Problem $1(25$ Points$)$

Link $\backslash (2(\backslash$overline$\{$O B$\})\backslash )$ in the figure, is $25$ mm wide and $12$ mm thick, and is cut from lowcarbon steel bar stock having a minimum yield strength of $160$ MPa and modulus of elasticity of $200$ GPa $.$ A concentrated force $\backslash (\backslash$mathrm$\{$F$\}=1000\backslash$mathrm$\{$~N$\}\backslash )$ is applied at point $\backslash ($ C $\backslash )$ on the link $3,$ as shown. The end$-$condition constant of the link $2$ is $\backslash ($ C$=1\mathrm{.}2\backslash )$ for both buckling in and out of the plane of the drawing.

a$)$ Draw the free body diagrams of links $2$ and $3,$ and based on static equilibrium, find the effective compressive load in the Link $\backslash (2(\backslash$overline$\{$O B$\})\backslash ).$ Notice that the two links are hinged together and at simply supported at their supports. $(3$ points$)$

b$)$ Determine the maximum compressive load in the link $2$ with $(2$ points$)$

c$)$ Considering a a design factor of $\backslash ($ n$\_\{$d$\}=4\backslash ),$ determine the critical in$-$plane buckling load, using the Johnson$/$Euler formulation $(10$ points$)$

d$)$ With design factor of $\backslash ($ n$\_\{$d$\}=4\backslash ),$ calculate the critical out$-$of$-$plane buckling load, using the Johnson$/$Euler formulation. $(10$ points$)$