
not to be taught as a part of shell
theory.
Anothct·
task
reader understanding
the mechanics of shells, from the formulation of the differential equations
to the discussion of the result of the analysis. Therefore, details of compu
tation have been de-emphasized, but all the diagrams displaying the
results of computations have been
retained. They will show the reader
how
a shell "feels" under a certain load, how i performs its load-carrying job.
The book may be divided into four parts. Chapter 1
contains
prelim
inary
 
 c o m p
PROPERTIES OF
1.1.2 Stress Resultants
1.2.1
1 :1 Transformation of :\[oments 17
Chapter 2
19
l
2.1.2 .Equilibrium of the Shell Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Axisymmetric Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2.1 Spherical Dome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2.2 Pressure Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2S
2.4.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4.3 Conical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.:U General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.3.2 Homogeneous Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 : ~
: - : i
Xl
 of th
E R T I E S
 OF  ST R
E SS SYS
t.heless, the
y is not ofte
ody are ca
ated for a cer
on  of a force
ua l ly transmit loads to s
up
ports
at
bo
nalyst does not envisage
r the cloth-and-r
  line,
e   and cons
cal surface", a
transm
sidere
e a
 SHELLS
compressiv
rce transm
itted b
alled the
t w here
a tensile forc
e N x
  w ould point
force
Nyx  
is
gn (Fig. 1.1
. I t changes when th
e po s
n
versed.
face
osi t ive s ig n of this force w
ill be def ined
case of 
olid
m
esultants as i
ese
in
. 1.2), th
this
al
ma
gnitude,
we
may
  disregard
ct ion
hich
e
 
6
CHAP. 1: STRESS S Y S T E ~ I S IN SHELLS
curvature, say rx. \Ve
(1.1 d-f)
"When we compare (1.1 b) and (1.1 e), we see that the equality of
the shearing stresses, •xu = •ux' does not imply the equality of the
shearing forces. The difference between Nxy
and
Nyx vanishes only if
rx = ry (e.g., for a sphere), or i f •xv does not depend on z. In a thin
shell t and z are small compared with the radii rx, ry; then
the difference
may often be neglected.
uniformly across the thickness
respect
Since these moments influence the equilibrium of the shell element,
>ve must consider them.
in a section
.-r
= const. is referred to a tangent to the line element dsu of the middle
surface. The moment is of differential magnitude and proportional
to dsy. I f
it
is designated by J.l1x dsy, the quantity .111x is finite and re
presents a moment per unit length of section. Consequently, it may
be measured
in such units as ft.lbjft or in·lbjft or others of the same
kind .Jfx is called the bending
moment of the section.
across the
anywhere in the plane of the cross
section and has a moment with respect to an axis
which is
moment is also proportional
to dsy and is
1.2 the relations
..6.J: xy
= - 'txy-r-.-z Z
-1/2 -1/2
which may be considered as the definitions of the bending and twisting
moments. The minus signs are arbitrary and fix the sign convention used
in this book (see also Figs. 5.1 b and 6.1 b).
When the same ideas are applied to a section y = const., another
bending moment and another twisting
moment
1lf l = - J a I r•r: z z dz ,
-1/2
+t/2
M
J
ar
forces N:x:y and Nu:x
 l
on
 t
ing forces, N
t
face. With
l to the axis
osely that th
 
r t
ul tants ,   only
  three are l
d
N.c
11
a t ing
ces
 b ending s
tresses are n
mption of
the membr
brane fo rces (i
y), we  do not imply th
at the normal
sent a pla
ace. Whe
~
culated for the sec
pass ing   through that point
, the ques
he fo
rces in  
r . The 
x cos 'X
N" =
  N
xsin
2
orces at a p
t
phical m etho
eful one,  a
raphical m et
r importance
ss·.
pattern
s.
We
consid
ear ing
forces  wh
hrough  this point. In a rectan g u l a
r coord inate
adius is
ions which p
, 
e
ent t
d down.
1.2.3 Obliq
ue Coordinat
=
p p r ox im at ion
  a parallelo
  in th
to the m iddle
o
ring forces g
e use a rec
forces  requ
ther refe
are , of course,  not equal since
equality  can
. Therefore, th e
di t ion of
moment equi l ib
 the shell:
sx .
 
atural descri
 a poin
e force tenso
. 1.6). On the
s ides d.s
r
g force" N

F
rom
t w e e n
the r thogon al
:
to R
u, we
 = const. :
gular shell ele
t fall
d
do
 other: 
gonal pair
 $, 'fJ of
s f o r m a tion fo
rmulas  need
e m e n t
having
rectangular
l to
 y (Fig.  1.8).
O R ~ I A
T I O N
S
17
ic
re
lation
~ s i n x ,
x ~ ,
x
; s i n x
t N;
red o
e r surf
u la r
are
chapter appears
Before we enter into the investigation of the
stress
Fig. ~ . 1 . Meridian of a shell of
revolution
of a plane curve about an axis.
in its plane. This generating curve is called a meritlian, and
an
arbitrary
the shell is described by specifying the
particular meridian on which it is found and by giving the value of a
second coordinate which varies along the meridian and is constant on
a circle around the axis of the shell, called a latitude circle.
We shall identify a meridian by the angular distance ()
of its plane
meridian and choose as second coordinate the
 
lut ion
op
force
 N  
0
, an
d  
t
ents m
ake an 
L EQUATIOX
urface of
ppl ied to all
ing this into
CHAP. 2: SHELLS OF REVOLUTION
There is still a third way of formulating the fundamental equations,
using rectangular coordinates r, z in the plane of the meridian (Fig. 2.1).
From (2.4b) we find that
a . A- a
(2.6a, b), we find
-arr
(2.8a, b)
There is some advantage in using this form of the equations, if the
shape of the meridian is given
by
nates
cartesian equation
2.2 Axisymmetric Loads
2.2.1 Differential E(tuations
In many practical problems the external forces have the same sym
metry as the shell itself. Then the stresses are independent of (),
and
this
d
1
Equation (2.6b) becomes independent of the other two and contains
only the shear:
8
treated
our
obtain
a
t m
be co
e
 (F i
gs. 2.5
resent
1
1
n c J > d c J > = -pa
 
.9b) we find
egative
thr
 meridiona
itted to a su
 
ylindrical 
y they
ord we shall
sure
p,
hen we let
pq, = 0, p,. =
q,
q,
Nq, = ~
~ r
1
r 9 p c o s < f > s i n < f
> d < J > = ~ ~
d independ e
hape of the
get
t
e 
fin
d
soid as
 boiler end
u l tan ts
I C   LOAm;
s
o of i ts axes,
 an equa tori
ssion. The el
al deformatio
Chapters 5 
s not, a shel
su l tan t s prod
uced by the
weight of the
droppin
0
- (co
e formulas
yield N
numerat
or
mes zero. T
r ibut ion is 
erical
 
udden local
oes n
ot repr
ad  
rve ab
out an
o typical ca
oroidal shells
the
broken
line. The meridian of each part begins and ends with a horizontal tan
gent. Therefore, the meridional forces acting at each edge do not have
a vertical component and cannot
transmit any
of pressure p, this pressure has a downward resultant on
the
inner half
and an upward resultant of the same magnitude on the outer halL
and neither part can be in equilibrium under
the
p and
the forces on its edges. It follows that a membrane stress system
with
0
Then
(2.10)
gives
N.p = ( . </>paR) . < >
asm + sm
the
at
Np = . ~ a
additional bending, because it again
leads to an incompatibility of deformations which we shall discuss on
p.94.
2.2.2.5 Tanks
Our next example we choose in the domain of steel tanks. Fig. ~ - 9
shows a spherical tank, as used for storing water or gas. It is a complete
sphere, supported along one of its parallel circles, AA. The essential
load for a water
It is
tank
.
By a simple integration, we find from (2.10) the meridional force
N.p = : ~ ~
sm 'Y
CH.\P. 2: SHELLS OF REVOLUTION
At the vertex cJ> = 0 the denominator vanishes. To obtain a finite value
of N.; the factor in brackets must
also become zero. This leads to
C = 1 6,
These formulas are valid above the supporting circle cJ> = c/>
0
part of the shell we have to apply another value
of C, which
= n. It is C = 5/6 and hence we have
N = y a 5 - 5 cos cf> + 2 cos2 cf>
.; 6 1 - coscf> '
V y a
~
The distribution of these forces is shown in Fig. 2.9.
l iJ.C. : .U. Spherical water tank; support
at
<i o = 1 0°
The location of the supporting circle does not influence the two
values of C. I f we give it a higher or lower position, only the domains
of validity of the two pairs of formulas are changed·. The corresponding
changes in the stress resultants are indicated by dotted lines in Fig. 2.9.
They show that a position of the support below cJ>
= 120° leads to com
pressive forces in the meridian, which in a thin-walled structure like
this one should be avoided,
and that
the
peak
which determines the wall thickness, but of course it leads
to a larger
ard , w
ll has 
dge, the
equate to
ll.
ect stress sy
stem   is not
possible in the
 bending will
tom
·w
hen  
we
assu
ed force app
l ied at
ne
in  
th
n com bined a
s shown in F
 solution
CHAP. 2: SHELLS OF REVOLUTION
The constant follows from a condition at the outer edge s = l of the
From of of the and
loads which
per
can
I t
F
The
the
boundary
condition
is therefore N. = - Pjsin<X. This determines C and then
ycota . Pl
the
difference
cone and N in the sphere.
This
is needed for structural
purposes, if the dimensions of the shells are so chosen that
the
thrusts
sphere balance each other. This condition can, of course,
be fulfilled only for a certain load, e.g. that one belonging to the highest
water level in the tank.
2.3 Shells of Constant Strength
A shell dome looks almost like a three-dimensional arch structure.
This raises the question whether or not for a given load there also exists
a
best
This question shows plainly the fundamental difference between
the
from bending moments;
its
equilibrium.
contrary
is true for a shell dome. We have seen how we
can have equilibrium without bending in almost
any
is of about
indeterminate funicular arch.
From this situation it follows that we can ask for more than absence
of bending. We can try to find a shape of the shell such that the mem
brane stress a has
of
own weight. The problem is a simple one if the
dome
 
  S T R E ~ G T H
39
te, the load pe r
 unit area of the 
coscf>.
I
f> (r) :
y be so lved by n u m erical in
tegration, b
e
he shell.
ine th
e meri
dian in
cartesian coor
 transfor
the problem of a sh ell of c
onstan t stres
ll there   exist
en ajy
  th
,
with
o th er shells
 numer
ica
4l
N+
t</>· Prn),
dN+On
 
. -
ir difference
Pon- P+n + n ~ i n c : s 4 l
Prn),
(Pon- P+n- n :::
]
- a   P+n + P
on- sin4l Prn sm
n
2
  load. It h as
d
as
lee side
half of
des.
fo
s to t
nd perform
the in-·
-
 
nce,.
  factor cos() or sin():
d from
ants assum e
1
e the
O
1
m 
of these force
represen ted
 recogn
  with 
resp
 
With
the
must be applied
to
external force
apply
in
 
a l to
ure , (
2.14).
goes
approx
imate
ly  in
The stress trajectories are,
middle surface of
the shell. Fig. 2.23 is a stereographic projection of these lines. This
particular projection was chosen because
it
between the curves.
I t be may seen in the figure how part of the trajec-
Fig. 2.: 3. Hemisphere on
four point supports;
from the supports, while others leave the free edge at
angles of 45°. At each point of the edge one of these trajectories carries
tension and one compression, since there the shell is in a state of pure
shear.
The trajectories may convey some idea of the stress pattern, but
they may also be misleading. In this particular case they overemphasize
the deviation from perfect axial symmetry
in the
upper part
of the
rather small
angle. Therefore one family of trajectories looks like rounded squares
in a region where the stress system is almost exactly that of a con
tinuously supported dome.
1
boundary
Bn must
of
done numerically.
of the
force N at the
edge or at the edges: but they yield also shearing forces
N ~
0
, and their distribution is beyond control, since no further free
constants are
just as they
appear
and have to provide a stiffening ring of sufficient strength against
bending in its own plane. This result is
not
 perfo
cal ly d
e te rm
in ate , in  
a
rizontal w
idth 2n
  mapping is
  identical w
ith M
y parts of
olutions will
gular points
eriod
2
ing it in
 (} ' 
 sin()
+ N
  must
ay e
 
 AXIAL SYl\U
  Point Suppo
,
m iddle s
urface of the
or the weigh
 m eeting at one point may
be comb
 
 
.41).
e find
orre
we
no
the forces due
he
This l eads to
 of all these
solution s look
ca l evaluat io
  terms
 has
a
  stro
ng 
singula
 
 A XIA
L SnHI
tart
fro
  the
fo
llowin
Again introducing the loads and the stress
resultants in the
form (2.23),
(2.24), we find the n-th harmonic of the hoop force N
0
,. immediately
dN,.
(2,44a, b)
may be solved one
after the other. Equation
Naon = -exp ( - J
2
sider
the
mushroom-shaped
Ps = Po = 0,
p,. = -psin x cos().
\Ye have to use our formulas with n = 1 and easily find
1V
0
= -p8COSCL.cos(),
N,o =-
8
1
2
external
1
1Y
8
AL SYMMETR
omes infinit e
lar i ty of
lways be inf
inite stress s
,. are
izontal force
  P and
Fig. 2.31.
n. Together
couple 
wit
h
e cone.
ngularity at the top does not
co
sion
whe
ther
n containing suc
h a singularit
e
t
ion to
a
tions
  o
f
shells. The
shell in
 Fig. 2.32a
aJleX, (b) with r
part the hom o
not wan
neous so lutio
32b . Here 
A, = 0. Writing 8
we hav
t
  of the
se for
ces is
ith (2.6) to get the set
(2.2
~ n
+ rp
,, . cot</>,
2
cf>
> .  
r completely
  rest r ic t
67
8
equation is
+ Bz<l-nJ /2 ,
The second term becomes
infinite for z = 0 and is therefore not applicable to shells closed at
the top.
From U, we find the n-th harmonic of the meridional force
of the shear
)\' u.
of
N</Jn =A i- Vz"-
2
Non=- A----====
2Va + 4z
~
0, corresponding to
a horizontal concentrated load as shown in Fig. 2.20 for a sphere. Fot·
n = 2, the stress resultants approach finite limits, and for n > 2 they
vanish at the top of the shell.
The results given here for a parabolic shell show the same
general
features
as
those found on p. 47 for the sphere. In the vicinity of
the
resultants
in any other
shell which there has a finite curvature equal to that of
the paraboloid. This proves
5*
T I O ~
ller.
J
ust
as  
does
t
ation 
(2.4
p
rests on eight supports of an angular width of 10°.
The
of Fig. 2.21. Results
00
represent the axisymmetric stress system of a shell with a con
tinuous support. I t is seen that the influence of the higher harmonics
caused
by
"Fig. 2.35. Pointed shell on eight supports
-6
-4
-2
10
3
lb/ft
Fig. 2.36. Stress resultants in the shell shown in Fig. 2.35
The
higher
the
solution
takes
on the character of a local disturbance along the edge of the shell. The
engineer's interest is always limited to the zone in which the forces
have appreciable magnitude; this zone
may
be so small that we can
safely neglect the variability of the coefficients in the differential equa
tions (2.26)
replace them by average values, say those at the center
of the
N ~ o n =
porta
sign,
t, we
need s
om e
he ge
g
0 ±wand  brin
ent
shaded
e shown   in
Fig. 2.40.  On
have the
e
ngle   i
'
nder
approach
to a general solution. We need not submit the given edge
load
our
not
find from
them the forces at any point of the shell from the equilibrium of a tri
angular
But the formulas (2.50), (2.51) give still more
than
curvature.
·when a spherical shell has two
edges, we can prescribe N at both of them and then have to accept
the shear N
8
which results. \Ve can also try to determine A,., B . so
that, for example, at the outer edge both forces N ~ and N ~
8
assume
to unduly
the
the
loaded
edge and the other increases. This increase is very pronounced if the
order
or if the
other. I f the shell is closed at
the
vertex, this increase leads to infinite
forces at this point, as we have seen. This indicates that such a set of
boundary conditions is not appropriate
and
Quite .to
and N
Fig. 2.43. l\Ieridional section
of a
bellshaped
shell
result at the other edge. The engineer would certainly prefer to do the
same as in
a stiffening ring at each edge to take care of
the
this
procedure is, so to speak, against the nature of"the shell. Our for
mulas show
this
quite clearly. I f nw at the lower edge is an integer mul
tiple ofn, then a normal force at the waist circle produces only a
normal
force at the lower edge, and we cannot assume both independently. A
shear
at one edge produces a pure shear at the other,
and
we must
prescribe one of them to make the problem determinate. Such a result,
of course, also appears
s S,.
ists of the
betw een
s the
ltants exists an
pends on  the
the
mathe
owing angle betw
O S
analogou s
to the
r. Since
func t ion  
point of the shell.  T h
e quan
ti ty
 
e te rminate .
 the shell
(w). \Ye
v, w the
y from
cili tate
if we
  simple nota
r ime and on
ent. If th er
as s l ight ly di
fferent direct
.
isplacem ent w
rder an d  th
= v· dcf> + w
ncrease to
>
ller th
CHAP. 2: SHELLS OF REVOLUTION
The relations (2.56a-c) enable us to find the strains when the displace
of
to
have
already
been determined and we want to know the displacements.
Then HooKE's law (2.54) will give us the left-hand sides of (2.56), and
these
equations
differential
equations
for
the
dis
placements u, v, w. The study of these equations is our next objective.
2.5.2 Inextensional Deformation
r2
=0.
(2.57a-c)
They
and
under what circumstances such a deformation may occur and how we
have to fix the edge of
the
Since the coefficients in (2.57) do not depend
o
(), we may write
the solution as a FoURIER series in(), and the n-th harmonic will be
u = U
11
a t a system of ordinary differential equations:
+ w"
=0,
nu
11
+ 'V
11
cos</> +
W
11
sin</>
11
rl
=0 .
(2.59a-c)
11
cos</>
(2.60)
and,
an equation in which only
the meridional component V
11
t ions .
e te spher
eous equ
ake the deform
ne
dis
be all th re
e of the she
ll is fixed   in
condi t io ns
p of bending mome
not a l
w a y s w o rk . Then a st
udy of the bending
ny
c
s, and all  t
 
ge
ig. 2.45
its one  inexten
 A and
e
the
um of al l
ear equ
are somewhat
con·
0
1
2
four terms on the right-hand
side:
0
solution is again a solution of the homogeneous equation (2.63).
Putting b
1
111
having two free constants b
01
W
11
= - v ~ = -
x
k ~ o
For the
third component, un, no new series need be computed, since (2.60)
yields
1l
appear
in a shell which
is closed at its apex. We have then only one constant of integration for
every harmonic n, and the inextensional
deformation is
Of course we can use the constant C
which
condition
for the displacement,
if we refrain from prescribing forces at the boundary; but never can all
three
the mem
we encountered in the
,., N
0
, N,.
0
w s i n < f > = r
2
com plete
u la r solution
ary different
2
Eon
sin</>,
\Ve
1 - c
oscf> the
ssumes the f
2
It may be solved
00
F(x) =
1
or
x
2
symmetry
of the shell, it will be half of an odd in
teger if n is even and an integer i f
n is odd, just as we found for x
1
I f we now assume vn
in the form (2.64) but with the same value of x
as appears in
we may
particular
factor is equivalent to adding an inextensional
deformation. I t enables us to fulfill one boundary condition for the dis
placements.
Such a power series method may be expected to yield fairly good
results in vicinity the the shell. But for greater values
of x
the
the
have
then
the advantage of finding at once that solution which is regular
at cJ> = 0. However, i n is great, the regular solution will assume
per
ceptible values only near the edge of the shell, and then it will be pre
ferable to start the
the
the
stresses
tion need not
does not
split it
this high degree of symmetry
and an inextensional deformation such as we have already treated in
detail.
For
assume the simpler form:
 
l \ I A T
  shell
i
xis nd mus
t be determ
ll o
f arbitra
At five points of
indicated
appears that the major part of the shell moves essentially
vertically downward. The slight inward component is due
to the
springing line, where the hoop force is
positive, the deflection is outward, and at the springing line it is hori-
V
dead load
vertical deflection w,.. For p = 58 lbjft
2
tw,.
in
hori
2.5.4 Toroidal Shell
\Ve have already encountered some difficulties in treating the stress
resultants
in toroidal shells. Here we shall see that the deformation
of these shells also has some peculiarities. On p. 31 we found that a
system of membrane forces is
not
always possible in such a shell, even
under conditions which would be sufficient in other shells. But we dia
at least find
jected
to an internal gas pressure. We shall see now that even in this
case membrane forces are possible only in restricted parts of the
shell,
result.
To simplify the mathematical representation, we assume here v = 0.
Then the stress resultants as given on p. 31 produce
the
strains
From
 
 a shell elem e
nt (Fig. 2.2) w
ork,
because
at ion are  dis
e zero. T he
in
spac
and we have o
 to its deform
t does no
ement r
ate an unesse
n t by assum i
ng that
th e
 sides r
ional
hen only  on
1
  up, we see
that the who
oport iona l
ent and is
of). b
n ). increases
  from 0
ored in
rgy and is ca
+ . L V ~ o Y
 
CHAP. 2: SHELLS OF REVOLUTION
taining harmonic constituents of orders n = 8, 16, ... , and each of
these
same order of the actual stress
system. On the left-hand side the first integral is still equal
to w,.,
where
all
functions N•,
three
equations
(2.6),
Occasionally we
regularity
of the stress system, but this too can be interpreted as a condition of
equilibrium of a particular shell element.
However, cases exist in which equilibrium conditions are not suffi
cient
to
determine the stress resultants in a shell. vVe see this
best
by
to
The
in
con
nection with (2.27). I t contains two constants of integration, one of
which, An, is determined by a condition of regularity. I f either
N.pn
or N•on is given at the boundary</>= n/2, this fact supplies an equation
for Bn, as we have seen in the slightly different case of the shell on iso
lated
supports. These problems are statically determinate.
The situation is different if the shell rests with the whole circum
ference of
The
bound
ary condition which will help us to find B,. is then a condition
of
zero
the
usual
method
of
the edge support consists
of two separate structural elements. The first one is a circular stiffening
ring, not deformable
normal
to this plane. This ring absorbs the shearing forces N•
6
vertical
the
I f we cut
of external
 
at B
e, bu
t t
102 CHAP. 2: SHELLS OF REVOLUTION
have to be done numerically. The boundary forces on the left-hand side
of the
2 N<o>
- E t •n •n on on ' •on •on 'f ' ' f '•
0
I f we again apply (2.74) but this time introduce N ~ > ... for both N • ...
and N:
0
;r/2
or v,., and they cannot
even be expected to do so, because the displacements are not completely
determined
by
the stress resultants in the shell. Any one of them
may
other
one
by
B = 0,
an inextensional deformation we always have un = vn at the
edge of
by inextensional deformations, and so is dependent upon the stress
resultants only.
In our problem of a shell resting on an unyielding foundation, both
v.
+ u,.
and
mines an inextensional deformation,
the
form
and find from this
 
3.1 Statically Determinate Problems
straight
line along a curve
while maintaining it parallel to its original direction. I t follows from
t.his definition that through every point of the cylinder one may pass
a
straight
line which lies entirely on this surface. These lines are called
the
the generators are horizontal. All planes which are normal to
the
genera
cylinder in identical curves which are called profiles.
The cylinder is named after the shape of the profile, e.g. a circular or
a parabolic cylinder.
natural
net of
coordinate lines. We choose an arbitrary profile as the datum line and
from this measure the coordinate x along the generators, positive in
one direction
In
angle </> t;sed on
surfaces of revolution, we introduce here the angle </> which a tangent
to
the
the
zontal plane (Fig. 3.1).
 
104 CHAP. 3: CYLIXDRICAL SHELLS
Now let us consider a shell whose middle surface is a cylinder. lVe
cut from it an element by two generators cf> and
cf> + dcf> and by two adjacent profiles x and
+ dx (Fig. 3.2). The mem
brane
l 'lg. 3.2. Element of a cylindrical shell
to
into normal and shear com
ponents as shown. The forces per unit length of section are N,., lV >
(normal forces) and N,, =N.; • (shearing forces). The load per
unit
area
increasing x and cf>, respectively, and a radial (normal) component
p,,
positive outward.
The stress resultants N x , N.;, N x,P are of the same kind as those
appearing
of
equi
them
These conditions may easily be read from Fig. 3.2. The
equilibrium
aN. . aN.;.
to
a
aN.; aN .;
aq; dcf> · dx + ----a;- dx •rdcf> + P,; •d •r dcf> = 0.
At right
angles to
the external
 
nents , p ~
er w
ill no
, a truss , or
m and shal l
use th i s word to mean any p lane
stiffe
r which is c
l
a dia
ph r
  end  (Fig. 3.
3). If we
t ion
 tran
and
.V •
·of
nifo rmly dist
 
ending
all its load
to
the two diaphragms at the ends of the span l. Of course,
the
distribution
of the forces N x ~ and Nx over the cross section
cannot
the
by
shell is completely
resist not only shearing forces N ~ but also normal for
ces N x .
without
any
support
the boundary
N.r = 2r d<f>
This shell is supported like a cantilever beam, and, again, the spanwise
distributions of N x ~ and Nx are those of the shear and the bending
moment of the beam analogue.
The three-dimensional support of such a cantilever shell will scarcely
be accomplished by a solid wall, as shown
in
have
again two diaphragms of the usual type, which resist only shearing
forces but do not accept forces Nx from the shell.
The
overhanging
end.
Span
wise
distribution
 
R ~ I I N
A T E PROB
analog
ca l ly
c t ion
t of the
ion s im
cal s
hell su
pported by
d only 
be ne
R ~ I I N
 here r
eplace th
and
 f
2
+ l
2
) co
lk head
s , in
cal co
 
 
ical re
  we pu
t y =
0, a
nd th
e 
edges
3: CYLINDRICAL SHELLS
and distribution follows from the solution of a plate problem, with
not other
additional shell problem and may be treated by
the
studied
forces
were independent of x. 'Ve shall add here one case of a more general
nature,
shells.
}'ig. 3.10. Inclined cyllnllrical tank
Fig. 3.10 shows a circular cylinder whose axis is inclined at an a n g l e ~
from
the
The water
to
Px
and
p,. = 0
for x < a tan a coscf>.
In the part of the shell which lies above the water level, the stress
resultants are given
boundary
conditions that the upper edge is completely free, we have for
this
domain /
1
For the lower part of the shell we
have
1
and
 
115
be determined from the condition that the stress
resultants are continuous at the water level.
Equation (3.1 a) yields immediately
N</J = ya(x
cosx- a sinx cos</>),
and this fortunately is zero at the level x = a tana cos</>.
The derivative is
This vanishes at
put
and this yields the shearing force as
N.x</J = ya(a
tana cos</>- x)
N
x
= ~ [x
2
2
Nx
vanishes
at the water level, and this leads to the result
Nx=
~ [x
2
the
stress
resultants
tion of x, and it is therefore
not
along the span, respectively.
In Fig. 3.10 some diagrams for Nx are given, which show how differently
this force is distributed over different cross sections.
I f a
=
0, the formulas degenerate into the trivial results for a vertical
cylindrical tank. The other limiting case, a = n/2, does
not
the
Circular
Cylinder
Since (3.1) can easily be solved by quadratures, it seems unnecessary
to employ FouRIER series in so simple a problem. However, such solut ions
8*
~ D R I C
hen
 ends,
 or
there
ila r
st r
ess sys
tems a
s be
fore . F
resses
independent of
assumed
as
1
(3.12)
the stress resultants
again functions of rJ> alone.
I f we measure x from one end of the cylinder, then Nx
== 0 at both
N , , ~ does not vanish. The FoURIER series represent,
therefore, the solution for a cylinder which is supported at the ends
x = 0 and X = l (Fig. 3.6).
When we
introduce the::;e
equa
1 V ~ , . =
Pr r'
TNX/1 =---;:- ~
As an example of the application of these formulas we consider a
circular cylinder (r =a)
of length l which
This
x as shown
in Fig. 3.6.
I f pis the weight per unit of surface, the load
components
are
p ~
= p
sinrj>,
into the form (3.11), we must
~
yields in our case
nn
valid for odd n, while all the even-order coefficients are zero.
When we
n n
 
118 CHAP. 3: C Y L L ~ D R I C A L SHELLS
This result is, of course, identical with (3.16) below. For the simple
the
discuss bending stresses
theory
we consider the
case just treated, a tube of circular profile, supported as in Fig. 3.7, and
subjected to the
N ~ =
-pacoscf>,
the
gener
the
lower half of the
shell, the upper half n e ~ d not be supported at the straight edges and
may carry its weight freely between the diaphragms, just as
the
tubular
shells do. Such barrel vaults have been used as roof structures.
Fig. 3.11. Barrel vault shell
However, the straight edges of a barrel vault are
not
completely
~ = ±
edge
member
only
in tension (Fig. 3.11). Its axial force N is, of course, variable along the
span. I t can
 
A T E P
R O B L
equal
 
nalogue
disclose deta
  the construc
raight
ed
  happens w he
p, vanishes for
the cycloid, and they have, the
refore, b een
g tltis and th
ion o
dge th
here
f
e p ro
122
CHAP. 3: C Y L I ~ D R I C A L SHELLS
script()
Y x ~
and the displacements. These are (Fig. 3.14): the axial displacement u,
parallel
to
and
ing x, the
circumferential displacement 'V
profile
of the middle surface and positive in the direction of increasing </>, and
the radial
middle surface
Shear
The strain E,.
<>f both
its ends:
in the
again
sum
of the rotations of the two line elements dx and r d</>
(Fig. 3.15):
124
CHAP. 3: C Y L I ~ D R I C A L SHELLS
shall do so for the simple case represented by Fig. :t6. From the con
that NJ = 0 at ends x = ± we the functions /
1
and /
2
which are given on p. 107. We still have fa and /
4
which
may
be used to satisfy two conditions for the displacements. Since we assume
that
the diaphragms are perfectly flexible in the x direction (hence
Nx
= 0), we have nothing to say about u, but we should, of course,
like to have
the shell because of its
connection with
the diaphragms. But this is too much for only two free
ftmctions, and we have to make a choice. Now there are forces N x ~ at
the
to determine fa
so as to have
v = 0 at x = ± l/2 and to leave it to additional
bending stresses to fulfill a similar condition for w. In this
way wc arrive
d<J>
[ dF d ( 1 dN ) ] . . ,
- S (l - 4 X ) (2 + V) d<J> - 'V d<J> r -d<J> -:- 1
J.V ~ .
Upon introduction of r(<J>), N ~ , and F(</>) for a special case they yield
immediately the displacements for the assumed boundary conditions.
3.2.2 Circular Pipe
these formulas we shall now calculate the
deflection
of
of
specific weighty and supported by two rings as shown in Fig. 3.7.
The stress resultants are given by (3.8). Comparison
of
(3.8b) and (3.3a)
yields F(</>) = ya sin</>. Using this and N ~ from (3.8a), we find from
(3.22):
0
12
es t ing
 to co m
 
ere we
ear in g
em of the
 carries only
edge loads ,
the mem
0) and
brane forces
by the 
ons tan ts
Cm and D
s (3.19) and d
of th
d D,, for every
h edge. T
e d b
se, also 
ite the
a
Fo
ind
l
Etu,
~ - V ~
the arbitrary
formulas, but the series for v satisfies automatically the condition
v = 0
at x = 0 and x = l, while ·u is not restricted in these cross sections of
the cylinder. At first glance one might expect that even the condition
w = 0 is satisfied at the ends. This, however, is not so. The formula
for
w
in
from Prn without a factor n -
1
vergence of the Fourier series for p,
11
toward zero as x = 0 or x = l is
approached, the corresponding series is non-uniformly convergent, and
so are the series for N ~ n and
wn.
general equations (3.27)
in
a cylinder subjected to its own weight. We find the following ex
pressions for the displacemcnts:
The
terms
for P ~ n and p,,.
on p. 117.
3.3 Statically Indeterminate Structures
As we saw on p. 107 the membrane forces of pipes and barrel
vaults
have the same spanwise distribution as those of simple beams, provided
that there are not more than two
diaphragms to which the loads are
transmitted.
diaphragms,
is statieally indeterminate. The theory of deforma
tions presented in the preceding section furnishes the means to solve
it if
true
vaults.
As an example of this kind we consider a pipe of circular profile,
having two
sider the dead load
may
Nx we have to go back
to
the
set
find
1
a ' f ' ad<jJ
Introducing N ~ into (3.21), we find for the displacements the expressions
px2 ~ x2
E t u = 3a cos q, + V p a X cos q, -
d<jJ 2 a + I X + I
'
Etv ~ 12a2sm'f'-
(4 + 3v) 2 s m ' f ' ..J._ d</Ja 6a2 - dcp 2a
+ [ 2 ( 1 + v ) l t - : ~ ] x + l 4 ·
Xow we have to find four boundary conditions from which to find
1
1
, 1
2
-·-li------· ·[$-+-
diaphragms
u == 0. At the other end of
each
bay,
conditions follow
that both displacements v and w
be zero at both diaphragms would evidently
be
making
at x = l.
x
1
. [ l
5l
2
2
_ p . [
2
that the bracketed
They cannot, therefore, be expected to
 
e p ro v
 ribs
 
o n
)
+
-
anes cf>
e r line
ke N  x
g
all
secto
ts
erent,
and
we
oduces   b e n d
ing
moments
0
= 90°,
T h
ad on
 f>o = 90° and if
the
local
nt, we h a v e N+o = 0, an
d the
dome t
a
n
. 3.18, a
and
a
e form er com
  t
of the
ined by 
lane force
res
ultants
w
The in tegra
 in t e rms
  represents a
0 
, therefore, w
etr ic part of the
load.
We
sultants
o s - - =
x
2
•J:tkm
  r dcJ> 2
ected that
the functions
In
ord
e
p
at ion of the
lem
we
oments and
tructure.
between
. sin-=-)
= 0 .
: ~ : . ; x+
n
n
'+' n
Wh
e
  = -
uce
nt and
valid for ev
is in equi
ial force
rce. This fact
 by s imply w
> t
L S
following e
2
"
- x
4
tan ts in
ot expec t
st cases, it wi
tions to be e
ed load on t
k for
,
d side of (3.4
) = 0
_ s
innkfn
- si
nn/k  ·
T
as the
F<ml = -.-
A cot"- +
n
We may now go backwards through our equations and find from (3.38b)
I
A-
and
2
;;
- - - , ' -COS--+ . -Slll--.
cosnjn n
= 0 of the dome; the second
term
is regular everywhere unless we extend the shell to the point </> = n.
For k = 1 (first harmonic) we have A= 1, independent of n. In this
case the B solution corresponds to a loading of the dome by a horizontal
force P, the A solution to the application of such a force and
an
external
1
as shown in Fig. 2.24 for a spherical shell. These loads may
easily be determined by examining the equilibrium
of
1
top
are in equilibrium with each other, and so are the forces N ~ m ~ in a hori
zontal section through the shell.
Then no external force or couple is
required at this point.
So far the situation is analogous to that which we found on
p.
48
at the
in the cylindrical sectors. Because
of the
10
147
(3.36) for the loads and the stress resultants is no longer possible,
and
there is no regular load which might be dealt with more easily than
with the general case.
time, we
then deal with a set of 3n simultaneous differential equations.
In simple cases, where symmetry reduces the number of unknowns,
it may still be possible to solve the stress problem. For instance, this
is
the
the two structures shown
in Fig. 3.27. The first shell, Fig. 3.27a, has one plane of
symmetry
and
(a)
(b)
Fig. 3.2i. Nonregular polygonal domes
therefore only 3 different hip forces F(ml. In the sectors which are inter
sected by the plane
and there are only 2 func
tions fim> and 4 functions f ~ m ) to be determined. This makes a
total
of
9 unknown functions. In the other case, Fig. 3.27 b, we have 2 different
hip forces, 1 function flm) (in the small sectors) and 2 functions / ~ " ' ,
i.e.,
5 unknowns altogether. Simultaneous systems of this size may still be
handled numerically in a reasonable time, once their coefficients have
been determined.
3.5.1 Uniform Load
Two examples of folded plate structures are shown in Fig. 3.28,
a roof and a bridge. They consist of a number of plane plates forming
a prismatic surface. Each of the plates is much longer than wide. These
structures have some similarity with cylindrical shells, although an
essential feature, the curved surface, is absent. Their theory is best
understood against the background of shell theory and this justifies
its
inclusion in this book. Moreover, a folded structure with a large num-
10*
L SH E L
 
ertheless, the
theory
stem, a l thoug
h it does not te l l the en t i
re
story.
T
rips (Fig. 3.2
aphragms  w h
th strip
e used
as a
ips m and  
 
resultant load
 of the
hbors,
it
ple
beam
CHAP. 3:
C Y L I ~ D R I C A L SHELLS
phragms and subjected to
bending moment
force
(3.45b)
If
he
plate strip is slender (h,. ~ l), and this we shall assume,
the
bend
elementary beam
111
0
strip is
m-1
Fig. 3.31. Beam action on an isolated •trip
opposite sign. On the other hand, the two strips will, of course, exert
forces upon each other which we have not
yet
taken into account.
Since such additional forces must lie in the planes of both strips, they
can only be shearing forces
T,.
and distribution
are not
the
direction
in
which they will be considered positive in agreement with the sign con
vention
 
a ~ c y
 w hile
(3.45) ma
RICAL S HELL
portional to x,
uations, w
ord
se
equa
tions
Sh,. •
T ~
] x
Ym+lb
ect ion of the Ym axis is ra
ther sm
i formly loaded
(if at all).
edges. I t
ing forc
> m + l .
m, l
m - 1
 
154
CHAP. 3: C Y L I ~ D R I C A L SHELLS
Following
the
same line of thought, one easily arrives at the following
form of the three-shear equation (3.48):
(3.55)
is interesting to apply the preceding formulas to some more
or
for analogous cylindrical shells.
Fig. 3.33 shows a pipelike structure of octagonal cross section. The
best approximation to
framework of the present
1
, P
2
•.. P
8
where
p
is
the
stresses in each cross section will be distributed sym
metrically with respect to a vertical axis and antimetrically with respect
to a horizontal axis, it will suffice to consider only one
quarter of the
structure
and to write (3.48) for the edges 3 and 4 only.
They are
T ~ + T ~ +
ular y
ona l   tube by
n e d . Th
make the stress system approach that of
the
does occur can be seen from Fig. 3.36. Here the
number
Fig. 3.36. Prismatic barrel
vault with many edl(es
been doubled. The stress diagrams are rather irrregular, and it is clear
that the membrane
3.5.4 Limitations of the Theory
We
saw
membrane
roof is
the
and
its
In the prismatic roof no edge member is needed, but
the
limitations
Fig. 3.36
certainly does not represent a physical reality. The real structure will
level the peaks, and it will achieve this the help of a
system
moments and transverse shearing forces similar to those in cylindrical
shells (see Chapter 5). The bending stresses
are
Additionally, there is another source of bending stresses. The loads
acting
on
the
produce
plate
the
edges.
The bending stresses a connected with these moments may be con
siderable;
the
thinner
the
plates
mine the
a
supports
are
the
strip
supports
of a peculiar kind, the deflection of each support depending on the
reactions
chapters the membrane theory of shells has
been developed for two important types: shells of revolution and cyl
inders. In both cases the theory made use of every advantage which
the particular
thus arriving at the
lacking
y
in rectilinear coordinates
generality. I t
is the purpose of this chapter to develop a general mem
brane theory for shells of arbitrary shape and th{m to apply it to some
shells which
do not
the shell by a
of rectangular
coordinates (Fig. 4.1), assuming that z is given as a function of x
and y.
are
between the
points of the middle surface, they may be used as a pair
of curvilinear
const. and
ng this   surfa
ce wi th
  These lines
meet at
in Fig. 4 .2.
T w o  o
f them, N., and
le
nent paral le l
wnt and its projec
plane
s thr
g them
 by th
e length
 of th
may
b
 
s
dy
N
X S OF EQ
t 
t.  Alo
ng  wi
th  them
ject ion
= _ _ _ ( aN. + aN•
• ) .._z_ _ (a
 
se of (4.4a, b
2
z  _ _ _
  the
riable co
ions m
r ium
a, b), are   id
y,
(4.6)
erator L
s function is
r is obta ined
  ca
e p ro d
trary l
eng th .
ete for
 
on ,
t
n elliptic paraboloid
s
w
ain parabolas
but of
~ ~
- - - - - -
~ - - - -
~
; . .
  by two
planes ;r
g. 4.5). I f  we
  choose the
same load
as befo
f fened and supp
ential forc
 by the shel
 
hole boun
ro.
1, 3, 5, . . .
3ny +_ _cos 5ny -
) yields the
form u
The convergence of these series shows an interesting peculiarity.
Since the
have alternating signs, the factor 1/n is sufficient to
assure at least a feeble convergence. But there is still the quotient of
the
two hyperbolic functions. For a fixed value of x, 0 < x < af2, and
large values of
and
decreases exponentially. This gives an excellent convergence, if we do
not go all too close
to
and
the
is
from the factor ( - l ) (n+l l /
2
not
the
stress
The
z = j(x) + g(y).
sections x = const. are congruent to each
other,
curves
y = const. The surface may therefore be generated by subjecting one
of these curves to a transverse translation. Such surfaces are called
surfaces of translation, and
ated are their generators.
When a shell is formed as a surface of translation, a shell element
bounded
by
two
pairs
of
fore
the
the
boundary
11
vertical load. This is the reason that the shear
tends
toward
infinity
The
physical
membrane
forces
substantial
magni
tude
shell.
On p. 164 we used the double symmetry of the problem to replace
the actual
the
entire boundary. In cases where there is less symmetry, we
may
when a linear function of x and y is added
to
if>. One may, therefore,
 
to make
I f even that
is lacking, we are faced with the possibility
of combining the solution for zero boundary values with the solution
F T
___J_ ___ T
of the homogeneom; differential equation for the boundary values shown
in Fig. 4.6a.
I t yields the stress resultants Nx = N
1
= 0, Nxy = T and represents the
stress system produced by edge shears T as shown in Fig. 4.6b. Whether
such
depends
the edge members at the four corners of the structure.
4.2.:J Solution by Relaxation Method
I t is not always possible to solve the stress problem by such
simple
we may
think of the shell shown in Fig. 4.5, but with constant wall thickness.
Then the load P:, produced by its weight, is a constant, but Pz increases
toward the edges and still more toward
the corners, and (4.3), which
describes this increase, is
way tempting for analytical work.
In this and other cases a numerical method is needed, and
it
that it opens the way for the applica
tion of the relaxation method. Its use is
limited to differential equations
of
the elliptic type and this requires that the middle surface of the shell
have positive GAussian
Like all finite-difference methods, relaxation cannot handle sin
 
shells
the
corners. To avoid these singularities,
we need a load distribution with p = 0 at these points. vVe obtain it
by subtracting from
can
any problem involving a symmetric, vertical load.
4.3 Hyperbolic Problems
The surface which spans
and is called a hyperbolic paraboloid. Its intersections with vertical
,.
)(
....
the
reciprocal of a length, is the twist (J2z(ox oy of the
surface, i.e.
hyperbolic paraboloid yields a type of shells which has often
been used
with
Introducing
he fiber forces Nr,
r hyperbolic parn
 a combinat
the-
R B I T R A R Y
SHAP
E
sh
the top an
urse, equal to
e stiff  enoug
h to resist
that Nr =
r. One
lls meet at an angle, o ne might th ink
 tha
ose sh e
ecause
ges. Therefor
ssary th
at the
r id
h t
ahd
the
comple te l
wou
nd before,
ibr ium
  in th
10. It.s stress
equation
ar t icu la r p ro
blem:
2a2([
> 2a
  a m inus  sig
n here, ind ic
belo
to be used
r t icu la r solu
t ion of (4 .21)
. For the s im
ing them  to (4.5) for th
e stress resu l
in  
w
e
c
or the
ent ds s
  have forces
s
w
  and in the
load. W e
onents
the tw o
the sam
e way,
m external fo
B, we can r id it com
pletely of all external
 supp
es on
1
an
d
B
A
1
forces.
there
nce on ly  
e is free, and we may ch
oose it
cels t
r ip a long 
ay, we procee
ces on
n the
1
, F
2
symmetric
have
to
the shell, the forces resulting at F
1
and
F
2
B
'
F i ~ . 4.13. Transfer of edge loads In hyperbolic shells, depending on the relation of the edges to
the generators
that this shell may, for a load of a certain symmetry, be supported by
four diaphragms. A roof
ensuing stresses, then the simple and intuitive method of solving
the
permits
favor.
We shall now illustrate this method by some examples. We begin
with the shell shown in plan projection in Fig. 4.14, assuming that there
is a vertical load such that Pz = p = const.
Then the
inhomogeneous
solutions (4.22) are applicable, and we choose the second one. I t yields
the stress resultants
on the edges AB and CD, we apply
here tensile forces of the same magnitude and resolve them
in
corn-
In zone III all forces have the opposite sign:
In the zones
one compression,
the
the particular
solution (4.23)
by
·when
parallel
to
this
the
same.
- 2
N •
• ph,
- 2
N
In the shefl of Fig. 4.14
In shells of positive curvature we found such discontinuities only along
the edges, but when treating the hyperboloid of revolution (p. 75) we
encountered the
discontinuity
on
propagated
shell. This is a general feature of all stress problems
governed by
differential equations
of the
type, and
we see
from (4.6) that the membrane stress problem is exactly of this
type
if the shell has negative curvature. This indicates that for all such shells
the results of the membrane theory are
to
tion.
the
the
stress
resultants will be quite different if we shorten the shell in Fig. 4.14
by
Starting
from
the
find that they will not cancel the thrust at the
right
and
 
function
a'41> a
the other terms
the ratio
of the right-hand sides. This condition will be fulfilled if the loads
satisfy the relations
per
the
projected
area dx* · dy* of a shell element. The total force acting on an element
of
p:dx*dy*,
and it follows from (4.24) and (4.26) that the load components on
the
fizdxdy = A
The
loads
the
connected by
there the
relations (4.24),
(4.25), and (4.26) we find the corresponding relations for the projected
forces:
Nxy
= Nty,
(4.28)
the
on
the
line
element
and (4.28) we find its relation
to
shellS:
y ox*
xternal fo
e coordinate
in the
hells:
(4.29)
 and t
he formula
 
e
want
d uc e   the
d .
determina te , t
he same is
Its bendi
s
cases the
rs a re
e rm ina te by
them selves, an
d since redundant
p u t ed for ea
ch case acco
x = x*, 
which transforms one shell into
the other
()
() = ()*'
ds
8
The
}'ig. 4.18. Vertica l stretching
of
a
ratio 1 A., and the corresponding element on S is
therefore
to establish relations between
area of the shell elements limited by meridians and parallel
circles is, dA
= as I> • as
ratio
is
on
p l = Pt ( os
  FORCES IN A ~ ' F I X E S
HELLS
183
rt icular typ
sure vessel,

ied by differe
say a sphere
favor
of
there
n a s t roke of the
pen
The
roduce these
1.. Bn = -
" ' ' ~ u " - s i
fers to use the
184 CHAP. 4: SHELLS OF ARBITRARY SHAPE
As an example of the application of (4.34), we consider the water
tank shown in Fig. 4.19. I t consists of a cylindrical part closed by two
half
ellipsoids
is filled as indicated, the reaction
in
water
Fig. 4.1 n. Water tank, supported by four
columns attached to
the second step we use the formulas (4.34) to make
the
match.
Since no load is applied directly to the roof shell, we may assume,
for the start, that the stress resultants
in
For the
bottom,
we found a solution on p. 34. This solution would de
scribe the stress
force
In
the
ertical loads
ulas (3.3) for
e forces in  a
 there
the cylinder
sum of all extern
series. The shear distribution along
the
in the
FouRIER representation:
y a
11
sinn0.
the
over the values n = 4, 8, 12, 16, ... The abbrevia
tion T,. has been introduced to keep the following formulas compact,
but we shall get rid of it before we write the final result.
Fig. 4.21. Edge
For the roof ellipsoid we must put .d
..
order to avoid a mean
ingless singularity at </> = 0, but Bn may still be chosen freely. At the
edge </> = 90° we have
ional force
N ~ = BnJ..
cosnO. At the upper edge of the cylinder the
n-th harmonic of the shear is Nxo = T,. sinnO. vVe could easily remove
the discrepancy
therefore
still postpone the final decision on the value of Bn and remove all dis
-crepancies by applying an additional shear
N,.o
Nx = B,,J.. cosn ()
to the upper edge of the cylinder.
At the connection of the cylinder with the bottom we have to pro
ceed in a similar way. We apply the solution (4.34) to the bottom shell,
this time putting B
until
later. The
edge load of the half ellipsoid at </> = 90° will then be a shear
N
11
.<:
= h of the cylinder we must then apply the shear
N,.o =(An- Tn) sin
 
m ong them.
n we
 shea
r
must
qual
t
edge x =
other edge x = h, we fin
d
 
y a y
  w
be cal led
h lie on t
 sui table as a
ine coordinat
y.
zon tal s t r e t ch ing
of a shell of revo
lut ion
We may
easily exp
lement ds
 
ly by compo
 sin
2
cf>*.
r parts
 
in the shell
  stresses
.2.3.
T
eed t
he shell e lem
  from (4.27)
the
affine
lu
erties. W e m
r o m
rces
e essentia l lo
 (P:r = p
 all th
e co rr
cessor
y
m
ateria
from
a
angular
values as coordinates of the corresponding point
on the ellipsoid.
shells are
element dA * = ds; · ds: of the sphere has the following
projections
on the x, y plane: dA* coscf>*.
The projections of the corresponding element dA of the ellipsoid are
obtained by multiplying
When the ellipsoid is subjected to a constant internal pressure
p,
a force p dA acts on the shell element dA.
Its
projections
of
MBRAXE FORCES I ~ A F F I X E S
HELLS
193
According 
to
the
/>* sin B
s</>* +
pre
c
Pozsm.:;v'
P
"'
2 b a f
T he h
13
mulas for 
X
V
c
2
m for an  ell
sure
p.
Sinc
tor which
nts in an ell
  the stress r
possible for
any choice
is resul t dem
arly t
hat pre
ssure vessels
dxV'
  au
aw .
= 1
 sm
 x
COBJ.
X
X
AB
sm
 X •
cessive steps of l inear izat i
on
in
 
read
th
ore
a
+
y •
C
il l X
+  ay co
's law
 in term
CHAP. 4: SHELLS OF ARBITRARY SHAPE
as (2.54) and (2.55). To find it, we introduce the orthogonal coordi
nates
; , 1J shown in Fig. 4.30a. In these, HooKE's law has the standard
form
E t y ~ ; ~ = 2 ( 1
+v)N;,
1
(4.41)
When we subject a small piece of the shell subsequently to uniform
strains E ~ ; , E ~ , and Y<'l
and, from the motions of the points
Band
1
2
w.
On the other hand, we find from (1.9) with o: = 0
and o:'l = 90° - w
then
Et
Ex=
Nx-.-
1
Et E
kinematic relations (4.40) and HooKE's law (4.44) are six equa
tions, from which, for known stress resultants, the strains and
the
dis
to reduce these equations to
a single one for the deflection w.
As a first step, we solve (4.40a, b) for oufox and ovfoy and introduce
the result in (4.40c), which then reads
.
( ~ ~ ) ()
Yxy sm ( I ) + (Ex+ Ey) cos w = ay+ ox cos X cos
aw '() aw ()'
•
 
use (4.40
to
 the
follow
or w  
) __
 _: (_
  . ......) _
 :__ ( ~ ) .
  (
aXJ cos
side of {4.4
stress r
rite these
y conditions
ary condition
ense
to
ent u cosx + w sin X along
 the edge
ss these qua
s of
cribing there
O
ay
e.
This
yields
, the re
 
ns , we
  find tha
shell shown
rp ing o
e
ed
a
n Yx f
of the she
nsional , m ad
e possible  b
y a simul
ports
y e
In the preceding chapters dealing with the membrane theory of
shells, we often met questions which this theory could not answer.
This indicates that in certain cases the bending stiffness f the shell,.
although
small,
cannot
a bending theory. In such a theory all
the
stress
resultants
the
mathematical·
analysis of such stress systems is far from simple. Therefore, solutions.
have
types
and
we
did
for
the
membrane·
3.1 ), ;r being
to
angular
sarily the
the
dimen-
sionless coordinates x(a and cf> will here be indicated by primes and dots:·
a ~ = ( )'
ax '
a<P .
The shell element as determined by the choice of coordinates is shown•
in Fig. 5.1 a, b. The first of these figures contains all
the
external and
internal forces acting on this element and the second one contains the
moments, represented in the usual way
by
arrows.
conditions of equi
librium, three of them concerning the force components and the other·
three, the moments.
N ~
contribution
shear Q.p.
The two
forces Q.p dx make an angle d<f> with each other
and
have the tangential resultant Q ~ dx · d<f> which points in the direction of
<f>. The
(5.1 b)
The
third equation refers to the radial components of forces. In the
membrane theory it is extremely simple; here it contains contributions
of both transverse
d</> · dx and
(oQ,.(ox) dx. a d<f>. We have, therefore, the equation
Q ~ + ~
be
caused
had in the membrane
(b)
The equations for the equilibrium of moments are easily explained.
For
an axis of reference, coinciding with the vector Px in Fig. 5.1
a,
we
moment . J . l l x ~ and the couple for111ecl by
the two forces Q ~ dx:
(5.1 d)
t ion of
 
an
  = 0.
ion of
the c
yl inde
r  may
be de
nts
nt alo
R & ~ T I
siona l elasti
ion ,
2)
th
at
for
all
ki
nemati
c 
that whatever
strain,
the shell is thin.
basic assumptions we must add a third one, which is
needed to keep our equations linear. I t is
3)
that
curvature
their first derivatives, the slopes, are negligible compared with unity.
From these three assumptions we
establish the kinematic relations
of
the cylindrical shell. Fig. 5.2a shows a section along a generator. The
heavy horizontal line is the middle surface before deformation. After
wards,
= w'fa.
1 it follows that
angle w'fa. The
A
of point A is therefore equal to the displacement u of
point A
0
minus the distance A is shifted back by this rotation of A
0
A
Z
I
(5.3a)
1
wrse section through the shell. The point A
0
the middle surface. Since the normal A
0
the point A
rotation of the normal,
A does not change.
The difference of the normal displacements w and wA is then due
only
w fa and proportional to 1 - cos of these angles.
Because of the third assumption, this is negligible, and we have
WA=W .
(5.3 c)
The next step is to find the strains L,., «:.p, y, .p at the point A. They
describe the deformation of an element on the cylindrical surface
passing
the radius r there by a + z,
and the displacements u, v, w
by
AL EQUATIO
st pu
e performed
tensional rigid
bend ing  
he expres
 
ltants may be tre
dditional explana
tion. For
z
of tfa and
y K appea
ulas is  no
w h i c h repr
esent the elast
ic l aw
1 - '11 (
here
will
be 
unders
require that it maintain everywhere its contact
with
=
two opposite corners will come closer to a tangential plane, while the
other
a rectangle
second
When
elastic law (5.9), it
a
N _ D ( 1 - v ) _ K ( l - v ) ( · Yx<P)
J. x<P- 2
fflq,x = K (1 - v) ~ . t < P ,
_ ~ l f x . p = K ( 1 - v ) ( ~ x < P -
;:)
elastic law. I f we put K
= 0, the moments vanish altogether and for
the forces we obtain the simple formulas of the membrane theory: each
normal force proportional to the sum
of
the corre8ponding strain and v
times the other one and the shearing forces equal to each
other
and
'Vhen we now consider
the terms
with ~ in
the moment. But there is a term with E.r in M.r.
I t
is due
the
shell element are
trapezoids (see Fig. 1.2). Therefore the resultant of a uniform distribu
tion
of
stresses a r across these faces does not lie on the
middle surface
distribution
distributed stresses a
the deformation
of
and shearing forces similar ex
planations may be found,
the
values
of
the
stresses at z = 0 are not necessarily the average values of the stresses
across the thickness of the shell.
 
ing theo
ry  of
tions
5.1.2.3 in Membrane
3,
assumption that all bending and twisting moments were zero. \Ve
found that often
tion
cannot
be
might
be
in
We now have
the
real
x ,
N r ~ in a cylindrical pipe supported at both ends by diaphragms
and to
its 125
tion of a shell of extensional rigidity D
=
real shell of thickness t has a. finite rigidity K.
To
ask how much the existence of this bending rigidity will change the
deformation would be equivalent
the bending problem, i.e. of the differential equations
(5.13)
on
(3.24),
The
law (5.9). We simplify this procedure without losing something essential
by assuming v = 0.
and
pt•
Mx =
24
a
2
(8a
2
+ l
2
- 4x
2
) cos</>,
pt•
of
equilibrium
Q ~ = O ,
a• xcos</>.
These shearing forces have been neglected in the conditions of equi
librium used
besides a
of the
the
that of the shearing force N ~ x may
be found
·
of the the a of
the
greater
order of magnitude is enforced there.
5.1.3 Differential Equations for the Displacemcnts
The conditions of equilibrium in the form (5.2) and the elastic law (5.9)
together are 12 equations for 11 unknowns: 8 stress resultants and
3 displacements.
got
one
problem is overdetermined. But the
surplus
of
In
set (5.1), is an immediate consequence of the relation <x.;
= <.px·
It
resultants by
become three differential equations for u, v, w, the displacements
of the middle surface:
u
~ u
-- -
P . p U ~ 0
= ,
2
=0 .
Here the dimensionless quantity k stands as an abbreviation for
(5.14)
Equations (5.13) are the differential equations of the bending theory
of a circular cylindrical shell. I f we write them in operator notation,
it
is
rather
demonstrate this
process for
C A L
ay be tween
ations, we  ha
 
rder, but sinc
imultaneous
et, derived f
ate elastic l
aw (5.12). Si
ions of
.:) + P;a2  0
erms} are ex
approximation
of the solution to be expected from the bending theory.
The main objective of the bending
theory
ment of these membrane solutions but a
study
certain edge loads which do not fit into the general
pattern
of
the
membrane theory. This will be done in much detail in Sections 5.3
through
5.5.
There exist, however, occasions where it is desirable to have
a
particular
solution of (5.13) for a given surface load Px• p<l>,
p, . We
may easily find such a solution if the loads are distributed according
to the following formulas:
a
. "' . A.x
a
(5.21)
Prmn are three constants which may be given in
dependently.
When
we see that
AX l
to be determined from the differential equations.
Introducing (5.22)
into (5.13), we may drop the trigonometric factors and arrive at the
following set
vmn• w,.n:
1 -VA - 11. r Am
Wmn- - y jPxm l l
[-
k,A2m] Vmn
ical valu es of umn• v,.,., m
ay be
111
,.co
sm'f'sin
7
a·
'f
' a
a
a
Q
- K
  [,( ,3
1+,, ]
aa ' ' +
AZ
se which must be zero i f the edge o
f the shell
iaphragm, i.
lane but off
ers no resista
= 0. We
rane solut ion do
t ion and not at all co m p l
icated. To un
ical results.
w n
tions a
Let us 
hich th
ulas
(3.2
mal and
N••I,l = 2p
in
a
  rath
er 
thic
heory w hen ap
s and t r ansv
erse shear in
g forces are ,
10-3p
1
a
2
iderable here
 but is  
N E O U S PROBLEl\1
221
Now
 let
us
consi
ceding
exam
ple.
to-
3
p
10
a
2
/D
mat ion due t
  forces
are 
10
a ,
m
are:
3
p
10
a
2
n are
~
ce even   dis
ads
may
be
certain usefu
l s
y
ents first incr
m ·
enon
  j
enient to  avo
id th e
mbrane solu
lowing sectio
ut
. In
 the 
sim
sider the 
ST. 223
p . 116
o m (5.
N,.
= 1;
, I
 
A1
ST.
u ,
6
e-ip
1
xta)
ine Ai
y
es very far
xpec t th
  exponential
ditions at x = 0.
  two
by
unctions
inary
argument
nd
e ma
e rest in
g to i
lP 1 + 2
st par
 
ive rgent, they have  a h
or izonta l res
ch yie ld  the twisting  
' 'ab
sin
) Pa - 2 1 P.t Ptl
si
 
rd. Th
eir dif
late
x
be arbitrarily given. But they are
five, Nr, Nx<i>' Mx, 1l'Ix<1>' Qx, whereas we have only four constants of
integration, G
Since they are equivalent to the corn bined action of
three
of
them
and
they
Gi.
I t becomes evident that the introduction of the effective edge for
ces S r and T x is
an
essential feature of the bending theory as it is re
presented in this book. It is beyond the scope
of
boundary
zone
of interest when a small hole is drilled in
the
shell. For this problem,
we have to abandon our basic assumptions and to replace them
by
something better. In all other cases boundary conditions on structures
and machine parts are
border problems a possible object of
investigation.
to
there
is a smooth transition between them, as it frequently happens in con
crete, cast iron,
the shell ends and something else begins.
5.3.3 Cooling Tower
The theory developed here may be applied to structures like the
cooling tower shown in Fig. 5.5. I f
the
shell were put directly on a con
tinuous foundation, there would be no problem at all. The weight of
the shell would cause compressive forces N x , increasing from zero at the
top to, say, N.= - P at the base and distributed uniformly over any
horizontal cross section of the tower. Now, in order to allow the cool
air to enter, the shell stands on a number of columns, and in the space
between them no force Nx is allowed to act on the edge. Therefore,
an
edge load must be superimposed on the simple stress system just de
scribed, and this edge load must be self-equilibrating and must cancel
the edge load N x = - P along the free parts o f the edge. A part of this
load is shown in Fig. 5.6. It may be expanded in ·a FouRIER series (5.25)
in which only
assume for
quantities must be known along
the
not much that might restrict the rotation
of
between
it
 
other two
depend
on the size of the stiffening ring which will be provided
at
the edge. For a numerical example wc consider the extreme case
that
w
Section A-B
the shell is high enough, the solution (5.33) for the semi-infinite cylinder
may be applied, and then it is not necessary to have another
set
of
,;,=8
I
I
6
(b)
(c)
and
second
NST .
233
one on the
 w ay towar
d a numerical
d {Ji for i =
articular
set
of
tribution of
s of the
5.5
When these eq
ua t ions
have been s
A L SHELLS
t the edge x =
.. become less
  is practical l
onents , incr
  m
 at some dist
 
of
N"'
t ~  
f;;
1.0
1.0
. 5.5
n that the s
 where they ar
a ne
wall supporte
eight. Since
  our bound
ary conditio
  compressive
stresses
rt
of
all this is re
ibuted. In F
ig. 5.8 bot
ower and- su
en we co nsid
=   CONST.
235
presence of
 forces Q+ a
nd then necess
arily also be
character and that m= 16 makes
.a.
much
larg
on to the ho
of
e
problem
 large r o
order k-
 order
d
wm(O) =
CHAP. 5: C£RCULAR CYLINDRICAL SHELLS
I f the constants cl'". (J4 are all of the same order of magnitude, the
first term of w;,. will be much larger than the second, while the opposite
is true for
we have a look
1
f1-
1
are markedly preponderant in M x, Qx, S,., but the
terms
with
3
, C are large enough, also in T x . The
second half of the solution (5.33) is therefore especially fit to
satisfy
a
pair of boundary conditions concerning N., or u and T
x or v, and those
are exactly the conditions we already were able to impose on the mem
brane solution. This part of the bending solution is
not
i f
we use
the
conditions concerning Mx or w' and Sx
or w. I f we replace w here by the hoop strain w + v', these are exactly
the
of
the solution is therefore the essential complement to the membrane theory_
The situation changes for higher values of m. There the splitting
of (5.29)
1
, "2'
conditions, and it is not
possible to anticipate part of these conditions when writing the mem
brane solution for the given loads.
5.4 Loads Applied to the Edges cJ> =
const.
in
Sec
tion 5.3 a complete solution of the bending problem of the
circular
the
in the case of barrel vaults, this solution is
not
prescribing arbitrary boundary conditions along edges
cJ> = const. I t has so far not been possible, and
probably
never will
be with simple mathematical means, to find a solution which can satisfy
any desired set of
conditions along all four edges o.f a rec
tangular panel cut from a cylindrical shell. However, we may exchange
the roles of
coordinates x and cJ> and find a solution which does
the
same
at
of s
that
6
+ [6A.
4
- 2
omplex and ma
par t icu la r so
lution of the
G of each
o ns (5.40), in
term ine Ai, Bi
as
wher
e
a
ne
ma
relations must
ho ld :
ax = as =
a bar represe
ular solutio
8 x 10
ered  in
sh ell
. F ig .
 5.9 y
ie lds
uch
in  
th
e
preced
  evi
dently
  are
unsui
  of course, f
stic
law
l cf> +
 (al 0
2- a2
(5.43)
coeffici
bscripts
1
To solv
ceed  
i
0) for Ai, B;
  values of th
ose forces and
r 0
 which we may w
consid
is no   force F
e shear N<
Table 5.2. Cylinder Loaded along a Generator
f
I
c
a,
u
1
< 1
V
1
p1
w
1
1
w
1
- "1
2
Nq,.
J. P1 + kJ.
2
.il'fz
Kfa
2
- A
2
- -
2a
2
2
P
1
2
a
1
2a
3
(1 -
v)J.(x
1
P1
+111Pzl
(x1
5.4.1.3 Symmetric Stress System
The second special case of practical importance is that of a shell
having boundaries at cJ>
the stress system will be symmetric with respect to the
generator cJ>
= I?
combine the real exponential functions to hyperbolic sines and cosines,
writing
1
cf>
cl>
= CONST.
243
<X a
2kxtf.lt
sym.
sin
<%1) - J. Pa
anti.
cos
From symmetry we now conclude that the coefficients of the
antimetric
the following form:
Wn = Gl Cosh"
2
1>
To find
and
find these expressions:
+ (a
3
p,
2
<I>,
VII =
-(Pl cl+ P2C2) Sinhxl <J>cosp,l <I>+ (Pl c 2 - P2Cl) Coshxl<I> sinp,l <I>
- (Pa Ca + P4 C4)
3
) Coshx
2
2
cp.
When we go back to (5.9) and (5.1 d, e) we find that some of the stress
resultants (the symmetric group) are expressed
by
I = c [(al cl + a2
C2)
c 2 - a2 Cl) Sinh XI <I> sin Ill <I>
+ (a
3
C
3
+ a
4
C
4
) Coshx
2
1=
-c [(al
cl+ a2 C2) Sinh xl <I> cos Ill <I>- (al c 2 - a2 Cl) Cosh xl <I> sin Ill <I>
+ (a
3
C
3
+ a
4
C
4
quantity belongs to the symmetric or to the
antimetric
group.
The
.coefficients
a
3
5.4.2 Barrel Vaults
The theory
siderably in
the
case of barrel vaults, in which l is much greater than
the
radius a of the shell. In Chapter 3 we saw that an edge load applied
to
the
straight
cannot
membrane forces N.p. Bending moments M.p and transverse forces Q.p
are needed to carry the load away from the edge,
and
Nx<l>
are needed to transmit the load to the ends o the span. However, the
deformation produced, in particular the lengthwise curvature o
2
wf