## Strength of Soils

The property that enables a material to remain in equilibrium when its surface is not level is known as its shear strength. Soils in liquid form have virtually no shear strength and even when solid have shear strengths of relatively small magnitudes compared with those exhibited by steel or concrete.

To appreciate this section some knowledge of the relevant strength of materials is useful. A brief summary of this subject is set out below.

### 3.1 Friction

Consider a block of weight W resting on a horizontal plane (Fig. 3.1a). The vertical reaction, R, equals W, and there is consequently no tendency for the block to move. If a small horizontal force, H, is now applied to the block and the magnitude of H is such that the block still does not move, then the reaction R will no longer act vertically but becomes inclined at some angle, a, to the vertical.

By considering the equilibrium of forces, first in the horizontal direction and then in the vertical direction, it is seen that:

Horizontal component of R = H = R sin a

(a) No horizontal (b) Horizontal force applied force applied a (=0 at sliding)

(a) No horizontal (b) Horizontal force applied force applied

### Fig. 3.1 Friction.

The angle a is called the angle of obliquity and is the angle that the reaction on the plane of sliding makes with the normal to that plane. If H is slowly increased in magnitude a stage will be reached at which sliding is imminent; as H is increased the value of a will also increase until, when sliding is imminent, a has reached a limiting value, <j>. If H is now increased still further the angle of obliquity, will not become greater and the block, having achieved its maximum resistance to horizontal movement, will move (cp is known as the angle of friction). The frictional resistance to sliding is the horizontal component of R and, as can be seen from the triangle of forces in Fig. 3.1b, equals N tan 4> where N equals the normal force on the surface of sliding (in this case N = W).

As a only achieves the value <f> when sliding occurs, it is seen that the frictional resistance is not constant and varies with the applied load until movement occurs. The term tan <fr is known as the coefficient of friction.

### 3.2 Complex stress

When a body is acted upon by external forces then any plane within the body will be subjected to a stress that is generally inclined to the normal to the plane. Such a stress has both a normal and a tangential component and is known as a compound, or complex, stress (Fig. 3.2).

### Principal plane

A plane that is acted upon by a normal stress only is known as a principal plane, there is no tangential, or shear, stress present. As is seen in the next section dealing with principal stress, only three principal planes can exist in a stressed mass.

### Principal stress

The normal stress acting on a principal plane is referred to as a principal stress. At every point in a soil mass, the applied stress system that exists can be resolved into three principal stresses that are mutually orthogonal. The principal planes corresponding to these principal stresses are called the major, intermediate and minor principal planes and are so named from a consideration of the principal stresses that act upon them. The largest principal stress, <7i, is known as the major principal stress and acts on the major principal plane. Similarly the intermediate principal stress, 02, acts on the intermediate principal plane whilst the smallest principal stress, 03, called the minor principal stress, acts on the minor principal plane. Critical stress values and obliquities generally occur on the two planes normal to the intermediate plane so that the effects of 02 can be ignored and a two-dimensional solution is possible.

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