Bearing Capacity and Shallow Foundations - Elements of Soil Mechanics - page 260

45 ° + φ
2

45 ° + φ
2

45 ° − φ
2

45 ° − φ
2

III III

90 ° I

90 °

II II

45 ° − φ
2

Fig. 9.5 General shear failure.

(2) Local shear failure
The failure pattern developed is of the same form as for general shear failure but only the slip surfaces
immediately below the foundation are well defined. Shear failure is local and does not create the large
zones of plastic failure which develop with general shear failure. Some heaving of the soil around the
foundation may occur but the actual slip surfaces do not penetrate the surface of the soil and there
is no tilting of the foundation.
(3) Punching shear failure
This is a downward movement of the foundation caused by soil shear failure only occurring along the
boundaries of the wedge of soil immediately below the foundation. There is little bulging of the surface
of the soil and no slip surfaces can be seen.
For both punching and local shear failure, settlement considerations are invariably more critical than
those of bearing capacity so that the evaluation of the ultimate bearing capacity of a foundation is
usually obtained from an analysis of general shear failure.

Prandtl's analysis

Prandtl ( 1921 ) was interested in the plastic failure of metals and one of his solutions (for the penetration
of a punch into metal) can be applied to the case of a foundation penetrating downwards into a soil with
no attendant rotation.
The analysis gives solutions for various values of φ , and for a surface footing with φ = 0, Prandtl obtained:

q = 5 14 . c

u u

Terzaghi's analysis

Working on similar lines to Prandtl's analysis, Terzaghi ( 1943) produced a formula for q u which allows for
the effects of cohesion and friction between the base of the footing and the soil and is also applicable
to shallow (z/B ≤ 1) and surface foundations. His solution for a strip footing is:

q = cN + γ zN + 0 5 . γ BN (6)

u c q γ

The coefficients N c , N q and N γ depend upon the soil's angle of shearing resistance and can be obtained
from Fig. 9.6. When φ = 0°, N c = 5.7; N q = 1.0; N γ = 0.

⇒ q = 5 7 . c + γ z

u

or q u = 5.7c for a surface footing.