Page 317 - Physical Principles of Sedimentary Basin Analysis

9.5 The deflection of a plate under compression 299

F w F

Figure 9.11. The buckling of a horizontal plate by a horizontal compressive force F.

q(x) minus the counteracting buoyancy pressure gw. A differential equation for the
deflection is found after differentiating twice, and we get
4
2
d w d w
D + F + gw = q(x) (9.71)
dx 4 dx 2
(see Exercise 9.1 for how to carry out the first of the two differentiations). Equation (9.71)
gives the deflection from the combined action of a surface load q(x) and a horizontal
force F, when the plate is supported by a buoyancy pressure w. We have already seen
in Section 9.4 that a periodic surface load produces a periodic deflection, and we will see
that the deflection from horizontal compression depends on the wavelength. The impact of
the horizontal force F is therefore studied in the case of the same periodic surface load as
in Section 9.4,
q(x) = q 0 cos(2πx/λ), (9.72)
where λ is the wavelength of the load. A constant surface load q 0 could have been added
to the periodic load, in order to make it oscillate between 0 and q 0 , but it is left out since
it only adds a constant subsidence q 0 /( g) to the deflection. A periodic surface load can
also be expressed using the wave number k = 2π/λ, which simplifies the expressions. The
deflection in the case of the surface load (9.72) is found by guessing that the solution has
the same form as the surface load
w(x) = w 0 cos(kx). (9.73)
By inserting solution (9.73) into equation (9.71) we get

4 2
w 0 Dk − Fk + g = q 0 (9.74)
or
q 0
w 0 =
. (9.75)
2
2
k 2 (Dk + g/k ) − F
We first notice that zero compression (F = 0) leads to the same deflection as in Section 9.4
for a periodic load. The next thing we notice is that the deflection increases with an increas-
ing compressive force, but the compressive force cannot increase beyond the limiting
force
2
2
F c = Dk + g/k , (9.76)
which would imply an infinite deflection. The upper bound for the compressive force F c
depends on the wave number (or alternatively on the wavelength). The critical force F c is
seen to increase with decreasing wave numbers (increasing wavelengths), which means that