a simplified model for dynamic response analysis...

Research ArticleA Simplified Model for Dynamic Response Analysis of FramedSelf-Centering Wall Structures under Seismic Excitations

Xiaobin Hu 12 Chen Lu 1 and Xiaoqing Zhu 1

1School of Civil Engineering Wuhan University Wuhan 430072 China2Engineering Research Center of Urban Disasters Prevention and Fire Rescue Technology of Hubei Province Wuhan 430072China

Correspondence should be addressed to Xiaoqing Zhu whuzxqwhueducn

Received 9 January 2019 Revised 26 February 2019 Accepted 26 March 2019 Published 15 April 2019

Academic Editor Salvatore Russo

Copyright copy 2019 Xiaobin Hu et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-is paper presents a simplified model for dynamic response analysis of the framed self-centering wall (FSCW) structure underseismic excitations In the analysis model the frame is equivalent as a single-degree-of-freedom system and collaborates with theself-centering (SC) wall to resist lateral loads By way of pushover analysis of a typical FSCW structure the proposed analysismodel is validated by comparing the analysis results with those obtained from the finite element analysis method Using theanalysis model motion equations of the FSCW structure under seismic excitations are established and solved through numericalsimulations Finally a comprehensive parametric study is conducted to investigate the effects of a variety of design parameters onseismic responses of the FSCW structure It shows that improving the yield force or elastic stiffness of the frame can help greatlylessen seismic responses of the FSCW structure in terms of the rotation angle of the SC wall

1 Introduction

In recent decades the self-centering (SC) structures haveattracted a lot of attention from the earthquake engineeringcommunity After strong ground shaking they undergoalmost no residual deformation and can resume normalservice with little or no rehabilitation Several types of SCstructural systems have been proposed such as SC wallstructures SC frame structures and SC braced frame struc-tures As a typical SC structure the SC wall is mainly com-posed of three parts wall posttensioned (PT) tendons anddampers It gains the self-centering ability usually by makinguse of the gap-opening behavior of the horizontal connectionsat the base or along the height of the wall [1]

Extensive studies have been conducted on the dynamicbehavior or seismic performance of the SC wall by theo-retical numerical and experimental methodologies Twotypes of analysis models ie the rigid body model and finiteelement (FE) model have been proposed in the theoreticalor numerical investigations For the former one the SC wallis treated as a rigid body the PT tendon is represented by anelastic spring and the dampers are simulated by elastoplastic

springs [2] Using this model as well as referring to the workconcerning rocking response analysis of the rigid blockunder ground motions [3ndash7] Hu et al [8] investigatedrocking responses of the SC wall under seismic excitationsDifferently for the FE model the wall is simulated by fiberbeam-column elements [9 10] or diagonal struts [11 12]while the rocking behavior at the base is simulated by agroup of contact springs In addition to the above studiessome scholars investigated experimentally or numericallythe seismic performance of the SC precast concrete walls[13ndash22] and SC confined masonry walls [11 12] -e re-search outcomes generally demonstrate that the SC wallexhibits minor damage with little or no residual displace-ments under cyclic loading or seismic excitations Moreoverthe addition of dampers can substantially enhance the en-ergy dissipation capacity of the SC wall

Recently the framed self-centering wall (FSCW) struc-ture has emerged by combining the SC wall with the frameTwo types of FSCW structures are mainly concernedie one adopting the rocking wall [23ndash26] and the otherone adopting the stepping wall [27ndash29] as shown in Figure 1For the latter one only a few studies have been carried out on

HindawiShock and VibrationVolume 2019 Article ID 2360150 14 pageshttpsdoiorg10115520192360150

its seismic performance mainly through the numericalanalysis method For example Ajrab et al [27] performednonlinear time-history analyses of the FSCW structureequipped with additional PT tendons and dampers -e re-sults show that the interstory drifts of the FSCW structure aremore uniform than those of the framed fixed-based wallstructure In addition Zibaei and Mokari [28] conducted acomparative study on seismic performance of RC frames RCframes with shear wall and RC frames with controlledrocking wall by means of pushover analysis -e results showthat for the RC frames with rocking wall system not onlyplastic hinge formation and interstory drifts are well dis-tributed but also energy dissipation and displacement duc-tility are increased Besides Hu et al [29] compared seismicresponses of the RC frame before and after retrofitted by SCwalls through nonlinear time-history analysis It shows thatthe SC wall can effectively reduce seismic responses of theoriginal RC frame structure while incurring little or no re-sidual deformation More recently Grigorian et al [30]proposed a methodology for developing earthquake-resilientstructures by introducing a rocking core-moment frame asthe lateral resisting component of a gravity-resisting structurethat is detailed to recenter while sustaining large lateraldeformations

From the above literature review it can be seen that thelimited studies regarding the FSCW structure usually in-volve nonlinear FE analysis It may be time-consumingespecially in case of analyzing the complex structure Fur-thermore there are still uncertainties in establishing the FEmodel for example how to determine the contact param-eters to realistically replicate the gap-opening behavior of theconnections In view of these issues this paper is mainlyaimed to develop a simplified analysis model of the FSCWstructure under lateral loading After validated by the FEanalysis method this model is further employed to establishmotion equations of the FSCW structure under seismicexcitations By means of numerical simulation seismic re-sponses of the FSCW structure are obtained Finally acomprehensive parametric study is carried out to investigatethe influences of a variety of design parameters whichmainly include those concerning the PT tendon thedampers and the frame on seismic responses of the FSCWstructure

2 Simplified Analysis Model ofFSCW Structures

21GeneralAnalysisModel As illustrated in Figure 2(a) theFSCW structure mainly comprises three parts the SC wallthe frame and the linking beam In this study only themoment frame is considered It is assumed that these threecomponents are lumped in the plane along the loadingdirection as shown in Figure 2(b) Furthermore the PTtendon is supposed to be installed vertically along the centerline of the SC wall and the dampers are placed at two ends ofthe bottom of the wall

Since the frame can be represented as a multi-degree-of-freedom (MDOF) system a general analysis model of theFSCW structure can thus be formed as seen inFigures 2(c)ndash2(d) Depending on whether the constraintsbetween the linking beam and the SC wall are considered ornot the analysis model can be categorized into two typesie the pin-jointed system (Figure 2(c)) and the rigid-jointedsystem (Figure 2(d)) In the figures h b and l represent theheight the width and the diagonal length of the SC wallrespectively with α being the angle between the wall diagonaland height In addition kj and hj are the stiffness and theheight relative to the ground of the jth (j 1 n) storey ofthe frame respectively where n is the number of storeys

22 Simplified Analysis Model

221 Formulation of the Simplified Model According to thedisplacement-based seismic design philosophy [31 32] theMDOF system can be equivalent as a single-degree-of-freedom (SDOF) system as shown in Figure 3 where mjFj and δj denote the mass horizontal force and lateraldisplacement of the jth storey of the MDOF system -eparameters concerning the equivalent SDOF system aredetermined as follows

heq 1113936

nj1 mjδjhj1113872 1113873

1113936nj1 mjδj1113872 1113873

(1a)

meq 1113936

nj1 mjδj1113872 1113873

δeq (1b)

(a) (b)

Figure 1 Illustration of two types of FSCW structures (a) One adopting the rocking wall (b) One adopting the stepping wall

2 Shock and Vibration

δeq 1113936

nj1 mjδ

2j1113872 1113873

1113936nj1 mjδj1113872 1113873

(1c)

where heq meq and δeq represent the equivalent heightequivalent mass and equivalent displacement of the SDOFsystem respectively

By converting the MDOF system into a SDOF system(Figure 3) the general analysis model of the FSCW structure(Figures 2(c) and 2(d)) can be further simplified as shown inFigure 4 Furthermore considering that the lateral dis-placement curve of the frame is approximately linear alongthe height [27ndash29] we can have

δj θhj j 1 2 n (2)

where θ represents the storey drift which is constant alongthe height of the frame -ereby for the nth storey it has

δn θhn (3)

Substituting equations (2) and (3) in equations (1a)ndash(1c)can yield

heq 1113936

nj1 mjh

2j1113872 1113873

1113936nj1 mjhj1113872 1113873

(4a)

meq 1113936

nj1mjhj1113872 1113873

2

1113936nj1 mjh

2j1113872 1113873

(4b)

δeq 1113936

nj1 mjh

2j1113872 1113873

1113936nj1 mjhj1113872 1113873

δn

hn

(4c)

From the above equations it can be clearly seen that heqmeq and δeq regarding the equivalent SDOF system rely only

kn

kj

k1

hl

b

αhj

kn

kj

k1

hl

b

α hj

Damper

PT tendon

Lumped SC wall

Lumped frameLinking beam

(a) (b)

(c) (d)

SC wall

Figure 2 General analysis model of the FSCW structure (a) 3D view of a typical FSCW structure (b) 2D representation of a FSCWstructure (c) Pin-jointed system (d) Rigid-jointed system

meq

δeq

heq

mn

mj

m1

δn

δj

δ1

Fn

Fj

F1

hj

Vb

Figure 3 Conversion of a MDOF system into an equivalent SDOFsystem

meq

heq

D1 D2

(a)

meq

heq

D1 D2

(b)

Figure 4 Simplified analysis model of the FSCW structure(a) Pin-jointed system (b) Rigid-jointed system

Shock and Vibration 3

on the values of hjmj (j 1 n) and δn corresponding tothe original MDOF system

222 Nonlinear Analysis Parameters of the Frame Toconduct nonlinear analysis of the FSCW structure underlateral loading the nonlinear behavior of the frame needs tobe explicitly taken into account For simplification purposethe hysteresis model of the frame is assumed to be bilinear asshown in Figure 5 where kf fyf and η represents the elasticstiffness the yield force and the postyield stiffness coefficientof the frame Actually the bilinear hysteresis is usuallyadopted to represent the general structural behavior espe-cially for the SDOF system irrespective of the materialsinvolved

In order to obtain nonlinear parameters of the frame thefollowing procedures can be implemented (1) Conductpushover analysis of the frame and obtain the capacity curveie the base shear vs top displacement curve as shown inFigure 6(a) Although various lateral load patterns such as theinverted triangular load uniform load or concentrated loadcan be considered in pushover analysis the one that makesthe lateral displacement approximately linear along the heightis recommended here (2) Convert the capacity curve into theforce vs displacement curve corresponding to the equivalentSDOF system as shown in Figure 6(b) by utilizing equation(4c) as well as the assumption that the frame and theequivalent system have the same base shear (3) Extract thevalues of kf fyf and η directly from the curve approximately bythe plotting method as seen in Figure 6(b)

3 Verification of the Proposed Analysis Model

31Details of theAnalyzedFSCWStructure A representativesix-storey concrete FSCW structure is analyzed in thissection Figure 7 gives the geometry dimensions and re-inforcement details of the FSCW structure In addition thethickness of the SC wall is 240mm -e compressivestrength and elastic modulus of the concrete are 30MPa and30times104MPa respectively -e yield strength and elasticmodulus of the longitudinal rebars in the frame and thelinking beams are 400MPa and 2times105MPa respectively-e PT tendon with an effective radius of 108mm isarranged along the center line of the wall -e elasticmodulus of the tendon and its initial stress are 2times105MPaand 1395MPa respectively For the dampers the yield forceand elastic stiffness are set to be 100 kN and 45 kNmmrespectively

32 FEModeling of the FSCW Structure In this section twodifferent FE models of the FSCW structure are established inthe software ABAQUS [33] one involves the original FSCWstructure and another one uses the proposed simplifiedanalysis model

321 General FE Model Figure 8(a) shows the overall FEmodel of the original FSCW structure -e SC wall ismodeled by the shell element S4R and its material is set to be

elastic since the wall usually exhibits minor or no damageduring lateral loading -e foundation is simulated by thesolid element C3D8R and its material is also set to be elasticNote that the uplift of the foundation may occur due to thecompliance effects of the underlying material and have agreat impact on the rocking behavior of the SC wallhowever it is beyond the scope of this study and thusneglected here -e interaction between the SC wall and thefoundation is considered by defining the surface-to-surfacecontact as shown in Figure 8(b) -e dampers are simulatedby axial connectors and the elastic-perfectly plastic con-stitutive relation is adopted Besides the PT tendon issimulated by the beam element B31 with one end tied to thetop of the wall and another one anchored directly to thefoundation and its material is also set to be elastic

-e frame and the linking beams are modeled using thebeam element B31 -e concrete constitutive relation shownin Figure 8(c) which ignores its tensile strength [34] is usedhere By way of the user material subroutine [33] the aboveconcrete constitutive relation is programmed into ABAQUS-e longitudinal rebars are simulated by adding the keywordlowastrebar in the input file and the elastic-perfectly plasticconstitutive relation is adopted

It is worth noting that the connection between the SCwall and the linking beam needs to be modeled properlyaccording to the type of the FSCW structure For the rigid-jointed system the SC wall and the linking beams are di-rectly merged where they intersect for the pin-jointedsystem the above merging operation is first conductedand then the keyword lowastrelease is used to release the con-straints between the SC wall and the linking beams

322 Simplified FE Model Figure 9 gives the simplified FEmodel of the FSCW structure where the frame is repre-sented by an equivalent SDOF system and simulated by atranslator connector To obtain nonlinear analysis param-eters of the frame pushover analysis of the frame is firstcarried out and the resulted capacity curve is shown inFigure 10 Following the procedure stated in Section 222the parameters of the SDOF system can be obtained asfollows meq 473times104 kg heq 128m kf 4375 kNmfyf 400 kN and η 01

33 Comparison between the Analysis Results Utilizing theabove two FE models pushover analyses of the FSCWstructure are conducted respectively To save space onlypin-jointed connection between the linking beam and the SC

d

fffyf

kf

ηkf

Figure 5 Hysteresis model of the frame


wall is considered here In addition it is supposed that theconcentrated force is applied laterally on the top of the SCwall

Figure 11 shows the capacity curves obtained using thetwo models It can be seen that on the whole the two curvesare close especially during the elastic stage Beyond theelastic stage although obvious difference can be observedthe maximum errors between the base shears obtained fromthe two models are less than 11 -e errors may mainlyoriginate from the inaccuracy of the parameters related tothe equivalent SDOF system By adjusting these parametersbetter matches between the results obtained from the twomodels can be available -erefore it can be concluded tosome extent that the proposed simplified analysis model ofthe FSCW structure is capable of approximately capturingthe nonlinear behavior of the FSCW structure under lateralloading while having the appealing advantage of simplicity

4 Dynamic Response Analysis of theFSCW Structure

In this section the above simplified analysis model is furtherutilized to establish motion equations of the FSCW structure

under seismic excitations Without loss of generality onlythe pin-jointed system shown in Figure 4(a) is taken intoaccount Based on the motion equations the numericalsimulation model is constructed using the MATLABSimulink software and the dynamic responses are obtained

41 Motion Equations of the FSCWStructure -e SC wall ofthe FSCW structure is treated as a rigid body which is widelyused in the previous relevant studies and has been verified bysome experiments (eg among others [21 22]) In additionit is supposed that the SC wall only rotates about points O orOprime without sliding under seismic excitations as shown inFigures 12(a) and 13(a) In the two figures θ is the rotationangle of the SC wall which is assumed to be positive for thewall rotating about point O and negative in case of rotatingabout Oprime Besides the damping associated with the system isneglected for simplification

411 Rotating about Oprime When the SC wall rotates aboutpoint Oprime as shown in Figure 12(a) the following momentequilibrium equation can be established according to theDrsquoAlembert principle

V

δn

(a)

δeq

V

fyf

kf

ηkf

δyf

(b)

Figure 6 Nonlinear parameters of the frame (a) Capacity curve of the frame (b) Force vs displacement curve of the equivalent SDOFsystem

6000 6000 6000 60003000

3000

3000

3000

3000

3000

22

1 1

3000

2 20550

2502-2

ϕ8200

ϕ8200

3 18

3 20

2 18

3 18

450

4501-1

22

Figure 7 Details of the analyzed FSCW structure (length unit mm)


MI + FItl

2+ mg

l

2sin[α + θ(t)] + Fp

b

2cos minus

θ(t)

21113888 1113889

+ f1 x1 _x1( 1113857b cos minusθ(t)

21113888 1113889 + P(d _d)heq 0

(5)

where MI and FIt are the inertial moment and the tan-gential inertial force of the SC wall respectively m is themass of the wall Fp is the force in the PT tendonf1(x1 _x1) is the restoring force of the right damper wherex1 is the displacement of the damper and _x1 is the cor-responding velocity and P(d _d) is the lateral forceinteracting between the SC wall and the frame where d is

the lateral displacement of the frame and _d is the cor-responding velocity

It is worth noting that only the lateral displacement ofthe linking beam end connected to the SC wall is con-sidered to facilitate the above formulation It is approxi-mately valid in case of small rotation of the SC wall as wellas considering that the width of the wall is usually smallerthan its height Under this circumstance the linking beamprimarily plays a role of delivering the lateral force betweenthe SC wall and the frame In addition the hypothesis thatthe SC wall rocks about its tips which arises from theaforementioned rigid body assumption may seem to beunrealistic as reported in some references (see among

Linking beam Frame

SC wall

PT tendon

Dampers

Foundation

yx

xy

zy

xAxial Axial

Tie constraints

Axial connector

Surface-to-surface contact

y

x

y

xAxial Axial

y

x

y

xAxialAxial

σ

ε

(a)

(c)

(b)

Figure 8 General FE model (a) Overall FE model (b) Definition of the interactions (c) Constitutive relation of the concrete in the frameand linking beams


others [35]) Although some modications to this as-sumption considering real conditions may yield moreprecise or meaningful results it is beyond the scope of thisstudy and can be further investigated in the future

As shown in Figure 12(a) the tangential inertialforce and the inertial moment of the SC wall can bedetermined as

FIt m minuseuroθlminus euroug(t)cos[α + θ(t)] (6)

MI minusIcgeuroθ(t) (7)

where Icg is the moment of inertia of the wall about itscentroid Icg (112)ml2 In addition using the geometryrelation the force Fp in the PT tendon can be easily obtainedas

Fp Fp0 minus kpb sinθ(t)2 (8)

where Fp0 and kp are the initial force and the elastic stinessof the tendon respectively

As shown in Figure 12(b) the horizontal force appliedon the frame can be obtained as

P(d _d) meq minuseurodminus euroug(t)[ ]minusF(d _d) (9)

where F(d _d) is the restoring force of the frameSubstituting equations (6)ndash(9) in equation (5) we can

have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial

Figure 9 Simplied FE model

00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)

Figure 10 Capacity curve of the frame

00 01 02 03 040

100

200

300

400

500

General FE modelSimplified FE model

V (k

N)

δn (m)

Figure 11 Capacity curves of the FSCW structure obtained usingthe two models

MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)

Figure 12 Mechanical analysis of the FSCW structure when the SCwall rotates about point Oprime (a) SC wall (b) frame

O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l

2sin[minusαminus θ(t)] minusFp0

b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

minusf1 x1 _x1( 1113857b cosθ(t)

2

+ heq meqheqeuroθ(t) + meq euroug(t) + F(d _d)1113960 1113961

minusmeuroug(t)l

2cos[minusαminus θ(t)]

(10)

where I0 is the moment of inertia of the wall about point OprimeI0 Icg + m(l24)

412 Rotating about O When the SC wall rotates aboutpoint O as shown in Figure 13(a) the following momentequilibrium equation can be established similarly

MI + FItl

2+ mg

l

2sin[αminus θ(t)] + f2 x2 _x2( 1113857b cos

θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)

where f2(x2 _x2) is the restoring force of the left damper x2is the displacement of the damper and _x2 is the corre-sponding velocity Under this circumstance equations(6)ndash(9) turn into

FIt m euroθ(t)l

2+ euroug(t)cos[αminus θ(t)]1113896 1113897 (12)

MI Icgeuroθ(t) (13)

Fp Fp0 + kpb sinθ(t)

2 (14)

P(d _d) meqeurod + euroug(t)1113960 1113961 + F(d _d) (15)

respectively From equations (11)ndash(15) we can have

I0euroθ(t) + mg

l

2sin[αminus θ(t)] + Fp0

b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)

Equations (10) and (16) can be combined into the fol-lowing compact form

euroθ(t) minus1

I0 + meqh2eq

1113896mgl

2sin[α sgn[θ(t)] minus θ(t)]

+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)

+ sgn[θ(t)] ε[minusθ(t)]f1(x _x) + ε[θ(t)]f2(x _x)1113864 1113865b cosθ(t)

2

+ heqF(d _d) + euroug ml

2cos[α sgn[θ(t)]minus θ(t)] + meqheq1113896 11138971113897

(17)

where sgn() is the sign function and ε() is the unit stepfunction as follows

ε(x) 0 xlt 0

1 xge 01113896 (18)

Equation (17) gives the basic equation governing dy-namic responses of the FSCW structure under seismicexcitations

42 Mathematical Description of the Hysteresis Fromequation (17) it can be seen that the restoring forces of thedampers and frame ie f1(x1 _x1) f2(x2 _x2) and F(d _d)need to be expressed explicitly so as to obtain the solutionAgain a bilinear hysteresis shown in Figure 5 is adopted forthe frame and the elastic-perfectly plastic hysteresis is usedfor the damper

Note that the elastic-perfectly plastic hysteresis can beseen as a special bilinear hysteresis provided that thepostyield stiffness is set to zero For the bilinear hysteresisillustrated in Figure 14 where fy xy ks and η represent theyield force the yield displacement the elastic stiffness andthe postyield stiffness coefficient respectively the restoringforce can be written mathematically as follows [36]

f(x _x) ηksx +(1minus η)ksz (19a)

_z _x 1minus ε( _x)ε zminus xy1113872 1113873minus ε(minus _x)ε minuszminusxy1113872 11138731113960 1113961 (19b)

where z is the hysteresis displacementFor the dampers when the wall rotates about point Oprime

the displacements of the two dampers are

x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)

When the wall rotates about point O the displacementsturn into

x1 0 (21a)

x2 2b sinθ(t)

2 (21b)


For the frame the following expression is approximatelyvalid

d heqθ(t) (22)

43 Consideration of the Impact When the SC wall returnsto the original position ie θ 0 an impact between the walland the ground occurs It is assumed that the rotation of thewall changes continuously during the impact According tothe principle of conservation of moment of momentum therelationship between the angular velocity of the SC wallbefore and after impact can be determined approximately asfollows [5]

I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)

where _θ1 and _θ2 are the angular velocity of the wall beforeand after the impact respectively From the above equationit can be derived that

k _θ2_θ1

1minus32sin2 α (24)

where k is the ratio between the angular velocity of the SCwall after and before the impact Note that k is generallysmaller than 10 since energy is always dissipated during theimpact

44 Numerical Simulation -e above sections present thecomplete description of dynamic behavior of the FSCWstructure under seismic excitations Due to its highlynonlinear nature the MATLABSimulink software isemployed here to obtain the numerical solutions [37] -eSimulink model is built as shown schematically in Figure 15A total of eight subsystems or modules ie the groundexcitation module the SC wall subsystem the PT tendonsubsystem the frame subsystem the damper subsystem twointegrator modules and the output module are involved inthemodel-e ground excitationmodule is used to input theseismic excitations -e main parameters related to thesubsystems include (i) b h and m (the SC wall subsystem)(ii) Fp0 and kp (the PT tendon subsystem) (iii) fyf kf and η(the frame subsystem) and (iv) fy and kd (the dampersubsystem) -e Integration 1 module is responsible tosimulate the impact between the SC wall and the ground by

modifying the angular velocity after the impact through theGain module For the integration modules the solver type ischosen to be ode23tb (stiffTR-BDF2) [37] the time intervalis set to be variable and the relative tolerance is set to be0001

-e established Simulink model is further applied toanalyze seismic responses of the FSCW structure with thesame parameters as given in Section 322 -e EL Centrorecord shown in Figure 16 is selected as the ground exci-tation and its peak value is adjusted to 510 cms2 [38] -edynamic responses of the FSCW structure including thetime-history curve of the rotation angle of the wall andhysteresis curves of the dampers and the frame are obtainedand plotted in Figures 17(a)ndash17(d) respectively It can beseen that the hysteresis of the dampers is elastic-perfectlyplastic and one of the frames is bilinear showing that theresults can properly replicate the hysteresis characteristics ofthe FSCW structure -erefore the Simulink model can beused approximately to predict dynamic responses of theFSCW structure under seismic excitations

5 Parametric Study

In this section a comprehensive parametric study is con-ducted using the above Simulink model to investigate theinfluence of a variety of design parameters on dynamicresponses of the FSCW structure under seismic excitations-e rotation angle of the SC wall is selected to representdynamic responses of the FSCW structure since it canapproximately give some other responses such as the lateraldisplacement of the frame (equation (22))

51 Ground Motion Records A total of seven earthquakerecords which include six real earthquake records and anartificial one designated as EQ01-EQ07 respectively areemployed here to conduct the time-history analysis -eproperties related to the selected earthquake records aresummarized and listed in Table 1 Again all the records arescaled to have a peak value of 510 cms2 [38]

52 Parameter Values Involved in ltis Study Six in-dependent design parameters are taken into account whichinclude the initial force and elastic stiffness of the PT tendonthe yield force and elastic stiffness of the damper and theyield force and elastic stiffness of the frame -e parametervalues involved in this study are listed in Table 2 while theother parameters have the same values as in Section 322

6 Results and Discussions

61 Effect of Parameters concerning the PT Tendon To in-vestigate the effect of the parameters regarding the PTtendon on seismic responses of the FSCW structure theparametric analyses are conducted with the values of Fp0 andkp altered according to Table 2 respectively -e otherparameter values are set as follows Fy 1879 kN kd

19504 kNmm fyf 500 kN and kf 47 kNmm

f (x x)

ks

xyx

fyηks

Figure 14 Bilinear hysteresis model


Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-

e PT tendom subsystem

e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s

Figure 15 Simulink model of the FSCW structure

0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )

Figure 16 EL Centro record used in the analysis

0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued


Figures 18(a) and 18(b) show the maximum rotationangle of the SC wall averaging over the selected records forvarious values of Fp0 and kp respectively In the two cases kpand Fp0 are set as 3932 kNmm and 50808 kN respectivelyIt can be seen that the maximum rotation angle of the SCwall decreases obviously with the increase of Fp0 whilechanging little with the increase of kp -erefore improvingthe initial force of the PT tendon has a beneficial effect onrestraining seismic responses of the FSCW structure interms of the rotation angle of the SC wall while the elasticstiffness of the PT tendon has little effect

62 Effect of Parameters concerning the Dampers In theparametric analyses regarding the dampers the valuesof Fy and kd are changed according to Table 2 respec-tively -e other parameter values are taken as followsFp0 50808 kN kp 3932 kNmm fyf 500 kN and kf

47 kNmm

Figures 19(a) and 19(b) show the maximum rotationangle of the SC wall averaging over the selected records forvarious values of Fy and kd respectively For the two casesthe values of kd and Fy are set as 19504 kNmm and1879 kN respectively It can be seen that on the whole themaximum rotation angle of the SC wall decreases as Fy or kdincreases -erefore increasing the yield force or the elasticstiffness of the damper can help lower seismic responses ofthe FSCW structure in terms of the rotation angle of the SCwall

63 Effect of Parameters concerning the Frame In theparametric analyses regarding the frame the values of fyf andkf are varied according to Table 2 respectively -e otherparameter values are taken as follows Fp0 50808 kNkp 3932 kNmm Fy 1879 kN and kd 19504 kNmm

Figures 20(a) and 20(b) show the maximum rotationangle of the SC wall averaging over the selected records forvarious values of fyf and kf respectively For the two casesthe values of kf and fyf are set as 47 kNmm and 500 kNrespectively It can be seen that the maximum rotation angleof the SC wall falls abruptly as fyf or kf increases For ex-ample as fyf changes from 300 kN to 700 kN the maximumrotation angle drops by about 4080 -erefore increasingthe yield force or the elastic stiffness of the frame can re-markably reduce seismic responses of the FSCW structure interms of the rotation angle of the SC wall

7 Conclusions

In this paper a simplified analysis model of the FSCWstructure is first proposed After validated by the FE analysismethod the model is further utilized to establish motionequations of the FSCW structure under seismic excitationsBy way of numerical simulation seismic responses of theFSCW structure are obtained Finally a comprehensiveparametric study is conducted to study the influence of avariety of design parameters on seismic responses of theFSCW structure Based on the findings of this study thefollowing conclusions can be drawn

0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)

Figure 17 Dynamic responses of the FSCW structure under seismic excitations (a) Time-history curve of the rotation angle of the wallhysteresis curve of (b) the left damper (c) the right damper and (d) the frame

Table 1 Earthquake records involved in this study

No Record Duration(s)

PGA(cms2)

EQ01 Imperial Valley 1940 EL Centro 3998 45003EQ02 Imperial Valley 1940 EL Centro 3998 66288EQ03 Loma Prieta 1989 Gilroy 3998 65249EQ04 Loma Prieta 1989 Gilroy 3998 95093EQ05 Northridge 1994 Sylmar 5998 80144EQ06 North Palm Springs 1986 5998 99943EQ07 mdash 4000 510

Table 2 Parameter values used in the study

Parameter ValuesPTtendon

Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794

Damper Fy (kN) 7245 10875 1879 24645 31735kd (kNmm) 9644 13924 19504 23076 2862

Frame fyf (kN) 300 400 500 600 700kf (kNmm) 29 38 47 56 65


θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)

Figure 19 Effects of the parameters regarding the dampers (a) Fy (b) kd

θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)

Figure 20 Effects of the parameters regarding the frame (a) fyf (b) kf

200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)

Figure 18 Effects of the parameters regarding the PT tendon (a) Fp0 (b) kp


(1) -e proposed simplified analysis model of the FSCWstructure can be approximately used to capture thenonlinear behavior of the FSCW structure under lateralloading while having the great advantage of highcomputational efficiency Based on the simplifiedanalysis model seismic responses of the FSCW struc-ture can be obtained by solving numerically themotionequations of the FSCW structure under seismicexcitations

(2) Within the parameter values considered in thisstudy improving the yield force or elastic stiffness ofthe frame can remarkably mitigate seismic responsesof the FSCW structure in terms of the rotation angleof the SC wall In addition increasing the initial forceof the PT tendon as well as the yield force or elasticstiffness of the damper has also a beneficial effect

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Disclosure

Any opinions findings and conclusions or recommenda-tions expressed in this study are those of the authors and donot necessarily reflect the views of the National ScienceFoundation of China

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China under grant no 51578429 -e fi-nancial support is gratefully acknowledged

References

[1] Y C Kurama Seismic analysis behavior and design ofunbonded post-tensioned precast concrete walls PhD dis-sertation Lehigh University Bethlehem PA USA 1997

[2] N S Armouti Seismic performance of precast concretestructural walls PhD dissertation Lehigh UniversityBethlehem PA USA 1993

[3] G W Housner ldquo-e behavior of inverted pendulum struc-tures during earthquakesrdquo Bulletin of the Seismological Societyof America vol 53 no 2 pp 403ndash417 1963

[4] C-S Yim A K Chopra and J Penzien ldquoRocking response ofrigid blocks to earthquakesrdquo Earthquake Engineering ampStructural Dynamics vol 8 no 6 pp 565ndash587 1980

[5] J Zhang and N Makris ldquoRocking response of free-standingblocks under cycloidal pulsesrdquo Journal of Engineering Me-chanics vol 127 no 5 pp 473ndash483 2001

[6] N Makris and J Zhang ldquoRocking response of anchoredblocks under pulse-type motionsrdquo Journal of EngineeringMechanics vol 127 no 5 pp 484ndash493 2001

[7] A Di Egidio D Zulli and A Contento ldquoComparison be-tween the seismic response of 2D and 3D models of rigid

blocksrdquo Earthquake Engineering and Engineering Vibrationvol 13 no 1 pp 151ndash162 2014

[8] X Hu Q Lu and Y Yang ldquoRocking response analysis of self-centering walls under ground excitationsrdquo MathematicalProblems in Engineering vol 2018 Article ID 437158512 pages 2018

[9] Y C Kurama S Pessiki R Sause L W Lu andM T El-Sheikh ldquoAnalytical modeling and lateral loadbehavior of unbonded post-tensioned precast concrete wallsrdquoResearch Report NoEQ-96-02 LehighUniversity BethlehemPA USA 1998

[10] Y C Kurama ldquoSeismic design of unbonded post-tensionedprecast concrete walls with supplemental viscous dampingrdquoACI Structural Journal vol 97 no 4 pp 648ndash658 2000

[11] L A Toranzo lte use of rocking walls in confined masonrystructures a performance-based approach PhD dissertationUniversity of Canterbury Christchurch New Zealand 2002

[12] L A Toranzo J I Restrepo J B Mander and A J CarrldquoShake-table tests of confined-masonry rocking walls withsupplementary hysteretic dampingrdquo Journal of EarthquakeEngineering vol 13 no 6 pp 882ndash898 Jul 2009

[13] F J Perez Experimental and analytical lateral load responseof unbonded post-tensioned precast concrete walls PhDdissertation Lehigh University Bethlehem PA USA 2004

[14] F J Perez S Pessiki and R Sause ldquoLateral load behavior ofunbonded post-tensioned precast concrete walls with verticaljointsrdquo PCI Journal vol 49 no 2 pp 48ndash64 2004

[15] F J Perez S Pessiki and R Sause ldquoSeismic design ofunbonded post-tensioned precast concrete walls with verticaljoint connectorsrdquo PCI Journal vol 49 no 1 pp 58ndash79 2004

[16] F J Perez S Pessiki and R Sause ldquoExperimental lateral loadresponse of unbonded post-tensioned precast concrete wallsrdquoACI Structural Journal vol 110 no 6 pp 1045ndash1056 2013

[17] J I Restrepo and A Rahman ldquoSeismic performance of self-centering structural walls incorporating energy dissipatorsrdquoJournal of Structural Engineering vol 133 no 11 pp 1560ndash1570 2007

[18] B Erkmen and A E Schultz ldquoSelf-centering behavior ofunbonded post-tensioned precast concrete shear wallsrdquoJournal of Earthquake Engineering vol 13 no 7 pp 1047ndash1064 Sep 2009

[19] L Pakiding S Pessiki R Sause and M Rivera ldquoLateral loadresponse of unbonded post-tensioned cast-in-place concretewallsrdquo in Proceedings of Structures Congress 2015 Reston VAUSA 2015

[20] L Panian M Steyer and S Tipping ldquoPost-tensioned concretewalls for seismic resistancerdquo Journal of the Post-TensioningInstitute vol 5 no 1 pp 7ndash12 2007

[21] B J Smith Y C Kurama and M J McGinnis ldquoDesign andmeasured behavior of a hybrid precast concrete wall specimenfor seismic regionsrdquo Journal of Structural Engineeringvol 137 no 10 pp 1052ndash1062 2011

[22] B J Smith Y C Kurama and M J McGinnis ldquoBehavior ofprecast concrete shear walls for seismic regions comparisonof hybrid and emulative specimensrdquo Journal of StructuralEngineering vol 139 no 11 pp 1917ndash1927 2013

[23] C E Grigorian and M Grigorian ldquoPerformance controland efficient design of rocking-wall moment framesrdquoJournal of Structural Engineering vol 142 no 2 article04015139 2016

[24] M Grigorian and C Grigorian ldquoAn introduction to thestructural design of rocking wall-frames with a view to col-lapse prevention self-alignment and repairabilityrdquo Structural


Design of Tall and Special Buildings vol 25 no 2 pp 93ndash1112016

[25] A Wada Z Qu H Ito S Motoyui H Sakata and K KasaildquoSeismic retrofit using rocking walls and steel dampersrdquo inProceedings of Improving the Seismic Performance of ExistingBuildings and Other Structures Reston VA USA 2009

[26] Z Qu A Wada S Motoyui H Sakata and S Kishiki ldquoPin-supported walls for enhancing the seismic performance ofbuilding structuresrdquo Earthquake Engineering amp StructuralDynamics vol 41 no 14 pp 2075ndash2091 2012

[27] J J Ajrab G Pekcan and J B Mander ldquoRocking wall-framestructures with supplemental tendon systemsrdquo Journal ofStructural Engineering vol 130 no 6 pp 895ndash903 2004

[28] H Zibaei and J Mokari ldquoEvaluation of seismic behaviorimprovement in RC MRFs retrofitted by controlled rockingwall systemsrdquo lte Structural Design of Tall and SpecialBuildings vol 23 no 13 pp 995ndash1006 2014

[29] X Hu Y Zhang and N S Moghaddasi ldquoSeismic perfor-mance of reinforced concrete frames retrofitted with self-centering hybrid wallrdquo Advances in Structural Engineeringvol 15 no 12 pp 2131ndash2143 2012

[30] M Grigorian A S Moghadam H Mohammadi andM Kamizi ldquoMethodology for developing earthquake-resilientstructuresrdquo Structural Design of Tall and Special Buildingsvol 28 no 2 2018

[31] T Pauley and M J N Priestley Seismic Design of ReinforcedConcrete and Masonry Buildings John Wiley amp Sons IncHoboken NJ USA 2009

[32] M J N Priestley G M Calvi M J Kowalsky andG H Powell ldquoDisplacement-based seismic design of struc-turesrdquo Earthquake Spectra vol 24 no 2 pp 555ndash557 2007

[33] Dassault Systemes ldquoAbaqus analysis userrsquos guiderdquo 2016 httpdskipptpanpldocsabaqusv613booksusbdefaulthtm

[34] Ministry of Construction of the Peoplersquos Republic of ChinaGB50010-2010 Code for Design of Concrete Structures ChinaArchitecture amp Building Press Beijing China 2010 inChinese

[35] H A Spieth A J Carr A G Murahidy et al ldquoModelling ofpost-tensioned precast reinforced concrete frame structureswith rocking beam-column connectionsrdquo in Proceedings ofNew Zealand Society of Earthquake Engineering Conference2004 New Zealand 2004

[36] L D Lutes and S Sarkani Random Vibrations Analysis ofStructural and Mechanical Systems Elsevier Butterworth-Heinemann Burlington MA USA 2004

[37] Mathworks ldquoMATLABsimulinkrdquo 2014 httpswwwmathworkscomproductssimulinkhtml

[38] Ministry of Construction of the Peoplersquos Republic of ChinaGB50011-2010 Code for Seismic Design of Buildings ChinaArchitecture amp Building Press Beijing China 2010 in Chinese


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

its seismic performance mainly through the numericalanalysis method For example Ajrab et al [27] performednonlinear time-history analyses of the FSCW structureequipped with additional PT tendons and dampers -e re-sults show that the interstory drifts of the FSCW structure aremore uniform than those of the framed fixed-based wallstructure In addition Zibaei and Mokari [28] conducted acomparative study on seismic performance of RC frames RCframes with shear wall and RC frames with controlledrocking wall by means of pushover analysis -e results showthat for the RC frames with rocking wall system not onlyplastic hinge formation and interstory drifts are well dis-tributed but also energy dissipation and displacement duc-tility are increased Besides Hu et al [29] compared seismicresponses of the RC frame before and after retrofitted by SCwalls through nonlinear time-history analysis It shows thatthe SC wall can effectively reduce seismic responses of theoriginal RC frame structure while incurring little or no re-sidual deformation More recently Grigorian et al [30]proposed a methodology for developing earthquake-resilientstructures by introducing a rocking core-moment frame asthe lateral resisting component of a gravity-resisting structurethat is detailed to recenter while sustaining large lateraldeformations

From the above literature review it can be seen that thelimited studies regarding the FSCW structure usually in-volve nonlinear FE analysis It may be time-consumingespecially in case of analyzing the complex structure Fur-thermore there are still uncertainties in establishing the FEmodel for example how to determine the contact param-eters to realistically replicate the gap-opening behavior of theconnections In view of these issues this paper is mainlyaimed to develop a simplified analysis model of the FSCWstructure under lateral loading After validated by the FEanalysis method this model is further employed to establishmotion equations of the FSCW structure under seismicexcitations By means of numerical simulation seismic re-sponses of the FSCW structure are obtained Finally acomprehensive parametric study is carried out to investigatethe influences of a variety of design parameters whichmainly include those concerning the PT tendon thedampers and the frame on seismic responses of the FSCWstructure

2 Simplified Analysis Model ofFSCW Structures

21GeneralAnalysisModel As illustrated in Figure 2(a) theFSCW structure mainly comprises three parts the SC wallthe frame and the linking beam In this study only themoment frame is considered It is assumed that these threecomponents are lumped in the plane along the loadingdirection as shown in Figure 2(b) Furthermore the PTtendon is supposed to be installed vertically along the centerline of the SC wall and the dampers are placed at two ends ofthe bottom of the wall

Since the frame can be represented as a multi-degree-of-freedom (MDOF) system a general analysis model of theFSCW structure can thus be formed as seen inFigures 2(c)ndash2(d) Depending on whether the constraintsbetween the linking beam and the SC wall are considered ornot the analysis model can be categorized into two typesie the pin-jointed system (Figure 2(c)) and the rigid-jointedsystem (Figure 2(d)) In the figures h b and l represent theheight the width and the diagonal length of the SC wallrespectively with α being the angle between the wall diagonaland height In addition kj and hj are the stiffness and theheight relative to the ground of the jth (j 1 n) storey ofthe frame respectively where n is the number of storeys

22 Simplified Analysis Model

221 Formulation of the Simplified Model According to thedisplacement-based seismic design philosophy [31 32] theMDOF system can be equivalent as a single-degree-of-freedom (SDOF) system as shown in Figure 3 where mjFj and δj denote the mass horizontal force and lateraldisplacement of the jth storey of the MDOF system -eparameters concerning the equivalent SDOF system aredetermined as follows

heq 1113936

nj1 mjδjhj1113872 1113873

1113936nj1 mjδj1113872 1113873

(1a)

meq 1113936

nj1 mjδj1113872 1113873

δeq (1b)

(a) (b)

Figure 1 Illustration of two types of FSCW structures (a) One adopting the rocking wall (b) One adopting the stepping wall


δeq 1113936

nj1 mjδ

2j1113872 1113873

1113936nj1 mjδj1113872 1113873

(1c)



δj θhj j 1 2 n (2)


δn θhn (3)


heq 1113936

nj1 mjh

2j1113872 1113873

1113936nj1 mjhj1113872 1113873

(4a)

meq 1113936

nj1mjhj1113872 1113873

2

1113936nj1 mjh

2j1113872 1113873

(4b)

δeq 1113936

nj1 mjh

2j1113872 1113873

1113936nj1 mjhj1113872 1113873

δn

hn

(4c)


kn

kj

k1

hl

b

αhj

kn

kj

k1

hl

b

α hj

Damper

PT tendon

Lumped SC wall


(a) (b)

(c) (d)

SC wall


meq

δeq

heq

mn

mj

m1

δn

δj

δ1

Fn

Fj

F1

hj

Vb


meq

heq

D1 D2

(a)

meq

heq

D1 D2

(b)















d

fffyf

kf

ηkf










V

δn

(a)

δeq

V

fyf

kf

ηkf

δyf

(b)


6000 6000 6000 60003000

3000

3000

3000

3000

3000

22

1 1

3000

2 20550

2502-2

ϕ8200

ϕ8200

3 18

3 20

2 18

3 18

450

4501-1

22



MI + FItl

2+ mg

l


b

2cos minus

θ(t)

21113888 1113889


21113888 1113889 + P(d _d)heq 0

(5)




Linking beam Frame

SC wall

PT tendon

Dampers

Foundation

yx

xy

zy

xAxial Axial

Tie constraints

Axial connector


y

x

y

xAxial Axial

y

x

y

xAxialAxial

σ

ε

(a)

(c)

(b)













have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial


00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)


00 01 02 03 040

100

200

300

400

500


V (k

N)

δn (m)


MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)


O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


δeq 1113936

nj1 mjδ

2j1113872 1113873

1113936nj1 mjδj1113872 1113873

(1c)



δj θhj j 1 2 n (2)


δn θhn (3)


heq 1113936

nj1 mjh

2j1113872 1113873

1113936nj1 mjhj1113872 1113873

(4a)

meq 1113936

nj1mjhj1113872 1113873

2

1113936nj1 mjh

2j1113872 1113873

(4b)

δeq 1113936

nj1 mjh

2j1113872 1113873

1113936nj1 mjhj1113872 1113873

δn

hn

(4c)


kn

kj

k1

hl

b

αhj

kn

kj

k1

hl

b

α hj

Damper

PT tendon

Lumped SC wall


(a) (b)

(c) (d)

SC wall


meq

δeq

heq

mn

mj

m1

δn

δj

δ1

Fn

Fj

F1

hj

Vb


meq

heq

D1 D2

(a)

meq

heq

D1 D2

(b)















d

fffyf

kf

ηkf










V

δn

(a)

δeq

V

fyf

kf

ηkf

δyf

(b)


6000 6000 6000 60003000

3000

3000

3000

3000

3000

22

1 1

3000

2 20550

2502-2

ϕ8200

ϕ8200

3 18

3 20

2 18

3 18

450

4501-1

22



MI + FItl

2+ mg

l


b

2cos minus

θ(t)

21113888 1113889


21113888 1113889 + P(d _d)heq 0

(5)




Linking beam Frame

SC wall

PT tendon

Dampers

Foundation

yx

xy

zy

xAxial Axial

Tie constraints

Axial connector


y

x

y

xAxial Axial

y

x

y

xAxialAxial

σ

ε

(a)

(c)

(b)













have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial


00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)


00 01 02 03 040

100

200

300

400

500


V (k

N)

δn (m)


MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)


O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia














d

fffyf

kf

ηkf










V

δn

(a)

δeq

V

fyf

kf

ηkf

δyf

(b)


6000 6000 6000 60003000

3000

3000

3000

3000

3000

22

1 1

3000

2 20550

2502-2

ϕ8200

ϕ8200

3 18

3 20

2 18

3 18

450

4501-1

22



MI + FItl

2+ mg

l


b

2cos minus

θ(t)

21113888 1113889


21113888 1113889 + P(d _d)heq 0

(5)




Linking beam Frame

SC wall

PT tendon

Dampers

Foundation

yx

xy

zy

xAxial Axial

Tie constraints

Axial connector


y

x

y

xAxial Axial

y

x

y

xAxialAxial

σ

ε

(a)

(c)

(b)













have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial


00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)


00 01 02 03 040

100

200

300

400

500


V (k

N)

δn (m)


MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)


O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









V

δn

(a)

δeq

V

fyf

kf

ηkf

δyf

(b)


6000 6000 6000 60003000

3000

3000

3000

3000

3000

22

1 1

3000

2 20550

2502-2

ϕ8200

ϕ8200

3 18

3 20

2 18

3 18

450

4501-1

22



MI + FItl

2+ mg

l


b

2cos minus

θ(t)

21113888 1113889


21113888 1113889 + P(d _d)heq 0

(5)




Linking beam Frame

SC wall

PT tendon

Dampers

Foundation

yx

xy

zy

xAxial Axial

Tie constraints

Axial connector


y

x

y

xAxial Axial

y

x

y

xAxialAxial

σ

ε

(a)

(c)

(b)













have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial


00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)


00 01 02 03 040

100

200

300

400

500


V (k

N)

δn (m)


MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)


O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


MI + FItl

2+ mg

l


b

2cos minus

θ(t)

21113888 1113889


21113888 1113889 + P(d _d)heq 0

(5)




Linking beam Frame

SC wall

PT tendon

Dampers

Foundation

yx

xy

zy

xAxial Axial

Tie constraints

Axial connector


y

x

y

xAxial Axial

y

x

y

xAxialAxial

σ

ε

(a)

(c)

(b)













have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial


00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)


00 01 02 03 040

100

200

300

400

500


V (k

N)

δn (m)


MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)


O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












have

Linking beam

SDOF system

SC wall

PT tendon

Dampers

Foundation

xy

zy

xx

y

zy

xAxial Axial


00 01 02 03 040

100

200

300

400

V (k

N)

δn (m)


00 01 02 03 040

100

200

300

400

500


V (k

N)

δn (m)


MI

FΙt

Oprime O

Fp f1 (x1 x1)

ug (t)

mg

P (d d)

θ

(a)

ug (t)

P (d d)

heq

meq d

(b)


O

mg

MI

FΙt

Oprime θ

f2 (x2 x2) Fp

P (d d)

ug (t)

(a)

P (d d)

meqd

heq

ug (t)

(b)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)


2


minusmeuroug(t)l


(10)



MI + FItl

2+ mg

l


θ(t)

21113888 1113889

+ Fpb

2cos

θ(t)

21113888 1113889 + P(d _d)heq 0

(11)


FIt m euroθ(t)l




2 (14)



I0euroθ(t) + mg

l


b

2cos

θ(t)

2+ kp

b2

4sin θ(t)

+ f2 x2 _x2( 1113857b cosθ(t)

2


minusmeuroug(t)l

2cos[αminus θ(t)]

(16)


euroθ(t) minus1

I0 + meqh2eq

1113896mgl


+ sgn[θ(t)]Fp0b

2cos

θ(t)

2+

kpb2

4sin θ(t)


2



(17)


ε(x) 0 xlt 0

1 xge 01113896 (18)








x1 minus2b sinθ(t)

2 (20a)

x2 0 (20b)


x1 0 (21a)

x2 2b sinθ(t)

2 (21b)



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



d heqθ(t) (22)


I0_θ1 minusm _θ1b

l

2sin α I0

_θ2 (23)


k _θ2_θ1






5 Parametric Study







f (x x)

ks

xyx

fyηks



Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Integration 1

Initial condition

Gain

Ground excitation

Add

Out 1 In 1

In 2

In 1In 2

In 1

In 1

In 2

Out 1

Out 1

Out 1

Out 1

Out 2

Out 2

(0) x0

-K-


e frame subsystem

e damper subsystem

e SC wall subsytem

Integration 2

Output

1s

1s


0 5 10 15 20 25 30ndash6

ndash4

ndash2

0

2

4

6

t (s)

uuml g (m

s2 )


0 5 10 15 20 25 30ndash2

ndash1

0

1

2

3

4

t (s)

θ (1

0ndash3middotra

d)

(a)

0000 0005 0010 0015 0020 0025

ndash200

ndash100

0

100

200

f (kN

)

d (m)

(b)

Figure 17 Continued




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




47 kNmm




7 Conclusions


0000

f (kN

)

0001 0002 0003 0004

ndash200

ndash100

0

100

200

d (m)

(c)

f (kN

)

d (m)ndash002 ndash001 000 001 002 003 004

ndash600

ndash400

ndash200

0

200

400

600

800

(d)




PGA(cms2)




Fp0 (kN) 19846 3547 50808 69676 102602kp (kNmm) 1536 2744 3932 5392 794




θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


θ max

(10ndash3

middotrad)

Fy (kN)50 100 150 200 250 300 350

10

15

20

25

30

(a)

θ max

(10ndash3

middotrad)

kd (kNmm)100 150 200 250 300

10

15

20

25

30

(b)


θ max

(10ndash3

middotrad)

300 400 500 600 70010

15

20

25

30

fyf (kN)

(a)

θ max

(10ndash3

middotrad)

kf (kNmm)30 40 50 60 70

10

15

20

25

30

(b)


200 400 600 800 100010

θ max

(10ndash3

middotrad)

15

20

25

30

Fp0 (kN)

(a)

θ max

(10ndash3

middotrad)

kp (kNmm)10 20 30 40 50 60 70 80 90

10

15

20

25

30

(b)





Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Data Availability


Disclosure




Acknowledgments


References













































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


a simplified model for dynamic response analysis...

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