Consider a nine$-$storey building whose lateral resisting system comprises four columns. Assume shear building behaviour. The mass and stiffness properties of the building are given in the table below.

a$)$ Using Rayleigh method, estimate the natural frequency $($in Hz $)$ of the first mode. Assume a square root function for the trial mode shape, i$.$e$.,$ phi$($x$)=$sqrt$($x$//$H$)$ where x is the height from the ground and H is the total height of the building. Perform calculations in a new Excel worksheet.

b$)$ Using a first mode approximation with $2\%$ damping and the trial mode shape in part $($a$),$ estimate the shear forces and bending moments in all storeys of Column $1$ when subjected to ground motion XYZ$.$ You may make use of the Excel spreadsheet developed in Question $1.$

Floor Mass $[$tonnes$]$ Storey Height $[$m$]$ Stiffness $[$kN$/$mm$]$ Column $1$ Column $2$ Column $3$ Column $41130013\mathrm{.}54004004004002130023\mathrm{.}54004004004003130033\mathrm{.}54004004004004115043\mathrm{.}32703203202505115053\mathrm{.}32703203202506115063\mathrm{.}32703203202507100073\mathrm{.}32603003002608100083\mathrm{.}32603003002609100093\mathrm{.}3260300300260$