Rupture Surface. Leonards (1982) discusses the importance of accurate characterization of the slip surfaces in back-analysis. The example that is used here is taken from analyses completed recently for a staged construction stability assessment of a spoil pile on a foundation containing loose sands and soft clays adjacent to a guidewall at Marmet Lock. The criterion for end of construction safety factor was 1.1. The end-ofconstruction design strength of the spoil material was specified as 77 kPa (1600 psf ). Based on the results of limit equilibrium analyses using circular failure surfaces, this strength was believed to be adequate to meet the stability criterion for failure within the spoil material. However, during numerical modeling using the program FLAC developed by ITASCA (2002) to assess stability within the foundation, an interesting failure surface developed. Figure 3 shows a simplified cross-section of the system analyzed and the material properties used. The simulation included modeling the placement of the embankment in one foot "lifts," with failure occurring near the point of reaching full embankment height The apparent critical circular surface obtained from limit equilibrium using the program PC-STABL is shown on this figure and has a safety factor of 1.12. Also shown on the figure with shear strain contours is the failure surface obtained from numerical modeling, wherein the base of the rupture surface is located within the embankment and then is redirected to the foundation immediately below the embankment when approaching the toe of the slope. In hindsight, this seems logical because the lower vertical stresses near the toe leads to lower shear strengths in the frictional foundation than the assumed cohesive strength of the embankment in this region. However, this fact was not apparent initially and would not have been uncovered using limit equilibrium methods unless anticipated and rigorously investigated. The system that appeared to have a safety factor of 1.12, was in fact at a state of imminent failure. If the failure had occurred and was back analyzed by the conventional approach, not appreciating the subtleties of the true rupture surface, the strength would be found to be 63 kPa (1300 psf), or 81 percent of the actual strength. In this case, the design assumption was unconservative because the critical slip surface was not located. The unconservative design assumption leads to a conservative interpretation of strength in back analysis. Importantly, a limit equilibrium analysis of the surface predicted by
Table 1. Back-calculated strength.
Embankment Strength
Back Calculated Friction Angle (degrees) Lower Bound 22-24 Upper Bound (High Friction Angle) 16 Upper Bound (High Cohesion) 11 Average 18
numerical modeling leads to a safety factor of 1.0, provided the specific surface is evaluated or a non–circular search routine is adequately constrained to identify the surface.