Question

A soil sample is underlying a water column of height \( h_1 \), as shown in the figure. The vertical effective stresses at points A, B, and C are \( \sigma_A' \), \( \sigma_B' \), and \( \sigma_C' \), respectively. Let \( \gamma_{sat} \) and \( \gamma' \) be the saturated and submerged unit weights of the soil sample, respectively, and \( \gamma_w \) be the unit weight of water. Which one of the following expressions correctly represents the sum \( (\sigma_A' + \sigma_B' + \sigma_C') \)?

• A. \( (2h_2 + h_3) \gamma' \)
• B. \( (h_1 + h_2 + h_3) \gamma' \)
• C. \( (h_2 + h_3) (\gamma_{sat} - \gamma_w) \)
• D. \( (h_1 + h_2 + h_3) \gamma_{sat} \)

Hint:

In problems involving effective stress, remember that the effective stress depends on the total stress and the pore water pressure, and it can be affected by the height of the water column above the soil layer.

Answer 1

The Correct Option is A

To solve for the sum \( (\sigma_A' + \sigma_B' + \sigma_C') \), we need to understand how effective stress works and how it relates to the unit weight of the water and the soil.
Step 1: Understanding the Effective Stress The effective stress at any point within the soil is given by:
\[ \sigma' = \sigma - u \] where \( \sigma \) is the total stress, and \( u \) is the pore water pressure. The total stress at any point is determined by the weight of the overlying material, while the pore water pressure depends on the height of the water column.
Step 2: Stress at Point A At point \( A \), the total stress is influenced by the overlying water column of height \( h_1 \). The pore water pressure at point \( A \) is \( u_A = \gamma_w h_1 \), where \( \gamma_w \) is the unit weight of water. Thus, the effective stress at point \( A \) is:
\[ \sigma_A' = \sigma_A - u_A = (\gamma_{sat} h_2) - \gamma_w h_1 \] Step 3: Stress at Point B At point \( B \), the total stress is determined by the weight of the overlying saturated soil (height \( h_2 \)) and the water column above point \( B \). The pore water pressure at point \( B \) is \( u_B = \gamma_w h_1 \), so the effective stress at point \( B \) is:
\[ \sigma_B' = \sigma_B - u_B = (\gamma_{sat} h_2) - \gamma_w h_1 \] Step 4: Stress at Point C At point \( C \), the total stress is determined by the weight of the water column over the saturated soil column (height \( h_3 \)) and the unit weight of water. The pore water pressure at point \( C \) is \( u_C = \gamma_w h_1 \). The effective stress at point \( C \) is:
\[ \sigma_C' = \sigma_C - u_C = (\gamma_{sat} h_3) - \gamma_w h_1 \] Step 5: Sum of the Effective Stresses Now, we can sum the effective stresses at points \( A \), \( B \), and \( C \):
\[ \sigma_A' + \sigma_B' + \sigma_C' = (\gamma_{sat} h_2) - \gamma_w h_1 + (\gamma_{sat} h_2) - \gamma_w h_1 + (\gamma_{sat} h_3) - \gamma_w h_1 \] Simplifying this expression:
\[ = 2 \gamma_{sat} h_2 + \gamma_{sat} h_3 - 3 \gamma_w h_1 \] Now, we notice that the result corresponds to the expression in Option A, \( (2h_2 + h_3) \gamma' \), where \( \gamma' = \gamma_{sat} - \gamma_w \), the submerged unit weight of the soil. Thus, the correct answer is (A).