Transactions, SMiRT-24

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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.

REFERENCES

Bielak, J. (1978), ‘Dynamic response of non-linear building-foundation systems’, Earthquake Engi-

neering & Structural Dynamics 6(1), 17–30.

Bielak, J., Loukakis, K., Hisada, Y. and Yoshimura, C. (2003), ‘Domain reduction method for

three–dimensional earthquake modeling in localized regions. part I: Theory’, Bulletin of the

Seismological Society of America 93(2), 817–824.

Boulanger, R. and Ziotopoulou, K. (2013), ‘Formulation of a sand plasticity plane-strain model for

earthquake engineering applications’, Soil Dynamics and Earthquake Engineering 53, 254–267.

Corigliano, M., Scandella, L., Lai, C. G. and Paolucci, R. (2011), ‘Seismic analysis of deep tunnels

in near fault conditions: a case study in southern Italy’, Bul letin of Earthquake Engineering

9(4), 975–995.

Dafalias, Y. F. and Manzari, M. T. (2004), ‘Simple plasticity sand model accounting for fabric

change eﬀects’, ASCE Journal of Engineering Mechanics 130(6), 622–634.

Dafalias, Y., Schick, D., Tsakmakis, C., Hutter, K. and Baaser, H. (2002), A simple model for

describing yield surface evolution, in ‘Lecture note in applied and computational mechanics’,

Springer, pp. 169–201.

El Ganainy, H. and El Naggar, M. (2009), ‘Seismic performance of three-dimensional frame struc-

tures with underground stories’, Soil Dynamics and Earthquake Engineering 29(9), 1249–1261.

Elgamal, A., Yan, L., Yang, Z. and Conte, J. P. (2008), ‘Three-dimensional seismic response

of humboldt bay bridge-foundation-ground system’, ASCE Journal of Structural Engineering

134(7), 1165–1176.

Fatahi, B. and Tabatabaiefar, S. H. R. (2013), ‘Fully nonlinear versus equivalent linear compu-

tation method for seismic analysis of midrise buildings on soft soils’, International Journal of

Geomechanics 14(4), 04014016.

Hashash, Y. M., Hook, J. J., Schmidt, B., John, I. and Yao, C. (2001), ‘Seismic design and analysis

of underground structures’, Tunnelling and underground space technology 16(4), 247–293.

Hu, L. and Pu, J. (2004), ‘Testing and modeling of soil-structure interface’, Journal of Geotechnical

and Geoenvironmental Engineering 130(8), 851–860.

Hughes, T. (1987), The Finite Element Method ; Linear Static and Dynamic Finite Element Anal-

ysis, Prentice Hall Inc.

Iguchi, M. and Luco, J. E. (1981), ‘Dynamic response of ﬂexible rectangular foundations on an

elastic halfâĂŘspace’, Earthquake Engineering and Structural Dynamics 9(3), 239–249.

Iida, M. (2012), ‘Three-dimensional ﬁnite-element method for soil-building interaction based on an

input wave ﬁeld’, International Journal of Geomechanics 13(4), 430–440.

Jeremi´c, B., Jie, G., Cheng, Z., Tafazzoli, N., Tasiopoulou, P., Abell, F. P. J. A., Watanabe,

K., Feng, Y., Sinha, S. K., Behbehani, F., Yang, H. and Wang, H. (2017), The Real ESSI

24th Conference on Structural Mechanics in Reactor Technology

BEXCO, Busan, Korea - August 20-25, 2017

Division V

Simulator System, University of California, Davis and Lawrence Berkeley National Laboratory.

http://real-essi.info/.

Jeremi´c, B., Jie, G., Preisig, M. and Tafazzoli, N. (2009), ‘Time domain simulation of soil–

foundation–structure interaction in non–uniform soils.’, Earthquake Engineering and Structural

Dynamics 38(5), 699–718.

Jeremi´c, B., Kunnath, S. and Xiong, F. (2004), ‘Inﬂuence of soil–foundation–structure interaction

on seismic response of the i-880 viaduct’, Engineering Structures 26(3), 391–402.

Jeremi´c, B. and Sture, S. (1997), ‘Implicit integrations in elasto–plastic geotechnics’, International

Journal of Mechanics of Cohesive–Frictional Materials 2, 165–183.

Mejia, L. and Dawson, E. (2006), Earthquake deconvolution for ﬂac, in ‘4th International FLAC

Symposium on Numerical Modeling in Geomechanics’, pp. 04–10.

Oprsal, I. and F¨ah, D. (2007), 1d vs 3d strong ground motion hybrid modelling of site, and pro-

nounced topography eﬀects at augusta raurica, switzerlandâĂŤearthquakes or battles, in ‘Pro-

ceedings of 4th International Conference on Earthquake Geotechnical Engineering June 25–28,

2007, Greece’.

Paolucci, R., Shirato, M. and Yilmaz, M. T. (2008), ‘Seismic behaviour of shallow foundations:

Shaking table experiments vs numerical modelling’, Earthquake Engineering & Structural Dy-

namics 37(4), 577–595.

Park, S. and Byrne, P. (2004), Practical constitutive model for soil liquefaction, in ‘Proc., 9th

Int. Symp. on Numerical Models in Geomechanics (NUMOG IX)’, CRC Press, Boca Raton, FL,

pp. 181–186.

Petersson, N. A. and Sj¨ogreen, B. (2017), ‘High order accurate ﬁnite diﬀerence modeling of seismo-

acoustic wave propagation in a moving atmosphere and a heterogeneous earth model coupled

across a realistic topography’, Journal of Scientiﬁc Computing pp. 1–34.

Romero, A., Galv´ın, P. and Dom´ınguez, J. (2013), ‘3d non-linear time domain fem–bem approach to

soil–structure interaction problems’, Engineering Analysis with Boundary Elements 37(3), 501–

512.

Rosakis, P., Rosakis, A., Ravichandran, G. and Hodowany, J. (2000), ‘A thermodynamic internal

variable model for the partition of plastic work into heat and stored energy in metals’, Journal

of the Mechanics and Physics of Solids 48(3), 581–607.

Shahrour, I., Khoshnoudian, F., Sadek, M. and Mroueh, H. (2010), ‘Elastoplastic analysis of the

seismic response of tunnels in soft soils’, Tunnelling and underground space technology 25(4), 478–

482.

Sinha, S. K. and Jeremi´c, B. (2017), Modeling of dry and saturated soil-foundation contact, Tech-

nical Report UCD–CompGeoMech–01–2017, University of California, Davis.

Spyrakos, C., Patel, P. and Kokkinos, F. (1989), ‘Assessment of computational practices in dynamic

soil-structure interaction’, Journal of computing in civil engineering 3(2), 143–157.

Yang, H. and Jeremi´c, B. (2017), ‘Energy dissipation analysis of elastic-plastic structural elements

[manuscript in preparation]’.

Yang, H., Sinha, S. K., Feng, Y., McCallen, D. B. and Jeremi´c, B. (2017), ‘Energy dissipation

analysis of elastic-plastic materials [in print]’, Computer Methods in Applied Mechanics and

Engineering .

Yang, Z., Elgamal, A. and Parra, E. (2003), ‘Computational model for cyclic mobility and associated

shear deformation’, Journal of Geotechnical and Geoenvironmental Engineering 129(12), 1119–

1127.

Yang, Z. and Jeremi´c, B. (2003), ‘Numerical study of the eﬀective stiﬀness for pile groups’, Inter-

national Journal for Numerical and Analytical Methods in Geomechanics 27(15), 1255–1276.