CIV301 Soil Mechanics Shear Strength Handout

CIV 301 Dr. Reem Sabouni

Source: J. Cernica, Geotechnical Engineering, Soil Mechanics, 1995. of 18

Shear Strength of Soil 1. INTRODUCTION

In many soil mechanics problems, the shear strength of the soil emerges as one of its most

important characteristics. Indeed, this problem group may include:

1. Stability of slopes (e.g., hillsides, cuts, embankments, earth dams)

2. Ultimate bearing capacity of a soil

3. Lateral pressure against retaining walls, sheeting, or bracing

4. Friction developed by piles

Only a small number of construction problems are not in some manner related to the shear strength

of the soil.

The shear strength of the soil may be attributed to three basic components:

1. Frictional resistance to sliding between solid particles

2. Cohesion and adhesion between soil particles

3. Interlocking and bridging of solid particles to resist deformation

It is neither easy nor practical to clearly delineate the effects of these components on the shear

strength of the soil. This becomes more apparent when one relates these components to the many

variables that directly or indirectly influence them, not to mention the lack of homogeneity and

uniformity of the characteristics that typify most soil masses. For example, these components may

be influenced by changes in the moisture content, pore pressures, structural disturbance, fluctuation

in the groundwater -table; underground water movement, stress history, time, perhaps chemical

action, or environmental conditions.

There are a number of tests, both laboratory and field, that are used to obtain a

measure of the shear strength of a given soil. These are coupled with various theoretical

considerations. A combination of-the theoretical with the experimental efforts provides

the background and tools for the basic understanding of and dealing with many of the

phenomena related to the shear strength of the soils.



2. BRIEF REVIEW OF SOME ELEMENTS FROM BASIC MECHANICS

With some appropriate modifications, many of the concepts used in some of the basic

mechanics courses can be employed to explain various phenomena in soils. In this

chapter we shall focus on the soil strength and some related parameters.

Friction Between Solid Bodies

A block of weight W is placed on a horizontal plane as shown in Fig. 1. The forces

acting on the block in. Fig. 1a are its gravity and the reaction of the plane. Although a

friction force is "available," it does not come into play since there is no horizontal

force applied to the block. Suppose now that a horizontal force, PI, is applied to the

block, as shown in Fig. 1b. If P1 is small, the block will not move; the resultant, Ra,

makes an angle, α, commonly referred to as the angle of obliquity of force P1. For

equilibrium, a resisting horizontal force, F1, must therefore exist to balance P1. For this

case, the angle α is less than the angle of friction denoted by φ. N is the component of

the weight normal to the plane.

If the applied horizontal force is increased to P2, as shown in Fig.1c, the friction

force also increases to a value of F2 It reaches a maximum value when the angle of

obliquity α is equal to the angle of friction φ. One notes that at this point a state of

impending motion (slip) exists. For the range of values 0 < α < φ no slip or sliding

occurs. At the point of impending motion, the maximum or available frictional force F could

be related to the normal resisting force by a coefficient -of friction μ as given by Eq. 1:

(1)

The coefficient of friction µ is independent of the area of contact. It is, however, strongly

dependent on the nature of the surface in contact—the type of material, the condition of

the surface, and so on. Furthermore, in most materials the coefficient of static friction is

somewhat larger than the kinetic coefficient.



If the horizontal force is increased to a value of P3 as shown in Fig.1d, such that the

angle α is greater than φ, the block will start sliding. The frictional force cannot exceed

the value given by F = µN, and therefore the block will accelerate to the right.

A similar analogy is indicated in Fig.2. The block resting on a plane inclined at an

angle α1 will remain at rest provided the angle α1 is less than the friction angle, φ.

Figure 1 Development of friction on a horizontal plane. (a) No friction. (b) Development of friction. (c) Friction, impending motion. (d) Friction, motion (acceleration).



Figure 2 Development of friction on inclined plane. (a) Friction, no motion. (b) Friction, impending motion. (c) Friction, motion (acceleration). The frictional force F is only as large as necessary to balance the component of the

weight parallel to the plane. On a steeper slope, as indicated in Fig.2b, we have a case

of impending motion when the angle of the plane, α2, equals the angle of friction, φ. It

is equal to the coefficient of friction times the normal force, as indicated by Eq. 10-1.

Finally, if the plane is inclined at an angle α3 that is greater than angle φ, the block

will start sliding. The maximum available frictional force is smaller than the

component of the weight parallel to the plane.

From either Fig.1c or Fig.2b, both conditions of impending motion, one notes that

tan φ FN

μ (2)

Although the frictional resistance in sands and other cohesionless granular materials resembles

that discussed above, there are some important differences. In such soils the frictional

resistance consists of both sliding and rolling friction, coupled with a certain degree of

interlocking of the solid particles. In addition, various other factors, such as degree of

saturation, particle size and shape, consistency, or intergranular pressure, have a pronounced effect

on the shear strength of the soil.



Stress at a Point One may recall from mechanics that, in general, both normal and shear stresses act on planes

passing through a point in an element of mass that has been subjected to external loading. Figure

3 depicts these stresses on a small element containing point A, which is located within the mass

shown in Fig.3a. The stresses normal to a plane are designated by σ and those parallel to any

given plane by τ. Thus, nine components of stress define the state of stress at a point. If such

stresses on all possible planes through a point are investigated, three planes at right angles to each

other will be found to be subjected to normal stresses only, as shown in Fig.3c. These stresses, in

our case σl , σ2, and σ3, are known as principal stresses, and the planes on which they act are

known as principal planes. Sometimes these planes are further defined as the major, the

intermediate, and the minor principal planes, tied to the magnitudes of the respective stresses

acting on them. Of course, such planes are at a different orientation from those shown in Fig.

10.3b since shear stresses do not exist on principal planes.

It is convenient, and in the case of soil studies usually sufficient, to illustrate some of the above-

mentioned relationships by considering the case of biaxial stresses, as shown in Fig.4. The element

in Fig.4b is an enlarged version of the element indicated in Fig.4a. As mentioned previously, the

general case of stress consists of both normal and shear stresses. Now let us assume the element

to be cut by a plane passing through m-n, and place the left segment in equilibrium, as shown

in Fig.4c. If the inclined plane is oriented θ degrees relative to the y axis, and if ΔAn

represents the area on the inclined plane, the areas of the vertical and horizontal faces of the

elements are ΔAncos θ and ΔAnSin θ, respectively. The components of these forces, parallel

and perpendicular to the m-n plane, are shown by the dashed lines. Summing the force in the

direction normal to the plane, we have

∆ ∆ cos cos ∆ cos sin ∆ sin sin

∆ sin cos



Figure 3 Stress at a point; case of triaxial stresses. (a) Point A in element of mass

subjected to load. (b) General state of stress at point A. (c) Principal stresses at point A.

which reduces to

cos sin 2 sin cos



From the trigonometric identities,

coscos 2θ 1

2 ; cos sinsin 2θ2 ; sin

cos 2θ 12

Thus, after rearranging terms, in terms of double angles, the above equation becomes

cos 2θ τ sin 2θ (3)

Figure 4 Stresses at a point; case of biaxial stresses.



Summing forces in the direction parallel to the plane, we have

∆ ∆ cos sin ∆ cos cos ∆ sin cos

∆ sin sin

which reduces to

sin cos cos sin

and, in terms of double angles,

sin 2θ τ cos 2θ (4)

The orientation of the principal plane is determined by setting the derivative dσ�/dθ=0. From

Eq.3,

sin 2 2 cos 2θ 0

from which τ

σ σ /2

Mohr's Circle for Stress

The normal and shear stresses given by Eqs. 3 and 4, respectively, may be represented graphically by

an extremely useful device known as Mohr's circle for stress. It is named after the German

engineer Otto Mohr, who devised it in 1882. The Mohr circle represents these equations in a

manner that makes them more easily understood and remembered and brings out their physical

significance more lucidly.

It is a rather simple matter to show that these are the equations of a circle in σn-τ plane. For

convenience, Eqs. 3 and 4 are rewritten below, Eq. 3 having been slightly rearranged:

cos 2θ τ sin 2θ (3)

sin 2θ τ cos 2θ (4)



Squaring both sides of each equation, we get

2 2 cos 2θ τ cos 2θ sin 2θ τ sin 2θ

and

2 sin 2θ τ cos 2θ sin 2θ τ cos 2θ

Hence., adding the two equations,

2 2 τ

This is the equation of a circle of the form

a R

where the radius is

R 2 τ

and . The center of the circle lies at a point , 0 . Note that the center of the circle

always lies on the σn axis.

The above discussion may be more meaningfully complemented and expanded



Figure 5 (a) General case of stress at a point. (b) Mohr circle for the stress condition in (a).

by an actual construction of a Mohr's circle. As a start, some basic steps shall be outlined:

1. The normal stresses are plotted as horizontal coordinates. The tensile stresses are considered

positive (plotted to the right of the origin); the compressive stresses are considered negative

(plotted to the left of the origin). However, in soil mechanics it is customary to indicate

compressive stresses as positive.

2. The shear stresses are plotted as vertical coordinates. Positive shear stresses

are plotted above the origin; negative shear stresses

are plotted below the origin.



3. Positive angles on the circle are obtained when measured in the counterclockwise sense;

negative angles on the circle are obtained in the clockwise sense. An angle of 2θ on the circle

corresponds to an angle of θ on the element.

Figure 5b shows Mohr's circle for the assumed stresses given in Fig.5a at the stressed point,

assuming that σx> σy > 0. The normal and shear stresses on vertical plane P give the coordinates of

one point on the circle, as indicated in Fig.5b by point D. Plane R, also vertical and 180° from P,

is located at 2θ = 360° on the circle from plane P, resulting in the same point on the circle.

Similarly, the stresses on horizontal planes Q and S plot a different single point E. This

shows that the stresses on parallel planes of an element oriented at any angle are corre-

spondingly equal. The point of intersection of a line DE connecting these two points and the

horizontal axis σn gives the location of the center of the circle. The diameter of the circle is

the length of line DE.

Figure 6 Mohr's circle for case of principal stresses.

σ3



The circle proves to be indeed a valuable visual aid. For example, the maximum shear

stress is readily apparent to be equal to the radius of the circle. The orientation of this

stress with respect to the original plane can either be measured or easily calculated with

the help of trigonometry. The principal stresses σmax and σmin can likewise be spotted

and determined easily, and the orientation can easily be measured or calculated.

Furthermore, on the principal planes (at σmax and σmin) the shear stresses are zero.

Figure 7 shows a number of additional basic relationships; which can be obtained

from Mohr's circle, such as that for the element of Fig. 10.6.

Figure 7 Some basic relationships, Mohr's circle for element of Fig.6a.

Observed directly from Fig.7, some of these basic relationships are:

1. The maximum shear stress is equal to the radius of the circle. Furthermore, the

maximum shear stresses are on planes that make an angle of 45° with the principal

planes.

2. The resultant stress on any plane, e.g., S, has a magnitude of √ . Its magnitude is

equal to the distance 0S shown in Fig.7. Furthermore, its angle of obliquity is equal to

tan .

3. The maximum angle of obliquity is constructed by the line 0T in Fig.7, such that the

line is tangent to the circle at point T. Correspondingly, the stresses, normal and shear,

that correspond to point T on the circle, represent the stresses on the plane of maximum

obliquity. Note that the shear stress on this plane is less than τmax. Interestingly, one

notes that slippage occurs at the point of maximum obliquity and not at the angle

where τmax occurs. It is, therefore, αf that assumes the more prominent position

regarding slip failure.



3 MOHR'S THEORY OF FAILURE

The shear strength of soil is generally regarded as the resistance to deformation by

continuous shear displacement of soil particles along surfaces of rupture. That is, the shear

strength of the soil is not regarded solely in terms of its ability to resist peak stresses, but it

must be viewed in the context of deformation that may govern its performance. In that light,

therefore, shear failure is necessarily viewed as the state of deformation when the functional

performance of the soil mass is impaired.

There are a number of different theories as to the nature and extent of the state of stress

and deformation at the time of failure. Failure of a soil mass, particularly cohesionless soil,

which develops its strength primarily from solid frictional resistance between the

interlocking of grains, appears to be best explained by Mohr's rupture theory. According to

Mohr's theory, the shear stress in the plane of slip reaches at the limit a maximum value

that depends on the normal stress acting in the same planes and the properties of the

material. This represents the combination of normal and shear stresses, which results in a

maximum angle of obliquity αf (see Fig.7).

Hence, based on Mohr's theory, the shear strength of a given soil is a function of the

normal stress and the soil properties (e.g., strength parameters c and φ). For

cohesionless soils, for example, the angle of obliquity reaches a maximum or limiting

value αf equal to the friction angle φ as, indicated in Fig.8. Hence, for this special case,

where the cohesion intercept is zero (c=0), the shear strength could be expressed as

φ ( 7 )

where s represents a convenient and widely used symbol to denote shear strength in soils.

One may note that for αf > 0, the value for s is always less than τmax.

Figure 9a depicts a general s-σ relationship, while Fig.9b illustrates a linear s-σ

relationship. The linearity may not be quite correct (for example, see Fig.17), but is

reasonably accurate and mathematically quite convenient.



Figure 8 Relationship between shear strength, normal stress, and angle of obliquity α.

Now let us assume that the function s = f (σ) represents the shear strength of the soil at

impending failure. This may be represented by the heavy line in Fig.10, designated as

Mohr's envelope. Based on Mohr's theory, any combination of stresses that falls within this

envelope represents a stable condition. All five Mohr circles represented in Fig.10 fall in

this category. It is to be noted, however, that circles I—IV are tangent to the Mohr

envelope, thereby depicting a condition of impending failure. The corresponding points of

tangency represent the resultant's shear and normal stresses for the respective angles of

obliquity. Also, the corresponding shear strengths for the various cases are the shear

stresses at the points of tangency. Circle V, on the other hand, is well within Mohr's

envelope. The maximum shear stress for this condition is below the shear strength of the

soil.

The friction angle decreases slightly in most soils, and particularly granular soils, with

increasing confining stresses. Hence, Mohr's envelope is slightly curved, a beast-fitted curve,

as indicated in Fig.10. However, the variation from a straight line in most instances is

relatively small. Thus, it is mathematically convenient to represent the Mohr envelope by

a straight line, sloped at an angle φ. This is shown in Fig.9b as first proposed by

Coulomb in 1776 in connection with his investigations of retaining walls.

Figure 11c shows a Mohr circle and the corresponding strength envelope for an

element stressed as shown in Fig.11a. For this case the expression for the



Figure 9 Shear strength according to (a) Mohr and (b)

Coulomb.

Mohr's envelope

Tension Compression s= f(σ)

Figure 10 Mohr's circles for various cases of stress: I, simple tension; II, pure shear; III, simple

compression; IV and V, biaxial compression.



shear strength, s, may be given as

φ (8)

where

c = cohesion or cohesion intercept

φ = angle of internal friction

Also from Fig.11c, the orientation of the failure plane with reference to the plane of major

principal stresses in a granular soil may be readily established as

Figure 11 (a) Element, σ1> σ3. (b) Normal and shear stresses on plane of failure. (c) Mohr's



circle for stress condition shown in (a) .

2 90 φ or

45 φ (9)

The normal and shear stresses on the plane of failure are shown in Fig.11 b.

Although it is difficult to measure it very accurately, pore-water pressure has long been

recognized as an important factor in shear-strength evaluation. Hvorslev's work

provided relevant data to support the use of effective stress parameters in describing the

shear strength of soils. Sometimes referred to as the Mohr-Coulomb-Hvorslev

equation, the relationship is

(10)

Where

c’= effective cohesion value

σ’= effective normal stress = σ-u

u = pore-water pressure

θ’= effective angle of internal friction

Equation 10 does not imply, however, any unique relationship between effective stress and

shear strength. Many other factors enter into the picture. Void ratio at failure, time, stress

history, grain structure (i.e., flocculated or dispersed), environmental conditions (i.e., pore

water, water table fluctuation, temperature), degree of saturation, conditions that

formulate the formation of soil, capillary tension, and effective stresses in the direction

normal to the plane of greatest shear distortion are some of the factors that may have a

significant effect, particularly for clay soils.



4 DETERMINATION OF THE SHEAR STRENGTH OF SOILS

In the preceding section the strength envelope was developed primarily from theoretical

considerations. Little was said about the practical and the experimental aspects. That is,

while the theoretical presentation serves as a satisfactory tool in defining failure in soils,

its validity and usefulness must be closely tied to the parameters observed

experimentally and under field conditions. Indeed, to foundation designer the very

essence of the Mohr envelope is experimental data. The failure criteria for soils,

subsequent to due considerations of the related parameters, are usually determined

experimentally by one or a combination of the following laboratory tests:

1. Direct shear test

2. Triaxial compression test

3. Unconfined compression test

As a general comment the typical soil stratum is neither continuous, homogeneous, nor

isotropic. Hence, under the best of conditions, a limited number of tests give but a

rough approximation of the stratum characteristics. It is thus rather apparent that a

soil sample being tested should be as representative as possible of the sample in the

field, and as undisturbed as possible. Also, the loads, restraints, and other conditions

of testing should correspond to a reasonable degree to those in situ. Yet it is

recognized that the more act of obtaining a sample from a natural deposit radically

alters the state of stress and induced strains.

CIV301 Soil Mechanics Shear Strength Handout

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