Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
§2 stress in soil
• Stress due to soil weight
• Contact stress
• Stress due to loading
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
§2.2 Stress due to Self-weight
•General situation
z  z
 x  k0  z
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Three Cases:
-For Many Layers of soil, the vertical stress due to self-weight of
soil is given as following.
n
 z   1  h1   2  h2  ......   n  hn    i  hi
i 1
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
-With uniform surcharge on infinite land surface
 z    z  p   ( z  h)
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
•Effective Vertical Stress due to Self-Weight of Soil
- Consider a soil mass having a horizontal surface and with the
water table at surface level. The total vertical stress (i.e. the total
normal stress on a horizontal plane) at depth z is equal to the
weight of all material (solids + water) per unit area above that
depth ,i.e.
σv = γsat z
- 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 = γw z
- Hence the effective vertical stress at depth z will be:
σ’v=σv- u
=( γsat - γw)z= γ’z
where γ’ is the buoyant unit weight of the soil.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
-Under water
Table
γ1h1
Sand
Water table
γ1'h2
γwh2
Clay (watertight)
σz=γ1h1+γ1'h2+γwh2+γsat3h3
 z   1  h1   '1h2   w  h2   sat2  h3
General, for sand below water table, the γ’ is used; but for clay
below water table, it is difficult to determine which one(γ’ or γsat)
is suitable.
We often choose the buoyant unit weight when the index of
liquid LI>=1; the saturated unit weight when the index of liquid
LI<=0. When 0<LI<1, the disadvantageous one is choosen.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Example
The stratum’s conditions and the related physical characteristics
parameters of a foundation are shown in Fig below. Calculate the
stress due to self-weight at a,b,c. Draw the stress distribution.
w=15.6% e=0.57
γs=26.6kN/m3
w=22% wL=32% wp=23%
γs=27.3kN/m3
Civil Engineering Department of Shanghai University
For silver sand, the buoyant unit
weight (γ‘)is used :
 s  w
26.6  10
 

 9.9kN / m3
1 e
1  0.67
'
1
I L
For saturated clay,
Soil Mechanics Chapter 2
w  wp
wL  w p

22  23
 0( Semisolid )
32  23
So, the clay can be considered
the watertight; then the saturated unit weight (γsat)is used :
Gs w
Sr 
1
e
 e  Gs w  27.3 / 10  0.22  0.6
 sat 2 
 s  e w
1 e
27.3  0.6 10

 20.8kN / m 3
1  0.6
Civil Engineering Department of Shanghai University
a
σz=0
b
σz(upper)=γ’1h1=9.9×2=19.8kPa
Soil Mechanics Chapter 2
σz(Down)=γ’1h1+ γw(h1+hw)=9.9×2+10×(2+1.2)=51.8kPa
c
σz=γ’1h1+ γw(h1+hw)+ γsat2h2
= 9.9×2+10×(2+1.2)+20.8×3=114.2kPa
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Exercise 2-2
The stratum’s conditions and the related physical characteristics
parameters of a foundation are shown in Fig below. Calculate the
stress due to self-weight at 10m depth. Draw the stress distribution.
Note: For saturated clay, both cases (watertight and non-watertight)
need to consider.
w=8% e=0.7
γs=26.5kN/m3
e=1.5
γs=27.2kN/m3
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
§ 2.3 Contact stress
2.3.1 Concept of Contact pressure
♦ A foundation is the interface between a structural load and the
ground. The stress p applied by a structure to a foundation is
often assumed to be uniform. The actual pressure then applied
by the foundation to the soil is a reaction, called the contact
pressure p and its distribution beneath the foundation may be far
from uniform.
♦ This distribution depends mainly on:
· stiffness of the foundation.
i.e. flexible → stiff → rigid.
· compressibility or stiffness of the soil.
· loading conditions – uniform or point loading.
2.3.2 Contact pressure – uniform loading
The effects of the stiffness of the foundation (flexible or rigid) and
the compressibility of the soil (clay or sand) are illustrated in Figure
below..
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Figure The distribution of contact pressure
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
2.3.3 Stiffness of foundation
• A flexible foundation has no resistance to deflection
and will deform or bend into a dish-shaped profile
when stresses are applied. An earth embankment
would comprise a flexible structure and foundation.
• A stiff foundation provides some resistance to
bending and well deform into a flatter dish-shape so
that differential settlements are smaller. This forms the
basis of design for a raft foundation placed beneath the
whole of a structure.
• A rigid foundation has infinite stiffness and will not
deform or bend, so it moves downwards uniformly. This
would apply to a thick, relatively small reinforced
concrete pad foundation.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
2.3.4 Stiffness of soil
• The stiffness of a clay will be the same under all parts of
the foundation so for a flexible foundation a fairly uniform
contact pressure distribution is obtained with a dish-shaped
(sagging) settlement profile. For a rigid foundation the
dish-shaped settlement profile must be flattened out so the
contact pressure beneath the centre of the foundation is
reduced and beneath the edges of the foundation it is
increased. Theoretically, the contact pressure increases to a
very high value at the edges although yielding of the soil
would occur in practice , leading to some redistribution of
stress.
• The stiffness of a sand increase as the confining pressures
around it increase so beneath the centre of the foundation
the stiffness will be smaller. A flexible foundation of the
sand will, therefore, produce greater strains at the edges
than in the centre so the settlement profile will be dishshaped but upside-down (hogging) with a fairly uniform
contact pressure. For a rigid foundation this settlement
profile must be flattened out so the contact pressure beneath
the centre would be increased and beneath the edges it
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
2.3.5 Contact pressure-point loading
• An
analysis for contact pressure
beneath a circular raft with a point
load W at its centre resting on the
surface of an incompressible soil
(such as clay) has been provided by
Borowicka,1939.
•
This shows that the contact pressure
distribution is non-uniform irrespective
of the stiffness of the raft or foundation.
For a flexible foundation the contact
pressure is concentrated beneath the
point load which is to be expected and
for a stiff foundation it is more uniform.
For a rigid foundation the stresses
beneath
the
edges
are
very
considerably increased and a
pressure distribution similar
to the distribution produced
by a uniform pressure on a
clay
is
obtained.
This
suggests that a point load at
the centre of a rigid
foundation is comparable to a
uniform pressure.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
2.3.6 Stress distribution
•
The stresses that already exist within the ground due
to self-weight of the soil are discussed in Chapter 4.
• When a load or pressure from a foundation or structure
is applied at the surface of the soil this pressure is
distributes throughout the soil and the original normal
stresses and shear stresses ate altered. For most civil
engineering applications the changes in vertical stress
are required so the methods given below are for
increases in vertical stress only.
-Central Loading
P=P/F
P
B
p=P/F
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
-Eccentric Loading
p1 P Pe P
6e
 
 (1  )
p2 F W F
B
L
e P
(0<e<=B/6)
B
p2
p1
0<e<B/6
p1
e=B/6
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
e P
2P
p' 
B
3(  e) L
2
e P
(e>B/6)
p2
p1
p1
L
e>B/6
B
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
h
p
pa
2.3.7 Additional pressure on the bottom of foundation
¦ Ãh
B
pa=p-γh
Where pa-additional pressure;
p-contact pressure;
γ-natural unit weight of soil( bouyant unit weight
if below water surface);
h- buried depth of the foundation.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Exercise 2-3
A load of 1200kN as well as a moment of 1500 kNm is
carried on a rectangular foundation (axb=6x4m) at a
shallow depth in a soil mass as shown in Fig below.
(1) Determine the maximum & minimum contact stresses on
the bottom of foundation.
(2) If the stress is tensional, Where does the width of
foundation extend assuming the stress at right tip is zero
and the position & value of loading, the length of the
foundation a do not change?
Calculate the contact stress after the extension.
M=1500kNm
N=1200kN
x
4m
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
§ 2.4 Stress due to loading
2.4.1 Stresses beneath point load（点荷载）
• Boussinesq published in 1885 a solution for the stresses
beneath a point load on the surface of a material which
had the following properties:
• semi-infinite – this means infinite below the surface
therefore providing no boundaries of the material apart
from the surface
• homogeneous – the same properties at all locations
• isotropic –the same properties in all directions
• elastic
–a linear stress-strain relationship.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Linear elastic assumption
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
P
X
r
β
x
y
σz
R
Y
z
τzy
τyx
Z
M
σy
(x,y,z)
τyz
τzx
xz
τ τx y
σx
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
3P Z 3
z 
 5
2 R
Substitute Z/R for Cos β,
3 p cos 3 
z 

2
R2
z 
3P

2
2Z
1
 r
1   
  Z 

2



5
2

3
 r
2 1   
  Z 
2



5
2
P
Z2
Properties
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
p
x
z
below point a
below direction Z
along horizontal surface
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
P
σz
Bulbs of pressure(压力泡)
Isoline(等值线)——Lines or Contours
of equal stress increase
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Soil Mechanics Chapter 2
P2
P1
σz2
σz1
σz1 +σz2
Z
Z
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
８
2.4.2 Stresses due to line load（线荷载）
pdy
X
Y
p
X
８
Z
τzx
σx
τxz
Z
M(X,Z)
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
3Z 3 pdy 3 pdy
Z3
d z 


5
2R
2
X 2 Y 2  Z2


 z   d z 

2 pZ 3
 X  Z
2




2
1


   X 2 
1    
  Z  

2 2
2
p
 
Z

5
2
Civil Engineering Department of Shanghai University
-polar coordinates
Z
R  X  Z , cos  
R0
2
0
2
2
2p
3
z 
cos 
R0
2p
x 
cos  sin 2 
R0
2p
 xz 
sin  cos 2 
R0
Soil Mechanics Chapter 2
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
８
2.4.3 Stresses due to uniform vertical loading
on an infinite strip （条形荷载）
b/2
b/2
Po dξ
Po
X(ξ)
dξ
ξ
Z
σz
τzx
σx
X
Z
τxz
M(X,Z)
Civil Engineering Department of Shanghai University
d z 
z  
b2
b / 2

2Z 3 p0 d
 X     Z
2
d z  
b2
b / 2

Soil Mechanics Chapter 2

2 2
2 Z 3 p 0 d
 X     Z
2

2 2

b
b 


b
b
Z X  
Z X   

X
X
p0 
2
2 


2  arctg
2

arctg

2
2

 
Z
Z
b
b




2
2
Z X  
Z X   

2
2  



   p0
Civil Engineering Department of Shanghai University

Soil Mechanics Chapter 2
-polar coordinates
b
dx
P0
A
β1
R0 d
dx 
cos 
X
B
0
β
β2
R0
Z
β
σz
dβ
Z
M
R0 d 


2 p0 
cos  
2 p0 dx
2 p0

3
3
d z 
cos  
cos  
cos 2 d
R0
R0

Civil Engineering Department of Shanghai University
1
 z   d z 
2
2 p0

1

Soil Mechanics Chapter 2
cos 2 d
2
p0 
1
1








sin
2





sin

2

1
1
2
2 

 
2
2

p0 
1
1





x 


sin
2





sin
2

1
1
2
2 
 
2
2

p0
cos 2  2  cos 2 1 
 xz 
2
-Principal stress
2
1  X   Z




2
Z 

  X
   xz
3
2
2


Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
 1 p0
  sin  

3 
ψ  β1  β2
(Out) or
tg 2  tg ( 1   2 ),
 max
then
ψ  β1  β2

1   2
p0
1
  1   3  
Sin
2

Properties
2
(In)
(1) Isoline(等值线)
(2) Direction of principal stress
(3) The value of maximum shear stress
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
b
P0
A
Χ
B
β2
β1
θ
ψ/2
ψ
Μ
σ3
Ζ
σ1
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
2.4.4 Stresses due to Linearly increasing vertical
loading on an infinite Strip
Y
∝
b
ξP0
b
-X
dξ
P0
Ο
X
dξ
ξ
Ζ
Χ-ξ
σΖ
X
Z
Μ( X , Z )
Civil Engineering Department of Shanghai University
d z 

2Z 3
 X     Z
2
2Z 3 p0
z 
b
p0  X


  b
   p0
b

2 2
p0d

b
d
  X   
0
Soil Mechanics Chapter 2
2
Z

2 2

X
X b
Z  X  b  
 arctg  arctg

 2
2 

Z
Z
Z  Z  b  

Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
2.4.5 Stresses due to a uniform loaded rectangular area
-central point method（中点法）
a/2
a/2
dn
r
b/
2
o
dξ
n
b/
2
ξ
R
z
σΖ
Y
m( x.y.z )
z
x
Civil Engineering Department of Shanghai University

R     Z
2
2
Soil Mechanics Chapter 2

2 12
3Z 3 p 0 dd
z   
 a 2 b 2
2R 5
2 p0 
ab
2abZ a 2  b 2  8Z 2


arctg
 
2Z a 2  b 2  4Z 2
a 2  4Z 2 b 2  4Z 2 a 2  b 2  4Z 2
 a0 p0
a 2 b2








Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
-Corner method（角点法）
a/2
a/2
po
b/
2
ξ
dξ
R
dn n
b/
2
x
POd
o
C
po
z
σΖ
Y
M( x.y.z )
z
3Z 3 p0 dd
Z   
 a / 2 b / 2
2R 5

p0 
abZ a 2  b 2  2 Z 2
ab

 arctg
 2

2
2
2
2
2
2
2
2
2
2  a  Z b  Z a  b  Z
Z a  b  Z 
  c  p0
a/2

b/2






mn(m 2  2n 2  1)
m
 arctg
 2

2
2
2
2
2
2
n m  n  1 
 (m  n )(1  n ) m  n  1
a
z
Where, m  , n 
b
b
1
c 
2
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Principle of superposition
For stresses beneath
points other than the
corner of the loaded
area the principle of
superposition should be
used, as described in
Fig below.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Stresses beneath flexible area of any shape
 Stresses beneath flexible area of any shape (Figure 5.9)
 Newmark (1942) devised charts to obtain the vertical stress at any
depth, beneath any point (inside or outside) of an irregular shape.
Use of the charts is explained in Figure 2.9.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Stresses beneath a flexible rectangle
♦ Stresses
beneath a flexible
rectangle (figure 2.7)
 The vertical stressσ y at a
depth z beneath the corner
of a flexible rectangle
supporting a uniform
pressure q has been
determined using:
σ v= qI
and influence factors I. given
by Giroud (1970) are
presented in Figure 5.7. they
are for an infinite soil
thickness.
 These curves are equivalent
to the commonly used charts
of Fadum (1948) but are
easier to use.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Stresses beneath a rigid rectangle
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Exercise 2-4
A point load of P is applied at the ground surface.
Calculate the stress and plot the variation:
(1) Vertical stress due to load P along vertical
line(r=0) at the different depths Z=1,2,3,4,5m.
(2) Vertical stress due to load P from different
distances (r=0,1,2,3,4,5m, respectively) along
horizontal surface (at the depth Z=2m).
(3) Vertical stress due to load P along vertical
line(r=1.5m) at the different depths
Z=0,0.5,1,2,3,4,5,6m.
Civil Engineering Department of Shanghai University
Soil Mechanics Chapter 2
Exercise 2-5
A uniform load of p(=20kpa) is applied at the ground
surface as shown in Fig below.
Calculate the stress due to load p beneath point A at 5m
depth.