Internat. J. Hath. & Math. Sci.
(1986) 161-174
Vol. 9 No.
161
THIN CIRCULAR PLATE UNIFORMLY LOADED OVER A
CONCENTRIC ELLIPTIC PATH AND SUPPORTED ON COLUMNS
W. A. BASSALI
Departme,t of Mathematics
Faculty of Science
Kuwa it UniVersity
Kuwait, P.O. Box 5969
(Received April 29, 1985)
ABSTRACT.
Within the limitations of
the classical thin plate theory expressions are
obtained for the small deflections of a thin isotropic circular plate uniformly load-
ed
over
ellipse
c,ncentric
a
and
four
supported by
columns
the
mo,aents
the vertices
at
recta,g[e whose sides are parallel to the axes of the ellipse.
of
a
Formulae are given for
and shears at the centre of the plate and on the edge.
Limiting cases are
invest igated.
KEY WORDS AND PIIRASES.
of
Deflections
circular
plates,
four point supports,
free
boundary, uniform loading on a concentric ellipse
1980 AMS SUBJECT CLASSIFICATION CODE. 73N.
|.
INTRODUCTION.
The technological importance of thin elastic plates is sufficiently well estabThin slabs of material are structures which are
require no elaboration.
lished
to
widely
used
engineering work
in
and
their
flexure has
transverse
been extensively
studied by many authors both theoretically and experimentally when the boundary of the
Support of circular discs at a discrete
simply supported or free.
slab is clamped,
number of points is of interest to the designer of reflecting surfaces and receivers,
particularly
when
structures.
The
the
surfaces
deflection
are
surface
parts
of
a
of
thin
astronomical
circular
and
plate
aeronautical
subjected
to
a
symmetrical loading and supported by equally spaced point columns along the periphery
of the plate has been considered by Nadai
and Woinowsky-Krieger
[2].
[I] whose results are quoted by Timoshenko
When the boundary of the circular plate is free and the
plate is supported at interior points and acted upon by two types of normal loadings
distributed over an eccentric circular patch the solutions have been obtained by the
author
[3,4], using the complex variable approach of Muskhelishvili, references
previous work are given at the ends of these papers.
point
supports
Kirte[n,
Pel[,
have
also
Woolley
been discussed
aad
Davis
[7]
Thin circular plates on multi-
by Yu and Pan
Kirstein
to
and
[5],
Wooley
Leissa and
[8,9]
Chantaramungkorn, Karasudhi and Lee ill] and Williams and Brinson [12].
We[Is
Vaughn
[6],
[I0]
There is good
a,reement between the theoretical results of [3] and the experimental results from the
162
W.A. BASSALI
tests reported in
[7].
ost of the results in [5-I0,12] are special cases of [B].
of an ela,;tically restrained circular plate subject to uniform,
.
parabolic loadings over
method
is
In
f papers [I-15] complex variable methods were applied to study th,, bending
qarieq
,f
concentric ellipse.
linearly vac/ing and
Frishbeir and Lucht
[16] used the //L//
complex potentials to derive the solution for a clamped circular plate which
transv_.rsely and uniformly loaded over the area of a polygon.
expressions
are
for
obtained
the deflection at any point
In this paper,
of a thin circular plate
which i ,niformly load,’.d over a concentric elliptic patch and supported by four equal
concentrated forces loc,ted at the corners of a rectangle whose sides are parallel to
the axes of the ellipse.
Formulae are given for the boundary and central values of
the
The limiting cases in which the radius of the plate
m,,ents
and
shears.
the eccentricity of the ellipse
0
or its minor axis
2.
BASIC EQUATIONS AND BOUNDARY CONDITIONS.
If
2h
Let
C
0
denote the boundary of a thin circular plate of centre
is the constant thickness of the plate,
oo,
are investigated.
an radius c.
0
then its flexural rigidity
D
is
given by [2]
2Eh3/3(I- 2
D
where
E
(2.1)
is the modulus of elasticity and
of the plate.
is Poisson’s ratio for the material
The mid-plane of the plate is chosen as the plane
to the classical small
Z
bending theory of thin plates the deflection
plane in the downward direction OZ
0
of a
and the notation used is that of [2]. According
O(x,y,Z)
rectangular Cartesian frame
at any point
x + iy
z
re
i0
w
of the mid-
satisfies the
biharmonic equation
DV
4w
p(z z),
(2.2)
where
61
V2
p(z,z)
and
x2
2
+
Oy
4
2
2
2
zz
r2
+
r
r
2
+
r
is the normal load intensity at the point
2
0
(2.3)
2
z.
The general solution
of (2.2) may be written as
z(z) +
w
where
the
(z), (z)
plate and
point
(r,v)
W
zi(i)
are functions of
+ ,(z) +
o(i)
+
W(z,z),
which are regular in the region occupied by
z
is a particular integral of (2.2).
The moments and shears at any
of the mid-plane of the plate are give by
M
r
-I
2 +
r d +
-D(vd
M
M
-D(d
F
Qr
2 +
(l-o)Dr
r-
(2.4)
2
vr-2d
d + r
-2
d
2
)w
)w
-D[vV 2
+ (l-v)d
2]w,
-D[V 2 + (v-l)d"]:;,
l(d-r -I )d’w
-Dd("2w)’ Qo
[2]
-Dr-ld’ (2w)’
(2.5a)
(2.5b)
(2.5c)
(’2.6)
THIN
CiECUI,AR
PLATE LOADED OVER A ELLIPTIC PATH AND SUPPORTED ON COLUMNS
d’
0/.Jr,
where
particular integral
51
+
r
we have
-SDz[
Qr-
+
02W/z],
=-4(l-v)D[z2"
+ 2iM
"
+
.(z), (z)
and
[3,p.730]
’
-4(;+ )D[2Re
Mj
M
bl
In terms of tLe complex potentials
0/3’
W(z,z)
163
+ z
(2.7a)
2(,,
+
W" )/ r 2 ],
(2.7b)
03W/3 z 2 O]/r
(’2.8)
where accents denote differentiation with respect to z
The condition- for the circ,lar edge
(Ir)r=c
( Qr
O, (V r) r=c
C
[2]
(2.9)
0
S
[
to be free are
r=c
Substitution from (2.5a,c) and (2.6) in (2.9) leads to
[f (d d’)w]
r
0
r=c
[Fr(d
d’)w] r=c
d + vr
-2d,2
(2.10)
0
whe re
f (d,d’)
d
F (d,d’)
r
d
r
2
+ or
3 +
r
-1
-ld2 r-2d
+ (2-v)r
(2.11a)
-1
dd’
2
+ (-3)r
-3d,2
(2.lib)
From (2.)a) and the first equation of (2.9) we see that
(-l)(d2w) r=c
v($2w) r=c
v[d’(V2w)] r=c
(-l)[d’d2w] r=c
Equation (2.5b) and the second equation of (2.6) then give
2
(M)
l-__y__
r=c
D[d2w] r=c
(Qo) r=c
I-.__
_D
u
c
[d,d2w] r=c
(2.12)
from which we deduce that
(1 + ) c
(Qv)r=c
(d’Mo)r= c
(2.13)
This relation and the second equation of (2.9) serve
to determine the periphery
values of the shears in terms of the moment values.
3.
STATEMENT OF THE PROBLEM.
The problem to be solved consists of determining the
deflection surface of a
thin circular plate of centre 0 and radius c
subjected to the following conditions:
(1)
The boundary
(2)
The total normal load
C
of the plate is free.
L
Poab
is uniformly distributed over the area of the
ellipse
2
x a + y b
1(0 _< b
a
c)
z/
2/
(3.14)
(3. lb)
W.A. BASSALI
164
(3)
L/4,
The plate is supported by four eqval concentrated forces, each of magnitude
P(z%
and located at tle points
I
’
2
Y3
se
s
;,2,3,4), where 0
c,
/2
The four points of support
2
2
2
2
2
s cos y/ a + s-sin y/b
is less
,%
0
y
’
Y4
0
we have two supports at
-Y
lie in tIe loaded or unloaded region according as
or greater than I.
For
y
(0,_+s).
we have two supports at
Symmetry with respect to both axes
See Figure I.
show tlmt it is sufficient to find the deflection
Deflections,
quadrant.
4.
at any
w
point
moments and shears at the four points
in the positive
z
+/-
+/-z,
are the same.
METHOD AND SOLUTION.
Iet
F
denote the boundary of the ellipse (3..b) and let the indices
ruler to the loaded region inside
respectively.
F
and the unloaded region between
The particular integrals
uniform intensities of normal loading
W
(z,)
W
point on
F
Pl
P0
pOz22/64D, W2(E,)
0.
C,
0
P2
may be taken as
(/.l)
moments and shears at any
slopes,
[2w/z]2
[Fw/Oz] 2
[$3w/z2]2
0
It was proved [13, p.105] that these transition conditions along
F
satisfied by
96--[(z)] 2 =-6----
-
abf2
2
+
(z2_f2),
Z
wher e
Introducing (.io),
2
+f2) Z
+ 6z In
+
ab
z
fg
a
2
b
f2]
2
d
a cos
a
2
+ b
2
using (3.14a)
-
8D/L
+ ib sin
are
(4.17)
(4.18)
(,. l)
(.21)
is the eccentric angle of any point
z
2
In
+
./; and .18) in (2.4)we get,
k
where
@
and
of (2.2) corresponding to the
2
and
F
and 2
lead to
[w] 2
along
W
and
The continuity requirements for the deflections,
if
/2
while for
(is,0)
,
Z
z
b cos
on
F
then
+ ia sin
and it is checked that the expression between the square brackets in (4.20) vanishes
(z 2 -f 2 is not uniform in
as it should.
It is to be noted that Z
along
region
while it is uniform in region 2.
change when
Thus,
z
traces a closed path
the terms containin
Z
to,
In fact,
any of the two loci (if,0)
in (4.20) should appear in
is also known that the singular part of the deflection
w
w
2
of the ellipse.
and not in
at any
point
inter-
Z
the two branches of
P
w
I.
near a
It
THIN CIR(’ULAR PLATE LOADED OVER A ELLIPTIC PATH AND SUPPORTED ON COLUMNS
F
downward concentrated force
is
F
W
R
sln
2
in R
aa the point of application of the force
P
is the distance between
R
where
Guided by tllese remarks and using (-.20),
4
r
ab
kw
ab
(
r
2
+--g
f
cos 2
165
we assume that
2 4
)
d r
6abf
(aF ab) rcos
+
cos 20 +
2
4u
2 2n
cos 2n0
-S + x(A + C r )r
n
n
0
(r
kw2
2
+
d
2) I-I
z+Z
in
z 2 -59[) (2 ]j
’d2z
r2z Z
[2f2 \7
+
Re
)r2ncos
S + (A + C r
n
n
0
(4.23)
(4.24)
2ng
whe re
4
s=.z
tt2
r
2
+ s
2
R
2
in
x
R,
(4.25)
2sr cos (0-y),
R
2
2
r
2
+ s
2
+ 2st cos
(0+)
(4.26)
2
2
+ s
2
R3
r
and
A ,C (n
n
n
R
+ 2sr cos (’,-y),
0, I,2,...)
r2 + s2
2sr cos (0+)
seen that the expressions (423) and (4.24) for
equation (Z.Z) corresponding
F
the required transition conditions along
n
0,
w
to the load intensities
behaviour at the four points of support
C (n
It is now easily
are real constants to be determined.
and
w
2
satisfy the biharmonic
and
PO
Pl
An (n
Assuming that
wlere,
and
A O.
The
To achieve
all the terms in (4.24) will be explicitly expressed in terms of biharmoand
r
r-
(z f)
1,2
will now be determined from the boundary conditions (2.10).
2 ...)
nic functions of
*
satisfy
and exhibit the appropriate singular
The unkown constants
condition of zero deflection at any point of support serves to find
this goal,
O,
P2
n cos
of the types
and
n6
sln
_I
z
2+/-n cos
+
f2)-
(
with the usual notation
_l_
(n )
Integrating (4.27a),
(-1)
sin
-
nO
(z di;2-f2))
,(z,f)= In
(zZ-f 2) -1/2
r
n
z
(n)
0
we have
( f2)n ( Igl )
z
(4.27a)
2-2n /2n
\ n)
using (4.27b) and noting that
(4.27b)
(z,0)
in(2z)
we get
W. A. BASSALI
166
Thus we have
z
Re
in
a+b
bf
r
+ 2
r2z(
+
(f/2r)
n+!
writing
Similarly,
f
in
( 2Tz
+Z
Z
6
Z
f2 j
2n
2n+l
(l-z2/f2)
r
2
r
[2-T-l
if
2
d
(, d2_2f2 )
[z[
2n-I
2
d
zl
+
21
+
(4.28b)
f).
f:
d
)
2
cos 2n0 (r
cos 2’0
’4.29a)
f).
we obtain the following
expanding,
and
( 2n2nr2
n/[--[
(r/2f)2n+l
Z
sin
r
7
4(n+2)
(4.28a)
sin(2n+l)0(r
r
2
n+3
+
expression for the left side of (4.29a) when
r
3f
cos 2nO (r e f),
r
by the binomial theorem we find for
2
cos 40
a+b2r 21 Z!
k
n/\’/2n+l
Z0 ’! (2n(
In
z+Z
7
2f
r
a+- -z+Z
(l-f2/z2)
Expanding
+
In
f
2n-5
d
2(2n-3)
2
sin(2n+l) (r
f).
(4.29b)
Substitution from (4.28a,b) and (4.29a,b) in (4.24) gives
3d2
kw2
16
(f/2r)2ncs
-S + Z
kw
12
r +
2
d
2
)In
(r2
2
2
4
+
b
(r/2f)
2n+l
a+b
f2
(2nn) (2r
n-22)
\2n-I
(2 )
il
(2nn) (2n2d_ )
d
sin 0 +
+
r
2
(r +
R
1
+ g
n
2
d
2
(1/2
r
+I
2
In
s2
in r + s
2
sr
s2) /s)n
k
/r 2
\n-Zi
r2)r2ncos
d2r 4
-----6f4 cos
2n8 (r
"_;
4u
f)
S
Z(An+Cnr2)r2ncos 2n8
(r < f).
0
(4.30b)
s
2
s
+21n r +
2
2r
(
)
cos (0-y)
;
n(-,},)
cos
n-+l
i
cos 20 +
(4.30a)
f
sin(2n+l)8 +
r
2
d
+ Z (A + C
n
n
0
2
r
n+l
4n2-1
in
r
2nO
It can be easily shown that for
Ra
2
2r
([
r
+ 2
(2
+
r
(4.31a)
and
(r2+s)2
S
For
r
s
en
2
we
interchange
s2) ()2n
/ r2
\2n--’
2
in r + s + Y:
r
2n-+
and
s
cos 2ny cos 2n (r
s).
(4.31b)
in (4.31a,b)
Introducing (4.31b) in (4.30a) yields
kw
r -> the greater of
where
L
A
O(r)
0
X L (r) cos 2n0
n
O
2
A
A
0
O’
+ B
s
+
-
O’ln
r +
f
(4.32)
and
s
2
(n
Cr Ln(r) A’rn 2n + B’n r-2n + C’rn 2+2n + D,r2-2n
n
I),
(4.33)
+In
a+bfl
BO
d
s
C
O
C
0
+ in
a+---
(4.34)
167
THIN CIRCULAR PLATE LOADED OVER A ELLIPTIC PATH Ab’D SUPPORTED ON COLUMNS
C’
AI
A’n
A (n
3)
B’n
C’n
C (n->2)
D’n
2f
n
2
C
3f
n
Applying
2
d
A 2 + 6f4
2
d
A
2n+2
s
A2
2
2
d (f/2)
2n
cos
2n
2n
4n(n+l (n+2)
2n(2n+l
s
(4.35)
2n
cos
2n
(f/2)
2n(n+l (.’n-I)
+
2n(2n-I
I)
(n
n
(4.36)
(n
(4 37)
I).
differential operators (2.11) and (’2.12) to the functions in (4.32)
t,e
gives
f (d d’)
r
tl.u(r)}
fr(d’d’) {Ln(r)
(n)
+
o
(<-i)C6env
cos
+
+
-4n
2
n ’)
(1
2n
n
Ar2n-2
(2n-l)
0
(2n++l)
0
(4.38)
-2n-2
(2n+l)B’r
n
(2n-<+l) D r
n
(i n)(2n--,)C’r
r2n-2
(n)-/n]
n
rF (d d’) iL (r) cos 2n0}
r
+
(2n-l).X,n
2n
in +
(2n+-I) C’r
+
r(d,d’) tLo(r)}
B
r
(4.39)
cos 2n0
(2n+l) B’r
n
Dr -2n]
-2n-2
(4.40)
cos 2n0
where
v,
o
(3 + v)/(
)
(4.4|)
Inserting (4.32) in the boundary conditions (2.0), using (4.38)-(4.40), equating
the coefficients of cos 2n0 (n
0, I,2,...) in the resulting identities to zero and
solving the obtained systems of linear equations we find
C
2
B(v
u
c
A’n
4u
2-2n
[{
’2n
C’n
u
2)
C
4n
2
+
0
iS(v
2
2+2 -1, u2ncos
2n(2n-l)J
2n
u2ncos
2n+l/
4u
2)
+ in
a+b
(4.42a)
(2nn) I4n2+<2-1
----]
(f/2c)2n
(2) \n+2 )]
2n +
2n
+
u
s/c
(f/2c)2n
2n+2
2n+|
[.
n+2
(n >1),
(4.42b)
(n- !),
n+l
v2}]
(4.42c)
where
(l-v)/(l+v),
3
d/c
v
(4.43)
lhen the values of
A 1, A
and C
are replaced by their values in terms of
2
by means of (4.35) it is found that (4.23) and (4.30a) take the forms
and C
2(a+b)
+ A
kw
(
a-b
kwi
+ C
0
3
d
2
+ A
where
C
0
r
2
2
r
+ C
0
O
r
O
r
2
2
)
2
r cos 20 +
+ Y. (A’ +
n
2)
C’r
n
(
d
2
+
+
r
2
(A’
n
+
C’II
2
r
r < f
the
4{
-
r2ncos
)
in
2)
r
is given by (4.4Za) and
region 2 at which
r
2r
2n
(a-b 2cos
+
\-]
40
S
gab
2n0
(4 44)
Z
+
(2:)
cos 2n0 (r
(f/2r) 2n
w
2
(2r 2
1) \2-i
2
n-2)
d
f),
A’n’ C’n are given by (4
deflection
}
A I, A
2
cos 2n0-S
(4 45)
42b c)
is furnished by (4.30b).
For points of
It is easily
168
W. A. BASSALI
seen that such points exist only if
if the eccentricity of the
i.e.,
b,
f
can always
The constant A
is given by (4.24).
In any case, w
ellipse e 2/2.
0
2
be determined from the condition that the deflection vanishes at any of the four
If all these points lie in the loaded region then (4.44) gives
points of support.
A
ab + s
0
2
sin
2
s2
2(a+b)a-b( )
/
+ cos
in sin
cos
2
2
su,oort ,oints
can be used to determine
A
s
0
d
2
[sin2
2
(
4
but if
A0
-< f
s
sin2
d24
f
In a+b
Z (A
0, A 0
u
then either
s
according as
2
f
s
or
f.
2
(4u -v
in(4s cos y) +
2n-12s2 )
/
2)
s2ncos
CnS2) (4.46)
and S
ab
lie in the unloaded region,
+ 2
2
+ cos y In cos
in sin
I )s
2
24
+
b
s
+ C s )s
n
2
r in r
(4.45) or (4.30b)
s -> f
If
we have
2
s2_s 2n
(A’n + C’n
}
cos 2ny
(4.47a)
2n c os
+ In
sin y + 2
Z1
8
+
_4u
2
v
2
(2:) (n-is22d2
2n-3]
(s/2f)2n+l
4n2-I
sin(2n+l)y
(4 47b)
2n
the deflection at the centre of the plate is
In any case,
W
0
ab-
0
8D
where the appropriate value of
5.
O
s
(d2nnl)
(2:) n2s) t(f/2s)2n
2
n
r (A’n
cos 4y
we have
2
(4u -v
+
then (4.30b) gives
2
s
2
A0
in[Z,s sin ) + cos
In
+
8s
In
+
/
For a single support at. the centre
If all the
In cos
A
S
0
in
(4 48)
S
is taken.
BOUNDARY ANt) CENTRAL VALUES OF MOMENTS AND SHEARS
It can be easily shown that the deflections (4.23) and (424) may be written in
the forms
w
w
2
2Re
[zj(z)
+
l(z)]
2Re
[z.2(z)
+
2(z)]
wherc
23
2k.Zl (z)
2k00
6abf
I>
2(z)
z
2
d
4
in
4
2
"ab
l(z)
2k2(z)
2k
d z
4
I
+
z+Z
a+b
,2zZ [z2
[3f2 \f2
Z
In
Z
W!(z,z)
+
(549)
(550)
+ Z C z
n
2n+
(5.51a)
0
) z4 (1/4 z2)
( z2 )
Z
3
2 +
5
2]
Z
+ in
+
+
ab
z+Z’. +
-,
4
Z
4
Z
in
+ Z A z 2n
n
0
ZIn Z
Z
’
+ Z C z 2n+l
0 n
4
z ZIn Z
+ I
0
A z 2n
n
(5.51b)
(5.52a)
(5.52b)
169
THIN CIRCULAR PLATE LOADED OVER A ELLIPTIC PATH AND SUPPORTED ON COLUMNS
z
Z%
An,
and the real constants
z%
C
have been determined in the previous
n
The moments and shears at any point of the plate can be obtained either by
substitution from (4.15), (5.Sla,b) and (5.52a,b) in (2.7a,b) and (2.8) or by introducing (4.44), (4.45), (4.30b) in (2.5a,b,c) and (2.6), noting that S is defined by
section.
s and interchanging r,s in
(4.25), (4.26) and its expansion is (4.31b) if r
After extensive algebraic manipulation, it is found that
(4.3b) if r -< s.
as expected and
(Mr)r=c=O
(+)L
4
(M)r=c
v
+ E
-u
(f/2c)2n 2n+;
n+2
+
n+
[(
.L.
(Mr0)r= c
/’2n’i
+ 2
2n
n+l
(Q)r=c
2 (2n +
E
kn
+1
2n
cos 2n
n*2
-2nu2)u2ncos
v
2n sin
2}
(5.54a)
cos 2nO
2nO
n(2n+;) v 2
+
(5.53b)
sin 2nO
n(2n+l)
+
2n +
n+;
(5.53a)
cos 2n0
2n+l 2
v
n+2
n
2n
2ny cos 2n0
os 2ny sin 2n0
2-2n +- l)u2ncos
/n(/n
n]
*
2n+<+;
n
-!
2
n+l
(2nu
2
+-’u2nc
2-2
(f/2c)
n]
L
(Qr)rc
2u
v
k
u2ncos
+--
2-2u
sin 2n@
n+2
(5.54b)
It is easily seen that (5.53b), (5.54a) satisfy the second boundary condition in (2.9)
and (5.53a), (5.54b) satisfy (2.13)
All the infinite series appearing in this section and in section 4 are convergent
in the intervals
-
mentioned and some of them will be summed in section 8.
The following formulae are obtained for the moments and shears at the centre:
(Mr) 0
(0)0
8--
(;+)
.a-b +u
a+--’
2)
cos 2y +
-+ (-)
f2 (v2
2
2
cos 2y
3)
cos 20
(5 55a)
8Kc
811
cos
2
(Qr)O
6.
+ 2 In
+
v
u
( 2 + 3- 2u
(l-) L
(Mr0) 0
S
a-b
+
u
(Qo)o
-
(v2-<2-3)
(<+3-2u)cos 2 +
8<c
sin 2.
(5.55b)
(5.56)
0
INFINITE PLATE UNIFOILMLY LOADED OVER AN ELLIPTIC PATCH AND SUPPORTED ON COLUMNS.
Letting
A
kw
+
O
r__
SaD
in (4.44), (4.45) and (4.30b) leads to
c
ab
S
’I
k
a-b
\a+b/
+ in
cos 40
a+b
r
2
+
a-b
2(a+b)
r
k
7
2
r c os 20
(6.57)
170
kw
W. A. BASSALI
A0 +
2
+ Z
kw
+ in
4n(n+)
<1/2
A0 +
2
6f
)
2
ab + s
0
+
+b)
d2
+
S
cos 2n0 (r
f
in
+
<
-
f),
(2
r
2n+l
4
+
r
2
)
b
2
)sin
r
)2
(
0 +
r
2
+
d
2
cos 20
2
sin (2n+;)O (r
7]
2n-3
f),
(6.SSb)
2
in sin y +cos y in cos y-
s
(6.58a)
2dZ
4n2_;
s in2y
a-b
2
r
(r/2f)
S + 2 Z
4,
where
A
n
2n-
+ in
r
4
In r
+
s
cos 2y-
2
{
8s2
+ in
a+
a_b2
+
k7
cos 4y
(6.59)
if the supports lie in te loaded region;
A
s
0
2
+
{$
4(n+;)
s
;Isin2y
"
2
s
<
s
in sin
)
2 + 4 2
b
2
cos y in cos y+ in
7 (s d)
s
sin y
(f/2s)2n+l
+ 2 Z
+
4n2-I
2
+
s2
2d2
(2nn) ( n’i
dk
cos 2ny
2n-
if the supports lie in the unloaded region and
A0
2
2
2
sin y in(4s siny) + s cos 2 y ln(4s cosy)
2n-3
)
s e f
8s2
3 +
In
2s
(6.60a)
-
| + d2
In a+b
f
2J
f
d2s
4
cos 2y +----- cos 4y
6f 4
sin(2n+l)y
if the supports lie in the unloaded region and
s
(6.60b)
f.
-
At the centre of the ellipse the moments (5.55a,b) reduce to
(M)
r 0
(Mr0)O
7.
(l+v)
g-
(M),0
(l-v)L
8
+ 2 in
-j
a-b|
cos 2y
+/-
(l-v)
cos 2y
sin 20.
cos 20
(6.6|a)
(6.6b)
THIN CIRCULAR PLATE UNDER A VARIABLE LINE LOADING ALONG A DIAMETER AND SUPPORTED
ON COLUMNS.
0
When the minor axis of the loaded elliptic patch
variable line loading extending along
and
PO
distance
corresponding
b
orel,
p!
x-axis from
x
we have the case of a
-a
to
x
a.
If
0
b
then the intensity of this line loading at a
2/a2
).
Deflections, moments and shears
p| /(|-x
to this case can be dedcued from those for region 2 in sections 4, 5
from the centre equals
x
by setting
tn columns
2bP0
such [hat
tL,e
O, d
f
a, L
apl
and
noting that separate expressions are
=ro,pvpqnt (,r,.O aqcord,ing as
lie outside or inside the circle
r
a
r
is greater or less than
a
and
THIN CIRCULAR PLATE LOADED OVER A ELLIPTIC PATH AND SUPPORTED ON COLUMNS
171
THIN CIRCULAR PLATE UNIFORMLY LOADED OVER A CONCENTRIC CIRCULAR PATCH AND
8.
SUPPORTED ON COLUMNS.
Setting
A
kw
b
+
a 2{
0
A
2
3
+
0
g
a
a/c
where
A
c
2-2n 2n
{
u
a
u
}
2n (2n-1
sin2y
+ s
2[ sin2y
r
4
+- S
8a
22}r2(
4n2+ 2
2
n
s
2
2
(8.62)
(t -2u
in a +
(A’ +
A0
r
2n0
C’n r2)r2ncs
+
2#
+
r
r
in
a 2 )-a
2
S
2n0
(8.63)
and
2nK
0
i
+
(A’n
+
A’n
2
[(t 2 -2u
+ in a +
2n
C’r2)r
c os
n
+ X (A’ +
n
kw
in (4.44), (4.45) and (4.42a,b,c) we get
0
a, f
cos 2ny,
c
C’n
-2nu2n
in siny + cos y in cosy
CLs2)s2ncos
+ in
n
2)s2nc os
2
4s-2
a
+
cos 2ny
(8.64)
B(2u2-t 2)
(8.65a)
2
n
2nu
-]
2ny(s -< a)
in siny + cos y in cosy + in(4s) +
(A’ + C’s
(
.B(2u2-t 2) ]- a2
z
-T
(
In
+
a)
2ny(s
)
(8.65b)
The inflnite series in (8.62) and (8.b3) equals
2
2
2
r +s +(K-l)c
c
2
2 2
2
t,/_(,)
ru
Jo(4’2 )) --7- {Jl (
(’ l)c 2
/_1(,2)}+ 8
J(’)
{J0 (’’ *)
+
+
+j
+
(
2
)
d(’2 )}
(8.66)
where
,
rs/c
_2n
COS
dO(,
0
91
y
2n9
_2nlcos2n+12n
__1
L
l _Zn211_1
cos
lg,
cos
(; )
4 22cos
in (I +
2n
dl(-,,)
0 + y
+2
cos
4--
2n9
g
2n
cos 9 in
1+2+2cs
+42_ 2cos
l+g2+2gcos
l+g2-2gcos
+ 2 sine tan
-i
2in
i-
2 sin
tan-
2
(8.67b)
2sin
_g2
(8.67c)
Re
Re 2
n
in
(8.67a)
2)
n
0
The last function is the dilogaritnm studied in the last three references of
[7].
The deflection at the centre is given by
kw
s
2
sin
2y
4s
2
s
In sin y+ cos y In cos y+ ina
+
(u
-t2
c
2o
(s
2
8a
a)
2
(8.69a)
W. A. BASSALI
172
kw
sin2y In
s
0
-
+ cos2y
sin
a
where
(2
6
+ 2u
2
I)(
22 ,/_l(U
kw
s
0
Z
3
u
2
5
(+
l+u2,
21 a l_j
u
16n (4nl-.
2
(u
I
+
(
2
4
+
0) -u
4-
4 +
+
B(2u2-t 2)
1"5
+ in
{K2
-1 +
+
(I-K
+2u2-1)In(l+u 4)
i
2
\
u4
2
2n
2
4
a)
(8.69b)
._L In
2
dl(U 2,2Y)
+
2
2,2)}
c26
+
l+u
2u
(K-I) td,u ,0) + d(u
+
s
I( 2u2_ 2
16n2+K2-1
i
2u
>
(
-u
2
2
(8.70)
a),
s
(K2+3)u
+
2}
c26
in (l+u
4 u 2 tan- 2u
2
(s
(8.71a)
a)
(8.71b)
2)
(i-u2)ln(1-u
It is verified that (8.71a,b) agree with (3.14) and (3.6a) of
(8.72)
[3],
noting the
There are misprints in equation (3.5a), p.738 of
difference in notation.
}
ta
whe r e
6’
c26 (s
s
)
-
s
Z-l) + in
a
in Z
In
+ In 4 + 8(2u2-t 2)
4
in(l-u )I
,2y)
we obtain
n
s
0
,2) +
n/4
Setting
kw
2_
2do(u
in cos y
[3] and
cos a which appears twice in this equation must be replaced by cos s
Putting
b
obtained we get
(l+v)L
4n
(bl0)r=c
closed formulae
u
2
2
+(
2
4
(l-u )(l-u )(I
+
in (5.53a,b), (5.54a,b) and surning the infinite series
0
f
a,
tt,e
I
t -u
2)
(+l)in(lll 2)
-I
+
(8.73a)
12
2
2
(MrU)r=c-
tan
u
2
(I -u
L
(qr)r=c
(Qo)r=c
Z
+
2
l-u cos
(sinii
2i
1-+ (l-u
(l+<)u
+ tan
2
2+
+
2)(1
l-u2cos 22
(8.73b)
12
+
u4
{sin 21
",,u2(-uZ)(_, ’)
12
sin 2,?
+
lu 2)
sin
(I
-!
+
:1
22,/
sin
+-
2
12
(8.74b)
TttIN CIRCULAR PLATE LOA3ED OVER A ELLIPTIC PATH AND SUPPORTED ON COLUMNS
173
where
I
+ u
4
2u
2
cos
(8.75)
1,2)
’2v: (j
It is easily verified that (8.73a), (.74b) satisfy (2.13) and (8.73b),(8.74a)
satisfy the second equation in (2.9)
Formulae (5.55a,b) and (5.56) for the moments at the centre reduce to
(8.76a)
(M
(I-)L
r-a 0
For
u
+-
8r
T/4
(
2
+ 3- 2u
2
cos
2
sin 20
(8.76b)
it is checked that the formulae (8.73)-(8.76) are in agreement with
those obtained by applying equations (2.44)-(2.46) of
[3],
which were derived by a
different metiod.
b
f
a
P
4
FIG.
ACKNOWLEDGEMENT.
The author is indebted to tlrs. Eman S. Ai-Khammash who typed the
manuscript
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I.
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2.
TIMOSHENKO, S.P., and OINOWSK’.’-KX!EGER, S., Theory of Plates and Shells, 2nd Ed.
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3.
points,
_’roc.
Ca,b.
’hil._So_c. _53_, 728-743 (1957).
W. A. BASSALI
174
4.
BASSALI, W.A., Problems Concerning the Benditg of Isotropic Thin Elastic Plates
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6.
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CHANTAtaMUNGKOI, K., KARASUDHI, P. and LEE, S.L., Eccentrically Loaded Circular
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13.
14.
BASSALI, W.A,, Bending of a Thin Circular Plate Under Hydrostatic Pressure Over
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15.
BASSALI, W.A.,
16.
FRISCHBIER, R., and LUCHT, W., The Circular Plate Subject to Constant Polygonal
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