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A concentrated load of 50t act vertically at a point on the soil surface. If Boussinesq’s 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
  • a)
    0.36
  • b)
    0.60
  • c)
    1.66
  • d)
    2.77
Correct answer is option 'D'. Can you explain this answer?
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A concentrated load of 50t act vertically at a point on the soil surfa...
Verticle stress below point load 

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A concentrated load of 50t act vertically at a point on the soil surfa...
Solution:

Given: Load = 50t, Depth1 = 3m, Depth2 = 5m

Boussinesq's equation is given by:

σz = [(1 + μ)/(2π)] [(qz)/(z^2 + r^2)] [(z^2 + r^2)/(z^2 + 2r^2)]

Where,

σz = vertical stress at depth z

μ = Poisson's ratio

qz = pressure exerted by the load at depth z

r = radius of the loaded area

Let's calculate the pressure exerted by the load at depths 3m and 5m.

At depth 3m:

q3 = (50t)/(π × 1m^2) = 50/π kN/m^2

At depth 5m:

q5 = (50t)/(π × 1.6m^2) = 50/(π × 1.6^2) kN/m^2

Now, let's calculate the vertical stresses at depths 3m and 5m using Boussinesq's equation.

At depth 3m:

σz1 = [(1 + μ)/(2π)] [(q3)/(3^2 + 1^2)] [(3^2 + 1^2)/(3^2 + 2×1^2)]

At depth 5m:

σz2 = [(1 + μ)/(2π)] [(q5)/(5^2 + 1^2)] [(5^2 + 1^2)/(5^2 + 2×1^2)]

Taking the ratio of the above two equations, we get:

σz2/σz1 = [(q5/q3) × (3^2 + 2×1^2)/(5^2 + 2×1^2)] = 2.77

Therefore, the ratio of vertical stresses at depths of 3m and 5m, respectively, vertically below the point of application of load will be 2.77 (Option D).
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A concentrated load of 50t act vertically at a point on the soil surface. If Boussinesq’s 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 bea)0.36b)0.60c)1.66d)2.77Correct answer is option 'D'. Can you explain this answer?
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