**Shear Strength of Soils **

The shear strength of a soil is equal to the maximum value of shear stress that can be mobilised within a soil mass without failure taking place.

The shear strength of a soil is a function of the stresses applied to it as well as how these stresses are applied. A knowledge of the shear strength of soils is necessary to determine the bearing capacity of foundations, the lateral pressure exerted on retaining walls, and the stability of slopes.

**Mohr Circle of Stresses**

In soil testing, cylindrical samples are commonly used in which radial and axial stresses act on principal planes. The vertical plane is usually the minor principal plane, whereas the horizontal plane is the major principal plane. The radial stress (s_{r}) is the minor principal stress (s_{3}), and the axial stress (a_{a}) is the major principal stress (s_{1}).

A graphical representation of stresses called the Mohr circle is obtained by plotting the principal stresses. The sign convention in the construction is to consider compressive stresses as positive and angles measured counter-clockwise also positive.

Draw a line inclined at angle θ with the horizontal through the pole of the Mohr circle to intersect the circle. The coordinates of the point of intersection are the normal and shear stresses acting on the plane, which is inclined at angle θ within the soil sample.

**Normal stress**

**Shear stress**

**The plane inclined at an angle of 45**^{0}to the horizontal has acted on it the maximum shear stress equal to , and the normal stress on this plane is equal to .**The plane with the maximum ratio of shear stress to normal stress is inclined at an angle of the horizontal, where a is the slope of the line tangent to the Mohr circle and passing through the origin.**

**Mohr-Coulomb Failure Criterion**

When the soil sample has failed, the shear stress on the failure plane defines the shear strength of the soil. Thus, it is necessary to identify the failure plane. Is it the plane on which the maximum shear stress acts, or is the plane where the ratio of shear stress to normal stress is the maximum?

It can be assumed that a failure plane exists, and it is possible to apply principal stresses and measure them in the laboratory by conducting a triaxial test. Then, the Mohr circle of stress at failure for the sample can be drawn using the known values of the principal stresses.

If data from several tests carried out on different samples up to failure is available, a series of Mohr circles can be plotted. It is convenient to show only the upper half of the Mohr circle. A line tangential to the Mohr circles can be drawn and is called the Mohr-Coulomb failure envelope.

Suppose the stress condition for any other soil sample is represented by a Mohr circle that lies below the failure envelope. In that case, every plane within the sample experiences shear stress which is smaller than the shear strength of the sample. Thus, the point of tangency of the envelope to the Mohr circle at failure gives a clue to the determination of the inclination of the failure plane. The orientation of the failure plane can be finally determined by the pole method.

Mohr-Coulomb failure criterion can be written as the equation for the line that represents the failure envelope. The general equation is

Where = shear stress on the failure plane *c* = apparent cohesion

= normal stress on the failure plane

f = angle of internal friction

The failure criterion can be expressed in the relationship between the principal stresses. From the geometry of the Mohr circle,

Rearranging,

where

**Methods of Shear Strength Determination**

**Direct Shear Test**

The test is carried out on a soil sample confined in a metal box of square cross-section, which is split horizontally at mid-height. A small clearance is maintained between the two halves of the box. The soil is sheared along a predetermined plane by moving the top half of the box relative to the bottom half. The box is usually square in the plan of size

60 mm x 60 mm. A typical shear box is shown.

If the soil sample is fully or partially saturated, perforated metal plates and porous stones are placed below and above the sample to allow free drainage. If the sample is dry, solid metal plates are used. A load normal to the plane of shearing can be applied to the soil sample through the box's lid.

Tests on sands and gravels can be performed quickly and are usually performed dry as it is found that water does not significantly affect the drained strength. For clays, the shearing rate must be chosen to prevent excess pore pressures from building up.

As a normal vertical load is applied to the sample, shear stress is gradually applied horizontally by causing the two halves of the box to move relative to each other. The shear load is measured together with the corresponding shear displacement. The change of thickness of the sample is also measured.

Several samples of the soil are tested, each under different vertical loads, and the value of shear stress at failure is plotted against the normal stress for each test. Provided there is no excess pore water pressure in the soil, the total and effective stresses will be identical. From the stresses at failure, the failure envelope can be obtained.

The test has several **advantages:**

- It is easy to test sands and gravels.
- Large samples can be tested in large shear boxes, as small samples can give misleading results due to imperfections such as fractures and fissures or may not be truly representative.
- Samples can be sheared along predetermined planes when the shear strength along fissures or other selected planes are needed.

The **disadvantages **of the test include:

- The failure plane is always horizontal in the test, which may not be the weakest plane in the sample. Failure of the soil occurs progressively from the edges towards the centre of the sample.
- There is no provision for measuring pore water pressure in the shear box, so it is impossible to determine effective stresses from undrained tests.
- The shear box apparatus cannot give reliable undrained strengths because it is impossible to prevent localised drainage away from the shear plane.

**Triaxial test**

- The triaxial test is carried out in a cell on a cylindrical soil sample having a length to diameter ratio of 2.
- The usual sizes are 76 mm x 38 mm and 100 mm x 50 mm. Three principal stresses are applied to the soil sample, out of which two are applied water pressure inside the confining cell and are equal.
- The third principal stress is applied by a loading ram through the top of the cell and is different to the other two principal stresses.

A typical triaxial cell in 2D is shown as

The soil sample is placed inside a rubber sheath sealed to a top cap and bottom pedestal by rubber O-rings. For tests with pore pressure measurement, porous discs are placed at the bottom and sometimes at the top of the specimen. Filter paper may be provided around the outside of the specimen to speed up the consolidation process. Pore pressure generated inside the specimen during testing can be measured using pressure transducers.

The triaxial compression test consists of two stages:

A soil sample is set in the triaxial cell and confining pressure is then applied.**First stage:**In this, additional axial stress (also called deviator stress) is applied, which induces shear stresses in the sample. The axial stress is continuously increased until the sample fails.**Second stage:**

The applied stresses, axial strain, and pore water pressure or change in sample volume can be measured during both stages.

**Test Types**There are several test variations, and those used mostly in practice are:

Cell pressure is applied without allowing drainage. Then keeping cell pressure constant, deviator stress is increased to failure without drainage.*UU (unconsolidated undrained) test:***CU (consolidated undrained) t***est*Drainage is allowed during cell pressure application. Then without allowing further drainage, deviator stress is increased, keeping cell pressure constant.**:**This is similar to the**CD (consolidated drained) test:****CU test**except that as deviator stress is increased, drainage is permitted. The loading rate must be slow enough to ensure no excess pore water pressure develops.

If pore water pressure is measured in the UU test, the test is designated by .

In the CU test, if pore water pressure is measured in the second stage, the test is symbolised as .

Significance of Triaxial Testing

The first stage simulates in the laboratory the in-situ condition that soil at different depths is subjected to different effective stresses. Consolidation will occur if the pore water pressure that develops upon the confining pressure is allowed to dissipate. Otherwise, the effective stress on the soil is the confining pressure (or total stress) minus the pore water pressure which exists in the soil.

During the shearing process, the soil sample experiences axial strain, and either volume change or development of pore water pressure occurs. The magnitude of shear stress acting on different planes in the soil sample differs. When at some strain the sample fails, this limiting shear stress on the failure plane is called the shear strength.

The triaxial test has many **advantages** over the direct shear test:

- The soil samples are subjected to constant stresses and strains.
- Different combinations of confining and axial stresses can be applied.
- Drained and undrained tests can be carried out.
- Pore water pressures can be measured in undrained tests.
- The complete stress-strain behaviour can be determined.

**Total Stress Parameters**

**UU Tests**

All Mohr circles for the UU test plotted in terms of total stresses have the same diameter.

The failure envelope is a horizontal straight line and hence

The equation can represent it:

**CU & CD Tests:**

For tests involving drainage in the first stage, the diameter increases with the confining pressure when Mohr circles are plotted in terms of total stresses. The resulting failure envelope is an inclined line with an intercept on the vertical axis.

It is also observed that c_{CU }¹ c_{CD} and f_{CU} ¹ f_{C}_{D}

It can be stated that the failure envelope is not unique for identical soil samples tested under different triaxial conditions of UU, CU and CD tests.

**Effective Stress Parameters**

Suppose the same triaxial test results of UU, CU and CD tests are plotted in terms of effective stresses taking into consideration the measured pore water pressures. In that case, it is observed that all the Mohr circles at failure are tangent to the same failure envelope, indicating that shear strength is a unique function of the effective stress on the failure plane.

This failure envelope is the shear strength envelope which may then be written as

where *c'* = cohesion intercept in terms of effective stress *f'* = angle of shearing resistance in terms of effective stress

If is the effective stress acting on the rupture plane at failure,is the shear stress on the same plane and is therefore the shear strength.

The relationship between the effective stresses on the failure plane is

### Stress-Strain Behaviour of Sands

Sands are usually sheared under drained conditions as they have relatively higher permeability. This behaviour can be investigated in direct shear or triaxial tests. The two most important parameters governing their behaviour are the **relative density (I _{D})** and the magnitude of the

**effective stress (σ**The relative density is usually defined in percentage as

^{,}).where * e_{max}* and

*are the maximum and minimum void ratios that can be determined from standard tests in the laboratory, and*

**e**_{min}*is the current void ratio. This expression can be re-written in terms of dry density as*

**e**where **g***_{dmax}* and

**g**are the maximum and minimum dry densities, and g

_{dmin}*is the current dry density. Sand is generally referred to as dense if*

_{d}*> 65% and loose if < 35%.*

**I**_{D}The influence of relative density on the behaviour of saturated sand can be seen from the plots of CD tests performed at the **same effective confining stress. **There would be no induced pore water pressures existing in the samples.

For the dense sand sample, the deviator stress reaches a peak at a low value of axial strain and then drops down, whereas for the loose sand sample, the deviator stress builds up gradually with axial strain. The behaviour of the medium sample is in between.

The following observations can be made:

• All samples approach the same ultimate conditions of shear stress and void ratio, irrespective of the initial density. The denser sample attains higher peak angle of shearing resistance in between.

• Initially dense samples expand or dilate when sheared, and initially loose samples compress.

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