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3-D Non-Linear Earthquake Soil-Structure Interaction Modeling of Embedded Small Modular Reactor (SMR)


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Presented here is a state of the art simulation methodology for the seismic response of embedded Small Modular Reactor (SMR). With this new methodology, the modeling uncertainty of whole soil structure interaction (SSI) system is greatly reduced. 3D realistic ground motion over large geological region was developed first. The local SSI system has also been properly modeled with refined mesh and full consideration of nonlinear behaviors. The realistic 3D motion was input into this high fidelity SSI system through Doamin Reduction Method. Transient seismic analysis then was conducted by RealESSI, which is a high performance Earthquake Soil Structure Interaction Simulator developed at UC Davis. The effects of nonlinear SSI behavior are investigated by comparing the results of nonlinear model with linear elastic model. It turns out that the acceleration response of the structure decreases and high frequency acceleration component is damped out after considering the nonlinear SSI effects.
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Transactions, SMiRT-24
BEXCO, Busan, Korea- August 20-25, 2017
Division V
3-D Non-Linear Earthquake Soil-Structure Interaction
Modeling of Embedded Small Modular Reactor (SMR)
Hexiang Wang1, Han Yang1, Sumeet K. Sinha1, Chao Luo4, Boris Jeremi´c2,3
1Graduate Student, Department of Civil and Environmental Engineering, UC Davis, CA, USA
2Professor, Department of Civil and Environmental Engineering, UC Davis, CA, USA
3Faculty Scientist, Earth Science Devision, LBNL, Berkeley, CA, USA
4Graduate Student, College of Civil Engineering, Tongji University, Shanghai, China
Presented here is a state of the art simulation methodology for the seismic response of embedded
Small Modular Reactor (SMR). With this new methodology, the modeling uncertainty of whole
soil structure interaction (SSI) system is greatly reduced. 3D realistic ground motion over large
geological region was developed first. The local SSI system has also been properly modeled with
refined mesh and full consideration of nonlinear behaviors. The realistic 3D motion was input into
this high fidelity SSI system through Doamin Reduction Method. Transient seismic analysis then
was conducted by RealESSI, which is a high performance Earthquake Soil Structure Interaction
Simulator developed at UC Davis. The effects of nonlinear SSI behavior are investigated by com-
paring the results of nonlinear model with linear elastic model. It turns out that the acceleration
response of the structure decreases and high frequency acceleration component is damped out after
considering the nonlinear SSI effects.
The seismic performance of nuclear facilities should be carefully analyzed considering the severe
public safety and social security problems the failure could bring. The structure investigated here is
a special kind of nuclear facility called Small Modular Reactor (SMR). Different from the prototype
Nuclear Power Plant (NPP) which consists of a surface structure and a beneath shallow foundation,
SMR is a deep-embedded structure (36 meters) and the part above the ground surface is 14 meters.
Because of the special configuration of SMR, effects of dynamic Soil Structure Interaction (SSI) on
its seismic response can be more significant and should be modeled more accurately.
In recent years, many researchers (Spyrakos et al. (1989), El Ganainy and El Naggar (2009),
Iida (2012)) made great efforts to realistic modeling of dynamic SSI sytem and seismic response
of underground structure. Romero et al. (2013) coupled FEM and BEM method to model wave
propagation in elastic foundation and corresponding dynamic response of the structure. Fatahi and
Tabatabaiefar (2013) investigated the seismic performance of midrise buildings on soft soils using
existing earthquake records. An elastoplastic SSI analysis was conducted by Shahrour et al. (2010)
to explore the seismic response of tunnels in soft soils. However, some inherent uncertainties still
existed in these previous studies and were not well addressed:
The most important uncertainty comes from the ground motion. For surface structures, peo-
ple usually use historical earthquake records and simplified 1-D horizontal excitation into SSI
system (Paolucci et al. (2008)). The vertical geound motion was totally neglected. However,
Oprsal and F¨ah (2007) has emphasized the necessity to use 3D ground motion by showing
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the big difference between 1D and 3D computation result. The uncertainty of input motion
for seismic modeling of underground structure is unfortunately even higher. Due to the lack
of ground motion observations along the depth, deconvolution method was adopted in many
studies (Elgamal et al. (2008)) to get the excitation motion at certain depth. The decon-
volution procedure is only a 1D linear inverse analysis, which is seemingly simple but will
unavoidably introduce considerable confusion and uncertainties to the modeling system(Mejia
and Dawson (2006)).
Another uncertainty comes from the method that people use to input seismic motion into
SSI system. The free field motion are directly imposed to the structure without considering
SSI. This is especially common for underground structures where simplified static loads are
directly imposed and these structures are simply designed to accommodate the estimated free
field deformation(Hashash et al. (2001)).
Nonlinear effect is also a very important factor that is neglected or simplified in many existing
studies. Actually there are two kinds of potential nonlinear behaviors in SSI system: One is
the elastoplasticity of surrounding soil and another is the slip behavior at the soil-structure
interface. At early 1980s, different structural behaviors have been found when the elasto-
plasticity of surrounding soil is considered (Bielak (1978), Iguchi and Luco (1981)). Also
Jeremi´c et al. (2004) reported that SFS interaction can have both beneficial and detrimental
effects on structural behavior and is dependent on the characteristics of the earthquake mo-
tion. Regarding the nonlinear interface behavior, Hu and Pu (2004) stressed that its accurate
modeling is a key part to get realistic solutions of SSI system.
Due to computational limitations and complicated nature of SSI problems mentioned above,
there is only few high-fidelity SSI simulations have been done for bridges (Jeremi´c et al. (2009)) and
tunnels (Corigliano et al. (2011)). To the author’s knowledge, there is no available realistic modeling
for embedded SMR structure. In this paper, we present high fidelity modeling of SMR with state-
of-the-art SSI techniques. 3D realistic free field motion is modeled by solving the wave propagation
equations over a large scale geological model. The free field motion is then input into SSI sytem
using Domain Reduction Method (Bielak et al. (2003)). Then Modeling Description section presents
nonlinear modeling details about elastoplastic surrounding soils and nonlinear interface behavior.
The nonlinear modeling result and its comparison with primitive linear elastic modeling result are
summarized in Simulation Results section. Combining all the modeling techniques together, the
modeling uncertainties listed above are greatly reduced and the nonlinear SSI effects are illustrated.
Energy dissipation is a widely-used indicator of material damage in elastic plastic materials. A
common misconception between plastic work and plastic energy dissipation has been observed in
a number of publications. Correct evaluation of energy dissipation should follow the principles of
thermodynamics that incoorperates plastic free energy (Rosakis et al., 2000, Dafalias et al., 2002).
The thermomechanical framework presented by Yang et al. (2017), Yang and Jeremi´c (2017) is
implemented in the Real ESSI (Real Earthquake Soil Structure Interaction) Simulator Jeremi´c
et al. (2017), which is used to perform energy analysis on the SMR model in this paper. Features of
energy dissipation in the SMR model discussed with insights on the safety and economy of deeply
embedded structures under earthquake loading.
Inputting seismic motions into finite element model is an indispensable step for the simulation of
soil structure interaction. The method we used here is called Domain Reduction Method, which was
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developed by Bielak et al. (2003). It is a modular, two-step dynamic procedure aimed at reducing
the large computation domain to a more manageable size. Firstly, a relatively coarse but large-
scale geological model can be built without local features(strutures, elastoplatic site effects). The
free field motion can be computed by solving wave propagation equation in this large background
geological model. We capture the time-series free filed motion on all the nodes of a single special
layer called DRM (Domain Reduction Method) layer. Then in the second step, the localised soil-
struture system with a refined mesh inside DRM layer is modeled. The purpose of the DRM layer
is to use the free field motion obtained from first step and generate equivalent earthquake force to
conduct dynamic nonlinear SSI analysis during the second step.
In this study, we use a fourth order accuare finite difference programme developed at Lawrence
Livermore National Labs called SW4 (Petersson and Sj¨ogreen (2017)) to simulate the propagation
of fault rupture in a huge geological model (9km ×6km ×20km). The magnitude of simulated
earthquake is 5.5. The shear wave velovity of soils in surface layer (500 meters thick) is 500 m/s.
In order to aviod outputting huge amount of ground motion data by SW4, we put a ESSI Box
(300m×300m×200m) into the big geological model of SW4. The ESSI box is made up of a bunch
of aligned ESSI nodes spacing at 5 meters. Then while running SW4 only the time series 3D free
field motion at all these ESSI nodes are recorded and output. After that a motion interpolation
program called SW42DRM was written to put DRM layer of the model inside the ESSI box by
specifying three translations and three rotations. The motions at ESSI nodes are interpolated to
DRM nodes and corresponding DRM motion input file is generated. This DRM motion input file is
further used to calculate equivalent earthquake force for DRM analysis in the Real ESSI Simulator.
The characteristic ground motions recorded by ESSI nodes are plotted in figure 1. The peak
ground acceleration (PGA) in x and y direction is about 1g. Apart from that, significant amount
of vertical motions with PGA 0.5gis also observed. The peak ground displacement (PGD) is
about 0.1m in horizontal direction. Since ESSI box is located in the foot wall of the reverse fault,
the permanent ground subsidence about 6cm is recorded. Fourier transformation and response
spectrum of the motions are shown in figure 2. The frequency range of the motion is within
15Hz. The dominant frequrncy of the motion is around 5 Hz. In response spectrum, we also see
significant resonance effects for structure whose fundamental period is around 0.2s corrsponding to
5 Hz fundamnetal frequency.
Figure 1: Acceleration and Displacement Time Series of Motion
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Figure 2: Strong Motion Fourier Transform and Response Spectrum
Figure 3: FEM model of SMR
In order to reduce our modeling size using DRM method, we simplify our target modeling system
into 6 layers. As shown in figure 3, the innermost part is structure layer, which is surrounded
by a soil layer. Following that, there is a DRM layer used to apply equivalent earthquake force.
Outside DRM layer three damping layers are placed. These damping layers are designed to add
high Rayleigh damping so that the outgoing wave can be adsorbed. Table 1 shows the damping
parameters we used. Finally, the size of whole FEM model is 72m×72m×56m. There are 177,806
nodes, 20172 27-node brick elements and 3,177 contact elements (modeling the interface between
soil and embedded structure). The average mesh size is about 3 meters. Newmark time integration
method is used in this study with parameters γ=0.7 and β=0.36. In order to capture the wave
propagation in FEM model, mesh size should be strictly controlled so that there is no artificial
filtering to motions above certain frequency. Hughes (1987) pointed out that 10 linear interpolation
finite elements and 2 quadratic interpolation elements are needded per wave wavelength. Since we
use second order 27 node brick element here, the minimum wave length our model can capture is
6 meters. Considering shear wave velovity vs= 500m/s, the maximum frequency calculated by
equation 1 is 83 Hz. Even if the material will become softer due to plastification, our model is still
goood enough considering the ground motion fmax 15Hz.
fmax =vsmin (1)
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Embeded Nuclear Structure
The nuclear facility we modeled here is called Small Modular Reactor (SMR). It is a 4 storied
reinforced concrete structure with total height 50 meters and 36 meters embedded in the ground.
The length and width of the structure are both 30 meters. The whole structure is modeled using
27-node solid brick element with linear elastic material. The Young’s modulus is selected as 30GPa
and Poisson ratio 0.2.
Soil Model
The depth of the soil surrounding the structure modeled here is 45m. The soil is assumed to be
saturated soil with undrained behavior during the earthquake. In order to considering nonlinear
site effects, the soil is modeled with elastoplastic material. In the past 20 years, many constitutive
models Yang et al. (2003), Dafalias and Manzari (2004), Park and Byrne (2004), Boulanger and
Ziotopoulou (2013) have been put forward to simulate the complicated stress-strain behavior of
soils. Yang and Jeremi´c (2003) found out von Mesis model can be approximately used to model
the undrained behaviors. Hence von Mises elastoplastic material with linear kinematic hardening
rule is adopted here. The material parameters can be seen in table 1. Backward Euler implicit
algorithm (Jeremi´c and Sture (1997)) is adopted for the iterations in constitutive level.
Table 1: Modeling parameters
shear wave velocity [m/s] 500
Young’s modulus [GPa] 1.25
Poisson ratio 0.25
von Mises radius [kPa] 60
Material parameters
kinematic hardening rate [MPa] 0
initial normal stiffness [N/m] 1e9
hardening rate [/m] 1000
maximum normal stiffness [N/m] 1e12
tangential stiffness [N/m] 1e7
normal damping [N/(m/s)] 100
tangential damping [N/(m/s)] 100
Contact parameters
friction ratio 0.25
structure layer 5%
surrounding soil 15%
DRM layer 20%
outside layer 1 20%
outside layer 2 40%
Damping parameters
outside layer 3 60%
Soft Contact Element
In order to model the slip behavior of the interface between structure and its surrounding soil,
node-to-node penalty based soft contact (interface) element (Sinha and Jeremi´c (2017)) is used
here. In soft contact, the normal stiffness exponentially grows as the relative displacement between
two contact nodes increases and finally truncated by maximum normal stiffness. 3,177 contact
elements are placed at the soil-structure interface. To ensure the stability of the numerical solution,
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the penalty stiffness in normal direction was chosen 2-3 order magnitude greater than the stiffness
of the soil. The contact parameters are also shown in Table 1.
Simulation Procedure
The nonlinear earthquake SSI analysis was conducted using RealESSI (Jeremi´c et al. (2017)) de-
veloped at UC Davis. Two SMR simulation cases were simulated. One case uses linear elastic
surrounding soil without contact element and another case uses nonlinear surrounding soil with
contact element as mentioned above. In both cases, two loading stages were modeled: First loading
stage is self weight by adding a uniform gravity field. By doing this we get the initial stress state
of structure and surrounding soil before earthquake comes. Then second loading stage is DRM
transient analysis. By adding equivalent earthquake forces at all boundary nodes of DRM, the
seismic performance of SMR under 3D earthquke loading (lasting for 20s) was investigated. The
whole analysis was run in parallel on 10 CPUs.
Figure 4: Time Series Acceleration Response
Figure 5: Acceleration Response in frequency domain
Figure 4 showes time series acceleration response of top center of SMR. The “Elastic” legend
represents the result of simulation case where the surrounding soil is modeled using linear elastic
material and no contact elements in soil-structure interface. The “Inelastic” legend represents the
simulation case where the surrounding soil is modeled using von Mises elastoplastic material with
linear kinamatic hardening. In addtion, contact elements are placed at the soil-structure interface
so that relaitve slip of two different materials can happen and interface gap can open and close
while shaking. Significant acceleration decreases can be seen in inelastic case. The horizontal peak
acceleration values reduce by almost 30% to 40%. This is because in inelastic case the surrounding
soil can palstify during shaking and dissipate energy so that it behaves like a layer of seismic
isolator surrounding the structure. The acceleration difference in z direction is less significant
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than horizontal direction. Also, in figure 5 the Fourier magnitude of high frequency component of
horizontal acceleration was seen significant decrease in inelastic case. This is reasonable considering
the plastification of surrounding soil that happens during the shaking.
Figure 6 shows the distribution of plastic strain in surrounding soil. There are two main plastic
zones near two bottom corners of the structure. Also the plastic strain at the soil-structure interface
is higher than adjacent area. It is interesting to note that there is an elastic zone at the beneath
of the structure. The shape of the elastic zone is like a tray. It happens because of soil-structure
interaction: The stiffness of the structure is much higher than its surrounding soil. Therefore, the
bottom of the structure has a tendency to keep as a plane under seismic loading, which result in
pressure redistribution in beneath soil. The pressure at two corners of the bottom plate is much
higher than the pressure at the middle part. This is why we see this tray-shaped elastic zone under
the structure.
Figure 6: Distribution of the magnitude of plastic strain
According to the thermomechanical framework presented by Yang and Jeremi´c (2017), the energy
dissipation in any decoupled material undergoing isothermal process can be expressed as:
Φ = σij ˙ij σij ˙el
ij ρ˙
ψpl 0 (2)
where Φ is the rate of change of energy dissipation per unit volume (or dissipation density), σij
and ij are the stress and strain tensors respectively, el
ij is the elastic part of the strain tensor, ρ
is the mass density of the material, and ψpl is the plastic free energy per unit volume (or plastic
free energy density). Equation 2 ensures the energy balance and nonnegative energy dissipation
conditions that correspond to the first and second law of thermodynamics.
With Equation 2, the energy balance of a SSI system is simply given by:
WInput =ES tored +EDissipated =KE +SE +P F +P D (3)
where WInput is the input work due to external loading, KE is the kinetic energy, SE is the elastic
strain energy, P F is the plastic free energy, and P D is the energy dissipation due to material
plasticity. Formulation for each energy component can be found in Yang et al. (2017). Note that
in Equation 3, it is assumed that no other forms of energy dissipation exists in the system.
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0 5 10 15 20
Energy Density [J/m3]
Time [s]
Figure 7: Energy dissipation in SMR model: (a) Plastic dissipation density field at the end of
simulation; (b) Evolution of energy components at location A.
Figure 7 (a) shows the distribution of plastic dissipation density in the SMR model at the end
of simulation. The case presented in this section is elastic plastic soil without contact element.
Note that the structure is modeled with elastic material, so they do not dissipate any energy. As
expected, more seismic energy is dissipated around the corners and edges of the structure due to
stress concentration. It can be observed that there are several elastic regions around the boundaries
of the structure, which means that the soil there does not plastify much and moves together with
the structure. Economy of the design can be improved by better utilizing the strength of soil around
these locations.
Figure 7 (b) show the evolution of energy components at location A. It can be observed that
the amount of plastic energy dissipation is much larger than the other forms of energy, indicating
that the nonlinear effect is quite significant in deeply embedded structure. Another interesting
observation is the small amount of plastic free energy whose quantity largely depends on material
hardening parameters and loading conditions. It should be pointed out that even if it is small,
plastic free energy should never be neglected so that the condition of nonnegative incremental
energy dissipation can be upheld.
The seismic response of an embedded SMR has been modeled with high fidelity. Using state-of-the-
art nonlinear SSI simulation techniques, many uncertainties in whole SSI system has been eliminated
and replaced with more realistic modeling. The methodology shown here is also applicable to
many other SSI problems. The simulation result of SMR shows that the acceleration response of
the structure decreases with nonlinear effects properly modeled. In addition, the high frequency
component of acceleration is damped out in inelastic case due to soil plastification.
Energy dissipation analysis shows that the soil close to the edge of the SMR structure dissipates
large amount of seismic energy during shaking. Such observation also indicates significant nonlinear
effect when elastoplastic material is used for soil modeling. Several elastic regions are identified
where design can be improved so that soil strength at these locations can contribute to the safety
of the SSI system.
There are still some other uncertain factors which are not included in this study, such as
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different geological and topological factors, the embedded depth of structure and magnitude of the
earthquake. More numerical experiments should be carried out to get comprehensive conclusion
about the nonlinear effects on soil structure interaction.
The authors appreciate the funding provided by United States Department of Energy. The 3D
realistic motions provided by Arthur Rodgers in Lawrence Livemore National Laboratory are also
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... To formulate stochastic effective forces using the above stochastic DRM, it is required to develop stochastic bedrock input motions. For a given site, several different ways have been proposed, e.g., DRM forces can be computed from physics-based simulated ground motions over regional bedrock geology (Graves et al., 2011;Wang et al., 2017;Abell et al., 2018). However, these physics-based ground motion simulations can become computationally intractable when uncertainties in seismic source or bedrock geology need to be considered. ...
Presented here is a modular methodology for time-domain stochastic seismic wave propagation analysis. Presented methodology is designed to analyse uncertain seismic motions as an input, propagating through uncertain material. Traditional approach for uncertain wave propagation relies on models that include deep bedrock, local soil site, and their random process and random field information. Such models can become quite large and computationally intractable. The modular approach proposed herein features two step approach that allows separate consideration of the deep bedrock and local site along with corresponding random field information. The first step considers an auxiliary stochastic motions problem in the bedrock. Stochastic local site response can then be simulated in a reduced domain within certain depth from the surface. Application of uncertain seismic motions at depth, for local uncertain site response is done using stochastic effective forces developed through the Domain Reduction Method. By using Hermite polynomial chaos expansion to represent the non-Gaussian random field of material parameters and non-stationary random process of seismic motion, the proposed modular methodology is formulated using intrusive stochastic Galerkin approach, as seen in the Stochastic Elastic–Plastic Finite Element Method (SEPFEM). Developed modular methodology is illustrated using a 1-D stochastic seismic wave propagation analysis with three cases, and simulation results are also verified with results from conventional approach.
... forces. In addition, developed Wave Potential Formulation -Domain Reduction Method (WPF-DRM) offers advantages for solving locally inhomogeneous and nonlinear SSI problems under inclined seismic excitations[39][40][41]. ...
Presented is an application of wave potential formulation (WPF) together with domain reduction method (DRM) to modeling earthquake soil structure interaction (ESSI) behavior in horizontally layered ground under inclined incident seismic waves. Wave potential formulation is used to develop a spatially varying, inclined seismic wave field from incident Primary (P) and Secondary (S) waves that propagate through layered ground. Developed seismic wave field is then used to develop effective forces for Domain Reduction Method that are then used for analyzing ESSI response of a soil structure system. Developed methodology, called WPF-DRM, is verified using analytic solution for a free field response of layered ground subjected to inclined incident waves. Developed WPF-DRM methodology is illustrated through analysis of an ESSI response of a deeply embedded structure, a small modular reactor (SMR) subjected to incident S wave polarized in vertical plane (SV) with variation in inclinations and frequencies. Presented example highlights the influences of incident wave inclination and frequency on ESSI response of analyzed SMR.
... It is noted that the formulation of DRM does not restrict the material behavior of interior soil. 3D inelastic constitutive behavior of soils under 3C seismic excitations can be considered using WP-DRM for realistic nonlinear SSI modeling ( Wang et al., 2017). ...
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Presented is a methodology to stress test soil structure interaction (SSI) systems using spatially varying three components (3C) seismic motions. The main idea is to develop seismic motions that excite broadband of frequencies, a range of intensities and incident wave angles, for variation in body and surface waves. In this study, it is acceptable to numerically fail the nuclear installation (NI) system, to develop high accelerations,damage structural components, and push NI system far beyond design states. Insights into possible damages and failure modes give engineers useful information that can lead to improving design of new and retrofitting existing NIs. The complete 3C seismic motions from inclined plane body waves and surface waves with variation in incident angle, frequency and intensities are developed using wave potential formulation. The 3C seismic excitations are then applied to SSI system by domain reduction method (DRM). Dynamic response of SSI system is simulated through finite element analysis. High frequency stress test motions are obtained analytically using Thompson-Haskell propagator matrix technique. Frequency content is only limited by the mesh size of finite element model of SSI system. Proposed methodology is illustrated through stress tests of a deeply embedded small modular reactor (SMR). Different stress test responses of SMR are compared.Suggestions for improved design of NIs are given.
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Modeling and simulation of earthquake soil-structure interaction (ESSI) requires a number of sophisticated modeling and simulation approaches to reduce modeling uncertainty and improve the accuracy of results. The superstructure can be supported by either shallow or deep foundation and in the dry, partially saturated, or fully saturated soil. An interface element is thus required to accurately model the interaction of dry as well as partially saturated soil with the foundation. The current modeling techniques mostly assume a hard normal contact behavior i.e. normal contact stiffness is constant with penetration. However, a more physical contact stress expected between the soil-foundation interface is non-linear. The normal contact stiffness increases with penetration until the soil surface becomes hard. At this state, any further penetration can be assumed to be of hard contact. In this thesis, a soft contact formulation is presented to model the non-linear stiffness at soil-foundation interface. The cyclic shear behavior of the soil-structure is highly non-linear and sophisticated. It includes hysteresis, hardening, softening (dilation) and particle breakage. Depending upon the normal stress or confinement, the shear behavior of the interface can have hardening until a peak shear strength is attained and then softening to the critical or residual shear strength. In this thesis, apart from the most popular Elastic Perfectly Plastic shear model, two additional shear models with nonlinear hardening and non-linear hardening/softening are proposed with minimum modeling parameters to model the monotonic as well as cyclic shear behavior at soil-foundation interface. In partially or fully saturated conditions, during dynamic events (seismic shaking) pore fluid pressures in soil adjacent to foundations will change dynamically. Moreover, for strong shaking, the structure might rock, and foundation-soil interface might develop gaps and create suction pressure pulling the water up in tension. A coupled element is developed to model the changes in dynamic pore-fluid pressures and effective stress at soil-foundation interface for submerged conditions. An extensive verification for all the components of the proposed elements is also performed.
Conference Paper
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Design earthquake ground motions for dynamic analyses are typically specified as outcrop motions, which may have to be modified for input at the base of a FLAC model. Often a ‘deconvolution’ analysis using a 1-D wave propagation code, such as the program SHAKE, is performed to obtain the appropriate input motion at depth. This seemingly simple analysis is often the subject of considerable confusion. In this paper the theory and operation of the program SHAKE and input requirements of FLAC are reviewed, and the application of SHAKE for adapting design earthquake motions for FLAC input is described. Numerical examples illustrating typical cases are presented, and several questions that commonly arise are addressed.
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Presented is an energy dissipation analysis framework for granular material that is based on thermodynamics. Theoretical formulations are derived from the second law of thermodynamics, in conjunction with a few plausible assumptions on energy transformation and dissipation. The role of plastic free energy is emphasized by a conceptual experiment showing its physical nature. Theoretical formulation is adapted in order to be applied in elastic-plastic finite element method (FEM) simulations. Developed methodology is verified through comparison of input work, stored energy, and energy dissipation of the system. Separation of plastic work into plastic free energy and energy dissipation removes a common mistake, made in a number of publications, where energy dissipation can attain negative values (energy production) which is impossible.
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We develop a numerical method for simultaneously simulating acoustic waves in a realistic moving atmosphere and seismic waves in a heterogeneous earth model, where the motions are coupled across a realistic topography. We model acoustic wave propagation by solving the linearized Euler equations of compressible fluid mechanics. The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. The motion is coupled by imposing continuity of normal velocity and normal stresses across the topographic interface. Realistic topography is resolved on a curvilinear grid that follows the interface. The governing equations are discretized using high order accurate finite difference methods that satisfy the principle of summation by parts. We apply the energy method to derive the discrete interface conditions and to show that the coupled discretization is stable. The implementation is verified by numerical experiments, and we demonstrate a simulation of coupled wave propagation in a windy atmosphere and a realistic earth model with non-planar topography.
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The paper presents the derivation of a fully implicit Newton algorithm for direct integration of constitutive equations, in extended stress-internal variable space, involving hardening or softening of a general dilatant isotropic elastoplastic geomaterial. All relevant derivatives are provided in tensor notation, thus facilitating implementation. The consistent, algorithmic tangent stiffness tensor is derived. The relative accuracy of a template algorithm is assessed on a number of examples by means of iso-error maps. We present a rather simple, one-increment example concerning convergence properties of the Newton iterative scheme at the global, finite element level, associated with the consistent tangent stiffness tensor for integrating the weak form of the equilibrium equations. © 1997 by John Wiley & Sons, Ltd. Mech. Cohesive-frictional mater. 2, 165–183 (1977)
A three-dimensional (3D) FEM for examining the soil-building interaction based on an input seismic wave field is proposed. A seismic wave field means seismic waves propagating in a 3D medium. An input seismic wave field is employed with the goal of adequately treating seismic surface waves trapped by a deep (several kilometers) underground structure in a soil-building interaction system. As the first stage of the proposed method, a simple linear method is constructed. The linear method was applied to estimate seismic responses of low- to high-rise RC model buildings during a large earthquake at a soft-soil site in Mexico City where surface waves are dominant. At the soft-soil site, all the buildings with and without piles vibrated together with the ground, probably suppressing the pile damage. The proposed method qualitatively provided us with more realistic building responses, compared with a conventional interaction analysis based on an input base motion. When a considerable amount of surface waves are present at a soft-soil site, the proposed method needs to be employed. (C) 2013 American Society of Civil Engineers.
In this study, the accuracy of a fully nonlinear method against an equivalent linear method for dynamic analysis of soil-structure interaction is investigated comparing the predicted results of both numerical procedures. Three structural models, including 5-story, 10-story, and 15-story buildings, are simulated in conjunction with two soil types with shear-wave velocities less than 600 m=s. The aforementioned frames were analyzed under three different conditions: (1) fixed-base model performing conventional time history dynamic analysis under the influence of earthquake records, (2) flexible-base model (considering full soil-structure interaction) conducting equivalent linear dynamic analysis of soil-structure interaction under seismic loads, and (3) flexible-base model performing fully nonlinear dynamic analysis of soil-structure interaction under the influence of earthquake records. The results of these three cases in terms of average lateral story deflections and interstorey drifts are determined, compared, and discussed. It is concluded that the equivalent linear method of the dynamic analysis underestimates the inelastic seismic response of mid-rise moment resisting building frames resting on soft soils in comparison with the fully nonlinear dynamic analysis method. Therefore, a design procedure using the equivalent linear method cannot adequately guarantee the structural safety for mid-rise building frames resting on soft soils.
Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.
In saturated clean medium-to-dense cohesionless soils, liquefaction-induced shear deformation is observed to accumulate in a cycle-by-cycle pattern cyclic mobility. Much of the shear strain accumulation,occurs rapidly during the transition from contraction to dilation near the phase transformation,surface,at a nearly constant low shear stress and effective confining pressure. Such a stress state is difficult to employ as a basis for predicting the associated magnitude of accumulated permanent shear strain. In this study, a more convenient,approach,is adopted,in which,the domain,of large shear strain is directly defined by strain space parameters. The observed cyclic shear deformation is accounted for by enlargement and/or translation of this domain in deviatoric strain space. In this paper, the model,formulation,details involved,are presented,and,discussed. A calibration phase,is also described,based,on data from,laboratory sample,tests and dynamic,centrifuge experiments,for Nevada sand at a relative density of about 40%. DOI: 10.1061/ASCE1090-02412003129:121119 CE Database subject headings: Liquefaction; Constitutive models; Cyclic plasticity; Soil dynamics; Centrifuge models.