Most critical circle, Swedish method

In the Swedish (slip circle) method, the failure surface is assumed to be circular. The factor of safety is calculated for many trial circles, and the critical circle is the one with the minimum factor of safety.

Procedure

1. Draw the slope section to scale, marking the toe, crest and any soil boundaries.
2. Select a trial centre $O$ (above the slope, using Fellenius' guide lines or a grid of points) and a radius $R$ so that the circle passes through the toe (or just below it).
3. Compute the factor of safety $F$ for this circle:
   • $\phi=0$ soil: $F = \dfrac{c_u L_a R}{Wx}$;
   • $c$-$\phi$ soil: divide the mass into slices and use the method of slices.
4. Keep the same centre and change $R$ to find the minimum $F$ for that centre.
5. Repeat for other centres arranged in a grid, writing the minimum $F$ at each centre.
6. Draw contours of equal factor of safety through the grid values. The centre of the smallest contour is the centre of the most critical circle, and its value is the minimum factor of safety.

   O1*   O2*   O3*
       F=1.8 F=1.5 F=1.7    <- grid of centres
   O4*   O5*   O6*
       F=1.6 F=1.3 F=1.5
   (O5 gives lowest F: critical circle)

For homogeneous soil with $\phi=0$, the position of the critical centre can also be found from Fellenius' construction (direction angles at the toe and crest) or Taylor's charts.