Question

A cylindrical concrete silo of 6 m internal diameter and 24 m height is filled with rough rice having a bulk density of 635 kg/m\(^3\). The angle of friction between the concrete wall and rough rice is 30°. The ratio between lateral and vertical pressure is 0.4. The ratio of lateral pressure at 10 m depth to the 5 m depth is _________. (Rounded off to 2 decimal places)

Hint:

To calculate the ratio of lateral pressure at different depths, first calculate the vertical pressure at each depth and then use the given ratio to find the corresponding lateral pressures.

Answer 1

We are given the following parameters:
Diameter of the silo \( D = 6 \, {m} \),
Height of the silo \( H = 24 \, {m} \),
Bulk density of rice \( \rho = 635 \, {kg/m}^3 \),
Angle of friction between concrete and rice \( \theta = 30^\circ \),
The ratio of lateral pressure to vertical pressure \( \frac{P_{{lateral}}}{P_{{vertical}}} = 0.4 \).
Step 1: Calculate the vertical pressure at different depths
The vertical pressure at a given depth in the silo is calculated using the formula: \[ P_{{vertical}} = \rho g h \] Where:
\( \rho \) is the bulk density of the rice,
\( g \) is the acceleration due to gravity (approximately \( 9.81 \, {m/s}^2 \)),
\( h \) is the depth.
At a depth of 5 m: \[ P_{{vertical, 5m}} = 635 \times 9.81 \times 5 = 31241.75 \, {Pa} \] At a depth of 10 m: \[ P_{{vertical, 10m}} = 635 \times 9.81 \times 10 = 62483.5 \, {Pa} \] Step 2: Calculate the lateral pressure at different depths
The lateral pressure is related to the vertical pressure by the ratio given: \[ P_{{lateral}} = 0.4 \times P_{{vertical}} \] At a depth of 5 m: \[ P_{{lateral, 5m}} = 0.4 \times 31241.75 = 12496.7 \, {Pa} \] At a depth of 10 m: \[ P_{{lateral, 10m}} = 0.4 \times 62483.5 = 24993.4 \, {Pa} \] Step 3: Calculate the ratio of lateral pressure at 10 m depth to the 5 m depth
Now, calculate the ratio of lateral pressure at 10 m depth to the lateral pressure at 5 m depth: \[ {Ratio} = \frac{P_{{lateral, 10m}}}{P_{{lateral, 5m}}} = \frac{24993.4}{12496.7} = 1.41 \] Thus, the ratio of lateral pressure at 10 m depth to the 5 m depth is 1.41.