Earthquake response analysis of rock-fall models by discontinuous deformation analysis

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1 c Earthquake response analysis of rock-fall models by discontinuous deformation analysis T. Sasaki, I. Hagiwara & K. Sasaki Rock Engineering Lab., Suncoh Consultants Co. Ltd., Tokyo, Japan R. Yoshinaka Saitama University, Saitama, Japan Y. Ohnishi & S. Nishiyama School of Urban & Environment Engineering, Kyoto University, Kyoto, Japan Keywords: rock-fall, DDA, seismic response analysis, dynamic problem ABSTRACT: This paper describes rock-fall models of rock slopes caused by earthquakes using discontinuous deformation analysis (DDA). It introduces viscous damping and the velocity-energy ratio for the rock-fall models based on DDA theory. This paper proposes a method of inputting the earthquake accelerations in DDA and examines the vibration characteristics of the rock slopes and the rock-falls of homogenous and two-layered models. The results show qualitative agreement, and so the methods are applicable to physical phenomena. INTRODUCTION Earthquakes are known to trigger rock-falls at rock slopes and earthquakes are common in Japan. However, it is very difficult to distinguish which rocks will fall in investigations before earthquakes and it depends on the experience and judgment of the investigator. An earthquake response analysis by DDA by Shi (00) evaluated the stability of Yucca Mountain tunnels combined with block theory. In this case, the earthquake acceleration is input directly for each block by inertia as the body forces. Hatzor et al. (00) compared an analytical solution with the block DDA model, and showed that the analytical error of DDA ranges from 5% to 0%. They also recommended introducing a 5% damping coefficient in the Mount Masada monument model. Zaslavsky et al. (00) pointed out that the spectral response of an earthquake at the parts where the topography changes, such as the edge of a cliff or the entrance of a tunnel, is different from that estimated using a point of the usual base rock. Ishikawa et al. (00) analyzed dynamic characteristics of the ballast of railway foundations and compared the results with experiments. Tsesarsky et al. (00) analyzed the frequency characteristics of a single block and compared the results with an experiment using a single block in the slope model. Nishiyama et al. (003) analyzed the stability of a masonry-type retaining wall by dynamic response analysis. These researches can be classified into two categories by the input method of an earthquake wave: the acceleration time history or the displacement time history. The present study investigates optimum values of the parameters in some actual rock-fall problems of natural large rock slopes by using DDA (Shi, 984), focusing on the input of earthquake acceleration time history, and estimates the energy of the falling rocks depending on the efficiency of countermeasure structures. OUTLINE OF THEORY The governing equation of the potential energy sys Π on large deformations of continuous and discontinuous elastic bodies is given by: Π Π Π Π n n m sys block i i i j ( ) + PL, i i j () The first term on the right side of equation () is the potential energy of the continuum part, and the second term is the potential energy of the contact between blocks. The first term is given by: i Π F x, F( x, * [ ij ij ijδ( ik kj k, i k j) ] c ρ ( 0 τ δd σ D D v v, V ρ Γ t udγ F( x, V dv [ ρ( b& u&& ) cu& ] dv () 67

2 The first term of equation () is the strain energy of velocity field, the second term is the surface traction energy, and the third term is the energy of the inertia force and damping force, where, F ( x, : shape function,ρ 0 : density before deformation, ρ c : density after deformation τ * ij : Kirchhoff stress velocity,dij : deformation velocity tensor, σ ij : Cauchy stress, &&u : acceleration, &u : velocity, ρ : unit mass, b: body force, c: viscosity coefficient, t :surface traction force, V: volume of body, and Γ : area of body. The second term on the right side of equation () is the potential energy of the contact between discontinuous planes, and is evaluated by the least squares method by using a penalty as follows: i, j j i Π PL k N [( u u) n] kt[ ut j u i T ] (3) where, k N : penalty coefficient of the normal direction, k T : penalty coefficient of the shear direction, j i ( u u) n : amount of penetration in the normal direction, u T : amount of slip in the shear direction, n: direction cosine of the contact plane. DDA (Shi, 984) is formulated from equation () using the kinematic equations based on Hamilton s principle and minimized potential energy expressed by: Mu&& + Cu& + Ku F (4) where, M: mass matrix, C: viscosity matrix, K stiffness matrix, F external force vector, &&u acceleration, &u velocity, andudisplacement of a block center. The viscosity matrix C in equation (4) can be rewritten as follows in terms of viscosity η and mass matrix M: C η M The physical meaning of viscosityη is the damping of the rock itself, the viscosity of air around the rock surfaces and the vegetation on the surface of a rock slope. The kinematic equation (4) is solved by Newmark s and method (Hilbert,993) by using parameters 0.5 and.0, and the algebraic equation for the increase in displacement is solved for each time increment by the following three equations: ~ ~ K u F (6) where, ρ [ 0 K ρ 68 ~ K η M + M + e K s c + ] (7) ~ F M u& + ( F σdv) Mα ( t) (8) where, u incremental displacement, K e stiffness matrix of linear term,k s initial stress matrix caused by rigid rotations α(t) time history of earthquake acceleration. The relations between displacements, velocities and accelerations at an arbitrary point of a block at time t in step i are expressed by the following three equations, respectively. u i u [ ] [ D( t)] t [ D ( t )] t [ D ( t )] + t [ D( t )] t [ ] [ ] u& & i i u&& i (9) (0) [ D ( t )] t [ ] u& [ ] i 3 NUMERICAL EXAMPLES [ D ( t )] + t () The purpose of numerical studies of rock-fall models is to distinguish the candidates of falling rocks on the rock slopes and to evaluate the applicability of the earthquake response analysis by using DDA, and the method of inputting earthquake accelerations. There are two methods in DDA of inputting the earthquake response as described above: displacement time response and acceleration time response, as indicated by instruments. Analytically, it is simpler to use the displacement time response record for input in DDA, however, generally, it is (5) the acceleration time response of an earthquake that is recorded. The authors examined and analyzed both input methods and compared the results. They propose giving the acceleration response at the base points of the rock slope block, which corresponds with the large virtual mass at the base block of DDA models. 3.Homogeneous Rock Slope Model Figure shows the rock-fall models on a rock slope in an earthquake. The height of the slope is assumed to be 00 m, and ten rock blocks exist at the top of the slope. Table shows the analytical conditions and material properties. The time interval used for the numerical calculation is 0.00 second. The input

earthquake wave is EL-Centro East-West and Up-Down at the bottom point of the base block of the rock slope at the same time.

Two values of the surface friction angle of the rock slope are assumed, 35 and 0 degrees, and the results for both are compared.

The characteristic frequency of the rock slope is about Hz, and that of rock-falls ranges from 0 to 0 Hz.

The result depends on the penalty value, and in this case was equivalents 0% of the elastic modulus of the block.

3 earthquake wave is EL-Centro East-West and Up-Down at the bottom point of the base block of the rock slope at the same time. The boundary conditions are defined such that the horizontal roller is at the bottom and free at both sides of the base block. Two values of the surface friction angle of the rock slope are assumed, 35 and 0 degrees, and the results for both are compared. The elastic modulus of the rock slope is assumed to be GPa and the penalty coefficient to be 0 GN/m 3. The characteristic frequency of the rock slope is about Hz, and that of rock-falls ranges from 0 to 0 Hz. These are approximately the same as the single mass analytical solutions and the influence of the penalty coefficient value shifts towards the high-frequency side. The result depends on the penalty value, and in this case was equivalents 0% of the elastic modulus of the block. In order to get the same value between input and output acceleration wave proportions, the mass of the base block was defined as 0,000 times to avoid the influences for the response by an additional mass of base block, and the body force acting downward was set to zero to eliminate up-down free vibrations of the slope block as shown in Table. Static analysis of this model shows that the blocks on the 0-degree slope do not move because the friction angle of the rock surface is defined 35 degrees. block of EL-Centro earthquake. Figure 4 shows the acceleration Fourier spectrum at the input point, which perfectly coincides with the input acceleration proportion. Figure.Acceleration time history of EL-Centro earthquake Figure 3.Acceleration response at input point of base block Figure. Rock slope model of homogeneous strata Table.Material properties of the models Time interval Name of input earthquake 0.00sec EL-Centro-EW,UD Elastic modulus, Poisson s Ratio Friction angle of the rock surface GPa,ν 0.5 φ 35 and 0 degrees Penalty coefficient Viscosity coefficient Velocity / Energy Ratio 0GN/m Unit mass of the base block 5000kN/m 3 (Virtual) Unit mass of the rock blocks 5kN/m 3 (Actual) Figure and Figure 3 show the acceleration time history and response at the input point on the base Figure 4. Fourier spectrum at the input point Figure 5. Rock-fall after 5 seconds (φ 35 degrees) 69

Figure 5 shows the result when the rock surface friction angle is 35 degrees, 5 seconds after the start of earthquake motion.

Figure 6 shows the position of falling rocks after 0 seconds.

The blue and red traces in the figure show a falling rock (3) contacting the representative slope in the central part of a group of falling rocks, and a falling rock (9) from the top.

4 Figure 5 shows the result when the rock surface friction angle is 35 degrees, 5 seconds after the start of earthquake motion. Blocks fall on the 0-degree slope and receive acceleration from the base block for. seconds. After that, about 50% of blocks move on the 45-degree slope repeated with collisions as shown in Figure5. Figure 6 shows the position of falling rocks after 0 seconds. The front of the group of falling rocks arrives at the bottom of the rock slope, and all the following falling rocks also arrive there on a steep slope side. The blue and red traces in the figure show a falling rock (3) contacting the representative slope in the central part of a group of falling rocks, and a falling rock (9) from the top. Figure 7 shows the position of falling rocks after 5 seconds. Most of the falling rocks arrive at a flat area of a lower part of the slope. In addition, a large mass of a falling rock (3) reaches the top. Figure 8 shows the acceleration response of the falling block (9) at the start of the earthquake. Figure 9 shows the velocity response of the falling rock (9). The vertical axis shows the falling rock velocity, and the horizontal axis shows the X coordinate along the slope in the horizontal direction. The falling blocks receive horizontal and vertical accelerations from the base rock slope block before sliding on the low-angle slope at first. When a falling rock begins to slide on a slope, the falling movement by gravity influences it, and the influence of the earthquake wave diminishes. In addition, acceleration by the earthquake from the slope is propagated again when the falling rock arrives at the flat area of the end of slope. In the EL-Centro wave, the level of the vertical acceleration component is large in the front and back parts, but the horizontal acceleration component reduces after 0 seconds to around /3, hence the response of falling rocks reflects the acceleration after 0 seconds. The speed of block rotation is lower than 0% of the plumb speed on the steep slope side and is comparatively small, because the block is flat and stable, but there is an influence of the earthquake wave from the slope on the low-angle slope before sliding and the flat area of the end of slope and the speed of the rotation component increases to 30% compared with natural rock-fall. Figure 0 shows the positions of falling rocks after 5 seconds in the case of 0-degree friction angle. In this case, the arrival distance in the flat area of the falling rocks is large compared with the case of 35 degrees, and so the jump at the bottom of the slope is small. 3.Two-layered Model Generally, a bedrock slope is not uniform and tends to present a multilayered structure. Figure shows the positions of falling rocks after five seconds in a model when the base is on a slope. The earthquake wave is input at the center of the bottom end of the base block. Figure 6. Rock-falls after 0 seconds ( 35 degrees) Figure 7. Rock-falls after 5 seconds ( 35 degrees) Figure 8. Acceleration response of block (9) Figure 9.Velocity along horizontal coordinate of block (9) 70

5 As for the dimensions of the model, the right and left size are doubled to give a symmetrical part of case. The input earthquake wave and bedrock properties of matter are more similar to the single strata case. Figures and 3 show the state after 0 seconds and the arrival position of falling rocks after 5 seconds. The arrival distance of the falling rocks is larger than in the single strata case. Figure 4 shows the vibration property for the slope of block No. itself. This result assumes a damping coefficient of %, and the characteristic frequency of the slope is around 3 Hz, which almost matches a one-spring mass model of the theoretical solution. Figure 5 shows the vibration property of a falling rock of block No.0 under gravity. The characteristic frequency of the falling rocks is around 0 0 Hz. Figure 6 shows the acceleration response of the slope block (). Figures 7 and 8 show the velocity response of the falling rock. In comparison with the single strata case, the velocities are about 0% faster. In addition, the velocity is affected by the vibration of the slope block as shown in Figure 6 while a falling rock passes by on the slope block. As a result, the vertical vibration is large, and so the arrival distance of the falling rocks increases after arrival at the bottom of the slope. Figure 3. Rock-falls after 5 seconds Figure 4.Vibration characteristics of the slope under gravity Figure 0. Rock-falls after 5 seconds ( 0 degrees) Figure 5.Vibration characteristics of a rock under gravity Figure. Rock-falls after 5 seconds Figure. Rock-falls after 0 seconds Figure 6. Acceleration response of the slope of block () 7

6 Figure 7.Velocity along horizontal coordinates of block (4) property of the slope block in two levels. In future, we will clarify the frequency characteristic of an input earthquake wave and the vibration characteristics, and the relation with the material property of the slope and falling rocks. The authors examined the basic vibration characteristics of a slope model of multi-layer ground, and its applicability, by using the presented methods. The results must be confirmed with actual vibration characteristics by earthquake wave records. ACKNOWLEDGMENT The authors thank Dr. Gen Hua Shi for many informative discussions. Figure 8.Velocity along horizontal coordinates of block (0) 4 CONCLUSIONS In this study, the authors presented an earthquake response analysis method and used it to analyze the slope stability of two kinds of models by DDA. An external earthquake force was shown to trigger rock-falls. In order to get the same response between input and output acceleration of earthquake record, the authors employed a large number of virtual mass for the base block to avoid the influences by the additional mass of the base block. And to adjust the base block mass, we can be control the characteristics of vibrations of rock slope with zero gravity force. The vibration characteristics of the falling blocks are governed by its mass and the gravity forces principally and the secondary, the frequency characteristics of earthquake accelerations and the friction angle of the block surfaces are influenced for the block motions during earthquakes. The authors also compared the characteristic behavior of falling rocks with varying friction angle in the model, and showed that the model can qualitatively express physical phenomena reasonably well. In the case of the two strata model, the velocities of the falling rocks and arrival distance were larger than with the single strata model due to the vibration REFERENCES Committee of Rock Mechanics, 999. Site Investigation and Stabilization Methods for Rock SlopesJapan Society for Civil Engineering, pp Hatzor Y. H., Arzi A. A. & Tsesarsky M., 00. Realistic dynamic analysis of jointed rock slopes using DDA, Proc. of ICADD-5, BALKEMA, pp Hilbert, L. B. JR, et. al., 993. A new discontinuous finite element method for interaction of many deformable bodies in geomechanics, U. C. Berkeley. Ishikawa T., Sekine E. & Ohnishi Y., 00. Shaking table tests of coarse granular materials with discontinuous analysis, Proc. of ICADD-5, BALKEMA, pp Nishiyama S., Ohnishi Y. et al Study on stability of retaining wall of masonry type by using discontinuous deformation analysis, Proceedings of Japan Society for Computational & Engineering Science, Vol. 8, pp Ohnishi Y., Yoshinaka R., Sasaki T. & Ishii D., 995. Stability analysis of rock foundation by discontinuous deformation analysis, Proc. ISRM Tokyo Congress, Technical Session (Rock foundation). Sasaki T., Sasaki K., & Yoshinaka R., 00. Study of rock stone falling by discontinuous deformation analysis, Proceedings of Japan Society for Computational & Engineering Science, Vol. 7, pp Shi G. H., 989. Block system modeling by discontinuous deformation analysis, Univ. of California, Berkeley, Dept. of Civil Eng. August. Shi G. H., 00. Single and multiple block limit equilibrium of key block method and discontinuous deformation analysis, Proc. of ICADD-5, BALKEMA, pp Tsesarsky M. & Hatzor Y. H., 00. Dynamic block displacement prediction validation of DDA using analytical solutions and shaking table experiments, Proc. of ICADD-5, BALKEMA, pp Zaslavsky Y., A. Shapira & Arzi, A. A., 00. Earthquake site response on hard rock-empirical study, Proc. of ICADD-5, BALKEMA, pp

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