To solve this problem, we need to analyze the forces acting on the semi-circular arc and determine the tension in cable BD. Here's a step-by-step approach:

Given:

• Radius of the semi-circular arc, $R = 5 \, \text{m}$
• Length of cable BD, $L = 7.5 \, \text{m}$
• Distance between points A and D, $AD = 10.0 \, \text{m}$
• Force acting at point B, $F = 18 \, \text{kN}$

Assumptions:

• The arc is symmetrical.
• The tension in the cable is uniform.
• The system is in static equilibrium.

Solution:

1. Identify Points and Angles:

   • Points A and D are at the base of the semi-circular arc.
   • Point B is at the top of the arc.
   • Point C is where the cable BD meets the arc.
   • The angle subtended by the arc (from A to D) is 180 degrees.

2. Analyze Geometry:

   • The semi-circular arc has a radius of 5 m. The distance from B to the arc's center is also 5 m.
   • Since $AD = 10 \, \text{m}$, the center of the arc is at the midpoint between A and D.
   • The length of the cable BD is 7.5 m.

3. Calculate the Angles:

   • Use the Law of Cosines in triangle BCD to find the angle $\theta$ between BD and the vertical.

   $\cos \theta = \frac{BC^2 + CD^2 - BD^2}{2 \cdot BC \cdot CD}$ Where:

   • $BC = 5 \, \text{m}$ (radius)
   • $CD = 5 \, \text{m}$ (radius)
   • $BD = 7.5 \, \text{m}$

4. Set Up Equilibrium Equations:

   • Resolve the forces vertically and horizontally:
   • Vertical forces: $T \sin \theta = 18 \, \text{kN}$
   • Horizontal forces: $T \cos \theta = \text{horizontal component of the reaction at A and D}$

5. Solve for Tension $T$:

Let's calculate these values now.The tension $T$ in cable BD is approximately 18.14 kN. The angle $\theta$ between cable BD and the vertical is about 97.18 degrees.

This tension accounts for balancing the vertical force of 18 kN applied at point B while considering the geometry of the arc and cable.

Would you like more details, or do you have any questions?

Here are 5 related questions that might help expand your understanding:

1. How would the tension change if the length of cable BD was different?
2. What would happen to the equilibrium if the force at B was not vertical?
3. How can we compute the horizontal components of forces at points A and D?
4. What would be the impact on tension if the arc radius changed?
5. How do we use vector resolution for complex force systems like this?

Tip: When solving problems involving tension and angles, always consider the geometry and use trigonometric identities for accurate calculations.