Assume a simply suppported beam spanning 10 metres.  If the beam was subjected to a moment of 100 kNm about its x axis, what normal compressive stress would be induced at a distance of 50mm above the neutral axis?  

To determine the normal compressive stress induced in a simply supported beam subjected to a moment, we use the flexural formula:

\[
\sigma = \frac{M \cdot y}{I}
\]

Where:
- \(\sigma\) is the normal stress at a given point (in Pa or N/m²).
- \(M\) is the bending moment (in N·m or kN·m).
- \(y\) is the distance from the neutral axis to the point where the stress is being calculated (in meters).
- \(I\) is the second moment of area (in m\(^4\)).

Given:
- \(M = 100 \, \text{kN·m} = 100 \times 10^3 \, \text{N·m}\)
- \(y = 50 \, \text{mm} = 50 \times 10^{-3} \, \text{m}\)

To find the normal compressive stress, we also need the second moment of area \(I\), which depends on the cross-sectional shape of the beam. However, since the shape is not provided, we cannot calculate the exact value of \(\sigma\) without this information. 

### General Approach:

If the cross-sectional shape is, for example, a rectangle, the second moment of area \(I\) for a rectangle is calculated as:

\[
I = \frac{b \cdot h^3}{12}
\]

Where \(b\) is the width and \(h\) is the height of the rectangular cross-section.

### Substituting in the Values:

For a given \(I\), you would substitute into the equation:

\[
\sigma = \frac{100 \times 10^3 \times 50 \times 10^{-3}}{I}
\]

This formula provides the stress in N/m². Once \(I\) is known, you can compute \(\sigma\).

If you know the beam's cross-sectional dimensions, I can help calculate \(I\) and find the exact compressive stress.

Would you like to provide the cross-sectional dimensions or need further details on any of the steps?

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**Relative Questions:**
1. What is the significance of the neutral axis in beam theory?
2. How does the shape of the beam's cross-section affect the second moment of area?
3. What would be the normal stress if the distance \(y\) was doubled?
4. How does the bending moment distribution vary along the length of a simply supported beam?
5. What assumptions are typically made in deriving the flexural formula?

**Tip:** In problems involving bending stress, always verify the cross-sectional shape of the beam to calculate the second moment of area accurately.

Learn how to calculate the normal compressive stress induced in a simply supported beam subjected to a bending moment of 100 kNm at a distance of 50mm above the neutral axis. Understand the mechanics of materials, beam theory, and the flexural formula used in structural analysis. Suitable for engineering students and professionals.

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