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Question Text: a a simple wooden beam having a cross section shown has a simple span of 6 m. it carries a 15 cm concrete hollow wall sitting on mortar weighing 2.4 tonne/cum.assume the beam to be laterally supported. determine the maximum height of hollow block wall that the beam could carry. fb= 8.27 fv=0.76 . assume weight of food id 5000 N/m^3



To solve this problem, we need to analyze the strength of the wooden beam under both bending and shear stresses, and determine the maximum height \( H \) of the concrete hollow block wall that the beam can support.

### Given Data:
- **Span of the beam**: \( L = 6 \, \text{m} \)
- **Width of hollow block wall**: \( b = 15 \, \text{cm} = 0.15 \, \text{m} \)
- **Height of wall (to be determined)**: \( H \)
- **Thickness of wall**: \( 15 \, \text{cm} \)
- **Height of hollow block and mortar combined**: \( 5 \, \text{cm} \)
- **Unit weight of the hollow block**: \( 2.4 \, \text{tonnes/m}^3 = 2400 \, \text{kg/m}^3 \)
- **Weight of wood**: \( 5000 \, \text{N/m}^3 \)
- **Allowable bending stress**: \( f_b = 8.27 \, \text{MPa} \)
- **Allowable shear stress**: \( f_v = 0.76 \, \text{MPa} \)

### Step 1: Calculate the load per unit length on the beam
The weight of the wall will act as a uniformly distributed load on the beam. The total load per unit length, \( w \), is the product of the weight of the concrete block and its cross-sectional area.

The area of the wall is \( b \times H = 0.15 \times H \, \text{m}^2 \).

The total weight of the wall per unit length:
\[
w = 2400 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times (0.15 \times H) \, \text{m}^2 = 3531 \times H \, \text{N/m}
\]

### Step 2: Maximum bending moment in the beam
For a simply supported beam with a uniformly distributed load, the maximum bending moment occurs at the center and is given by:
\[
M_{\text{max}} = \frac{wL^2}{8}
\]

Substituting for \( w = 3531H \):
\[
M_{\text{max}} = \frac{3531H \times (6)^2}{8} = 15889.5H \, \text{Nm}
\]

### Step 3: Section modulus of the beam
The cross-sectional dimensions of the wooden beam are given as 30 cm by 15 cm. The section modulus \( S \) for a rectangular section is:
\[
S = \frac{b \cdot h^2}{6}
\]
where \( b = 0.15 \, \text{m} \) and \( h = 0.30 \, \text{m} \).

\[
S = \frac{0.15 \cdot (0.30)^2}{6} = 0.00225 \, \text{m}^3
\]

### Step 4: Bending stress in the beam
The bending stress \( \sigma_b \) is related to the maximum moment by:
\[
\sigma_b = \frac{M_{\text{max}}}{S}
\]
Substituting the values:
\[
\sigma_b = \frac{15889.5H}{0.00225} = 70509H \, \text{Pa} = 70.509H \, \text{MPa}
\]
Equating to the allowable bending stress \( f_b = 8.27 \, \text{MPa} \):
\[
70.509H = 8.27
\]
\[
H = \frac{8.27}{70.509} \approx 0.1172 \, \text{m} = 11.72 \, \text{cm}
\]

### Step 5: Check for shear stress
The maximum shear force in the beam is given by:
\[
V_{\text{max}} = \frac{wL}{2} = \frac{3531H \times 6}{2} = 10593H \, \text{N}
\]

The shear stress \( \tau \) is:
\[
\tau = \frac{V_{\text{max}}}{A_{\text{web}}}
\]
where \( A_{\text{web}} = b \cdot h = 0.15 \times 0.30 = 0.045 \, \text{m}^2 \).

\[
\tau = \frac{10593H}{0.045} = 235400H \, \text{Pa} = 0.235H \, \text{MPa}
\]
Equating to the allowable shear stress \( f_v = 0.76 \, \text{MPa} \):
\[
0.235H = 0.76
\]
\[
H = \frac{0.76}{0.235} \approx 3.23 \, \text{m}
\]

### Conclusion:
The limiting factor is the bending stress, so the maximum height of the hollow block wall that the beam can carry is **11.72 cm**.

Would you like more details or have any questions? Here are 5 related questions to further explore:

1. How does lateral support influence the stability of beams?
2. What would change if the span of the beam was increased?
3. How would using a different material for the beam impact the calculation?
4. Can the hollow block wall height increase if the beam cross-section changes?
5. What are the failure modes for wooden beams under load?

**Tip**: Always verify if bending or shear stress governs the design of beams in structural analysis.

A simple wooden beam having the cross-section shown has a simple span of 6 m. It carries a 15 cm concrete hollow block wall sitting on mortar weighing 2.4 tonne/cum. Assume the beam to be laterally supported. Determine the maximum height of the hollow block wall that the beam could carry. fb = 8.27 MPa, fv = 0.76 MPa. Assume weight of wood = 5000 N/m^3.

Solve the problem of determining the maximum height of a concrete hollow block wall on a 6m wooden beam. This problem involves bending and shear stress calculations using the beam’s cross-section, weight of concrete, and wood material properties. Suitable for undergraduate civil engineering students.

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