Stress
Singularities
Resulting
Frolll
Vari-
ous
Boundary
Conditions
in
Angular
Corners
of
Plates
in
Extension
BY
M.
L.
WILLIAMS,
1
PASADENA,
CALIF.
As
an
analog
to
the
bending
case
published
in
an
earlier
paper,
the
stress
singularities
in
plates
subject~d
to
ex-
tension
in
their
plane
ai:e
discussed.
Three
sets
of
bound-
ary
conditions
on
the
radial
edges
are
investigated,:
free-
free,
clamped-clamped,
and
clamped-free.
Providing
lhe
vertex
angle
is
less
than
180
degrees,
it
is
found
that
un-
bounded
stresses
occur
at
the
vertex
only
in
the
case
of
the
mixed
boundary
condition
with
the
strength
of
the
singularity
being
somewhat
stronger
than
for
the
similar
bending
case.
For
vertex
angles
between
180
and
360
de-
grees,
all
the
cases
considered
may
have
stress
singularities.
In
amplification
of
some
work
of
Southwell,
it
is
shown
that
there
are
certain
analogies
between
the
characteris-
tic
equations
governing
the
stresses
in
extension
and
bending,
respectively,
if
JI,
Poisson's
ratio,
is
replaced
by
-JI.
Finally,
the
free-free
extensional
plate
behaves
locally
at
the
origin
exactly
the
same
as
a
clamped-
clamped
plate
in
bending,
independent
of
Poisson's
ratio.
In
conclusion,
it
is
noted
that
the
free-free
case
analy-
sis
may
be
applied
to
stress
concentrations
in
V-shaped
notches.
INTRODUCTION
I
N
an
earlier
paper
2
the
stress singularities
at
the
vertex of a
sector plate in bending were discussed, where
it
was shown
that
unbounded stresses
may
occur for certain vertex angles
if
the
radial edges of
the
sector were ( 1) simply supported-simply
supported, (2) clamped-free, (3) simply supported-free, or (4)
clamped-simply supported.
The
analogous problem for a
thin
plate in a
state
of plane stress or strain
is
now considered for
three cases,
(a)
clamped-clamped, (b) free-free,
and
(c) clamped-
free,
and
it
will be shown
that
of these three cases, only
the
last,
namely,
the
clamped-free
boundary
condition,
may
have un-
bounded stresses
at
the
corner for certain vertex angles.
METHOD
OF
SOLUTION
Following
the
extensional development for generalized plane
stress in
thin
plates as given
by
Coker
and
Filon
3
the
polar dis-
1
Assistant
Professor,
Guggenheim
Aeronautical
Laboratory,
Cali-
fornia
Institute
of
Technology.
2
"Surface
Singularities
Resulting
From
Various
Boundary
Condi-
tions
in
Angular
Corners
of
Plates
Under
Bending,"
by
M.
L.
Wil-
liams,
U. S.
National
Congress
of
Applied
Mechanics,
Illinios
Insti-
tute
of
Technology,
Chicago,
Ill.,
June,
1951.
a
"A
Treatise
on
Photoelasticity,"
by
E.
G.
Coker
and
L.
G.
N.
Filon,
Cambridge
University
Press,
1931,
p.
163.
Presented
at
the
West
Coast
Conference
of
the
Applied
Mechan-
ics
Division,
Los Angeles, Calif.,
June
26-28,
1952, of THE
AMERJ
CAN
SOCIETY
OF
MECHANICAJ.
ENGINEERS.
Discussion
of
this
paper
should
be
addressed
to
the
Secretary,
ASME,
29
West
39th
Street,
New
York,
N.
Y.,
and
will
be
accepted
until
January
10, 1953, for
publication
at
a
later
date.
Discussion
received
after
the
closing
date
will
be
returned.
NOTE:
Statements
and
opinions
advanced
in
papers
are
to
be
understood
as
individual
expressions of
their
authors
and
not
those
of
the
Society.
Manuscript
received
by
ASME
Applied
Mechanics
Division,
February
28. 1952.
526
placements in
the
radial
and
circumferential directions,
U,
and
Uu,
respectively,
may
be
written
in terms of
the
biharmonic stress
function
X
and
the
harmonic function
1/11
in
the
form
ox
01/;1
2µU,
= - - + (1 -
u)r
-
0
............
[1]
or o
1
ox
01/;1
2µUo = - - -
0
+ (1 -
u)r
2
-
••••••••••
[2]
r 0 or
whereµ
is
the shear modulus
and
u . J1/(l + JI),
JI
being Pois-
son's ratio. Also,
the
stresses
may
be expressed as
o2x
U9 =
~·····················
(4]
1 0
2
x 1
ox
Tro
r
oroO
+
-;:;-
oO
..............
[
5
]
Xis
related
to
1/11
by
v
2
x =
:r
(r
~~1).
................
[6]
The
boundary
conditions will be considered on two radial edges
of a sector including a variable vertex angle
a;
the
boundary
con-
ditions along
the
circumferential edge will be unspecified inasmuch
as
the
stresses near
the
vertex will be locally determined solely
by
the
boundary
conditions along
the
radial edge providing
the
cir-
cumferential
boundary
is greater
than
several plate thicknesses
from
the
vertex.
Let
us assumec:mlutions for x
and
1/;1in
the
form
x =
rH
1
[b1
sin
(A
+
1)0
+ b
2
cos
(A
+
1)0
+basin
(A
- 1)0
+
b4
cos
(A
- 1)0]
==
rH
1
F(O;
A)
.....................................
[7]
1/;
1
=
rm(a
10
cos
mO
+ ll2 sin
mO)
}
.......................
[
8
]
==
rmG(
;
m)
·
On account of
Equation
[6],
it
must
follow
that
A.
= m + 1 in
order
that
the
powers of r be compatible
and
further
4 4
a1
= -
"--
b3;
a2
=
"--
b
•..........
[9]
/\-1
/\-l
by
equating coefficients of like trigonometric terms.
Substitution of Equations
[7]
and
[8]
into [I],
through
Equa-
tion
[5]
utilizing
Equation
[9]
gives
2µU,
=
r>-[-(A
+
I)F(O)
+ (1 - u)G'(O)]
.....
[101
WILLIAMS-STRESS
SINGULARITIES
RESULTING
FROM
VARIOUS
BOUNDARY
CONDITIONS
527
ao.----.--.--,,--.---,--r--,--.--.-.,--,--------------,
CASE
NO.
BOUNDARY
CONDITION
FREE-FREE
CLAMPED-
CLAMPED
5.0
f---+--+-~-l-+---l--+--+--t----+--+---t----1
I
2
3 CLAMPED-FREE
\
4.0
1---1-1+\-+--+--+-+---+--+--+--+---t-""1
_.l
CD
Re/'\-1
'1"-r
0
40°
80°
120°
160°
200°
240°
2eo•
360°
VERTEX ANGLE
-«
Fm.
1
VARIATION
OF
MINIMUM
REAL
PART
OF
EmEN
VALUE
WITH
VERTEX
ANGLE
2µUe =
r>-[-F'(O)
+ (1 -
cr)(X
- l)G(O)]
.....
(11]
er,
= r>--
1
[F"(0)
+
(X
+ l)F(O)J
............
(12]
cr
8
= r>--
1
[X(X
+ l)F(O)J
...................
(13]
T,9
=
r>--l[-XF'(O)J
.......................
(14]
Having
therefore a representation of
the
stresses
and
displace-
ments
for
the
sector
plate
in extension,
the
effects of different
homogeneous
boundary
conditions
may
be investigated.
FREE-FREE
EDGES
If
both
radial edges
are
pref)cribed
to
be free,
cre(O)
=
r,11(0)
= 0,
cr
8
(a)
=
r,
11
(a)
= 0 which
by
Equations
(13]
and
(14] require
F(O) = F'(O) =
F(a)
=
F'(a)
= 0 where
the
primes indicate dif-
ferentiation
with
respect
to
0.
While F(O) contains four unde-
termined
constants
b.(i =
1,
2, 3, 4),
and
four
boundary
conditions
are
prescribed,
the
constants
cannot
be determined uniquely be-
cause
the
four equations
are
homogeneous. ,
For
there
to
be a
solution,
the
determinant
must
vanish.
In
the
case of
the
plate
with free-free edges,
this
eigen
equation
is
sin
z = C
1
z
.....................
[15]
where z =
Xa
and
C
1
=
±(sin
a)/a.
Equation
(15]
determines
the
value
or
values of
the
eigen value X as a function of a.
Inasmuch
as
the
continuity
of displacements requires X > 0,
any
value of
Equation
(15]
satisfying
this
restriction will give
an
admissible solution for
the
extensional problem.
Furthermore,
as
may
be seen from
Equations
(12]
to
(14], a value of X such
that
O < X < 1 will give
unbounded
stresses
near
the
vertex of
the
sector.
The
purpose of
this
paper
is
to
investigate
the
occurrence
of
any
values of X
(or
its real
part
if Xis complex) which lie in this
region.
Stated
alternatively,
it
is desired
to
find
the
minimum
root
of
Equation
(15]
with
the
restriction
that
X > 0.
The
results, showing min
Re
X as a function of a for Poisson's
ratio
v = 0.3,
(er
=
3
/
13
)
are
f>resented in Fig.
1.
CLAMPED-CLAMPED
EDGES
Fixed edges
at
0 = 0
and
0 = a require
that
U,(O) =
U11(0)
=
O,
U,(a)
= U
8
(a)
=
0.
Upon using
Equations
(10]
and
(11).
and
.again formulating
the
four simultaneous homogeneous equat10ns,
the
eigen
equation
in
this
case again becomes
where z =
Xa
and
sin z =
C2z
.....................
(16]
sin a
C2
= ±
----
(3 - 4cr)a
The
min
ReMa)
for this case are also shown in Fig. 1 for
'II
= 0.3.
CLAMPED-FREE
The
mixed
boundary
condition of a clamped-free
plate
in exten-
sion requires
cr11(0)
=
r,11(0)
=
U,(a)
= UfJ(a) =
0.
Using
Equations
(10), (11], (13],
and
(14]
the
eigen
equation
is fountl
to
be of s.?mewhat different form, namely
sin
2
z
=
[4(1
-
cr)2]-[
sin
2
a J z
2
••••••
3 -
4cr
(3 - 4cr)a
2
[17]
The
minimum values of
the
eigen
parameter
for this' case
are
also shown in Fig. 1 as a function of vertex angle for
'II
= 0.3.
COMPARISON
OF
RESULTS
A
study
of Fig. 1 reveals
that
of
the
three cases considered, only
the
plate
with
mixed
boundary
conditions, namely, clamped-free,
may
have
a singularity if
the
vertex angle lies between 63
and
180
degrees.
As
discussed elsewhere,
2
this does
not
mean
that
a singu-
larity
must
occur, because
other
higher eigenvalues from
Equa-
tion (17],
at
90 deg, for example, will
not
introduce singular be-
havior
at
the
origin. On
the
other
hand,
an
arbitrary
loading
along
the
circumferential
boundary
formed
by
a linear combina-
tion of
the
eigen functions in
the
manner
suggested
by
the
author
4
will contain
the
eigen function giving rise
to
the
singularity
and
hence locally in
the
vicinity of
the
origin introduce unbounded.
stresses.
For
vertex angles between 180
and
360
degrees, all
the
cases
considered
may
have
stress singularities.
'"Theoretical
and
Experimental
Effect of Sweep
Upon
the
Stress
and
Deflection
Disfribution
in
Aircraft
Wings of
High
Solidity.
Part
6-The
Plate
Problem
for a
Cantilever
Sector
of Uniform
Thickness,"
by
M.
L. Williams,
AFTR
5761,
November,
1950.
528
JOURNAL OF
APPLIED
MECHANICS
DECEMBER,
1952
TABLE
1
EIGEN
EQUATIONS
FOR
BENDING
AND
EXTENSION
Case
no.
2
Bound.ary
condition
Free-free
Clamped-clamped
Clamped-free
Eigen
equation
sinz
= Ciz
Hinz
C2z
Anot.her conclwiion cJ.n be
drawn
from
a comparison of
the
ex-
tensional
and
bending
results.
For
easy
reference a
table
giving
the
eigen
equations
for
both
extension
and
bending
is
herein
as
Table
1.
It
if;l
seen
that
the
eigen
equations
for
the
free-free ex-
tensional case
are
exactly
the
same
as
for
the
clamped-clamped
bending
case which means
that
locally a clamped
bent
plate
be-
haves as a free extensional plate.
For
the
cases of generaliz9d
plane
stress
the
eigen
equations
for
the
free-free bending
and
clamped-clamped extension cases,
and
for
the
clamped-free
bending
and
extension cases
are
the
same
if
v is replaced
by
-v.
Southwell
noted
this
association of Pois-
Loading
Constants
Bending
Ci =
±
1_-
v
sin
a
3 + v a
Extension
Ci
=
±
s_in
a
a
Bending
C2
=
±
~~-~
a
Bxtension
c.
=
±
l_t_v~~
3-
v
"'
Bending
Ki
=
±
[(3·+·0~1-=~5]
i1z
Kz
=
±
[}~-~]
i1z
si~~
Extension
son's
ratio
between
the
bending
and
extension cases• and, further-
more, derived
the
correspondence of
the
boundary
conditions.
The
point
to
be
observed here
is
that
the
characteristic
equation
for
the
eigen values of
the
stress
and
deformation
behavior are
also analogous in
the
two
case::i.
In
conclusion
it
is sugge,:;ted
that
the
analysis for
the
free-free
caoe
be
applied
to
determine
the
type
of stress
variation
at
the
base of V-shaped
notches
in
plates
or
rods.
6
"On
the
Analogues
Rein.ting
Flexure
and
Extension
of
·Flat
Plates,"
by
R.
V.
Southwell,
Quarterly
Journal
of
.Mechanics and
Applied
Mathematics, vol. 3,
part
3.
September,
1950,
pp.
257-270.
