Probability-Based Design of Reinforced Rock Slopes Using Coupled FORM and Monte Carlo Methods

The FORM extends the Hasofer–Lind (1974) index (for correlated normal variates) to deal with correlated non-normal random variates, and hence includes the earlier Hasofer–Lind index as a special case. The classical intricate FORM procedure in the rotated u space is mathematically elegant, as explained (or discussed) in commendable details in Ditlevsen (1981), Shinozuka (1983), Ang and Tang (1984), Der Kiureghian and Liu (1986), Madsen et al. (1986), Melchers (1987), Tichy (1993), Haldar and Mahadevan (2000), Rackwitz (2001), Baecher and Christian (2003), Kottegoda and Rosso (2008), and Melchers and Beck (2018), for example.

This study on the probability-based design of rock slopes uses the more intuitive and efficient Low and Tang (2007) spreadsheet-automated algorithm, which obtains the same solutions as the mathematically intricate classical FORM procedure.

In FORM, the reliability index β can be written as follows:where R is the correlation matrix, and μ_i^N and σ_i^N are equivalent normal mean and equivalent normal standard deviation values, which can be calculated by the Rackwitz–Fiessler (1978) transformation:where x is the original non-normal variate, Φ⁻¹[.] is the inverse of the cumulative probability (CDF) of a standard normal distribution, F(x) is the original non-normal CDF evaluated at x, ϕ{.} is the probability density function (pdf) of the standard normal distribution, and f(x) is the original non-normal probability density ordinates at x. Equation 2 was used in the Low and Tang (2004) Excel-automated FORM procedure. One can regard the computation of the FORM β by Eq. 2 as that of finding the smallest equivalent hyper-ellipsoid (centered at the equivalent normal mean-value point μ^N and with equivalent normal standard deviations σ^N) that is tangent to the LSS. Hence, for correlated non-normals, the ellipsoidal perspective still applies in the original coordinate system, except that the non-normal distributions are replaced by an equivalent normal ellipsoid.

An alternative FORM computational approach was given in Low and Tang (2007), summarized in Fig. 1, which uses the following equation for the reliability index β:where n is the dimensionless vector defined by the bracketed term in Eq. 2(a). The above equation can be entered easily as an Excel array formula using Excel matrix functions mmult, transpose and minverse. For each value of the vector n tried by the Excel Solver in the Low and Tang (2007) FORM method, a short Excel VBA function code x_i(DistributionName, para, ni), shown in Fig. 9 of the Appendix, automates the computation of x_i from n_i, for use in the constraint g(x) = 0, via the following equation:in which F(x_i) is the original non-normal CDF of x_i, and Φ(n_i) is the standard normal CDF.

Figure 1 provides an illustration involving three correlated non-Gaussian variables, for which the performance function is g(x) = VW − Z. Failure is reached when the resistances V and W decrease from their mean values to their most probable failure values of 33.12 and 40.12, and the load Z increases from its mean value to its most probable failure value of 1329. The most probable point (MPP) of failure (33.12, 40.12, 1329) is the point where the expanding 3D equivalent dispersion ellipsoid first touches the LSS. This MPP is a failure state because 33.12 × 40.12 = 1329. Much valuable information and insights are provided by the MPP of failure (the x* values) and the sensitivity indicators (the n* values), as discussed in Low (2021; 2022), and Low and Bathurst (2022). The focus in this paper is different, namely on the integrated use of FORM, direct MCS, Importance sampling (IS), and SORM, where the design point located by FORM becomes the logical center of IS, and the “exact” failure probability from MCS, IS, or SORM allows a revised design much closer to the target failure probability. In this way, the merits of FORM, MCS, IS, and SORM complement one another.

One needs to appreciate the ease of moving from the n space (e.g., Fig. 1) to the rotated u space (in the classical FORM method) to understand the simple extensions of FORM to MCS, IS and SORM in the examples below. The vectors n and u can be obtained, one from the other, using the equations below (e.g., Low et al. 2011):andin which L is the lower triangular matrix of the Cholesky decomposition of the correlation matrix R. The lower triangular matrix L is related to R by \({\mathbf{LL}}^{T} = {\mathbf{R}}\), where superscript T denotes the transpose of a matrix. The matrix L can be obtained easily using a very short Excel VBA code for Cholesky decomposition which is available in the public domain.

Only when the random variables are uncorrelated is \({\mathbf{n}} = {\mathbf{u}}\), because then L⁻¹ = L = I (the identity matrix). In general \({\mathbf{n}}\) is not equal to \({\mathbf{u}}\).

This study shows that one can couple any of the following three methods (easily extended from the Excel FORM template) to obtain near-exact P_f in a revised design. The Appendix explains the extension for the simple g(x) = VW − Z case in Fig. 1. The procedure is the same for complicated cases like the reinforced pentahedral blocks in later sections.

MCS and IS for near-exact P_f estimation are easily extended from the Excel FORM template, with simple VBA codes, while SORM requires more VBA codes. In return, their near-exact P_f values enable a much more accurate re-design via FORM, in the single loop FORM–MCS–FORM design procedure for cases with multiple failure domains. The much faster IS and SORM can be used instead of MCS for cases with a dominant single failure mode, as illustrated next in an interesting case where an inclined load is revealed by FORM as actually playing the role of resistance.

Figure 2a shows the forces acting on a foundation on rock containing a planar discontinuity dipping out of slope face. The factor of safety against sliding on the discontinuity plane is the ratio of available resistance on the discontinuity plane, \(cA + N^{\prime}\tan \phi\), to the sliding force Q_sliding:where N′ (the force perpendicular to the discontinuity plane) and the destabilizing Q_sliding force are determined by the two equations below:in which the symbols are as defined by the annotated inset at the top right of Fig. 2.

The deterministic analysis in Fig. 2b indicates that when an active bolt force T of 8 MN is applied over a 5 m width (out of plane), the factor of safety against sliding is 1.69, in agreement with Wyllie (1999, p195). If bolt force is zero, the factor of safety is 1.28.

For the given bolt force, Excel Solver indicates that a maximum F_s of 1.87 is obtained when ψ_T is 202.9°, which means 22.9° above horizontal, that is, 17.1° above the discontinuity plane. Wyllie aptly noted that it is, however, easier to drill and grout in a direction below horizontal.

The global (or lumped) factor of safety as defined by Eq. 6 are typically based on mean values of the parameters (c, ϕ, Q₁, Q₂, a, W, ψ_p, …), and conveys no information on the chance of failure (defined by F_s ≤ 1.0). As an alternative, RBD via FORM can be conducted, based not only on the mean values but also the uncertainty of the input parameters. That the outcome of probability-based design depends on the inputs characterizing uncertainties is discussed in Sect. 6.

Figure 3a shows the simple template for RBD via FORM using the Low and Tang (2007) computational approach. The ten random variables are Q_1V, Q_1H, Q₂, T, a, W, h_w, c, \(\phi\) \(\psi_{p}\) ,which have been defined in the top-right inset of Fig. 2a. Bounded non-Gaussian distributions are used.

In the bounded beta distribution BetaDist(λ₁, λ₂, min, max) used for seven of the ten random variables in Fig. 3a, λ₁ and λ₂ are shape parameters which can define various shapes for the beta distribution. When λ₁ = λ₂, the beta distribution is symmetric (as in the unbounded normal distribution).

The following established relationships are useful to keep in mind:

When λ₁ = λ₂ = 7.5 in the beta distribution BetaDist(λ₁, λ₂, min, max),where c.o.v. is the coefficient of variation (= σ/μ).

The μ and σ values are shown under the eponymous columns in the middle of Fig. 3a. Figure 3b compares BetaDist (7.5, 7.5, 18, 42) for the friction angle ϕ, for which μ = 30 and σ = 3 by Eqs.9a and 9b, with the normal distribution N(30, 3).

The 3-parameter PERT (also called beta-PERT) distribution, PERTDist(min, mode, max), is used for random variables Q₂, a and h_w in Fig. 3a. The mean μ and standard deviation σ of PERTDist(.) can be calculated from the following established equations:

Figure 3b compares the PERT distribution with the triangular distribution for the inclined load Q₂.

The loads Q_1V and Q_1H are assumed to be positively correlated with a correlation coefficient equal to 0.5, and cohesion c and friction angle ϕ of the discontinuity plane are negatively correlated with a correlation coefficient equal to − 0.5, as shown in the correlation matrix R. Also, the weight W will be larger when the discontinuity inclination angle ψ_p (Fig. 2a) is smaller, hence W and ψ_p are assumed negatively correlated with ρ_Wψp = -0.5. The area (A) of discontinuity plane in Fig. 2b is 190 m² when ψ_p = 40°. In probabilistic analysis, area A will vary according to \(A = 190 \times {{\sin 40^\circ } \mathord{\left/ {\vphantom {{\sin 40^\circ } {\sin \psi_{p} }}} \right. \kern-0pt} {\sin \psi_{p} }}\) as ψ_p changes, assuming the discontinuity plane connects with the tension crack at the same constant elevation.

The RBD-via-FORM procedure in Fig. 3a is similar to Fig. 1. It is easily linked to the deterministic set-up of Fig. 2b by replacing the numerical values of Q_1V, Q_1H, Q₂, T, a, W, h_w, c, \(\phi\), \(\psi_{p}\) in Fig. 2b with cell addresses which point to the cells under the x* column in Fig. 3a. The deterministic formulation is also needed in the performance function g(x) at the top of Fig. 3a, expressed as:Or

Equations 11a and 11b are mathematically equivalent when g(x) = 0.

Initially the n* column values in Fig. 3a were zeros. Excel Solver was then invoked (as in Fig. 1), to minimize the β cell, by changing the n* column values, subject to the constraint that the g(x) cell is equal to zero. A mean bolt force (μ_T) of 7.32 MN (over a 5 m width) was found to be required to achieve a β of 3.001 (≈ 3.0, typical target for ULS), as shown in Fig. 3a under the column labeled μ.

The ten x* values represent the first point of contact with the limit state surface (defined by g(x) = 0) when an equivalent hyper-ellipsoid expands from its equivalent normal mean-value point, as explained in connection with Eq. 2. The point represented by the x* values which render g(x) = 0 is the design point (or checking point), also referred to as the most probable point (MPP) of failure.

The following are noteworthy:

The failure probability based on FORM reliability index (Eq. 1) is approximate when the LSS is nonplanar and/or the random variables obey non-normal distributions. Having obtained a required design of mean bolt force μ_T = 7.32 MN (over a 5 m width) for a β of 3.001 in Fig. 3a, it is desirable to check the accuracy of \(P_{f} \approx \Phi \left( { - 3.001} \right)\) = 0.134, and to get a new design mean bolt force of μ_T that satisfies the target P_f of 0.13% more closely, as illustrated below.

Section 1.2 and Fig. 10 in the Appendix present the simple extension of the FORM template to MCS, IS, and SORM, each of which can be coupled with FORM to converge to a target failure probability in the single loop FORM–MCS–FORM. Of particular efficiency are IS and SORM (instead of MCS) for cases with a dominant single failure mode (e.g., Fig. 3), as illustrated in the steps below:

\({\text{new }}\beta_{2} = \Phi^{ - 1} \left( {1 - \frac{0.134\% }{{0.0766\% }} \times 0.13\% } \right) = 2.84\), with the aim of target P_f of 0.13%.

In Microsoft Excel, this is \(\beta_{2} \,\, = \,\,NormSInv\left( {1 - \eta p_{T} } \right)\) η = 0.134/0.0766.

The above-coupled FORM and near-exact P_f methods are summarized by the following expressions:

The P_f1 in the denominator of Eq. 12a can be evaluated by one of three methods, as follows:

Option 1 must be used (i.e., using MCS to determine P_f1) if there are multiple failure domains, as in Sects. 4.2 and 5.1 later where sliding mode can be along both planes, or along one of the two planes.

Options 2 and 3 are faster than Option 1, and either can be used if there is a single dominant failure mode, as in Fig. 2 where the sliding mode is down the discontinuity plane. (Option 3, in which SORM is used to determine the P_f1 in Eq. 12a, was suggested in Low and Einstein (2013, Eq. 15)).

The verification (Eq. 12b) can be done using the faster IS or SORM (as in Fig. 3c) if there is a single dominant failure mode. Otherwise MCS must again be used for P_f2, as in Sect. 5.1.

Figure 3c shows that three Importance Sampling each of 8,000 trials show much smaller scatter than three direct MCS each of 300,000 trials. This is because the center of importance sampling is at the FORM design point (i.e., the MPP of failure located by FORM), whereas in direct MCS the samples emanate from the mean-value point, and require larger sample size than IS.

The next section applies the coupled FORM and importance sampling method to design the reinforcing force for a potentially unstable rock slope in Hong Kong. Other derisking measures and the final option adopted will be mentioned.

Figure 5 shows the deterministic questions on a pentahedral block from Hoek and Bray (1981, p348 & 351), in S.I. units. The solutions for questions 1(a) and 1(b), F = 1.1379 (when planes 1, 2, and 3 are filled with water), and F = 1.7360 (when the three planes are dry), obtained using the first 80 equations in H&B Comprehensive formulations, are identical to the F values given in Hoek and Bray (1981).

The solution for question 2 of Fig. 5, obtained readily using the Excel Solver constrained optimization tool, yields the same F_min of 1.04 and the same worst direction (ψ_e =  − 1.61°, α_e = 173.03°) as those in Hoek and Bray (1981, Eqs. 81–98). In the efficient Excel Solver approach, the initial trial worst direction was (plunge ψ_e = 0, trend α_e = 185°), where 185° was the dip direction of the slope face. Excel Solver was used to minimize the F cell, by changing (automatically) both the ψ_e and α_e cells. An F value of 1.037 was obtained as the minimum, together with a worst direction of (ψ_e =  − 1.611°, α_e = 173.03°).

The solution for question 3 of Fig. 5 was also obtained efficiently using the Excel Solver constrained optimization tool. Initially the value in the T cell was zero and plunge angle ψ_t = 0, and the initial trial trend direction (α_t) of T is taken to be opposite to the trend (α_i) of the line of intersection (Eq. 47 in Fig. 5), that is, initial trial value of α_t = 157.73 + 180, or 338°. The Excel Solver tool was invoked, to minimize the T × 1 formula cell, by changing (automatically) the three cells of T, ψ_t and α_t, subject to the constraint that the F cell value is 1.50. Excel Solver first reported a converged solution, then, when run a second time, found a solution (T = 15265 kN, ψ_t =  − 6.99°, α_t= 349.42°) which is virtually identical to that in Hoek and Bray (Hoek and Bray, 1981, Eqs. 99–113).

(Note: The plunge ψ_I and trend α_i of the line of intersection are calculated from arcsine and arctangent functions, respectively, as given by Eqs. 46 and 47 in Hoek and Bray (1981, Appendix 2) and Wyllie (2018, Appendix III). Arcsine and arctangent can return two principal values. Nevertheless, one can decide the correct trend direction for a downward plunging line of intersection by requiring the trend to be between a right-angle quadrant to the left and right of the dip direction of a non-overhanging slope face (i.e., Plane 4 in Fig. 5), or within ± 90° opposite to the dip direction of an overhanging slope face.)

The deterministic case of the Hoek and Bray pentahedral block of Fig. 5 is extended in this section to reliability-based design of the bolt force T for a target probability of failure of 0.1%.

The 11 random variables in Fig. 7b are those related to (i) geometry: H₁, dips (ψ₁, ψ₂) and dip directions (α₁, α₂) of the two discontinuities, (ii) water pressure (u₁,u₂) and shear strength parameters of the discontinuities (c₁, c₂, ϕ₁, ϕ₂). Their mean values μ (first column) are the same as those in the deterministic example in Fig. 7a. The values of their standard deviations σ (second column) are illustrative but realistic and within the typical range. All are assumed to follow the 4-parameter bounded BetaDist(λ₁, λ₂, min, max), which is symmetric when λ₁ = λ₂, and, when λ₁ = λ₂ = 7.5, obeys the relationships min = μ − 4σ and max = μ + 4σ. We already encountered an example of BetaDist (7.5, 7.5, min, max) in Fig. 3b. These bounded symmetric beta distributions are intentionally selected as approximations of the symmetric but unbounded normal distributions, which are not used here because occasional extremely large or extremely small (or negative) values of random numbers generated from normal distributions may cause numerical errors (during MCS involving huge sample size of hundreds of thousands) in the H&B comprehensive vectorial procedure which has around 100 equations. The deterministic set-up of Fig. 7a is easily linked to the FORM template of Fig. 7b by replacing 11 numerical inputs in the former with cell addresses reading values from the x* column in the latter. The performance function g(x) is “ = F − 1”. The β cell contains an array formula as explained in Fig. 1. The 11 random variables are assumed uncorrelated in this illustrative example.

The bolt force T (next to the load E from the foundations of a pylon) in Fig. 7a was zero. It is now required to find the (active) bolt force T for a target failure probability of 0.1%. The bolt force T is taken to act horizontally (ψ_t = 0) with a trend direction of 59°, that is, opposite to the trend ψ_i (= 239.4°) of the line of intersection. A first-design T value of 19.3 kN was required to achieve a β of 3.1 using FORM. The FORM P_f of 0.1% by Eq. 1 is approximate. The P_f1 by MCS was 0.168%, as shown in Fig. 7b bottom left. A revised β₂ of 3.251 was computed from Eq. 12a, which is satisfied by a revised T of 31.6 kN. Three MCS runs each with sample size of 400,000 obtained a P_f2 of 0.112%, which suggested a β₃ of 3.282, satisfied by a revised T of 34 kN (Fig. 7b). The optional P_f3 by Excel MCS in 1.2 million trials is 0.101%, which agrees with Pf₃ of 0.099% via MATLAB MCS using the same number of trials.

Measurements obtained from instrumentation and monitoring such as load cells for active anchors (Boon et al. 2015b, 2019) may suggest realistic coefficient of variation (c.o.v.) if the uncertainty in the bolt force is to be modeled. (In Figs. 3 and 6, the c.o.v. of the bolt force T is assumed to be 0.1). Alternatively, numerical analysis may be used to obtain c.o.v. of bolt forces (Boon et al. 2015a).

For cases with multiple failure modes, direct MCS should be done first despite its time-consuming nature and large sample-size requirement when failure probability is small, because MCS can sample into the failure domains of the different failure modes. For the pentahedral case in Figs. 5 and 6, MCS must be used for P_f1 because of multiple failure domains. The faster IS can be used for verifying P_f2 only after MCS indicates all failure modes are sliding on both planes. The single dominant failure mode of Fig. 6 should not be presumed, as direct MCS for P_f1 revealed that for the tetrahedral block in Fig. 7, the number of sliding failures on plane 1 is more than twice that of sliding on both planes, even though the failure mode is sliding on both planes when the random variables are at their mean values. More about this phenomenon in the next section.

For the pentahedral block in Fig. 5, the mean-value mode (sliding on both planes) was revealed to be the single dominant mode in the MCS of Fig. 6. For the tetrahedral block in Fig. 7, although the mean-value mode is also sliding along both planes, the more likely failure mode in MCS is sliding on plane 1, with other modes also contributing to failure probability. In contrast, for the tetrahedral block in Fig. 8, the governing sliding mode is along plane 2 for the given water pressures u₁ and u₂, but the sliding mode can change to a different one (along both planes) when the water pressures are different.

The deterministic data in Fig. 8 are the same as Priest (1993, Example 8.5), which compute the F_s using equations and the graphical stereographic projection method. The F = 0.8493 in Fig. 8, obtained using the H&B Comprehensive vectorial procedure, is identical to the F value reported by SWedge (RocScience 2022, Verification Problem #7). The “F₂ = 0.8493” in Fig. 8 means that the mode of sliding is along plane 2, sliding away from the line of intersection and from plane 1. One may note the factors contributing to this single-plane sliding mode: discontinuity 1 is steep (ψ₁ = 74°), and its water pressure (u₁ = 25 kPa) is high relative to the water pressure (15 kPa) on the less steep discontinuity 2 (ψ₂ = 41°).

Had the slope been dry, with u₂ = u₂ = 0, the F_s is 2.3716, and the sliding mode is along both planes.

For other pressures of u₁ and u₂ between the dry scenario and the scenario represented by the data in Fig. 8 (where u₁ = 25 kPa, u₂ = 15 kPa), the governing sliding mode could be either sliding on both planes, or on plane 2 only. For example, with u₂ = 15 kPa, the both-planes mode (i.e., sliding along the line of intersection) governs when u₁ ≤ 21.91 kPa), and Plane 2 mode governs when u₁ > 21.91 kPa. There is an abrupt drop of F at the transition from both-planes mode to the Plane 2 mode, caused by the sudden loss of cohesive resistance c₁A₁ when the block detaches from plane 1. For cases with multiple failure modes (i.e., multiple failure domains), direct MCS (with sampling emanating from the mean-value point) must be used when it is possible that more than one failure mode will contribute toward the total failure probability.

Simpson et al. (2011) reviewed case histories of failures caused by water pressure and the safety provisions related to water pressure in some existing geotechnical codes, including Eurocode 7.