Q$1.$ An idealized two$-$pier bridge modeled as a shear frame is shown below. The piers are separated by a $10$ cm gap from a fully rigid body $($with no relative displacement within the body$).$ Use the TBDY$2018$ design spectrum values for Soil Class ZD $($Ss $=1\mathrm{.}296$g$,$ S$1=0\mathrm{.}361$g$,$ Fs$=1\mathrm{.}0,$ F$1=1\mathrm{.}939,$ Sds $=1\mathrm{.}296$g$,$ Sd$1=0\mathrm{.}70$ g $)$ to answer the following questions

i$)$ Calculate the maximum displacement of the frame. Is the $10$ cm gap sufficient to prevent contact? Explain your reasoning.

ii$)$ Determine the maximum shear force in each pier of the bridge.

iii$)$ Determine the maximum bending moment in each pier of the bridge.

Given Data: Mass $\backslash (=\backslash \{\backslash$mathbf$\{200\}\backslash \}\backslash )$ tons, Elasticity modulus, $\backslash (\backslash$mathrm$\{$E$\}=3\mathrm{.}0\backslash$mathrm$\{$e$\}7\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ),$ Pier heights: $2$ H and H where $\backslash (\backslash$mathrm$\{$H$\}=5\backslash )$ and Piers' Diameter: $\backslash (\backslash \{65\backslash \}\backslash )$ cm