$2.-$ A shear stress ${}_{xy}$ acts in a two$-$dimensional field in which the maximum allowable

shear stress is denoted by ${}_{\text{m}\text{a}\text{x}}$ and the major principal stress by ${}_{\text{I}}.$

Derive, using the geometry of Mohr's circle of stress, expressions for the maximum

values of direct stress which may be applied to the $x$ and $y$ planes in terms of the three

parameters given above.

Hint: ${}_x=O{P}_1=OB-BC+C{P}_1$

Ans.

${}_x={}_{\text{I}}-{}_{\text{m}\text{a}\text{x}}+\sqrt[\phantom{2}]{{}_{\text{m}\text{a}\text{x}}^2-{}_{xy}^2}$

${}_y={}_{\text{I}}-{}_{\text{m}\text{a}\text{x}}-\sqrt[\phantom{2}]{{}_{\text{m}\text{a}\text{x}}^2-{}_{xy}^2}$