Geoscience Reference
In-Depth Information
Table 5.1
Deflection on an elastic plate by a line load V at
x
=
0
Intact plate
Broken plate
3
8
D
3
4
D
Vα
Vα
w
0
, deflection at
x
=
0
w
b
, deflection of forebulge
−
0
.
0432
w
0
−
0
.
0670
w
0
x
b
d
w
(
x
)
d
x
= 0at
x
=
x
b
3
πα
4
πα
3
πα
4
π
2
x
0
(
w
(
x
) = 0at
x
=
x
0
)
of the forebulge d
w
0,so, by differentiating Eq. (5.61), the horizontal
distance from the origin (
x
/
d
x
=
x
b
, can be
calculated. Once
x
b
is known, Eq. (5.61)isused to obtain the height of the fore-
bulge
w
(
x
b
). The value
x
b
provides an estimate for the half-width of the depres-
sion. An alternative smaller estimate may be provided by finding
x
0
, the value
of
x
for which
w
(
x
0
)
=
0) to the top of the forebulge,
x
=
0(Table5.1). It is important to note that the magnitude of
the load,
V
, controls only the depth of the depression, not the width. The width of
the depression depends upon the flexural rigidity of the plate. This fact provides
away to use the widths of basins to calculate the elastic thicknesses of the plates
upon which they lie. Three steps are required:
=
1. the half-width, or width, of the depression (
x
0
,
x
b
)isused to obtain a value for
α
, the
flexural parameter (Table 5.1,Fig. 5.15);
2. the value for
α
is used to calculate the flexural rigidity
D
(Eq. (5.63)); and
3. the value for
D
is used to calculate the elastic thickness
h
(Eq. (5.57)).
Consider, for example, a depression with a half-width of 150 km. Using
Fig. 5.15 and Table 5.1,wecan estimate the flexural parameter
as 64 km. Then,
with Eqs. (5.63) and (5.57), the elastic thickness of the lithosphere is calculated
as 24 km when
α
10
3
kg m
−
3
,
10
3
kg m
−
3
,
g
10 m s
−
2
,
ρ
m
=
3.3
×
ρ
w
=
1.0
×
=
10
22
Nm,
E
D
=
0.25.
If the elastic plate is assumed to be broken at
x
9.6
×
=
70 GPa and
σ
=
=
0 rather than being intact,
w
0
e
−
x
/α
cos(
x
3
the solution to Eq. (5.60)is
w
(
x
)
(4
D
).
This deflection of a broken plate is narrower and deeper than the deflection of an
intact plate with the same rigidity (Table 5.1).
Similar but more complex analyses have been used to estimate the elastic thick-
ness of the Pacific plate under the Hawaiian-Emperor island chain (Fig. 5.13).
(The problem is more complex than the simple solution given here because the
island chain has a finite width and so cannot be treated as a line force acting at
x
=
/α
), with
w
0
=
V
α
/
0. Also, the age of the Pacific plate and therefore its thickness change along
the length of the island chain.)
The bending of the oceanic lithosphere at a subduction zone can also be
modelled by Eq. (5.56). In this case, it is necessary to include a load
V
at one
end (
x
=
=
0) of the plate and a horizontal bending moment
M
per unit length. The
