A cylindrical concrete silo (6 m internal diameter, 24 m height) is filled with rough rice (bulk density 635 kg/m³). Wall-friction angle = 30°; lateral/vertical pressure ratio = 0.4. Find the ratio of lateral pressure at 10 m depth to that at 5 m depth. (Rounded to 2 decimal places)
Official Answer Key:
1.44 to 1.48
Step-by-Step Solution & Explanation
• **Governing Equation (Janssen\'s Theory for Deep Silos):**
Lateral pressure $L(z)$ at grain depth $z$ is given by:
$L(z) = k \cdot \left( \frac{\rho g R_h}{k \mu\'} \right) \left[ 1 - \exp\left( -\frac{k \mu\' z}{R_h} \right) \right] = L_{\infty} \left[ 1 - \exp\left( -\frac{z}{z_0} \right) \right]$
• **Parameters:**
- Internal diameter $D = 6\text{ m} \implies \text{Hydraulic radius } R_h = \frac{D}{4} = \frac{6}{4} = 1.5\text{ m}$.
- Coefficient of wall friction $\mu\' = \tan(\phi_w) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.57735$.
- Lateral-to-vertical pressure ratio $k = 0.4$.
• **Characteristic Depth $z_0$:**
$z_0 = \frac{R_h}{k \mu\'} = \frac{1.5}{0.4 \times 0.57735} = \frac{1.5}{0.23094} = 6.4952\text{ m}$
• **Ratio of Lateral Pressure at $10\text{ m}$ to $5\text{ m}$:**
$\frac{L(10)}{L(5)} = \frac{1 - e^{-10 / z_0}}{1 - e^{-5 / z_0}} = \frac{1 - e^{-10 / 6.4952}}{1 - e^{-5 / 6.4952}} = \frac{1 - e^{-1.5396}}{1 - e^{-0.7698}} = \frac{1 - 0.21447}{1 - 0.46310} = \frac{0.78553}{0.53690} = 1.463$
• **Conclusion:** The ratio is $1.46$ (Official range: 1.44 to 1.48).
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