Nonlinear Time-Domain Analysis of a Sliding Block on a Plane

Missouri University of Science and Technology Scholars' Mine

International Conferences on Recent Advances 2010 - Fifth International Conference on Recent in Geotechnical Earthquake Engineering and Advances in Geotechnical Earthquake Soil Dynamics Engineering and Soil Dynamics

27 May 2010, 4:30 pm - 6:20 pm

Nonlinear Time-Domain Analysis of a Sliding Block on a Plane

Mohamed G. Arab Arizona State University, Tempe, AZ

Edward Kavazanjian Jr. Arizona State University, Tempe, AZ

Neven Matasovic Geosyntec Consultants, Huntington Beach, CA

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Recommended Citation Arab, Mohamed G.; Kavazanjian, Edward Jr.; and Matasovic, Neven, "Nonlinear Time-Domain Analysis of a Sliding Block on a Plane" (2010). International Conferences on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. 23. https://scholarsmine.mst.edu/icrageesd/05icrageesd/session04b/23

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NONLINEAR TIME-DOMAIN ANALYSIS OF A SLIDING BLOCK ON A PLANE

Mohamed G. Arab Edward Kavazanjian Jr. Neven Matasovic Arizona State University Arizona State University Geosyntec Consultants Tempe, AZ-USA 85287-5306 Tempe, AZ-USA 85287-5306 Huntington Beach, CA-USA 92648

ABSTRACT

A time domain finite difference numerical model of a sliding rigid block on a plane is developed using a simple elastic-perfectly plastic Mohr-Coulomb interface model. The model is shown to accurately predict the slip-stick and slip-slip behavior deduced from an analytical solution for behavior of a sliding block on a horizontal plane and the results of physical model tests of a block on both horizontal and inclined planes subject to harmonic and non-uniform excitation provided the appropriate interface strength is employed. Back analyses of the physical model tests show that for some geosynthetic interfaces, the interface shear strength depends upon the velocity of sliding. The numerical model developed herein provides a basis for rigorous evaluation of several important problems in geotechnical earthquake engineering, including the cumulative permanent seismic deformation of landfills, embankments, slopes, and retaining walls and the stresses induced by seismic loading in geosynthetic elements of landfill liner and cover systems.

INTRODUCTION

This paper describes time domain analyses of the response of damage to geosynthetic liner and cover systems at landfills. a block on a plane to subject to harmonic and earthquake-like The application of Newmark analysis to quantify the seismic base motions. The interface between the block and the plane displacement of a gravity retaining wall is conceptually illu- is modeled in a nonlinear manner using an elastic-perfectly strated in Fig. 1. plastic constitutive relationship. Results of the analysis are compared to previous numerical analyses and physical model In engineering practice, a Newmark analysis is often assumed tests of the response of a block on a horizontal plane subject to to provide a realistic assessment of cumulative permanent harmonic excitation to validate the numerical model. The seismic deformation. However, several investigators have validated model is then used to investigate the response of a demonstrated that the displacement calculated in a Newmark block on an inclined plane. These analyses are conducted as a analysis is in general only an index of the permanent seismic prelude to more complex analyses of the response of geosyn- displacement of a geotechnical structure, e.g. Rathje and Bray thetic-lined landfills to earthquake ground motions. However, (2000). In landfill engineering, the displacement calculated in the analytical model has even broader applications to a wide a Newmark analysis is generally recognized as merely an in- range of problems in geotechnical earthquake engineering. dex of seismic performance (Augello et al., 1995, Matasovic et al., 1995), though it is used to evaluate the potential for The sliding of a block on a plane is widely used in geotechnic- damage to geosynthetic elements of landfill containment sys- al earthquake engineering as an analog for evaluating the cu- tems quantitatively based upon correlation with observed per- mulative permanent displacement of geotechnical structures formance of landfills in earthquake (Anderson and Kavazan- subject to strong ground shaking in an earthquake. This type jian, 1995; Kavazanjian et al., 1998). of seismic displacement analysis is often referred to as a Newmark analysis as it was first described in the earthquake The primary reason that a Newmark analysis generally yields engineering literature by Newmark (1965). In earthquake only an index of permanent seismic displacement is that it engineering practice, Newmark analyses are used to estimate fails to consider the impact of the relative displacement along the sliding displacement of rigid gravity and semi-gravity re- the slip surface on the response of the overlying mass. The taining walls, the cumulative displacement of unstable slopes, nonlinear time domain analysis presented in this paper offers lateral spreading in liquefied ground, and the potential for the potential for considering the impact of slip among the fail-

Paper No. 4.08b 1

ure surface on the response of the overlying mass, thereby diately beneath the base of the sliding mass. The block and providing a more accurate assessment of the permanent seis- the plane initially move together and the block is assumed to mic displacement of this mass. The method of analysis em- begin moving relative to the plane when the acceleration of ployed herein also offers the potential for evaluating the the plane exceeds the yield acceleration (i.e., friction coeffi- stresses and strains in both the sliding mass itself and in the cient) of the block. The block ceases movement after the ac- interface between the sliding mass and the underlying materi- celeration of the plane falls below the yield acceleration of the al, e.g., in the geosynthetic components of a landfill liner sys- block and subsequent deceleration of the block brings the ve- tems subject to strong ground shaking. locity of the block relative to the plane back to zero. The relative displacement between the block and the plane is calculated by double integrating the acceleration of the block relative to the plane.

Displacements in a Newmark analysis are typically calculated considering excursions above the yield acceleration on only one side of the acceleration time history to account for a pre- ferred direction of sliding. However, in some cases the analy- d b a sis may consider excursions above the yield acceleration on Rigid Block both sides of the acceleration time history. Cases where ex- b a cursions above the yield acceleration on both sides of the case d history are considered include cases where the yield acceleration is the same on both sides of the time history and cases where the yield acceleration in one direction is greater than the yield acceleration in the other direction. The case where the a a yield acceleration is the same in both directions pertains to frictional seismic base isolation and is analogous to a block on b a horizontal plane. The case where the yield acceleration dif- b fers on either side of the time history may represent a slope or retaining wall and is analogous to a block on an inclined plane.

c c The acceleration time history used in a Newmark analysis may be an actual recorded acceleration time history, chosen on the basis of magnitude, distance, or other characteristics of the design earthquake and scaled to an appropriate peak ground acceleration, or it may be taken from the results of a seismic d d response analysis for the geotechnical structure of interest. When the results of a seismic response analysis are used, they time time are often from a response analysis that ignores the relative displacement along the slip surface, e.g., from an equivalent- Acceleration Displacement linear seismic response analysis. This type of Newmark analysis is referred to as a decoupled analysis in that the response of Fig.1. Cumulative seismic deformation of a gravity retaining the overlying geotechnical structure is decoupled from (or wall (after AASHTO, 2007). calculated independently of) its permanent (slip) displacement. Decoupled analysis can lead to significant errors in the displacements calculated in a Newmark analysis (Rathje and NEWMARK ANALYSIS IN GEOTECHNICAL Bray, 2000). The only way to eliminate the decoupled as- EARTHQUAKE ENGINEERING sumption from a Newmark analysis and thereby avoid the associated errors is to employ a non-linear time domain seis- In a Newmark analysis of the seismic displacement of a land- mic response analysis. fill, slope, embankment, or retaining wall the sliding block is analogous to the displacing waste mass, earth, or retaining wall, the plane on which the block sits is analogous to the slip PREVIOUS STUDIES surface from a limit equilibrium analysis, and the coefficient of friction between the sliding block and the underlying plane Rigid Block is analogous to the yield acceleration from the limit equilibrium analysis (i.e., to the seismic coefficient for a pseudo- Several researchers have used the block-on-a-plane model to static factor of safety of 1.0). The acceleration time history of investigate the feasibility of using geosynthatic materials for the underlying plane is assumed to be equal to the earthquake frictional base-isolation. This work has included shaking table ground motion acceleration time history of the ground imme- and centrifuge tests on rigid blocks lined with geosynthetic

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materials (e.g., Kavazanjian et al. 1991; Yegian and Lahlaf reversal of the direction of motion when the relative velocity 1992; Yegian et al. 1998; Yegian and Kadakal 1998a; 1998b; between the block and the plane is zero). This pattern of dis- 2004). The block-on-a-plane model has also been used by placement, illustrated in Fig 9(b), was referred to as the slip- several investigators to study the seismic deformation of em- slip condition. bankments and geosynthetically-lined landfills (e.g., Elgamal et al. 1990; Kavazanjian and Matasovic, 1995; Wartman, Matasovic et al. (1998) used an analytical model of a rigid 1999; and Wartman et al., 2003; 2005). Both Elgamal et al. mass on an inclined plane to investigate factors influencing the (1990) and Wartman et al. (2003) conducted shaking table seismic deformation of geosynthetic landfill cover systems. tests with a rigid block on an inclined plane and used the These investigators studied the influence of two way sliding, Newmark procedure to model the test results. In both of these the vertical acceleration component, and degradation of the studies, the shaking table was excited with a simple harmonic yield acceleration on permanent seismic displacement using a (sinusoidal) wave and/or earthquake like motions. Both El- decoupled model. Their analyses indicated that the effect of gamal et al. (1990) and Wartman et al. (2003) found that the two way sliding and the vertical component of the earth- Newmark analysis accurately predicts the relative displace- quake ground motion are, for most practical purposes, negligi- ment between the rigid block and the plane if the appropriate ble. However, the study revealed that the degradation of the dynamic interface friction coefficient is used. They further yield acceleration may be an important factor affecting the demonstrated that slip at the interface between the block and accuracy of a conventional Newmark analysis as neglecting of the plane limits the transmission of energy to the sliding mass this phenomenon may result in a conservative assessment of due to its stick–slip dynamic response. Wartman et al. (2003) the maximum calculated permanent seismic displacements. concluded that when a deformable (compliant) block was employed the Newmark procedure underestimates the shaking- Deformable (Compliant) Block induced displacements when the predominant frequency of the excitation is somewhat lower than, or near, the natural fre- Makdisi and Seed (1978) first proposed modifying the New- quency of the sliding mass. These investigators also concluded mark analysis to account for the deformability of earthen that the Newmark procedure generally over-predicts shaking- structures. The Makdisi and Seed procedure, in which the site induced displacements when the predominant frequency of the response and seismic deformation calculations are decoupled, input motion is significantly greater than the natural frequency is still widely used for seismic slope stability analysis. How- of the sliding mass. ever, the fact that the relative displacement across the sliding interface is ignored in the seismic response analysis results in Westermo and Udwadia (1983) presented an analytical solu- significant limitations for this method of analysis. tion for a rigid mass resting on a flat horizontal surface with an interface friction coefficient, μ, excited with a simple har- Kramer and Smith (1997) considered the effects of both the monic motion of amplitude a0. They showed that the system compliance of the overlying mass and the relative displace- experienced two different slippage conditions that were inde- ment along the slip surface on the results of a Newmark analy- pendent of the frequency of the harmonic motion and de- sis. These investigators analyzed the response of a lumped pended only on a non-dimenssional parameter η, where: mass linear visco-elastic single degree of freedom system rest- µ g ing on an inclined plane. Results using the visco-elastic model η = (1) were compared to results using a rigid block. Based upon this ao comparison, Kramer and Smith concluded that ignoring the and g is the acceleration of gravity. The quantity µg is the compliance of the overlying mass could be unconservative for yield acceleration of the block, i.e., the acceleration at which very thick and/or very soft failure masses. the block begins to slide relative to the plane. Rathje and Bray (2000) performed nonlinear coupled analyses Two different slippage conditions were identified by Wester- of the stick-slip behavior of a deformable one-dimensional (1- mo and Udwadia (1983). These two conditions were referred D) mass. The Rathje and Bray model incorporated nonlinear to as the slip-stick and slip-slip conditions. At the end of a soil properties and assumed one-directional sliding. The seis- sliding cycle the block will either stick before moving again or mic response of the sliding mass was modeled using equiva- immediately start to slide in the opposite direction. Westermo lent-linear viscoelastic analysis to capture the nonlinear re- and Udwadia showed that if η was greater than 0.53, the block sponse of the overlying earth and waste materials. The results will stick before the base acceleration exceeds the threshold from this model were shown to compare favorably with those for movement (μg) for the next half cycle. In this case the from a nonlinear tome domain analysis with the program D- motion of the block will consist of two sliding and two non- MOD2000 (Matasovic, 1993; www.GeoMotions.com). sliding intervals per cycle, as illustrated in Fig. 9(a). This mode of motion was referred to as slip-stick behavior. How- Rathje and Bray (2001) compared results of dynamic response ever, for η equal to or less than 0.53, the end of the slip inter- analyses from 1-D and two-dimensional (2-D) models to eva- val converges on the time of initiation of sliding in the oppo- luate the seismic response of solid waste landfills. Results site direction. In this case, the block begins to slide in the op- indicate that 1-D analysis provides a reasonably conservative posite direction immediately and the block never comes to a estimate of the seismic loading and earthquake-induced per- complete stop (except for the instantaneous moment during manent displacement for deep sliding surfaces but less seismic

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loading and permanent displacement than that predicted by 2- based upon the static interface strength of the interface as D analysis for shallower sliding surfaces. However in their measured in a tilt-table test and then adjusted as necessary to analysis, they did not model the geosynthetic interface in ei- find the best fit with the experimental results. A shear stress ther the 2-D or 1-D analysis mode. time history was applied to the base of the mesh to model the shaking table input motion.

DEVELOPEMENT OF THE NUMERICAL MODEL Fig. 3 shows a comparison between the block acceleration measured in one of the uniform sinusoidal loading tests and A time domain finite difference numerical model of a rigid the results calculated in the non-linear finite difference time block on a plane with a frictional interface was developed us- domain analysis. The test was conducted using a geomem- ing the large strain formulation coded in FLAC 6.0 (Itasca, brane/geotextile interface with a measured static coefficient of 2008; www.itasca.com). The model was developed with the friction equal to 0.16. However, analysis conducted with a ultimate goal of modeling the seismic behavior of geosynthet- frictional coefficient of 0.15 resulted in better agreement be- ically-lined landfills. The frictional interface was modeled tween the measured values and the numerical analysis results. using a simple elastic-perfectly plastic stress-strain relation- The best-fit numerical analysis results calculated using a fric- ship and the Mohr-Coulomb failure criterion. The numerical tion coefficient of 0.15 in the Mohr-Coulomb interface model model was validated using a series of shaking table tests of a are shown in Fig. 3. These results suggest that the dynamic sliding block on a horizontal plane conducted by Kavazanjian coefficient of friction for the interface may be slightly differ- et al. (1991) to demonstrate the ability of a layered geosynthet- ent than the static interface friction coefficient. The same ic system to provide frictional base isolation. In these tests, a conclusion was reached by Kavazanjian et al. (1991) based rigid block with one geosynthetic material glued to its bottom solely on the experimental results. side was placed on a shaking table that had a second geosyn- 0.8 thetic material secured to it. Four different combinations of Base Acceleration (Experimental) 0.6 Base Acceleration (Numerical Analysis) geotextile and geomembrane materials were subject to series 0.4 BLock Acceleration (Experimental) of uniform sinusoidal motions of varying amplitude. Three of 0.2 Block Acceleration (Numerical Analysis) the geosynthetics combinations were also subjected to a non- 0 uniform earthquake-like motion based upon the S90W com- -0.2 -0.4 ponent of the 1940 El Centro acceleration. The acceleration of Acceleration (g) -0.6 the block and the displacement of the block relative to the 0 0.5 1 1.5 2 2.5 3 shaking table were monitored during these tests. Time (Sec)

Fig. 3. Comparison between best-fit numerical analysis and To model the Kavazanjian et al. (1991) shaking table tests in acceleration response measured by Kavazanjian et the time domain finite difference analyses, the two-layer, nine al. (1991) uniform sinusoidal excitation. macro-element mesh shown in Fig. 2 was used. The upper layer represents the rigid block and the lower layer represents Fig. 4 presents a comparison of the relative displacement of the shaking table. the rigid block as measured in the shaking table test and as calculated in the numerical analysis. The small discrepancies Y between the experimental results and the numerical analysis Interface Elements may be attributed to the somewhat asymmetrical behavior in X the experimental results: note the somewhat non-uniform ac- Rigid Block celeration behavior of the block in Fig. 3 and the slight drift towards increasing peak displacement in the negative direction in Fig. 4.

Base 8 6 Block Relative Displacement 4 (Experimental) 2 Block Relative Displacement 0 (Numerical Analysis) -2 -4 -6 -8 Relative Disp. Disp. (cm) Relative Fig.2. Finite difference model (macro elements shown) of the 0 0.5 1 1.5 2 2.5 3 Time (sec) horizontal shaking table test. Fig. 4. Comparison between best-fit numerical analysis and Values of bulk and shear modulus representative of structural displacements measured by Kavazanjian et al. (1991) steel were used to model both the rigid block and shaking ta- during uniform sinusoidal loading. ble. An interface element was used between the upper and lower layers of the mesh to model the geosynthetic The numerical model illustrated in Fig. 2 was also employed /geosynthetic interface. The interface was assigned an elastic to reproduce the slip-stick and stick-stick modes of behavior stiffness approximately equal to ten times the bulk modulus of predicted analytically by Westermo and Udwadia (1983). An the mesh elements. The interface shear strength was initially interface friction angle corresponding to a friction coefficient

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of 0.3 was employed in the numerical analyses and the numer- tive velocity decreases to zero, at which point the block im- ical model was excited with a harmonic sinusoidal motion of a mediately begins sliding in the other direction and never frequency of 1 Hz using two different peak accelerations. A comes to rest on the plane. peak acceleration of 0.4g, corresponding to a value of η = 0.75, was used in one analysis and a peak acceleration of 0.9 The time domain numerical model was used to simulate tests g, corresponding to a value of η = 0.33, was used in a second conducted by Yegian et al. (1998) and Yegian and Kadakal analysis. In the first case, since 0.53 < η < 1.0, stick-slip beha- (2004) that used earthquake time histories as input. These in- vior is expected. In the second case, since η < 0.53, slip-slip vestigators conducted shaking table tests of rigid blocks with a behavior is expected. The results of the numerical analysis are geotextile/geomembrane interfaces using earthquake motions shown in Fig. 5 Fig 5(a) shows the calculated acceleration, as input. Yegian et al. (1998) used the Los Angeles University velocity and displacement of the rigid block for the slip-stick Hospital Grounds record from the 1994 M 6.7 Northridge case (0.4 g peak acceleration, η = 0.75). Looking at the rela- earthquake scaled to 0.9 g. Yegian and Kadakal (2004) used tive velocity of the block, the block experiences 3 modes of the Corralitos, Capitola, and Santa Cruz records from 1989 M displacement per cycle. The block first sticks to the base until 7.1 Loma Prieta earthquakes scaled to peak accelerations of the acceleration exceeds µg. The block then starts to slide and 0.1 g to 0.4 g. Because earthquake motions are asymmetric, the relative velocity increases until the acceleration of the base both the shaking table tests and the numerical analyses yield a reaches µg. At this point the relative velocity decreases until it residual permanent displacement. Fig. 6 shows a comparison reaches zero and the block sticks to the plane again. Fig. 5(b) between the residual permanent displacement reported by Ye- illustrates the slip-slip behavior for the second case (0.9 g peak gian et al. (1998) and Yegian and Kadakal (2004) and the acceleration, η = 0.33). In this case the block slides in one permanent displacement calculated using the numerical model. direction until the block passes the µg threshold and the rela- 1 ao 0.5 0.8 ao Base Acceleration Base Acceleration (b) 0.4 (a) 0.6 μg 0.3 0.4 μg

0.2 0.2

0.1 0

0 -0.2 -0.1

Acceleration (g) -0.4

Acceleration (g) -0.2 -0.6 Block -0.3 -0.8 Acceleration -0.4 Block -1 -0.5 Acceleration Base Velocity 200 80 Relative Velocity 150 60 Block Velocity Block Velocity 100 40 50 20 0 0 -50 -20

Velocity (cm/s) -100 Velocity (cm/s) -40 Relative Velocity Base Velocity -150 -60 -200 -80 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 Time (s) Time (s) Relative Displacement Relative Displacement 25 2 20 1.5 15 1 10

0.5 5

0 0 -5 -0.5 -10 Displacement (cm) Displacement (cm) -1 -15 -1.5 η = 0.75 -20 -2 -25 η = 0.33 < 0.53 0.53< η<1 Slip Slip Slip Stick Slip Stick Slip

Fig. 5. The influence of η on the response of a rigid block on a horizontal plane (a) slip-stick response, (b) slip-slip response.

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The comparison of residual permanent displacement from the To model the Elgamal et al. (1990) shaking table tests in the experimental results of Yegian et al. (1998) and Yegian and time domain analyses, a two-layer, nine macro element mesh, Kadakal (2004) and the numerical analysis in Fig. 6 shows similar to the one used to simulate the horizontal shaking table generally good agreement. The primary discrepancy is for the tests, was used. As in the horizontal plane model shown in Corralitos record from the Loma Prieta event. The calculated Fig. 2, in the inclined plane model, illustrated in Fig. 7, the permanent displacement for the Corralitos record is less than upper layer represents the rigid block and the lower layer the displacement measured experimentally in the 0.1 g to 0.25 represents the shaking table. The same values for bulk and g range. However, both calculated and experimental results shear modulus of the block and plane and for the elastic stiff- show a similar trend of increasing displacement with increas- ness of the interface used in the model in Fig.2 were assigned ing of the base peak acceleration. One interesting aspect of the to the inclined plane model in Fig. 7. A shear stress time histo- results is that in some cases the permanent displacement de- ry was applied to the base of the mesh to model the shaking creases as base acceleration increases. table input motion.

25 Capitola (FLAC) Y Santa Cruz (FLAC) \ Interface Elements Y \ 20 Corralitos (FLAC) X Northridge (FLAC) X Santa Cruz (Experimental) 15 Capitola (Experimental) Corralitos (Experimental) Northridge (Experimental) Rigid Block 10

5 Base Displacement (cm)

0 β

-5 0 0.2 0.4 0.6 0.8 1 Peak Base Acceleration (g) Fig. 7. Finite difference model (macro elements shown) of an Fig. 6. Residual permanent displacement from the numerical inclined shaking table test. model compared to the experimental results of Yegian

et al. (1998) and Yegian and Kadakal (2004). Fig. 8 shows a comparison between the block acceleration measured in the uniform sinusoidal loading test of Elgamal et Despite small differences between experimental and numerical al. (1990) and the results calculated in the time domain finite results, the analyses shown in Fig. 3 through Fig. 6 suggest difference analysis using a friction angle based upon the static that the elasto-plastic Mohr-Coulomb interface model is capa- friction coefficient of 0.36 reported by Elgamal et al. (1990). ble of reproducing the dynamic behavior of a block on a hori- 0.8 zontal plane with a frictional interface if the appropriate inter- Base Acceleration Experimental Elgamal et al. (1990) face shear strength is employed. 0.6 Block Accleration Expermintal Elgamal et al. (1990) 0.4 Base Acceleration (Numerical Analysis) Block Acceleration (Numerical Analysis) 0.2 0 RIGID BLOCK ON AN INCLINED PLANE -0.2 -0.4

Acceleration(g) -0.6 Sand-Sand Interface -0.8 0 0.5 1 1.5 2 Time (Sec.) Elgamal et al. (1990) conducted a shaking table test of a block on a plane inclined at 10o to assess the accuracy of the New- Fig. 8. Acceleration from the numerical model compared to mark (1965) procedure. An inclined plane coated with sand- the experimental results of Elgamal et al. (1990). paper was mounted on a shake table and a solid-metal block Fig. 9 presents a comparison of the relative displacement of with sand glued to the base was placed on the plane. Accele- the rigid block as measured in the shaking table test and as rometers were attached to both the sliding block and the shak- calculated in the numerical analysis. The primary discrepancy ing table. A harmonic sinusoidal excitation at a frequency of 1 between the experimental results and the numerical analysis is Hz with an amplitude sufficient to induce the block to slide at the start of shaking, when there is a delay in the initiation of down slope was applied to the base for 12 seconds. In addition relative displacement. Note that in the experimental results to the acceleration of the block and the plane, the relative slip the block sticks for a little less than 2 seconds, i.e., for the first between the block and the plane was measured over the course one and a half cycles of motion (the excitation frequency was of the test. 1 Hz) before the block starts to move relative to the plane. This delay in the initiation of relative movement suggests that there may have been some initial adhesion at the interface

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(perhaps due to interlocking of sand grains) that had to be and one earthquake-like motion. The input type, excitation overcome before the block began to slide. However, once the frequency, peak acceleration of the input motion, and cumula- block began to move, relative displacement accumulated at tive relative displacement between the block and the plane for approximately the same rate in the numerical analysis as ob- these tests are presented in Table 1. served in the model, as evidenced by the nearly parallel displacement curves in Fig. 9. Table. 1. Input motions used in the tests reported by Wartman 80 et al. (2003) Block Relative Displacement(Elgammal) 70 Block Relative Displacement(Numerical Analysis) Type Frequency Input Motion Relative Dis- 60 (Hz) Peak Accelera- placement tion (g) (cm) 50 Uniform 1.33 0.01 0.19 40 Uniform 2.66 0.09 0.12 30 Uniform 2.66 0.15 2.42 20 Uniform 2.66 0.2 12.64

Relative Displacement (cm) 10 Uniform 4.0 0.07 0.12 0 Uniform 4.0 0.13 2.51 0 2 4 6 8 10 12 14 Uniform 4.0 0.17 2.51 Time (Sec.) Fig. 9. Displacement from the numerical model compared to Uniform 4.0 0.25 6.60 the experimental results of Elgamal et al. (1990). Uniform 6.0 0.07 0.45 The generally good agreement shown in Fig 8 and Fig. 9 be- Uniform 6.0 0.1 0.83 tween the model test results of Elgamal et al. (1990) and the numerical analysis suggests that in the numerical model is Uniform 6.0 0.12 1.39 capable of reproducing the behavior of a rigid block on an Uniform 6.0 0.19 3.65 inclined plane subject to dynamic excitation. Furthermore, the good agreement shown in these two figures suggests that in Uniform 8.0 0.04 0 this case the dynamic sliding resistance is similar to the static Uniform 8.0 0.07 0.13 sliding resistance. In this respect, the sand-sand interface tested by Elgamal et al. (1990) is different from the geomem- Uniform 8.0 0.11 0.54 brane-geotextile interface tested by Kavazanjian et al. (1991), which showed a difference between the dynamic and static Uniform 8.0 0.16 1.20 sliding resistance. Uniform 10.92 0.05 0.17 Geosynthetic-Geosynthetic Interface Uniform 10.92 0.06 0.21 Uniform 10.92 0.09 0.46 Wartman (1999) and Wartman et al. (2003) report on shaking o table tests of a rigid block on a plane inclined at 11.37 . A Uniform 10.92 0.21 3.20 steel block, 2.54 cm thick, with a cross sectional area of 25.8 cm2 and mass of 1.6 kg was positioned on top of the plane. Uniform 12.8 0.07 0.20 The interface between the block and the plane was a smooth Uniform 12.8 0.23 3.01 high density polyethylene (HDPE) – non woven geotextile interface similar to one that might be found in a geosynthetic Sweep - 0.25 3.43 liner system for a landfill. Fig. 10 illustrates the test setup Sweep - 0.35 4.90 employed by these investigators. Accelerometers were fitted to both the sliding block and the inclined plane. Displacement Sweep - 0.58 11.78 transducers were employed to measure the absolute displacement of the sliding block and absolute displacement of the Earthquake Kobe 2.0 8.22 shaking table. Relative displacement was calculated from the like two absolute displacement measurements. The input motions used in the tests described by Wartman (1999) and Wartman et al. (2003) were a suite of 22 uniform The uniform sinusoidal loading tests were conducted for dura- sinusoidal motions, three sinusoidal frequency sweep motions, tion of 6 seconds at frequencies varying from 1.33 to 12.76 Hz

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with acceleration amplitudes varying from 0.01 g to 0.23 g. The numerical model shown in Fig. 7 was used to simulate the Instability of the shaking table when the full amplitude of mo- shaking table tests reported in Table. 1. For each test, the in- tion was used over the entire duration of testing led to the use terface friction angle was varied until a calculated cumulative of a motion in these so-called uniform loading tests that linear- relative displacement approximately equal to the one reported ly ramped up to the peak amplitude for 1.5 seconds followed in the test was achieved. The open symbols in Fig. 12 show by the full amplitude of motion for 3 seconds and then 1.5 the best-fit friction angle from the numerical analysis plotted seconds of ramping down. The frequency sweep motion li- versus average sliding velocity. The average sliding velocity nearly increased in both frequency (from 1 to 16 Hz) and acce- was calculated as the cumulative relative displacement divided leration over a 5 second duration. by the total sliding time for the block. The total sliding time was determined assuming that the block was sliding when the relative velocity exceeded 0.01cm/s. As illustrated in Fig. 11, Y Y\ \ the friction angle back calculated from the numerical analysis X X was less consistently than the friction angle reported by Wartman et al. (2003) for these tests. Rigid Block 20 o Inclined Plane Numerical Analysis β=11.37 Experimental small disp. (<1.2mm) Wartman (2003) 19 Experimental large to med. Disp. Wartman (2003) Shaking table 18 )

Ф 17

accelerometer horizontal 16 LVDT horizontal 15 accelerometer parallel to inclined plane friction angle ( angle friction LVDT parallel to inclined plane 14

13 Fig. 10. Schematic representation of the shaking table setup near static and the instrumentation used by Wartman (1999) and (Wartman et al. (2003) (after Wartman,1999). 12 0.01 0.1 1 10 average sliding velocity (cm/s)

To back analyze his test results, Wartman (1999) employed Fig. 11. Interface friction angle as a function of average slid- the computer program YSLIP_PM developed by Matasovic et ing velocity of the block ( after Wartman et al., 2003). al. (1998) for Newmark analysis. The interface friction angle was varied in the YSLIP_PM analysis until calculated accele- The discrepancy shown in Fig. 11 between the numerical ration and displacement time histories of the block matched analysis results and the friction angle reported by Wartman et the measured values as closely as possible. However, interface al. (2003) may be attributed to the use by Wartman et al. friction angles back calculated in these analyses were consis- (2003) of the acceleration parallel to the plane rather than the tently greater than the value of 12.7o ± 0.7o determined by tilt acceleration in the horizontal direction to predict the friction tests reported in Wartman et al. (2003). angle between the block and the plane. Yan et al. (1996) present the following equation for the horizontal acceleration Wartman (1999) postulated that the friction angle of the inter- at yield at the initiation of movement in the downslope direc- face during dynamic loading was controlled by two factors: 1) tion of a rigid block on a plane subject to horizontal accelera- the amount of displacement; and 2) sliding velocity. Fig. 11 tion: shows the friction angle back calculated from the table accele-  ration at the initiation of sliding plotted versus the average X y = g (tan(φ - β )) (2) sliding velocity of the block as presented by Wartman et al. (2003). The calculated friction angles were in the range of where X is the horizontal acceleration at yield, φ is the fric- 14o–19o and appeared to increase linearly with increasing av- y erage sliding velocity. Wartman (1999) cites an increase in the tion angle between the block and the plane, and β is the angle threshold acceleration for the initiation of sliding in the sweep of the plane. Prior to yield, the vertical acceleration of the frequency test with an amplitude of 0.58 g from approximately block is zero and the acceleration of the block parallel to the 0.1 g at the beginning of the test to 0.2 g at the end the test as  plane at yield, X ' y , is given by Eq. 3: additional evidence of an interface friction angle proportional to sliding velocity for the HDPE / non-woven geotextile inter-   face that was tested. X ' y = X y cosβ (3)

Paper No. 4.08b 8

However, once the block yields there is a downward vertical acceleration induced in the block and the acceleration of the block parallel to the plane increase accordingly. Thus, using the acceleration parallel to the plane of a yielding block to predict the yield acceleration according to Eq. 2 and Eq. 3 results in overprediction of the block friction angle.

Y \ Y \ X X  m. X y .sin β

 FN.tan(Ф) m. X y

 m.g .cos β m. X y .cos β m.g m.g .sin β β

Fig. 12. Free body diagram of a block on an inclined plane Fig. 13. Frequency sweep motion (a) Horizontal block acce- subject to horizontal excitation at the initiation of leration (b) Block acceleration normal to the plane downslope sliding. (c) Block acceleration parallel to the plane. This phenomenon is illustrated in Fig. 13, which shows results Fig. 14 presents a comparison of the relative displacement of from the numerical model for a frequency sweep test. Figure the rigid block as measured in the shaking table test and as 13(a) shows the calculated horizontal acceleration of the block calculated in the numerical analysis for the Kobe acceleration from the frequency sweep analysis. Fig. 13(b) shows the cor- input. Once again, the interface friction angle in the numerical responding calculated vertical acceleration of the block while model was varied until the maximum relative displacement matched the measured value. The best fit interface friction Fig. 13(c) shows the calculated acceleration of the block paral- o lel to the plane. Note that Fig. 13(a) shows that the horizontal angle from this analysis was 16.2 . The relative displacement acceleration at yield in this numerical analysis is constant and time history from the numerical analysis closely matches the equal to 0.10g, the value that would be predicted from Eq. 2 relative displacement time history observed in the laboratory using the input friction angle. However, Fig. 13(c) shows the test, including episodes of upslope relative displacement at around 5 seconds. Furthermore, the best fit friction angle of acceleration of the block parallel to the plane increases from o 0.1g to 0.15 g as the horizontal acceleration of the base in- 16.2 and the average sliding velocity between 5 and 8 creases over the duration of the test. seconds of 2.67 cm/s plots in the middle of the band of results from the numerical analysis of the uniform sinusoidal loading In the shaking table test conducted by Wartman (1999) that tests reported in Fig. 11. corresponds to the numerical analysis shown in Figure 13, an increase in the acceleration parallel to the plane from 0.1g to 0.2g was observed. The additional 0.05 g in acceleration at yield parallel to the plane observed in the shaking table test may be attributed to the increase in average velocity of the block as the frequency and amplitude of the excitation increases.

Wartman (1999) also reports on shaking table tests conducted using the same physical model described above and a scaled version of a Hyogoken Nanbu (Kobe) earthquake recording (Kobe Port Island array, depth = 79 m) as input to the shaking table. The peak acceleration of the scaled record was 2.0 g. The time-domain finite difference numerical model of a rigid block on a plane was also used to model this test.

Fig. 14. Measured and calculated block relative displacement for the Kobe Earthquake acceleration input.

Paper No. 4.08b 9

Stick-Slip and Slip-Slip Behavior CONCLUSIONS

The numerical model was also used to investigate the potential A time-domain finite difference model of a rigid block sliding for slip-stick and slip-slip conditions for an inclined plane, on a plane has been developed using a simple elastic-perfectly where the threshold acceleration is different in the downslope plastic constitutive model and the Mohr-Coulomb failure crite- and upslope directions. A plane inclined at 11.37o with an rion to characterize the load-displacement behavior of the in- interface friction coefficient of 0.29 was used in the analysis, terface between the block and the plane. This numerical mod- corresponding to a downslope yield acceleration of 0.08 g and el has been shown to accurately reproduce the slip-stick and an upslope yield acceleration of 0.52 g. Fig. 15(a) shows the slip-slip behavior described by Westermo and Udwadia (1983) calculated block acceleration, relative velocity and relative for frictional sliding of a rigid block on a horizontal plane. The displacement for an input sinusoidal acceleration with an am- numerical model has also been shown to accurately predict plitude of 0.2g and a frequency of 2.66 Hz. In this case, block shaking table tests of a sliding block on horizontal and in- response is in a stick-slip mode. Fig. 10(b) shows the calcu- clined planes subject to uniform and non-uniform motions lated block acceleration, relative velocity and relative dis- provided the appropriate friction angle is used to characterize placement between the base and the block for an input sinu- the interface. Comparison of physical model test results to the soidal acceleration with an amplitude of 2.0g and a frequency results of best-fit numerical analyses demonstrates that the of 2.66Hz. In this case, a slip-slip mode of response is ob- appropriate friction angle depends upon the velocity of sliding. served, but the upslope movement is relatively small com- However, the rate dependence appears to be slightly less than pared to the downslope movement. deduced in a previous study using a simpler back analysis.

2.50 0.25 \ X-Acceleration 0.20 X-Acceleration X -Acceleration 2.00 (a) Base \ (b) 0.15 Base Block 1.50 X -Acceleration 0.10 1.00 Block 0.05 0.50 0.00 0.00 -0.05 -0.50

Acceleration (g) -1.00 X-Acceleration

Acceleration (g) -0.10 -0.15 -1.50 Block -0.20 -2.00 X-Acceleration X-Velocity Y-Acceleration -0.25 Y-Acceleration -2.50 X-Velocity Block Base Block Block 2.50 0.15 Base Relative Velocity Block 0.10 Relative Velocity 2.00 Block 0.05 1.50

0.00 1.00 X-Velocity -0.05 0.50 Velocity (m/s) Velocity Velocity (m/s) Velocity Block

-0.10 X-Velocity 0.00 Block -0.15 -0.50 Upward 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Time (S) Time (S) 8.00 3.40 X\-Relative Displacement \ 3.20 7.00 X -Relative Displacement Block Block 3.00 6.00 2.80

5.00 2.60 Upward Upward 2.40 4.00 Displacement (m) Displacement (cm) 2.20 3.00 2.00

2.00 1.80

Slip Slip Slip Slip Slip Stick Slip Fig. 15. The calculated acceleration, velocity and the displacement of the rigid block on an inclined plane (a) slip-stick response, (b) slip-slip response

Paper No. 4.08b 10

The model described herein provides a basis for analysis of a Kavazanjian, E., Jr., Hushmand, B., and Martin, G.R. [1991]. variety of important problems in geotechnical earthquake en- Frictional Base Isolation Using a Layered Soil-Synthetic Liner gineering in a more rigorous manner than currently employed System", Proc., 3rd U.S. Conference on Lifeline Earthquake in engineering practice. The model is easily extended to con- Engineering, ASCE Technical Council on Lifeline Earthquake duct fully coupled analyses of the seismic response of com- Engineering Monograph No. 4, 1139-1151. pliant geotechnical structures with well-defined sliding surfaces. Problems that can be addressed using the extended model Kavazanjian, E., Jr., Matasovic, N. and Caldwell, J. (1998), include the cumulative seismic displacement of landfills, “Damage Criteria for Solid Waste Landfills,” Proc. Sixth U.S. slopes, embankments, and gravity retaining walls and the National Conference on Earthquake Engineering (on CD stress induced in geosynthetic elements of landfill liner and ROM), Seattle, Washington, 31 May – 4 June. cover systems by ground shaking. Kavazanjian, E., Jr. and Matasovic, N. [1995], “Seismic Anal- ysis of Solid Waste Landfills”, In: Geoenvironment 2000, ASCE Geotechnical Special Publication No. 46, ACKNOWLEDGMENT Vol. 2, pp. 1066-1080.

The work in this paper is part of a joint Arizona State Univer- Kramer, S. L., and Smith, M. W. [1997]. ‘‘Modified Newmark sity / Ohio State University / Geosyntec Consultants research model for seismic displacements of compliant slopes’’, J. program titled GOALI: Collaborative Research: The Integrity Geotech. Engrg., ASCE, 123(7), 635–644. of Geosynthetic Elements of Waste Containment Barrier Sys- tems Subject to Large Settlements or Seismic Loading. This Makdisi, F. I., and Seed, H. B. [1978]. ‘‘Simplified procedure project is funded by the Geomechanics and Geotechnical Sys- for estimating dam and embankment earthquake-induced de- tems, GeoEnvironmental Engineering, and GeoHazards Miti- formations’’, J. Geotech. Engrg. Div., ASCE, 104(7), 849– gation program of the National Science Foundation Division 867. of Civil, Mechanical, and Manufacturing Innovation under grant number CMMI-0800873. The authors are grateful for Matasovic N. [1993]. “Seismic Response of Composite Hori- this support. The authors also wish to thank Prof. Joseph zontally-Layered Soil Deposits”, Ph.D. Dissertation, Civil Wartman of Drexel University for providing the shaking table Engineering Dept., Univ. of California, Los Angeles, Calif., results in digital form. 483 p.

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