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This image shows a semi-circular arc with a radius of 5 m and a tension force in cable \( BD \) to be determined. The arc is supported by a 7.5 m long cable, and there is an external force of 18 kN acting vertically downward at point \( B \) at the top of the arc. The arc spans horizontally 10 m between points \( A \) and \( D \), with point \( C \) being the center of the circle.

We are tasked with determining the tension in cable \( BD \).

### Known Data:
- Radius of the semi-circular arc, \( r = 5 \, \text{m} \).
- Length of cable \( BD = 7.5 \, \text{m} \).
- Force acting vertically downward at point \( B \), \( F_B = 18 \, \text{kN} \).
- Horizontal distance between points \( A \) and \( D \), \( AD = 10 \, \text{m} \).

### Approach:
1. **Force Equilibrium**: We will apply static equilibrium conditions. The forces in the horizontal and vertical directions should sum to zero.
2. **Geometry**: We'll use trigonometry to resolve the geometry, determining angles and components of the forces.
3. **Tension in Cable \( BD \)**: The tension force can be determined by resolving forces along the cable and considering the equilibrium of point \( B \).

Let me proceed with the calculations.The tension in cable \( BD \) is approximately \( 25.46 \, \text{kN} \).

Would you like a detailed explanation of the steps, or do you have any further questions?

Here are 5 related questions to expand on the topic:
1. How does changing the length of cable \( BD \) affect the tension in it?
2. What would happen to the tension if the force at point \( B \) increased to 25 kN?
3. Can we calculate the horizontal and vertical components of the tension force in cable \( BD \)?
4. How does the radius of the semi-circle impact the tension in the cable?
5. How would we calculate the reaction forces at points \( A \) and \( D \)?

**Tip**: When analyzing forces in structures, always check if the system is in equilibrium by summing forces and moments in both horizontal and vertical directions.

For the semi-circular arc shown in Figure 14.1 with radius of 5 m, supported by a 7.5 m long cable, determine the tension in cable BD.

Solve for the tension in cable BD for a semi-circular arc supported by a 7.5 m cable with a downward force of 18 kN at the top. Learn to apply static equilibrium and trigonometry to calculate the tension. Ideal for engineering students studying statics.

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