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The problem involves designing the dimensions of a simply supported steel beam subjected to a uniformly distributed live load. Here's a summary of the problem details:

- **Length of the beam (L):** 6.00 meters
- **Uniformly distributed load (w):** 2900 kg/m
- **Maximum allowable stress in steel (σc):** 200 kg/cm²
- **Yield stress of steel (fy):** 4200 kg/cm²
- **Steel percentage (p):** \( p = p_{\text{max}} \) (Maximum steel percentage)

### Steps to solve the problem:
1. **Determine the maximum moment (Mmax):**
 For a simply supported beam with a uniformly distributed load, the maximum moment occurs at the center and is given by:
 \[
 M_{\text{max}} = \frac{w L^2}{8}
 \]
 where \( w \) is the load per unit length, and \( L \) is the span of the beam.

2. **Calculate the required section modulus (S):**
 The section modulus is related to the maximum moment and the allowable stress by:
 \[
 S = \frac{M_{\text{max}}}{\sigma_c}
 \]
 where \( \sigma_c \) is the allowable stress in the material.

3. **Select or design a beam cross-section that provides the required section modulus.**

Let me calculate these values step by step for you.

### Step 1: Calculate Maximum Moment (Mmax)
Given:
- \( w = 2900 \text{ kg/m} \)
- \( L = 6.00 \text{ m} \)

\[
M_{\text{max}} = \frac{2900 \times 6.00^2}{8}
\]

Let me calculate that.The maximum moment \( M_{\text{max}} \) is **13,050 kg·m**.

### Step 2: Calculate the Required Section Modulus (S)

The section modulus \( S \) can be calculated using the formula:

\[
S = \frac{M_{\text{max}}}{\sigma_c}
\]

where:
- \( M_{\text{max}} = 13,050 \text{ kg·m} \)
- \( \sigma_c = 200 \text{ kg/cm}^2 \)

Since \( M_{\text{max}} \) is in kg·m, we need to convert it to kg·cm:

\[
M_{\text{max}} = 13,050 \times 100 = 1,305,000 \text{ kg·cm}
\]

Now, let's calculate the section modulus \( S \).The required section modulus \( S \) is **6525 cm³**.

### Next Steps:
1. **Select a steel section**: Choose a standard steel section (I-beam, H-beam, etc.) that has a section modulus equal to or greater than 6525 cm³.
2. **Verify other criteria**: Ensure that the chosen section meets other design criteria, such as deflection limits, shear strength, and stability.
3. **Check steel percentage**: Given the maximum percentage of steel, verify that the selected section satisfies this condition.

Would you like further assistance in selecting a suitable steel section or any additional details?

### Relative Questions:
1. How do you calculate the maximum deflection for a simply supported beam under a uniform load?
2. What factors influence the selection of a steel section beyond just the section modulus?
3. How does the yield stress of steel impact the overall design of the beam?
4. What are common methods to increase the moment capacity of a beam without changing the material?
5. What role does the maximum allowable stress play in determining the safety of a structure?

### Tip:
Always consider both bending and shear stresses when designing beams to ensure a safe and economical design.

Diseñar las dimensiones de una viga simplemente apoyada que tiene un claro de 6.00 m con una carga viva de 2900 kg/m y utilizaremos el porcentaje máximo de acero.

This problem involves designing the dimensions of a simply supported beam with a 6-meter span and a uniformly distributed live load of 2900 kg/m. The solution includes calculating the maximum moment and the required section modulus, critical for structural engineers. Suitable for undergraduate students in Civil or Structural Engineering.

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