Information Technology Reference
In-Depth Information
with
k
D
u
D
T
ED
u
=
.
In the case of rectangular flat elements (Chapter 2, Example 2.5) with
constant
D,
.
∂ν∂
y
+
y
∂
(
1
−
ν
2
∂
x
..................... .....................
∂∂
x
+
y
∂
(
1
−
ν)
2
∂
y
x
x
k
D
=
D
(13.19a)
.
∂
(
1
−
ν)
2
∂
(
1
−
ν)
2
∂ν∂
+
∂
∂∂
+
∂
y
x
x
y
y
y
x
x
For circular plates with in-plane loading and constant
D,
2
r
φ
∂∂
φ
∂
+
r
2
.
∂
r
∂
φ
+
1
−
ν
1
r
1
r
2
∂
r
+
r
∂
+
φ
∂
(
1
−
ν)
2
r
2
1
−
∂
φ
∂∂
r
+
∂
φ
+
r
r
r
r
r
k
D
=
D
........................
....................................
2
r
r
∂
φ
+
φ
∂
r
∂
r
.
φ
∂
−
2
r
∂∂
r
2
−
ν
1
r
2
1
1
r
1
r
1
1
r
2
1
1
r
φ
∂
+
∂∂
φ
−
∂
φ
+
−
∂
−
∂
+
r
r
r
(13.19b)
These relationships form the basis for the development of stiffness matrices. See Chapter 6,
Section 6.4.1 for the use of
k
D
in the formulation of a solution.
13.2
Transverse Deformation of a Plate
In plate theory, the equations for a three-dimensional continuum are to be referred to the
middle surface of a plate. We will begin by establishing the kinematic, material law, and
equilibrium relations for the transverse deformation of a plate. These relationships will
then be specialized for a plate without shear deformation effects, the so-called
Kirchhoff
1
plate
. Finally, a plate with shear deformation effects will be studied.
13.2.1 Kinematical Relationships
The displacements
u
x
,u
y
,
and
u
z
of the plate are to be expressed by the deflection
w
and
rotations
y
of the plate middle surface. See Fig. 13.4 for positive displacements.
We choose to use definitions of displacements and forces that are traditional with plate
theory, rather than the definitions consistent with the coordinate directions used elsewhere
in this work. For example, note how the definition of
θ
x
and
θ
θ
x
differs from that of Chapter 1,
where
x
of this chapter. The middle surface is assumed to remain
unstrained. Similar to beam theory, it is assumed that plane sections normal to the middle
surface before bending remain plane after bending. When shear deformation is taken into
account, the plane will not necessarily remain normal to the middle surface. With the plane
section remaining plane assumption, the displacements appear as
θ
=
θ
y
corresponds to
θ
u
x
u
y
u
z
z
00
0
z
0
001
θ
x
=
θ
(13.20)
y
w
1
Gustav Robert Kirchhoff (1824-1887) was a German physicist who followed Bunsen, of Bunsen burner fame, to
a lengthy stay as a professor at the University of Heidelberg. Kirchhoff was one of the German scientists credited
with applying quality scientific methodology for overcoming the headstart in the industrial revolution of the
English and French. Kirchhoff's law was fundamental to the thermodynamics of radiation, and, as interpreted
by Planck, fundamental to quantum physics. His work on chemical elements with Bunsen led to the method of
spectral analysis. A midcareer accident forced him to use crutches or a wheelchair and to discontinue experimental
research.
Search WWH ::
Custom Search