Design for flexure a W beam of 40 ft span. The beam has a uniform dead load of 1.5 kip/ft through
the entire length and a uniform live load of 2.5 kip/ft over the last 32 ft. There is also a
concentrated dead load 30 kips at 8 ft away from the left support. Select the most economical
section. The beam has adequate lateral support. Use A50 steel.

To design a W beam for flexure with the given specifications, let's go through the analysis step-by-step. The main steps involved are calculating the maximum moment and then selecting a W section that satisfies the strength and economy requirements.

### Given Data:
- **Span, \( L \):** 40 ft
- **Uniform dead load, \( w_D \):** 1.5 kip/ft (applies over the entire length of the beam)
- **Uniform live load, \( w_L \):** 2.5 kip/ft (applies over 32 ft from the right end)
- **Concentrated dead load, \( P_D \):** 30 kips (located 8 ft from the left support)
- **Material:** A50 steel (yield strength \( F_y = 50 \) ksi)
- **Adequate lateral support:** No lateral-torsional buckling concerns.

### Step 1: Calculate Maximum Moment Due to Dead and Live Loads

#### 1. Dead Load Moment
1. **Uniform Dead Load \( w_D = 1.5 \) kip/ft** over 40 ft:
   \[
   M_{D, \text{uniform}} = \frac{w_D \cdot L^2}{8} = \frac{1.5 \cdot (40)^2}{8} = 300 \text{ kip-ft}
   \]

2. **Concentrated Dead Load of 30 kips at 8 ft:**
   \[
   M_{D, \text{concentrated}} = P_D \cdot a \cdot \left(1 - \frac{a}{L}\right) = 30 \cdot 8 \cdot \left(1 - \frac{8}{40}\right)
   \]
   \[
   = 30 \cdot 8 \cdot 0.8 = 192 \text{ kip-ft}
   \]

3. **Total Dead Load Moment:**
   \[
   M_{D, \text{total}} = M_{D, \text{uniform}} + M_{D, \text{concentrated}} = 300 + 192 = 492 \text{ kip-ft}
   \]

#### 2. Live Load Moment
1. **Uniform Live Load \( w_L = 2.5 \) kip/ft** over the last 32 ft:
   \[
   M_L = \frac{w_L \cdot L_{\text{live}}^2}{8} = \frac{2.5 \cdot (32)^2}{8} = 320 \text{ kip-ft}
   \]

#### 3. Total Factored Moment
Using the load factors (for AISC LRFD), typically:
\[
M_u = 1.2 M_D + 1.6 M_L
\]
\[
M_u = 1.2 \cdot 492 + 1.6 \cdot 320 = 590.4 + 512 = 1102.4 \text{ kip-ft}
\]

### Step 2: Select a W-Shape Based on Required Moment Capacity

To resist the factored moment \( M_u \), the required section modulus \( S_{req} \) can be calculated as:
\[
S_{req} = \frac{M_u}{\phi F_y}
\]
Where:
- \( \phi = 0.9 \) (resistance factor for bending in AISC)
- \( F_y = 50 \) ksi

\[
S_{req} = \frac{1102.4 \times 12}{0.9 \times 50} = \frac{13228.8}{45} = 293.97 \text{ in}^3
\]

### Step 3: Choose the Most Economical W-Shape
Using the **AISC Steel Manual**, we would look for a W shape with \( S_x \geq 293.97 \, \text{in}^3 \) and select the lightest section.

**Example Selection:**  
The W27x94 has:
- \( S_x = 302 \, \text{in}^3 \) (adequate for our design moment)
- Weight: 94 lb/ft (an economical choice based on strength and weight)

### Conclusion
The most economical section that satisfies the bending requirements is **W27x94**. 

Would you like further details on this design or have any questions?

---

1. How would you determine deflection limits for this beam?
2. What other checks would be necessary if lateral-torsional buckling was a concern?
3. How can you verify the shear capacity of this W-beam under the given loads?
4. What would change if A36 steel was used instead of A50?
5. How does the load combination factor affect the selection of beam size?

**Tip:** Always verify shear and deflection limits after selecting a section to ensure comprehensive safety and performance.

Design for flexure a W beam of 40 ft span. The beam has a uniform dead load of 1.5 kip/ft through the entire length and a uniform live load of 2.5 kip/ft over the last 32 ft. There is also a concentrated dead load of 30 kips at 8 ft away from the left support. Select the most economical section. The beam has adequate lateral support. Use A50 steel.

This guide explains the flexural design of a W beam for a 40 ft span with detailed calculations for maximum moments under dead and live loads, including uniform and concentrated loads. Step-by-step solutions cover selecting an economical beam section and ensuring adequate strength with A50 steel, suitable for civil engineering students and professionals.

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