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In this article we will discuss about:- 1. Introduction to the Determination of Stresses on any Plane in a Soil Mass 2. Principal Stresses and Principal Planes 3. Mohr’s Circle of Stresses 4. Mohr-Coulomb Failure Theory 5. Stress Path.

**Introduction to the Determination of Stresses on any Plane in a Soil Mass****: **

Soil is a particulate system with individual particles bonded together by weak or negligible cementing or cohesive forces. Hence, when external loads are applied on a soil mass from the structure, the soil particles tend to undergo displacement relative to one another.

Thus, soils resist external loads by shear, although direct shear forces are not applied on the soil. Most of the loads from earth or earth-supported structures on soils cause compression of the soil. A compression failure of a soil mass is in reality a shear failure along a failure plane. The tensile strength of the soil is almost zero and the compressive strength is not relevant because it is the shear strength that controls the stability of the structures supported on a soil mass.

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There is a fundamental difference between soil and other construction materials such as steel or concrete. In case of steel or concrete, there is strong bonding between different particles that helps in resisting loads in various forms such as compression, tension, flexure, shear, and torsion. Failure in such materials occurs by overcoming this bond strength along a fracture line.

In contrast to this, soil is a particulate system with or without a small cohesion between the particles. When the applied stresses overcome the cohesive forces between the soil particles or particle groups, there is a relative movement or displacement between particles, which is considerably higher than that in other materials, before failure is actually initiated. The frictional force arising out of this relative movement between particles is an important component of the soil strength in resisting loads.

Thus, because of the particulate nature of soil, external loads applied on a soil mass from Earth or earth-supported structures induce shear stresses in a soil mass. The resistance offered by the soil on any plane at any point in a soil mass to the induced shear stresses is called the shear strength of the soil.

The shear strength of a soil is not same on all planes in a soil mass and is dependent on several parameters. Similarly, shear stresses also depend on the direction of the plane with respect to the applied stresses. Failure occurs on the plane on which applied shear stress reaches and just exceeds the potential shear strength of the soil on that plane; such a plane is called the critical failure plane.

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Thus, critical failure plane represents the unique combination of applied shear stress and potential shear strength on which the former just equals the latter. It may be noted that the potential shear strength of the soil may be less on other planes than that on failure plane, but failure does not occur on these planes, as long as the component of shear stress on these planes is less than the shear strength. Similarly, there may be planes on which the induced shear stress may be more than that on failure plane.

Shear strength is the principal engineering property of a soil mass that controls the safety and stability of the Earth or earth-supported structures. It is necessary to understand and determine the shear strength of the soil in order to ensure the stability of these structures. Significant progress has been made in the understanding of shear strength of soil.

However, because of non-homogeneous nature of the soil in terms of composition, particle size and structure, and several other variables involved, shear strength is a complex engineering property and research is still underway for complete theoretical analysis of soil problems involving shear strength.

The soil possesses certain inherent shear strength and if the shear stress on the critical plane is less than this shear strength, the soil mass is in equilibrium. When the induced shear stress on any plane is equal to the shear strength of the soil on that plane, failure will be immanent and such a plane will be called the failure plane or critical plane.

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It is therefore required to determine the stresses on any plane at any point in a soil mass in order to determine the safety and stability of a structure supported on a soil mass. This is done with the help of Mohr’s circle of stresses.

**Principal Stresses and Principal Planes: **

The stresses acting on any plane at a point in a soil mass can be resolved into normal and tangential (shear) components. The plane on which the shear stress is zero is called the principal plane and the normal stress on the principal plane is called the principal stress. There are three principal planes at any point in a soil mass, which are mutually perpendicular, and hence three principal stresses (σ_{1}, σ_{2} and σ_{3}) on these principal planes.

The maximum principal stress is called the major principal stress, designated by the symbol σ_{1}.The minimum principal stress is called the minor principal stress, designated by the symbol σ_{3}. The principal stress that has value in between the major and minor principal stresses is called the intermediate principal stress and is designated by the symbol σ_{2}.

Thus, every point in a soil mass is subjected to three principal stresses due to external loads. In soil mechanics, the effect of intermediate principal stress is usually neglected for simplifying the analysis for cases, such as continuous footing or a retaining wall, where the intermediate principal stress acts along the length. This assumption may not introduce serious errors when the length of the structure is large compared to the other two dimensions; this is known as plane strain problem.

**Mohr’s Circle of Stresses****: **

Mohr’s circle of stresses enables to determine the stresses on any plane at any point in a soil mass. Mohr’s circle is a graphical representation of stresses at any point in a soil mass with normal stress on the x-axis and shear stress on the y-axis. Consider a small soil element subject to stresses shown in Fig. 13.1, where σ_{1} is the major principal stress and σ_{3} is the minor principal stress. Hence, plane AB and DC are the major principal planes and planes AD and BC are the minor principal planes.

**The stresses on any plane EF inclined at an angle ****θ**** with the major principal plane can be determined by constructing Mohr’s circle, shown in Fig. 13.2, in the following steps: **

1. Draw the x- and y-axes with O as origin. The x-axis represents the normal stress and the y-axis represents the shear stress.

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2. Plot OA = σ_{3} on the x-axis. Similarly, plot OB = σ_{1} on the x-axis.

3. Bisect AB to locate the midpoint at C.

4. With C as center and AC = BC as radius, draw a circle, which is the Mohr’s circle.

5. The coordinates of every point on the Mohr’s circle gives the stresses on a plane. Thus, the coordinates of point A (σ_{3}, 0) give the stresses on the minor principal plane. Hence, point A represents minor principal plane. Similarly, point B with coordinates (σ_{1}, 0) represents major principal plane.

6. Now, to determine the stresses on plane EF, which makes an angle θ with the major principal plane, draw the line CD, making an angle 2θ with CB to intersect the Mohr’s circle at point D. The coordinates of point D give the stresses on the plane EF

x-coordinate of point D = Normal stress on plane EF –

Similarly,

y-coordinate of point D = Shear stress on plane EF; τ = DE

Since angle θ can take any value, Mohr’s circle helps us to determine the stresses on any plane inclined at an angle θ with the major principal plane.

**Mohr-Coulomb Failure Theory****: **

Among several failure theories available, Mohr’s failure theory is found to be applicable to soils. Mohr’s theory has ultimately evolved into Mohr-Coulomb failure theory.

**Mohr’s Failure Theory****: **

Mohr’s failure theory states that a material essentially fails by shear, and the critical shear stress on the failure plane causing failure is a function of the normal stress on that plane, in addition to the properties of the material. Mohr’s theory can be mathematically expressed as –

τ_{f }= *f* (σ_{f}) …(13.3)

where τ_{f} is the shear stress on the failure plane at failure and σ_{f} the normal stress on the failure plane at failure.

When the normal stress on the failure plane increases, the shear strength of the soil also increases and vice versa. As per Mohr’s theory, the relation between the shear stress on failure plane and the corresponding normal stress is nonlinear. A plot can be made between the normal stress on the x-axis and the shear stress on failure plane on the y-axis; the curve so obtained is known as Mohr’s failure envelope, as shown in Fig. 13.4.

**Coulomb’s Equation for Shear Strength****: **

Coulomb slightly modified the Mohr’s theory stating that there is a linear relationship between the shear stress and the normal stress on the failure plane. Coulomb gave the following equation for the function* f*(σ_{f}) –

τ_{f} = c + σ_{f} tan*ɸ *…(13.4)

where c is cohesion, also called unit cohesion, and *ɸ* the angle of internal friction, also called angle of shearing resistance.

Graphical representation of Eq. (13.4) gives a straight line known as Coulomb’s failure envelope, as shown in Fig. 13.5. There is a unique failure envelope for each soil. The y-intercept of the Coulomb’s failure envelope is cohesion (c) and the slope of the Coulomb’s failure envelope is the angle of internal friction (ɸ). Together, cohesion and angle of internal friction are called shear parameters.

Mohr’s circle represents the stresses on all planes at a point in a soil mass. Mohr’s circle can be drawn representing stress conditions at any point in a soil on the same plot that contains Coulomb’s failure envelope.

A soil is in equilibrium as long as the Mohr’s circle of stress is below the Coulomb’s failure envelope. Failure will occur in the soil at that point if the Mohr’s circle touches the failure envelope, as shown in Fig. 13.5.

The tangent point, where the Mohr’s circle touches the failure envelope, represents the failure plane and the x- and y-coordinates of the point give the normal stress and the shear stress on the failure plane, respectively. Mohr’s circle cannot intersect and cross failure envelope because the moment Mohr’s circle touches the failure envelope, the soil will fail in shear and shear stress cannot increase beyond failure shear stress. Sometimes, the suffix f is omitted and Eq. (13.4) can be rewritten as –

τ = c + σ_{n} tanɸ …(13.5)

where σ_{n} is the normal stress. Both σ and σ_{n} are used interchangeably to represent normal stress.

**Modified Mohr-Coulomb Equation****: **

Terzaghi stated that the shear strength of a soil is a function of effective normal stress on the failure plane but not the total stress. Incorporating Terzaghi’s effective stress principle into Eq. (13.5), the modified Mohr-Coulomb’s equation for shear strength may be written as –

τ = c’ + σ_{n}‘ tanɸ’ …(13.6)

where σ_{n}‘ is the effective normal stress on the failure plane = σ_{n }– u, and u the pore pressure on the failure plane, and c’ and ɸ’ are the effective shear strength parameters.

It may be noted that the shear stress on the failure plane is the shear strength of the soil. Therefore, Eq. (13.6) can be used to determine the shear strength of the soil on any plane.

Later research established that the parameters c’ and ɸ’ are not unique or fundamental properties of a soil but may vary for the same soil depending on the drainage conditions in the soil mass. The shear parameters are, therefore, considered as mathematical parameters that represent failure condition under the given drainage condition.

The parameter c is, therefore, preferably called cohesion intercept and the parameter *ɸ* is called angle of shearing resistance. The angle of shearing resistance gives the rate at which the shear stress on failure plane increases with increase in the normal stress.

**Stress Path: **

A stress path is a curve that shows the changes in stresses as the load acting on the soil specimen changes. Lambe’s stress path is a locus of points of maximum shear stress acting on the soil specimen, as the load is changed. Figure 13.47 shows the Mohr’s circle with the maximum shear stress point.

Figure 13.48 shows Mohr’s circles that represent the stress conditions on a soil specimen as the deviator stress on the soil specimen increases keeping the cell pressure constant. The line joining the points of maximum shear stresses (points 1, 2, and 3) is the stress path. The direction of arrow on the stress path indicates the direction of stress changes. It is not necessary to draw the complete stress circles. Only points 1, 2, 3, etc. corresponding to maximum shear stresses are plotted, as shown in Fig. 13.49.

It is more convenient to draw the stress path on a p-q plot.

**The values of p and q depend on the type of stress path as follows: **

1. Effective stress path (ESP), which is a plot between –

2. Total stress path (TSP), which is plotted between the total stresses –

3. Total stress minus static pore pressure path (TSSP), which is a plot between –

where u is the static pore water pressure.

In the usual triaxial compression test, the static pore water pressure is zero and hence the stress paths TSP and TSSP coincide. In the field, the static pore water pressure depends on the depth of the point under consideration below the GWT.

**If the vertical stress (σ _{v}) and the horizontal stress (σ_{h}) are principal stresses, the three stress paths will be plots between the following: **

where u is the static pore water pressure.

**The following observations may be made with reference to the stress path: **

i. TSP to the right of ESP indicates positive pore water pressure and the opposite a negative pore water pressure.

ii. A negative value of q indicates that σ_{h} > σ_{v} and a positive value of q indicates σ_{h} < σ_{v}.

iii. A line drawn at 45° to the axis from any point on the stress path cuts the x-axis at a stress equal to σ_{h}.

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