Page 316 - Physical Principles of Sedimentary Basin Analysis

298 Flexure of the lithosphere

Exercise 9.6 Let the surface load q(x) be given as a piecewise linear curve, where the load
at position x in interval i from x i to x i+1 is

q i+1 − q i
q(x) = c i (x − x i ) + q i where c i = (9.67)
x i+1 − x i
and where q i is the load at position x i . The piecewise linear surface load can be represented
∞

a
as a Fourier series q(x) = n=1 n sin(k n x) where k n = πn/L. Show that the Fourier
coefficients are
2 x 1
(q i − c i x i ) x i+1
N
a n = c i cos(k n x) + sin(k n x) − cos(k n x) (9.68)
L k n k 2 k n
i=1 n x i
b
when N is the number of intervals and [ f (x)] is used as notation for f (b) − f (a).
a
Solution: Each interval i has the contribution
2 x i+1
I n,i = q(x) sin(k n x)dx (9.69)
L
x i

to each Fourier coefficient a n . Useful integral: x sin(ax)dx =−(x/a) cos(ax) +
2
(1/a ) sin(ax).
Comments: The basis function sin(k n x) imposes the boundary condition w = 0atthe
boundaries x = 0 and x = L. The Fourier solution is in this respect different from the
superposition of point load solutions (see Note 9.3), because the point load solutions have
w = 0at x =±∞. The Fourier series solution is more computationally demanding than
the corresponding point load solution, because of the sum over n in addition to the sum
over intervals i. The infinite sum over n can be truncated at a sufficiently large number,
depending on the wanted accuracy, and could typically be 100.
Exercise 9.7 Show that coefficients in the Fourier series (9.65) are zero for every even n,
(n = 2, 4,...) when the load q(x) is symmetric around L/2.

9.5 The deflection of a plate under compression
An already bent lithospheric plate will buckle further under the action of an external com-
pressive horizontal force (see Figure 9.11). We therefore start by extending the equation for
the deflection of a plate by adding the bending moment (torque) from the horizontal force.
The bending moment from a horizontal force F becomes Fw when the vertical deflec-
tion is w. The torque balance, where the internal bending moment is equal to the external
bending moment, is
2
d w ∞

D = (x − x) q(x ) − gw(x ) dx − Fw (9.70)
dx 2 x
where the internal bending moment is the left-hand side, and the external bending moment
is the right-hand side. The pressure on the plate in the vertical direction is the surface load