Research ArticleNew Equivalent Linear Impact Model for Simulation of SeismicIsolated Structure Pounding against Moat Wall
Yang Liu,1 Wen-Guang Liu,1 Xin Wang,2 Wen-Fu He,1 and Qiao-Rong Yang1
1 Department of Civil Engineering, Shanghai University, Shanghai 200072, China2 School of Mechanicals, Shanghai Dianji University, 1350 Ganlan Road, Shanghai, China
Correspondence should be addressed to Wen-Guang Liu; [email protected]
Received 4 July 2014; Accepted 18 September 2014; Published 24 November 2014
Base-isolated buildings subjected to extreme earthquakes or near-fault pulse-like earthquakes can exceed their design gap distanceand impact against the surrounding moat wall. Based on equating energy dissipation and maximum collision compressiondeformation of isolated structurewith theHertz-dampmodel andKevin-Voigtmodel in the process of collision, an equivalent linearimpact model (ELIM) is proposed to better predict impact response of seismic isolated structure. The formula of the equivalentlinear stiffness of ELIM is theoretically derived.The effectiveness of ELIM is verified by comparing the results of numerical analyseswith the results of pounding experiments. Four near-fault earthquakes are selected to validate rationality and accuracy of theproposed model using numerical analysis. The results indicate that the proposed linear model can nearly capture impact behaviorof isolated structure in simulating the pounding-involved structural response.
1. Introduction
Base-isolated structure can be subjected to impact withsurrounding retaining walls during strong earthquake shak-ing containing low-frequency energy if there is insufficientclearance between these elements. It may be the major causeof seismic damage or even collapse of the buildings duringearthquakes. In the February 2011 Christchurch earthquake,6∼12%of the surveyed buildingswere observed to have severedamage resulting from pounding [1].
Impact problem of a base-isolated structure has gainedthe interest of researchers. Nagarajaiah and Sun studied abase-isolated fire command and control building in LosAngeles during 1994 Northridge earthquake. It was observedthat the response of base-isolated buildings has been alteredsignificantly due to the occurrence of impact [2]. Matsagarand Jangid examined the case of seismically isolated multi-degree-of-freedom (MDOF) structures pounding with var-ious types of isolation systems using Newmark’s step-by-step iteration method [3]. They also investigated the seismicresponse of a single-story asymmetric structure supported on
various base isolation systems during the impact with adja-cent structures [4]. Through parametric analyses, Komodro-mos et al. investigated the pounding effects of a seismicallyisolated building with the surrounding moat wall, whichrevealed the detrimental effects of the structural impactingon the effectiveness of seismic isolation [5, 6]. Agarwalet al. examined the case of a seismically isolated buildingpounding with an adjacent fixed-supported building [7].Polycarpou and Komodromos investigated the effects ofpotential pounding incidences on the seismic response ofa typical seismically isolated building subjected to variousearthquake excitations by numerical simulations [8, 9]. Pantand Wijeyewickrema studied seismic pounding of a typicalfour-story base-isolated reinforced concrete (RC) buildingwith retaining walls at the base and a four-story fixed-baseRC building using three-dimensional finite element analyses[10]. Masroor and Mosqueda conducted a series of earth-quake simulation experiments to assess the performance ofultimate state of seismically isolated buildings under strongground motions including pounding against a moat wall[11]. Moustafa and Mahmoud assessed the pounding of
Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 151237, 10 pageshttp://dx.doi.org/10.1155/2014/151237
2 Shock and Vibration
adjacent buildings with fixed-base and isolated-base usinginput energy, dissipated energy, and damage indices [12].Previous studies indicated that the responses of base-isolatedbuildings were substantially influenced by seismic pound-ing.
Engineering structural pounding is a complex phe-nomenon involving plastic deformations at contact points,local cracking or crushing, fracturing due to impact, friction,and so forth. There are generally two different approaches tomodel structural pounding [14].The first one applies the clas-sical theory of impact, stereomechanics, which is based on thelaws of conservation of energy and momentum and does notconsider transient stresses and deformations in the impactingbodies [15–17]. The second approach to model poundingis to directly simulate the pounding force during impact.The pounding force between colliding structures is usuallymodeled by elastic or viscoelastic impact elements whichbecome active when the contact is detected. Most proposedelements represent local force penetration of two objects atthe contact point without considering the vibration aspect ofcollision. Typical models are linear spring-damper (Kelvin-Voigt) models for viscoelastic behavior [10] and nonlinearspring elements based on Hertz contact law [17, 18]. Relatedto this work, Komodromos et al. [5] investigated the use ofcommon force-based impact models to simulate pounding ofconcrete slabs in isolated buildings to moat walls, includinga linear viscoelastic model with permanent deformation toavoid tensile impact forces during restitution. Polycarpou etal. [19] proposed a newnonlinear inelastic force-based impactmodel to describe the behavior of rubber dumpers underimpact loading, as an alternative in mitigating the impactforce. Pant and Wijeyewickrema [20] developed modifiedKelvin-Voigt (MKV) model, which had been extended tosimulate pounding with friction of a base-isolated buildingwith retainingwalls at the base. Khatiwada et al. [21] proposeda generic model that can have either a linear or nonlinearforce-deformation relationship for the analysis of buildingpounding. Xu et al. [22] investigated the random responseof vibrating system with inelastic impact which is describedthrough a modified Hertzian contact model by an equivalentnonlinearization technique. Abdel Raheem [23] studied theeffect of impact using linear and nonlinear contact forcemodel for different separation distances and compared themwith nominal model without pounding consideration. Mostcommercial software packages provide the linear springelement with a gap to model impact, but linear modelparameter is hard to determine. For simplification, the totalstiffness of the contact elements placed on any side of the baseslab is assumed to be equal to the axial stiffness of the slab,but it lacks some theory foundations. Although numericalanalysis results using the nonlinear Hertz spring model havea good agreement with experimental results, the nonlinearHertz springmode cannot be directly used in the commercialsoftware packages [24].Muthukumar [13] showed that a bilin-ear spring can provide a reasonable substitute. However, it isrelated to a key parameter 𝑎. Thus, a new linear model whoseparameters can be easily determined in practice engineeringis urgently needed to study for accuracy simulation of impactbehavior.
In this paper, an equivalent linear impactmodel (ELIM) isproposed to study the behavior of isolated structure poundingwith moat wall. Formula of the equivalent linear stiffness ofELIM is theoretically derived, based on equating energy dissi-pation and maximum collision compression deformation ofisolated structure using the Hertz-damp model and Kevin-Voigt model in the process of collision. Subsequently, theresults of numerical simulations have been compared withthe results of an impact experiment conducted by droppingballs. Finally, simulation results and sensitivity analyses arepresented to check the accuracy of the proposedmodel whichcaptures the behavior of isolated structure pounding withmoat wall.
2. New Equivalent LinearImpact Model (ELIM)
Figure 1 shows the model of impact between two collidingbodies. The Hertz-damp model is used in the first impactsystem, and linear viscoelastic impact model is used inthe second impact system. According to stereomechanicalmodel, the energy lossΔ𝐸 during the impact can be expressedin terms of the coefficient of restitution 𝑒 and the approachingvelocities V
1, V2of two colliding bodies as follows [25]:
Δ𝐸 =
1
2
𝑚
1𝑚
2
𝑚
1+ 𝑚
2
(1 − 𝑒
2) (V1− V2)
2
. (1)
The Hertz contact law [17] was originally proposed forstatic contact of two bodies, in which stresses and deforma-tions near the contact point were described as a function ofthe geometric and elastic properties of the bodies.The contactforce is related to the relative indentation of two bodies witha nonlinear spring of stiffness Hertz-damp impact modelwhich can be described as [25, 26]
𝐹
𝑐= 𝑘
ℎ𝛿
3/2+ 𝑐
ℎ̇
𝛿; 𝛿 ≥ 0,
𝐹
𝑐= 0; 𝛿 < 0.
(2)
The nonlinear damping coefficient is taken as follows [27]:
𝑐
ℎ= 𝜉
ℎ𝛿
3/2. (3)
The contact force is related to the relative indentationof two bodies with a nonlinear spring having a stiffness 𝑘
ℎ
calculated as [26]
𝑘
ℎ=
4
3𝜋
(
1
𝜆
1+ 𝜆
2
)√
𝑅
1𝑅
2
𝑅
1+ 𝑅
2
,
𝜆
𝑖=
1 − V2𝑖
𝜋𝐸
𝑖
,
𝑅
𝑖=
3√
3𝑚
𝑖
4𝜋𝜌
𝑖
.
(4)
The linear viscoelastic impact model, also known asKelvin-Voigt model, is one of the most commonly used
Shock and Vibration 3
kh
chm1 m2
�1 �2
gp
(a)
m1m2
�1 �2kk
ck
gp
(b)
Figure 1: Equivalent model of impact between two colliding bodies: (a) system 1 withHertz-dampmodel and (b) system 2with Kelvinmodel.
Impa
ct fo
rce
Displacement
kk
gp
(a)
Displacement
Impa
ct fo
rce
kh
gp
(b)
Figure 2: Contact force-displacement relationship for various impact models: (a) Kelvin model and (b) Hertz-damp model [13].
models in structural pounding and consists of a linear impactspring and a viscous impact dashpot.TheHertz-dampmodeland Kelvin-Voigt model are shown in Figure 2. Owing to itssimplicity, the linear viscoelastic model has been widely usedfor the simulation of structural pounding. The impact force𝐹
𝑐is provided by the expression [25]
𝐹
𝑐= 𝑘
𝑘𝛿 + 𝑐
𝑘̇
𝛿; 𝛿 ≥ 0,
𝐹
𝑐= 0; 𝛿 < 0.
(5)
The impact of element damping can be obtained from theformula
𝑐
𝑘= 𝜉
𝑘𝛿
𝑛. (6)
The damping ratio is [27]
𝜉
𝑘=
3𝑘
𝑘(1 − 𝑒)
2𝑒 (V1− V2)
. (7)
Considering themomentum and energy balance betweenthe start and the end of the compression phase, we have
1
2
𝑚V21+
1
2
𝑚V22= ∫
𝛿max
0
𝑘
ℎ𝛿
𝑛𝑑𝛿 + ∫
𝛿max
0
𝜉
ℎ𝛿
𝑛̇
𝛿𝑑𝛿
+
1
2
(𝑚
1+ 𝑚
2) 𝑉
2
=
𝑘
ℎ𝛿
𝑛+1
max𝑛 + 1
+ ∫
𝛿max
0
𝜉
ℎ𝛿
𝑛̇
𝛿𝑑𝛿
+
1
2
(𝑚
1+ 𝑚
2) 𝑉
2,
(8)
𝑚V1+ 𝑚V2= (𝑚
1+ 𝑚
2) 𝑉. (9)
Substituting (9) into (8) yields
𝑘
ℎ𝛿
𝑛+1
max𝑛 + 1
+ ∫
𝛿max
0
𝜉
ℎ𝛿
𝑛̇
𝛿𝑑𝛿 =
1
2
𝑚
1𝑚
2
𝑚
1+ 𝑚
2
(V1− V2)
2
.(10)
4 Shock and Vibration
Impa
ct fo
rce
Impa
ct fo
rce
ΔE ΔE
gpgp
Figure 3: Equal energy dissipated for various impact models.
In the equation, the second formula is energy dissipatedby the damping force.
The energy dissipated by the damping force can beevaluated as
Δ𝐸
1= ∫
𝛿max
0
𝜉
ℎ𝛿
𝑛̇
𝛿𝑑𝛿.(11)
The energy dissipated by the damping force in thecompression and restitution phase
Δ𝐸
2= 𝑒Δ𝐸
1, (12)
Δ𝐸 = Δ𝐸
1+ Δ𝐸
2. (13)
Substituting (13) into (12) yields
𝑘
ℎ𝛿
𝑛+1
max𝑛 + 1
+
1
2
𝑚
1𝑚
2
𝑚
1+ 𝑚
2
(1 − 𝑒) (V1− V2)
2
=
1
2
𝑚
1𝑚
2
𝑚
1+ 𝑚
2
(V1− V2)
2
.
(14)
This energy loss can be expressed by [28]
Δ𝐸 =
𝑘
ℎ𝛿
𝑛+1
max (1 − 𝑒
2)
𝑒 (𝑛 + 1)
.
(15)
This energy loss of impact bodies using Kevin-Voigt canbe expressed by
Δ𝐸 =
1
3
𝜉
𝑘(V1− V2) 𝛿
2
max (1 + 𝑒) . (16)
Based on equal energy dissipated by the damping forceand maximum of two systems using Kelvin-Voigt model andHertz-damp model (see Figure 3), impact stiffness of Kelvin-Voigt model can be obtained from (7), (15), and (16):
𝑘
𝑘=
4𝑘
ℎ
5
√𝛿max. (17)
m
h
y(t)
Figure 4: Numerical model of the ball.
3. Experimental Verification ofPounding Force Models
In order to verify the accuracy of ELIM of structural pound-ing, the results of the numerical analysis have been comparedwith the results of the experiments conducted by Jankowski[29]. It was carried out by dropping balls onto a rigid planesurface of concrete material. In the numerical analysis, themodel of pounding between the falling ball and a stationaryrigid surface, shown in Figure 4, is used.
The dynamic equation of motion for such a model can bewritten in the form
𝑚
..
𝑦(𝑡) + 𝑓
𝑝(𝑡) = 𝑚𝑔, (18)
where 𝑚 is the mass of a ball, 𝑦(𝑡) is its vertical acceleration,𝑔 stands for the acceleration of gravity, and 𝑓
𝑝(𝑡) is the
pounding force which is equal to zero when 𝑦(𝑡) ≤ ℎ (ℎ isa drop height) and can be calculated by (5) when 𝑦(𝑡) > ℎ,where deformation is 𝛿(𝑡) expressed as
𝛿 (𝑡) = 𝑦 (𝑡) − ℎ. (19)
Shock and Vibration 5
0.0 0.1 0.2 0.3 0.4 0.5 0.6−500
0
500
1000
1500Im
pact
forc
e (N
)
Time (ms)
Experiment Variable damping Constant damping
Figure 5: Comparison of pounding force time histories betweennumerical results and experimental results.
Table 1: Comparison of impact forces and deformations of pound-ing model with and without damping.
Case Peak impact force (N) Peak displacement (mm)With damping 1192 2.17 × 10−2
Without damping 1315 2.38 × 10−2
A time-stepping integration procedure with constanttime step 𝑡 = 1×10
−7 s has been applied to solve the equationof motion (18) numerically. Using (17), the stiffness values ofELIM have been calculated. The pounding force time historymeasured during the experiment and the histories receivedfrom the numerical analysis for the first impact are presentedin Figure 5. Figure 5 also compares calculation results withconstant damping coefficient and variable damping coeffi-cient. It indicates that there is a good agreement betweenthe numerical and experimental results. In addition, it isfound that the physical characteristics of numerical resultsby using constant damping coefficient are inconsistent withexperimental results. This is because in the case of the linearviscoelastic model, the negative force can be observed justbefore separation.However,maximum impact force obtainedfrom numerical analysis using constant damping coefficientis close to the experimental result.
In practice, some commercial software packages providethe linear spring element with a gap to model impact butcannot give the damping coefficient of impact model. Fur-thermore, it is complex for engineers to calculate the dampingcoefficient using (5).Thus, the accuracy of results using ELIMwith no damping should be assessed by comparative analysis.Table 1 shows themaximum impact force and deformation ofequivalent linear impact model with and without damping. Itshows that, regardless of the damping effect, the impact forceand impact compression deformation are obviously larger
gp
kb
mb
kw
fp
mw
cb
Figure 6: Simplified truss contact element of base-isolated structurefor impact simulation.
than calculation results considering the damping effect, butthe simulation errors are within 30%. The damping effectcan reduce the overall vibration response of the structure,so ignoring the effect of damping coefficient cannot lead tounsafe results for performance evaluation of the structure.
4. Selection of Near-Fault Pulse-Like GroundMotion Records
Code for seismic design of buildings has prescribed generalguidelines but they do not provide specifics for selecting thetype of earthquake records for nonlinear dynamic analysis(Katsanos, 2006). In general, it is better to choose the recordsbased on real recordswhich are from strongmotions databaseto consider statistical characteristics of strong motions. Cur-rently, the key ground motions for assembling the databaseare from the Japanese K-Net strong motion network (NIED),PEER/NGA, and CESMD (http://www.strongmotioncenter.org/). Because of large isolator displacements due to longperiod pulses associated with the near-fault motion, fourdifferent earthquake records (Table 2) are used in order toexamine the effects of the characteristics of the excitationon the seismic response of the seismically isolated buildingduring poundings using the proposed model.
All selected large pulse-like earthquake records are char-acterized by low-frequency content, in order to induce largedisplacements of the seismically isolated building. Thesemotions cover amomentmagnitude range from 6.0 to 7.6 anda rupture distance (closest distance from site to fault ruptureplane) range from 0 to 10 km. The peak ground velocities of4 ground motion records are more than 30 cm/s. The peakaccelerations (PGA) corresponding to earthquakes of majorlevels are specified to be 0.4 g.
5. The Simplified Base-Isolated Structure andMoat Wall Model
The seismic response of a base-isolated structure duringimpact with the surrounding moat wall is investigated. The
6 Shock and Vibration
Table 2: Near-fault pulse-like record data.
Number Event 𝑅rup Mechanism V30
𝑇
𝑝
N1 EL CENTRO ARRAY #10 4.0 Strike-slip 205.6 4N2 EL CENTRO ARRAY #6 0.6 Strike-slip 210.5 4.2, 4.5N3 459 6.3 Strike-slip 545.0 0.86, 1.8N4 802 7.0 Reverse 513.7 4.9
0 10 20 30 40−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Time (s)
Acce
lera
tion
(m/s
2 )
Hertz-damp modelProposed model
(a)
Time (s)
0 10 20 30 40−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
Acce
lera
tion
(m/s
2 )
Hertz-damp modelProposed model
(b)
Hertz-damp modelProposed model
Time (s)
0 10 20 30−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
Acce
lera
tion
(m/s
2 )
(c)
Time (s)
0 10 20 30−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
Acce
lera
tion
(m/s
2 )
Hertz-damp modelProposed model
(d)
Figure 7: Acceleration time history of isolated structure (a) N1; (b) N2; (c) N3; (d) N4.
numerical studies are carried out using amathematicalmodelof the base-isolated structure to calculate the response ofinterest such as the peak absolute acceleration and the peakrelative displacement of the isolation layer. The total weightis 200 t. The period of base-isolation structure is 3 s. Hertzimpact stiffness equals 4.15 × 10
6N/m3/2. The equivalentdamping ratio of isolated structure is 10%. Simplified 2-DOFmodel of base-isolated structure pounding againstmoatwall is shown in Figure 6. In Figure 6, 𝑥
𝑏is displacement of
isolated structure, 𝑥𝑤is displacement of moat wall, and the
dynamic equations are
𝑚
𝑏
..
𝑥
𝑏+ 𝑘
𝑏𝑥
𝑏+ 𝑐
𝑏�̇�
𝑏+ 𝑓
𝑝(𝑡) = −𝑚
𝑏
..
𝑥
𝑔,
𝑚
𝑤
..
𝑥
𝑤+ 𝑘
𝑤𝑥
𝑤− 𝑓
𝑝(𝑡) = 0,
(20)
where..
𝑥
𝑔is seismic acceleration; 𝑚
𝑏, 𝑘
𝑏, and 𝑐
𝑏are mass,
stiffness, and damping of isolated structure. 𝑘
𝑤and 𝑐
𝑤are
stiffness and damping of moat wall; 𝑓𝑝(𝑡) is impact force.
Shock and Vibration 7
0 5 10 15 20 25 30 35 40−0.10
−0.05
0.00
0.05
0.10
Disp
lace
men
t (m
)
Time (s)
Hertz-damp model
Proposed model
(a)
0 5 10 15 20 25 30 35 40−0.10
−0.05
0.00
0.05
0.10
Hertz-damp model
Proposed model
Disp
lace
men
t (m
)
Time (s)
(b)
Hertz-damp model
Proposed model
0 5 10 15 20 25 30−0.10
−0.05
0.00
0.05
0.10
Disp
lace
men
t (m
)
Time (s)
(c)
Hertz-damp model
Proposed model
0.10
0 5 10 15 20 25 30−0.10
−0.05
0.00
0.05
Disp
lace
men
t (m
)
Time (s)
(d)
Figure 8: Displacement time history of isolated structure (a) N1; (b) N2; (c) N3; (d) N4.
6. Numerical Results
Thenumerical procedure has been programmedbyMATLABsoftware. Acceleration, displacement, and impact force ofseismically isolated structure are plotted in Figures 7, 8, and 9,respectively.The prediction of acceleration and displacementresponses using ELIM is an agreement with the result usingHertz-damp model, and large acceleration responses underfour near-fault earthquakes are captured in the numericalmodel within few errors. By comparing Figures 7 and 9, itcan be seen that the maximum values of the accelerationresponses of the isolated structure were amplified whenimpact occurred. The acceleration response for N1 is 0.14 g,and corresponding impact force is 140.2 kN. Although theimpact at some points under N4 cannot be well predicted,
numerical simulations using ELIM capture the dominantcharacteristics of the contact force, indicating that the effect ofthe impact on isolated structure response is largely dependingon the earthquake characteristics. The numerical modelincluding ELIM can reproduce the seismic response of base-isolated structure impacting against a moat wall.
7. Sensitivity Analysis
Sensitivity analyses were conducted to examine the effectsof model parameters on the simulated structural peakresponses. The Hertz stiffness (𝑘
ℎ), moat wall spring stiffness
(𝑘𝑤), and isolation layer spring stiffness (𝑘
𝑏) are considered
here. Changes in these three parameters over a range of−50%,+50% of their base value were examined for resulting peak
8 Shock and Vibration
0 5 10 15 20 25 30 35 40−150
−100
−50
0
50
100
150
200
Impa
ct fo
rce (
kN)
Time (s)
Hertz-damp model
Proposed model
(a)
0 5 10 15 20 25 30 35 40−200
−150
−100
−50
0
50
100
150
Impa
ct fo
rce (
kN)
Time (s)
Hertz-damp model
Proposed model
(b)
0 5 10 15 20 25 30−100
−50
0
50
100
Impa
ct fo
rce (
kN)
Time (s)
Hertz-damp model
Proposed model
(c)
0 5 10 15 20 25 30−100
−50
0
50
100Im
pact
forc
e (kN
)
Time (s)
Hertz-damp model
Proposed model
(d)
Figure 9: Impact force time history (a) N1; (b) N2; (c) N3; (d) N4.
response of the structure. Ratios between peak responsesusing ELIM and Hertz model are
Ratioacc =
𝐴
𝑘− 𝐴
ℎ
𝐴
ℎ
× 100%,
Ratiodis =
𝑑
𝑘− 𝑑
ℎ
𝑑
ℎ
× 100%,
Ratio𝑓
=
𝑓
𝑘− 𝑓
ℎ
𝑓
ℎ
× 100%,
(21)
where𝐴
ℎ, 𝑑ℎ, and𝑓
ℎare peak absolute acceleration, displace-
ment, and impact force responses of the isolated structureusing Hertz-damp model, respectively; 𝐴
𝑘, 𝑑
𝑘, and 𝑓
𝑘are
peak absolute acceleration, displacement, and impact forceresponses of the isolated structure using ELIM, respectively.
Each separate parameter was varied one at a time, and theresponse of the structure was compared with correspondingresults using base values. The base values are obtained fromthe procedure explained in Section 6 of this paper. Figures 10,11, and 12 show the sensitivity of the three parameters on thebase level peak acceleration and displacement, respectively.It can be seen that moat wall spring stiffness parameter hasthe largest effect on impact force of the isolated structureresulting in a maximum 16% response change in comparisonwith base values for a 50% change inmodel parameter values.Changes in all responses are very small because of changesin Hertz stiffness and moat wall spring stiffness parameters.Figure 12 shows that the isolated structure spring stiffness hasthe largest effect on the impact force. Impact force obtainedusing ELIM is underestimated for a 50% change in springstiffness of isolated structure, so accurately calculating impact
Shock and Vibration 9
−50 −25 0 25 50−2
0
2
4
6
8
10
12
14
Ratio
(%)
Acceleration Displacement Impact force
kh change from base value (%)
Figure 10: Effect of changing Hertz stiffness on structural response.
−50 −25 0 25 50−12
−8
−4
0
4
8
12
16
Ratio
(%)
Acceleration Displacement Impact force
kw change from base value (%)
Figure 11: Effect of changingmoat wall spring stiffness on structuralresponse.
−50 −25 0 25 50−10
−5
0
5
10
15
20
25
30
Ratio
(%)
Acceleration Displacement Impact force
kb change from base value (%)
Figure 12: Effect of changing isolation layer spring stiffness onstructural response.
force using ELIM is not recommended if spring stiffness ofisolated structure is varied largely. The maximum change inall structure response is less than 15% due to 50% changein the three parameters. The sensitivity analysis shows thatELIM is a reliablemodel considering uncertainty in assigningparameter. However, the drawback of ELIM is that isolatedstructure model parameters should be taken into account forELIM to obtain more accurate simulation results.
8. Conclusions
In the Hertz contact model with nonlinear damping, theexpression has been validated by shaking table test, but itcannot be directly used in the commercial software packages.To simplify it, an approximate formula of ELIM is theoreti-cally derived based on equating energy dissipation andmaxi-mum collision compression deformation of isolated structureusing the Hertz-damp model and Kevin-Voigt model in theprocess of collision. By comparing numerical analysis andexperimental results of dropping balls, the correctness offormula (17) has been verified. This model can be used intocommercial software packages by inputting linear impactstiffness and damping. As it is easy for engineers to calculatelinear impact stiffness and damping of ELIM based onHertz-damp model, ELIM can be directly adopted in engineeringpractice. ELIM is recommended to be used in actual projectsdue to its inexpensive computational overhead.
In addition, ELIM can be used to predict impact responseand capture impact behavior of base-isolated structurepounding against moat wall subjected to near-fault pulse-like earthquakes.The numerical simulations indicate that theacceleration and displacement responses of isolated structureand surrounding wall can be captured by comparing Hertz-damp and ELIM. It is beneficial for engineers to design areasonable base-isolated structure pounding against moatwall. Sensitivity analysis conducted shows that the maximumchange in structure response is less than 16% due to a 50%change in impact model parameters. These studies demon-strate that ELIM can provide reasonable results consideringuncertainty in assigning model parameters. Therefore, reli-able results of pounding simulation in structural engineeringcan be obtained from ELIM.
Notations
𝑚
1, 𝑚2: Masses of impact bodies
V1, V2: The velocities before impact
𝑘
ℎ, 𝑐ℎ: The nonlinear impact spring stiffness
and damping𝑔
𝑝: Gap width
𝑅
1, 𝑅2: Radii of two isotropic spheres
𝜆
1, 𝜆2: Material parameters
]𝑖and 𝐸
𝑖: Poisson’s ratio and modulus of elasticityof sphere 𝑖, respectively
𝑚
𝑖: Colliding mass
𝜌
𝑖: The density of concrete
𝑉: A common velocity of the two bodies𝛿max: Maximum penetration.
10 Shock and Vibration
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
The authors gratefully acknowledge the financial support ofNational Natural Science Foundation of China nos. 51278291and 51308331.
References
[1] G. L. Cole, R. P. Dhakal, and F. M. Turner, “Building poundingdamage observed in the 2011 Christchurch earthquake,” Earth-quake Engineering and Structural Dynamics, vol. 41, no. 5, pp.893–913, 2012.
[2] S. Nagarajaiah and X. Sun, “Base-isolated FCC building: impactresponse in Northridge Earthquake,” Journal of Structural Engi-neering, vol. 127, no. 9, pp. 1063–1075, 2001.
[3] V. A. Matsagar and R. S. Jangid, “Seismic response of base-isolated structures during impact with adjacent structures,”Engineering Structures, vol. 25, no. 10, pp. 1311–1323, 2003.
[4] V. A. Matsagar and R. S. Jangid, “Impact response of torsionallycoupled base-isolated structures,” Journal of Vibration andControl, vol. 16, no. 11, pp. 1623–1649, 2010.
[5] P. Komodromos, P. C. Polycarpou, L. Papaloizou, and M. C.Phocas, “Response of seismically isolated buildings consideringpoundings,” Earthquake Engineering & Structural Dynamics,vol. 36, no. 12, pp. 1605–1622, 2007.
[6] P. Komodromos, “Simulation of the earthquake-inducedpounding of seismically isolated buildings,” Computers andStructures, vol. 86, no. 7-8, pp. 618–626, 2008.
[7] V. K. Agarwal, J. M. Niedzwecki, and J. W. van de Lindt,“Earthquake induced pounding in friction varying base isolatedbuildings,” Engineering Structures, vol. 29, no. 11, pp. 2825–2832,2007.
[8] P. C. Polycarpou and P. Komodromos, “Earthquake-inducedpoundings of a seismically isolated building with adjacentstructures,” Engineering Structures, vol. 32, no. 7, pp. 1937–1951,2010.
[9] P. C. Polycarpou and P. Komodromos, “On poundings of aseismically isolated building with adjacent structures duringstrong earthquakes,” Earthquake Engineering and StructuralDynamics, vol. 39, no. 8, pp. 933–940, 2010.
[10] D. R. Pant and A. C. Wijeyewickrema, “Structural perfor-mance of a base-isolated reinforced concrete building subjectedto seismic pounding,” Earthquake Engineering and StructuralDynamics, vol. 41, no. 12, pp. 1709–1716, 2012.
[11] A. Masroor and G. Mosqueda, “Impact model for simulationof base isolated buildings impacting flexible moat walls,” Earth-quake Engineering and Structural Dynamics, vol. 42, no. 3, pp.357–376, 2013.
[12] A.Moustafa and S.Mahmoud, “Damage assessment of adjacentbuildings under earthquake loads,” Engineering Structures, vol.61, pp. 153–165, 2014.
[13] S. Muthukumar, A contact element approach with hysteresisdamping for the analysis and design of pounding in bridges [Ph.D.thesis], Georgia Institute of Technology, 2003.
[14] R. Jankowski, “Non-linear viscoelastic modelling of earth-quake-induced structural pounding,” Earthquake Engineeringand Structural Dynamics, vol. 34, no. 6, pp. 595–611, 2005.
[15] R. DesRoches and S. Muthukumar, “Effect of pounding andrestrainers on seismic response of multiple-frame bridges,”Journal of Structural Engineering, vol. 128, no. 7, pp. 860–869,2002.
[16] A. Ruangrassamee and K. Kawashima, “Relative displacementresponse spectra with pounding effect,” Earthquake Engineeringand Structural Dynamics, vol. 30, no. 10, pp. 1511–1538, 2001.
[17] W. Goldsmith, Impact: The Theory and Physical Behavior ofColliding Solids, Dover, New York, NY, USA, 2001.
[18] K. T. Chau, X. X. Wei, X. Guo, and C. Y. Shen, “Experimentaland theoretical simulations of seismic poundings betweentwo adjacent structures,” Earthquake Engineering & StructuralDynamics, vol. 32, no. 4, pp. 537–554, 2003.
[19] P. C. Polycarpou, P. Komodromos, and A. C. Polycarpou, “Anonlinear impact model for simulating the use of rubber shockabsorbers for mitigating the effects of structural poundingduring earthquakes,” Earthquake Engineering and StructuralDynamics, vol. 42, no. 1, pp. 81–100, 2013.
[20] D. R. Pant and A. C. Wijeyewickrema, “Performance ofbase-isolated reinforced concrete buildings under bidirectionalseismic excitation considering pounding with retaining wallsincluding friction effects,” Earthquake Engineering & StructuralDynamics, vol. 43, no. 10, pp. 1521–1541, 2014.
[21] S. Khatiwada, N. Chouw, and J. W. Butterworth, “A genericstructural pounding model using numerically exact displace-ment proportional damping,” Engineering Structures, vol. 62-63,pp. 33–41, 2014.
[22] M. Xu, Y. Wang, X. Jin, and Z. Huang, “Random vibrationwith inelastic impact: equivalent nonlinearization technique,”Journal of Sound and Vibration, vol. 333, no. 1, pp. 189–199, 2014.
[23] S. E. Abdel Raheem, “Mitigation measures for earthquakeinduced pounding effects on seismic performance of adjacentbuildings,” Bulletin of Earthquake Engineering, vol. 12, no. 4, pp.1–20, 2014.
[24] A. Masroor and G. Mosqueda, “Experimental simulation ofbase-isolated buildings pounding against moat wall and effectson superstructure response,” Earthquake Engineering and Struc-tural Dynamics, vol. 41, no. 14, pp. 2093–2109, 2012.
[25] R. Jankowski, “Analytical expression between the impact damp-ing ratio and the coefficient of restitution in the non-linearviscoelastic model of structural pounding,” Earthquake Engi-neering and Structural Dynamics, vol. 35, no. 4, pp. 517–524,2006.
[26] S. Muthukumar and R. DesRoches, “A Hertz contact modelwith non-linear damping for pounding simulation,” EarthquakeEngineering and Structural Dynamics, vol. 35, no. 7, pp. 811–828,2006.
[27] K. Ye, L. Li, and H. Zhu, “Modified kelvin pounding analyticalmodel,” Engineering Mechanics, vol. 12, no. 26, pp. 245–248,2009 (Chinese).
[28] K. Ye, L. Li, and H. Zhu, “A note on the Hertz contact modelwith nonlinear damping for pounding simulation,” EarthquakeEngineering and Structural Dynamics, vol. 38, no. 9, pp. 1135–1142, 2009.
[29] R. Jankowski, “Comparison of numerical models of impactforce for simulation of earthquake-induced structural pound-ing,” in Computational Science—ICCS 2008, vol. 5101 of LectureNotes in Computer Science, pp. 710–717, Springer, Berlin, Ger-many, 2008.
