Residual factor as a variable in slope reliability analysis

Abstract

In the past, residual factor R in strain-softening soil slopes has been included, either directly or indirectly, as a deterministic variable in both deterministic and probabilistic studies. This paper discusses the uncertainties associated with R and outlines a systematic approach for the reliability analysis of a natural slope in which shear strength parameters and pore pressure ratio are random variables, each assumed with a lognormal probability distribution. For the residual factor R, seven probability distribution options under the generalized beta-distribution system are considered. Slope reliability is computed based on the first order reliability method (FORM) and validated against Monte-Carlo simulation (MCS). Results obtained from two illustrative examples indicate that the probability of failure, with R as one of six random variables, can be orders of magnitude higher than that based on five random variables with R considered as a deterministic parameter. The magnitude of influence of R as a random variable is, however, highly dependent on its probability distribution, the left-skewed triangular distribution having the most significant influence in both the examples. Results of sensitivity analyses reveal that, for almost all of its assumed probability distributions, R is the most dominant among the six random variables. Effects of variation of some of the statistical and correlation properties of the other random variables, viz. the shear strength parameters and the pore pressure ratio, on the results of reliability analyses are also studied.

Appendices

Appendix 1

Overall or average residual factor for a slip surface of arbitrary shape — general case

In general, shear strength, being proportional to the normal effective stress, would vary from point to point along a potential slip surface, and hence, the local residual factor, would also vary. Therefore, it is very useful to consider an expression for average residual factor, R, which represents the whole of a potential slip surface, as follows:

$$ R\kern0.36em =\kern0.48em \frac{s_p\kern0.36em -\kern0.36em {s}_{av}}{s_p\kern0.36em -\kern0.36em {s}_r} $$

(A.1)

in which

s_p:

Average peak strength

s_r:

Average residual strength, and

s_av:

Average current shear strength

Let us consider an arbitrary slip surface of total length L being subdivided into n infinitesimally small segments of lengths Δl_i (i = 1, 2, …, n) such that the corresponding values of residual factor R_i (i = 1, 2, …, n) do not vary within a segment of length Δl_i. For a slope in perfectly brittle soils, the most general case would be when strain-softening to residuals have taken place at m nos. of segments (m<n) whose total length is L_r, while the remaining (n – m) segments of total length (L – L_r) are still at their peak shear strengths. This means, at the current state, s_i = s_ri for i = 1 to m, while, s_i = s_pi for i = (m+1) to n. In such a situation, the three average strengths in Eq. (A.1), namely, s_r, s_p, and s_av are obtained as follows:

$$ {s}_r\kern0.36em =\kern0.48em \frac{\sum_{i=1}^m{s}_{r i}\kern0.24em \Delta {l}_i}{\sum_{i=1}^m\Delta {l}_i}\kern0.36em =\kern0.36em \frac{\sum_{i=1}^m{s}_{r i}\kern0.24em \Delta {l}_i}{L_r}=\frac{SUM1}{L_r}\kern0.24em (say) $$

(A.2)

$$ {s}_p\kern0.36em =\kern0.48em \frac{\sum_{i= m+1}^n{s}_{p i}\kern0.24em \Delta {l}_i}{\sum_{i=1}^m\Delta {l}_i}\kern0.36em =\kern0.36em \frac{\sum_{i= m+1}^n{s}_{p i}\kern0.24em \Delta {l}_i}{L-{L}_r}=\frac{SUM2}{L-{L}_r}\kern0.24em (say) $$

(A.3)

$$ {s}_{av}\kern0.36em =\kern0.48em \frac{\sum_{i=1}^n{s}_i\kern0.24em \Delta {l}_i}{\sum_{i=1}^n\Delta {l}_i}\kern0.36em =\kern0.36em \frac{\sum_{i=1}^n{s}_i\kern0.24em \Delta {l}_i}{L}=\frac{\sum_{i=1}^m{s}_i\kern0.24em \Delta {l}_i+\sum_{i= m+1}^n{s}_i\kern0.24em \Delta {l}_i}{L}\kern0.36em =\frac{SUM_1+{SUM}_2}{L} $$

(A.4)

Substituting the above in Eq. (A.1),

$$ {R}_{av}\kern0.36em =\kern0.48em \frac{s_p\kern0.36em -\kern0.36em {s}_{av}}{s_p\kern0.36em -\kern0.36em {s}_r}=\frac{SUM2/\left( L-{L}_r\right)-\left( SUM1+ SUM2\right)/ L}{SUM2/\left( L-{L}_r\right)- SUM1/{L}_r}\kern0.36em =\kern0.36em \frac{L_r}{L} $$

(A.5)

which agrees with Skempton’s definition.

Appendix 2

Factor of safety for a curved slip surface — modified Bishop simplified method

The expression for the factor of safety, F, associated with a curved slip surface of circular shape for a simple slope, based on the Bishop simplified method, has been modified for a strain-softening soil, by including the residual factor R. The modified expression is as follows:

$$ F=\frac{\sum \left[\left\{{c}_{r f}^{\prime}\; b+ W\left(1-{r}_u\right)\times \tan {\varphi}_{r f}^{\prime}\right\}/{m}_{\alpha rf}\right]}{\sum W \sin \alpha} $$

(B.1)

where, b is the slice width, W is the slice weight, r_u is the non-dimensional pore water pressure ratio at slice base, and α is the inclination of slice base. Further,

$$ {c}_{r f}^{\prime }={R{ c}^{\prime}}_r+\left(1- R\right){c}_p^{\prime } $$

(B.2)

$$ \tan {\varphi}_{r f}^{\prime }= R \tan {\varphi}_r^{\prime }+\left(1- R\right) \tan {\varphi}_p^{\prime } $$

(B.3)

where, R is the overall or average residual factor for the entire length of the curved slip surface (assumed to be an arc of a circle in this case).

The factor m_αrf is given by:

$$ {\mathrm{m}}_{\alpha rf}=\left(1+\frac{ \tan \alpha \tan {\varphi}_{rf}^{\prime }}{F}\right) \cos \alpha $$

(B.4)

The commonly used expression for factor of safety based on the Bishop simplified method (no strain-softening) is given by:

$$ F=\frac{\sum \left[\left\{{c}^{\prime } b+ W\left(1-{r}_u\right)\times \tan {\varphi}^{\prime}\right\}/{m}_{\alpha}\right]}{\sum W \sin \alpha} $$

(B.5a)

where, ${\mathrm{m}}_{\alpha}=\left(1+\frac{ \tan \alpha \tan {\varphi}^{\prime }}{F}\right) \cos \alpha$ .(B.5b)

It may be noted that Eq. (B.1) is analogous to Eq. (B.5a) except that c′ is replaced by ${c}_{rf}^{\prime }$ given by Eq. (B.2), tan ϕ′ is replaced by tan ${\varphi}_{rf}^{\prime }$given by Eq. (B.3), and m_α is replaced by m_αrf given by Eq. (B.4).