Research Article Free and Forced Vibration Analysis of an Infilled …downloads.hindawi.com/journals/sv/2014/439591.pdf · 2019-07-31 · Research Article Free and Forced Vibration

Research ArticleFree and Forced Vibration Analysis of an Infilled Steel FrameExperimental Numerical and Analytical Methods

Mohammad Amin Hariri-Ardebili1 Hamid Rahmani Samani2 and Masoud Mirtaheri3

1 Department of Civil Environmental and Architectural Engineering University of Colorado at Boulder (UCB)Boulder CO 80309-0428 USA

2Department of Civil Engineering Islamic Azad University Pardis Branch Tehran 13516555 Iran3Department of Civil Engineering KN Toosi University of Technology Tehran Iran

Correspondence should be addressed to Mohammad Amin Hariri-Ardebili mohammadhaririardebilicoloradoedu

Received 20 February 2014 Revised 27 July 2014 Accepted 7 August 2014 Published 28 August 2014

Academic Editor Nuno M Maia

Copyright copy 2014 Mohammad Amin Hariri-Ardebili et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Structural frames with masonry infill panels make up a significant portion of the buildings constructed in earthquake-prone areasprior to the developing of the seismic design standards In this paper the effects of masonry panels on the vibration response of aninfilled steel-frame building are investigated Various ambient and steady state forced vibration tests are carried out to realize thedynamic characteristics of the system 3D finite element models of the building with and without infill panels are provided based onmarcomodeling theorem A set of analytical approximate formulas are also derived to estimate the vibrational period The naturalfrequencies of the building are computed using numerical analytical and experimental methods The results show that neglectingthe effect of infill panels leads to considerable error Moreover it is shown that there is good agreement among the results obtainedby the three methods considering the effect of infill panels

1 Introduction

Steel-frame buildings with masonry infill walls are a typeof construction widely used throughout many developingcountries Hollow clay tile blocks hollow concrete blocksand normal bricks are used in infill walls (Figure 1(a)) Theinfill walls being traditionally nonengineered have as-builtproperties which at the design stage are almost impossibleto estimate reliably andor to specify and at the constructionstage are hard to control [1] However the effects of infillwalls on the structural properties of a building have beenrecognized by engineers and studied for a long time [2] Theinfill walls are supposed to increase the building stiffnessThey also may have some undesirable effects on the buildingperformance such as enhancing the soft storey mechanismor causing short column effects Chaker and Cherifati [3]suggested that plane stress finite elements provide a betterrepresentation of the in-plane initial stiffness of the infillpanels under the small strain condition In order to simulatethe behavior of the infill wall different types of models can beused [4]

(i) micromodeling in which the effect of mortar jointsare considered as a discrete element In this approachthe brick and mortar are modeled as continuumelements and interface between the brick and mortaris modeled by an interface joint element (Figure 1(b))

(ii) mesomodeling in which the bricks are modeled bycontinuum elements but the mortar joint and itsinterface with bricks are modeled together as aninterface element (Figure 1(c))

(iii) macromodeling in which a single numerical modelrepresents the infill panel effect [2] Macromodelingis considered in two levels that is (1) homogenizedmodel in which the brick mortar and the interfaceare modeled as one continuum element (Figure 1(d))and (2) strut model in which the infill panel is mod-eled by one (Figure 1(e)) or more struts (Figure 1(f))in each direction

Micromodeling is relatively time-consuming especiallyfor analysis of large structures [5] Differentmechanisms have

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 439591 14 pageshttpdxdoiorg1011552014439591

2 Shock and Vibration

Generic model

(a)

BrickInterfaceMortar

(b)

BrickInterface

(c)

Continuum element(s)

(d)

Strut

(e)

Strut(s)

(f)

Figure 1 Modeling the masonry infill panels (a) generic model (b) micromodeling (c) mesomodeling (d) homogenized macromodeling(e) single-strut macromodeling and (f) multistrut macromodeling

been used for the micro- and mesomodeling for examplenonlinear orthotropic model [6] smeared crack model [7]and dilatant interface constitutive model [8] The infill panelismacromechanicallymodeled as homogeneousmaterial andthe average properties are used for the whole panel [9]Modeling the infill panel as an equivalent strut is the mostconventional technique Table 1 summarizes different strutmodeling approaches [4] In addition all the macro- meso-and micromodels for simulating the presence of the infillpanels are affected by a strong sensitivity to the parametersof the model Sensitivity and (aleatory andor epistemic)uncertainty analysis of infilled frames were investigated byPanagiotakos and Fardis [10] Uva et al [11] Celarec et al [12]and Celarec and Dolsek [13]

On the other hand the vibrational behavior of the framestructures was investigated by different researchers Foutchstudied the vibrational characteristics of a nine-story rein-forcement concrete frame [30] Jain studied the ImperialCounty Services Building analytically and compared theresponse with those recorded by strong motion accelero-graphs during the 1979 Imperial Valley earthquake [31] How-ever the fundamental frequencies from the numerical resultsand the field measurements were not in close agreementbecause of the foundation flexibility Jennings et al [32] testeda forced vibration experiment on a twenty-two story steel-framed structure and extracted first six natural frequenciesFoutch [33] compared the natural frequencies and 3D modeshapes of a twenty story steel structure using forced vibrationtest and finite element method He concluded that theproperties of the primarily translational modes are predictedreasonably well but adequate prediction of torsional motionsis not obtained Trifunac [34] compared ambient and forced

vibration techniques for testing full-scale structures undersmall level excitations He concluded that both methodsprovide mutually consistent results

Beside the different experimental and numerical tech-niques that are used for free and forced vibration analysis ofstructures the dynamic behavior of the buildings and towerscan be estimated by analytical methods Free vibration ofbeam-like structures was studied by several researchers [35ndash37] Most of the previous studies have been oriented towardsthe closed-form solutions of the free flexural vibration ofa beam Li et al [38] derived the exact solution for freeflexural vibration of the stepped cantilever beams subjected toaxial loads They used the exact solution of a prismatic beamaccompanied by the transfermatrixmethodHowever effectsof the soil-structure interaction (SSI) shear deformation andthe joint flexibility were not considered Lin and Chang [39]investigated free vibration analysis of a multispan beam witharbitrary number of flexible constraints They assumed thateach span of the continuous beam obeys Timoshenko beamtheory Firouz-Abadi et al [40] studied the transverse freevibrations of typical truncated nonuniform Euler-Bernoullibeams by using a WKB global approximation Sina et al[41] proposed a new beam theory for free vibration analysisof functionally graded beams The beam properties wereassumed to be varied through the thickness following a powerlaw distribution Also it was assumed that the lateral normalstress of the beam is zero Carrera et al [42] proposed a hier-archical finite element method based on the Carrera UnifiedFormulation for free vibration analysis of beamwith arbitrarysection geometries Ghasemzadeh et al [43] proposed a set ofapproximate formulas to determine the natural frequencies ofthe structures considering the panel zone flexibility and SSI

Shock and Vibration 3

Table 1 Summary of different strut modeling approaches [4]

Model Researchers year [Reference] Number of strut(s) Modeling outlines

1 Holmes 1961 [14] One Modeling the ultimate strength stiffness and deflectionat failure of the infill

2 Stafford-Smith 1962 [15] One Modeling effective width of the equivalent (linear) strut

3 Stafford-Smith and Carter 1969 [16] One Modeling stiffness ultimate strength and cracking loadof the infill

4 Mainstone and Weeks 1970 [17] One Modeling stiffness and strength of the infill5 Mainstone 1971 [18] One Modeling stiffness and strength of the infill

6 Klingner and Bertero 1978 [19] One Modeling the hysteretic response strength and stiffnessof the infill

7 Te-Chang and Kwok-Hung 1984 [20] One Modeling stiffness and strength of the infill

8 Syrmakezis and Vratsanou 1986 [21] FiveConsidering the effect of the contact length on themoment distribution of the frame as well as strengthand stiffness of the infill

9 Zarnic and Tomazevic 1988 [22] One Modeling the lateral strength and stiffness of the infill

10 Schmidt 1989 [23] Two Modeling the frame-infill interaction as well as strengthand stiffness of the infill

11 Chrysostomou 1991 [24] ThreeModeling the frame-infill interaction as well as thehysteretic seismic response of the infilled frameconsidering stiffness and strength degradation

12 Saneinejad and Hobbs 1995 [25] One Modeling nonlinear force-displacement response ofinfill up to the ultimate load

13 Crisafulli 1997 [26] TwoModeling the frame-infill interaction as well asaccounting for compressive and shear strength of theinfill

14 Flanagan and Bennett 1999 [27] One Modeling corner crushing strength and stiffness of theinfill

15 El-Dakhakhni et al 2003 [28] Three Modeling the frame-infill interaction as well as cornercrushing failure mechanism

16 Dolsek and Fajfar 2008 [29] One Modeling the force-displacement response of the infillby a tri-linear response including post-peak response

effects Akgoz and Civalek [44] studied vibration responseof the nonhomogenous and nonuniform microbeams inconjunction with Bernoulli-Euler beam and modified couplestress theorem Li et al [45] derived a closed form solution forfree vibration analysis of axially inhomogeneous beams withdifferent end conditions

In this paper an old steel building in which the outerframes have been filled by masonry materials is selectedas case study Various steady state forced vibration tests aswell as ambient vibration test are carried out for realize theframersquos dynamic characteristics Also a set of experimentsare used to find out the compressive and shear strengthof the masonry infill panels Some approximate formulasare proposed for free vibration analysis of the steel framein which the structure is idealized as prismatic cantileverflexural-shear beam In addition 3D finite element model ofthe building is developed and the macromodeling approachis used for simulation of the infill panels Finally the naturalfrequencies resulting from different methods are compared

2 Description of the Case Study Building

The selected structure is a 36-year-old five-story steel-frame building located in Tehran Iran The outer frames

of the building are filled by brick and masonry materials(Figure 2(a)) The dimensions are 4070m times 1470m Thebuilding is designed according to the Iranian National Build-ing Code (INBC) [46] for a very high seismic zone Thebuilding is located on soil type III (where the average shearwave velocity to a depth of 30mwould be 180ms to 360ms)The foundation is strip footing with the depth of 80 cmand the width of 150 cm The structural system is momentresisting frame in the E-W direction and concentricallybraced frame in the N-S direction A rigid diaphragm can beassumed according to the roof system used in this buildingYoungrsquos modulus E and the Poissonrsquos ratio 120592 of the materialfor the main steel frame are 21E5MPa and 03 respectivelyDead load equal to DL = 650 kgm2 for all stories exceptthe roof level (DL = 600 kgm2) and live load equal to LL= 350 kgm2 for all stories except the second floor (LL =750 kgm2) are considered based on the as-built drawingsThe typical plan view of the building for all the stories isshown in Figure 2(b)

3 Experimental Tests

31 Instruments The generated vibrations in structural sys-tem should be recorded using the appropriate instruments for


(a)

104 m

26

3 m

22

55 m

(b)

Figure 2 Case study building (a) general view (b) typical plan view

(a) (b)

Figure 3 Instruments (a) data logger (b) accelerometer

further analyses Instruments used for measurement in thepresent study are data logger and accelerometers (Figure 3)EDX-1500A is a sixteen-canal data collector which is ableto measure variable frequencies up to 10KHz (Figure 3(a))Furthermore it includes a powerful monitoring system andcan be used for revision and process of collected data EDX-1500A includes four signal conditioner cards in which one ofthem is VAQ-60A and the others are CDV-60A Each cardis used for connecting four accelerometers CDV-60A cardis used just for miniature accelerometers and VAQ-60A cardis used just for connection of the server accelerometers It isnoteworthy that all the accelerometers should be calibratedbefore using in the final measurements For this purposeVAQ-test is required for ensuring the proper performance ofthe accelerometers

There are various methods for shaking of the structurebased on purpose of the research One of the simplest andinexpensive methods is based on the rotation of out of centermasses using amotor For this purpose a shaker was designedby the authors (Figure 4(b)) The shaker basically includestwo same mass-points rotating in opposite the directionswith the same velocity Layout and theory of this method areshown in Figure 4(a) As it is clear the resultant force in thismethod has a sinusoidal nature just in one direction

32 Steady State Forced Vibration Test In this test shakeris installed on the roof of the building approximately nearthe center of stiffness In addition uniaxial accelerometers

are installed in all the stories near the center of the stiffnessBy shaking the structure in N-S direction and analyzing therecorded data the natural frequency of building is measuredas (120596119899)N-S = 270Hz By the same method for E-W direction

(120596119899)E-W = 250HzThe shape of the entire frequency-response wave is

controlled by the amount of damping in the system so itis possible to derive the damping ratio from many differentproperties of the curve The most conventional method isbandwidth or half-powermethod inwhich the damping ratiois determined from the frequencies at which the response is1radic2 times the peak responseWith thismethod the dampingratio is computed as about 6 for this structure

33 Ambient Vibration Test Considering that all sides of theselected building are free it is possible to apply ambient vibra-tion test without using any external shaker For this purposedata loggers are installed near the center of the stiffness ineach story inN-S and E-Wdirections and results are recordedon a stormy day Resulting frequencies are the same as thoseobtained from forced vibration test except that the wind isused as a natural shaker in the ambient vibration test

4 Numerical Model

In this section finite element model of the structure isprepared and the natural frequencies are extracted For this


Shaft Rotating arms Bearing

Bearing

Pulley

Angle section

x

zx

y120579120579

F0 = m1205962F0 = m1205962

Fmax = 2F0Fmin = minus2F0

(a) (b)

Figure 4 Designed shaker (a) layout and theory of the shaker (b) general view under the operation

5 cm

1 cm

5 cm

10 cm 20 cm

Figure 5 Mortar shear stress test on the infill panel

purpose all the properties of the infill panels should beextracted based on the standard tests

41 Material Properties of Infill Panels All the infill panelsshould be in the well condition without any cracking Thenbased on Iranian rehabilitation guideline (2007) chapter 7-6-1-1-2 shear strength of the mortar should be evaluated asfollows [46] ldquoAt least one test is required per 3001198982 of the infillpanels At least 8 tests are required for each building to obtainthe reliable results Moreover the dispersion of the test locationsshould be selected in a way that covers all parts of the buildingrdquoImplementation of this experiment is shown in Figure 5 Themortar average bed-joint shear strength is calculated as

120590119905119900=

119881test119860119887

minus 120590119888 (1)

where119881test is the loadmagnitude due to first movement of themasonry unit 120590

119888is the stress due to gravity loads at the test

location and 119860119887is total net area of the bed joints above and

below the test unitThe results of the tests and correspondingcalculations are summarized in Table 2 Based on this tablethe average shear stress in themortar is 225 kgcm2 howeverthe acceptable value for the mortar shear stress should becalculated using the probabilistic method chapter 7-6-1-1-2in a way that 80 of the test results should have greater valueof the final shear stress [47]Thus the final value of themortarshear stress is 193 kgcm2 in the present case Consideringthat the final value is less than 20 kgcm2 the minimum

Table 2 Mortar shear stress of infill panels

Number 119881test119860119887 (kgcm2) 120590

119888(kgcm2) 120590to (kgcm

2)V1 32 119 201V2 48 123 357V3 36 103 257V4 27 123 147V5 34 111 229V6 27 077 193V7 27 054 216V8 38 114 266Average 225

requirements for the mortar strength are not satisfied andthere is need for rehabilitation and strengthens

Based on the conducted tests and considering Iranianrehabilitation guideline [46] and FEMA356 [48] recommen-dations for calculation of the compressive strength of themortar the basic parameters are as follows brick com-pressive and tensile strength 1198911015840cb and 119891

1015840

tb are 385 kgcm2

and 55 kgcm2 respectively and the mortar compressivestrength 1198911015840

119895 is 100 kgcm2 In addition the thickness of

the brick and the mortar are h = 50 cm and j = 10 cmrespectivelyThus the compressive strength can be calculatedas

1198911015840

119898=

1198911015840

cb15

times

((11989541ℎ) times 1198911015840

119895) + 1198911015840

tb

((11989541ℎ) + 1) times 1198911015840

tb (2)

The resulting 1198911015840

119898is 2666 kgcm2 Based on the FEMA356

[48] the factor to translate lower-bound masonry propertiesto the expected strength masonry properties is 13 Also itrecommends a factor to translate compressive strength tothe modules of elasticity The final value for the compressivestrength and the modulus of elasticity are 34658 kgcm2 and19061 kgcm2 respectively

42 DevelopingMacromodels Asmentioned before there arevarious methods in order to simulate the infill panelsrsquo effects


Detachmentframe-infill

Diagonalcompression

area

x

y

z

yx

hco

l

hin

f

lbeam

linf

120579

a

tinf

rinf

Figure 6 Macromodeling of the infill panel based on the equivalent strut approach

In the present study both the macromodeling based oncontinuum elements (Figure 1(d)) and the compressive-strutelement (Figure 1(e)) are used All the beams and columnsare modeled using ldquoBeamrdquo elements while all the beam-column connections are assumed to be rigid SSI effect isconsidered by modeling the foundation and its surroundingenvironment as massless medium using ldquoSolidrdquo elementsThe massless foundation is extended about three times ofthe building height in all directions It should be noticedthat there is no need to apply the nonreflecting boundaryconditions on the far-end boundaries of the foundationmodel because no seismic analysis is performed in this paperIn the macromodeling based on the continuum elementsthe infill panel is directly modeled by 3D ldquoShellrdquo elementsconsidering appropriate properties of the main panel In themacromodeling based on equivalent strut approach the infillpanel is replaced with a diagonal compressive-strut elementof width 119886 and the same thickness and modulus of elasticityof infill panel using ldquoLinkrdquo elements This is a two-nodepinned-end element which connects two opposite cornersof the frame The element is capable of transferring onlycompressive pressure in the axial direction In this methodthe equivalent width 119886 for the elastic in-plane behavior priorto cracking can be calculated based on Figure 6 [46]

119886 = 0175[1205821ℎcol]minus04

119903inf (3)

where

1205821= [

119864me119905inf sin 21205794119864fe119868colℎinf

]

025

(4)

in which ℎcol is the column height between centerlines ofbeams ℎinf is the height of infill panel 119864fe is the expectedmodulus of elasticity of frame material 119864me is the expectedmodulus of elasticity of infill material 119868col is the moment ofinertia of column 119903inf is the diagonal length of infill panel

119905inf is the thickness of infill panel and equivalent strut 120579 is theangle whose tangent is the infill height-to-length aspect ratioand 120582

1is the coefficient used to determine the equivalent

width of the strutConsequently three different models are provided in this

study for simulation of the building

(i) The structure is modeled without infill panels effects(reference case)

(ii) The structure is modeled considering the infill panelsas a compressive struts

(iii) The structure is modeled considering the infill panelsas 3D shell elements

5 Analytical Approximate Solution

51 Fundamentals and Assumptions In order to derive a setof appropriate approximate formulas for analytical vibrationanalysis of the structure a doubly symmetric structure in planis selected as a sample which is subjected to uniformly dis-tributed gravity loads at story levels as shown in Figure 7(a)It is also assumed that both the beams and the columnshave uniform sections throughout the height of the buildingFurthermore floor slabs are considered as rigid diaphragmsin their own plane so that the relative displacements betweenframes are restricted

The structure is idealized as a prismatic cantilever beamwith flexural rigidity EI shear rigidity GA axial distributedcompression force 119873 and mass per unit length 119898 as shownin Figure 7(b) Floor masses and gravity loads at story levelsare also replaced by concentratedmasses119898

119894and concentrated

forces 119873119894 respectively SSI is modeled using the axial and

torsional springs The general algorithm for the analyticalsolution is summarized in Figure 8 and will be explained indetail in the next subsections


L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)

Figure 7 Analytical model (a) schematic drawing of a doubly symmetric structure (b) equivalent beam model of the structure

52 Concept of Energy Method

521 Kinetic and Potential Energy Assuming that the frameis vibrating freely the kinetic energy 119879 is given by [43]

119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)

in which 119906(119909 119905) is transverse displacement of the frame dotsindicate the differentiationwith respect to time 119905 119899 is the totalnumber of stories and 119871 is the height of the structure Thetransverse displacement of the structure is expressed as

119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)

where119906119861and119906119878are contribution of beam (frame) and springs

(soil) in the lateral displacement respectively 119906119891is portion

of the displacement induced by bending deformation and119906sh is portion of the displacement induced by the sheardeformation of the beam 119906

119886is transverse displacement due

to foundation sliding and 119906119903is transverse displacement due

to foundation rotationThe potential energy of the system Π consists of the

strain energy 119880 and the work done by external loads 119881

Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)

where 120583 is the shear shape factor of the cross-section consid-ering the effects of uneven distribution of shear deformation

over the cross-section 119870119886is the stiffness coefficient against

sliding of the foundation119870119903is the stiffness coefficient against

rotation of the foundation and 120579119903is rotation of the torsional

spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)

in which V(119909 119905) represents the axial shortening due to lateraldeformation and is expressed as

V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)

522 Energy Equation Assuming that the structure is vibrat-ing freely in simple harmonic motion the transverse dis-placement 119906(119909 119905) and the velocity of vibration (119909 119905) areexpressed as

119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)

where 120596 is the natural vibration frequency The maximumpotential energy of the system over a vibration cycle isassociated with the maximum displacement that is

Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T

Potential ΠStrain energy U

External work V

Frame

Soil

Energy equation Define

Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity

Moment resisting frame

Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS

mlowast klowast klowastGTmax = Πmax

Figure 8 Proposed algorithm for analytical vibration analysis of the structure

and also the maximum kinetic energy of the system over avibration cycle is associated with maximum velocity

119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)

The principle of energy conservation states that the totalenergy in a freely vibrating system without damping isconstant thus 119879max must be equal to Πmax which yields

1205962=

119896lowastminus 119896lowast

119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)

As seen (14) is of the same form as the frequency expressionfor a single degree of freedom (SDOF) system In otherwordsby restricting the lateral displacement of the structure to asingle shape function 119906(119909) which defines the mode shapethe structure behaves as a generalized SDOF system [47]


Parameters 119898lowast and 119896lowast are generalized mass and generalizedstiffness of the system respectively and 119896

lowast

119866is generalized

geometric stiffness of the system which considers the effectof axial loads on the natural frequency Assuming that all thestories have equal height of ℎ

119904 all floor slabs have equal mass

of 1198980 and gravity loads at story levels are equal to119873

0 Thus

(15) can be simplified as

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)

523 Proper Shape Function The accuracy of frequency ofvibration depends on the assumed shape function 119906(119909) Themore accurate the shape function is the more accurate theresult would be It is worth mentioning that 119906(119909) is not aparticular function and any shape function satisfying thegeometric boundary conditions can be used By using thedeflected shape of the equivalent beam due to a selected set offorces the geometric boundary conditions are automaticallysatisfied Several methods are available to determine thedeflected shape of the structure but arguably the mostpromising one is the principle of virtual work

119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)

where 119876 and 119872 are the real shear and moment functionsalong the length of the structure respectively and 119902 and 119898]are the virtual force functions in the structure arising fromthe application of unit load119876

119887and119872

119887are the real shear and

moment at the base of the structure respectively and finally119902119887and 119898

119887are the virtual forces at the base of the structure

from the application of unit load respectively

53 Calculation of the Natural Frequency Considering theSSI effects the deflected shape of the structure 119906(119909) underlateral uniform distributed load can be written as

119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)

The subscript 119865 represents the fixity of the lowermostelevation and subscript 119878 represents the effects of soil

flexibility Using (14) for extracting the natural frequencies inthe present case new parameters can be derived as follows

119898lowast= 119898lowast

119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)

Different terms are defined as follows in which the subscripts119889 and 119888 represent the portions associated with distributedaxial load and concentrated axial forces respectively

119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)

54 Properties of Equivalent Beam

541 Flexural Rigidity The modulus of elasticity 119864 for theequivalent beam should be the same as the modulus ofelasticity of the real structure The moment of inertia 119868 ofthe equivalent beam is

119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)

in which 119860119888119894is the cross-sectional area of the 119894th column 119910

119894

is the distance of the 119894th column from 119911 axis and 119911119894is the

distance of the 119894th column from 119910 axis

542 Shear Rigidity

(1) Moment Resisting Frame In order to calculate the shearrigidity per unit web frame 119866119860120583 a subassemblage isextracted from the frame assuming that inflection pointsoccur at midspan of the beams on either side of the joint andat midheight of the columns above and below the joint Thissubassemblage is shown in Figure 9 It is further assumed thatthe beams on either side of the joint are of the same sectionand length and that the columns above and below the joint arethe same section and length 119866119860120583 is obtained by equalizingthe displacement of the subassemblage to displacement of a

db

hs

l

dc

Figure 9 Subassemblage for calculation of the shear rigidity

shear element Considering completely rigid joints the shearrigidity is computed as follows

(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)

in which ℎ119904is the height of column (story height) 119897 is the

length of the beam (bay width) 119889119887and 119889

119888are depth of the

beam and depth of the column respectively 119868119887and 119868119888are

moment of inertia of the beam and the column respectivelyand 119860

119887and 119860

119888are cross-sectional area of the beam and the

column respectively

(2) Braced Frame The equivalent shear rigidity of doublebracings is

(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)

where 119860br and 119864br are the cross-sectional area and Youngrsquosmodulus of the brace respectively For a single and tension-only bracing the coefficient 2 in (25) must be replaced with1

(3) Infilled Frame In order to consider the effect of infillpanel shear rigidity of the infill panel should be added to theequivalent shear rigidity of the system

(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)

(4) Dual System The equivalent shear rigidity of the dualsystem is

(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results

In this section the results obtained from numerical simu-lations experimental tests analytical approximate formulasand the codified proposed values for natural frequencies arecompared Table 3 summarizes all the methods used in thisstudy and also the corresponding abbreviations Based on thistable two experimental methods six numerical models withand without infill panel effect on rigidflexible foundationfour analytical approximate formulations and finally threecodified-based methods are used in order to estimate thenatural frequencies

Figure 10 shows the natural frequencies of the consideredbuilding in N-S and E-W directions using various methodsAs mentioned before the results of the natural frequenciesbased on forced vibration and ambient vibration tests havethe same values and are 270Hz in N-S and 250Hz in E-Wdirections Based on the numerical models it can be con-cluded that neglecting infill panel effect leads to decreasingthe frequency meaningfully Also modeling the foundationas a rigid medium leads to increasing the frequency a littlein all models Modeling infill panel as a compressive strutincreases frequencies in both directions The differencesbetween N3 and T1 are about 11 for N-S direction and 10in E-W direction Macromodeling based on shell elementsincreases the natural frequencies than to macromodelingbased on the compressive-strut theorem In this conditionthe differences between N5 and the experimental tests are56 and 6 for N-S and E-W directions respectively Asseen modeling the infill panel based on the continuumelement theorem leads to realistic behavior than to modelingit as a compressive strut

Table 4 summarizes the percentage of the error betweenthe numerical approximate formulas and the codified basedmethods with the exact values obtained from the experimen-tal testsUsing the proposed approximate formulas neglectingthe infill panels leads to almost the same results as thoseobtained from numerical simulations On the other handusing the proposed technique considering both infill paneland SSI effects leads to very close results to the experimentaltest In this condition the percentage of the errors between F4and T1 are only 15 and 24 for N-S and E-W directionsrespectively Once again it should be mentioned that theproposed formulation does not account for the torsionaldeformation effect That means that the term GJ119897 is notincorporated in the formulas Therefore when a building isunsymmetrical the error of the formula is meaningful Thevalue of the error depends to the distance between the centerof mass and the center of rigidity

Finally the results of the experimental tests are comparedwith codified proposed formulas All three design codes areproposed an empirical formulation in the form of 119879 = 119862 times

119867120573 in which 119867 is the height of the structure and 119862 and 120573

are the constant values which differ by the type of the lateralresisting system Considering that the lateral resisting systemin the current case is a combination of moment resistingand braced and infill panel systems the value of 119862 is 005based on INBC and 00488 based on UBC and ASCE [48]codes The value of the 120573 is 075 based on all three codes (all

Table 3 Different methodsmodels for frequency analysis

Abbreviation DescriptionT1 Steady state forced vibration testT2 Ambient vibration testN1 Numerical model without infill panel + SSI

N2 Numerical model without infill panel + rigidfoundation

N3 Numerical model with infill panel (compressivestrut) + SSI

N4 Numerical model with infill panel (compressivestrut) + rigid foundation

N5 Numerical model with infill panel (shell element)+ SSI

N6 Numerical model with infill panel (shell element)+ rigid foundation

F1 Analytical formula without infill panel + rigidfoundation

F2 Analytical formula without infill panel + SSI

F3 Analytical formula with infill panel + rigidfoundation

F4 Analytical formula with infill panel + SSIC1 Codified based on INBCC2 Codified based on UBCC3 Codified based on ASCE

000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3

Figure 10 Natural frequencies of the building in two orthogonaldirections

values are presented in SI system) Using the codified-basedmethod leads to the same frequency for both directions ofthe building because this method does not directly accountfor the direction Also it seems that using UBC and ASCEmethods leads to a little better result in comparison withINBC

Figure 11 shows the first mode shape of the consideredbuilding in two orthogonal directions using experimentaltest and numerical models These mode shapes are extractedbased on the relative lateral displacements of the structure innumerical models and also values of the resonance test whichdescribe average peak of Fourier spectrum in different storiesAs seen in both directions N5 model has the closest modeshape to the actual mode shape of building and N1 model hasthe least similarity to the real mode shape


Table 4 Percentage of the errors between numerical analytical and codified methods with experimental tests

Direction Numerical methods Approximate formulas Codified-based methodsN1 N2 N3 N4 N5 N6 F1 F2 F3 F4 C1 C2 C3

N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght

Normalized displacement

T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)

Figure 11 Mode shapes of the building (a) N-S direction (b) E-W direction

7 Conclusions

This paper presents vibration analysis of a steel-frame build-ing considering the infill panelsrsquo effect through experimentalnumerical and analytical approaches In order to extractthe actual natural frequencies of the building various steadystate forced and ambient vibration experimental tests werecarried out It was found that the results of two methodsmeaning the forced and ambient vibration are the same Inaddition a set of appropriate tests were carried out to realizethe compressive and shear strength of the masonry materialsof the infill panels It was observed that the materials arenot in good condition and this old building needs to berehabilitated

A set of approximate formulas were proposed for freevibration analysis of steel structures with various structuralsystems where the frame is idealized as prismatic can-tilever flexural-shear beam In the proposed technique soil-structure interaction is considered as axial and torsionalsprings whose potential energy is formulated and incorpo-rated into overall potential energy of the structure

Moreover a set of three-dimensional finite element mod-els of the building were provided In these models mainstructural systems are modeled by ldquobeamrdquo elements whilethe infill panels are simulated macromechanically by eithercontinuum elements or equivalent struts

It was found that modeling infill panels has significanteffect on vibration characteristics of the structures andneglecting their effects can leads to errors about 35ndash45in natural frequencies Considering soil-structure interac-tion leads to more flexibility in the system and decreasesfrequency of system Generally modeling infill panels bycontinuum elements lead to higher frequencies and decreasethe percentage of the errors between the finite element modeland experimental tests It was concluded that using detailedfinite element model of the structure modeling infill panelsusing shell elements and also considering the soil-structure-interaction may leads to more accurate result

It was shown that the proposed approximate formulashave good capability in estimation of the natural frequencyof steel structures They can be used for computing thefrequency of various structural systems by less computationalefforts In the present study there is only 15ndash25 errorwhen using the approximate formulas compared to the forcedvibration test Also it was shown that the codified proposedformulas have a close estimation to the test results and leadto errors of about 2ndash12 in considered building

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] M N Fardis S N Bousias G Franchioni and T B Pana-giotakos ldquoSeismic response and design of RC structures withplan-eccentric masonry infillsrdquo Earthquake Engineering andStructural Dynamics vol 28 no 2 pp 173ndash191 1999

[2] K A Ghassan Non-ductile behavior of reinforced concreteframes with masonry infill panels subjected to in-plane loading[PhD thesis] University of Illinois at Chicago USA ChicagoIll USA 1998

[3] A A Chaker and A Cherifati ldquoInfluence of masonry infillpanels on the vibration and stiffness characteristics of RC framebuildingsrdquo Earthquake Engineering amp Structural Dynamics vol28 no 9 pp 1061ndash1065 1999

[4] S Sattar Influence of masonry infill walls and other buildingcharacteristics on seismic collapse of concrete frame buildings[PhD thesis] University of Colorado Boulder Colo USA2013

[5] P B Shing H R Lofti A Barzegarmehrabi and J BunnerldquoFinite element analysis of shear resistance of masonry wallpanels with and without confining framesrdquo in Proceedings ofthe 10thWorld Conference on Earthquake Engineering pp 2581ndash2586 A A Balkema Rotterdam The Netherlands 1992

[6] M Dhanasekar and A W Page ldquoInfluence of brick masonryinfill properties on the behaviour of infilled framesrdquoProceedingsof the Institution of Civil Engineers vol 81 no 2 pp 593ndash6051986

[7] T C Liauw and C Q Lo ldquoMultibay infilled frames withoutshear connectorsrdquoACI Structural Journal vol 85 no 4 pp 423ndash428 1988

[8] A B Mehrabi and P B Shing ldquoFinite element modeling ofmasonry-infilled RC framesrdquo Journal of Structural Engineeringvol 123 no 5 pp 604ndash613 1997

[9] H R Lotfi and P B Shing ldquoAn appraisal of smeared crackmod-els for masonry shear wall analysisrdquo Computers and Structuresvol 41 no 3 pp 413ndash425 1991

[10] T B Panagiotakos and M N Fardis ldquoSeismic response ofinfilled RC frames structuresrdquo in Proceedings of the 11th WorldConference on Earthquake Engineering Paper No 225 Aca-pulco Mexico 1996

[11] G Uva F Porco and A Fiore ldquoAppraisal of masonry infill wallseffect in the seismic response of RC framed buildings a casestudyrdquo Engineering Structures vol 34 no 1 pp 514ndash526 2012

[12] D Celarec P Ricci and M Dolsek ldquoThe sensitivity of seis-mic response parameters to the uncertain modelling variablesof masonry-infilled reinforced concrete framesrdquo EngineeringStructures vol 35 pp 165ndash177 2012

[13] D Celarec and M Dolsek ldquoThe impact of modelling uncer-tainties on the seismic performance assessment of reinforcedconcrete frame buildingsrdquo Engineering Structures vol 52 pp340ndash354 2013

[14] M Holmes ldquoSteel frame with brickwork and concrete infillingrdquoICE Proceedings vol 19 pp 473ndash478 1961

[15] B Stafford-Smith ldquoLateral stiffness of infilled framesrdquo Journalof Structural Division vol 88 pp 183ndash199 1962

[16] B Stafford-Smith and C Carter ldquoA method of analysis forinfilled framesrdquo ICE Proceedings vol 44 pp 31ndash48 1969

[17] R J Mainstone and G A Weeks ldquoThe influence of a boundingframe on the racking stiffness and strengths of brick wallsrdquo inProceedings of the 2nd International Brick Masonry Conference(SIBMAC rsquo70) Building Research Station England UK 1970

[18] R J Mainstone ldquoOn the stiffness and strengths of infilledframesrdquo ICE Proceedings vol 49 no 2 p 230 1971

[19] R E Klingner and V V Bertero ldquoEarthquake resistance ofinfilled framesrdquo Journal of the Structural Division vol 104 no6 pp 973ndash989 1978

[20] L Te-Chang and K Kwok-Hung ldquoNonlinear behaviour of non-integral infilled framesrdquo Computers and Structures vol 18 no3 pp 551ndash560 1984

[21] C A Syrmakezis and V Y Vratsanou ldquoInfluence of infill wallsto RC frames Responserdquo in Proceedings of the 8 th EuropeanConference on Earthquake Engineering (EAEE rsquo86) pp 47ndash53European Association for Earthquake Engineering IstanbulTurkey 1986

[22] R Zarnic and M Tomazevic ldquoAn experimentally obtainedmethod for evaluation of the behavior of masonry infilledRC framesrdquo in Proceedings of the 9th World Conference onEarthquake Engineering pp 163ndash168 1988

[23] T Schmidt ldquoAn approach of modelling masonry infilled framesby the FE method and a modified equivalent strut methodDarmstadt ConcreterdquoAnnual Journal on Concrete and ConcreteStructures pp 185ndash194 1989

[24] C Z Chrysostomou Effects of degrading infill walls on thenonlinear seismic response of two-dimensional steel frames [PhDthesis] Cornell University Press Ithaca NY USA 1991

[25] A Saneinejad andBHobbs ldquoInelastic design of infilled framesrdquoJournal of Structural Engineering vol 121 no 4 pp 634ndash6501995

[26] F J Crisafulli Seismic behavior of reinforced concrete structureswith masonry infills [PhD thesis] University of CanterburyChristchurch New Zealand 1997

[27] R D Flanagan and R M Bennett ldquoArching of masonry infilledframes comparison of analytical methodsrdquo Practice Periodicalon Structural Design and Construction vol 4 no 3 pp 105ndash1101999

[28] WW El-DakhakhniM Elgaaly andA AHamid ldquoThree-strutmodel for concrete masonry-infilled steel framesrdquo Journal ofStructural Engineering vol 129 no 2 pp 177ndash185 2003

[29] M Dolsek and P Fajfar ldquoThe effect of masonry infills on theseismic response of a four-storey reinforced concrete framemdashadeterministic assessmentrdquo Engineering Structures vol 30 no 7pp 1991ndash2001 2008

[30] D A Foutch Study of the vibration characteristics of two multi-story building [PhD thesis] California Institute of TechnologyPasadena Calif USA 1977

[31] S K Jain ldquoContinuum models for dynamics of buildingsrdquoJournal of Engineering Mechanics vol 110 no 12 pp 1713ndash17301984

[32] P C Jennings R B Matthiesen and J Brent Hoerner ldquoForcedvibration of a tall steel-frame buildingrdquo Earthquake Engineeringand Structural Dynamics vol 1 pp 107ndash132 1972

[33] D S A Foutch ldquoThe vibrational characteristics of a twelve-storey steel frame buildingrdquo Earthquake Engineering and Struc-tural Dynamics vol 6 no 3 pp 265ndash294 1978

[34] M D Trifunac ldquoComparisons between ambient and forcedvibration experimentsrdquo Earthquake Engineering and StructuralDynamics vol 1 no 2 pp 133ndash150 1972

[35] N M Auciello and G Nole ldquoVibrations of a cantilever taperedbeam with varying section properties and carrying a mass atthe free endrdquo Journal of Sound and Vibration vol 214 no 1 pp105ndash118 1998


[36] J Wu and C Chen ldquoAn exact solution for the natural frequen-cies and mode shapes of an immersed elastically restrainedwedge beam carrying an eccentric tip mass with mass momentof inertiardquo Journal of Sound and Vibration vol 286 no 3 pp549ndash568 2005

[37] D-W Chen and J-S Wu ldquoThe exact solutions for the naturalfrequencies and mode shapes of non-uniform beams withmultiple spring-mass systemsrdquo Journal of Sound and Vibrationvol 255 no 2 pp 299ndash322 2003

[38] Q S Li H Cao and G Li ldquoAnalysis of free vibrations of tallbuildingsrdquo Journal of Engineering Mechanics vol 120 no 9 pp1861ndash1876 1994

[39] H Lin and S C Chang ldquoFree vibration analysis of multi-spanbeams with intermediate flexible constraintsrdquo Journal of Soundand Vibration vol 281 no 1-2 pp 155ndash169 2005

[40] R D Firouz-Abadi H Haddadpour and A B Novinzadeh ldquoAnasymptotic solution to transverse free vibrations of variable-section beamsrdquo Journal of Sound and Vibration vol 304 no3ndash5 pp 530ndash540 2007

[41] S A Sina H M Navazi and H Haddadpour ldquoAn analyticalmethod for free vibration analysis of functionally gradedbeamsrdquoMaterials and Design vol 30 no 3 pp 741ndash747 2009

[42] E CarreraM Petrolo and P Nali ldquoUnified formulation appliedto free vibrations finite element analysis of beams with arbitrarysectionrdquo Shock and Vibration vol 18 no 3 pp 485ndash502 2011

[43] H Ghasemzadeh H Rahmani-Samani and M MirtaherildquoVibration analysis of steel structures including the effect ofpanel zone flexibility based on the energy methodrdquo EarthquakeEngineering and Engineering Vibration vol 12 pp 587ndash5982013

[44] B Akgoz and O Civalek ldquoFree vibration analysis of axiallyfunctionally graded tapered Bernoulli-Euler microbeams basedon themodified couple stress theoryrdquoComposite Structures vol98 pp 314ndash322 2013

[45] X-F Li Y-A Kang and J-X Wu ldquoExact frequency equationsof free vibration of exponentially functionally graded beamsrdquoApplied Acoustics vol 74 no 3 pp 413ndash420 2013

[46] Instruction for Seismic Rehabilitation of Existing BuildingsCode No 360 Management and Planning Organization (Officeof Deputy for Technical Affairs) 2007

[47] A K ChopraDynamics of Structures PrenticeHall EnglewoodCliffs NJ USA 1995

[48] ASCE-FEMA ldquoPre-standard and commentary for the seismicrehabilitation of buildingsrdquo FEMA 356 2000

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Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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Journal of

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Generic model

(a)

BrickInterfaceMortar

(b)

BrickInterface

(c)

Continuum element(s)

(d)

Strut

(e)

Strut(s)

(f)

Figure 1 Modeling the masonry infill panels (a) generic model (b) micromodeling (c) mesomodeling (d) homogenized macromodeling(e) single-strut macromodeling and (f) multistrut macromodeling

been used for the micro- and mesomodeling for examplenonlinear orthotropic model [6] smeared crack model [7]and dilatant interface constitutive model [8] The infill panelismacromechanicallymodeled as homogeneousmaterial andthe average properties are used for the whole panel [9]Modeling the infill panel as an equivalent strut is the mostconventional technique Table 1 summarizes different strutmodeling approaches [4] In addition all the macro- meso-and micromodels for simulating the presence of the infillpanels are affected by a strong sensitivity to the parametersof the model Sensitivity and (aleatory andor epistemic)uncertainty analysis of infilled frames were investigated byPanagiotakos and Fardis [10] Uva et al [11] Celarec et al [12]and Celarec and Dolsek [13]

On the other hand the vibrational behavior of the framestructures was investigated by different researchers Foutchstudied the vibrational characteristics of a nine-story rein-forcement concrete frame [30] Jain studied the ImperialCounty Services Building analytically and compared theresponse with those recorded by strong motion accelero-graphs during the 1979 Imperial Valley earthquake [31] How-ever the fundamental frequencies from the numerical resultsand the field measurements were not in close agreementbecause of the foundation flexibility Jennings et al [32] testeda forced vibration experiment on a twenty-two story steel-framed structure and extracted first six natural frequenciesFoutch [33] compared the natural frequencies and 3D modeshapes of a twenty story steel structure using forced vibrationtest and finite element method He concluded that theproperties of the primarily translational modes are predictedreasonably well but adequate prediction of torsional motionsis not obtained Trifunac [34] compared ambient and forced

vibration techniques for testing full-scale structures undersmall level excitations He concluded that both methodsprovide mutually consistent results

Beside the different experimental and numerical tech-niques that are used for free and forced vibration analysis ofstructures the dynamic behavior of the buildings and towerscan be estimated by analytical methods Free vibration ofbeam-like structures was studied by several researchers [35ndash37] Most of the previous studies have been oriented towardsthe closed-form solutions of the free flexural vibration ofa beam Li et al [38] derived the exact solution for freeflexural vibration of the stepped cantilever beams subjected toaxial loads They used the exact solution of a prismatic beamaccompanied by the transfermatrixmethodHowever effectsof the soil-structure interaction (SSI) shear deformation andthe joint flexibility were not considered Lin and Chang [39]investigated free vibration analysis of a multispan beam witharbitrary number of flexible constraints They assumed thateach span of the continuous beam obeys Timoshenko beamtheory Firouz-Abadi et al [40] studied the transverse freevibrations of typical truncated nonuniform Euler-Bernoullibeams by using a WKB global approximation Sina et al[41] proposed a new beam theory for free vibration analysisof functionally graded beams The beam properties wereassumed to be varied through the thickness following a powerlaw distribution Also it was assumed that the lateral normalstress of the beam is zero Carrera et al [42] proposed a hier-archical finite element method based on the Carrera UnifiedFormulation for free vibration analysis of beamwith arbitrarysection geometries Ghasemzadeh et al [43] proposed a set ofapproximate formulas to determine the natural frequencies ofthe structures considering the panel zone flexibility and SSI



























(a)

104 m

26

3 m

22

55 m

(b)


(a) (b)









4 Numerical Model




Bearing

Pulley

Angle section

x

zx

y120579120579

F0 = m1205962F0 = m1205962


(a) (b)


5 cm

1 cm

5 cm

10 cm 20 cm




120590119905119900=

119881test119860119887

minus 120590119888 (1)






Number 119881test119860119887 (kgcm2) 120590

119888(kgcm2) 120590to (kgcm




1015840

tb are 385 kgcm2




1198911015840

119898=

1198911015840

cb15

times

((11989541ℎ) times 1198911015840

119895) + 1198911015840

tb

((11989541ℎ) + 1) times 1198911015840

tb (2)







Diagonalcompression

area

x

y

z

yx

hco

l

hin

f

lbeam

linf

120579

a

tinf

rinf



119886 = 0175[1205821ℎcol]minus04

119903inf (3)

where

1205821= [


]

025

(4)
















L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)




119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)


119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)








Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)





spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)


V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)


119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)


Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































(a)

104 m

26

3 m

22

55 m

(b)


(a) (b)









4 Numerical Model




Bearing

Pulley

Angle section

x

zx

y120579120579

F0 = m1205962F0 = m1205962


(a) (b)


5 cm

1 cm

5 cm

10 cm 20 cm




120590119905119900=

119881test119860119887

minus 120590119888 (1)






Number 119881test119860119887 (kgcm2) 120590

119888(kgcm2) 120590to (kgcm




1015840

tb are 385 kgcm2




1198911015840

119898=

1198911015840

cb15

times

((11989541ℎ) times 1198911015840

119895) + 1198911015840

tb

((11989541ℎ) + 1) times 1198911015840

tb (2)







Diagonalcompression

area

x

y

z

yx

hco

l

hin

f

lbeam

linf

120579

a

tinf

rinf



119886 = 0175[1205821ℎcol]minus04

119903inf (3)

where

1205821= [


]

025

(4)
















L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)




119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)


119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)








Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)





spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)


V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)


119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)


Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

104 m

26

3 m

22

55 m

(b)


(a) (b)









4 Numerical Model




Bearing

Pulley

Angle section

x

zx

y120579120579

F0 = m1205962F0 = m1205962


(a) (b)


5 cm

1 cm

5 cm

10 cm 20 cm




120590119905119900=

119881test119860119887

minus 120590119888 (1)






Number 119881test119860119887 (kgcm2) 120590

119888(kgcm2) 120590to (kgcm




1015840

tb are 385 kgcm2




1198911015840

119898=

1198911015840

cb15

times

((11989541ℎ) times 1198911015840

119895) + 1198911015840

tb

((11989541ℎ) + 1) times 1198911015840

tb (2)







Diagonalcompression

area

x

y

z

yx

hco

l

hin

f

lbeam

linf

120579

a

tinf

rinf



119886 = 0175[1205821ℎcol]minus04

119903inf (3)

where

1205821= [


]

025

(4)
















L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)




119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)


119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)








Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)





spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)


V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)


119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)


Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Bearing

Pulley

Angle section

x

zx

y120579120579

F0 = m1205962F0 = m1205962


(a) (b)


5 cm

1 cm

5 cm

10 cm 20 cm




120590119905119900=

119881test119860119887

minus 120590119888 (1)






Number 119881test119860119887 (kgcm2) 120590

119888(kgcm2) 120590to (kgcm




1015840

tb are 385 kgcm2




1198911015840

119898=

1198911015840

cb15

times

((11989541ℎ) times 1198911015840

119895) + 1198911015840

tb

((11989541ℎ) + 1) times 1198911015840

tb (2)







Diagonalcompression

area

x

y

z

yx

hco

l

hin

f

lbeam

linf

120579

a

tinf

rinf



119886 = 0175[1205821ℎcol]minus04

119903inf (3)

where

1205821= [


]

025

(4)
















L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)




119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)


119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)








Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)





spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)


V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)


119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)


Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Diagonalcompression

area

x

y

z

yx

hco

l

hin

f

lbeam

linf

120579

a

tinf

rinf



119886 = 0175[1205821ℎcol]minus04

119903inf (3)

where

1205821= [


]

025

(4)
















L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)




119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)


119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)








Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)





spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)


V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)


119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)


Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










L

y

x

xn

xi

x2x1

(a)

Nmn

mi

m2

m1

Nn

Ni

N2

N1

(b)




119879 =

1

2

int

119871

0

119898(119909 119905)2119889119909 +

1

2

119899

sum

119894=1

119898119894(119909119894 119905)2 (5)


119906 (119909 119905) = 119906119891(119909 119905) + 119906sh (119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119861(119909119905)

+ 119906119886(119909 119905) + 119906

119903(119909 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119878(119909119905)

(6)








Π = 119880 + 119881 (7)

where

119880 =

1

2

int

119871

0

119864119868[11990610158401015840

119891(119909 119905)]

2

119889119909

+

1

2

int

119871

0

119866119860

120583

[1199061015840

sh (119909 119905)]2

119889119909

+

1

2

119870119886[119906119886(119909 119905)]

2

+

1

2

119870119903[120579119903(119905)]2

(8)





spring Consider

119881 = minus

1

2

int

119871

0

119873V (119909 119905) 119889119909 minus1

2

119899

sum

119894=1

119873119894V (119909119894 119905) (9)


V (119909 119905) =1

2

int[1199061015840(119909 119905)]

2

119889119909 (10)


119906 (119909 119905) = 119906 (119909) sin120596119905

(119909 119905) = 120596119906 (119909) cos120596119905(11)


Πmax =1

2

[int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh (119909)]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

minusint

119871

0

119873V (119909) 119889119909 minus119899

sum

119894=1

119873119894V (119909119894)]

(12)


Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Energy in system

Kinetic T


External work V

Frame

Soil


Shape function

Natural frequency

Equivalent beam

Flexural rigidity

Shear rigidity


Braced frame

Infilled frame

Dual system

u(x t)

uf

ua

ush

uB

uS

uS

uF

ur

mlowastF

mlowastS

mlowastFd

mlowastFc

mlowastSd

mlowastSc

mlowast

klowast

klowastG

u(x)

u(x)

klowastF

klowastS

klowastGFd

klowastGFc

klowastGSd

klowastGSc

klowastGF

klowastGS




119879max =1

2

1205962[int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

] (13)


1205962=


119866

119898lowast

(14)

where

119898lowast= int

119871

0

119898[119906(119909)]2119889119909 +

119899

sum

119894=1

119898119894[119906(119909119894)]2

119896lowast= int

119871

0

119864119868[11990610158401015840

119891(119909)]

2

119889119909

+ int

119871

0

119866119860

120583

[1199061015840

sh]2

119889119909 + 1198701198861199062

119886+ 1198701199031205792

119903

119896lowast

119866=int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 +

119899

sum

119894=1

119873119894int [1199061015840(119909119894)]

2

119889119909

(15)




lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











lowast





0 Thus


119898lowast= int

119871

0

119898[119906(119909)]2119889119909 + 119898

0

119871ℎ119904

sum

119894=1

1199062(119909119894)

119896lowast

119866= int

119871

0

(119873int [1199061015840(119909)]

2

119889119909) 119889119909 + 1198730

119871ℎ119904

sum

119894=1

int

119894ℎ119904

0

[1199061015840(119909119894)]

2

119889119909

(16)


119906 (119909) = int

119872119898V

119864119868

119889119909 + int120583

119876119902

119866119860

+

119872119887119898119887

119870120593

+

119876119887119902119887

119870119904

(17)


119887and119872






119906 (119909) = 119906119891+ 119906sh⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119906119865

+ 119906119878 (18)

where

119906119865=

119875

2119864119868

(minus

119871

3

1199093+

1198712

2

1199092+

1199094

12

) +

120583119875

119866119860

(119871119909 minus

1199092

2

)

119906119878=

119875119871

119870119886

+

1198751198712

2119870119903

119909

(19)




119865119889+ 119898lowast

119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119865

+ 119898lowast

119878119889+ 119898lowast

119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119898lowast

119878

119896lowast

119866= 119896lowast

119866119865119889+ 119896lowast

119866119865119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119865

+ 119896lowast

119866119878119889+ 119896lowast

119866119878119888⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896lowast

119866119878


119865+ 119896lowast

119878

(20)


119898lowast

119865119889= 119898(

2

15

12058321198995ℎ5

119904

(119866119860)2+

37

840

120583

1198997ℎ7

119904

119866119860119864119868

+

13

3240

1198999ℎ9

119904

(119864119868)2)

119898lowast

119865119888= 1198980(

1

120

1205832119899ℎ4

119904

(161198994+ 15119899

3minus 1)

(119866119860)2

+

1

5040

times 120583119899ℎ6

119904

(2221198996+ 315119899

5+ 70119899

4+ 28119899

2minus 5)

119866119860 times 119864119868

+

1

51840

times 119899ℎ8

119904

(2081198998+ 405119899

7+ 180119899

6+ 20119899

2minus 3)

(119864119868)2

)

119898lowast

119878119889= 119898(

1

12

1198997ℎ7

119904

1198702

119903

+

13

360

1198998ℎ8

119904

119864119868119870119903

+

5

24

120583

1198996ℎ6

119904

119866119860119870119903

+

1198993ℎ3

119904

1198702

119886

+

1

10

1198996ℎ6

119904

119864119868119870119886

+

2

3

120583

1198994ℎ4

119904

119866119860119870119886

+

1

2

1198995ℎ5

119904

119870119903119870119886

)

119898lowast

119878119888= 1198980(

1

24

1198995ℎ6

119904

(21198992+ 3119899 + 1)

1198702

119903

+

1198993ℎ2

119904

1198702

119886

+

1

2

ℎ4

119904(1198995+ 1198994)

119870119886119870119903

+

1

1440

1198994ℎ7

119904

(521198994+ 90119899

3+ 35119899

2+ 3)

119864119868119870119903

+

1

24

1198994ℎ5

119904120583

(51198992+ 6119899 + 1)

119866119860119870119903

+

1

360

1198992ℎ5

119904

361198994+ 45119899

3+ 91198992minus 1

119864119868119870119886

+

1

6

1198992ℎ3

119904120583

41198992+ 3119899 minus 1

119866119860119870119886

)


119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119896lowast

119866119865119889= 119873(

1

4

12058321198994ℎ4

119904

(119866119860)2+

1

18

120583

1198996ℎ6

119904

119866119860 times 119864119868

+

1

160

1198998ℎ8

119904

(119864119868)2)

119896lowast

119866119865119888= 1198730(

1

12

12058321198992ℎ3

119904

(31198992+ 2119899 minus 1)

(119866119860)2

+

1

180

times 1205831198992ℎ5

119904

(101198994+ 91198993minus 1)

119866119860 times 119864119868

+

1

30 240

1198992ℎ7

119904

times

(1891198996+ 270119899

5+ 70119899

4+ 21119899

2minus 10)

(119864119868)2

)

119896lowast

119866119878119888= 119873(

1

3

120583

1198995ℎ5

119904

119866119860119870119903

+

1

20

1198997ℎ7

s119864119868119870119903

+

1

8

1198996ℎ6

119904

1198702

119903

)

119896lowast

119866119878119889= 1198730(

1

720

1198993ℎ6

119904

(361198994+ 45119899

3+ 10119899

2minus 1)

119864119868119870119903

+

1

12

1205831198993ℎ4

119904

41198994+ 3119899 minus 1

119866119860119870119903

+

1

8

ℎ5

119904

1198996+ 1198995

1198702

119903

)

119896lowast

119865=

1198995ℎ5

119904

20119864119868

+ 120583

1198993ℎ3

119904

3119866119860

119896lowast

119878=

1198712

119870119886

+

1198714

4119870119903

(21)



119868119911= sum119860

119888119894(119910119894)2

119868119910= sum119860

119888119894(119911119894)2 (22)


119894



542 Shear Rigidity


db

hs

l

dc



(

119866119860

120583

)

frame=

119864ℎ119904

119878119889

(23)

where

119878119889=

(119897 minus 119889119888)3

12119868119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)3

12119868119888

+

119864

119866

[

(119897 minus 119889119888)

119860119887

ℎ2

119904

1198972+

(ℎ119904minus 119889119887)

119860119888

]

(24)






119887and 119860


column respectively


(

119866119860

120583

)

brace= 2119860br119864br

119897

1198972+ ℎ2

119904

(25)



(

119866119860

120583

)

infilled frame= (

119866119860

120583

)

frame+ (

119866119860

120583

)

infill panel (26)


(

119866119860

120583

)

dual system= (

119866119860

120583

)

frame+ (

119866119860

120583

)

brace (27)


6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6 Results


















000510152025303540

T1 T2 N1 N2 N3 N4 N5 N6 F1 F2 F3

Nat

ural

freq

uenc

y (H

z)

Vibration method

N-SE-W

F4 C1 C2 C3







N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












N-S 40 36 11 5 55 107 39 44 63 15 115 93 93E-W 44 40 10 56 6 112 44 47 72 24 44 2 2

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(a)

0

02

04

06

08

1

00 02 04 06 08 10

Nor

mal

ized

hei

ght


T1N1

N3N5

(b)


7 Conclusions









References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References




















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Free and Forced Vibration Analysis of an Infilled …downloads.hindawi.com/journals/sv/2014/439591.pdf · 2019-07-31 · Research Article Free and Forced Vibration