To determine the magnitude of the internal bending moment at point $C$ in the beam, we will follow these steps:

Recap of the Situation:

• The beam is subjected to a uniformly distributed load (UDL) $w = 38 \, \text{kN/m}$ over a length of $2 \, \text{m}$.
• The beam is simply supported at points $A$ and $B$, with point $C$ at the midpoint between $A$ and $B$.
• The distances between points $A$, $C$, and $B$ are each $2 \, \text{m}$.

Steps to Calculate the Bending Moment at Point $C$:

1. Calculate the reaction forces at the supports: Since the beam and loading are symmetric, the reactions at $A$ and $B$ are each $38 \, \text{kN}$.

2. Calculate the bending moment at point $C$: To find the bending moment at point $C$, we can consider the left section of the beam from $A$ to $C$.

   • The reaction force at $A$ is $38 \, \text{kN}$.
   • The distance from $A$ to $C$ is $2 \, \text{m}$.
   • The UDL over this section creates a uniformly distributed load with a total force of $76 \, \text{kN}$ (acting over $2 \, \text{m}$).

   The bending moment at point $C$ due to the reaction force at $A$ and the UDL can be calculated as follows:

   Moment due to the reaction force at $A$: $M_C = R_A \times \text{distance from } A \text{ to } C = 38 \, \text{kN} \times 2 \, \text{m} = 76 \, \text{kN} \cdot \text{m}$

   Moment due to the UDL between $A$ and $C$: The UDL creates a load of $76 \, \text{kN}$, which acts at the midpoint of the loaded section, i.e., $1 \, \text{m}$ from $A$. $M_C = - \left( w \times \text{length} \times \frac{\text{distance from } A \text{ to the UDL center}}{2} \right)$ $M_C = - \left( 38 \, \text{kN/m} \times 2 \, \text{m} \times 1 \, \text{m} \right) = - 76 \, \text{kN} \cdot \text{m}$

3. Total Bending Moment at Point $C$: The total bending moment at point $C$ is the sum of the moments due to the reaction force and the UDL: $M_C = 76 \, \text{kN} \cdot \text{m} - 76 \, \text{kN} \cdot \text{m} = 0 \, \text{kN} \cdot \text{m}$

Conclusion:

The magnitude of the internal bending moment at point $C$ in the beam is $0 \, \text{kN} \cdot \text{m}$.

Would you like to explore more on this topic or have any other questions?

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Here are 5 related questions to consider:

1. What would be the bending moment at the midpoint between $C$ and $B$?
2. How would the bending moment change if the UDL were extended over the entire length of the beam?
3. What is the significance of a zero bending moment at a particular point in a beam?
4. How would the bending moment distribution look for a beam with different lengths between supports?
5. Can the shear force at a point in a beam be zero while the bending moment is non-zero? Explain.

Tip: The bending moment is often highest where the shear force changes sign. Analyzing the shear force distribution can provide insight into potential points of maximum bending moment.