$($b$)$ Using the above stiffness matrix and a three$-$element representation as indicated in Figure Q$1,$ determine the vertical displacement distribution. Assume that the length of the pile is $l=18[m]$ and the cross$-$sectional area $A=0\mathrm{.}003[m^r].$ Young's modulus is specified as $E=200,000[k\frac{N}{m^2}]$ while the traction, which acts along the length of the pile, is described by the elastic stiffness $c=10[k\frac{N}{m}].$ The load is given in terms of the point load $P=100[kN]$ on the upper end of the pile.

$[12$ marks$]$

Geometry:

$l=18[m]$

$A=0\mathrm{.}003[m^2]$

Material properties:

$E=200,000[k\frac{N}{m^2}]$

$c=10[k\frac{N}{m}]$

Loading:

$P=100[kN]$

Figure Q$1.$ Vertical displacement of a pile: Geometry, material characteristies and loading conditions