ARTICLE IN PRESS International Journal of Rock Mechanics & Mining Sciences 41 (2004) 813–832 Dynamic stability analysis of jointed rock slopes using the DDA method: King Herod’s Palace, Masada, Israel Y.H. Hatzora,*, A.A. Arzib, Y. Zaslavskyc, A. Shapirad a Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel b Geotechnical Consulting, 80 Krinizi st., Suite 23, Ramat-Gan 52601, Israel c Seismology Division, Geophysical Institute of Israel, Lod, Israel d The International Seismological Center, Pipers Lane, Berkshire RG194NS, UK Accepted 19 February2004 Abstract A dynamic, two-dimensional, stability analysis of a highly discontinuous rock slope is demonstrated in this paper. The studied rock slope is the upper terrace of King Herod’s Palace in Masada, situated on the western margins of the seismicallyactive Dead Sea Rift. The slope consists of sub-horizontallybedded and sub-verticallyjointed, stiff, dolomite blocks. The dynamicdeformation of the slope is calculated using a fullydynamicversion of DDA in which time-dependent acceleration is used as input. The analytically determined failure modes of critical keyblocks in the jointed rock slope are clearly predicted by DDA at the end of the dynamic calculation. It is found however that for realistic displacement estimates some amount of energy dissipation must be introduced into the otherwise fullyelastic, un-damped, DDA formulation. Comparison of predicted damage with actual slope performance over a historic time span of 2000 years allows us to conclude that introduction of 2% kinetic damping should suffice for realistic damage predictions. This conclusion is in agreement with recent results of Tsesarskyet al. (In: Y.H. Hatzor (Ed.), Stability of Rock Structures: Proceedings of the Fifth International Conference of Analysis of Discontinuous Deformation, Balkema Publishers, Lisse, 2002, pp. 195–203) who compared displacements of a single block on an inclined plane subjected to dynamic loading obtained byDDA and byshaking table experiments. Using dynamic DDA it is shown that introduction of a simple rock bolting pattern completely stabilizes the slope. r 2004 Elsevier Ltd. All rights reserved. Keywords: Rock slope stability; DDA; Earthquake engineering; Site response; Damping 1. Introduction DDA predictions with actual terrace performance over historic times. Thus, the amount of energydissipation In this paper, we applya dynamicdiscontinuous and related effects associated with shaking of a real deformation analysis (DDA) to a real jointed rock slope jointed rock slope maybe estimated, and the appro- which has sustained a relativelywell known record of priate values can be used for realistic dynamic modeling earthquakes over the past 2000 years—the upper terrace of jointed rock slopes using DDA. of King Herod’s Palace in Masada, situated along the western margins of the seismicallyactive Dead Sea rift valley. Since we know the terrace has sustained tremors 1.1. The numerical DDA within a reasonablyestimated range of intensityin documented historic events, we have a good constraint The DDA method [1], is similar in essence to the finite on dynamic DDA predictions, from the field. In element method (FEM). It uses a finite element type of particular, the amount of required energydissipation mesh but where all the elements are real isolated blocks, in DDA, or ‘‘damping’’, can be explored bycomparing bounded bypre-existing discontinuities. DDA however is more general since blocks can be of anyconvex or *Corresponding author. Tel.: +972-8-647-2621; fax: +972-8-647- concave shape. When the blocks are in contact 2997. Coulomb’s law applies to the contact interface and the E-mail address: [email protected] (Y.H. Hatzor). simultaneous equilibrium equations are formulated and 1365-1609/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2004.02.002 ARTICLE IN PRESS 814 Y.H. Hatzor et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 813–832 solved for each loading or time increment. In the FEM where method the number of unknowns is the sum of the 2 3 ðy À y0Þ degrees of freedom of all nodes. In the DDA method the 6 10Àðy À y0Þðx À x0Þ 0 7 6 2 7 number of unknowns is the sum of the degrees of ½Ti¼ : 4 ðx À x Þ 5 freedom of all the blocks. Therefore, from a theoretical 01ðx À x Þ 0 ðy À y Þ 0 0 0 2 point of view, the DDA method is a generalization of the FEM [2]. ð3Þ DDA considers both statics and dynamics using a This equation enables the calculation of displace- time-step marching scheme and an implicit algorithm ments at anypoint ðx; yÞ within the block when the formulation. The difference between static and dynamic displacements are given at the center of rotation and analysis is that the former assumes the velocity as zero in when the strains (constant within the block) are known. the beginning of each time step, while the latter inherits In the two-dimensional formulation, the center of the velocityof the previous time step. rotation with coordinates ðx0; y0Þ coincides with the The formulation is based on minimization of potential block centroid with coordinates ðxc; ycÞ: Assuming n energyand uses a ‘‘penalty’’method to prevent blocks constitute the block system, the simultaneous penetration or tension between blocks. Numerical equilibrium equations are written in a matrix form as penalties in the form of stiff springs are applied at the follows: contacts to prevent either penetration or tension 2 32 3 2 3 between blocks. Since tension or penetration at the K11 K12 K13 ÁÁÁK1n D1 F1 6 76 7 6 7 contacts will result in expansion or contraction of the 6 76 7 6 7 6 K21 K22 K23 ÁÁÁK2n 76 D2 7 6 F2 7 springs, a process that requires energy, the minimum 6 76 7 6 7 6 K31 K32 K33 ÁÁÁK3n 76 D3 7 6 F3 7 energysolution is one with no tension or penetration. 6 76 7 6 7 6 ÁÁÁÁÁÁÁ76 Á 7 ¼ 6 Á 7; ð4Þ When the system converges to an equilibrium state, 6 76 7 6 7 6 76 7 6 7 however, there are inevitable penetration energies at 6 ÁÁÁÁÁÁÁ76 Á 7 6 Á 7 6 76 7 6 7 each contact, which balance the contact forces. Thus, 4 ÁÁÁÁÁÁÁ54 Á 5 4 Á 5 the energyof the penetration (the deformation of the Kn1 Kn2 Kn3 ÁÁÁKnn Dn Fn springs) can be used to calculate the normal and shear contact forces. where each coefficient Kij is defined bythe contacts Shear displacement along boundaries is modeled in between bocks i and j: Since each block ðiÞ has six DDA using the Coulomb–Mohr failure criterion. By degrees of freedom defined bythe components of Di adopting first-order displacement approximation the above, each Kij is in itself a 6  6 sub-matrix. Also, each DDA method assumes that each block has constant Fiand Di are 6  1 sub matrices where Di represents the stresses and strains throughout. deformation variables ðd1i; d2i; d3i; d4i; d5i; d6iÞ of block i; and Fi is the loading on block i distributed to the six deformation variables. Sub matrix ½Kii depends on the 1.2. Basic mathematical formulation material properties of block i and ½Kii; where iaj; is defined bythe contacts between block i and j: For complete mathematical formulations refer to Shi The system of equations above (4) can also be [1,2]; here onlythe most basic equations are summar- represented in a more compact form as KD ¼ F where ized. The displacements ðu; vÞ at anypoint ðx; yÞ in a K is a 6n  6n stiffness matrix and D and F are 6n  1 block i can be related in two dimensions to six displacement and force vectors, respectively. Hence, in displacement variables: total, the number of displacement unknowns is the sum of the degrees of freedom of all the blocks. Note that in T Di ¼ðd1;i; d2;i; d3;i; d4;i; d5;i; d6;iÞ essence the system of equations (4) is similar in form to T ¼ðu0; v0; r0; ex; ey; gx;yÞ ; ð1Þ that in the FEM [3]. The solution of the system of equations is constrained bythe systemof inequalities associated with block where ðu0; v0Þ are the rigid bodytranslations at a specific kinematics (no penetration and no tension between point ðx0; y0Þ within the block, r0 is the rotation angle of blocks) and Coulomb friction for sliding along block the block with a rotation center at ðx0; y0Þ; and ex; ey; interfaces, which is the main source of energyconsump- and gxy are the normal and shear strains in the block. The complete first-order approximation of block dis- tion. The final displacement variables for a given time placements proves to take the following form [1,2]: step are obtained in an iterative process (see [1–3] for ! details). u The equilibrium equations [1,2] are established by ¼½Ti½Di; ð2Þ minimizing the total potential energy P produced bythe v forces and stresses. The ith row of Eq. (4) consists of six ARTICLE IN PRESS Y.H. Hatzor et al. / International Journal of Rock Mechanics & Mining Sciences 41 (2004) 813–832 815 linear equations: MacLaughlin [7], the dynamic solution for a single block qP on an incline subjected to constant gravitational ¼ 0; r ¼ 1-6; ð5Þ acceleration was repeated using the new version of qdri DDA [5]. For a slope inclination of 22.6, four dynamic where the dri is the deformation variable of block i: For displacement tests were performed for interface friction block i the equations: angle values of 5,10,15, and 20. The agreement qP qP between the analytical and DDA solutions was within ¼ 0; ¼ 0 ð6Þ qu0 qv0 1–2%.