A modified stereographic projection approach and a free software tool for kinematic analysis of rock slope toppling failures

Toppling failure is one of the common instability modes of rock slopes. Up to date, the stereographic projection approach is still an indispensable tool for kinematic toppling failure analysis of rock slopes, since it is very convenient to perform kinematic analysis that is based on all discontinuity orientations, not just some representative values, and it visually shows the analysis results. This paper first discussed the limitations of the Goodman’s graphical approach for kinematic analysis of rock slope toppling failures, and then proposed a modified graphical approach and developed practical software in a spreadsheet of Microsoft Excel for practitioners. The discrepancy between the results obtained from Goodman’s approach and our modified approach was investigated with respect to different limit values of the intersection angle between the strikes of the slope and discontinuity (denoted as γ_Tlim), cut slope dip angles, and discontinuity friction angles. The results showed that: (1) when γ_Tlim = 30°, the designed slope according to the Goodman’s graphical approach might be conservative or unsafe, which is related to the cut slope angle and the friction angle of discontinuity; (2) when γ_Tlim = 30° and cut slope dip angle ≤60°, with respect to discontinuity friction angle = 10°, 20°, 30° or 40°, the designed slope according to the Goodman’s graphical approach might be conservative; (3) when γ_Tlim = 30° and cut slope dip angle ≥70°, with respect to discontinuity friction angle = 10°, 20°, 30° or 40°, the designed slope according to the Goodman’s graphical approach is possibly unsafe; (4) when γ_Tlim = 30° and cut slope dip angle ≤60°, with the discontinuity friction angle increasing from 10° to 40°, the discrepancy of the critical regions determined by the two approaches increases with respect to the same values of cut slope dip angle; (5) when γ_Tlim = 30° and cut slope dip angle ≥70°, with the discontinuity friction angle increasing from 10° to 40°, the discrepancy of the critical regions determined by the two approaches first increases and then decreases with respect to the same values of cut slope dip angle; and (6) therefore, when performing kinematic analysis of rock slope toppling failures, the modified graphical approach proposed in this paper is recommended.