4.2 Bearing capacity of a shallow strip footing (plane strain) To illustrate the factors on which the bearing capacity depends a simple (not accurate)lower bound approach is considered below. Footing Surcharge qs ↓ Soil at state of Soil at state of Active Failure Passive with Gv>Gh Failure with Frictionless Gh>Ov Discontinuity When the soil is at a state of failure the Mohr-Coulomb criterion is satisfied,and we have o,=N。o:+2cVNg N。=g,+ccot o ccoto Under the stress is given by Oy =qr +YZ At failure N,=9+yz ccot g.+ccotφ hence N (q:+yz+ccot)-cco In the region away from the footing the horizontal stress will be greater than the vertical stress. At any depthzthe vertical stress is given by Gy qr YZ and at failure N。= Gn+ccot qs +yz ccot and n=N (qs +Yz+ccot)ccot For equilibrium the horizontal forces from the two failure zones must be equal4.2 Bearing capacity of a shallow strip footing (plane strain) To illustrate the factors on which the bearing capacity depends a simple (not accurate) lower bound approach is considered below. When the soil is at a state of failure the Mohr-Coulomb criterion is satisfied, and we have 1 = N 3 + 2  c N or N c c      = + + 1 3 cot cot Underneath the footing the vertical stress will be greater than the horizontal stress, that is v = 1 and h = 3, and in the limit these will be related by the Mohr-Coulomb criterion. At any depth z the vertical stress is given by   v qf = + z At failure N q z c c f h      = + + + cot cot hence      h f N = q + z + c − c 1 ( cot ) cot In the region away from the footing the horizontal stress will be greater than the vertical stress. At any depth z the vertical stress is given by   v qf = + z and at failure N c q z c h s      = + + + cot cot and h = N (qs +  z + c cot ) − c cot  For equilibrium the horizontal forces from the two failure zones must be equal qf Footing Surcharge qs Soil at state of Active Failure with v > h Soil at state of Passive Failure with Frictionless h > v Discontinuity H