Methods for Nonhomogeneous Slopes

The application of earth pressure theory can be illustrated by the simple example shown in Fig. 12.1. By assuming the active force, P _A , and the passive force, P _p as horizontal, it can be easily proved by Rankine’s or Couloumb’s theory that the failure plane inclines at an angle of 45° + ø/2 for the active wedge and 45° − ø/2 for the passive wedge. From basic soil mechanics 12.1$${{P}_{A}}\,=\,\frac{1}{2}\,\gamma \,\mathop{H}_{A}^{2}\,{{\tan }^{2}}\,\left( 45{}^\circ \,-\,\frac{\phi }{2} \right)\,-\,2c{{H}_{A}}\,\tan \,\left( 45{}^\circ \,-\,\frac{\phi }{2} \right)$$12.2$${{P}_{P}}\,=\,\frac{1}{2}\,\gamma \,\mathop{H}_{P}^{2}\,{{\tan }^{2}}\,\left( 45{}^\circ \,-\,\frac{\phi }{2} \right)\,-\,2c{{H}_{P}}\,\tan \,\left( 45{}^\circ \,-\,\frac{\phi }{2} \right)$$ in which H _A and H _p are the height of active and passive wedges, respectively. The factor of safety can be determined by 12.3$$F\,=\,\frac{cl\,+\,W\,\tan \,\phi }{{{P}_{A\,}}\,-\,{{P}_{P\,}}}$$ in which l is the length of failure surface at middle block, and W is the weight of middle block.