Effect of Non-Uniform Gravitational Field on Seismically-Induced Ground Movements in Centrifuge Models Antonios Vytiniotis Andrew J.

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1 NSF GRANT # CMS ana NSF PROGRAM NAME: Seismic Risk Management for Port Systems Effect of Non-Uniform Gravitational Field on Seismically-Induced Ground Movements in Centrifuge Models Antonios Vytiniotis PhD Student, Massachusetts Institute of Technology Andrew J. Whittle Professor, Massachusetts Institute of Technology Abstract: One of the elements for the NEES-GC project on Seismic Risk Reduction for Port Facilities involves validation of numerical models against physical experiments using the geotechnical centrifuge. For example, tests have been performed to demonstrate the effectiveness of Prefabricated Vertical (PV) drains in mitigating liquefaction risks. This paper examines the effect of the non-uniform gravitational field on the seismically-induced ground movements in these experiments, by means of 2-D non-linear, time domain finite element analyses with coupled deformation and seepage. Results show that, even for beam centrifuge devices with very large arm radius, the effects of nonuniform gravitational field have a significant effect on ground deformations, and should always be taken into account in numerical simulations. 1. Introduction: Use of centrifuge models for validating the predictions of numerical models has been a very significant issue in geotechnical earthquake engineering research. However, in many cases large discrepancies exist between predicted and measured displacements under basal shaking (Arulanandan et al, 1993). These discrepancies have been attributed to factors such as the variation of soil permeability during and after a shaking event. It has been suggested that during shaking, grains loose contact from each other. This phenomenon alters the soil s pore shape factor, increasing the permeability during shaking, and increasing settlements (Arulanandan et al, 1993, Popescu & Prevost, 1993). Another important mechanism is considered here: the effect of non-uniformity of the gravitational field. The acceleration field inside modern centrifuge devices varies both horizontally and vertically inside the centrifuge model, as seen in Figure 1. Increasing the arm radius improves the uniformity of the acceleration field. The radial variation of the gravitational field has been discussed and taken into account by Schofield (1980). Typical results show that the variation of the vertical stress field is quite small (e.g. 2% for models, with depth to radius d/r=10%) but its effect on ground displacements is not discussed. This analysis also does not account for the very important effects on the Figure 1 Non-Uniform Gravitational Acceleration field Figure 2 Empirical Correction for Non-Uniform Gravitational Acceleration field (Arulanandan et al, 1992)

2 horizontal stress field. It is also commonly suggested that the radial vectors of the gravitational acceleration field can be assumed to be parallel, when the ratio of model width to centrifuge arm length (w/r) is less than 10% (Wood, 2004). During the test the laminar box moves with constant angular velocity, ω. Inside the laminar box, acceleration of gravity (G i ) has a horizontal (G x i ) and a vertical component (G y i ): (1) The distance of a point R i from the represented as: (2) (3) center of rotation is (4) x y Where r i and r i are the horizontal and vertical components of the position vector r i relative to the center of rotation: (5) This effect has been well known and has been taken into account in indirect ways. A simple case scenario is shown in Figure 2. If the initial soil surface before the test is a horizontal surface, after shaking and occurrence of liquefaction, the top surface of the centrifuge model tends to move towards an equipotential surface. In the limiting case, if no shear resistance is left in the soil, this surface will become cylindrical with radius the same as the radius of the arm of the centrifuge. As seen in Figure 2, a curve is fitted so that areas below and above (A and B respectively) the surface compared to the initial surface are the same. Thus the predicted data are corrected, so that they can be used to validate numerical analyses performed with uniform vertical acceleration field (Arulanandan et al, 1993). This approach has two errors. It assumes that the radius of the surface curvature formed by the end of the shaking event is equal to the length of the arm of the centrifuge. The final curvature in reality depends on the nature of soil, the existence of water in the model, the degree to which the sand layers have liquefied, the particular model geometry, and other factors. On the other hand, this empirical approach cannot be used to predict the variation of this correction with time. This paper considers the effects of the non-uniformity and radial components of the gravitational acceleration field inside a centrifuge test. The discussion is based on a centrifuge test performed at UC Davis (Kamai et al, 2008) performed in order to evaluate the use of PVmitigation technique. This Drains as a liquefaction risk test has been selected as a suitably complicated geometry that should be used to validate the complicated effects in question. A comparison has been made between analyses that include only uniform vertical acceleration components and radially varying gravitational acceleration fields. 2. Numerical Simulation of a Centrifuge Test: The OpenSees finite element framework has been used in the numerical analyses of the centrifuge tests. OpenSees can be used to perform dynamic coupled-pore pressure displacement analyses using advanced elasto-plastic models. The current analysess use the u-p approximation (Zienkiewicz et al, 1999) to simulate the events of interest, as it has been shown to be a computationally efficient and accurate assumption for most geotechnical earthquake engineering problems (Zienkiewicz et al, Figure 3 Geometry of the Numerical Model

3 1980). More advanced approximations such as the u-p- for even faster w approximation are mostly needed events such as blasting (López-Querol et al, 2008), or more accurate predictions along boundaries (Zienkiewicz et al, 1999). Openseess has been validated against simple one-dimensional problems and results show excellent agreement between numerical and analytical solutions (Vytiniotis, 2009). Also, OpenSees has been compared against the finite difference software Flac with good agreement (Mejia, 2007). Table 1 Model Properties for PD-MYS02 Constitutive Model Dense Parameter Nevada Sand (D r = 80%) ρ (Mgr/m 3 ) 2.07 G ref (kpa) K ref (kpa) φ ( ) 36.5 γ peak 0.1 p ref (kpa) 80 ψ PT ( ) 26.0 c c d d Loose Nevada Sand (D r =40%) The centrifuge model chosen for this comparison is the SSK01 test (Kamai et al., 2008) performed at UC Davis. UC Davis centrifuge device has a radius of 9.1m to bucket floor, a maximum payload mass of 4500 kg, and an available bucket area of 4.0 m 2. It is considered to be one of the largest geotechnical centrifuges in the world. It can produce 75g's of centrifugal acceleration at its effective radius of 8.5m. ( Figure 3 shows the geometry and properties of the scale model. It consists of two facing slopes of Yolo loam, that cap a layer of medium-loose Nevada sand. The left side of the model is treated with PV drains made of nylon, in order to acceleratee the dissipation of excess pore pressure during liquefaction. The model dimensions give a ratio of model depth to effective radius, d/r=5.1% and a ratio of model width to effective radius, w/r=19.4%. Thus the approximation of parallel vectors of gravitational acceleration is not sufficient in this model. Table 2 Model Properties for PI-MYS Constitutive Model Parameter Input ρ (Mgr/m 3 ) 1.3 G ref (kpa) K ref (kpa) C (kpa) 6.0 γ peak 0.1 Figure 3 also illustrates the main features of the Opensees finite element model used to simulate centrifuge test SSK01. Coupled flow and deformation in the high permeability sand layers are represented by QuadUP elements which include 4-nodes for bilinear interpolation of displacements and pore pressures. The effective stress-strain-strength properties of the Nevada sand are characterized by the pressure dependent multiyield surface model (PD-MYS02; Yang et al., 2002), Figure 4 Validation of Numerical Model assuming Uniform Gravitational Acceleration against Centrifuge Test

4 Figure 5 Effect of Gravitational Field on Predictions of Pore Pressures Figure 6 Effect of Gravitational Field on Predictions of Horizontal Accelerations Table 1. The capping layer of low permeability, total

5 with separate sets of input parameters for the dense and loose Nevada sand layers. The model properties used are shown in Table 1, based on data from Arulmoli (1992). The capping layer of low permeability, compacted Yolo loam is represented by 4-noded Quad elements (bilinear displacement interpolation using total stresses), and its undrained mechanical properties are represented by the pressure independent multi-yield surface model (PI-MYS), a model based on the conventional multi-surface plasticity framework (Iwan, 1967), with parameters shown in Table 2. The constitutive models have been ested and shown to predict reasonably the stress strain characteristics of the test soils in element tests (Vytiniotis, 2009). The centrifuge model is built within a laminar box. This design ensures equal horizontal displacements at the lateral boundaries of the centrifuge model. These periodic boundary conditions are represented in the finite element model by constraining nodes at the left and right boundaries with equal displacement degrees of freedom, while the mass of the plates is added to these boundary nodes. The sand layers are continuous across the model and hence, remain in contact with the box walls throughout shaking. In contrast, the Yolo loam forms a partial cap and is absent in the central channel. During shaking events, the loam can separate from the walls of the laminar box. This behavior is modeled by introducing zero-thickness, no-tension elements between the Yolo loam and the walls of the box. The array of PV drains is installed on the left-side of the centrifuge model and is represented in the finite element model by a series of uniformly-spaced line elements. The elements are used predict the uncoupled mechanical deformation and flow inside the drains. Flow is assumed to be laminar, and parameters are selected using the Darcy-Weisbach equation (Vytiniotis, 2009). The water table is located at the top of the sand layer in the centrifuge model, while the PV drains discharge at the ground surface (i.e. above the Yolo loam). Hence, there is a significant storage effect as described by Pestana et al (1997). This effect can also be represented by the utilized drain elements. The finite element approximation of planar flow (vs. radial flow in the centrifuge model) is represented using an equivalent hydraulic conductivity (Hird et al., 1992) to match the average degree of consolidation within the surrounding soil mass. Seismic loading is represented by applying a uniform basal excitation (acceleration) across all nodes. The load history used in SSK01 mode comprises of five sequences of basal shaking at different intensities. The first 2 cycles of very low intensity (a max /g=0.01, 0.03) are used to test the model and the monitoring equipment. The last three shaking events (a max /g=0.07, 0.11, 0.3) constitute the part of the experiment that is used in the current validation. The simulations were performed at prototype scale. Figure 7 Effect of Gravitational Field on Predictions of Horizontal Displacements

6 3. Results: Figure 4 compares typical predictions of the numerical model assuming a uniform vertical field of gravitational acceleration, with experimental measurements. These comparisons show that the analysis describes realistically the excess pore pressures in the untreated side (point E) but underestimates pore pressures on the treated side of the model (point B). There is good agreement with measured accelerations on the treated side (point A). However, the analysis does not predict correctly the de-amplification associated with liquefaction on the untreated side of the model (point D). Further investigation of these discrepancies is the subject of on-going research, as the relative density and permeability of the sand have not been measured accurately. Inadequacies of the soil model can also significantly affect the predictions. After this initial validation of the numerical model, a non-uniform radially varying gravitational field was applied to the numerical model and the same analysis was repeated. The Coriolis effect was ignored. As we can see from the comparison of predicted pore pressures in Figure 5 the change in gravitational field has no practical influence on the generation of excess pore pressures, at point A and B for three loading events. In Figure 6, the comparison of predicted horizontal accelerations also shows small differences due to changes in the gravitational field. Smaller spikes of high acceleration are observed in the non-uniform case because the gravity counteracts the rotation mechanism observed on the right side of the model. In Figure 7 we see differences in the predicted horizontal displacements. Much smaller displacements are observed in the non-uniform case since the radially changing gravity field reduces the tendency of the slope to move towards the central channel. A similar comparison of vertical displacements (Figure 8) shows a significant effect for large base shaking (a max /g=0.3) only on the untreated side of the model, where liquefaction occurs. Figure 9 (a) shows the differences in computed permanent vertical displacements for each phase of shaking under the assumptions of uniform and non- uniform gravitational field. It is clear that as the level of shaking increases, and as time elapses during a shaking event, these discrepancies become larger and larger. Figure 9 (b) shows the permanent surface settlements from the third shaking event. During shaking, the sides of the model tend to heave and the center of the model tends to settle due to the actual shape of the gravitational field. Small discrepancies exist on the left side of the model since the PV drains limit significantly the vertical settlements, due to both the drainage action and their vertical stiffness. The discrepancies between the two approximations are dependent on the specific model characteristics. So, there is no empirical approach that could allow for accurate correction of the centrifuge results for the non-uniformity of the stress field. Figure 8 Effect of Gravitational Field on Predictions of Vertical Settlements

7 4. Conclusions: The effect of non-uniform gravitational acceleration field should always be considered when simulating the response of liquefiable sands in centrifuge tests. From the numerical analyses, it has been found that the non-uniformity of the acceleration field affects significantly both the horizontal and the vertical components of deformations. Numerical models should always take into account the exact gravitational acceleration field, otherwise their results can only be qualitatively correct. This phenomenon is very important and easier to quantify than other mechanisms affecting liquefaction induced settlements. Figure 9 Effect of Gravitational Field on Predictions of Vertical Settlement Profiles 5. Acknowledgements: The authors would like to acknowledge support from NSF grant No. CMS , Seismic Risk Management for Port Systems. Antonios Vytiniotis would like to also acknowledge support from the Alexander S. Onassis Public Benefit Foundation. 6. References: [1] K. Arulanandan, J.J. Subico, Post liquefaction settlement of sands, Proceedings of the Wroth Memorial Symposium, England: Oxford University, [2] K. Arulmoli, K.K. Muraleetharan, M. M. Hosain, L. S. Fruth VELACS Laboratory Testing Program, Soil Data Report, The Earth Technology Corporation, Irvine, California, Report to the National Science Foundation, Washington D.C., March, [3] C.C. Hird, I.C. Pyrah, D. Russell, Finite Element Modelling of Vertical Drains beneath Embankments on Soft Ground, Géotechnique, vol. 42, no. 3, pp , [4] W.D. Iwan, On a class of models for the yielding behavior of continuous and composite systems, J. Applied Mechanics, ASME, vol. 34, pp , [5] R. Kamai, S. Kano, C. Conlee, A. Marinucci, R. Boulanger, E. Rathje, Evaluation of the Effectiveness of Prefabricated Vertical Drains for Liquefaction Remediation: Centrifuge Data Report for SSK01, Center for Goetechnical Modeling, University of California at Davis, [6] S. López-Querol, J.A. Fernandez-Merodo, P. Mira, M. Pastor, Numerical Modelling of Dynamic Consolidation on Granular Soils, International Journal for Numerical and Analytical Methods in Geomechanics, vol. 32, pp , [7] S. Mazzoni, F. McKenna, G.L. Fenves, OpenSees Command Language Manual, [8] L. Mejia, Z. Yang, Critical Assessment of OpenSees for Seismic Design of Pile Foundations Subject to Liquefaction Hazards, Report Prepared for PEER Center, College of Engineering, University of California, Berkeley, [9] J.M. Pestana, C.E. Hunt, R.R. Goughnour, A.M. Kammerer, Effect of Storage Capacity on Vertical Drain Performance in Liquefiable Sand Deposits, [10] R. Popescu, J.H. Prevost, Numerical Class A Predictions for Models Nos 1, 2, 3, 4a, 4b, 6, 7, 11 &12, Verifications of Numerical Procedures for the Analysis of Soil Liquefaction Problems, Balkema, Rotterdam, pp , [11] A.N. Schofield, Cambridge geotechnical centrifuge operations, Géotechnique, Vol. 30, no. 3, pp , [12] A. Vytiniotis, Numerical Simulation of the Response of Sandy Soils Treated with Pre-Fabricated Vertical Drains, Master s Thesis, Massachusetts Institute of Technology, [13] D.M. Wood, Geotechnical Modelling, Taylor and Francis, [14] Z. Yang, A. Elgamal, E. Parra, Computational Model for Cyclic Mobility and Associated Shear Deformation, Journal of Geotechnical and Geoenvironmental Engineering, vol. 129, no. 12, pp , [15] O. Zienkiewicz, A. Chan, M. Pastor, B. Schrefler, T. Shiomi, Computational Geomechanics, Chichester: John Wiley & Sons Ltd., [16] O.C. Zienkiewicz, C.T. Chang, P. Bettess Drained, Undrained, Consolidating, and Dynamic Behavior Assumptions in Soils, Géotechnique, vol. 30, no. 4, pp , 1980.