Analytical Model for Deflections of Bonded Posttensioned ...4...

Research ArticleAnalytical Model for Deflections of Bonded PosttensionedConcrete Slabs

Min Sook Kim1 Joowon Kang2 and Young Hak Lee1

1Department of Architectural Engineering Kyung Hee University 1732 Deogyeong-daero Yongin Republic of Korea2School of Architecture Yeungnam University 280 Daehak-ro Gyeongsan Republic of Korea

Correspondence should be addressed to Young Hak Lee leeyhkhuackr

Received 13 December 2016 Accepted 7 February 2017 Published 5 March 2017

Academic Editor Francesco Caputo

Copyright copy 2017 Min Sook Kim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a finite element analysis approach to evaluate the flexural behavior of posttensioned two-way slabs dependingon the tendon layout A finite element model was established based on layered and degenerated shell elements Nonlinearities ofthe materials are considered using the stress-strain relationships for concrete reinforcing steel and prestressing tendons Flexuraltesting of the posttensioned two-way slabs was conducted to validate the developed analytical process Comparing the analyticalresults with the experimental results in terms of deflections it showed generally good agreements Also a parametric study wasperformed to investigate the effects of different types of tendon layout

1 Introduction

Posttensioned concrete slabs have many advantages suchas rapid construction reduction of overall member depthand reduced materials In addition posttensioned concreteslabs with proper posttensioning show little deflection andfew cracks under service loads Although posttensionedconcrete slabs have many advantages their performance isstill not fully understood and the behaviors of two-wayslab systems are more difficult to determine than thoseof one-way slabs To evaluate posttensioned concrete slabsseveral experimental studies have been performed Burnsand Hemakom [1] observed the strength and behavior ofposttensioned flat plates They applied the banded tendonlayout in column strips in the x-direction and distributedsingle tendons in the y-direction on the slabs Through thisstudy they found that the banded and distributed tendonlayout improved the flexural and shear capacities Kosut et al[2] experimentally evaluated the behavior of posttensionedflat plates with distributed and banded tendon arrangementsin each direction They found that banded tendons on thecolumn strip can resist punching shear failure and dis-tributed tendons can improve flexural capacity Roschke andInoue [3] tested prestressed concrete flat slabs to investigate

strain distribution in regions adjacent to the transverse post-tensioning bands

To analyze complex posttensioned concrete slabs effi-ciently some researchers have proposed finite elementapproaches Van Greunen and Scordelis [4] researched anumerical procedure for the materials and a geometricnonlinear analysis for prestressed concrete slabs Wu et al[5] proposed a tendon model based on the finite elementmethod that can represent the interaction between tendonsand concrete They verified the accuracy of their proposedequation against existing experimental data El-Mezaini andCitipitioglu [6] developed quadratic and cubic finite elementswith movable nodes to predict the behavior of different bondconditions for the tendons Kang and Huang [7] proposednonlinear finite element models to evaluate the behaviorof unbonded posttensioned slab-column connections Thespring elements and contact formation were applied to themodel to consider the interface between the concrete andprestressing tendon Kang et al [8] compared the structuralperformance of the bonded and unbonded posttensionedconcrete members through experiment and analysis Ghallab[9] suggested using simple equations to predict the pre-stressing tendons at ultimate stage of continuous concretebeams The simple equations were verified by comparing

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 3037946 7 pageshttpsdoiorg10115520173037946

2 Advances in Materials Science and Engineering

with existing experimental data Although much analyticalresearch has been performed to evaluate the behavior ofbonded and unbonded posttensioned concrete membersa relatively limited number of studies have been reportedfor the prediction of flexural behavior considering tendonlayouts

Two-way slab systems offer several possible arrangementsfor the tendon layout [10] banded distributed or a mixedlayout Posttensioned slabs are used for long spans and heavylive loads so flexural strength and ductility are importantFlexural strength usually governs the behavior of the interiorpanel in two-way slabs In other words the distribution oftendons can affect the flexural behavior and ductility of theinterior panel of the two-way slabsThoughmany researchershave focused on the development of finite element modelslittle information is available on the flexural behavior of post-tensioned two-way slabs with different tendon layouts In thisstudy examined was the flexural behavior of posttensionedtwo-way slabs depending on the tendon layout

The objective of this paper is to present an efficientnumerical analysis approach for the materials and a geomet-ric nonlinear analysis for the posttensioned two-way slabs Inthis study developed was a nonlinear finite element modelthat can simulate the behavior of posttensioned two-wayslabs The reinforced concrete was modeled as combinationof concrete steel and prestressing tendons The test resultswere compared with those from the finite element model aswell

2 Finite Element Model

21 Layered Element Formulation A finite element modelformulated using layered and degenerate shell elements canbe used in a three-dimensional global analysis of structuresEight-node isoparametric degenerated shell elements wereformulated following the procedure ofHinton andOwen [11]It was assumed that plane cross-sections remain both planeand normal during bending Layered elements were appliedto account for the behaviors of the reinforced concrete mem-bers which exhibited different properties in the thicknessdirection because of the placement of the reinforcementsEach element is divided into layers and each layer has oneintegration point on its midsurface Each layer was com-posed of different materials concrete steel reinforcementand prestressing tendon are defined separately The strainsand stresses are calculated at midpoint of each layer Thestrain-displacement matrix and the constitutive matrix arecalculated at the midpoint of each layer Stress resultants areevaluated by integrating the corresponding stress Normalforces and bending moments can be obtained by

119873119909(119910) = intℎ2minusℎ2

120590119909(119910)119889119911 = ℎ2119899sum119894=1

120590119894119909(119910)Δ120577119894

119872119909(119910)(119909119910) = minusintℎ2minusℎ2120590119909(119910)(119909119910)119911 119889119911

= minusℎ24119899sum119894=1

120590119897119909(119910)(120591119909119910)120577119894Δ120577119894

(1)

k = 067 +62

k = 1

0

fc

fcmaxfcmax

Figure 1 Stress-strain relationship in compression by Collins andPorasz (1989)

where 119873119909 is normal force 119872119909 is bending moment 120590 isnormal stress ℎ is layer depth and 119899 is number of layers

The reinforcement layers were used tomodel the in-planereinforcement Transverse reinforcement can be specified asa property of a concrete layer The prestressing tendons at asingle depth were grouped with the same prestressing forceinto one layer [12] The concrete steel reinforcement andprestressing tendons were assumed to be perfectly bondedThe perfect bond is applicable to the analysis of reinforcedconcrete and posttensioned concrete with bonded tendonsThe same degrees of freedom were assigned to concrete andreinforcement nodes occupying a single location

22 Constitutive Models In this paper the failure of concretetwo-way slabs is considered to be tension cracking or plasticyielding of reinforcement Uncracked concrete was assumedto be a linear elasticmaterial After cracking the concrete wastreated as an orthotropic material The total material matrixconsists of concrete steel reinforcement and prestressingtendons

Figure 1 shows the stress-strain relationship for concretein compression [13] The compressive stress of concrete canbe calculated by

119891119888 = 1198911198882max (1198991205761205761015840119888)(119899 minus 1) + (1205761205761015840119888)119899119896 119891119888max = 119891101584011988808 minus 034 (1205761198881205761015840119888) le 1198911015840119888

119899 = 08 + 1198911198882max17 (2)

where119891119888 is the concrete stress1198911198882max is the compressive stressof cracked concrete 120576119888 is the concrete strain 1205761015840119888 is the concretestrain corresponding to peak compressive stress and1198911015840119888 is thecompressive cylinder strength of concrete

After cracking the stiffness of the reinforced concretedecreases but it does not drop to zero because the intact

Advances in Materials Science and Engineering 3

minus 1

fcr

ft

Ec

Ec

cr ccr

Figure 2 Average stress-strain relationship proposed by Lin andScordelis (1975)

fy

fs = fy

fs = Ess

y

Figure 3 Stress-strain relationship of steel reinforcement

concrete between adjacent cracks can still carry some tensilestress due to the bond between the reinforcement and sur-rounding concrete In this paper tension stiffening wasmodeled as a constitutive model [14] as shown in Figure 2

The steel reinforcement was considered as steel layerswith uniaxial behavior A bilinear model was adopted for theelastoplastic stress-strain relationship as shown in Figure 3The stress-strain relationship was characterized by Youngrsquosmodulus 119864119904 and the uniaxial yield stress 119891119910

For the prestressing tendonMenegotto and Pintorsquosmodel[15] was adopted (Figure 4) The mathematical expression isgiven as follows

120590119901 = 119864119901120576119875[[[119876 + 1 minus 119876

[1 + (119864119901120576119901119870119891119901119910)119873]1119873]]]

119876 = 119891119901119906 minus 119870119891119901119910119864119901120576119901119906 minus 119870119891119901119910 (3)

where 120590119901 is the stress of a prestressing tendon 119864119901 is Youngrsquosmodulus of the prestressing tendon 120576119901 is the strain of the

fps

fpu

fpy

py pu

Figure 4 Constitutive model of prestressing tendon proposed byMenegotto and Pinto (1973)

prestressing tendon 119891119901119910 is the yield stress 119891119901119906 is the ultimateyield stress 120576119888119906 is the ultimate strain and 119873 119870 and 119876 areempirical parameters whose values were recommended byNaaman [16] as 606 10325 and 000625 respectively

23 Analysis Procedures Directmethod is applied as solutionalgorithm [11] In each iterative step the full load is applied tothe structures The obtained unknowns are the full displace-ments In the first iteration the materials have linearly elasticbehavior and the initial displacements are zero After thatthe model can calculate the new stiffness matrix consideringthe appropriate material constitutive models Full load isreapplied to the model then the stiffness matrix is updatedand displacement can be found The steps of evaluation andupdate of stiffness matrix are repeated until the satisfiedconvergence condition

The displacement criterion was selected as the conver-gence conditionThe displacement convergence is as follows

radicΣ(119863119886 minus 119863119901)2Σ1198632119886 times 100 lt 119879 (4)

where 119863119886 is current step displacement 119863119901 is previous stepdisplacement and 119879 is tolerance

Large tolerance value can lead to inaccurate results andthe tolerance was set to 08

3 Experimental Program

To validate the suggested finite element model flexuraltesting was performedThe designed compressive strength ofthe concrete used for the fabrication of the specimens was35MPa The average compressive strength measured at 28days was 367MPa Deformed steel bars with a diameter of13mm and 10mm were used for longitudinal reinforcementsand stirrups respectivelyTheir tensile strength andmodulusof elasticity were 400MPa and 200GPa respectively Sevenwire steel-strand tendons with 127mm of diameter wereused Their nominal ultimate tensile strength was 1860MPa


Tendon350 D13280

D10

rei

nfor

cem

ent

25016882

3000 mm300

0m

m

(a) PT-x specimen

Tendon350

Tend

on

350

D13280

D10

rei

nfor

cem

ent

25016882

3000 mm

300

0m

m

(b) PT-xy specimen

Figure 5 Details of the specimens

Table 1 Details of specimens

Specimen 119860 119904119909 (mm2) 119860 119904119910 (mm2) 119891119901119890 (MPa) 119891119901119890119891119901119906 120588119901119909 () 120588119901119910 () 120588119904119909 () 120588119904119910 () 119889119901 (mm) 1198911015840119888 (MPa)PT-x 4935 mdash 1488 08 0198 mdash 0185 0185 168 367PT-xy 4935 4935 1488 08 0198 0198 0185 0185 168

Two posttensioned two-way slabs were manufactured forthe test One specimen had tendons distributed in only thex-direction (PT-x) and the other had tendons distributed inboth the x- and y-directions (PT-xy) Details of the specimensare presented in Figure 5 and summarized in Table 1 Thesize of the test specimens was 3000mm times 3000mm with250mm thickness Both specimens were posttensioned witha constant eccentricity of 43mm A prestressing force of1488MPa was applied corresponding to approximately 80of the tensile strength of the tendon

Load was applied to each specimen using a hydraulic jackwith maximum capacity of 5000 kNThe test specimens weresimply supported along the four sidesThe force generated bythe hydraulic jack was transmitted to a loading plate placedat the middle of the specimen The distance from support toloading point was 125m giving a shear span to depth ratio of6

4 Numerical Modeling and Discussion

In order to evaluate the accuracy of the analytical model thetheoretical value is compared with test results Comparisonis made in terms of load-deflections curves The analyticalmodel sizewas set tomatch the posttensioned specimens andthe model contained 100 elements It was divided into ninelayers in the direction of different thickness The thicknessesof the first and last layers were determined by consideringthe concrete cover The steel reinforcements were placed atthe 2nd and 8th layers in the direction of the thicknessThe prestressing tendons were placed at the 6th layer in the

direction of the thickness In this paper Youngrsquos modulus ofconcrete (119864119888) was determined according toACI 318 [17] Pois-sonrsquos coefficient (]) is 015 and the equation to compute themodulus of rupture of concrete is 032radic1198911015840119888 (Table 2) To sim-ulate the experimental support conditions four sides of slabsare simply supported and the load is applied to the center ofthe slab

Themodel values were comparedwith load-displacementcurves obtained from the posttensioned two-way slabs underflexural loads The comparisons between the test results andthe analytical results are shown in Figures 6 and 7 As shownin the Figures both test and analytical results indicated thatthe load-displacement curves exhibit three stages elasticcracking and plastic Overall the analytical model predictedthe deflection of the posttensioned specimens in a relativelyaccurate manner However in all cases the models showa slightly stiffer response at the cracking stage becauseprestressing loss and slip were not considered in the finite ele-ment analysis Both the testing and analytical results showedthat the one-way and two-way prestressing tendon layoutdid not significantly affect the maximum load capacity anddeflection

The validated finite elementmodel was used to investigatethe effects of changes in span length height and concretestrength In total nine posttensioned slabs were analyzed asshown in Table 3 To perform the parametric analysis thesame geometry and material properties were used same asin the verification of the proposed model along with theone-way tendon layout The same load to each model was


Table 2 Material properties used in the finite element analysis

Concrete Steel and prestressing tendon1198911015840119888 (MPa) 119891119905 (MPa) 119864119888 (MPa) 120576119888119903 ]119888 119891119910 (MPa) 119864119904 (GPa) ]119904367 38 284729 0002 015 360 210 025

Table 3 Analytical dimensions and results of parametric study

Model Span Height Compressive strength of concrete Displacement(mm) (mm) (MPa) (mm)

PT1 3000 250 35 4177PT2 4500 250 35 4854PT3 6000 250 35 5613PT4 7500 250 35 6720PT5 3000 180 35 4362PT6 3000 350 35 4014PT7 3000 450 35 3732PT8 3000 250 50 4033PT9 3000 250 65 3816

PT-xPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)

10 20 30 40 500Displacement (mm)

Figure 6 Load-displacement relations of PT-x specimen

applied and compared the results in terms of deflectionsFigure 8ndash10 show the displacement according to span lengthslab height and the compressive strength of the concreteThedisplacement increased with the span length and decreasedas the height and concrete strength increased The resultsof the parametric analysis indicate that the deflection ofposttensioned two-way slab is more affected by the spanlength than by the other variables

5 Conclusions

In this paper the flexural behavior of posttensioned concretetwo-way slab was analytically investigated A finite elementanalysis model was proposed to predict the flexural behaviorof specimens depending on the tendon layout and conductedflexural tests to evaluate the validity and applicability of the

PT-xyPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)


Figure 7 Load-displacement relations of PT-xy specimen

750060003000 4500Span (mm)

35

40

45

50

55

60

65

70

Max

imum

disp

lace

men

t (m

m)

Figure 8 Effect of span length on displacement


36

39

42

45

Max

imum

disp

lace

men

t (m

m)

180 350 450250

Height (mm)

Figure 9 Effect of height of slabs on displacement

Compressive strength of concrete (MPa)

36

39

42

45

Max

imum

disp

lace

men

t (m

m)

655035

Figure 10 Effect of compressive strength of concrete on displace-ment

proposed model The following specific conclusions weredrawn from this study

(1) A finite element analysismodel has been developed toevaluate the flexural behavior of posttensioned two-way slabs considering tendon layout The proposedfinite element model which considers the nonlinearbehavior of concrete and reinforcement and neglectsthe bond slip and loss of prestressing force givesrelatively good predictions for the load-deflectioncurves

(2) The increase in themaximum load capacity was unaf-fected by tendon layouts Regardless of the directionof the tendon the load-displacement curves indicatedsimilar responses The analytical response is slightlystiffer than the test results at the cracking stagespossibly because of bond slip and the loss of prestress-ing force Consideration of bond slip and the lossof prestressing force could improve the model accu-racy However the difference in deflection betweenthe analytical and test results is relatively smallThe proposed finite element analysis demonstrates

the rationality of the posttensioning two-way slabsmodel

(3) To investigate the effects of span length slab heightand concrete strength on posttensioned two-wayslabs a parametric study was conducted The displa-cement increased with the span length and decreasedas the concrete strength and member height inc-reasedThe span length height and concrete strengthall contribute to the flexural strength

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by a National Research Foundationof Korea (NRF) grant funded by the Korean Government(MSIP) (NRF-2013R1A2A2A01067754)

References

[1] N H Burns and R Hemakom ldquoTest of post-tensioned flat platewith banded tendonsrdquo Journal of Structural Engineering vol 111no 9 pp 1899ndash1915 1985

[2] G M Kosut N H Burns and C V Winter ldquoTest of four-panelpost-tensioned flat platerdquo Journal of Structural Engineering vol111 no 9 pp 1916ndash1929 1985

[3] P N Roschke andM Inoue ldquoEffects of banded post-tensioningin prestressed concrete flat slabrdquo Journal of Structural Engineer-ing vol 117 no 2 pp 563ndash583 1991

[4] J Van Greunen and A C Scordelis ldquoNonlinear analysis of pre-stressed concrete slabsrdquo Journal of Structural Engineering vol109 no 7 pp 1742ndash1760 1983

[5] X-H Wu S Otani and H Shiohara ldquoTendon model fornonlinear analysis of prestressed concrete structuresrdquo Journalof Structural Engineering vol 127 no 4 pp 398ndash405 2001

[6] N El-Mezaini and E Citipitioglu ldquoFinite element analysis ofprestressed and reinforced concrete structuresrdquo Journal ofStructural Engineering vol 117 no 10 pp 2851ndash2864 1991

[7] T H K Kang and Y Huang ldquoNonlinear finite element analysesof unbonded post-tensioned slab-column connectionsrdquo PTIJournal vol 8 no 1 pp 4ndash19 2012

[8] T H-K Kang Y Huang M Shin J D Lee and A S CholdquoExperimental and numerical assessment of bonded and un-bonded post-tensioned concrete membersrdquo ACI StructuralJournal vol 112 no 6 pp 735ndash748 2015

[9] A Ghallab ldquoCalculating ultimate tendon stress in externallyprestressed continuous concrete beams using simplified formu-lasrdquo Engineering Structures vol 46 pp 417ndash430 2013

[10] B O Alami ldquoLayout of post-tensioning and passive reinforce-ment in floor slabsrdquo PTI Technical Notes 8 1999

[11] E Hinton and D R J Owen Finite Element Software for Platesand Shells Pineridge Press Swansea UK 1984

[12] J Chern C You andZ P Bazant ldquoDeformation of progressivelycracking partially prestressed concrete beamsrdquo PCI Journal vol37 no 1 pp 74ndash85 1992

[13] M P Collins and A Porasz ldquoShear design for high strengthconcreterdquo in Proceeding of the Workshop on Design Aspects ofHigh Strength Concrete pp 75ndash83 1989


[14] C-S Lin and A C Scordelis ldquoNonlinear analysis of rc shells ofgeneral formrdquo Journal of the Structural Division vol 101 no 3pp 523ndash538 1975

[15] MMenegotto and P E Pinto ldquoMethod of analysis for cyclicallyloaded R C Plane frames including changes in geometryand non-elastic behavior of elements under combined normalforce and bendingrdquo in Proceedings of the IABSE Symposium onResistance and Ultimate Deformability of Structures Acted on byWell-Defined Repeated Loads pp 15ndash22 Lisbon Portugal 1973

[16] A E Naaman ldquoA new methodology for the analysis of beamsprestressed with external or unbonded tendonsrdquo External Pre-stressing in BridgesACI SP120-16 AmericanConcrete InstituteDetroit Mich USA 1990

[17] American Concrete Institute (ACI) ldquoBuilding code require-ment for reinforced concrete and commentaryrdquo ACI 318-14American Concrete Institute (ACI) Farmington Hills MichUSA 2014

Submit your manuscripts athttpswwwhindawicom

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CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


with existing experimental data Although much analyticalresearch has been performed to evaluate the behavior ofbonded and unbonded posttensioned concrete membersa relatively limited number of studies have been reportedfor the prediction of flexural behavior considering tendonlayouts

Two-way slab systems offer several possible arrangementsfor the tendon layout [10] banded distributed or a mixedlayout Posttensioned slabs are used for long spans and heavylive loads so flexural strength and ductility are importantFlexural strength usually governs the behavior of the interiorpanel in two-way slabs In other words the distribution oftendons can affect the flexural behavior and ductility of theinterior panel of the two-way slabsThoughmany researchershave focused on the development of finite element modelslittle information is available on the flexural behavior of post-tensioned two-way slabs with different tendon layouts In thisstudy examined was the flexural behavior of posttensionedtwo-way slabs depending on the tendon layout

The objective of this paper is to present an efficientnumerical analysis approach for the materials and a geomet-ric nonlinear analysis for the posttensioned two-way slabs Inthis study developed was a nonlinear finite element modelthat can simulate the behavior of posttensioned two-wayslabs The reinforced concrete was modeled as combinationof concrete steel and prestressing tendons The test resultswere compared with those from the finite element model aswell

2 Finite Element Model

21 Layered Element Formulation A finite element modelformulated using layered and degenerate shell elements canbe used in a three-dimensional global analysis of structuresEight-node isoparametric degenerated shell elements wereformulated following the procedure ofHinton andOwen [11]It was assumed that plane cross-sections remain both planeand normal during bending Layered elements were appliedto account for the behaviors of the reinforced concrete mem-bers which exhibited different properties in the thicknessdirection because of the placement of the reinforcementsEach element is divided into layers and each layer has oneintegration point on its midsurface Each layer was com-posed of different materials concrete steel reinforcementand prestressing tendon are defined separately The strainsand stresses are calculated at midpoint of each layer Thestrain-displacement matrix and the constitutive matrix arecalculated at the midpoint of each layer Stress resultants areevaluated by integrating the corresponding stress Normalforces and bending moments can be obtained by

119873119909(119910) = intℎ2minusℎ2

120590119909(119910)119889119911 = ℎ2119899sum119894=1

120590119894119909(119910)Δ120577119894

119872119909(119910)(119909119910) = minusintℎ2minusℎ2120590119909(119910)(119909119910)119911 119889119911

= minusℎ24119899sum119894=1

120590119897119909(119910)(120591119909119910)120577119894Δ120577119894

(1)

k = 067 +62

k = 1

0

fc

fcmaxfcmax

Figure 1 Stress-strain relationship in compression by Collins andPorasz (1989)

where 119873119909 is normal force 119872119909 is bending moment 120590 isnormal stress ℎ is layer depth and 119899 is number of layers

The reinforcement layers were used tomodel the in-planereinforcement Transverse reinforcement can be specified asa property of a concrete layer The prestressing tendons at asingle depth were grouped with the same prestressing forceinto one layer [12] The concrete steel reinforcement andprestressing tendons were assumed to be perfectly bondedThe perfect bond is applicable to the analysis of reinforcedconcrete and posttensioned concrete with bonded tendonsThe same degrees of freedom were assigned to concrete andreinforcement nodes occupying a single location

22 Constitutive Models In this paper the failure of concretetwo-way slabs is considered to be tension cracking or plasticyielding of reinforcement Uncracked concrete was assumedto be a linear elasticmaterial After cracking the concrete wastreated as an orthotropic material The total material matrixconsists of concrete steel reinforcement and prestressingtendons

Figure 1 shows the stress-strain relationship for concretein compression [13] The compressive stress of concrete canbe calculated by

119891119888 = 1198911198882max (1198991205761205761015840119888)(119899 minus 1) + (1205761205761015840119888)119899119896 119891119888max = 119891101584011988808 minus 034 (1205761198881205761015840119888) le 1198911015840119888

119899 = 08 + 1198911198882max17 (2)

where119891119888 is the concrete stress1198911198882max is the compressive stressof cracked concrete 120576119888 is the concrete strain 1205761015840119888 is the concretestrain corresponding to peak compressive stress and1198911015840119888 is thecompressive cylinder strength of concrete

After cracking the stiffness of the reinforced concretedecreases but it does not drop to zero because the intact


minus 1

fcr

ft

Ec

Ec

cr ccr


fy

fs = fy

fs = Ess

y





120590119901 = 119864119901120576119875[[[119876 + 1 minus 119876

[1 + (119864119901120576119901119870119891119901119910)119873]1119873]]]

119876 = 119891119901119906 minus 119870119891119901119910119864119901120576119901119906 minus 119870119891119901119910 (3)


fps

fpu

fpy

py pu











Tendon350 D13280

D10

rei

nfor

cem

ent

25016882

3000 mm300

0m

m

(a) PT-x specimen

Tendon350

Tend

on

350

D13280

D10

rei

nfor

cem

ent

25016882

3000 mm

300

0m

m

(b) PT-xy specimen

















PT-xPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)




5 Conclusions


PT-xyPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)



750060003000 4500Span (mm)

35

40

45

50

55

60

65

70

Max

imum

disp

lace

men

t (m

m)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

180 350 450250

Height (mm)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

655035







Competing Interests


Acknowledgments


References


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




minus 1

fcr

ft

Ec

Ec

cr ccr


fy

fs = fy

fs = Ess

y





120590119901 = 119864119901120576119875[[[119876 + 1 minus 119876

[1 + (119864119901120576119901119870119891119901119910)119873]1119873]]]

119876 = 119891119901119906 minus 119870119891119901119910119864119901120576119901119906 minus 119870119891119901119910 (3)


fps

fpu

fpy

py pu











Tendon350 D13280

D10

rei

nfor

cem

ent

25016882

3000 mm300

0m

m

(a) PT-x specimen

Tendon350

Tend

on

350

D13280

D10

rei

nfor

cem

ent

25016882

3000 mm

300

0m

m

(b) PT-xy specimen

















PT-xPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)




5 Conclusions


PT-xyPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)



750060003000 4500Span (mm)

35

40

45

50

55

60

65

70

Max

imum

disp

lace

men

t (m

m)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

180 350 450250

Height (mm)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

655035







Competing Interests


Acknowledgments


References


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Tendon350 D13280

D10

rei

nfor

cem

ent

25016882

3000 mm300

0m

m

(a) PT-x specimen

Tendon350

Tend

on

350

D13280

D10

rei

nfor

cem

ent

25016882

3000 mm

300

0m

m

(b) PT-xy specimen

















PT-xPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)




5 Conclusions


PT-xyPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)



750060003000 4500Span (mm)

35

40

45

50

55

60

65

70

Max

imum

disp

lace

men

t (m

m)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

180 350 450250

Height (mm)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

655035







Competing Interests


Acknowledgments


References


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials









PT-xPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)




5 Conclusions


PT-xyPT-x_FEA

0

100

200

300

400

500

600

700

800

900

Appl

ied

load

(kN

)



750060003000 4500Span (mm)

35

40

45

50

55

60

65

70

Max

imum

disp

lace

men

t (m

m)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

180 350 450250

Height (mm)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

655035







Competing Interests


Acknowledgments


References


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




36

39

42

45

Max

imum

disp

lace

men

t (m

m)

180 350 450250

Height (mm)



36

39

42

45

Max

imum

disp

lace

men

t (m

m)

655035







Competing Interests


Acknowledgments


References


























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Analytical Model for Deflections of Bonded Posttensioned ...4...

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