Geoscience Reference
In-Depth Information
Consider the oceanic lithosphere deforming under an applied vertical load
V
(
x
) and no horizontal force. Water fills the resulting depression in the seabed.
However, there is a net hydrostatic restoring force of (
ρ
m
−
ρ
w
)
gw
per unit area.
This restoring force acts because the deformed lithosphere is not in isostatic
equilibrium: a thickness
w
of mantle with density
ρ
m
has effectively been replaced
by water with density
ρ
w
. Thus, for the oceanic lithosphere, Eq. (5.56)is
D
d
4
w
d
x
4
=
V
(
x
)
−
(
ρ
m
−
ρ
w
)
gw
(5.58)
In the case of the deformation of continental lithosphere, when the depression
is filled with sediment, the hydrostatic restoring force is (
ρ
m
−
ρ
c
)
gw
, since
mantle with density
ρ
m
has been replaced by crust with density
ρ
c
.For continental
lithosphere Eq. (5.56)is
D
d
4
w
d
x
4
=
V
(
x
)
−
(
ρ
m
−
ρ
c
)
gw
(5.59)
These differential equations must be solved for given loads and boundary con-
ditions to give the deflection of the plate as a function of horizontal distance. In
the particular case in which the load is an island chain (assumed to be at
x
=
0),
Eq. 5.58 is
D
d
4
w
d
x
4
+
(
ρ
m
−
ρ
w
)
gw
=
0
(5.60)
The solution to this equation for a line load
V
at
x
=
0is
w
(
x
)
=
w
0
e
−
x
/α
[cos(
x
/α
)
+
sin(
x
/α
)]
,
x
≥
0
(5.61)
where the deflection at
x
=
0is
3
V
α
w
0
=
(5.62)
8
D
α
and the parameter
, called the
flexural parameter
,isgivenby
1
/
4
4
D
(
ρ
m
−
ρ
w
)
g
α
=
(5.63)
Figure 5.15 shows the deflection given by Eq. (5.61)asafunction of
x
. Notice
the clear arch, or forebulge, on either side of the central depression. At the top
Figure 5.15.
Deflection of
an elastic plate by a line
load at
x
= 0. The
deflection is normalized
with respect to
w
0
, the
deflection at
x
0, which
is determined by the load
and the physical
properties of the plate.
Deflection is symmetrical
about
x
=
0. Solid line,
intact plate; dashed line,
plate broken at
x
= 0.
=
