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SHEAR STRENGTHSHEAR STRENGTH
In general, the shear strength of any material is the load per unit
area or pressure that it can withstand before undergoing
shearing failure.
When you hear “Shear Failure” you probably think of Shearing
Pins or Bolts.How do these fail?
Shearing Pins can be used to fasten together two steel plates:With high enough plate forces in
opposite directions…
Each pin has sheared into two pieces.
So shear forces are those that tend to cause shear failure.
The failure plane for metals will be parallel to the external shear forces.
The internal shear stress, τ is simply the shear force, T acting on the failure plane divided by the area, A of the failure plane:
Area, A
AT
τ
If the shear force causes failure, then the shear stress that results, τf is the shear strength of the material.
The Shear Force that acts on the failure plane is resisted by the
strength of the material.
Since the external force is acting parallel to the failure plane, the
internal strength of the material is thought of as its internal friction, F. This is the material’s reaction to the external shear force, T.
to overcome the friction force, F on the plane where the object
rests
that must be applied to an object of known weight, W
The tangent of the friction angle, is the ratio of F to W which is also known as the coefficient of friction.Friction problems in mechanics
determine the external force, T
WW
and thereby cause the object to move.
The object’s weight vector, W acts normal to the failure plane.
Vector addition gives the resultant vector, R which acts at an angle of
WRT the normal to the plane.
SHEAR STRENGTH SHEAR STRENGTH IN SOILSIN SOILS
The loading of a material that undergoes shear failure is not always parallel to the failure
plane.
Consider and element of soil within a large soil mass:
Soil Surface
Bedrock
Soil Mass
If the soil is loaded (yet sober):
Soil Element
The load transmits stress to the element by inter-particle contacts.
This is the major principal stress distribution, designated σ1 due to
the load.
For visual simplicity we replace the distributed load with an
equivalent point load.
σ1
The soil below the element will react with a stress of equal
magnitude but directed upwards so it too is designated σ1.
The element squeezed vertically will tend to bulge horizontally to which
the soil reacts with confining pressures σ2 and σ3 in the other
principal directions.
Since we assume the soil is isotropic, the confining lateral pressure will be the same in all directions and so σ2 =
σ3 allowing us to view it in 2 dimensions.
σ1
σ3
σ1
σ3
σ2
σ2
But what has this got to do with SHEAR STRENGTH?
Soil undergoes shear failure when one portion moves relative to the
rest.
For this to happen, a failure plane develops within the soil.
σ1
σ1
2-D
The friction force on this failure plane is overcome by the external
forces and viola:
Θ
SHEAR FAILURE!
The Shear Stress at failure, τf, is the pressure required to
overcome the friction on the surface of the failure plane (a.k.a.
Shear Strength).
The angle of internal friction, characterizes the shear strength of the soil and is one of its shear
strength parameters.
There are 3 basic laboratory tests that can be performed on soil samples to evaluate the shear
strength parameters:
σf
1. Direct Shear Test2. Triaxial Compression Test3. Unconfined Compression
Test
Θ τf
Rf
DIRECT SHEAR TEST
Can be performed on all types of soil, moist or dry.
Measures shear stress at failure on failure plane for various
normal stresses.Failure plane is controlled
(parallel to direction of applied load).
DIRECT SHEAR TEST
A shear box has three parts:
a basea top extensionand a normal load pistonThe prepared soil sample is placed in the box.
A normal (90 to the horizontal) load is applied to the soil.
Then the top and base are pushed in opposite directions
This forces failure to occur on a horizontal plane between the
top and base:
The horizontal force is increased until the sample shears in
two:
The procedure is repeated two more times using successively heavier
normal loads.
DIRECT SHEAR TESTIn the CV504 labs, the inside dimensions
of the shear box are 60 mm by 60 mm.
This means the failure plane has an area of 3600 mm2.
The shear force at failure (maximum) and normal load, both in Newtons are divided by this plane area to
find the shear stress at failure and the normal stress in MPa.
The shear force required to shear the sample increases in
proportion to the normal load.
The shear strength of the soil therefore is not constant but changes with the confining
pressure.
For this reason, the soil’s shear strength is characterized by shear strength parameters:
(c,).
DIRECT SHEAR TESTPlotting the shear stress versus normal
stress:
Sh
ear
Str
ess
, τ
(kPa)
Normal Stress, σn(kPa)
First Test
τf
Second Test
τf
Third Test
τf
Fitting a best fit line through these points:we have an estimate of Coulomb’s failure envelope
The τ axis intercept is the apparent cohesion, c of the soil.
c
The slope angle of this line is the angle of internal friction, of the
soil.
The equation of Coulomb’s failure envelope: τf = c + σntan .
TRIAXIAL COMPRESSION TESTCan be performed on all types of
soil, moist or dry and can consolidate sample to in situ conditions by tracking pore
water pressures.Measures vertical stress applied to
soil sample and confining pressure.
Shear stress on failure plane must be calculated from principal
stresses.
TRIAXIAL COMPRESSION TESTCylindrical specimens are prepared from sampled soil.
Preparation varies with material properties (clay vs sand vs cohesive
granular).
Specimens are weighed and dimensions measured first.
diameter
len
gth
The specimen is mounted between 2 platens and then inserted into a latex
sleeve.
The specimen is then placed in a plexiglas chamber.
TRIAXIAL COMPRESSION TESTThe specimen is mounted on the
pedestal of the chamber base as shown.
Then the chamber is placed on the base and locked into place.
The assembly is then mounted on the compression testing
machine.
specimen
For an undrained test, the drain valve is closed.
For a drained test the drain valve is opened and pore water
collected.
latex sleeve
porous disc
loading cap
pedestal
plexiglas chamber
loading ram
drainage or pore water pressure measurement
water supply for cell (confining)
pressure
air release valve
Water is forced into the cell with the supply valve open as well as the air release
valve.
Once the cell is filled with water, the air release valve is closed and the cell pressure is
increased to the desired value for the test.
TRIAXIAL COMPRESSION TESTThe effect of the cell pressure on
the specimen is illustrated below:
The goal is to simulate the stresses confining the specimen in the
ground.
Then a vertical axial load is applied to the loading ram creating compressive stresses or the deviator stress ∆σ :
Plan View of Specimen
The cell pressure, σ3, is also known as the Minor Principal Stress.
The Major Principal Stress, σ1, is the combination of the deviator stress
and cell pressure:
Side View of
Specimen
σ3
σ3
σ3
∆ σ
∆ σ
31 σΔσσ
But how can we find τf and σf from σ1 and σ3 ?
Enter Christian Otto Mohr:
Source: “commons.wikimedia.
org”
TRIAXIAL COMPRESSION TEST
for any material, the internal shear and normal
stresses acting on ANY plane within the material,
caused by external stresses or loads
can be determined using a trigonometric transformation of
the external stresses.
Herr Mohr was born in Germany on 1835-10-08 and was a
renowned Civil Engineer and professor until his death on
1918-10-02.While contemplating the symmetry of his name, Otto
started tinkering with the properties of the circle when he
discovered that...
In other words, he discovered MOHR’S
CIRCLE.
TRIAXIAL COMPRESSION TESTRemember the plot of Shear Stress versus
Normal Stress?If you plot σ1 and σ3 on the σn axisthen fit one circle through these points
Normal Stress, σn(kPa)
σ1σ3
τf
c
Shear
Str
ess
, τ
(kPa)
then you’ve got a Mohr’s circle!During the test, this circle starts as one point
at σ3 and then grows to the right as axial stress, ∆σ increases but σ3 remains
constant.
∆σ σ1 σ1∆σ ∆σ
Ultimately, the test ends when shear failure occurs and the circle has become tangent to the failure envelope.
The point of tangency of the circle and failure envelope defines the shear strength, τf and normal stress, σf.
σf
TRIAXIAL COMPRESSION TESTBut how do we find the failure envelope
from a triaxial compression test?Geometrically, you need at least two circles
in order to define a line tangent to both.This means that you need to perform the
test at least twice on the same material but at different cell pressures.
Normal Stress, σn(kPa)
c
Shear
Str
ess
, τ
(kPa)
But how can you be sure one of them isn’t bogus?A third test at yet another cell pressure
would help to confirm the validity of the failure envelope.
As with most lab measurements, the ideal (one line tangent to all three circles) is
difficult to achieve.
If one line cannot be drawn tangent to all three circles, a best fit is made as long as one circle is not out to lunch compared to
the others.
TRIAXIAL COMPRESSION TESTOnce we have the shear strength
parameters, and c defining the failure envelope,
then for each test, the shear strength, τf and normal stress, σf can be found.
Normal Stress, σn(kPa)
Rσ3
c
Shear
Str
ess
, τ
(kPa)
Remember the deviator stress, ∆σ = σ1 - σ3, which is the diameter of the Mohr’s
Circle.
So the radius of the Mohr’s Circle, R is half the diameter or:
C
R
σ1
R
Instead of doing this graphically, we can use trigonometry to find equations for τf and σf using the angle of the failure plane, Θ and the values of σ1 and σ3
The Centre of the Mohr’s Circle, C is then:
Θ
specimen
failure planeΘ
2σσ
R 31
2σσ
31 C
TRIAXIAL COMPRESSION TESTTo follow the trig we label the vertices: ’s EBC & BCF are both isosceles & EBF is
90.
Normal Stress, σn(kPa)σ3
τf
c
Shear
Str
ess
, τ
(kPa)
ABC = 90 so ACB = 90 - and DBC =
DCB = 180 – 2Θ = 90 -
C σ1
EFB = 90 – Θ & BCF =180 – 2(90- Θ) = 2Θ
Rearranging:
σf
2ΘΘ
Θ2σσ
R 31
2σσ
31 C
A
B
DE F
245θ
TRIAXIAL COMPRESSION TESTIn DBC, side BD is the same as τf .
using Θ and the σ1 & σ3 values for each trial, τf and σf can be found for each
trial.
Normal Stress, σn(kPa)σ3
τf
c
Shear
Str
ess
, τ
(kPa)
Also in DBC, side DC = Rcos(180-2Θ) So… knowing you can find Θ
and
C
R
σ1
σf = C – Rcos(180-2Θ) or C + Rcos(2Θ)
σf
2ΘΘ
Θ2σσ
R 31
2σσ
31 C
A
B
DE F
245θ
sin2θσσ21
2θ180Rsin 31f
4.3) (Eqn. sin2θσσ21
31f
4.4) (Eqn. cos2θσσ21
σσ21
3131f
TRIAXIAL COMPRESSION TESTWhat happens when the pore water is not
allowed to drain (UNDRAINED TEST)?
As the external pressure increases, the internal pore water pressure (acting in the
opposite direction to the external) increases to match (and trivialize) the
effect.
Typically, the deviator stress at failure is fairly constant for each different cell
pressure.
Normal Stress, σn(kPa)
cu = τf u 0
Shear
Str
ess
, τ
(kPa)
Therefore, the failure envelope is typically a horizonal line and u = 0.
And, the apparent cohesion, cu will be the same for each trial and equal to the
shear strength, τf
The normal stress, σf for each trial will then be σ3 + cu
(The radii are all the same)
σf σf σf
One final word on nomenclature…All stress symbols used in DRAINED
tests are usually primed…σ1’,σ3’,σf’ and f’ indicating that they are in terms of EFFECTIVE STRESS and the shear strength parameters are denoted (’,c’).
All stress symbols used in UNDRAINED tests are not
primed…σ1,σ3,σf and f indicating that they are in
terms of TOTAL STRESS and the shear strength parameters are
denoted (u,cu)
UNCONFINED COMPRESSION TEST
Is performed mainly on cylindrical, moist clay specimens sampled
from bore holes.
Measures vertical stress applied to soil sample with no confining
pressure.Shear stress on failure plane is determined similarly to undrained
triaxial compression test.
The axial load starts at 0 and increases steadily as in the triaxial compression
test.
Instead of calling it the deviator stress, σ, it is called the unconfined compressive
stress, qu.
The Mohr’s circle continues to grow until failure occurs either when the specimen’s shear strength is reached or 15% strain.
Normal Stress, σn(kPa)
ququ
c = τf
Shear
Str
ess
, τ
(kPa)
If a qu does not maximize before 15% strain is reached then the qu at 15% strain is used to define the
unconfined compressive strength of the specimen, quf
If the qu does maximize before 15% strain, then the maximum qu value is used as
quf.
qu qu ququ qu
This is analogous to the circle becoming tangent to the failure
envelope when shear failure occurs.
The point of tangency of the circle and failure envelope defines the shear strength, τf and normal stress, σf.
σf
UNCONFINED COMPRESSION TEST
qu
f
Because σ3 = 0 and quf is the diameter of the circle, the shear strength, τf and normal stress at failure, σf are
both estimated to be half of quf.