A wooden plank of negligible thickness is 1.2m long. Its width changes uniformly from 30cm at one end to 15cm at the other end. The plank is laid flat on soft soil and carries a uniform load of 1800N distributed evenly along its length.
(i) Show that the local width w(x) at a distance x (in metres) from the wider end is given by w(x)=0.30−0.125x(m).
(ii) Derive an expression for the elemental normal force dF acting on a strip of length dx of the plank and hence obtain an expression for the elemental pressure p(x) on the soil.
(iii) Integrate your expression over the entire length to find the average pressure exerted on the soil.
(iv) Comment qualitatively on whether the narrower end is more likely to sink into the soil compared to the wider end.