A concentrated vertical load of 3000 kN is applied on a horizontal ground surface. Points P and Q are at depths 1 m and 2 m below the ground, respectively, along the line of application of the load. Considering the ground to be a linearly elastic, isotropic, semiinfinite medium, the ratio of the increase in vertical stress at P to the increase in vertical stress at Q is _______ . (in integer)Correct answer is '4'. Can you explain this answer?

Question

Answer

Introduction:
In this problem, we are analyzing the increase in vertical stress at two different depths below the ground surface when a concentrated vertical load is applied. The ground is assumed to be a linearly elastic, isotropic, semi-infinite medium. We need to find the ratio of the increase in vertical stress at point P (1 m depth) to the increase in vertical stress at point Q (2 m depth).

Analysis:
To solve this problem, we can use the Boussinesq's formula, which relates the increase in vertical stress at a point below the ground surface to the applied load. According to this formula, the increase in vertical stress at a point below the ground surface is inversely proportional to the square of the distance from the point of application of the load.

Step 1: Calculate the increase in vertical stress at point P:
Using the Boussinesq's formula, the increase in vertical stress at point P can be calculated as follows:

Δσ_P = (q / π) * [(z + d) / √((z + d)^2 + r^2)]^2

Where:
Δσ_P = Increase in vertical stress at point P
q = Applied load (3000 kN)
z = Depth of point P (1 m)
d = Depth of load application (0 m, as the load is applied at the ground surface)
r = Horizontal distance from the point of application of the load to point P (0 m, as both are on the same line)

Plugging in the values, we get:

Δσ_P = (3000 / π) * [(1 + 0) / √((1 + 0)^2 + 0^2)]^2

Simplifying the equation, we get:

Δσ_P = (3000 / π) * (1 / √1)^2
Δσ_P = (3000 / π) * (1 / 1)
Δσ_P = 3000 / π

Step 2: Calculate the increase in vertical stress at point Q:
Similarly, using the same formula, the increase in vertical stress at point Q can be calculated as follows:

Δσ_Q = (q / π) * [(z + d) / √((z + d)^2 + r^2)]^2

Where:
Δσ_Q = Increase in vertical stress at point Q
q = Applied load (3000 kN)
z = Depth of point Q (2 m)
d = Depth of load application (0 m, as the load is applied at the ground surface)
r = Horizontal distance from the point of application of the load to point Q (0 m, as both are on the same line)

Plugging in the values, we get:

Δσ_Q = (3000 / π) * [(2 + 0) / √((2 + 0)^2 + 0^2)]^2

Simplifying the equation, we get:

Δσ_Q = (3000 / π) * (2 / √4)^2
Δσ_Q = (3000 / π) * (2 / 2)^2
Δσ_Q = (3000 / π) * 1

Step 3: Calculate the ratio of

Answer

Similar Civil Engineering (CE) Doubts

A concentrated load of 50t act vertically at a point on the soil surface. If Boussinesqs equation is applied for computation of stress, then the ratio of vertical stresses at depths of 3 m and 5 m, respective, vertically below the point of application of load will be

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