Let's start by addressing each question based on the provided image.

Question 1: Maximum Principal Stress

The given stress state involves normal stresses $\sigma_x = 100$ MPa, $\sigma_y = 60$ MPa, and a shear stress $\tau_{xy} = 48$ MPa. To determine the maximum principal stress, we use the principal stress formula:

$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left( \frac{\sigma_x - \sigma_y}{2} \right)^2 + \tau_{xy}^2}$

Step-by-Step Calculation:

1. Calculate the average normal stress: $\frac{\sigma_x + \sigma_y}{2} = \frac{100 + 60}{2} = 80 \, \text{MPa}$

2. Calculate the difference in normal stresses: $\frac{\sigma_x - \sigma_y}{2} = \frac{100 - 60}{2} = 20 \, \text{MPa}$

3. Calculate the principal stress difference: $\sigma_{1,2} = 80 \pm \sqrt{20^2 + 48^2} = 80 \pm \sqrt{400 + 2304} = 80 \pm \sqrt{2704} = 80 \pm 52 \, \text{MPa}$

Thus, the two principal stresses are:

$\sigma_1 = 80 + 52 = 132 \, \text{MPa} \quad \text{and} \quad \sigma_2 = 80 - 52 = 28 \, \text{MPa}$

The maximum principal stress is 132 MPa.

Answer for Question 1: 132 MPa

Question 2: Shear Stress After Rotation

To determine the shear stress after a counterclockwise rotation of $30^\circ$, we use the transformation equations for stresses, particularly the shear stress transformation equation:

$\tau' = \frac{\sigma_x - \sigma_y}{2} \sin(2\theta) + \tau_{xy} \cos(2\theta)$

where:

• $\theta = 30^\circ$
• $\sigma_x = 100 \, \text{MPa}$, $\sigma_y = 60 \, \text{MPa}$, $\tau_{xy} = 48 \, \text{MPa}$

Step-by-Step Calculation:

1. Calculate $\sin(2\theta)$ and $\cos(2\theta)$ for $\theta = 30^\circ$: $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \quad \text{and} \quad \cos(60^\circ) = 0.5$

2. Apply the values to the shear stress equation: $\tau' = \frac{100 - 60}{2} \times 0.866 + 48 \times 0.5$ $\tau' = 20 \times 0.866 + 24 = 17.32 + 24 = 41.32 \, \text{MPa}$

So, the shear stress after the rotation is approximately 41.3 MPa.

Answer for Question 2: 41.3 MPa

Do you want more details on any part of the calculations?

Here are 5 related questions that might interest you:

1. How do you calculate the minimum principal stress in a stress state?
2. What is the physical meaning of principal stresses?
3. How does the direction of rotation affect the stress transformation?
4. Can we always assume the same transformation formulas for any rotation angle?
5. How does shear stress vary with different angles of rotation?

Tip: Always double-check angle conventions (clockwise vs. counterclockwise) when applying stress transformation equations to avoid errors.