3D DDA vs. analytical solutions for dynamic sliding of a tetrahedral wedge

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1 ICADD9, Nanyang Technological University, Singapore 5 7 November 9 3D DDA vs. analytical solutions for dynamic sliding of a tetrahedral wedge D.Bakun-Mazor, Y.H. Hatzor,, and S.D. Glaser 3 Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel Department of Structural Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel. 3 Department of Civil and Environmental Engineering, University of California Berkeley, CA. USA. ( of corresponding author: bakun@bgu.ac.il) INTRODUCTION In this research, the validity of the three dimensional Discontinuous Deformation Analysis (3D-DDA) ] is examined using analytical solutions for three dimensional problems involving two different failure modes: ) dynamic sliding of a single block on an inclined plane, and ) dynamic sliding of a tetrahedral wedge simultaneously on two faces. From the early nineties, researchers in the DDA community have documented the accuracy of the original two-dimensional method (D-DDA) by performing validation studies with respect to analytical solutions, by comparison with results of other numerical techniques, and from laboratory and field data. A paper by MacLaughlin et al. ] contains a summary of nearly published quantitative validation studies. With respect to dynamic loading and response, several works e.g. 3, 4] have calibrated D-DDA results with respect to the Newmark method 5] and the Goodman and Seed 6] solution. Wartman et al. 7] investigated the analytical implementation of the Newmark method and Goodman and Seed solution with laboratory tests, using physical tests of a block sliding on a tilting and shaking table. Tsesarsky et al. 8] used Wartman s data to explore the validity of the D-DDA results for dynamic loading. As expected from numerical forward modelling analysis, the input parameters, such as the contact spring stiffness, the boundary conditions, and time interval, have a decisive influence on the accuracy of the output results. Recently the validity and accuracy of 3D-DDA has been explored, yet only preliminary or partial work on this subject has been published to date 9-4]. The reason may be due to the difficulty in developing a complete contact theory that governs the interaction of many 3D blocks 4]. Considering 3D validations, Shi ] reports very high accuracy for two examples of block sliding modelled with 3D-DDA, subjected to gravitational load only. Moosavi et al. ] compare 3D-DDA results for dynamic block displacement with an analytical solution. Yeung et al 3] validate the wedge stability analysis method using physical models and field case histories, and report a good agreement between physical and numerical results in terms of both the effective failure mode and the block displacement history, although no quantitative comparison between 3D-DDA and lab test results is reported. In this study, an independent mathematical solution for dynamic block sliding in 3D is developed based on the vector analysis (VA) formulation presented by Goodman and Shi 5]. The developed 3D solution employs static formulation of the force balance on the block at each time-step, according to the assumed sliding mode. The incremental sliding force or acceleration thus calculated is integrated numerically twice to yield the three displacement components (x, y, z) versus time t 6]. We first compare the developed VA and existing Newmark solutions, and then proceed with the developed 3D VA solution in validation of 3D-DDA ]. ANALYTICAL FORMULATIONS OF VECTOR ANALYSIS SOLUTION. Limit Equilibrium Equations The static limit equilibrium equations formulated for each time step are discussed in this section for both single face and double face sliding. Note that the expected failure mode must be known in advance to formulate these equations. Furthermore, in all cases studied here the static and dynamic resultant forces are applied to the centroid of the sliding block, this is slightly in contrast to the physical reality where the input motion is applied to the foundation upon which the block rests... Single Face Sliding A typical model of a block on an incline is illustrated in Fig. (a). The dip and dip direction angles are, α = and β = 9, respectively. Although it is a simple D problem, the model is plotted as if it were 3D to

2 ICADD9, Nanyang Technological University, Singapore 5 7 November 9 demonstrate the robust VA solution. For this purpose, a Cartesian coordinate system (x,y,z) is defined where X is horizontal and points to east, Y is horizontal and points to north, and Z is vertical and points upward. The normal vector of the inclined plane is: nˆ nx, ny, nz ], where: nx sin( )sin( ) n sin( )cos( ) () n y z cos( ) Fig. : Typical models of sliding blocks in 3D coordinate system. (a) Block on single face inclined to /9, (b) a tetrahedral wedge sliding on two faces, the line of intersection is inclined 3 below North and inclination of the boundary faces are 5/63 and 5/96. The force equations presented below refer to a block with a unit mass. Hence, these equations can be discussed in terms of accelerations. The resultant force vector that acts on the system at each time-step is r rx, ry, rz ]. The driving force vector that acts on the block ( m ), namely the projection of the resultant force vector on the sliding plane, at each time step is: m ( nˆ r) nˆ () The normal force vector that acts on the block at each time step is: p ( nˆ r) nˆ (3) At the beginning of a time step, if the velocity of the block is zero then the resisting force vector due to the interface friction angle is: tan( ) p mˆ, tan( ) p m f (4) m, else where mˆ is a unit vector in direction m. If, at the beginning of a time step the velocity of the block is not zero, then: f tan() p vˆ (5) where vˆ is the direction of the velocity vector. In an unpublished report 6], the author refers only to the case of a block subjected to gravitational load, where the block velocity and the driving force have always the same sign. However, in dynamic input loading cases, momentary driving force could be opposite to the block velocity... Double Face Sliding Double face sliding, or wedge analysis stability is a classic problem in rock mechanics that has been studied by many authors 7-9]. A typical model of a wedge is shown in Fig. (b). The normal to plane is n n, n, n ] and the normal to plane is n n, n, n ]. Consider a block sliding simultaneously on two ˆ x y z ˆ x y z boundary planes along their line of intersection Î, where: Iˆ ˆ ˆ n n (6) The resultant force in each time step is as before r r, r, r ], and the driving force in each time step is: m ( r Iˆ ) Iˆ (7)

3 ICADD9, Nanyang Technological University, Singapore 5 7 November 9 The normal force acting on plane in each time step is p p, p, p ], and the normal force acting on plane in each time step is q q, q, q ], where: p q (( r nˆ ˆ n ) I ˆ ) ) Iˆ ˆ ) (8) (( r nˆ n (9) As in the case of single face sliding, the direction of the resisting force ( f ) depends upon the direction of the velocity of the block. Therefore, as before, in each time step: tan( ˆ ) p tan( ) q m, V and tan( ) p tan( ) q m f m, V and tan( ) p tan( ) q m () tan( ) tan( ) ˆ p q v, V. Dynamic equations of motion The sliding force, namely the block acceleration during each time step, is s sx s y sz ] and is calculated as the force balance between the driving and the frictional resisting forces: s m f () The block velocity and displacement vectors are V Vx, Vy, Vz ] and D Dx, Dy, Dz ], respectively. At t =, the velocity and displacement are zero. The average acceleration for time step i is: S i s i s i () The velocity for time step i is therefore: Vi Vi Sit (3) It follows that the displacement for time step i is: Di Di Vi t Sit (4) Due to the discrete nature of the VA algorithm, sensitivity analyses were performed to discover the maximum value of the time increment for the trapezoidal integration method without compromising accuracy. The results are found to be sensitive to the time interval size as long as the friction angle is greater than the slope inclination. We find that the time increment can not be larger than. sec to obtain accurate results. 3 RESULTS The validity of the 3D VA formulation presented in section is tested using the classical Newmark solution for the dynamics of a block on an inclined plane. Once the validity of the VA approach is confirmed we proceed to check the validity of 3D-DDA using the VA approach. 3. Single face sliding The typical Newmark solution requires condition statements and is solved using a numerical time steps algorithm as discussed for example by Kamai and Hatzor 4]. We relate here to the Newmark's procedure as the 'analytical solution', to distinguish between the analytical approach and the VA and DDA solutions. Fig. (A) shows a comparison between the analytical (Newmark), VA, and 3D-DDA solutions for a plane with dip and dip direction of = o and = 9 o, respectively, and friction angle of = 3 o. We use for dynamic loading a sinusoidal motion in the horizontal X axis, so the resultant input acceleration vector is r rx ry rz ].5g sin(t) g]. The accumulated displacements are calculated up to cycles (t f = π sec). The input horizontal acceleration is plotted as a shaded line and the acceleration values are shown on the right hand-side axis. The theoretical mechanical properties as well as the numerical parameters for the 3D DDA simulations are listed in Table. For both the Newmark and VA methods the numerical integration is calculated using a time increment of t =. sec. For the 3D methods (VA and 3D-DDA), the calculated displacement vector is normalized to one dimension along the sliding direction. An excellent agreement is obtained between the VA and analytical solutions throughout the first two cycles of motion. There is a small discrepancy at the end of the second cycle which depends on the numerical

4 ICADD9, Nanyang Technological University, Singapore 5 7 November 9 procedures and will decrease whenever the time increment decreases. The relative error of the VA and 3D-DDA methods with respect to the existing Newmark solution is shown in the lower panel of Fig. A, where the relative error is defined as: DNewmarkl Dnumerical E rel % (5) DNewmarkl The relative errors for both VA and 3D DDA are found to be less the 3% in the final position. Relative Error (%) A Newmark Solution 3D-DDA Input Motion Erel, VA Erel, DDA Horizontal Input motion (m/s).. Relative Error (%) B D-DDA Input Motion (x) Input Motion (y) 4 6 Fig. : Block displacement vs. time for the case of a block on an incline subjected to gravitational and cycles of horizontal sinusoidal loading. (A) Comparison between the Newmark solution, VA and 3D-DDA for D horizontal input motion along the X axis. The relative error for the numerical solutions is plotted in the lower panel where the Newmark solution is used as a reference. (B) Comparison between VA and 3D-DDA for D horizontal input motion along the X and Y axes simultaneously. The relative error is plotted in the lower panel where the VA is used as a reference Horizontal Input motion (m/s) After the verification procedure with respect to the existing Newmark solution has been successfully completed, the VA algorithm is found to be suitable to serve as a reference solution for 3D dynamic problems that are examined using 3D DDA. Fig. (B) shows a comparison between the VA solution and 3D DDA results for block sliding on an inclined plane as presented in Fig. (a) and subjected to two components of dynamic, horizontal, input loading. The resultant input acceleration vector is r r r r ].5g sin(t).5g sin(5t) g], and the friction angle is again = 3 o. The two components of the input horizontal acceleration are plotted as shaded lines and the acceleration values are shown on the right- hand side axis. Note that the relative error presented in the lower panel now refers to the VA solution and defined here as: DVectorAnalysis D3 D DDA E rel % (6) D VectorAnalysis The relative error in the final position in this simulation is approximately 8%. 3. Double faces sliding A comparison between VA and 3D-DDA for the dynamic sliding of a wedge is shown in Fig. 3(A) using cumulative displacement versus time. The input acceleration is now defined by a sinusoidal curve on the horizontal Y axis, and gravitational load on the vertical Z axis, so the resultant dynamic load vector is r rx ry rz ].5g sin(t) g]. The line of intersection between the two planes is inclined 3 below the Y axis, and the orientations of the bounding planes are 5/63 and 53/96, as illustrated in Fig. (b). The studied

5 ICADD9, Nanyang Technological University, Singapore 5 7 November 9 friction angles of the planes are = = ; all other numerical control parameters are listed in Table. The relative error is calculated using equation 6 and plotted in the lower panel of Fig. 3(A). The response of the modelled wedge to the Imperial Valley earthquake recorded as measured in El-Centro, CA, is studied and presented in Fig. 3(B). The lower panel presents the three components of the recorded signal multiplied by a factor of 5 to obtain meaningful displacements. The modelled friction angles of the boundary planes here are = = 3 so that the block is at rest under gravity load only. Two different numerical spring stiffness values are studied and the obtained results are plotted with comparison to the VA solution. Relative Error (%) A D-DDA Input Motion (y) Horizontal Input motion (m/s) Input Motion (m/s ) az ay ax B D-DDA ; k= MN/m 3D-DDA ; k= MN/m Fig. 3: Dynamic sliding of a wedge: comparison between 3D-DDA and VA solutions. (A) Wedge response to one component of horizontal sinusoidal input motion and self weight. lower panel presents the relative error calculated according to equation 6. (B) Wedge response to 3D loading using data from the Imperial Valley earthquake (the three components, multiplied by a factor of 5, are shown in the lower panel) Table : Numerical parameters for all 3D DDA forward modelling simulations and VA algorithm. Model Type (Figure) Single plane (Fig. A,B) Wedge, sine curve (Fig. 3A) Wedge, El Centro (Fig. 3B) Mechanical Properties: Elastic Modulus, MPa Poisson's Ratio Density, kg/m 3 Friction angle, Degrees 3 3 Numerical Parameters: Dynamic control parameter Number of time steps Time interval, Sec..5. Assumed max. disp. Ratio, m..5. Penalty stiffness, MN/m 5 and Max. time step for VA, Sec... 4 CONCLUSIONS The newly developed VA algorithm can be utilized to validate 3D numerical solutions, for example dynamic sliding of a block on a single plane and along two planes simultaneously. Since the resisting frictional force direction depends upon the sliding direction a set of condition statements must be implemented in the VA solution to obtain the correct solution at each time step. We report here very good agreement between VA solution and results obtained with the existing D Newmark solution for dynamic sliding of a block on an incline.

6 ICADD9, Nanyang Technological University, Singapore 5 7 November 9 3D DDA is validated here using the VA solution for cases of dynamic sliding and an excellent agreement is found using both synthetic and real earthquake records as dynamic input for both single and double plane sliding. We want to note here that the resultant input acceleration in this study is always applied to the centre of mass of the sliding block, in all Newmark, VA, and 3D-DDA types of analyses. In the physical reality, however, the dynamic input is applied to the foundations and the block responds dynamically to the induced vibrations at the foundations. Further physical tests must be conducted, for example using carefully monitored shaking table experiments, to explore wave propagation behaviour from the shaking foundation to the responding block. 5 ACKNOWLEDGEMENTS Financial support from the U.S. - Israel Bin-national Science Foundation (BSF) through contract 4 is gratefully acknowledged. 6 REFERENCES ] Shi G. H., "Three dimensional discontinuous deformation analysis", 38th US Rock Mechanics Symposium, Washington, DC,, pp ] MacLaughlin M. M., Doolin D. M., "Review of validation of the discontinuous deformation analysis (DDA) method", Int J Numer Anal Met, 3, 4, 6, pp ] Hatzor Y. H., Feintuch A., "The validity of dynamic block displacement prediction using DDA", Int J Rock Mech Min, 38, 4,, pp ] Kamai R., Hatzor Y. H., "Numerical analysis of block stone displacements in ancient masonry structures: A new method to estimate historic ground motions", Int J Numer Anal Met, 3,, 8, pp ] Newmark N. M., "Effects of earthquakes on dams embankments", Geotechnique, 5, 965, pp ] Goodman R. E., Seed H. B., "Earthquake induced displacements in sands and embankments", J Soil Mech Foundation Div ASCE, 9(SM), 966, pp ] Wartman J., Bray J. D., Seed R. B., "Inclined plane studies of the Newmark sliding block procedure", J Geotech Geoenviron, 9, 8, 3, pp ] Tsesarsky M., Hatzor Y. H., Sitar N., "Dynamic displacement of a block on an inclined plane: Analytical, experimental and DDA results", Rock Mech Rock Eng, 38,, 5, pp ] Jiang Q. H., Yeung M. R., "A model of point-to-face contact for three-dimensional discontinuous deformation analysis", Rock Mech Rock Eng, 37,, 4, pp ] Liu J., Kong X. J., Lin G., "Formulations of the three-dimensional discontinuous deformation analysis method", Acta Mechanica Sinica,, 3, 4, pp 7-8. ] Moosavi M., Jafari A., Beyabanaki S., "Dynamic three-dimensional discontinuous deformation analysis (3-D DDA) validation using analytical solution", The Seventh international conference on the analysis of discontinuous deformation (ICADD-7), Honolulu, Hawaii, 5, pp ] Wang J., Lin G., Liu J., "Static and dynamic stability analysis using 3D-DDA with incision body scheme", Earthquake Engineering and Engineering Vibration, 5,, 6, pp ] Yeung M. R., Jiang Q. H., Sun N., "Validation of block theory and three-dimensional discontinuous deformation analysis as wedge stability analysis methods", Int J Rock Mech Min, 4,, 3, pp ] Yeung M. R., Jiang Q. H., Sun N., "A model of edge-to-edge contact for three-dimensional discontinuous deformation analysis", Computers and Geotechnics, 34, 3, 7, pp ] Goodman R., Shi G., "Block theory and its application to rock engineering, Prentice-Hall Englewood Cliffs, NJ, 985, 338 pp. 6] Shi G. H., "Technical manual and verification for Keyblock codes of dynamic Newmark Method", Unpublished technical report, DDA Company, Belmont, CA, ] Goodman R. E., "Methods of Geological Engineering in Discontinuous Rocks, West Publishing Company, San Francisco, 976, 47 pp. 8] Hatzor Y. H., Goodman R. E., "Three-dimensional back-analysis of saturated rock slopes in discontinuous rock-a case study", Geotechnique, 47, 4, 997, pp ] Hoek E., Bray J. W., "Rock slope engineering, Institution of Mining and Metallurgy, London, 98, 358 pp.