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The task presented is a structural analysis problem for a beam with given point loads \( F_1 \), \( F_2 \), \( F_3 \), \( F_4 \), and \( F_5 \) and a supported span from \( A \) to \( B \). The goal is to calculate the support reactions at points \( A \) and \( B \) for the following cases:

1. **Given loads**: Calculate reactions at supports \( A \) and \( B \) with the given point loads.
2. **Remove load \( F_5 \)**: Calculate reactions at supports \( A \) and \( B \) when the point load \( F_5 \) is removed.
3. **Reduce \( F_1 \) and \( F_2 \) by 10%**: Calculate reactions at supports \( A \) and \( B \) when the point loads \( F_1 \) and \( F_2 \) are reduced by 10%.

### Input dataThe task presented is a structural analysis problem for a beam with given point loads \( F_1 \), \( F_2 \), \( F_3 \), \( F_4 \), and \( F_5 \) and a supported span from \( A \) to \( B \). The goal is to calculate the support reactions at points \( A \) and \( B \) for the following cases:

1. **Given loads**: Calculate reactions at supports \( A \) and \( B \) with the given point loads.
2. **Remove load \( F_5 \)**: Calculate reactions at supports \( A \) and \( B \) when the point load \( F_5 \) is removed.
3. **Reduce \( F_1 \) and \( F_2 \) by 10%**: Calculate reactions at supports \( A \) and \( B \) when the point loads \( F_1 \) and \( F_2 \) are reduced by 10%.

### Input 

- \( F_1 = 2.5 \, \text{kN} \) at 1.00 m from \( A \)
- \( F_2 = 1.8 \, \text{kN} \) at 3.50 m from \( A \) (1.00 m + 2.50 m)
- \( F_3 = 3.0 \, \text{kN} \) at 6.10 m from \( A \) (1.00 m + 2.50 m + 2.60 m)
- \( F_4 = 2.0 \, \text{kN} \) at 8.60 m from \( A \) (1.00 m + 2.50 m + 2.60 m + 2.50 m)
- \( F_5 = 4.0 \, \text{kN} \) at 9.60 m from \( A \) (1.00 m + 2.50 m + 2.60 m + 2.50 m + 1.00 m)

The beam is simply supported, with reaction forces \( A_y \) and \( B_y \) at supports \( A \) and \( B \), respectively.

### Step-by-step process:
We will use static equilibrium conditions for the beam:
- **Sum of vertical forces (force equilibrium)**:
  \[
  A_y + B_y = F_1 + F_2 + F_3 + F_4 + F_5
  \]
- **Sum of moments about one support (moment equilibrium)**:
  Taking moments about \( A \):
  \[
  \sum M_A = 0: \quad B_y \times 9.60 = F_1 \times 1.00 + F_2 \times 3.50 + F_3 \times 6.10 + F_4 \times 8.60 + F_5 \times 9.60
  \]

Would you like to proceed with calculating these values for part (a), or do you have any specific preference on where to start?

Feel free to ask any questions, and I can also provide further clarification on the calculations! 

Here are 5 related questions for further exploration:
1. How do we calculate the bending moment at any point along the beam?
2. What happens to the reaction forces when a load is removed?
3. How is shear force affected by the changes in the loading conditions?
4. What is the influence of reducing both \( F_1 \) and \( F_2 \) by 10% on the overall deflection of the beam?
5. What methods can be used to verify the support reactions in complex beam structures?

**Tip**: When solving beam problems, always ensure that the static equilibrium conditions are satisfied (sum of forces and moments).

Berechnen Sie zum Einfeldträger 3.16 die Auflagerkräfte A und B: a) bei der vorgegebenen Belastung, b) bei Wegfall der Einzellast F_5, c) bei Verminderung der Einzellasten F_1 und F_2 um 10%.

Learn how to calculate the reactions at supports A and B for a simply supported beam with various point loads. The problem covers static equilibrium equations and involves scenarios such as given loading, removal of load F_5, and reduction of loads F_1 and F_2 by 10%.

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