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The problem on the board asks for the design of the number of 20 mm diameter bars to adequately support a given load with certain parameters. Let's break down and analyze the details:

### Given Information:
1. **Design the number of 20 mm bars** for adequate support.
2. **Ultimate loads** are to be supported.
3. **\( f_c' = 28 \, \text{MPa} \)** (concrete strength).
4. **\( f_y = 414 \, \text{MPa} \)** (steel yield strength).
5. **\( \phi_s = 10 \, \text{mm} \)** (stirrup diameter).
6. **Clear cover (CC) = 40 \, \text{mm}**.
7. There is a **50 kN point load**.
8. Length and spacing details are drawn: 
   - Lengths: 2 m, 4 m.
   - UDL (uniformly distributed load) of **40 kN/m** on the 4 m section.

### Steps to Proceed:
1. **Determine the total load** acting on the beam:
   - The total load includes the **50 kN point load** and the **UDL of 40 kN/m** over the 4 m length.
   \[
   W_{\text{total}} = 50 \, \text{kN} + (40 \, \text{kN/m} \times 4 \, \text{m}) = 50 \, \text{kN} + 160 \, \text{kN} = 210 \, \text{kN}
   \]

2. **Shear and moment analysis**:
   - Given the placement of the point load and UDL, we can calculate the shear forces and bending moments in the beam to ensure the design is safe.

3. **Reinforcement design**:
   - We need to compute the number of steel bars required using the formula for the area of steel \( A_s \), which can be derived from the moment capacity and axial loads.

4. **Cross-section details**:
   - From the drawing, it appears the section is 300 mm wide by 400 mm deep.

5. **Safety and strength checks**:
   - Using the given strengths \( f_c' \) and \( f_y \), we can proceed with checking the concrete and steel design limits.

Do you want a detailed breakdown of any specific part of this problem (e.g., shear force analysis, moment analysis, or reinforcement calculation)?

### Next steps to expand on:
1. How do we calculate the required reinforcement area \( A_s \) from the given loads?
2. What is the method to compute bending moment for this beam?
3. How do we apply the load distribution to get the internal forces?
4. How does the clear cover and bar diameter influence the spacing and number of bars?
5. What is the relation between concrete strength and steel reinforcement in beam design?

**Tip:** Always ensure the bar spacing and placement provide enough support against shear and bending forces to avoid failure.

Design the number of 20 mm bars to adequately support the given ultimate loads. Use f'c = 28 MPa, fy = 414 MPa, φs = 10 mm, and CC = 40 mm. Pa = 50 kN. Beam lengths are 2 m and 4 m, with a 40 kN/m UDL on the 4 m section.

This problem involves calculating the required number of 20 mm diameter reinforcement bars to support a total load of 210 kN, combining point and uniformly distributed loads. Using concrete strength f'c = 28 MPa and steel yield strength fy = 414 MPa, the solution involves shear force and moment analysis for proper structural design.

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