# Shear Strength of Soils

DOI: 10.4018/978-1-4666-6505-7.ch005

## Abstract

Shear strength is a very important property of soils. The concept is used by geotechnical engineers in estimating the bearing capacity of foundations and in assessing the stability of retaining walls, slopes, and embankments and the design and construction of highway and airfield pavements. The shear strength of a soil can be regarded as its intrinsic capacity to resist failure when forces act on the soil mass. The strength is a function of the type as well as the physico-chemical make-up of the soil. This chapter explores the strength theories that are used in geotechnical engineering and which derive essentially from the classical theories of elasticity and plasticity. Such theories are usually modified when used by soil engineers in recognition of the discrete and multi-phase nature of soil deposits. It is important therefore to understand some of the fundamental concepts or hypothesis of continuum mechanics to be able to appreciate the limitations and applicability of strength theories in geotechnical engineering.
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## 5.1 Stress Concepts

Stress is defined as force per unit area. As was explained in chapter 4, the stress at a point is uniquely defined by specifying the magnitude and direction of a single vector having the units of force per unit elemental area, Chapter 4, Figure 1(a). In engineering, it is found more useful to define stress at a point in terms of the direct and shear components of stress acting normally and tangentially to the six faces of an elemental prism. The components of stress are then three normal stresses σxx, σyy, σzz and three pairs of reciprocally equal tangential stresses, σxy = σyx; σxz = σzx; σyz = σzy, Chapter 4, Figure 1(b).

Figure 1.

### 5.1.1 Principal Stresses

As stated in Chapter 4, it is possible to have a prism which is oriented in such a way that the shear components of stress are zero. The normal stresses so obtained are called principal stresses. The relationship between the principal stresses and the corresponding conjugate stresses can be obtained by considering the equilibrium of forces acting on an oblique plane which is arbitrarily inclined to the faces of the parallelopiped as in Figure1 in Chapter 4 and with direction cosines l, m and n. This can be derived more easily for a two-dimensional state of stress.

Let us consider a 2-dimensional state of stress as shown in Figure 1. It is assumed that the stresses normal to the plane of this paper are zero i.e. σz = 0; τxz = 0; τzy = 0. Consider the equilibrium of forces on the plane which makes an angle with the vertical, Figure 2.

(5.1a)
Figure 2.

Plane stresses in different orientations

where

ΔAn = Area of the inclined plane.

If we divide by ΔAn, Equation 5.1a becomes

(5.1b)

From trigonometry, we have the following identities

;

Substituting the above identifies in Equation 5.1b and rearranging we have

(5.1c)

Considering forces parallel to the inclined plane we have

Therefore

Dividing by ΔAn gives

Using trigonometric identities we have

(5.2a)

Differentiate Equation 5.1c with respect to θ

i.e.

(5.2b)

This is the plane at which σn is maximum or minimum i.e. the principal plane. The principal stresses represent the maximum and minimum values of all the stresses at the given point: the axes are commonly designated so that σ1 ≥ σ2 ≥ σ3.

#### 5.1.1.1 Principal Planes and Principal Stresses

For the plane stress loading, the Principal Planes are the two planes where the normal stress (σ) is the maximum or minimum. Usually

• There are no shear stresses on principal planes.

• The two planes are mutually perpendicular.

• The orientations of the planes (θp) are given by:

5.3

which gives two values (θp1 and θp2)

The Principal stresses are the normal stresses (σ) acting on the principal planes and are given as

(5.4a)
(5.4b) where

(5.4c)

#### 5.1.1.2 Maximum Shear (τmax)

Similarly it is possible to have three planes on which the shear rather than the normal stresses are at maximum. Such stresses are called the maximum shear stresses. For plane stress loading, the maximum shear stress occurs on two mutually perpendicular planes and the orientations of the two planes (θs) are given by:

(5.5) which gives two values (θs1 and θs2) i.e.

τmax = R(5.6a)

The planes are related by

The values of τmax can also be obtained in terms of principal stresses using equation 5.2

(5.6b)

For a 3-D case, the principal shear stresses are given by

;
;
(5.6c)

The angular distortions on the planes of maximum shear are

;
;
(5.6d)
• Problem 5.1: Stress at a Point.

The principal stresses on a prism of soil are 20 kN/m2 and 12 kN/m2. Determine the normal and shear stresses on a plane which makes an angle of 30o with the horizontal plane.

Solution:

Using Equation 5.1c we have

Similarly Equation 5.2 gives

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