Upper and lower bounds to the factor of safety and f(x) by the lower bound method

The previous extremum principle assumes the factor of safety to be different among different slices. The extremum principle can also be formulated assuming a single factor of safety by utilizing the Morgenstern–Price method which is based on the force and moment equilibrium with an assumption of f(x). Then the bounds to the actual factor of safety will be given by the upper

ζ_i=Ficos β_itan φ_vi+ Cvi

Fisin βi

, 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1/FOS

0.2 0.4 0.6

Ailc Aglc

0.8 1

x

Figure 2.20 Local factor of safety along the interfaces for the problem in Figure 2.9.

and lower bounds of the factor of safety arising from all combinations of f(x).

If a pattern search is used where 10 combinations are assigned for each f(x), a problem with 11 slices will require 10¹⁰ combinations with tremendous computation and has hence never been tried in the past. This approach appears to be impossible until the modern artificial intelligence-based optimization methods are developed, which will be discussed in Chapter 3.

To determine the bounds of the factor of safety and f(x), the slope shown in Figure 2.18 can be considered. For a failure surface with n slices, there are n – 1 interfaces and hence n – 1 control variables representing f(x_i). f(x) will lie within the range 0–1.0, while the mobilization factor l and the objec-tive function FOS based on the Morgenstern–Price method will be deter-mined for each set of f(x_i). The maximum and minimum factors of safety of a prescribed failure surface satisfying force and moment equilibrium will then be given by the various possible f(x_i) which requires the use of modern global optimization methods with the requirement given by eq. (2.77),

Maximize (or minimize) FOS subject to 0 ≤f(x_i) ≤1.0 for all i (2.77) In carrying out the optimization analysis as given by eq. (2.77), the con-straints from the Mohr–Coulomb relation along the interfaces between slices as given by eq. (2.78) should be considered.

(2.78) where L is the vertical length of the interface between slices. The con-straint by eq. (2.78) can have a major impact on λbut not the FOS, and this will be illustrated by the numerical examples in the following section.

Since other than f(x) the Morgenstern–Price method is totally governed by the force and moment equilibrium, the maximum and minimum factors of safety from varying f(x) will provide the upper and lower bounds to the factor of safety of the slope which is not possible with the classical approach.

Cheng (2003) and Cheng and Yip (2007) have applied the simulated annealing method complying with eqs (2.77) and (2.78) to evaluate the bounds to the factor of safety and have coded the method into a general purpose commercially available program, SLOPE 2000. Consider the cases shown in Figures 2.4 and 2.9; the bounds to the factor of safety are given as 1.032/1.022 (Figure 2.4) and 1.837/1.826 (Figure 2.9) if eq.

(2.78) is enforced. It is noticed that while for normal problems with no soil nail or external loads, the upper and lower bounds to the factor of safety are usually close so that f(x) has a negligible effect on the analysis;

the results for Figure 2.9 is extreme in that there is significant difference between the upper and lower bounds of the factors of safety. Based on lots of trial tests, the authors have found that this situation is rare but is not uncommon. The f(x) associated with the maximum and minimum

V≤ Ptanφ⁰+ c⁰L,

extrema can be approximated by the relations shown in Figure 2.21, where f(x) is plotted from the toe of slope to the crest of slope along the increasing x direction. It should be noted that this figure applies only to a simple slope passing through the toe of slope, and the slope has a level instead of inclined back. For a general slope with external load and soil nails, the use of a simple inter-slice force function is difficult, and the use of the numerical method available in SLOPE 2000 is recommended. A worked example in evaluating f(x) by the lower bound method is given in the Appendix of this book.

The previous studies on convergence by Baker (1980) or by Cheng et al.

(2008a) are mainly concerned with the numerical results instead of investi-gating the fundamental importance of f(x). For a problem with a set of con-sistent and acceptable internal forces, the FOS must exist as it can be determined explicitly if the internal forces are known. Failure to converge will not occur if the double QR method is used, though the use of the iteration method may fail to converge due to the limitation of the mathematical method. If no FOS can be determined from the double QR method, this is equivalent to a consistent set of internal forces under the specified f(x) not existing. If a problem fails to converge for a particular f(x), a FOS can usually be found by tuning f(x). Physically, it means that f(x) cannot be arbitrarily assigned to a slope. If f(x) is not associated with a consistent set

Max. extremum

0.2 0

0.2 0.4 0.6 0.8 1

0.4 0.6

x

0.8 1

Min. extremum

f (x) = arc cot(ax + b)/c

f (x)

Figure 2.21 Simplified f(x) for the maximum and minimum extrema determination.

of internal forces, then that f(x) is not acceptable. That means that f(x) can-not be randomly specified or else there will be no consistent internal forces (and hence FOS) associated with the f(x). The present approach provides a systematic way to determine f(x) for an arbitrary problem, and convergence is virtually eliminated in the analysis. The basic trend of f(x) shown in Figure 2.21 for the two extrema established by Cheng et al. (2007d) is good enough for practical purposes.

For the two extrema from the present analysis, the authors view that the maximum extremum should be taken as the factor of safety of the pre-scribed failure surface. As discussed, the internal forces within the soil mass should re-distribute until the maximum resisting capacity of the soil mass is fully mobilized, which is the lower bound approach. The present definition also possesses an advantage in that it is independent of the definition of f(x).

It is well known that there are cases where f(x) may have a noticeable influ-ence on the factor of safety. There is no clear guideline on the acceptance of the FOS due to the use of different f(x). The use of the maximum extremum can also avoid this dilemma which has been neglected in the past.

Using the lower bound approach, f(x) is not an arbitrary function and can be uniquely determined, so the question on f(x) can be viewed as settled as far as the lower bound theorem is concerned.