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To solve for the forces in the cables \(AB\), \(BC\), \(BD\), and \(DE\) in this structure, we need to apply **static equilibrium equations** to the joints where multiple cables meet.

Here’s how we approach the problem:

1. **Equilibrium Conditions**: For each joint, the sum of forces in both the horizontal and vertical directions must equal zero because the system is in equilibrium.

   \[
   \sum F_x = 0 \quad \text{(sum of horizontal forces)}
   \]
   \[
   \sum F_y = 0 \quad \text{(sum of vertical forces)}
   \]

2. **Joint Analysis**: Start with joints where fewer cables are attached, as this will simplify solving for the forces step-by-step.

### Step-by-Step Breakdown:

- **Joint D**: There’s a 100 kN vertical force applied at D, and cables \(BD\) and \(DE\) are connected to it. The angle between cable \(DE\) and the horizontal is 60°. We will resolve the forces into components here.
  
  Let:
  - \(T_{BD}\) be the tension in cable \(BD\)
  - \(T_{DE}\) be the tension in cable \(DE\)

  Resolve the forces in the x and y directions:
  - Vertical equilibrium at D: 
    \[
    T_{DE} \sin 60^\circ = 100 \, \text{kN}
    \]
  - Horizontal equilibrium at D: 
    \[
    T_{BD} = T_{DE} \cos 60^\circ
    \]

- **Joint B**: At B, there are three cables attached: \(AB\), \(BC\), and \(BD\), forming a triangle with given side lengths (3-4-5 triangle).

  We know the geometrical relations from the 3-4-5 triangle (a common Pythagorean triplet) that will help us determine the angles. Then, resolve the forces into components.

Let me first solve for the force in \(T_{DE}\) and \(T_{BD}\) using the equilibrium equations at joint D.

I will compute that now.The forces in the cables at joint D are:

- \( T_{DE} = 115.47 \, \text{kN} \)
- \( T_{BD} = 57.74 \, \text{kN} \)

Now, let's proceed to joint B, where we have cables \(AB\), \(BC\), and \(BD\) forming a triangle. The next step will involve resolving the forces at this joint using the triangle's geometry.

Would you like to continue with the analysis of joint B or need further details on these results? Let me know if you have any specific questions.

Here are some related questions:
1. How are forces resolved into horizontal and vertical components?
2. What is the significance of the angles between cables in determining tension forces?
3. How does a 3-4-5 triangle help simplify angle calculations in trusses?
4. How do we ensure equilibrium at each joint in a truss structure?
5. What are the differences in force distribution when external loads are applied?

**Tip:** When analyzing trusses, always start with the joint that has the fewest unknowns.

Analyze and solve for the forces in cables AB, BC, BD, and DE using static equilibrium principles. This detailed example involves force resolution at different joints in a truss system with angles and external loads. Suitable for undergraduate students studying statics and structural analysis.

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