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NUMERICAL STUDIES OF SEISMICALLY INDUCED SLOPE DEFORMATION USING SMOOTHED PARTICLE HYDRODYNAMICS METHOD
ABSTRACT
There has been growing interest in improving current procedures for estimating seismically-induced deformations of natural and man-made slopes due to recently frequent earthquake events and the resulted damaged to infrastructure systems. The aim of this study is to develop a numerical model to effectively and reliably assess seismically-induced slope deformations that typically involve large deformations and complex soil constitutive behaviors. A numerical model based on the meshfree Smoothed Particle Hydrodynamics (SPH) method has been developed by implementing various advanced constitutive models into the SPH formulations. The developed model is validated by two readily available and well-documented experiments: axisymmetric collapses of granular columns and model slope tests on a shaking table. For the former, the non-dilatant Drucker-Prager (D-P) constitutive relationship with perfect plasticity is used. The developed model precisely reproduces the experimentally-observed three regimes of flow patterns based on the initial aspect ratio of the granular column. In addition to the flow patterns, the simulated final deposit height and run-out distance along with the non-deformed region after the collapse of granular columns are in excellent agreement with experimental data in the literature. For the latter, a constitutive model that combines the strain-softening viscoplasticity and Modified Kondner and Zelasko (MKZ) rule is implemented and utilized to account for the effects of wave propagation in the sliding mass, cyclic nonlinear behavior of soil, and progressive reduction in shear strength during sliding, which are not explicitly considered in various Newmark-type analyses widely used in the current research and practice in geotechnical earthquake engineering. The initiation of slope failure and subsequent progressive development of the sliding surface are successfully captured by the developed SPH model. A localized shear band along the failure surface and a bulge near the toe of the model slope are observed in the simulations, showing a good agreement with the experimental observations. The simulated failure mode, displacement time histories, and acceleration response spectra at several monitor locations along the model slope also agree well with the experimental recordings.
Based on the validated SPH model, a parametric study is followed to investigate the effects of spatial parameters including both particle spacing and smooth length on the accuracy of SPH simulations. The parametric study also investigates the effects of material strength and shear modulus along with boundary conditions on the seismicallyinduced slope deformations, providing insights into the mechanisms of earthquakeinduced slope deformations. It is thus suggested that the proposed SPH model is an effective tool for assessing the seismic performance of soil slopes. It may be also used to advance the computational capability of modeling geotechnical engineering phenomena involving large deformations.
TABLE OF CONTENTS
LIST OF FIGURES…………………………………………………………………………………………….. vii
ACKNOWLEDGEMENTS………………………………………………………………………………….. xii
Chapter 1 Introduction…………………………………………………………………………………………… 1
1.1 Motivation……………………………………………………………………………………………….. 1
1.2 Pseudostatic Analysis……………………………………………………………………………….. 3
1.3 Permanent Displacement Analysis…………………………………………………………….. 5
1.3.1 Newmark rigid-block analysis…………………………………………………………… 5
1.3.2 Derivatives of Newmark analysis………………………………………………………. 8
1.4 Stress-Deformation Analysis and Advanced Computational Methods……………… 9
1.4.1 Arbitrary Lagrangian Eulerian method (ALE)…………………………………… 10
1.4.2 Discrete Element Method (DEM)…………………………………………………….. 12
1.4.3 Mesh-free Methods………………………………………………………………………… 14
1.5 Dissertation Scope and Layout…………………………………………………………………. 18
References……………………………………………………………………………………………………. 20
2.1 Introduction……………………………………………………………………………………………. 27
2.2 Numerical Implementation………………………………………………………………………. 29
2.2.1 Governing and constitutive equations for soil……………………………………. 29
2.2.2 SPH formulations…………………………………………………………………………… 32
2.2.3 Time integration…………………………………………………………………………….. 37
2.3 Model Validation……………………………………………………………………………………. 39
2.4 Numerical Simulation for 3-D Granular Flows…………………………………………… 42
2.4.1 Granular flow patterns……………………………………………………………………. 44
2.4.2 Final runout distance………………………………………………………………………. 49
2.4.3 Final deposit height………………………………………………………………………… 50
2.4.4 Non-deformed region……………………………………………………………………… 51
2.5 Conclusions……………………………………………………………………………………………. 54
References……………………………………………………………………………………………………. 56
Chapter 3 Simulation of Earthquake-induced Slope Deformation Using SPH Method…. 59
3.1 Introduction……………………………………………………………………………………………. 60
3.2 SPH Formulations…………………………………………………………………………………… 65
3.3 Constitutive Model………………………………………………………………………………….. 71
3.3.1 Finite strain consideration……………………………………………………………….. 72
3.3.2 Isotropic strain softening viscoplasticity…………………………………………… 74
3.3.3 Nonlinear cyclic behavior of soil……………………………………………………… 82
3.4 Explicit Time Integration…………………………………………………………………………. 89
3.5 Model Calibration…………………………………………………………………………………… 91
3.5.1 Calibration of viscous parameters in the Peirce viscoplastic model……… 92
3.5.2 Calibration of strain softening parameters…………………………………………. 98
3.5.3 Calibration of curve-fitting parameters in MKZ model…………………….. 102
3.5.4 Calibration of rate-dependent stiffness of clay…………………………………. 104
3.6 Model Validation………………………………………………………………………………….. 108
3.6.1 Slope failure mode……………………………………………………………………….. 109
3.6.2 Acceleration response spectra………………………………………………………… 113
3.6.3 Slope deformation………………………………………………………………………… 115
3.7 Conclusions………………………………………………………………………………………….. 119
References………………………………………………………………………………………………….. 122
Chapter 4 Parametric Study………………………………………………………………………………… 131
4.1 Introduction………………………………………………………………………………………….. 131
4.2 Viscoplastic Parameters…………………………………………………………………………. 132
4.3 Spatial Parameters in SPH Method………………………………………………………….. 136
4.3.1 Particle-spacing effects…………………………………………………………………. 136
4.3.1.1 Rate-independent model……………………………………………………. 137
4.3.1.2 Rate-dependent model………………………………………………………. 144
4.3.2 Smoothing-length effects………………………………………………………………. 147
4.3.2.1 Rate-independent model……………………………………………………. 148
4.3.2.2 Rate-dependent model………………………………………………………. 149
4.4 Effects of Soil Properties on Slope Deformations……………………………………… 150
4.4.1 Residual strength and plastic modulus……………………………………………. 151
4.4.2 Peak strength……………………………………………………………………………….. 160
4.4.3 Shear modulus…………………………………………………………………………….. 161
4.5 Non-reflecting Boundary……………………………………………………………………….. 164
4.5.1 Mathematical formulation…………………………………………………………….. 167
4.5.2 Validation of quiet boundary against FEM……………………………………… 171
4.5.3 Modeling slope test with quiet boundary…………………………………………. 175
4.6 Conclusions………………………………………………………………………………………….. 178
References………………………………………………………………………………………………….. 180
Chapter 5 Conclusions and Recommendations………………………………………………………. 182
5.1 Conclusions………………………………………………………………………………………….. 182
5.2 Recommendations for Future Work…………………………………………………………. 186
5.2.1 Solid-fluid couplings in saturated soils……………………………………………. 186
5.2.2 Correction factors for Newmark analysis………………………………………… 187
5.2.3 Coupled SPH-FEM method for soil-structure interactions………………… 188
References………………………………………………………………………………………………….. 189
Appendix A SPH Simulation of Collapse of Two-dimensional Granular Column……… 190
Appendix B SPH Simulation of an Elastic Cantilever Beam…………………………………… 195
CHAPTER 1 INTRODUCTION
1.1 Motivation
Landslides are one of the most damaging hazards generated by earthquakes,
particularly in hilly and mountainous terrains. Large earthquakes are capable of triggering thousands of landslides within land areas approaching 100,000 km2 around the quake (Keefer 1984). For example, the 1994 earthquake (M 6.7 ) in Northridge, California, triggered more than 11,000 landslides over an area of about 10,000 km2 (Harp and Gibson 1995). The recent 2008 Wenchuan earthquake ( M 8.0 ) reportedly triggered more than 50,000 geohazards in the forms of landslides, rockfalls, and debris flows (Huang and Xu 2008). Earthquake-induced landslides pose a significant threat to human lives and infrastructure systems (e.g., roads, pipelines, utilities). Therefore, predicting the location and shaking conditions needed to trigger landslides is a key element in regional seismic hazard assessment (Jibson et al. 1998).
Methods for assessing the performance of a slope during earthquakes fall into three categories (Jibson 2011): (1) pseudostatic analysis, (2) permanent displacement analysis, and (3) stress-deformation analysis. Pseudostatic analysis, used mainly for preliminary or screening analysis due to its crude characterization of the physical processes (Jibson 2011), can only indicate the possibility of slope failure. As a significant improvement to pseudostatic analysis, permanent displacement analysis can estimate the consequences after the incidence of slope failure. It provides a more quantitative measure to evaluate the performance of slopes during earthquakes. Both pseudostatic analysis and permanent displacement analysis are based on highly simplified geometric and material models. However, these two methods are unable to reliably evaluate earthquake-induced slope deformations under complex geological conditions. To fill these needs, stressdeformation analysis, usually performed using computational methods such as the Finite Element Method (FEM) and Finite Difference Method (FDM), has been extensively applied to critical projects and complex earth structures. This approach can account for complex soil behaviors and geometric conditions. The continuum-scale grid based methods (e.g. FEM and FDM) have difficulties in modeling large deformations due to severe grid distortion and entanglement. As a result, stress-deformation analysis is currently limited to estimating relatively small seismically-induced slope deformations (Jibson 2011). The drawback of grid-based methods in dealing with large deformations therefore considerably impedes their application in the analysis of earthquake-induced slope deformations. Therefore, the permanent displacement analysis prevails in the current geotechnical engineering practice.
In order to overcome these drawbacks in stress-deformation analysis, various methods have been proposed. Arbitrary Lagrangian Eulerian (ALE) method is a remedy to FEM in dealing with highly deforming materials on a continuum scale. In ALE method, the mesh moves in an arbitrary manner which is independent of the motion of the material being analyzed. Since the mesh is not connected to the material, state variables on ALE grids must be properly mapped from the original grids. This arbitrary meshing scheme can effectively avoid severe distortion of purely Lagrangian grid.
Nevertheless, this method encounters tremendous challenges when applied to modeling ultimate deformations under dynamic loading (Liu and Liu 2004) and when complex material models are used. Another method to handle large deformations is the Discrete Element Method (DEM) (Cundall and Strack 1979) which is a particle method formulated on a micro-structural scale. As opposed to FEM that treats material as a continuum, DEM treats material as discrete particles and is widely accepted as an alternative to FEM in solving engineering problems in granular and discontinue materials. DEM is a computationally intensive method and, at present, is restricted to small-scale simulations.
This study is motivated by the lack of an efficient, effective and reliable method capable of modeling earthquake-induced slope deformations. A more detailed review of current approaches in evaluating seismic slope stability is presented during this chapter.
1.2 Pseudostatic Analysis
As the earliest attempt at analyzing seismic effects on slopes, pseudostatic analysis has been commonly used in engineering practice. Pseudostatic analysis calculates a factor of safety (FS) using a limit equilibrium method in which the seismic shaking is represented by a constant inertial force applied on a sliding mass. As shown in Figure 11, the horizontal inertial force is expressed by the product of pseudostatic coefficient k and weight of the sliding mass W . The pseudostatic coefficient is defined as
ah (1-1) k
g
where ah is the horizontal acceleration of ground motion and g is the acceleration of gravity.
Figure 1-1. Pseudostatic slope stability analysis
A common procedure in pseudostatic analysis is to iteratively conduct limit equilibrium analysis with different values of k until the FS approaches to unity. The resulting pseudostatic coefficient is called the yield coefficient k y . The slope is considered unsafe if the horizontal acceleration of ground motion exceeds the yield acceleration (i.e., ay kyg ).
A major weakness of pseudostatic analysis is that it assumes the seismic force is constant and acts in one direction. This method tends to be over conservative in many situations. However, it has been shown that pseudostatic analysis is not conservative for soils that are susceptible to pore pressure build up or losing more than 15% of their peak shear strength during seismic shaking (Kramer 1996). Another limitation of pseudostatic analysis is that it is unable to predict the consequences of slope instability. The analysis can be used to evaluate the stability of slopes, but it is unable to assess the movement of sliding mass after limit equilibrium is exceeded.
1.3 Permanent Displacement Analysis
For engineering practice, it is of particular interest to predict the deformation of natural and man-made slopes under seismic shaking. The predicted seismically-induced permanent deformation is a useful design index as it indicates the potential damage to the slope or an earth structure founded on the slope. In order to overcome the drawbacks of pseudostatic analysis, Newmark (1965) proposed a displacement-based procedure to evaluate the serviceability of slopes under earthquake shaking. Newmark analysis and its various derivatives are widely used in the current geotechnical engineering practice to evaluate earthquake-induced slope deformations.
1.3.1 Newmark rigid-block analysis
The Newmark rigid-block method analogizes an earth mass sliding along a shear failure surface to a rigid block sliding over an inclined plane. This analogy is illustrated in Figure 1-2. The permanent slope deformation induced by earthquakes is estimated by the permanent displacement of the rigid block sliding along the inclined plane under a base acceleration. The block has a yield or critical acceleration ay . This yield acceleration can be determined through the aforementioned pseudostatic analysis.
Figure 1-2. Analogy between potential sliding mass and block on an inclined plane
Figure 1-3 presents the essential idea of Newmark rigid-block analysis. Figure 1-3(a) shows an acceleration time history. The sliding is initiated after the ground acceleration exceeds the critical acceleration as shown in Figure 1-3(a). The acceleration values of the record that exceed the yield acceleration are integrated to produce the relative velocity time history of the sliding mass as shown in Figure 1-3(b). The relative velocity time history is subsequently integrated as a function of time to obtain the cumulative displacement shown in Figure 1-3(c). Displacement continues and the permanent displacement accumulates until the inertial forces fall below the yield resistance, and the velocities of the sliding mass and the slip surface coincide. The Newmark integration routine has been implemented into readily available computer software (e.g., Jibson and Jibson 2003).
Newmark analysis is based on the following simplifying assumptions: (1) the soil behaves in a rigid (i.e., neglecting wave propagation in the soil), perfectly plastic manner; (2) displacement occurs along a single, well defined slip surface; and (3) the soil does not undergo degradation of strength and stiffness during shaking (Wartman et al. 2003). Therefore, Newmark-type analyses are more applicable to shallow landslides in more brittle materials rather than to deeper landslides in softer materials (Jibson 2007). As shallow landslides are the predominant failure mechanism in earthquake-induced landslides, Newmark-type analysis has been widely used to estimate earthquake-induced slope deformations.
Figure 1-3. Illustration of Newmark analysis: (a) acceleration time history; (b) sliding velocity time history; (c) sliding displacement time history
1.3.2 Derivatives of Newmark analysis
Various modifications to the sliding block procedure have been proposed. The method proposed by Makdisi and Seed (1978) is a widely used analysis that accounts for internal deformations and wave propagation in sliding masses. This method consists of two steps. First, a one-dimensional dynamic analysis of the earth slope is performed by representing the slope as a multiple degree-of-freedom system. Based on the calculated horizontal acceleration histories at different degrees of freedom, an average acceleration history of the sliding mass is developed. This average acceleration is commonly referred to as the horizontal equivalent acceleration (HEA). Second, the resulting HEA is used as the acceleration time history in Newmark’s rigid-block analysis for estimating permanent displacements of a sliding mass above the potential failure surface. This method, in which the dynamic response analysis and displacement analysis are conducted
independently, is referred to as decoupled analysis.
On the other hand, the dynamic response and sliding displacements of a slope are modeled in a simultaneous manner in the coupled analysis method which was proposed by Lin and Whitman (1983) and Rathje and Bray (1999, 2000). It models the sliding mass above a potential sliding surface as a multiple degree-of-freedom system. The sliding effects on the dynamic system are accounted for by introducing the sliding force at the sliding interface into the dynamic motions. As a result, the permanent displacements of the sliding mass are obtained directly by integrating the 1-D dynamic equations.
The Newmark method and its various derivatives discussed above have been
validated against laboratory tests performed on shaking tables or centrifuge (e.g. Yegian and Lahlaf 1992; Wartman 1999, Wartman et al. 2003, 2005), and against case histories of earthquake-induced landslides (e.g., Pradel et al. 2005). After comparing Newmark predicted slope deformation with a well documented case history of earthquake-induced landslide movement during the Northridge earthquake ( M 6.7 ), Pradel et al. (2005) concluded that Newmark-type sliding block analysis can result in reasonable estimates of seismic displacements for landslides using site-specific geotechnical analyses. However, uncertainties in groundwater level, ground motion characteristics, and shear strength of on-site soil should be considered when performing this type of analysis to estimate seismic slope performance (Pradel et al. 2005). Therefore, deformation predicted by Newmark-type analysis is merely a useful index of how a slope is likely to perform during seismic shaking (Jibson et al. 1998).
1.4 Stress-Deformation Analysis and Advanced Computational Methods
Using highly simplified geometry and material models, the analytical procedures described above are not physically precise and unable to reliably evaluate slope performance under complex geological conditions (e.g., topography, soil profile, and seismic shaking). To overcome these limitations, computational techniques such as the finite element method (FEM) (e.g., Finn et al. 1986; Crosta et al. 2005) and finite difference method (FDM) (e.g., Bathurst and Simac 1994) have recently been used to evaluate earthquake-induced slope deformations. Adaptive for complex geological conditions and capable of simulating complex soil behaviors, numerical simulation can provide insight into the mechanism of earthquake-induced slope deformations, particularly the initiation and subsequent progressive movement of seismic slope failure. To simulate these phenomena with high fidelity, it is imperative to establish a numerical model with the capability of handling post-failure large deformation which is a key feature in earthquake-induced slope failures. The majority of current stress-deformation analysis of seismic slope deformation is limited to small deformation cases due to the limitation of classical FEM and FDM in solving large deformations.
Computational methods that are widely used to simulate large-deformation problems can be grouped into three categaries: (1) grid-based continuum scale method improved by adaptive techniques, such as adaptive meshing and Arbitrary Lagrangian Eulerian (ALE); (2) particle-based mirco-structure scale method such as DEM; and (3) mesh-free continuum scale method, such as the smoothed particle hydrodynamics (SPH) method and element free Galerkin Method (EFG).
1.4.1 Arbitrary Lagrangian Eulerian method (ALE)
Grid-based Lagrangian numerical methods formulated at a continuum scale, such as FEM and FDM, generally have difficulty in modeling large deformations because of a severely distorted and entangled mesh that may result in numerical error and failure of convergence. An adaptive remeshing procedure (e.g., Khoei and Lewis 1999) has been used to remediate this issue in the Lagrangian approach. This remeshing technique completely rezones the model, which is computationally expensive and technically formidable for three dimensional problems. Another alternative is the description of material movement in an Eulerian viewpoint in which the mesh is stationary and the material flows through the mesh. Eulerian approach is largely used in computational fluid dynamics and is well suited for high-deformation flows. This method, however, becomes problematic in modeling free surface flow and tracing material response. The Level Set Method (LSM) (e.g., Osher and Fedkiw 2002) has recently been used to capture freesurface and interfaces of fluid-like materials on a fixed Eulerian grid. A surface function dependent on material velocity in LSM enables this method to produce more smooth and realistic solutions on free surface flows than conventional Eulerian methods.
Arbitrary Lagrangian-Eulerian (ALE) method (e.g., Belytschko et al. 2000; Donea et al. 1982; Huges et al. 1981) is a very effective alternative and is currently becoming a standard numerical approach for solving large-deformation problems. ALE allows the mesh to move independently from the motion of material. Although the mesh may move in an arbitrary fashion, it typically deforms with the material while incrementally smoothing the distorted mesh. The state variables are then mapped from the distorted mesh to the smoothed mesh. As a result, the ALE algorithm is usually performed in conjunction with the remeshing procedure. Figure 1-4 shows how ALE can improve the simulation of an impact of a metal bar by allowing mesh smoothing. Only a quarter of the metal bar impacting a rigid wall is shown. As shown in Figure 1-4 (b), the mesh at the impacting region is highly distorted in the updated Lagrangian method. It will yield inaccurate results and may lead to convergence difficulties. As compared to the updated Lagrangian method, the mesh at the impacting region in the ALE analysis remains wellshaped and will significantly improve solution convergence and quality. The remeshing in ALE is distinct from earlier adaptive remeshing techniques. As opposed to complete remeshing techniques, ALE does not alter the topology of the mesh, maintaining element type and connectivity. As a consequence, the application of ALE is usually limited to geometries where the material motion is relatively predictable. To preserve high-quality mesh upon extreme deformation is still a challenging task for ALE. Another hurdle of ALE method is in modeling the path dependent behavior of plastic flow. As plastic behaviors are path or history dependent, the relative motion between mesh and material must be properly accounted for in the material constitutive equations, which imposes a significant burden on the application of ALE to advanced material modeling.
Figure 1-4. High speed impact of a metal bar: (a) A quarter model of a metal bar impacting a rigid wall; (b) Updated Lagrangian solution; (c) ALE solution
1.4.2 Discrete Element Method (DEM)
Another method widely used in geotechnical engineering to solve large-deformation problems is the discrete element method (DEM) (Cundall and Strack 1979) that approximates geomaterials at a micro-mechanical level. In this approach, the material (e.g., sand and rock) is modeled as a collection of discrete particles, each of which represents an individual grain of the material. The motion of each discrete particle is calculated separately using Newton’s Second law, and the interaction forces between each pair of the particles are obtained through simple mechanical contact models utilizing springs, dashpots and frictional sliders. DEM analysis, which can capture discontinuous and heterogeneous material behaviors from a micro-scale perspective, has been used to simulate a wide variety of problems involving granular materials. Successful application of DEM in landslides has been documented in literatures (e.g., Cleary & Campbell 1993; Campbell et al. 1995). However, as each particle in DEM represents a real solid grain, this method requires a tremendous number of discrete particles for a large-scale problem, and is hence usually restricted to small-scale and short-duration simulations.
1.4.3 Mesh-free Methods
Continuum-scale numerical methods that do not require a mesh (i.e., meshfree) are considered as more desirable and efficient for the simulation of large-deformation and large-scale problems such as seismically-induced slope failures in geotechnical engineering. The major difference between the classical methods (e.g. FEM and FDM) and meshfree methods is the absence of grids in the latter. One advantage of meshfree methods is the elimination of mesh reliance by constructing approximations entirely in terms of nodes that have no topological connection among them. Meshfree methods can be viewed as an extension of classical grid-based methods to scattered node configurations without fixed connectivity. A common feature of all meshfree methods is the use of an influence domain as shown in Figure 1-5. Unlike in a grid-based method where variable approximations are dependent on the mesh, computational nodes in a meshfree method interact with each other on the basis of influence domain. Each node carries an influence domain throughout the entire calculation, and the nodes that fall in the influence domain are updated at each timestep. Therefore, large deformations can be easily handled in a meshfree method due to its adaptive nature.
One of the earliest meshfree methods is the Smoothed Particle Hydrodynamics (SPH) method developed by Lucy (1977) and Gingold and Monaghan (1977) for astrophysical applications. SPH utilizes the principle of inverse distance weighting to approximate field quantities in an influence domain. The weighting function in SPH serves the similar purpose as the shape function in FEM. The SPH method has been widely used to simulate free surface flows and multiphase flows (e.g., Monaghan 1994; Monaghan et al. 2003; Monaghan and Kocharyan 1995) and flow through porous media (e.g., Zhu et al. 1999).
More recently, SPH method has been used to simulate the elastic response of solids (e.g.,
Libersky et al. 1993; Gray et al. 2001) and elasto-plastic behavior of geomaterials (e.g., Bui et al. 2008; Chen and Qiu 2012a, 2012b).
(a) (b)
Figure 1-5. Comparison of approximations using (a) grid-based method; (b) mesh-free
method
It was found by Swegle et al. (1995) that SPH usually suffers tension instability when particles are under tensile stress situations. Substantial improvements have been made to remediate this numerical instability, enabling this method to be applied more broadly. Libersky et al. (1997) successfully modeled high-velocity impact of nonlinear solids by using a conservative smoothing technique. Gray et al. (2001) proposed an artificial stress method to reduce numerical instability in the simulation of elastic largedeformation problems. More recently, total Lagrangian (Bonet and Kulasegaram 2001) and updated Lagrangian corrections (Vidal et al. 2007) in the SPH method have been proven stable and robust for solid mechanics. Figure 1-6 shows an application of ALE and SPH to the simulation of a metal bar impacting a rigid wall. The comparison of solutions provided by SPH and ALE shows that SPH is capable of producing satisfactory results in modeling large deformation problems in solids. SPH method has also been applied to simulate geotechnical engineering problems involving large deformations in a phenomenological manner (e.g., Bui et al. 2008). However, very simple plastic constitutive models were used in these studies and the accuracy of SPH method in simulating complex soil behaviors remains unknown.
Other meshfree methods that have been widely used in solid mechanics include the element free Galerkin (EFG) (Belytschko et al. 1994) and Reproducing Kernel Particle Method (RKPM) (Liu et al. 1995). These methods are formulated based on weak forms of differential equations, and are not truly meshfree as a background mesh is required for numerical integrations. They are consequently more computationally expensive than SPH method.
(a)
(b)
Figure 1-6. High speed impact of a metal bar with contours showing the Von Mises stress in the bar: (a) ALE solution; (b) SPH solution
1.5 Dissertation Scope and Layout
The objective of this research is to develop and validate a new numerical model for simulating seismically-induced slope deformations. The numerical model presented in this study is intended to advance our capability in simulating dynamic slope failures and to provide insights into slope deformations caused by earthquakes. As a relatively mature and efficient meshfree method, the SPH method is utilized in this research to treat large deformations in geomaterials. The motivation of conducting this study and a brief review of current literature were presented previously in Chapter 1.
Chapter 2 presents the development of a 3-D SPH model and its application to the simulation of granular materials under large deformations. The developed model is validated against well-documented experiments of axisymmetric collapse of granular columns. This chapter is based on a paper published in the International Journal of Geomechanics, ASCE, 2012.
Chapter 3 presents the development, calibration, and validation of a SPH model for the simulation of seismically-induced slope deformations under undrained condition. The capability of SPH method in modeling complex material behaviors is examined in this chapter. An advanced constitutive model that combines the strain softening viscoplasticity and cyclic nonlinearity is implemented into the 3-D SPH code presented in Chapter 2. The developed SPH model accounts for the cyclic nonlinear behavior of soil, progressive reduction in shear strength, and strain-rate dependency of soil properties (e.g. stiffness and strength) during dynamic loading. The developed SPH model is then used to simulate a readily available and well-documented model slope test on a shaking table.
This chapter is based on a manuscript submitted to the International Journal for Numerical and Analytical Methods in Geomechanics.
Chapter 4 is an extension of the previous chapter, investigating the effects of several key parameters in the SPH model on seismically-induced slope deformations. These parameters encompass both SPH parameters and material properties, such as the particle spacing, radius of the SPH influence domain, peak and residual strengths, as well as fundamental frequency. The parametric study discusses (1) how the spatial parameters in the SPH method (e.g., particle spacing and influence domain) impact the accuracy of SPH simulations, and (2) how the soil properties (e.g., peak and residual strengths) influence earthquake-induced slope deformations. A non-reflecting boundary condition in the SPH method is also presented in this chapter.
Chapter 5 draws final conclusions and provides recommendations for future research on modeling dynamic behaviors of geomaterials under large deformations.
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NUMERICAL STUDIES OF SEISMICALLY INDUCED SLOPE DEFORMATION USING SMOOTHED PARTICLE HYDRODYNAMICS METHOD
