Q$2.$ Figure $2$ shows a structural system consisting of two vertical bars AB and DE $,$ and a horizontal beam AD $.$ All structural members have an identical circular cross section with a diameter of $d$ and a material Young's modulus of $E,$ and the critical slenderness ratios are ${}_5=15,$ and ${}_P=40.$ A vertical load of $P=3E\frac{d^2}{800}$ is applied at C $.$ The yield strength of the column material is ${}_s=\frac{E}{100},$ and the corresponding factor of safety is $n_s=2.$ The factor of safety associated with buckling is ${}_{st}=5.$ The lengths of all structural members are measured using $l=10d.$

$(1)$ Calculate the axial forces in AB and DE $,$ and draw the bending moment diagram of AD $.$

$(2)$ Check the safety of each structural member.

$(3)$ Determine the allowable applied load $P.$