Section 6.2
Solid Mechanics Part II
Kelly
125
6.2
The Moment-Curvature Equations
6.2.1 From Beam Theory to Plate Theory
In the beam theory, based on the assumptions of plane sections remaining plane and that
one can neglect the transverse strain, the strain varies linearly through the thickness.
In
the notation of the beam, with
y
positive up,
R
y
xx
/
−
=
ε
, where
R
is the
radius of
curvature
,
R
positive when the beam bends “up” (see Part I, Eqn. 4.6.16).
In terms of the
curvature
R
x
v
/
1
/
2
2
=
∂
∂
, where
v
is the deflection (see Part I, Eqn. 4.6.35), one has
2
2
x
v
y
xx
∂
∂
−
=
(6.2.1)
The beam theory assumptions are essentially the same for the plate, leading to strains
which are proportional to distance from the neutral (mid-plane) surface,
z
, and expressions
similar to 6.2.1.
This leads again to linearly varying stresses
xx
σ
and
yy
(
zz
is also
taken to be zero, as in the beam theory).
6.2.2 Curvature and Twist
The plate is initially undeformed and flat with the mid-surface lying in the
y
x
−
plane.
When deformed, the mid-surface occupies the surface
)
,
(
y
x
w
w
=
and
w
is the elevation
above the
y
x
−
plane, Fig. 6.2.1.
Fig. 6.2.1: Deformed Plate
The slopes of the plate along the
x
and
y
directions are
x
w
∂
∂
/
and
y
w
∂
∂
/
.
Curvature
Recall from Part I, §4.6.10, that the curvature in the
x
direction,
x
κ
, is the rate of change
of the slope angle
ψ
with respect to arc length
s
, Fig. 6.2.2,
ds
d
x
/
=
.
One finds that
x
y
•
•
initial
position
w