A Stress-Strain Model for Unconfined Concrete in...

Research ArticleA Stress-Strain Model for Unconfined Concrete inCompression considering the Size Effect

Keun-Hyeok Yang Yongjei Lee and Ju-Hyun Mun

Architectural Engineering Kyonggi University Suwon Republic of Korea

Correspondence should be addressed to Ju-Hyun Mun mjhkguackr

Received 29 September 2018 Revised 24 January 2019 Accepted 17 February 2019 Published 12 March 2019

Academic Editor Hugo C Biscaia

Copyright copy 2019 Keun-Hyeok Yang et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this study a stress-strain model for unconfined concrete with the consideration of the size effect was proposede compressivestrength model that is based on the function of specimen width and aspect ratio was used for determining the maximum stress Inaddition in stress-strain relationship a strain at the maximum stress was formulated as a function of compressive strengthconsidering the size effect using the nonlinear regression analysis of data records compiled from a wide variety of specimens edescending branch after the maximum stress was formulated with the consideration of the effect of decreasing area of fractureenergy with the increase in equivalent diameter and aspect ratio of the specimen in the compression damage zone (CDZ) modele key parameter for the slope of the descending branch was formulated as a function of equivalent diameter and aspect ratio ofthe specimen concrete density and compressive strength of concrete Consequently a rational stress-strain model for unconfinedconcrete was proposed is model reflects trends that the maximum stress and strain at the peak stress decrease and the slope ofthe descending branch increases when the equivalent diameter and aspect ratio of the specimen increase e proposed modelagrees well with the test results irrespective of the compressive strength of concrete concrete type equivalent diameter and aspectratio of the specimen

1 Introduction

e stress-strain relationship for unconfined concrete is afundamental material property for the design and analysis ofstructural elements [1ndash3] Generally in stress-strain re-lationship the ascending and descending branches are de-pendent on concrete type and compressive strength as wellas the maximum diameter of aggregate specimen width ordiameter and aspect ratio in the descending branch from thecrack propagation in the fracture zone [4ndash7]e slope of theascending branch commonly increases with the increase incompressive strength of concrete and the decrease inspecimen width or diameter and aspect ratio while that ofthe descending branch increases with concrete compressivestrength specimen width or diameter and aspect ratio Tostudy this trend researchers [5 8ndash10] proposed concretecompressive strength models with the size effect in variousapproaches based on the fracture energy theory Bazant andPlanas [8] reported that concrete compressive strength was

considerably affected by the specimen width or diameterindicating that it decreased by 10 when the specimenwidth or diameter increased twice Sim et al [5] emphasizedthat the size effect on compressive strength for lightweightconcrete (LWC) was more notable than that for normalweight concrete (NWC) In particular because cracks at thefailure zone for LWC pass through lightweight aggregateparticles the crack band zone is more localized in LWCthan NWC Hence the size effect of concrete on the peakstress and descending branch behavior that is directly relatedto crack propagations in the failure zone could be morenotable in LWC than those in NWC [5] However for thesize effect of concrete on the descending branch from thecrack propagation in a localized crack band zone only fewstudies have been conducted In particular very little lit-erature on the size effect in LWC is available Furthermorethe existing proposed models [6 7] for the stress-strainrelationship regarding the size effect of concrete on thedescending branch are typically determined from NWC test

HindawiAdvances in Materials Science and EngineeringVolume 2019 Article ID 2498916 13 pageshttpsdoiorg10115520192498916

data records rather than from LWC with limited ranges ofvariables

Markeset and Hillerborg [6] generalized the compres-sion damage zone (CDZ) model to consider the size effect ofconcrete on the descending branch using a function of straindissipated by the shear band including the fracture energyHowever the descending branch behavior in Markeset andHillerborgrsquos model are drawn from the limited ranges ofvariables In addition this model is limited for a practicalequation because the specific information about the strain ofthe starting point for the softening behavior is not availableSamani and Attard [7] considered the size effect on thedescending branch in stress-strain relationship proposed byAttard and Setungersquos model [11] using the CDZ model [6]As the Samani and Attardrsquos [7] model has an identical strainmodel of shear band as that of Markeset and Hillerborgrsquos [6]information regarding the material property factors isnecessary to predict the stress-strain relationship In addi-tion the descending branch of these models does not fullyconsider the effect of aggregate property on crack propa-gation and the localized fracture zone For example in LWCthe contribution of stress transfer at the crack plane foraggregate interlocking action is little [5] because most of thecracks at the failure plane pass through lightweight aggregateparticles In addition the strength and elastic modulus ofaggregates such as magnetite in heavyweight concrete(HWC) are typically higher than those in NWC which cancause a wider fracture zone by crack propagation Hence thesize effect of LWC and HWC could be different from that ofNWC because the size effect on concrete depends on theareas of failure and crack propagation in the fracture zoneHowever as the existing models [6 7] incorporating the sizeeffect of concrete on the descending branch were derivedfrom limited NWC test data records a limited number ofstudies for the size effect of concrete using lightweight andheavyweight aggregates are available

e objective of this study is to propose a model for thestress-strain curve considering the size effect of variousconcrete types In this model the basic formula for thestress-strain curve and the key parameter that determinesthe slopes of the ascending and descending branchesestablished by Yang et al [1] was used to generate a completenonlinear curve e concrete compressive strength modelproposed by Sim et al [5] that considers the size effect wasused for the peak stress e strain at the peak stress wasgeneralized with a simple equation using the regressionanalysis of data records compiled from specimens withvarious ranges of equivalent diameter and aspect ratio ofspecimen compressive strength and density In the soft-ening behavior of the descending branch in the stress-strainrelationship the size effect and fracture energy were con-sidered usingMarkeset and Hillerborgrsquos CDZmodel [6]ekey parameter for the softening behavior was determinedfrom the secant modulus joining the origin and 05fSEprime where fSEprime is the compressive strength of concrete consid-ering the size effect e strain model at 05fSEprime to determinethe key parameter was generalized with various ranges ofvariables using nonlinear regression analysis Finally the keyparameter that determines the slope of the descending

branch was formulated as a function of equivalent diameterand aspect ratio of specimen concrete density and concretecompressive strength using parametric numerical analysise accuracy of the proposed model was evaluated using anormalized root-mean-square error obtained from thecomparisons of predicted curves with the test results

2 Database

To formulate the material properties Yang et al [1] com-piled 3295 data records for the elastic modulus of concrete(Ec) 415 data records for strain at the peak stress (ε0) and 96data records for strain at 50 of the peak stress (ε05) in thedescending branch In the compiled data records thenumbers available for the concrete compressive strength(fcprime) and density (ρc) were 3295 and 3274 respectively andvaried from 84MPa to 170MPa and from 1200 kgm3 to4500 kgm3 respectively To apply the effect of concretetype to the empirical formulations the data records weredivided into LWC NWC and HWC according to ρc econcrete density (ρc) varied from 1200 kgm3 to 2000 kgm3

for LWC from 2000 kgm3 to 2500 kgm3 for NWC andfrom 2500 kgm3 to 4500 kgm3 for HWC e data recordscompiled by Yang et al [1] were all measured from a cylinderof diameter 100mm and height 200mm Hence the datarecords compiled by Yang et al [1] lack the test results fordifferent specimen size To compensate for this additionaltest results for the size effect of concrete were compiled ecompiled additional test results [4 5 12ndash15] for the stress-strain relationship were 38 data records for LWC and26 data records for NWC as shown in Table 1 To considervarious specimen shapes an equivalent diameter (deq asymp

(4B Dπ)1113968

)was introduced whereB andD are the sectionalwidth and depth of prism specimen respectively assumingthat the area of the prism specimens is identical to that ofcylinder specimen For example for prism specimens with a B

of 300mm andD of 400mm deq corresponds to 3989mm Inthe data records deq aspect ratio (hdeq) fcprime and ρc werevaried from 50mm to 800mm 05 to 8 4mm to 20mm171MPa to 902MPa and 1500 kgm3 to 2464 kgm3 re-spectively e specimen equivalent diameter (deq) and hdeqwere varied from 100mm to 350mm and 1 to 2 for LWC and50mm to 800mm and 05 to 8 for NWC respectively

3 Model Generalization

31 Basic Approach e stress-strain relationship for un-confined concrete in compression is a parabola with as-cending and descending branches and a vertex at the peakstress [1ndash3] is shape can be generalized using the fol-lowing equation [1]

y β1 + 1( 1113857x

xβ1+1 + β1 (1)

where y fcfcprime is the normalized stress x εcε0 is thenormalized strain fc and εc are the concrete stress andstrain at some point in stress-strain curve respectively andβ1 is the key parameter that determines the slopes of theascending and descending branches e ascending branch

2 Advances in Materials Science and Engineering

can be determined from Ec which is defined as the slope ofthe line joining the origin and 40 of the peak stress [16] Inaddition the descending branch can be determined from thesecant modulus joining the origin and 05fcprime [1] In accor-dance with Yang et alrsquos model [1] the equation for the keyparameter β1 that determines the slopes of the ascending anddescending branches can be expressed in the following forms

04 Xa( 1113857β1+1

+ 04minusXa( 1113857β1 minusXa 0 for εc le ε0

Xd( 1113857β1+1

+ 1minus 2Xd( 1113857β1 minus 2Xd 0 for εc gt ε0(2)

where Xa 04fcprimeEcε0 and Xd ε05ε0Bazant [9] proposed the crack band zone by a crack

width with microcrack propagation for concrete failure Inaddition Bazant [9] idealized the crack band width as afunction of da assuming that microcracks propagated theinterfaces between aggregates and pastes Sim et al [5]proposed a smaller area of the crack band zone in LWC thanNWC as shown in Figure 1 based on the crack band theory[9] is model includes the effect of reduced area of thecrack band zone caused by the cracks at the failure zonepassing through lightweight aggregate particles and alsoconsiders the size effect on concrete due to the decrease inαdeq as deq increases where α is the crack length e aboveprevious models clearly revealed that the size effect exists incompressive strength of concrete due to different propa-gations of longitudinal splitting cracks at different specimensizes although further elaborated analytical approach wouldbe needed to account for the effect of splitting cracks on thesize effect in different concrete types Sim et al [5] derivedthe equation for the compressive strength of concrete (fSEprime )considering the size effect from the energy balance up to thepeak stress in the crack band zone idealizing the propagationof longitudinal splitting cracks in that the strain energy forconcrete deformation dissipated by the crack band zoneequals the total energy consumed by the band of the axialmicrosplitting cracks e proposed model is as follows

fSEprime A1

(1k) hdeq1113872 1113873X4

1113969

1 + B1 deqdX1a1113872 1113873 ρc2300( 1113857

minusX21113872 11138731113960 111396105

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦fcprime (3)

where A1 is(nk1X3F2)(1minus (EcEt))

1113968 n is the number of

microcracks in the band B1 is F1F2 F1 is zFzα1 F2 is

zFzα2 F is f(α1 α2) α1 and α2 are the modification factorsto account for the volume of the crack band zone Et is thestrain-softening modulus and X1 X2 X3 and X4 are theexperimental constants In equation (3) fcprime indicates thecompressive strength of concrete measured in a referencespecimen with d of 150mm and hdeq of 2 From equation(3) this indicates that fcprime is considerably affected by thefunctions of deq hdeq and ρc [5 8ndash10] Sim et al [5] de-termined the functions of A1 B1 dX1

a k X2 and X4 inequation (3) from 1509 data records with LWC and NWCtest results and proposed as follows

fSEprime 09

hdeq1113872 1113873minus06

1113969

1 + 0017deq ρc2300( 1113857minus1

1113960 111396105 + 063⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦fcprime (4)

In the model of Sim et al [5] equation (4) considers afunction of ρc that reflects the trend that the size effect ismore notable in LWC than that in NWC e stress-strainrelationship is shown in Figure 2 using equation (4) for thepeak stress In Figure 2 the key parameter β1 can be de-termined with the following equations

04 Xa1( 1113857β1+1

+ 04minusXa1( 1113857β1 minusXa 0 for εc le εSE (5)

Xd1( 1113857β1+1

+ 1minus 2Xd1( 1113857β1 minus 2Xd1 0 for εc gt εSE (6)

whereXa1 04fSEprime EcεSEXd1 εSE05εSE εSE is the strain atthe peak stress considering the size effect and εSE05 is thestrain at 05fSEprime after the peak stress In equation (1) the keyparameter β1 that determines the ascending and descendingbranches requires information about the functions of Ec εSEand εSE05 as expressed in equations (5) and (6)

32 Determination of β1 in Ascending Branch e size effecton concrete is based on the crack band theory indicating thecrack width with the propagation of microcracks as shownin Figure 3 It implies that the behavior of the points of ldquoOrdquoand ldquoArdquo without any cracks in the graph is not affected bythe size effect In addition Taylor and Broms [17] reportedthat the bond cracks can be definitely observed between 38and 42 of the peak stress at the ascending branch of thestress-strain curve As a result the strain at the ascendingbranch begins to increase rapidly with the propagation of thebond cracks showing the nonlinear curve as pointed out byNeville [16] According to fracture mechanics the size effecton concrete is primarily caused by the crack propagationus the present study considered that Ec representing thequasi-linear relationship at the ascending branch is mar-ginally affected by the size effect Yang et al [1] formulatedthe empirical formulation for Ec on the basis of the re-gression analysis of 2680 data records for NWC 370 datarecords for LWC with ρc ranging from 1200 to 2000 kgm3and 245 data records for HWC with ρc ranging from 2500 to4450 kgm3 Noguchi et al [18] reported that Ec can beexpressed as a function of fcprime and ρc and included correctionfactors to account for the effects of aggregate type andpresence of supplementary cementitious materials (SCMs)In Yang et alrsquo model [1] the various unusual aggregates such

Table 1 Distribution of parameters in the data records for stress-strain curves

fcprime(MPa)

Range lt20 20ndash40 40ndash60 60ndash80 80ndash100 TotalLWC 4 26 8 0 0 38NWC 2 3 16 0 5 26

deq(mm)

Range lt50 50ndash100 100ndash200 200ndash300 gt400 TotalLWC 0 34 1 2 1 38NWC 3 19 3 0 1 26

hdeq

Range 05-1 1-2 2-3 3ndash5 5ndash8 TotalLWC 3 35 0 0 0 38NWC 4 8 3 6 5 26

da(mm)

Range 4 4ndash10 10ndash15 15ndash20 gt20 TotalLWC 1 2 2 33 0 38NWC 0 14 0 12 0 26

Advances in Materials Science and Engineering 3

as articial lightweight aggregates and heavyweight mag-netite particles were implicitly considered in the use of theparameter of ρc In addition the correction factor for SCMscan be implicitly included when the empirical constants areobtained from regression analysis resulting in negligibleerrors Yang et alrsquos model reected the trend that a lowerincreasing rate in Ec than that in fcprime was considered by usinga power function of fcprime to represent the nonlinear re-lationship between the two parameters Consequently Yanget alrsquos model [1] was used for Ec as follows

Ec 8470 fcprime( )13ρc

2300( )

117 (7)

Meanwhile as shown in Figure 2 equation (4) can beexpressed as follows from the relation of fSEprime and E0

εSE 09

hdeq( )

minus06radic

1 + 0017deq ρc2300( )minus1[ ]05 + 063

fcprime

E0 (8)

Consequently εSE can be expressed as follows

εSE fSEprime

E0 (9)

where E0 is the secant modulus joining the origin and thepeak stress Equation (9) shows that εSE is fully aected bythe functions of fSEprime and E0 However data records orpredicted model for E0 is not available in the literatureHence in this study E0 can be proposed as follows using acertain relation with Ec as shown in Figure 2

εSE χfSEprime

Ec( ) (10)

where χ is a coecient to account for the relation of E0 andEc which can be determined from the test results From thedata records compiled in this study equation (10) can beproposed as follows (Figure 4)

εSE 00016 exp 220fSEprime

Ec( )[ ] (11)

e key parameter β1 that determines the slope of theascending branch can be solved by substituting equations (7)and (11) into equation (5)e key parameter β1 was calculatedusing the NewtonndashRaphson method identical to Yang et alrsquosmodel [1] e β1 values determined for dierent concreteproperties need to be formulated as a simple equation forpractical application of the proposed stress-strain relationshipof concrete us the present research involved a parametricstudy to generalize β1 under the comprehensive ranges ofparameters as follows fcprime between 10MPa and 180MPa ρcbetween 1400 kgm3 and 4000kgm3 deq between 50mm and500mm and hdeq between 05 and 5 Note that the geo-metrical conditions of specimens were considered in theparametric study because the given parameters in equations (4)and (11) are aected by the equivalent width and aspect ratio ofthe specimen From the regression analysis using the solutions

y=

f cfprime SE

(Xa1 04)(Xd1 05)

Esec

E0

Ec

Specimen(deq = 150 mm hdeq = 2)

Specimen(deq lt 150 mm hdeq lt 2)

Specimen(deq gt 150 mm hdeq gt 2)

0

02

04

06

08

1

12

1 2 3 40x = εcεSE

Figure 2 Generalization of the compressive stress-strain curve ofconcrete considering the size eect

αdeq gt α2deqα

2deq

Confined region

hα

hα

Crack band zone

(a) (b)

Stress relief strip

Confined region

deq deq

ωc ωc

ωc

Figure 1 Idealized crack band zone at peak stress based on concrete fracture mechanics (a) NWC (b) LWC


obtained from the NewtonndashRaphson method the key pa-rameter β1 to account for the slope of the ascending branch canbe simply formulated as follows (Figure 5)

β1 033 exp 042fSEprime

10( )

2300ρc

( )15

for εc le εSE

(12)

Equation (12) shows that the slope of the ascendingbranch increases with the increase in fSEprime or decrease in ρcConsequently the slope of the ascending branch includes thesize eect of concrete with the generalization of a function offSEprime considering the parameters deq and hdeq

33 Determination of β1 in Descending Branch e com-pressive failure behavior of unconned concrete is commonlycharacterized by the mode I (pure tensile) and mode III(sliding displacement due to diagonal shear) in estimating thefracture zone As pointed out by Markeset and Hillerborgmicrosplitting cracks (mode I) are formulated due to Poissonrsquoseect under pure compressive stresses whereas sliding dis-placement along diagonal tensile cracks (mode III) occurs at

45-degree slope relative to the principal normal stresses whenthe maximum shearing stresses reach the shear capacity ofconcrete Hence Markeset and Hillerborg assumed thatlongitudinal crack zones of concrete relate to the tensilefracture energy zone whereas diagonal shear crack zones after

fc

A

C

D εcO

BBprime

Cprime

Internal work forspecimen (deq gt150mm hdeq gt 2)

Internal work forspecimen (deq = 150mm hdeq = 2)

f primec

(a)

Total energyElastic energyfc

B

A

C

O εc

Bprime

C prime

Total energy

Specimen(deq = 150mm hdeq = 2)

Elastic energySpecimen(deq gt150mm hdeq gt 2)

f primec

(b)

Figure 3 Fracture energy for concrete failure

0001

00015

0002

00025

0003

00035

0004

00045

0005

0 0001 0002 0003 0004 0005

LWCNWCHWC

Best fit curvey = 00016exp[220(x)]

ε SE

f primeSEEc

Figure 4 Nonlinear regression analysis for εSE

0

10

20

30

40

50

0 2 4 6 8 10 12

Best fit curvey = 033 texp[042(x)]

R2 = 097

β 1

( f primeSE10)(2300ρc)15

f primec = 10‒180 MPaρc = 1200‒4000 kgm3

deq= 50‒500 mmhdeq= 05‒5

Figure 5 Formulation of key parameter β1 in the ascending branchin stress-strain relationship


the peak stress identify the shear fracture energy zone (Fig-ure 6) In fact most of cracks in concrete are caused by tensilestresses rather than compressive stresses because the tensileresistance of concrete is highly lower than the compressiveresistance Overall the compressive failure of concrete iscommonly caused by the tensile fracture due to Poissonrsquoseffect e descending branch behavior after the peak stress isdetermined from the localized deformation developed in thedamaged or failure zone while the undamaged zone elasticallyunloads [6] Hence the undamaged zone is generated onlywhen h is greater than the damaged zone height (hd) Forcompressive behavior Markeset and Hillerborg [6] idealizedthe CDZ model generated from the longitudinal micro-splitting cracks and localized diagonal tensile shear band crackin the damaged zone (Figure 6) In CDZmodel microsplittingcracks can be idealized as a crack band zone with severalmicrosplitting cracks because their propagation requires en-ergy releaseus the size effect of concrete on compression isexpected as demonstrated by lots of previous researchesBazant and Planas also idealized the fracture energy zone ofconcrete under compression considering the tensile anddiagonal shear cracks to consider the size effect According toMarkeset andHillerborg [6] the total strain εc in the softeningbehavior is the sum of the strain during the unloading regionafter the peak stress in the undamaged zone the strain whilemicrosplitting crack occurs in the longitudinal direction andthe strain caused by the diagonal shear band (ωh)

εc ε + εdhd

h1113888 1113889 +

ωh

for hgt hd (13a)

εc ε + εd +ωh

for hle hd (13b)

where hd is the height of the region propagated by thelongitudinal microsplitting cracks ω is the localized de-formation assumed between 04 and 07 for NWC and lessthan 03 for LWC e longitudinal microsplitting cracks inunconfined concrete under compression commonly developat approximately 75sim90 of the peak stress From thisfinding in the CDZ model the amount of energy (Win)released in the unloading zones can be obtained using thefollowing equation

Win

GF

c(1 + k) (14)

where GF is the fracture energy and k is the factor basedon the material properties c is the factor to account foraverage spacing between longitudinal spitting microcracksAssuming that the strain (εd) in the region propagated by thelongitudinal microsplitting cracks is proportional to thetensile fracture energy (GF) it can be proposed as follows

εd 2kGF

c(1 + k)fcprime1113890 1113891

fcprime minusfc

fcprime1113888 1113889

08

(15)

As expressed in equation (15) the CDZ model proposedby Markeset and Hillerborg [6] includes a function of GF inthe descending branch However GF k and ω requirecalibration according to various concrete types because they

are based on the material properties which are too de-manding for a practical application Furthermore because c

is proposed only for a da of 16mm the use of a practicalequation is limited for other specimens with larger aggre-gate Hence to obtain information about these factors acomprehensive test is required with various influencingparameters including concrete type deq hdeq and da Toimprove Markeset and Hillerborgrsquos model [6] the key pa-rameter β1 by Yang et al [1] was applied to the descendingbranch behavior e peak stress from equation (4) by Simet al [5] and εSE05 from equations (13a) and (13b) are used toproduce the following equation

εSE05 εSE minusfSEprime

Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889

middotfSEprime minus 05fSEprime

fSEprime1113888 1113889

08hd

h+ωh

for hgt hd

(16a)


Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08

+ωh

for hle hd

(16b)

where εSE Ec and fSEprime are known values In equations (16a)and (16b) the first term in the right is moved to the left sideand can be arranged as follows

εSE05 minus εSE minusfSEprime

Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873hd

h+ωh

for hgt hd

(17a)


Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873 +ωh

for hle hd

(17b)

where the values of k and c are experimental constants inpredicting the softening in the CDZ model On the basis oftest results Markeset and Hillerborg assumed the value of k

as 30 for NWC and 10 for LWC However the value of k forHWC is still unknown because of the lack of test dataMarkeset and Hillerborg also introduced the factor c toaccount for the average spacing of the longitudinal micro-splitting cracks due to the primary tensile stresses eyassumed the value of c as 125 for the maximum aggregatesize of 16mm However there is no further information onthe value of c for different aggregate sizes although thespacing of the longitudinal microsplitting cracks can besignificantly affected by the aggregate size due to the ag-gregate interlock action In addition the value of c dependson the equivalent width and aspect ratio of the specimenbecause the energy release at the crack band zone is affectedby the spacing of the longitudinal microsplitting cracks

e present study conducted the regression analysis oftest data on εSE05 to simply generalize the right-hand side ofequations (17a) and (17b) including the factors k and c Test


results of εSE05 minus (εSE minus (fSEprime Ec)) according to deq hdeqand ρc are shown in Figure 7 In Figure 7(a) εSE05 minus(εSE minus (fSEprime Ec)) nonlinearly decreased with the increase indeq indicating that it decreased by approximately 18 forspecimens with hdeq of 2 and approximately 31 forspecimens with hdeq of 1 when deq increased by 3 timesεSE05 minus (εSE minus (fSEprime Ec)) almost linearly decreased with hdeqincreased irrespective of fcprime εSE05 minus (εSE minus (fSEprime Ec)) rangedbetween 00026 and 00049 for specimens with hdeq of 2 and00018 for specimens with hdeq of 55 indicating that itdecreased by 43 when hdeq increased by 3 times Inaddition εSE05 minus (εSE minus (fSEprime Ec)) increased with the increasein ρc and its increasing rate according to fcprime was almostconstant εSE05 minus (εSE minus (fSEprime Ec)) ranged between 0002 and00038 for LWC (ρc less than 2000 kgm3) and 00029 to00044 for HWC (ρc more than 2500 kgm3) ese implythat the descending branch behavior in the stress-strainrelationship for unconned concrete is considerably af-fected by the functions of deq hdeq and ρc Based on thisanalysis εSE05 minus (εSE minus (fSEprime Ec)) was generalized as func-tions of GF fSEprime hdeq and ρc (Figure 8) using regressionanalysis from the test results [4 5 12ndash15 19ndash29] for 45 datarecords for LWC 91 data records for NWC and 24 datarecords for HWC


Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)

where the CEB-FIP model [30] for GF that includes func-tions of da and fcprime was expressed as follows

GF GF0fcprime

10( )

07

(19)

where GF0 is 0025Nmm 003Nmm and 005Nmm for daof 8mm 16mm and 32mm respectively Overall thepresent model can predict the softening performance ofconcrete with dierent parameters including the compressivestrength and density of concrete equivalent width and aspectratio of specimens and aggregate sizes even though the valuesfor k and c are not determined from test specimense valueof β1 in the descending branch can be solved using equations(11) and (18)e solution of β1 in the descending branch wasalso calculated using the NewtonndashRaphson method as in theascending branch Finally the key parameter β1 was for-mulated using the analytical parametric study In the ana-lytical parametric study fcprime deq hdeq da and ρc wereselected from 10MPa to 180MPa 50mm to 500mm 05 to 54mm to 25mm and 1400 kgm3 to 4000 kgm3 respectivelyFrom the analytical results statistical optimization was per-formed to generalize the key parameter β1 that determines theslope of the descending branch as follows (Figure 9)

β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)

Finally the stress-strain relationship for unconnedconcrete can be proposed as follows

fc β1 + 1( ) εcεSE( )xεcεSE( )β1+1 + β1

fSEprime (21)

where εSE is given by equation (11) fSEprime is given by equation(4) and key parameter β1 is given by equation (12) or (20)e proposed stress-strain relationship for unconned

h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)

Confined region Unloading region

Unloading region

Longitudinalsplitting crack

(mode I)

Win

(deq = 150 mm hdeq = 20)

Win

(deq gt 150 mm hdeq gt 20)Stressrelief strip

Stress relief strip

Crack band zone

(a) (b)

Diagonal shear crack(sliding mode III)

Confined region

hd

f primec

f primec

f primec

Ws = kWin

(deq gt 150 mm hdeq gt 20)

Figure 6 Compression damage zone (CDZ) model (a) Ascending branch (b) Descending branch


concrete can consider the size eect on concrete in theascending and descending branches using the powerfunctions of the key parameters β1 and fSEprime

4 Comparisons with Test Results

e test results compiled from the available literatures[4 5 12ndash15 19ndash29] were compared with predictions of this

study and the existing models [1 6 7 11] e existingmodels for the strain-stress relationship proposed byMarkeset and Hillerborg [6] and Samani and Attard [7] wereselected as summarized in Table 2 Figure 10 shows com-parisons of the predicted and measured stress-strain curves[4 19ndash25] e comparative analysis focused on the eect ofdeq hdeq ρc and fcprime on the stress-strain curve Table 3summarizes the normalized root-mean-square error(NRMSE) obtained from the comparisons of test results withpredictions In Table 3 cm and cs are the mean and standard

Best fit curve(f primeSE = 235ndash544 MPa hdeq = 2)


00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)

Best fit curve(f primeSE= 376ndash90 MPa ρc = 2234ndash2473 kgm3)

0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)

Figure 7 Variation in εSE05 in softening behavior (a) Equivalent diameter deq (b) Aspect ratio hdeq (c) Concrete density ρc

0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC

Best fit curvey = 00027exp[19x]

ε SE0

5ndash

(εSE

ndashfprime S

E E c

)

1000 times [GF04(f primeSE

14deq06)(hdeq)ndash12 (ρc2300)2]

Figure 8 Nonlinear regression analysis to determine εSE05

00

10

20

30

40

50

0 1 2 3 4

Best fit curvey = 083(x)13

R2 = 094

(f primeSE10)065(deq150)02(hdeq)04(2300ρc)12

β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm

Figure 9 Formulation of key parameter β1 in the descendingbranch in stress-strain relationship


Table 2 Summary of stress-strain models considering the size eect

Researcher Stress-strain relationship of concrete

Markeset andHillerborg [6]

Ascending branch mdash

Descending branch

εh ε0 minus (fcprimeEc) + εd((25d)h) + (ωh) for hgt 2dεh ε0 minus (fcprimeEc) + εd + (ωh) for hle 2dεd (2kGFc(1 + k)fcprime)((fcprime minusfc)fcprime)

08

GF 000097fcprime + 00418

Samani andAttard [7]

Ascending branchfc [(A(εcε0) + B(εcε0)

2)(1 + ((Aminus 2)(εcε0)) + ((B + 1)(εcε0)2))]fcprime

A Ecε0fcprime B ((Aminus 1)2055)minus 1

Ec (3320fcprimeradic

+ 6900)(ρc2300)15

Descending branch

fc [(ficfcprime)((εh minus ε0)(εic minus ε0))

2]fcprime

εic [276minus 035 ln(fcprime)]ε0 fic [141minus 017 ln(fcprime)]fcprimeεh ε0 + ((εc minus ε0)(h0h)) + (((fc minusfcprime)Ec)[1minus (h0h)]) + εd((2deq minus h0)h) for hgt 2deqεh ε0 + ((εc minus ε0)(h0h)) + (((fc minusfcprime)Ec)[1minus (h0h)]) + εd(1minus (h0h)) for hle 2deq

εd (2kGFc(1 + k)fcprime)((fcprime minusfc)fcprime)08

GF 000097fcprime + 00418ε0 (fcprimeEc)(μ1

fcprime

4radic

)μ1 426 for crushed aggregates and 378 for gravel aggregates

is study

fc (((β1 + 1)(εcεSE)x)((εcεSE)β1+1 + β1))fSEprime

fSEprime (((09(hdeq)

minus06radic

)[1 + 0017deq(ρc2300)minus1]05) + 063)fcprimeεSE 00016 exp[220(fcprimeEc)]

Ec 8470(fcprime)13(ρc2300)

117

Ascending branch β1 033 exp[042(fSEprime 10)(2300ρc)15]

Descending branch β1 083[(fSEprime 10)062(d150)02(hd)035(2300ρc)

12]13

fc fcprime fSEprime and Ec are in MPa deq h h0ω and c are in mm GF is in Nmm ρc is in kgm3

Strain

NWC HWCLWC0005

0 0 0 0 0 0 0

ExperimentsPredictions

hdeq = 2 hdeq = 1hdeq = 4 2 1

hdeq = 05 1 2

Yang and Sim [20]f primeSE = 302ndash444 MPa

Wang et al [19](f primeSE = 232 MPa

hdeq = 2

Vonk [12]f primeSE = 399ndash488 MPa

Jansen and Shah [4]f primeSE = 448ndash450 MPahdeq = 2 25 35 45

Mertol et al[21]f primeSE = 3268 MPa

hdeq = 2

Nilson [25]f primeSE = 825 MPa hdeq = 2]

Yang et al [23]f primeSE = 289ndash504 MPa

hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued


deviation of the NRMSE respectively It is noteworthy thatthe comparisons of test results with the predictions ofMarkeset and Hillerborg [6] were conducted only in thedescending branch because Markeset and Hillerborgrsquosmodel [6] provides the equations only for the descendingbranch behavior

Markeset and Hillerborg [6] idealized the CDZ modelconsidering fracture energy and proposed the descendingbranch behavior in the stress-strain relationship In the CDZmodel the total strain is a combination of the strain (ε) in theregion where the undamaged zone elastically unloads afterthe peak stress the strain (εd) in the region propagated by thelongitudinal microsplitting cracks and the strain (ωh) by

the diagonal tensile band crack In the model εd is based onthe assumption that inelastic deformation in the damagedzone determines the descending branch behavior To con-sider inelasticity in the descending branch εd introduces((fcprime minusfc)fcprime)

08 as that expressed in equation (8) never-theless the descending branch is predicted as a virtuallylinear curve (Figure 10(a)) However the shapes of thedescending branch measured in the existing test results areprimarily curved rather than linear Hence the accuracy ofMarkeset and Hillerborgrsquos model [6] according to theconcrete type uctuates with large deviations e values ofcm obtained by Markeset and Hillerborgrsquos model [6] are 039for LWC 029 for NWC and 029 for HWC In particular


StrainNWC HWCLWC0005

0 0 0 0 0 0 0


hdeq = 05 1 2


Wang et al [19]f primeSE = 232 MPa

hdeq = 2

Zhang and Gjoslashrv [22]f primeSE = 895 MPa hdeq = 2




hdeq = 2


Yang et al [23]f primeSE = 289ndash504 MPa]

hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2


Vonk [12]f primeSE = 399ndash488 MPa]

Jansen and Shah [4]f primeSE = 448ndash450MPahdeq= 2 25 35 45


hdeq = 2

Nilson [25]f primeSE = 825 MPa hdeq = 2


hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)

Figure 10 Typical comparisons of predicted and measured stress-strain curves (a) Markeset and Hillerborg [6] (b) Samani and Attard [7](c) is study


Markeset and Hillerborgrsquos model [6] underestimates thecompressive stress of HWC is is because this model doesnot consider the decreasing effect of the slope for HWC inthe descending branch because the factor of k to determineεd according to concrete type is 3 and 1 only for NWC andLWC respectively without considering HWC

Samani and Attard [7] applied the size effect to theequations for the descending branch proposed by Attard andSetungersquos model [11] As summarized in Table 2 thedescending branch behavior proposed by Samani and Attard[7] is also based on the CDZ model and its strain equation issimilar to that of Markeset and Hillerborgrsquo s model [6] Inaddition Markeset and Hillerborgrsquos model [6] for εd com-posed of functions of k c and GF is used without modi-fication e descending branch behavior in Samani andAttardrsquos model [7] however is different from that inMarkeset and Hillerborg [6] indicating that it is predicted asa curve with an inflection point As shown in Figure 10(b)the ascending branch behavior composed of functions fcprimeand ρc in Samani and Attardrsquos model [7] is identical to that inAttard and Setungersquos model [11] In the descending branchbehavior of Samani and Attardrsquos model [7] the increasingeffect of the decreasing slope is explained well with theincrease in hdeq as shown in Figure 10(b) HoweverSamani and Attardrsquos model [7] underestimates the com-pressive stress for HWC is is because in this model thefactor k related with the material property does not considerthe decreasing effect of the descending slope for HWC isimplies that the factor k in Samani and Attardrsquos model [7]requires calibration using additional test results In additionbecause Samani and Attardrsquos model [7] does not consider thesize effect on the peak stress it overestimates the com-pressive strength of concrete for LWC e overestimationincreased with the increase in deq e values of cm obtainedby Samani and Attardrsquos model [7] are 041 for LWC 023 forNWC and 073 for HWC

e proposed model in this study shows better agree-ment with test results irrespective of deq hdeq concretetype and fcprime e values of cm and cs are 024 and 016 forLWC 019 and 012 for NWC and 010 and 001 for HWCrespectively e results are lower than those of the modelsof Markeset and Hillerborg [6] and Samani and Attard [7]e proposed values of cm and cs are 019 and 013 re-spectively which are the lowest among other models Basedon the CDZ model a rational stress-strain model for un-confined concrete considering the size effect is proposedusing the key parameter β1 formulated by functions of fSEprime deq hdeq and ρc Note that most tests to investigate thestress-strain curves of concrete in compression were con-ducted using standard cylindrical specimens of 100times

200mm or 150times 300mm Moreover very few specimenswith the equivalent diameter exceeding 200mm are available inthe literatures because of the capacity limitation of the testingmachine us the proposed models need to be further ex-amined in LWC and HWC specimens with a larger size

5 Conclusions

From the proposed stress-strain relationship model forvarious unconfined concrete types considering the size effectbased on the CDZ model the following conclusions werederived

(1) Although the concrete commonly has microcracksthe elastic modulus of concrete typically defined as04fcprimewhere bond cracks occurred in was not affectedby the size effect whereas the strain at the peak stresswas affected by the size effect because of propagationof cracks

(2) εSE05 minus (εSE minusfSEprime Ec) that closely related to theslopes of descending branches in stress-strain re-lationship decreased averagely by 25 and 43

Table 3 Comparisons of normalized root-mean-square error

Concrete type Aspect ratio Statistical valueResearcher

Markeset and Hillerborg [6] Samani and Attard [7] is study

LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013


respectively when equivalent diameter and aspectratio increased by 3 times e corresponding valuesfor lightweight concrete (LWC) were lower thanthose for normal weight concrete (NWC) andheavyweight concrete (HWC)

(3) e key parameter β1 determining the slope of theascending and descending branches could be proposedas an exponential function of (fSEprime 10)(2300 ρc)

15

and (fSEprime 10)065(deq150)02 (hdeq)04(2300ρc)12

respectively(4) e proposed model of the stress-strain relationship

for unconfined concrete showed good agreementswith the test results irrespective of equivalent di-ameter and aspect ratio of specimen concretedensity and compressive strength

Notations

deq Equivalent diameterda Maximum size of aggregateh Height of specimenhdeq Aspect ratio of specimenh0 Reference height of specimenhd Damage zone height of specimenE0 Secant modulus at the peak stressEc Elastic modulus of concreteEt Strain-softening modulusfc Stress at stress-strain curvefcprime Compressive strength of concrete

measured in the standard specimenfic Inflection stress in descending branchfSEprime Compressive strength concrete

considering the size effectGF Fracture energyk Factor relating to material propertyk1 Conversion coefficientn Number of microcracks in the bandWin Amount of energy released in the

unloading zonesX1 X2 X3 and X4 Experimental constantsα1 α2 Modification functions to account for

the volume of the crack band zoneβ1 Key parameter that determines the slope

of ascending and descending branchesc Factor relating to height of specimencm Mean of normalized root-mean-square

errorcs Standard deviation of normalized root-

mean-square errorε Strain induced from elastically

unloading in the undamaged zoneε05 Strain at 05fcprime after the peak stressε0 Stain at the peak stressεc A strain at stress-strain curveεd Strain in the damaged zone relating to

longitudinal microsplitting crackingεh Total strain occurred in compression

damage zone (CDZ)

εic Strain at inflection stress in descendingbranch

εSE Strain at the peak stress considering thesize effect

εSE05 Strain at 05fSEprime after the peak stressμ1 Factor relating to type of coarse

aggregateρc Concrete densityχ Coefficient to account for the relation

of E0 and Ecω Localized deformationωh Strain of diagonal tensile shear band

Data Availability

e data records used to support the findings of this studyare included within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research was supported by the research grant fromKyonggi University through the Korea Agency for Infra-structure Technology Advancement funded by theMinistry ofLand Infrastructure and Transport of the Korean Govern-ment (Project no 19TBIP-C126470-03)

References

[1] K H Yang J H Mun M S Cho and T H-K Kang ldquoStress-strain model for various unconfined concretes in compres-sionrdquo ACI Structural Journal vol 111 no 4 pp 819ndash8262014

[2] T H Almusallam and S H Alsayed ldquoStress-strain re-lationship of normal high-strength and lightweight concreterdquoMagazine of Concrete Research vol 47 no 170 pp 39ndash441995

[3] B-I Bae H-K Choi B-S Lee and C-H Bang ldquoCompressivebehavior and mechanical characteristics and their applicationto stress-strain relationship of steel fiber-reinforced reactivepowder concreterdquo Advances in Materials Science and Engi-neering vol 2016 Article ID 6465218 11 pages 2016

[4] D C Jansen and S P Shah ldquoEffect of length on compressivestrain softening of concreterdquo Journal of Engineering Me-chanics vol 123 no 1 pp 25ndash35 1997

[5] J-I Sim K-H Yang H-Y Kim and B-J Choi ldquoSize andshape effects on compressive strength of lightweight con-creterdquo Construction and Building Materials vol 38pp 854ndash864 2013

[6] G Markeset and A Hillerborg ldquoSoftening of concrete incompressionmdashlocalization and size effectsrdquo Cement andConcrete Research vol 25 no 4 pp 702ndash708 1995

[7] A K Samani and M M Attard ldquoA stressndashstrain model foruniaxial and confined concrete under compressionrdquo Engi-neering Structures vol 41 pp 335ndash349 2012

[8] Z P Bazant and J Planas Fracture and Size Effect in Concreteand Other Quasibrittle Materials CRC Press New York NYUSA 1998


[9] Z P Bazant ldquoSize effect in blunt fracture concrete rockmetalrdquo Journal of Engineering Mechanics vol 110 no 4pp 518ndash535 1984

[10] J-K Kim and S H Eo ldquoSize effect in concrete specimens withdissimilar initial cracksrdquo Magazine of Concrete Researchvol 42 no 153 pp 233ndash238 1990

[11] M Attard and S Setunge ldquoStress-strain relationship ofconfined and unconfined concreterdquo ACI Materials Journalvol 93 no 5 pp 432ndash442 1996

[12] R Vonk Softening of Concrete Loaded in Compression PhDthesis Eindhoven University of Technology EindhovenNetherlands 1992

[13] K H Lee K H Yang J H Mun and S J Kwon ldquoMechanicalproperties of concrete made from different expanded light-weight aggregatesrdquo ACI Materials Journal 2018 In press

[14] J-S Mun J-H Mun K-H Yang and H Lee ldquoEffect ofsubstituting normal-weight coarse aggregate on the work-ability and mechanical properties of heavyweight magnetiteconcreterdquo Journal of the Korea Concrete Institute vol 25no 4 pp 439ndash446 2013 in Korean

[15] G Muciaccia G Rosati and G Di Luzio ldquoCompressivefailure and size effect in plain concrete cylindrical specimensrdquoConstruction and Building Materials vol 137 pp 185ndash1942017

[16] A Neville Properties of Concrete Pearson Education LimitedHarlow UK 5th edition 2011

[17] M A Taylor and B B Broms ldquoShear bond strength betweencoarse aggregate and cement paste or mortarrdquo ACI Pro-ceedings vol 61 no 8 pp 939ndash958 1964

[18] T Noguchi F Tomosawa K M Nemati B M Chiaia andA P Fantilli ldquoA practical equation for elastic modulus ofconcreterdquo ACI Structural Journal vol 106 no 5 pp 690ndash6962009

[19] P T Wang S P Shah and A E Naaman ldquoStress-straincurves of normal and lightweight concrete in compressionrdquoACI Journal Proceedings vol 75 no 11 pp 603ndash611 1978

[20] K H Yang and J I Sim ldquoModeling of the mechanicalproperties of structural lightweight concrete based on sizeeffectsrdquo Technical Report Department of Plant∙ArchitecturalEngineering Kyonggi University Suwon Republic of Korea2011 in Korean

[21] H C Mertol S J Kim A Mirmiran S Rizkalla and P ZialdquoBehavior and design of HSC members subjected to axial-compression and flexurerdquo in Proceedings of the 7th Inter-national Symposium on Utilization of High-StrengthHighndashPerformance Concrete (SP-228) H G Russell Edpp 395ndash420 American Concrete Institute Washington DCUSA 2005

[22] M H Zhang and O E Gjoslashrv ldquoMechanical properties of highstrength lightweight concreterdquo ACI Materials Journal vol 88no 3 pp 240ndash247 1991

[23] K H Yang J S Mun and H Lee ldquoWorkability and me-chanical properties of heavyweight magnetite concreterdquo ACIMaterials Journal vol 111 no 3 pp 273ndash282 2014

[24] T H Wee M S Chin and M A Mansur ldquoStress-strain re-lationship of high-strength concrete in compressionrdquo Journal ofMaterials in Civil Engineering vol 8 no 2 pp 70ndash76 1996

[25] A H Nilson ldquoHighndashstrength concrete an overview of Cornellresearchrdquo in Proceedings of Symposium on Utilization ofHighndashStrength Concrete pp 27ndash37 Stavanger Norway June1987

[26] S-T Yi J-K Kim and T-K Oh ldquoEffect of strength and ageon the stress-strain curves of concrete specimensrdquo Cementand Concrete Research vol 33 no 8 pp 1235ndash1244 2003

[27] T C Liu StressndashStrain Response and Fracture of Concrete inBiaxial Compression Cornell University Ithaca NY USA1971

[28] A Tomaszewicz ldquoBetongens arbeidsdiagramrdquo SINTEF Re-port No STF 65A84065 SINTEF Trondheim Norway 1984

[29] J H Mun J S Mun and K H Yang ldquoStressndashstrain re-lationship of heavyweight concrete using magnetite aggre-gaterdquo Journal of Architectural Institute of Korea vol 29 no 8pp 85ndash92 2013 in Korean

[30] Comite Euro-International du Beton (CEB-FIP) StructuralConcrete Textbook on Behaviour Design and PerformanceInternational Federation for Structural Concrete (fib) Lau-sanne Switzerland 1999


CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018


Journal of

Chemistry

Analytical ChemistryInternational Journal of


ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of



Advances in Condensed Matter Physics


International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of


NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of


High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in



ChemistryAdvances in


Advances inPhysical Chemistry


BioMed Research InternationalMaterials

Journal of


Na

nom

ate

ria

ls


Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

data records rather than from LWC with limited ranges ofvariables

Markeset and Hillerborg [6] generalized the compres-sion damage zone (CDZ) model to consider the size effect ofconcrete on the descending branch using a function of straindissipated by the shear band including the fracture energyHowever the descending branch behavior in Markeset andHillerborgrsquos model are drawn from the limited ranges ofvariables In addition this model is limited for a practicalequation because the specific information about the strain ofthe starting point for the softening behavior is not availableSamani and Attard [7] considered the size effect on thedescending branch in stress-strain relationship proposed byAttard and Setungersquos model [11] using the CDZ model [6]As the Samani and Attardrsquos [7] model has an identical strainmodel of shear band as that of Markeset and Hillerborgrsquos [6]information regarding the material property factors isnecessary to predict the stress-strain relationship In addi-tion the descending branch of these models does not fullyconsider the effect of aggregate property on crack propa-gation and the localized fracture zone For example in LWCthe contribution of stress transfer at the crack plane foraggregate interlocking action is little [5] because most of thecracks at the failure plane pass through lightweight aggregateparticles In addition the strength and elastic modulus ofaggregates such as magnetite in heavyweight concrete(HWC) are typically higher than those in NWC which cancause a wider fracture zone by crack propagation Hence thesize effect of LWC and HWC could be different from that ofNWC because the size effect on concrete depends on theareas of failure and crack propagation in the fracture zoneHowever as the existing models [6 7] incorporating the sizeeffect of concrete on the descending branch were derivedfrom limited NWC test data records a limited number ofstudies for the size effect of concrete using lightweight andheavyweight aggregates are available

e objective of this study is to propose a model for thestress-strain curve considering the size effect of variousconcrete types In this model the basic formula for thestress-strain curve and the key parameter that determinesthe slopes of the ascending and descending branchesestablished by Yang et al [1] was used to generate a completenonlinear curve e concrete compressive strength modelproposed by Sim et al [5] that considers the size effect wasused for the peak stress e strain at the peak stress wasgeneralized with a simple equation using the regressionanalysis of data records compiled from specimens withvarious ranges of equivalent diameter and aspect ratio ofspecimen compressive strength and density In the soft-ening behavior of the descending branch in the stress-strainrelationship the size effect and fracture energy were con-sidered usingMarkeset and Hillerborgrsquos CDZmodel [6]ekey parameter for the softening behavior was determinedfrom the secant modulus joining the origin and 05fSEprime where fSEprime is the compressive strength of concrete consid-ering the size effect e strain model at 05fSEprime to determinethe key parameter was generalized with various ranges ofvariables using nonlinear regression analysis Finally the keyparameter that determines the slope of the descending

branch was formulated as a function of equivalent diameterand aspect ratio of specimen concrete density and concretecompressive strength using parametric numerical analysise accuracy of the proposed model was evaluated using anormalized root-mean-square error obtained from thecomparisons of predicted curves with the test results

2 Database

To formulate the material properties Yang et al [1] com-piled 3295 data records for the elastic modulus of concrete(Ec) 415 data records for strain at the peak stress (ε0) and 96data records for strain at 50 of the peak stress (ε05) in thedescending branch In the compiled data records thenumbers available for the concrete compressive strength(fcprime) and density (ρc) were 3295 and 3274 respectively andvaried from 84MPa to 170MPa and from 1200 kgm3 to4500 kgm3 respectively To apply the effect of concretetype to the empirical formulations the data records weredivided into LWC NWC and HWC according to ρc econcrete density (ρc) varied from 1200 kgm3 to 2000 kgm3

for LWC from 2000 kgm3 to 2500 kgm3 for NWC andfrom 2500 kgm3 to 4500 kgm3 for HWC e data recordscompiled by Yang et al [1] were all measured from a cylinderof diameter 100mm and height 200mm Hence the datarecords compiled by Yang et al [1] lack the test results fordifferent specimen size To compensate for this additionaltest results for the size effect of concrete were compiled ecompiled additional test results [4 5 12ndash15] for the stress-strain relationship were 38 data records for LWC and26 data records for NWC as shown in Table 1 To considervarious specimen shapes an equivalent diameter (deq asymp

(4B Dπ)1113968

)was introduced whereB andD are the sectionalwidth and depth of prism specimen respectively assumingthat the area of the prism specimens is identical to that ofcylinder specimen For example for prism specimens with a B

of 300mm andD of 400mm deq corresponds to 3989mm Inthe data records deq aspect ratio (hdeq) fcprime and ρc werevaried from 50mm to 800mm 05 to 8 4mm to 20mm171MPa to 902MPa and 1500 kgm3 to 2464 kgm3 re-spectively e specimen equivalent diameter (deq) and hdeqwere varied from 100mm to 350mm and 1 to 2 for LWC and50mm to 800mm and 05 to 8 for NWC respectively

3 Model Generalization

31 Basic Approach e stress-strain relationship for un-confined concrete in compression is a parabola with as-cending and descending branches and a vertex at the peakstress [1ndash3] is shape can be generalized using the fol-lowing equation [1]

y β1 + 1( 1113857x

xβ1+1 + β1 (1)

where y fcfcprime is the normalized stress x εcε0 is thenormalized strain fc and εc are the concrete stress andstrain at some point in stress-strain curve respectively andβ1 is the key parameter that determines the slopes of theascending and descending branches e ascending branch



04 Xa( 1113857β1+1


Xd( 1113857β1+1




fSEprime A1

(1k) hdeq1113872 1113873X4

1113969

1 + B1 deqdX1a1113872 1113873 ρc2300( 1113857

minusX21113872 11138731113960 111396105

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦fcprime (3)






fSEprime 09

hdeq1113872 1113873minus06

1113969

1 + 0017deq ρc2300( 1113857minus1

1113960 111396105 + 063⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦fcprime (4)


04 Xa1( 1113857β1+1


Xd1( 1113857β1+1





fcprime(MPa)


deq(mm)


hdeq


da(mm)





2300( )

117 (7)


εSE 09

hdeq( )

minus06radic

1 + 0017deq ρc2300( )minus1[ ]05 + 063

fcprime

E0 (8)


εSE fSEprime

E0 (9)


εSE χfSEprime

Ec( ) (10)



Ec( )[ ] (11)


y=

f cfprime SE

(Xa1 04)(Xd1 05)

Esec

E0

Ec




0

02

04

06

08

1

12

1 2 3 40x = εcεSE


αdeq gt α2deqα

2deq

Confined region

hα

hα

Crack band zone

(a) (b)

Stress relief strip

Confined region

deq deq

ωc ωc

ωc





10( )

2300ρc

( )15

for εc le εSE

(12)




fc

A

C

D εcO

BBprime

Cprime



f primec

(a)


B

A

C

O εc

Bprime

C prime

Total energy



f primec

(b)


0001

00015

0002

00025

0003

00035

0004

00045

0005

0 0001 0002 0003 0004 0005

LWCNWCHWC


ε SE

f primeSEEc


0

10

20

30

40

50

0 2 4 6 8 10 12


R2 = 097

β 1



deq= 50‒500 mmhdeq= 05‒5




εc ε + εdhd

h1113888 1113889 +

ωh

for hgt hd (13a)

εc ε + εd +ωh

for hle hd (13b)


Win

GF

c(1 + k) (14)


εd 2kGF

c(1 + k)fcprime1113890 1113891

fcprime minusfc

fcprime1113888 1113889

08

(15)





Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08hd

h+ωh

for hgt hd

(16a)


Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08

+ωh

for hle hd

(16b)



Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873hd

h+ωh

for hgt hd

(17a)


Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873 +ωh

for hle hd

(17b)







Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)


GF GF0fcprime

10( )

07

(19)


β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)



fSEprime (21)


h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)


Unloading region


(mode I)

Win

(deq = 150 mm hdeq = 20)

Win


Stress relief strip

Crack band zone

(a) (b)


Confined region

hd

f primec

f primec

f primec

Ws = kWin










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





04 Xa( 1113857β1+1


Xd( 1113857β1+1




fSEprime A1

(1k) hdeq1113872 1113873X4

1113969

1 + B1 deqdX1a1113872 1113873 ρc2300( 1113857

minusX21113872 11138731113960 111396105

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦fcprime (3)






fSEprime 09

hdeq1113872 1113873minus06

1113969

1 + 0017deq ρc2300( 1113857minus1

1113960 111396105 + 063⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦fcprime (4)


04 Xa1( 1113857β1+1


Xd1( 1113857β1+1





fcprime(MPa)


deq(mm)


hdeq


da(mm)





2300( )

117 (7)


εSE 09

hdeq( )

minus06radic

1 + 0017deq ρc2300( )minus1[ ]05 + 063

fcprime

E0 (8)


εSE fSEprime

E0 (9)


εSE χfSEprime

Ec( ) (10)



Ec( )[ ] (11)


y=

f cfprime SE

(Xa1 04)(Xd1 05)

Esec

E0

Ec




0

02

04

06

08

1

12

1 2 3 40x = εcεSE


αdeq gt α2deqα

2deq

Confined region

hα

hα

Crack band zone

(a) (b)

Stress relief strip

Confined region

deq deq

ωc ωc

ωc





10( )

2300ρc

( )15

for εc le εSE

(12)




fc

A

C

D εcO

BBprime

Cprime



f primec

(a)


B

A

C

O εc

Bprime

C prime

Total energy



f primec

(b)


0001

00015

0002

00025

0003

00035

0004

00045

0005

0 0001 0002 0003 0004 0005

LWCNWCHWC


ε SE

f primeSEEc


0

10

20

30

40

50

0 2 4 6 8 10 12


R2 = 097

β 1



deq= 50‒500 mmhdeq= 05‒5




εc ε + εdhd

h1113888 1113889 +

ωh

for hgt hd (13a)

εc ε + εd +ωh

for hle hd (13b)


Win

GF

c(1 + k) (14)


εd 2kGF

c(1 + k)fcprime1113890 1113891

fcprime minusfc

fcprime1113888 1113889

08

(15)





Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08hd

h+ωh

for hgt hd

(16a)


Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08

+ωh

for hle hd

(16b)



Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873hd

h+ωh

for hgt hd

(17a)


Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873 +ωh

for hle hd

(17b)







Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)


GF GF0fcprime

10( )

07

(19)


β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)



fSEprime (21)


h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)


Unloading region


(mode I)

Win

(deq = 150 mm hdeq = 20)

Win


Stress relief strip

Crack band zone

(a) (b)


Confined region

hd

f primec

f primec

f primec

Ws = kWin










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






2300( )

117 (7)


εSE 09

hdeq( )

minus06radic

1 + 0017deq ρc2300( )minus1[ ]05 + 063

fcprime

E0 (8)


εSE fSEprime

E0 (9)


εSE χfSEprime

Ec( ) (10)



Ec( )[ ] (11)


y=

f cfprime SE

(Xa1 04)(Xd1 05)

Esec

E0

Ec




0

02

04

06

08

1

12

1 2 3 40x = εcεSE


αdeq gt α2deqα

2deq

Confined region

hα

hα

Crack band zone

(a) (b)

Stress relief strip

Confined region

deq deq

ωc ωc

ωc





10( )

2300ρc

( )15

for εc le εSE

(12)




fc

A

C

D εcO

BBprime

Cprime



f primec

(a)


B

A

C

O εc

Bprime

C prime

Total energy



f primec

(b)


0001

00015

0002

00025

0003

00035

0004

00045

0005

0 0001 0002 0003 0004 0005

LWCNWCHWC


ε SE

f primeSEEc


0

10

20

30

40

50

0 2 4 6 8 10 12


R2 = 097

β 1



deq= 50‒500 mmhdeq= 05‒5




εc ε + εdhd

h1113888 1113889 +

ωh

for hgt hd (13a)

εc ε + εd +ωh

for hle hd (13b)


Win

GF

c(1 + k) (14)


εd 2kGF

c(1 + k)fcprime1113890 1113891

fcprime minusfc

fcprime1113888 1113889

08

(15)





Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08hd

h+ωh

for hgt hd

(16a)


Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08

+ωh

for hle hd

(16b)



Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873hd

h+ωh

for hgt hd

(17a)


Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873 +ωh

for hle hd

(17b)







Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)


GF GF0fcprime

10( )

07

(19)


β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)



fSEprime (21)


h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)


Unloading region


(mode I)

Win

(deq = 150 mm hdeq = 20)

Win


Stress relief strip

Crack band zone

(a) (b)


Confined region

hd

f primec

f primec

f primec

Ws = kWin










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






10( )

2300ρc

( )15

for εc le εSE

(12)




fc

A

C

D εcO

BBprime

Cprime



f primec

(a)


B

A

C

O εc

Bprime

C prime

Total energy



f primec

(b)


0001

00015

0002

00025

0003

00035

0004

00045

0005

0 0001 0002 0003 0004 0005

LWCNWCHWC


ε SE

f primeSEEc


0

10

20

30

40

50

0 2 4 6 8 10 12


R2 = 097

β 1



deq= 50‒500 mmhdeq= 05‒5




εc ε + εdhd

h1113888 1113889 +

ωh

for hgt hd (13a)

εc ε + εd +ωh

for hle hd (13b)


Win

GF

c(1 + k) (14)


εd 2kGF

c(1 + k)fcprime1113890 1113891

fcprime minusfc

fcprime1113888 1113889

08

(15)





Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08hd

h+ωh

for hgt hd

(16a)


Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08

+ωh

for hle hd

(16b)



Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873hd

h+ωh

for hgt hd

(17a)


Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873 +ωh

for hle hd

(17b)







Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)


GF GF0fcprime

10( )

07

(19)


β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)



fSEprime (21)


h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)


Unloading region


(mode I)

Win

(deq = 150 mm hdeq = 20)

Win


Stress relief strip

Crack band zone

(a) (b)


Confined region

hd

f primec

f primec

f primec

Ws = kWin










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





εc ε + εdhd

h1113888 1113889 +

ωh

for hgt hd (13a)

εc ε + εd +ωh

for hle hd (13b)


Win

GF

c(1 + k) (14)


εd 2kGF

c(1 + k)fcprime1113890 1113891

fcprime minusfc

fcprime1113888 1113889

08

(15)





Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08hd

h+ωh

for hgt hd

(16a)


Ec1113888 1113889 +

2kGF

c(1 + k)fSEprime1113888 1113889


fSEprime1113888 1113889

08

+ωh

for hle hd

(16b)



Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873hd

h+ωh

for hgt hd

(17a)


Ec1113888 1113889

2kGF

c(1 + k)fSEprime1113888 1113889 0508

1113872 1113873 +ωh

for hle hd

(17b)







Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)


GF GF0fcprime

10( )

07

(19)


β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)



fSEprime (21)


h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)


Unloading region


(mode I)

Win

(deq = 150 mm hdeq = 20)

Win


Stress relief strip

Crack band zone

(a) (b)


Confined region

hd

f primec

f primec

f primec

Ws = kWin










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






Ec( ) 00027 exp19 times 1000

G04F

fprime14SE d06eq

h

deq( )

minus12

middotρc

2300( )

2

(18)


GF GF0fcprime

10( )

07

(19)


β1 083fSEprime

10( )

065 deq150( )

02h

deq( )

04 2300ρc

( )12

13

for εc gt εSE

(20)



fSEprime (21)


h

σc

σc σc

σc

σc

ε

Ws = kWin

(deq = 150 mm hdeq = 20)



εd

ω (mm)


Unloading region


(mode I)

Win

(deq = 150 mm hdeq = 20)

Win


Stress relief strip

Crack band zone

(a) (b)


Confined region

hd

f primec

f primec

f primec

Ws = kWin










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls










00000000050001000015000200002500030000350004000045

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

100 200 300 400 500 600 700 800 9000deq (mm)

(a)


0000

0001

0002

0003

0004

0005

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

2 4 6 8 100hdeq

(b)



00010

00015

00020

00025

00030

00035

00040

00045

00050

ε SE0

5ndash

(εSE

ndashfprime S

EE c

)

1500 2000 2500 3000 3500 40001000ρc (kgm3)

(c)


0

0001

0002

0003

0004

0005

0006

0007

0 01 02 03 04

LWCNWCHWC


ε SE0

5ndash

(εSE

ndashfprime S

E E c

)




00

10

20

30

40

50

0 1 2 3 4


R2 = 094


β 1


deq= 50‒500 mmhdeq= 05‒5da= 4‒25 mm







Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls








Descending branch


08

GF 000097fcprime + 00418






+ 6900)(ρc2300)15

Descending branch


2]fcprime




fcprime

4radic


is study



minus06radic



117



12]13


Strain

NWC HWCLWC0005

0 0 0 0 0 0 0



hdeq = 05 1 2



hdeq = 2




hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(a)

Figure 10 Continued








0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls










0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120St

ress

(MPa

)

(b)



0 0 0 0 0 0 0


hdeq = 05 1 2



hdeq = 2





hdeq = 2



hdeq = 2

0

20

40

60

80

100

120

Stre

ss (M

Pa)

(c)







5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls








5 Conclusions







LWC

1 cm 050 045 021cs 035 011 010

2 cm 028 038 027cs 024 017 024

Subtotal cm 039 041 024cs 013 029 016

NWC

05sim10 cm 032 024 013cs 012 007 002

2sim35 cm 029 022 017cs 016 016 011

40sim80 cm 024 034 039cs 003 003 005

Subtotal cm 029 023 019cs 014 014 012

HWC 20 cm 029 073 010cs 014 009 001

Total cm 032 034 019cs 019 021 013




15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






15




Notations











damage zone (CDZ)






Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




A Stress-Strain Model for Unconfined Concrete in...

Documents

Transcript of A Stress-Strain Model for Unconfined Concrete in...