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Learning outcomes
Understand the assumptions and limitations of four soil models: Coulomb, Mohr-Coulomb, Mohr-Coulomb, Tresca Tresca and Taylor. Know Know how to select the appropriate soil model to interpret soil test data.
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Importance
All models make assumptions. assumptions. You must understand these assumptions to know the limitations of a selected model. The response of soils depends on many factors factors including the drainage drainage condition, the history of loading and the stress path. You must be b e able to select and use the appropriate model that best best represents the expected expected soil condition. Poor choice and use could lead to misrepresentation and failure.
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Importance
All models make assumptions. assumptions. You must understand these assumptions to know the limitations of a selected model. The response of soils depends on many factors factors including the drainage drainage condition, the history of loading and the stress path. You must be b e able to select and use the appropriate model that best best represents the expected expected soil condition. Poor choice and use could lead to misrepresentation and failure.
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Key terms
applied shearing Shear strength of a soil is the maximum internal resistance to applied forces.
Effective friction angle, ′, is a measure of the shear strength of soils due to friction.
Cementation, ccm, is a measure of the shear strength of a soil from forces that
cement the particles.
Soil tension, ct , is a measure of the apparent shear strength of a soil from soil suction
(negative (negative pore-wat pore -water er pressures or o r capillary stresses).
Cohesion, co, is a measure of the intermolecular forces.
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Key terms
Undrained shear strength, su, is the shear strength of a soil when sheared at constant
volume.
Apparent cohesion, C, is the apparent shear strength at zero normal effective stress. Critical state is a stress state reached in a soil when continuous shearing occurs at
constant shear stress to normal effective stress ratio and constant volume.
Dilation is a measure of the change in volume of a soil when the soil is distorted by
shearing.
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MODELS TO INTERPRET SHEAR STRENGTH
A soil model is an idealized representation of the soil to allow us to understand its response to loading and other external events. A soil model should not be expected to capture all the intricacies of real soil behavior. Each soil model may have a different set of assumptions and may only represent one or more aspects of soil behavior.
Popular soil models Coulomb Mohr-Coulomb Tresca
Simple
Some other soil models Taylor Critical state
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COULOMB’S SOIL MODEL
Soils, in particular granular soils, are endowed by nature with slip planes. Each contact of one soil particle with another is a potential micro-slip plane. Loadings can cause a number of these micro slip planes to align in the direction of least resistance. Thus, we can speculate that a possible mode of soil failure is slip on a plane of least resistance.
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COULOMB’S SOIL MODEL FOR UNCEMENTED,SOILS Soil fails by impending frictional sliding on a plane LINEAR FAILURE ENVELOPE
Soils at critical state: ′ = ′cs, = p = 0 f (n ) f
tan cs
CURVED FAILURE ENVELOPE Soils at peak state: ′ = ′p, = p > 0
f (n ) f
tan p (n ) f
tan cs
p
′cs ′p
is the dilation angle (a measure of the soil’s ability to expand – > increase in volume)
cs is a constant for a given soil and is a fundamental soil property; p is not a constant for a given soil normal effective stress and the ability of the s oil to dilate. ′
′
it depends on the
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WHAT IS DILATANCY?
Dilation is not a peculiarity of soils, but occurs in many other materials, for example, rice and wheat. The ancient traders of grains were well aware of the phenomenon of volume expansion of grains. However, it was Osborne Reynolds (1885) who described the phenomenon of dilatancy and brought it to the attention of the scientific community..
Dilation can be seen in action at a beach.
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COULOMB’S SOIL MODEL FOR CEMENTED SOILS
f ccm (n ) f
tan o
ccm is the cementation strength and o is the apparent friction angle. Neither ccm nor o is a fundamental soil parameter. Adding the cementation strength to the apparent frictional strength is not strictly correct since they are not mobilized at the same shear strains.
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ISSUES WITH AND USE OF THE COULOMB’S MODEL
ISSUES
Coulomb’s model applies strictly
to two rigid bodies with a common potential sliding plane. It is a limiting force model (force at impending frictional sliding ) It does not consider soil deformation. It is independent of the loading history of the soil.
USE
It can be used for failures that occur along a slip plane, such as a joint or the interface of two soils or the interface between a structure and a soil. Stratified soil deposits such as overconsolidated varved clays (regular layered soils that depict seasonal variations in deposition) and fissured clays are likely candidates for failure following Coulomb’s
model, especially if the direction of shearing is parallel to the direction of the bedding plane.
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KEY POINTS REGARDING COULOMB’S MODEL
f n f tan cs p ,
f
c cm
n
f
tan
o
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MOHR – COULOMB (MC) FAILURE CRITERION Soil fails by frictional sliding on a plane of maximum stress obliquity
MC failure criterion defines failure when the maximum principal effective stress ratio, (1 ) f (3 ) f
,
called the maximum effective stress obliquity, is achieved and not when the maximum shear stress [(1
3 )/2]max
is achieved.
The failure shear stress is then less than the maximum shear stress.
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MOHR – COULOMB (MC) FAILURE CRITERION
Friction angle 1 f 3 f sin
OB OA
2
1 f 3 f
(1 ) f
(3 ) f (1 ) (3 ) f
2
Inclination of failure plane to the plane of the major principal stress 45
2
4
2
Maximum shear stress [(1
3 )/2]max
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MOHR – COULOMB (MC) FAILURE CRITERION
Failure stresses for uncemented soils
(n )
f
f
1 3 2
1 3 2
1 3 2
cos
sin
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MC FAILURE CRITERION
Uncemented soils
at critical state 1 3 1 3 cs
sincs =
cs
1 3 2
cos cs
At peak state 1 3 1 3 p
sin p =
3 p = 1 cosp 2 p
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MC FAILURE CRITERION
Unsaturated, cemented, cohesive soils
1 3 sino 2C coto + 1 3 f = C
1 2
tan o 1 1 sin o 3 1 sin o
s s e r t s r a e h S
o C Normal effective stress, n
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ISSUES WITH AND USE OF THE MC MODEL
ISSUES
MC model applies strictly to two rigid bodies with a common potential sliding plane.
It is a limiting stress model.
It does not consider soil deformation. Soil deformation is important in real soils.
It is independent of the loading history of the soil. The strength of real soils is dependent on loading history.
The shear strength in compression and extension is the same. Real soils show different strengths in compression and extension. Usually, the extension strength is lower than the compressive strength.
USE
It can be used for long term (drained condition) stability calculations and to interpret the long term strength of overconsolidated finegrained and dense coarsegrained soils.
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KEY POINTS: MC FAILURE CRITERION
Coupling Mohr’s circle with Coulomb’s frictional law allows us
to define shear failure based on the stress state of the soil.
Failure occurs, according to the Mohr – Coulomb failure criterion, when the soil reaches the maximum principal effective stress obliquity.
The maximum shear stress is not the failure shear stress. Information on the deformation or the initial stress state of the soil is not needed to interpret soil strength using the MC failure criterion.
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TRESCA’S MODEL Soil fails when the shear stress is one-half the principal stress difference
Tresca’s failure criterion is used to
interpret the undrained shear strength. su
(1 ) f
( 3 ) f 2
( 1 ) f
( 3 ) f 2
The shear strength under undrained loading depends only on the initial void ratio or the initial water content or initial confining pressure. An increase in initial normal effective stress, sometimes called confining pressure, causes a decrease in initial void ratio and a larger change in excess porewater pressure when a soil is sheared under undrained condition.
The result is that the Mohr’s circle of total
stress expands and the undrained shear strength increases. Thus, su is not a fundamental soil property.
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TRESCA’S MODEL
The value of su depends on the magnitude of the initial confining pressure or the initial void ratio (or initial water content). Analyses of soil strength and soil stability problems using su are called total stress analyses (TSA).
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ISSUES WITH AND USE OF THE TRESCA’S MODEL
ISSUES
It is a yield criterion for solid bodies that has been adopted as a failure criterion for soils (a deformable body).
It is a limiting stress criterion.
It does not consider soil deformation. Soil deformation is important in real soils.
It is independent of the loading history of the soil. The strength of real soils is dependent on loading history.
Compression and expansion strength is the same. Real soils show different strengths in compression and in expansion
USE
Short term (undrained condition) stability calculations and to interpret the undrained shear strength of fine-grained soils.
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KEY POINTS TRESCA’S FAILURE CRITERION
For a total stress analysis, which applies to fine-grained soils, the shear strength parameter is the undrained shear strength, su. Tresca failure criterion is used to interpret the undrained shear strength of fine grained soils
The undrained shear strength depends on the initial void ratio or initial water content or initial confining pressure. It is not a fundamental soil shear strength parameter. Information on the deformation of the soil is not needed to interpret soil strength using Tresca failure criterion.
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TAYLOR’S FAILURE CRITERION The shear strength comes from sliding friction and the interlocking of soil particles
Taylor (1948) used an energy method to derive a simple soil model. He showed that the shear strength of soil is due to sliding friction from shearing and the interlocking of soil particles.
Unlike Coulomb failure criterion, Taylor failure criterion does not require the assumption of any physical mechanism of failure, such as a plane of sliding. It can be applied at every stage of loading for soils that are homogeneous and deform under plane strain conditions similar to simple shear.
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TAYLOR’S FAILURE CRITERION: FORMULATION The shear strength comes from sliding friction and the interlocking of soil particles Equilibrium:
d f z d z d z
Simplification: d f z z d
Critical state:
d z d
0.
f tan cs ; tan cs z cs
Peak:
tan cs z p
tan p
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ISSUES WITH AND USE OF THE TAYLOR’S MODEL ISSUES
Applies to two-dimensional stress systems.
An extension of Taylor failure criterion to account for three-dimensional stress is presented in Chapter 11.
Neither Taylor nor Coulomb failure criterion explicitly considers the rotation of the soil particles during shearing. Gives a higher peak dilation angle than Coulomb failure criterion.
USE
Long term stability calculations of homogeneous soils.
Cannot be applied to soils that fail along a joint or an interface between two soils.
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DIFFERENCES AMONG THE THREE POPULAR FAILURE CRITERIA
Name
Failure criteria
Test data interpretation*
Soil treated as
Best used for
Rigid, frictional material
Layered or fissured Direct shear overconsolidated soils or a soil where a prefailure plane exists
Coulomb
Failure occurs by impending, frictional sliding on a slip plane.
Mohr – Coulomb
Failure occurs by Rigid, frictional impending, frictional material sliding on the plane of maximum principal effective stress obliquity.
Tresca
Failure occurs when one- Homogeneous solid Short term (undrained half the maximum condition) strength of fineprincipal stress grained soils difference is achieved.
Long term (drained Triaxial condition) strength of overconsolidated finegrained and dense coarsegrained soils Triaxial
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SUMMARY OF EQUATIONS FOR THE THREE POPULAR FAILURE CRITERIA Name
Peak
Coulomb
p ( n ) f
p unsaturated, cemented soils: f C (n ) f tan o C co ct ccm
Mohr – Coulomb
sin p (3) p (1 ) p
Critical state
tan ( cs p ) ( n ) f tan
1 3 1 3 p 1 sin p tan 2 45 p 1 sin p 2
Cemented soils: sino
C co ct
cs (n ) f
sin cs (3 ) cs (1 )cs
tan
cs
1 3 1 3 cs 1 sin cs tan 2 45 cs 1 sin cs 2
1 3 2C coto + 1 3
ccm
Inclination of the failure plane to the plane on which the major principal effective stress acts. Inclination of the failure plane to the plane p on which the major principal effective stress o p =45 + o 2 acts. cs 45 cs 2
Tresca
( su ) p
1 3 p 2
( su )cs
1 3 cs 2
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RANGES OF FRICTION ANGLES AND DILATION ANGLES FOR SOILS Ranges of Friction Angles for Soils (degrees) p cs Soil type Gravel 30 – 35 35 – 50
r
Mixtures of gravel and sand with fine-grained soils 28 – 33 30 – 40 Sand 27 – 37* 32 – 50 Silt or silty sand 24 – 32 27 – 35 Clays 15 – 30 20 – 30 5 – 15 *Higher values (32° – 37°) in the range are for sands with significant amount of feldspar (Bolton, 1986). Lower values (27° – 32°) in the range are for quartz sands.
Typical Ranges of Dilation Angles for Soils Soil type
p
Dense sand Loose sand Normally consolidated clay
(degrees) 10 – 15º <10º 0º
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TYPICAL VALUES OF S U FOR SATURATED FINE-GRAINED SOILS Description
Very soft Soft Medium stiff Stiff Very stiff Extremely stiff
su (kPa(
su (psf)
< 10 10-25 25 – 50 50 – 100 100 – 200 > 200
<200 200 - 500 500 - 1000 1000 - 2000 2000 - 4000 > 4000
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Quiz 1 Which failure criterion (model) is best suited to analyze the potential failure of the soil mass shown? 1.
Mohr-Coulomb
2.
Coulomb
3.
Tresca
4.
None of the above
Dense sand
Loose silty sand
Stiff overconsolidated clay
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Quiz 2 The critical state friction angle of a soil is 30 degrees. If the normal effective stress imposed by a building is 100 kPa, the shear stress (kPa) to cause failure is most nearly 1.
86.6
2.
100
3.
50
4.
57.7
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Quiz 3 The critical state friction angle of a soil is 30 degrees. The ratio of the major principal effective stress to the minor principal effective stress to cause failure is most nearly 1.
0.5
2.
1
3.
2
4.
3
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PRACTICAL IMPLICATIONS OF FAILURE CRITERIA
Region I. Impossible soil
states. A soil cannot have soil states above the boundary AEFB.
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PRACTICAL IMPLICATIONS OF FAILURE CRITERIA
Region II. Impending instability (risky
design).
Region AEFA is characteristic of dilating soils that show peak shear strength and are associated with the formation of shear bands. The shear bands consist of soils that have reached the critical state and are embedded within soil zones with high interlocking stresses due to particle rearrangement. These shear bands grow as the peak shear strength is mobilized and as the soil strain-softens subsequent to the critical state.
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PRACTICAL IMPLICATIONS OF FAILURE CRITERIA
Region III. Stable soil states (safe
design).
One of your aims as a geotechnical engineer is to design geotechnical systems on the basis that if the failure state were to occur, the soil would not collapse suddenly but would continuously deform under constant load. This is called ductility. Soil states that are below the failure line or failure envelope AB would lead to safe design. Soil states on AB are failure (critical) states
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KEY POINTS
There are several failure criteria for soils. Each criterion has application to certain soil conditions. The three popular failure criteria (Coulomb, MohrCoulomb and Tresca) assume that soil is a rigid-plastic material with no deformation prior to failure.
The Coulomb and MohrCoulomb failure criteria are applicable to estimate long term failure. The Mohr-Coulomb failure criterion also assume that failure shear strength of soil in compression and extension is the same. In reality, the shear strength at failure in extension is less than in compression.
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KEY POINTS
Soil states above the peak shear strength boundary are impossible. Soil states within the peak shear strength boundary and the failure line (critical state) are associated with brittle, discontinuous soil responses and risky design. Soil states below the failure line lead to ductile responses and are safe.
You should not rely on ′p in geotechnical design, because the amount of dilation one measures in laboratory or field tests may not be mobilized by the soil under construction loads. You should use ′cs unless experience dictates otherwise. A higher factor of safety is warranted if ′p rather than ′cs is used in design.
