Question

Two equivalent descriptions of the state of stress at a point are shown in the figure. The normal stresses $\sigma_1$ and $\sigma_2$ as shown on the right must be, respectively,

• A. $\tau_0$ and $-\tau_0$
• B. $-\tau_0$ and $\tau_0$
• C. $\dfrac{\tau_0}{\sqrt{2}}$ and $-\dfrac{\tau_0}{\sqrt{2}}$
• D. $-\dfrac{\tau_0}{\sqrt{2}}$ and $\dfrac{\tau_0}{\sqrt{2}}$

Hint:

For pure shear, the principal stresses are equal in magnitude but opposite in sign. The orientation of principal planes is always $45^\circ$ from the shear planes.

Answer 1

The Correct Option is A

Step 1: Interpret the stress state.
On the left diagram, the point is subjected to pure shear stress $\tau_0$ (positive on top surface, negative on side surface). This is the classical case of a pure shear stress element.
Step 2: Principal stresses in pure shear.
For pure shear, the principal stresses are: \[ \sigma_1 = +\tau_0, \sigma_2 = -\tau_0 \] The directions of these principal stresses are rotated by $45^\circ$ relative to the original coordinate axes.
Step 3: Match with figure on the right.
On the right-hand rotated square (diamond orientation), the stresses are shown along directions rotated $45^\circ$ from the original axes. This corresponds to the principal axes. Therefore, \[ \sigma_1 = \tau_0, \sigma_2 = -\tau_0 \] Final Answer: \[ \boxed{\tau_0 \text{ and } -\tau_0} \]