Equivalent continuum analyses of jointed rockmass: Some case studies

Transcription

1 Volume 5, Number, March 9, pp.9-5 [INVITED PAPER] Equivalent continuum analyses of jointed rockmass: Some case studies T. G. SITHARAM Professor, Department of Civil Engineering, Indian Institute of Science, Bangalore-56, India. Received 6 8; accepted 9 ABSTRACT The paper presents summary of the work carried out by the author on the modelling of jointed rock mass with some field applications. The equivalent continuum analyses approach presented attempts to use statistical relations, which are simple and obtained after analyzing a large data from the literature on laboratory test results of jointed rock masses. Systematic investigations were done including laboratory experiments to develop the methodologies to determine the equivalent material properties of rock mass and their stress-strain behaviour, using a hyperbolic approach. Present study covers the development of equivalent continuum model for rock mass, implementation of the model in FLACD for -dimensional applications and subsequently verification leading to real field application involving jointed rocks. The model was rigorously validated by simulating jointed rock specimens. Element tests were conducted for both uniaxial and triaxial cases and then compared with the respective experimental results. The numerical test program includes laboratory tested cylindrical rock specimens of different rock types, from plaster of Paris representing soft rock to granite representing very hard rock. The results of the equivalent continuum modelling were also compared with explicit modelling results where joints were incorporated in the model as interfaces. Several case studies have been presented with the developed model. Keywords: Jointed rock mass, equivalent continuum analyses, modeling, field applications, explicit modeling of joints, power house caverns, large slopes. BACKGROUND Rock is distinguished from other engineering materials by the presence of inherent discontinuities such as joints, bedding planes and faults that control its behavior. Hence, the prediction of the response of rocks and rock masses derives largely from their discontinuous and variable nature. Reliable characterization of the strength and deformation behavior of jointed rocks is important for the safe and economic design of civil and mining structures such as arch dams, bridge piers, tunnels, slopes and large caverns. Realistic evaluation of the shear strength and deformation characteristics presents formidable theoretical and experimental difficulties due to the complex geometry and behavior of jointed rock. A numerical approach to treating the rock mass with equivalent material properties for obtaining the overall response has been advocated in recent years (Sitharam, 7; Sitharam et al., ; Sitharam and Latha, ). Several numerical methods have been developed by various researchers to model jointed rock masses using various techniques. Singh (97) has presented continuum characterization methods for jointed rock masses and expressions were presented to estimate the elastic moduli of the equivalent continuum anisotropic rock mass. Zienkiewicz et al. (977) has used the equivalent continuum approach, referred to as a multi-laminate model to simulate a discontinuous rock mass. Gerrard (98) has also used an equivalent continuum approach by expressing the compliance of an element as the sum of the compliances of the intact rock and that of the individual joint sets. Cai and Horii (99) proposed a constitutive model that presents the effects of density, orientation and connectivity of joints, as well as the property of the joints themselves. The constitutive equation is derived from the relation between the average stress and strain over a representative volume, which consists of many fractures, on the basis of micromechanics. This analysis is valid for small displacements before the failure of the joints. In the other micromechanics based model of Yoshida and Horii (), the jointed rock mass is replaced with an equivalent continuum body whose constitutive equation is obtained from the relation between the average stress and strain over a representative volume element. The constitutive equations directly reject the orientation and spacing of the dominant joints and can incorporate strong induced anisotropy. In the crack tensor model of Oda et al. (99) the compliance tensor of the jointed rock mass is given as the sum of the elastic compliance tensor of the base rock and that corresponding to the crack deformation. The crack tensor is defined in terms of the size of the joints and components of the unit vector normal to the joints. The damage tensor model developed by Kawamoto et al. (988) is based on the concept that the effective cross section of the material is reduced due to the damage. When the number of joints in the rock mass is many and it is not possible to obtain information about all of them, thus, it is not possible to deal with each joint individually. It is necessary to replace the JCRM All rights reserved.

2 4 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp.9-5 jointed rock mass with an equivalent continuum body for analysis with an appropriate associated constitutive model. However, the constitutive models which are discussed above and many others, are very complex and need much input data from experimental or field testing in order to carry out the analysis. So, there is a need for a simple technique where the equivalent continuum method can capture sufficiently well the behavior of a jointed rock mass using minimal input from the field or from tests and experiments. Considering the inherently inhomogeneous nature of rock masses, the practical equivalent continuum approach attempts to use statistical relations, which are simple and obtained after analyzing a large set of data from the literature on laboratory test results relating to jointed rock masses. The material properties are represented by a set of relations which express the properties of the jointed rock mass as a function of a joint factor and the properties of the intact rock. The joint factor for a given jointed rock is estimated based on the joint fabric. The tangent elastic modulus of the intact rock is represented by a confining stress-dependent hyperbolic relation. The jointed rock has been represented as an equivalent continuum using the statistical relations arrived at from the analysis of the experimental data of Roy (99), Arora (987), Yaji (984), Brown and Trollope (97), VidyaBhushan Maji (7) and Einstein and Hirschfield (97). These relations express the tangent modulus of the jointed medium as a function of the joint factor and the tangent modulus of the intact rock. The results have been presented in the form of stress strain curves for the jointed rocks and compared with the experimental results. These results are also compared with explicit modeling of rock mass where the joints are modeled using interface element. The developed model called as Practical Equivalent Continuum Model has been applied to the analyses of well-documented field case studies of large rock excavations and slopes.. EMPIRICAL RELATIONSHIPS An effort has been made to arrive at empirical relations, which express the strength and deformation of jointed rock as a function of intact rock properties and a joint factor. These relations are determined by statistical analysis of a large amount of experimental data from Brown and Trollope (97), Einstein and Hirschfeld (97), Yaji (984), Arora (987), VidyaBhushan Maji (7), and Roy (984). The experimental data covers a wide range of rocklike material and rocks namely plaster of Paris, different kinds of sandstone, granite and gypsum plaster with paralled and unparalled joints with different joint fabric for different confining pressures. Thus, it is hoped that the developed equations for jointed rock mass may have a reasonably wide applicability. Based on the statistical analysis, the uniaxial compressive strength and elastic modulus obtained from uniaxial compressive tests and triaxial tests of jointed rock at different confining pressures are expressed as the function of the joint factor and intact rock properties. So, knowing the intact rock properties and the joint factor, the jointed rock properties can be estimated. Statistical analyses were carried out to arrive at possible empirical relations for the tangent modulus at different confinements and uniaxial compressive strength. A large amount of experimental data of uniaxial compressive strength ratio and elastic modulus ratio versus joint factor of the jointed rock specimens was digitally filtered to reduce the scatter in the data. Linear and nonlinear relationships between the uniaxial compressive strength, tangent elastic modulus at different confinements, and joint factor have been arrived at by using least-squares fitting for linear relationships and Lorentzian minimization for nonlinear relationships. Least-squares minimization assumes that the x values are accurately determined and that an error exists only in the dependant variable y. The errors are assumed to map a Gaussian profile and are normally distributed. Lorenzian minimization is very robust when the data is noisy and also converges quite rapidly. The correlation coefficient and the standard error of the fitted relationship is defined below. The correlation coefficient, r of a relationship fitted to x, y data is expressed as n i r [ y ( y ) ] n i i ( y y) i p i where y i = y value for a given x; y data and the (y p ) i = value of y computed using the relationship fitted; and y = mean of y values. The standard error of the relation fitted is computed as S () n i [ yi ( yp) i] () n In the present analysis the variable y is either uniaxial compressive strength ratio (σ cr ) or modulus ratio (E r ) and variable x is joint factor J f. Both linear and nonlinear relations are obtained for σ cr and E r as a function of a joint factor J f using the procedure described above. The experimental data is compiled in the form of a joint factor J f, the uniaxial compressive strength ratio σ cr, and elastic modulus ratio E r. The experimental data used in the analysis covers a wide range of rocks like plaster of Paris, different kinds of sandstone, granite, and gypsum plaster with filled and unfilled joints with different joint fabric for different confining pressures. The experimental data is digitally filtered to reduce the scatter in the data for a better fit. The values of correlation coefficient for the fitted equations are given for the digitally filtered data set and the standard error of the fit is calculated (Sridevi, ) with respect to the actual experimental data. Rank of the relationship fitted is in the order of minimum standard error. The relation for which the standard error is minimum is ranked as and the next relation is ranked,, and 4 based on the increase in the standard error. Based on the statistical analysis of the data, empirical relationships for the uniaxial compressive strength ratio as a function of joint factor (J f ) are derived. Figure shows the plot of the uniaxial compressive strength ratio versus the joint factor of the experimental data with the empirical relationship fitted for the equation with rank. The elastic modulus ratio from unconfined compressive strength test, from triaxial test, with different confining pressures is considered in the analysis. Linear and nonlinear relationships between the elastic modulus ratio at the three different confining pressures and the joint factor are derived using the experimental data. Much more unconfined compressive strength test data is available compared to triaxial test data. Hence the relationships arrived for elastic modulus using unconfined compressive test data are more reliable than those obtained using triaxial test data. Sample plot of the experimental data with the fitted relationship is shown in Figure. All the 4

3 Ratio of Elastic modulus at = Uniaxial compressive strength ratio T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp statistical relations arrived in the analysis are for < J f < 8. For intact rock, i.e., for J f =, σ cr and E r =. (σ cr ) Plaster of Paris Jamrani sandstone Arora (987) Agra sandstone Plaster of Paris Kota sandstone Yaji (984) Granite Plaster of Paris Singh and Dev (988) Kota sandstone Sharma (989) Einstein and Hirschfeld (97) Brown and Trollope (97) Brown (97) Present experimental data Joint Factor Figure. Uniaxial compressive test data and the fitted relation between J f and σ cr. Data from Arora (987), Yaji (984), Einstein and Hirschfeld(97), Brown and Trollope (97), Brown (97), Roy (99). (E r ) σ Plaster of Paris Agra sandstone Jamrani sandstone Plaster of Paris Kota sandstone Granite Einstein and Hirschfeld (97) Brown and Trollope (97) Brown (97) Joint Factor J f cr e.64 Arora (987) Yaji (984) E r e.86 J f Figure. Experimental data and the fitted relation between J f and E r. Data from Arora (987), Yaji (984), Einstein and Hirschfeld (97), Brown and Trollope (97), Brown (97), and Roy (99). Extensive laboratory testing of intact and jointed specimens of different grades of plaster of Paris, sandstone and granite in uniaxial and triaxial compression revealed that the important factors which influence the strength and modulus values of the jointed rock are (i) joint frequency, J n, (ii) joint orientation, β, with respect to the major principal stress direction and (iii) joint strength. The joint factor (Ramamurthy, 99) for a given jointed rock is estimated using the following equation: J J n f () nr. where, J n is number of joints per meter depth, n is the inclination parameter depending on the orientation of the joint β, r is the roughness or joint strength parameter depending on the joint condition. The value of n is obtained by taking the ratio of log (strength reduction) at β = 9 o to log (strength reduction) at the desired value of β. The joint strength parameter r is obtained from a shear test along the joint and is given as r = τ j /σ nj where τ j is the shear strength along the joint and σ nj is the normal stress on the joint. This inclination parameter is independent of joint frequency. The values of n are given for various orientation angles and the joint strength parameter r is given for various uniaxial compressive strengths of intact rock and they are presented by Ramamurthy (99), based on extensive laboratory testing of rocks. So, knowing the intact rock properties and the joint factor, the jointed rock properties can be estimated using the following groups of empirical/ statistical relations. Group : The jointed rock is modeled using the statistical relations given as below. cj J.4.89exp f cr (4) 6. ci E j J E a b f r exp (5) E c i These relationships arrived are based on the statistical analysis of a large amount of experimental data. E r is the tangent modulus ratio, E j is the tangent modulus of jointed rock, E i is the tangent modulus of intact rock, J f is the joint factor, a, b, and c are statistical constants. The values of these constants are given in Table for,. and 5. MPa confining pressures. For the confining pressures other than listed in the Table the values of a, b, and c have been linearly interpolated or extrapolated based on the value of a, b, and c for 5. MPa confining pressure. Table. Values of empirical constant a, b, c for different confining pressure. Confining Pressure, MPa Value of a in Eq. (5) Value of b in Eq. (5) Value of c in Eq. (5) Group : The jointed rock is modeled using the following empirical relations given by Ramamurthy (99). cj c r = =exp-.8 Jf (6) ci Ej Er = J E σ - =exp-.5 f σ i Tangent elastic modulus of jointed rock for any other σ is derived from the tangent elastic modulus of jointed rock at = using the formula given below. Ej Ej ( ) cj -exp -. Equation (7) is valid for =. For different confining pressures, the elastic modulus of jointed rock is calculated using Equation (8), where j is obtained from Equation (6) and E j at = is obtained from Equation (7). These relations were given by Ramamurthy and Arora (994) based on laboratory studies on numerous artificial joints. (7) (8) 4

4 4 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp.9-5 Group : The jointed rock is modeled by a set of statistical relations after Sitharam (7) as given below. cj c r = =exp-.65j (9) f ci i Ej Er = =exp a J E f () These relations are arrived at Equations (9) and () by fitting the equations of the same form as Ramamurthy (99) to the experimental data using statistical analysis. The value of the empirical constant a is given in tabular form Table for different confining pressures. For confining pressure other than those listed in the Table the value of a can be interpolated or extrapolated based on the value of a for to 7. MPa confining pressure. Table. Value of empirical constant a for different confining pressure. Confining Pressure, MPa Value of a HYPERBOLIC STRESS-STRAIN RELATIONSHIPS The constitutive relations for the intact and jointed rock used in the equivalent continuum analysis are described in detail in this section. The stress strain behavior of rocks over a wide range of stress field is nonlinear and dependent upon confining pressure. The nonlinear elastic confining stressdependent model following a hyperbolic relation proposed by Duncan and Chang (97) is used in the present study. The material behavior of intact rock is modeled using the following non-linear relation: d( ) E ( ) E R E ti r f ti d( a) ( ) f ar f Eti f () n E K P P () ti a a ( ) f R f () ( ) ult Where, E t is the tangent modulus computed at 5 percent of failure stress for the intact rock, E ti is the initial tangent modulus, is the major principal stress, is the confining pressure, a is axial strain, K is the modulus number, n is modulus exponent and P a is atmospheric pressure. ( ) f is the failure stress and ( ) ult is the asymptotic value of stress. The value of modulus exponent K and modulus number n are determined from the plots of tangent modulus versus confining pressure for the intact rock. For brittle failure in rocks the failure stress and the asymptotic value of stress are almost same so R f =. For heavily jointed rocks which have a similar stress-strain relationship of soils, the asymptotic value of stress and failure stress are slightly different and so R f is taken as.9. For the linear stress-strain relation an R f =. is taken. 4. EQUIVALENT CONTINUUM MODEL First implementation of this equivalent continuum model has been done with finite element (Sitharam, ), and the numerical model has been developed from an existing finite element code for a non-linear soil structure interaction program (NLSSRIP). As mentioned, the jointed rock is represented as an equivalent continuum whose properties have been derived from the intact rock properties and a joint factor based on the joint fabric. Nonlinearity in the finite element analysis has been incorporated in the form of material non-linearity of both the intact and jointed rock. The incremental method is used for the solution of the non-linear problem by the finite element method. The load is increased in a series of steps or increments. Each increment is analyzed twice: the first time using the moduli values for the elements based on the stresses at the beginning of the increment; and the second time using moduli value based on the average stresses during the increment. The changes in stress and strain of the elements and changes in the nodal displacements during each increment are added to the values at the beginning of the increment. At the beginning of each new increment of loading, an appropriate modulus value is selected for each element on the basis of the values of stress or strain in that element. The non-linear stress strain behavior is approximated by a series of straight lines. The displacements at each increment of loading are accumulated to give the total displacement at any stage of loading, and the incremental process is repeated until the total load is applied. The principal advantages of this procedure are its complete generality and its ability to provide a relatively complete description of the load deformation behavior. Initial stresses may be readily accounted for as the tangent modulus is expressed in terms of the stresses only. In the equivalent continuum approach, the discontinuous rock body is modeled using four-node quadrilateral elements, the properties of each element being defined in terms of some combination of the properties of the intact rock and those of the joints. Non-linearity in the finite element analysis has been incorporated in the form of material non-linearity of both the intact and jointed rock. During loading, if a rock element is found to fail in shear, this is noted, but no changes are effected, and the element is allowed to follow the hyperbolic relation as before, in keeping with the non-linear elastic formulation of the problem. During analysis, it has been found that some elements which fail at one stage recoup at a subsequent higher loading stage. If the elements fail in tension, they are assigned very small values of the elastic modulus for the subsequent loads. In the case of explicit representation of joints, the intact rock is represented by four-node quadrilateral elements and the joints are represented by a two dimensional gap and friction elements. This element is a two-node non-linear interface element used to model node-to-node contact between two bodies with or without friction. It is represented by a pair of coupled non-linear orthogonal springs in the normal and tangential directions to the interface, which are assumed to be very stiff, relative (with stiffness, K n and K t ) to the bodies they are attached to. The Coulomb law is used for friction. Frictionless contact may be modeled by specifying a zero coefficient of friction. The element may assume open or 4

5 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp a closed status on relative displacement in the normal direction. The closed status may be sticking or sliding depending on whether the friction limit μ f n is reached, where μ is the coefficient of friction and f n is the normal compressive force in the gap. Later the model has also been incorporated in the commercial finite difference code First Langrangian Analysis of Continua FLAC (Sitharam and Madhavi Latha, ). A FISH function has been written to incorporate joint factor model with Duncan-Chang non-linear hyperbolic relationships in FLAC. The work verifies the validity of proposed model for different field case studies, namely two large power station caverns, one in Japan and the other in Himalayas and Kiirunavara mine in Sweden. Sequential excavation was simulated in the analysis by assigning null model available in FLAC to the excavated rock mass in each stage. The settlement and failure observations reported from field studies for these different cases were compared with the predicted observations from the numerical analysis in this study. The results of numerical modeling applied to these different cases are systematically analyzed to investigate the efficiency of the numerical model in estimating the deformations and stress distribution around the excavations. Results indicated that the model is capable of predicting the settlements and failure observations made in field fairly well. Results from this study confirmed the effectiveness of the practical equivalent continuum approach and the joint factor model used together for solving various problems involving excavations in jointed rocks. Recently, the applicability of the model has been extended by incorporating the model in the FLAC-D (Sitharam, 5). Initially, it has been validated with simple element tests i.e by simulating -D triaxial testing of rock specimen and is compared with experimental results as well as explicit modeling. Later, the applicability of the model for field problems is investigated by undertaking numerical modeling of the Nathpa Jhakri powerhouse cavern, India, considering -dimensional geometry and stresses along with stages of excavation. 4. Validation of the model 4.. Equivalent continuum model For validation purposes, the finite element analysis has been carried out for single-, multiple- and block jointed rocks (Figure ) of sandstone, granite, Agra sandstone and gypsum plaster using the proposed model. The jointed rock properties are expressed using Equations (6)-(8). The properties of the intact rocks used for the simulation are presented in Table. The results have been plotted in the form of stress strain curves and compared with the experimental results. The stress strain curves compare well with the experimental results for single-, multiple- and block jointed specimens with different confining pressures and joint inclination angles. Some sample stress strain plots for multiple-jointed specimens of Agra Sandstones is shown in Figure 4 and block-jointed specimens of gypsum plaster for different confining pressures shown in Figure 5 along with the experimental results. It can be seen from Figures 4 and 5 that the equivalent continuum model provides a good approximation of the jointed rock behavior. 4.. Comparison of the equivalent continuum model with explicit modeling of joints The results obtained using explicit modeling of joints for single-jointed specimens of sandstone and granite and multiple-jointed specimens of Agra sandstone are compared with the results obtained for the same specimens using equivalent continuum analysis. The comparison of stress strain curves for single-jointed specimens of sandstone is given in Figure 7. The experimental results of Yaji (984) and Arora (987) are also plotted in the figure for comparison. Though only a few comparison plots are shown here, the trend of the results is consistent for different rocks, different joint inclination angle and confining pressure. Figure. Jointed rock specimens and corresponding finite element. Table. Properties of intact rocks used for the numerical modeling. Property Jamrani sandstone Agra Sandstone Gypsum Plaster Mass density, kn/m UCS, MN/m 7 Mod. No. K Mod.Exp. n Cohesion, MN/m Friction, o Classification Hard Rock Very hard rock Soft Rock 4. Validation of the FLAC model 4.. Equivalent continuum model Element tests of jointed rock mass with one to four joints with different orientation and subjected to different confining 4

6 Deviatoric stress (MPa) Deviatoric Stress(MPa) Deviatoric stress (MPa) 44 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp.9-5 pressure have been simulated using the proposed equivalent continuum model. To validate the results of equivalent continuum model, it is compared with the actual laboratory Experimental, joint Experimental, joints Experiemntal, joints Corresponding FLACD-ECM results Corresponding FLACD-explicit results One joint Two joints Three joints 4.. Explicit modeling using FLAC Explicit modeling of jointed rock sample has also been done to know the efficiency of Numerical modeling with FLAC-D. Introduction of interfaces in cylindrical sample FLAC-D is complicated job and creating more than 4-5 interfaces is not recommended. Here, up to four joints have been tried. Figure 6 shows the cylindrical sample with one and three joints as interfaces. The material properties for explicit modeling, assumed to be elastic which is given in Table Axial strain (%) Figure 4. Stress strain plot for different confining pressures (experimental data after Arora, 987). m Block separation to Interfaces introduce interface Figure 6. Cylindrical samples with interfaces σ =.8 MPa σ =6.9 MPa σ =.4 MPa σ =.4 MPa Table 4. Properties of rocks for explicit modeling. Properties Sandstone Agra Sandstone Density, kn/m.5.7 UCS, MPa 7 Cohesion, MPa. 9. Friction, degrees Elas. Mod., GPa 5. Classification Hard rock Very Hard rock Axial Strain (%) Figure 5. Stress strain plot for different confining pressures (experimental data after Brown and Trollope, 97) Experimental, σ =5MPa Experimental, σ =.5MPa Experimental, σ =MPa FLACD Equivalent continuum FLACD Explcit σ =5MPa σ =.5MPa experiment results and also by explicit modeling of joints in rock. Equivalent continuum analysis has been carried out for two types of rocks namely Jamrani sandstone and Agra sandstone for both intact and jointed specimens (Arora, 987; Yaji, 984). The properties of the intact rocks used for the simulation are presented in Table 6. The jointed rock properties are expressed using Equations (), () and (). The stress-strain curves for each load increment has been studied for different values of confining pressure for both intact and jointed specimens with different orientations of joints. The experimental values (Arora, 987) then compared with the numerical results. However, testing was performed in three different confining pressures namely,.5 and 5 MPa. Various Plot for deviatoric stress vs axial strain, at respective confining pressure, for both experimental as well as numerical testing results are shown in Figure 7. It can be seen from the results that the equivalent continuum model developed for the jointed rockmass matches well with the actual experimental test results Axial strain (%) σ =MPa Figure 7. Comparison of numerical and experimental results of sandstone with single joint at 6 degree inclination for,.5 and 5 MPa confining pressure. 5. FIELD CASE STUDIES 5. Finite Element analysis of Shiobara power house cavern The equivalent continuum model developed has been applied for the analysis of a large-scale cavern in jointed rock mass for the Shiobara power station in Japan (Horii et al., 999). This case study was selected for modeling using the equivalent continuum model because the complete field 44

7 Displacement (cm) Displacement (cm) Displacement (cm) T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp BI BI BI5 BI7 BI9 m 5m 8m m 4.7MPa.5MPa 5 BI BI BI4 BI6 BI8 Figure 8 Figure 9 Figure 8. Cross section of cavern and location of displacement transducer Figure 9. Finite element mesh for the cavern and the surrounding rock. the value of the joint inclination parameter n =.5 and the joint strength parameter r is based on the uniaxial compressive strength of the rock and is taken as r =.9. The cavern, along with the surrounding rock, has been analyzed using the above approach at the completion of the whole excavation. The results of the finite element analysis have been plotted in terms of relative displacement at the completion of the whole excavation along the each measurement line. The measured values of relative displacement given by Horii et al. (999 and ) are also plotted for comparison. The relative displacement versus distance from the cavern wall along the measurement lines BI, BI, BI7 and BI8 are as shown in Figures () to () along with the measured values. information, including displacement data, was available for possible analysis and comparison with the proposed model. The Shiobara power station cavern constructed by the Tokyo Electric Company is a large cavern for a pumped storage power station with a maximum output of 9 MW. The rock mass surrounding the cavern was characterized mainly as rhyolite consisting of platy and columnar joints. The cavern was located at a depth of m below the ground level. The cavern (Figure 8) measured 8 m in width, 5 m in height and 6 m in length. The amount of jointed rock mass excavated due to opening of the cavern was estimated to about 9, m. The three in situ principal stresses were recorded as 5.,.9 and.8 MPa. The reported average intact rock compressive strength and elastic modulus were 8. MPa and 4. GPa, respectively. The elastic modulus of the jointed rock mass was in the range of 5 GPa and the strength parameters measured for the jointed rock mass were c = MPa and = 45 o. Multi-point bore hole extensometer (MPBX) data were available at several locations along different measurement lines BI to BI9 (Figure 8) around the cavern for a possible comparison with our model. The cross section of the cavern along with the location of the MPBXs is as shown in Figure 8. The measurement lines along which the field displacements are available are also shown in Figure 8. The jointed rock mass surrounding the cavern has been analyzed by the finite element method using the proposed equivalent continuum approach. Equivalent material properties for jointed rock were modeled using the statistical relations given in Equations (4) and (5). The values for a, b, and c constants used here are the same as the ones listed in Table. Figure 9 shows the finite element mesh ( m m) used for the above problem (Sitharam, ). The problem was analyzed with the initial stresses existing in the surrounding rock. The joint properties, the dip angle and the average spacing of the three dominant joints present in the surrounding rock are given in Table 5. The joint factor for the above problem has been estimated as follows. The number of joints per meter depth in the surrounding rock which have an influence on the cavern determine the joint frequency value J n as 5. Since the most critical joint has a joint orientation of o, Table 5. Properties of joint sets for Shiobara power house cavern. Joint set Dip angle spacing Joint frequency (J n) Joint inclination parameter (n) Joint strength parameter (r) Joint factor (J f) J n/(nr) I 6 R cm II 6 L cm III L cm Observed Values (after Horii et.al) Eq Continuum resullts 5 5 Distance from Cavern wall (m) Figure. Measured (after Horii et al., 999 and ) and calculated relative displacements along the measurement line BI. 5 4 Observed Values (after Horii et.al) Eq Continuum resullts 5 5 Distance from Cavern wall (m) Figure. Measured (after Horii et al., 999 and ) and calculated relative displacements along the measurement line BI 5 4 Observed Values (after Horii et.al) Eq Continuum resullts 5 5 Distance from Cavern wall (m) Figure. Measured (after Horii et al., 999 and ) and calculated relative displacements along the measurement line BI7. 45

8 Displacement (cm) 46 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp Observed Values (after Horii et.al) Eq Continuum resullts minor horizontal virgin stress (Perpendicular to the ore body) in MPa and z m is the mine level in meters (z-axis positive upwards). 5 5 Distance from Cavern wall (m) Figure. Measured (after Horii et al., 999 and ) and calculated relative displacements along the measurement line BI8. 5. FLAC analysis of Kiirunavaara mine,sweden. The Kiirunavaara mine, which is 4 m long, with an average width of 9 m, is located 44 kilometers north of the Arctic Circle in the city of Kiruna in North Sweden and currently produces Mt of magnetic iron ore annually. The mine was initially mined as open pit, starting at the top of Kiirunavaara mountain. Underground mining using sublevel caving is started in 95s. The mine strikes nearly northsouth and dips, on average 6 to the east. The ore body is relatively strong surrounded by competent quartz porphyry on the hanging wall and syenite porphyry on the footwall. The rock mass is relatively jointed, with two to three joint sets occurring in 8 different structural domains of the mine. One joint set is oriented roughly parallel to the ore body as the other two strike obliquely to the ore body. All dip fairly steeply (6 to 9). A representative section of the mine, namely section Y is selected for numerical modeling. The geometry of the mine at this section showing different mining levels is given in Figure 4. The mine has experienced large-scale stability problems. Because of sub level caving, the hanging wall continuously caves in as the ore is mined. Due to this, large subsidence of the ground surface is observed. Instabilities and large-scale failures were observed both in hanging wall and footwall of the mine. Signs of failure were first observed in the footwall in the year 985, with more widespread cracking underground in year 989. The locations where first cracks were observed in 985 are mapped by Sjoberg (999) and are presented in Figure 5. Failure surface can be drawn connecting these failure observations, which can be compared using numerical simulations from present study. Qualitative and quantitative estimation of this instability is very important as the city of the Kiruna and the railroads are located on the hanging wall side of the mine. Several researchers attempted to predict the instabilities using different approaches. Sjoberg (999) simulated the failure using numerical model incorporating Hoek and Brown failure criterion, which involves exact evaluation of various parameters for the equivalent continuum model. The input parameters are selected for the numerical analysis from the observations reported by Lupo (996) as given in Table 6. The input for the virgin stress distribution is taken from the average stress components presented by Sjoberg (999) as: Table 6. Input parameters for numerical modeling of Kiirunavaara mine (after Lupo, 996). Parameter Hanging Foot wall wall Orebody Uniaxial Compressive strength, MPa 4 5 Elastic Modulus, MPa Poisson's ratio..7.5 Density, kg/m Cohesion, MPa..6.6 Angle of internal friction Results from the numerical analysis showed that the type of failure occurred in this model are primarily shear failure. In continuum type of model, since the displacements are continuous, we can not see the development of cracks as observed in the field. Failure can only be observed from the concentration of shear strains in the model. The path of concentrated shear strains represents the failure surface in the model. The failure thus simulated by the numerical model using practical equivalent continuum approach can be compared with the failure observations in the field. Shear failure was observed in the footwall of the model while excavating for the mining level of -586 m, agreeing with the field observations as reported by Sjoberg (999). Typical failure surface for the mining step of 586 m is presented in Figure 5. It can be observed from the figure that failure develops along a curved surface within the foot wall. The figure is zoomed between the excavation steps -65 m to -586 m so as to clearly visualize the failure surface. The shear strains are concentrated near the slope, at a depth of 55 m, which is comparable with the field observations presented in Figure 5. The predicted failure surface matched fairly well with the field observations. Ground Surface -6 m Open pit -9 m - m Hanging Wall -65 m m m 984 Foot Wall -586 m m m 99-7 m Ore body Figure 4. FLAC model for Kiirunavaara mine showing different mining levels. v = -.7z m -.6, H = -.6z m -.6, h = -.4z m -.4. Where v is the vertical virgin stress, H is the major horizontal virgin stress (Parallel to the ore body), h is the 46

9 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp overburden is taken into consideration by applying equivalent amount of pressure at the top. Analysis procedure was similar to FLAC-D as mentioned above accept that here the - dimensinal geometry has been considered and the displacements values measured were consistent with the actual experimented values. Figure 5. Shear strains for FLAC model for 586 m mining level. 5. FLAC Analysis: Naptha Jhakri Power House Cavern Nathpa-Jhakri hydropower project in the state of Himachal Pradesh in India involved a major opening for powerhouse cavern of dimensions 6 m m 49 m (length width height) at a depth of 6.5 m below the ground level (Figure 6). The opening is located in the left bank at about 5 m from the Sutlej river. The rock mass around the cavern is mainly quartz mica schist with joints and discontinuities (Bhasin 996). The measured in-situ stresses are 4.7 and 6.4 MPa in E-W and N-S and directions respectively and 5.89 MPa in vertical direction. The properties of intact rock and joints are given in Tables 7 and 8 respectively (Varadarajan, ). The cavern is excavated in stages. The sequence of excavation with the elevation of each excavation step and the locations of installation of multi point borehole extensometers (MPBX) for displacement measurements are shown in Figure 7. Since the cavern is symmetric, only half of the cavern is analyzed (Sitharam and Madhavi Latha, ). The finite difference grid used for the analysis is of size m 45 m with rectangular zones. The excavation steps are simulated in the numerical analysis and the locations of the installation of extensometers are identified for obtaining the displacements for comparison with the measured displacements from instrumentation of the cavern in field. The variation of displacements with time is also obtained from numerical analysis by solving for equilibrium after each excavation step. Comparison of the observed and predicted deformations along the measurement line at different locations for various excavation levels after the completion of excavation is presented in Table 9. In all the cases, the minimum value of deformation corresponds to the measuring point of the MPBX positioned close to the cavern wall and the maximum value corresponds to the farthest measuring point from the wall. It can be observed from Table 9 that the numerical model is efficient in predicting the deformations around the cavern wall. The variation of displacement with time for a particular excavation step and with progress of excavation steps is well represented in the analysis. Analysis of the cavern has also been extended to FLAC- D for better understanding of stress and deformational behavior of the cavern. For the -D numerical simulation, the grid selected is shown in Figure 7. As shown in Figure 7, only 4 m overburden is taken into model, rest m Figure 6. Excavation sequence and locations of extensometers for Nathpa-Jhakri powerhouse cavern. Table 7. Properties of intact rock for Nathpa-Jhakri powerhouse cavern. Modulus number K 45 Modulus exponent n.4 Cohesion, MPa 6.8 Poisson ratio.9 Angle of internal friction, degrees Table 8. Properties of joints for Nathpa-Jhakri powerhouse cavern. Frequency of joints J n 5 Orientation of critical joint 55 Inclination parameter n.8 Joint strength parameter r.6 Joint factor J f.9 47

10 89m 45m 48 T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp.9-5 Table 9 Measured and predicted deformations at for Nathpa- Jhakri cavern. Excavation Deformation along the Location of level line (mm) From Stage To MPBX EL, at EL, Observed Predicted EL, m m m (A) (B) (B) -. to (C) (D) (E) Figure 7. D Grid of the power house cavern with FLAC-D (All dimensions are in meters). 5.4 Seismic Stability Analysis of A River Abutment Slope The Chenab river forms about 6 m deep gorge in a V- shaped valley in the area between Bakkal and Kauri villages (Figure 8). The railway link between Katra and Laole section in Jammu and Kashmir has been planned to have a steel arch bridge on Chenab river with around 95 m span at a height of 6 m. The rocks present at the bridge site are heavily jointed. The stability analysis of the right abutment slope at Kauri side of Chenab river between Katra and Laole, in Jammu and Kashmir, India was simulated using FLAC as a plane strain problem. Seismic stability analyses have been done using pseudostatic approach and also by applying complete time history of a real earthquake. The assessment of slope stability of large slopes in jointed rocks requires the use of non-linear material models. Here, the concept of joint factor (J f ) along with hyperbolic model has been used for the study. The subsurface essentially consists of dolomitic limestone with different degrees of weathering and fracturing. The rock mass ratings for surface weathered and fractured dolomitic limestone are given in Table. The intact rock properties of the right abutment of the slope are presented in Table. Calculated equivalent rock mass properties, cohesion.785 MPa, friction angle o, Young s modulus 4.4 GPa, and Poisson s ratio.5. The Hoek and Brown Parameter m for broken rock.59 where as the s value calculated for RMR of 4 is.7. Table. The ratings for surface weathered and fractured dolomitic limestone (NIRM Report 4). Parameter Fractured Surface weathered dolomitic dolomitic limestone limestone RQD 49 Q 6..5 RMR 48 5 GSI 4 7 J v 6 Table. Intact rock properties for the slope at right abutment (NIRM Report 4). Property Average value Density, kg/m 76 Young s modulus, GPa 65 Poisson s ratio.5 UCS, MPa 5 Cohesion, MPa Friction, degrees.76 Hoek & Brown parameter, m.5 The slope is simulated here, with FLAC in plane strain using small strain mode. Relatively high discretization with approximately 6 zones for a slope height of 59 m is considered in the analyses. Horizontal displacements are fixed for nodes along the left and right boundaries while both horizontal and vertical displacements are fixed along the bottom boundary. 6m Kauri side Right abutment Bakkal side Left abutment 8m 69m Figure 9 Numerical mesh of the slope using FLAC. Figure 8. Slope at the Chenab river site. Slope stability analysis was done with the shear strength reduction technique, where the simulations are run for a series of trial factor of safety F trial with c and adjusted according to the equations: 48

11 Acceleration (g) T. G. SITHARAM / International Journal of the JCRM vol.5 (9) pp c trial trialc (4) F.5.5 trial tan trial tan F (5).5.5 Corrected transverse component of acceleration The value F trial at which slope fails is found efficiently in FLAC using bracketing and bisection method. The static factor of safety for the slope with rock mass properties of cohesion.785 MPa and friction o is found to be.86. Analyses represent the effects of earthquake shaking by pseudostatic accelerations that produce inertial forces, which act through the centroid of the failure mass. The magnitude of the horizontal pseudo static force is a W F h h k W g h (6) Where, a h is horizontal pseudostatic accelerations, k h is dimensionless horizontal pseudostatic coefficients, and W is the weight of the failure mass. The horizontal pseudostatic forces are assumed to act in directions that produce positive driving moments. With the values of cohesion.785 MPa and friction o, the factor of safety for the peak a h =. was found.. Figure shows the maximum shear strain rate for the slope along with velocity vectors. Failure surface is almost circular from top of the slope to the toe. Figure shows the plasticity indicator plot of the slope. They reveal those zones in which the stresses satisfy the yield criterion. A failure indicated by the contiguous line of active plastic zones that join two surfaces. The yielded zone due to shear can be seen in the Figure and as observed, it also follows a circular/non-circular failure surface. Plastic Zone In tension Elastic Zone Figure Figure Figure. Critical failure surface. Figure. Plasticity indicator plot depicting the yielded zones. To carry out the full dynamic analysis with the real earthquake records, the corrected transverse component of acceleration time history was applied (Figure ) at the base of the slope. Free field boundary has been used to minimize the wave reflection. It has been observed that the overall effect of continuous inertial forces may lead to accumulation of the displacement of a particular section of the slope. Once the applied ground motions generated due to inertia forces have ceased, no further deformation has occurred, as there is no marked loss in the strength of rock. The variation in displacements along the slope values were recorded and studied for different Rayleigh damping values Time (s) Figure The corrected transverse component of acceleration time history of Uttarkashi earthquake, Oct,99 :5 IST, used in the study (Data from Dept. of Earthquake Engg., IIT Roorkee). 6. CONCLUDING REMARKS The paper presents the summary of the work on jointed rock mass along with development of equivalent continuum model, validation and field case studies. The statistical equations for representing the jointed rock mass properties are very simple and give a fair estimate of jointed rock properties in the absence of reliable experimental data. The accuracy of the estimation of the jointed rock properties depends upon the estimation of joint factor. Joint factor is very important for the description of jointed rock and to find the correlation between the parameters and the geological data. Since the database covers a wide range of rock properties the statistical relationships arrived more or less give a good estimate of uni-axial compressive strength and elastic modulus for all rock types. Jointed rock has been successfully modeled as an equivalent continuum whose properties represent the properties of the jointed rock. It can be seen from the results that the equivalent continuum model developed works well for single-, multiple- and block-jointed rock with different joint fabric and joint orientation under a wide range of confining pressures. In the equivalent continuum model, the jointed rock properties are expressed as a simple function of intact rock properties and joint properties. As the joints are not modeled separately, the analysis becomes simpler as highly complex joint fabrics can be modeled using a simple finite element mesh / finite difference (FLAC) unlike explicit modeling. The major advantage of explicit modeling of joints is that the mode of failure in the jointed rock mass can be reasonably estimated and the zones most susceptible to failure can be identified which is not possible in the equivalent continuum modeling. Explicit modeling of joints is efficient only when the jointed rock mass has few major joints. Equivalent continuum analysis can be applied to a single-jointed rock mass through to heavily jointed rock mass effectively without compromising the accuracy of the results. Moreover, the input data required for the equivalent continuum model are minimal. It can be inferred from the results that the equivalent continuum model developed provides a reasonable estimate of rock mass behavior in the absence of detailed experimental data. The only input data required for the analysis are the properties of intact rock and the joint properties for estimating the joint factor. It can be concluded from the analysis of various case studies that the 49