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To solve this problem, we need to determine the maximum height \( H \) of the concrete hollow block (CHB) wall that the beam can safely carry. The problem gives us the dimensions, materials, and allowable stresses. Here's how we approach it:

### Given Data
1. **Span of the beam**: \( L = 6 \text{ m} \)
2. **Concrete hollow block density**: \( 2.4 \text{ tonnes/m}^3 = 2400 \text{ kg/m}^3 = 24000 \text{ N/m}^3 \)
3. **Allowable bending stress**: \( f_b = 8.27 \text{ MPa} \)
4. **Allowable shear stress**: \( f_v = 0.176 \text{ MPa} \)
5. **Weight of wood**: \( 5000 \text{ N/m}^3 \)

### Beam and Wall Cross-Section Details
1. Width of the beam: \( 15 \text{ cm} = 0.15 \text{ m} \)
2. Overall height of the cross-section: \( 30 \text{ cm} = 0.3 \text{ m} \)
3. Height of the CHB wall: \( H \text{ cm} \) (variable to be found)
4. Thickness of CHB wall and mortar: \( 5 \text{ cm} = 0.05 \text{ m} \)

### Step 1: Determine the Load
The load from the CHB wall per unit length of the beam is:
\[
w_{\text{CHB}} = \text{density} \times \text{width} \times H
\]
\[
w_{\text{CHB}} = 24000 \text{ N/m}^3 \times 0.15 \text{ m} \times H \text{ m}
\]
\[
w_{\text{CHB}} = 3600H \text{ N/m}
\]

### Step 2: Determine the Bending Moment Capacity
For a simply supported beam, the maximum bending moment \( M \) occurs at the center:
\[
M = \frac{wL^2}{8}
\]
Where \( w \) is the total uniformly distributed load.

### Step 3: Moment of Inertia (I) Calculation
The beam’s cross-section is rectangular:
\[
I = \frac{b h^3}{12}
\]
Where:
- \( b = 0.15 \text{ m} \) (width of the beam)
- \( h = 0.3 \text{ m} \) (height of the beam)

\[
I = \frac{0.15 \times (0.3)^3}{12} = 3.375 \times 10^{-4} \text{ m}^4
\]

### Step 4: Calculate the Bending Stress
The maximum bending stress is given by:
\[
f_b = \frac{M \times c}{I}
\]
Where \( c \) is the distance from the neutral axis to the outermost fiber (0.15 m in this case).

Rearranging to solve for \( M \):
\[
M = \frac{f_b \times I}{c}
\]
\[
M = \frac{8.27 \times 10^6 \times 3.375 \times 10^{-4}}{0.15}
\]
\[
M \approx 18.6 \text{ kNm}
\]

### Step 5: Equate the Bending Moment to the Load
\[
\frac{w_{\text{CHB}} \times L^2}{8} \leq 18.6 \text{ kNm}
\]
Substituting \( w_{\text{CHB}} = 3600H \):
\[
\frac{3600H \times (6)^2}{8} \leq 18.6 \times 10^3
\]
\[
16200H \leq 18.6 \times 10^3
\]
\[
H \leq \frac{18.6 \times 10^3}{16200}
\]
\[
H \leq 1.15 \text{ m}
\]

### Final Result
The maximum height \( H \) of the concrete hollow block wall that the beam can support is approximately **1.15 meters**.

Would you like me to provide more details on any of the steps or calculations?

### Related Questions
1. How would the solution change if the span of the beam was increased?
2. What is the effect of using a different material with a different density for the wall?
3. How does the shear stress limit impact the maximum height calculation?
4. If the beam were not laterally supported, how would the maximum height change?
5. How does changing the width of the beam affect its load-carrying capacity?

### Tip
Always check both bending and shear stresses when designing beams to ensure that neither limit is exceeded for a safe structure.

A simple beam having a cross section shown has a simple span of 6.0 m. It carries a 15 cm concrete hollow block wall sitting on mortar weighing 2.4 tonnes/m^3. Assume the beam to be laterally supported. Determine the maximum height of the concrete hollow block wall that the beam could carry. fb = 8.27 MPa, fv = 0.176 MPa, Assume weight of wood = 5000 N/m^3.

Learn how to calculate the maximum height of a concrete hollow block wall on a 6m span beam. This solution involves structural mechanics, bending moment formulas, and allowable stress theory. Given data includes a wall weight of 2.4 tonnes/m^3, beam span of 6m, and material properties like fb = 8.27 MPa and fv = 0.176 MPa.

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