Advanced intelligent computing theories and applications ...
61179126 Theories and Applications of Plate Analysis 0471429899
-
Upload
pandal-raj -
Category
Documents
-
view
220 -
download
0
Transcript of 61179126 Theories and Applications of Plate Analysis 0471429899
8/10/2019 61179126 Theories and Applications of Plate Analysis 0471429899
http://slidepdf.com/reader/full/61179126-theories-and-applications-of-plate-analysis-0471429899 1/4
194 Energy and Variational Methods for Solution of Lateral Deflections
The maximum deflection at x = y = a/ 2, using the first three terms (m = n = 1;m = 1, n = 3; m = 3, n = 1 ) of the series expression (4.2.28) is
wmax =
7.9289 p0a4
= 0.002625
p0a4
.π7D D
A comparison with the corresponding result of a more exact solution, wmax =0.00263 p0a
4 /D, given in Ref. [2], indicates only an insignificant error. This
example illustrates the high accuracy obtainable by the Ritz method, providedthat proper shape functions are used. Considering more terms, even this rela-tively small error can be eliminated.
ILLUSTRATIVE EXAMPLE II
Determine the maximum deflection of the clamped rectangular plate shown in
Fig. 4.2.2 by the Ritz method. Assume that the plate is subjected to constant
lateral load and use a/b = 1.5 span ratio.
b
O
X
A A
b
aa
Y
(a)
p z = p0 = const
X
Section A− A
(b)
Figure 4.2.2 Rectangular plate with fixed edges.
To utilize the apparent symmetry of the deflected plate surface, we take the
coordinate axes through the middle of the plate parallel to the sides. In this case,
the deflection given by Eq. (2.5.30) becomes∞
∞
mπ x
nπ y
8/10/2019 61179126 Theories and Applications of Plate Analysis 0471429899
http://slidepdf.com/reader/full/61179126-theories-and-applications-of-plate-analysis-0471429899 2/4
W mnw(x, y) = m n 4
1 − ( −1 )m cos a 1 − ( −1 )
n cos b
for m, n = 1, 3, 5, . . .. (4.2.30)
8/10/2019 61179126 Theories and Applications of Plate Analysis 0471429899
http://slidepdf.com/reader/full/61179126-theories-and-applications-of-plate-analysis-0471429899 3/4
Ritz’s Method
This series expression satisfies the given boundary conditions
(w) x =±a = 0,
∂w
= 0;∂x
x =±a ( 4.2.3
(w)y =±b = 0,∂w
= 0.∂y
y =±b
For the sake of simplicity, let us consider only the first term (m = n = 1 ) in Eq.(4.2.30). Thus, we can write
W π x
π y
w=
11
1 + cos 1 + cos . ( 4.2.31a )4
a b
Substitution of this expression into Eq. (4.2.6) gives the strain energy of the plate in
bending; therefore
D a b
Dπ4W
23
b 3a2 . ( 4.2.32 )
Ub = ( ∇2 w)
2 d x d y =
11
+ +2
−a
32 a3 b3 ab
−b
Similarly, from Eq. (4.1.9), the potential of the external forces is computed:
V = − p0 +a
+b
w(x, y) d x d y = − p0W 11ab.
−a −b
Minimization of the total potential,
∂(Ub + V )
=
0,∂W 11
yields
W 11 =
16 p0a4
1
.Dπ4
3 + 3(a4 /b
4 ) + 2(a
2
/b2 )
( 4.2.33 )
( 4.2.34 )
If a/b = 1.5 and ν = 0.3, themaximum deflection at x =
y = 0 is calculatedfrom Eqs.(4.2.31a) and
8/10/2019 61179126 Theories and Applications of Plate Analysis 0471429899
http://slidepdf.com/reader/full/61179126-theories-and-applications-of-plate-analysis-0471429899 4/4
A comparison with the
“exact” solution of the
problem [2], which is
shows that the approximate
solution is accurate enough
for most practical pur-poses.
By considering more terms
in the series representation
of the deflections, a more
accurate solution can beobtained.