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
ABSTRACT
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.
INTRODUCTION
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.
DOMAIN REDUCTION METHOD
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.
3D FREE FIELD MOTIONS
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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d
Figure 2: Strong Motion Fourier Transform and Response Spectrum
MODELING DESCRIPTION
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 =vs/λmin (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.
SIMULATION RESULTS
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
ENERGY DISSIPATION
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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A
C
D
B
(a)
0
500
1000
1500
2000
2500
3000
0 5 10 15 20
Energy Density [J/m3]
Time [s]
KE+SE+PF+PD
SE+PF+PD
PF+PD
PD
(b)
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.
CONCLUSION
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.
ACKNOWLEDGEMENTS
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
appreciated.
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