Flexural behaviour of rebar-reinforced ultra-high

Flexural behaviour of rebar-reinforcedultra-high-performance concrete beams

Shiming ChenProfessor, College of Civil Engineering, Tongji University, Shanghai, China

Rui ZhangPhD student, College of Civil Engineering, Tongji University, Shanghai,China

Liang-Jiu JiaAssistant Professor, College of Civil Engineering, Tongji University,Shanghai, China (corresponding author: [email protected])

Jun-Yan WangProfessor, Key Laboratory of Advanced Civil Engineering Materials, TongjiUniversity, Ministry of Education, Shanghai, China

A detailed experimental programme was conducted to investigate the structural behaviour of rebar-reinforcedultra-high-performance concrete (UHPC) beams subjected to pure bending and combined bending and shear.Four flexural bending tests and four bending–shear tests were conducted and four different reinforcementratios (1·09%, 2·75%, 3·60% and 4·99%) were used. The bending characteristics of the rebar-reinforced UHPCbeams under static loading were assessed, including cracking, failure patterns, deflection, ductility and flexuralcapacity. The influence of the reinforcement ratio on the flexural performance of the rebar-reinforced UHPCbeams was observed and analysed. A model was developed for predicting the ultimate flexural capacity ofrebar-reinforced UHPC beams. The adequacy of the prediction model was assessed using experimental data fromthe current study and the published literature, and the results showed that the model can be used to estimatethe ultimate flexural capacity of rebar-reinforced UHPC members accurately. The results of this study providevaluable information for a comprehensive understanding of the structural behaviour of rebar-reinforced UHPCmembers.

NotationAs area of longitudinal rebarb width of beam cross-sectiond effective depth of beam cross-sectiondf diameter of fibreds diameter of steel rebarEc elastic modulus of UHPCEs elastic modulus of steel rebarfbt stress at a crack width of 0·3 mmfc compressive strength of UHPCftj post-cracking direct tensile strengthfu ultimate strength of steel rebarfy yield strength of steel rebarh depth of beam cross-sectionk tensile strength coefficientl0 test spanlf length of fibreMcr cracking momentMs maximum bending moment in shear testMu ultimate momentMy yield momentPcr cracking loadPu ultimate loadPy yield loadVf fibre volume contentVs shear force corresponding to maximum bending

moment in shear testVu ultimate shear capacity

x depth of compressive zoneΔcr mid-span deflection at cracking loadΔp mid-span deflection at peak loadΔu mid-span deflection at ultimate loadΔy mid-span deflection at yield loadλ shear span ratioμp, μu ductility indicesρ reinforcement ratioρsv stirrup ratio

IntroductionConcrete is currently the most widely used constructionmaterial, but has limitations such as low tensile strength andpoor ductility. High-performance concrete reinforced withsteel fibres may be able to overcome these limitations, andmajor research efforts have focused on the study of fibre-reinforced concrete (FRC) (Abadel et al., 2015; Ismail et al.,2017; Lee et al., 2017; Reddy and Subramaniam, 2017).Fibres have been found to be capable of controlling crackpropagation, preventing large crack widths and increasingthe ultimate flexural strength, toughness and stiffness ofconcrete beams. One of the most important functions ofsteel fibres in concrete is to transfer tensile stress across acracked section, providing residual strength after cracking.Barros et al. (2004) proposed a method for modelling thepost-cracking behaviour of FRC. In addition, differentbending test methods have been specified to evaluate the

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Magazine of Concrete Research

Flexural behaviour of rebar-reinforcedultra-high-performance concrete beamsChen, Zhang, Jia and Wang

Magazine of Concrete Researchhttp://dx.doi.org/10.1680/jmacr.17.00283Paper 1700283Received 27/06/2017; revised 24/10/2017; accepted 26/10/2017Keywords: fibre-reinforced concrete/reinforcement/testing, structural elements

ICE Publishing: All rights reserved

mailto:[email protected]

flexural performance of FRC (ASTM, 1997; JCI, 2003;Rilem, 2002).

Ultra-high-performance concrete (UHPC) is a new class ofconcrete that has been developed over recent decades (Fehlinget al., 2014; FHWA, 2006, 2013; JSCE, 2004). As an advancedcementitious composite, UHPC consists of a dense high-strength matrix and steel fibres. Because UHPC has a very lowwater-to-binder ratio and contains high-fineness admixtures andhigh volume contents of steel fibres (mostly 2% by volume), itexhibits excellent properties such as high strength (compressivestrength >150 MPa and tensile strength >8 MPa), high energyabsorption capacity, excellent durability and long-term stability(AFGC, 2013). UHPC therefore shows great application pro-spects in engineering structures, especially those structures sub-jected to tensile and bending loads.

At the material level, numerous studies have been conducted toinvestigate the mechanical properties of UHPC. The stress–strain property and the post-cracking behaviour of UHPC havebeen investigated and analysed (Graybeal, 2007; Graybeal andDavis, 2008; Hassan et al., 2012; Kang et al., 2010; Willeet al., 2014; Yoo et al., 2015). The tensile properties of UHPCare distinct from those of conventional concrete, and this isattributed to the increased cracking capacity of the cementi-tious composite matrix and the crack-bridging behaviour ofthe reinforcing fibres. In contrast to FRC, UHPC exhibitsimproved post-cracking behaviour before crack localisation,fibre pull-out and loss of tensile capacity. Under the assump-tion that an identical UHPC matrix is used, the high ductilityof UHPC is mainly a result of the bridging effect of fibres at acracked section (Soutsos et al., 2012). The material, geometryand volume contents of fibres should thus be very carefullydetermined to improve the flexural performance of UHPC(Barnett et al., 2010; Dong et al., 2011; Yoo et al., 2013).

However, practical applications of UHPC in structures havebeen limited, mainly due to the complexity of its preparationand construction techniques, and its high cost compared withconventional concrete. It is still a common belief that UHPC isnot an ideal cost-efficient construction material, but is a usefulsupplement to modern engineering materials. The practical useof UHPC structural components may thus require structuraloptimisation to take full advantage of its advanced mechanicaland durability properties.

To obtain efficient and reliable structural performance, struc-tures made using UHPC generally include reinforcements suchas steel bars. The combination of a cementitious matrix andfibres allows for a much shorter embedded or developmentlength of the reinforcing bar, which provides the potential forthe redesign of some structural systems such as field-cast con-nections between prefabricated bridge elements (Yuan andGraybeal, 2015). In such cases, the structural performance ofrebar-reinforced UHPC components and the principle of

synergistic action between a steel bar and UHPC need to beclarified. Unfortunately, however, to the best of the authors’knowledge, only a few experimental results on the flexuralcapacity and behaviour of rebar-reinforced UHPC beams atthe structural level have been reported (Graybeal, 2008; Kimet al., 2012; Wang et al., 2015, 2017; Yang et al., 2010, 2011;Yoo et al., 2015) and the design reinforcement ratio of all thesetest beams was below 2%. More information is thereforerequired for a comprehensive understanding of the structuralbehaviour of rebar-reinforced UHPC members.

The purpose of the work reported in this paper was to investi-gate the structural behaviour of rebar-reinforced UHPC beamssubjected to pure bending and combined bending and shear.Bending characteristics such as cracking, failure patterns,deflection, ductility and flexural capacity of rebar-reinforcedUHPC beams under static loading were assessed and a modelwas developed for predicting the ultimate flexural capacity ofrebar-reinforced UHPC beams. The averaged actions of thefibres to the tensile properties of UHPC were taken intoaccount by introducing the tensile strength coefficient k. Theadequacy of the proposed model was demonstrated usingexperimental data from the current study and the publishedliterature.

Experimental programme

Materials and mix proportionsThe UHPC used in this study was a type of ultra-high-performance fibre-reinforced cementitious composite. Unlikereactive powder concrete, this type of composite can be curedat normal temperature without the need for steam curing,which greatly broadens the scope of its engineeringapplication.

The constituent material proportions were determined basedon optimisation of the granular mixture, which allowed for afinely graded and highly homogeneous concrete matrix. Themix proportion used in this study, as given by weight ratios, isshown in Table 1. The fine aggregate was natural river sandwith a fineness modulus of 2·4; no coarse aggregates were usedin the mixture. The filler was a slag powder with a specificsurface area greater than 750 m2/kg. The smallest particles ofthe UHPC components, silica fume, had a specific surface areagreater than 20 000 m2/kg, which was small enough to fillmicropores. Polycarboxylate superplasticiser was also added toprovide suitable workability and adequate viscosity for uniformfibre dispersion. Steel micro-fibres of length 13 mm and diam-eter 0·2 mm were incorporated as 2% of the total volume. Theproperties of the fibre are summarised in Table 2.

Strength properties of UHPCNine dog-bone shaped specimens with a rectangular cross-section of 50� 100 mm and length of 500 mm were fabricatedand tested as tensile specimens. The specimen shape is shown

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Magazine of Concrete Research Flexural behaviour of rebar-reinforcedultra-high-performance concrete beamsChen, Zhang, Jia and Wang

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in Figure 1(a). The gripping setup, in which four linear vari-able differential transformers (LVDTs) were employed tomeasure the tensile strain of UHPC, is shown in Figure 1(b).The tensile coupon tests were divided into preloading andformal loading. The loading rate was 1 mm/s until the instantthe load reached 0·5 kN, and then it was changed to 0·5 mm/suntil rupture of the specimen. The average tensile stress–straincurve of the specimens is shown in Figure 2.

The compressive strength of the UHPC was obtained throughcompressive tests on cube specimens of three different sizes(70 mm, 100 mm and 150 mm). Each group comprised sixcube samples. The specimens were cured under standard

conditions (temperature of 22± 2°C and humidity ≥95%) untiltesting. Details of the test setup are shown in Figure 3. Loadwas applied through displacement control at a rate of0·1 mm/min using a universal testing machine with amaximum capacity of 3000 kN.

The strength values of each group of specimens are shown inTable 3. Similar to ordinary concrete and high-strength con-crete, UHPC also shows a size effect; that is, compressivestrength is inversely proportional to the sample size. Assumingthat the standard specimen is a 150 mm cube, the size conver-sion factor for the 100 mm specimen was determined to be125·6/141·5= 0·89, which is lower than the recommended

Table 1. Mix proportions of the UHPC mixture by weight ratio

WaterPowder material (cement,

slag powder and silica fume) Fine aggregate Superplasticiser Vf: %

0·18 1·0 0·95 0·025 2

Table 2. Properties of steel fibre

Length: mm Diameter: mm Aspect ratio, lf/df Density: kg/m3 Tensile strength: MPa Elastic modulus: GPa

13 0·2 65 7800 2750 200

Thickness= 100 nm

100

10015

0

500

150 10

0

50

R62·5

(a) (b)

Figure 1. Setup for tension coupon test of UHPC: (a) configuration of coupon (dimensions in mm); (b) test setup

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value of 0·95 (GB/T 50081 (CMC, 2002)). This indicates thatthe size effect of UHPC is more significant than that of ordin-ary concrete, which is consistent with other results in the litera-ture (Graybeal, 2007); that is, the incorporation of fibres in a

UHPC matrix may result in a decrease in the coefficient ofvariation (CoV) of compressive strength results.

Details of beam specimensThe experimental programme included tests on four beamspecimens with rectangular cross-sections, the details of whichare presented in Table 4. All the beams had a cross-sectionalarea of 150 mm� 220 mm and were 2000 mm long. Theheight to width ratio of the beams was chosen as 1·5. In orderto ensure bending failure rather than shear failure for all thebeams, shear reinforcement was designed and consisted ofclosed stirrups with a diameter of 8 mm.

The specimens had varying reinforcement ratios, obtained byadjusting the amount of rebars and the number of layers.Three different rebar diameters were used (ds = 14 mm (D14),22 mm (D22) and 25 mm (D25)), leading to four differentreinforcement ratios (ρ=1·09%, 2·75%, 3·60% and 4·99%) cal-culated by As/bd. The properties of the steel rebars are sum-marised in Table 5.

Preparation of test specimensTo fabricate the beam specimens, wood forms were first manu-factured. Rebars were then positioned and strain gauges wereinstalled at mid-span in order to measure the tensile strain ofthe rebar. Next, UHPC was mixed using a concrete mixer.After 10 min of mixing, the UHPC mixture was identicallyand carefully placed at mid-span and allowed to flow in orderto provide similar fibre orientation and dispersion in all thespecimens. The UHPC exhibited excellent flowing and self-consolidating characteristics, so vibrating compaction was notrequired. Finally, the beam specimens were covered with wetsacking to prevent water loss. All the specimens weredemoulded 3 d after concrete pouring and cured in the labora-tory until testing.

Test setup and instrumentationFlexural tests were conducted under four-point loading con-ditions, as shown in Figures 4 and 5. During the test, the loadwas incrementally applied using the load-controlled technique.A 500 kN hydraulic jack was used and the load was appliedon a steel beam to spread the load on the test specimen.

14

12

10

8

6

4

2

00 0·2 0·4

Axial strain: %

0·6 0·8 1·0

Axi

al t

ensi

le s

tren

gth:

MPa

Figure 2. Tensile stress–strain curve of UHPC

Cube specimen(150 × 150 × 150 mm)

Figure 3. Compressive strength test setup

Table 3. Compressive strength test results of UHPC cubes of different size

Compressive strength: MPa

70�70�70 mm 100�100�100 mm 150�150�150 mm

UHPC-1 154·2 134·4 113·3UHPC-2 156·8 137·7 123·9UHPC-3 160·0 140·5 125·5UHPC-4 163·0 142·1 128·9UHPC-5 164·8 145·7 128·9UHPC-6 167·7 148·7 133·3Mean 161·1 141·5 125·6

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Table 4. Details of experimental beams B-1 to B-4

B-1 B-2 B-3 B-4

Cross-section: mm

25 100 25

150

3222

0

25 100 25

150

3622

0

25 100 25

150

3822

0

25 100 25150

3868

220

Longitudinal rebars Two D14 Two D22 Two D25 Two D25 + one D221 layer 1 layer 1 layer 2 layers

Rebar area: mm2 307·9 760·3 981·7 1361·9Rebar ratio: % 1·09 2·75 3·60 4·99Support rebars Two D8 Two D8 Two D8 Two D8Stirrups D8@150 D8@150 D8@100 D8@60

D represents the rebar diameter (mm)

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Support rollers were installed 100 mm from both ends of thebeam. The load was increased in predefined steps and loadingwas terminated when the beam was observed to be softened.Figure 5 shows a test specimen in the loading frame.

Five strain gauges were attached on the side surface of thebeam at the mid-span section (Figure 4) to measure strain atdifferent heights and two strain gauges were glued onto the topsurface and the bottom surface of the specimen. In addition,vertical deflections of each specimen were measured at variouslocations (at the mid-span, two loading points and two sup-ports, as shown in Figure 4) using LVDTs. Finally, a forcesensor was bonded to the actuator to measure the load appliedto the beam. All test data were recorded via a computer-con-trolled data logger.

Test results and discussion

Load–deflection relationshipThe mid-span deflections of the beams measured at eachloading step were compared and the load–mid-span deflectioncurves of the tested specimens are shown in Figure 6. Table 6

lists the first cracking load, yield load and peak load for eachspecimen.

UHPC has a multiple-cracking property and it is difficult toobserve the exact occurrence of the first crack visually.Therefore, the crack load was defined as the load at the end ofthe initial linear stage in the load–displacement curve. Theyield load was defined as the load when the steel yielded, asdetected by the strain gauge attached to the steel rebar. Thepeak load was defined as the maximum load measured duringthe test. Before crack initiation, deflection of the beamincreased linearly. After the crack initiation, deflectionincreased non-linearly until the maximum load was reached.

The flexural cracking stiffness of the specimens was not signifi-cantly decreased because the crack propagation rate was effec-tively restrained by the steel fibres and rebars. When yieldingof the rebars occurred, the curvature of the section and deflec-tion of the beam began to increase rapidly. It was observedthat the strengthening and toughening effect of the steel fibresat the cracked section was fully developed, and some fibreshad been pulled out. A popping sound was also heard duringthe test. Ultimately, obvious plastic deformation graduallydeveloped in the compression zone until the UHPC wascrushed, and then the specimen failed.

The initial cracking load (Pcr) of each beam varied between 30and 50 kN. Beam B-4 had the smallest cracking load (30 kN)and beam B-3 had the largest cracking load (50 kN). Themeasured loads at yielding of the beams (Py) were 139·9 kN,

Table 5. Properties of deformed steel rebars

Type ds: mm As: mm2 Es: GPa fy: MPa fu: MPa

D14 14 154 200 461 632D22 22 380 200 417 616D25 25 491 200 456 588

Rebar

P/2 P/2

100 600 200 200

2000

200 600 100

LVDT

Strain gauge

C1C2

C3

C4

C5

C6C7

30

7011

015

019

0

220

Figure 4. Loading and instrumentation for displacement and strain measurements (dimensions in mm)

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230·4 kN, 276·3 kN and 312·9 kN for B-1, B-2, B-3 and B-4,respectively; the ultimate loads at the end of the tests (Pu)were, respectively, 144·2 kN, 238·0 kN, 301·3 kN and353·1 kN.

DuctilityConventionally, the ductility of a reinforced concrete (RC)beam is qualitatively evaluated by a ductility index in terms ofcurvature ductility or deflection ductility. In this paper, thedeflection ductility is used, which is derived from the mid-spandeflection at the peak load divided by the deflection at theinstant of steel yielding, as expressed by

1: μp ¼ Δp=Δy

Shin et al. (1989) proposed Equation 2 to evaluate the deflec-tion ductility when a RC beam continues to bear load wellafter reaching the peak flexural load.

2: μu ¼ Δu=Δy

Different definitions for deflection at the limit state have beengiven and used previously. Hadi and Elbasha (2007) definedthe ultimate deflection as the deflection corresponding to 80%of the maximum load capacity recorded over the yield deflec-tion. Yang et al. (2010) defined the ultimate deflection at theinstant when the load–deflection curve starts to decreasesharply. The latter definition of the ultimate state was adoptedin this study considering the ultimate state of the test beams.The ductility index was calculated using the deflection at theultimate state as well as the deflection at the yielding load state.

The ductility indices obtained from Equations 1 and 2 for eachbeam are summarised in Table 6. The mean value of the duct-ility index was 3·42 for beam B-1 with a rebar ratio of 1·09%,3·49 for beam B-2 with a rebar ratio of 2·75%, 4·58 for beamB-3 with a rebar ratio of 3·60% and 3·23 for beam B-4 with arebar ratio of 4·99%. These results reveal that the ductility ofthe beam was affected by the reinforcement ratio and all of thebeams exhibited a ductility index greater than 3·2. After reach-ing the peak load, the load decreased slowly, accompanied bywidening of the main crack. The crack width increased slowly,not abruptly, indicating that failure of the specimen was ductile.

For the beam with the lowest reinforcement ratio (B-1),tensile cracks in the beam propagated very quickly. The time

Steel reaction beam

Load spread beam

Hinged support

Specimen

Hydraulic cylinder

Figure 5. Beam flexural test setup

400

350

300

250

200

150

100

50

00 10 20

Mid-span deflection: mm

30 40 50

Load

: kN

B-1

B-2

B-3

B-4

Figure 6. Load–mid-span deflection curves of test beams

Table 6. Bending test results of rebar-reinforced UHPC beams

Beam

Initial cracking Yielding state Peak stateUltimatedeflection Ductility index

Pcr: kN Δcr: mm Mcr: kN.m Py: kN Δy: mm My: kN.m Pu: kN Δp: mm Mu: kN.m Δu: mm Δp/Δy Δu/Δy

B-1 35 0·69 10·5 139·9 5·56 42·0 144·2 8·91 43·3 19·03 1·60 3·42B-2 40 1·16 12·0 230·4 8·54 69·1 238·0 16·15 71·4 29·81 1·89 3·49B-3 50 1·10 15·0 276·3 9·21 82·9 301·3 12·50 90·4 42·17 1·36 4·58B-4 30 0·60 9·0 312·9 10·50 93·9 353·1 14·91 105·9 33·95 1·42 3·23

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from initial cracking to fibre pull-out was relatively short andthe effect of the fibre bridging capacity could not be fullydeveloped. Therefore, the beam exhibited relatively low duct-ility. However, for the beam with the highest reinforcementratio (B-4), the tensile crack width was very small when thebeam was close to failure and the tensile zone of the UHPCwas not fully effective, which also resulted in low ductility.Only for moderate reinforcement ratios (B-2, B-3) was themain crack in the tensile zone of UHPC fully developed andthe fibre bridging effect sufficiently achieved during thecracking process from the instant of crack initiation to steelfibre pull-out. Therefore, these specimens showed betterductility.

Cracking and failure patternsThe tensile cracking behaviour and fibre pull-out behaviour ofthe UHPC specimens under flexural load were observed. Itwas found that the cracking and failure patterns of the rebar-reinforced UHPC beams were characterised by multiple micro-cracks and localised macro-cracks.

In general, cracks were not observed when the load increasedlinearly at the early stage of the test. When the tensile strain ofthe UHPC reached the ultimate tensile strain, several micro-cracks occurred and started to develop at the bottom surfaceof the beam within the region between the two loading pointswhere the beam was in pure bending. New cracks appeared ina sparse and symmetrically distributed pattern due to the highhomogeneity of the UHPC matrix. As the test progressed,new micro-cracks gradually formed between existing cracks.Most of the cracks continued to develop up towards thetop surface and, from visual examination, the cracks did notappear to widen. The height of crack propagation wasrestricted by the bridging capacity of the fibres, and tightcracks perpendicular to the flexural tensile stress formed at thelower surface of the beam, as shown in Figure 7. Before theyield load, the number of cracks increased continuously, whilethe crack width did not increase significantly. The test resultsshowed that, in a rebar-reinforced UHPC beam, the UHPC iscapable of redistributing stress and developing multi-cracksbefore fibre pull-out, which is similar to the results of otherresearch (Yang et al., 2010).

Theoretically, tensile failure of UHPC occurs when steel fibresare pulled out from the matrix, and pull-out will occur whenthe load carried by an individual fibre exceeds the bondstrength provided by the UHPC matrix, which is influenced bythe shape and aspect ratio of the fibre. At the peak load of thebeam, the fibres in one specific cracked section started to showextensive pull-out. This crack became significantly wider thanany other cracks in the beam, as shown in Figure 8. Thereafter,failure of the rebar-reinforced UHPC beam was caused bylocal bond failure between the fibres and the UHPC matrix aswell as rebar yielding.

Load–steel strain relationshipCurves of load against steel strain are shown in Figure 9;the strain was measured by the strain gauge attached to thesurface of the steel bar (Figure 4). In the whole process ofthe bending test, the load–steel strain curves are similar to theload–displacement curves and can be divided into three stages.Before the appearance of cracks, deformations of the UHPCand the steel bar were coordinated under tension and the loadwas linearly related to the steel strain. After initial cracking,the stress in the tensile zone was borne by the steel bar and thefibres crossing cracked sections. The steel strain continued toincrease linearly with load, but the slope of the curve was lessthan that in the elastic stage before cracking and the slope wasproportional to the reinforcement ratio. After yielding of thesteel bar, the steel strain increased rapidly and deformation ofthe specimen also increased, while the corresponding loadremained almost constant.

Load–concrete strain relationshipThe concrete strain was measured by strain gauges positionedon the side surface of the beam at mid-span (Figure 4).Typical load–strain curves of specimen B-4 are shown inFigure 10, where positions C1 and C2 developed compressivestrains (negative) and positions C4, C5, C6 and C7 developedtensile strains (positive) as the load increased. S1 represents thestrain of steel rebar of specimen B-4.

Position C1 corresponds to the upper surface of the cross-section (Figure 4). When the specimen reached the ultimatebearing capacity, the strain was 3846 με, slightly larger thanthe peak compressive strain of UHPC ( fc/Ec). Therefore, inanalysis of the flexural capacity of the rebar-reinforced UHPCbeams, it is reasonable to model the compressive behaviour ofUHPC by a simplified linear stress–strain curve (AFGC,2013). Position C3 was located at one-third point over thebeam section depth from the top surface of the beam. Thestrain at C3 changed from compression to tension becausethe propagation of cracks from the bottom to the top surfacewould cause the neutral axis to move upwards. The strain atC7 developed linearly at first, followed by a non-linear region.The load value at the end of the linear portion of the load–strain curve was almost the same as the corresponding value inthe load–deflection curve. The strain at C7 did not show asudden change even after cracking, which is different from thetensile strain in conventional concrete. This might be attributedto the fact that, after cracking of the UHPC, the developmentof tensile strain was effectively restrained by the fibre bridgingeffect at the crack surfaces.

Analysis of the plane-section assumptionThe plane-section assumption – a basic assumption in theanalysis of RC structures – is a necessary prerequisite for asimplified theory of RC structures (Wight and MacGregor,2012). The strain distribution at the mid-span section of

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(a) (b)

(c) (d)

(e) (f)

Figure 7. Typical crack propagation pattern (beam B-4): (a) crack step 1; (b) crack step 2; (c) crack step 3; (d) crack step 4;(e) crack step 5; (f) crack step 6

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specimens was investigated in this study. The concrete strainwas obtained by strain gauges C1–C7 attached to the beamsurfaces (Figure 4). With an increase in load level, both the

strain value and sectional curvature increased accordingly,along with upward movement of the neutral axis, as shown inFigure 11. However, more critical is the fact that the strainmeasured at the mid-span section was approximately pro-portional to the distance from the point to the neutral axisunder different loads. Therefore, the plane-section assumptionwas satisfied during the bending process of the rebar-reinforcedUHPC beams, which provides a theoretical basis for simplifiedcalculation of the bearing capacity of a normal section.

Flexural analysis of rebar-reinforcedUHPC beamsIn traditional flexural theory of RC members (Park andPaulay, 1975), the tensile strength of concrete is generally neg-lected in calculating the ultimate flexural capacity of RCbeams due to the fact that the tensile strength of concrete ismuch less than the compressive strength. However, for UHPC,the main benefits of the fibres are activated and effective aftermatrix cracking, as the fibres crossing a cracking surface guar-antee stress transfer between the cracked surfaces. Therefore,taking into account the contribution of fibres, the tensilestrength of UHPC must be considered in flexural analysis ofUHPC beams (Stroeven, 2009; Stroeven and Hu, 2006).

Various models describing the non-linear flexural response ofUHPC beams were developed by Yang et al. (2012) and Yooand Yoon (2015), taking into account both the multi-crackingbehaviour and the propagation of discrete cracks at a microlevel. However, in the structural design of reinforced UHPCbeams, the ultimate bending capacity is usually of mostconcern. Therefore, a simple and applicable method is proposedto predict the ultimate flexural capacity of a rebar-reinforcedUHPC beam, in which the averaged actions of all fibres to thetensile properties of UHPC are taken into account.

Stress–strain relationship of materialsTo evaluate the post-cracking behaviour of UHPC, a tensilestress–crack opening displacement (σ–w) model can be derivedbased on inverse analysis of curves of load against crackmouth opening displacement obtained from three-pointbending tests on notched prism specimens (Baby et al., 2012,2013; Yoo et al., 2014). In the sectional analysis of UHPCbeams, the σ–w model needs to be converted to a stress–strain(σ–ε) curve. For this, a series of conversion formulas was pro-posed by the Association Française du Génie Civil (AFGC)and the material models of UHPC under compression andtension were suggested according to the length and type offibre used (AFGC, 2013). In the AFGC recommendation(AFGC, 2013), the compressive behaviour of UHPC is simplymodelled as a linear stress–strain curve up to the peakstrength. In this study, the average compressive strain ofUHPC at the upper surface was 3398 με when the test beamreached its ultimate bearing capacity, which was approximatelyequal to the peak compressive strain of UHPC under axial

Figure 8. Localised macro-crack under bending failure

400

350

300

250

200

150

100

50

00 3000 6000

Steel strain: × 10–6

9000 12 000 15 000

Load

: kN

B-4

B-3

B-2

B-1

Figure 9. Load–steel strain curves of test beams

400

350

300

250

200

150

100

50

0

Load

: kN

–4000 –2000 20000

Strain: × 10–6

4000 6000 8000 10 000

C-1C-3C-5C-7

C-2C-4C-6S-1

Figure 10. Load–concrete strain relationship (beam B-4)

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compression ( fc/Ec). However, whether the ultimate compres-sive strain of UHPC is affected by the size effect of beamspecimens or the compressive strength of UHPC has yet to bestudied. The tensile response of UHPC was classified into twolaws based on the capacity of tensile resistance to crackopening displacement: (a) the strain-softening law ( ftj > fbt)and (b) the strain-hardening law ( ftj < fbt), as shown in Figure12. For steel rebars, a bilinear stress–strain curve was simplyassumed, as shown in Figure 13. The properties of the steelrebars are summarised in Table 5.

Ultimate moment of rebar-reinforced UHPC beamThe stress and strain distributions at failure are shown inFigure 14. The distribution of compressive stress in concrete issimplified without taking into account any horizontal portionor descending portion of the stress–strain relationship, which ispermissible according to Fehling et al. (2014). The stress resul-tant (Fcd) for a cross-section with a rectangular compressionzone therefore lies at the one-third point from the top. The

resultant tensile force in the steel (Fsd) acts at the level of thecentroid of the reinforcement. Acting in addition to this coupleis the resultant of the tensile stress in the fibre-reinforcedUHPC (Ffd). The distribution of the tensile stress transferredacross the crack by the fibres is obtained directly from thestress–crack width diagram, depending on the actual localcrack width. For simplicity, a tensile strength coefficient k isintroduced, and the tensile stress of the UHPC in the tensionzone is represented by a rectangular stress block, as shown inFigure 14(c). In addition, the height of the tension zone is mul-tiplied by a reduction factor of 0·9 due to the fact that theheight of the elastic uncracked zone of UHPC in the tensionzone is very small when the specimen approaches the ultimateflexural capacity (Fehling et al., 2014).

In this study, the compressive strength of a 100 mm cube wastaken for the value of fc (see Table 3) and the post-crackingdirect tensile strength ( fbt) was taken as approximately equival-ent to the peak tensile strength obtained from tension coupontests of the UHPC (see Figure 2). As reaching the elastic limit

–4000 –2000–1000–3000 20001000 30000

Strain: × 10–6

4000–2000 –1000–3000 20001000 30000

Strain: × 10–6

–2000 –1000 20001000 30000

Strain: × 10–6

–2000 –1000 20001000 30000

Strain: × 10–6

(d)(c)

(b)(a)

4000

240

200

160

120

80

40

0

Hei

ght:

mm

240

200

160

120

80

40

0

Hei

ght:

mm

240

200

160

120

80

40

0

Hei

ght:

mm

240

200

160

120

80

40

0

Hei

ght:

mm

20 kN40 kN60 kN80 kN100 kN120 kN140 kN

50 kN

100 kN

150 kN

200 kN

250 kN

300 kN

40 kN70 kN100 kN140 kN170 kN200 kN230 kN

50 kN

100 kN

150 kN

200 kN

250 kN

300 kN

350 kN

Figure 11. Strain distribution at mid-span section: (a) B-1; (b) B-2; (c) B-3; (d) B-4

11



jyguo

高亮

for a steel rebar with a distinctive yield point generally initiallyleads to the localisation of further deformations at the crackgoverning the load-carrying capacity, the stress in the steelrebar when applying the proposed design model should belimited to the yield point, fy. Alternatively, when exploiting thestrain hardening of the steel rebar up to its ultimate tensilestrength, owing to the already wide cracks, the contribution ofthe fibres can be ignored, which is on the safe side.

Therefore, the equilibrium conditions known from conventionalRC theory were as follows when taking the contribution of thefibres into account.

Equilibrium of moments is

3: ΣM ¼ 0 ¼ Mu � Fsd d � x3

� �� Ffd h� 0�45ðh� xÞ � x

3

� �

and equilibrium of forces is

4: ΣN ¼ 0� Fsd � Fcd þ Ffd

where

5: Fcd ¼ 12bxfc

6: Fsd ¼ Asfy

7: Ffd ¼ 0�9bðh� xÞkfbt

By substituting the measured ultimate moment (Mu) intoEquation 3 and solving in conjunction with Equations 4–7, thetensile strength coefficients (k) of the test specimens were cal-culated and are listed in Table 7.

The ultimate flexural capacity could then be derived from

8: Mu ¼ fyAs d � x3

� �þ 0�425fbtbðh� xÞ 0�55hþ 7

60x

� �

Comparison with test resultsTable 8 shows a comparison of experimental ultimate flexuralcapacity results (Mu,exp) with results predicted (Mu,pre) usingthe suggested model (Equation 8) and conventional flexuraltheory without considering fibres (Wight and MacGregor,2012). It can be seen that the results based on the suggestedmodel agree well with the experimental results of the

Stress

Strainε 'c

εlim ε1% ε0·3 εe

Ec

f1%

fc

fc

fbtftj

Stress

Strainε 'c

εlim ε1% ε0·3 εe

Ec

f1%

ftjfbt

(a)

(b)

Figure 12. Material models for UHPC: (a) strain-softening;(b) strain-hardening

Stress

Strain

fy

Es

ε y εu

Figure 13. Material model for steel rebar

12



rebar-reinforced UHPC beams, with a mean Mu,pre/Mu,exp of1·00 and a CoV of zero. The conventional flexural theory con-siderably underestimated the ultimate flexural capacity of therebar-reinforced UHPC beams, with Mu,pre/Mu,exp in the range0·60–0·89 and a mean value of 0·77. The proposed model canthus predict the test results obtained in this work well.

To extend the validity of the suggested method for calculatingthe ultimate flexural capacity of rebar-reinforced UHPCbeams, predictions were also made for beams tested in otherstudies (Yang et al., 2010; Yoo and Yoon, 2015). The dimen-sions and properties of these beams are summarised inTable 9. The predicted and measured ultimate moments arealso shown in Table 9, and it can be seen that the predictionmodel agrees well with the experimental results, with a meanand CoV of Mu,pre/Mu,exp of 0·891 and 0·003, respectively.

These results suggest that for predicting the flexural capacity ofrebar-reinforced UHPC beams, averaging the actions of all thefibres to the tensile properties of the UHPC by introducing atensile strength coefficient (k) is simple and effective at themacro level.

Bending–shear behaviour of rebar-reinforcedUHPC beamsAccording to the failure modes of the specimens in the four-point bending tests, it was found that the failures were mainlyconcentrated in the pure bending region with only a smallamount of oblique cracks in the bending and shear span,where there remained good integrity. To study the likely influ-ence of the shear span to depth ratio on the performance ofthe rebar-reinforced UHPC members and also to make full useof the specimens, a single-point loading test was carried out onthis bending shear span to investigate the failure mode andultimate bearing capacity of the rebar-reinforced UHPC beamsunder combined bending and shear.

Single-point loading test

Details of specimensThe UHPC, steel rebar, section size and other parameters ofthe specimen have already been introduced; further details forthe single-point loading test are shown in Table 10. The

h

b

d

Ash – x

x

εs

εc

φ

λ (h – x) fy

kfbt

Fsd

Fcd

fd

fc

M

(a) (b) (c) (d)

Figure 14. Strain and stress distribution over cross-section of rebar-reinforced UHPC beam at failure: (a) cross-section; (b) straindistribution; (c) equivalent stress distribution; (d) equilibrium of forces

Table 7. Tensile strength coefficients of test beams

Beam Tensile strength coefficient, k

B-1 0·46B-2 0·49B-3 0·49B-4 0·45Mean 0·47

Table 8. Comparison of measured ultimate moments (Mu,exp) and predicted ultimate moments (Mu,pre)

Beam Mu,exp: kN.m

Mu,pre: kN.m Mu,pre/Mu,exp

Without considering fibres Present method Without considering fibres Present method

B-1 43·3 26·05 43·71 0·60 1·01B-2 71·4 55·18 70·93 0·77 0·99B-3 90·4 75·40 89·78 0·83 0·99B-4 105·9 93·70 106·48 0·89 1·01Mean — — — 0·77 1·00CoV — — — 0·02 0·00

13



Table 10. Details of beams used in single-point loading tests

Test beam Longitudinal rebar ρ: % Stirrup ρsv: % Test span: mm λ

B-1-S Two D14 1·09 D8@150 0·45 600 1·36B-2-S Two D22 2·75 D8@150 0·45 600 1·36B-3-S Two D25 3·59 D8@100 0·67 600 1·36B-4-S Two D25 + one D22 4·97 D8@60 1·11 600 1·36

Table 9. Summary of beam details and comparison of experimental and predicted ultimate moments

Specimen Fibre type Vf: % lf/df fc: MPa b: mm h: mm ρ: % fy: MPa Mu,exp: kN.m Mu,pre: kN.m Mu,pre/Mu,exp

Yang et al. (2010)R12-1 Straight 2 65 191 180 270 0·60 500 87·01 71·18 0·818R12-2 Straight 2 65 191 180 270 0·60 500 83·28 71·18 0·855R13-1 Straight 2 65 192 180 270 0·90 500 97·52 82·94 0·851R13-2 Straight 2 65 192 180 270 0·90 500 106·56 82·94 0·778R13C-1 Straight 2 65 192 180 270 1·20 500 92·15 82·94 0·900R14-1 Straight 2 65 196 180 270 1·20 500 116·50 100·52 0·863R14-2 Straight 2 65 196 180 270 1·20 500 116·84 100·52 0·860R22-1 Straight 2 65 191 180 270 1·31 500 107·01 95·19 0·890R22-2 Straight 2 65 191 180 270 1·31 500 105·71 95·19 0·900R23-2 Straight 2 65 196 180 270 1·96 500 131·65 125·06 0·950

Yoo and Yoon (2015)S13-1 Smooth 2 65 211·8 150 220 0·94 495 39·29 39·40 1·003S13-2 Smooth 2 65 211·8 150 220 1·5 510 55·85 51·16 0·916S19·5-1 Smooth 2 97·5 209·7 150 220 0·94 495 41·99 39·09 0·931S19·5-2 Smooth 2 97·5 209·7 150 220 1·5 510 56·34 50·85 0·903S30-1 Smooth 2 100 209·7 150 220 0·94 495 43·16 39·71 0·920S30-2 Smooth 2 100 210 150 220 1·50 510 56·07 51·45 0·918T30-1 Twisted 2 100 232 150 220 0·94 495 43·47 40·06 0·921T30-2 Twisted 2 100 232 150 220 1·50 510 60·26 51·89 0·861

Mean 0·891CoV 0·003

LVDT LVDT

LVDT

P

100 600 100

800

Figure 15. Loading and instrumentation of displacement measurement (dimensions in mm)

14



bending shear span in the bending test, from the loading pointto the support section (600 mm long) was used as the test spanfor the shear tests. The shear span ratio, λ= l0/h, was 1·36.

Test setup and instrumentationThe test was conducted using a hydraulic pressure testingmachine with a maximum capacity of 2000 kN. The test speci-men was placed on a steel beam that was placed on the lowerplaten of the testing machine. Hinged supports were installedat a distance of 100 mm from both ends of the beam. In orderto prevent instability of the test beam, a screw jack wasarranged at the cantilevered part of the beam. Displacementmeasuring devices (LVDTs) were installed at the mid-span andsupports of the beam, as shown in Figure 15.

The load was applied in predetermined steps. The load wasthen held and data were collected. This procedure continued

Test span

Screw jack Steel beam Hinged support

Figure 16. Single-point loading test setup

(a) (b)

(c) (d)

Figure 17. Failure mode and crack patterns of test beams: (a) B-1-S; (b) B-2-S; (c) B-3-S; (d) B-4-S

15



until softening of the specimen was observed. Figure 16 showsa tested specimen in the testing machine.

Test results and discussion

Cracking and failure patternsAll the test beams suffered bending failure. The crack distri-bution and failure patterns are shown in Figure 17. For speci-men B-1-S, with the lowest reinforcement ratio, only onebending crack appeared from initial loading to final failureand the UHPC in the compression zone did not crush. Thefailure mode was characterised by a main bending crack ofwidth greater than 1·5 mm. The failure patterns of specimensB-2-S, B-3-S and B-4-S were similar. In the initial stage ofloading, a bending crack appeared first, followed by a smallnumber of diagonal cracks. As the load continued to increase,the main crack gradually widened until the UHPC crushed inthe compression zone and the specimen reached its ultimatebearing capacity.

It appears that the flexural capacity of the beams increasedproportionally with an increase in reinforcement ratio.Similarly, the shear capacity of the beams increased with anincrease in the stirrup ratio. When the maximum shear forcecontrolled by bending failure did not reach the shear bearingcapacity of the beam, the beam experienced bending failurerather than shear failure. Table 11 lists the peak load, peakdeflection and failure mode of the tested beams.

Load–deflection relationshipThe load–deflection curves of test beams B-1-S to B-4-S areshown in Figure 18. At the initial stage of loading, thebending stiffness of the specimen was small and the curve wasconvex to the deflection axis due to the existence of the initialmicro-cracks that developed in the bending test. With anincrease in load, the initial crack gradually closed under theaction of pressure and the curve became approximately linear.The bending stiffness of the specimen began to decrease andthe mid-span deflection increased non-linearly after theappearance of bending cracks and shear diagonal cracks. Withgradual widening of the main bending crack and crushing ofthe UHPC in the compression zone, the specimen reached itsultimate bearing capacity. The measured loads at the ultimatestate (Pu) were 517·0 kN, 637·2 kN, 783·1 kN and 902·3 kNfor B-1-S, B-2-S, B-3-S and B-4-S, respectively.

Bending–shear bearing capacityThe unified theory of RC (Hsu, 1993) indicates that the longi-tudinal rebars and stirrups required for the design of RCbeams are related to each other; that is, there is a correlationbetween the flexural bearing capacity and shear bearingcapacity of the beam. Elfren et al. (1976) considered that therelationship of RC members under bending moment, shearforce and torque could be expressed as a spatial envelope.When the torque is zero, a correlation curve of the momentand shear force of the member can be obtained, as a parabolicrelationship.

Table 12 compares the experimental results of the tested beamsunder bending tests and shear tests. In the table, Vu is the ulti-mate shear capacity of the beam, calculated according toCECS 38:2004 (CECS, 2004) and Ms and Vs are themaximum bending moment and the corresponding shear forceobtained by the shear test, respectively.

It can be concluded from Figure 19 that the existence of shearstress increased the flexural capacity of the UHPC beams ascompared with the ultimate bending moment (Mu) of thebending test. When the tested beam suffered bending failureunder the combined action of bending and shear, the ultimate

Table 11. Experimental results for peak load and failure mode in single-point loading tests

Beam specimen Pu: kN Mu: kN.m Δu: mm Failure mode

B-1-S 517·0 77·2 2·49 Bending failureB-2-S 637·2 95·2 8·62 Bending failureB-3-S 783·1 117·1 11·60 Bending failureB-4-S 902·3 135·0 8·91 Bending failure

Note: Calculation of the peak moment took into account the cantilevered end of the test beam

1000

800

600

400

200

00 2 4 6 8

Mid-span deflection: mm

10 12 14 16

Load

: kN

B-1-S

B-2-S

B-3-S

B-4-S

Figure 18. Load–mid-span deflection curves of test beams

16



moment Ms was 1·28–1·75 times the value of Mu for specimensB-1-S to B-4-S. With an increase of Vs/Vu (i.e. as the shearforce applied on the tested specimen gradually approached itsultimate shear capacity), the effect of shear stress on theimprovement of flexural capacity reduced gradually.

ConclusionsA detailed experimental investigation of the structural behav-iour of rebar-reinforced UHPC beams subjected to purebending and combined bending and shear was conducted. Thefollowing conclusions were drawn from the test resultsobtained.

(a) All specimens exhibited bending failure. The UHPC wasfound to be capable of redistributing stress andundergoing multiple cracking before fibre pull-out. Thecracking and failure patterns of the reinforced UHPCbeams were characterised by multiple micro-cracks andlocalised macro-cracks. The flexural capacity of the beamincreased with an increase in reinforcement ratio.

(b) At the peak load carried by the beam, the compressivestrain of the UHPC was 3846 με, slightly larger than thepeak compressive strain of UHPC ( fc/Ec). Therefore, inanalyses of the flexural capacity of rebar-reinforcedUHPC beams, it is reasonable to model the compressive

behaviour of UHPC by a simplified linear stress–straincurve.

(c) The plane-section assumption was satisfied during thebending process of the rebar-reinforced UHPC beams,which provides a theoretical basis for simplifiedcalculation of the bearing capacity of the normal section.

(d ) A model for predicting the ultimate flexural capacity ofrebar-reinforced UHPC beams was proposed. Theaveraged actions of the fibres to the tensile properties ofthe UHPC were taken into account by introducing thetensile strength coefficient k. The results predicted by themodel were found to agree well with test data obtained bythe authors and other researchers, demonstrating that theproposed model can be used to predict the ultimateflexural capacity of rebar-reinforced UHPC beamsaccurately.

(e) The existence of shear stress increased the flexuralcapacity of the rebar-reinforced UHPC beams ascompared with the ultimate bending moment (Mu) of thepure bending test. Under combined bending and shear,the ultimate moment Ms was 1·28–1·75 times the value ofMu for specimens B-1-S to B-4-S. The effect of shearstress on the improvement in flexural capacity graduallyreduced as the shear force borne by the test specimengradually approached its shear capacity.

AcknowledgementsThe authors greatly appreciate the support of the Science andTechnology Commission of Shanghai Municipality throughHigh Peak Discipline Research. The authors also acknowledgewith thanks the provision of UHPC by Shanghai RoyangInnovative Material Technologies (China Co., Ltd) and theassistance of technicians at the National Key Laboratory ofDisaster Reduction in Civil Engineering of Tongji University.

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2·0

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Flexural behaviour of rebar-reinforced ultra-high