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ARMA 13-486 Probabilistic-Numerical Modeling of Stability of a Rock Slope in Amasya-Turkey Gheibie, S., Duzgun, S.H.B. Mining Engineering Department, Middle East Technical Univesity, Ankara, Turkey Akgun, A. Geological Engineering Department, Karadeniz Technical University, Trabzon, Turkey Copyright 2013 ARMA, American Rock Mechanics Association th This paper was prepared for presentation at the 47 US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, USA, 23-26 June 2013. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. ABSTRACT: Developments in computation facilities and increase in application of sophisticated mathematical methods in engineering problems have affected the methods of slope stability analysis. In recent years, the numerical modeling methods have extensively applied instead of limit equilibrium methods. Also, the probabilistic methods are considered in rock slope designs to quantify the uncertainties involved in the slope stability problem. In this paper, a probabilistic-numerical approach is proposed, which is based on integration of three dimensional Distinct Element Method (DEM) and probabilistic approach for stability analysis of rock slopes. Barton models were used to model the behavior of rock discontinuities and the shear strain was considered as failure indicator of discontinuities. The proposed methodology was applied to a rock slope in Amasya, Turkey. The effect of basic friction angle and cohesion of joints infilling material and its strength reduction due to weathering were included in the analysis. The shearing behavior of fourteen discontinuities and the failure probability of each block were investigated, by using Reliability Index (β) obtained for each discontinuity. 1. INTRODUCTION Rock mass parameters involve various uncertainties. Utilization of probabilistic methods in rock engineering allows rational treatment of these uncertainties that significantly influence the safety of a rock slope. Moreover, probabilistic approaches offer systematic way of alleviating uncertainties and quantifying the reliability of a design [1]. There are several applications of probabilistic limit equilibrium methods in the literature [2, 3, 4, 5, 6, 7 and 8]. However, according to Krahn [9] the fundamental shortcoming of limit equilibrium methods which only satisfy static equation is that they do not consider strain and displacement compatibility. Therefore, it would be better if the probabilistic numerical methods are adopted. The examples of probabilistic numerical slope stability analyses are: [10, 11, 12, 13 and 14]. In this paper, a probabilistic- numerical approach was proposed, which is based on integration of Distinct Element Method and probabilistic approach for stability analysis of discontinuous rock slopes. In contrast to other examples of probabilistic-numerical slope stability analyses mentioned, the proposed approach is in 3D and focuses more on shear behavior of discontinuities. Barton models were used to model the behavior of discontinuities and the shear strain was considered as their failure indicator. The proposed methodology was applied to a rock slope in Amasya. 2. PROPOSED METHODOLOGY For analyzing the stability of rock slopes, different numerical methods are applied, however, the commonly acceptable method for discontinuous rock slopes is Distinct Element Method. The shear strain or shear displacement of rock discontinuities was considered as the failure indicator in this study. The flowchart in Fig. (1) indicates the process for development of the proposed V Realization of random varibles scuh as I cohesion, friction angle, Model construction JCS and JRC II III FISH function FISH function C, ϕ, Ks Average normal stress VI IV δ< δpeak Survival δ> δpeak Failure FISH function Solving Shear displacement (δ) VII Fitting appropraite distribution function for shear displacement obtained for each discontinuity VIII IX Calculating failure probability as the area of region where δ> δpeak Calculating Reliability Index Figure . 1.1- The process of development of proposed probabilistic numerical approach probabilistic numerical approach for analyzing of rock slope stability. As indicated in Fig. (1), at the first step the geometry of slope is constructed in 3DEC. In the proposed methodology, the Barton models were used to model the rock discontinuities. The Barton model was not included as material model in 3DEC library hence it was applied indirectly to the model. Barton suggested instantaneous cohesion and friction angle concepts by which the nonlinear behavior of normal and shear stress ( -σn) relation can be equalized by drawing tangents to the -σn curve for defined σn values. The instantaneous cohesion and friction angle are obtained from Eqs. (1) to (3): ⎛ ⎞ πJRC ∂τ JCS = tan⎜⎜ JRClog10 + φb ⎟⎟ − ∂σ n σn ⎝ ⎠ 180 ln10 ⎡ 2⎛ ⎞ ⎤ JCS + φb ⎟⎟ + 1⎥ ⎢ tan ⎜⎜ JRClog10 σn ⎢⎣ ⎝ ⎠ ⎥⎦ ⎛ ∂τ ⎝ ∂σ n φi = arctan⎜⎜ ci = τ − σ n ⎞ ⎟⎟ ⎠ tan φ i (1) (2) (3) By using Eqs. (1), (2) and (3) the relevant cohesion and friction angle for a definite stress level and consequently for any discontinuity were calculated. Therefore, the calculated values for cohesion and friction angle were applied by using Coulomb slip model (Stages III). The joint material parameters required to apply Coulomb slip are Joint Normal Stiffness (kn), Joint Shear Stiffness (ks), Friction Angle (Jfriction), Cohesion (Jcohesion), Joint Tensile Strength (Jten) and Dilatancy Angle (dil). According to Eqs. (2) and (3), both the cohesion and friction angle are function of normal stress applied on discontinuity surface. Therefore, a FISH function was coded to calculate the average normal stress on each plane (Stages II). The failure criterion in this methodology is shear displacement; and the Joint Shear Stiffness (ks) is one of the most important factors that directly controls the shear displacement. According to Barton and Choubey [15] the shear displacement δ (peak) required to reach the peak shear strength determines the stiffness of joints in shear. Barton and Choubey [15] stated that joint shear stiffness is extremely important input data in finite element analyses of joints, since joints are very deformable in shear compared to normal direction and compared to intact rock. The reliable method of estimating shear strength was developed for any given values of JCS, JRC, Φr and σn. The next step is to estimate the δ (peak) for an estimate of Ks. Barton and Choubey [15] assumed δ (peak) as 1% of joint length (L) and estimated the Ks based on relations given in Eqs. (4) and (5): (4) φ (5) Barton et al. [16] suggested Eq. (6) for estimation of the δ (peak) value as: 1 δpeak = Ln 500 × JRCn 3 L (6) Therefore, it is possible to estimate the δ (peak) and then Ks value by substitution of Eq. (6) into Eq. (4). In 3DEC the joint parameters must be assigned to relevant location or discontinuity. Commonly, the models are complex and the material properties should be assigned by FISH coding. In stage V, the realization of random variables are selected from their distribution and input to the model. These variables are transformed to rock discontinuity properties using FISH function. Then the model is executed and the shear displacement (δ) of each discontinuity is recorded (Stages IV). According to Fig. (1), in stage VI, the shear displacement obtained in stage IV is compared to the peak shear displacement estimated by Eq. (6). In the proposed methodology, it is assumed that if the shear displacement is greater than the estimated peak shear displacement (δpeak) it is called as failure. The boundary of the failure and stability is called the limit state condition in the proposed approach. This methodology was developed for one or more random variables. However, when the number of random variables is more than one, for different realization of random variables the model is run and the shear displacement of each discontinuity is recorded and according to stage VI the failure state is obtained. Then an appropriate distribution function is fitted to the shear displacement. (Stages VII) Then, area for which δ> δpeak is the failure probability and the corresponding Reliability Index is obtained from Pf=1-Φ (β), Where Φ() is the cumulative distribution function of the standard normal variate. (Stages VIII and IX in Fig. 1). 3. IMPLEMENTATION OF PROPOSED METHODOLOGY The case study was selected as the Kings Rock Grave in Amasya, Turkey, which was carved on a rock mass containing bedding planes and joints. The host rock is limestone and discontinuities cut the Grave by forming a rock slope. The structures and slope are suffering from rock fall and rock slides and there are considerably potential block to fall or slide. Fig. (2) indicates fallen blocks and potential regions in Amasya. Fig. (3) indicates the constructed model of Kings Rock Grave. Fourteen numbers of discontinuities were considered to behave plastically. The other joints behaved elastically and no slip was permitted. Fig. (4) indicates the ID’s of the discontinuity to be analyzed as their material number. It is to be noted that the material number 20 was not permitted to slip however, other joints were model based on Coulomb slip model. The stability was assessed in two different cases. In one of the cases, the basic friction angle was considered to be 330 and the analyses were performed for JCS=50 MPa and 70 MPa and different values for JRC and cohesion. In the other case, the basic friction angle was considered to be 300 and the analyses were performed for JCS=70 MPa and different values for JRC and cohesion. Fig. 2. Failed and potential blocks to fail in region friction angle and joint wall compressive strength increase the probability of failure and increases the displacement. Fig. (5) indicates the reduction of failure probability due to increase in cohesion value for both basic friction angle values of 330 and 300. The infilling material in discontinuities is calcite which has high potential to be weathered such that the infilling calcite has been washed and smooth to rough joint surfaces are seen in some discontinuities, therefore based on Fig. (5), the structure will considerably suffer from reduction of cohesion. Also the reduction of JCS caused by weathering during time will considerably affect the stability of the structure. Fig. 3. The constructed model of Kings Rock Grave 120 Friction=30 7 10 2 9 3 11 Failure probability % 1 Friction=33 100 8 80 60 40 20 12 4 0 13 0 5 6 100 200 300 400 Cohesion of infilling material (KPa) 14 Fig. 5. Relation of cohesion and failure probability of discontinuities Fig. 4. Material numbers of fourteen discontinuities to be analyzed 4. RESULTS AND DISCUSSION The model was run about thirty five times for different cohesion, basic friction angle, JCS and JRC values. The shear displacements of each discontinuity for all of the runs were fitted appropriate distribution functions and the probability of δ>δpeak was calculated that is the probability of failure of corresponding discontinuity. Since for different cohesion, basic friction angle, JCS and JRC values the δpeak changes, the average, minimum and maximum value of δpeak were used to calculate the probability of failure and its corresponding reliability index. Table (1) indicates the average, minimum and maximum value of δpeak, the probability of failure and its corresponding reliability index for each discontinuity. The β<1 was considered as failure state therefore, the discontinuities no. 1, 2, 10, 11 and 13 have failed and other discontinuities are in safe condition. The analysis indicated that the increase of cohesion decreases the displacements and failure probability of the structure as expected. Also, reduction of basic Table 1. Failure probability and corresponding reliability index (β) for average, minimum and maximum value of δpeak for each discontinuity The ID of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 discontinuities δPeak_Avreage (cm) 0.52 3.22 3.42 3.1 3.1 3.1 6.6 6.3 5.6 2.6 2.4 2.9 2.0 2.0 δPeak_min (cm) 0.3 2 2 2 2 2 4 4 3.7 1.7 1.6 1.9 1.3 1.3 δPeak_max (cm) 0.7 4 4 4 3.9 3.9 8.3 8 7.1 3.14 3 3.7 2.5 2.5 Pf(δ>δPeak_Avreage) % 47.67 21.3 3.8 4.45 4.6 0 4.8 5 0 11.8 12.5 10.6 18 0 Pf(δ>δPeak_max) % 50.5 23.8 9.2 9.2 9.2 0 2.9 2.9 0 16.9 17.8 15.4 22.6 0 Pf(δ>δpeak_min) % 46.2 20 2.86 2.86 3 0 3.8 3.9 0 9.8 10.2 8.4 16.7 0 β_Ave 0.06 0.8 1.77 1.7 1.7 INF 1.7 1.6 INF 1.2 1.2 1.25 0.9 INF β_min - 0.71 1.3 1.3 1.3 INF 1.9 1.9 INF 0.96 0.9 1.0 0.75 INF β_max 0.097 0.84 1.9 1.9 1.9 INF 1.8 1.8 INF 1.3 1.27 1.4 0.97 INF 5. COLCLUSION REFRENCES In this paper, three dimensional distinct element method using 3DEC software was combined with probabilistic approach for analyzing rock slope stability and a probabilistic-numerical approach was propose. To follow the methodology practically, a slope containing a historical grave in Amasya Turkey was analyzed. The model was given different realization of random variables such as JRC, cohesion, JCS and friction angle. Thirty five realizations of random variables were run; the shear displacements of each discontinuity for all thirty five realizations were fitted appropriate distribution functions. For shear displacements lower than estimated peak shear displacement by Barton formula the failure probability and the corresponding reliability index (β) were obtained. By assuming β<1 as failure, the discontinuities number 1, 2, 10 and 11 are in failure state. Also, the results indicated that the discontinuities number 6, 9 and 14 are totally safe. 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