Saturday, November 17, 2012

Mechanism of Failure: Terzaghi for a Strip Footing.

The shapes of the failure surfaces under ultimate loading conditions are given in Fig. 12.6. The zones of plastic equilibrium represented in this figure by the area gedcfmay be subdivided into

1 . Zone I of elastic equilibrium
2. Zones II of radial shear state
3. Zones III of Rankine passive state

When load qu per unit area acting on the base of the footing of width B with a rough base is transmitted into the soil, the tendency of the soil located within zone I is to spread but this is counteracted by friction and adhesion between the soil and the base of the footing. Due to the existence of this resistance against lateral spreading, the soil located immediately beneath the base remains permanently in a state of elastic equilibrium, and the soil located within this central Zone I behaves as if it were a part of the footing and sinks with the footing under the superimposed load.

The depth of this wedge shaped body of soil abc remains practically unchanged, yet the footing sinks. This process is only conceivable if the soil located just below point c moves vertically downwards. This type of movement requires that the surface of sliding cd (Fig. 12.6) through point c should start from a vertical tangent. The boundary be of the zone of radial shear bed (Zone II) is also the surface of sliding. As per the theory of plasticity, the potential surfaces of sliding in an ideal plastic material intersect each other in every point of the zone of plastic equilibrium at an angle (90° - Ø). Therefore the boundary be must rise at an angle Ø to the horizontal provided the friction and adhesion between the soil and the base of the footing suffice to prevent a sliding motion at the base.

Figure 12.6 General shear failure surface as assumed by Terzaghi for a strip footing

The sinking of Zone I creates two zones of plastic equilibrium, II and III, on either side of the footing. Zone II is the radial shear zone whose remote boundaries bd and af meet the horizontal surface at angles (45° - Ø/2), whereas Zone III is a passive Rankine zone. The boundaries de and fg of these zones are straight lines and they meet the surface at angles of (45° - Ø/2). The curved parts cd and cf in Zone II are parts of logarithmic spirals whose centers are located at b and a respectively.

Ultimate Bearing Capacity of Soil 
Equations for Square, Circular, and Rectangular Foundations
Ultimate Bearing Capacity for Local Shear Failure
Ultimate Bearing Capacity qu in Purely Cohesionless and Cohesive Soils Under General Shear Failure