Stresses 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
z2
z3
∗
∗ ∗
∗ ∗ ∗
3Dr. Abdulmannan Orabi IUST

With uniform surcharge on infinite land
surface
Con version land surface
Origina
lland
surface
∗

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

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

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

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.

 Pressure, u
Stresses in Saturated Soil with Downward
Seepage
Pore water
Total
stress
Effective
stress
Dept
h
Dept
h
Dept
h

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.

• 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

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.

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.

Vertical stress due to a concentrated load
•Boussinesq’s Formula
•Wastergaard Formula

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

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.

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

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).

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.

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.

Distribution on a horizontal plane
The vertical stress distribution on a
horizontal plane at depth of z below the
ground surface
U
5
5

The vertical stress
distribution on a
vertical plane at a
distance of r from the
point load
.
Distribution on a vertical
plan
e
O

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'

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
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.
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

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

Stresses in soil

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