Page 311 - Physical Principles of Sedimentary Basin Analysis

9.2 Flexure from a point load 293

From the symmetry condition dw/dx = 0at x = 0 it follows that d 1 = d 2 . Finally, the
force balance (9.32)gives

∞

V 0
x
x
= gd 1 α w(ˆ)d ˆ = gd 1 α (9.47)
2 0
which implies that d 1 = V 0 /(2 gα), where the following integral is used:

e −ˆ x (cos ˆ + sin ˆ)d ˆ =−e −ˆ x cos ˆ. (9.48)
x
x
x
x
4
The coefficient d 1 is sometimes rewritten using that g = 4D/α , which follows from
3
definition (9.34)of α. The alternative version d 1 then becomes d 1 = V 0 α /(8D).
Note 9.3 Semi-numerical solution. It has already been mentioned that a general load can
be represented by a series of point loads and that the solution for the deflection then
becomes the sum of the solutions for each point load. Such a superposition of solu-
tions is possible because the partial differential equation for the deflection is linear (see
Exercise 9.5). A general surface load q(x) can be turned into a series of point loads by
representing the surface by discrete intervals. The weight of the load in the interval i from
x i to x i+1 becomes the point load
x i+1
1

V i = q(x)dx ≈ q(x i ) + q(x i+1 ) (x i+1 − x i ). (9.49)
2
x i
We let each point load V i act at the center of interval i, which has the position x c,i =
(x i + x i+1 )/2. The solution for the deflection from N point loads can then be written as
the sum
N

w(x) = V i f (|x − x c,i | /α) (9.50)
i=1
where f is the function f (u) = e −u (cos u + sin u). Figure 9.8 shows an example of a
wide load that is represented by three point loads, and where the deflection is given as the
superposition of the three solutions for the point loads. The solution is compared with an
analytical solution based on Fourier series for the same load, and they are almost equal.
How the deflection can be represented by a Fourier series is shown in Note 9.5. A finer
discretization than only three point loads implies an even better approximation of the exact
solution. The figure also shows the solution when the entire load is represented by just one
point, which is inaccurate for the shown load. The superposition of point loads is a fast and
accurate algorithm that is simple to implement.

Exercise 9.5 Show that if w 1 (x) and w 2 (x) are both solutions of equation (9.14)for the
deflection, then it follows that c 1 w 1 (x) + c 2 w 2 (x) is also a solution for any constant
coefficients c 1 and c 2 .