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Stress Distribution
in soil
Stress distribution in soils
Stress in soil can be caused by the
following:
 Stress in soil due to self-weight
 Stress in soil due to surface load
 i hi
i1
z  1 h1 2 h2 ......n hn
Stresses due to self-weight
Stresses in a Layered Deposit
The stresses in a deposit consisting of layers
of soil having different densities may be
determined as
Vertical stress at depth
z1
Vertical stress at depth
z2
Vertical stress at depth
z3
∗
∗ ∗
∗ ∗ ∗
3Dr. Abdulmannan Orabi IUST
With uniform surcharge on infinite land
surface
Con version land surface
Origina
lland
surface
∗
4Dr. Abdulmannan Orabi IUST
Vertical Stresses
Vertical stresses due to self weight increase
with depth,
There are 3 types of geostatic stresses:
a.Total Stress, σtotal
b.Effective Stress, σ'
c.Pore Water Pressure, u
Total Stress = Effective stress + Pore Water
Pressure
σtotal = σ' + u
∗ 5IUST
Total vertical stress
Consider a soil mass having a horizontal
surface and with the water table at surface
level. The total vertical stress at depth z is
equal to the weight of all material (solids +
water) per unit area above that depth
σ = γ. z
6Dr. Abdulmannan Orabi IUST
Pore water pressure
If the pores of a soil mass are filled with
water and if a pressure induced into the
pore water, tries to separate the grains, this
pressure is termed as pore water pressure
The pore water pressure at any depth will be
hydrostatic since the void space between the
solid particles is continuous, therefore at
depth z:
U= γ. z
7Dr. Abdulmannan Orabi IUST
Effective vertical stress due to self-weight of
soil
The pressure transmitted through grain to grain
at the contact points through a soil mass is
termed as effective pressure.
The difference between the total stress (σtotal )
and the pore pressure (u) in a saturated soil has
been defined by Terzaghi as the effective stress
(σ').
σ'total = σtotal-u
8Dr. Abdulmannan Orabi IUST
Stresses in Saturated Soil
If water is seeping, the effective stress at any
point in a soil mass will differ from that in
the static case.
It will increase or decrease, depending on
the direction of seepage.
The increasing in effective pressure due
to the flow of water through the pores
of the soil is known as seepage pressure.
9Dr. Abdulmannan Orabi IUST
 Pressure, u
Stresses in Saturated Soil with Downward
Seepage
Pore water
Total
stress
Effective
stress
Dept
h
Dept
h
Dept
h
10Dr. Abdulmannan Orabi IUST
Stresses due to surface load
Introduction
To analyze problems such as
compressibility of soils, bearing capacity
of foundations, stability of
embankments, and lateral pressure on
earth retaining structures, we need to
know the nature of the distribution of
stress along a given cross section of
the soil profile.
11Dr. Abdulmannan Orabi IUST
• When a load is applied to the soil surface, it
increases the vertical stresses within the
soil mass. The increased stresses are
greatest directly under the loaded area, but
extend indefinitely in all directions.
• Allowable settlement, usually set by
building codes, may control the allowable
bearing capacity.
• The vertical stress increase with depth must
be determined to calculate the amount of
settlement that a foundation may undergo
12Dr. Abdulmannan Orabi IUST
Foundations and structures placed on
the surface of the earth will produce
stresses in the soil.
These stresses will decrease with the
distance from the load.
How these stresses decrease depends
upon the nature of the soil bearing the
load.
13Dr. Abdulmannan Orabi IUST
Stress Due to a Concentrated Load
Individual column footings or wheel
loads may be replaced by equivalent
point loads provided that the stresses
are to be calculated at points
sufficiently far from the point of
application of the point load.
14Dr. Abdulmannan Orabi IUST
Stress Due to a Concentrated Load
Vertical stress due to a concentrated load
•Boussinesq’s Formula
•Wastergaard Formula
15Dr. Abdulmannan Orabi IUST
Stress Due to a Concentrated Load
Boussinesq’s Formula for Point
LoadsJoseph Valentin Boussinesq (13 March 1842 – 19 February 1929) was
a French mathematician and physicist who made significant
contributions to the theory of hydrodynamics, vibration, light, and
heat.
In 1885, Boussinesq developed the mathematical relationships for
determining the normal and shear stresses at any point inside a
homogenous, elastic and isotropic mediums due to a concentrated
point loads located at the surface
16Dr. Abdulmannan Orabi IUST
Assumptions:
 The soil mass is elastic, isotropic (having identical
properties in all direction throughout), homogeneous
(identical elastic properties) and semi-infinite depth.
 The soil is weightless.
 The distribution of σz in the elastic medium is apparently
radially symmetrical.
 The stress is infinite at the surface directly beneath the
point load and decreases with the square of the depth.
 At any given non-zero radius, r, from the point of load
application, the vertical stress is zero at the surface,
increases to a maximum value at a depth where E
39.25°
, approximately, and then decreases with depth.
Stress Due to a Concentrated Load
According to Boussinesq’s analysis, the vertical
stress increase at point A caused by a point load of
magnitude P is given by
D
∆
∆ M
∆ N
O
P
Q
.
P
D
1
18Dr. Abdulmannan Orabi IUST
Stresses due to concentrated loa
•Let P (x, y, z) be the point in the
soil mass where vertical stresses
are to be determined due to applied
load Q on the ground surface.
• By Boussinesq’s solution polar radial stress
at P(x, y, z) is
Where,
R=polar distance between the origin O and
point P
Β=angle which line PQ makes with vertical
................(i
)
 The vertical stress at point P,
 With substituting equation (i)
OR
Where,
= Boussinesq influence coefficient
for the vertical stress.
Equation shows that the vertical stress is
 Directly proportional to the load
 Inversely proportional to the depth squared, and
 Proportional to some function of the ratio (
r/z).
24Dr. Abdulmannan Orabi IUST
Stress Due to a Concentrated Load
It should be noted that the expression for z
is independent of elastic modulus (E) and
Poisson’s ratio (µ), i.e. stress increase with
depth is a function of geometry only.
25Dr. Abdulmannan Orabi IUST
Pressure Distribution Diagram
 Equation may be used to draw three types of
pressure distribution diagram. They are:
 The vertical stress distribution on a horizontal plane at
depth of z below the ground surface
 The vertical stress distribution on a vertical plane at a
distance of r from the load point, and
 The stress isobar.
26Dr. Abdulmannan Orabi IUST
Distribution on a horizontal plane
The vertical stress distribution on a
horizontal plane at depth of z below the
ground surface
U
5
5
27Dr. Abdulmannan Orabi IUST
The vertical stress
distribution on a
vertical plane at a
distance of r from the
point load
.
Distribution on a vertical
plan
e
O
28Dr. Abdulmannan Orabi IUST
Stress isobar:
 An isobar is a line which connects all points
of equal stress below the ground surface. In
other words, an isobar is a stress contour. We
may draw any number of isobars as shown in
Fig. for any given load system.
 Each isobar represents a fraction of the load
applied at the surface. Since these isobars
form closed figures and resemble the form of a
bulb, they are also termed bulb of pressure or
simply the pressure bulb.
 Normally isobars are drawn for vertical,
horizontal and shear stresses. The one that is
most important in the calculation of
settlements of footings is the vertical pressure
isobar.
LINE LOADS:
 By applying the principle of the above theory, the stresses at any
point in the mass due to a line load of infinite extent acting at the
surface may be obtained.
 The state of stress encountered in this case is that of a plane
strain condition. The strain at any point P in the Y-direction
parallel to the line load is assumed equal to zero. The stress бy
normal to the XZ-plane is the same at all sections and the shear
stresses on these sections are zero.
 The vertical бz stress at point P may be written in rectangular
coordinates as
 where, / z is the influence factor equal to 0.637 at x/z =0.
32
STRIP LOADS
 Such conditions are found for structures extended very much in one
direction, such as strip and wall foundations, foundations of
retaining walls, embankments, dams and the like.
34
 Fig. shows a load q per unit area acting on a strip of infinite length
and of constant width B. The vertical stress at any arbitrary point
P due to a line load of qdx acting at can be written from Eq. as
 Applying the principle of superposition, the total stress бz at point
P due to a strip load distributed over a width B(= 2b) may be
written as
 The non-dimensional values can be expressed in a more convenient
form as
35
Vertical Stress due to a uniformly loaded circular
area
6
6
Q
6'
37Dr. Abdulmannan Orabi IUST
At point A we can calculate the vertical stress. Assume small element with
area rdφ.dr of the uniform load q from Boussinesq’s theory
dQ = qdr.rdφ
σz = q
σz = q . A
This equation when the point A lies under C.G of uniform load
To calculate vl stress to point I which has distance equal r
σz = q (A + B)
Vertical Stress Caused by a Rectangular loaded
area
39Dr. Abdulmannan Orabi IUST
Newmark (1935) has derived an
expression for the vertical stress at a
point below the corner of a
rectangular area loaded uniformly as
shown in Figure. The following is the
popular form of Newmark's equation
for σz : which is widely used for the
calculation purpose.
3
Iq
Z
L
Z
B


n
m
I3 is a function of m and
n
NEWMARK’S INFLUENCE CHARTS
 The Newmark’s Influence Chart is useful for the
determination of vertical stress(σ) at any point below
the uniformly loaded area of any shaped.
 This method is based on the concept of the vertical
stress at point below the centre of uniformly loaded
circular area A charts, consisting of number of circles
and radiating lines, is so prepared that the influence of
each area unit is the same at the centre of the circles,
i.e. each area unit causes the equal vertical stress at the
centre of the circle.
Stress = (IF).q.M
Here
IF = Influence Factor, which we have
taken equal to 0.005
q = pressure intensity at top.
M = Number of elements of the chart
covered by the prepared plan.
B
B +
z
2
1
z
"
O
Approximate Method
2V:1H method
A simple but approximate method is sometimes
for calculating the stress change at various
a result of the application of a pressure at the
surface.
45Dr. Abdulmannan Orabi IUST
The transmission of stress is
assumed to follow outward
fanning lines at a slope of 1
horizontal to 2 vertical.
B +
z
L
z
B
1
2
2V:1H method
Stress on this
plane
B
j
"
d ∗ 
Stress on this plane at
depth z, Rectangular
footing
, d - , 
-
" d

B +
z
46Dr. Abdulmannan Orabi IUST
Approximate method for rectangular loads
In preliminary analyses of vertical stress increase under the
center of rectangular loads, geotechnical engineers often use an
approximate method (sometimes called the 2:1 method).
The vertical stress increase under the
center of the load is
z =
(B  z)(L  z)
qsBL
4
7

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Stresses in soil

  • 2. Stress distribution in soils Stress in soil can be caused by the following:  Stress in soil due to self-weight  Stress in soil due to surface load
  • 3.  i hi i1 z  1 h1 2 h2 ......n hn Stresses due to self-weight Stresses in a Layered Deposit The stresses in a deposit consisting of layers of soil having different densities may be determined as Vertical stress at depth z1 Vertical stress at depth z2 Vertical stress at depth z3 ∗ ∗ ∗ ∗ ∗ ∗ 3Dr. Abdulmannan Orabi IUST
  • 4. With uniform surcharge on infinite land surface Con version land surface Origina lland surface ∗ 4Dr. Abdulmannan Orabi IUST
  • 5. Vertical Stresses Vertical stresses due to self weight increase with depth, There are 3 types of geostatic stresses: a.Total Stress, σtotal b.Effective Stress, σ' c.Pore Water Pressure, u Total Stress = Effective stress + Pore Water Pressure σtotal = σ' + u ∗ 5IUST
  • 6. Total vertical stress Consider a soil mass having a horizontal surface and with the water table at surface level. The total vertical stress at depth z is equal to the weight of all material (solids + water) per unit area above that depth σ = γ. z 6Dr. Abdulmannan Orabi IUST
  • 7. Pore water pressure If the pores of a soil mass are filled with water and if a pressure induced into the pore water, tries to separate the grains, this pressure is termed as pore water pressure The pore water pressure at any depth will be hydrostatic since the void space between the solid particles is continuous, therefore at depth z: U= γ. z 7Dr. Abdulmannan Orabi IUST
  • 8. Effective vertical stress due to self-weight of soil The pressure transmitted through grain to grain at the contact points through a soil mass is termed as effective pressure. The difference between the total stress (σtotal ) and the pore pressure (u) in a saturated soil has been defined by Terzaghi as the effective stress (σ'). σ'total = σtotal-u 8Dr. Abdulmannan Orabi IUST
  • 9. Stresses in Saturated Soil If water is seeping, the effective stress at any point in a soil mass will differ from that in the static case. It will increase or decrease, depending on the direction of seepage. The increasing in effective pressure due to the flow of water through the pores of the soil is known as seepage pressure. 9Dr. Abdulmannan Orabi IUST
  • 10.  Pressure, u Stresses in Saturated Soil with Downward Seepage Pore water Total stress Effective stress Dept h Dept h Dept h 10Dr. Abdulmannan Orabi IUST
  • 11. Stresses due to surface load Introduction To analyze problems such as compressibility of soils, bearing capacity of foundations, stability of embankments, and lateral pressure on earth retaining structures, we need to know the nature of the distribution of stress along a given cross section of the soil profile. 11Dr. Abdulmannan Orabi IUST
  • 12. • When a load is applied to the soil surface, it increases the vertical stresses within the soil mass. The increased stresses are greatest directly under the loaded area, but extend indefinitely in all directions. • Allowable settlement, usually set by building codes, may control the allowable bearing capacity. • The vertical stress increase with depth must be determined to calculate the amount of settlement that a foundation may undergo 12Dr. Abdulmannan Orabi IUST
  • 13. Foundations and structures placed on the surface of the earth will produce stresses in the soil. These stresses will decrease with the distance from the load. How these stresses decrease depends upon the nature of the soil bearing the load. 13Dr. Abdulmannan Orabi IUST
  • 14. Stress Due to a Concentrated Load Individual column footings or wheel loads may be replaced by equivalent point loads provided that the stresses are to be calculated at points sufficiently far from the point of application of the point load. 14Dr. Abdulmannan Orabi IUST
  • 15. Stress Due to a Concentrated Load Vertical stress due to a concentrated load •Boussinesq’s Formula •Wastergaard Formula 15Dr. Abdulmannan Orabi IUST
  • 16. Stress Due to a Concentrated Load Boussinesq’s Formula for Point LoadsJoseph Valentin Boussinesq (13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. In 1885, Boussinesq developed the mathematical relationships for determining the normal and shear stresses at any point inside a homogenous, elastic and isotropic mediums due to a concentrated point loads located at the surface 16Dr. Abdulmannan Orabi IUST
  • 17. Assumptions:  The soil mass is elastic, isotropic (having identical properties in all direction throughout), homogeneous (identical elastic properties) and semi-infinite depth.  The soil is weightless.  The distribution of σz in the elastic medium is apparently radially symmetrical.  The stress is infinite at the surface directly beneath the point load and decreases with the square of the depth.  At any given non-zero radius, r, from the point of load application, the vertical stress is zero at the surface, increases to a maximum value at a depth where E 39.25° , approximately, and then decreases with depth.
  • 18. Stress Due to a Concentrated Load According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by D ∆ ∆ M ∆ N O P Q . P D 1 18Dr. Abdulmannan Orabi IUST
  • 19. Stresses due to concentrated loa •Let P (x, y, z) be the point in the soil mass where vertical stresses are to be determined due to applied load Q on the ground surface. • By Boussinesq’s solution polar radial stress at P(x, y, z) is Where, R=polar distance between the origin O and point P Β=angle which line PQ makes with vertical ................(i )
  • 20.  The vertical stress at point P,  With substituting equation (i) OR Where, = Boussinesq influence coefficient for the vertical stress.
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  • 24. Equation shows that the vertical stress is  Directly proportional to the load  Inversely proportional to the depth squared, and  Proportional to some function of the ratio ( r/z). 24Dr. Abdulmannan Orabi IUST
  • 25. Stress Due to a Concentrated Load It should be noted that the expression for z is independent of elastic modulus (E) and Poisson’s ratio (µ), i.e. stress increase with depth is a function of geometry only. 25Dr. Abdulmannan Orabi IUST
  • 26. Pressure Distribution Diagram  Equation may be used to draw three types of pressure distribution diagram. They are:  The vertical stress distribution on a horizontal plane at depth of z below the ground surface  The vertical stress distribution on a vertical plane at a distance of r from the load point, and  The stress isobar. 26Dr. Abdulmannan Orabi IUST
  • 27. Distribution on a horizontal plane The vertical stress distribution on a horizontal plane at depth of z below the ground surface U 5 5 27Dr. Abdulmannan Orabi IUST
  • 28. The vertical stress distribution on a vertical plane at a distance of r from the point load . Distribution on a vertical plan e O 28Dr. Abdulmannan Orabi IUST
  • 29. Stress isobar:  An isobar is a line which connects all points of equal stress below the ground surface. In other words, an isobar is a stress contour. We may draw any number of isobars as shown in Fig. for any given load system.  Each isobar represents a fraction of the load applied at the surface. Since these isobars form closed figures and resemble the form of a bulb, they are also termed bulb of pressure or simply the pressure bulb.  Normally isobars are drawn for vertical, horizontal and shear stresses. The one that is most important in the calculation of settlements of footings is the vertical pressure isobar.
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  • 32. LINE LOADS:  By applying the principle of the above theory, the stresses at any point in the mass due to a line load of infinite extent acting at the surface may be obtained.  The state of stress encountered in this case is that of a plane strain condition. The strain at any point P in the Y-direction parallel to the line load is assumed equal to zero. The stress бy normal to the XZ-plane is the same at all sections and the shear stresses on these sections are zero.  The vertical бz stress at point P may be written in rectangular coordinates as  where, / z is the influence factor equal to 0.637 at x/z =0. 32
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  • 34. STRIP LOADS  Such conditions are found for structures extended very much in one direction, such as strip and wall foundations, foundations of retaining walls, embankments, dams and the like. 34
  • 35.  Fig. shows a load q per unit area acting on a strip of infinite length and of constant width B. The vertical stress at any arbitrary point P due to a line load of qdx acting at can be written from Eq. as  Applying the principle of superposition, the total stress бz at point P due to a strip load distributed over a width B(= 2b) may be written as  The non-dimensional values can be expressed in a more convenient form as 35
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  • 37. Vertical Stress due to a uniformly loaded circular area 6 6 Q 6' 37Dr. Abdulmannan Orabi IUST
  • 38. At point A we can calculate the vertical stress. Assume small element with area rdφ.dr of the uniform load q from Boussinesq’s theory dQ = qdr.rdφ σz = q σz = q . A This equation when the point A lies under C.G of uniform load To calculate vl stress to point I which has distance equal r σz = q (A + B)
  • 39. Vertical Stress Caused by a Rectangular loaded area 39Dr. Abdulmannan Orabi IUST Newmark (1935) has derived an expression for the vertical stress at a point below the corner of a rectangular area loaded uniformly as shown in Figure. The following is the popular form of Newmark's equation for σz : which is widely used for the calculation purpose.
  • 41. NEWMARK’S INFLUENCE CHARTS  The Newmark’s Influence Chart is useful for the determination of vertical stress(σ) at any point below the uniformly loaded area of any shaped.  This method is based on the concept of the vertical stress at point below the centre of uniformly loaded circular area A charts, consisting of number of circles and radiating lines, is so prepared that the influence of each area unit is the same at the centre of the circles, i.e. each area unit causes the equal vertical stress at the centre of the circle.
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  • 44. Stress = (IF).q.M Here IF = Influence Factor, which we have taken equal to 0.005 q = pressure intensity at top. M = Number of elements of the chart covered by the prepared plan.
  • 45. B B + z 2 1 z " O Approximate Method 2V:1H method A simple but approximate method is sometimes for calculating the stress change at various a result of the application of a pressure at the surface. 45Dr. Abdulmannan Orabi IUST The transmission of stress is assumed to follow outward fanning lines at a slope of 1 horizontal to 2 vertical.
  • 46. B + z L z B 1 2 2V:1H method Stress on this plane B j " d ∗ Stress on this plane at depth z, Rectangular footing , d - , - " d B + z 46Dr. Abdulmannan Orabi IUST
  • 47. Approximate method for rectangular loads In preliminary analyses of vertical stress increase under the center of rectangular loads, geotechnical engineers often use an approximate method (sometimes called the 2:1 method). The vertical stress increase under the center of the load is z = (B  z)(L  z) qsBL 4 7