A Review on Structural Behavior, Design, and Application of Ultra-High-Performance Fiber-Reinforced Concrete

3.1 Bond Behavior Between UHPFRC and Reinforcements

In order to practically apply a newly developed UHPFRC in the structures, bond performance with reinforcements should be examined. Many researchers (Jungwirth and Muttoni 2004; Ahmad Firas et al. 2011; Yoo et al. 2014a, b, 2015a) have investigated the bond behavior of internal steel and fiber-reinforced polymer (FRP) reinforcements with UHPFRC. Jungwirth and Muttoni (2004) carried out pullout test of deformed steel reinforcing bar using a 160 mm cube. Various bond lengths ranging from 20 to 50 mm and two different bar diameters of 12 and 20 mm were adopted. In their study, the average bond strength of steel bars embedded in UHPFRC was found to be 59 MPa, approximately 10 times higher than the bond strength of steel bars embedded in ordinary concrete, and the theoretical development length of deformed steel bars in UHPFRC was suggested by l _b  = f _y d _b /4τ _max, where f _y is the yield strength of steel bar, d _b is the nominal diameter of steel bar, and τ _max is the bond strength. Yoo et al. (2014c) examined the effects of fiber content and embedment length on the bond behavior of deformed steel bars embedded in UHPFRC. For this, Yoo et al. (2014c) performed a number of pullout tests by modifying the test method, proposed by RILEM recommendations (RILEM 1994); the 150 mm cubic specimens with a single bar embedded vertically along the central axis were fabricated and used for testing. The embedment lengths were determined by 1 and 2 times the bar diameter, instead of using 5d _b , as suggested by the RILEM recommendations. The bond strength was insignificantly affected by the fiber content and embedment length, but it clearly correlated with the compressive strength. The CEB-FIP Model Code (MC90) (CEB-FIP 1993), which defined τ _max as 2.0f _c ′^0.5, substantially underestimated the bond strength of steel bars in UHPFRC because the parameters were suggested based on test data from previous concretes. Thus, Yoo et al. (2014c) proposed modified coefficients for the bond strength of steel bars in UHPFRC, based on a number of test data, as follows (Fig. 1):

$$ \tau_{\hbox{max} } = 5.0\sqrt {f_{c}^{\prime } } $$

(1)

In addition, CMR model (Cosenza et al. 1995), which sets τ = τ _max × (1 − e^−s/sr ) ^β , was found to be appropriate for simulating the ascending bond stress versus slip behavior of steel bars embedded in UHPFRC, and the parameters were proposed as τ _max = 5.0f _c ′^0.5, s _r  = 0.07, and β = 0.8, where s _r and β are coefficients based on the curve fitting of test data.

Ahmad Firas et al. (2011) experimentally investigated the bond performance between carbon-fiber-reinforced polymer (CFRP) bars and UHPFRC according to the surface treatment, embedment length, bar diameter, and concrete age. Based on the test data, it was noted that the bond strength was insignificantly affected by the surface treatment of the glass-fiber-reinforced polymer (GFRP) bar; similar bond strengths for smooth bars and sand-coated bars were obtained. On the other hand, a decrease in bond strength was obtained by increasing both bar diameter and embedment length. The ultimate bond strength of CFRP bars in UHPFRC was insignificantly changed by age after 3 days, because it was primarily affected by the shear strength of the connection between the core and the outer layer of the CFRP bars. Ahmad Firas et al. (2011) suggested a development length for sand-coated bars of approximately 40d _b , and a development length for a smooth bar of longer than 40d _b . Yoo et al. (2015b) also examined the local bond behavior of GFRP bars embedded in UHPFRC. The average bond strengths of GFRP bars in UHPFRC were found to be from 16.7 to 22.8 MPa for a d _b of 12.7 mm, and from 19.3 to 27.5 MPa for a d _b of 15.9 mm, which are approximately 73 and 66 % less, respectively, than the bond strengths of deformed steel bars. Similar to the case of CFRP bars in UHPFRC (Ahmad Firas et al. 2011), bond failure was generated by the delamination of the resin and fiber in the bar. Based on a database of 68 pullout test results for GFRP bars in UHPFRC, Yoo et al. (2015b) suggested an equation for the relationship between normalized bond strength and development length by using regression analysis and by assuming no influence of the normalized cover parameter on bond strength, as follows (Fig. 2b):

$$ \frac{u}{{\sqrt {f_{c} '} }} = 1.05 + 0.85\frac{{d_{b} }}{{L_{e} }} $$

(2)

where u is the bond strength (=τ _max), d _b is the bar diameter, and L _e is the embedment length.

The American Concrete Institute (ACI) 440.1R model (ACI 2006) was inappropriate for UHPFRC; it significantly overestimated the test data (normalized bond strength), as shown in Fig. 2a.

Schäfers and Seim (2011) performed experimental and numerical investigations on the bond performance between timber and UHPFRC. The glued-laminated timber was bonded to sandblasted and ground UHPFRC with the “Sikadur 330” epoxy resin. Regardless of the bond length and surface treatment, most of specimens showed failure of the bond in the timber close to the bond-line. Based on the Volkersens theory, Schäfers and Seim (2011) suggested a bond length of 400 mm for standard test method to evaluate the bond strength of timber-concrete composites and noted that the effect of tensile stresses, orthogonal to the bond-line, can be neglected when the bond length is beyond 300 mm.

3.2 Flexural Dominated Reinforced UHPFRC Beams, Girders, and Composite Structures

Due to its excellent post-cracking tensile performance with multiple micro-cracks occurred, UHPFRC has attracted attention from engineers for application in structural elements subjected to bending. Several international recommendations (AFGC-SETRA, JSCE, and KCI) from France, Japan, and South Korea (AFGC-SETRA 2002; JSCE 2004; KCI 2012) thus provide stress–strain models for compressive and tensile stress blocks in the cross-section, as well as the detailed process of predicting the ultimate capacity of UHPFRC elements under flexure. Since strain (and stress) distribution in the cross-section varies according to the curvature of a beam, multilayer sectional analysis (Yoo and Yoon 2015) is required to calculate an appropriate neutral axis depth and moment capacity at a certain curvature level.

Yoo and Yoon (2015) first reported test results of a number of reinforced UHPFRC beams to investigate the effects of steel fiber aspect ratio and type on flexural performance. Since a portion of the tensile stress after cracking was resisted by the steel fibers, low reinforcement ratios (percent) of 0.94 and 1.50 % were selected. In order to prevent brittle shear failure, stirrups were conservatively designed based on the specimens made of ultra-high-performance concrete (UHPC) without fibers. From the test results (Fig. 3), the beams made by UHPFRC with 2 % by volume of steel fibers exhibited much higher post-cracking stiffness and ultimate load capacity, compared to those made by UHPC without fiber, called ‘NF’. In addition, the use of long straight or twisted steel fibers (S19.5, S30, and T30) led to a higher ductility than the use of short straight steel fibers (S13), which are applied for commercial UHPFRC available in North America (Graybeal 2008), at the identical fiber volume fraction. However, it is very interesting to note that much lower ductility indices were obtained by including steel fibers. This is caused by the fact that due to the very high bond strength between UHPFRC and steel rebar and its crack localization behavior, the steel rebar ruptured at a relatively smaller mid-span deflection, as compared with UHPC beams without fiber. Thus, Yoo and Yoon (2015) concluded that the strain-hardening behavior of UHPFRC was unfavorable to the ductility of reinforced beams.

In order to establish reasonable design codes for UHPFRC, Yang et al. (2010) carried out several four-point flexural tests for UHPFRC beams having reinforcement ratios less than 0.02. Test variables were the amount of steel rebar and the placement method. From their test results, placing concrete at the ends of the beams yielded better performance than when concrete was placed at the mid-length because of better fiber orientation to the direction of beam length at the maximum moment zone. In addition, they reported that all test beams showed a ductile response with the ductility index ranging from 1.60 to 3.75 and were effective in controlling cracks. However, the meaning of ‘ductile response’ could be incorrectly delivered to readers because no test results of reinforced UHPC beams without fiber were reported. In accordance with the test results by Yoo and Yoon (2015), reinforced UHPFRC beams exhibited lower ductility indices compared to beams without fiber due to the crack localization behavior, and Dancygier and Berkover (2016) also reported that the inclusion of steel fibers resulted in a decrease of flexural ductility of beams with low conventional reinforcement ratios.

Yang et al. (2011) examined the flexural behavior of large-scale prestressed UHPFRC I-beams. They indicated that the high volume content of steel fibers in UHPFRC effectively controlled the increase in crack widths, and led to multiple micro-cracks due to the fiber bridging at crack surfaces. The flexural strength of prestressed UHPFRC I-beams was insignificantly affected by the presence of stirrups. Graybeal (2008) also investigated the flexural behavior of a full-scale prestressed UHPFRC I-girder (AASHTO Type II girder) containing 26 prestressing strands. Based on the experimentally observed behavior, he reported that a UHPFRC I-girder shows larger flexural capacities than that of a conventional concrete girder with similar cross-sectional geometry. In addition, an inversely proportional relationship between crack spacing and maximum tensile strain was experimentally observed, as shown in Fig. 4, and the following equation was suggested:

$$ \varepsilon = 450 + \frac{2520}{{\sqrt {s_{cr} } }} + \frac{25{,}800}{{\sqrt {s_{cr}^{2} } }}\quad \left( {\text{in mm}} \right) $$

(4)

where ε is the tensile strain and s _cr is the crack spacing.

In recent years, several studies (Ferrier et al. 2015; Yoo et al. 2016) have been carried out to develop a new type of high-performance lightweight beams by applying UHPFRC and FRP rebar. Ferrier et al. (2015) investigated the structural behavior of I-shaped UHPFRC beams reinforced with CFRP and GFRP rebar, according to the rebar axial stiffness ranging from 9 MN to 30 MN. Experimental results indicated that the CFRP rebar was effective in increasing the bending stiffness, which results in a lower mid-span deflection, as compared with the case of the GFRP rebar due to the higher elastic modulus of the former. Thus, they concluded that the axial stiffness of the FRP reinforcement is the most influential parameter of bending stiffness of beams. Yoo et al. (2016) also examined the flexural behavior of UHPFRC beams reinforced with GFRP rebar and hybrid reinforcements (steel + GFRP rebar), according to the axial stiffness ranging from 13 to 95.5 MN. Hybrid reinforcements were considered in their study because it has been considered as one of the most promising methods to overcome the large service deflection problems of conventional FRP-reinforced concrete beams reported by several researchers (Lau and Pam 2010; Yoon et al. 2011). Due to the strain-hardening characteristics of UHPFRC, all tested beams provided very stiff load versus deflection response even after the formation of cracks (Fig. 5), which is distinctive response with conventional FRP-reinforced concrete beams, and satisfied the service crack width criteria of the Canadian Standards Association (CAN/CSA) S806 (CAN 2002). Furthermore, the deformability factors suggested by Jaeger et al. (1995) were higher than the lower limit (D _f  = 4) of CAN/CSA-S6 (CAN 2006) for all test beams. Therefore, it was noted that the use of UHPFRC could be a new solution for solving the major drawbacks limiting the practical application of FRP rebar instead of steel rebar. An increase in the GFRP reinforcement ratio led to an improvement in the flexural performance, such as higher post-cracking stiffness, load carrying capacity, and ductility. However, the application of hybrid reinforcements to UHPFRC nullified the main advantage of using FRP to solve the corrosion problem and showed insignificant improvement in the structural performance. Synthetically, Yoo et al. (2016) recommended the use of GFRP rebar with UHPFRC, rather than the use of hybrid reinforcements.

Ferrier et al. (2009) also examined the flexural behavior of a new type of hybrid beam, made of glued-laminated wood and UHPFRC planks, including steel and FRP rebar. They mention that structural efficiency was obtained by using the hybrid beams, as a consequence of the increased bending stiffness due to the high elastic modulus of UHPFRC planks. In addition, the inclusion of steel and FRP rebar in the lower UHPFRC plank significantly increased the ultimate load capacity of the hybrid beams, as compared with when only pure wood elements were used. These advantages of using hybrid beams lead to the potential for reducing the beam depth or increasing the span length of the beam, compared with conventional timber structures.

To practically apply UHPFRC in real architectural and civil structures, appropriate design technique should be suggested based on the material models. Several international recommendations (AFGC-SETRA 2002; JSCE 2004; KCI 2012) thus provide material models for designing flexural members made of UHPFRC. Based these recommendations, many researchers have already precisely predicted the flexural behaviors of reinforced UHPFRC beams (Yang et al. 2010; Yang et al. 2011; Ferrier et al. 2015; Yoo and Yoon 2015; Yoo et al. 2016). In particular, the UHPFRC beams without stirrups were well predicted by AFGC-SETRA recommendations without consideration of fiber orientation coefficient (K = 1) (Yang et al. 2011; Yoo et al. 2016) because the fiber alignment in the direction of beam length was insignificantly disturbed by the internal rebars. However, Yoo and Yoon (2015) recently reported that the fiber orientation coefficient that is proposed by the AFGC-SETRA recommendations (i.e., K = 1.25), should be considered for simulating the flexural behavior of reinforced UHPFRC beams with stirrups, since the fiber orientation was clearly disturbed by the stirrups (Fig. 6). In the case of FRP-reinforced concrete elements, it is well known that the service deflection prediction is the most important parameter for designing such structures, because of the larger service deflection than that of beams reinforced with steel rebar. Ferrier et al. (2015) and Yoo et al. (2016) successfully predicted the load versus deflection curves of FRP-bar-reinforced UHPFRC beams by sectional analysis, in which they considered compressive and tensile stress blocks in the cross-section, similar to the method used for the above steel-bar-reinforced beams. Yoo and Banthia (2015) also accurately predicted the service deflection of UHPFRC beams reinforced with GFRP rebar and hybrid reinforcements (steel + GFRP rebar), based on a micromechanics-based finite element (FE) analysis; the average ratios of the serviceability deflections from predictions and experiments were found to be 0.91 with a standard deviation of 0.07.

3.3 Shear Resistance of Structural UHPFRC Beams, Girders, and Bridge Decks

Baby et al. (2013b) carried out shear tests of eleven 3-m long UHPFRC I-shaped girders with various shear reinforcements (stirrups and/or steel fibers, or neither) combined with longitudinal prestressing or passive steel bars. To examine the actual fiber orientation effect on the shear performance, the three-point flexural tests were performed by using notched prism specimens extracted from both of the undamaged ends of I-girders at different inclination angles. Test results, as shown in Fig. 7, clearly indicated that the fiber orientation significantly influenced the mechanical (flexural) performance; thus, they noted that the actual fiber orientation needs to be taken into account for shear design, as recommended by AFGC-SETRA recommendations (AFGC-SETRA 2002). By including 2.5 % steel fibers, an almost 250 % increase in shear strength was observed (Baby et al. 2013c). The stirrups yielded first, while localization of the shear crack took place significantly later, as shown in Fig. 8. Thus, crack localization is primarily influenced by the strain capacity of the UHPFRC, and the contributions of the fiber bridging and the stirrups up to their yield strength seem to be effective only when the tensile strain capacity of the UHPFRC is much higher than the yield strain of the stirrups. In their study (Baby et al. 2013c), the AFGC-SETRA recommendations were conservative for the shear-cracking strength, but reasonable for the ultimate shear strength prediction of UHPFRC I-girders. Baby et al. (2013a) also examined the feasibility of applying the modified compression field theory for the shear capacity of reinforced or prestressed UHPFRC beams. Based on their analytical results, the modified compression field theory was determined to be applicable for predicting the shear behavior with an effective estimation of the reorientation of the compressive struts with an increase in the load.

Voo et al. (2010) investigated the shear strength of prestressed UHPFRC I-beams without stirrups, according to the shear span-to-depth ratio (a/d) and the type of steel fibers. They indicated that a higher shear strength was obtained by using a higher fiber volume content and a lower a/d. The theory of the plastic shear variable engagement model presented a good basis for their shear design and a good relationship to the experimental results; the ratio of shear strengths obtained from experiment and theory was found to be 0.92, with a coefficient of variation of 0.12. In addition, Bertram and Hegger (2012), Yang et al. (2012), and Tadepalli et al. (2015) mentioned that the shear strength increased with an increase in the fiber content and a decrease in the a/d ratio. For instance, the inclusion of 2.5 % steel fibers led to a 177 % higher ultimate load than that without fiber, and by changing the a/d ratio from 3.5 to 4.4, the shear capacity was reduced by 10 % (Bertram and Hegger 2012). Bertram and Hegger (2012) also noted that the size effect on shear strength was more substantially affected by the beam height as compared with the web thickness, and that about 12–14 % higher shear capacity was obtained when the effective prestressing force increased by 20 %. By comparing the test results with computed values, Yang et al. (2012) noticed that the predictions using the AFGC-SETRA and JSCE recommendations provided accurate estimates of the shear strength of UHPFRC I-beams (Fig. 9).

In order to replace the open-grid steel decks from moveable bridges, which have several drawbacks, such as poor rideability, high noise levels, susceptibility to fatigue damage, and high maintenance costs, Saleem et al. (2011) examined the structural performance of lightweight UHPFRC bridge decks reinforced with high-strength steel rebar. They properly designed and proposed UHPFRC waffle decks to satisfy the strength, serviceability, and self-weight requirements for moveable bridges. The governing failure mode was shear, and in the multi-unit decks, shear failure was followed by punching shear failure at close to the ultimate state. However, the shear failure was less abrupt and catastrophic as compared with the commonly seen shear failure mode. Thus, Xia et al. (2011) recommended the ductile shear failure with higher post-cracking shear resistance of UHPFRC beams containing high-strength steel rebar as an acceptable failure mode, rather than including transverse reinforcements, because of their economic problems. The use of 180° hooks at both ends of the steel rebar, recommended by ACI 318 (ACI 2014), was also effective in avoiding bond failure, compared with specimens without end anchorage. Based on a thorough analysis of the experimental results, Saleem et al. (2011) noted that although the proposed UHPFRC waffle deck system exhibits shear failure mode, it has great potential to serve as an alternative to open-grid steel decks, which are conventionally used for lightweight or moveable bridges.

3.4 Torsional Behavior of Structural UHPFRC Beams and Girders

Empelmann and Oettel (2012) examined the effect of adding steel fibers (v _f of 1.5 and 2.5 %) on the torsional behavior of UHPFRC box girders. They experimentally observed that the inclusion of steel fibers led to a better cracking performance such as smaller crack widths and multitudinous cracks, higher ultimate and cracking torque, and improved torsional stiffness. Interestingly, the angle of the diagonal cracks was found to be approximately 45° for all test series, regardless of the steel fiber contents. Yang et al. (2013) also investigated the torsional behavior of UHPFRC beams reinforced with mild steel rebars. In order to estimate the effects of steel fiber content and transverse and longitudinal rebar ratios, thirteen UHPFRC beams were fabricated and tested. Based on their test results (Yang et al. 2013), an improvement in the initial cracking and ultimate torque were obtained by increasing the fiber volume fractions (Fig. 10a), which is consistent with the findings from Empelmann and Oettel (2012). Moreover, higher ultimate torque was found with increases in the ratio of stirrups with longitudinal rebar (Figs. 10b and 11c). In addition, the torsional stiffness after initial cracking was also improved by increasing the ratio of stirrups, as shown in Fig. 10b. In contrast to Empelmann and Oettel’s findings, Yang et al. (2013) reported that the angle of the diagonal compressive stress ranged from 27° to 53°, and was affected by the number of stirrups and longitudinal rebar. For example, the angle of localized diagonal cracks increased with an increase in the number of stirrups, as illustrated in Fig. 11. Fehling and Ismail (2012) also reported similar test results for the torsional behavior of UHPFRC elements. They specifically said that the inclusion of steel fibers was effective in improving the torsional performance, such as cracking and ultimate torsional capacities, torsional ductility, post-cracking stiffness, and toughness. The use of longitudinal reinforcements and stirrups also obviously improved the torsional performance.

3.5 Performance of Structural UHPFRC Beams, Slabs, and Columns Under Extreme Loadings

Fujikake et al. (2006a) and Yoo et al. (2015a, c) examined the impact resistance of reinforced or prestressed UHPFRC beams by testing a number of specimens using a drop-weight impact test machine. In their studies (Fujikake et al. 2006a), an increase in the maximum deflection of UHPFRC beams was observed by increasing the drop height while maintaining the weight of the hammer, owing to the increase of kinetic energy. The initial stiffness in the UHPFRC beams was insignificantly affected by the impact damage because of the excellent fiber bridging capacities after matrix cracking, and the residual load–deflection (or moment–curvature) curves, shifted based on the maximum deflection by impact, exhibited quite similar behaviors with those of the virgin specimens without impact damage. Hence, Fujikake et al. (2006a) mentioned that the maximum deflection response can be used as the most rational index for estimating the overall flexural damage of reinforced UHPFRC beams. Yoo et al. (2015a) reported that better impact resistance, i.e., lower maximum and residual deflections and higher deflection recovery, was obtained by increasing the amount of longitudinal steel rebars, and the maximum and residual deflections of reinforced UHPFRC beams decreased significantly by adding 2 % (by volume) of steel fibers, leading to a change in the damage level from severe to moderate, whereas slight decreases in the maximum and residual deflections were found by increasing the fiber length at identical volume fractions (Yoo et al. 2015c). A higher ultimate load capacity was also obtained for the beams under impact loading, compared to those under quasi-static loading, and the residual load capacity after impact damage improved by including 2 % steel fibers and using the longer steel fibers. Fujikake et al. (2006a) and Yoo et al. (2015a) successfully predicted the mid-span deflection versus the time response of structural UHPFRC beams by using the sectional analysis and single- (or multi-) degree-of-freedom model. Improved mechanical compressive and tensile strengths according to the strain-rate were considered in the analysis by using the equations for the dynamic increase factor (DIF) of the UHPFRC, as suggested by Fujikake et al. (2006b, 2008).

Aoude et al. (2015) investigated the blast resistance of full-scale self-consolidating concrete (SCC) and UHPFRC columns under various blast-impulse combinations based on a shock-tube instrumentation. They verified that the steel bar-reinforced UHPFRC columns showed substantially higher blast resistance than the reinforced SCC columns in terms of reducing the maximum and residual deflections, enhancing damage tolerance, and eliminating secondary blast fragments. Based on the single-degree-of-freedom (SDOF) model and lumped inelasticity approach, Aoude et al. (2015) predicted the inelastic deflection-time histories. From the numerical results, several important findings were obtained as follows; (1) since the numerical predictions are sensitive to the choice of DIF, as given in Fig. 12a, further study needs to be done to develop the strain-rate models for using in the blast analysis of UHPFRC columns, and (2) the plastic hinge length (L _p ) seems to be reduced in UHPFRC columns from L _p  = d (column effective depth), which has been used for the analysis in conventional reinforced concrete columns, as shown in Fig. 12b. Astarlioglu and Krauthammer (2014) numerically simulated the response of normal-strength concrete (NC) and UHPFRC columns subjected to blast loadings based on SDOF models using the dynamic structural analysis suite (DSAS) and reported that the UHPFRC columns presented lower mid-span displacement and sustained more than four times the impulse as compared with the NC columns.

Wu et al. (2009) carried out a series of blast tests of NC and UHPFRC slabs w/ and w/o reinforcements to examine their blast resistance. When the similar blast loads were applied, the UHPFRC slabs without reinforcement exhibited less damage than the NC slabs with reinforcements, and thus, they noticed that the application of UHPFRC is effective in blast design. The UHPFRC slab with passive reinforcements was superior to all other slab specimens, and the strengthening of NC slabs with external FRP plates in the compressive zone was efficient in improving the blast resistance. Yi et al. (2012) examined the blast resistance of the reinforced slabs made of NC, ultra-high-strength concrete (UHSC), and RPC, which is identical to UHPFRC. By analyzing the crack patterns and maximum and residual deflections, they indicated that RPC has the best blast-resistant capacity, followed by UHSC and then NC. For example, the maximum deflections of NC, UHSC, and RPC slabs from 15.88 ANFO charge were found to be 18.57, 15.14, and 13.09 mm, respectively. Mao et al. (2014) investigated the capability of modeling the impact behavior of UHPFRC slabs using the commercial explicit FE program, LS-DYNA (2007). Through FE analysis, they also studied the effects of steel fibers and rebar on the blast resistance of UHPFRC slabs. Importantly, they observed that the K&C model (mostly used for simulating the blast behavior of concrete structures) with automatically generated parameters provided a much better ductile response than the actual behavior, and thus, a modified parameter b ₂ from 1.35 to −2 should be applied for UHPFRC. After verifying the numerical modeling with test data, a parametric study was carried out, and some useful results were obtained: (1) the additional use of steel fibers and rebar provide similar influence in the form of extra resistance to the UHPFRC panel under far field blast loading, and (2) under near field blast loading, the resistance of the UHPFRC panels increased substantially with steel rebar, as shown in Fig. 13.

4.1 AFGC-SETRA Recommendations

4.1.1 Material Models

In AFGC-SETRA recommendations, UHPFRC is referred to as a cementitious material with a compressive strength in excess of 150 MPa, possibly obtaining 250 MPa, and including steel (or polymer) fibers to provide a ductile tensile behavior. The parameters of the design strength were suggested based on the mechanical test results of Ductal^® (Orange et al. 1999), as follows: f _ck  = 150–250 MPa, f _tj  = 8 MPa, and E _c  = 55 GPa, where f _ck is the compressive strength, f _tj is the post-cracking direct tensile strength, and E _c is the elastic modulus. A partial safety factor γ _bf is also introduced, with γ _bf  = 1.3 in the case of fundamental combinations and γ _bf  = 1.05 in the case of accident combinations. To consider the fiber orientation effect on the tensile behavior, three different fiber orientation coefficients were suggested, as follows: K = 1 for placement methods validated from test results of a representative model of actual structure, K = 1.25 for all loading other than local effects, and K = 1.75 for local effects.

In the case of compressive model, a bilinear stress–strain model can be used, as shown in Fig. 14, with the design strength and strain parameters—σ _bcu  = 0.85f _ck /θγ_b and ε _u  = 0.003.

The tensile stress–crack opening displacement (σ–w) model is recommended to be first derived based on an inverse analysis. To apply the σ–w model into the tensile stress block in the cross-section, it needs to be transformed to the tensile stress–strain (σ–ε) model by using the characteristic length, l _c , which is l _c  = 2/3 × h for the case of rectangular or T-beams, where h is the beam height.

To obtain the tensile σ–ε model, the elastic tensile strain and strains at crack widths of 0.3 mm and 1 % of beam height need to be calculated based on the following equations:

$$ \varepsilon_{e} = \frac{{f_{tj} }}{{E_{c} }} $$

(5)

$$ \varepsilon_{0.3} = \frac{{w_{0.3} }}{{l_{c} }} + \frac{{f_{tj} }}{{\gamma_{bf} E_{c} }} $$

(6)

$$ \varepsilon_{1\% } = \frac{{w_{1\% } }}{{l_{c} }} + \frac{{f_{tj} }}{{\gamma_{bf} E_{c} }} $$

(7)

where ε _e is the elastic strain, w _0.3 is the crack width of 0.3 mm, ε _0.3 is the strain at crack width of 0.3 mm, w _1% is the crack width corresponding to 0.01H (H is the specimen height), and ε _1% is the strain at crack width corresponding to 0.01H.

The ultimate tensile strain is expressed by ε _lim = L _f /4l _c , where ε _lim is the ultimate tensile strain and L _f is the fiber length.

The stresses at two different characteristic points (at w _0.3 and w _1%) are expressed as follows:

$$ f_{bt} = \frac{{f\left( {w_{0.3} } \right)}}{{K\gamma_{bf} }} $$

(8)

$$ f_{1\% } = \frac{{f\left( {w_{1\% } } \right)}}{{K\gamma_{bf} }} $$

(9)

where f _bt is the stress at a crack width of 0.3 mm and f _1% is the stress at a crack width corresponding to 0.01H.

The completed material models under compression and tension are given in Fig. 14. Based on the capacity of tensile resistance with crack opening displacement, the AFGC-SETRA recommendations classify the tensile response by two different laws: (1) the strain-softening law (f _tj  > f _bt ) and (2) the strain-hardening law (f _tj  < f _bt ), as shown in Fig. 14.

Other important material properties of UHPFRC with heat treatment are given by AFGC-SETRA recommendations, as follows:

• Poisson’s ratio: 0.2.

• Thermal expansion coefficient (×10⁻⁶/°C): 11.0.

• Long-term creep coefficient: 0.2.

• Total (autogenous) shrinkage: 550 × 10⁻⁶.

4.1.2 Flexural Design

A chapter for the flexural design of UHPFRC structures is not included in the AFGC-SETRA recommendations. However, based on the suggested material models in Fig. 14, the design of a UHPFRC element subjected to bending can be performed by using the sectional analysis (Fujikake et al. 2006a; Yang et al. 2011; Ferrier et al. 2015; Yoo and Yoon 2015; Yoo et al. 2016). A schematic description of a multi-layered cross-section with strain and stress distributions and an algorithm for sectional analysis are shown in Fig. 15 (Yoo and Yoon 2015). The cross-section of the elements is first divided into a number of layers along the height. After that, the compressive and tensile stresses at each layer can be calculated by assuming that the plane section remains a plane at a given curvature. Then, the neutral axis depth can be calculated from the force equilibrium condition. Lastly, the moment is calculated. The calculation was repeated until the ultimate strain of the steel rebar was reached.

4.2 JSCE Recommendations

4.2.1 Material Models

In JSCE recommendations, UHPFRC is defined as the concrete with f _c ′ ≥ 150 MPa, f _crk  ≥ 4 MPa, and f _tk  ≥ 5 MPa, where f _c ′ is the compressive strength, f _crk is the cracking strength, and f _tk is the tensile stress at crack width of 0.5 mm. JSCE recommendations were proposed based on Ductal^® (Orange et al. 1999), which is commercially available UHPFRC with heat treatment and 2 vol.% of steel fibers having d _f  = 0.2 mm and L _f  = 15 mm, and provided the strength properties used for structural design as follows; f _c ′ = 180 MPa, f _crk  = 8 MPa, f _tk  = 8.8 MPa, and E _c  = 50 GPa. Importantly, they suggested a bilinear stress–strain curve for compression (Fig. 16a) and a bilinear tension-softening curve (TSC) with f _tk  = 8.8 MPa, w _1k  = 0.5 mm, and w _2k  = 4.3 mm for tension (Fig. 16b). In order to obtain the structural safety, γ _c  = 1.3 was proposed for the partial safety factor. To take into account the suggested TSC for tensile stress block in the cross-section, the crack opening displacement should be transformed to a strain by using the equivalent specific length, L _eq , as follows

$$ L_{eq} = 0.8h\left[ {1 - \frac{1}{{\left( {1.05 + {\raise0.7ex\hbox{${6h}$} \!\mathord{\left/ {\vphantom {{6h} {l_{ch} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${l_{ch} }$}}} \right)^{4} }}} \right] $$

(16)

where h is the overall depth of beam, l _ch is the characteristic length (=G _F E _c /f _tk ² ), and G _F is the fracture energy.

By using the equivalent specific length, the tensile stress–strain model is obtained by the following equations, based on the TSC:

$$ \varepsilon_{cr} = \frac{{f_{tk} }}{{\gamma_{c} E_{c} }} $$

(17)

$$ \varepsilon_{1} = \varepsilon_{cr} + \frac{{w_{1k} }}{{L_{eq} }} $$

(18)

$$ \varepsilon_{2} = \frac{{w_{2k} }}{{L_{eq} }} $$

(19)

where ε _cr is the factored elastic strain, ε ₁ is the strain at the end of the initial plateau, w _1k is the crack opening displacement for which a certain stress level is retained after the first crack, ε ₂ is the strain at zero tensile stress, and w _2k is the crack opening displacement at zero tensile stress.

Other material properties of the UHPFRC, proposed by the JSCE recommendations, are as follows:

• Poisson’s ratio: 0.2.

• Coefficient of thermal expansion (×10⁻⁶/°C): 13.5.

• Thermal conductivity (kJ/mh °C): 8.3.

• Thermal diffusivity (×10⁻³m²/h): 3.53.

• Specific heat (kJ/kg °C): 0.92.

• Total shrinkage: 550 × 10⁻⁶.

• Creep coefficient: 0.4.

• Density (kN/m³) for calculating dead load: 25.5.