Problem $1(1$D boundary value problem$)$

A force of $100KN$ is applied on a column as shown. The column is made from two materials. The top one is a functionally graded material with a linearly varying modulus and densities. Its length is $2$ meter. The density and elastic modulus of the top material at point A are $2700\frac{kg}{m^3}$ and $72$Gpa, respectively. The density and modulus of the top material at point B are $3000\frac{kg}{m^3}$ and $100$Gpa. The bottom material is made from steel $($density $=7800\frac{kg}{m^2}$ and modulus $=200$GPa $).$ The length of the bottom material is $1m.$ The cross$-$sections of both materials comprising the column are cylindrical with a diameter of $0\mathrm{.}5m.$

Using equilibrium $1$D $)=(0$ and while considering the weight and the applied force determine:

a$)$ The stress distribution in both members

b$)$ The displacement distribution in both members.

c$)$ Plot the displacement distribution in both members $($in a single plot$)$

d$)$ Plot the stress distribution in both members $($in a single plot$)$