CONTENT

Basic Concepts

Mohr-Coulomb failure theory

Methods of determining shear strength parameters

Factors affecting shear strength of soils

Shear behavior of granular soils

Shear behavior of cohesive soils

Other methods of determining shear strength parameters

Shear stress paths

Stress-strain characteristics of soils

Other theories of failure

Importance of Shear strength of soils

Mohr circle of stresses in 2D space

Mohr circle of stresses in 3D space

BASIC CONCEPTS

One of the most important and the most controversial engineering properties of soil is its shear strength or ability to resist sliding along internal surfaces within a mass.

The stability of a cut, the slope of an earth dam, the foundations of structures, the natural slopes of hillsides and other structures built on soil depend upon the shearing resistance offered by the soil along the probable surfaces of slippage.

There is hardly a problem in the field of engineering which does not involve the shear properties of the soil in some manner or the other.

Soils like any other materials fail at some point when they are subjected to increasing shear stresses.

They cannot withstand a shear stress larger than their shear strength and deform extensively when the applied shear stress approaches their shear strength.

MOHR CIRCLE AND SHEAR STRESSES

In 2D space (e.g., on the s1s2 , s1s3, or s2s3 plane), the stresses on a soil mass at failure can be factored into the normal stress (sn) and the shear stress (ss) on the failure plane surface.

Note: The equations are given for a 2D space are in the s1s2 plane, where s1 is greater than s2.

If we were dealing with the s2s3 plane, then the two principal stresses would be s2 and s3.

GENERALIZED BIAXIAL STRESS SYSTEM

NORMAL STRESS

The normal stress, sn:

In parametric form the equation becomes:

where

p is the center, which lies on the normal stress axis (x-axis), which is also the mean stress;

q is the diameter, which is also the deviatoric stress;

a is equal to 2q, where q is the rupture angle, measured from point p to the failure point.

qssss

s 2cos22

2121

n

s cos2

qpn

SIGN CONVENTIONS

The common sign conventions followed are:

sn is compressive when it is “+”, i.e., when sn>0.

sn is tensile when it is “-”, i.e., when sn< 0.

It can be observed that:

sn = s1 at q 0o (a maximum)

sn = s2 at q 90o (a minimum)

There are no shear stress on the three principal planes (perpendicular to the principal stresses).

SHEAR STRESS

The shear stress, tq:

In parametric form the equation becomes:

In parametric form the equation becomes:

tq > 0 represents left-lateral shear

tq < 0 represents right-lateral shear

tq = 0 at q 0o or 90o or 180o (a minimum)

tq = (s1s2)/2 at q 45o (maximum shear stress)

The maximum ss is 1/2 the differential stress.

qss

tq 2sin2

21

s sin2

qs

MOHR CIRCLE FOR STRESSES

.

SPECIAL STATES OF STRESS – UNIAXIAL STRESS

Uniaxial Stress (compression or tension) One principal stress (s1 or s3) is non-zero, and the

other two are equal to zero.

Uniaxial compression Compressive stress in one direction (i.e. s1 > s2=s3 =

0) The Mohr circle is tangent to the ordinate at the origin

(i.e., s2=s3= 0) on the + (compressive) side.

Uniaxial tension Tension in one direction (i.e. 0 s1 = s2 > s3)

The Mohr circle is tangent to the ordinate at the origin on the - (i.e., tensile) side.

SPECIAL STATES OF STRESS - AXIAL STRESS

Axial (confined) compression: s1 > s2 = s3 > 0

Axial extension (extension): s1 = s2 > s3 > 0

The Mohr circle for both of these cases are to the right of the origin (non-tangent).

Uniaxial Compression Mohr Circle

General Compression /Tension Mohr Circle

SPECIAL STATES OF STRESS – BIAXIAL STRESS

Biaxial Stress:

Two of the principal stresses are non-zero and the other is zero.

Pure Shear:

s1 = -s3 and is non-zero (equal in magnitude but opposite in sign)

s2 = 0 (i.e. a biaxial state).

The normal stress on planes of maximum shear is zero (pure shear).

The Mohr circle is symmetric with respect to the ordinate (center is at the origin).

Biaxial stresses on a plate element

Biaxial stresses Mohr Circle

Pure shear

SPECIAL STATES OF STRESS – TRIAXIAL STRESS

Triaxial Stress:

s1, s2, and s3 have non-zero values

s1 > s2 > s3 and can be tensile or compressive

This is the most general state in nature, and the

Mohr circle has three distinct circles.

No simple method exists for drawing Mohr circles to represent the general case, in which both normal and shear stresses act on all six faces of the cubical element.

There are two simple cases, however, which can be represented by three Mohr circles:

MOHR CIRCLE IN 3D

1. A cubical element which has only normal stresses (i.e. principal stresses) acting on the six faces.

MOHR CIRCLE IN 3D

2. A cubical element which has only normal stress (a principal stress) acting on one pair of opposite parallel faces, but has both normal and shear stresses acting on both the remaining pairs of faces.

ISOTROPIC STRESS The 3D, isotropic stresses are equal in magnitude in all

directions (as radii of a sphere).

Magnitude of the mean of the principal stresses:

When principal stresses are equal (isotropic condition):

i.e., it is an invariant (does not depend on a specific coordinate system). No need to know the principal stress; any one can be used.

Leads to dilation; but no shape change.

If v1 and vo are final and original volumes.

Pm

3

321 ssss

321 sss P

oo

o

vv

v

v

vv

1

Formulation of the theory

Mohr-Coulomb theory in s – t space

Mohr-Coulomb theory in p – q space

Remarks on the Mohr-Coulomb theory

MOHR – COULOMB FAILURE CRITERIA:

A point of Mohr’s circle defines the normal stress and the shear stress on a certain plane. The stresses on all planes together form the circle, because when the plane rotates the stress points traverse the circle.

The ratio of shear stress to normal stress varies along the circle, i.e. this ratio is different for different planes. It is possible that for certain planes the failure criterion is satisfied.

The failure criterion has also been indicated, in the form of two straight lines, making an angle ϕ with the horizontal axis. Their intersections with the vertical axis is at distances c.

The term yield indicates the onset of irreversible (or plastic) deformations. Accordingly, the yield criterion is the condition for the development of irreversible deformations.

In order to underline that failure of a soil is determined by the effective stresses, the stresses in this figure have been indicated as 0.

There are two planes, defined by the points C and D, in which the stress state is critical. On all other planes the shear stress remains below the critical value.

Thus it can be conjectured that failure will start to occur whenever Mohr’s circle just touches the Coulomb envelope. This is called the Mohr-Coulomb failure criterion.

If the stress circle is completely within the envelope no failure will occur, because on all planes the shear stress remains well below the critical value.

Circles partly outside the envelope are impossible, as the shear stress on some planes would be larger than the critical value.

The failure envelop is defined by the equation:

st tan c

The Mohr envelope may be assumed as a straight line although it is curved under certain conditions.

The Mohr circle which is tangential to the shear strength line is called the Mohr circle of rupture.

Thus, the Mohr envelope constitutes a shear diagram and is a graph of the Coulomb equation for shearing stress.

This is called the Mohr-Coulomb Failure Theory.

MATHEMATICAL FORMULATION OF THE MOHR–COULOMB FAILURE:

Substituting for the values of σn and τ in the Mohr-Coulomb equation and solving for σ1 we obtain:

The plane with the least resistance to shearing along it will correspond to the minimum value of σ1 which can produce failure. i.e. when the denominator in the second term of the equation is at a maximum.

qqq

sss

qssss

qss

tancoscossin

tan

tan2cos22

2sin2

2

3

31

313131

c

c

Differentiating, and simplifying:

Substituting and simplifying:

where Nϕ is called as the flow value.

( )

245

0tancoscossin 2

q

qqqq

o

c

d

d

ss

ss

NcN

c oo

2

245tan2

245tan

31

2

31

MOHR–COULOMB YIELD ON THE p–q SPACE:

Recall that:

The failure envelop on this space can be given as:

where:

(Derive!)

)(2

)(31

31 ssss

qp

tanpaq

sintancos ca

The σ–τ and p–q failure lines have different slopes and intercepts but have a common intersection point F with the horizontal axis.

In-terms of c and ϕ, the expression of the yield criterion in the p–q space will be:

)0 qfor (sin3

cos6

sin3

sin6

)0 qfor (sin3

cos6

sin3

sin6

<

>

cpq

cpq

REMARKS:

The Mohr-Coulomb criterion is a rather good criterion for the failure state of sands.

For such soils the cohesion usually is practically zero, c = 0, and the friction angle usually varies from = 30o to = 45o, depending upon the angularity and the roundness of the particles.

Clay soils usually have some cohesion, and a certain friction angle, but usually somewhat smaller than sands.

Great care is needed in the application of the Mohr-Coulomb criterion for very small stresses.

For clay one might find that a Mohr’s circle would be possible in the extreme left corner of the diagram, with tensile normal stresses.

It is usually assumed that this is not possible, and therefore the criterion should be extended by a vertical cut-off at the vertical axis.

To express that the cohesion of soils does not necessarily mean that the soil can withstand tensile stresses, the property is sometimes denoted as apparent cohesion, indicating that it is merely a first order schematization.

The Mohr-Coulomb criterion can also be used, at least in a first approximation, for materials such as rock and concrete.

In such materials the cohesion may be quite large, at least compared to soils. The contribution of friction is not so dominant as it is in soils.

Also it often appears that the friction angle is not constant, but decreases at increasing stress levels.

In some locations, (offshore coastal areas) calcareous soils are found. These are mostly sands, but the particles have been glued together (presence of calcium). Such materials have high values of the cohesion c, which may easily be destroyed, however, by a certain deformation.

This deformation may occur during the construction of a structure, for instance driving of piles. During the exploration of the soil this may have been found to be very strong, but after installation much of the strength has been destroyed.

An advantage of true frictional materials is that the friction usually is maintained, also after very large deformations. Soils such as sands may not be very strong, but at least they maintain their strength.

For clays the Mohr-Coulomb criterion is reasonably well applicable, provided that proper care is taken of the influence of the pore pressures, which may be a function of time, so that the soil strength is also a function of time. Some clays have the special property that the cohesion increases

with time during consolidation. This leads to a higher strength because of overconsolidation.

For very soft clays the Mohr-Coulomb criterion may not be applicable, as the soil behaves more like a viscous liquid.

Laboratory tests

Direct Shear Test

Triaxial Test

Field tests

For granular soils

For cohesive soils

SHEAR STRENGTH PARAMETERS:

The shear strength parameters c and ϕ of soils either in the undisturbed or remolded states may be determined by any of the following laboratory methods:

Direct or box shear test

Triaxial compression test

Field methods are also available, and depend upon the type of the soil to be investigated.

The laboratory or the field method that has to be chosen in a particular case depends upon the type of soil and the accuracy required.

Wherever the strength characteristics of the soil in-situ are required, laboratory tests may be used provided undisturbed samples can be extracted from the stratum.

However, soils are subject to disturbance either during sampling or extraction from the sampling tubes in the laboratory even though soil particles possess cohesion.

It is practically impossible to obtain undisturbed samples of cohesionless soils and highly pre-consolidated clay soils.

Soft sensitive clays are nearly always remolded during sampling.

Laboratory methods may, therefore, be used only in such cases where fairly good undisturbed samples can be obtained.

DIRECT SHEAR TEST:

The original form of apparatus for the direct application of shear force is the shear box. The box shear test, though simple in principle, has certain shortcomings which are identified later on.

The apparatus consists of a square brass box split horizontally at the level of the center of the soil sample, which is held between metal grills. Vertical load is applied to the sample as shown in the figure and is held constant during a test.

A gradually increasing horizontal load is applied to the lower part of the box until the sample fails in shear. The shear load at failure is divided by the cross-sectional area of the sample to give the ultimate shearing strength.

The test may be repeated with a few more samples having the same initial conditions as the first sample. Each sample is tested with a different vertical load.

REFERENCES: Das, B. M. “Principles of Foundation Engineering.” 7th ed,

Cengage Learning, 2011.

Holtz R.D. and Kovacs W.D. “An Introduction to Geotechnical Engineering.” Prentice Hall, 1981.

Lambe T.W. and Whitman R.V. “Soil Mechanics.” Wiley, 1969.

Lancellota, Renato. “Geotechnical Engineering.” 2nd ed, 2009.

Murthy, V.N.S. “Geotechnical Engineering: Principles and Practices of Soil Mechanics and Foundation Engineering.” Marcel Dekker, 1999.

Terzaghi K., Peck R.B. and Mesri G. “Soil Mechanics in Engineering Practice.” 3rd ed. Wiley; 1996.

Verruijt, A. “Soil Mechanics.”, Delft University of Technology, Dordrecht, 2001.