The initial stresses at a certain depth in a clay soil is $100$ kPa on the horizontal plane and $50$ kPa on the vertical plane. Construction of a surface foundation induces an additional stress consisting of a vertical stress of $45$ kPa, a lateral stress of $20$ kPa, and a counterclockwise shear stress of $40$ kPa on the vertical plane. $($a$)$ Plot Mohr$$s circle for the initial stresses; $($b$)$ Plot Mohr$$s circle after the construction of the foundation; $($c$)$ Determine the magnitude of change of principal stresses; $($d$)$ Determine the magnitude of change of maximum shear; $($e$)$ Determine the change in orientation of the principal planes resulting from the foundation construction.