CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
EFFECT OF SHEAR IN SYMMETRICALLY
LOADED SHALLOW SPHERICAL SHELLS
A project submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
Dean-Yuan Liu
January 1981
The Project of Dean-Yuan Liu is approved:
Professor
Sem~'
am: engarajan
, Chairman
California State University, Northridge
ii
ACKNOWLEDGEMENT
I would like to thank my committee, Professors T. M.
Lee, D. Schwartz and S. Rengarajan for their time and
advice throughout this study.
I am particularly indebted to Professor T. M. Lee for
guidance and assistance throughout the project.
I would like to express my appreciation to my parents
for their encouragement throughout this advance study in
engineering.
iii
TABLE OF CONTENTS
Page
....................................ii
Acknowledgement ••• .... ... ... .... ... ... .. . ......... .iii
Table of Contents. . ... . ..... .... .. .. ... ............ .iv
Abstract. .. ........... ..... ............. .............v
List of Symbols. ...... ...... ... .......... .......... .vi
Chapter I
Introduction. .. .. .. . .... . . .............. •• 1
Approval ••••••••
Chapter II
Chapter III
Differential Equations and Stress
Boundary Conditions ••••••••••••
• ••• 4
Reduction of Differential Equation in
Terms of Two Unknown Functions •••••••• 9
.............................14
Chapter V Conclusions. .... ......... . ... ......... . ..15
Bibliography •••• ............. ........... ....... .... .16
List of Figures. .............. ... .. ............ .... .17
List of Tables •• ..... ...... ......... ............... .21
Chapter IV
Solution •••
iv
ABSTRACT
EFFECT OF SHEAR IN SYMMETRICALLY LOADED
SHALLOW SPHERICAL SHELLS
by
Dean-Yuan Liu
Master of Science in Engineering
This paper presents a comprehensive report of shear
effect on shallow spherical shells of uniform thickness.
It is applied to shells with or without a central hole and
is based on series representation of normal displacement
and stress function.
The result of the effect of shear deformation is
compared with the classical bending theory.
It shows the
influence of geometric parameters on the result.
For a
certain class of shell geometries the effect of shear
deformation is shown to be significant.
v
LIST OF SYMBOLS
,
Mrr
Pr
t.rr
}(yr
N66l
Direct stress resultants in the
meridional and circumferential
directions
Vr
Transverse shear stress resultant
' Me&-
' Pn
J
te.~
Stress couples in the meridional and
circumferential directions
Components of the load intensity in
the meridional and normal directions
Strains tangent to the middle surface
) K&&
Change in curvature
u, w
Components of displacement
F
Stress function
R
Radius of curvature of the middle
surface
t,
Thickness
-r
Specified location
ro
Radius of interior hole
a
Rim radius
ro
Ratio of
-
r
Ratio of
r
to
a
o(.
Ratio of
a
to
R
l'L
Ratio of
t
K
Br
a
~to
0
to
a
Shear coefficient
Rotation of a line perpendicular to
the middle surface
vi
E
Young's modulas
y
L)
Poisson's ratio
Flexural rigidity
vii
CHAPTER I
introduction
When the applied static loads have rotational
symmetry, the determination of the effect of shear on
small, elastic deformations of shallow spherical shells of
uniform thickness is the purpose of this study.
It intends
to determine the geometric parameters influence on the
shear effect.
The differential equations of spherical shells,
without shear deformation, are given by FlUgge, Ref. 1.
These equations are reduced to the case of shallow
spherical shells by Reissner, Ref. 2, who neglected the
transverse shear forces in the in-plane equilibrium
equations.
These equations can be used for the anlysis of
shell segments of thin, elastic, spherical shells.
A
segment will be called shallow if the ratio of its height
to base diameter is less than, say, 1/8.
The results
obtained on the basis of this assumption will also be
applicable to shells which are not shallow when the loads
are such that the stresses are effectively restricted to
I
shallow zones.
The equations of shallow spherical shells with shear
deformation are derived by Naghdi, Ref. 3, in tensor
notation.
A system of differential equation is deduced
1
~or
2
with small displacements, which include the effect of
transverse shear deformation.
The corresponding equation
of the classical theory, where the effect of transverse
shear deformation is neglected, are contained in the works
of Marguerre, Ref. 4, and Green and Zerna, Ref. 5.
The
effect of rotatory inertia can be easily added to these
equations for vibration problems of shallow shells.
Kalnins, Ref. 6, has specified these equations in
polar coordintes to study vibration problems of shallow
shells.
It deals with vibrations of shallow spherical
shells and the coupled effects of longitudinal, transverse,
as well as the thickness-shear modes and are valid for a
higher frequency range than those shown in the work by A.
Kalnins and P.M. Naghdi, Ref. 7.
Danny Stewart Litt, Ref. 8, following the approach of
Ref. 2, leads to two simultaneous differential equations
for the solution of a stress function and the transverse
displacement.
One of the equations is formed by
consideration of transverse equilibrium as in the Kirchhoff
theory for plate tending, while the second equation is
obtained by consideration of cqmpatability of in-plane
deformation, in a manner similar to the theory of
stretching of plates.
The shear effect and the influence of some chosen
parameters are investigated in this report.
error o
It is intended
3
range when shear effect is not taken into account.
CHAPTER II
Differential Equations and Stress
Boundary Conditions
The differential equations and boundary conditions,
Danny Stewart Litt, Ref. 8, used in the determination of
small displacements of shallow spherical shells of uniform
thickness are considered here.
A very satisfactory
approximate theory can be developed by making the following
assumptions:
1.
The material is assumed to be homogeneous,
isotropic, and linearly elastic.
2.
The shell thickness
to
is sufficiently small
compared to the rim radius a, which in turn
is sufficiently small compared to the radius
of curvature R of the middle surface.
The relations are those used by Kalnins, Ref. 6.
They can be readily obtained from Naghdi's work, Ref. 3,
with the addition of appropriate inertia terms which inelude the effect of transverse shear deformation in shallow
shells.
Here we use them for the case of static loading.
Let
Mrr
be the stress couples of the
stresses tangent to the middle surface,
transverse shear stress resultant, and
Vr
Nrr
the
-
N~
stress resultants tangent to the middle surface.
4
the
We
5
intensity acting in, or normal to the tangent plane of the
middle surface,
u, w
the components of the displacement
of the middle surface in the meridional and normal
directions, Fig. 1,
By
the rotation of a line
t=rr
perpendicular to the middle surface, Fig. 2;
the strains tangent to the middle surface, and
K99
Kyy
the change in curvatures.
Taking a similar approach used in Ref. 8, we now
introduce the following dimensionaless symbols:
r
=
rt
=
w
=
F
=
Y'
=
0.
(
w
)'
d
dr
u
=
Vy
MY>'
=
I{
=
Vr
6
a.[}
Pr = to£
Br =
p
=
Br
Substituting above dimentionaless symbols into the
equations from Ref. 2 and Ref. 6 for the case of static,
rotationally symmetric loading, we get
C
YNrr ) ' -NetJ +Y'fr
=
2.1
0
-o<yctfrr +~)
cv-Vr- )'
+rpY\
=o
2.2
2.3
The stress strain relations, Ref. 2, are written in
following form:
Crr
=
~
= eN&~
Mrr =
Ne&
=
clfr.r -Y!C& )
-vfrrr)
r[
/J( 1- j).l)
tL
2.4
/:J. ( 1- }.);)
2.5
c J<ry +j}Kee )
2.6
+Jl(r'f')
2.7
C Ket>
7
The components of direct and bending strain are taken
in terms of meridional and normal components of
displacement
u
and
w,
in the form, Ref. 2.
l.rY = u' +o< VJ
2.8
£:&~ =
fu.
2.9
Kyy
=
a;
2.10
Kl'"
=
I
-B.,
r
2.11
By
= -w
I
+ o<VJ
K v;y
+ :J.(I+V)
2.12
The value for the shear coefficient,
be 5/6.
K , is taken to
If the effect of shear deformation is neglected
the problem reduces to the classical theory given by
Reissner.
Reduction of the problem to two simultaneous
equations for two functions.
The two simultaneous
equations are for a stress function,
' component of displacement
w.
F
and for the
The stress potential is
introduced, following the approach of Ref. 2, page 82
N yr
=
+
F
1
+
IVYY' L
2.13
8
= F"
where
Note:
2.14
NyyL
The symbols
2.15
M,
N, etc., in this paper are reserved
for the appropriate non-dimensional quantities, while in
Ref. 2 they are referred to with an added bar,
etc.
M,
N,
CHAPTER III
Reduction of Differential Equation in Terms
of Two Unknown Functions
Let
F'
3.1
= w'
3.2
X =
Y
so that eqs. 2.13, 2.14 become
3.3
=
3.4
x'
Substituting eqs. 3.1, 3.2, 3.3, 3.4 into eq. 2.2
yields
(y
c(
\I
V'(
I
)
-
I
o< y ( T X
+lfrrL.
Vr / - co<x +oZrx)
+o<. Y')()
I
in WhiCh
( o/..)(
+ '(
=
+X /
c~ -
(o{
rx )
I
)
+
rfn
=
0
3 • 5a
o<NrrL ) = o
3 .5b
This results in the
transverse equilibrium equation
3.5c
Integrating this equation with respect to
9
r
yields
10
jco< Vr- o<.rxfcl Y
.::} ( o(
where
VY C0
o(
+
Jr cP.,-a~!lrtLJdr =fob
) +]r (p, -o(NrrL.) Jr
r )(
=
c.
3.6a
3.6b
is an arbitrary constant.
Substituting eq. 3.2 into eq. 2.82 gives
Br
= -
Vr
= < Br
y
+
d. (I+ J))
·-K--···
-ty );p.(
Vr
-------
) -t-
Jl
3.7a
------
)
3.7b
Further substitution of eq. 3.7b into 3.6b gives the
equation
3.8
Combining eqs. 2.3, 2.6, 2.7, 2.10, 2.11, 2.12 and
.Set eq. 3.9b into eq. 3.6b gives the differential equation
!of transverse equilibrium
11
3.10
where
Nr;L
is given by eq. 2.15.
From eq. 2.9 we have
t(
/
ul =
< € fJfJ - ~ W ) + < E f!Jt;
ul =
( (ft;f)
c
where
- o{ vJ
/
)
3.11a
/
)
y Etu?
I
- o<uJ-o<.YW
= t_ss -+ YE 1919-
3.11b
/
)
I
The compatability relation is formed by combining
eqs. 2.8, 3.11b and 3.2.
This gives the compatability
relation
)I-
~rr-o<xy= o
3.12
Set eqs. 3.3, 3.4 into eqs. 2.4, 2.5, we have
tr}'" = c
fFJs
=
t
I
1!
/
-p x-+ /VrrL- J) X. )
x')- J)
3.13
and
c-1: X +lv'r-rL >]
3.14
Combining eqs. 3 .12, 3,.13 and 3 .14 results the
differential equation
ol..
I
,lj/
/ (HV)l/
y = ( X -+ rI X / -?X
-)'!YrrL - r /YYrL
H
ng eq.
o eq. 3.
)
gives the
3.15
12
differential equation of compatability
Equations 3.10 and 3.16 form'two simultaneous fourth
order ordinary differential equations.
GO
arbitrary constant
However, the
appeared in eq. 3.10, requires a
total of five boundary conditions to obtain the unknown
function
x
and
conditions for
For the case of stress boundary
B
shells with a central hole, we need to
consider the following three quantities:
flry ~
-f. x + Nrr L
3.3
The remaining two quantities can be derived by first
substituting eqs. 2.10, 2.11 into eq. 2.6 to obtain
M,;
<
I
t!B r
Br_,.r
3.17
)
and then combining eqs. 2.3, 2.6, 2.7, 2.10 and 2·.18 to
yield
V,.=
I
If
IJ.( J-1/~
<
8,
-1-
I
78;- -
f
t.a.
Br
)
3.18
13
p
prescribed on one rim.
On the other rim, Mtr and one of the two quanti ties
\ly
A(yy
can be prescribed with the remaining one
being defined by equilibrium.
These are the required five
relations.
For a closed shell finite values of ;vf,rare obtained
at
Y= o
X (0)
=
only if
3.19
0
while continuity of the slope
B ( ( 0)
=
B
requires
0
3 • 20
Myr
These two conditions, together with specification of
and one of the two quanti ties
N
yr
or
Y
from equation 3.8.
V y" on the other
rim define the solution.
We can solve for function
This permits the determination of the deflection
integration of eq. 3.2.
The deflection
u
w
by
can be solved
by the use of eq. 2.9 without further integration.
'
CHAPTER IV
Solution
The variables
3.80 and 3.86.
X
and
By can be found from equations
They are series solutions in the form of
Bessel functions of complex arguments.
can be obtained from equation 3.8.
X
and
Bt •
Constant
differential equation.
conditions.
The variable
Y
It has the same form as
C0 is a lead team in the
It can be determined from boundary
It takes the same form as the Bessel function
series representation.
The particular loading conditions
considered here permit the particular solution having the
same form as the homogeneous portion, Ref. 8.
The
solutions are
4.1
.ll\-:5
Y
=I.cilcr
x~1
+.L.
K=-1
.J.k-~
Br = ..r:
bK r
K=l
4.3
14
CHAPTER V
Conclusions
The influence of the geometric parameters
and Y,: on the transverse displacement
w
01.,
1"{,_;
can be obtained
by varying the parameters individually in the range,
o./ ~ fl. ~ o, b
Ref. 8.
The maximum percentage difference in the transverse
displacement was evaluated for each geometric configuration and loading condition, by comparing the
displacement
w
to that calculated from the classical
theory.
The geometric parama ter
do. , has very little effect
on the percentage difference although not shown in Tables
1-3 here.
Solution for various values of
~
and
~
were
considered in order to determine the sample of validity of
classical solution, taking a difference of 5% as a maximum
permissible value.
For the four types of loading
conditions shown in Fig. 3, it was determined that the
.classical SQlution is valid whenever
15
and
BIBLIOGRAPHY
Flligge, w., Stresses in Shells, New York, SpringerVerlag, 1973, Chapter 6.
Green, A.E. and W. Zerna, "Theoretical Elasticity,"
Oxford Clarendon Press, 1954, Chapter XI.
Kalnins, A., "On Vibrations of Shallow Spherical
Shells," The Journal of the Acoustical Society of
America, Vol. 33, No. 8, p. 1102-1107, August,
1961.
Kalnins, A. and P.M. Naghdi, The Journal of the
Acoustical Society of America, Vol. 32, p. 342-347,
1960.
Litt, Danny S., Shear in Symmetrically Loaded Shallow
Spherical Shells, UCLA, 1970.
Marguerre, K, Zur, Applied Mechanical 93-101, 1938.
Naghdi, P.M., "Note on the Equation of Shallow Elastic
Shells," Quarterly of Applied Mathematics, Vol. 14,
p. 331-336, October, 1956 •.
Reissner, E., "Stresses and Small Displacement of
Shallow Spherical Shells," Journal of Mathematics
and Physics, Vol. 25, p. 80-85, 1946.
Timoshenko, S. and Woinowsky-Krieger, s., "Theory of
Plates and Shells," Second Edition, New York,
McGraw-Hill, p. 558-561, 1959.
Wybe, c. Roy, "Advanced Engineering Mathematics," Fourth
Edition, New York, McGraw Hill, 1975.
'
I
16
LIST OF FIGURES
Figure
I.
Deflections, Loads, Transverse Shear and
Tangential Stresses Resultants, and
Bending Moments
II.
Shear Deformation
III.
Applied Loading Conditions
17
Mrr
Figure I
Deflections, Loads, Transverse
Shear and Tangential Stress
Resultants, and Bending Moments
18
Figure II
Shear Deformations
19
'
Uniform Load
Parabolic Load
Horizontal Force
~------~)
0
Figure III
Bending Moment
Applied Loading Conditions
20
"-...____/
'
LIST OF TABLES
Table
I.
II.
III.
Uniform Load
Parabolic Load
Horizontal Force
21
PERCENTAGE DIFFERENCE
~
o. I
I
P.Jl6 X
D.7tf%
0. 2.
o.65%
0,3
o. .SS%
0. 0
o.
o. 3
3.4-6%
I? It:{
?,&JZ%
/."l,
.). 9$'%
/D.
S,o J'/o
;. 76(J/o
3
.,h
I3
~.61%
~ ,/? Dlo
o<
=-
Transverse Displacement
Table I
tf
0 ,.2
0,0..2
uJ = o
at
Uniform Load
22
0,
!3.17%
ff%
4-f'"%
r=r;
PERCENTAGE DIFFERENCE
~
0,
I
.:2.
7° fo
o, {,6%
o.tJ%
0.3
o. s-t%
o.
D
o, I
/)
0.
0,2
o.
0,3
d..ilt
{ifo%_
//. o2bx
~.6.2%
5": 1J%
S:31%
/0,
~~
:;; x
4.'s%
tf.
lf.j.
~- -3~ i
~ .o -3 ~~.
o< ::: o.o.2
Transverse Displacement
Table II
t.f
w
=0
Parabolic Load
23
at
sot
Yo
r = ;;,
PERCENTAGE DIFFERENCE
~
o.o
o. I
o.at%
13. J7%'
I%
s.tl'i
/0.
1
o. I
D.77%
3.4-b :{
3 11%
D.2
/l._/;_s%
..:l. 6
.3
(),54-%_
. )__,/ J%
D,
OiJ
(f.
e><=o.o..2.
c;.:;.%
w = ()
Transverse Displacement
Table III
o.4
0.3
?. jp Yo
7. Yo
0 ..2.
Horizontal Force
24
at
/~.'1-.r%
*lX
f. 7S"/o
r=t;
I
I