$1.)$ Use step$-$by$-$step limit analysis to analyze the structure shown in Fig. $1$ and determine:

The $P-$ relationship curve.

The $-$ relationship curve of each plastic hinge.

The plastic hinge rotation demand $({}_p{)}_{\text{d}\text{e}\text{m}\text{a}\text{n}\text{d}in}$ each plastic hinge to allow the beam

to develop ductility ratio $\frac{u}{}=5.$

The location of the inflection point from point $A$ at each analysis step.

The moment envelop curve along the beam span.

$H70030013{24}_{\text{:}}{I}_x=2\mathrm{.}01{10}^{9}m{m}^4$;$Z_x=6\mathrm{.}46{10}^{6}m{m}^4$;$F_Y=345$MPa;

$E=200$GPa.

Note: Neglect self$-$weight of the beam. $(50\%)$

Fig. $1$ Fixed$-$end beam subjected to concentrated load