Chapter 7.2, Problem 64P

The Mohr's circle shown corresponds to the state of stress given in Fig. 7.5a and b. Noting that σ_x’ = OC + (CX') cos (2θ_p – 2θ) and that τ_x’y’ = (CX') sin (2θ_p – 2θ), derive the expressions for σ_x’ and τ_x’y’ given in Eqs. (7.5) and (7.6), respectively. [Hint: Use sin (A + B) = sin A cos B + cos A sin B and cos (A + B) = cos A cos B – sin A sin B.]

Fig. P7.64

Fig. 7.5 (repeated) State of plane stress: (a) referred to {xyz}, (b) referred to {x’y’z’}.