Seismic rock slope stability charts based on limit analysis methods

Transcription

1 Available online at Computers and Geotechnics 36 (2009) Seismic rock slope stability charts based on limit analysis methods A.J. Li a, *, A.V. Lyamin b, R.S. Merifield a a Centre for Offshore Foundations Systems, The University of Western Australia, WA 6009, Australia b Centre for Geotechnical and Materials Modelling, The University of Newcastle, NSW 2308, Australia Received 2 October 2007; received in revised form 5 January 2008; accepted 5 January 2008 Available online 5 March 2008 Abstract Earthquake effects are commonly considered in the stability analysis of rock slopes and other earth structures. The standard approach is often based on the conventional limit equilibrium method using equivalent Mohr Coulomb strength parameters (c and /) in a slip circle slope stability analysis. The purpose of this paper is to apply the finite element upper and lower bound techniques to this problem with the aim of providing seismic stability charts for rock slopes. Within the limit analysis framework, the pseudo-static method is employed by assuming a range of the seismic coefficients. Based on the latest version of oek Brown failure criterion, seismic rock slope stability charts have been produced. These chart solutions bound the true stability numbers within ±9% or better and are suited to isotropic and homogeneous intact rock or heavily jointed rock masses. A comparison of the stability numbers obtained by bounding methods and the limit equilibrium method has been performed where the later was found to predict unconservative factors of safety for steeper slopes. It was also observed that the stability numbers may increase depending on the material parameters in the oek Brown model. This phenomenon has been further investigated in the paper. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. Keywords: Safety factor; Earthquake; Pseudo-static; Seismic coefficient; Failure criterion. Introduction * Corresponding author. Tel.: ; fax: addresses: anjui@civil.uwa.edu.au (A.J. Li), andrei.lyamin@- newcastle.edu.au (A.V. Lyamin), merifield@civil.uwa.edu.au (R.S. Merifield). In seismically active regions, earthquakes are a major trigger for instability of natural and man-made slopes. Therefore, seismic effects are essential design considerations for slope stability, retaining walls, bridges and other engineering structures. Currently, the conventional pseudo-static (PS) approach is still widely accepted as a means for evaluating slope stability. In the PS method, the earthquake effects are simplified as horizontal and/or vertical seismic coefficients ( and k v ). The magnitude of the coefficients is expressed in terms of a percentage of gravity acceleration. Due to the simplicity of the PS approach, it has drawn the attention of a number of investigators [ 6]. In particular, Baker et al. [4] and Loukidis et al. [6] have adopted the PS method in limit equilibrium analysis and limit analysis, respectively, to provide chart solutions for soil slopes. It should be noted that by using complicated dynamic response analysis coupled with appropriate constitutive laws, a more precise seismic evaluation for slopes could be obtained. owever, the PS method is still recommended as a screening procedure to identify the requirement for more sophisticated dynamic analyses. The pseudo-static approach has certain limitations [7,8], but this methodology is considered to be generally conservative, and is the one most often used in current practice. Since Taylor [9] proposed a set of stability charts for soil slopes, chart solutions have been presented by many researchers [6,0 3] and are still widely used as design and teaching tools. Unfortunately, most of the existing charts are proposed for estimating the stability of soil slopes. This is most likely due to the fact that assessing rock X/$ - see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:0.06/j.compgeo

2 36 A.J. Li et al. / Computers and Geotechnics 36 (2009) mass strength is difficult and most geotechnical software is written in terms of the conventional Mohr Coulomb failure criterion. ence, there are very few stability charts which can be used for rock slopes that are based on rock mass yield criteria. Fortunately, attractive finite element upper and lower bound approaches have been developed by Lyamin and Sloan [4,5] and Krabbenhoft et al. [6]. These techniques can be used to bracket the true stability solutions for geotechnical problems from above and below, and are suited to assigning many typical failure criteria. One classical rock yield criterion, the oek Brown failure criterion [7], can be incorporated into the numerical upper and lower bound formulations. This criterion has been successfully implemented and applied to bearing capacity and slope stability problems by Merifield et al. [8] and Li et al. [9], respectively. This paper aims to investigate the seismic effects on rock slope stability where the earthquake inertia force is simulated by using the pseudo-static method. Moreover, as performed by Li et al. [9], the advantage of limit theorems will be exploited to bracket the true solutions for rock slope stability numbers and to provide a range of seismic stability charts for rock slopes. In this study, both of the upper and lower bound techniques developed by Lyamin and Sloan [4,5] and Krabbenhoft et al. [6] are employed. The PS analyses in this study do not account for the effects of pore pressure, and the strength of rock masses is assumed to be unaffected during earthquake excitation. As a means of comparison, the limit equilibrium method will then be used in conjunction with equivalent Mohr Coulomb parameters for the rock and compared with the solutions obtained from the numerical limit analysis approaches. 2. Applicability 2.. Applicability of the generalised oek Brown failure criterion Predicting the strength of large-scale rock masses is a classical challenge for geotechnical engineers. Fortunately, the empirical failure criterion proposed by oek and Brown [7,20] is widely accepted as a means of estimating the strength of rock masses which are assumed to be isotropic. It is important to point out that the assumption of isotropy means the oek Brown yield criterion is unsuitable for slope stability problems where shear failures are governed by a preferential direction imposed by a singular discontinuity set or combination of several discontinuity sets (e.g. sliding over inclined bedding planes, toppling due to near-vertical discontinuity, or wedge failure over intersecting discontinuity planes). Therefore, as pointed out by oek [2], the oek Brown yield criterion is applicable to intact rock and heavily jointed rock masses, such as the GROUP I and GROUP III shown in Fig.. The slopes that belong to GROUP II have highly anisotropic rock properties where the oek Brown failure criterion is not applicable. The anisotropy can be induced by weathering effects or naturally occurring discontinuities such as joints, faults and bedding planes. In this paper, the rock masses of all slopes have been assumed as either intact or heavily jointed rocks as GROUP I and GROUP III so that the oek Brown failure criterion is applicable. The applicability and limitations of the GSI system has been discussed in detail by Marinos et al. [22]. After oek [20], the oek Brown failure criterion can be described by the following equations: GROUP I INTACT ROCK GROUP II SINGLE DISCONTINUITIES TWO DISCONTINUITIES SEVERAL DISCONTINUITIES GROUP III JOINTED ROCK MASS Jointed Rock σ ci, GSI,, γ Fig.. Applicability of the oek Brown failure criterion to slope stability.

3 A.J. Li et al. / Computers and Geotechnics 36 (2009) r 0 ¼ r0 3 þ r r 0 a 3 ci m b þ s ðþ r ci where GSI 00 m b ¼ exp 28 4D GSI 00 s ¼ exp 9 3D a ¼ 2 þ 6 e GSI=5 e 20=3 From the above equations, it can be seen that the magnitudes of m b, s and a rely on the geological strength index (GSI) which describes the rock mass quality. Guidelines on how to determine the value of GSI and other parameters within the oek Brown failure criterion can be found in [23], and will not be discussed in this paper. In addition, D is defined as a disturbance coefficient that ranges from 0 for undisturbed in situ rock masses and for disturbed rock mass properties. For the analyses presented in this paper, a value of D = 0 has been adopted to simulate natural slopes and can be compared to the results without seismic effects from Li et al. [9] Applicability of the pseudo-static method In general, the seismic coefficients are determined from experience by using the maximum horizontal acceleration or peak ground acceleration of a design earthquake. It should be noted that, current design using PS analysis is often based on a horizontal seismic coefficient ( ). Therefore, this study is primarily focused on investigating the earthquake effects on rock slope stability by using a range of horizontal seismic coefficients. With reference to the magnitude of, Seed [2] suggested that the PS method is applicable in assessing the performance of embankments constructed of materials which do not suffer significant strength loss during earthquakes. It is recommended to utilise k = 0. for earthquakes of Richter s magnitude 6.5, and k = 0.5 for earthquakes of Richter s magnitude 8.5. For both cases, a safety factor F P.5 is required for design. The suggestion proposed by ynes-griffith and Franklin [24] is one of the widely used and accepted methods for determining an appropriate value of. They recommended that a PS analysis can be used for preliminary evaluation of slope stability, where a seismic coefficient equal to one-half the measured bedrock acceleration is adopted. Provided the obtained factor of safety is greater than.0, the slope design can be accepted. For factors of safety of less than, ynes-griffith and Franklin [24] proposed that a more thorough numerical analysis need to be performed. owever, due to the determination of magnitude is related to the measured bedrock acceleration as discussed above, the PS method may not account for the site amplification induced by the underlain stratum [25] or topography [26], etc. Pseudo-Static coefficient, In order to select an appropriate PS coefficient for a given site, a diagram (Fig. 2) summarized by the California Division of Mines and Geology [27] provides the recommendations in regards to the seismic coefficient ( ), versus a required factor of safety. From this diagram, it can be seen that the recommended values do not exceed Therefore, the range of the seismic coefficients adopted in the present paper will be between = 0.0 and Problem definition ynes & Franklin (984) M 8.25 Seed (979) Recommended Pseudo-Static Safety Factor A plane strain illustration of the upper and lower bound slope stability analyses is shown in Fig. 3 where the jointed rock mass has an intact uniaxial compressive strength r ci, geological strength index GSI, intact rock yield parameter, and unit weight c. All the quantities are assumed constant throughout the slope. In the limit analyses, the seismic force is assumed as a horizontal internal body force and its magnitude is represented by the horizontal seismic coefficient ( ). The direction of W, is considered to be positive when acting outward with respect to the slope (see Fig. 3). For given slope geometry (, b) and rock mass (r ci GSI, ), the optimised solutions of the upper bound and lower bound programs can be carried out with M 6.5 Fig. 2. Design recommendations for pseudo-static analysis. Toe Rigid Base W Fig. 3. Problem definition. W Jointed Rock σ ci, GSI,, γ d

4 38 A.J. Li et al. / Computers and Geotechnics 36 (2009) respect to the unit weight (c). In this study, slope inclinations of b = 30,45,60 and 75 are analysed. The nondimensional stability number proposed by Li et al. [9] has been adopted, and is given as N ¼ r ci =cf ð2þ where F is the safety factor of the slope. 4. Previous studies To overcome the problem of estimating rock slope strength and stability governed by the complicated failure mechanisms, Jaeger [28] and Goodman and Kieffer [29] had outlined several simple methods and emphasized their limitations. Buhan et al. [30] found that the rock mass scale-effect may influence the results of stability analysis. oek and Bray [] and Zanbak [3] proposed chart solutions for stability and toppling problems of rock slopes, respectively. owever all of the above studies were based on a static analysis. This means that the seismic effects were not taking into account. In order to estimate rock slope stability under earthquake effects, Seed s [2] recommendation of seismic coefficient based on the PS analysis can be used. Newmark [] applied and extended the PS method to evaluate the ground movement induced by an earthquake. This approach has been accepted and extensively used to study earthquake triggered landslides and rockslides [26,32,33]. Pradel et al. [34] in particular, obtained a good agreement of slope crest displacement between the calculated and observed results. In their study, the strength parameters used in analyses are determined by repeated direct shear testing and back analysis. Bhasin and Kaynia [35] utilised the distinct element method to investigate a rock slope progressive failure under initial static loading, climatic effects, and dynamic loading. Based on finite element and limit equilibrium analyses, Luo et al. [36] found that ground water may significantly reduce slope stability during earthquake excitation where the obtained maximum seismic coefficient change by up to 60%. Sepúlveda et al. [37] pointed out that the topographic amplification effects such as slope orientation and seismic wavelength may influence the rock slope stability assessment. In the case study of Chen et al. [38], the vertical ground acceleration was found to be an important factor leading to rockslide under near field conditions. Reliability analysis was employed by Shou and Wang [39], Wang et al. [40] and ack et al. [4] to predict the failure risk of rock slopes and investigate the influence of earthquakes. Currently, practising engineers typically use stability charts originally derived for soils when attempting to predict the stability of rock slopes (i.e oek and Bray []). Zanbak [3] proposed a set of stability charts for rock slopes susceptible to toppling, whilst Siad [42] produced charts based on the upper bound approach that can be used for rock slopes with earthquake effects. owever, the design charts presented by these authors require conventional Mohr Coulomb soil parameters, cohesion (c) and friction angle (/), as input. Recently, Collins et al. [43] and Drescher and Chrostopoulos [44] and Yang et al. [45] adopted tangential strength parameters (c t and / t ) from nonlinear failure criteria to estimate the slope stability. owever, only the study of Yang et al. [45] were based on the latest version of the oek Brown yield criterion and obtained optimised stability factors using the upper bound approach and a failure mechanism originally formulated by Chen [46]. A review of the literature reveals that few numerical slope stability analyses have been performed which are based on the native form of the oek Brown failure criterion (where strength parameters r ci, GSI, and D are input variables). The only exception is the stability charts proposed by Li et al. [9]. owever, in their study, the seismic effects were not taken into consideration. Therefore, the purpose of this paper is to provide a set of rock slope stability charts which incorporate the earthquake effects and can be used by engineers in the assessment of seismic rock slope stability. 5. Results and discussion 5.. Numerical limit analysis solutions Figs. 4 6 present three sets of stability charts obtained from the numerical upper and lower bound formulations with horizontal seismic coefficients of = 0., 0.2 and 0.3, respectively. The magnitude of in current design codes is generally within this range. For all analyses performed in this study, the maximum difference between the bound solutions was found to be less than ±9%. As a consequence, average values of the upper and lower bound stability numbers will be used in the following discussions. Referring to Figs. 4 6, it can be observed that the stability number N decreases when GSI or increases, as expected. Remembering from Eq. (2) that the stability number N is proportional to the inverse of the factor of safety F. A lower factor of safety equates to a higher stability number and viseversa. The direction of increasing stability is shown by the arrow in Fig. 4a. A decrease in stability number with increasing GSI or is not unexpected because, based on the definition of the oek Brown yield criterion, the larger magnitudes of GSI or signify that the rock masses have greater overall strength for any given normal stress. owever, one exception to this trend is shown in Fig. 6d where N increases slightly with increasing. This phenomenon will be discussed in more detail below. In addition, the chart solutions can be presented as an alternative form which is a function of the slope angle (b) as shown in Fig. 7 where N can be seen to increase when the inclination of a slope increases. For a given slope inclination, the stability number can be obtained by estimating GSI and. For the same rock mass properties of a slope, the difference in stability numbers between various slope angles can provide the corresponding variation in factor

5 A.J. Li et al. / Computers and Geotechnics 36 (2009) = 30, = = 45, = 0. 0 SLIDE-oek-Brown Model Tangential Method SLIDE-oek-Brown Model Tangential method 0 /γ F /γ F Increasing Stability = 60, = 0. SLIDE-oek-Brown Model Tangential Method GSI=0 GSI=00 GSI=0 /γ F /γ F = 75, = 0. SLIDE-oek-Brown Model Tangential Method GSI=0 GSI=00 GSI=0 GSI=00 0. GSI= Fig. 4. finite element limit analysis solutions of stability numbers ( = 0.). of safety. For instance, it can be observed in Fig. 7 that decreasing slope angle from b = 75 to 60 can increase the safety factors by 50 00% for = 5. This trend is similar to the results of Li et al. [9] in which reducing the slope angle was found to increase the factor of safety by more than 50%.

6 40 A.J. Li et al. / Computers and Geotechnics 36 (2009) = 30, SLIDE-oek-Brown Model Tangential Method 00 = 45, SLIDE-oek-Brown Model Tangential method 0 GSI=0 /γ F GSI=0 /γ F GSI=00 GSI= = 60, SLIDE-oek-Brown Model Tangential Method 000 = 75, SLIDE-oek-Brown Model Tangential Method 00 GSI=0 0 GSI=0 /γ F /γ F 0 GSI=00 0. GSI= Fig. 5. finite element limit analysis solutions of stability numbers ( ). An analysis of static rock slope stability using the oek Brown criterion has been recently performed by Li et al. [9]. Figs. 8 and 9 display the stability numbers from the PS analyses compared to the results of these static analyses. The average lines shown in these figures with = 0. represent the stability numbers obtained when the PS force

7 A.J. Li et al. / Computers and Geotechnics 36 (2009) = 30, = 45, SLIDE-oek-Brown Model 00 SLIDE-oek-Brown Model 0 N= σ ci /γ F N= σ ci /γ F = 60, SLIDE-oek-Brown Model GSI=0 GSI=00 GSI=0 N= σ ci /γ F N= σ ci /γ F = 75, SLIDE-oek-Brown Model GSI=0 GSI=00 GSI=0 GSI=00 GSI= Fig. 6. finite element limit analysis solutions of stability numbers ( ). acts toward the rock slope face. Understandably, when the PS force acts toward the slope, this was found to increase the stability of the slope as seen in Fig. 8. In general, the rock slopes and the potential epicentres such as faults or volcanos can scatter around in seismically active region, so adopting the more critical W direction (away form the slope) in the analyses is more appropriate from a design perspective.

8 42 A.J. Li et al. / Computers and Geotechnics 36 (2009) N= σ ci /γ F 0 0. GSI=80, = 0. = 75 = 60 = 45 = /γf =30, GSI =00 Decreasing Stability =45, GSI =70 = 0. = 0.0 = -0., N= σ ci /γ F 0 = 75 = 60 = 45 = 30 /γ F 0. Decreasing Stability = 0. = 0.0 = -0. N= σ ci /γ F GSI=20, = 75 = 60 = 45 = 30 Fig. 7. finite element limit analysis solutions of stability numbers. Referring to Figs. 8 and 9, it can be found that N increases with increasing. This means that the factors of safety will be smaller as the earthquake loading 0.0 Fig. 8. Comparisons of stability numbers between the static and pseudostatic analyses (b = 30 and 45 ). increases. In addition, from these figures, the factor of safety is found to decrease by 30% ( = 5) or more when increases by 0. for all slope angles. Fig. 0 shows the several of the observed upper bound plastic zones for different GSI. In general, the modes of failure consisted of shallow toe type mechanisms for all analyses. It can be noticed that the depth of slip surface increases only slightly with increasing GSI. But this phenomenon is not observed when the slope angle b A similar trend is found where the depth of slip surface increases slightly with the reduction of. Moreover, Fig. indicates that the depth of the plastic zones is almost unchanged for various seismic coefficients. This means that the shape of the potential failure surface is almost independent of the magnitude of for a given geometry and rock strength parameters Analytical upper bound solutions Using the generalised tangential technique proposed by Yang et al. [45] in conjunction with the assumed failure mechanism proposed by Chen [46] shown in Fig. 2, the optimised height of a slope with oek Brown rock

9 A.J. Li et al. / Computers and Geotechnics 36 (2009) =60, GSI =50 0 Decreasing Stability /γf 0. = 0. = 0.0 = -0. GSI = 0 =75, GSI =20 /γ F 00 0 Decreasing Stability = 0. = 0.0 GSI = 50 = -0. Fig. 9. Comparisons of stability numbers between the static and pseudostatic analyses (b = 60 and 75 ). strength parameters can be obtained using tangential parameters (c t and / t ). Yang et al. [45] proposed that, for a given friction angle (/ t ), c t can be expressed as ða= aþ tan / t c t ¼ cos / t m b að sin / t Þ þ sin / t r ci 2 2 sin / t m b a m ð= aþ bað sin / t Þ þ s tan / 2 sin / t m t ð3þ b where s, m b and a are from the latest version of oek Brown yield criterion shown in Eq. (). Therefore, the new tangential Mohr Coulomb yield criterion can be expressed as s r ci ¼ r n r ci tan / t þ c t r ci In addition, to transfer the yield surface from the (s/ r ci r n /r ci ) plane to the major and minor principal stress plane (r /r ci r 3 /r ci ), we can use Eq. (5) r ¼ 2ðc=r ciþ cos / r ci sin / þ þ sin / sin / r 3 ð5þ r ci where c and / are the cohesion and friction angle in the s r n plane. It should be noted that / t in Eq. (4) is ð4þ GSI = 00 Fig. 0. Comparisons between the upper bound plastic zones and failure surfaces of the tangential method for various GSI ( = 0. and = 0). unknown and determined by the optimisation that the smallest slope height is obtained which is given as [46]: sin b 0 ¼ 2c sinðb 0 aþ tan / t c t fexp ½2ðh h h 0 Þ tan / t Š g ðf f 2 f 3 f 4 Þþ ðf 5 f 6 f 7 f 8 Þ fsinðh h þ aþ exp ½ðh h h 0 Þ tan / t Š sinðh 0 þ aþg ð6þ Eq. (6) has been formulated in FORTRAN and optimised in this study using the ooke Jeeves algorithm which is a function of four variables, namely h h, h 0 and b 0 for specify-

10 44 A.J. Li et al. / Computers and Geotechnics 36 (2009) Shear stress (τ) φ t c t = 0. Normal stress (σ n ) (a) Tangential method θ o θ h α W k v W (b) Logarithmic spiral failure mechanism Fig. 2. Illustration of adopted tangential method and failure mechanism. Fig.. Comparisons between the upper bound plastic zones and failure surfaces of the tangential method for various (GSI = 50 and = 5). ing the assumed failure mechanism and / t for specifying the location of the tangency point. More details on how the parameters (h h, h 0 and b 0 ) are determined and the definitions of the parameters f f 8 can be found in Chen [46] and Yang et al. [45]. To examine the results of PS analysis employed in the present study, the newly obtained average lower and upper bound solutions are compared with the optimised upper bound solutions using Eq. (6) in Figs. 4 and 5. The tangential upper bound solutions are shown as cross symbols which are stability numbers obtained from Eq. (2) for GSI = 0, 50 and 00. It can be observed that most of the cross symbols plot below the newly obtained average lower and upper bound solutions. This means that using the tangential approach may overestimate the factor of safety for the rock slope. The difference in stability numbers between the average bounding solutions and the results of the tangential method is found to increase sharply when GSI and slope inclination increase. For the case of b =75 and GSI = 00 (Figs. 4d and 5d), the difference can be by up around 80% which is quite significant and cannot be ignored. Referring to these two figures, the value of N obtained using the tangential method with GSI = 50 is found to be smaller than that of newly obtained lower and upper bound average solutions with GSI = 60. In particular, for larger ( P 25), they are close to the average bound solutions with GSI = 70. It was found that the stability number estimate of tangential technique tends to be unconservative, particularly for larger GSI. In order to determine the source of overestimation in the factor of safety when using the tangential method, the failure surface information from the numerical lower bound results is observed more closely. The stress components in the vicinity of the lower bound failure zone/mechanism have been extracted and the location of each point on the oek Brown yield envelope can be observed. This is shown in Fig. 3 for b =75 and = 0.. For comparison

11 A.J. Li et al. / Computers and Geotechnics 36 (2009) σ /σ ci σ /σ ci σ /σ ci 0.0 oek-brown (GSI=0, =0) Tangential method Lower bound ( =0.) Lower bound fitted yield surface σ =5.784 (σ 3 ) σ 3 oek-brown (, =0) Tangential method 0. Lower bound ( =0.) Fitted yield surface σ =3.276 (σ 3 ) σ oek-brown (GSI=00, =0) Tangential method Lower bound ( =0.) Lower Bound Fitted yield surface σ =6.047 (σ 3 ) σ 3 Fig. 3. Comparisons of yield surfaces between numerical lower bound and tangential method (b = 75 ). purposes, linear regression and Eq. (5) are also employed here to obtain the magnitudes of / t and c t /r ci from the lower bound solutions. Table displays optimised / t and c t /r ci values from the tangential approach. Compared with the lower bound linear regression values, the differences in / t and c t /r ci are significant, except for the case of GSI = 0. Due to the fact that / t and c t /r ci are almost the same for GSI = 0 and Table Comparisons of tangential Mohr Coulomb parameters for various quality rocks (b =75 and = 0.) GSI Tangential method Lower bound linear regression / t c t /r ci / t c t /r ci = 0, the difference in stability numbers between these two approaches is small, as shown in Fig. 4d. Referring to Fig. 3, based on Eq. (5) and Table, the yield surface adopted in tangential method can be plotted in the r /r ci r 3 /r ci plane. The difference between the tangential method and native oek Brown yield surfaces is found to increase significantly with increasing GSI. Therefore, the difference in stability numbers between the average bounding solutions and results of tangential method, as mentioned previously, can be explained. For the case of GSI = 50 and 00, it is apparent in Fig. 3 that the yield surface from the lower bound solutions plots below that from the results using the tangential technique. In particular, for GSI = 00, the yield surface of the tangential method is significantly above the oek Brown failure surface which will lead to a larger shear strength and thus a larger factor of safety. This explains the large differences observed in Figs. 4 and 5 and the unconservative nature of the tangential technique for these cases. The predicted slip surfaces by using the tangential method (bold line) are compared with the upper bound plastic zones in Figs. 0 and. It can be observed that the difference between the obtained slip surfaces increases significantly with increasing GSI, which agrees with the difference in the yield surfaces shown in Fig. 3. InFig., by using the tangential method, the variation of slip surface is also found to be insignificant for various Limit equilibrium method (LEM) solutions In this study, the commercial limit equilibrium software SLIDE [47] and Bishop s simplified method [48] have been employed to make comparisons with the average lower and upper bound solutions. A limit equilibrium analysis which adopts the oek Brown strength parameters can be performed when using SLIDE. More information in how SLIDE calculates and determines the shear strength of each individual slice can be found in Li et al. [9] and oek [23]. Referring to Figs. 4 6, the displayed triangular points are the stability numbers obtained using the oek Brown strength parameters based on the LEM. It can be seen that most of the results from SLIDE are remarkably close to the average lines of the upper and lower bound limit analysis solutions. owever, the exception to this observation can be found in the case of b =75, shown in Figs. 4d and 6d. In these figures, for lower GSI values (GSI 6 50), the stability numbers obtained using the LEM have been underestimated by 8 30%, compared to the average

12 46 A.J. Li et al. / Computers and Geotechnics 36 (2009) bound solutions. Although comparisons between the upper bound and LEM show an underestimation of between 0 22%, the difference in stability numbers between the limit analysis and the LEM is still quite significant. Due to the true stability numbers being bounded by the upper and lower bound limit analysis solutions, results obtained from the LEM tend to be unconservative. This means that, for rock slope of high slope inclination (b P 75 ), unsafe factors will be obtained by using the PS limit equilibrium analyses. 00 = 75, = 0.4 SLIDE-oek-Brown Model GSI=0 GSI= Investigation of stability numbers increasing with increasing As mentioned above, Fig. 6d indicated that for high slope angles (b P 75 ) and high lateral coefficients ( P 0.3), the stability number N was found to actually increase slightly with increasing. This implies that the stability of the slope is essentially independent of the material shear strength. This observation requires a more thorough investigation. It is evident from Fig. 6d that the LEM results do not exhibit the same response compared to the numerical bounds for b =75 and. Therefore, additional limit equilibrium analyses for slopes with b =75 and = 0.4 have been performed (using SLIDE) with the results illustrated in Fig. 4. Now we can observe that the stability number N increases with increasing when = 0.4 is adopted. This trend is similar to the results of the bounding methods shown in Fig. 6d where. With this being the case we can conclude that the observed phenomenon is real and needs further investigation. In doing so, the lower bound stress conditions in the region of the failure plane were extracted and observed more closely. The obtained information is displayed in Fig. 5 along with the oek Brown yield envelope for each rock material. When GSI = 00, = 35 and = 0.0, most of stress points extracted along the slip surface, and therefore much of the slip surface length itself, are in a state of compression. In contrast, for the case of GSI = 00, = 35, and (square symbols), most of the stress points extracted along the slip surface fall in the region with small r /r ci r 3 /r ci and are in tension. This observation is also true when GSI = 00, = 5, and (triangular symbols). This suggests that the tensile strength of the material will tend to dictate the overall stability. It is found in Fig. 5 that when = 5 the tensile strength is larger than that for = 35. This means that when collapse of the rock slope is due to tensile failure, rock masses with a smaller can provide high strength and therefore are more stable. This would explain the decrease in stability number N with increasing seen in Fig. 6d. It should be noted that in Fig. 5, for the rock masses with lower GSI, although the tensile strength is relative small, it still play an important role. Therefore, the phenomenon that stability number N increases slightly with increasing exists as well. N= σ ci /γ F 0 GSI=70 GSI=00 Fig. 4. Stability numbers from the limit equilibrium analyses ( = 0.4). σ The observed stress conditions on the base of each slice in limit equilibrium analysis using SLIDE are shown in Fig. 6 for b =75, = 35, GSI = 00 and = 0.4. The square points represent the stresses of each slice for this case. It can be clearly seen that these stresses locate intensively in the region with the small r n /r ci and comparing to the case with = 5, the shear stress is actually lower. This means that, based on the oek Brown analytical model, the shear strength of rock masses with =5 GSI=00 =35, Lower bound ( = 0.0) GSI=00 =35, Lower bound ( ) GSI=00 =5, Lower bound ( = 0.0) GSI=00 =5, Lower bound ( ) Tensile Compressive σ 3 GSI=00 =35 GSI=00 =5 =5 Fig. 5. The oek Brown failure criterion for variable GSI and values and observed lower bound results.

13 A.J. Li et al. / Computers and Geotechnics 36 (2009) τ and = 35 will be similar for small ratios of r n /r ci. While considering the case with = 0.0, larger r n /r ci was observed on the base of the slices, where shear strength for = 5 is obviously less than that of = 35. It was found that, under the strong earthquake loading, rock slopes tend to fail due to tensile stresses. It should be noted that this phenomenon only occurs for steep slopes combined with a high seismic coefficient. owever, tensile strength of rocks or rock masses can be small. Therefore, a rock slope stability based design is recommended to avoid this situation. 6. Conclusions Tensile GSI=00 =35 GSI=00 =5 GSI=00 =35, SLIDE ( = 0.0) GSI=00 =35, SLIDE ( = 0.4) GSI=00 =5, SLIDE ( = 0.4) Compressive σ n Fig. 6. The observed stress conditions along the slip plane from the limit equilibrium analyses. Using the numerical upper and lower bound techniques, rigorous bounds on the seismic rock slope stability have been presented as chart solutions. These stability charts, including the earthquake effects, are based on the oek Brown failure criterion and can be used for estimating seismic rock slope stability in the initial design phase. This study follows the general consideration of the earthquake effects which only takes the horizontal seismic coefficient ( ) into account. A range of magnitudes is also included, which are consistent with most design codes. Although the vertical seismic coefficient (k v ) is often ignored in practice, it is still worth investigating its influence on the stability of rock slopes in further studies. Based on this study, the following conclusions can be made. A narrow range of stability numbers (N) have been bounded using pseudo-static (PS) upper bound and lower bound solutions within ±9% or better for all considered cases. The stability analysis of rock slopes using the limit equilibrium method was found to overestimate the factors of safety for the cases with higher slope angles and lower GSI. Compared with the upper bound solutions alone, this overestimate can be as high as 22% for the steep slopes (i.e b = 75 ) with GSI values less than approximately 50. When the horizontal seismic coefficient ( ) increases by a factor of 0., the safety factor of a rock slopes may decrease by more than 30%. But reducing the angle of slope by 5 can increase the safety factor by at least 50%. By using the tangential method proposed by Yang et al. [45], the overestimates of rock slope stability increase with GSI increasing and can be up to 80%. In addition, an inaccurate prediction of slip surface would be obtained. It was found that for the cases with high slope angle and significant seismic coefficient the stability numbers for rock slopes obeying oek Brown yield criterion will increase with increasing. 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