## What is shear strength?

The term shear strength is often associated with the soil. Shear strength is defined as the resistance to the deformation of soil particles due to the continuous action of shear stress. In short, shear strength is the maximum shear stress that a soil can sustain.

The determination of stress-strain characteristics of soil is important to obtain detailed information about the soil. Shear strength, compressive strength, and tensile strength of soil can be determined using theoretical methods and laboratory methods. The various methods for determining shear strength are as follows.

## Theoretical methods for determining shear strength

### Mohr-Coulomb theory

Mohr-Coulomb's theory in geotechnical engineering is used to determine the shear strength of soil at different effective and principal stresses. This theory is applicable for all types of saturated and unsaturated soils. As per this theory, shear stress on a failure plane is a linear function of normal stress. Mathematically, it can be stated as

${\tau}_{f}=f\left({\sigma}_{n}\right)$

Where ${\tau}_{f}$ is the shear stress at failure and σ_{n} is the normal stress.

Analyzing the soil, it was observed by Mohr-Coulomb that the relationship between shear strength and normal stress can be represented by a straight line. Hence, the following strength equation was derived by Mohr-Coulomb.

${\tau}_{f}=c+{\sigma}_{n}.\mathrm{tan}\varphi $

Where $c$ is the cohesion and $\varphi $ is the angle of internal resistance.

The maximum shear stress or the shear strength of the soil can be calculated using Mohr’s stress circle. As per Mohr’s stress circle, the analytical equations for normal stress and shear stress are given as

${\sigma}_{n}=\left(\frac{{\sigma}_{1}+{\sigma}_{3}}{2}\right)+(\frac{{\sigma}_{1}-{\sigma}_{3}}{2}.\mathrm{cos}2\alpha )$

Where ${\sigma}_{1}$ is the major principal stress, ${\sigma}_{3}$ is the minor principal stress, and $\alpha $ is the angle of failure.

### Terzaghi's effective stress principle

According to the father of soil mechanics, Terzaghi, effective normal stresses control the shearing resistance of the soils. Hence, Terzaghi revised Mohr-Coulomb's theory in terms of the effective stress principle. Mathematically, this principle states that

${\tau}_{f}=c+\sigma \text{'}.\mathrm{tan}\varphi \text{'}\phantom{\rule{0ex}{0ex}}{\tau}_{f}=c+(\sigma -u).\mathrm{tan}\varphi \text{'}\phantom{\rule{0ex}{0ex}}$

Where $c$ and $\varphi $' are effective cohesion and effective angle of friction respectively, and u is the pore-water pressure.

## Effect of particle size on strength of soil

Particle size plays a very important role in determining strength. If the particle size is finer, strength will be more due to a strong cohesive bond between particles, whereas if the particle size is coarse, strength will be less as a cohesive force between particles will be lesser.

## Laboratory methods for determining shear strength

### Direct shear tests

The direct shear tests are the simple and the most commonly used tests for determining the shear strength of the soil. The test can be used for both saturated and unsaturated soils. The direct shear test is performed as per ASTM D3080. The tests are generally preferred for coarse-grained cohesionless soils and not for cohesive soils having fine-grained particle sizes. As per the recommendations of ASTM D3080, three or four samples should be tested for a single soil. In direct shear tests, an apparatus called direct shear box apparatus is used. The shear box consists of two parts of square or circular cross-section. The lower half portion of the shear box is held rigidly and rested over slides or rollers which can push the lower side at a constant rate using the geared jacks. The upper half portion of the shear box butts against a proving ring. The chosen soil specimen is compacted in the shear box and held between metal grids, porous grids, and plates. The normal force is applied to the soil specimen through the pressure pad and the shear force is applied by moving the lower shear box with the help of geared jacks. The readings are noted for the normal force required and shear force required for deforming, that is shearing of soil specimen, and the normal stress and shear stress are calculated by multiplying the corresponding force value with the area of cross-section of the soil.

As per ASTM D3080, direct shear tests can be performed for all three types of drainage conditions. For drained direct shear tests, perforated plates are used. The procedure for this test is the same but the shearing takes a long time as the drainage is very slow. For consolidated undrained direct shear tests, perforated grids are used. First, the normal stress is applied to allow sample consolidation and after the completion of the consolidation, the sample is sheared quickly in 5 to 10 minutes. For performing undrained direct shear tests, plain plates are used where no water is allowed to drain.

The demerits of direct shear tests are as follows:

- In direct shear tests, the stress distribution across the soil sample is very complex.
- The orientation of the failure plane is always horizontal, which is not always correct. Hence, shear strength obtained by direct shear tests may not be always accurate.

### Triaxial compression tests

Triaxial compression tests are the most common tests performed on the soil to determine the strength of the soil. The test can be used for both saturated and unsaturated soils. The apparatus consists of a high-pressure cylinder cell, consisting of three-outlet connections connected to the cell fluid inlet, pore water outlet, and drainage outlet. A separate compressor is used to apply fluid pressure in the cell. A cylindrical specimen is put in a rubber membrane. A stainless steel piston running through the center of the top cap applies deviator stress on the sample.

A particular confining pressure or cell pressure (${\sigma}_{3}$) is applied to the cell and the value of ${\sigma}_{1}$ at deformation is noted down. The following relation between effective cohesion and effective angle of internal friction is derived from the triaxial compression tests-

$\stackrel{-}{{\sigma}_{1}}=\stackrel{-}{[{\sigma}_{3}}.{\mathrm{tan}}^{2}.(45+\frac{\varphi \text{'}}{2})]+[2c\text{'}.\mathrm{tan}.(45+\frac{\varphi \text{'}}{2})]$

where, $\stackrel{-}{{\sigma}_{1}}$ is the effective major principle stress, $\stackrel{-}{{\sigma}_{3}}$ is the effective minor principle stress, $c\text{'}$ is the effective cohesion, and $\varphi \text{'}$ is the effective angle of friction.

Using various values of ${\sigma}_{1}$ and ${\sigma}_{3}$ obtained from the test, the values of effective cohesion and effective angle of internal friction can be calculated.

Also, another important relation is derived from the triaxial compressive tests

${\sigma}_{d}={\sigma}_{1}-{\sigma}_{3}$

Where ${\sigma}_{d}$ is the deviator stress.

### Unconfined compression tests

The unconfined compression test is a special type of triaxial compression test. In this test, ${\sigma}_{3}=0$, or the cell pressure, also known as confined pressure is absent. Hence, the specimen is subjected to only major principal stress ${\sigma}_{1}$, until the specimen fails.

### Vane shear tests

Vane shear test is a quick test performed to find the shear strength of cohesive soil. The vane shear test can be performed in the laboratory or on the in-situ soil on the field. The apparatus consists of four thin steel plates welded to a steel rod. A calibrated torque string is attached to the apparatus to measure the torque. The vanes are gently pushed into the soil specimen and the soil and the vanes are rotated at a uniform speed, which is measured using the dial attached to the apparatus. The vanes then shear the soil specimen and the corresponding torque value is noted down by multiplying the dial reading with the spring constant. The shear strength is then calculated after calculations.

Using the value of torque, the undrained cohesion of the soil is calculated using the following formula,

${c}_{u}=\frac{T}{\mathrm{\pi}.{\mathrm{d}}^{2}.[{\displaystyle \frac{\mathrm{H}}{2}}+{\displaystyle \frac{\mathrm{d}}{6}}]}$ for double shearing

${c}_{u}=\frac{T}{\mathrm{\pi}.{\mathrm{d}}^{2}.[{\displaystyle \frac{\mathrm{H}}{2}}+{\displaystyle \frac{\mathrm{d}}{12}}]}$ for single shearing

Where $T$ is the torque applied, $H$ is the height of the vane, $d$ is the overall diameter of the vane.

## Advancements in testing

The field of soil mechanics is advancing every day and new methods are being developed for determining new factors. Every year expert lectures like Géotechnique lectures are held on an international level to discuss and present ideas about researches.

## Context and Applications

The test for determination of shear strength is useful for the students undergoing the following courses:

- Bachelors of Technology in Civil Engineering
- Masters of Technology in Geotechnical Engineering

## Practice Problems

1. What does Mohr-Coulomb's theory state?

- Shear stress is a function of Major principal stress
- Shear stress is a function of Normal stress
- Shear stress is a function of Tangential stress
- Shear stress is a function of Minor principal Stress

Answer: Option b

Explanation: According to Mohr-Coulomb's theory, shear stress is a function of normal stress.

2. Who is considered the father of soil mechanics?

- Terzaghi
- Vanapalli
- Mohr
- Coulomb

Answer: Option a

Explanation: Terzaghi is considered the father of soil mechanics.

3. In which test, the cell pressure is equal to zero?

- Vane shear test
- Unconfined compression test
- Direct shear test
- Unsaturated soils test

Answer: Option b

Explanation: In an unconfined compression test, the cell pressure is equal to zero.

4. What is the alternate name of confining pressure?

- Deviator stress
- Cell pressure
- Normal pressure
- Unsaturated soils pressure

Answer: Option b

Explanation: The alternate name of confining pressure is cell pressure.

5. What is the formula for calculating deviator stress?

a) ${\sigma}_{d}={\sigma}_{1}-{\sigma}_{3}$

b) ${\sigma}_{d}={\sigma}_{1}+{\sigma}_{3}$

c) ${\sigma}_{d}={\sigma}_{3}-{\sigma}_{1}$

d) ${\sigma}_{d}={\sigma}_{1}/{\sigma}_{3}$

Answer: Option a

Explanation: The formula for calculating deviator stress is ${\sigma}_{d}={\sigma}_{1}-{\sigma}_{3}$.

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