# 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).

- kP / z
- kP / z
^{3} - kP / z
^{2} - kP / √z

**Answer:** Option C

**Explanation:**

Using Boussinesq's equation,

σ_{z} =

^{2}

^{2}/z

^{2}}

^{5/2}

σ

_{z}= k

^{2}

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 (σ_{v}) at a depth of 2m and a radial distance of 1m from the point of application of the load is _____ .

- 47.5 kPa
- 47.6 kPa
- 47.7 kPa
- 47.8 kPa

**Answer:** Option D

**Explanation:**

Using Boussinesq's equation,

σ_{z} =

^{2}

^{2}/z

^{2}}

^{5/2}

Here, Q = 700 kN, z = 2m, r = 1m.

Substituting the above values, σ

_{z}= 47.8 kPa

## 3. A 25kN point load acts on the surface of an infinite elastic medium. The vertical pressure intensity in kN/m^{2} at a point 6m below and 4m away from the load will be _____ .

- 132
- 13.2
- 1.32
- 0.132

**Answer:** Option D

**Explanation:**

Using Boussinesq's equation,

σ_{z} =

^{2}

^{2}/z

^{2}}

^{5/2}

Here, Q = 25 kN, z = 6m, r = 4m.

Substituting the above values, σ

_{z}= 0.132 kN/m

^{2}

## 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/m^{2}. What will be the vertical stress in kN/m^{2} below the centre of a 4m x 6m rectangular footing at the same depth and the same load intensity?

- 25
- 100
- 200
- 400

**Answer:** Option D

**Explanation:**

Considering Fig (a), given that the vertical stress at some depth below the corner due to certain load intensity is 100 kN/m

^{2}

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/m

^{2}

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

- 50
- 75
- 80
- 100

**Answer:** Option A

**Explanation:**

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

Average vertical stress, σ

_{z}=

Here, z = 1m,= L = 2m, B = 1m, Q = 150 kN/m

^{2}

Average vertical stress, σ

_{z}=

^{2}

## 6. The vertical stress at point P_{1} due to the point load Q on the ground surface as shown in figure is σ_{z}. According to Boussinesq's equation, the vertical stress at the point P_{2} shown in figure will be _____ .

- σ
_{z}/2 - σ
_{z} - 2σ
_{z} - 4σ
_{z}

**Answer:** Option D

**Explanation:**

According to Boussinesq's equation,

σ_{z} =

^{2}

^{2}/z

^{2}}

^{5/2}

When r = r/2 and z = z/2,

Vertical stress =

^{2}

^{2}/(z/2)

^{2}}

^{5/2}

⇒ Vertical stress = 4σ

_{z}

## 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 _____ .

- 0.25
- 0.5
- 2.0
- 4.0

**Answer:** Option C

**Explanation:**

Boussinesq's Equation for line load, σ_{z} =

^{2}/z

^{2}}

^{2}

Here, q= 500 kN/m

Below the line load, x = 0

Thus, σ

_{z}∝

Now,

_{z @2m}/ σ

_{z @4m}

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 _____ .

**Answer:** 190.8

**Explanation:**

Here, P = 5000 kN, z = 5m and r = √(2

^{2}+ 2

^{2}) = 2√2 m

Using Boussinesq's equation,

σ

_{z}=

^{2}

^{2}/z

^{2}}

^{5/2}

Here, Q = P = 5000 kN.

Substituting the known values of Q, z and r, σ

_{z}= 190.8 kPa.

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

- When the water content of soil lies between its liquid limit and plastic limit, the soil is said to be in plastic limit.
- Boussinesq’s theory is used for the analysis of stratified soil.
- The inclination of stable slope in cohesive soil can be greater than its angle of internal friction.
- 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 i_{c} respectively. For initiation of quicksand condition, which one of the following statements are TRUE?

- σ′ = 0 and i = i
_{c} - σ′ ≠ 0 and i = i
_{c} - σ = 0 and i = i
_{c} - σ′ ≠ 0 and i ≠ i
_{c}

**Answer:** Option A

**Explanation:**

Necessary condition for Quicksand condition is

i) Effective stress, σ′ = 0

ii) Hydraulic gradient, i = Critical Hydraulic gradient, i_{c}