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May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
42
A VARIATIONAL APPROACH TO STATIC ANALYSIS OF A THIN
RECTANGULAR ORTHOTROPIC PLATE SUBJECTED TO
UNIFORMLY DISTRIBUTED LOAD
Emma J.B.1, Sule, S.
2
Department of Civil Engineering, University of Nigeria, Nsukka, Enugu State.
1
Department of Civil and Environmental Engineering, University of Port Harcourt, Rivers State2
Abstract
In this paper, the values of numerical factors for deflection of a thin rectangular isotropic plate
subjected to uniformly distributed load with different boundary conditions all round fixed, all
round simply supported and two opposite sides fixed, and the other two simply supported are
determined using variational approach. The results obtained using variation approach is
compared with those obtained from literature. It is shown among other findings that variational
solution does not yield a satisfactory result most especially for all round simply supported plates
but produced satisfactory results for all round fixed and two opposite sides fixed and the other
two simply supported plates
Keywords: numerical factors, rectangular orthotropic plate, boundary conditions, variational
approach.
1.0 Introduction
A plate as an engineering structure is defined as a
body in the shape of a prism with thickness, small in
comparison with its other dimensions. They are
commonly referred to as slabs or thin-walled
structure. It is used in modern structures to transmit
lateral and or in-plane load to adjacent support. It is
one of the most important components employed in
the main branches of engineering construction,
building, civil engineering, hydraulic engineering,
naval architecture and air-craft construction [Gould,
1999; Iyengar, 1988; Mansfield, 1989]. The classical
method that leads to exact solution is not only
rigorous and time consuming but proves in many
cases quite laborious and almost impossible due to its
mathematical difficulties [Charlton, 1961; Biot,
1972].The problems encountered in thin plate theory
can be solved with the aid of various approximate
methods such as energy method, finite element
method, finite difference method and fourier series
[Bares, 1969; Chajes, 1974; El Nachie, 1990].
However, common problems are encountered. For
example, numerical methods (FDM and FEM) lead to
an algebraic equation of large matrix size demanding
large computer memories, thereby making the
analysis cumbersome and time wasting. Navier’s
method is regarded as the widely used approximate
method for the analysis of thin plates. Nevertheless, it
is noted that the double trigonometric series in the
method are not convenient for numerical
computations if higher derivatives of the function
“w” are involved. Besides, satisfactory solution by
Navier method is only obtained for simply supported
thin plates. This study highlights the use of
polynomial function in the analysis thin rectangular
orthotropic plate subjected to a centre point loading
considering three different boundary conditions.
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
43
3.0 Derivation of Governing Equations
The strain energy U of bending of plate is given by:
dxdyyx
w
y
w
x
w
y
w
x
wDU
lx
o
ly
o
22
2
2
2
22
2
2
2
2
122
(1)
The potential energy of uniformly distributed load over the plate is given by:
x yl l
dxdyyxqwV0 0
, (2)
The total potential energy of the system, VUT (3)
Substitution of equations [1] and [2] into equation [3] gives
x yx y l ll l
T dxdyyxqwdxdyyx
w
y
w
x
w
y
w
w
wD
0 00 0
22
2
2
2
22
2
22
2
2
,.122
(4)
Let
The coefficient Ai be considered as the coordinate defining the shape of the deflection surface.
xl length of plate in the x-direction
yl length of plate in the y-direction
and denotes the position of point load at any giving point in x and y coordinates respectively.
To evaluate equation [1], let the deflection ),( yxx be given as
)(.)(.),( yxAyxw (5)
Where
)(x Represent x –coordinate
)(y Represent y-coordinate
Let
4
4
3
3
2
210)( xaxaxaxaax (a)
4
4
3
3
2
210)( xbxbxbxbby (b) (6)
By differentiating, equations [6(a)] and [6(b)], we have the following derivatives:
For x – direction:
4
4
3
3
2
210 xaxaxaxaax
3
4
2
321
/ 432 xaxaxaax (7)
2
432
// 1262 xaxaax
xaax 43
/// 246
Likewise, for y – direction:
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
44
4
4
3
3
2
210 ybybybybby
3
4
2
321
/ 432 ybybybby
2
432
// 1262 ybybby (8)
ybbx 43
/// 246
Determination of the coefficients using boundary conditions
Figure 1: All round fixed supported plate
The boundary conditions are:
00
100
/0
/
2
0
x
xx
l
ll
(9)
At x = 0:
)(0
)(0
10/
00
ba
aa
(10)
At x = lx:
)(4320
)(0
3
4
2
321
/
4
4
3
3
2
210
blalalaa
alalalalaa
xxx
xxxx
(11)
(21)
At x = 2
xl:
16842
14
4
3
3
2
210
xxxx lalalalaa (12)
Substituting [10 (a,b)] into [11(b)] gives
)(
0
432
4
4
3
3
2
2
xx
xxx
laala
lalala
(13)
Substituting equation [13] into [11(b)] gives
x
x
y y
ly
lx
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
45
x
xxxx
laa
lalalaal
43
3
4
2
343
2
2
43)(200
(14)
Substituting equation [10 (a,b),13 and [14] into [12] gives
168
22
4001
4
4
4
444
3
xxxx
x lalalala
l
16164421
4
4
4
4
4
4
4
4
4
4 xxxxx lalalalala
44
16
xla (15)
Substituting equation [15] into [14], gives
343
32162
xx
x
ll
la
(16)
Substituting equation [15] and [16] into [13] gives
2432
161632
xx
x
x
xll
l
lla
(17)
Therefore,
44332210
16,
32,
16,0,0
xxx la
la
laaa (18)
Substituting equation [18] into [6] with respect to yandx respective we obtain
4
4
3
32
4
4
3
3
2
2 216
163216
xxxxxx l
x
l
x
l
x
l
x
l
x
l
xx (19)
Likewise
4
4
3
3
2
2
4
4
3
3
2
2 216
163216
yyyyyy l
y
l
y
l
y
l
y
l
y
l
yy (20)
ly
y
y
x
x lx
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
46
Figure 2: All round simply supported plate.
The boundary conditions are:
00
100
//0
//
2
0
x
xx
l
ll
(21)
At x = 0:
)(20
)(0
220//
00
baa
aa
(22)
At x = lx:
)(12620
)(0
2
432
//
4
4
3
3
2
210
blalaa
alalalalaa
xxl
xxxxl
x
x
(23)
At x = 2
xl :
16842
14
4
3
3
2
210
2
xxxx
l
lalalalaa
x
(24)
Substituting equation [22 (b)] into [23 (b)] gives:
x
xx
laa
lala
43
2
43
2
1260
(25)
Substituting equation [22(a,b] and [25] into [23(a)] gives:
3
41
4
4
4
41 2000
x
xxx
laa
lalala
(26)
Substituting equation [22(a,b), 25 and 26] into [24] gives:
16
5
168
20
201
4
4
4
4
4
4
4
4 xxxx lalalala (27)
44
5
16
xla
Substituting equation [27] into [25] gives:
343
5
32
5
162
x
x
x ll
la
(28)
Substituting equation [27] into [26] gives:
x
x
x ll
la
5
16
5
16 3
41 (29)
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
47
Therefore,
44332105
16,
5
32,0,
5
16,0
xxx la
laa
laa (30)
Substituting equation [30] into [6(a)] gives:
4
4
3
3
4
4
3
3 2
5
1616
5
32
5
16
xxxxxx l
x
l
x
l
x
l
x
l
x
l
xx (31)
Similarly,
4
4
3
3
4
4
3
3 2
5
16
5
16
5
32
5
16
yyyyyy l
y
l
y
l
y
l
y
l
y
l
yy (32)
Figure 3: Two opposite sides fixed and the other two, simply supported plate
From equations, obtained for all around fixed and all round simply supported plates, we deduced that,
4
4
3
3
2
24
3
3
2
2 216
4
163216
xxxxx l
x
l
x
l
x
l
x
l
x
l
xx (33)
4
4
3
3
4
4
3
3 2
5
16
5
16
5
32
5
16
yyyyyy l
y
l
y
l
y
l
y
l
y
l
yy (34)
All round fixed rectangular plate:
Substituting equations [19] and [20] into [5] we obtain,
4
4
3
3
2
2
4
4
3
3
2
2 216.
216.),(
xyyxxx l
y
l
y
l
y
l
x
l
x
l
xAyxw
4
4
3
3
2
2
4
4
3
3
2
2 22256.),(
xyyxxx l
y
l
y
l
y
l
x
l
x
l
xAyxw (35)
Hence,
4
4
3
3
2
2
4
2
322
2 212122256.
xyyxxx l
y
l
y
l
y
l
x
l
x
lA
x
w (36)
Similarly,
ly
y
x
y
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
48
4
4
3
3
2
2
4
2
322
2 212122256.
xxxyyy l
x
l
x
l
x
l
y
l
y
lA
x
w (37)
Expanding equation [36, and 37] we have:
24
22
43
4
33
3
23
2
42
4
32
3
22
2
2
2 12122412242256.
yxyxyxyxyxyxyx ll
yx
ll
xy
ll
xy
ll
xy
ll
y
ll
y
ll
yA
x
w
44
42
34
22 1224
yxyx ll
yx
ll
yx (38)
42
22
34
4
33
3
32
2
24
4
23
3
22
2
2
2 12122412242256.
yxyxyxyxyxyxyx ll
yx
ll
yx
ll
yx
ll
yx
ll
x
ll
x
ll
xA
y
w
44
24
44
23 1224
yxyx ll
yx
ll
yx (39)
Hence,
22
22
2
2
2
2
2
2
2
2
22
2
2
2 6126612621
2256.
yxyxxyxyxxyyyx ll
yx
ll
yx
l
x
ll
xy
ll
xy
l
x
l
y
l
y
ll
yA
x
w
22
22
2
2
2
2
2
2
2
2
22
2
2
2 6126612621
512
yxyxxyxyxxyyyx ll
yx
ll
yx
l
x
ll
xy
ll
xy
l
x
l
y
l
y
ll
yA
x
w
yxxyxyxxyyyx ll
yx
l
x
ll
xy
ll
xy
l
x
l
y
l
y
ll
yA
x
w2
2
2
2
2
2
2
22
22
22
2
2
2 19248724812641
512 (40)
3
3
42
42
4
4
4
2
33
32
3
3
3
3
22
22 724812192484288
xyxyxyyxyxyyx l
x
ll
yx
ll
xy
l
y
ll
yx
ll
xy
l
y
ll
yx
24
24
4
4
4
4
43
43
33
33
23
23
3
3 2161443672288432288
yxyxyyxyxyxyx ll
yx
ll
yx
l
x
ll
yx
ll
yx
ll
yx
ll
yx
44
44
34
34 36144
yxyx ll
yx
ll
yx (41)
Similarly,
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
49
2
2
2
2
2
2
2
22
22
22
2
2
2 19248724812641
512
yxyyxyxyxxyx ll
xy
l
y
ll
yx
ll
xy
l
y
l
x
l
x
ll
xA
y
W
3
3
24
24
4
4
4
4
23
23
3
3
3
3
22
22 724812192484288
yyxyxxyxyxxyx l
y
ll
yx
ll
yx
l
x
ll
yx
ll
yx
l
x
ll
yx
42
42
4
4
4
4
34
34
33
33
32
32
3
3 2161443672288432288
yxyxyyxyxyxyx ll
yx
ll
xy
l
y
ll
yx
ll
yx
ll
yx
ll
xy
44
44
43
43 36144
yxyx ll
yx
ll
yx (42)
According to Timoshenko and Woinowsky-Krieger (1959) for a polygonal plate if one of the boundary conditions is
either w = 0 or n
w
= 0 where n = direction normal to the edge. The third term is negligible. Thus the strain
energy equation (1)becomes :
dxdyy
w
x
wDU
x yl l
0 0
2
2
22
2
2
2 (43)
Hence, by substituting the derivatives of equations [41and 42] into equation [43], and integrating rigorously, we
have:
44
2
22
2
22050
7512..
2xy
yx
llll
AD
U (44)
Uniformly Distributed Load (UDL):
Substituting equations [35] into [2(a)] and integrating, we have:
225
64 yx llqAV (45)
Substituting equations [44, and 46] into [3], and making the total potential energy a minimum, gives
0
A
T
225
64
22050
7512..0 44
2
22
yx
xy
yx
yx
llqll
ll
llAD
Hence,
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
50
44
44003417968.0
xy
yx
llD
lqlA
(46)
Substituting equation [46]) into [5], we obtain:
yx
xy
yx
llD
lqlyxw )(44
44003418.0),(
(47)
All round simply supported plate:
Substituting equation [31] and [32] into [5]gives:
4
4
3
3
4
4
3
3 2
5
16.
2
5
16.),(
yyyxxx l
y
l
y
l
y
l
x
l
x
l
xAyxw
4
4
3
3
4
4
3
3 2.
2
25
256
yyyxxx l
y
l
y
l
y
l
x
l
x
l
xA (48)
Therefore,
4
4
3
3
4
3
3
2 2.
46
25
256
yyyxxx l
y
l
y
l
y
l
x
l
x
l
lA
x
w
4
4
3
3
4
2
32
2 2.
1212
25
256
yyyxx l
y
l
y
l
y
l
x
l
xA
x
w (49)
4
4
3
3
4
2
32
2 2.
1212
25
256
xxxyy l
x
l
x
l
x
l
y
l
yA
y
w (50)
Expanding equation [49] we have:
44
42
34
32
3
2
43
4
33
3
32
2 122412122412
25
256
yxyxxxxxyxyy ll
yx
ll
yx
ll
yx
ll
xy
ll
xy
ll
xyA
x
w
33
3
23
2
332
3
22
2
22
2 221
25
3072
yxyxxyxyxyyx ll
xy
ll
xy
l
x
ll
y
ll
y
lll
xyA
x
w (51)
Hence,
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
51
54
5
44
4
35
3
25
2
534
3
24
2
4
2
2
2
2
2 44482241
25
3072
yxyxyxyxxyxyxxyx ll
y
ll
y
ll
xy
ll
xy
l
x
ll
y
ll
y
lll
xyA
x
w
46
42
36
32
26
22
6
2
65
6
64
6
55
5
45
4 424288
yxyxyxxyxyxyxyx ll
yx
ll
yx
ll
yx
l
x
ll
xy
ll
y
ll
xy
ll
xy
66
62
56
524
yxyx ll
yx
ll
yx (52)
Similarly,
45
5
44
4
53
3
52
2
543
3
42
2
4
2
2
2
2
2 44482241
25
3072
yxyxyxyxyyxyxyyx ll
x
ll
x
ll
yx
ll
yx
l
y
ll
x
ll
x
lll
xyA
y
w
44
24
63
23
62
22
6
2
65
6
64
6
55
5
54
4 424288
yxyxyxxyxyxyxyx ll
yx
ll
yx
ll
yx
l
x
ll
yx
ll
x
ll
yx
ll
yx
66
26
65
254
yxyx ll
yx
ll
yx (53)
Substituting the derivatives of equations [52] and [53] into [43] and integrating, we obtain:
y
x
x
y
yxy
x
x
y
yx l
l
l
l
ll
DA
l
l
l
l
ll
DAU
33
22
2332
2 383184.12
18900
31
25
3072
2 (54)
Substituting equation [48] into [2 (a)] and integrating, we obtain:
625
256 yxllqAV (55)
Substituting equations [54 and 55] into [3], and making it a minimum we have:
0)(
A
AT
T
625
256383184.1220
33
22
yx
y
x
x
y
yx
llP
l
l
l
l
ll
DA
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
52
625766368.24
256
33
33
y
x
x
y
yx
l
l
l
lD
llqA (56)
Substituting equation [56] into [5], we obtain:
)(016538557.0
),(33
32
yx
l
l
l
lD
lqlyxw
y
x
x
y
yx
(57)
Two opposite sides fixed and the other side simply supported
Substituting equations [33 and 34] into [5] yields
4
4
3
3
2
2
4
4
3
3 216
2
5
16),(
yyyxxx l
y
l
y
l
y
l
x
l
x
l
xAyxw
4
4
3
3
2
2
4
4
3
3 22
5
256),(
yyyxxx l
y
l
y
l
y
l
x
l
x
l
xAyxw (58)
4
4
3
3
2
2
4
2
32
2 21212
5
256
yyyxx l
y
l
y
l
y
l
x
l
xA
x
w (59)
4
4
3
3
4
2
322
2 212122
5
256
xxxyyy l
x
l
x
l
x
l
y
l
y
lA
y
w (60)
Expanding equations [59 and 60], we obtain:
44
42
34
32
24
22
43
4
33
3
23
2
2
2 122412122412
5
256
yxyxyxyxyxyx ll
yx
ll
yx
ll
yx
ll
xy
ll
xy
ll
xyA
x
w (61)
44
24
43
23
4
2
34
4
33
3
324
4
23
3
22
2 122412122412242
5
256
yxyxyxyxyxyxyxyxyx ll
yx
ll
yx
ll
xy
ll
yx
ll
yx
ll
xy
ll
x
ll
x
ll
xA
y
w (62)
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
53
Hence,
2
2
2
4
23
2
2
2 61266126
2.
5
256
yxyxxyyyx ll
xy
ll
xy
l
x
l
y
l
y
ll
xyA
x
w
3
3
3
3
2
2
2
22
23
22
2
2
2 2881444322887221614436
5
512
yxyyxyxxyyyx ll
xy
l
y
ll
xy
ll
xy
l
x
l
y
l
y
ll
xyA
x
w
42
42
32
32
22
22
2
2
2
2
4
4
4
4 36144216144367236
yyyyyxyyyyxy ll
yx
ll
yx
ll
yx
ll
yx
l
x
ll
xy
l
y (63)
From equation [62], we obtain,
42
23
22
22
2
2
3
3
2
2
3
3
2
2
22
2 6126612621
5
2256
yxyxyyxyxyxxyx ll
yx
ll
yx
l
y
ll
yx
ll
yx
l
y
l
x
l
x
ll
xA
x
w
22
22
2
2
3
3
2
2
3
3
2
22
2
2
2
2
2 19248244812241
5
512
yxyyxyxyxxyx ll
yx
l
y
ll
yx
ll
yx
l
y
l
x
l
x
ll
xA
y
w
6
6
25
25
24
24
5
5
4
4
5
5
4
4
23
23 19219248484496
xyxyxyxyxxxyx l
x
ll
yx
ll
yx
ll
yx
ll
yx
l
x
l
x
ll
yx
36
36
35
35
34
34
33
33
32
32
3
3
26
26
6
6 72288288144288724812
yxyxyxyxyxxyxyx ll
yx
ll
yx
ll
yx
ll
yx
ll
yx
l
y
ll
yx
ll
yx
46
46
45
45
44
44
43
43
42
42
4
4 361441447214436
yxyxyxyxyxy ll
yx
ll
yx
ll
yx
ll
yx
ll
yx
l
y (64)
Substituting equation [63 and 64] into [43] and integration, we obtain:
y
x
x
y
yx l
l
l
l
llA
DU
6
31
525
1
5
512
2
332
2 (65)
Uniformly Distributed Load (UDL):
Substituting equation [58] into [2(a)] and integrating we obtain
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
54
750
256 yxllqAV (66)
Substituting equations [65 and 66] into [3], and making it a minimum, we obtain:
0)(
A
Ap
p
0750
256
6
31
525
1
5
512332
yx
y
x
x
y
yx
llq
l
l
l
l
llDA
Hence
y
x
x
y
yx
l
l
l
lD
lqlA
6
31
525
1
000032552.0
33
33
(67)
Substituting equation [67] into [5], we have:
)()(
6
31
525
1
000032552.0),(
33
33
yx
l
l
l
lD
lqlyxw
y
x
x
y
yx
(68)
4.0 Results and Discussion
An example for numerical study
Table of results obtain for all round fixed, all round simply supported and two opposite sides fixed and the
other two opposite side simply supported under transverse point and uniformly distributed load is presented below.
Taking the maximum deflection of the plate to be at the middle, for when the point load is applied at the
middle and when it is entirely uniformly distributed as well. Then we have 2
,2
yxl
yl
x .
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved
www.eaas-journal.org
55
Table 1: Table of values of numerical factor for deflection of a uniformly distributed load on ARF, ASS, and 2opp(F&SS) rectangular plates for various
values of span ratio xy ll .
xy ll Energy method Levy’s method Difference
ARF ASS 2opp (F&SS) ARF ASS 2OPP(F&SS) ARF ASS 2opp (F&S.S)
1.0
D
q437499904.0
D
q116935296.2
D
q709457643.0
D
q32256.0
D
q03936.1
D
q49152.0
0.1149399 04(35.6)
1.077575 296 (103.7)
0.217937 643 (44.3)
1.1
D
q519900659.0
D
q515648689.2
D
q966015156.0
D
q3384.0
D
q2416.1
D
q6425.0
0.135900 659 (35.4)
1.27404 8689 (102.6)
0.323455 156 (50.3)
1.2
D
q590317413.0
D
q856374954.2
D
q25298316.1
D
q44032.0
D
q44384.1
D
q81664.0
0.149997
413 (34.0)
1.412534
954 (97)
0.43634
9315(53)
1.3
D
q648086655.0
D
q135903581.3
D
q699501192.1
D
q48896.0
D
q63328.1
D
q99328.0
0.1591266
55 (32.5)
1.50262
3581 (92.0)
0.706221
192 (71.2)
1.4
D
q6914274467.0
D
q359393024.3
D
q86572596.1
D
q52992.0
D
q8048.1
D
q1776.1
0.164354
467 (30.5)
1.554593
024(86.1)
0.688125
96(58.4)
1.5
D
q730669942.0
D
q535500185.3
D
q966559893.1
D
q5632.0
D
q97632.1
D
q35936.1
0.167469
942 (29.7)
1.5591801
85 (79.2)
0.60719
9893 (44.7)
May 2013. Vol. 3, No. 3 ISSN2305-8269
International Journal of Engineering and Applied Sciences © 2012 EAAS & ARF. All rights reserved www.eaas-journal.org
56
5.0 Discussion of Results
Looking at Table 1, it can be seen that the maximum
coefficient value of deflection due to uniformly
distributed load for all round fixed plate in terms of
energy method is higher than the one obtained in
Levy’s method. It is observed that the percentage
error is high for square plate (35.6%). The error
diminishes as the span ratio increases, this indicates
that the energy method converges faster as the span
ratio increases, and increasing its number of terms
would yield a better approximate result. The basic
factor for this error is that the assumed polynomial
function chosen is not so close to the actual
deflection function (curve) as the Levy’s single
trigonometric function is.
In the case of all round simply supported
plate in Table 1, no close agreement is noted for the
maximum deflection coefficient values. The
difference between the two methods are significantly
outrageous, up to a maximum of 103.7%. However,
the percentage error decreases as the span ratio
increases. Nevertheless, it suggests that, the
deflection function chosen does not suit energy
method in terms of all round simply supported plate.
Hence, cannot be recommended for use for simply
supported plates.
In the case of plate with two opposite sides
fixed and the other two sides simply supported in
Table 1, the percentage error increases as the span
ratio increases up to 1.3 and then begins to decline.
From Table 1, it is observed that, the deflection
coefficients decrease as the span ratio increase in all
the three cases considered. Also, for two opposite
sides fixed and the other two simply supported in
Energy and Levy’s methods respectively their results
are closely the same.
6.0 CONCLUSION
The following conclusions are drawn from the study.
1. That the polynomial deflection function
considered in this study does not yield a
satisfactory result especially for all round
simply supported plates. However, for all
round fixed and two opposite sides fixed and
the other two simply supported plates, it can
be used. Because the results shown in Table
1 is for their first term, increasing the
number of terms would yield a considerably
upper bound approximate satisfactory result,
though will lead to over design and
consequently, uneconomical, yet durability
and strength will be achieved.
2. The Navier’s solution method and Levy’s
method stood better for solving simply
supported plate problems especially in the
light of the deflection function chosen.
3. The Energy method is best useful for in all
round fixed plates.
4. Even though Energy method has been
proved to be an excellent approximate
method, this study has proved that
unsatisfactory result can be obtained if an
unsatisfactory deflection function is chosen.
References
1. Bares, R., Tables for the analysis of Plates, slabs
and Diaphragms based on the elastic theory.
German & English Edition. Bauverlag, Berlin,
1969.
2. Biot M.A. “A New Approach to the Mechanics of
Orthotropic Multilayered Plates.” Int. J. Solid
Structure, 1972, Vol.9, pp.613 - 490, Pergamon
Press, Great Britain.
3. Chajes, A., Principles of structural stability
theory. Prentice-Hall, Inc. Englewood Cliffs,
New Jersey, 1974.
4. Charlton, T.M., Analysis of Statically
indeterminate framework, Longmans, London,
1961, pp. 102.
5. El Naschie, M.S., Stress, stability and Chaos in
structural Engineering: An Energy Approach.
McGraw-Hill Book Company, London, 1990.
6. Gould, P.L., Analysis of Shells and Plates.
Prentice Hall, Upper saddle River, New Jersey,
1999.
7. Iyengar, N. G.R., Structural Stability of Columns
and Plates. Ellis Horwood limited, Chichester,
1988.
8. Mansfield E.H., The bending and stretching of
plates. 2nd Ed. Cambridge University Press,
Cambridge, 1989.
