Problem $1$

A bar with cross$-$sectional area, A$,$ and Young's modulus, E$,$ is loaded with point loads at

B$,$ C$,$ and D as shown.

a$)$ Plot the stress distribution along the bar axis, $(x).$

b$)$ Suppose that the allowable normal stress is ${}_0,$ the material density is $,$ and the

length, $L,$ of the bar is fixed. Determine the most efficient cross$-$section

distribution, $A(x)$ to minimize the mass. Make a sketch $($to scale$)$ of your design.