Question $1(100$ points$)$

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The beam shown is made from a material with the yielding strength of $S_y=240$MPa. The intensity of the distributed load is $w=25k\frac{N}{m}$ and the length designated by L is $2\mathrm{.}0$ m : Determine the smallest size $a$ of the cross section if a factor of safety of $n=1\mathrm{.}6$ is desired based on the maximum shear stress $($MSS$)$ theory.

Hints:

Determine the normal and shear stress at the critical points.

For each case find the principal stresses and using the MSS theory with the given factor of safety find a value for size $"$a$".$

Pick the appropriate value of $a.$