Figure II$-2$ shows a beam with square cross sections. $(1)$ Draw the bending moment diagram. $(2)$ If the allowable tensile stress of the material iron is $[]=3{10}^5\frac{q}{a},$ and the allowable compressive stress is $[{}_c]=5{10}^5\frac{q}{a},$ check the beam safety according to the strength condition. $(3)$ If the size of the cross section can be adjusted, determine the minimum width $b$ of the square cross section in order to avoid strength failure.

Fig. II$-2$