$(5$ points$)$ A building frame with four identical columns that have an effective stiffness of $k$ and a rigid

floor of mass $m$ is shown in Figure $3.$ The natural period of vibration of the frame in the horizontal

direction is found to be $0\mathrm{.}45$ s $.$ When a heavy machine of mass $500$ kg is mounted $($clamped$)$ on the

floor, its natural period of vibration in the horizontal direction is found to be $0\mathrm{.}55$ s $.$ Determine the

effective stiffness $k$ and mass $m$ of the building frame.