Flexural Behavior and Design of Ultrahigh-Performance Concrete Beams

To conduct the experimental program described in this paper, one pretensioned bulb-tee bridge girder made with UHPC was tested under displacement-controlled loading in flexure to failure. The girder’s cross-sectional shape was based on the Precast Concrete Economical Fabrication (PCEF) Committee recommended sections, with geometrical modifications to reduce the thickness of the bottom flange and the width of the top flange. The standard drawings for the original PCEF sections can be found in the New Jersey DOT Design Manual for Bridges and Structures (NJDOT 2016). The cross-sectional dimensions and reinforcement details of the tested girder are shown in Fig. 2. The girder had a top and bottom flange width of 813 mm (32 in.), a web width of 178 mm (7 in.), and a height of 889 mm (35 in.); it was 18.90 m (62 ft) long and contained a total of 26 17.8-mm- (0.7-in.)-diameter straight steel strands, with 24 strands located in the bottom bulb and two strands in the top bulb, as indicated in Fig. 2. No mild steel reinforcement bars in the longitudinal or transverse directions were included in the girder. The 17.8-mm- (0.7-in.)-diameter strands were Grade 1860-MPa (270-ksi) low-relaxation, seven-wire strands [ASTM A416/A416M (ASTM 2018a)] with a cross-sectional area of $189.7{\mathrm{mm}}^2$ ($0.294{\mathrm{in.}}^2$) and a minimum ultimate strength of 1,860 MPa (270 ksi); they were pretensioned to 75% of their ultimate strength and placed at a minimum spacing of 50.8 mm (2 in.) center-to-center. The first, second, and third layers of strands were located at 50.8 mm (2 in.), 101.6 mm (4 in.), and 838 mm (33 in.) from the bottom of the section, respectively. The gross cross-sectional properties of the girder are an area of $3.72\times {10}^5{\mathrm{mm}}^2$ ($577{\mathrm{in.}}^2$), a moment of inertia about the centroid of $3.92\times {10}^{10}{\mathrm{mm}}^4$ ($9.42\times {10}^4{\mathrm{in.}}^4$), and a centroid located at 432 mm (17 in.) from the bottom of the section.

The UHPC product used in the girder was proprietary and was supplied in three primary constituents, namely, the preblended powder containing the granular constituents (e.g., cement, silica fume, ground quartz, and fine sand), liquid admixtures, and fibers. The mix design consisted of $\mathrm{2,182}\mathrm{kg}/{\mathrm{m}}^3$ ($\mathrm{3,678}\mathrm{lb}/{\mathrm{yd}}^3$) of preblended powder, $166\mathrm{kg}/{\mathrm{m}}^3$ ($280\mathrm{lb}/{\mathrm{yd}}^3$) of water, $73.4\mathrm{kg}/{\mathrm{m}}^3$ ($124\mathrm{lb}/{\mathrm{yd}}^3$) of liquid admixtures, and $157\mathrm{kg}/{\mathrm{m}}^3$ ($265\mathrm{lb}/{\mathrm{yd}}^3$) of brass-coated straight steel fibers (2% dosage by volume). The fibers had a length of 13 mm (0.51 in.), a diameter of 0.2 mm (0.0079 in.), and a supplier-reported minimum tensile strength of 2,600 MPa (377 ksi).

The specimen was fabricated at a precast plant in Florida using one of the plant’s central mixers. In the mixing process, all granular constituents were blended first, then the water and superplasticizers were added gradually. After the mixture turned from a powdered state into a viscous fluid, the fibers were dispersed into the mix. Before casting of each specimen, a flow table test was performed per ASTM C1856 (ASTM 2017) to ensure a flowable consistency corresponding to a spread equal to or greater than 203.2 mm (8 in.).

Two batches of UHPC of approximately $3.44{\mathrm{m}}^3$ ($4.50{\mathrm{yd}}^3$) were mixed to obtain the total volume of fresh UHPC needed to make the girder. The fresh UHPC of the first batch, Batch A, filled the bottom half of the girder’s rigid steel form, while the fresh UHPC of the second batch, Batch B, filled the top half of the form. There was no delay in placement either during or between batches. The placement of fresh UHPC was performed from one stationary discharge point located in the middle of the girder form. The material flowed to each end of the form as it filled the bottom bulb; it then rose upward through the web height before starting to flow again laterally as it filled the top bulb. The discharge point was then moved sequentially along the girder length to top off the filling of the form. Minor external vibration was applied to the sides of the form to facilitate the release of entrapped air. The girder was then covered with a plastic sheet and cured at outdoor ambient temperatures. The placement occurred during the month of June 2019 in which the average daily temperature ranged between 22°C (72°F) and 32°C (89°F). Five days after casting, the forms were removed and the strands were detensioned. The compressive strength required for detensioning was 96.5 MPa (14 ksi); the fabricator elected to wait until the average compressive strength was 130.4 MPa (18.9 ksi). The compressive strength was determined by performing compression tests on two cylindrical specimens (one specimen per batch) having a diameter of 76.2 mm (3 in.) and height of 152.4 mm (6 in.). The cylinders were tested by the precast plant quality control crew according to ASTM C39 (ASTM 2018b) with modifications listed in ASTM C1856 (ASTM 2017).

The 18.90-m- (62-ft)-long girder was tested in flexure with a span of 18.29 m (60 ft) at the Federal Highway Administration (FHWA) Turner-Fairbank Highway Research Center (TFHRC) structural testing facility. The age of the girder at the time of the test was 155 days after casting. An overview of the test setup is shown in Fig. 3. The girder was supported by a roller support at one end (west side) and by a partial pin support assembly bearing on a load cell and a hydraulic jack at the other end (east side); it was loaded in a four-point bending configuration with the middle reaction points each located at 0.46 m (1.5 ft) from the midspan, as shown in Fig. 4. The roller and partial pin supports each consisted of a 152-mm-(6-in.)-diameter steel cylinder inserted between two steel bearing plates having a thickness of 38.1 mm (1.5 in.), a width of 0.30 m (1 ft), and a length equal to the width of the bottom flange. The bearing plates of the pin support were grooved to prevent rolling, while the plates at the roller support were flat. To relieve frictional forces on the pin support generated by the loading jack line of action relative to the inclination of the bearing location, an additional steel bearing plate resting on a polytetrafluoroethylene (PTFE) sheet was included within the partial pin support assembly, as shown in Fig. 4. When the hydraulic jack applied load and deformation onto the east end of the girder, two reaction points near the midspan resisted the loading. These reaction points consisted of a pair of steel transfer plates reacting against the reaction frame through a set of three load cells with spherical bearing plates (two on the west side and one on the east side of the midspan), a spreader beam, and a greased spherical bearing, as shown in Fig. 3(b). Each of the middle transfer plates had a thickness of 102 mm (4 in.) and a width of 0.30 m (1 ft). All transfer and bearing plates were grouted to the girder’s top or bottom flanges and were long enough to fully support the width of the girder.

In addition to the four load cells, the girder was instrumented with six wire potentiometers (WPs) and 13 linear variable displacement transducers (LVDTs). The WPs (WP1–WP6) measured the vertical deflection at selected locations along the span, as shown in Fig. 4. The strain profile throughout the test was captured by mounting a pair of LVDTs, one at the top and one at the bottom of the girder, in five regions (Regions 1–5) covering the middle 2.44 m (8 ft) of the span where monitoring of flexural cracking was desired. Figs. 3(b) and 4 show the top (T1–T5) and bottom (B1–B5) LVDTs mounted on the girder in each of the five regions, with Region 3 being the constant moment region. These LVDTs measured the average longitudinal displacement that can be transformed into average strain by dividing the measured data by the LVDT gauge length. The top LVDTs had a gauge length of 610 mm (24 in.) and were mounted on one side of the top flange at a vertical distance below the top extreme fiber of 25.4 mm (1 in.) for LVDTs T1, T3, and T5, and of 76.2 mm (3 in.) for LVDTs T2 and T4. The bottom LVDTs (B1–B5) had a gauge length of 584 mm (23 in.) and were mounted on the bottom surface of the girder at a vertical distance of 50.8 mm (2 in.) below the girder. Finally, the slippage of the strands during the test was monitored through a set of three LVDTs mounted on three bottom-layer strands on the west end of the girder.

The load was applied by a hydraulic jack in 44.5-kN (10-kip) increments until the girder began to sustain inelastic damage and exhibit a reduced flexural stiffness. The loading was then switched to displacement control at a jack displacement rate of between $1.27\mathrm{mm}/\min$ ($0.05\mathrm{in.}/\min$) and $7.62\mathrm{mm}/\min$ ($0.3\mathrm{in.}/\min$). Periodically throughout the test, unloading/reloading cycles of 89 kN (20 kip) were performed to measure the residual stiffness of the girder. The loading continued until girder failure, which was defined as the formation of a dominant crack triggered by the pullout of the fiber reinforcement and the subsequent rupturing of the prestressing strands. Given that the girder deformation exceeded the stroke of the loading jack, the test was conducted by loading the girder in three separate test stages. At the end of each stage, the girder was completely unloaded to allow for the necessary preparations to reset the loading apparatus. The first test stage was halted when the load at the jack was 547 kN (123 kips) with a girder vertical displacement (WP6) of 216 mm (8.5 in.). The second stage was halted when the jack load was 812 kN (182.1 kips) with a girder vertical displacement (WP6) of 461 mm (18.1 in.). The third and last stage subjected the girder to a maximum jack load of 854 kN (192 kips), which occurred at a vertical displacement (WP6) of 517 mm (20.4 in.), on its way to complete failure (strand rupture) at a jack load of 761 kN (171 kips) and girder vertical displacement (WP6) of 610 mm (24.0 in.). Note that the stroke of the loading jack was reset once during the second test stage and twice during the third test stage. The reset was performed using additional hydraulic jacks to hold the applied load while the stroke of the loading jack was being retracted and spacer plates inserted between the jack and the girder.

A key objective of this study was to evaluate the predicted flexural response of the tested girder utilizing a proposed design methodology in which the mechanical properties had been obtained from independent material tests. For this reason, companion specimens were cast from each of the two batches used to make the girder to assess the UHPC’s density, compression, and tension properties at the time of the test. The material specimens were demolded at the time of detensioning of strands and stored in an ambient environment, generally alongside the girder, until the time of the test.

The average results of the material property tests are summarized in Table 1. The average density ${\overline{\omega }}_c$, modulus of elasticity ${\overline{E}}_c$, compressive strength ${\overline{f}}_c^{\prime }$, and strain at compressive strength ${\overline{\epsilon }}_{cu}$ of each UHPC batch were obtained from three cylindrical specimens tested according to ASTM C1856 (ASTM 2017). The cylindrical specimens had a diameter of 76.2 mm (3 in.) and a height of 152.4 mm (6 in.). The average tensile parameters, namely, the effective cracking stress ${\overline{f}}_{t,cr}$, the localization stress ${\overline{f}}_{t,\mathrm{loc}}$, and the localization strain ${\overline{\epsilon }}_{t,\mathrm{loc}}$, were obtained from prismatic specimens tested in uniaxial tension according to AASHTO T 397 (AASHTO 2022), which is based on the test method by Graybeal and Baby (2013). The tensile test captures the tensile load and associated strain over a 101.6-mm (4-in.) gauge length during a fixed-end, uniaxial displacement-controlled test. The prismatic specimens had a length of 432 mm (17 in.) and a square $50.8\mathrm{mm}\times 50.8\mathrm{mm}$ ($2\mathrm{in.}\times 2\mathrm{in.}$) cross section. The tensile parameters reported in Table 1 were obtained from the individual stress-strain curves of three and four prisms cast from Batches A and B, respectively. In determining the individual results, the effective cracking stress $f_{t,cr}$ was taken as the stress at the intercept of a line with a slope equal to the elastic modulus and a strain offset of 0.02% (Haber et al. 2018; El-Helou et al. 2022). The localization stress $f_{t,\mathrm{loc}}$ and strain, ${\epsilon }_{t,\mathrm{loc}}$, were visually determined as the first point in the stress-strain plot where the stress decreases continuously with increasing strain. This point indicates the onset of fiber pullout and the accumulation of the deformation into a single dominant crack. The average uniaxial stress-strain curves for each batch are shown in Fig. 5(a).

The stress-strain relationships for the prestressing strands were experimentally determined from six strand pullout tests performed according to ASTM A370 (ASTM 2020) and are plotted in Fig. 5(b). The strain data were continuously collected until a uniaxial strain of at least 0.03. The yield strength was measured at 1.0% extension under load and the total elongation value was determined by adding the 1.0% yield extension to the percent extension between the jaws gripping the strand at rupture, as specified by ASTM A416 (ASTM 2018a). The strands had an average yield strength ${\overline{f}}_{py}$ of 1,690 MPa (246 ksi), an average ultimate tensile strength ${\overline{f}}_{pu}$ of 1,932 MPa (281 ksi), and an average rupture strain ${\overline{\epsilon }}_{pu}$ of 0.0674. The coefficient of variation (COV) was 0.9% for ${\overline{f}}_{py}$ and ${\overline{f}}_{pu}$, and 5.1% for ${\overline{\epsilon }}_{pu}$. The strands modulus of elasticity $E_p$ was assumed to be 196.5 GPa (28,500 ksi). Based on these results, the strands conform to the mechanical property thresholds of ASTM A416 for Grade 1860 MPa (270 ksi) strands.

The applied moment versus deflection curves at the east and west point loads are shown in Fig. 6(a), with the unloading and reloading portions at the transitions between the three test stages removed for clarity. The presented applied moment $M$ is equal to the east and west load cells readings multiplied by the flexural span of 8.69 m (28.5 ft) and does not include the initial moment at the midspan $M_{\mathrm{ini}}$ induced by the self-weight of the girder and the loading apparatus (e.g., spreader beam, transfer plates, load cells). The initial moment was equal to 467 kN-m (345 kip-ft) and was calculated based on the difference in the load cell readings before and after the girder and loading apparatus were installed. The east and west reaction point deflections were calculated as the vertical displacement near the east and west load cells (WP2 and WP3) with respect to a reference line connecting the east and west supports. The data from the east and west load cells correlated well, as is shown in Fig. 6(a). The average curve in Fig. 6(a) represents the average applied moment versus the average deflection behavior of the constant moment region and was obtained by averaging the load and displacement results at the east and west reaction points throughout the test.

The average midspan moment-deflection response of the girder was linearly elastic until softening started to occur at an applied moment of approximately 3,655 kN-m (2,695 kip-ft) and a deflection of 65.9 mm (2.60 in.), indicating the initiation of flexural cracking in the girder. As the load continued to increase, the girder continued to sustain inelastic damage and exhibited a gradual decrease in the flexural stiffness. This behavior was likely due to the development of tight and closely spaced cracks within the tensile zone at the bottom half of the section. These cracks were periodically investigated by halting the loading protocol during the first two test stages, up to an applied moment of 6,920 kN-m (5,104 kip-ft), and by spraying the girder’s bottom flange and web within the constant moment region with an evaporative liquid (denatured alcohol). The liquid briefly penetrates the cracks while evaporating from the girder’s surface, temporarily making the cracks visible. However, none of the flexural cracks were visible to the naked eye during the first two test stages, indicating extremely tight cracks that were efficiently controlled by the fiber and strand reinforcements. The detection of cracks was not investigated during the third test stage due to safety concerns.

The flexural capacity of the girder was achieved at an applied moment of 7,271 kN-m (5,362 kip-ft) and a midspan deflection of 234 mm (9.21 in.). As the capacity was reached, a localized crack appeared at the midspan, initiating at the extreme bottom fiber and propagating swiftly up through the bottom half of the web. The localization of the crack resulted in an abrupt decrease of approximately 13.2% in the applied moment as shown in Fig. 6(a). A photo of the crack pattern at the midspan immediately after localization is shown in Fig. 7(a) (crack pattern shown for this reading is indicated by a star in Fig. 6). After localization, the girder began to hinge about the cross section at the localized crack. This behavior is unique to strain-hardening fiber-reinforced concrete beams, wherein: (1) the significant increase in curvature at the localized crack will contribute to a major portion of the overall deflection of the beam; and (2) the curvature at the nonlocalized cracks will be controlled by the unloading and reloading stiffness of these sections where the UHPC is exhibiting prelocalization behavior. The large deformations at the localized crack cross section locally strained the bottom strands until they ruptured, separating the girder into two pieces only connected by the top two strands. A photo of the girder after strand rupture is shown in Fig. 7(b). After the localization-induced applied moment reduction, the applied moment marginally increased with increasing displacement until the strands ruptured at an applied moment of 6,467 kN-m (4,769 kip-ft) and midspan deflection of 287 mm (11.3 in.).

The pair of top and bottom LVDTs that were installed in each of the five instrumented regions of the girders (Fig. 4) were used to create the average linear strain profiles in these regions throughout the duration of the test (assuming plane sections before bending remain plane after bending). Each strain profile was then utilized to determine the change in concrete strains at extreme top (compression) and bottom (tension) fibers within the respective region as a function of the average applied moment at the midspan, as shown in Fig. 6(b). The relationships in Fig. 6(b) confirm that, after the localized crack initiated within the constant moment region (Region 3) at the peak moment, the deformation of the cross section at the localized crack increased, as evidenced by the strain measurements at the top and bottom of the section, while the neighboring cracks closed as the applied load decreased. Note that the longitudinal deformation after the formation of a localized crack should be discussed in terms of crack opening rather than strain; the postpeak strain values of Region 3 in Fig. 6(b) reflect the observed longitudinal extensions within the gauge length.

The stress in the prestressing strands at the beginning of the flexural test, including all losses between strand release and time of test, was estimated by utilizing the concrete strains measured by four vibrating wire gauges (VWGs) cast into the midspan cross section. Two gauges were embedded between the top strands and two at the centroid of the bottom strands. The initial concrete strain profile was then constructed by comparing the concrete strains at the time of test to the reference strains taken prior to the detensioning of the strands. The difference in concrete strains at the level of each layer of strands was then determined assuming a perfect bond between the strands and the surrounding concrete. When these strains are multiplied by the modulus of elasticity of the strands, $E_p=196.5\mathrm{GPa}$ (28,500 ksi), the difference in the stresses in the strands between jacking and the time of test can be estimated. The effective prestress in the first (top), second, and third (bottom) layers of strands at the midspan at the start of the test was calculated to be $-\mathrm{1,336}\mathrm{MPa}$ ($-193.7\mathrm{ksi}$), $-\mathrm{1,157}\mathrm{MPa}$ ($-167.8\mathrm{ksi}$), and $-\mathrm{1,144}\mathrm{MPa}$ ($-166.0\mathrm{ksi}$), corresponding to strains of ${\epsilon }_{p1,\mathrm{ini}}=-0.00680$, ${\epsilon }_{p2,\mathrm{ini}}=-0.00589$, and ${\epsilon }_{p3,\mathrm{ini}}=-0.00582$, respectively (tension taken as a negative value).

The initial stress profile in the girder at the beginning of the test was obtained from an elastic analysis of the cross section at the midspan and using a concrete modulus of elasticity of 43.6 GPa (6,317 ksi), taken as the average of the two elastic modulus values reported for each batch (Table 1). The initial stresses at the midspan were ${\sigma }_{T,\mathrm{ini}}=0.94\mathrm{MPa}$ (0.136 ksi) at the extreme top fiber (compression taken as a positive value) and ${\sigma }_{B,\mathrm{ini}}=29.7\mathrm{MPa}$ (4.31 ksi) at the extreme bottom fiber, corresponding to strains of ${\epsilon }_{T,\mathrm{ini}}=0.000022$ and ${\epsilon }_{B,\mathrm{ini}}=0.000682$, respectively.

A summary of the test results at the midspan, including the effect of the prestressing force and considering the self-weight of the girder and loading apparatus, is calculated at the five points in flexural behavior, namely, the initiation of cracks; service stress limit of prestressing steel taken when the stress in the bottom layer of strands is equal to 80% of the strand yield stress (AASHTO 2020), i.e., $f_{p3}=0.80{\overline{f}}_{py}$; yielding of the bottom layer of strands, i.e., $f_{p3}={\overline{f}}_{py}$; crack localization; and strand rupture as presented in Table 2. In Table 2, $M_{\exp }$ is total moment corresponding to applied moment $M$ plus the self-weight moment $M_{\mathrm{ini}}$, as described in Eq. (1); ${\epsilon }_T$ and ${\epsilon }_B$ are the total concrete strains in the top and bottom layers calculated according to Eqs. (2) and (3), in which $\Delta {\epsilon }_T$ and $\Delta {\epsilon }_B$ are the change in strains captured in extreme top and bottom fibers during the test within Region 3; ${\epsilon }_{p3}$ is the strain in the bottom layer of strands calculated according to Eq. (4), in which ${\epsilon }_{p3,\mathrm{ini}}=-0.00582$ is the initial strain in the bottom layer of strands and $\mathrm{\Delta }{\epsilon }_{cp3}$ is the change in strain in the concrete at the level of the bottom layer of strands during the test within Region 3; $X$ is the neutral axis depth calculated from the strain profile formed by ${\epsilon }_T$ and ${\epsilon }_B$; and $\psi ={\epsilon }_T/X$ is the sectional curvature at the midspan. The 80% of yield and yield moment values reported in Table 2 and Fig. 6 were determined when ${\epsilon }_{p3}$ reached a strain equal to -0.00705 (corresponding to $0.80f_{py}$) and the yielding strain of the strands, ${\epsilon }_{py}=-0.01$, respectively

Finally, the slippage of selected strands during the test was monitored with three LVDTs mounted on three of the bottom-layer strands located within one half of the section on the west end of the girder. As expected with a beam of this length, these measurements demonstrated that no strand slippage occurred during the test.

A key starting point for the proposed flexural design framework is the delineation of the scope of the considered UHPC-class materials and the definition of the fundamental parameters. UHPC is generally defined as portland cement composite made of an optimized gradation of granular constituents, a water-to-cementitious materials ratio less than 0.25, and a high percentage of discontinuous internal steel fiber reinforcement. The specific mechanical properties include a minimum compressive strength of 124 MPa (18.0 Ksi), a minimum cracking strength of 5.0 MPa (0.73 Ksi), and the ability to sustain the cracking strength through a minimum localization strain of at least 0.0025.

The compressive properties, namely, the modulus of elasticity $E_c$, the compressive strength $f_c^{\prime }$, and the strain at compressive strength ${\epsilon }_{cu}$, are obtained from cylindrical specimens tested according to ASTM C39 (ASTM 2018b) and ASTM C469 (ASTM 2014) with provisions specific to UHPC described in ASTM C1856 (ASTM 2017). The uniaxial elastic-perfectly plastic model proposed by El-Helou et al. (2022) is adopted in this framework to determine the stress in the UHPC as a function of the compression strain in the section, as shown in Fig. 8(a). This model mimics the experimental uniaxial stress-strain response in the elastic region up to a reduced compressive stress of $\alpha f_c^{\prime }$, where $\alpha$ is a reduction factor on compressive strength reflecting the linearity limit of the compressive stress-strain response. After this stress level, the model sustains the reduced compressive resistance until the strain at the material’s compressive strength ${\epsilon }_{cu}$ is reached. El-Helou et al. (2022) suggested a reduction factor $\alpha$ value of 0.85, based on the experimental results of five commercially available UHPC products (Graybeal and Stone 2012; Haber et al. 2018).

The tension parameters, namely, the effective cracking strength $f_{t,cr}$, the localization stress $f_{t,\mathrm{loc}}$, and the localization strain ${\epsilon }_{t,\mathrm{loc}}$, are obtained from prismatic specimens tested in direct tension according to the method developed by Graybeal and Baby (2013). This method delivers the uniaxial stress-strain responses of the tested specimens, facilitating the identification of critical tension parameters. The effective cracking strength $f_{t,cr}$, is taken as the intercept of a line having a slope equal to the elastic modulus and a strain offset of 0.02%. The localization stress $f_{t,\mathrm{loc}}$ and strain ${\epsilon }_{t,\mathrm{loc}}$ are taken as the stress and strain of the data point where the stress begins to continuously decrease with increasing strain. As proposed by El-Helou et al. (2022), the constitutive law for UHPC in tension idealizes the experimental curve into an elastic-perfectly plastic relationship, as shown in Fig. 8(b), for material exhibiting a stress plateau after cracking, or a bilinear relationship, as shown in Fig. 8(c), for material exhibiting a continuous increase in stress after cracking and when the localization stress $f_{t,\mathrm{loc}}$ is at least 20% greater than the effective cracking stress $f_{t,cr}$. El-Helou et al. (2022) proposed that a reduction factor $\gamma$ be applied on the tensile stress parameters ($f_{t,cr}$ and $f_{t,\mathrm{loc}}$) to account for variability in the material behaviors; the value of the reduction factor $\gamma$ is not to exceed 0.85.

For the conventional steel reinforcement, an elastic-perfectly plastic stress-strain relationship is adopted, as shown in Fig. 9(a), in which $E_s$ is the modulus of elasticity of conventional steel; $f_{sy}$ and ${\epsilon }_{sy}$ are the yielding stress and strain, respectively; and ${\epsilon }_{su}$ is the strain at rupture of the bars. The idealized stress-strain relationships for prestressing strands can be obtained from existing nonlinear models, such as the power formula based on the work of Skogman et al. (1988) shown in Fig. 8(b). For instance, the stress in the prestressing steel $f_{ps}$ for 1,860 MPa (270 ksi) seven-wire low-relaxation strands can be computed as a function of the steel strain ${\epsilon }_{ps}$ according to Eq. (5). To obtain $f_{ps}$ in kilopounds per square inch in Eq. (5), replace 6,116 MPa with 887 ksi, 190,385 MPa with 27,613 ksi, and 1,860 MPa with 270 ksi (PCI 2014)

The flexural behavior of UHPC beams can be obtained analytically by employing a strain compatibility approach and utilizing the constitutive models for UHPC and steel reinforcement depicted in Figs. 8 and 9, respectively. For flexural members fully utilizing the UHPC tensile resistance, the nominal moment–sectional curvature ($M_n-\psi$) diagram can be idealized in four key points, as illustrated in Fig. 10, for a UHPC member with conventional steel reinforcement. The sectional curvature is defined as the ratio of the strain in the extreme compression layer ${\epsilon }_c$ divided by the depth of the neutral axis $X$: $\psi ={\epsilon }_c/X$. The first key point in the behavior (${\psi }_{cr}$, $M_{cr}$) represents the initiation of the first flexural tensile crack, occurring when the strain in the extreme tension layer reaches the cracking strain limit of UHPC, ${\epsilon }_{t,cr}=f_{t,cr}/E_c$. At this point, the compression and tensile stresses remain within the elastic region of their respective constitutive models (Fig. 8). The second point in the moment-curvature diagram (${\psi }_y$, $M_y$) coincides with the yielding of the extreme tension layer of the reinforcing steel, occurring when the steel strain is equal to the yielding strain of conventional steel ${\epsilon }_{sy}$ or prestressing strands ${\epsilon }_{py}$. Note that in Fig. 10, an intermediate point (${\psi }_{sl}$, $M_{sl}$) in behavior is shown corresponding to the chosen baseline sectional curvature and moment for ductility considerations as discussed later in the manuscript. The third key point (${\psi }_L$, $M_L$) indicates the flexural nominal capacity of the section ($M_n=M_L$) and coincides with the crack localization, where the tensile capacity of UHPC is fully utilized and the strain in the extreme tension layer is equal to the localization strain ${\epsilon }_{t,\mathrm{loc}}$. The cross-sectional strain and stress diagrams at the flexural crack localization point for a UHPC material exhibiting a stress plateau after cracking [Fig. 8(b)] are shown in Fig. 11, where the compression stresses remain elastic with the strain value in the extreme compression layer ${\epsilon }_c$, less than ${\epsilon }_{cp}=\alpha f_c^{\prime }/E_c$ [Fig. 11(a)], or become plastic when ${\epsilon }_{cp}<{\epsilon }_c\le {\epsilon }_{cu}$ [Fig. 10(b)]. In Fig. 10, it is assumed that the localization strain of UHPC ${\epsilon }_{t,\mathrm{loc}}$ is greater than the yielding strain of the reinforcement ${\epsilon }_{sy}$, and thus the yielding point (${\psi }_y$, $M_y$) occurs at a smaller curvature and moment values than those defining the localization point (${\psi }_L$, $M_L$). For prestressed members, the yielding point occurs before the localization point if the change in the prestressing steel strain between the effective prestress and yield (${\epsilon }_{py}-{\epsilon }_{pe}$) is greater than the UHPC localization strain ${\epsilon }_{t,\mathrm{loc}}$. After crack localization (${\epsilon }_t>{\epsilon }_{t,\mathrm{loc}}$), the UHPC can no longer contribute to the tension resistance of the member, leading to a reduction in the flexural capacity. The last point in the flexural behavior shown in Fig. 10 (${\psi }_c$, $M_c$) coincides with the compression failure of UHPC, occurring when the strain in the extreme compression layer reaches the ultimate compressive strain of UHPC ${\epsilon }_{cu}$. The cross-sectional strain and stress diagrams at compression failure and after the localization of cracks (${\epsilon }_t>{\epsilon }_{t,\mathrm{loc}}$) are depicted in Fig. 12 for a UHPC material exhibiting a stress plateau after cracking [Fig. 8(b)]. The shape of the postlocalization moment-curvature relationship is nonlinear (dashed line in Fig. 10) and depends on the UHPC material properties and the design parameters of the section. Intermediate points between localization and compression failure need to be computed if the exact shape of the postlocalization curve is desired.

In the determination of the moment-curvature diagram in Fig. 10, it is assumed that the strain in the steel reinforcement ${\epsilon }_s$ remains less than the strain at rupture of the bars ${\epsilon }_{su}$ when the compressed UHPC crushes. However, due to the high compressive strength of UHPC materials, the rupture of the steel reinforcement (conventional steel or prestressing strands) can occur before concrete crushing (i.e., when the strain in the extreme compression layer is less than ${\epsilon }_{cu}$). In an effort to engage the full compressive strength of UHPC in a flexural design, Shao and Billington (2019) demonstrated that the postlocalization moment capacity of UHPC beams can be carried through high-tensile strains until compression crushing ($M_c\ge M_L$) when the beam is heavily reinforced with conventional steel reinforcement and when the strain-hardening characteristics of the steel are both engaged and substantial. While such behavior is desirable because it significantly increases the ductility of UHPC beams, to date the approach has only received limited experimental investigation. When the tensile strain in the UHPC exceeds the localization strain at a crack (${\epsilon }_t>{\epsilon }_{t,\mathrm{loc}}$), the loss of the UHPC fiber-bridging resistance mechanism results in a change in the internal force-resisting mechanism (i.e., increased curvature, increased strains, and movement of the neutral axis). If the increased tensile strain results in increased tensile resistance due to the engagement of additional discrete reinforcement force-resisting mechanisms, such as the strain-hardening of steel or the further engagement of unbonded reinforcements, then a new stable moment-resisting state may be realized. For this reason, postlocalization analysis should take into consideration the width of the localized crack, the local straining of the reinforcement crossing the crack, and the resistance provided by the reinforcement. In the current proposal, the postlocalization flexural capacity is not recommended for use in design because more research is needed to quantify the relationship between the rupture strain of the bars after UHPC crack localization, the opening of the localized crack, and the development length of the reinforcement in the vicinity of the localized crack. The development length of the reinforcement is a critical consideration as it significantly influences the length of the bar or strand over which the increased tensile strain demand will be resisted.

For flexural members not fully utilizing the UHPC tensile resistance, the ultimate capacity occurs at compression crushing ($M_n=M_c$), when the UHPC in the extreme compression layer reaches its ultimate compressive strain limit (${\epsilon }_c={\epsilon }_{cu}$) while the strain in the extreme tensile fiber is less than the localization strain (${\epsilon }_t<{\epsilon }_{t,\mathrm{loc}}$). In these situations, the beam design does not take full advantage of the UHPC tensile performance characteristics, prompting a compression failure at reduced tensile strain values, such as could occur in sections having a very high reinforcement ratio and/or sections subjected to high axial compressive loads. The nominal flexural moment–sectional curvature ($M_n-\psi$) for this failure mode can be defined by straight lines, with either (1) three key points coinciding with the initiation of first flexural tensile crack (${\psi }_{cr}$, $M_{cr}$), yielding of the extreme tension layer of the reinforcing steel (${\psi }_y$, $M_y$), and the compression crushing of UHPC (${\psi }_c$, $M_c$) when the steel strain at crushing is greater than the yield strain; or (2) two key points coinciding with the initiation of cracks (${\psi }_{cr}$, $M_{cr}$) and compression crushing (${\psi }_c$, $M_c$) when the steel strain at crushing is less than or equal to the yield strain.

Based on the experimentally observed behaviors and analytical results, a flexural design framework for UHPC beams reinforced with conventional steel or prestressing strands is proposed in this section. The design framework is similar to portions of existing UHPC structural design guidance detailed in various international documents [NF P18-710 (AFNOR 2016a); SIA 2052 (SIA 2016); CSA S6:19 (CSA 2019)] and is tailored to generally parallel existing bridge design specifications in the United States (AASHTO 2020) while accounting for UHPC mechanical performance.

The first step in the structural design is the characterization of the material-level properties and the establishment of constitutive laws that describe the mechanical response when subjected to structural loads. For UHPC in compression, the parabolic stress-strain relationship typically used for conventional concretes is replaced with the idealized elastic-perfectly plastic model in Fig. 8(a). The compression model requires the modulus of elasticity $E_c$, compressive strength $f_c^{\prime }$, and strain at compressive strength ${\epsilon }_{cu}$ as input parameters, which can be experimentally obtained from compression tests performed according to ASTM C1856 (ASTM 2017). In lieu of experimental testing, the elastic modulus can be approximated by Eq. (6) and ${\epsilon }_{cu}$ can be taken as the greater of ${\epsilon }_{cp}=\alpha f_c^{\prime }/E_c$ and 0.0035, as proposed by El-Helou et al. (2022). The reduction factor $\alpha$ can be taken as 0.85

In tension, UHPC offers a postcracking resistance that is sustained through large tensile strains (${\epsilon }_t>0.0025$), delivering a beneficial contribution to the flexural capacity of a beam. The tensile constitutive models for UHPC can be idealized with the elastic-perfectly plastic or bilinear (when $f_{t,\mathrm{loc}}\ge 1.20f_{t,cr}$) relationships shown in Figs. 8(b) and 7(c), respectively. The tensile parameters, namely, the effective cracking stress $f_{t,cr}$ and the localization stress $f_{t,\mathrm{loc}}$ and strain ${\epsilon }_{t,\mathrm{loc}}$, can be obtained by executing direct tension tests proposed by Graybeal and Baby (2013).

The predicted moment-curvature behavior of the prestressed girder tested by the authors, and previously discussed in this paper, was obtained by discretizing its cross section into 2.5-mm-(0.1-in.)-thick horizontal layers through the height and assuming constant strains within each layer. The initial strains in each layer at the beginning of the test were determined from the UHPC strain measured by the VWGs embedded at the midspan and following the analytical considerations described previously. That is, the initial strain in the first (top), second, and third (bottom) layer of strands is equal to ${\epsilon }_{p1,\mathrm{ini}}=-0.00680$, ${\epsilon }_{p2,\mathrm{ini}}=-0.00589$, and ${\epsilon }_{p3,\mathrm{ini}}=-0.00582$, respectively (tension taken as a negative value). The initial strain in each layer of UHPC was calculated from the initial strain profile defined by the top strain, ${\epsilon }_{T,\mathrm{ini}}=0.000022$, and bottom strain, ${\epsilon }_{B,\mathrm{ini}}=0.000682$ (compression taken as a positive value). The total strain in the UHPC section was obtained by superposing the initial strain profile with an assumed strain profile, i.e., ${\epsilon }_i={\epsilon }_{i,\mathrm{ini}}+\mathrm{\Delta }{\epsilon }_i$, in which ${\epsilon }_i$, ${\epsilon }_{i,\mathrm{ini}}$, and $\mathrm{\Delta }{\epsilon }_i$ are the total, initial, and assumed strains in the UHPC layer $i$, as shown in Fig. 13. Similarly, the total strains in the strand layers, ${\epsilon }_{p1}$, ${\epsilon }_{p2}$, and ${\epsilon }_{p3}$, were calculated by superposing the initial steel strains, ${\epsilon }_{p1,\mathrm{ini}}$, ${\epsilon }_{p2,\mathrm{ini}}$, and ${\epsilon }_{p3,\mathrm{ini}}$, with the change in strain in the UHPC at the level of each steel layer, $\mathrm{\Delta }{\epsilon }_{cp1}$, $\mathrm{\Delta }{\epsilon }_{cp2}$, and $\mathrm{\Delta }{\epsilon }_{cp3}$, respectively. The stresses in each layer were calculated by employing the mechanical models shown in Figs. 8(a and b) and 9(b). The stresses were then converted into compression and tension forces, which were summed to check force equilibrium in the section. This process was repeated by assuming different strain profiles until equilibrium was satisfied, after which the flexural moment was computed. The procedure was repeated by increasing the values of assumed strains until the full moment–sectional curvature diagram was derived.

In calculating the UHPC stresses, the compression and tension parameters were informed from the average results of the material tests performed on companion specimens presented in Table 1. The UHPC modulus of elasticity was taken as the average of the results obtained from the two UHPC batches with $E_c=43.6\mathrm{GPa}$ (6,317 Ksi). The compressive strength and ultimate compression strain were taken from the results of Batch B, which filled the top half of the girder: $f_c^{\prime }=161\mathrm{MPa}$ (23.3 ksi) and ${\epsilon }_{cu}=0.00401$. The tensile parameters were taken from the results of Batch A, which filled the bottom half of the girder, and without a reduction factor (i.e., $\gamma =1.00$): $\gamma f_{t,cr}=\gamma f_{t,\mathrm{loc}}=-9.3\mathrm{MPa}$ ($-1.35\mathrm{ksi}$) [elastic-perfectly plastic model in Fig. 8(b)] and ${\epsilon }_{t,\mathrm{loc}}=-0.00497$. The tensile design model is compared to the uniaxial tensile test results in Fig. 5(a). The steel stress-strain model relationship for the 1,860 MPa (270 ksi) low-relaxation strands was determined according to Eq. (5) and depicted atop the experimental data from the strand pullout tests in Fig. 5(b). For strand stress values greater than 1,860 MPa (270 ksi) or strain values greater than 0.0272, a linear interpolation was implemented up to a rupture stress $f_{pu}$ of 1,932 MPa (281 ksi) occurring at a strain ${\epsilon }_{pu}$ of $-0.0674$, as shown in Fig. 5(b). Table 3 presents a summary of the strain compatibility analysis results at the five points in flexural behavior, namely, the initiation of cracks; strand service stress limit, i.e., $0.80f_{py}$, reached in the bottom layer of strands; yielding of the bottom layer of strands; crack localization; and compression crushing. These results represent the nominal design capacities of the girder and can be directly compared to the experimental data summarized in Table 2. The predicted-over-experimental flexural moment ratio, $M_n/M_{\exp }$, is 0.97, 1.09, 0.99, and 0.92 at first crack, 80% of yield, yield, and localization, demonstrating the capability of the proposed methodology to predict the flexural capacity of prestressed UHPC beams. The predicted nominal design curve is compared to the experimental results in Fig. 14, plotted as the relationship between the applied moment and the change in curvature during the test and simulation. The simulated design and experimental moments in Fig. 14 are taken as $M_n-M_{\mathrm{ini}}$ and $M_{\exp }-M_{\mathrm{ini}}$, respectively, where $M_{\mathrm{ini}}=467\mathrm{kN}-\mathrm{m}$ (345 kip-ft) is the initial moment in the beam at the beginning of the test. The change in curvature $\mathrm{\Delta }\psi$ is taken as $\mathrm{\Delta }{\epsilon }_T/\mathrm{\Delta }X$, where $\mathrm{\Delta }X$ is the depth of the neutral axis of the intermediate strain diagram shown in Fig. 13, a value calculated in the strain compatibility analysis and measured during the test. Note that the simulated design curve was obtained from 110 calculation points, which is greater than the minimum four key points proposed in the design methodology.

The experimental and predicted design responses in Fig. 14(a) are in good agreement, demonstrating the capability of the proposed method to predict the full moment–sectional curvature response of prestressed beams. The results also confirm that the shape of the moment–sectional curvature diagram can be idealized by four key points, as illustrated in Fig. 10. Because the compression strain limit observed during the test was approximately 20% larger than the ultimate compression strain specified in the simulations, the predicted failure mode of the girder was compression crushing instead of strand rupture, as observed in the test [Table 3 and Fig. 14(a)]. This result suggests that the UHPC in a structural element can undergo compression strains exceeding the strain at peak stress, as obtained from a uniaxial compression test, likely due to the confinement effects within the larger beam and strain gradient effect within a beam cross section with large curvature. Similarly, the localization strain of UHPC observed during the test (${\epsilon }_B=-0.00618$ in Table 2 at localization point) was approximately 24% greater than the localization strain obtained from direct tension testing [${\epsilon }_B={\overline{\epsilon }}_{t,\mathrm{loc}}=-0.00497$ in Table 1 (Batch A) and Table 3 at localization point], indicating that the primary tensile reinforcement and strain gradient effects may enhance the flexural tensile strain capacity of the UHPC.

The design curvature ductility ratio of the girder, ${\mu }_{\mathrm{sim}}=3.50$, can be calculated by dividing the sectional curvature at localization over the sectional curvature at 80% of yield reported in Table 3. Because the ductility ratio is greater than the specified ratio ${\mu }_{\min }$ of 3.0, the flexural resistance factor for this girder is equal to 0.90. Note that the experimental ductility ratio obtained from the results reported in Table 2 is greater than ${\mu }_{\mathrm{sim}}$ with ${\mu }_{\exp }=4.72$ because the cracks in the girder localized at a higher localization strain than the one used in the simulations. The factored moment-curvature design curve is compared to the nominal and experimental curves in Fig. 14(a).

To illustrate the key differences between the behavior of prestressed conventional concrete beams and similar UHPC beams, a strain compatibility analysis of the same girder was performed ignoring the tensile resistance of UHPC, i.e., $f_{t,cr}=f_{t,\mathrm{loc}}=0$, with the results plotted in Fig. 14(a). While this analysis resulted in a nominal moment capacity ($M_n=M_c$) lower than the experimental capacity, it both incorrectly predicted a concrete crushing failure mode at the top of the section, rather than localization of cracks at the bottom, and also significantly overestimated the curvature ductility ratio ${\mu }_{nt,\mathrm{sim}}$ compared to the experimental ductility ratio, i.e., ${\mu }_{nt,\mathrm{sim}}=2.67{\mu }_{\exp }$. This comparison demonstrates that conducting a capacity-based design that ignores the UHPC tensile resistance may inadvertently result in a structure that does not demonstrate the desired ductility.

Finally, if the localization strain measured during the girder test, ${\epsilon }_{t,\mathrm{loc}}=-0.00636$, is used in the model, the simulated nominal design curve is identical to the prelocalization experimental moment-curvature curve, as shown in Fig. 14(b), when the bilinear constitutive law in Fig. 8(c) is employed with the following calibrated input values for the cracking and localization stresses: $f_{t,cr}=-6.21\mathrm{MPa}$ (0.90 ksi) and $f_{t,\mathrm{loc}}=-15.5\mathrm{MPa}$ ($-2.25\mathrm{ksi}$). In this analysis, the estimated value of the cracking stress in the girder was approximately 33% lower than the one obtained from direct tension testing, while the localization stress was 49% higher than the localization stress value obtained from tension testing (Batch A in Table 1).

This section explores the experimental results from tests of two additional prestressed girders and 42 non-prestressed reinforced beams found in the literature and compares them to model predictions. The beams had varying cross-sectional shapes and geometries (rectangular shapes for non-prestressed, and I and pi shapes for prestressed). The height of the non-prestressed rectangular beams ranged between 220 mm (8.7 in.) and 381 mm (15 in.), while the height of the prestressed I- and pi-shaped beams were 838 mm (33 in.) and 914 mm (36 in.). The beams were made with different types of UHPC products with varying amounts of fiber volume content (1.0%, 1.5%, and 2%) and prestressing or conventional steel reinforcement (reinforcement ratio $\rho$ ranging between 0% and 4.99%). The fibers varied in shape and geometries, including smooth, twisted, straight, and hooked-end, with diameters and lengths ranging between 0.12 and 0.30 mm (0.0047 and 0.012 in.) and 10.0 and 30.0 mm (0.39 and 1.18 in.), respectively. The reported flexural behaviors of the beams are in line with the flexural behavior of UHPC members described previously in this paper: the maximum flexural moment was reached immediately before a single localized crack—initiated by the pullout of the crack-bridging fibers—occurred, followed by the subsequent rupture of the tensile reinforcement. Therefore, the analytical predictions presented in this section focus on determining the capacity of the beams at the onset of localization, as recommended in the proposed design methodology that is developed in this paper. The predicted results for the beams found in the literature are presented in Table 4. A portion of the input material parameters for the UHPC and reinforcing steel given in Table 4 were obtained from independent material tests that accompanied each beam, i.e., simulations of beams tested by Chen et al. (2018), Yoo et al. (2017), Qiu et al. (2020a), Yoo and Yoon (2015), Graybeal (2006b), and Graybeal (2009). The uniaxial tensile parameters ($\gamma f_{t,cr}$, $\gamma f_{t,\mathrm{loc}}$, and ${\epsilon }_{t,\mathrm{loc}}$ with $\gamma =1.00$) for the UHPC in these beams were obtained from uniaxial testing methods or inverse analysis techniques performed on the results of flexural beam tests. In cases where key material parameters were not reported, they were obtained by fitting the model results to the experimental data from one of the tested beams in a set, then comparing the predicted flexural capacities of the remaining beams within the same set to the experimental results without changing the estimated input parameters, i.e., simulations of beams tested by Yang et al. (2010), Meade and Graybeal (2010), and Hasgul et al. (2018), as indicated in Table 4. The predicted flexural moment capacity $M_n$ of the beams presented in Table 4 is in good agreement with an average predicted-over-experimental moment ratio, $M_n/M_{\exp }$, of 0.96 with a COV of 19.6% for all beams, providing another confirmation of the capability of the proposed methodology to predict the flexural capacity of prestressed and non-prestressed UHPC beams. The proposed closed-form model inherently depends on the input mechanical properties; more rigorous determination of these properties will result in a more accurate approximation of the flexural capacity.

Flexural design of concrete members in the United States is founded on mechanical models and sectional design methods specified in AASHTO LRFD BDS (AASHTO 2020) and ACI 318-19 (ACI 2019). In these documents, the analysis of the beam is based on the concepts of equilibrium and strain compatibility utilizing mechanical models to determine the stresses on strained cross sections. In compression, the stress-strain behavior of concrete at ultimate limit state may be simplified by a calibrated rectangular stress block. At ultimate loading, discrete longitudinal steel reinforcements are added and proportioned to carry all tensile stresses while ignoring the minimal tensile resistance offered by the concrete. When subjected to flexural loads, the tensile reinforcements span the concrete cracks and enable flexural tensile strains far greater than that at first concrete cracking. The ultimate capacity of the beam is commonly limited by the attainment of a flexural compression failure, defined when the strain in the concrete in compression reaches the assumed compression strain limit of 0.003.

Reinforced concrete flexural elements are described as (1) compression-controlled when the net tensile strain in the extreme tension steel at ultimate is less than a specified value, generally associated with the yield strain; (2) tension-controlled when the steel strain at failure is greater than a specified tensile strain limit value, e.g., 0.005 for non-prestressed reinforcement with a specified minimum yielding stress not exceeding 517 MPa (75 ksi) (AASHTO 2020); and (3) within the transition region when the net tensile strain in the steel at failure is between the specified thresholds of the compression- and tension-controlled sections. The steel tensile threshold value for the tension-controlled sections is expected to be sufficiently large to allow for excessive deformation and cracking, providing for a ductile behavior and ample warning of impending failure. In contrast, given the small tensile strains in compression-controlled sections, this failure mechanism may be brittle, with little warning of impending failure. Moreover, the failure of common compression members, i.e., columns, can lead to detrimental consequences to the stability of the entire structure. To address this issue, a reduced moment resistance factor is specified for compression-controlled sections, i.e., ${\varphi }_f=0.75$, compared to the resistance factor specified for tension-controlled sections, i.e., ${\varphi }_f=0.90$, for non-prestressed and prestressed components designed according to ACI 318-19 (ACI 2019), and ${\varphi }_f=0.90$ and 1.00 for non-prestressed components and prestressed components designed according to AASHTO LRFD BDS (AASHTO 2020), respectively.

In the French design standard, the resisting forces of the sections of a member subjected to flexure are calculated from a linear strain diagram included in a domain bounded by a compression strain limit, corresponding to UHPC crushing, and a reinforcing steel limit, corresponding to the rupture of reinforcing steel (AFNOR 2016a). For members with no steel reinforcing bars or prestressing strands, tensile strains are limited to the strain at peak tensile strength of UHPC. The cross-sectional forces are obtained by employing compression and tension mechanical stress-strain models to calculate the stresses corresponding to the assumed strains. The compression models include a detailed model based on parabolic representation of the stress-strain results obtained from compression tests and a simplified elastic-perfectly plastic model with reduced design stress and strain limits. The tension models include an elastic-perfectly plastic or bilinear stress-strain representation, limited by the strain at crack localization (hardening limit), for materials exhibiting a constant or increasing stress after cracking, respectively. The material compression parameters are obtained directly from uniaxial compression tests, while the tensile parameters are obtained indirectly from a prism flexure test using an inverse analysis technique. The inverse analysis back-calculates an approximation of the uniaxial tensile stress-strain response from the moment-deflection results of the prism test and includes specific procedures to infer the sectional curvatures of the beam corresponding to the measured deflections at each load step (AFNOR 2016b).

In the Swiss design recommendations, the ultimate bending resistance of a reinforced UHPC cross section is calculated by resolving the stresses corresponding to a linear strain diagram included in a domain bounded by the strains when the peak compression (crushing) and tensile (localization) strengths are reached (SIA 2016). The compression stress-strain model is linear elastic to failure, with the failure stress defined by the design compression strength obtained from compression testing of cylinders or cubes. In tension, a rectangular stress block is permitted with a stress equal to the design tensile strength (which is a proportion of the peak tensile strength) and extending to a distance of $0.9(h-X)$ from the extreme tension fiber, where $h$ is height of the beam section and $X$ is the depth of the neutral axis. If the tensile model is used to describe a UHPC member with steel reinforcing bars or strands as the primary flexural tensile reinforcement, the tensile strain limit (strain at peak tensile strength) may be increased by a factor of 2.0 due to simultaneous action of the fibers and reinforcing steel; in this case, the design tensile strength is taken as a reduced value of the elastic limit (cracking stress). The uniaxial tensile stresses and strains of UHPC are obtained either from a direct tension test or from an inverse analysis of the load-deflection results of flexural prism tests. The inverse analysis method assumes a stress distribution in the section at peak force and uses simplified procedures to obtain the strain-hardening portion of the UHPC stress-strain response based on the measured vertical deflection and sectional curvatures of the prism during the test. The applicability of the Swiss design recommendation document is limited to UHPC materials exhibiting a uniaxial stress-strain response showing a continuous increase in stress after cracking until the peak stress is reached, i.e., strain-hardening behavior.

In the Canadian design recommendations, the design capacity of flexural elements composed of strain-hardening fiber-reinforced concrete (SH-FRC), such as UHPC, is calculated by determining the sectional forces corresponding to a linear strain diagram bounded by the SH-FRC crushing strain limit in compression and the rupture of steel reinforcement in tension (Annex A8.1 of CSA S6:19). The compression stress-strain model is linear elastic when the strain value in the extreme compression layer is less than the limiting strain at crushing. When the extreme compression fiber of the member equals the ultimate compression strain (i.e., crushing limit), the compression model can be taken as the traditional rectangular stress block, which is compatible with a triangular stress distribution for materials with compression strength values exceeding 120 MPa (17.4 ksi). The tensile model at the ultimate limit state is a rectangular distribution extending over cross-sectional layers where the tensile strains are less than or equal to the limiting tensile strain (i.e., strain at peak tensile resistance). The compression parameters are obtained from uniaxial compression tests, while the uniaxial tensile parameters (i.e., peak stress and strain) are obtained either from a direct tension test or from an inverse analysis of the load-deflection results of flexural prism tests. The inverse analysis technique is based on empirical approximations of the stresses at specific vertical deflections and crack openings recorded during the testing of a flexural prism of specified dimensions. The Canadian design document specifies a flexural resistance method based on sectional curvature ductility. The maximum flexural resistance $M_1$ is taken as the maximum of the flexural moment (1) at yielding of the reinforcement in tension, (2) when the strain in the extreme tension fiber is equal to the limiting tensile strain, or (3) at concrete crushing. The curvature ductility ratio is defined as the ratio of the sectional curvature at $M_1$ to the sectional curvature at the yielding of the tensile flexural reinforcement. If the ductility ratio is greater than 2.0, the flexural resistance of the member is taken equal to the moment at yielding of reinforcement. In cases where the curvature ductility is less than 2.0, the flexural resistance of the member must be taken as the maximum of $0.50\times M_1$ or the flexural resistance calculated without considering the contribution of fibers. Fig. 1 presents a comparison of the cross-sectional stress conditions of UHPC members computed following the recommendations of the French, Swiss, and Canadian documents when the strain at the extreme tension layer is equal to the design tensile strain limit ${\epsilon }_{t,\lim }$ and with elastic stresses in compression.