Page 307 - Physical Principles of Sedimentary Basin Analysis

9.2 Flexure from a point load 289

where it is written as the sum of the deflection from the weight of the beam and the deflec-
tion from the end load. The solution can be used to find the maximum deflection at the free
end, which is
w(ˆ=1) = 3w 1 + 2w 2 , (9.29)
x
where the maximum deflection due to the weight of the beam is 3w 1 and the maximum
deflection due to the end load is 2w 2 .
The surface load becomes q 0 = W 0 g/l when the beam has the weight W 0 , and the end
load becomes F 0 = W b g when it is due to a weight W b . It is then seen that the weight of
the beam and the end load contribute equally to the deflection when the beam has 3/8of
the weight of the end load.
The force F 0 can also be represented as a load per unit length by use of the Dirac delta
function δ(x) as F 0 δ(x − l), which makes the total surface load per unit length q(x) =
q 0 + F 0 δ(x − l). The Dirac delta function is defined by the integral

∞

f (x )δ(x − x ) dx = f (x) (9.30)
−∞
and it is seen to be zero everywhere, except at x = 0 where it is infinite.
9.2 Flexure from a point load
The flexure of a 2D elastic plate under a point load has a solution that turns out to be
simple and useful. It is useful because any load q(x) can be discretized into a series
of discrete point loads, and the flexure from the full load q(x) can then be approxi-
mated by superposition of the solutions for the discrete point loads (see Note 9.3). A
point load in 2D is actually a line load in 3D because it extends along a line (normal
to the cross-section) through the point where it is applied. The point load V 0 is placed
at x = 0 and it is assumed that the plate is unaffected at large distances away from the
load, which gives w = 0for x →±∞. Equation (9.14) for the deflection of the plate
becomes
4
d w
D + gw = 0 (9.31)
dx 4
where = m − c , because the load is zero everywhere except at x = 0. It is a
fourth-order equation and we therefore need four boundary conditions. We already have
two and the third follows from symmetry around x = 0, which implies that dw/dx = 0
for x = 0. The fourth condition, which involves the point load, is the force balance in the
vertical direction. The load on the plate must be balanced by the buoyancy of the displaced
mantle, which is written as

∞ ∞

V 0
+ c g w(x)dx = m g w(x)dx. (9.32)
2 0 0