Exploring the mechanical properties of steel- and polypropylene-reinforced ultra-high-performance concrete through numerical analyses and experimental multi-target digital image correlation

Abstract

This study presents experimental and numerical investigations on the mechanical properties of ultra-high-performance concrete (UHPC) reinforced with single and hybrid micro- and macro-steel and polypropylene fibers. For this purpose, a series of cubic, cylindrical, dog-bone, and prismatic beam specimens (total fiber by volume = 1%, and 2%) were tested under compressive, tensile, and flexural loadings. A method, namely multi-target digital image correlation (MT-DIC) was used to monitor the displacement and deflection values. The obtained experimental data were subsequently used to discuss influential parameters, i.e., flexural strength, tensile strength, size effect, etc. Numerical analyses were also carried out using finite element software to account for the sensitivity of different parameters. Furthermore, nonlinear regression analyses were conducted to obtain the flexural load-deflection curves. The results showed that the MT-DIC method was capable of estimating the tensile and flexural responses as well as the location of the crack with high accuracy. In addition, the regression analyses showed excellent consistency with the experimental results, with correlation coefficients close to unity. Furthermore, size-effect modeling revealed that modified Bazant theory yielded the best estimation of the size-effect phenomenon compared to other models.

Graphical abstract

Keywords

UHPC / MT-DIC / flexural behavior / tensile behavior / steel fiber / polypropylene fiber

1 Introduction

With advancements in technology and inspection practices, a new class of cementitious materials known as ultra-high-performance concrete (UHPC) has emerged during the past few decades. The exceptional characteristics of UHPC in terms of mechanical and chemical properties, as asserted by many researchers [1–4], can be further improved by the inclusion of fibers, which resolves the brittleness issue in addition to enhancing the inherent properties of UHPC. However, despite these favorable characteristics, the inclusion of different kinds of fibers along with the nonlinear nature of concrete are obstacles in establishing unified constitutive models to characterize its behavior. This necessitates further experimental, analytical, and numerical models to contribute to the existing literature and help deepen the knowledge with respect to its behavior under different loading conditions. In this regard, in the following, the literature related to the behavior of ultra-high-performance fiber-reinforced concrete (UHPFRC) under various loading conditions is discussed.

Zhang et al. [5] studied the direct tension of rebar-reinforced UHPC dog-bone specimens. Two series of smooth fibers with l_{\mathrm{f}}/d_{\mathrm{f}}/f_{\mathrm{t}} (length, diameter, and tensile strength of fiber, respectively) equal to 8 mm, 0.12 mm, and 2475 MPa, respectively, and hooked-end (H) fibers with l_{\mathrm{f}}/d_{\mathrm{f}}/f_{\mathrm{t}} equal to 13 mm, 0.20 mm, and 2425 MPa, respectively, were incorporated in overall volumetric ratios of 2.0%, and 3.5%. 8-, 10-, and 12-mm steel bars were also used in ratios of 3.14%, 4.91%, and 7.07%, respectively. The cross-section of the area of interest in dog-bone specimens was 80 mm \times 80 mm. The results showed that despite the improved strain-hardening behavior in specimens with higher fiber contents, the overall response of the load−deflection curves was similar, i.e., the presence of steel bars greatly improved the tensile capacity by 2.1–5.8 times compared to its counterpart without steel bars. Similarly, but more notably, specimens containing steel bars were reported to possess a 39-fold improved tensile stiffness compared to plain UHPCs.

Deng et al. [6] studied the mixed effect of steel–polypropylene (PP) fibers and coarse aggregates on uniaxial compressive stress–strain curves of UHPCs with various dosages. They highlighted the twofold advantage of PP fiber incorporation: the contribution to both the interface, failure pattern and the improved bond with the matrix. By contrast, the presence of coarse aggregates only improved the strength and had negative effects on ductility. It was reported that the presence of PP reduces the negative effects of coarse aggregates.

A regression-based study was carried out by Ashkezari et al. [7] to quantify the compressive, tensile, splitting, and flexural strengths of UHPCs reinforced with 0%–3% steel-fiber content. They reported 81%, 66%, 228%, and 180% improvements in the compressive, splitting, and flexural strengths of the specimens, respectively, compared to the unreinforced counterparts by incorporating 3% steel fiber. The authors highlighted that these values are higher than those reported in the literature. Chun and Yoo [8] investigated the hybrid effect of micro-straight and macro-straight, H, and twisted steel fibers in overall ratios of 2% by volume. Replacing 1.5% of the macro-straight fibers with their micro counterparts increased the bond strength by 112% owing to a better interfacial bond with the matrix, while the converse was true for the hooked and twisted fibers. The efficiency of the macro fibers decreased when replaced with micro fibers. Twisted fibers showed very high pullout ratios (694% higher than straight fibers). Negative correlations were reported for multiple fiber pullout and tensile performance of the specimens, irrespective of the fiber type. A similar trend as that reported by Chun and Yoo [8] was observed by Kim et al. [9]; they found that the inclusion of PP fibers decreases the compressive strength and tensile strength of UHPFRCs, and their positive hybrid effect in terms of strain capacity was limited to values less than 0.5% by volume. Markić et al. [10] highlighted that there is a misconception concerning the positive contribution of steel fiber in longitudinally reinforced concrete beams in terms of ductility. This is evidenced by the superior performance of specimens without fibers compared to those reinforced with steel fibers, especially when members are lightly reinforced. To address this complexity, a model was proposed to determine whether fibers were the reason behind the failure of reinforced concrete members and/or crushing of concrete occurred beforehand.

Ríos et al. [11] studied the tensile fracture of steel-reinforced concrete specimens on the macro and micro scale using micro-computed tomography to relate the mechanical properties to the porosity of the mixes. It was observed that steel fibers serve as tiny shovels that help to mix concrete while reducing porosity. This reduction in porosity led to a higher compressive strength compared to the plain specimen. However, this effect was not observed for the elastic modulus. Shi et al. [12] reported a notable decrease in workability when the amount of straight and H steel fibers exceeded 2%. Exceeding this value increased the entrapped air in the specimen, made mixing very difficult, and induced balling in straight steel fibers. Different elastic moduli were reported in tension and compression experiment, with the latter being 1.5-fold higher than that of the unreinforced specimen. The fibers contributed more to ductility and toughness than to strength, especially during compression. The tensile and pullout behavior of UHPFRCs with straight, twisted, H, and half-H fibers with aspect ratios equal to 65, 100, 80, and 66.7, respectively, were studied by Yoo et al. [13]. The tensile performance of the straight steel fibers was the best, while the half-H fibers exhibited poor tensile performance and stress concentrations at the hooked end, necessitating further studies to validate the authors’ findings and other parametric analyses. Abbas et al. [14] incorporated steel fibers with l_{\mathrm{f}}= 8, 12, and 16 mm into UHPCs in 1%, 3%, and 6% ratios by volume, respectively, and carried out compressive, splitting, and flexural tests. Their results showed that the presence of the fibers changes the failure pattern from brittle to ductile, and the mechanical properties of the specimens increased with the increase in fiber dosage, with the short fibers having the best performance.

Sturm et al. [15] conducted an experimental study to address the lack of research with regard to blending fibers at the structural level and whether properties achieved at the material level can be further extended to the structural level. For this purpose, 200 mm × 260 mm × 4200 mm beams were tested under three-point loading using single and hybrid 35 mm H fibers and 13 mm micro steel (MS) fibers. Micro fibers exhibited the best performance in the serviceability limit while blending fibers contributed primarily to the post-peak response and decrease in the maximum deflection values. In a study on 75 mm × 75 mm × 285 mm beams with a notch width and depth of 2.5 and 12 mm, respectively, Isa et al. [16] used recycled-tire steel fibers and recycled-tire steel cords in UHPC beams as a more environmentally friendly alternative to investigate the post-cracking tensile cracking of UHPFRCs, for which practically no unified standard method exists. Their results showed that recycled-tire steel fibers perform better overall than their counterparts and that the approach adopted by Fib Model Code 2010 [17] largely overestimated the energy absorption and tensile strength of UHPCs by 76% and 31%, respectively. Hence, a new constitutive tensile σ−ε model was developed to capture the flexural properties of eco-efficient UHPC beams using reasonable mix proportions. The uniaxial tensile behavior of 30-mm-long steel fibers in ratios equal to 2.55% by volume was studied by Donnini et al. [18]. For this purpose, dog-bone specimens (length: 330 mm; cross-section: 30 mm × 45 mm) were tested using digital image correlation (DIC) as a tool to measure displacements, deformations, and the evolution of cracks. The results revealed that the fibers change the sudden brittle pattern to a ductile one with progressive opening of a single crack; strain-softening behavior was observed for fiber ratios below 1.25%, and a strain-hardening multi-micro cracking phase emerged for fiber ratios exceeding 1.9%. Larsen and Thorstensen [19] conducted a review study on the effect of steel fibers on the tensile and compressive properties of UHPCs. They reported that ASTM C1609/C1609 [20] was the most commonly used method for flexural testing, and despite the existence of standard guidelines for some UHPC parameters, researchers tended to use the relevant test methods given for conventional concrete. Geometry was reported to be a primary factor affecting the results. According to Larsen and Thorstensen [19], studies on the impacts of constituent materials, mix designs, and curing regimes were not focused and the optimum fiber type depended on the fiber content, as some exhibited poor performance in lower dosages but good performance in higher dosages. It was emphasized that, to minimize uncertainty and reach a consensus on the findings from a statistical point of view, previous studies must be replicated, and not a single test method should be relied on when various standards exist in this regard.

Another aspect of fiber-reinforced concrete (FRC) that requires due consideration is its size-dependent properties; size effects have been observed in compression and flexure [21,22], shear [23], etc. According to Ref. [21], with the increase in compressive strength of the mix, the impact of the size effect diminished for coarse-grained UHPC, while the opposite was true for fine-grained UHPC. For flexure, behavior similar to that of conventional concrete was reported. According to Ref. [22], UHPFRC beams with higher fiber content were more heavily influenced by size changes, which was attributed to the uneven fiber distribution in thicker elements. Shear capacity was less influenced by size changes, where this result was even more pronounced in specimens with higher fiber content [23], which is contrary to the results reported in Ref. [22]. For flexural loading. It is also noteworthy that numerical modeling becomes important when considering size effects, as the cost of fabricating large-scale FRC and UHPFRC specimens can be substantial. Furthermore, modeling is an important tool in the multi-scale simulation of concrete [24].

In summary, a holistic review of previous studies shows that further studies are required to validate and/or extend the findings of UHPFRC specimens under various conditions. For this purpose, single and hybrid MS, round-crimped (RC), crimped (C), H, and PP fibers were incorporated into UHPC in overall volumetric ratios of 1% and 2%, and the data obtained from different mechanical tests using a DIC-based method, namely, multi-target digital image correlation (MT-DIC), are discussed in detail. Furthermore, numerical analyses and size-effect studies for UHPFRC are comparatively less in the literature, and this review not only aims to address these shortcomings but also to extend the work and carry out parametric studies on the governing factors.

2 Multi-target digital image correlation

DIC is a very common measurement technique that makes it feasible to capture temperature fields [25], monitor crack evolution, characterize failure surfaces [26], and mechanical properties in 3D-printed materials [27], etc., most of which cannot be easily captured by linear variable displacement transducers (LVDTs). LVDTs are usually used to capture the displacement of a specific point on an object, which may pose the following limitations: (1) limited measurement range; (2) high cost; (3) susceptibility to fluctuations in voltage, which may disturb their efficiency; and (4) inability of horizontal LVDTs to capture vertical displacement in cases where a specimen is subjected to lateral loads. The third author of this article has successfully used the MT-DIC method used herein to capture the deformation and buckling of steel-plate shear walls [28].

2.1 Digital image correlation versus multi-target digital image correlation

In the DIC method, shots from the specimen surface before and after loading are used to calculate the deformation parameters. The initial image taken at the undeformed stage serves as the reference image. For better efficiency, a random scatter of speckled patterns is spread onto the specimen surface. A typically small square from the initial image is selected as the “reference subset”, and a larger square is chosen as a “searching subset” from the deformed image (Fig.1(a)).

A regular set of circular labels with a diameter of 7 mm, stuck to the surface of the specimen, is used instead of a speckle pattern in the MT-DIC method. Choosing a circular shape is advantageous over other shapes such as triangles and rectangular, because circles allow the deformation pattern to be captured in any direction with an equal amount of noise. Each label is tracked separately (Fig.1(b)). In other words, in lieu of the correlation between subset images, a correlation between single labels is extracted. Using a regular pattern of labels eliminates the errors originating from speckle size and the selection of subsets. In summary, the methodology to capture the deformed shape is as follows.

1) Identification of labels and their center (x_{(i,j)},y_{(i,j)}) as well as their center-to-center distance, a, in the reference image (Fig.1(b)). a is equal to 30 mm in this study.

2) A circular search area with a radius of is assumed for each label, the value of which can be determined by the user at the beginning. If the target label was not found in the specified search area, the search radius is gradually increased up to r=a/2 (Fig.1(b)).

3) In cases of tearing or buckling (e.g., for steel plates), some labels may disappear from view, preventing the program from tracking the label. In these cases, given the regular pattern of labels, the new coordinates of the missing label are assumed to be the average coordinates of the adjacent labels. Upon re-emergence, the missing label can be tracked again in the subsequent images.

In this study, the area of interest is the surface of the concrete specimen, and hence only 2D deformations are considered.

The evaluation of relative displacements of labels with respect to one another and in relation to the undeformed shape allows the calculation of deformations and other relevant parameters based on the finite element theory. The principles of conventional DIC and MT-DIC are given in Fig.1(a)–Fig.1(c). The specifications of the camera and laptop used for this purpose are given in Tab.1.

2.2 Limitations of multi-target digital image correlation

1) 3D deformations cannot be captured using this method.

2) In cases where more than one label disappears from view, for example, in the event of severe buckling or folding of a steel plate, error occurs. This method is poor when it comes to capturing fine cracks. Random cracks are better captured using the conventional DIC method.

3) Sensitivity of the model to light intensity of the atmosphere (constant light intensity is desirable).

4) In the case of large deformations, labels may not fit into the camera’s framework, rendering the tracking process impossible.

5) For large-scale specimens, it is time-consuming to prepare an organized layout of labels.

A large spacing (small number of labels) between circular labels leads to a loss of accuracy. The choice of the number of labels is based on trial and error and is closely related to the pixel size of the label; labels that are too small will cause error. Based on a trial-and-error method, labels larger than 5 mm in diameter were found suitable, and hence circles with a diameter of 7 mm were chosen. It is noteworthy that the resolution of the camera used in this study only allows the detection of macro cracks. For a detailed description of the method, calibration, accuracy, source of errors, etc., the reader is referred to Ref. [28].

3 Methods

3.1 Materials and mix design

Type II Portland Cement, local fine sand sifted through a No. 16 (1.18 mm) sieve, silica fume with a size of 229 nm (sieved before use), a polycarboxylate-ether-based high-range water-reducing agent used as a superplasticizer (i.e., AURAMIX), quartz powder, water, and various steel fibers were used to fabricate the UHPFRC specimens. The mix design of UHPFRC, chemical composition of silica fume, and properties of the steel fibers are given in Tab.2–Tab.5.

Concerning rheology, according to the recommendations of ASTM C230/C230M [29], a mini-slumped cone with a lower and upper diameter of 100 and 70 mm were used for this purpose. The height of the cone was 50 mm. Only slump values were recorded as the rheology indicator. Needless to say, the addition of PP fibers significantly diminishes workability. Therefore, slump values were not measured for specimens with PP fibers. Slump values for different specimens are given in Tab.6. Single and binary combinations of micro and macro fibers in overall ratios of 1% and 2% by volume were added to the concrete, where the digit following the specimen ID indicates the fiber content by volume; for example, “RC1” denotes concrete containing 1% RC fibers by volume.

3.2 Compression tests

Compression tests were carried out on 150 mm × 300 mm cylindrical specimens according to ASTM C39/C39M [30] in a displacement-controlled manner at a rate of 0.1 mm/min. The elastic moduli tests were carried out on three specimens, and the average value was used in calculations according to ASTM C469/C469M [31], as follows:

(1)E_{\text{c}}=\frac{0.4f_{\text{c}}^{\prime }-f_{\text{c1}}}{{\epsilon }_2-0.00005},

where E_{\mathrm{c}} is the elastic modulus, f_{\mathrm{c}}^{\prime } is the compressive stress corresponding to strain {\epsilon }_2, and f_{\mathrm{c}1} is the compressive stress corresponding to an axial strain of 0.00005. It is noteworthy that the MT-DIC method was not used in the compression and elastic moduli tests.

3.3 Tensile tests

Direct tensile tests were carried out on 28-d dog-bone specimens with an overall length of 325 mm, width of 80 mm, and thickness of 40 mm. The dimensions of the narrow cross-section were 40 mm × 40 mm. It is noteworthy that only the axial degree of freedom (DOF) was constrained in the tensile test method. Inclined compartments of the tensile setup only served as a means to grab the specimen, and thus no additional stress was imposed on the specimen (Fig.2). The load was applied in a displacement-controlled manner at a rate of 0.1 mm/min. As previously mentioned, the center of the colorful labels was tracked to record displacement values, and the average values of the displacements in the narrow section were used for calculations.

3.4 Flexural tests

Four-point-bending tests (4PBT) were performed at 28 d on 100 mm × 100 mm × 500 mm beams with a clear span of 450 mm according to ASTM C1609/C1609M [20]. Details of the 4PBT are given in Fig.3.

4 Numerical simulation

ATENA [32] and GID [33] pre-processor software were used to simulate the flexural behavior of the UHPFRC beams. Numerous researchers have used this software to model normal- and FRC [34–41]. Given this, a unified method to model FRC does not yet exist to authors. As such, an inverse analysis based on the procedure proposed by the developers of ATENA [32] was performed to obtain the post-cracking tensile stress vs. fracture strain of UHPFRCs. For this purpose, the NonlinearCementitious2User material model, which adopts a fracture–plastic approach, was used to model concrete, and a linear elastic steel was used for the loading plates and supports. The elastic moduli, tensile strength, and compressive strength of the specimens were used as input parameters. Eight-node hexahedral elements were used to model the concrete, steel supports, and loading plates. The mesh size was 10 mm. The load was applied in a displacement-controlled manner at a rate of 0.1 mm/step until failure occurred. Monitoring points were used to record the reaction values at the load plates and mid-span of the beam. The Newton–Raphson method and the line-search method were used to solve the nonlinear equations. The fracture strain used to define the tensile function is defined as

(2)\epsilon =\frac{w}{L_{\text{t}}},

where \epsilon is the fracture strain; w is the crack width; and L_{\mathrm{t}}, equal to 10 mm in this study, is the characteristic length defined in Fig.4 for the post-cracking tensile function.

The methodology used to simulate the behavior of fiber-reinforced composites is as follows.

1) The fracture strain and tensile stress of the first crack are defined as the x and y axes of the tensile function, respectively, based on the experimental results and/or the experience of the user.

2) The analysis is performed using by comparing the experimental and numerical results.

3) If the difference between two sets of results is negligible, the analysis is complete. Otherwise, Eq. (2) should be used to recalculate the fracture strain from the crack widths obtained from the analyses at distinct deflection values where the difference is considerable. Based on the new fracture strain, the initial tensile function should be modified, and the stress values should be multiplied by the ratio of the experimental-to-numerical results.

It should be emphasized that the accuracy of the tensile function and its concomitant results depend on the number of trial-and-error iterations and the accuracy required by the user.

5 Results and discussion

5.1 Compression results

Tab.7 shows the IDs, compressive strengths, and elastic moduli of all specimens at 28 d. The inclusion of fibers transformed the failure pattern from brittle to ductile. As the load increased, inclined cracks formed at the edges of the specimens and propagated toward the center of the specimen. None of the specimens lost their integrity, even at ultimate loads, except for PP1 and PP2. The failure pattern was mostly characterized by the localized spalling of concrete and exposure of fiber. This ductile failure contributed to increased compressive capacity. However, an increase in fiber content from 1% to 2% led to a negligible increase of 10% in the compressive strength for the RC, PP, C, H, and MS specimens and minimal influence on the elastic moduli of the specimens. Considering single macro fibers as the reference, the addition of MS fibers to the macro fibers caused the most notable increase in compressive capacity for RC1MS1 (20%), followed by C1MS1 (17%), and H1MS1 (16%). This result can be explained by the fact that the presence of high-strength micro fibers inhibit the growth of fine cracks at the initial stages of loading, increasing the load capacity. This explanation did not apply to the hybrid combination of low-strength PP and macro steel fibers, as this type of fiber mainly contributes to the durability characteristics of concrete rather than strength. Regarding elastic modulus, the trends were less pronounced.

5.2 Tension results

Fig.5 shows the load−displacement curves of dog-bone specimens. The location of the cracks is also given in Fig.6. It is clear from Fig.5 that increasing the fiber content from 1% to 2% led to a 62% improvement in the tensile strength of specimen MS, followed by 24% for RC and 20% for C, H, and PP. With regard to hybrid fibers, specimens RC1PP1, C1PP1, and H1PP1 outperformed both PP1 and PP2, with the least improvement for specimen H1PP1 (24%) and the most improvement (53%) for RC1PP1 compared to PP1 and PP2,. Nonetheless, the addition of PP fibers to the macro steel fibers degraded the mechanical properties by 11%–26%. This can be justified by the weak bond formed between the PP fibers and the concrete matrix, making the specimens with single PP fibers vulnerable to the crushing of the aggregate. This observation gives credence to the sharp decrease in the load−displacement curve shown in Fig.5. By contrast, the hybrid micro- and macro steel fibers outperformed their single counterparts, with specimen RC1MS1 having the highest tensile strength (12.5 MPa), which is 80% higher than that of specimen RC1. Similar observations were made for other specimens.

5.2.1 Nonlinear regression (load−displacement curves)

To quantify the direct tensile response, nonlinear regression analyses were carried out on the stress–strain curves of dog-bone specimens; the confidence interval was 95%. The overall trend of the fitting curves was obtained using Eq. (3):

(3)y=\frac{a+bx}{1+cx+d{x}^2},

where a, b, c, and d are the fitting parameters; x corresponds to the strain values; and y corresponds to the stress values. The results given for specimen RC1 (unless otherwise shown, in subsequent sections, only the results for specimen RC1 are presented for similarity and brevity) in Tab.8 are in good agreement with the experimental results. The fitting curve is given in Fig.7.

From a statistical point of view, it should be emphasized that a large number of specimens should be tested to obtain an acceptable regression-based equation, and the formulas given herein are only applicable to this study and/or should merely serve as a rough estimation of experimental results in similar research.

5.2.2 Nonlinear regression (energy absorption–displacement curves)

Energy absorption is a measure of ductility, and its determination can aid in distinguishing between a brittle and ductile failure. For this purpose, a similar nonlinear regression approach was adopted, resulting in Eq. (4):

(4)y=V\frac{x^{\eta }}{k^{\eta }+x^{\eta }},

where V, \eta, and k are the fitting parameters; x corresponds to the displacement values; and y represents the energy-absorption values in kN/mm². Similar to the previous section, Tab.9 and Fig.8 reveal that the consistency between the experimental and regression results is high, as evidenced by R² values above 0.90. Negligible inconsistencies arise in the initial part of the curve owing to the linear growth of energy absorption and nonlinear nature of Eq. (4).

5.3 Flexural results

5.3.1 4PBT

A comparison between the results of the MT-DIC method and the numerical results is given in Fig.9. It is evident from the figure that MT-DIC is a good tool for capturing the flexural-load deflection response of the beam. Similarly, the differences between the numerical simulations and experimental results are less than 10%, and hence the experimental response is deemed to provide a satisfactory estimation. Moreover, deviations between the numerical and experimental results in the ascending branch of the curve are attributed to the nonlinear nature, shape, type, and ratio of the fibers, making accurate simulation difficult. Increasing the fiber content from 1% to 2% led to a relatively good increase in the flexural response of the specimen: increases of 23%, 26%, 14%, 12%, and 22% were recorded for specimens RC, C, H, MS, and PP, respectively.

Needless to say, in hybrid fibers, MS fibers contributed much more to the improved flexural capacity. A similar observation was also made for the hybrid combination of macro steel and PP fibers. Specimen RC1MS1 had the highest flexural strength (54 MPa). Furthermore, the strain contours of the cracked specimens in Fig.10 show that the MT-DIC method is capable of identifying the locations of cracks. Fig.11 shows scanning electron microscopy (SEM) images of the MS fibers. It can be observed that under applied loads, the fiber debonds from the matrix, and the steel fiber fractures. Debonding occurs because the smooth surface of the steel fiber is not amenable to entanglement with the matrix. Besides, no large pores exist within the close vicinity of fibers, indicating that the structure of the mix is dense when micro fibers are used. Furthermore, as shown in Fig.11, the steel fibers inhibited the growth of cracks in the vicinity, compromising the integrity of the concrete matrix.

5.3.2 Nonlinear regression analysis (load−deflection curves)

Similar to the methodology adopted in Subsubsection 5.2.1, nonlinear regression analyses were performed on normalized load–normalized load deflection curves such that the load−deflection values were normalized with respect to the peak load and its corresponding deflection. Normalization obviously makes the obtained results comparable those of similar studies. Equation (3) yielded the best results for the flexural specimens; the x and y values denote the ratios of given deflection and load values relative to that of the peak point. Values of the fitting parameters and the respective normalized curves given in Tab.10 and Fig.12 reveal that the correlation between the experimental and numerical results is very high. The deviation of the fitting curves from the experimental curves, especially in the post-peak descending branch, is attributed to the complexities introduced into the concrete by different fibers with different geometries and contents.

5.3.3 Nonlinear regression (energy absorption–displacement curves)

A similar procedure as that described in Subsubsection 5.2.2 was adopted herein, resulting in Eq. (4) for the energy absorption of concrete beams. Tab.11 and Fig.13 compare the experimental and regression curves, revealing their good agreement.

5.3.4 Size effects

The dimensions of the beams used in the experiment were 100 mm × 100 mm × 500 mm, and size effects were only considered in the numerical study. Fig.14 shows the load−deflection curves of beams with different sizes. The results clearly indicate that as the size of the specimens increase, the initial stiffness and load capacity of the specimens also increase. Similarly, with the increase in size, the slope of the maximum crack width–deflection curve rises dramatically (in ATENA [32], it is possible to obtain crack width values and contours). Fig.14 and Fig.15 imply that as a beam enlarges, the existing macro crack widens at a much greater rate than the increase in the vertical deflection of the beam.

With reference to the modeling of the size effect, three well-known theories are available in the literature: (1) the statistical theory introduced by Weibull [42], which relates the size effect to the intrinsic random nature of material strength; (2) the fracture-based theory introduced by Bazǎnt and Chen [43] (Eq. (5)), which correlates the size effect with the release of fracture energy and stress distribution; and (3) Carpinteri and Chiaia’s theory [44] (Eq. (6)), which relates the size effect to crack fractality. In this study, the methods of Bazǎnt and Chen [43] and Carpinteri and Chiaia [44] will be considered owing to their suitability for quasi-brittle materials (the random nature of strength is not an objective of this study; hence, the Weibull approach is not utilized). For very large specimens, Eq. (5) tends to give zero-stress results, which are illogical. Therefore, Eq. (5) was modified by Kim and Yi [45] to yield Eq. (7).

(5){\sigma }_{\text{N}}=\frac{\beta f}{\sqrt{1+\beta }};\beta =\frac{d}{d_0},

(6){\sigma }_{\text{N}}=F\sqrt{1+\frac{l_{\text{ch}}}{d}}=\sqrt{A+\frac{B}{d}},

(7){\sigma }_{\text{N}}=\frac{\beta f}{\sqrt{1+\beta }}+\alpha f,

where {\sigma }_{\mathrm{N}} is the nominal stress; d is the effective depth of the beam; f is the tensile strength of the material; and F, l_{\mathrm{c}\mathrm{h}}, A, d_0, and \alpha are parameters that can be obtained by fitting the experimental results. Fig.16 clearly shows the size effect in the beams (i.e., the stress reduces with the increase in size). It can be inferred from Eq. (6) that for very small specimens, the model proposed by Carpinteri and Chiaia [44] is likely to give infinite strengths, while Bazǎnt and Chen’s model [43] shows an almost linear trend such that for large specimens, the stress tends toward zero. In addition, the results of the fitting parameters (Tab.12) show that the modified Bazant theory is better overall in capturing the size-effect phenomenon.

6 Conclusions

Experimental and numerical studies were carried out in this research to characterize the mechanical properties of UHPFRC beams reinforced with single and hybrid micro- and macro steel and PP fibers. A DIC-based method, i.e., MT-DIC, was used to capture the displacement, deflection, and strain contours of the specimens. The results were further validated by performing inverse analyses using finite-element software, and complementary parametric analyses were carried out to provide better insight into the investigated parameters. The salient findings of this study are as follows.

1) Hybrid combinations of micro- and macro steel fibers had the best overall performance, with specimen RC1MS1 (1% RC fiber + 1% MS fiber) demonstrating the highest tensile strength of 12.5 MPa.

2) Inverse analyses were carried out using finite-element software, which successfully captured the overall behavior of UHPFRC beams during flexure.

3) Nonlinear regression relationships were proposed for both the energy absorption, load−displacement, and load−deflection of both dog-bones and beams, which correlated very well with the experimental results, as evidenced by R^2 values close to unity.

4) Increasing the fiber content from 1% to 2% by volume of concrete improved all the investigated mechanical properties, with an insignificant influence on the compressive strength and elastic moduli of the specimen (less than 10%). The workability was notably reduced upon the addition of PP fibers to the mix.

5) Size-effect theories were applied to UHPFRC beams subjected to a four-point bending test. The results showed that the modified Bazant theory performs the best overall in capturing the size-effect phenomenon of the beams irrespective of the fiber type or ratio.

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