Chapter 4
2
• Applied loading is usually perpendicular to the plane of the element.
• The basic assumptions:
i.
All points across the thickness have the same transverse deflection.
ii.
(a)
Kirchhoff
theory: a plane section perpendicular to the midplane remains plane and
perpendicular to the midplane after deformations. This is the
classical thin plate theory
,
which neglects transverse shear deformations. Most existing analytical solutions for
plate bending fall into this category.
(b)
Mindlin-Reissner
theory
(Fig.1.2): a plane section (perpendicular to the midplane)
remains plane but not necessarily perpendicular to the midplane. This theory takes into
account
transverse shear deformations, and hence is suitable for the analysis of thick
plates.
.
a
b
a
'
b'
w
(
x
)
x
z
Original position
(a) Deformations at a section
u
o
z
- z
u
o
(x,y)
(b) Displacements
w
,
x
a
'
b
'
Mid-plane
θ
x
θ
x
w
,x
θ
x
w
,
x
Figure 1.2 Bending deformations - Mindlin theory
Thus, the displacements of a point (
x
,
y
,
z
) any where in a flat plate can be expressed as:
w
(
x
,
y
,
z
)
=
w
(
x
,
y
)
u
(
x
,
y
,
z
)
=
u
o
(
x
,
y
)
−
z
θ
x
(
x
,
y
)
v
(
x
,
y
,
z
)
=
v
o
(
x
,
y
)
−
z
θ
y
(
x
,
y
)
(1.1)
where
w
= transverse deflection (perpendicular to element surface)
u
o
,
v
o
= in-plane displacements at the point (
x
,
y
) of the midplane (
z
= 0). These
displacements are zero if the plate is not subject to in-plane forces, and if transverse loading
is not too great to cause stretching due to arching action.
θ
x
= rotation of the
x
-section at the point (
x
,
y
).
In the case of Kirchhoff theory where the section
a
'
b
'
remains perpendicular to the mid plane