Research ArticleDevelopment of Constitutive Model for Precast PrestressedConcrete Segmental Columns
M Hafezolghorani Esfahani1 F Hejazi1 R Vaghei1 E Nikbakht2 and D C J Tze1
1Department of Civil Engineering University of Putra Malaysia Serdang Malaysia2Department of Civil Engineering Universiti Teknologi Petronas Malaysia
Correspondence should be addressed to F Hejazi farzadfhejazicom
Received 1 December 2015 Accepted 28 February 2016
Academic Editor Anna Pandolfi
Copyright copy 2016 M Hafezolghorani Esfahani et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The interest of using precast segmental columns in construction of concrete bridges has significantly increased in recent years Oneresearch area of concrete bridges is the application of Precast Prestressed Concrete Segmental (PPCS) Column in any structuralanalysis software or FE program codeModeling a PPCS columnwhich consists of variousmaterials with interaction between themis complicated and time-consumingThis research attempts to formulate the stiffness matrix of PPCS columns in order to form theconstitutive model in linear form to evaluate the response of the columns A two-dimensional finite element model is presented inthe finite element package ANSYS Parametric studies are conducted by finite element models to verify the constitutive models forthe PPCS column with a different number of concrete segments Comparison between the constitutive model and the FE programresults indicates that the constitutive model is accurate enough to predict the deformation of the PPCS columns
1 Introduction
Construction of segmental bridge started in Europe in1950 The first attempt of cast-in-place segmental concretebridge was conducted across the Lahn River in BalduinsteinGermany in 1950 however the primary precast segmentalconcrete bridge was constructed in 1962 across the RiverSeine in France Later this construction method gainedworldwide recognition Bridge construction time facilitatingconstruction and minimizing the traffic disruption are themain advantages of precast construction in contrast to cast-in-situ construction Precast segmental concrete bridges arenormally constructed in low seismic areas Numerous precastsegmental concrete columns and pier constructions havebeen carried out in the US in low seismic regions such as thestates of Texas and California in the United States [1] Studiesof PPCS column can be categorized into two parts firstlywith bonded and secondly with unbounded posttensioningsystems
Different researchers presented various aspects of pre-stressing with bonded tendons in their research [2 3]In bonded posttensioned systems the lateral strength of
columns could be enhanced by the bonding between pre-stressing strands and surrounding concrete however itmightresult in tendon yielding To the best of our knowledgethere are just a few studies in the literature about unboundedtendons [4 5] Unbounded posttensioned steels might leadto reduced prestress loss during strong seismic excitationsThe segmental column can be restored to its original config-uration after earthquakes which cannot be done for bondedstructure It also maintains the continuity between columnsegments and foundation There are several componentsin precast segmental bridge columns which are combinedand associated with continuous posttensioning strands Theaction of these types of bridges against seismic loadingdiffers from conventional bridge columns besides segmentalbridges act with rocking mechanism which takes place whilethe segments opening is to be found and as a consequencethe resultant damage is much lower than that in the con-ventional monolithic bridge columns The minor cracks anddamage in this system make them more economical thanthe monolithic system due to the reparability of this systemafter severe earthquake loading The PRESS program hasbeen established (Precast Seismic Structural Systems) as an
Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2016 Article ID 9453649 11 pageshttpdxdoiorg10115520169453649
2 Modelling and Simulation in Engineering
innovative alternative solution for precast connections byusing prestressing strands and mild steel in order to indicateappropriate ductility compared to conventional monolithicsystems [6]
Prestressing strands have the capability of recenteringagainst earthquake which means they return the columnsat the original place over the unloading stages The mildsteel reinforcements are able to dissipate the earthquakeenergy The concept of a hybrid system has been introducedfor the first time and applied for posttensioning in beam-column connections [7] They claimed that the combinationof prestressing strands and mild steel reinforcements canappropriatelymeet the design codesrsquo requirementsMild steelreinforcement and posttensioned tendons have an influen-tial role in the ductility and strength of hybrid segmentalsystems due to their combination [8] The performance ofunbonded posttensioned prefabricated concrete segmentalbridge column subjected to lateral earthquake loading hasbeen analytically and experimentally investigated for fourprecast posttensioned columns with different thickness ofsteel jackets and the results indicated that these typesof bridge columns have appropriate performance againstseismic energy with negligible residual displacement Newcriteria for functional and survival limits for posttensioned(PT) precast segmental bridge columns have been proposed[9] At the functional level of earthquake the objective is tokeep the structure operational without occurrence of majordamage while at the survival limit the objective is to preventthe structure from collapsing They defined three criteria forthe functionality of structure First limit is yielding of theprestressing strandsThe second criterion is the displacementwhich leads to 1 residual drift and the third limit is 07 timesthe survival-level displacement The surviving-level limit isthe displacement at which the structure starts to collapseReduction of longitudinal reinforcements and increase inaxial load result in minimal residual displacement [10]Reduction by 86 in the residual displacement can be cap-tured by replacing half of the rebar with prestressing strandsand applying prestressing force that is equivalent to axial loaddue to dead loadThey also investigated the dynamic analysisof the unbounded center strand columns A combinationratio of energy dissipation of mild steel internal andorexternal supplemental energy dissipation and self-centeringposttensioning strands as one of the predominant designfactors in hybrid posttensioned bridges has been proposedto achieve appropriate energy dissipation and self-centeringcapability with less damage and residual displacement againstseismic loading [11 12] They used two rotational springsone representing the behavior of prestressed tendons withoutthe contribution of mild steel reinforcement and the otherrepresenting only the mild steel reinforcement contributionThey investigated five specimens with different combinationratios The experimental results proved the advantages ofenhanced performance of the hybrid system in comparisonwith traditional monolithic solutions The proposed hybridsystem is shown to have minor flexural cracking negligibleresidual displacement and a stable hysteretic behavior up tohigh ductility level so a combination ratio of prestressingstrands and mild steel reinforcements in design is proposed
Some researchers have also investigated the behavior ofhollow unbonded precast segmental bridge columns withrectangular box cross section in which the prestressingstrands were passed through the hollow ducts [12]
The actual simplicity at design and construction phaseof high performance and low-cost hybrid bridge piers werestudied by some researchers [13] They also applied shortlengths of unbondedmild steel reinforcements at the junctionof footing-first segment to prevent premature yielding oflongitudinal reinforcements They indicated that a certainamount of unbonded length of mild reinforcements providesmore energy dissipation and strength against earthquakeloading Furthermore they claimed that the unbonded pre-cast column could return to the undeformed position afterearthquakes Eight large-scale posttensioned precast columnswere carried out to validate the detailed finite element modelsubjected to cyclic tests which was developed by ABAQUSplatform and the outcomes showed a good agreement [14] A3D finite element model of precast walls and connection wasdeveloped using finite element model [15] A numerical ana-lytical model with nonlinear factors of prestressing precastconcrete bridge column systems which consists of a segmentmodel prestressing tendon and joint mode was developed[16] The detailed finite element model of prestressing bridgecolumn was carried out with the structural analysis softwareldquoOpen Seesrdquo and finally the finite element outcomes metthe experimental results The monotonic behavior of precastsegmental bridge columns has been investigated throughthree-dimensional finite element models [17 18] Later theystudied the accuracy of analytical results for predicting thedamage when subjected to lateral loading They also inves-tigated the effects of posttensioning forces the amount ofsegments and aspect ratio (the relation between the heightsof the column and the diameter of the column) Two types ofnumerical models for unbonded posttensioned (PT) precastconcrete segmental bridge were presented by the computerprogram PISA [19] A 3D nonlinear finite element model hasbeen proposed for hybrid posttensioned precast segmentalbridge columns to analyse the different prestressing strandlevels subjected to nonlinear static and lateral seismic loading[20 21]
Modeling a PPCS column with various materials andinteraction has been studied by finite element model (FEM)however the results show that this methodology is compli-cated and time-consuming This paper presents the stiffnessmatrix of unbonded prestressed precast segmental (PPCS)column to form the constitutive model Finite element for-mulations are derived in explicit form which is applicablein any structural analysis software or FEM program codeFinally three finite element models of Precast PrestressedConcrete Segmental (PPCS) Column with respect to onetwo and three segments called RC1SS RC2SS and RC3SS areinvestigated
2 Precast Prestressed ConcreteSegmental Column
21 Description of the Case Study and Material PropertiesVarious experiments on different large-scale specimens have
been carried out in 2002 [4] This study is chosen to developthe constitutivemodelThe geometry of the studied specimenis shown in Figure 1 Due to plane stress and plane strainanalysis two-dimensional view of the studied specimen hasbeen chosen and equivalent section area is considered for 2DFE modeling and analysis
Thematerial properties of the specimens are presented inTable 1
22 Constitutive Stress-Strain Relationships Concrete is aquasi-brittle material which means concrete behaves differ-ently against compression and tension In general concretersquostensile strength is only about 8 to 15 of the compressivestrength As shown in Figure 2(a) a stress-strain curve forconfined concrete is presented in which 119891
119888and 119891
119888119906are the
peak compressive and ultimate strengths and 1205760and 120576119888119906
arethe corresponding strains respectively [22]
Idealized nonlinear prestressing steel with stress-strainmodel for 7-wire low-relaxation prestressing strand fromASTM A722 was proposed [4] The curve which is shown inFigure 2(b) can be derived by
prestress steel limit of proportionality 120576119897119901= 00086
Although the developed constitutive law is applicable for bothlinear and nonlinear analysis linear analysis has been usedin this paper to codify the finite element program due to itssimplicity of application and verification
3 Development of Constitutive Law
The proposed constitutive model for precast segmentalcolumns comprises two stiffness matrixes for concrete andreinforcement Each concrete segmental is modeled as two4-node isoparametric elements as shown in Figure 3
The shape function for isoparametric 4-node element isshown in Figure 4
The displacement (119906 V) is assumed as a bilinear functionover the element and is given by the following [23]
Figure 4 Shape function of isoparametric four nodes
The force-displacement relation can be evaluated by
119865 = 119896119889 (6)
The strain-displacement relation is given by
120576 = 119861119889 (7)
The stress-strain relation can be obtained by
120590 = 119863120576 (8)
The discontinuous longitudinal reinforcement bar in theconcrete segments and tendons is modeled as two (2) noddedbar elements Each node of the truss is laterally restrained tothe adjacent node of segmented column The shape functionof reinforcement bar is evaluated as shown in Figure 5
120585 = minus1
N1 =1 minus 120585
2
120585 = 1
( ) N2 =1 + 120585
2( )
Figure 5 Shape function of bar element
The stiffness matrix of reinforcement can be obtained by
[119896119904] = int [119861]
119879
[119863] [119861] 119889V = int119897
[119861]119879
[119863] [119861]119860119889119909
= int119897
[119861]119879
[119863] [119861]119860 |119869| 119889120585
(9)
where
119889119883 = |119869| 119889120585 (10)
It is known as
120597119873119894
120597119909= [120597119909
120597120585]
minus1
times120597119873119894
120597120585
119869 =120597119909
120597120585
(11)
Generally the global stiffness of prestressed column in thisstudy is formulated by assembling stiffness matrix for eachconcrete segment prestressing tendon longitudinal bar andinterface element
4 Comparison of the ProposedConstitutive Model and FE Model
The contribution of tendons in precast concrete segmentsis investigated and new constitutive model is proposed andverified through modeling in FE software ANSYS
Solution ofmany engineering problems is based on linearapproximations and in this study the linear response isconsidered although developed constitutive law is applicablefor nonlinear analysis as well
Modelling and Simulation in Engineering 5
Longitudinal bar
KS1 KS2
Lateral load
Pin connection4 (0 0)
Axial load
Longitudinal bar
Concrete Concrete
2 (L2 H)
5 (L2 0)
3 (L H)
6 (L 0)
1 (0 H)
(a) Proposed constitutive model
Axial load
Pinnedconnection
Lateralload
XYZ
(b) FE model
Figure 6 One concrete segment model (RC1SS)
41 One Concrete Segment Figure 6(a) shows the constitutivemodel of one concrete segment where the longitudinal bars(KS1 KS2) are separated into sections equally and placed atboth sides of the concrete segment that is KS1 is sharingnode 1 and node 4 with the concrete segment in 119910 directionwhile KS2 is sharing node 3 and node 6 The length of thelongitudinal bar is the same as the height of single concretesegment The lateral load is applied at node 1 of concretesegment with load step of 100 kN up to 500 kN and 500 kNis vertically and equally applied at top nodes (ie 1 2 and 3)Figure 6(b) shows the finite element model for one concretesegment subjected to the lateral and axial loads
The stiffness matrix for one concrete segment with lon-gitudinal bar can be derived as shown in (12) The stiffnessmatrix variables 119860 119861 119862 119886 119887 119888 119889 119890 119891 and 119892 are defined inTable 2
A comparison between the constitutive model and FEmodel responses is carried out As can be observed inFigure 7 it can be proven that the proposed constitutivemodel for one concrete segment has a good agreement withthe FEmodel in plane stress analysis and plane strain analysisAs expected displacement in plane strain analysis is about80mm which is four times higher than the displacementin plane stress analysis and both approaches show linearbehavior
It can be seen from Table 3 that in plane stress analysis themaximum displacement by the proposed constitutive modelis close to that obtained by FE model with the former valuea little bit more than the latter one A maximum differenceof 46 is observed between constitutive model and FEresults In plane strain analysis the maximum displacement
is 7689mm in the proposed constitutivemodel however thecorrespondence value in FEmodel is 7394mmTable 3 showsabout four percent difference in plane strain analysis betweenproposed constitutivemodel and FE results for one segmentalconcrete whilst discrepancy percentage of aforementionedmethods in plane stress analysis is increased to 46
6 Modelling and Simulation in Engineering
0
100
200
300
400
500
600
0 20 40 60 80 100Displacement (mm)
Late
ral l
oadi
ng (k
N)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 7 Lateral load-deflection of the proposed constitutivemodeland FE model for RC1SS
42 Two Concrete Segments Two concrete segments areattached together as shown in Figure 8(a) with prestressingtendon placed at the middle of the concrete segment which
connects node 2 and node 8 in order to connect the concretesegments together (RC2SS)The intention of designing longi-tudinal reinforcement discontinuously is to avoid the fractureof mild steel at the critical joint opening when a huge lateralload is applied The developed constitutive model for RC2SSis examined under the following load condition as shownin Figure 8(b) The prestressing tendon role is to overcomeweakness of concrete in tension by providing the clampingload between the concrete segment and support
From (13) the stiffness matrix for the prestressing tendoncan be derived as
[119870119905] =
119864119905119860119905
119867
8119910 2119910
[
1
minus1
minus1
1]
119905 =119864119905119860119905
119867times
8119871119867(1 minus V2)119864119905
(plane stress)
119905 =119864119905119860119905
119867times16119871119867 (1 + V) 119861
119864119905(plane strain)
(13)
So by substituting these parameters in (14) the stiffnessmatrix can be derived as
Comparison between the constitutive and FE models for twoconcrete segments RC2SS is shown in Figure 9 From thisfigure it can be seen that there is a good agreement betweenthe proposed constitutive model results for RC2SS and theFE model results in plane stress and plane strain analysis
Consequently it can be determined that the maximumdisplacement in plane strain analysis is about 200mm whichis almost three times higher than the displacement in planestress analysis Linear behavior has been captured for bothaforementioned approaches
Modelling and Simulation in Engineering 7
Table 2 Stiffness matrix variables
Plane stress condition Plane strain condition
119860 =1
2(1 minus V) 119861 = (1 minus 2V)
119862 =119864119905
8119871119867(1 minus V2)119862 =
119864119905
16119871119867(1 + V)119861
119886 = 41198672
+ 1198601198712
119886 = 81198672
(1 minus V) + 1198712 (1 minus 2V)
119887 = 41198672
minus 1198601198712
119887 = 81198672
(1 minus V) minus 1198712 (1 minus 2V)
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
innovative alternative solution for precast connections byusing prestressing strands and mild steel in order to indicateappropriate ductility compared to conventional monolithicsystems [6]
Prestressing strands have the capability of recenteringagainst earthquake which means they return the columnsat the original place over the unloading stages The mildsteel reinforcements are able to dissipate the earthquakeenergy The concept of a hybrid system has been introducedfor the first time and applied for posttensioning in beam-column connections [7] They claimed that the combinationof prestressing strands and mild steel reinforcements canappropriatelymeet the design codesrsquo requirementsMild steelreinforcement and posttensioned tendons have an influen-tial role in the ductility and strength of hybrid segmentalsystems due to their combination [8] The performance ofunbonded posttensioned prefabricated concrete segmentalbridge column subjected to lateral earthquake loading hasbeen analytically and experimentally investigated for fourprecast posttensioned columns with different thickness ofsteel jackets and the results indicated that these typesof bridge columns have appropriate performance againstseismic energy with negligible residual displacement Newcriteria for functional and survival limits for posttensioned(PT) precast segmental bridge columns have been proposed[9] At the functional level of earthquake the objective is tokeep the structure operational without occurrence of majordamage while at the survival limit the objective is to preventthe structure from collapsing They defined three criteria forthe functionality of structure First limit is yielding of theprestressing strandsThe second criterion is the displacementwhich leads to 1 residual drift and the third limit is 07 timesthe survival-level displacement The surviving-level limit isthe displacement at which the structure starts to collapseReduction of longitudinal reinforcements and increase inaxial load result in minimal residual displacement [10]Reduction by 86 in the residual displacement can be cap-tured by replacing half of the rebar with prestressing strandsand applying prestressing force that is equivalent to axial loaddue to dead loadThey also investigated the dynamic analysisof the unbounded center strand columns A combinationratio of energy dissipation of mild steel internal andorexternal supplemental energy dissipation and self-centeringposttensioning strands as one of the predominant designfactors in hybrid posttensioned bridges has been proposedto achieve appropriate energy dissipation and self-centeringcapability with less damage and residual displacement againstseismic loading [11 12] They used two rotational springsone representing the behavior of prestressed tendons withoutthe contribution of mild steel reinforcement and the otherrepresenting only the mild steel reinforcement contributionThey investigated five specimens with different combinationratios The experimental results proved the advantages ofenhanced performance of the hybrid system in comparisonwith traditional monolithic solutions The proposed hybridsystem is shown to have minor flexural cracking negligibleresidual displacement and a stable hysteretic behavior up tohigh ductility level so a combination ratio of prestressingstrands and mild steel reinforcements in design is proposed
Some researchers have also investigated the behavior ofhollow unbonded precast segmental bridge columns withrectangular box cross section in which the prestressingstrands were passed through the hollow ducts [12]
The actual simplicity at design and construction phaseof high performance and low-cost hybrid bridge piers werestudied by some researchers [13] They also applied shortlengths of unbondedmild steel reinforcements at the junctionof footing-first segment to prevent premature yielding oflongitudinal reinforcements They indicated that a certainamount of unbonded length of mild reinforcements providesmore energy dissipation and strength against earthquakeloading Furthermore they claimed that the unbonded pre-cast column could return to the undeformed position afterearthquakes Eight large-scale posttensioned precast columnswere carried out to validate the detailed finite element modelsubjected to cyclic tests which was developed by ABAQUSplatform and the outcomes showed a good agreement [14] A3D finite element model of precast walls and connection wasdeveloped using finite element model [15] A numerical ana-lytical model with nonlinear factors of prestressing precastconcrete bridge column systems which consists of a segmentmodel prestressing tendon and joint mode was developed[16] The detailed finite element model of prestressing bridgecolumn was carried out with the structural analysis softwareldquoOpen Seesrdquo and finally the finite element outcomes metthe experimental results The monotonic behavior of precastsegmental bridge columns has been investigated throughthree-dimensional finite element models [17 18] Later theystudied the accuracy of analytical results for predicting thedamage when subjected to lateral loading They also inves-tigated the effects of posttensioning forces the amount ofsegments and aspect ratio (the relation between the heightsof the column and the diameter of the column) Two types ofnumerical models for unbonded posttensioned (PT) precastconcrete segmental bridge were presented by the computerprogram PISA [19] A 3D nonlinear finite element model hasbeen proposed for hybrid posttensioned precast segmentalbridge columns to analyse the different prestressing strandlevels subjected to nonlinear static and lateral seismic loading[20 21]
Modeling a PPCS column with various materials andinteraction has been studied by finite element model (FEM)however the results show that this methodology is compli-cated and time-consuming This paper presents the stiffnessmatrix of unbonded prestressed precast segmental (PPCS)column to form the constitutive model Finite element for-mulations are derived in explicit form which is applicablein any structural analysis software or FEM program codeFinally three finite element models of Precast PrestressedConcrete Segmental (PPCS) Column with respect to onetwo and three segments called RC1SS RC2SS and RC3SS areinvestigated
2 Precast Prestressed ConcreteSegmental Column
21 Description of the Case Study and Material PropertiesVarious experiments on different large-scale specimens have
been carried out in 2002 [4] This study is chosen to developthe constitutivemodelThe geometry of the studied specimenis shown in Figure 1 Due to plane stress and plane strainanalysis two-dimensional view of the studied specimen hasbeen chosen and equivalent section area is considered for 2DFE modeling and analysis
Thematerial properties of the specimens are presented inTable 1
22 Constitutive Stress-Strain Relationships Concrete is aquasi-brittle material which means concrete behaves differ-ently against compression and tension In general concretersquostensile strength is only about 8 to 15 of the compressivestrength As shown in Figure 2(a) a stress-strain curve forconfined concrete is presented in which 119891
119888and 119891
119888119906are the
peak compressive and ultimate strengths and 1205760and 120576119888119906
arethe corresponding strains respectively [22]
Idealized nonlinear prestressing steel with stress-strainmodel for 7-wire low-relaxation prestressing strand fromASTM A722 was proposed [4] The curve which is shown inFigure 2(b) can be derived by
prestress steel limit of proportionality 120576119897119901= 00086
Although the developed constitutive law is applicable for bothlinear and nonlinear analysis linear analysis has been usedin this paper to codify the finite element program due to itssimplicity of application and verification
3 Development of Constitutive Law
The proposed constitutive model for precast segmentalcolumns comprises two stiffness matrixes for concrete andreinforcement Each concrete segmental is modeled as two4-node isoparametric elements as shown in Figure 3
The shape function for isoparametric 4-node element isshown in Figure 4
The displacement (119906 V) is assumed as a bilinear functionover the element and is given by the following [23]
Figure 4 Shape function of isoparametric four nodes
The force-displacement relation can be evaluated by
119865 = 119896119889 (6)
The strain-displacement relation is given by
120576 = 119861119889 (7)
The stress-strain relation can be obtained by
120590 = 119863120576 (8)
The discontinuous longitudinal reinforcement bar in theconcrete segments and tendons is modeled as two (2) noddedbar elements Each node of the truss is laterally restrained tothe adjacent node of segmented column The shape functionof reinforcement bar is evaluated as shown in Figure 5
120585 = minus1
N1 =1 minus 120585
2
120585 = 1
( ) N2 =1 + 120585
2( )
Figure 5 Shape function of bar element
The stiffness matrix of reinforcement can be obtained by
[119896119904] = int [119861]
119879
[119863] [119861] 119889V = int119897
[119861]119879
[119863] [119861]119860119889119909
= int119897
[119861]119879
[119863] [119861]119860 |119869| 119889120585
(9)
where
119889119883 = |119869| 119889120585 (10)
It is known as
120597119873119894
120597119909= [120597119909
120597120585]
minus1
times120597119873119894
120597120585
119869 =120597119909
120597120585
(11)
Generally the global stiffness of prestressed column in thisstudy is formulated by assembling stiffness matrix for eachconcrete segment prestressing tendon longitudinal bar andinterface element
4 Comparison of the ProposedConstitutive Model and FE Model
The contribution of tendons in precast concrete segmentsis investigated and new constitutive model is proposed andverified through modeling in FE software ANSYS
Solution ofmany engineering problems is based on linearapproximations and in this study the linear response isconsidered although developed constitutive law is applicablefor nonlinear analysis as well
Modelling and Simulation in Engineering 5
Longitudinal bar
KS1 KS2
Lateral load
Pin connection4 (0 0)
Axial load
Longitudinal bar
Concrete Concrete
2 (L2 H)
5 (L2 0)
3 (L H)
6 (L 0)
1 (0 H)
(a) Proposed constitutive model
Axial load
Pinnedconnection
Lateralload
XYZ
(b) FE model
Figure 6 One concrete segment model (RC1SS)
41 One Concrete Segment Figure 6(a) shows the constitutivemodel of one concrete segment where the longitudinal bars(KS1 KS2) are separated into sections equally and placed atboth sides of the concrete segment that is KS1 is sharingnode 1 and node 4 with the concrete segment in 119910 directionwhile KS2 is sharing node 3 and node 6 The length of thelongitudinal bar is the same as the height of single concretesegment The lateral load is applied at node 1 of concretesegment with load step of 100 kN up to 500 kN and 500 kNis vertically and equally applied at top nodes (ie 1 2 and 3)Figure 6(b) shows the finite element model for one concretesegment subjected to the lateral and axial loads
The stiffness matrix for one concrete segment with lon-gitudinal bar can be derived as shown in (12) The stiffnessmatrix variables 119860 119861 119862 119886 119887 119888 119889 119890 119891 and 119892 are defined inTable 2
A comparison between the constitutive model and FEmodel responses is carried out As can be observed inFigure 7 it can be proven that the proposed constitutivemodel for one concrete segment has a good agreement withthe FEmodel in plane stress analysis and plane strain analysisAs expected displacement in plane strain analysis is about80mm which is four times higher than the displacementin plane stress analysis and both approaches show linearbehavior
It can be seen from Table 3 that in plane stress analysis themaximum displacement by the proposed constitutive modelis close to that obtained by FE model with the former valuea little bit more than the latter one A maximum differenceof 46 is observed between constitutive model and FEresults In plane strain analysis the maximum displacement
is 7689mm in the proposed constitutivemodel however thecorrespondence value in FEmodel is 7394mmTable 3 showsabout four percent difference in plane strain analysis betweenproposed constitutivemodel and FE results for one segmentalconcrete whilst discrepancy percentage of aforementionedmethods in plane stress analysis is increased to 46
6 Modelling and Simulation in Engineering
0
100
200
300
400
500
600
0 20 40 60 80 100Displacement (mm)
Late
ral l
oadi
ng (k
N)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 7 Lateral load-deflection of the proposed constitutivemodeland FE model for RC1SS
42 Two Concrete Segments Two concrete segments areattached together as shown in Figure 8(a) with prestressingtendon placed at the middle of the concrete segment which
connects node 2 and node 8 in order to connect the concretesegments together (RC2SS)The intention of designing longi-tudinal reinforcement discontinuously is to avoid the fractureof mild steel at the critical joint opening when a huge lateralload is applied The developed constitutive model for RC2SSis examined under the following load condition as shownin Figure 8(b) The prestressing tendon role is to overcomeweakness of concrete in tension by providing the clampingload between the concrete segment and support
From (13) the stiffness matrix for the prestressing tendoncan be derived as
[119870119905] =
119864119905119860119905
119867
8119910 2119910
[
1
minus1
minus1
1]
119905 =119864119905119860119905
119867times
8119871119867(1 minus V2)119864119905
(plane stress)
119905 =119864119905119860119905
119867times16119871119867 (1 + V) 119861
119864119905(plane strain)
(13)
So by substituting these parameters in (14) the stiffnessmatrix can be derived as
Comparison between the constitutive and FE models for twoconcrete segments RC2SS is shown in Figure 9 From thisfigure it can be seen that there is a good agreement betweenthe proposed constitutive model results for RC2SS and theFE model results in plane stress and plane strain analysis
Consequently it can be determined that the maximumdisplacement in plane strain analysis is about 200mm whichis almost three times higher than the displacement in planestress analysis Linear behavior has been captured for bothaforementioned approaches
Modelling and Simulation in Engineering 7
Table 2 Stiffness matrix variables
Plane stress condition Plane strain condition
119860 =1
2(1 minus V) 119861 = (1 minus 2V)
119862 =119864119905
8119871119867(1 minus V2)119862 =
119864119905
16119871119867(1 + V)119861
119886 = 41198672
+ 1198601198712
119886 = 81198672
(1 minus V) + 1198712 (1 minus 2V)
119887 = 41198672
minus 1198601198712
119887 = 81198672
(1 minus V) minus 1198712 (1 minus 2V)
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
been carried out in 2002 [4] This study is chosen to developthe constitutivemodelThe geometry of the studied specimenis shown in Figure 1 Due to plane stress and plane strainanalysis two-dimensional view of the studied specimen hasbeen chosen and equivalent section area is considered for 2DFE modeling and analysis
Thematerial properties of the specimens are presented inTable 1
22 Constitutive Stress-Strain Relationships Concrete is aquasi-brittle material which means concrete behaves differ-ently against compression and tension In general concretersquostensile strength is only about 8 to 15 of the compressivestrength As shown in Figure 2(a) a stress-strain curve forconfined concrete is presented in which 119891
119888and 119891
119888119906are the
peak compressive and ultimate strengths and 1205760and 120576119888119906
arethe corresponding strains respectively [22]
Idealized nonlinear prestressing steel with stress-strainmodel for 7-wire low-relaxation prestressing strand fromASTM A722 was proposed [4] The curve which is shown inFigure 2(b) can be derived by
prestress steel limit of proportionality 120576119897119901= 00086
Although the developed constitutive law is applicable for bothlinear and nonlinear analysis linear analysis has been usedin this paper to codify the finite element program due to itssimplicity of application and verification
3 Development of Constitutive Law
The proposed constitutive model for precast segmentalcolumns comprises two stiffness matrixes for concrete andreinforcement Each concrete segmental is modeled as two4-node isoparametric elements as shown in Figure 3
The shape function for isoparametric 4-node element isshown in Figure 4
The displacement (119906 V) is assumed as a bilinear functionover the element and is given by the following [23]
Figure 4 Shape function of isoparametric four nodes
The force-displacement relation can be evaluated by
119865 = 119896119889 (6)
The strain-displacement relation is given by
120576 = 119861119889 (7)
The stress-strain relation can be obtained by
120590 = 119863120576 (8)
The discontinuous longitudinal reinforcement bar in theconcrete segments and tendons is modeled as two (2) noddedbar elements Each node of the truss is laterally restrained tothe adjacent node of segmented column The shape functionof reinforcement bar is evaluated as shown in Figure 5
120585 = minus1
N1 =1 minus 120585
2
120585 = 1
( ) N2 =1 + 120585
2( )
Figure 5 Shape function of bar element
The stiffness matrix of reinforcement can be obtained by
[119896119904] = int [119861]
119879
[119863] [119861] 119889V = int119897
[119861]119879
[119863] [119861]119860119889119909
= int119897
[119861]119879
[119863] [119861]119860 |119869| 119889120585
(9)
where
119889119883 = |119869| 119889120585 (10)
It is known as
120597119873119894
120597119909= [120597119909
120597120585]
minus1
times120597119873119894
120597120585
119869 =120597119909
120597120585
(11)
Generally the global stiffness of prestressed column in thisstudy is formulated by assembling stiffness matrix for eachconcrete segment prestressing tendon longitudinal bar andinterface element
4 Comparison of the ProposedConstitutive Model and FE Model
The contribution of tendons in precast concrete segmentsis investigated and new constitutive model is proposed andverified through modeling in FE software ANSYS
Solution ofmany engineering problems is based on linearapproximations and in this study the linear response isconsidered although developed constitutive law is applicablefor nonlinear analysis as well
Modelling and Simulation in Engineering 5
Longitudinal bar
KS1 KS2
Lateral load
Pin connection4 (0 0)
Axial load
Longitudinal bar
Concrete Concrete
2 (L2 H)
5 (L2 0)
3 (L H)
6 (L 0)
1 (0 H)
(a) Proposed constitutive model
Axial load
Pinnedconnection
Lateralload
XYZ
(b) FE model
Figure 6 One concrete segment model (RC1SS)
41 One Concrete Segment Figure 6(a) shows the constitutivemodel of one concrete segment where the longitudinal bars(KS1 KS2) are separated into sections equally and placed atboth sides of the concrete segment that is KS1 is sharingnode 1 and node 4 with the concrete segment in 119910 directionwhile KS2 is sharing node 3 and node 6 The length of thelongitudinal bar is the same as the height of single concretesegment The lateral load is applied at node 1 of concretesegment with load step of 100 kN up to 500 kN and 500 kNis vertically and equally applied at top nodes (ie 1 2 and 3)Figure 6(b) shows the finite element model for one concretesegment subjected to the lateral and axial loads
The stiffness matrix for one concrete segment with lon-gitudinal bar can be derived as shown in (12) The stiffnessmatrix variables 119860 119861 119862 119886 119887 119888 119889 119890 119891 and 119892 are defined inTable 2
A comparison between the constitutive model and FEmodel responses is carried out As can be observed inFigure 7 it can be proven that the proposed constitutivemodel for one concrete segment has a good agreement withthe FEmodel in plane stress analysis and plane strain analysisAs expected displacement in plane strain analysis is about80mm which is four times higher than the displacementin plane stress analysis and both approaches show linearbehavior
It can be seen from Table 3 that in plane stress analysis themaximum displacement by the proposed constitutive modelis close to that obtained by FE model with the former valuea little bit more than the latter one A maximum differenceof 46 is observed between constitutive model and FEresults In plane strain analysis the maximum displacement
is 7689mm in the proposed constitutivemodel however thecorrespondence value in FEmodel is 7394mmTable 3 showsabout four percent difference in plane strain analysis betweenproposed constitutivemodel and FE results for one segmentalconcrete whilst discrepancy percentage of aforementionedmethods in plane stress analysis is increased to 46
6 Modelling and Simulation in Engineering
0
100
200
300
400
500
600
0 20 40 60 80 100Displacement (mm)
Late
ral l
oadi
ng (k
N)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 7 Lateral load-deflection of the proposed constitutivemodeland FE model for RC1SS
42 Two Concrete Segments Two concrete segments areattached together as shown in Figure 8(a) with prestressingtendon placed at the middle of the concrete segment which
connects node 2 and node 8 in order to connect the concretesegments together (RC2SS)The intention of designing longi-tudinal reinforcement discontinuously is to avoid the fractureof mild steel at the critical joint opening when a huge lateralload is applied The developed constitutive model for RC2SSis examined under the following load condition as shownin Figure 8(b) The prestressing tendon role is to overcomeweakness of concrete in tension by providing the clampingload between the concrete segment and support
From (13) the stiffness matrix for the prestressing tendoncan be derived as
[119870119905] =
119864119905119860119905
119867
8119910 2119910
[
1
minus1
minus1
1]
119905 =119864119905119860119905
119867times
8119871119867(1 minus V2)119864119905
(plane stress)
119905 =119864119905119860119905
119867times16119871119867 (1 + V) 119861
119864119905(plane strain)
(13)
So by substituting these parameters in (14) the stiffnessmatrix can be derived as
Comparison between the constitutive and FE models for twoconcrete segments RC2SS is shown in Figure 9 From thisfigure it can be seen that there is a good agreement betweenthe proposed constitutive model results for RC2SS and theFE model results in plane stress and plane strain analysis
Consequently it can be determined that the maximumdisplacement in plane strain analysis is about 200mm whichis almost three times higher than the displacement in planestress analysis Linear behavior has been captured for bothaforementioned approaches
Modelling and Simulation in Engineering 7
Table 2 Stiffness matrix variables
Plane stress condition Plane strain condition
119860 =1
2(1 minus V) 119861 = (1 minus 2V)
119862 =119864119905
8119871119867(1 minus V2)119862 =
119864119905
16119871119867(1 + V)119861
119886 = 41198672
+ 1198601198712
119886 = 81198672
(1 minus V) + 1198712 (1 minus 2V)
119887 = 41198672
minus 1198601198712
119887 = 81198672
(1 minus V) minus 1198712 (1 minus 2V)
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
Figure 4 Shape function of isoparametric four nodes
The force-displacement relation can be evaluated by
119865 = 119896119889 (6)
The strain-displacement relation is given by
120576 = 119861119889 (7)
The stress-strain relation can be obtained by
120590 = 119863120576 (8)
The discontinuous longitudinal reinforcement bar in theconcrete segments and tendons is modeled as two (2) noddedbar elements Each node of the truss is laterally restrained tothe adjacent node of segmented column The shape functionof reinforcement bar is evaluated as shown in Figure 5
120585 = minus1
N1 =1 minus 120585
2
120585 = 1
( ) N2 =1 + 120585
2( )
Figure 5 Shape function of bar element
The stiffness matrix of reinforcement can be obtained by
[119896119904] = int [119861]
119879
[119863] [119861] 119889V = int119897
[119861]119879
[119863] [119861]119860119889119909
= int119897
[119861]119879
[119863] [119861]119860 |119869| 119889120585
(9)
where
119889119883 = |119869| 119889120585 (10)
It is known as
120597119873119894
120597119909= [120597119909
120597120585]
minus1
times120597119873119894
120597120585
119869 =120597119909
120597120585
(11)
Generally the global stiffness of prestressed column in thisstudy is formulated by assembling stiffness matrix for eachconcrete segment prestressing tendon longitudinal bar andinterface element
4 Comparison of the ProposedConstitutive Model and FE Model
The contribution of tendons in precast concrete segmentsis investigated and new constitutive model is proposed andverified through modeling in FE software ANSYS
Solution ofmany engineering problems is based on linearapproximations and in this study the linear response isconsidered although developed constitutive law is applicablefor nonlinear analysis as well
Modelling and Simulation in Engineering 5
Longitudinal bar
KS1 KS2
Lateral load
Pin connection4 (0 0)
Axial load
Longitudinal bar
Concrete Concrete
2 (L2 H)
5 (L2 0)
3 (L H)
6 (L 0)
1 (0 H)
(a) Proposed constitutive model
Axial load
Pinnedconnection
Lateralload
XYZ
(b) FE model
Figure 6 One concrete segment model (RC1SS)
41 One Concrete Segment Figure 6(a) shows the constitutivemodel of one concrete segment where the longitudinal bars(KS1 KS2) are separated into sections equally and placed atboth sides of the concrete segment that is KS1 is sharingnode 1 and node 4 with the concrete segment in 119910 directionwhile KS2 is sharing node 3 and node 6 The length of thelongitudinal bar is the same as the height of single concretesegment The lateral load is applied at node 1 of concretesegment with load step of 100 kN up to 500 kN and 500 kNis vertically and equally applied at top nodes (ie 1 2 and 3)Figure 6(b) shows the finite element model for one concretesegment subjected to the lateral and axial loads
The stiffness matrix for one concrete segment with lon-gitudinal bar can be derived as shown in (12) The stiffnessmatrix variables 119860 119861 119862 119886 119887 119888 119889 119890 119891 and 119892 are defined inTable 2
A comparison between the constitutive model and FEmodel responses is carried out As can be observed inFigure 7 it can be proven that the proposed constitutivemodel for one concrete segment has a good agreement withthe FEmodel in plane stress analysis and plane strain analysisAs expected displacement in plane strain analysis is about80mm which is four times higher than the displacementin plane stress analysis and both approaches show linearbehavior
It can be seen from Table 3 that in plane stress analysis themaximum displacement by the proposed constitutive modelis close to that obtained by FE model with the former valuea little bit more than the latter one A maximum differenceof 46 is observed between constitutive model and FEresults In plane strain analysis the maximum displacement
is 7689mm in the proposed constitutivemodel however thecorrespondence value in FEmodel is 7394mmTable 3 showsabout four percent difference in plane strain analysis betweenproposed constitutivemodel and FE results for one segmentalconcrete whilst discrepancy percentage of aforementionedmethods in plane stress analysis is increased to 46
6 Modelling and Simulation in Engineering
0
100
200
300
400
500
600
0 20 40 60 80 100Displacement (mm)
Late
ral l
oadi
ng (k
N)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 7 Lateral load-deflection of the proposed constitutivemodeland FE model for RC1SS
42 Two Concrete Segments Two concrete segments areattached together as shown in Figure 8(a) with prestressingtendon placed at the middle of the concrete segment which
connects node 2 and node 8 in order to connect the concretesegments together (RC2SS)The intention of designing longi-tudinal reinforcement discontinuously is to avoid the fractureof mild steel at the critical joint opening when a huge lateralload is applied The developed constitutive model for RC2SSis examined under the following load condition as shownin Figure 8(b) The prestressing tendon role is to overcomeweakness of concrete in tension by providing the clampingload between the concrete segment and support
From (13) the stiffness matrix for the prestressing tendoncan be derived as
[119870119905] =
119864119905119860119905
119867
8119910 2119910
[
1
minus1
minus1
1]
119905 =119864119905119860119905
119867times
8119871119867(1 minus V2)119864119905
(plane stress)
119905 =119864119905119860119905
119867times16119871119867 (1 + V) 119861
119864119905(plane strain)
(13)
So by substituting these parameters in (14) the stiffnessmatrix can be derived as
Comparison between the constitutive and FE models for twoconcrete segments RC2SS is shown in Figure 9 From thisfigure it can be seen that there is a good agreement betweenthe proposed constitutive model results for RC2SS and theFE model results in plane stress and plane strain analysis
Consequently it can be determined that the maximumdisplacement in plane strain analysis is about 200mm whichis almost three times higher than the displacement in planestress analysis Linear behavior has been captured for bothaforementioned approaches
Modelling and Simulation in Engineering 7
Table 2 Stiffness matrix variables
Plane stress condition Plane strain condition
119860 =1
2(1 minus V) 119861 = (1 minus 2V)
119862 =119864119905
8119871119867(1 minus V2)119862 =
119864119905
16119871119867(1 + V)119861
119886 = 41198672
+ 1198601198712
119886 = 81198672
(1 minus V) + 1198712 (1 minus 2V)
119887 = 41198672
minus 1198601198712
119887 = 81198672
(1 minus V) minus 1198712 (1 minus 2V)
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
41 One Concrete Segment Figure 6(a) shows the constitutivemodel of one concrete segment where the longitudinal bars(KS1 KS2) are separated into sections equally and placed atboth sides of the concrete segment that is KS1 is sharingnode 1 and node 4 with the concrete segment in 119910 directionwhile KS2 is sharing node 3 and node 6 The length of thelongitudinal bar is the same as the height of single concretesegment The lateral load is applied at node 1 of concretesegment with load step of 100 kN up to 500 kN and 500 kNis vertically and equally applied at top nodes (ie 1 2 and 3)Figure 6(b) shows the finite element model for one concretesegment subjected to the lateral and axial loads
The stiffness matrix for one concrete segment with lon-gitudinal bar can be derived as shown in (12) The stiffnessmatrix variables 119860 119861 119862 119886 119887 119888 119889 119890 119891 and 119892 are defined inTable 2
A comparison between the constitutive model and FEmodel responses is carried out As can be observed inFigure 7 it can be proven that the proposed constitutivemodel for one concrete segment has a good agreement withthe FEmodel in plane stress analysis and plane strain analysisAs expected displacement in plane strain analysis is about80mm which is four times higher than the displacementin plane stress analysis and both approaches show linearbehavior
It can be seen from Table 3 that in plane stress analysis themaximum displacement by the proposed constitutive modelis close to that obtained by FE model with the former valuea little bit more than the latter one A maximum differenceof 46 is observed between constitutive model and FEresults In plane strain analysis the maximum displacement
is 7689mm in the proposed constitutivemodel however thecorrespondence value in FEmodel is 7394mmTable 3 showsabout four percent difference in plane strain analysis betweenproposed constitutivemodel and FE results for one segmentalconcrete whilst discrepancy percentage of aforementionedmethods in plane stress analysis is increased to 46
6 Modelling and Simulation in Engineering
0
100
200
300
400
500
600
0 20 40 60 80 100Displacement (mm)
Late
ral l
oadi
ng (k
N)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 7 Lateral load-deflection of the proposed constitutivemodeland FE model for RC1SS
42 Two Concrete Segments Two concrete segments areattached together as shown in Figure 8(a) with prestressingtendon placed at the middle of the concrete segment which
connects node 2 and node 8 in order to connect the concretesegments together (RC2SS)The intention of designing longi-tudinal reinforcement discontinuously is to avoid the fractureof mild steel at the critical joint opening when a huge lateralload is applied The developed constitutive model for RC2SSis examined under the following load condition as shownin Figure 8(b) The prestressing tendon role is to overcomeweakness of concrete in tension by providing the clampingload between the concrete segment and support
From (13) the stiffness matrix for the prestressing tendoncan be derived as
[119870119905] =
119864119905119860119905
119867
8119910 2119910
[
1
minus1
minus1
1]
119905 =119864119905119860119905
119867times
8119871119867(1 minus V2)119864119905
(plane stress)
119905 =119864119905119860119905
119867times16119871119867 (1 + V) 119861
119864119905(plane strain)
(13)
So by substituting these parameters in (14) the stiffnessmatrix can be derived as
Comparison between the constitutive and FE models for twoconcrete segments RC2SS is shown in Figure 9 From thisfigure it can be seen that there is a good agreement betweenthe proposed constitutive model results for RC2SS and theFE model results in plane stress and plane strain analysis
Consequently it can be determined that the maximumdisplacement in plane strain analysis is about 200mm whichis almost three times higher than the displacement in planestress analysis Linear behavior has been captured for bothaforementioned approaches
Modelling and Simulation in Engineering 7
Table 2 Stiffness matrix variables
Plane stress condition Plane strain condition
119860 =1
2(1 minus V) 119861 = (1 minus 2V)
119862 =119864119905
8119871119867(1 minus V2)119862 =
119864119905
16119871119867(1 + V)119861
119886 = 41198672
+ 1198601198712
119886 = 81198672
(1 minus V) + 1198712 (1 minus 2V)
119887 = 41198672
minus 1198601198712
119887 = 81198672
(1 minus V) minus 1198712 (1 minus 2V)
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 7 Lateral load-deflection of the proposed constitutivemodeland FE model for RC1SS
42 Two Concrete Segments Two concrete segments areattached together as shown in Figure 8(a) with prestressingtendon placed at the middle of the concrete segment which
connects node 2 and node 8 in order to connect the concretesegments together (RC2SS)The intention of designing longi-tudinal reinforcement discontinuously is to avoid the fractureof mild steel at the critical joint opening when a huge lateralload is applied The developed constitutive model for RC2SSis examined under the following load condition as shownin Figure 8(b) The prestressing tendon role is to overcomeweakness of concrete in tension by providing the clampingload between the concrete segment and support
From (13) the stiffness matrix for the prestressing tendoncan be derived as
[119870119905] =
119864119905119860119905
119867
8119910 2119910
[
1
minus1
minus1
1]
119905 =119864119905119860119905
119867times
8119871119867(1 minus V2)119864119905
(plane stress)
119905 =119864119905119860119905
119867times16119871119867 (1 + V) 119861
119864119905(plane strain)
(13)
So by substituting these parameters in (14) the stiffnessmatrix can be derived as
Comparison between the constitutive and FE models for twoconcrete segments RC2SS is shown in Figure 9 From thisfigure it can be seen that there is a good agreement betweenthe proposed constitutive model results for RC2SS and theFE model results in plane stress and plane strain analysis
Consequently it can be determined that the maximumdisplacement in plane strain analysis is about 200mm whichis almost three times higher than the displacement in planestress analysis Linear behavior has been captured for bothaforementioned approaches
Modelling and Simulation in Engineering 7
Table 2 Stiffness matrix variables
Plane stress condition Plane strain condition
119860 =1
2(1 minus V) 119861 = (1 minus 2V)
119862 =119864119905
8119871119867(1 minus V2)119862 =
119864119905
16119871119867(1 + V)119861
119886 = 41198672
+ 1198601198712
119886 = 81198672
(1 minus V) + 1198712 (1 minus 2V)
119887 = 41198672
minus 1198601198712
119887 = 81198672
(1 minus V) minus 1198712 (1 minus 2V)
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
119888 = 2119867119871 (V + 119860) 119888 = 2119867119871
119889 = 2119867119871 (V minus 119860) 119889 = 2119867119871 (4V minus 1)
119890 = 1198712
+ 41198672
119860 119890 = 21198712
(1 minus V) + 41198672 (1 minus 2V)
119891 = 1198712
minus 41198672
119860 119891 = 21198712
(1 minus V) minus 41198672 (1 minus 2V)
119892 =119864119904119860119904
2119867times8119871119867(1 minus V2)
119864119905119892 =119864119904119860119904
2119867times16119871119867(1 + V)119861
119864119905
Table 4 represents the percentage difference between twotypes of analysis on two concrete segments In plane stressanalysis maximum displacements of 7446mm for the pro-posed constitutive model and 7394mm in FE program areachieved subjected to applied load Likewise in plane strainanalysis applied forces produce 19121mm and 19419mmdisplacements by proposed constitutivemodel and FEmodelrespectively Table 4 shows almost 065 difference betweentwo approaches in plane stress analysis for different applied
load however this percentage difference in plane strainanalysis for two concrete segments is diminished from 673to 153 owing to the applied load from 100 kN to 500 kN asa lateral load and 500 kN as a vertical load
43 Three Concrete Segments The PPCS column consistingof three concrete segments is shown in Figure 10(a)The con-stitutivemodel has 12 nodes where each node has two degreesof freedom and resulting stiffness matrix size is 24 times 24 Theprestressing tendons are pulled between node 2 and node 11There are discontinuous bars at both sides of the concretesegment The stiffness matrix for three segmental precastcolumns is developed by assembling the stiffness matrixesand presented by (15) Based on the developed constitutivemodel the PPCS column with three concrete segments ismodeled by only 15 elements which reduce computation timeand convergence issues of analysis considerably
The analysis result of RC3SS is plotted in Figure 11 andtabulated in Table 5 It is clear from these plots that theconstitutive model has smaller displacement than the FEmodel As can be observed in Figure 11 there is a goodagreement between the proposed constitutive model and theFE model in plane stress and plane strain analysis of RC3SSConsequently it can be found that displacement in planestress analysis is about 200mm and it is almost doubled inplane strain
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 9 Lateral load-deflection of the proposed constitutivemodeland FE model for RC2SS
A comparison between the constitutive model and FE modelresults is carried out under plane stress and plane strainanalysis as shown in Table 5 The displacement increasesclearly with applied load for instance in plane stress anal-ysis load applied is the same as in previous steps andresults in 20502mm and 21269mm displacements in theproposed constitutive and FE models respectively Likewisein plane strain analysis applied force produces 38858mmand 41478mm displacements at node 1 in the proposedconstitutive and FE models respectively
A maximum difference of 36 is observed between theresults of the constitutive model and FE program for planestress analysis of RC3SS model Also it can be seen thatthe constitutive model provides reasonable correlation withthe FE program results under plane strain analysis for PPCScolumns with three concrete segments
Based on the outputs it is clear from the plots thatincreasing the load increases the maximum displacement inplane stress or plane strain analysis It can be determinedthat in plane stress analysis with the proposed constitutivemodel applying load incrementally up to 500 kNwith 100 kNas each load step in a lateral direction and 500 kN as avertical load results in almost 20mm 75mm and 200mmas maximum displacements which are close to the FEMresults with about 46 07 and 36 difference in onetwo and three concrete segments respectively Likewise theproposed constitutive model in plane strain analysis showsabout 77mm 191mm and 388mmdisplacement in one twoand three concrete segments subjected to imposed loadingwhile the FE program shows 74mm and 194mm to 414mmas maximum displacements by 4 153 and 6 differencerespectively
Short-term stress losses may happen due to wobble andcurvature frictions and anchorage slip On the other handlong-term stress losses include relaxation elastic shorteningand losses due to creep shrinkage and superimposed loads
Although in this study effects of losses are not consideredhowever as a matter of fact it can be easily incorporated intothe stiffnessmatrix of prestress concrete particularly in elastic
modulus of tendon and concrete individually as a degradedelastic modulus before forming global matrix (refer to (14)and (15))
Short-term losses also could be considered in stiffnessmatrix of prestressed concrete element as a multiplayer of theyield stress of tendon which varies between 07 and 09 Theeffective strand strain after transfer is slightly overestimatedbecause the model considered low-relaxation prestressingstrand and does not take into account the long-term prestresslosses due to concrete creep and shrinkage
5 Conclusions
Modeling of PPCS column is not easy using availablecommercial software packages because all the parts shouldbe modeled separately Moreover the interaction betweendifferent parts should be defined which not only increasescomputation time but also requires experts
In this study an attempt was made to develop theconstitutive model for PPCS columns in bridge structuresMoreover FEM model for one two and three segments wasdeveloped in order to perform linear and nonlinear analysisfor PPCS column subjected to static and dynamic loading
A special FEM program was codified by using developedPPCS columns and the accuracy of the developed consti-tutive model was evaluated by comparing analysis resultswith commercial software Using this method would not onlyreduce modeling and computation time but also consider-ably facilitate the convergence problems due to diminishingmeshing adaption issues by substituting meshing assemblageto one explicit matrix in which all characteristics of meshesare incorporated The PPCS column is applicable as a sup-plementary subroutine in any structural analysis software orcan be used to develop any FEM program code for analysis ofPPCS columns
] Poissonrsquos ratio119891119888 Stress of concrete in compression119891119905 Stress of concrete in tension119891119910119904 Yield stress of reinforcing bar
119891119910119901 Yield stress of prestressing steel
119891119888119906 Ultimate compressive strength of concrete
119896119888 Stiffness matrix of concrete119896119904 Stiffness matrix of reinforcing bar119896119905 Stiffness matrix of prestressing steel119861 Strain-displacement matrix119863 Constitutive matrix119873 Shape function119869 Jacobian matrix120576119897119901 Prestress steel limit
120576119901 Strain of prestressing steel120585 Natural coordinate
10 Modelling and Simulation in Engineering
Lateral load
Concrete
KS1
KS3
Prestressedtendon
Axial load
1
4
7
2 3
5 6
8 9
10 11 12
Concrete Concrete
Concrete Concrete
Concrete
Longitudinal bar
KS2
KS4
KS5 KS6
Pin connection(a) Proposed constitutive model
Pin
Lateral load Axial load
XYZ
2nd concretesegmental
3rd concretesegmental
1st concretesegmental
connection
Prestressedtendon
(b) FE model
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
Figure 10 Three concrete segmentsrsquo model (RC3SS)
0
100
200
300
400
500
600
0 100 200 300 400 500
Late
ral l
oadi
ng (k
N)
Displacement (mm)
FE modellowastlowastConstitutive modellowastlowastFE modellowastConstitutive modellowast lowastPlane stress
lowastlowastPlane strain
Figure 11 Lateral load-deflection of the proposed constitutivemodel and FE model for RC3SS
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Acknowledgments
This work received financial support from the Ministry ofHigher Education ofMalaysia under FRGS Research Projectsno 5524254 and no 5524256 and was further supported bythe University PutraMalaysia under Putra Grant no 9415100These supports are gratefully acknowledged
References
[1] Y-C Ou M Chiewanichakorn A J Aref and G C Lee ldquoSeis-mic performance of segmental precast unbonded posttensioned
concrete bridge columnsrdquo Journal of Structural Engineering vol133 no 11 pp 1636ndash1647 2007
[2] K Chang C H Loh H S Chiu et al Seismic Behavior ofPrecast Segmental Bridge Columns and Design Methodologyfor Applications in Taiwan Taiwan Area National ExpresswayEngineering Bureau Taipei Taiwan 2002 (Chinese)
[3] T-H Kim H-M Lee Y-J Kim and H M Shin ldquoPerformanceassessment of precast concrete segmental bridge columns witha shear resistant connecting structurerdquo Engineering Structuresvol 32 no 5 pp 1292ndash1303 2010
[4] J T Hewes andM N Priestley Seismic Design and Performanceof Precast Concrete Segmental Bridge Columns University ofCalifornia Oakland Calif USA 2002
[5] S L Billington and J K Yoon ldquoCyclic response of unbondedposttensioned precast columns with ductile fiber-reinforcedconcreterdquo Journal of Bridge Engineering vol 9 no 4 pp 353ndash363 2004
[6] M J Nigel Priestley ldquoOverview of PRESSS research programrdquoPCI Journal vol 36 no 4 pp 50ndash57 1991
[7] W C Stone G S Cheok and J F Stanton ldquoPerformanceof hybrid moment-resisting precast beam-column concreteconnections subjected to cyclic loadingrdquoACI Structural Journalvol 92 no 2 pp 229ndash249 1995
[8] Y Wang Z Y Bu and L Hu ldquoSeismic behavior of precastsegmental bridge columns with carbon fibre reinforcement asenergy dissipation barsrdquo Applied Mechanics and Materials vol157 pp 1148ndash1152 2012
[9] W-P Kwan and S L Billington ldquoUnbonded posttensionedconcrete bridge piers II Seismic analysesrdquo Journal of BridgeEngineering vol 8 no 2 pp 102ndash111 2003
[10] J Sakai and S A Mahin Analytical Investigations of NewMethods for Reducing Residual Displacements of Reinforced Con-crete Bridge Columns Pacific Earthquake Engineering ResearchCenter University of California at Berkeley Berkeley CalifUSA 2004
[11] A Palermo S Pampanin and G M Calvi ldquoConcept anddevelopment of hybrid solutions for seismic resistant bridgesystemsrdquo Journal of Earthquake Engineering vol 9 no 6 pp899ndash921 2005
Modelling and Simulation in Engineering 11
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
[12] Y-C Ou M-S Tsai K-C Chang and G C Lee ldquoCyclicbehavior of precast segmental concrete bridge columns withhigh performance or conventional steel reinforcing bars asenergy dissipation barsrdquo Earthquake Engineering amp StructuralDynamics vol 39 no 11 pp 1181ndash1198 2010
[13] A Palermo S Pampanin and D Marriott ldquoDesign modelingand experimental response of seismic resistant bridge piers withposttensioned dissipating connectionsrdquo Journal of StructuralEngineering vol 133 no 11 pp 1648ndash1661 2007
[14] M A Elgawady and A SharsquoLan ldquoSeismic behavior of self-centering precast segmental bridge bentsrdquo Journal of BridgeEngineering vol 16 no 3 pp 328ndash339 2011
[15] R Vaghei F Hejazi H Taheri M S Jaafar and A A Ali ldquoEval-uate performance of precast concrete wall to wall connectionrdquoAPCBEE Procedia vol 9 pp 285ndash290 2014
[16] Z Wang W Song Y Wang and H Wei ldquoNumerical analyticalmodel for seismic behavior of prestressing concrete bridgecolumn systemsrdquo Procedia Engineering vol 14 pp 2333ndash23402011
[17] H Dawood M ElGawady and J Hewes ldquoBehavior of seg-mental precast posttensioned bridge piers under lateral loadsrdquoJournal of Bridge Engineering vol 17 no 5 pp 735ndash746 2012
[18] Z-Y Bu and Y-C Ou ldquoSimplified analytical pushover methodfor precast segmental concrete bridge columnsrdquo Advances inStructural Engineering vol 16 no 5 pp 805ndash822 2013
[19] C-C Chou H-J Chang and J T Hewes ldquoTwo-plastic-hingeand two dimensional finite element models for post-tensionedprecast concrete segmental bridge columnsrdquo Engineering Struc-tures vol 46 pp 205ndash217 2013
[20] E Nikbakht K Rashid F Hejazi and S A Osman ldquoAnumerical study on seismic response of self-centring precastsegmental columns at different post-tensioning forcesrdquo LatinAmerican Journal of Solids and Structures vol 11 no 5 pp 864ndash883 2013
[21] E Nikbakht K Rashid F Hejazi and S A Osman ldquoApplicationof shape memory alloy bars in self-centring precast segmentalcolumns as seismic resistancerdquo Structure and InfrastructureEngineering vol 11 no 3 pp 297ndash309 2015
[22] J B Mander M J Priestley and R Park ldquoTheoretical stress-strain model for confined concreterdquo Journal of Structural Engi-neering vol 114 no 8 pp 1804ndash1826 1988
[23] G Strang andG J FixAnAnalysis of the Finite ElementMethodPrentice-Hall Englewood Cliffs NJ USA 1973
