A frame is loaded with two proportional loads, as shown in Fig. $1$a$.$ The relative plastic moment values in different frame members are shown in the figure, with Mp $=86$ kNm$.$

$(1)$ For the frame shown in Fig. $1$a$,$ find the value of the load factor $\backslash$lambda at collapse and prove that the yield condition is satisfied for the corresponding collapse mechanism.

$(2)$ If boundary conditions at A and D are fixed supports, as shown in Fig. $1$b$,$ find the value of the load factor $\backslash$lambda at collapse.

$(3)$ If plastic moment of beam BC is $0\mathrm{.}5$Mp$,$ as shown in Fig. $1$c$,$ find the value of the load factor $\backslash$lambda at collapse.

$(4)$ From the results obtained from $(1)-(3),$ discuss on the influence of load, boundary conditions and section sizes on the collapse load and mechanism of a frame.