Problem $1[25$ marks$]$

You are designing the long and slender machine element. It is a $9$ m long, solid, prismatic beam that is supported by a roller at the left$-$hand side $($LHS$)$ and a pin joint at the right$-$hand side $($RHS$).$ The beam is under a $\backslash (1\mathrm{.}5\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ uniformly distributed load on the first $6$ m from LHS and a $3$ kN point load at $3$ m from RHS$.$ The element is made of conventional steel.

a$)$ Determine the maximum force and moment acting on the system and display their distribution graphically.

$[6$ Marks$]$

b$)$ Assume a factor of safety of $2$ and demonstrate how two different cross$-$section of the element affect the maximum stress on the system and its location.

$[10$ Marks$]$

c$)$ Provide the material properties needed to complete this static analysis.

$[9$ Marks$]$

Fig. Q$1$: Structural element under concentrated and distributed loads