11
Critical State Soil Mechanics
1-Introduction To Critical State Soil Mechanism
2-Importance of CSSM
3-The assumption (premises) of CSSM.
4-Yield surface
5-Computing the CSL
6-The rules of stress paths in CSSM
7-Example of a CD triaxial test in lightly OC clay
8-Example of a CD triaxial test in heavily OC clay
9-Useful Relationships from CSSM
10-Predicting Mohr-Coulomb DRAINED Shear Strength
11-Predicting if an Over consolidated Soil Will Fail in Tension
12-Undrained Shear Strength Ratio for Isotropic Consolidation Test
13-Undrained Shear Strength Ratio for Direct Simple Shear
14-Estimating DSS Shear Strength from Axial Compression Testing
15-Relating Drained and Undrained Shear Strengths
16-Skempton Pore Pressure Parameter Estimation
18-References
Contents
Introduction
Critical State Soil Mechanics ( CSSM )
Critical State Soil Mechanics: the area of soil mechanics that encompasses the conceptual models that represent the mechanical behavior of saturated remolded soils based on the critical state concept.
In other words: it is an effective stress framework describing mechanical soil response.
The research of the CSSM started becoming important and hot in the 1970s until nowadays less frequently because of the fundamental theories have been established.
Example: if we are talking about the soil friction angle it will not be the peak fiction angle or the soften friction angle it will be the friction angle of the soil at that residual (critical) state .(doesn’t matter if the soil had high relative density or low relative density because the critical state friction angle is the same for that particular type of soil regardless of its initial relative density ).
Importance Of CSSM
Even a basic CSSM can help us predict many of the complex soil mechanics behaviors for example:
1 - normally consolidated (NC) and over consolidated (OC) shear and consolidation behavior.
2- undrained vs drained stress path in shear loading.
3 - positive vs negative porewater pressure development during shear
4 - volume change (dilation, contraction).
5 - undrained shear strength prediction.
6 - soil constitutive behavior. (stress-strain plots)
Basic Terms of CSSM
The Basic terms of the CSSM:
These terms can be obtained by laboratory testing and field testing or approximations.
: The friction angle.
: The slope of the virgin compression line. (the compression index of the soil).
: The slope of the consolidation curve on the rebound portion of the curve. (the recompression index).
: The void ratio of the soil.
: The effective vertical stress of the soil.
: The soil over the consolidation ratio. (preconsolidation stress)
CSSM Premises
Critical State Line ( CSL )
Premises
Invariants
Stress Path Space
Modified Failure Envelope
Compression
Extension
Stress path Space
Void ratio
Vs
Mean stress in
(log scale)
Void ratio
vs
Mean stress space
Mc: the soil is being sheared in compression. (critical state friction angle).
Mc
Me: the soil is being sheared in extension
Me
Additional Note
Yield Surface
plastic
elastic
Yield Surface
Elastically
The soil can fail :
1- if it reaches yield surface.
2- reaches the critical state line.
• The exception of the yield surface shape recovery is that if the stress path crosses the yield or the modified failure envelope, instead of expanding the yield surface contracts ( the circles start to shrink).
As long as the effective stress path (ESP) stays within the yield surface, the soil will behave elastically, once the (ESP) reaches the yield surface, the soil will begin to yield and accumulate permanent deformations and volume change (if drained).
Once the (ESP) moves beyond the initial yield surface, the surface will expand. the ESP must be within or on the yield surface .it can never exist outside the yield surface.
Computing the CSL
• The NCL and URL shouldn’t be calculated, those come from a regular 1-dimensional consolidation test.
Computing CSL
Plot
The Rules of Stress Paths in CSSM
The Rules of
Stress Paths
in CSSM
1) Movement of the ESP
(effective stress loading path)
in p’-q space translate directly to the movement of the stress path e-p’ space. (TSP: total stress loading path)
Rule 1
Rule 2
2) Shear Failure always occurs at (p’f , qf) which is the point where ESP touches the CSL(or the Mc line) .
This point corresponds to
(ef , p’f) on the CSL in e-p’ space.
Rule 3
3 - Under drained loading the
TSP=ESP
because there are no pressure pores exists, void ratio and volume can change in the e-p’ space (the soil can contract or dilate) .
Rule 4
4) Under undrained loading , the ESP deviates from the TSP in p’-q space
because of the pore water pressure generation
if negative porewater pressure generated , the soil is dilated
if positive pore water pressure generated , the soil is contractive
the void ratio remains constant during the undrained loading condition e-p’ space that means
the initial void ratio = final void ratio (e0=ef).
5)Any ESP inside of the yield surface corresponds to elastic behavior of the soil,
it also corresponds to movement along the URL in e-p’ space.
Rule 5
Rule 6
6 - once the ESP reaches the yield surface,
the effective p’c (the pre-consolidation stress) changes
and the yield surface change size.
If the ESP above the CSL , then the yield surface contracts.
if the ESP is below the CSL, then the yield surface expands.
(that means the p’c will increase to accommodate the expansion of the yield surface).
7 - the ESP must always lie within or on the yield surface,
the ESP can never pass the outside of the yield surface.
Rule 7
8 - in undrained loading,
all ESP movement within the yield surface (elastic zone) is on a vertical line (p’y (mean stress when it reaches the yield surface) =p’0 the initial mean stress ) ,
once the yield surface is reached , the ESP begins moving rapidly towards the CSL as the size of the yield surface adjusts .
Rule 8
9- in drained loading ,the void ratio changes with the movement of the ESP .
The stress path remains on the initial URL as long as the ESP in p’-q space is within the initial yield surface .
Once the ESP reaches the yield surface ,the surface changes size
because the effective p’c changes. This causes a vertical shift in the URL corresponding to the new effective p’c.
Rule 9
Rule 10
10) A normally consolidated soil has
p’0=p’c.
means that the ESP is already on the yield surface,
and any additional movement along the ESP will cause the yield surface to expand .
Rule 11
11)A lightly over consolidated soil has R0<2. A heavy over consolidated soil R0>2.
(R0 is the invariant version of the over consolidation ratio).
R0=P’c/P’0=OCR.
* THE SOIL MUST YIELD FIRST BEFORE IT CAN FAIL ON THE CSL.
IF IT TOUCHES THE CSL BEFORE IT YIELDS, THEN IT WILL NOT FAIL*
Plot
Steps to compute the ESP in p’- q and e-p’ spaces
for
a drained condition.
Steps to compute the ESP in p’- q and e-p’ spaces
for
a drained condition.
1)Given the initial stresses p’0 and q0, determine the mean effective stress and the deviatoric stress at initial yield (p’y and qy) by finding the coordinates of the intersection of the initial yield surface with the effective stress path.
*these equations are applicable for elliptical yield surface, there are other equations for triangle and circular yield surfaces.
THE (M can be Mc or Me depending on whether we are doing compression or tension to our loading)
The n0 is the slope of the total stress path. (in direct shear test it will not equal to 3)
Step 1
The n0 is the slope of the total stress path.
(in direct shear test it will not equal to 3)
2 - Calculate the mean and deviatoric stress of the ESP at failure (pf and qf)
Step 2
3 - For each (p’,q)combination after initial yield is reached ,
compute the corresponding effective pre-consolidation mean stress P’c as :
Step 3
4- for each effective
pre-consolidation mean stress,
compute the corresponding void ratio, e:
Step 4
Steps to compute the ESP in p’- q and e-p’ spaces
for
undrained condition.
Steps to compute the ESP in p’- q and e-p’ spaces
for
undrained condition.
1)Given the initial stresses p’0 and q0, determine the mean effective stress and the deviatoric stress at initial yield (p’y and qy) by finding the coordinates of the intersection of the initial yield surface with the effective stress path.
Step 1
2- Calculate the mean and deviatoric stress of the ESP at failure (pf and qf):
Step 2
3) divide the horizontal distance between the initial mean effective stress, P’0 (also the yield stress P’y),
and the failure mean effective stress ,P’f into a number of equal mean effective stress increments.
Step 3
4)for each p’ increment beyond the yield surface,
compute the pre-consolidation mean stress as:
Step 4
5)for each p’ increment beyond the yield surface, compute the q value of the ESP as:
Step 5
6 - for each p’ increment, compute the pore pressure
as the difference between
the
TSP AND ESP:
Step 6
Examples
In this example we can see that the point (o) its right at the R0 and this is a drained test so the slope of the total
stress path will be 3
at the point (D) the stress path starts to deviate and change and that the yield surface is going to grow and grow until we get the surface where the pre-consolidation stress will be somewhere on the normally consolidation line .
*At point (f) we reach the CSL that correspond with the CSL at the p-q space.
Example of a
CD
triaxial test in
lightly
OC clay
We can see that our starting point is to the left of the midpoint of the ellipse so its heavily over consolidated.
* We can see that the stress path passes the critical state line and goes up and reaches the yield surface at point D which corresponds to a stress path along the unload/reload line until it reaches its point d in the e-p’ space.
*at point D the surface starts o shrink because we are in the peak zone, so the stress path starts to go back that same line.
*because the surface is shrinking, the pre-consolidation stress (C<) begins moving to the left, so the URL will start moving up.
* it will keep going back the stress path until we have a yield surface like the one in picture (red ellipse). the unload /reload line will be shifted up until it reaches the point F and then we have failure the soil.
Example of a
CD
triaxial test in
heavily
OC clay
*the dashed line represents the total stress path and it has the slope of 3 ,
we can see the ESP as long as it’s in the yield surface it’s a vertical line which means that its correspond to the same void ratio and once it hits the yield surface it will start shifting to the left and it will keep shifting until it reaches the point F .
*At failure the horizontal distance between the total stress path and the effective stress path is equal to the difference in pore pressure, and since the ESP is on the left of the TSP that means that we have positive pore pressure meaning we have contraction during undrained loading.
Example of a
Cu
triaxial test in
lightly
OC clay
References
References
• Atkinson, J.H., D. Richardson and S.E. Stallebrass. 1990. Effect
• of recent stress history on the stiffness of overconsolidated
• soil. Géotechnique 40(4):531-541.
• Bailey, A.C., C.E. Johnson and R.L. Schafer. 1984. Hydrostatic
• compaction of agricultural soils. Transactions of the ASAE
• 27(4): 925-955.
• Chi, L. and R.L. Kushwaha. 1989. Finite element analysis of
• forces on a plane soil blade. Canadian Agricultural
• Engineering 31(2):135-140.
• Chi, L., R.L. Kushwaha and J. Shen. 1993. An elasto-plastic
• constitutive model for agricultural cohesive soil. Canadian
• Agricultural Engineering 35(4):245-251.
• Critical State Soil Mechanics Andrew Schofield and Peter Wroth Lecturers in Engineering at Cambridge University
Useful relationships from CSSM
This relationship can be used to estimate the drained shear strength at initial yield (i.e., when the soil starts to deform plastically).
Note that for a strain-softening soil, this strength corresponds to the PEAK drained shear strength.
Predicting Mohr-Coulomb DRAINED Shear Strength
If you want to use the residual (i.e., Large strain) drained strength instead, then that is the strength associated with the intersection point between the CSl and the ESP
Predicting Mohr-Coulomb DRAINED Shear Strength
You can tell if a soil is strain-hardening of strain-softening in drained shear by looking at pcs. If pcs > 1, the soil is strain-softening .
If pcs < 1, the soil is strain- hardening.
Predicting if an Over consolidated Soil Will Fail in Tension
Predicting if an Over consolidated Soil Will Fail in Tension
Undrained Shear Strength Ratio for Isotropic Consolidation Test
Undrained Shear Strength Ratio for Isotropic Consolidation Test
Undrained Shear Strength Ratio for Direct Simple Shear
Undrained Shear Strength Ratio for Direct Simple Shear
Estimating DSS Shear Strength from Axial Compression Testing
Estimating DSS Shear Strength from Axial Compression Testing
Relating Drained and Undrained Shear Strengths
Relating Drained and Undrained Shear Strengths
Skempton Pore Pressure Parameter Estimation