Previous Years GATE Questions - Stress Distribution in Soil

1.  The vertical stress at depth z, directly below the point load P is _____ (k is constant).

[GATE 1997: 1 Mark]
  1. k
    P / z
  2. k
    P / z3
  3. k
    P / z2
  4. k
    P / √z

Answer: Option C

Explanation:
Using Boussinesq's equation,
σz =

3P / 2Πz2
x 1 / { 1 + r2/z2}5/2
σz = k
P / z2

Where, k - Boussinesq influence factor.

2.  A point load of 700 kN is applied on the surface of thick layer of saturated clay. Using Boussinesq's elastic analysis, the estimated vertical stress (σv) at a depth of 2m and a radial distance of 1m from the point of application of the load is _____ .

[GATE 1998: 2 Marks]
  1. 47.5 kPa
  2. 47.6 kPa
  3. 47.7 kPa
  4. 47.8 kPa

Answer: Option D

Explanation:
Using Boussinesq's equation,
σz =

3Q / 2Πz2
x 1 / { 1 + r2/z2}5/2
Here, Q = 700 kN, z = 2m, r = 1m.
Substituting the above values, σz = 47.8 kPa

3.  A 25kN point load acts on the surface of an infinite elastic medium. The vertical pressure intensity in kN/m2 at a point 6m below and 4m away from the load will be _____ .

[GATE 2003: 1 Mark]
  1. 132
  2. 13.2
  3. 1.32
  4. 0.132

Answer: Option D

Explanation:
Using Boussinesq's equation,
σz =

3Q / 2Πz2
x 1 / { 1 + r2/z2}5/2
Here, Q = 25 kN, z = 6m, r = 4m.
Substituting the above values, σz = 0.132 kN/m2

4.  The vertical stress at some depth below the corner of a 2m x 3m rectangular footing due to a certain load intensity is 100 kN/m2. What will be the vertical stress in kN/m2 below the centre of a 4m x 6m rectangular footing at the same depth and the same load intensity?

[GATE 2007: 1 Mark]
  1. 25
  2. 100
  3. 200
  4. 400

Answer: Option D

Explanation:

Depicts the question relating to the variation of stress when the size of footing is doubled

Considering Fig (a), given that the vertical stress at some depth below the corner due to certain load intensity is 100 kN/m2
Now, For Fig (b), the vertical stress at the centre of 4m x 6m footing due to the same load = Sum of vertical stress below the corner of all 2m x 3m footing
⇒ The vertical stress at the centre of 4m x 6m footing due to the same load = 4 x 100 = 400kN/m2

5.  A footing 2m x 1m exerts a uniform pressure of 150 kN/m2 on the soil. Assuming a load dispersion of 2 vertical to 1 horizontal, the average vertical stress (kN/m2) at 1m below the footing is _____ .

[GATE 2008: 2 Marks]
  1. 50
  2. 75
  3. 80
  4. 100

Answer: Option A

Explanation:

Load dispersion from footing to a depth of 1m

As the load dispersion is 2 vertical to 1 horizontal, n =
1 / 2

Average vertical stress, σz =
Q x LB / (L + 2nz)(B+2nz)

Here, z = 1m,= L = 2m, B = 1m, Q = 150 kN/m2
Average vertical stress, σz =
150 x 2 x 1 / (2 + 1)(1 + 1)
=
300 / 6
= 50 kN/m2

6.  The vertical stress at point P1 due to the point load Q on the ground surface as shown in figure is σz. According to Boussinesq's equation, the vertical stress at the point P2 shown in figure will be _____ .

Depicts the position of points on which the vertical stress due to a concentrated load is to be determined
[GATE 2010: 2 Marks]
  1. σz/2
  2. σz
  3. 2σz
  4. 4σz

Answer: Option D

Explanation:
According to Boussinesq's equation,
σz =

3Q / 2Πz2
x 1 / { 1 + r2/z2}5/2
When r = r/2 and z = z/2,
Vertical stress =
3Q / 2Π(z/2)2
x 1 / { 1 + (r/2)2/(z/2)2}5/2
⇒ Vertical stress = 4σz

7.  A uniformly disturbed line load of 500 kN/m is acting on the ground surface. Based on Boussinesq's theory, the ratio of vertical stress at depth 2m to that at 4m, right below the line of loading, is _____ .

[GATE 2017: 1 Mark, I Set]
  1. 0.25
  2. 0.5
  3. 2.0
  4. 4.0

Answer: Option C

Explanation:
Boussinesq's Equation for line load, σz =

2q / Πz
x 1 / { 1 + x2/z2}2
Here, q= 500 kN/m
Below the line load, x = 0
Thus, σz ∝
1 / z

Now,
σz @2m / σz @4m
=
1/2 / 1/4
=
4 / 2
= 2
Thus, the ratio of vertical stress at depth 2m to that at 4m, right below the line of loading is 2.

8.  Consider a square-headed area ABCD on the ground with its centre at M as shown in the figure. Four concentrated vertical loads of P = 5000 kN are applied on this area, one at each corner. The vertical stress increment (in kPa, upto one decimal place) due to these loads according to the Boussinesq's equation , at a point 5m right below M is _____ .

Depicts the position of load and the position at which vertical stress is to be determined
[GATE 2017: 2 Marks, II Set]

Answer: 190.8

Explanation:

Stress Distribution in Soil - Depicts the position of load and the position at which vertical stress is to be determined

Here, P = 5000 kN, z = 5m and r = √(22 + 22) = 2√2 m
Using Boussinesq's equation,
σz =
3Q / 2Πz2
x 1 / { 1 + r2/z2}5/2
Here, Q = P = 5000 kN.
Substituting the known values of Q, z and r, σz = 190.8 kPa.

9.  Which one of the following statements is NOT correct?

[GATE 2018: 1 Mark, II Set]
  1. When the water content of soil lies between its liquid limit and plastic limit, the soil is said to be in plastic limit.
  2. Boussinesq’s theory is used for the analysis of stratified soil.
  3. The inclination of stable slope in cohesive soil can be greater than its angle of internal friction.
  4. For saturated dense fine sand, after applying overburden correction, if the Standard Penetration test value exceeds 15, dilatancy correction is to be applied.

Answer: Option B

Explanation:
Boussinesq's theory is based on the assumption that the soil is homogeneous and isotropic. Hence, it can not be applied for stratified soil.

10.  In a soil specimen, the total stress, effective stress, hydraulic gradient and critical hydraulic gradient are σ, σ′, i and ic respectively. For initiation of quicksand condition, which one of the following statements are TRUE?

[GATE 2019: 1 Mark, I Set]
  1. σ′ = 0 and i = ic
  2. σ′ ≠ 0 and i = ic
  3. σ = 0 and i = ic
  4. σ′ ≠ 0 and i ≠ ic

Answer: Option A

Explanation:
Necessary condition for Quicksand condition is
i) Effective stress, σ′ = 0
ii) Hydraulic gradient, i = Critical Hydraulic gradient, ic

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