Stress along different direction frtom axis direction

[Please insert the figure later, for better explanation]

 

<Normal stress>

$$\sigma_{x}=\frac{P}{A_{x}}$$

$$\sigma_{x}^{\prime}=\frac{P\cos\theta}{A_{x}^{\prime}}$$

$$A_{x}^{\prime}=\frac{A_{x}}{\cos\theta}$$

$$\sigma_{x}^{\prime}=\frac{P\cos\theta}{1}\times\frac{\cos\theta}{A_{x}}=\frac{P}{A_{x}}\cos^{2}\theta=\sigma_{x}\cos^{2}\theta$$

$$\sigma_{x}^{\prime}=\sigma_{x}\cos^{2}\theta$$

 

<Shear stress>

$$\tau_{x^{\prime}y^{\prime}}=\frac{1}{A_{x}^{\prime}}\times(-P\sin\theta)$$

$$\tau_{x^{\prime}y^{\prime}}=\frac{\cos\theta}{A_{x}}(-P\cos\theta)$$

$$\tau_{x^{\prime}y^{\prime}}=-\frac{P}{A_{x}}\cos\theta\sin\theta$$

$$\tau_{x^{\prime}y^{\prime}}=-\frac{P}{A_{x}}\cos\theta\sin\theta$$

$$\tau_{x^{\prime}y^{\prime}}=-\sigma_{x}\cos\theta\sin\theta$$