A Generalized Stochastic Framework for Analyzing Rock Slopes Stability and Designing Reliable Remedial Measures with Limited Investigation Data: Theory and Field Implementation

7 Availability of limited data for rock properties is a very frequently encountered issue for 8 the rock slopes along Himalayan highways due to problems like high costs, manual efforts, 9 geological complexities, difficult terrain etc. involved in rock testing and investigation. 10 Under these conditions, support estimation for rock slides mitigation using traditional 11 deterministic and reliability approaches becomes highly questionable due to inaccuracy in 12 the estimated statistical parameters of rock properties. To resolve this issue, this article 13 proposes a computationally efficient methodology which utilizes Advanced Re-Sampling 14 Reliability Approach (ARRA) along with deterministic approach and Target Reliability 15 Approach (TRA) to estimate required support for rock slides mitigation when limited field 16 and laboratory investigation data is available, with acceptable accuracy and confidence. 17 Proposed methodology was used to design the support measures to mitigate two massive 18 rock slides along a rock-slide prone highway i.e. Rishikesh-Badrinath National Highway 19 (NH-58) in India. It was observed from the analysis that availability of limited test data 20 induces high uncertainty in the statistical parameters (mean and standard deviation) and 21 probability distribution of rock properties. Support estimation carried out using traditional 22 deterministic and reliability approaches with this inaccurate probabilistic characterization 23 of rock properties, can lead to inaccurate support estimates for potential rock slides in the 24 presence of limited data; however these methods when coupled with ARRA can lead to 25 significant improvement in computa tional efficiency and the designer’s confidence for the 26 estimated support.


Introduction
Further in all these available limited studies, one major issue which is completely ignored 52 is that the accuracy of these traditional reliability methods depends significantly on the 53 accuracy of the estimated statistical parameters (mean, standard deviation (SD) and 54 Probability Distribution (PD))of rock properties, which in turn depend on the quality and 55 quantity of field and laboratory investigation data. Even if the quality of test data is 56 maintained by performing tests according to standard methods suggested by ISRM, 57 quantity (number of test data) of the test data in rock mechanics domain remains 58 inadequate mostly due to involvement of high costs; absence of any strict guidelines 59 regarding numbers; practical difficulties like site preparation, sample collection, sample 60 disturbance, data interpretation etc. involved in lab and in-situ testing (Duzgun et al., 2002;   This section describes the systematic steps involved in the support estimation for rock 183 slides mitigation using proposed methodology. Figure 2 shows the detailed flowchart of 184 the proposed methodology. Following section briefly outlines the steps to be followed in 185 the proposed methodology.

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Step 1 Derivation of Performance Function: Derive the performance function i.e. Factor Where , =Vector of random rock properties and random external forces; T =Support 191 force.

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Step 2 Estimation of Rock Properties: Estimate the input rock properties from standard 193 International Society of Rock Mechanics (ISRM) suggested field and laboratory testing 194 methods to generate the original data set for these rock properties.

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Step 3Estimation of Uncertainty in Rock Properties: Estimate statistical parameters   Table 1. Figure 5 shows the relevant dimensions provided in Table 1. Support design for 294 this potential rock slide is carried out as described in flowchart of Fig. 2 and section 3. to water pressure in the tension crack is 1 , and the uplift force due to water pressure along 305 the critical failure/discontinuity plane is 2 . An external stabilising bolt force is applied 306 normal to the slope face OA, which will be inclined at an angle to the failure plane OC. where, weight of sliding block OABC can be expressed in term of unit weight of rock (γ) 316 and slope geometrical parameters shown in Fig. 5 as:

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Water pressure along tension crack ( 1 ) and along rock discontinuity ( 2 ) can be 319 expressed in terms of unit weight of water ( )and slope dimensions as:

a) Estimation of Uncertainty in Rock Properties using Original Data
342 Table 3 shows the summary of the estimated mean, SD and best PD of , and r 343 from the original data provided in Table 2 as suggested in section 2.1a.   Further, probability of best fit distribution of a particular PD was also evaluated by taking 370 ratio of number of times a PD was selected as best fit during bootstrap sampling to the 371 total number of bootstrap samples i.e. =10000 for this case (Table 6). For , 372 Uniform distribution is found to be best-fit PDF with 94.43% probability. For , 373 Lognormal distribution is found to be best-fit PDF with 65.89% probability. Lognormal 374 distribution has high probability of being best-fit PDF for r , with 41.34% probability. 375 Most likely probability distribution for r is different from the best fit probability

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Deterministic analysis is carried out to estimate FOS by using mean values of geometrical 389 dimensions of slope, rock properties and external forces (Table 1 and Table 3) Table 8, support force to 420 fulfill the acceptability criterion of =2.0i.e. TTRA was found to be 3640 kN. approximately 0.50 (50%) which was considerably lower than unsupported slope (Fig. 10).

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Still the value of was much higher than the acceptable value of 0.01 (1%). Hence, 426 trials were made with increased support force and the support force required to fulfill the 427 acceptability criterion i.e. < 1 % was found to be 4520 kN (Fig. 10). Table 7 also 428 shows the significant improvement in confidence interval (CI) of the for the slope 429 supported with TFinal = 4520 kN as compared to slope under in-situ condition.

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It is suggested to distribute this support force along the slope face using 32 mm diameter 431 rock bolts of 100 kN tensile capacity with c/c vertical spacing of 4 m as shown in Fig. 11.

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All the rock bolts should be pre-tensioned to the 80% of their tensile capacity to restrain

Rock Slide 2 440
Similar to the rock slide 1, kinematic analysis was carried out for this potential rock slide 441 and it was observed that the possible failure mechanism for this potential rock slide is 442 wedge failure (Pain, 2012). The major rock type present at the site is Dolomite (unit weight 443 γ=25 kN/m 3 ). Geometrical dimensions of this potential rock slide are given in Table 9 and 444 Fig. 12 shows the pictorial representation of these dimensions. Support design for this 445 potential rock slide is carried out as described in Fig. 2 and section 3. This case study is        Table No  Caption   Table 1 Geometrical dimensions of the potential Rock-Slide 1 Table 2 Original data set for JRCn, JCSn and r estimated for critical discontinuity of potential Rock Slide 1 Table 3 Statistical parameters and best-fit probability distributions of critical discontinuity properties along with some external forces for potential Rock Slide 1 Table 4 Bootstrap mean and Bootstrap SD of sample mean and sample SD of JCSn, JRCn and r for critical discontinuity of potential Rock Slide 1 Table 5 Bootstrap mean and Bootstrap SD of AICD values for different probability distributions for JCSn, JRCn and r of critical discontinuity of potential Rock Slide 1 Table 6 Number of times a probability distribution is chosen as best-fit distribution for JCSn, JRCn and r of critical discontinuity during bootstrap sampling for potential Rock Slide 1 Table 7 Results of ARRA for potential Rock Slide 1 under in-situ and reinforced condition Table 8 Details of TRA iterative scheme of MPTP search to achieve =2 for potential Rock Slide 1 Table 9 Geometrical dimensions of the potential Rock-Slide 2 Table 10 Original data set for JRCn, JCSn and r estimated for both critical discontinuities of potential Rock Slide 2 Table 11 Statistical parameters and best-fit probability distributions of critical discontinuities properties for potential Rock Slide 2  Location of the potential rock slides Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors. Details of the geometry and forces acting on the potential rock slide 1 undergoing planar failure mode Probability distribution of FOSevaluated using traditional reliability analysis for potential Rock Slide 1 Figure 9 Probability distributions of (a) sample mean of FOS, (b) sample SD of FOS and(c) reliability index evaluated by ARRA for potential Rock Slide 1 Figure 10 Probability distribution of the reliability index evaluated from ARRA for potential rock slide 1 under different conditions Figure 11 Details of the remedial measures suggested to mitigate rock slide 1 Figure 12 Details of the geometry and forces acting on the potential rock slide 2 undergoing wedge failure mode  Details of the remedial measures suggested to mitigate rock slide 2