Inverse Analysis of R-UHPFRC Beams to Determine the Flexural Response under Service Loading and at Ultimate Resistance

The Swiss guidelines for UHPFRC, SIA 2052 (Swiss Society of Engineers and Architects 2017), specify four-point bending tests of rectangular plates to obtain the tensile properties of UHPFRC by inverse analysis of experimental results. Plates of width $b_m$ and thickness $h_m$ are tested under four-point bending over a span $l_m$ with force application points spaced at $l_m/3$ (Fig. 1). Force and midspan deflection are recorded during testing. The material properties are obtained by inverse analysis methods.

In this paper, 10 beams of 3 different types (Fig. 2) are discussed; each member was cast separately. Three beams of Type 1, three beams of Type 2, and two beams of Type 3 were tested under quasi-static loading until failure. Additionally, one beam of Type 2 and one beam of Type 3 were tested under loading–unloading cycles to investigate the structural behavior under service conditions. The number of specimens of each type is presented in Table 1.

Beams of Type 1 contained one rebar of diameter $Ø20\mathrm{mm}$, and cover thickness $c_{\mathrm{nom}}=10\mathrm{mm}$, i.e., $Ø/2$. Beams of Type 2 were reinforced with one rebar with $Ø34\mathrm{mm}$ and $c_{\mathrm{nom}}=17\mathrm{mm}$, i.e., $Ø/2$. Type 3 members contained one longitudinal rebar with $Ø20\mathrm{mm}$ and $c_{\mathrm{nom}}=10\mathrm{mm}$, and $\mathrm{\Omega }$-shaped $Ø6\mathrm{mm}$ stirrups (Fig. 2). Although Types 1 and 3 had the same longitudinal reinforcement ($Ø20\mathrm{mm}$), the rebars were fabricated and delivered separately, and thus they were treated separately.

All the beams were cast in horizontal position (as tested), pouring fresh UHPFRC from the top at one end. Six external vibrators attached to walls of the formwork assured good flow of the mix. After casting, the formworks were covered with foil for 7 days. Then the beams were unmolded, wrapped in foil, and transported to storage area.

The beams were subjected to quasi-static displacement-controlled four-point bending tests. The constant bending moment zone varied between 0.2 and 0.7 m (Fig. 2), to ensure bending failure mode rather than shear failure. The displacement was applied using a servohydraulic actuator and transmitted using a hinge and a steel beam. The resultant force was measured using the load cell of the actuator. For Groups 2 and 3, foil strain gauges (SGs) were glued on rebars at midspan and $\pm 200\mathrm{mm}$ from midspan before casting.

Commercially available UHPFRC mix Holcim710 (Holcim (Schweiz) AG (Zürich, Switzerland) was used, with 3.8% by volume 13-mm straight steel fibers with an aspect ratio of 65. The minimum age at the moment of testing was 3 months. The cement hydration in UHPFRC is in an advanced stage after 28 days and stops almost completely after 90 days (Habel et al. 2006), and thus it was assumed that the age had no influence on the material properties. To verify this, the companion plates were tested under four-point bending at 28 and 90 days after casting, which is discussed subsequently.

The mean compressive strength obtained by testing of $70\times 140\text{-}\mathrm{mm}$ cylinders in direct compression at 28 days according to the Swiss standard was $f_{Uc}=140.7\mathrm{MPa}$.

Both longitudinal reinforcement and stirrups were of type B500B according to the Swiss standard SIA 262 (SIA 2014) and Eurocode EC2 (BSI 2005), with theoretical characteristic yielding strength $f_{sk}=500\mathrm{MPa}$. The properties of longitudinal reinforcement obtained using direct tension test according to Swiss standard are presented in Table 2. The rebars used in Type 1 beams had higher strength; however, they still met the requirements of B500B reinforcement class.

Inverse analysis methods for an element under four-point bending are based on sectional stress distribution. Stress and strain values are sought for three points on the force–deflection curve [Fig. 3(a), A, B, and C].

Point A indicates the end of linearity of the force ($F$)–deflection ($\delta$) curve, implying loss of elasticity of the material. Sectional stress distribution and deflection at this point can be calculated using elasticity theory, obtaining Young’s modulus $E_U$ and elastic limit stress $f_{Ute}$. Alternatively, $E_U$ can be calculated for each $F\text{-}\delta$ pair. Point A is the point at which an irreversible decrease of $E_U$ occurs.

The ultimate resistance of the element is reached at Point C. Using a stress block in tension and elastic response in compression for UHPFRC, knowing both the position of the neutral axis and the acting bending moment, and employing the sectional force equilibrium, the tensile strength of UHPFRC, $f_{Utu}$, is found [Fig. 3(c)]. In the case of R-UHPFRC members, the contribution of the reinforcement bar is taken into account as well.

Point B marks the moment at which softening behavior of UHPFRC comes into play. It is detected with iterative methods to identify the loss of agreement between the experimental deflection–force curve and that obtained analytically using a simplified material model with stress cut off at $f_{Utu}$. The lack of agreement indicates loss of validity of the model without postpeak resistance, and thus beginning of the softening behavior contribution of UHPFRC in bending resistance.

Inverse analysis using FEM is based on finding a material model such that the computed and experimental structural response are in good agreement, and is used to verify results and assumptions of an analytical inverse analysis (Denarié et al. 2017). DIANA FEA 2017 software was used, as in Sadouki et al. (2017).

A two-dimensional (2D) model of the plate specimen subjected to four-point bending was built using plane-stress rectangular $5\times 5\text{-}\mathrm{mm}$ finite elements. The UHPFRC was modeled as a continuum. The elastic, strain-hardening, and bilinear softening material response was simulated using an elastic multidirectional fixed-crack model. The material remained elastic until $f_{Ute}$ was reached. Then the stress–deformation curve was defined for strain hardening and postpeak softening responses. The localization of a fictitious crack along the element was determined by dividing the constant bending moment area into 20-mm-wide vertical zones. One of them, corresponding to the location of the critical section in the tested specimen, was modeled using the nominal material properties, whereas the rest of the plate was modeled using the material model with the same modulus of elasticity $E_U$, elastic limit stress $f_{Ute}$, and hardening modulus $E_{UH}$, but with higher tensile strength $f_{Utu}$. The constitutive law of UHPFRC including the beginning of the softening branch was varied until obtaining a similar force–deflection response as in the experiment. By observation of the first part of the force–deflection curve, $E_U$ and $f_{Ute}$ were adjusted iteratively. Observing further segments of the curve, $E_{UH}$, $f_{Utu}$, and finally the softening response were determined. The experiment was modeled until the ultimate member resistance was reached using a nonlinear solver with variable loading steps. The load introduction as displacement by means of nonlinear springs that acted in compression only reflected the possibility of loss of contact between the testing machine and the plate.

A similar method was used for modeling the R-UHPFRC beams. The T-shaped cross-section was modeled by division of the 2D model into six horizontal parts. The uppermost part represented the flange, and the remaining five parts composed the web. The variable web thickness was modeled through stepwise variation of the thickness such that the error of the moment of inertia of the beam was below 1%. The longitudinal rebars were modeled as straight, horizontal, perfectly anchored bars. Additionally, to avoid crushing of singular elements over the supports, 200-mm-long and 20-mm-thick steel plates were added, respecting the theoretical static scheme of the beam. The finite-element size was same as for the plates, i.e., $5\times 5\mathrm{mm}$.

Only the material model of the UHPFRC was fitted. The material properties of the rebars were adopted as elastic-perfectly plastic using the average material properties obtained from testing for each type of beam (Table 2).

The tensile strength of UHPFRC with steel fibers can be determined using nondestructive testing of the relative magnetic inductance. The distribution of tensile strength $f_{Utu}$ in an UHPFRC element depends on local orientation and content of fibers (Naaman 2018; Oesterlee et al. 2009), and can be approximated for straight fibers as follows (Hannant 1978; Naaman 1972):where ${\tau }_f$ = average fiber pull-out stress; $V_f$, $l_f$, and $d_f$ = fiber volumetric content, length, and diameter, respectively; and ${\mu }_0$ and ${\mu }_1$ = fiber orientation and efficiency factors respectively.

The fiber orientation factor ${\mu }_0$ reflects the probability that a fiber crosses a given section. Under the assumption of homogenous fiber distribution, factor ${\mu }_0$ is determined as the ratio of the total area of fibers in the section to the fiber volume fraction (Krenchel 1975; Stroeven 2009)

The fiber efficiency factor ${\mu }_1$ considers the effect of angles between the fibers and a cross section of the composite (Oesterlee 2010; Wille et al. 2014) on the fibers’ pull-out efficiency, and is dependent on ${\mu }_0$ (Bastien-Masse et al. 2016). Factor ${\mu }_0$ can be obtained precisely using image analysis of orthogonal surfaces of specimens extracted from structural members or accompanying elements for material testing (Oesterlee et al. 2009; Wuest 2007). It also can be assessed using nondestructive testing for sake of applicability in practice. The factor ${\mu }_1$ can be estimated on the basis of ${\mu }_0$ (Bastien-Masse et al. 2016; Nunes et al. 2017).

Because most UHPFRCs use steel fibers, their magnetic inductance $L$ can be exploited to establish ${\mu }_0$ and $V_f$ directly, and ${\mu }_1$ indirectly (Nunes et al. 2016, 2017). After measuring the magnetic inductance on the element surface in two directions, and the inductance of the air, ($L_x$, $L_y$, and $L_{\mathrm{air}}$, respectively) the magnetic permeability is found as

Using the linear dependence of mean magnetic permeability $[{\mu }_{r,\mathrm{mean}}=({\mu }_{r,x}+{\mu }_{r,y})/2]$ on fiber content and since the matrix with no fibers ($V_f=0\%$) has the permeability ${\mu }_{r,\mathrm{mean}}=0$, the slope of linear regression is determined. This slope is dependent on the type of fiber used (Pimentel and Nunes 2016), and subsequently is used to calculate the local $V_f$ at the measurement point. Slope values ranging from 3.8 to 4.55 are given in the literature (Nunes et al. 2016; Shen and Brühwiler 2020).

Using the fiber orientation factor (${\rho }_x-{\rho }_y$), ${\mu }_0$ and ${\mu }_1$ are obtained (Nunes et al. 2017)

Because the magnetic permeability of UHPFRC depends on the fiber type (Nunes et al. 2016), the calibration for the UHPFRC mix used in this research was performed. The sensor used in this research consisted of a ferritic U-shaped core with $28\times 30\text{-}\mathrm{mm}$ cross section, 93 mm width, and 76 mm height. Two coils were made of 0.5-mm copper wire with about 170 turns each and were connected together (Fig. 5). The electrical inductance was measured using a LCR meter. Due to the circular magnetic flux produced in the U-shaped sensor, a certain effective depth is penetrated depending on the power of the magnetic field. For a sensor similar to the one used in the present study, but with smaller operating voltage (0.1 V), the effective depth was found to be equal to 25 mm (Li et al. 2018). Because according to Lenz’s law the inductance is proportional to electromotive force, it can be expected that the effective depth increases with higher voltage in the coil. If the effective depth of the sensor is higher than the thickness of element used for calibration, the method cannot be used for elements of a different thickness. In this study, 2 V current was chosen, as in Nunes et al. (2016); thus it was decided to perform the calibration on specimens specially prepared for this purpose instead of the plates used for material testing (30-mm thick).

Four plates $40\times 200\times \mathrm{1,500}\mathrm{mm}$ were cast vertically to mock-up the web of the T-shaped beam (Fig. 6). To assure nonuniformity of fiber alignment and distribution, and to obtain more calibration points for the method, four different methods of casting were applied: (1) casting from the top at one end, with no vibrating; (2) casting from the top at one end, with vibrating after casting; (3) casting from the top at one end, with vibrating during and after material placing; and (4) casting from the top at two ends, with vibrating during and after material placing.

After 28 days, the magnetic inductance was measured along the plates in two directions. The plates were cut to obtain six specimens from each plate (Fig. 6, a–f) with dimensions similar to the bending tensile test specimen according to Swiss standard SIA 2052 (Swiss Society of Engineers and Architects 2017), which were tested under four-point bending (Fig. 1) to obtain $f_{Utu}$ by means of inverse analysis, as described previously. Despite the greater thickness than that of the standard specimens, the same stress distribution at ultimate resistance was assumed (Shen et al. 2020).

Assuming an average $V_f=3.8\%$ in all specimens, the slope of the regression curve was determined according to Nunes et al. (2016) to be equal to 4.59. Because the magnetic measurements on webs of T-shaped beams under the same assumption yielded average slopes of 4.64, it was accepted that

Importantly, no correlation between the web thickness of T-beams and fitted slopes could be found. This confirms that the effective depth of the sensor was smaller than the thickness of plate used for calibration.

After finding the local $V_f$ and the factors ${\mu }_0$ and ${\mu }_1$ for measurement points at the critical section of tested plates, fiber pull-out shear stress for this kind of mix was estimated. Using Eq. (16), ${\tau }_f=7.5\mathrm{MPa}$, which corresponds well to values obtained in pull-out test of fibers with the same diameter (6.9–10 MPa) (Orange et al. 2000; Wuest 2007). The method therefore was calibrated for the present UHPFRC mix, and could be used for $f_{Utu}$ calculation. The results obtained with destructive and nondestructive values during calibration for all 24 specimens are presented in Fig. 7. The rather large scatter of $f_{Utu}$ was due to the different casting methods, and should not be associated with material variation in the T-shaped beams.