Problem $3.$ A rigid beam $BG$ is supported by a smooth pin $G$ and by rods $(1)$ and $(2),$ that are attached to the beam at points $B$ and $D,$ respectively. Both rods are identical with stiffness $k.$ The beam is horizontal when only rod $(1)$ is attached and $P=0.$ During fabrication, rod $(2)$ was made $$ shorter than needed, as illustrated in the figure below $($where kEFPCrod$(1)F_1{F}_2$rod you can assume that both rods have the same stiffness $k.$

$(a)$ After assembly $by$ forcing point $Eto$ fit into point $F,$ a vertical force $Pis$ applied $atCas$ indicated. Sketch a suitable free$-$body diagram and use the diagram $to$ obtain the relation between internal force $in$ rod$(1)F_1$ and the internal force $in$ rod $(2)F_2.(2$ points$)$

$(b)$ Sketch a suitable configuration $of$ the deformed structure and use $itto$ write one equation that relates the elongation $of$ rod$(1){}_1$ with the vertical displacement $of$ point $B,{}_B$; and one equation that relates the elongation $of$ rod $(2){}_2$ with the vertical displacement $of$ point $D{}_D,$ and another equation $to$ relate ${}_B$ and ${}_D.(4$ points$)$

$(c)$ Identify what $is$ missing $so$ far among the three components: $(1)$ equilibrium equation; $(2)$ constitutive relation; $(3)$ geometric constraints, and write down the equations with respect $to$ the missing component. $(2$ points$)$

$(d)$ Obtain $an$ expression for the strain $in$ rod $(1),lo{n}_1.$ Your answer should $be$ a function $ofP,,L$ and $k.(2$ points$)$

$(e)$ BONUS Knowing that the pin $atDisin$ double shear and $is$ made $of$ a material with ultimate shear stress ${}_u,$ determine the required diameter $of$ the pin. $(+3$ points$)$