Principal Stresses and Mohr's Circle — Worked Examples

A point is under $\sigma_x=80\text{ MPa}$ tension with $\sigma_y=0$ and $\tau_{xy}=0$. Determine the in-plane principal stresses and maximum in-plane shear stress.

A point has $\sigma_x=60\text{ MPa}$, $\sigma_y=20\text{ MPa}$, and $\tau_{xy}=0$. Determine the principal stresses.

A point is in pure shear with $\tau_{xy}=50\text{ MPa}$ and $\sigma_x=\sigma_y=0$. Determine the principal stresses.

Given $\sigma_x=50\text{ MPa}$, $\sigma_y=-10\text{ MPa}$, and $\tau_{xy}=40\text{ MPa}$, determine $\sigma_1$ and $\sigma_2$.

A point has $\sigma_x=80\text{ MPa}$, $\sigma_y=-40\text{ MPa}$, and $\tau_{xy}=25\text{ MPa}$. Determine principal stresses and maximum in-plane shear.

A point has $\sigma_x=100\text{ MPa}$, $\sigma_y=20\text{ MPa}$, and $\tau_{xy}=0$. Determine $\sigma_{x'}$ and $\tau_{x'y'}$ for $\theta=30^\circ$.

A point has $\sigma_x=80\text{ MPa}$, $\sigma_y=20\text{ MPa}$, and $\tau_{xy}=30\text{ MPa}$. Determine a principal-plane angle.

A point has $\sigma_x=120\text{ MPa}$, $\sigma_y=40\text{ MPa}$, and $\tau_{xy}=30\text{ MPa}$. Determine maximum in-plane shear and the normal stress acting on those planes.

The three principal stresses at a point are $\sigma_1=90\text{ MPa}$, $\sigma_2=30\text{ MPa}$, and $\sigma_3=0$. Compare the in-plane shear based on $\sigma_1$ and $\sigma_2$ with the absolute maximum shear.

An element is rotated physically by $15^\circ$. Through what angular magnitude is the corresponding radius rotated on Mohr's Circle?

For $\sigma_x=100\text{ MPa}$, $\sigma_y=20\text{ MPa}$, and $\tau_{xy}=0$, a $30^\circ$ transformation gives $\sigma_{x'}=80\text{ MPa}$. Determine $\sigma_{y'}$ using the invariant sum of normal stresses.