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.