Question

According to Janssen theory, the lateral pressure exerted by the grain on the wall is influenced by -
(A) Pressure coefficient
(B) Shape of grain
(C) Bulk density of grain
(D) Depth of grain
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only.
• B. (C) and (D) only.
• C. (A), (C) and (D) only.
• D. (A), (B), (C) and (D).

Hint:

Remember that in Janssen's equation, as the depth (\(h\)) becomes very large, the lateral pressure reaches a constant maximum value: \(L_{\text{max}} = \frac{w \cdot R}{\mu'}\). This is a distinct difference from hydrostatic pressure, which increases linearly with depth without limit.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Janssen's theory is a classical model used to estimate the pressures exerted by stored granular materials (like grains) on the walls of deep bins or silos.
Unlike liquid storage tanks, a portion of the grain weight is supported by friction against the silo walls, which reduces the rate at which vertical and lateral pressures increase with depth.
Key Formula or Approach:
The lateral pressure (\(L\)) at any depth (\(h\)) is given by Janssen's equation: \[ L = \frac{w \cdot R}{\mu'} \left[ 1 - e^{-\frac{k \cdot \mu' \cdot h}{R}} \right] \] where:
\(w\) = bulk density of the grain

\(R\) = hydraulic radius of the bin (cross-sectional area divided by perimeter)

\(\mu'\) = coefficient of friction between the grain and the wall material

\(k\) = pressure coefficient (ratio of lateral to vertical pressure)

\(h\) = depth of the grain column

Step 2: Detailed Explanation:
Let us analyze which factors influence lateral pressure according to the equation:

(A) Pressure coefficient (\(k\)): This term appears explicitly in the exponential term of the equation. It dictates how much of the vertical pressure is transferred laterally to the bin wall. Thus, (A) is a key factor.

(B) Shape of grain: The shape of the grain is not a variable in Janssen's equation. While grain shape can indirectly affect bulk density or friction coefficients, it is not a parameter used in the mathematical model. Thus, (B) is not a direct factor.

(C) Bulk density of grain (\(w\)): This is a multiplier in the equation, directly influencing the scale of the pressure. Thus, (C) is a key factor.

(D) Depth of grain (\(h\)): The depth of the grain column appears in the exponential term. As depth increases, the term \((1 - e^{-kh\mu'/R})\) asymptotically approaches 1, meaning lateral pressure increases with depth up to a maximum limit. Thus, (D) is a key factor.
Therefore, lateral pressure is directly influenced by (A), (C), and (D).

Step 3: Final Answer:
The correct option is (C), representing (A), (C), and (D) only.