At a point in a strained material, the principle stresses are 100mpa and 180mpa both tensile, find the normal and shear stresses at a section inclined at 101 degree with the axis of the minor principle stress using both analytical and graphical methods.

To solve this problem, we will determine the normal stress (\(\sigma_\theta\)) and shear stress (\(\tau_\theta\)) on a plane inclined at 101° to the axis of the minor principal stress. The principal stresses are given as \(\sigma_1 = 180 \, \text{MPa}\) (major principal stress) and \(\sigma_2 = 100 \, \text{MPa}\) (minor principal stress). 

### Analytical Method:

We can use the stress transformation equations to find the normal and shear stresses on an inclined plane. The normal and shear stresses at an angle \(\theta\) (in this case, \(\theta = 101^\circ\)) are given by the following equations:

#### Normal Stress (\(\sigma_\theta\)):
\[
\sigma_\theta = \frac{\sigma_1 + \sigma_2}{2} + \frac{\sigma_1 - \sigma_2}{2} \cos(2\theta)
\]

#### Shear Stress (\(\tau_\theta\)):
\[
\tau_\theta = \frac{\sigma_1 - \sigma_2}{2} \sin(2\theta)
\]

Given that \(\sigma_1 = 180 \, \text{MPa}\), \(\sigma_2 = 100 \, \text{MPa}\), and \(\theta = 101^\circ\), we will now substitute these values into the equations.

#### Step 1: Calculate Normal Stress
\[
\sigma_\theta = \frac{180 + 100}{2} + \frac{180 - 100}{2} \cos(2 \times 101^\circ)
\]

#### Step 2: Calculate Shear Stress
\[
\tau_\theta = \frac{180 - 100}{2} \sin(2 \times 101^\circ)
\]

I'll calculate these values now.Using the analytical method, the stresses on the plane inclined at \(101^\circ\) with respect to the minor principal stress are:

- **Normal Stress (\(\sigma_\theta\))**: \(102.91 \, \text{MPa}\)
- **Shear Stress (\(\tau_\theta\))**: \(-14.98 \, \text{MPa}\)

### Graphical Method:

In the graphical method (Mohr's Circle), the following steps are performed:

1. **Construct the Circle**:
   - The center of Mohr's circle is located at \(\frac{\sigma_1 + \sigma_2}{2}\).
   - The radius of the circle is \(\frac{\sigma_1 - \sigma_2}{2}\).

2. **Determine the Angle**:
   - On Mohr's Circle, the angle \(2\theta\) corresponds to the angle from the principal stress direction to the inclined plane.
   - Plot the point on the circle corresponding to \(2\theta = 202^\circ\) (since the angle on Mohr's Circle is doubled).

From the Mohr’s Circle construction, you can determine the same values for the normal and shear stresses as calculated analytically.

Would you like more details on the graphical construction? Here are some related questions to explore further:

1. What is the significance of Mohr’s Circle in stress analysis?
2. How do normal and shear stresses change with the angle of inclination?
3. What happens to the stress components when the plane is parallel to the principal stress direction?
4. How would the results change if one of the principal stresses were compressive?
5. Can Mohr's Circle be used to find the principal stresses given stress components on an arbitrary plane?

**Tip:** Mohr’s Circle is not only useful for graphical stress analysis but also for visualizing the relationship between normal and shear stresses on different planes.

At a point in a strained material, the principal stresses are 100MPa and 180MPa both tensile, find the normal and shear stresses at a section inclined at 101 degrees with the axis of the minor principal stress using both analytical and graphical methods.

Solve for normal and shear stresses on a plane inclined at 101° to the axis of the minor principal stress. This solution involves stress transformation equations and the graphical method using Mohr's Circle. Given principal stresses of 100 MPa and 180 MPa, the problem is designed for engineering students.

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