Research Article Flexural Strength Evaluation of...

Research ArticleFlexural Strength Evaluation of Reinforced Concrete Memberswith Ultra High Performance Concrete

Baek-Il Bae1 Hyun-Ki Choi2 and Chang-Sik Choi3

1Research Institute of Industrial Science Hanyang University 17 Haengdang-Dong Seongdong-Gu Seoul 04763 Republic of Korea2Department of Fire and Disaster Prevention Engineering Kyungnam University Gyeongsangnam-do 51767 Republic of Korea3Department of Architectural Engineering Hanyang University 17 Haengdang-Dong Seongdong-Gu Seoul 04763 Republic of Korea

Correspondence should be addressed to Hyun-Ki Choi chk7796kyungnamackr

Received 25 September 2015 Revised 5 December 2015 Accepted 7 December 2015

Academic Editor Stefano Sorace

Copyright copy 2016 Baek-Il Bae 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

Flexural strength evaluation models for steel fiber reinforced ultra high strength concrete were suggested and evaluated with testresults Suggested flexural strength models were composed of compression stress blocks and tension stress blocks Rectangularstress block triangular stress block and real distribution shape of stress were used on compression side Under tension rectangularstress block distributed to whole area of tension side and partial area of tension side was used The last model for tension side isrealistic stress distribution All these models were verified with test result which was carried out in this study Test was conductedby four-point loading with 2000 kN actuator for slender beam specimen Additional verifications were carried out with previousresearches on flexural strength of steel fiber reinforced concrete or ultra high strength concrete Total of 21 test specimens wereevaluated As a result of comparison for flexural strength of section neutral axis depth at ultimate state models with triangularcompression stress block and strain-softening type tension stress block can be used as exact solution for ultra high performanceconcrete For the conservative and convenient design of section modified rectangular stress block model can be used with strainsoftening type tension stress block

1 Introduction

Usually flexural strength of normal strength concrete mem-bers is designed using rectangular stress block parametersCurrent design codes provide the rectangular stress blockparameters for simplified design methodology Howeverthese stress blocks are determined by tests of reinforcedconcrete columns and they have apparent limitations Rect-angular stress block can be used because the shape of stress-strain relation of concrete is similar to the trapezoid How-ever shape of stress-strain relationship of concrete changedinto triangle as increase of compressive strength of concreteFor this reason rectangular stress block parameters dependon the compressive strength of concrete For example thecurrent ACI code [1] suggests that higher value of com-pressive strength of concrete can be used as 085 times thespecified compressive strength of concrete And the depthof rectangular stress block has the lower bound of 065 at76MPa of compressive strength of concrete Ultimate strain

of concrete is suggested by value of 0003 These values aredetermined from test results of normal strength concreteHowever depending on the compressive strength mechan-ical properties and failure type of concrete are changed

Generally after experiencing peak stress sudden dropof load resistance can be observed Ultra high strengthconcrete also failed with this failure mode Making brittlefailure of ultra high strength concrete matrix more ductileunder compression steel fiber can be included in the matrixInclusion of steel fiber can change the explosive failure of ultrahigh strength concrete and provide higher tensile strengthand deformability So steel fiber is usually used for ultra highstrength concrete matrix

Ultra high performance concrete usually has muchhigher compressive strength and tensile strength than normalstrength concrete generally ranging from 100 to 200MPaShape of stress distribution in compression side of sectionand tensile strength of concrete shall be considered insection design Design guidelines for ultra high performance

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 2815247 10 pageshttpdxdoiorg10115520162815247

2 Advances in Materials Science and Engineering

concrete suggested the way to design the section of membersuggested stress-strain relation However stress-strain rela-tion for ultra high performance concrete needs specific testresults not using stress blocks or assumptions Therefore inthis study various types of compression and tension stressblock combinations were evaluated with experimental resultand previous research results for easy and safe design of ultrahigh performance concrete members

2 Review of Current Design Codesfor Flexural Strength of Ultra HighPerformance Concrete

Reinforced concrete members using normal strength con-crete are designed with an assumption that stress distributioncan be shapedwith rectangle and concrete cannot transfer thetensile stress However these assumptions cannot be appliedto flexural strength calculation of ultra high performanceconcrete members Since ultra high performance concretehas much higher compressive strength than normal strengthconcrete and usually reinforcedwith steel fiber shape of stressdistribution in compression side will be changed and tensilestress distribution in tension side should be considered inorder to calculate the flexural strength of section Some ofdesign guidelines for high strength concrete or steel fiberreinforced concrete have different assumptions for flexuralstrength calculationThey can be categorized into two groupsone uses stress block parameters and the other uses specifiedstress-strain relation of concrete

Current design code ACI318 [1] suggests that flexuralstrength of reinforced concrete section can be calculated by

119872119899= 119860119904119891119910(119889 minus

119886

2) (1)

In this equation 119886 depth of rectangular stress blockcan be determined by using stress block parameter 120573

1 For

compressive strength of concrete between 17 and 28MPa 085can be used as the value of 120573

1 1205731shall be decreased linearly

a rate of 005 for each 7MPa of compressive strength ofconcrete above 28MPa of compressive strength of concreteThe smallest value of 120573

1is 065

As can be seen in ACI318 [1] current design code provi-sions did not consider the effect of steel fiber Some of designguidelines suggested the way to calculate flexural strength ofsteel fiber reinforced concrete section ACI 544 committee [2]provides the flexural strength equations by adopting researchresults of Henager and Doherty [3] especially for rectangularsection member

119872119899= 119860119904119891119910(119889 minus

119886

2) + 120590119905119887 (ℎ minus 119890) (

ℎ

2+

119890

2minus

119886

2) (2)

where 119872119899is nominal flexural strength of section 119891

119910is yield

strength of steel rebar 119889 is effective depth of section 119886 isdepth of stress block ℎ is height of section 119890 = (120576

119904(fibers) +

0003)(1198880003) 120576119904is strain in tension side 120576

119904(fiber) = 120590

119891119864119904

119888 is neutral axis depth and tensile strength of steel fiberreinforced concrete can be calculated using

120590119905= 000772

119897119891

119889119891

120588119891119865be (3)

where 119897119891is length of steel fiber 119889

119891is diameter of steel fiber 120588

119891

is percent by volume of steel fiber and 119865be is bond efficiencyfactor

Imam et al [4] suggested the modified ACI 544 [2]model which can be used as steel fiber reinforced concretewith high strength matrix Imam et al investigated the bondstress between steel fiber and matrix They suggested thattensile stress block height coefficient should be changed into002 According to this modification tensile strength can becalculated using

120590119905= 2119865 119865 =

119897119891

119889119891

119881119891120578119891 (4)

where119881119891means volume fraction of steel fiber (= 120588

119891100) and

120578119891is fiber factor (10sim12) Moment capacity of section can be

determined according to ACI 544 [2] (2)Lim et al [5] suggested that stress block parameters

should be reevaluated with change of matrix and steel fiberThey use 120572

1as 090 because steel fiber can provide more

ductility under compression either Tensile strength of steelfiber reinforced concrete can be determined using

120590119905119906

= 1205781015840

01205781119881119891119897119891

120591119906

2119903 (5)

where 1205781015840

0is steel fiber orientation factor 120578

1is length efficiency

factor 120591119906is average ultimate bond stress at the fiber-matrix

interface and 119903 is the ratio of the fiber cross-sectional areato its perimeter Since Lim et al [5] developed their modelwith plasticity approach they use whole area over the neutralaxis as compressive stress block Neglecting cover thicknessand considering tensile stress block in tension side of sectionneutral axis depth 119909 can be calculated using

119909 =

119889120590119905119906

+ 119891119910119887

1205721120590119888119906

+ 120590119905119906

(6)

where 120590119888119906

is compressive strength of concrete 119887 is width ofsection and 119891

119910is yield strength of reinforcement From (6)

internal moment arm can be calculated

ℎ = 119889 minus119909

2 (7)

where 119889 is effective depth of section Using (5) (6) and (7)flexural capacity of section can be calculated by using

119872119906= 119891119910ℎ + 120590119905119906

119887

2(ℎ2

minus1199092

4) (8)

Although stress block approach is easy to use for flexuralstrength calculation it cannot consider the difference ofconcrete with higher strength matrix or other characteristics

Advances in Materials Science and Engineering 3

⟨Strain distribution⟩ ⟨Stress distribution⟩

Compression

Tension

ACI318 ACI544 Lim et al

Figure 1 Previously suggested stress block combination

Flexural strength calculation models for normal strengthconcrete and steel fiber reinforced concrete were illustratedin Figure 1 The main difference between normal strengthconcrete model and steel fiber reinforced concrete model isexistence of tensile stress block Difference among steel fiberreinforced concrete models is the range of tensile stress dis-tribution However they are not exact models because stressdistribution might be changed with compressive strength ofmatrix and tensile stress distribution is more comprehensivethan used in Figure 1

For the exact solution for flexural strength of sectioncomprehensive stress-strain relations are directly applied tocalculate the flexural strength of section The representativemodels considering real stress distribution are provided byRILEM 120590 minus 120576 method [9] EC2 flexural analysis [10] andAFGC-Setra guideline [11] They can provide more accuratevalue than flexural strength model made up of stress blocksHowever they need more comprehensive computation pro-cess and some material test

3 Flexura Strength Calculation Model

According to the material test about ultra high performanceconcrete most of stress-strain relation shapes are triangularunder compression Therefore under compression triangu-lar stress block may be used for the design of ultra highperformance concrete flexural members Previous research[12] suggested rectangular stress block parameters for highstrength and ultra high strength concrete However most ofcode provisions use the rectangular stress block parametersbecause they mainly focused on the use for normal strengthconcrete They consider the shape of stress-strain relationusing various value of 120573

1 depending on compressive strength

of concrete Rectangular stress block slightly overestimatesthe flexural strength of concrete member especially for highreinforcement ratio and compressive strength of concrete Ascan be seen in Section 2 tensile stress block for steel fiberreinforced concrete has been shown in various shape andsize Therefore designing ultra high performance concretemembers stress block parameters should be reorganized

In this study three types of stress block parameterswere considered ACI stress block parameters stress blockparameters from UHPC member design guideline and tri-angular stress block determined by maximum compressive

b

dh

As

e c

120576cu fcu

120576f

120576s

120576tuftb

ft

fy

Section Assumed Assumedstrain distribution stress distribution

Figure 2 Strain and stress distribution of ultra high performanceconcrete section

strength and corresponding strain resulting from materialtests Tensile behavior of steel fiber reinforced concrete wasdivided into strain hardening strain softening and fullyplastic behavior three types In this study tensile stress blockswere composed of these three types of tensile behavior of steelfiber reinforced concrete

Strain and stress distribution of ultra high performanceconcrete section were shown in Figure 2 In this study threetypes of stress blocks were used under compression andtension respectively Total of 9 types of flexural strengthmodels were investigated These models were illustrated inFigure 3 The most important design parameter for flexuralstrength is neutral axis depth Neutral axis depth for 9 typesof flexural strength model was developed as follows

1198881198881

=

119860119904119891119910minus 120574ℎ119891

119905119887

051198911015840119888119887 minus 05 (120578 minus 1) 119891

119905119887 minus 120574120578119891

119905119887

1198881198882

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

051198911015840119888119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198883

=

119860119904119891119910+ 120574119891119905ℎ119887

051198911015840119888119887 + 120574120578119891

119905119887

1198881198884

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887


fcu

ftb

ft

fy

(a) Type 1

fcu

ftb

ft

fy

(b) Type 2

fcu

fy

1205741ft

(c) Type 3

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(d) Type 4

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(e) Type5

fy

1205741ft

1205731A

CIc

1205721ACIfcu

(f) Type 6

ftb

ft

fy

1205731U

HPC

c1205721UHPCfcu

(g) Type 7

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu

(h) Type 8

fy

1205741ft

1205731U

HPC

c

1205721UHPCfcu

(i) Type 9

Figure 3 Stress block models

1198881198885

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198886

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721ACI119891

1015840

1198881205731ACI119887 + 120574120578119891

119905119887

1198881198887

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887

1198881198888

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198889

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721UHPC119891

1015840

1198881205731UHPC119887 + 120574120578119891

119905119887

(9)

where 1198911015840

119888is compressive strength of concrete 119860

119904is area of

tensile rebar 119891119910is yield strength of steel rebar 119891

119905is ultimate

tensile strength of concrete 120574 is ratio between post crackingtensile strength and ultimate tensile strength 120578 can be definedby 120576119891120576119888119906

+ 1 1205721and 120572

1UHPC are rectangular stress blockparameter for compressive strength of concrete for normalstrength concrete andUHPC respectively 120573

1and 1205731UHPC are

stress block depth parameter for normal strength concreteand UHPC respectively 119887 is width of section and 119889 is

effective depth of section 120576119891for 120578 is strain corresponding

to ultimate tensile strength and 120576119888119906

is ultimate compressivestrain of concrete

The most difficult and controversial part is the determi-nation of tensile strength of fiber reinforced concrete Swamyand Al-Tarsquoan [13] suggested the equation according to thecomposite theory as follows

119891119905= 0970119891

119903(1 minus 119881

119891) + 2119881

119891

119871119891

119863119891

(10)

where 119891119903is modulus of rupture of concrete and other

variables are fiber geometry defined in Section 2Using neutral axis depth defined in (9) considering

shape of stress block nominal flexural strength of ultra highperformance concrete members can be calculated as follows

Case 1 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(11)


Table 1 Mix proportions

119908119887

Weight ratio Steel fiber Admixture 119891119888119896

Cement Water Silica fume Sand Filler (Micro Silica)() (kg) (MPa)

017 1 021 024 104 031 2 108 200

Case 2 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

3) + 119860

119904119891119910(119889 minus 119888)

(12)

Case 3 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 120574119891

119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(13)

Cases 4 7 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(14)

Cases 5 8 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

2) + 119860

119904119891119910(119889 minus 119888)

(15)

Cases 6 9 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 120574119891119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(16)

4 Flexural Behavior of Ultra HighPerformance Concrete Members

41 Test Plan In order to verify the applicability of sug-gested models ultra high performance concrete beam wastested Average ultimate compressive strength of standardcylinder was 216MPa Splitting strength of standard cylinderis distributed between 72sim195MPa Mix proportions forultra high performance concrete are summarized in Table 1Mechanical properties of concrete and rebar used in thisstudy were summarized in Tables 2 and 3 respectively

Table 2 Mechanical characteristics of rebar

MaterialsYield

strength(MPa)

Yieldstrain(120576119910)

Tensilestrength(MPa)

Poissonrsquosratio

D25 422 00021 621 028D10 384 00019 568 027

Preventing premature shear failure of specimen total spanof test specimen is reinforced by stirrups with spacing of150mm Stirrups were not located between two loadingpoints 2000 kN actuator was used for test and shear-span todepth ratio was 65 In order to verify the neutral axis depthcalculation model which was shown in (9) strain gages forconcrete were mounted at the concrete surface Strain gagesfor steel also can be attached to reinforcement at the centerof test specimen Details of test specimens were illustrated inFigure 4

42 Test Results Test specimens have shown the flexuralfailure pattern Because of inclusion of steel fiber cracklocalization did not occur until crushing of concrete occurredat extreme compression fiber After initial crack occurredcrackswere spread to outside themaximumbendingmomentarea After yielding of reinforcement diagonal tension crackwas not observed and cracks were spread to supports Atdeflection 98mm crushing of concrete occurred and crackswere propagated to crushing area with opening of initialcrack Final stage of failure and load-deflection relation wereshown in Figures 5(a) and 5(b) respectively Maximumload was 179 kN which occurred after yielding of reinforce-ment Analyzing strain gauges attached to concrete andreinforcement neutral axis depth was 123mm at peak loadstage

Neutral axis depth is important index for reinforcedconcrete members because flexural strength and ductilitycan highly depend on the neutral axis depth of sectionNeutral axis depth can be measured by test using the valueof strain gauge attached to compression fiber and tensionreinforcements Since strain gauge attached to extreme tensileand compression fiber failed before experiencing peak loadstrain of compression and tension reinforcement were usedCurvature at first yield of tension reinforcement was 00122(1m) and peak load curvature was 0019 (1m) Neutral axisdepth was 925mm and 1228mm from extreme compressionfiber respectively As shown in Figure 6 change of neutralaxis depth occurred after yielding of tension reinforcementAfter experiencing peak load neutral axis depth did notchange until crushing of concrete in compression side ofsection occurred


Table 3 Mechanical characteristics of concrete

Stress state Ultimate strength(MPa) Youngrsquos modulus (MPa) Ultimatecracking

strain (permil) Poissonrsquos ratio

Compression 216 54306 3738 (120576119888119906) 026

Tension 98 0221 (120576119905)

350

300

50

150 1900 1900

5-D25

2-D10 Load Load

500 150

Strain gage

Strain gage

D10150 A

A

concrete

rebar

(a) Setting and measurement planSection A-A

5-D25

D10

200

240

6050

350

(b) Section

Figure 4 Details of test specimen

Table 4 Comparison between test results and assumed model

Model 119888 120576119904

119872119899

119875119899

(mm) (permil) (kNm) (kN)Type 1 9088 123 363 202Type 2 8598 13 296 164Type 3 9503 118 425 235Type 4 6876 163 356 197Type 5 7872 142 297 164Type 6 8739 128 433 240Type 7 9130 122 316 175Type 8 9813 114 308 170Type 9 10228 110 353 197Test results 93 910 322 179119888 neutral axis depth 120576

119904 strain at tensile reinforcement atmid length of beam

119872119899 nominal flexural strength of section (predicted value) and 119875

119899 load for

119872119899

43 Validation of Flexural StrengthModel Verifying suitabil-ity of flexural strength models test results were comparedwith assumed flexural strength model Comparison resultswere listed in Table 4 As expected model type 1 which hastriangular stress block has shown comparatively high accu-racy However this model overestimated moment capacity ofsection Overestimation of this model was caused by largearea of tensile stress block and higher value of momentarm Higher value of moment arm can be derived by theexistence of residual strength Model type 2 also has shownacceptable accuracy but this model underestimated momentcapacity Underestimation of this model was caused by therelatively low level of tensile stress block area and lower valueof moment arm for tensile stress block This smaller momentarm was derived by the end of the stress block Model type3 which has rectangular tensile stress block has shown lowaccuracy and overestimation

Generally rectangular stress block has shown deeperneutral axis depth from compression fiber than triangularstress block moment arm has lower value than the casesof triangular stress block types Model types with ACI

rectangular stress block have low accuracy on neutral axisdepth tensile strain of reinforcements andmoment capacityThis phenomenonwas caused by the larger area of stress blockthan triangular stress block However Models with ultrahigh performance concrete stress block parameters whichare derived from test results [12] have shown relatively highaccuracy with all types of tensile stress block This modelespecially predicts neutral axis depth more accurately thanother models As discussed above the most accurate tensilestress block was also stress block including residual strengthof concrete

Considering low ductility which is caused by explosivefailure of ultra high strength concrete the most appropriatemodel for design of flexural strength is type 8whichwasmod-elled by UHPC-rectangular stress block parameters undercompression and tension softening considering model undertension Type 7 also can be used for design purposes butin this case strength reduction factor is carefully consideredwith material properties

5 Validation of Flexural Strength Models withPrevious Researches

For the verification of wide range applicability of assumedmodel existing test results [6ndash8] of ultra high performanceconcrete members were compared with assumed modelsBecause a few number of specimens exist only 22 test resultswere compared with suggested model Specifications forcollected test results were shown in Table 5 Test results fromeach research were summarized in Table 5 All test specimensexperienced yielding of reinforcement before reaching peakload They failed with flexural failure at the center of thespecimens Collected test specimens have 027 to 236 oftensile reinforcement ratio and 80sim200MPa of compressivestrength of concrete Fiber contents were distributed from05 to 20

According to the test results of Ashour et al [6] effect ofthe fibers to flexural strength is independent of the amount ofreinforcement but additional moment is proportional to con-crete compressive strength Therefore compressive strength


(a) Final stage of failure

Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)

Peak strengthType 1 202 kN

Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu

(b) Load-deflection relation

Figure 5 Test results

Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)

Yielding ofreinforcement

93mm Peak load

Extreme tensile fiber

Extreme compression fiber

Measurement 1 compression steelMeasurement 2 compression fiber

Type 7 973mm

Figure 6 Change of neutral axis depth

of concrete should be considered in flexural strength modelDancygier and Savir [7] investigated flexural strength of fiberreinforced concrete with conventional reinforcement ratiolower than 1 Fiber length which affects reinforcing index(RI = 119881

119891119871119891119863119891 where119881

119891is fibre volume fraction 119871

119891is fiber

length and119863119891is fiber diameter) directly changes the flexural

strength of section Generally fiber length changes the failuremode of fiber reinforced concrete under bending especiallyafter peak load Therefore material behavior under tensionshould be considered for flexural strength calculation Yang etal [8] investigated about placing method and reinforcementratio especially for ultra high performance concrete whichhave compressive strength around 200MPa According tothis research reinforcement ratio also affects the flexuralstrength and ductility although members have much higher

compressive strength of concrete Some of test results did notevaluate prediction methods previously investigated [2 4 5]

Therefore all assumed models were examined with testresults Comparison of test results with flexural strengthmodels was shown in Figures 7(a)ndash7(d) and descriptive sta-tistical data were shown in Table 6 Existing flexural strengthmodel which was reviewed in this paper only has accuracywhen using comparatively lower compressive strength ofconcrete (80sim100MPa) as shown in Figure 7(a)The varianceof this model increases with increase of compressive strengthof concrete As shown in Figures 7(b) and 7(c) types 1to 6 still have overestimated the flexural strength of ultrahigh performance concreteHowever when tension softeningmodel was applied accuracy comparatively increased Types7 to 9 which use the newly developed rectangular stressblock show more accuracy and safety rather than the other9 models Types 7 and 8 especially have high accuracy overthe range of ultra high performance concrete Because type8 which uses tension softening model underestimates theflexural capacity it would be used for safe design of ultra highperformance concrete members

6 Conclusion

The following conclusions can be made from the statisticalinvestigation analytical works and tests for FRUHSC forflexural strength

(1) Stress distribution of ultra high performance concreteunder compression shaped as triangle Under tensilestress stress distribution of ultra high performanceconcrete can be varied with fiber contents or shape

(2) Nine-flexural strengthmodels were evaluated For theconservative estimation ultimate strain at compres-sion fiber was assumed to be 0003 These flexuralstrength models were evaluated using assumed stressdistribution


Table 5 Previous test results

Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906

(mm) (mm) (mm) (mm2) () (MPa) () (mm) (mm) (MPa) (kN)Ashour et al [6]

B-05-M2 200 250 215 509 118 82 05 08 60 530 491B-10-M2 200 250 215 509 118 87 1 08 60 530 542B-05-M3 200 250 215 763 178 82 05 08 60 530 695B-10-M3 200 250 215 763 178 87 1 08 60 530 714B-05-M4 200 250 215 1018 237 82 05 08 60 530 881B-10-M4 200 250 215 1018 237 87 1 08 60 530 897B-05-H2 200 250 215 509 118 107 05 08 60 530 485B-10-H2 200 250 215 509 118 111 1 08 60 530 537B-05-H3 200 250 215 763 178 107 05 08 60 530 696B-10-H3 200 250 215 763 178 111 1 08 60 530 741B-05-H4 200 250 215 1018 237 107 05 08 60 530 891B-10-H4 200 250 215 1018 237 111 1 08 60 530 935

Dancygier and Savir [7]H5-F2-1 35 200 300 273 151 028 129 075 09 35 480 236H5-F2-1 60 200 300 273 151 028 124 075 09 60 480 273H8-F2-1 35 200 300 273 302 055 124 075 09 35 480 389H8-F2-1 60 200 300 273 302 055 122 075 09 60 480 372H5-F2-1 35 3 200 300 273 151 028 122 075 09 35 616 281

Yang et al [8]R12-1 180 270 235 253 060 191 2 02 13 400 770R13-1 180 270 235 380 090 192 2 02 13 400 863R14-1 180 270 235 507 120 197 2 02 13 400 1031R23-2 180 270 220 760 120 196 2 02 13 400 1165119887 beam width ℎ beam height 119889 effective depth of beam 119860

119904119905 tensile reinforcement area 120588 reinforcement ratio 119891

119888119906 compressive strength of concrete 119881

119891

volume fraction of steel fiber119863119891 fiber diameter 119871

119891 fiber length 119891

119910 yield strength of reinforcement and 119875

119906 test results (load at ultimate failure)

Table 6 Descriptive statistics on collected test data(testprediction)

ID Mean Median SD Var COV IAEACI544 124 106 034 0115 027 1720Imam 122 101 045 0201 037 1983Lim 113 104 026 0068 023 1415Type 1 081 090 020 0040 024 3356Type 2 118 107 029 0086 025 1672Type 3 070 076 016 0024 022 4654Type 4 087 091 014 0021 016 1563Type 5 118 106 030 0091 025 1662Type 6 069 075 015 0023 022 4918Type 7 090 095 015 0021 016 1030Type 8 111 107 024 0060 022 1142Type 9 073 077 014 0021 020 3558SD standard variation Var variance COV coefficient of variation and IAEintegrated absolute error

(3) The most accurate model of ultra high performanceconcrete under compression is triangular Becauseultra high performance concrete has large elastic areafailure occurred with the same time experiencing

ultimate strength However for the safe design ofsection flexural strength model which uses modifiedrectangular stress blocks considering mechanicalcharacteristics of UHPC should be used

(4) Evaluating validation process using test result ofthis study and previous researches existing flexuralstrength calculation models cannot accurately andsafely predict the flexural strength of ultra highstrength concrete specimens especially for compres-sive strength larger than 100MPa However flexuralstrength model suggested in this study can pro-vide conservative and highly accurate (10 of error)results

Conflict of Interests

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

Acknowledgments

This research was supported by a grant (15CTAP-C097356-01) from Creative Challenge Research Program fundedby Ministry of Land Infrastructure and Transport of


Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00

Compressive strength of concrete (MPa)80 100 120 140 160 180 200

Underestimation

Overestimation

ACI544mdashAshour (1)ImammdashAshour (1)LimmdashAshour (1)ACI544mdashAshour (2)ImammdashAshour (2)

LimmdashAshour (2)ACI544mdashYangImammdashYangLimmdashYang

(a) Existing modelTe

st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation

Type 1mdashAshourType 2mdashAshourType 3mdashAshourType 1mdashDancygierType 2mdashDancygier

Type 3mdashDancygierType 1mdashYangType 2mdashYangType 3mdashYang

(b) Assumed types 1sim3

Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation



(c) Assumed types 4sim6

Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation



(d) Assumed types 7sim9

Figure 7 Applicability verification for prediction model

Korean government and National Research Foundation ofKorea(NRF) funded by the Ministry of Science ICT andFuture Planning (no NRF-2014R1A2A1A11051049)

References

[1] ACI Committee ldquoBuilding code requirements for structuralconcrete and commentaryrdquo ACI 318-11 American ConcreteInstitute Farmington Hills Mich USA 2011

[2] American Concrete Institute Committee 544 ldquoDesign con-siderations for steel fiber reinforced concreterdquo InternationalConcrete Abstracts Portal vol 85 no 5 pp 563ndash579 1988

[3] C H Henager and T J Doherty ldquoAnalysis of reinforced fibrousconcrete beamsrdquo Journal of the Structural Division vol 102 no1 pp 177ndash188 1976 ASCE Proceedings

[4] M Imam L Vandewalle and F Mortelmans ldquoShear-momentanalysis of reinforced high strength concrete beams containingsteel fibresrdquo Canadian Journal of Civil Engineering vol 22 no3 pp 462ndash470 1995

[5] T Y Lim P Paramasivan and S L Lee ldquoShear and momentcapacity of reinforced steel fiber concrete beamsrdquo Magazine ofConcrete Research vol 39 no 140 pp 148ndash160 1987

[6] S A Ashour F F Wafa and M I Kamal ldquoEffect of theconcrete compressive strength and tensile reinforcement ratio


on the flexural behavior of fibrous concrete beamsrdquo EngineeringStructures vol 22 no 9 pp 1145ndash1158 2000

[7] A N Dancygier and Z Savir ldquoFlexural behavior ofHSFRCwithlow reinforcement ratiosrdquo Engineering Structures vol 28 no 11pp 1503ndash1512 2006

[8] I H Yang C Joh and B-S Kim ldquoStructural behavior ofultra high performance concrete beams subjected to bendingrdquoEngineering Structures vol 32 no 11 pp 3478ndash3487 2010

[9] RILEM and Final recommendations of TC 162-TDF ldquoTest anddesign methods for steel fibre reinforced concrete 120590-120576 designmethodrdquoMaterials and Structures vol 36 pp 560ndash565 2003

[10] European Committee for Standardization Eurocode 2 (EC2)Design of Concrete Structures 2004

[11] AFGCGroupe de Travail BFUP SETRA-AFGCUltra High Per-formance Fiber-Reinforced Concretes Interim Recommendation2002

[12] B I Bae C S Choi and H K Choi ldquoEvaluation of rectangularstress block parameters of reactive powder concrete membersrdquoJournal of the Architectural Institute of Korea vol 28 no 7 pp3ndash11 2012

[13] R N Swamy and S A Al-Tarsquoan ldquoDeformation and ultimatestrength in flexure of reinforced concrete beamsmadewith steelfibre concreterdquo Journal of the American Concrete Institute vol78 no 5 pp 395ndash405 1981

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


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


concrete suggested the way to design the section of membersuggested stress-strain relation However stress-strain rela-tion for ultra high performance concrete needs specific testresults not using stress blocks or assumptions Therefore inthis study various types of compression and tension stressblock combinations were evaluated with experimental resultand previous research results for easy and safe design of ultrahigh performance concrete members

2 Review of Current Design Codesfor Flexural Strength of Ultra HighPerformance Concrete

Reinforced concrete members using normal strength con-crete are designed with an assumption that stress distributioncan be shapedwith rectangle and concrete cannot transfer thetensile stress However these assumptions cannot be appliedto flexural strength calculation of ultra high performanceconcrete members Since ultra high performance concretehas much higher compressive strength than normal strengthconcrete and usually reinforcedwith steel fiber shape of stressdistribution in compression side will be changed and tensilestress distribution in tension side should be considered inorder to calculate the flexural strength of section Some ofdesign guidelines for high strength concrete or steel fiberreinforced concrete have different assumptions for flexuralstrength calculationThey can be categorized into two groupsone uses stress block parameters and the other uses specifiedstress-strain relation of concrete

Current design code ACI318 [1] suggests that flexuralstrength of reinforced concrete section can be calculated by

119872119899= 119860119904119891119910(119889 minus

119886

2) (1)

In this equation 119886 depth of rectangular stress blockcan be determined by using stress block parameter 120573

1 For

compressive strength of concrete between 17 and 28MPa 085can be used as the value of 120573

1 1205731shall be decreased linearly

a rate of 005 for each 7MPa of compressive strength ofconcrete above 28MPa of compressive strength of concreteThe smallest value of 120573

1is 065

As can be seen in ACI318 [1] current design code provi-sions did not consider the effect of steel fiber Some of designguidelines suggested the way to calculate flexural strength ofsteel fiber reinforced concrete section ACI 544 committee [2]provides the flexural strength equations by adopting researchresults of Henager and Doherty [3] especially for rectangularsection member

119872119899= 119860119904119891119910(119889 minus

119886

2) + 120590119905119887 (ℎ minus 119890) (

ℎ

2+

119890

2minus

119886

2) (2)

where 119872119899is nominal flexural strength of section 119891

119910is yield

strength of steel rebar 119889 is effective depth of section 119886 isdepth of stress block ℎ is height of section 119890 = (120576

119904(fibers) +

0003)(1198880003) 120576119904is strain in tension side 120576

119904(fiber) = 120590

119891119864119904

119888 is neutral axis depth and tensile strength of steel fiberreinforced concrete can be calculated using

120590119905= 000772

119897119891

119889119891

120588119891119865be (3)

where 119897119891is length of steel fiber 119889

119891is diameter of steel fiber 120588

119891

is percent by volume of steel fiber and 119865be is bond efficiencyfactor

Imam et al [4] suggested the modified ACI 544 [2]model which can be used as steel fiber reinforced concretewith high strength matrix Imam et al investigated the bondstress between steel fiber and matrix They suggested thattensile stress block height coefficient should be changed into002 According to this modification tensile strength can becalculated using

120590119905= 2119865 119865 =

119897119891

119889119891

119881119891120578119891 (4)

where119881119891means volume fraction of steel fiber (= 120588

119891100) and

120578119891is fiber factor (10sim12) Moment capacity of section can be

determined according to ACI 544 [2] (2)Lim et al [5] suggested that stress block parameters

should be reevaluated with change of matrix and steel fiberThey use 120572

1as 090 because steel fiber can provide more

ductility under compression either Tensile strength of steelfiber reinforced concrete can be determined using

120590119905119906

= 1205781015840

01205781119881119891119897119891

120591119906

2119903 (5)

where 1205781015840

0is steel fiber orientation factor 120578

1is length efficiency

factor 120591119906is average ultimate bond stress at the fiber-matrix

interface and 119903 is the ratio of the fiber cross-sectional areato its perimeter Since Lim et al [5] developed their modelwith plasticity approach they use whole area over the neutralaxis as compressive stress block Neglecting cover thicknessand considering tensile stress block in tension side of sectionneutral axis depth 119909 can be calculated using

119909 =

119889120590119905119906

+ 119891119910119887

1205721120590119888119906

+ 120590119905119906

(6)

where 120590119888119906

is compressive strength of concrete 119887 is width ofsection and 119891

119910is yield strength of reinforcement From (6)

internal moment arm can be calculated

ℎ = 119889 minus119909

2 (7)

where 119889 is effective depth of section Using (5) (6) and (7)flexural capacity of section can be calculated by using

119872119906= 119891119910ℎ + 120590119905119906

119887

2(ℎ2

minus1199092

4) (8)

Although stress block approach is easy to use for flexuralstrength calculation it cannot consider the difference ofconcrete with higher strength matrix or other characteristics



Compression

Tension










b

dh

As

e c

120576cu fcu

120576f

120576s

120576tuftb

ft

fy





1198881198881

=

119860119904119891119910minus 120574ℎ119891

119905119887

051198911015840119888119887 minus 05 (120578 minus 1) 119891

119905119887 minus 120574120578119891

119905119887

1198881198882

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

051198911015840119888119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198883

=

119860119904119891119910+ 120574119891119905ℎ119887

051198911015840119888119887 + 120574120578119891

119905119887

1198881198884

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887


fcu

ftb

ft

fy

(a) Type 1

fcu

ftb

ft

fy

(b) Type 2

fcu

fy

1205741ft

(c) Type 3

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(d) Type 4

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(e) Type5

fy

1205741ft

1205731A

CIc

1205721ACIfcu

(f) Type 6

ftb

ft

fy

1205731U

HPC

c1205721UHPCfcu

(g) Type 7

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu

(h) Type 8

fy

1205741ft

1205731U

HPC

c

1205721UHPCfcu

(i) Type 9


1198881198885

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198886

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721ACI119891

1015840

1198881205731ACI119887 + 120574120578119891

119905119887

1198881198887

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887

1198881198888

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198889

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721UHPC119891

1015840

1198881205731UHPC119887 + 120574120578119891

119905119887

(9)

where 1198911015840


119904is area of


119905is ultimate


+ 1 1205721and 120572








119891119905= 0970119891

119903(1 minus 119881

119891) + 2119881

119891

119871119891

119863119891

(10)




Case 1 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(11)



119908119887



017 1 021 024 104 031 2 108 200

Case 2 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

3) + 119860

119904119891119910(119889 minus 119888)

(12)

Case 3 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 120574119891

119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(13)

Cases 4 7 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(14)

Cases 5 8 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

2) + 119860

119904119891119910(119889 minus 119888)

(15)

Cases 6 9 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 120574119891119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(16)




MaterialsYield

strength(MPa)



Poissonrsquosratio

D25 422 00021 621 028D10 384 00019 568 027








Compression 216 54306 3738 (120576119888119906) 026

Tension 98 0221 (120576119905)

350

300

50

150 1900 1900

5-D25

2-D10 Load Load

500 150

Strain gage

Strain gage

D10150 A

A

concrete

rebar


5-D25

D10

200

240

6050

350

(b) Section



Model 119888 120576119904

119872119899

119875119899




119899 load for

119872119899










Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)


Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu



Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)


93mm Peak load




Type 7 973mm



119891119871119891119863119891 where119881


119891is fiber





6 Conclusion






Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Compression

Tension










b

dh

As

e c

120576cu fcu

120576f

120576s

120576tuftb

ft

fy





1198881198881

=

119860119904119891119910minus 120574ℎ119891

119905119887

051198911015840119888119887 minus 05 (120578 minus 1) 119891

119905119887 minus 120574120578119891

119905119887

1198881198882

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

051198911015840119888119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198883

=

119860119904119891119910+ 120574119891119905ℎ119887

051198911015840119888119887 + 120574120578119891

119905119887

1198881198884

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887


fcu

ftb

ft

fy

(a) Type 1

fcu

ftb

ft

fy

(b) Type 2

fcu

fy

1205741ft

(c) Type 3

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(d) Type 4

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(e) Type5

fy

1205741ft

1205731A

CIc

1205721ACIfcu

(f) Type 6

ftb

ft

fy

1205731U

HPC

c1205721UHPCfcu

(g) Type 7

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu

(h) Type 8

fy

1205741ft

1205731U

HPC

c

1205721UHPCfcu

(i) Type 9


1198881198885

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198886

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721ACI119891

1015840

1198881205731ACI119887 + 120574120578119891

119905119887

1198881198887

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887

1198881198888

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198889

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721UHPC119891

1015840

1198881205731UHPC119887 + 120574120578119891

119905119887

(9)

where 1198911015840


119904is area of


119905is ultimate


+ 1 1205721and 120572








119891119905= 0970119891

119903(1 minus 119881

119891) + 2119881

119891

119871119891

119863119891

(10)




Case 1 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(11)



119908119887



017 1 021 024 104 031 2 108 200

Case 2 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

3) + 119860

119904119891119910(119889 minus 119888)

(12)

Case 3 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 120574119891

119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(13)

Cases 4 7 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(14)

Cases 5 8 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

2) + 119860

119904119891119910(119889 minus 119888)

(15)

Cases 6 9 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 120574119891119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(16)




MaterialsYield

strength(MPa)



Poissonrsquosratio

D25 422 00021 621 028D10 384 00019 568 027








Compression 216 54306 3738 (120576119888119906) 026

Tension 98 0221 (120576119905)

350

300

50

150 1900 1900

5-D25

2-D10 Load Load

500 150

Strain gage

Strain gage

D10150 A

A

concrete

rebar


5-D25

D10

200

240

6050

350

(b) Section



Model 119888 120576119904

119872119899

119875119899




119899 load for

119872119899










Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)


Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu



Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)


93mm Peak load




Type 7 973mm



119891119871119891119863119891 where119881


119891is fiber





6 Conclusion






Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




fcu

ftb

ft

fy

(a) Type 1

fcu

ftb

ft

fy

(b) Type 2

fcu

fy

1205741ft

(c) Type 3

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(d) Type 4

ftb

ft

fy

1205731A

CIc

1205721ACIfcu

(e) Type5

fy

1205741ft

1205731A

CIc

1205721ACIfcu

(f) Type 6

ftb

ft

fy

1205731U

HPC

c1205721UHPCfcu

(g) Type 7

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu

(h) Type 8

fy

1205741ft

1205731U

HPC

c

1205721UHPCfcu

(i) Type 9


1198881198885

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721ACI119891

1015840

1198881205731ACI119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198886

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721ACI119891

1015840

1198881205731ACI119887 + 120574120578119891

119905119887

1198881198887

=

119860119904119891119910+ 120574119891119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 120574120578119891

119905119887

1198881198888

=

119860119904119891119910+ 05 (1 + 120574) 119891

119905119887ℎ

1205721UHPC119891

1015840

1198881205731UHPC119887 minus 05 (120578 minus 1) 119891

119905119887 + 05120578 (1 + 120574) 119891

119905119887

1198881198889

=

119860119904119891119910+ 120574119891119905ℎ119887

1205721UHPC119891

1015840

1198881205731UHPC119887 + 120574120578119891

119905119887

(9)

where 1198911015840


119904is area of


119905is ultimate


+ 1 1205721and 120572








119891119905= 0970119891

119903(1 minus 119881

119891) + 2119881

119891

119871119891

119863119891

(10)




Case 1 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(11)



119908119887



017 1 021 024 104 031 2 108 200

Case 2 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

3) + 119860

119904119891119910(119889 minus 119888)

(12)

Case 3 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 120574119891

119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(13)

Cases 4 7 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(14)

Cases 5 8 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

2) + 119860

119904119891119910(119889 minus 119888)

(15)

Cases 6 9 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 120574119891119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(16)




MaterialsYield

strength(MPa)



Poissonrsquosratio

D25 422 00021 621 028D10 384 00019 568 027








Compression 216 54306 3738 (120576119888119906) 026

Tension 98 0221 (120576119905)

350

300

50

150 1900 1900

5-D25

2-D10 Load Load

500 150

Strain gage

Strain gage

D10150 A

A

concrete

rebar


5-D25

D10

200

240

6050

350

(b) Section



Model 119888 120576119904

119872119899

119875119899




119899 load for

119872119899










Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)


Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu



Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)


93mm Peak load




Type 7 973mm



119891119871119891119863119891 where119881


119891is fiber





6 Conclusion






Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119908119887



017 1 021 024 104 031 2 108 200

Case 2 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 119891

119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

3) + 119860

119904119891119910(119889 minus 119888)

(12)

Case 3 Consider

119872119899= (

119891119888119896119888119887

2)

2

3119888 + 120574119891

119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(13)

Cases 4 7 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(14)

Cases 5 8 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 119891119905(119890 minus 119888) 119887

2

3(119890 minus 119888)

+ 120574119891119905(ℎ minus 119890) 119887 (119890 +

ℎ minus 119890

2) + 119860

119904119891119910(119889 minus 119888)

(15)

Cases 6 9 Consider

119872119899= (11988611198911198881198961205731119888119887)

119888

2+ 120574119891119905(ℎ minus 119890) (119890 minus 119888 +

ℎ minus 119890

2)

+ 119860119904119891119910(119889 minus 119888)

(16)




MaterialsYield

strength(MPa)



Poissonrsquosratio

D25 422 00021 621 028D10 384 00019 568 027








Compression 216 54306 3738 (120576119888119906) 026

Tension 98 0221 (120576119905)

350

300

50

150 1900 1900

5-D25

2-D10 Load Load

500 150

Strain gage

Strain gage

D10150 A

A

concrete

rebar


5-D25

D10

200

240

6050

350

(b) Section



Model 119888 120576119904

119872119899

119875119899




119899 load for

119872119899










Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)


Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu



Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)


93mm Peak load




Type 7 973mm



119891119871119891119863119891 where119881


119891is fiber





6 Conclusion






Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







Compression 216 54306 3738 (120576119888119906) 026

Tension 98 0221 (120576119905)

350

300

50

150 1900 1900

5-D25

2-D10 Load Load

500 150

Strain gage

Strain gage

D10150 A

A

concrete

rebar


5-D25

D10

200

240

6050

350

(b) Section



Model 119888 120576119904

119872119899

119875119899




119899 load for

119872119899










Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)


Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu



Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)


93mm Peak load




Type 7 973mm



119891119871119891119863119891 where119881


119891is fiber





6 Conclusion






Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Load

s (kN

)

220

200

180

160140120100806040200

160

140

120

100

80

60

40

20

0

Deflection (mm)


Type 7 175kN

Type 1 Type 7

fcu

ftb

ft

fy

ftb

ft

fy

1205731U

HPC

c

1205721UHPCfcu



Neu

tral

axis

(mm

)

350

300

250

200

150

100

50

0

000

000

001 002

002

003 004

004

005

Curvature (1m)


93mm Peak load




Type 7 973mm



119891119871119891119863119891 where119881


119891is fiber





6 Conclusion






Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





Specimen 119887 ℎ 119889 119860119904119905

120588 119891119888119906

119881119891

119863119891

119871119891

119891119910

119875119906







119891












Acknowledgments



Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






References























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Test

valu

eth

eorit

ical

val

ue35

30

25

20

15

10

05

00


Underestimation

Overestimation




st va

lue

theo

ritic

al v

alue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation




Test

valu

eth

eorit

ical

val

ue

35

30

25

20

15

10

05

00


Underestimation

Overestimation






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



Research Article Flexural Strength Evaluation of...

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Transcript of Research Article Flexural Strength Evaluation of...