A single$-$story warehouse $($labeled as "original structure" below$)$ has been converted to an artist's loft after installing a mezzanine level $($see "renovated structure" shown below$).$ Assume that both the roof slab $($mass $5M)$ and mezzanine level slab $($mass $M)$ are rigid and are the only elements with substantial mass in the structure. Consider response in flexure only; ignore axial and shear deformations in the columns. Note that the mezzanine slab is connected to the columns rigidly, i$.$e$.$ it prevents them from rotating at th$\text{'}$ me:'zanine slab level. The relative values of modulus of elasticity for the steel columns are shown on the sketc'i. Note that the new column that has modulus of clasticity I has a pin$-$support at its ground level end.

a$.$ Find the natural periods of the renovated structive. $($Do not use Rayleigh's quotient approach.$)$ ilow much has the fundamental mode per id changed from the original structure?

b$.$ Find and sketch the modeshapes of the renovated structure.