Research ArticleNonlinear Earthquake Analysis of Reinforced Concrete Frameswith Fiber and Bernoulli-Euler Beam-Column Element

Muhammet Karaton

Civil Engineering Department Engineering Faculty Fırat University 23119 Elazig Turkey

Correspondence should be addressed to Muhammet Karaton mkaratonfiratedutr

Received 20 September 2013 Accepted 29 October 2013 Published 22 January 2014

Academic Editors A El-Shafie A K Gupta and D-F Lin

Copyright copy 2014 Muhammet KaratonThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamic analysis of reinforcedconcrete (RC) structural element Stiffness matrix of this element is obtained by using rigidity method A solution technique thatincluded nonlinear dynamic substructure procedure is developed for dynamic analyses of RC frames A predicted-corrected formofthe Bossak-120572method is applied for dynamic integration scheme A comparison of experimental data of a RC column element withnumerical results obtained from proposed solution technique is studied for verification the numerical solutions Furthermorenonlinear cyclic analysis results of a portal reinforced concrete frame are achieved for comparing the proposed solution techniquewith Fibre element based on flexibility method However seismic damage analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed mass and load Damage region propagation and intensities according toboth approaches are researched

1 Introduction

The realistic modeling of the nonlinear static or dynamicbehavior of RC structures is a more sophisticated prob-lem due to the inelastic behavior of concrete plasticityof reinforcement interface debonding of these materialsand so on In the last thirty years researchers have mademore effort for the numerical modeling of RC structuresand most of the state-of-the-art on this problem dealswith two main approaches lumped plasticity modeling [1ndash3] and distributed-inelasticity modeling (ie the so-calledfibre beam-column elements FBCE) [4ndash8] In the firstapproach nonlinear springs based on moment-rotation andforce-displacement curves are used and nonlinear volumeis assumed to be lumped in the specific location of theelement Mohr et al [1] were used a series of polynomialshape functions for strain distributions of the vertical andshear on cross-section Li et al [2] was proposed a simplifiedlumped hinge for determination of failure mode of staticallyindeterminate structure Reshotkina and Lau [3] was useda concentrated plasticity approach for axial-flexure-shearinteractions on the inelastic behavior of reinforced concretemembers under the seismic loads In the second approach

the nonlinear behaviors of concrete and reinforcement mate-rials are separately calculated This method divided into twoas approaches was based on flexibility and rigidity Tauceret al [9] were developed a flexibility method for nonlin-ear dynamic analysis of reinforced concrete element Thismethod based onBernoulli-Euler hypothesis included biaxialbending and axial force conditions Nonlinear behavior ofconcrete and steel were computed by using uniaxial stress-strain relationships Force-displacement interpolation func-tions are used for the obtaining of element flexibility matrixElement stiffness matrix is achieved by inverse of flexibilitymatrix Furthermore Ceresa et al [10] were developed abeam-column element based on flexibility matrix by using tothe Timoshenko beam theory under cyclic loading Damagesin the element are taken into account with coaxial rotatingcrackmodel Lu et al [4]were usedmultilayered shell elementand fibre beam-column element based on flexibility methodformodeling of the different frame and framewith shear wallHowever an important research report about methods basedon rigidity was performed by Kawano et al [6] In the reportnonlinear analyses of reinforced concrete space frames withmultistory and multibay are obtained under the earthquake

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 905963 15 pageshttpdxdoiorg1011552014905963

2 The Scientific World Journal

loads Cubic Hermitian polynomials for transversal displace-ments and the linear shape functions for axial displacementsare used for obtaining of stiffnessmatrixHowever theGauss-Lobatto integration rule was used for obtaining element massand stiffness matrices Element damage location was simplypresented and element was not divided into subelementcalled as ldquosegmentrdquo Some calculation problems in heavilydamage location was shown and they are reported to arisefrom flexural deformation Furthermore strain rate effects inresponse to reinforced concrete frames was investigated byIribarren [7] Used formulations are based on rigiditymethodand a strain rate dependentmaterial formulation is developedfor both the concrete and steel constitutive response Brum[8] was obtained nonlinear dynamic analyses of frame andmasonry structures by using a method which was based onstiffness matrix In the research a uniaxial constitutivemodelfor concrete and masonry is proposed for compression andtension regions of the materials under the cyclic loadings

In this study a beam-column element based on theEuler-Bernoulli beam theory is presented for the fiber RCelement Element stiffness matrices are obtained by usingrigidity methodThe beam or column element is divided intosubelements called ldquosegmentrdquo [11] Furthermore the internalfreedoms of this segment are dynamically condensed toexternal freedoms at the end of the element Thus nonlineardynamic analysis of high RC building can be obtained withina short time This procedure requires that nested loops ofthe element and structure are obtained at the same time (theso-called nonlinear dynamic substructure) However thiscondensation procedure is not used in the modeling of thefibre element approach (FEA) of the RC element In additionuniform or trapezoidal loads of the segment are assumed tobe zero or condensed to the external freedoms at the end ofthe element in the FEA [11] This case is not preferred dueto the required redistribution of loading However in thisstudy external loading of the segment is taken into accountby considering the damage occurred in the element

The present research is organized as follows (i) pre-senting of fiber Bernoulli-Euler beam-column element basedon rigidity method (ii) developing of nonlinear dynamicsubstructure technique of RC frames (iii) verification of theconstitutive model with respect to experimental result of acolumn structural element and dynamic analysis results offibre RC element of a portal frame (iv) obtaining of seismicdamage analyses of an 8-story RC frame with soft-story fordistributedlumped mass and load case and (v) results

2 Fiber and Bernoulli-Euler Approach(FBEA) for Reinforced Concrete BeamColumn Element

In this section theory of Fiber and Bernoulli-Euler elementwhich is based on stiffness matrix is firstly presented forreinforced concrete section Elements are divided into subele-ments called segment and they are assumed as substructuresof element Furthermore cross-section of the each segmentis also subdivided into a number of fiberslayers In the

next section nonlinear dynamic substructuresmethod is alsomentioned to be applied to freedoms in element ends

21 Obtaining Segment Stiffness andMassMatrices Thestraindistribution on a cross-section of beam is assumed to beuniform due to axial forces and linear due to only bendingaccording to the Bernoulli-Euler approach Furthermore if aplane section before bending is plane after bending and if thestrain is assumed to be small and shear stresses are omittedthe strain of a point on a cross-section in the axial directioncan be written as

120576119909119909=

119889119906

119889119909

minus

1198892V

1198891199092119910 (1)

where 119906 and V are the displacement of the axial and verticaldirections of element respectively (Figure 1) However ifcross-section of element is divided into fiberslayers thisequation for each fiberlayer in local axis can be rewritten as

120576120585119899= 1 minus120578

1015840

119899

119889119906

119889120585

1198892V1198891205852

(2)

where 119889120576120585119899

is incremental axial strain of a fiberlayer ona segment in 120585 local axis direction Therefore if the cubicHermitian polynomials for transversal displacements and thelinear shape functions for axial displacements are used [12](2) can be written as

120576120585119899= 1 minus120578

1015840

119899

119889 [119873 (120585)]

119889120585

1198892

[119873 (120585)]

1198891205852

119902 = 1 minus1205781015840

119899 [119861] 119902 (3)

where 119902 is displacement vector including displacementand rotation of a segment [119861] is strain-displacement matrixThus incremental stress in each fiberlayer can obtained as

119889120590120585119899= 119864119879119899119889120576120585119899= 1198641198791198991 minus120578

1015840

119899 [119861] 119902 (4)

where 119864119879119899 determined by using uniaxial stress-strain rela-

tionship of using materials is tangent elasticity modulusof each fiber Therefore total strain energy of a segmentsubelement is obtained as

Π =

1

2

int

119871Seg

120590119879

120576119860Seg119889119909

=

1

2

int

1

minus1

119902119879

[119861]119879

[

1

minus1205781015840

119899

119864119879119899119860 1 minus120578

1015840

119899] [119861] 119902 119889120585

=

1

2

int

1

minus1

119902119879

[119861]119879

times

[

[

[

[

119873

sum

119899=1

119864119879119899119860119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

119873

sum

119899=1

119864119879119899119860119899(1205781015840

119899)

2

]

]

]

]

[119861] 119902 119889120585

(5)

The Scientific World Journal 3

y

120585 u

120578

Columnelement Beam element

Concrete fibers Steel fibers

120585 u

120578

120585 u

120578

z

x120577 w

120577 w

(a)

120590

120576

+=

Concrete fibers

Confined

Unconfined

Steel fibers

120590

120576

(b)

Figure 1 Concrete and steel fibers and global and local axes for FBEA

where 119871Seg and 119860Seg are expressed to be length and cross-section area of segment respectively 119860

119899is also area of a

fiberlayer Thus element stiffness matrix can be obtained byusing minimum potential energy principle Stiffness matrixof a segment can be written as

[119870Seg]

=

1

2

int

1

minus1

[119861]119879[

[

[

[

119873

sum

119899=1

119864119879119899119860119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

119873

sum

119899=1

119864119879119899119860119899(1205781015840

119899)

2

]

]

]

]

[119861] 119889120585

(6)

The segment stiffness matrix is obtained by using areascoordinates and tangent elasticity modulus of fiberlayer forreinforced concrete sections In this study linear superpo-sition rule is used for different material properties of thefiberlayer in the section (Figure 1) Furthermore a fiberconcrete or reinforced bar on the cross-section may becracked or damaged due to external loading For this reasona relationship between the damaged and undamaged casesmust be obtained for the solutions Thus damage in eachconcretereinforcement fiber can be written as [13]

119889119899= 1 minus

119864119879119899

119864119874119899

(7)

where 119864119874119899

and 119864119879119899

are undamaged and damagedtangentelasticity module of the 119899th fiber respectively 119889

119899 damage

intensities are obtained separately under tensile and compres-sive stress for each fiber

However total kinetic energy of particle velocities on thecross-section of an element throughout its neutral axis can bewritten as

119879 =

1

2

int

Ω

V120585120588V120585119889Ω (8)

Segment

External node Internal nodes

y

External node

Segment Segment Segment

z

x

Figure 2 Segments and external and internal nodes for FBEA

where120588 ismass density and V120585is particle velocity If fiberlayer

elements are used for (8) the mass matrices of an element onthe local axis are obtained as

[119872Seg] = int1

minus1

119899119891119894119887

sum

119899=1

([119873 (120585)]119879

120588119899119860119899[119873 (120585)]) 119889120585 (9)

where [119873(120585)] is the element shape functions matrix on thelocal axis and 120588

119899is also mass density of the 119899th fiber Stiffness

and mass matrices of a segment obtained by local axis aretransformed to global axes by using transformation matrix

22 Obtaining with Nonlinear Dynamic Substructures Tech-nique of Element Stiffness and Mass Matrices In FBEA cubicHermitian shape functions for rotation and shear strain andlinear shape functions for the axial deformation are usedrespectively The Gauss-Lobatto integration rule is generallyused for obtaining element mass and stiffness matrices alongwith 120585 local axis [6ndash8] However substructure procedures aregenerally preferred for solutions of high buildings Howeverthis technique in the previous study is not used due tonumerical difficultly Element stiffnessflexibility matrix isobtained by using cross-section properties on the integrationpoints Therefore effects of shape functions on the solutionare very important In this study an element is divided

4 The Scientific World Journal

Loading point

SD345D6

D1012

80m

m

900

450

24320

SD345

D6

D10

24

320

1265

Figure 3 Experimental setup and geometrical properties of RCcolumn

into subelements which are called ldquosegmentrdquo to removedisadvantage effects of the shape functions The freedoms ofsegment are condensed to the end freedoms at the ends ofthe element Thus if freedoms at ends and internal regionsof element are called as external and internal freedomsrespectively (Figure 2) these external and internal freedomsfor stiffness matrices can be written as

[119870119864119864] 119906119864 + [119870

119864119868] 119906119868 = 119865

119864 (10a)

[119870119868119864] 119906119864 + [119870

119868119868] 119906119868 = 119865

119868 (10b)

where [119870119864119864] and [119870

119868119868] are stiffness matrices which include

external and internal freedoms respectively [11] 119906119864 and

119906119868 are the displacement vectors referred by these freedom

119865119864 and 119865

119868 are also the external loads of these freedoms

When applying the substructure procedure to (10a) and (10b)the external load and stiffnessmatrices of a frame element canbe rewritten as

[119870119864119864119865

] 119906119864 = 119865

119864119865 (11a)

[119870119864119864119865

] = [119870119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119870119868119864] (11b)

119865119864119865 = 119865

119864 minus [119870

119864119868] [119870119868119868]minus1

119865119868 (11c)

However if the Rayleigh method may be used to obtainelement damping matrices this matrix is defined as

[

[119862119864119864] [119862119864119868]

[119862119868119864] [119862119868119868]]

= 120572dam [[119872119864119864] [119872

119864119868]

[119872119868119864] [119872

119868119868]] + 120573dam [

[119870119864119864] [119870119864119868]

[119870119868119864] [119870119868119868]]

(12)

where 120572dam and 120573dam are Rayleigh damping coefficients withrespect to mass and stiffness matrices respectively [14]

However if substructure procedure in (11a) (11b) and (11c)is applied to mass and damping matrices [15]

[119872119864119864119865

] = [119872119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119872119868119864]

minus [119872119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119872119868119868] [119870119868119868]minus1

[119870119868119864]

(13a)

[119862119864119864119865

] = [119862119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119862119868119864]

minus [119862119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119862119868119868] [119870119868119868]minus1

[119870119868119864]

(13b)

mass and damping matrices can be obtained for the externalfreedom Global stiffness damping and mass matrices maybe achieved by using the stiffness and mass matrices belong-ing to external freedomThus global stiffness mass matricesand external load vector can be calculated as

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

] [119862119878] =

nelemsum

119896=1

[119862119864119864119865

]

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

] 119865119878 =

nelemsum

119896=1

119865119864119865

(14)

Therefore the dynamic equilibrium equations of the struc-ture may be given as

[119872119878] 119886119864119905+Δ119905

+ [119862119878] V119864119905+Δ119905

+ [119870119878] 119906119864119905+Δ119905

= 119865119878gr119905+Δ119905

+ 119865119878stat

(15)

where subscripts gr and stat indicate that the quantity isrelated to ground acceleration and static loads

23 Bossak-120572 Form of the Equation of Motion The equationof motion for the RC frame is given in (15) In this study theBossak-120572 integration method presented by Wood et al [16]is used for the solution of the equation in the time domainIntegration scheme of the method retains the Newmarkmethod Furthermore the Bossak-120572 integration method isrequired to be modified to (15) in the time domain as follows

(1 minus 120572119861) [119872119878] 119886119878119905+Δ119905

+ 120572119861[119872119878] 119886119878119905

+ [119862119878] V119878119905+Δ119905

+ 119865119878res119905+Δ119905

= (1 minus 120572119861) 119865119878gr119905+Δ119905

+ 120572119861119865119878gr119905

+ 119865119878stat

(16)

where 120572119861

is the Bossak parameter used for controllingthe numerical dissipation The Bossak parameter should bechosen as in (17) for unconditional stability and second-orderaccuracy

120572119861le

1

2

120573 =

1

4

(1 minus 120572119861)2

120574 =

1

2

minus 120572119861 (17)

The Scientific World Journal 5

ETc =120590oc

120576ce[minus120572119898(120576119888minus120576119900119888 )]

minus120576 120576c 120576oc

120590ot

120590oc

120576t120576ot

ETt =120590ot

120576te[minus120572119898(120576119905minus120576119900119905 )]120590

120576

(a)

120576

120590

minusfy

fy 120572t

120576

(b)

Figure 4 Stress-strain relationships of (a) concrete and (b) steel

Load

(kN

)

Displacement (m)

Numerical (FBEA)Experimental

80

40

0

minus40

minus80minus008 minus004 0 004 008

Figure 5 Comparison of experimental and numerical responses ofRC column

In this study 120572119861is selected as minus010 To solve the nonlinear

dynamic equation of motion for the RC frame the Newton-Raphson method is used in conjunction with the predictor-corrector technique The Bossak-120572 time integration algo-rithm is given byWood et al [16] Predicted displacement andvelocity vectors for the time step 119905 + Δ119905 are obtained by usingdisplacement and velocity vectors at the time step 119905 which isknown Thus they can be calculated as

119878119905+Δ119905

= 119906119878119905+ Δ119905V

119878119905+

1

2

Δ1199052

(1 minus 2120573) 119886119878119905+Δ119905

(18a)

V119878119905+Δ119905

= V119878119905+ Δ119905 (1 minus 120574) 119886

119878119905 (18b)

where 120573 and 120574 are Newmarkrsquos coefficients The displacementand velocity vectors [17] of the RC frame can be written interms of the predicted vectors shown in (18a) and (18b) Thevectors may be defined as

119906119878119905+Δ119905

= 119878119905+Δ119905

+ 120573Δ1199052

119886119878119905+Δ119905

(19a)

V119878119905+Δ119905

= V119878119905+Δ119905

+ Δ119905120574119886119878119905+Δ119905

(19b)

These relations can be substituted into (16) and a timemarching algorithm can be applied to this equation as givenin the Appendix

3 Numerical Applications

31 Comparison of Experimental and Numerical Analysis of aRC Column In this section experimental cyclic test resultsof a RC column with numerical solutions obtained fromproposed solution method are compared The experimentaltest result includes response under uniaxial static and lateralcyclic forces Loading and material properties of the experi-mental study are given by Takahashi [18] The experimentalsetup is shown in Figure 3 Exponential decreasing functionsfor the softening region of tensile and compressive strengthsof the concrete for the constitutive model of FBEA areused These functions are shown in Figure 4(a) The bilinearkinematic hardening rule is used for nonlinear behavior ofthe steel for the two approaches (Figure 4(b)) Static loadsare converted to masses which are condensed to the elementends for all solutions and displacement values obtaineddue to the loads being considered as the initial conditionAcceleration data a sinus wave shape and time varying areused for all dynamic solutions This dynamic load is appliedto the horizontal direction at 1280mm height of RC columnTangent stiffness matrix is used for the solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Load-displacement curves of experimental and numer-ical results are given in Figure 5 Maximum and minimumvalues of cyclic displacement obtained fromnumerical resultsare approximately between minus0055 and 0055m Numericalresponse of RC element which is obtained with FBEA isshown as similar to envelope curve of experimental resultsAll values of the horizontal displacements are on the envelopecurve of the experimental result It is said that this solutiontechnique and material models of concrete and steel can beused for the solution of the RC structural element under thedynamic loading

32 Nonlinear Dynamic Analyses of a Portal RC FrameStructure In this section nonlinear dynamic analyses of aportal RC frame are obtained for the comparing of proposedsolution technique with model based on flexibility Seismo-Structure program [19] is used for the flexibility based solu-tions (with fibre element method) Finite element meshes areshown in Figure 6 for both approaches Finite element meshof proposed method is divided into segment for comparingwith analysis results of Seismo-Structure Program ACI 318-02 [20] code is used for the material properties of concrete

6 The Scientific World Journal

1 2

3 4300

0

6000mm

1

3

2

(a)

1

2

3 4

5

6

7 8

9 1011 12 13

1

3

4

11 12

8

7

6

5

9

10

2

(b)

Figure 6 Finite element mesh for (a) FEBA and (b) FEA

15

1

05

0

minus05

minus1

minus150 02 04 06 08 1 12 14 16 18 2

Time (s)

Ag

Figure 7 Time history graphs of sinus acceleration

and steel Cross-section and material properties of the beamand column of selected portal RC frame are given in Table 1The tensile softening region in the fibre element method isnot taken into account while the compressive behavior of theconcrete is used In Seismo-Structure program if the tensilestresses obtained from the solutions reach the tensile strengthof the concrete tensile strength suddenly drops to zero In thiscase a sudden collapse of the structure occurs

In this study exponential decreasing functions in thesoftening region of tensile and compressive strengths of theconcrete for constitutive model of FBEA are used Thesefunctions are shown in Figure 4(a) The bilinear kinematichardening rule is used for nonlinear behavior of the steel forthe two approaches (Figure 4(b)) Static loads are convertedto masses which are condensed to the element ends for allsolutions and displacement values obtained due to the loadsbeing considered as the initial condition Acceleration dataand a sinus wave shape are used for all dynamic solutions(Figure 7) This dynamic load is applied to the horizontaldirection

Displacement time history graphs of node 3 obtainedfrom FBEA and FEA are shown in Figure 8 Displacementtime amplitude values are approximately similar until time of075 sec and after this time displacement amplitude values

Disp

lace

men

t (m

m)

Time (s)

FEAEFBEA

4

3

2

1

0

minus1

minus2

minus3

minus40 02 04 06 08 1 12 14 16 18

Figure 8 Displacement time history graphs of node 3 for FEA andFBEA

obtained from the FEA are bigger than those obtained fromthe FBEA This case arises from not taking into accountthe concrete tensile softening region for the fibre elementmethod However failure of structure is not seen for twoapproaches in all times

Accumulated tensile damage cases obtained for bothapproaches are given in Figure 9 The first damage zones areobtained at the whole of end regions of the beams and atoutside surfaces of bottom regions of the columns for theFBEA Damages are shown at the whole cross-section ofbottom regions of the columns for the FEA Furthermoreobtained damage zone regions in the beam and column forthe FBEA and FEA are similar at 119905 = 10 sec However somedifferences between both approaches are seen from the pointof intensities of the damage at this time However damageintensities at end regions of the beam according to FBEAare extended and damage regions are propagated to middleregion of the beam at the 119905 = 15 sec Damage intensitiesat end regions of the column are also extended and damageregions are propagated tomiddle region fromupper region ofthe column at the 119905 = 15 sec Additionally damage intensities

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 The Scientific World Journal

loads Cubic Hermitian polynomials for transversal displace-ments and the linear shape functions for axial displacementsare used for obtaining of stiffnessmatrixHowever theGauss-Lobatto integration rule was used for obtaining element massand stiffness matrices Element damage location was simplypresented and element was not divided into subelementcalled as ldquosegmentrdquo Some calculation problems in heavilydamage location was shown and they are reported to arisefrom flexural deformation Furthermore strain rate effects inresponse to reinforced concrete frames was investigated byIribarren [7] Used formulations are based on rigiditymethodand a strain rate dependentmaterial formulation is developedfor both the concrete and steel constitutive response Brum[8] was obtained nonlinear dynamic analyses of frame andmasonry structures by using a method which was based onstiffness matrix In the research a uniaxial constitutivemodelfor concrete and masonry is proposed for compression andtension regions of the materials under the cyclic loadings

In this study a beam-column element based on theEuler-Bernoulli beam theory is presented for the fiber RCelement Element stiffness matrices are obtained by usingrigidity methodThe beam or column element is divided intosubelements called ldquosegmentrdquo [11] Furthermore the internalfreedoms of this segment are dynamically condensed toexternal freedoms at the end of the element Thus nonlineardynamic analysis of high RC building can be obtained withina short time This procedure requires that nested loops ofthe element and structure are obtained at the same time (theso-called nonlinear dynamic substructure) However thiscondensation procedure is not used in the modeling of thefibre element approach (FEA) of the RC element In additionuniform or trapezoidal loads of the segment are assumed tobe zero or condensed to the external freedoms at the end ofthe element in the FEA [11] This case is not preferred dueto the required redistribution of loading However in thisstudy external loading of the segment is taken into accountby considering the damage occurred in the element

The present research is organized as follows (i) pre-senting of fiber Bernoulli-Euler beam-column element basedon rigidity method (ii) developing of nonlinear dynamicsubstructure technique of RC frames (iii) verification of theconstitutive model with respect to experimental result of acolumn structural element and dynamic analysis results offibre RC element of a portal frame (iv) obtaining of seismicdamage analyses of an 8-story RC frame with soft-story fordistributedlumped mass and load case and (v) results

2 Fiber and Bernoulli-Euler Approach(FBEA) for Reinforced Concrete BeamColumn Element

In this section theory of Fiber and Bernoulli-Euler elementwhich is based on stiffness matrix is firstly presented forreinforced concrete section Elements are divided into subele-ments called segment and they are assumed as substructuresof element Furthermore cross-section of the each segmentis also subdivided into a number of fiberslayers In the

next section nonlinear dynamic substructuresmethod is alsomentioned to be applied to freedoms in element ends

21 Obtaining Segment Stiffness andMassMatrices Thestraindistribution on a cross-section of beam is assumed to beuniform due to axial forces and linear due to only bendingaccording to the Bernoulli-Euler approach Furthermore if aplane section before bending is plane after bending and if thestrain is assumed to be small and shear stresses are omittedthe strain of a point on a cross-section in the axial directioncan be written as

120576119909119909=

119889119906

119889119909

minus

1198892V

1198891199092119910 (1)

where 119906 and V are the displacement of the axial and verticaldirections of element respectively (Figure 1) However ifcross-section of element is divided into fiberslayers thisequation for each fiberlayer in local axis can be rewritten as

120576120585119899= 1 minus120578

1015840

119899

119889119906

119889120585

1198892V1198891205852

(2)

where 119889120576120585119899

is incremental axial strain of a fiberlayer ona segment in 120585 local axis direction Therefore if the cubicHermitian polynomials for transversal displacements and thelinear shape functions for axial displacements are used [12](2) can be written as

120576120585119899= 1 minus120578

1015840

119899

119889 [119873 (120585)]

119889120585

1198892

[119873 (120585)]

1198891205852

119902 = 1 minus1205781015840

119899 [119861] 119902 (3)

where 119902 is displacement vector including displacementand rotation of a segment [119861] is strain-displacement matrixThus incremental stress in each fiberlayer can obtained as

119889120590120585119899= 119864119879119899119889120576120585119899= 1198641198791198991 minus120578

1015840

119899 [119861] 119902 (4)

where 119864119879119899 determined by using uniaxial stress-strain rela-

tionship of using materials is tangent elasticity modulusof each fiber Therefore total strain energy of a segmentsubelement is obtained as

Π =

1

2

int

119871Seg

120590119879

120576119860Seg119889119909

=

1

2

int

1

minus1

119902119879

[119861]119879

[

1

minus1205781015840

119899

119864119879119899119860 1 minus120578

1015840

119899] [119861] 119902 119889120585

=

1

2

int

1

minus1

119902119879

[119861]119879

times

[

[

[

[

119873

sum

119899=1

119864119879119899119860119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

119873

sum

119899=1

119864119879119899119860119899(1205781015840

119899)

2

]

]

]

]

[119861] 119902 119889120585

(5)

The Scientific World Journal 3

y

120585 u

120578

Columnelement Beam element

Concrete fibers Steel fibers

120585 u

120578

120585 u

120578

z

x120577 w

120577 w

(a)

120590

120576

+=

Concrete fibers

Confined

Unconfined

Steel fibers

120590

120576

(b)

Figure 1 Concrete and steel fibers and global and local axes for FBEA

where 119871Seg and 119860Seg are expressed to be length and cross-section area of segment respectively 119860

119899is also area of a

fiberlayer Thus element stiffness matrix can be obtained byusing minimum potential energy principle Stiffness matrixof a segment can be written as

[119870Seg]

=

1

2

int

1

minus1

[119861]119879[

[

[

[

119873

sum

119899=1

119864119879119899119860119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

119873

sum

119899=1

119864119879119899119860119899(1205781015840

119899)

2

]

]

]

]

[119861] 119889120585

(6)

The segment stiffness matrix is obtained by using areascoordinates and tangent elasticity modulus of fiberlayer forreinforced concrete sections In this study linear superpo-sition rule is used for different material properties of thefiberlayer in the section (Figure 1) Furthermore a fiberconcrete or reinforced bar on the cross-section may becracked or damaged due to external loading For this reasona relationship between the damaged and undamaged casesmust be obtained for the solutions Thus damage in eachconcretereinforcement fiber can be written as [13]

119889119899= 1 minus

119864119879119899

119864119874119899

(7)

where 119864119874119899

and 119864119879119899

are undamaged and damagedtangentelasticity module of the 119899th fiber respectively 119889

119899 damage

intensities are obtained separately under tensile and compres-sive stress for each fiber

However total kinetic energy of particle velocities on thecross-section of an element throughout its neutral axis can bewritten as

119879 =

1

2

int

Ω

V120585120588V120585119889Ω (8)

Segment

External node Internal nodes

y

External node

Segment Segment Segment

z

x

Figure 2 Segments and external and internal nodes for FBEA

where120588 ismass density and V120585is particle velocity If fiberlayer

elements are used for (8) the mass matrices of an element onthe local axis are obtained as

[119872Seg] = int1

minus1

119899119891119894119887

sum

119899=1

([119873 (120585)]119879

120588119899119860119899[119873 (120585)]) 119889120585 (9)

where [119873(120585)] is the element shape functions matrix on thelocal axis and 120588

119899is also mass density of the 119899th fiber Stiffness

and mass matrices of a segment obtained by local axis aretransformed to global axes by using transformation matrix

22 Obtaining with Nonlinear Dynamic Substructures Tech-nique of Element Stiffness and Mass Matrices In FBEA cubicHermitian shape functions for rotation and shear strain andlinear shape functions for the axial deformation are usedrespectively The Gauss-Lobatto integration rule is generallyused for obtaining element mass and stiffness matrices alongwith 120585 local axis [6ndash8] However substructure procedures aregenerally preferred for solutions of high buildings Howeverthis technique in the previous study is not used due tonumerical difficultly Element stiffnessflexibility matrix isobtained by using cross-section properties on the integrationpoints Therefore effects of shape functions on the solutionare very important In this study an element is divided

4 The Scientific World Journal

Loading point

SD345D6

D1012

80m

m

900

450

24320

SD345

D6

D10

24

320

1265

Figure 3 Experimental setup and geometrical properties of RCcolumn

into subelements which are called ldquosegmentrdquo to removedisadvantage effects of the shape functions The freedoms ofsegment are condensed to the end freedoms at the ends ofthe element Thus if freedoms at ends and internal regionsof element are called as external and internal freedomsrespectively (Figure 2) these external and internal freedomsfor stiffness matrices can be written as

[119870119864119864] 119906119864 + [119870

119864119868] 119906119868 = 119865

119864 (10a)

[119870119868119864] 119906119864 + [119870

119868119868] 119906119868 = 119865

119868 (10b)

where [119870119864119864] and [119870

119868119868] are stiffness matrices which include

external and internal freedoms respectively [11] 119906119864 and

119906119868 are the displacement vectors referred by these freedom

119865119864 and 119865

119868 are also the external loads of these freedoms

When applying the substructure procedure to (10a) and (10b)the external load and stiffnessmatrices of a frame element canbe rewritten as

[119870119864119864119865

] 119906119864 = 119865

119864119865 (11a)

[119870119864119864119865

] = [119870119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119870119868119864] (11b)

119865119864119865 = 119865

119864 minus [119870

119864119868] [119870119868119868]minus1

119865119868 (11c)

However if the Rayleigh method may be used to obtainelement damping matrices this matrix is defined as

[

[119862119864119864] [119862119864119868]

[119862119868119864] [119862119868119868]]

= 120572dam [[119872119864119864] [119872

119864119868]

[119872119868119864] [119872

119868119868]] + 120573dam [

[119870119864119864] [119870119864119868]

[119870119868119864] [119870119868119868]]

(12)

where 120572dam and 120573dam are Rayleigh damping coefficients withrespect to mass and stiffness matrices respectively [14]

However if substructure procedure in (11a) (11b) and (11c)is applied to mass and damping matrices [15]

[119872119864119864119865

] = [119872119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119872119868119864]

minus [119872119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119872119868119868] [119870119868119868]minus1

[119870119868119864]

(13a)

[119862119864119864119865

] = [119862119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119862119868119864]

minus [119862119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119862119868119868] [119870119868119868]minus1

[119870119868119864]

(13b)

mass and damping matrices can be obtained for the externalfreedom Global stiffness damping and mass matrices maybe achieved by using the stiffness and mass matrices belong-ing to external freedomThus global stiffness mass matricesand external load vector can be calculated as

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

] [119862119878] =

nelemsum

119896=1

[119862119864119864119865

]

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

] 119865119878 =

nelemsum

119896=1

119865119864119865

(14)

Therefore the dynamic equilibrium equations of the struc-ture may be given as

[119872119878] 119886119864119905+Δ119905

+ [119862119878] V119864119905+Δ119905

+ [119870119878] 119906119864119905+Δ119905

= 119865119878gr119905+Δ119905

+ 119865119878stat

(15)

where subscripts gr and stat indicate that the quantity isrelated to ground acceleration and static loads

23 Bossak-120572 Form of the Equation of Motion The equationof motion for the RC frame is given in (15) In this study theBossak-120572 integration method presented by Wood et al [16]is used for the solution of the equation in the time domainIntegration scheme of the method retains the Newmarkmethod Furthermore the Bossak-120572 integration method isrequired to be modified to (15) in the time domain as follows

(1 minus 120572119861) [119872119878] 119886119878119905+Δ119905

+ 120572119861[119872119878] 119886119878119905

+ [119862119878] V119878119905+Δ119905

+ 119865119878res119905+Δ119905

= (1 minus 120572119861) 119865119878gr119905+Δ119905

+ 120572119861119865119878gr119905

+ 119865119878stat

(16)

where 120572119861

is the Bossak parameter used for controllingthe numerical dissipation The Bossak parameter should bechosen as in (17) for unconditional stability and second-orderaccuracy

120572119861le

1

2

120573 =

1

4

(1 minus 120572119861)2

120574 =

1

2

minus 120572119861 (17)

The Scientific World Journal 5

ETc =120590oc

120576ce[minus120572119898(120576119888minus120576119900119888 )]

minus120576 120576c 120576oc

120590ot

120590oc

120576t120576ot

ETt =120590ot

120576te[minus120572119898(120576119905minus120576119900119905 )]120590

120576

(a)

120576

120590

minusfy

fy 120572t

120576

(b)

Figure 4 Stress-strain relationships of (a) concrete and (b) steel

Load

(kN

)

Displacement (m)

Numerical (FBEA)Experimental

80

40

0

minus40

minus80minus008 minus004 0 004 008

Figure 5 Comparison of experimental and numerical responses ofRC column

In this study 120572119861is selected as minus010 To solve the nonlinear

dynamic equation of motion for the RC frame the Newton-Raphson method is used in conjunction with the predictor-corrector technique The Bossak-120572 time integration algo-rithm is given byWood et al [16] Predicted displacement andvelocity vectors for the time step 119905 + Δ119905 are obtained by usingdisplacement and velocity vectors at the time step 119905 which isknown Thus they can be calculated as

119878119905+Δ119905

= 119906119878119905+ Δ119905V

119878119905+

1

2

Δ1199052

(1 minus 2120573) 119886119878119905+Δ119905

(18a)

V119878119905+Δ119905

= V119878119905+ Δ119905 (1 minus 120574) 119886

119878119905 (18b)

where 120573 and 120574 are Newmarkrsquos coefficients The displacementand velocity vectors [17] of the RC frame can be written interms of the predicted vectors shown in (18a) and (18b) Thevectors may be defined as

119906119878119905+Δ119905

= 119878119905+Δ119905

+ 120573Δ1199052

119886119878119905+Δ119905

(19a)

V119878119905+Δ119905

= V119878119905+Δ119905

+ Δ119905120574119886119878119905+Δ119905

(19b)

These relations can be substituted into (16) and a timemarching algorithm can be applied to this equation as givenin the Appendix

3 Numerical Applications

31 Comparison of Experimental and Numerical Analysis of aRC Column In this section experimental cyclic test resultsof a RC column with numerical solutions obtained fromproposed solution method are compared The experimentaltest result includes response under uniaxial static and lateralcyclic forces Loading and material properties of the experi-mental study are given by Takahashi [18] The experimentalsetup is shown in Figure 3 Exponential decreasing functionsfor the softening region of tensile and compressive strengthsof the concrete for the constitutive model of FBEA areused These functions are shown in Figure 4(a) The bilinearkinematic hardening rule is used for nonlinear behavior ofthe steel for the two approaches (Figure 4(b)) Static loadsare converted to masses which are condensed to the elementends for all solutions and displacement values obtaineddue to the loads being considered as the initial conditionAcceleration data a sinus wave shape and time varying areused for all dynamic solutions This dynamic load is appliedto the horizontal direction at 1280mm height of RC columnTangent stiffness matrix is used for the solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Load-displacement curves of experimental and numer-ical results are given in Figure 5 Maximum and minimumvalues of cyclic displacement obtained fromnumerical resultsare approximately between minus0055 and 0055m Numericalresponse of RC element which is obtained with FBEA isshown as similar to envelope curve of experimental resultsAll values of the horizontal displacements are on the envelopecurve of the experimental result It is said that this solutiontechnique and material models of concrete and steel can beused for the solution of the RC structural element under thedynamic loading

32 Nonlinear Dynamic Analyses of a Portal RC FrameStructure In this section nonlinear dynamic analyses of aportal RC frame are obtained for the comparing of proposedsolution technique with model based on flexibility Seismo-Structure program [19] is used for the flexibility based solu-tions (with fibre element method) Finite element meshes areshown in Figure 6 for both approaches Finite element meshof proposed method is divided into segment for comparingwith analysis results of Seismo-Structure Program ACI 318-02 [20] code is used for the material properties of concrete

6 The Scientific World Journal

1 2

3 4300

0

6000mm

1

3

2

(a)

1

2

3 4

5

6

7 8

9 1011 12 13

1

3

4

11 12

8

7

6

5

9

10

2

(b)

Figure 6 Finite element mesh for (a) FEBA and (b) FEA

15

1

05

0

minus05

minus1

minus150 02 04 06 08 1 12 14 16 18 2

Time (s)

Ag

Figure 7 Time history graphs of sinus acceleration

and steel Cross-section and material properties of the beamand column of selected portal RC frame are given in Table 1The tensile softening region in the fibre element method isnot taken into account while the compressive behavior of theconcrete is used In Seismo-Structure program if the tensilestresses obtained from the solutions reach the tensile strengthof the concrete tensile strength suddenly drops to zero In thiscase a sudden collapse of the structure occurs

In this study exponential decreasing functions in thesoftening region of tensile and compressive strengths of theconcrete for constitutive model of FBEA are used Thesefunctions are shown in Figure 4(a) The bilinear kinematichardening rule is used for nonlinear behavior of the steel forthe two approaches (Figure 4(b)) Static loads are convertedto masses which are condensed to the element ends for allsolutions and displacement values obtained due to the loadsbeing considered as the initial condition Acceleration dataand a sinus wave shape are used for all dynamic solutions(Figure 7) This dynamic load is applied to the horizontaldirection

Displacement time history graphs of node 3 obtainedfrom FBEA and FEA are shown in Figure 8 Displacementtime amplitude values are approximately similar until time of075 sec and after this time displacement amplitude values

Disp

lace

men

t (m

m)

Time (s)

FEAEFBEA

4

3

2

1

0

minus1

minus2

minus3

minus40 02 04 06 08 1 12 14 16 18

Figure 8 Displacement time history graphs of node 3 for FEA andFBEA

obtained from the FEA are bigger than those obtained fromthe FBEA This case arises from not taking into accountthe concrete tensile softening region for the fibre elementmethod However failure of structure is not seen for twoapproaches in all times

Accumulated tensile damage cases obtained for bothapproaches are given in Figure 9 The first damage zones areobtained at the whole of end regions of the beams and atoutside surfaces of bottom regions of the columns for theFBEA Damages are shown at the whole cross-section ofbottom regions of the columns for the FEA Furthermoreobtained damage zone regions in the beam and column forthe FBEA and FEA are similar at 119905 = 10 sec However somedifferences between both approaches are seen from the pointof intensities of the damage at this time However damageintensities at end regions of the beam according to FBEAare extended and damage regions are propagated to middleregion of the beam at the 119905 = 15 sec Damage intensitiesat end regions of the column are also extended and damageregions are propagated tomiddle region fromupper region ofthe column at the 119905 = 15 sec Additionally damage intensities

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 3

y

120585 u

120578

Columnelement Beam element

Concrete fibers Steel fibers

120585 u

120578

120585 u

120578

z

x120577 w

120577 w

(a)

120590

120576

+=

Concrete fibers

Confined

Unconfined

Steel fibers

120590

120576

(b)

Figure 1 Concrete and steel fibers and global and local axes for FBEA

where 119871Seg and 119860Seg are expressed to be length and cross-section area of segment respectively 119860

119899is also area of a

fiberlayer Thus element stiffness matrix can be obtained byusing minimum potential energy principle Stiffness matrixof a segment can be written as

[119870Seg]

=

1

2

int

1

minus1

[119861]119879[

[

[

[

119873

sum

119899=1

119864119879119899119860119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

minus

119873

sum

119899=1

1198641198791198991198601198991205781015840

119899

119873

sum

119899=1

119864119879119899119860119899(1205781015840

119899)

2

]

]

]

]

[119861] 119889120585

(6)

The segment stiffness matrix is obtained by using areascoordinates and tangent elasticity modulus of fiberlayer forreinforced concrete sections In this study linear superpo-sition rule is used for different material properties of thefiberlayer in the section (Figure 1) Furthermore a fiberconcrete or reinforced bar on the cross-section may becracked or damaged due to external loading For this reasona relationship between the damaged and undamaged casesmust be obtained for the solutions Thus damage in eachconcretereinforcement fiber can be written as [13]

119889119899= 1 minus

119864119879119899

119864119874119899

(7)

where 119864119874119899

and 119864119879119899

are undamaged and damagedtangentelasticity module of the 119899th fiber respectively 119889

119899 damage

intensities are obtained separately under tensile and compres-sive stress for each fiber

However total kinetic energy of particle velocities on thecross-section of an element throughout its neutral axis can bewritten as

119879 =

1

2

int

Ω

V120585120588V120585119889Ω (8)

Segment

External node Internal nodes

y

External node

Segment Segment Segment

z

x

Figure 2 Segments and external and internal nodes for FBEA

where120588 ismass density and V120585is particle velocity If fiberlayer

elements are used for (8) the mass matrices of an element onthe local axis are obtained as

[119872Seg] = int1

minus1

119899119891119894119887

sum

119899=1

([119873 (120585)]119879

120588119899119860119899[119873 (120585)]) 119889120585 (9)

where [119873(120585)] is the element shape functions matrix on thelocal axis and 120588

119899is also mass density of the 119899th fiber Stiffness

and mass matrices of a segment obtained by local axis aretransformed to global axes by using transformation matrix

22 Obtaining with Nonlinear Dynamic Substructures Tech-nique of Element Stiffness and Mass Matrices In FBEA cubicHermitian shape functions for rotation and shear strain andlinear shape functions for the axial deformation are usedrespectively The Gauss-Lobatto integration rule is generallyused for obtaining element mass and stiffness matrices alongwith 120585 local axis [6ndash8] However substructure procedures aregenerally preferred for solutions of high buildings Howeverthis technique in the previous study is not used due tonumerical difficultly Element stiffnessflexibility matrix isobtained by using cross-section properties on the integrationpoints Therefore effects of shape functions on the solutionare very important In this study an element is divided

4 The Scientific World Journal

Loading point

SD345D6

D1012

80m

m

900

450

24320

SD345

D6

D10

24

320

1265

Figure 3 Experimental setup and geometrical properties of RCcolumn

into subelements which are called ldquosegmentrdquo to removedisadvantage effects of the shape functions The freedoms ofsegment are condensed to the end freedoms at the ends ofthe element Thus if freedoms at ends and internal regionsof element are called as external and internal freedomsrespectively (Figure 2) these external and internal freedomsfor stiffness matrices can be written as

[119870119864119864] 119906119864 + [119870

119864119868] 119906119868 = 119865

119864 (10a)

[119870119868119864] 119906119864 + [119870

119868119868] 119906119868 = 119865

119868 (10b)

where [119870119864119864] and [119870

119868119868] are stiffness matrices which include

external and internal freedoms respectively [11] 119906119864 and

119906119868 are the displacement vectors referred by these freedom

119865119864 and 119865

119868 are also the external loads of these freedoms

When applying the substructure procedure to (10a) and (10b)the external load and stiffnessmatrices of a frame element canbe rewritten as

[119870119864119864119865

] 119906119864 = 119865

119864119865 (11a)

[119870119864119864119865

] = [119870119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119870119868119864] (11b)

119865119864119865 = 119865

119864 minus [119870

119864119868] [119870119868119868]minus1

119865119868 (11c)

However if the Rayleigh method may be used to obtainelement damping matrices this matrix is defined as

[

[119862119864119864] [119862119864119868]

[119862119868119864] [119862119868119868]]

= 120572dam [[119872119864119864] [119872

119864119868]

[119872119868119864] [119872

119868119868]] + 120573dam [

[119870119864119864] [119870119864119868]

[119870119868119864] [119870119868119868]]

(12)

where 120572dam and 120573dam are Rayleigh damping coefficients withrespect to mass and stiffness matrices respectively [14]

However if substructure procedure in (11a) (11b) and (11c)is applied to mass and damping matrices [15]

[119872119864119864119865

] = [119872119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119872119868119864]

minus [119872119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119872119868119868] [119870119868119868]minus1

[119870119868119864]

(13a)

[119862119864119864119865

] = [119862119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119862119868119864]

minus [119862119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119862119868119868] [119870119868119868]minus1

[119870119868119864]

(13b)

mass and damping matrices can be obtained for the externalfreedom Global stiffness damping and mass matrices maybe achieved by using the stiffness and mass matrices belong-ing to external freedomThus global stiffness mass matricesand external load vector can be calculated as

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

] [119862119878] =

nelemsum

119896=1

[119862119864119864119865

]

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

] 119865119878 =

nelemsum

119896=1

119865119864119865

(14)

Therefore the dynamic equilibrium equations of the struc-ture may be given as

[119872119878] 119886119864119905+Δ119905

+ [119862119878] V119864119905+Δ119905

+ [119870119878] 119906119864119905+Δ119905

= 119865119878gr119905+Δ119905

+ 119865119878stat

(15)

where subscripts gr and stat indicate that the quantity isrelated to ground acceleration and static loads

23 Bossak-120572 Form of the Equation of Motion The equationof motion for the RC frame is given in (15) In this study theBossak-120572 integration method presented by Wood et al [16]is used for the solution of the equation in the time domainIntegration scheme of the method retains the Newmarkmethod Furthermore the Bossak-120572 integration method isrequired to be modified to (15) in the time domain as follows

(1 minus 120572119861) [119872119878] 119886119878119905+Δ119905

+ 120572119861[119872119878] 119886119878119905

+ [119862119878] V119878119905+Δ119905

+ 119865119878res119905+Δ119905

= (1 minus 120572119861) 119865119878gr119905+Δ119905

+ 120572119861119865119878gr119905

+ 119865119878stat

(16)

where 120572119861

is the Bossak parameter used for controllingthe numerical dissipation The Bossak parameter should bechosen as in (17) for unconditional stability and second-orderaccuracy

120572119861le

1

2

120573 =

1

4

(1 minus 120572119861)2

120574 =

1

2

minus 120572119861 (17)

The Scientific World Journal 5

ETc =120590oc

120576ce[minus120572119898(120576119888minus120576119900119888 )]

minus120576 120576c 120576oc

120590ot

120590oc

120576t120576ot

ETt =120590ot

120576te[minus120572119898(120576119905minus120576119900119905 )]120590

120576

(a)

120576

120590

minusfy

fy 120572t

120576

(b)

Figure 4 Stress-strain relationships of (a) concrete and (b) steel

Load

(kN

)

Displacement (m)

Numerical (FBEA)Experimental

80

40

0

minus40

minus80minus008 minus004 0 004 008

Figure 5 Comparison of experimental and numerical responses ofRC column

In this study 120572119861is selected as minus010 To solve the nonlinear

dynamic equation of motion for the RC frame the Newton-Raphson method is used in conjunction with the predictor-corrector technique The Bossak-120572 time integration algo-rithm is given byWood et al [16] Predicted displacement andvelocity vectors for the time step 119905 + Δ119905 are obtained by usingdisplacement and velocity vectors at the time step 119905 which isknown Thus they can be calculated as

119878119905+Δ119905

= 119906119878119905+ Δ119905V

119878119905+

1

2

Δ1199052

(1 minus 2120573) 119886119878119905+Δ119905

(18a)

V119878119905+Δ119905

= V119878119905+ Δ119905 (1 minus 120574) 119886

119878119905 (18b)

where 120573 and 120574 are Newmarkrsquos coefficients The displacementand velocity vectors [17] of the RC frame can be written interms of the predicted vectors shown in (18a) and (18b) Thevectors may be defined as

119906119878119905+Δ119905

= 119878119905+Δ119905

+ 120573Δ1199052

119886119878119905+Δ119905

(19a)

V119878119905+Δ119905

= V119878119905+Δ119905

+ Δ119905120574119886119878119905+Δ119905

(19b)

These relations can be substituted into (16) and a timemarching algorithm can be applied to this equation as givenin the Appendix

3 Numerical Applications

31 Comparison of Experimental and Numerical Analysis of aRC Column In this section experimental cyclic test resultsof a RC column with numerical solutions obtained fromproposed solution method are compared The experimentaltest result includes response under uniaxial static and lateralcyclic forces Loading and material properties of the experi-mental study are given by Takahashi [18] The experimentalsetup is shown in Figure 3 Exponential decreasing functionsfor the softening region of tensile and compressive strengthsof the concrete for the constitutive model of FBEA areused These functions are shown in Figure 4(a) The bilinearkinematic hardening rule is used for nonlinear behavior ofthe steel for the two approaches (Figure 4(b)) Static loadsare converted to masses which are condensed to the elementends for all solutions and displacement values obtaineddue to the loads being considered as the initial conditionAcceleration data a sinus wave shape and time varying areused for all dynamic solutions This dynamic load is appliedto the horizontal direction at 1280mm height of RC columnTangent stiffness matrix is used for the solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Load-displacement curves of experimental and numer-ical results are given in Figure 5 Maximum and minimumvalues of cyclic displacement obtained fromnumerical resultsare approximately between minus0055 and 0055m Numericalresponse of RC element which is obtained with FBEA isshown as similar to envelope curve of experimental resultsAll values of the horizontal displacements are on the envelopecurve of the experimental result It is said that this solutiontechnique and material models of concrete and steel can beused for the solution of the RC structural element under thedynamic loading

32 Nonlinear Dynamic Analyses of a Portal RC FrameStructure In this section nonlinear dynamic analyses of aportal RC frame are obtained for the comparing of proposedsolution technique with model based on flexibility Seismo-Structure program [19] is used for the flexibility based solu-tions (with fibre element method) Finite element meshes areshown in Figure 6 for both approaches Finite element meshof proposed method is divided into segment for comparingwith analysis results of Seismo-Structure Program ACI 318-02 [20] code is used for the material properties of concrete

6 The Scientific World Journal

1 2

3 4300

0

6000mm

1

3

2

(a)

1

2

3 4

5

6

7 8

9 1011 12 13

1

3

4

11 12

8

7

6

5

9

10

2

(b)

Figure 6 Finite element mesh for (a) FEBA and (b) FEA

15

1

05

0

minus05

minus1

minus150 02 04 06 08 1 12 14 16 18 2

Time (s)

Ag

Figure 7 Time history graphs of sinus acceleration

and steel Cross-section and material properties of the beamand column of selected portal RC frame are given in Table 1The tensile softening region in the fibre element method isnot taken into account while the compressive behavior of theconcrete is used In Seismo-Structure program if the tensilestresses obtained from the solutions reach the tensile strengthof the concrete tensile strength suddenly drops to zero In thiscase a sudden collapse of the structure occurs

In this study exponential decreasing functions in thesoftening region of tensile and compressive strengths of theconcrete for constitutive model of FBEA are used Thesefunctions are shown in Figure 4(a) The bilinear kinematichardening rule is used for nonlinear behavior of the steel forthe two approaches (Figure 4(b)) Static loads are convertedto masses which are condensed to the element ends for allsolutions and displacement values obtained due to the loadsbeing considered as the initial condition Acceleration dataand a sinus wave shape are used for all dynamic solutions(Figure 7) This dynamic load is applied to the horizontaldirection

Displacement time history graphs of node 3 obtainedfrom FBEA and FEA are shown in Figure 8 Displacementtime amplitude values are approximately similar until time of075 sec and after this time displacement amplitude values

Disp

lace

men

t (m

m)

Time (s)

FEAEFBEA

4

3

2

1

0

minus1

minus2

minus3

minus40 02 04 06 08 1 12 14 16 18

Figure 8 Displacement time history graphs of node 3 for FEA andFBEA

obtained from the FEA are bigger than those obtained fromthe FBEA This case arises from not taking into accountthe concrete tensile softening region for the fibre elementmethod However failure of structure is not seen for twoapproaches in all times

Accumulated tensile damage cases obtained for bothapproaches are given in Figure 9 The first damage zones areobtained at the whole of end regions of the beams and atoutside surfaces of bottom regions of the columns for theFBEA Damages are shown at the whole cross-section ofbottom regions of the columns for the FEA Furthermoreobtained damage zone regions in the beam and column forthe FBEA and FEA are similar at 119905 = 10 sec However somedifferences between both approaches are seen from the pointof intensities of the damage at this time However damageintensities at end regions of the beam according to FBEAare extended and damage regions are propagated to middleregion of the beam at the 119905 = 15 sec Damage intensitiesat end regions of the column are also extended and damageregions are propagated tomiddle region fromupper region ofthe column at the 119905 = 15 sec Additionally damage intensities

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 The Scientific World Journal

Loading point

SD345D6

D1012

80m

m

900

450

24320

SD345

D6

D10

24

320

1265

Figure 3 Experimental setup and geometrical properties of RCcolumn

into subelements which are called ldquosegmentrdquo to removedisadvantage effects of the shape functions The freedoms ofsegment are condensed to the end freedoms at the ends ofthe element Thus if freedoms at ends and internal regionsof element are called as external and internal freedomsrespectively (Figure 2) these external and internal freedomsfor stiffness matrices can be written as

[119870119864119864] 119906119864 + [119870

119864119868] 119906119868 = 119865

119864 (10a)

[119870119868119864] 119906119864 + [119870

119868119868] 119906119868 = 119865

119868 (10b)

where [119870119864119864] and [119870

119868119868] are stiffness matrices which include

external and internal freedoms respectively [11] 119906119864 and

119906119868 are the displacement vectors referred by these freedom

119865119864 and 119865

119868 are also the external loads of these freedoms

When applying the substructure procedure to (10a) and (10b)the external load and stiffnessmatrices of a frame element canbe rewritten as

[119870119864119864119865

] 119906119864 = 119865

119864119865 (11a)

[119870119864119864119865

] = [119870119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119870119868119864] (11b)

119865119864119865 = 119865

119864 minus [119870

119864119868] [119870119868119868]minus1

119865119868 (11c)

However if the Rayleigh method may be used to obtainelement damping matrices this matrix is defined as

[

[119862119864119864] [119862119864119868]

[119862119868119864] [119862119868119868]]

= 120572dam [[119872119864119864] [119872

119864119868]

[119872119868119864] [119872

119868119868]] + 120573dam [

[119870119864119864] [119870119864119868]

[119870119868119864] [119870119868119868]]

(12)

where 120572dam and 120573dam are Rayleigh damping coefficients withrespect to mass and stiffness matrices respectively [14]

However if substructure procedure in (11a) (11b) and (11c)is applied to mass and damping matrices [15]

[119872119864119864119865

] = [119872119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119872119868119864]

minus [119872119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119872119868119868] [119870119868119868]minus1

[119870119868119864]

(13a)

[119862119864119864119865

] = [119862119864119864] minus [119870

119864119868] [119870119868119868]minus1

[119862119868119864]

minus [119862119864119868] [119870119868119868]minus1

[119870119868119864]

+ [119870119864119868] [119870119868119868]minus1

[119862119868119868] [119870119868119868]minus1

[119870119868119864]

(13b)

mass and damping matrices can be obtained for the externalfreedom Global stiffness damping and mass matrices maybe achieved by using the stiffness and mass matrices belong-ing to external freedomThus global stiffness mass matricesand external load vector can be calculated as

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

] [119862119878] =

nelemsum

119896=1

[119862119864119864119865

]

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

] 119865119878 =

nelemsum

119896=1

119865119864119865

(14)

Therefore the dynamic equilibrium equations of the struc-ture may be given as

[119872119878] 119886119864119905+Δ119905

+ [119862119878] V119864119905+Δ119905

+ [119870119878] 119906119864119905+Δ119905

= 119865119878gr119905+Δ119905

+ 119865119878stat

(15)

where subscripts gr and stat indicate that the quantity isrelated to ground acceleration and static loads

23 Bossak-120572 Form of the Equation of Motion The equationof motion for the RC frame is given in (15) In this study theBossak-120572 integration method presented by Wood et al [16]is used for the solution of the equation in the time domainIntegration scheme of the method retains the Newmarkmethod Furthermore the Bossak-120572 integration method isrequired to be modified to (15) in the time domain as follows

(1 minus 120572119861) [119872119878] 119886119878119905+Δ119905

+ 120572119861[119872119878] 119886119878119905

+ [119862119878] V119878119905+Δ119905

+ 119865119878res119905+Δ119905

= (1 minus 120572119861) 119865119878gr119905+Δ119905

+ 120572119861119865119878gr119905

+ 119865119878stat

(16)

where 120572119861

is the Bossak parameter used for controllingthe numerical dissipation The Bossak parameter should bechosen as in (17) for unconditional stability and second-orderaccuracy

120572119861le

1

2

120573 =

1

4

(1 minus 120572119861)2

120574 =

1

2

minus 120572119861 (17)

The Scientific World Journal 5

ETc =120590oc

120576ce[minus120572119898(120576119888minus120576119900119888 )]

minus120576 120576c 120576oc

120590ot

120590oc

120576t120576ot

ETt =120590ot

120576te[minus120572119898(120576119905minus120576119900119905 )]120590

120576

(a)

120576

120590

minusfy

fy 120572t

120576

(b)

Figure 4 Stress-strain relationships of (a) concrete and (b) steel

Load

(kN

)

Displacement (m)

Numerical (FBEA)Experimental

80

40

0

minus40

minus80minus008 minus004 0 004 008

Figure 5 Comparison of experimental and numerical responses ofRC column

In this study 120572119861is selected as minus010 To solve the nonlinear

dynamic equation of motion for the RC frame the Newton-Raphson method is used in conjunction with the predictor-corrector technique The Bossak-120572 time integration algo-rithm is given byWood et al [16] Predicted displacement andvelocity vectors for the time step 119905 + Δ119905 are obtained by usingdisplacement and velocity vectors at the time step 119905 which isknown Thus they can be calculated as

119878119905+Δ119905

= 119906119878119905+ Δ119905V

119878119905+

1

2

Δ1199052

(1 minus 2120573) 119886119878119905+Δ119905

(18a)

V119878119905+Δ119905

= V119878119905+ Δ119905 (1 minus 120574) 119886

119878119905 (18b)

where 120573 and 120574 are Newmarkrsquos coefficients The displacementand velocity vectors [17] of the RC frame can be written interms of the predicted vectors shown in (18a) and (18b) Thevectors may be defined as

119906119878119905+Δ119905

= 119878119905+Δ119905

+ 120573Δ1199052

119886119878119905+Δ119905

(19a)

V119878119905+Δ119905

= V119878119905+Δ119905

+ Δ119905120574119886119878119905+Δ119905

(19b)

These relations can be substituted into (16) and a timemarching algorithm can be applied to this equation as givenin the Appendix

3 Numerical Applications

31 Comparison of Experimental and Numerical Analysis of aRC Column In this section experimental cyclic test resultsof a RC column with numerical solutions obtained fromproposed solution method are compared The experimentaltest result includes response under uniaxial static and lateralcyclic forces Loading and material properties of the experi-mental study are given by Takahashi [18] The experimentalsetup is shown in Figure 3 Exponential decreasing functionsfor the softening region of tensile and compressive strengthsof the concrete for the constitutive model of FBEA areused These functions are shown in Figure 4(a) The bilinearkinematic hardening rule is used for nonlinear behavior ofthe steel for the two approaches (Figure 4(b)) Static loadsare converted to masses which are condensed to the elementends for all solutions and displacement values obtaineddue to the loads being considered as the initial conditionAcceleration data a sinus wave shape and time varying areused for all dynamic solutions This dynamic load is appliedto the horizontal direction at 1280mm height of RC columnTangent stiffness matrix is used for the solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Load-displacement curves of experimental and numer-ical results are given in Figure 5 Maximum and minimumvalues of cyclic displacement obtained fromnumerical resultsare approximately between minus0055 and 0055m Numericalresponse of RC element which is obtained with FBEA isshown as similar to envelope curve of experimental resultsAll values of the horizontal displacements are on the envelopecurve of the experimental result It is said that this solutiontechnique and material models of concrete and steel can beused for the solution of the RC structural element under thedynamic loading

32 Nonlinear Dynamic Analyses of a Portal RC FrameStructure In this section nonlinear dynamic analyses of aportal RC frame are obtained for the comparing of proposedsolution technique with model based on flexibility Seismo-Structure program [19] is used for the flexibility based solu-tions (with fibre element method) Finite element meshes areshown in Figure 6 for both approaches Finite element meshof proposed method is divided into segment for comparingwith analysis results of Seismo-Structure Program ACI 318-02 [20] code is used for the material properties of concrete

6 The Scientific World Journal

1 2

3 4300

0

6000mm

1

3

2

(a)

1

2

3 4

5

6

7 8

9 1011 12 13

1

3

4

11 12

8

7

6

5

9

10

2

(b)

Figure 6 Finite element mesh for (a) FEBA and (b) FEA

15

1

05

0

minus05

minus1

minus150 02 04 06 08 1 12 14 16 18 2

Time (s)

Ag

Figure 7 Time history graphs of sinus acceleration

and steel Cross-section and material properties of the beamand column of selected portal RC frame are given in Table 1The tensile softening region in the fibre element method isnot taken into account while the compressive behavior of theconcrete is used In Seismo-Structure program if the tensilestresses obtained from the solutions reach the tensile strengthof the concrete tensile strength suddenly drops to zero In thiscase a sudden collapse of the structure occurs

In this study exponential decreasing functions in thesoftening region of tensile and compressive strengths of theconcrete for constitutive model of FBEA are used Thesefunctions are shown in Figure 4(a) The bilinear kinematichardening rule is used for nonlinear behavior of the steel forthe two approaches (Figure 4(b)) Static loads are convertedto masses which are condensed to the element ends for allsolutions and displacement values obtained due to the loadsbeing considered as the initial condition Acceleration dataand a sinus wave shape are used for all dynamic solutions(Figure 7) This dynamic load is applied to the horizontaldirection

Displacement time history graphs of node 3 obtainedfrom FBEA and FEA are shown in Figure 8 Displacementtime amplitude values are approximately similar until time of075 sec and after this time displacement amplitude values

Disp

lace

men

t (m

m)

Time (s)

FEAEFBEA

4

3

2

1

0

minus1

minus2

minus3

minus40 02 04 06 08 1 12 14 16 18

Figure 8 Displacement time history graphs of node 3 for FEA andFBEA

obtained from the FEA are bigger than those obtained fromthe FBEA This case arises from not taking into accountthe concrete tensile softening region for the fibre elementmethod However failure of structure is not seen for twoapproaches in all times

Accumulated tensile damage cases obtained for bothapproaches are given in Figure 9 The first damage zones areobtained at the whole of end regions of the beams and atoutside surfaces of bottom regions of the columns for theFBEA Damages are shown at the whole cross-section ofbottom regions of the columns for the FEA Furthermoreobtained damage zone regions in the beam and column forthe FBEA and FEA are similar at 119905 = 10 sec However somedifferences between both approaches are seen from the pointof intensities of the damage at this time However damageintensities at end regions of the beam according to FBEAare extended and damage regions are propagated to middleregion of the beam at the 119905 = 15 sec Damage intensitiesat end regions of the column are also extended and damageregions are propagated tomiddle region fromupper region ofthe column at the 119905 = 15 sec Additionally damage intensities

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 5

ETc =120590oc

120576ce[minus120572119898(120576119888minus120576119900119888 )]

minus120576 120576c 120576oc

120590ot

120590oc

120576t120576ot

ETt =120590ot

120576te[minus120572119898(120576119905minus120576119900119905 )]120590

120576

(a)

120576

120590

minusfy

fy 120572t

120576

(b)

Figure 4 Stress-strain relationships of (a) concrete and (b) steel

Load

(kN

)

Displacement (m)

Numerical (FBEA)Experimental

80

40

0

minus40

minus80minus008 minus004 0 004 008

Figure 5 Comparison of experimental and numerical responses ofRC column

In this study 120572119861is selected as minus010 To solve the nonlinear

dynamic equation of motion for the RC frame the Newton-Raphson method is used in conjunction with the predictor-corrector technique The Bossak-120572 time integration algo-rithm is given byWood et al [16] Predicted displacement andvelocity vectors for the time step 119905 + Δ119905 are obtained by usingdisplacement and velocity vectors at the time step 119905 which isknown Thus they can be calculated as

119878119905+Δ119905

= 119906119878119905+ Δ119905V

119878119905+

1

2

Δ1199052

(1 minus 2120573) 119886119878119905+Δ119905

(18a)

V119878119905+Δ119905

= V119878119905+ Δ119905 (1 minus 120574) 119886

119878119905 (18b)

where 120573 and 120574 are Newmarkrsquos coefficients The displacementand velocity vectors [17] of the RC frame can be written interms of the predicted vectors shown in (18a) and (18b) Thevectors may be defined as

119906119878119905+Δ119905

= 119878119905+Δ119905

+ 120573Δ1199052

119886119878119905+Δ119905

(19a)

V119878119905+Δ119905

= V119878119905+Δ119905

+ Δ119905120574119886119878119905+Δ119905

(19b)

These relations can be substituted into (16) and a timemarching algorithm can be applied to this equation as givenin the Appendix

3 Numerical Applications

31 Comparison of Experimental and Numerical Analysis of aRC Column In this section experimental cyclic test resultsof a RC column with numerical solutions obtained fromproposed solution method are compared The experimentaltest result includes response under uniaxial static and lateralcyclic forces Loading and material properties of the experi-mental study are given by Takahashi [18] The experimentalsetup is shown in Figure 3 Exponential decreasing functionsfor the softening region of tensile and compressive strengthsof the concrete for the constitutive model of FBEA areused These functions are shown in Figure 4(a) The bilinearkinematic hardening rule is used for nonlinear behavior ofthe steel for the two approaches (Figure 4(b)) Static loadsare converted to masses which are condensed to the elementends for all solutions and displacement values obtaineddue to the loads being considered as the initial conditionAcceleration data a sinus wave shape and time varying areused for all dynamic solutions This dynamic load is appliedto the horizontal direction at 1280mm height of RC columnTangent stiffness matrix is used for the solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Load-displacement curves of experimental and numer-ical results are given in Figure 5 Maximum and minimumvalues of cyclic displacement obtained fromnumerical resultsare approximately between minus0055 and 0055m Numericalresponse of RC element which is obtained with FBEA isshown as similar to envelope curve of experimental resultsAll values of the horizontal displacements are on the envelopecurve of the experimental result It is said that this solutiontechnique and material models of concrete and steel can beused for the solution of the RC structural element under thedynamic loading

32 Nonlinear Dynamic Analyses of a Portal RC FrameStructure In this section nonlinear dynamic analyses of aportal RC frame are obtained for the comparing of proposedsolution technique with model based on flexibility Seismo-Structure program [19] is used for the flexibility based solu-tions (with fibre element method) Finite element meshes areshown in Figure 6 for both approaches Finite element meshof proposed method is divided into segment for comparingwith analysis results of Seismo-Structure Program ACI 318-02 [20] code is used for the material properties of concrete

6 The Scientific World Journal

1 2

3 4300

0

6000mm

1

3

2

(a)

1

2

3 4

5

6

7 8

9 1011 12 13

1

3

4

11 12

8

7

6

5

9

10

2

(b)

Figure 6 Finite element mesh for (a) FEBA and (b) FEA

15

1

05

0

minus05

minus1

minus150 02 04 06 08 1 12 14 16 18 2

Time (s)

Ag

Figure 7 Time history graphs of sinus acceleration

and steel Cross-section and material properties of the beamand column of selected portal RC frame are given in Table 1The tensile softening region in the fibre element method isnot taken into account while the compressive behavior of theconcrete is used In Seismo-Structure program if the tensilestresses obtained from the solutions reach the tensile strengthof the concrete tensile strength suddenly drops to zero In thiscase a sudden collapse of the structure occurs

In this study exponential decreasing functions in thesoftening region of tensile and compressive strengths of theconcrete for constitutive model of FBEA are used Thesefunctions are shown in Figure 4(a) The bilinear kinematichardening rule is used for nonlinear behavior of the steel forthe two approaches (Figure 4(b)) Static loads are convertedto masses which are condensed to the element ends for allsolutions and displacement values obtained due to the loadsbeing considered as the initial condition Acceleration dataand a sinus wave shape are used for all dynamic solutions(Figure 7) This dynamic load is applied to the horizontaldirection

Displacement time history graphs of node 3 obtainedfrom FBEA and FEA are shown in Figure 8 Displacementtime amplitude values are approximately similar until time of075 sec and after this time displacement amplitude values

Disp

lace

men

t (m

m)

Time (s)

FEAEFBEA

4

3

2

1

0

minus1

minus2

minus3

minus40 02 04 06 08 1 12 14 16 18

Figure 8 Displacement time history graphs of node 3 for FEA andFBEA

obtained from the FEA are bigger than those obtained fromthe FBEA This case arises from not taking into accountthe concrete tensile softening region for the fibre elementmethod However failure of structure is not seen for twoapproaches in all times

Accumulated tensile damage cases obtained for bothapproaches are given in Figure 9 The first damage zones areobtained at the whole of end regions of the beams and atoutside surfaces of bottom regions of the columns for theFBEA Damages are shown at the whole cross-section ofbottom regions of the columns for the FEA Furthermoreobtained damage zone regions in the beam and column forthe FBEA and FEA are similar at 119905 = 10 sec However somedifferences between both approaches are seen from the pointof intensities of the damage at this time However damageintensities at end regions of the beam according to FBEAare extended and damage regions are propagated to middleregion of the beam at the 119905 = 15 sec Damage intensitiesat end regions of the column are also extended and damageregions are propagated tomiddle region fromupper region ofthe column at the 119905 = 15 sec Additionally damage intensities

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 The Scientific World Journal

1 2

3 4300

0

6000mm

1

3

2

(a)

1

2

3 4

5

6

7 8

9 1011 12 13

1

3

4

11 12

8

7

6

5

9

10

2

(b)

Figure 6 Finite element mesh for (a) FEBA and (b) FEA

15

1

05

0

minus05

minus1

minus150 02 04 06 08 1 12 14 16 18 2

Time (s)

Ag

Figure 7 Time history graphs of sinus acceleration

and steel Cross-section and material properties of the beamand column of selected portal RC frame are given in Table 1The tensile softening region in the fibre element method isnot taken into account while the compressive behavior of theconcrete is used In Seismo-Structure program if the tensilestresses obtained from the solutions reach the tensile strengthof the concrete tensile strength suddenly drops to zero In thiscase a sudden collapse of the structure occurs

In this study exponential decreasing functions in thesoftening region of tensile and compressive strengths of theconcrete for constitutive model of FBEA are used Thesefunctions are shown in Figure 4(a) The bilinear kinematichardening rule is used for nonlinear behavior of the steel forthe two approaches (Figure 4(b)) Static loads are convertedto masses which are condensed to the element ends for allsolutions and displacement values obtained due to the loadsbeing considered as the initial condition Acceleration dataand a sinus wave shape are used for all dynamic solutions(Figure 7) This dynamic load is applied to the horizontaldirection

Displacement time history graphs of node 3 obtainedfrom FBEA and FEA are shown in Figure 8 Displacementtime amplitude values are approximately similar until time of075 sec and after this time displacement amplitude values

Disp

lace

men

t (m

m)

Time (s)

FEAEFBEA

4

3

2

1

0

minus1

minus2

minus3

minus40 02 04 06 08 1 12 14 16 18

Figure 8 Displacement time history graphs of node 3 for FEA andFBEA

obtained from the FEA are bigger than those obtained fromthe FBEA This case arises from not taking into accountthe concrete tensile softening region for the fibre elementmethod However failure of structure is not seen for twoapproaches in all times

Accumulated tensile damage cases obtained for bothapproaches are given in Figure 9 The first damage zones areobtained at the whole of end regions of the beams and atoutside surfaces of bottom regions of the columns for theFBEA Damages are shown at the whole cross-section ofbottom regions of the columns for the FEA Furthermoreobtained damage zone regions in the beam and column forthe FBEA and FEA are similar at 119905 = 10 sec However somedifferences between both approaches are seen from the pointof intensities of the damage at this time However damageintensities at end regions of the beam according to FBEAare extended and damage regions are propagated to middleregion of the beam at the 119905 = 15 sec Damage intensitiesat end regions of the column are also extended and damageregions are propagated tomiddle region fromupper region ofthe column at the 119905 = 15 sec Additionally damage intensities

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 7

Table1Materialand

cross-sectionprop

ertie

softhe

portalfram

eexample

Material

2Φ14

25

550

600

125 30

0m

m

3Φ14

25 25

125

25

Beam

2Φ12

20

300

130

300

mm

3Φ12

3Φ12

20

20

130

130 20130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

040

040

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

t = 18 s

00 02 04 06 08 10

t = 18 s

t = 15 s t = 15 s

t = 10 st = 10 s

FBEA

t = 05 s

FEA

t = 05 s

00 02 04 06 08 10

Figure 9 Accumulated tensile damage zones of the portal frame

Node 9

300

0

2600

0

600018000mm

6000 6000

300

0300

0300

0300

0300

0300

0500

0

Figure 10 Finite element mesh of 8-story RC frame with soft-story

at end regions of the beam according to FEA are not extendedat this time Damage intensities at end regions of the columnare also extended and damage regions are propagated tomiddle region from upper region of the column after thistime damage zones are extended but propagation of damagesare not seen according to both approaches at the 119905 = 18 sec

33 Seismic Damage Analyses of an 8-Story RC Frame Struc-ture with Soft-Story In this numerical application nonlineardynamic analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumpeddistributed massand load ACI 318-02 [20] code is used for the materialproperties of concrete and steel Cross-section and materialproperties of the beam and column of selected RC frame aregiven in Table 2 Finite elementmesh and gravity loading caseare shown in Figure 10 Uniaxial stress-stain relationships ofthe concrete and steel are seen in Figures 4(a) and 4(b)respectively Static loads and displacements are consideredas initial conditions in all solutions Spectrum acceleration

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

Table2Materialand

cross-sectionprop

ertie

sof8-story

RCfram

e

Material

2Φ16

20 660700

115

3Φ18

250

mm115

2020

20

Beam

s

2Φ16

20

400

230

400

mm

3Φ16

3Φ16

20

20 180 20180

130

Colum

ns

Con

crete

Elastic

ityMod

ulus

(119864119888M

Pa)

25742960

25742960

Com

pstr

ength(119891119888M

Pa)

3000

3000

Tensile

strength(119891119905M

Pa)

274

274

Massd

ensity(120588ton

m3 )

24

24

Steel

Elastic

ityMod

ulus

(119864119904M

Pa)

2100

002100

00Yield

strength(119891119910M

Pa)

4200

4200

Tangentm

odulus

(120572119905)

005

005

Massd

ensity(120588ton

m3 )

7951

7951

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

Time (s)

Ag

04

03

02

01

0

minus01

minus02

minus030 1 2 3 4 5 6 7 8 9 10

Figure 11 Time history graphs of synthetic acceleration

Time (s)0 1 2 3 4 5 6 7 8 9 10

Disp

lace

men

t (m

m)

Distributed massloadLumped massload

50

40

30

20

10

0

minus10

minus20

minus30

minus40

Figure 12 Horizontal displacement time history graphs of node 9for lumpeddistributed mass and load cases

curve given by Z1 type soil in the Turkish Regulation Codeon Building in Disaster Areas [20] is selected as the targetspectrum curve Synthetic earthquake acceleration data formaximum amplitude 03 g are produced This syntheticacceleration-time graph is shown in Figure 11 and it is affectedon the horizontal direction of the RC frame structure Tan-gent stiffness matrix is used for the all solution and dampingmatrix is also assumed to be proportional to stiffness matrix

Displacement time history graphs of node 9 obtainedfrom nonlinear dynamic analyses results for lumpeddistributed mass and load cases are shown in Figure 12Displacement amplitude values are approximately similaruntil time of 108 sec and after this time displacementamplitude values obtained from the distributedmass and loadcase are bigger than those obtained from the lumped caseAbsolute maximum horizontal displacements for the lumpedand distributed cases are obtained as 226 and 414mmrespectively Thus absolute maximum displacement value

according to distributed case is approximately 83 of ratiobigger than that of lumped case This result arises fromthe redistribution of loading in the internal regions ofthe elements for the distributed approach However theredistribution of loading in these internal regions is onlyobtained at the end regions of elements for the lumpedapproach The redistribution procedure for loading inthese internal regions requires more iterative steps for adistributed approach However numerical dissipation is notshown for both solutions and Newton-Raphson proceduresfor all element solutions which are converged for the twoapproaches The Bossak-120572 dynamic integration algorithm isapplied successfully to the nonlinear dynamic solutions

Accumulated tensile damage regions obtained for bothmassload case are given in Figures 13 and 14 Damage zonesin the lumped case are obtained at the end of the beamof all floors and at the whole of the beam cross-sectionfor the time = 100 sec Damage zones occurred at upperparts of the beams at the beam-column join region andat lower parts of the mid region of the beams until thistime for the distributed case Furthermore damage zonesseen at lower parts of first-floor columns occurred for bothapproaches but intensities and regions of these damagesaccording to distributed approach are bigger than those oflumped massload case The damage intensities are someincreased and damage propagations are remained at the sameregions for both approaches until 119905 = 300 sec after this timedamage intensities according to two approaches are increasedat between 119905 = 300 and 800 sec and no change for thedamage zones is obtained However increases in the damageintensities achieved a minimal level between 119905 = 800 and1000 sec It is said that the damage zones obtained accordingto the distributedlumped case had important differences

4 Conclusions

In this study a beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamicanalysis of reinforced concrete (RC) structural elementStiffness matrix of this element is obtained by using rigid-ity method A solution technique that included nonlineardynamic substructure procedure is developed for dynamicanalyses of RC frames A predicted-corrected form of theBossak-120572 method is applied to dynamic integration schemeA comparison of experimental data of a RC column elementwith numerical results obtained from proposed solutiontechnique is studied for verification of the numerical solu-tions Furthermore nonlinear cyclic analysis results of a por-tal reinforced concrete frame are achieved for comparing ofthe proposed solution technique with fibre element based onflexibilitymethodHowever seismic damage analyses of an 8-story RC frame structure with soft-story are investigated forcases of lumpeddistributed mass and load Damage regionpropagation and intensities according to both approaches areresearched Results obtained from this study are presented tobe itemized as follow

(i) Constitutive model based on rigidity method is usedin obtaining the stiffness matrixThe beam or column

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 11

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 13 Accumulated damage zones in building for distributed massloading case

element is divided into a subelement called thesegment The internal freedoms of this segment aredynamically condensed to the external freedoms atthe ends of the element FBEA requires less freedomthan the FEA due to the nonlinear dynamic sub-structure Thus nonlinear dynamic analysis of high

RC building can be obtained within short times Thiscondensation procedure in previous study is not usedfor the modeling of the FBEA of RC element

(ii) Numerical response of RC element which obtainedwith FBEA is similar to envelope curve of experimen-tal results All values of the horizontal displacements

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 The Scientific World Journal

(a) 119905 = 100 sec (b) 119905 = 300 sec (c) 119905 = 400 sec (d) 119905 = 500 sec

(e) 119905 = 600 sec (f) 119905 = 700 sec (g) 119905 = 800 sec

00 02 04 06 08 10

(h) 119905 = 1000 sec

Figure 14 Accumulated damage zones in building for lumped massloading case

are on the envelope curve of the experimental resultIt is said that this solution technique and materialmodels of concrete and steel can be used for thesolution of the RC structural element under thedynamic loading

(iii) Accumulated tensile damage cases obtained for bothFBEA and FEA approaches are similar but somedifferents for displacement values obtained accordingto both methods are seen These differents are to besmall for little values of tensile damage intensities

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 13

and propagation regions of damage and are to be bigdepending on increasing of the damage intensitiesand propagation regions of damage However tensiledamage regions are obtained in detail according to theFBEA

(iv) When the results obtained from lumped and dis-tributed approaches are investigated absolute max-imum displacement values for distributed approachare seen to be bigger than those for lumped approachThis case arises from the redistribution of loads in theinternal regions of the elements for the distributedapproach However the redistribution of loads inthese internal regions is not used for the lumpedapproachThe redistribution procedure for loading inthese internal regions requires more iterative steps

(v) Obtained damage zones for the lumped massloadcase are seen at the end of the beam of all floors andat the whole of the beam cross-section but obtaineddamage regions for the distributed massload caseshowed up at upper parts of the beams at the beam-column join region and at lower parts of the midregion of the beams Furthermore additional damagezones at lower parts of first-floor columns occurredfor both approaches but intensities and regions ofthese damages according to distributed approach arebigger than those of lumped approach

(vi) However numerical dissipation is not shown for allsolutions andNewton-Raphsonprocedures for all ele-ment solutions are converged for the two approachesThe Bossak-120572 time integration algorithm and pro-posed solution technique are successfully applied tothe nonlinear dynamic solutions

Appendix

Time Marching Algorithm

The algorithm applied to (16) is as follows

(1) Compute integration parameters

1198601=

1

120573Δ1199052 119860

2=

120574

120573Δ119905

(A1)

(2) 119906119878119905+Δ119905

V119878119905+Δ119905

and 119886119878119905+Δ119905

are known set globaliteration counter 119894 = 1

(3) Predict response at 119905 + Δ119905

119906119878119894

119905+Δ119905= 119878119905+Δ119905

(A2a)

V119878119894

119905+Δ119905= V119878119905+Δ119905

(A2b)

119886119878119894

119905+Δ119905= 0 (A2c)

(4) Set element counter nel = 1

(5) Set element iteration counter 119895 = 1Subtract incremental displacement vector of externaljoints from global displacement vectors

Δ119906119864119895

119905+Δ119905= SUB (Δ119906

119878119894

119905+Δ119905) (A3)

(6) Compute the element stiffness mass damping matri-ces and element external and internal force vectors

(7) Compute iterative and incremental displacement vec-tors of internal joints

Δ119906119868119895+1

119905+Δ119905= [119870119868119868]119895

119905+Δ119905((Δ119865

119868gr119895+1

119905+Δ119905

minus Δ119865119868res119895+1

119905+Δ119905)

minus[119870119868119864]119895

119905+Δ119905Δ119906119864119894

119905+Δ119905)

(A4a)

119906119868119895+1

119905+Δ119905= 119906119868119895

119905+Δ119905+ Δ119906

119868119895+1

119905+Δ119905 (A4b)

(8) Check for convergence of iteration processΔ119906119868119895+1

119905+Δ119905 (Euclidian norm of the unbalanced

internal displacement vector at time step 119905 + Δ119905 andelement iteration 119895) using an element displacementtolerance (Tolelem)

(a) If Δ119906119868119895+1

119905+Δ119905sum119895+1

119898=1Δ119906119868119898

119905+Δ119905 le Tolelem

convergence is achievedSet

119906119868119905= 119906119868119895+1

119905+Δ119905 V

119868119905= V119868119895+1

119905+Δ119905 119886

119868119905= 119886119868119895+1

119905+Δ119905

(A5a)

[119870119864119864119865

]119895

119905+Δ119905= [119870119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5b)

119865119864119865119894+1

119905+Δ119905= 119865119864119895+1

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

119865119868119895+1

119905+Δ119905

(A5c)

[119872119864119864119865

]119895

119905+Δ119905= [119872119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119872119868119864]119895

119905+Δ119905

minus [119872119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

times [119872119868119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5d)

[119862119864119864119865

]119895

119905+Δ119905= [119862119864119864]119895

119905+Δ119905minus [119870119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119864]119895

119905+Δ119905

minus [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

+ [119862119864119868]119895

119905+Δ119905([119870119868119868]119895

119905+Δ119905)

minus1

[119862119868119868]119895

119905+Δ119905

times ([119870119868119868]119895

119905+Δ119905)

minus1

[119870119868119864]119895

119905+Δ119905

(A5e)

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 The Scientific World Journal

[119870119878] =

nelemsum

119896=1

[119870119864119864119865

]119895

119905+Δ119905 [119862

119878] =

nelemsum

119896=1

[119862119864119864119865

]119895

119905+Δ119905

[119872119878] =

nelemsum

119896=1

[119872119864119864119865

]119895

119905+Δ119905 119865

119878 =

nelemsum

119896=1

119865119864119865119895

119905+Δ119905

(A5f)

(b) If global convergence is not achieved set 119895 = 119895+1 and return to Step 5

(9) Compute the effective dynamic stiffness matrix andthe vector of unbalanced forces

[119878]

119894

119905+Δ119905

= (1 minus 120572119861) 1198601[119872119878] + 1198602[119862119878]119894

119905+Δ119905+ [119870119878]119894

119905+Δ119905

(A6a)

Δ119865119878119894+1

119905+Δ119905= (1 minus 120572

119861) 119865119878119905+Δ119905

+ 120572119861119865119878gr119905

minus Δ119865119878res119894+1

119905+Δ119905

+ 119865119878stat + [119872119878]

times (minus (1 minus 120572119861) 119886119878119894+1

119905+Δ119905+ 120572119861119886119878119894

119905)

minus [119862119878]119894

119905+Δ119905V119878119894+1

119905+Δ119905

(A6b)

where [119870119878]119894

119905+Δ119905and Δ119865

119878res119894+1

119905+Δ119905are the tangent stiff-

nessmatrix and the incremental restoring force vectorof external joints on the global axis of the structurerespectively which are assembled from the elementcontributions

(10) Solve incremental displacements in the global axis

[119878]

119894

119905+Δ119905

Δ119906119878119894+1

119905+Δ119905= Δ119865

119878119894+1

119905+Δ119905 (A7)

(11) Update displacement velocity and acceleration vec-tors

119906119878119894+1

119905+Δ119905= 119906119878119894

119905+Δ119905+ Δ119906

119878119894+1

119905+Δ119905 (A8a)

V119878119894+1

119905+Δ119905= V119878119905+Δ119905

+ 1198602(119906119878119894+1

119905+Δ119905minus 119878119905+Δ119905

) (A8b)

119886119878119894+1

119905+Δ119905= 1198601(119906119878119894

119905+Δ119905minus 119878119905+Δ119905

) (A8c)

(12) Check for convergence of the iteration processΔ119865119878119894+1

119905+Δ119905 (Euclidian norm of the unbalanced force

vector at time step 119905 + Δ119905 and global iteration 119894) usinga force tolerance (Tolglo)

(a) If Δ119865119864i+1t+Δtsum

119894+1

119899=1Δ119865119864119899

119905+Δ119905 le Tolglo con-

vergence is achievedSet 119906

119878119905= 119906119878119894+1

119905+Δ119905 V119878119905= V119878119894+1

119905+Δ119905 and 119886

119878119905=

119886119878119894+1

119905+Δ119905

(b) If global convergence is not achieved set 119894 = 119894+1and return to Step 4

(13) Set 119905 = 119905 + Δ119905 and return Step to 2

Abbreviations

FEA Fibre element approachFBEA Fiber and Bernoulli-Euler approachRC Reinforced concrete

Symbols

119860119899 Area of the 119899th fiberlayer

119886119878 Acceleration vector of structure

119860Seg Cross-section area of segment[119861] Strain-displacement matrix[119862119864119864] Damping matrix for external freedoms

[119862119864119864119865

] Frame damping matrix for externalfreedoms

[119862119864119868] Damping matrix for external-internal

freedoms[119862119868119864] Damping matrix for internal-external

freedoms[119862119868119868] Damping matrix for internal freedoms

119864119874119899

Initial elasticity module of the 119899thfiberlayer

119864119879119899 Tangent elasticity module of the 119899th

fiberlayer119865119864 External load vector for external freedoms

119865119868 External load vector for internal freedoms

119865119868gr Dynamic external force vector for internal

freedoms119865119868res Dynamic restoring force vector for

internal freedoms119865119878gr Dynamic external force vector of structure

119865119878res Dynamic restoring force vector of

structure119865119878stat Static external force vector of structure

[119870119864119864] Stiffness matrix for external freedoms

[119870119864119864119865

] Frame stiffness matrix for externalfreedoms

[119870119864119868] Stiffness matrix for external-internal

freedoms[119870119868119864] Stiffness matrix for internal-external

freedoms[119870119868119868] Stiffness matrix for internal freedoms

[119870Seg] Stiffness matrix of the segment119871Seg Length of segment[119872119864119864] Mass matrix for external freedoms

[119872119864119864119865

] Frame mass matrix for external freedoms[119872119864119868] Mass matrix for external-internal

freedoms[119872119868119864] Mass matrix for internal-external

freedoms[119872119868119868] Mass matrix for internal freedoms

[119872Seg] Mass matrix of the segment119873 Number of total fiberlayer on the fiber

cross-section[119873(120585)] The element shape functions matrix119902 Displacement vector119879 Total kinetic energy of particle velocities

on the cross-section of an elementthroughout its neutral axis

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 15

Tolelem Tolerance value of element for Newton-Raphson procedure

Tolglo Tolerance value of structure for Newton-Raphson procedure

119906 Displacement in the axial directions of ele-ment local axis

119906119864 Displacement vector for external freedoms

119906119868 Displacement vector for internal freedoms

119906119878 Displacement vector of structure

119878 Predicted displacement vector of structure

V Displacement in the vertical directions ofelement local axis

V119878 Velocity vector of structure

V119878 Predicted velocity vector of structure

V120585 Particle velocity

120572119861 Bossak parameter

120572dam Rayleigh damping coefficient for mass matrix120573 First Newmarkrsquos coefficient120573dam Rayleigh damping coefficient for stiffness

matrix120576119909119909 Strain of a point on a cross-section in the axial

direction of element119889120576120585119899 Incremental axial strain of the 119899th fiberlayer

in 120585 local axis direction of a segment120574 Second Newmarkrsquos coefficient1205781015840

119899 Vertical coordinates of the 119899th fiberlayer 120578 in

local axis direction of a segment120588 Mass densityΠ Total strain energy of a segment

Conflict of interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Mohr J M Bairan and A R Marı ldquoA frame element modelfor the analysis of reinforced concrete structures under shearand bendingrdquo Engineering Structures vol 32 no 12 pp 3936ndash3954 2010

[2] Y Li X Lu H Guan and L Ye ldquoAn improved tie force methodfor progressive collapse resistance design of reinforced concreteframe structuresrdquo Engineering Structures vol 33 no 10 pp2931ndash2942 2011

[3] S S Reshotkina andD T Lau ldquoModeling damage-based degra-dations in stiffness and strength in the post-peak behaviour inseismic progressive collapse of reinforced concrete structuresrdquoin Proceedings of the 15th World Conference on EarthquakeEngineering Lisboa Portugal September 2012

[4] X Lu X Lu H Guan and L Ye ldquoCollapse simulation ofreinforced concrete high-rise building induced by extremeearthquakesrdquo Earthquake Engineering and Structural Dynamicsvol 42 no 5 pp 705ndash7723 2012

[5] OpenSees ldquoOpen system for earthquake engineering sim-ulationrdquo Pacific Earthquake Engineering Research CenterUniversity of California Berkeley Calif USA 2013 httpopenseesberkeleyedu

[6] A Kawano M C Griffith H R Joshi and R F Warner ldquoAnal-ysis of behaviour and collapse of concrete frames subjected to

severe groundmotionrdquo Research Report No R163 Departmentof Civil and Environmental Engineering Adelaide UniversitySouth Australia Australia 1998

[7] B S Iribarren Progressive collapse simulation of reinforced con-crete structures influence of design and material parameters andinvestigation of the strain rate effects [PhD thesis] PolytechnicFaculty Faculty of Applied Sciences Bruxelles Royal MilitaryAcademy Universite Libre de 2010

[8] J F S Brum A model for the non linear dynamic analysis ofreinforced concrete andmasonry framed structures [PhD thesis]Universitat Politecnica de Catalunya 2010

[9] F F Taucer E Spacone and F C Filippou ldquoA Fibre beam-column element for seismic response analysis of reinforced con-crete structuresrdquo EERC Report-9117 Earthquake EngineeringResearch Center Berkeley Calif USA 1991

[10] P Ceresa L Petrini R Pinho and R Sousa ldquoA fibre flexure-shear model for seismic analysis of RC-framed structuresrdquoEarthquake Engineering and Structural Dynamics vol 38 no5 pp 565ndash586 2009

[11] S Krishnan ldquoThree-dimensional nonlinear analysis of tallirregular steel buildings subject to strong ground motionrdquoEERL-2003-01 Earthquake Engineering Research LaboratoryCalifornia Institute of Technology Berkeley Calif USA 2003

[12] T R Chandrupatla and A D Belegundu Introduction to FiniteElements in Engineering Prentice Hall New Jersey NJ USA2012

[13] F Legeron P Paultre and J Mazars ldquoDamage mechanicsmodeling of nonlinear seismic behavior of concrete structuresrdquoJournal of Structural Engineering vol 131 no 6 pp 946ndash9552005

[14] K J Bathe Finite Element Procedures in Engineering AnalysisPrentice hall Englewood Cliffs NJ USA 1982

[15] R J Guyan ldquoReduction of stiffness and mass matricesrdquo AIAAJournal vol 3 no 2 article 380 1965

[16] W L Wood M Bossak and O C Zienkiewicz ldquoA alphamodification of Newmarkrsquos methodrdquo International Journal forNumerical Methods in Engineering vol 15 pp 1562ndash1566 1981

[17] I Miranda R M Ferencz and T J R Hughes ldquoAn improvedimplicit-explicit time integrationmethod for structural dynam-icsrdquo Earthquake Engineering and Structural Dynamics vol 18no 5 pp 643ndash653 1989

[18] Y TakahashiDevelopment of high seismic performance RC pierswith object-oriented structural analysis [PhD thesis] Faculty ofEngineering of Kyoto University 2002

[19] ldquoEarthquake engineering software solutionrdquo SeismoStruc Ver 62013 httpwwwseismosoftcom

[20] ACI ldquoBuilding code requirements for structural concreterdquo ACI318-02 American Concrete Institute Farmington Hills MichUSA 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of