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Citation: Yang, F.; Zhao, H.; Ma, T.;
Bao, Y.; Cao, K.; Li, X.
Three-Dimensional Numerical
Analysis of Seismic Response of Steel
Frame–Core Wall Structure with
Basement Considering Soil–Structure
Interaction Effects. Buildings 2024,14,
3522. https://doi.org/10.3390/
buildings14113522
Academic Editor: Harry Far
Received: 16 September 2024
Revised: 26 October 2024
Accepted: 27 October 2024
Published: 4 November 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
buildings
Article
Three-Dimensional Numerical Analysis of Seismic Response
of Steel Frame–Core Wall Structure with Basement Considering
Soil–Structure Interaction Effects
Fujian Yang 1,2,3 , Haonan Zhao 4, Tianchang Ma 5, Yi Bao 1, Kai Cao 1and Xiaoshuang Li 1,*
1
School of Urban Construction, Changzhou University, Changzhou 213164, China; fjyang@outlook.com (F.Y.);
yibao10@126.com (Y.B.); caokai0213@163.com (K.C.)
2School of Civil Engineering and Architecture, Tianjin University, Tianjin 300072, China
3Jujiang Construction Group, Jiaxing 314599, China
4School of Environmental Science and Engineering, Changzhou University, Changzhou 213164, China;
hnzhao99@163.com
5School of Architecture & Civil Engineering, Shenyang University of Technology, Shenyang 110870, China;
ma.tc@sut.edu.cn
*Correspondence: xsli2011@126.com
Abstract: In recent years, numerous studies highlighted the crucial role of the soil–structure inter-
action (SSI) in the seismic performance of basement structures. However, there remains a limited
understanding of how this interaction affects buildings with basement structures under varying site
conditions. Based on the three-dimensional (3D) numerical analysis method, the influence of the SSI
on the seismic response of high-rise steel frame–core wall (SFCW) structures situated on shallow-box
foundations were investigated in this study. To further investigate the effects of the SSI and site
conditions, three types of soil profiles—soft, medium, and hard—were considered, along with a
fixed-foundation model. The results were compared in terms of the maximum lateral displacement,
inter-story drift ratio (IDR), acceleration amplification coefficient, and tensile damage for the SFCW
structure under different site conditions, with both fixed-base and shallow-box foundation configura-
tions. The findings highlight that the site conditions significantly affected the seismic performance of
the SFCW structure, particularly in the soft soil, which increased the lateral deflection and inter-story
drift. Moreover, compared with non-pulse-like ground motion, pulse-like ground motion resulted in
a higher acceleration amplification coefficient and greater structural response in the SFCW structure.
The RC core wall–basement slab junction was a critical region of stress concentration that exhibited a
high sensitivity to the site conditions. Additionally, the maximum IDRs showed a more significant
variation at incidence angles between 20 and 30 degrees, with a more pronounced effect at a seismic
input intensity of 0.3 g than at 0.2 g.
Keywords: soil–structure interaction; shallow-box foundation; underground basement; 3D numerical analysis;
artificial boundary condition
1. Introduction
Historically, assessments of building seismic performance were frequently based on
the premise of rigid foundation models. Such methodologies either ignore the substantial
impact of the soil–structure interaction (SSI) or attempt to account for it by adjusting
the period and damping ratio of the fixed-base building to reflect the flexibility of the
foundation [
1
]. The SSI is a sophisticated phenomenon that occurs in the coupling process
between the underground foundations and soil media, particularly for stiff and large
structures constructed on relatively soft ground, and has become one of the most significant
disciplines in the field of earthquake engineering [2–4]. The influence of soil on structural
behavior is primarily manifested through the bearing stiffness, damping properties, and
dynamic interactions between the soil and the structure, collectively referred to as the
Buildings 2024,14, 3522. https://doi.org/10.3390/buildings14113522 https://www.mdpi.com/journal/buildings
Buildings 2024,14, 3522 2 of 21
site effect. This study investigated the influence of the SSI on the seismic response of a
high-rise steel frame–core wall (SFCW) structure located on a shallow-box foundation using
a three-dimensional (3D) numerical analysis method across three site conditions.
When an earthquake occurs, the superstructure is subjected to strong inertial forces,
while the seismic performance of the non-subterranean structure is different from that of the
superstructure because it is limited by the surrounding geotechnical constraints. Moreover,
the site effects impact the propagation path and amplification of seismic waves, significantly
influencing the dynamic response of both the superstructure and underground structures, thus
affecting the overall seismic performance of the entire
soil–foundation–structure system [5]
.
The SSI and local site effects are generally more pronounced in soft-soil formations and
high-rise structures, significantly altering the free-field input motion; the dynamic char-
acteristics of the building; and ultimately, its response. Changes in the soil properties are
crucial for understanding the SSI effects, as different soil types (such as sand and clay)
respond differently to dynamic loads. The stiffness of the soil directly influences the natural
frequency and vibration modes of the structure. Therefore, a comprehensive evaluation of
the seismic performance of the entire soil–foundation–structure system under different site
conditions is required.
The study of SSI effects can be traced back to the late 19th century, and research on
this problem developed rapidly in the second half of the 20th century with the advent of
computers and the need for improved seismic performance in modern buildings and indus-
tries [
6
]. Previous studies and available experimental data have made it clear that ignoring
the soil types or failing to consider the coupling between the soil and structure can have a
detrimental effect on the seismic performance analysis of buildings [
7
–
10
]. Consequently,
the SSI has garnered significant attention from researchers in recent years, particularly in
the context of assessing the seismic performance of both above-ground and below-ground
structures. Due to the SSI, the dynamic response of a structure under fixed-foundation
conditions is different from that when it is located in the soft-soil site. This conclusion was
already confirmed by many scholars. Nasab et al. [
11
] previously conducted nonlinear
dynamic analyses to examine the seismic performance of soft-story buildings, noting a
significant increase in the likelihood of structural failure when considering the SSI effect
compared with fixed foundations. This was particularly evident in site classes with a low
shear wave velocity. In the study of Hamidia et al. [
12
], different soil types (soft and hard)
and other factors were considered to evaluate the seismic collapse resistance of the struc-
ture. Far [
13
] developed a computational model for 5-story and 15-story unsupported steel
frames, which was analyzed under two different boundary conditions: a fixed foundation
and a flexible foundation (considering the SSI). The results indicate that the IDR of the
structure on the soft-soil site increased significantly (more than 2 times) when considering
the SSI and was more obviously affected by seismic action. Additionally, many scholars
incorporated the SSI and site effects into their studies on the seismic vulnerability assess-
ment of buildings. The findings consistently demonstrate that these factors significantly
influence the structural performance [14–18].
A 3D numerical study on the influence of the SSI on the seismic response of high-
rise buildings with wall–frame structural systems was conducted by Scarfone et al. [
19
],
highlighting the role of different foundation systems and the very soft-soil layers present in
the supporting soil. The numerical results show that the existence of a soft-soil layer has an
important influence on the dynamic interaction between the soil and structure. Arboleda-
Monsalve et al. [
20
] developed a numerical model of a 40-story building using OpenSees
to study the SSI effects on the seismic performance of high-rise buildings.
Mata et al. [21]
investigated a 3D regular concrete rectangular frame building with 4, 8, 12, 16, 20, or 24
floors supported by cushion foundations on sandy and cohesive soil with shear wave
velocities below 250 m/s. The findings reveal that the SSI significantly impacts the seismic
response of RC moment-resisting frame structures. The fragility curve illustrated that a
building with a flexible foundation has a greater likelihood of collapse compared with one
with a fixed foundation. Kant et al. [
22
] conducted a series of 3D finite element analyses
Buildings 2024,14, 3522 3 of 21
using ABAQUS software to investigate the effects of the number of floors, foundation
size on lateral story displacements, IDR, natural periods, and shear pressures in high-rise
buildings. To study the impact of the seismic SSI on a 20-story building, Bagheri et al. [
23
]
performed 3D numerical simulations. In contrast to previous studies, their research featured
three different foundation systems, including shallow, deep-buried, and pile foundations
under two distinct soil profiles. The results indicate that as the deformation capacity of the
foundation increases, the maximum base shear decreases, while the degree of foundation
stiffening and rotation increases.
Although the shaking table test is complex, it remains an effective method for study-
ing the dynamic SSI under seismic conditions. Recently, numerous shaking table tests
were conducted to investigate the role of the SSI in the seismic analysis of buildings. Lei
et al. [
24
] designed a cylindrical 3D laminar shear soil container for shaking table tests
to effectively simulate the dynamic SSI under a 3D seismic input. El Hoseny et al. [
25
]
conducted scale modeling tests on a 15-story steel structure using shaking table tests and
numerical simulations to investigate the effects of variable embedment depths, a fixed
foundation, and a flexible foundation on the seismic performance of high-rise buildings.
The results demonstrated that the SSI effect leads to amplified lateral displacements, po-
tentially jeopardizing the safety of the structure. For the seismic performance of buildings
constructed on soft soils, Oz et al. [
26
] selected 40 existing buildings in Turkey and devel-
oped a nonlinear model that accounted for fixed foundations, as well as rigid-, medium-,
and soft-soil conditions.
On top of the previous foundation, to cope with the complexity of current building
structures, a number of scholars researched the derivation of the SSI effects. For example, if
other structures are situated nearby, the response of a given structure can be altered due to
scattered waves generated by each surrounding structure. This situation is known as the
structure–soil–structure interaction (SSSI). Some researchers pointed out that considering
the SSSI may change the response of the structure, either positively or
negatively [27–31]
.
At the same time as the density of buildings increases, people have to find ways to increase
the use of space, and the use of buildings gradually develops from expanding upward to
expanding downward, so basement structures are becoming more common. At present,
most high-rise buildings have basement structures, and the underground space of high-rise
buildings, as a special kind of underground space, is surrounded by a large amount of soil,
which also affects the dynamic characteristics of the soil–structure system. Therefore, it
is necessary to study the seismic performance of high-rise buildings with basement struc-
tures under seismic action. Several scholars also researched this issue. When a building
has a basement, the foundation is more deeply embedded in the soil, leading to more
pronounced kinematic and inertial interactions. In buildings with basements, kinematic
interaction occurs due to the difference in stiffness between the building foundation and
the surrounding soil. This causes the seismic waves to change as they pass through the
structure, impacting the motion at the foundation level [
32
]. Pinto et al. [
33
] proposed a
combined experimental–numerical method to evaluate the effect of high-rise building su-
perstructures on the basement seismic lateral earth pressure. Zhang and Far [
34
] developed
an SSI numerical model with a basement in ABAQUS software to investigate the effect of
the SSI on high-rise frame–core tube structures. The study results show that compared
with the fixed foundation case, the maximum lateral shift and inter-story drift of almost all
structures with flexible foundation models are amplified to varying degrees, regardless of
the aspect ratio, foundation type, and soil type.
In summary, changes in the soil properties not only affect the behavior of structures
under seismic or other loads but are also directly related to the accurate simulation and
evaluation of the SSI effects. In the study of the SSI effect, soil characteristics analysis is
an important link that cannot be ignored. Although the above results provide many new
insights into the seismic performance of buildings with basement structures, site conditions
were not taken into account in the above studies. At this stage, the problem of the interaction
between the two media has been widely emphasized in the field of engineering. Therefore,
Buildings 2024,14, 3522 4 of 21
there are obvious shortcomings in the simple analysis of above-ground structures, so it is
urgent to evaluate the seismic performance of the whole soil–foundation-structure system
to explore the influence of different soil conditions on the seismic performance of building
structures with basements.
Addressing the existing challenges in current research, this study focused on a 16-story,
three-bay steel frame–core wall (SFCW) building structural system with a basement. It
considered three types of soil profile (soft soil, medium soil, and hard soil), and investigated
the effects of the SSI on the dynamic response of the high-rise SFCW structure. In this
paper, a 3D numerical analysis method based on a viscous-spring artificial boundary and
the seismic wave input method is proposed, and the method was validated using ABAQUS
software. Additionally, two widely used seismic records, the near-fault pulse-like motion
recorded at the Sylmar Converter Sta Station during the 1994 Northridge seismic event,
and the near-fault non-pulse-like motion recorded at the El Centro Array Station #9 during
the 1940 Imperial Valley seismic event, were used as seismic inputs. The effects of the site
conditions and incident angle on the seismic performance of the SFCW structure with a
fixed-base or shallow-box foundation were investigated by analyzing and comparing the
inter-story drift ratios, maximum lateral displacements, acceleration amplification factors,
and tensile damage.
2. Development and Validation of 3D Numerical FEM Model
2.1. Three-Dimensional Viscous-Spring Artificial Boundary
When using limited space to simulate semi-infinite soil, the application of the boundary
conditions and seismic inputs significantly influences the accuracy of the calculation. Many
scholars [
35
,
36
] extensively researched this area. Among them, the viscous-spring artificial
boundary (VSAB) model studied by Liu et al. [
35
] is widely used in the SSI finite element
calculations due to its high accuracy and practicality. The VSAB is a continuous parallel-
spring damping system on the artificial truncated boundary. It realizes the absorption of
external waves from the finite domain to the infinite domain by employing similar springs
and dampers on the discrete boundary grid nodes. Figure 1illustrates the application
of the VSAB in the 3D numerical analysis. The surrounding soil material determines the
mechanical parameters of the springs and dampers, and the specific formulas are as follows:
Buildings2024,14,xFORPEERREVIEW5of22
s2(1 )
E
cv
(3)
p
(1 )
2(1 )(1 2)
Ev
cvv
(4)
whereEandνaretheelasticmodulusandPoisson’sratioofthemedium,respectively.
Node
Node control area
y
x
z
l
y
x
K
l1
K
l2
K
l3
C
l3
C
l1
C
l2
l
A
l
Figure1.Mechanicalmodelofthe3DVSAB.
2.2.SeismicWaveInputMethodBasedonVSAB
Inthesiteseismicresponseandsoil–foundation–structuredynamicinteractionanal-
ysis,thereasonableinputmethodofexternalseismicwavesdeterminetheaccuracyand
reliabilityofthefinalcalculationresults[37,38].BasedontheVSABmodel,LiuandLu
[39]achievedseismicwaveinputbyconvertingtheseismicwaveproblemintoawave
sourceproblem,i.e.,convertingtheseismicwaveintoequivalentnodalloadsthatacton
theboundaryinaconcentratedforce.AccordingtoLiuandLu[39],theequalnodeforce
Fliindirectioni(=x,y,z,representingthex-direction,y-direction,andz-direction,respec-
tively)ofanynodelontheboundaryundertheactionoftheincidentseismicwavecanbe
expressedas
f
ff
li li li li li l li
FKuCu A
(5)
whereKliandCliarethecoefficientmatricesofthespringanddashpotsintheVSAB,re-
spectively.Fortheleftandrightboundaries,(,,)
nli diag K K KK
and
(,, )
nli diag C C CC
;forthefrontandrearboundaries,(,, )
li n
diag K KK
K
and
(,, )
li n
diag C CCC
;fortheboomboundary,(, ,)
nli diag K K KK
and
(,,)
nli diag C C CC
,inwhichdiag(·)meansthediagonalmatrix.Alisthecontrolareaof
theboundarynodel,whereAl=(A1+A2+A3+A4)/4.
f
li
u
,
f
li
u
,and
f
li
representthedis-
placement,velocity,andstressfieldsatnodel,respectively.Theli
F
canbecalculated
accordingtothelawofwavepropagationandthestressingstateofthewavefield[40].
BasedonthetheoryoftheVSABandseismicwaveinputmethod,aPythonscript
wasdevelopedbytheauthortotransformthedisplacementandvelocitytimehistoryof
theSVwavesintoanodalforcetimehistoryandtransferallthenodalforcetimehistory
tothesoftwareABAQUS(seeFigure2).
Figure 1. Mechanical model of the 3D VSAB.
In the tangent direction,
Kτ=ατ
G
R,Cτ=ρcs(1)
In the normal direction,
Kn=αnG
R,Cn=ρcp(2)
where K
τ
and K
n
are the divergent and the standard spring stiffness coefficients, respec-
tively; C
τ
and C
n
are the damper coefficients of the divergent and traditional springs,
respectively; and
ατ
and
αn
are the correction coefficients of the VSAB. For the 3D system,
Buildings 2024,14, 3522 5 of 21
their values were taken from reference [
35
], where
αt
and
αn
were 0.667 and 1.333, respec-
tively. Ris the distance from the wave source to the point of the artificial boundary;
ρ
and G
are the mass density of the medium and the shear modulus, respectively, in which
G=ρc2
s
and
λ+
2
G=ρc2
p
;c
s
and c
p
are the shear wave velocity and longitudinal wave velocity,
respectively, which can be expressed as
cs=sE
2ρ(1+v)(3)
cp=sE(1+v)
2ρ(1+v)(1−2v)(4)
where Eand νare the elastic modulus and Poisson’s ratio of the medium, respectively.
2.2. Seismic Wave Input Method Based on VSAB
In the site seismic response and soil–foundation–structure dynamic interaction anal-
ysis, the reasonable input method of external seismic waves determine the accuracy and
reliability of the final calculation results [
37
,
38
]. Based on the VSAB model, Liu and Lu [
39
]
achieved seismic wave input by converting the seismic wave problem into a wave source
problem, i.e., converting the seismic wave into equivalent nodal loads that act on the
boundary in a concentrated force. According to Liu and Lu [
39
], the equal node force F
li
in direction i (=x, y, z, representing the x-direction, y-direction, and z-direction, respec-
tively) of any node l on the boundary under the action of the incident seismic wave can be
expressed as
Fli =Kliuf
li +Cli
.
uf
li +Alσf
li (5)
where K
li
and C
li
are the coefficient matrices of the spring and dashpots in the VSAB, respec-
tively. For the left and right boundaries,
Kli =diag(Kn
,
Kτ
,
Kτ)
and
Cli =diag(Cn
,
Cτ
,
Cτ)
;
for the front and rear boundaries,
Kli =diag(Kτ
,
Kτ
,
Kn)
and
Cli =diag(Cτ
,
Cτ
,
Cn)
;
for the bottom boundary,
Kli =diag(Kτ
,
Kn
,
Kτ)
and
Cli =diag(Cτ
,
Cn
,
Cτ)
, in which
diag(
·
) means the diagonal matrix. A
l
is the control area of the boundary node l, where
Al= (A1+ A2+ A3+ A4)/4. uf
li ,.
uf
li ,
and
σf
li
represent the displacement, velocity, and stress
fields at node l, respectively. The
Fli
can be calculated according to the law of wave
propagation and the stressing state of the wave field [40].
Based on the theory of the VSAB and seismic wave input method, a Python script was
developed by the author to transform the displacement and velocity time history of the SV
waves into a nodal force time history and transfer all the nodal force time history to the
software ABAQUS (see Figure 2).
Buildings2024,14,xFORPEERREVIEW6of22
FEM numerical model
Geost ress in itializ ation
Boundary nodes
3D numerical analyses
Geometry of soil–foundation–
structure system Material properties Displacement and velocity
of incident SV waves
Nodal and element parameters
Equivalent area A
l
, transfer matrix T,
velocities c
s
and c
p
VSAB m odel Nodal velocity,
displacement, and stress
Nodal velocity,
displacement, and stress
Parameters
Modeli ng in ABAQ US MATLAB and Python script in ABAQUS
Figure2.Schematicflow-chartoftheimplementationofSVwavesintoABAQUS.
2.3.ValidationoftheVSABandSeismicWa veInputMethod
TheprecisionoftheVSABandseismicwaveinputmethodsignificantlyinfluences
theaccuracyofnumericalanalysisresultswhenconsideringthedynamicSSI.Therefore,
thisstudyutilizedanillustrativeexampleofexternalsourcefluctuationstovalidatethe
precisionandaccuracyoftheVSABmethodandseismicwaveinputbycomparingthe
numericalandanalyticalsolutions.Tothisend,acalculationregionwithdimensionsof
100m(length)×100m(width)×50m(height)wasconsideredwithinthe3Dinfiniteelastic
half-spaceforthenumericalanalysis.Thematerialpropertiesofthefiniteregionwere
definedasfollowswithinthe3Dinfiniteelastichalf-space:themassdensitywas2000
kg/m3,theelasticmoduluswas200MPa,Poisson’sratiowas0.25,andtheshearwave
velocitywas200m/s.TheschematicdiagramofthecalculationmodelisshowninFigure
3,andtheobservationpointsselectedfortheanalysisaredenotedasPA, PB,PC,andPD.
Infiniteelementnumericalanalysis,toensuretheaccuracyandstabilityofthecalcu-
lation,itisnecessarytocontrolthesizeofthesmallestelement.AssuggestedbyLiaoand
Liu[41],themaximalsize(∆x)ofelementsisrelatedtothewavelengthandshouldmeet
thefollowingformula:
,min
11
~
10 8 s
x
(6)
,min
max
s
v
s
V
f
(7)
whereisthesmallestwavelength.VsvistheSVwavevelocityandfmaxisthehighest
frequencyoftheinputseismicwave,bothofwhichwereneededinthisstudy.Accord-
ingly,the8-nodeC3D8RelementoftheABAQUSprogramwasusedtomodelthisfinite
medium.TheFEMmeshsizeofthemediumwastakenas2m×2m×2m.
Apulse-likeSVwavesignalwasverticallyincidentattheboomofthemodel.The
finitedifferencefunctionofthefollowingformulaapproximatedtheinputsignal:
044444
113
() 16 () 4 6 4 ( 1)
424
FFGGGGG
(8)
Figure 2. Schematic flow-chart of the implementation of SV waves into ABAQUS.
Buildings 2024,14, 3522 6 of 21
2.3. Validation of the VSAB and Seismic Wave Input Method
The precision of the VSAB and seismic wave input method significantly influences
the accuracy of numerical analysis results when considering the dynamic SSI. Therefore,
this study utilized an illustrative example of external source fluctuations to validate the
precision and accuracy of the VSAB method and seismic wave input by comparing the
numerical and analytical solutions. To this end, a calculation region with dimensions of
100 m (length)
×
100 m (width)
×
50 m (height) was considered within the 3D infinite
elastic half-space for the numerical analysis. The material properties of the finite region
were defined as follows within the 3D infinite elastic half-space: the mass density was
2000 kg/m
3
, the elastic modulus was 200 MPa, Poisson’s ratio was 0.25, and the shear wave
velocity was 200 m/s. The schematic diagram of the calculation model is shown in Figure 3,
and the observation points selected for the analysis are denoted as PA, PB, PC, and PD.
Buildings2024,14,xFORPEERREVIEW7of22
whereG4(τ)=τ3H(τ),τ=t/T0,F0=0.5m,T0isthepulseduration,andH(τ)istheHeaviside
function.Inthisstudy,T0=0.25s,totalduration=1.0s,andtimestep=0.005s.Thedis-
placementtimehistoryoftheinputsignalisshowninFigure4.
VSAB VSAB
Free-field surface
50 m
50 m
26 m
PA
PB
PC
PD
y
xz
Incident
wave
Figure3.Schematicdiagramofseismicwaveincidencecalculationmodel.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Displacement (m)
Time (s)
Figure4.Displacement–timehistorycurveofanincidentwave.
Figure5comparesthedisplacementtimehistoryobtainedusingtheVSABandseis-
micinputmethodbasedonthewavesuperpositiontechniquewiththeanalyticalsolution.
ThiscomparisonwasmadeattheboomobservationpointPBandthefreesurfaceobser-
vationpointPAwithinahomogeneoushalf-spacefinite-domainmodel.Theresultsshow
thattheSVwave,whichpropagatedvertically,reachedthefreesurfaceofthegroundafter
0.375s,accompaniedbythesuperpositionofthereflectedwave.Atthefreesurfaceobser-
vationpointPA,themaximumhorizontaldisplacementresponsewastwicetheamplitude
oftheincidentwave,anditswaveformcorrespondedtotheincidentwave.By0.50s,the
shakingofthefreesurfaceceased,andthedisplacementresponsebecamezero.Thisindi-
catesthatthereflectedwavewasfullyabsorbedbytheVSABuponreachingtheboom
artificialboundary,withoutbeingreflectedtotheground.Figure5alsopresentsthenu-
mericalcalculationresultsofthedisplacementtimehistoryofthefree-fieldsurfaceobser-
vationpointPAandtheboomobservationpointPB,consistentwiththeanalyticalsolu-
tion.Theresultsofthesenumericalsimulationswereinlinewiththetheoryofwaveprop-
agationlaws,therebyprovingthecorrectnessandvalidityoftheVSABseingandseismic
inputproceduresdiscussedinthispaper.
Figure 3. Schematic diagram of seismic wave incidence calculation model.
In finite element numerical analysis, to ensure the accuracy and stability of the calcula-
tion, it is necessary to control the size of the smallest element. As suggested by Liao and
Liu [
41
], the maximal size (
∆
x) of elements is related to the wavelength and should meet
the following formula:
∆x=1
10 ∼1
8λs,min (6)
λs,min =Vsv
fmax (7)
where
λs,min
is the smallest wavelength. V
sv
is the SV wave velocity and f
max
is the highest
frequency of the input seismic wave, both of which were needed in this study. Accordingly,
the 8-node C3D8R element of the ABAQUS program was used to model this finite medium.
The FEM mesh size of the medium was taken as 2 m ×2 m ×2 m.
A pulse-like SV wave signal was vertically incident at the bottom of the model. The
finite difference function of the following formula approximated the input signal:
F(τ) = 16F0G4(τ)−4G4τ−1
4+6G4τ−1
2−4G4τ−3
4+G4(τ−1)(8)
where G
4
(
τ
) =
τ3
H(
τ
),
τ
=t/T
0
,F
0
= 0.5 m, T
0
is the pulse duration, and H(
τ
) is the
Heaviside function. In this study, T
0
= 0.25 s, total duration = 1.0 s, and time step = 0.005 s.
The displacement time history of the input signal is shown in Figure 4.
Buildings 2024,14, 3522 7 of 21
Buildings2024,14,xFORPEERREVIEW7of22
whereG4(τ)=τ3H(τ),τ=t/T0,F0=0.5m,T0isthepulseduration,andH(τ)istheHeaviside
function.Inthisstudy,T0=0.25s,totalduration=1.0s,andtimestep=0.005s.Thedis-
placementtimehistoryoftheinputsignalisshowninFigure4.
VSAB VSAB
Free-field surface
50 m
50 m
26 m
PA
PB
PC
PD
y
xz
Incident
wave
Figure3.Schematicdiagramofseismicwaveincidencecalculationmodel.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Displacement (m)
Time (s)
Figure4.Displacement–timehistorycurveofanincidentwave.
Figure5comparesthedisplacementtimehistoryobtainedusingtheVSABandseis-
micinputmethodbasedonthewavesuperpositiontechniquewiththeanalyticalsolution.
ThiscomparisonwasmadeattheboomobservationpointPBandthefreesurfaceobser-
vationpointPAwithinahomogeneoushalf-spacefinite-domainmodel.Theresultsshow
thattheSVwave,whichpropagatedvertically,reachedthefreesurfaceofthegroundafter
0.375s,accompaniedbythesuperpositionofthereflectedwave.Atthefreesurfaceobser-
vationpointPA,themaximumhorizontaldisplacementresponsewastwicetheamplitude
oftheincidentwave,anditswaveformcorrespondedtotheincidentwave.By0.50s,the
shakingofthefreesurfaceceased,andthedisplacementresponsebecamezero.Thisindi-
catesthatthereflectedwavewasfullyabsorbedbytheVSABuponreachingtheboom
artificialboundary,withoutbeingreflectedtotheground.Figure5alsopresentsthenu-
mericalcalculationresultsofthedisplacementtimehistoryofthefree-fieldsurfaceobser-
vationpointPAandtheboomobservationpointPB,consistentwiththeanalyticalsolu-
tion.Theresultsofthesenumericalsimulationswereinlinewiththetheoryofwaveprop-
agationlaws,therebyprovingthecorrectnessandvalidityoftheVSABseingandseismic
inputproceduresdiscussedinthispaper.
Figure 4. Displacement–time history curve of an incident wave.
Figure 5compares the displacement time history obtained using the VSAB and seismic
input method based on the wave superposition technique with the analytical solution. This
comparison was made at the bottom observation point PB and the free surface observation
point PA within a homogeneous half-space finite-domain model. The results show that the
SV wave, which propagated vertically, reached the free surface of the ground after 0.375 s,
accompanied by the superposition of the reflected wave. At the free surface observation
point PA, the maximum horizontal displacement response was twice the amplitude of the
incident wave, and its waveform corresponded to the incident wave. By 0.50 s, the shaking
of the free surface ceased, and the displacement response became zero. This indicates that
the reflected wave was fully absorbed by the VSAB upon reaching the bottom artificial
boundary, without being reflected to the ground. Figure 5also presents the numerical
calculation results of the displacement time history of the free-field surface observation
point PA and the bottom observation point PB, consistent with the analytical solution. The
results of these numerical simulations were in line with the theory of wave propagation
laws, thereby proving the correctness and validity of the VSAB setting and seismic input
procedures discussed in this paper.
Buildings2024,14,xFORPEERREVIEW8of22
0.0 0.2 0.4 0.6 0.8 1.0
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
Displacement (m)
Time (s)
Analytical so lution
Numerical solution
0.0 0.2 0.4 0.6 0.8 1.0
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
Displacement (m)
Time (s)
Analytical solution
Numerical solution
(a)PA. (b)PB.
Figure5.DisplacementofdifferentobservationpointsundertheverticalinputSVwave.
Figure6demonstratesthattheequivalentinputseismicloadobtainedthroughthe
wavesuperpositionmethodalignedwiththeextractedboundarynodereactionsfromthe
numericalanalysis.Thisalignmentdemonstratedthattheseismicwavesattheartificial
boundarywereunaffectedbyreflectedwaves,whichfurthervalidatedtheaccuracyofthe
seismicwaveinputmethod.Additionally,theverificationofthesoil–foundation–struc-
ture3Dnumericalmodelcanbesupportedbyreferencesintheliterature[7].
0.0 0.2 0.4 0.6 0.8 1.0
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
Force (×10
7
N)
Time (s)
Analytical solution
Numerical solution
0.0 0.2 0.4 0.6 0.8 1.0
−4.0
−3.0
−2.0
−1.0
0.0
1.0
2.0
3.0
4.0
Force (×10
7
N)
Time (s)
Analytical solution
Numerical solution
(a)PB.(b)PC.
Figure6.Equivalentinputseismicloadsinthex-directionwereobtainedfromdifferentobservation
points.
3.SteelFrame–CoreWallStructureUnderInvestigation
3.1.DescriptionofStructureandFoundationSystem
A16-story,three-baysteelframebuildingwithreinforcedconcrete(RC)corewall
structuresystemsdesignedaccordingtotheChinesebuildingcode[42]andrepresenting
conventionalhigh-risebuildingswastakenasareferenceinthisstudy.Thiscasebuilding
wasregularalongitsheight(y-direction),withatotalheightof57.6mabovegroundlevel
and18mwideinboththex-directionandz-direction.AsshowninFigure7,thebuilding
featuredaregulartwo-way(i.e.,thex-directionandz-direction)symmetricalsteelframe–
corewallstructuralsystem,inwhichthesteelcolumnsandbeamswerecoupledwithRC
corewallsthroughtheRCslabs.Specifically,theRCcorewallservedasanelevatorshaft
withtwodoorsinthez-directionandthegeometricsizeofthesedoorswas2.0m×2.7m
(width×height).Thecross-sectionofthesteelcolumnswasasquaresteeltubewithdi-
mensionsof500mm×500mm×25mm(height×width×thickness).Thecross-sectionof
thesteelbeamswasanH-beamwithasizeof400mm×300mm×20mm×16mm(height
×width×thicknessoftheflangeandweb).
Figure 5. Displacement of different observation points under the vertical input SV wave.
Figure 6demonstrates that the equivalent input seismic load obtained through the
wave superposition method aligned with the extracted boundary node reactions from the
numerical analysis. This alignment demonstrated that the seismic waves at the artificial
boundary were unaffected by reflected waves, which further validated the accuracy of the
seismic wave input method. Additionally, the verification of the soil–foundation–structure
3D numerical model can be supported by references in the literature [7].
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0.0 0.2 0.4 0.6 0.8 1.0
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
Displacement (m)
Time (s)
Analytical so lution
Numerical solution
0.0 0.2 0.4 0.6 0.8 1.0
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
Displacement (m)
Time (s)
Analytical solution
Numerical solution
(a)PA. (b)PB.
Figure5.Displacementof