As shown in Figure $2,$ we will use a simple beam structure using H$-$beam. From the left point A $($x$=0)$ in the horizontal direction $($x$-$direction$),$ a force of $1,000$ kN is applied in the vertical down direction at x $=5,10,$ and $15$ m positions. Refer to Figure $2$ for the bending moment distribution diagram caused by external force$\mathrm{.}1)$ If the width of the H$-$beam cross$-$section is b $=600$ mm$,$ the plate thickness is t $=60$ mm$,$ and the depth is h$,$ express the cross$-$sectional secondary moment $($Ib$)$ as an equation for h$.$

$2)$ Based on the bending moment distribution diagram provided in Fig. $2,$ find the distribution diagram by position in the horizontal direction of the maximum axial stress acting on the tensile edge of the H$-$beam cross section

$3)$ It is intended to be designed so that the maximum axial stress of the tensile podium is $200$ MPa or less. Find the required depth $($hb $)$ and the cross$-$sectional area $($Ab$)$ of the H$-$beam section.

$4)$ When the depth of all cross sections is unified to the required depth $($hb$)$ obtained above $($i$.$e$.,$ using a pristine beam$),$ obtain the total amount of material $($volume$,$ Vb$)$ required for the construction of the H$-$beam bridge.