(a) Rankine’s Analysis
Rankine considered the equilibrium of two soil elements, one immediately below the foundation (element I) and the other just beyond the edge of the footing (element II), but adjacent to element
I. When the load on the footing increases, and approaches a value qf, a state of plastic equilibrium is reached under the footing. For the shear failure of element I, element II must also fail by lateral thrust from element I.
Now, for element I, the major principal stress p1 from vertical direction is
According to Rankine’s active earth pressure theory the resulting stress p2 (called the minor stress) in the horizontal direction is given by
(i.e. minor principal stress=major principal stress x ka)
where ka = co-efficient of active earth pressure
where Ø is the angle of repose for the soil.
For element II, the vertical stress p3 is evidently equal to the weight of overburden = γ D. However, the stress p2 in the horizontal directfon is the same as found in (i) above. However, since p2 is much more than p3, major stress on element II is p2 and minor stress is p3. From Rankine’s earth pressure theory, minor principal stress = ka x (major principal stress)
Substituting the values of p2 and p3, we get
Eq. (2.7) gives the bearing capacity of cohesionlcss solis as zero at the ground surface. This is not consisten with the general experience. However, Eq. 2.6 may be used in the following form to get the minimum depth of foundation.:
where q= intensity of loading.
(b) Terzaghi’s Analysis*
An analysis of the condition of complete bearing capacity failure, usually termed as general shear failure was made by Terzaghi by assumlng that the soil behaves like an ideally plastic material. Fig. 2.20 (a) shows a shallow footing in which the depth D iS equal to or less than the width B of the footing. The loaded soil fails along a composite surface ABCB1A1.
FIG. 2.20 TERZAGHI'S ANALYSIS
Terzaghi gave the following equations:
where Nc, Nq and Nγ are the dimensionless numbers, called the bearing capacily factors, the values of whích can be obtained from Table 2.1. The above analysis corresponds to general shear failure in which the soil properties are such that a slight downward movement of footing develops fully plastic zones and the soil bulges out [Fig. 2.20 (c)]. In case of fairly soft or loose and compressíble soil, large deformation may occur below the footing before the failure zones are fully developed. Such a failure is known as local shear failure [Fig. 2.20 (d)] which is associated with considerable vertical soil movement before soil bulging takes place. The bearing capacity factors corresponding to the local shear failure are indicated with dashes, i.e. Nc’ , Nq’ and Nγ’ (Table 2.1). Terzaghi gave the following equation for local shear failure:
TABLE 2.1 TERZAGHI'S BEARING CAPACITY FACTORS.