Information Technology Reference
In-Depth Information
FIGURE 13.14
Polar coordinates. Positive displacement
,
moment
m
, shear force
V
, and applied loading
p
z
. Sign
Convention 1.
w
, slope
θ
13.3.2
Circular Plates
For circular plates, it is convenient to switch from a rectangular to a polar coordinate system
(Fig. 13.14). Equation (13.41) can simply be employed in polar coordinates with
=
∂
2
1
r
∂
∂
r
2
∂
1
2
∂φ
2
∇
r
2
+
r
+
(13.67)
∂
2
To derive this expression and other similar useful formulas in polar coordinates, begin with
the relationships in Eq. (13.7). Derivatives are found by the chain rule of differentiation. For
example,
∂w
∂
x
=
∂w
r
∂
x
+
∂w
∂φ
∂
x
=
∂w
r
∂w
r
1
φ
−
φ
cos
sin
∂
∂
∂φ
∂
∂φ
r
(13.68)
sin
2
sin
2
2
2
2
2
∂
x
2
=
∂
w
w
2
∂
w
∂φ∂
sin
φ
cos
φ
+
∂w
∂
φ
2
∂w
∂φ
sin
φ
cos
φ
+
∂
w
∂φ
φ
cos
2
φ
−
+
∂
∂
r
2
r
r
r
r
r
2
r
2
Substitution of the derivatives of Eq. (13.68) and similar expressions into the governing
equations of the previous subsections leads to the local form of the governing differential
equation
p
z
K
4
∇
w
=
(13.69a)
4
2
2
, where
2
Et
3
2
with
∇
=∇
∇
∇
is defined in Eq. (13.67) and
K
=
/
[12
(
1
−
ν
)
]
.
Equation
(13.69a) can be written as
∂
∂
2
2
2
2
1
r
∂
∂
r
2
∂
1
1
r
∂
∂
r
2
∂
1
p
z
K
r
2
+
r
+
r
2
+
r
+
w
=
(13.69b)
∂
∂φ
2
∂
∂φ
2
Search WWH ::
Custom Search