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.