Transcript Consolidation - Sepuluh Nopember Institute of Technology

Shear Strenght
of a Soil Mass
updated May 29, 2007
Haryo Dwito Armono, M.Eng, Ph.D


How to know the state of stress
at one or more points in the
problem being considered?
z
A
Point A at depth z
A soil mass at A is enlarged and
assumed as forces at point A act
in two dimensional plane
F1
F2
F3
F5
N =  x 1
O
F4

C

T =  x 1
N = x sin 
Area = 1
Area = 1 sin 

B
A
Area = 1 cos 
V = y cos 
We could resolve these
forces into component on
a small element at any
point such as point O
Sign convention
+
 Compressive stresses are positive
 Tensile stresses are negative
-
 Decrease in length is positive
 Increase in length is negative
+
-
 Positive shear forces produce
clockwise moment about a point
outside the element
 Opposite of structural mechanics
+
+


N =  x 1
Fh  N  T cos   N sin   0
Fv  V  T sin   N cos   0
C
T =  x 1
N = x sin 
Area = 1
Area = 1 sin 
 x sin     cos     sin   0
 y cos     sin     cos   0

B
A
Area = 1 cos 
V = y cos 
Solving simultaneously for  and  :
    x sin    y cos  
2
2
   ( x   y ) sin  cos  
x  y
2
 x  y
2

 x  y
2
sin 2
cos 2
 
 


x  y
2
 x  y
2

 x  y
2
cos 2
sin 2
Square both equation and add together to get
equation of a Cirlcle  MOHR CIRCLE
Which represent the state of stress at a point at
equilibrium
radius 
center 
 x  y
2
x  y
2

1   3
2
 3
 1
2
Principal stresses:
 x   1  major principal stress (largest)
 y   3  minor principal stress (smallest)
 z   2  intermediate principal stress (not shown as 2D derivation)
act on a plane where  = 0
 
1   3

1   3
2
2
 
   1 3 sin 2
2
cos 2
 Any straight line drawn through the pole will intersect the Mohr
circle at a point which represent the state of stress on a plane
inclined at the same orientation in space as the line
Example 10.2



1.
Stresses on an element as shown.
Found the normal stress  and shear stress 
on the plane inclined at  = 35o from reference
plane which rotated 20o from horizontal.
Solution:
Plot Mohr circle with
1   3
52  12
 20 kPa
2
2
   3 52  12
center  1

 32 kPa
2
2
radius 
2.
3.
4.

Find the pole of circle.
Draw a line paralel to a plane on which you
know the stresses.
This plane is inclined at an angle of 20o to the
horizontal.
Start at point A, where the line intersect the
Mohr circle defines the pole P.
Find the stresses on  plane, which is inclined
at 35o to the base of the element (Point C)
Scale off th coordinates of Point C to
determine   39 kPa and  = 18.6 kPa
Example 10.3
A


B


1.
2.
3.
4.
5.
The stress shown
Find  and  when   30o and
determine orientation of major and
minor principle of stresses
Find max and orientation of plane
upon which it acts
Solution:
Plot known Points A and B and
drawn circle
Draw line parallel to plane upon
which Point A acts to determine Pole
Draw line at   30o and reference
plane.
  1.8 kPa and  = 5.3 kPa
From circle. Determine 1 and 3
graphically : 1 = 6.4 kPa and 3 = 4.4 kPa and their orientation : 1 =
11o and 3 = - 101o
For max we can compute from
 
equation    x y sin 2
2
where   90o or graphically max =
5.4 kPa
What this all means?
 If you can determine the shear and normal stress
acting on a potential failure plane, then you can
determine whether or not failure has occurred…
 If     c   n tan  where    n
then soil fails
  c   n tan  is Mohr - Coloumb Failure Criterion





A


c
3

1
Furthermore…


 f  (45  )
failure
2
available
  max   failure ; this is due to the fact

the  available   max
max
c’

45o
3f
failure
1f
 for over consolidated clay (c  0)
1  sin  2 c cos 
1   3

1  sin  1  sin 
 for sand (c' = 0) and normally consolidated clay (c=0)
 1 1  sin 


 tan 2 (45  )
 3 1  sin 
2
If c’=0
c  cohesion
  angle of internal friction


failure
available
max
3
f
failure

1
Shear Strength and Behavior
1 – 3
 Cohesionless Soils: i.e, gravel, sand, silt
dense

Loose sand (high void ratio)
shearing
loose
Volume
decrease
1

Dense sand (low void ratio)
shearing
Volume
Expansion
increase
dense

1
loose
Compression
A critical void ratio exist at which no volume
change occurs upon shearing
Type of shear strength test
1.
2.
3.
4.
5.
Direct shear test
Triaxial compression test
Unconfined compression test
Penetration test
Vane shear test
1
2
3
4
5
(lab)
(lab)
(lab)
(field)
(field)
Direct Shear Test (cohesionless material)
Triaxial Compression Test
1
2 = 3c
3
Soil cylinders is subjected to different
normal and shear stress
Mohr circle plot gives c and  (shear
strength parameters)
Unconfined Compression Test
Only for cohesive soil
Cylinders of soil ested with 3 = 0
 3= 0

failure
 0

3
1
1 to cause failure is found
Penetration Test
Measures resistance to penetration
 Dynamic Penetration : Standard Penetration Test (SPT)

Disadvantages
 Uncertainties of energy transmission in blows
 Slow test

Advantages
 Soil sample is obtained
 Static Penetration Test : Cone Penetration Test (CPT)

Advantages
 Point resistance, side friction and pore water pressure mesured
 No energy transmission problem
 Fast

Disadvantages
 No soil sample is obtained
Vane Shear Test
dr
T
r
Top and bottom of
soil cylinder
h
d
The torque required to rotate the
vane in the soil is measured
T  area of blue cylinder acting at d/2 x Shearforce
+ perimeter of hatched area (top & bottom cylinder)
acting at the center distance (r) x Shearforce
The shear strength is mobilised
near to the cylinder surface also
in the circular surfaces at top and
bottom of the vane
2
d
T   .d .h  S   2  2 rdr.S .r
2
0
d


2

T = max torque applied (kNm)
n = vane height (m)
d = vane diameter (m)
S = shear strength of soil
d
2
d hS  4 S  r 2 dr
2
0
d
 r3  2
 d hS  4 S  
2
 3 0

2
d3
2
6

d3 
 S  d 2h 
6 
2

d 2 hS   S
Factors affect shear stresses of sand
 Lower void ratio or higher relative density
 Increasing angularity
 Better graded
 Greater surface roughness
Bibliography
Holtz, R.D and Kovacs, W.D., “An Introduction to Geotechnical
Engineering”
Das, B.M., “Soil Mechanics”
Das, B.M., “Advanced Soil Mechanics”

failure
available
max
c’

45o


3f
failure
1f

failure
available
max

3
failure
1
F1
F2
F3
F5
N =  x 1
O
F4

C
T =  x 1
N = x sin 
Area = 1
Area = 1 sin 

B
A
Area = 1 cos 
V = y cos 
1 – 3
Volume
Expansion
dense
dense
loose

1
loose
Compression

1