GATE CE 2020 Set 1 — Question 45

Given data:
Surcharge $q = 70$ kN/m$^2$
Unit weight of sand $\gamma_{sand} = 18$ kN/m$^3$
Saturated unit weight of silty clay $\gamma_{sat, silty\_clay} = 20$ kN/m$^3$
Saturated unit weight of clay $\gamma_{sat, clay} = 21$ kN/m$^3$
Unit weight of water $\gamma_w = 9.81$ kN/m$^3$
Water table is at 3 m below ground level.We need to find the effective stresses at points P and Q immediately after preloading.Key Concept: Immediately after an instantaneous application of surcharge on a saturated clay layer, the entire surcharge is initially carried by the pore water as excess pore water pressure. Therefore, the effective stress in the clay layer does not change instantaneously. For sand, which is highly permeable, consolidation is considered instantaneous, meaning the effective stress changes immediately.From the figure:

• Point P is at a depth of 3 m, at the interface of sand and silty clay, and also at the water table.
• Point Q is at a depth of $3 + 4 = 7$ m, within the silty clay layer.

Effective stress at Point P (at 3 m depth):
Point P is at the top of the silty clay layer. Since it's a clayey soil, immediately after preloading, the effective stress at P will be the initial effective stress (before surcharge).Initial effective stress at P ($\sigma'_{P,initial}$):
$\sigma'_{P,initial} = \text{Effective stress due to sand layer above P}$
$\sigma'_{P,initial} = \gamma_{sand} \times 3 \text{ m}$ (since P is at the water table, pore pressure is 0 at this point, and sand is above WT)
$\sigma'_{P,initial} = 18 \text{ kN/m}^3 \times 3 \text{ m} = 54 \text{ kPa}$So, immediately after preloading, the effective stress at P is 54 kPa.Effective stress at Point Q (at 7 m depth):
Point Q is within the silty clay layer. Immediately after preloading, the effective stress at Q will also be the initial effective stress (before surcharge).Initial effective stress at Q ($\sigma'_{Q,initial}$):
$\sigma'_{Q,initial} = \text{Effective stress due to sand layer} + \text{Effective stress due to silty clay layer up to Q}$
$\sigma'_{Q,initial} = (\gamma_{sand} \times 3) + (\gamma_{sat, silty\_clay} - \gamma_w) \times 4$
$\sigma'_{Q,initial} = (18 \text{ kN/m}^3 \times 3 \text{ m}) + (20 \text{ kN/m}^3 - 9.81 \text{ kN/m}^3) \times 4 \text{ m}$
$\sigma'_{Q,initial} = 54 \text{ kPa} + (10.19 \text{ kN/m}^3) \times 4 \text{ m}$
$\sigma'_{Q,initial} = 54 \text{ kPa} + 40.76 \text{ kPa}$
$\sigma'_{Q,initial} = 94.76 \text{ kPa}$Rounding off to the nearest integer, the effective stress at Q is 95 kPa.Therefore, immediately after preloading, the effective stresses at points P and Q are 54 kPa and 95 kPa, respectively.The final answer is $\boxed{D}$