The tensile behavior of ultrahighperformance fiberreinforced concrete (UHPFRC) depends on the dispersion and orientation of steel fibers within the concrete matrix. The uneven dispersion of randomly oriented steel fibers in concrete may cause differences in the tensile behavior between material testing specimens and beams. Therefore, in this study, the tensile behavior was investigated by fitting the analysis result of the momentcurvature curve to the experimental result of a UHPFRC beam. To this end, three UHPFRC mixtures with different compressive strengths were fabricated to test the material properties and flexural behavior of UHPFRC beams. Both a single type of steel fiber and a combination of steel fiber types were used with volume fractions of 1.0% and 1.5%, respectively, in the three mixtures. Based on the design recommendations, the material properties of UHPFRC were modeled. The results ultimately show that by fitting the analysis results to the experimental results of the momentcurvature curves, the tensile strength of UHPFRC beams can be reasonably estimated.
Concrete is a brittle material with a low tensile strength; furthermore, increasing the compressive strength of concrete will increase its brittleness. The advent of ultrahighperformance fiberreinforced concrete (UHPFRC) represents the outcome of continuous research to improve the performance of highstrength concrete under tension. Following the improvements in its mechanical properties, UHPFRC has become suitable for many applications requiring long spans, such as in stadia, bridges, and docks. Therefore, many researchers have conducted numerous studies to explore the tensile and flexural behaviors of UHPFRC [
The most important factor affecting the tensile behavior of UHPFRC is the inclusion of steel fiber. The addition of steel fiber to UHPFRC makes it more ductile, increases its strength, and improves its resistance to cracking [
However, the orientation and distribution of steel fiber in a concrete matrix are random; unfortunately, this randomness can considerably affect the flexural response of a UHPFRC beam. Consequently, many researchers have attempted to control the orientation and distribution of steel fibers in UHPFRC members. For instance, Abrishambaf et al. [
Therefore, the objective of this study is to estimate the tensile behavior of UHPFRC beams subjected to flexure. The experimental parameters included the compressive strength of UHPFRC and the volume content of a combination of steel fibers. Target compressive strengths of 120, 150, and 180 MPa were considered for the UHPFRC; moreover, both a single type of steel fiber and a combination of steel fiber types with volume fractions of 1.0% and 1.5%, respectively, were used in this study. The material behavior of UHPFRC was modeled based on material testing results, and flexural tests were carried out on nine UHPFRC beams. Finally, the tensile strength of the UHPFRC beams was estimated by fitting the analysis results with the testing results of the momentcurvature curves.
In this study, the UHPFRC mixtures included straight steel fibers at volume fractions of 1.0% and 1.5%. Ordinary Portland cement (OPC) was used as the cementitious material, and fine aggregate with a diameter of 0.5 mm or less was used. The waterbinder ratios
Three fiberreinforced concrete mixtures, namely, FRC120, FRC150, and FRC180, were fabricated, where the number in each mixture label represents the target compressive strength of the resultant concrete. A single type of straight fiber with a length of 19.5 mm was used in the FRC120 mixture at a volume fraction of 1.0%, while a combination of fibers was utilized in the FRC150 and FRC180 mixtures at a volume fraction of 1.5%. The combination of fibers used in the FRC150 and FRC180 mixtures constituted 0.5% straight fibers with a length of 16.5 mm and 1.0% straight fibers with a length of 19.5 mm. The straight fibers had a diameter of 0.2 mm, a unit weight of 7500 kg/m^{3}, and a tensile strength of 2500 MPa. The detailed mixing proportions are given in Table
Mixing proportions of the UHPFRC.
Mixture 


Binders 


Steel fiber  

OPC  Zr  BFS  Fiber volume content, 
Diameter,  
FRC120  0.22  209  770  58  135  847  231  1.0 = 1.0 (19.5 mm)  0.2 
FRC150  0.18  180  788  99  99  867  236  1.5 = 1.0 (19.5 mm) + 1.0 (16.5 mm)  0.2 
FRC180  0.18  178  783  196  —  862  235  1.5 = 1.0 (19.5 mm) + 1.0 (16.5 mm)  0.2 
OPC: ordinary Portland cement; Zr: zirconium; BFS: blastfurnace slag; S: sand; F: filler; W: water.
The compressive strength of UHPFRC was obtained through compressive testing on cylindrical specimens with a height of 200 mm and a diameter of 100 mm. Three linear variable displacement transducers (LVDTs) were installed around the cylindrical specimens to measure the displacement during the loading step. The stressstrain curve of the UHPFRC was calculated by using the loaddisplacement relationship, which was obtained from the compressive strength test. In addition, the modulus of elasticity was calculated from the UHPFRC stressstrain curve [
The mean compressive strengths of the FRC120, FRC150, and FRC180 specimens were 133.7, 148.8, and 181.2 MPa, respectively. In addition, the mean elastic moduli of the FRC120, FRC150, and FRC180 specimens were 40150, 43220, and 45140 MPa, respectively.
To evaluate the tensile behavior of UHPFRC, including its postcracking behavior, prismatic specimens of each mixture were fabricated and tested. The prismatic specimens, which had a height of 100 mm, a width of 100 mm, and a length of 400 mm, had a notch with a depth of 10 mm cut into the tensile zone. The crack mouth opening displacement (CMOD) was measured by using a clip gauge attached to both edges of the notch.
A threepoint loading test was performed to obtain the tensile behavior of the UHPFRC. The clear span length between the specimen supports was 300 mm, and the load was measured during its application. To measure the deflections of the prismatic specimens, three LVDTs were attached at the midheight of each specimen. The experimental setup for measuring the CMOD of the prismatic specimens is shown in Figure
CMOD test setup.
LoadCMOD relationship curves.
FRC120 specimens
FRC150 specimens
FRC180 specimens
The UHPFRC stressstrain relationship was modeled based on the current design recommendations [
The tensile behavior of the UHPFRC can be estimated by performing an inverse analysis based on the testing results of the loadCMOD relationship curve. Accordingly, the tensile stressCMOD relationship was derived from the loadCMOD relationship through inverse analysis, after which the tensile stressstrain curve was obtained from the tensile stressCMOD relationship. The tensile strengths of the FRC120, FRC150, and FRC180 mixtures estimated from the loadCMOD relationship were 7.21, 9.07, and 7.76 MPa, respectively; these results show that the tensile strength of the FRC180 mixture was less than that of the FRC150 mixture. At the beginning of the test, it was expected that the tensile strength of UHPFRC would increase with the compressive strength; however, this is not consistent with the tensile test results. The tensile and compressive stressstrain relationships are shown in Figure
Stressstrain relationships.
Compression
Tension
The compressive stressstrain relationships for the FRC120, FRC150, and FRC180 mixtures are shown in Figure
Modeling of the compressive behavior of UHPFRC.
Modeling and estimation of the tensile behavior of UHPFRC.
FRC120 series
FRC150 series
FRC180 series
Material modeling of the UHPFRC.
Mixture  Compressive behavior  Tensile behavior  









 
FRC120  133.7  0.0033  40,150  7.21  6.98  0.00018  0.00198  0.00618  0.02925 
FRC150  148.8  0.0034  43,220  9.07  9.07  0.00021  0.00201  0.00621  0.02685 
FRC180  181.2  0.0040  45,140  7.76  7.76  0.00017  0.00197  0.00617  0.02625 
The terms in the table are shown in Figure
A total of nine beams with rectangular crosssectional dimensions of 200 × 250 mm and a length of 3300 mm were fabricated and tested. Figure
Dimensions of the beam.
Details of the beams.
Beams  Target compressive strength (MPa)  Mean compressive strength (MPa)  Fiber volume content, 
Beam dimensions  Rebar  

Width of section, 
Height of section, 
Effective depth of beam, 
Number  Area, 
Rebar ratio, 
Yielding strength,  
FRC120R2  120  133.7  1.0  200  250  213.5  2D13  258  0.59  420 
FRC120R3  200  250  213.5  3D13  387  0.89  420  
FRC120R4  200  250  213.5  4D13  516  1.19  420  


FRC150R2  150  148.8  1.5  200  250  213.5  2D13  258  0.59  420 
FRC150R3  200  250  213.5  3D13  387  0.89  420  
FRC150R4  200  250  213.5  4D13  516  1.19  420  


FRC180R2  180  181.2  1.5  200  250  213.5  2D13  258  0.59  420 
FRC180R3  200  250  213.5  3D13  387  0.89  420  
FRC180R4  200  250  213.5  4D13  516  1.19  420 
The beams were tested under a fourpoint loading system, as shown in Figure
Instrumentation used for the flexural tests of the beams.
Three LVDTs were placed within the constant moment region of each beam to measure the deflection during the test. To obtain the strain of both the concrete and the rebar, electrical resistance strain gauges were also attached to each beam. The strain in the concrete was measured by five strain gauges attached on the sides of the beams at the midspan, and the strain in the rebar was measured by four strain gauges attached on the surface of the rebar.
The cracking and failure patterns of the UHPFRC beams are shown in Figure
Cracking and failure of the UHPFRC beams.
Rebar ratio of 0.59% (R2)
Rebar ratio of 0.89% (R3)
Rebar ratio of 1.19% (R4)
The loaddeflection curves of the beams with different rebar ratios are shown in Figure
Loaddeflection relationship curves.
Rebar ratio of 0.59% (R2)
Rebar ratio of 0.89% (R3)
Rebar ratio of 1.19% (R4)
Experimental results of the beams.
Beams  Initial cracking stage  Yielding stage  Ultimate stage  

Initial cracking load, 
Initial cracking moment, 
Initial cracking deflection, Δ_{cr} (mm)  Yielding load, 
Yielding moment, 
Yielding deflection, Δ_{y} (mm)  Ultimate load, 
Ultimate moment, 
Ultimate deflection, Δ_{u} (mm)  
FRC120R2  32.0  19.2  4.1  63.5  38.1  6.7  66.7  40.0  22.2 
FRC120R3  60.1  36.1  4.0  117.2  70.3  15.3  123.3  74.0  20.9 
FRC120R4  54.6  32.8  4.2  123.8  74.3  15.8  127.3  76.4  26.4 
FRC150R2  41.0  24.6  2.5  71.5  42.9  11.0  76.0  45.6  17.9 
FRC150R3  48.4  29.0  2.6  107.7  64.6  31.7  118.6  71.2  34.6 
FRC150R4  44.1  26.5  3.8  123.9  74.3  16.1  139.9  83.9  39.5 
FRC180R2  41.1  24.7  1.9  105.3  63.2  12.8  111.5  66.9  17.8 
FRC180R3  40.8  24.5  1.8  122.0  73.2  14.4  131.3  78.8  21.2 
FRC180R4  41.0  24.6  1.7  126.9  76.1  12.0  147.2  88.3  25.5 
With regard to the beams with a rebar ratio of 0.59% (R2), the deflections at the ultimate states of the FRC120R2, FRC150R2, and FRC180R2 beams were 22.2, 17.9, and 17.8 mm, respectively. The deflection of the FRC120R2 beam was greater than that of the other beams, and the deflections at the ultimate states of the FRC150R2 and FRC180R2 beams were similar. With regard to the beams with a rebar ratio of 0.89% (R3), the deflections at the ultimate states of the FRC120R3, FRC150R3, and FRC180R3 beams were 20.9, 34.6, and 21.2 mm, respectively. At the ultimate state, the FRC150R3 beam showed the greatest deflection and the deflections of the FRC120R3 and FRC180R3 beams were similar. With regard to the beams with a rebar ratio of 1.19% (R4), the ultimate deflection of the FRC150R4 beam was the greatest. Therefore, the deflection at the ultimate state of a UHPFRC beam is not directly affected by the compressive strength of UHPFRC, while the bending strength of a UHPFRC beam is approximately proportionally affected by the compressive strength of concrete.
The loaddeflection relationship of a beam subjected to bending depends on the curvature along the axis of the beam. In a lowloading state, because the widths of the cracks in the constant moment region are small and uniform, the curvature in this region is relatively uniform; this results in deflection, which depends on the constant curvature. However, if a major crack forms, the curvature in the constant moment region will no longer be uniform; thus, the midspan deflection of the beam will not depend on the constant curvature. As shown in Figure
In this study, a sectional analysis was performed by using a multilayer section to predict the bending strength of UHPFRC beams. The cross section of the test beam is divided into several layers along the height, and it is assumed that the compression and tensile strain are both linear throughout the cross section. The compressive strain at the top layer and the tensile strain at the bottom layer of the cross section are calculated by using two variables: the crosssectional curvature (
The compressive strain at the top face and the tensile strain at the bottom face are calculated as follows:
After the strains in the top and bottom layers are determined, the strain in the other layers can be obtained, as shown in Figure
Strain compatibility and equilibrium forces in sectional analysis.
Cross section
Strain distribution
Stress distribution
Equilibrium of forces
To predict the bending behavior of UHPFRC beams, two analytical approaches were used in this study. In the first approach, which is shown schematically in Figure
Approaches for estimating the tensile behavior of a UHPFRC beam.
Approach I
Approach II
Prediction of momentcurvature curves (FRC120 series beams).
FRC120R2 beam
FRC120R3 beam
FRC120R4 beam
Prediction of momentcurvature curves (FRC150 series beams).
FRC150R2 beam
FRC150R3 beam
FRC150R4 beam
Prediction of momentcurvature curves (FRC180 series beams).
FRC180R2 beam
FRC180R3 beam
FRC180R4 beam
The analysis result of the bending moment of the FRC120R2 beam overestimated the experimental result, as shown in Figure
Analysis results.
Beams  Tensile strength (MPa)  Bending strength (kN·m)  Ratio  Difference between tensile strengths 


CMOD test result ( 
Estimation from curve fitting ( 
Beam test result (1)  Prediction using the tensile strength from the CMOD test (2)  (2)/(1)  
FRC120R2  7.21  2.83  40.0  61.7  1.54  154.8 
FRC120R3  7.21  5.42  74.0  73.0  0.99  33.0 
FRC120R4  7.21  5.50  76.4  84.4  1.10  31.1 
FRC150R2  9.07  4.59  45.6  66.6  1.46  97.6 
FRC150R3  9.07  7.79  71.2  75.8  1.06  16.4 
FRC150R4  9.07  8.44  83.9  85.0  1.01  7.5 
FRC180R2  7.76  8.90  66.9  61.5  0.92  12.8 
FRC180R3  7.76  9.38  78.8  71.2  0.90  17.3 
FRC180R4  7.76  9.39  88.3  80.9  0.92  17.4 
For the FRC150 beam series, the analysis result of the bending moment of the FRC150R2 beam overestimated the test result greatly; thus, the ratio of the analysis result to the experimental result for the bending strength is 1.46. Meanwhile, the analysis results of the bending moments of the FRC150R2 and FRC150R3 beams were in good agreement with the experimental results; thus, the ratios of the analysis results to the experimental results are 1.06 and 1.01, respectively.
In contrast to the FRC120 and FRC150 beam series, the analysis results of the FRC180 beam series were underestimated. The analysis results of the bending strengths of the FRC180R2, FRC180R3, and FRC180R4 beams were less than the corresponding experimental results, with analyticalexperimental result ratios of 0.92, 0.90, and 0.92, respectively.
The most important parameter for predicting the bending momentcurvature curve of UHPFRC is the tensile strength. It is well known that an accurate prediction of the bending strength of a UHPFRC beam is dependent on the tensile strength of the UHPFRC. The deviation of the analysis result from the experimental result of the bending strength may primarily be due to the difference between the tensile strength obtained from the material test with the notched specimen and the actual tensile strength of the test beam.
Therefore, a second approach was employed to estimate the actual tensile strength of each test beam. In the second approach, a sectional analysis was performed for the momentcurvature curve to ensure that the bending strength of the analytical momentcurvature curve was in good agreement with that of the experimental result. The fitting of the bending strengths from the analytical momentcurvature curves and from the experimental work is illustrated in Figures
Fitting of momentcurvature curves between the experimental and analysis results (FRC120 series beams).
FRC120R2 beam
FRC120R3 beam
FRC120R4 beam
Fitting of momentcurvature curves between the experimental and analysis results (FRC150 series beams).
FRC150R2 beam
FRC150R3 beam
FRC150R4 beam
Fitting of momentcurvature curves between the experimental and analysis results (FRC180 series beams).
FRC180R2 beam
FRC180R3 beam
FRC180R4 beam
The tensile strength of UHPFRC is a major parameter influencing the bending capacity of a UHPFRC beam. If the bending strength of the analysis result is almost identical to the experimental result, the tensile properties used as input for the sectional analysis can be assumed to be the actual tensile properties for the test beam.
The tensile strengths obtained from the material testing with the notched specimens and estimated by fitting the bending strengths of the analytical momentcurvature curves to the experimental result (shown in Figures
From the test of the notched prismatic specimen, the tensile strength of the FRC120 beam series was 7.21 MPa, while the estimated tensile strengths of the FRC120R2, FRC120R3, and FRC120R4 beams were 2.83, 5.42, and 5.50 MPa, respectively. The tensile strength of the FRC150 beam series obtained from the test on the prismatic specimen was 9.07 MPa, while the estimated tensile strengths of the FRC150R2, FRC150R3, and FRC150R4 beams were 4.59, 7.79, and 8.44 MPa, respectively. Meanwhile, the tensile strength of the FRC180 beam series obtained from the tensile test was 10% less than that estimated by fitting the momentcurvature curve.
An experimental study on the flexural behavior of UHPFRC is presented in this study, and the tensile strength of UHPFRC beams is estimated by fitting the analysis result of the momentcurvature curve to the experimental result of the bending strength. Based on the experimental and analysis results, the following conclusions can be made:
The width of one of the welldistributed cracks in the constant moment zone was significantly widened, and the crack became localized. The localization of cracks within the constant moment region caused the bridging effect to weaken, and steel fibers were eventually pulled out of the matrix in the region of localized cracking.
The bending strengths of the FRC150 and FRC180 beam series were greater than those of the FRC120 beam series, indicating that the flexural strength of the UHPFRC beams increased with the steel fiber content.
The bending strength obtained from the sectional analysis using the material test result was compared with the beam test result. This comparison showed that the bending strengths of several beams differed from the corresponding test results, thereby indicating that the tensile strength obtained from the material test would be significantly different from that estimated from the beam test results. The estimated tensile strengths of the FRC120R2, FRC120R3, and FRC120R4 beams were 60.7, 24.8, and 23.7% less than the tensile strength obtained from the material test, respectively. Similarly, the estimated tensile strengths of the FRC150R2, FRC150R3, and FRC150R4 beams were 49.4, 14.1, and 6.9% less than the tensile strength obtained from the material test, respectively. Meanwhile, the estimated tensile strengths obtained from the FRC180 beam series were 14.7∼21.0% greater than the tensile strength obtained from the material test.
The tensile strengths of the UHPFRC beams were estimated reasonably by fitting the analysis results of the momentcurvature curve to the beam test results. The estimated tensile strength of the UHPFRC increased with its compressive strength.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This research was supported by a grant (13SCIPA02) from the Smart Civil Infrastructure Research Program funded by the Ministry of Land, Infrastructure and Transport (MOLIT) of the Korean government and the Korea Agency for Infrastructure Technology Advancement (KAIA).