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Let's break down the problems and work through each one step by step. I will start with problem 1, and we can move sequentially.

### Problem 1:
*Consider the beam in the figure below. Determine the formulas for the shear force \( V(x) \) and bending moment \( M(x) \) functions for the beam.*

**Given:**
- The beam is supported at \( A \) (a pin support) and \( B \) (a roller support).
- Two concentrated loads \( P \) are applied symmetrically, each at \( \frac{L}{4} \) from the center.

#### Step 1: Determine reactions at supports
Since the loads are symmetric and the beam is statically determinate (with two support reactions), the vertical reaction at both supports \( A \) and \( B \) will be equal. The total load is \( 2P \), and it is evenly distributed between the two supports.

\[
R_A = R_B = \frac{2P}{2} = P
\]

#### Step 2: Shear force function \( V(x) \)
We will divide the beam into three sections for calculating the shear force:

1. **For \( 0 \leq x < \frac{L}{2} \)** (left side):
   \[
   V(x) = P \quad \text{(since there are no loads to the left of \( P \))}
   \]

2. **For \( \frac{L}{2} \leq x < \frac{3L}{4} \)** (between the loads):
   At \( x = \frac{L}{2} \), the shear force will drop by \( P \) because of the load:
   \[
   V(x) = P - P = 0
   \]

3. **For \( \frac{3L}{4} \leq x \leq L \)} (right side):
   At \( x = \frac{3L}{4} \), the shear force will drop again by \( P \) due to the second load:
   \[
   V(x) = -P
   \]

#### Step 3: Bending moment function \( M(x) \)
The bending moment can be calculated by integrating the shear force function or by using moment equilibrium for each segment.

1. **For \( 0 \leq x < \frac{L}{2} \)**:
   \[
   M(x) = P \cdot x
   \]

2. **For \( \frac{L}{2} \leq x < \frac{3L}{4} \)}:
   Between \( \frac{L}{2} \) and \( \frac{3L}{4} \), the moment decreases:
   \[
   M(x) = P \left( \frac{L}{4} \right) - P(x - \frac{L}{2})
   \]

3. **For \( \frac{3L}{4} \leq x \leq L \)}:
   In the final section, the moment further reduces.

---
### Problem 2:
*Consider the beam in the second figure. Determine the formulas for the shear force \( V(x) \) and bending moment \( M(x) \) functions for the beam.*

This is a beam with a uniformly distributed load of \( 4 \, \text{N/m} \) across a length of 8 meters, with an overhanging portion extending 2 meters beyond the right support.

We will analyze this problem by determining reactions at supports and computing the shear force and moment distributions in two segments: \( 0 \leq x \leq 8 \) (loaded part) and \( 8 \leq x \leq 10 \) (overhang).

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Would you like me to continue with the solution for Problem 2 or go into more detail for Problem 1? Also, feel free to ask for clarification at any point!

**Follow-up Questions:**
1. How does the shear force diagram relate to the bending moment diagram for these types of beams?
2. How would you use boundary conditions to find constants in the deflection curve for beams?
3. What is the significance of symmetry in calculating reactions in statics problems like these?
4. Could you explain the relationship between distributed loads and shear force variation?
5. What methods can be used to calculate deflection in beams subjected to loads?

**Tip:** For statically determinate beams, summing moments and forces at supports simplifies the process of finding reactions.

Determine the shear force and bending moment for beams with point loads and distributed loads, and solve related beam deflection problems.

Learn how to solve shear force and bending moment problems for beams with point and distributed loads. Explore beam deflection using differential equations and find the maximum slope and deflection. This content is suitable for undergraduate engineering students.

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