Test: Vertical Stress Distribution - Civil Engineering (CE) MCQ

Concept:
Newmark's influence chart for stress distribution
It is a graphical method used to compute vertical, horizontal and shear stress due to uniformly distributed load over an area of any shape or geometry below any point that lies either below or outside the loaded area by equation called Boussinesq's equation.

Stress below any point is calculated as :-

Where, m = No. of concentric circles, n = No. of radial lines, q = Intensity of load, N = Equivalent number of areas covered by plan area.
The product of  is called Influence Factor.
Given
m = 8, n = 10
Therefore, Influence factor (I.F)

I.F. = 0.0125

Assumptions made in Westerguard Theory:

1. Soil is Homogeneous, Anisotropic (as the physical properties along the different directions are different ), and Elastic.
2. Soil is considered as Cohesive as clay.
3. The soil profile is Layered.
4.  Poisson's Ratio is considered zero for all practical purposes.
5. The load will act as a point load on the surface.

As per Westergaard’s  equation, the vertical stress due to point load at any point Z below soil strata is calculated using the following formula:

Where,
Q is the concentrated Load
Z is the depth where stress needs to be calculated
I_w is the influence factor, which is given as:

r is the radial distance from a point load to depth Z

Statement 1: True
The following are the assumptions in Boussinesq’s theory:

1. The soil is homogeneous and isotropic.
2. The soil mass is semi-infinite; that is, it extends infinitely in all directions below a level surface.
3. The soil is elastic i.e. it obeys Hooke’s law; that is, the stress-strain relationship is linear.
4. The soil is weightless and unstressed before the application of the load.
5. The load is applied at the ground surface and it is a point load.

Statement 2: False
The typical contact pressure distribution under a rigid footing like RCC  in cohesionless soil like sand is given below:

Based on the above, the following can be inferred about contact pressure:

1. It is non-uniform.
2. It is maximum at the center and minimum at the corners.

• Westergaard’s Theory Assumptions: Elastic medium of semi-infinite extent but containing numerous, closely spaced horizontal sheets of a negligible thickness of an infinite rigid material which permits only downward deformation as a whole without allowing it to undergo any lateral strain.
• Westergaard's assumptions are more close to the field reality, especially for over-consolidated and laminated sedimentary or stratified soils, which exhibit marked anisotropy.
• Based on this criterion of no lateral displacement, Westergaard derived the following equation for a point load, Q, at a depth z from the surface:
  
  Westergaard’s analysis for stress distribution beneath loaded areas is more suitable to stratified or layered soils.

Westergaard’s Theory Assumptions: 

Elastic medium of semi-infinite extent but containing numerous, closely spaced horizontal sheets of a negligible thickness of an infinite rigid material which permits only downward deformation as a whole without allowing it to undergo any lateral strain.
Westergaard's assumptions are more close to the field reality, especially for over-consolidated and laminated sedimentary or stratified soils, which exhibit marked anisotropy.
Based on this criterion of no lateral displacement, Westergaard derived the following equation for a point load, Q, at a depth z from the surface:

∴ Westergaard’s analysis for stress distribution beneath loaded areas is applicable to stratified soils.

Concept:
Boussinesq Equation:

Concept:
Boussinesq’s equation for vertical stress


Where,
σ_Z = vertical stress in kN/m²
Q = concentrated load in kN
Z = point below the ground level where stress has to be calculated
r = radial distance
when we need to calculate the vertical stress exactly below the concentrated point load then,
i.e. r = 0

Calculation:
Given, Z = 6 m
Q = 2000 kN
We have to calculate the vertical stress exactly below the point load,
∴ r = 0

Westergaard’s Theory Assumptions: 
Elastic medium of semi-infinite extent but containing numerous, closely spaced horizontal sheets of a negligible thickness of an infinite rigid material which permits only downward deformation as a whole without allowing it to undergo any lateral strain.
Note:
Westergaard's assumptions are more close to the field reality, especially for over-consolidated and laminated sedimentary or stratified soils, which exhibit marked anisotropy.
Based on this criterion of no lateral displacement, Westergaard derived the following equation for a point load, Q, at a depth z from the surface:

∴ Westergaard’s analysis for stress distribution beneath loaded areas is applicable to stratified soils.

Concept:
Total stress and effective stress can be given as,
Total stress(T) = γ_sat × Z
Effective stress(E) = γ_sub × Z
γ_sub = γ_sat − γ_w

Calculation:
γ_sat = 2.1t/m²
γ_w = 1t/m²
γ_sub = 2.1 − 1 = 1.1t/m²
T = 2.1 x 4 = 8.4
E = 1.1 x 4 = 4.4

Concept:
Vertical pressure distribution on the vertical line:

• The figure above shows the variation of vertical stress distribution on a vertical line at a distance r from the axis of loading.
• The vertical stress first increases, then attains a maximum value, and then decreases.
• It can be shown, that the maximum value of σ_Z on a vertical line is obtained at the point of intersection of the vertical plane with the radial line at β = 13° 15' through the point load, as shown in the figure.

When r = 1 m, we get
Z = 1/0.817 = 1.225
And corresponding to this Z value, the maximum σ_z will be