This document summarizes stress distribution in soil due to concentrated and uniform loads. It presents Westergaard and Boussinesq equations to calculate vertical stress below a concentrated load. It also discusses the approximate and elastic theory methods to calculate vertical stress below a uniform load on circular and rectangular areas using influence coefficients. Several examples are provided to illustrate calculating vertical stress at different depths and locations below concentrated and uniform loads.

1. 1

2. Department of Civil Engineering
Prepared by:
Engr. Mamoon Kareem
University of Wah, Wah Cantt.
BSc (Civil), UET Taxila
MS (Water), NUST Isb.
Lecture 01
Chapter No 01
Stress Distribution in Soil

3. 1. Concentrated Load
2. Uniform Load
Prepared by:
Engr. Mamoon Kareem
Department of Civil Engineering
University of Wah, Wah Cantt.
Stress Distribution in Soil

4. Outline
• Introduction
• VERTICAL PRESSURE BELOW A CONCENTRATED LOAD
• Westergaard Equation
• Boussinesq Equation
• VERTICAL PRESSURE BELOW A UNIFORM LOAD
• Approximate Method
• Theory based on Elastic Theory
• Uniform Load on Circular Area
• Uniform Load on Rectangular Area
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5. Introduction
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Distribution of Pressure
The pressure’s magnitude
decreases with
increasing depth.

6. Vertical Pressure below a Concentrated Load
1. Westergaard Equation
p =
P
1 − 2μ
2 − 2μ
2πz2 1 − 2μ
2 − 2μ
+
r
z
2
3
2
where p = vertical stress at depth z
P = concentrated load
µ = Poisson’s ratio
z = depth
r = horizontal distance from point of application to point at which
p is desired
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7. Vertical Pressure below a Concentrated Load
1. Westergaard Equation
• p is sometimes referred to as the vertical stress increment because it
represents stress added by the load to the stress existing prior to
application of the load.
• The stress existing prior to application of the load is the overburden
pressure.
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8. Vertical Pressure below a Concentrated Load
1. Westergaard Equation
• If Poisson’s ratio taken to be zero,
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9. Vertical Pressure below a Concentrated Load
2. Boussinesq Equation
𝑝 =
3𝑃
2𝜋𝑧2 1 +
𝑟
𝑧
2
5
2
• These equations give stress ‘p’ as a function of both the vertical
distance z and horizontal distance r.
• For low r/z ratios, the Boussinesq equation gives higher values of p
than those resulting from the Westergaard equation.
• The Boussinesq equation is more widely used.
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10. Vertical Pressure below a Concentrated Load
Equations in terms of Stress Influence Factors
• Westergaard Equation
• Boussinesq Equation
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11. Values of Iw and IB
for different values
of r/z can also be
determined from
the graph.

12. Vertical Pressure below a Concentrated Load
Example 01:
• Given:
A concentrated load of 250 tons is applied to the ground surface.
• Required:
The vertical stress increment due to this load at a depth of 20 ft directly below
the load.
• Solution:
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13. Vertical Pressure below a Concentrated Load
Example 02:
• Given:
A concentrated load of 250 tons is applied to the ground surface.
• Required:
The vertical stress increment due to this load at a point 20 ft below the ground
surface and 16 ft from the line of the concentrated load
• Solution:
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14. Vertical Pressure below a Concentrated Load
Approximate Method
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15. Influence Coefficients for Points under Uniformly Loaded Circular Area

16. Vertical Pressure below a Concentrated Load
Uniform Load on a Rectangular Area
The influence coefficient is multiplied
by the uniform load applied to the
rectangular area to determine the
pressure at depth z below each corner
of the rectangle.
For influence coefficient, read Table
using m and n.
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𝑝 = 𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 × 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑙𝑜𝑎𝑑

17. Influence Coefficients for Points under Uniformly Loaded Rectangular Area

18. Vertical Pressure below a Concentrated Load
Uniform Load on a Rectangular Area
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It is sometimes necessary to
determine the pressure below a
rectangular loaded area at points
other than directly below a corner
of the rectangular area.
For example, it may be necessary to
determine the pressure at some depth
directly below the center of a
rectangular area or at some point
outside the downward projection of
the rectangular area.

19. Vertical Pressure below a Concentrated Load
Example 03:
• Given:
A 15-ft by 20-ft rectangular foundation carrying a uniform load of 4000 lb/ft2
is applied to the ground surface.
• Required:
The vertical stress increment due to this uniform load at a point 10 ft below
the corner of the rectangular loaded area.
• Solution:
Department of Civil Engineering,Geotechnical Engineering II
𝑝 = 𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 × 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑙𝑜𝑎𝑑
For influence coefficient,
read Table using m and n.

20. Vertical Pressure below a Concentrated Load
Example 04:
• Given:
A 20-ft by 30-ft rectangular foundation carrying a uniform load of 6000 lb/ft2
is applied to the ground surface.
• Required:
The vertical stress increment due to this uniform load at a point 20 ft below
the center of the loaded area.
• Solution:
Department of Civil Engineering,Geotechnical Engineering II
𝑝 = 𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 × 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑙𝑜𝑎𝑑
For influence coefficient,
read Table using m and n.

21. Vertical Pressure below a Concentrated Load
Example 05:
• Given:
1. A rectangular loaded
area ABCD shown in
plan in Figure.
2. The load exerted on
the area is 80 kN/m2.
• Required:
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Vertical stress increment due
to the exerted load at a depth
of 3 m below point G.

22. Vertical Pressure below a Concentrated Load
Example 05:
• Solution:
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𝑝 = 𝑖𝑛𝑓𝑙𝑢𝑒𝑛𝑐𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 × 𝑢𝑛𝑖𝑓𝑜𝑟𝑚 𝑙𝑜𝑎𝑑
For influence coefficient, read Table
using m and n.
Load on 𝐴𝐵𝐶𝐷
= Load on 𝐷𝐸𝐺𝐼
− 𝐴𝐸𝐺𝐻 − 𝐶𝐹𝐺𝐼 + 𝐵𝐹𝐺𝐻

23. Thank you …
… for paying attention