Introduction
Ultra high performance fiber reinforced cementitious composites (UHPFRCs) are a relatively new class of building materials (
Naaman 2018). They are composed of a cementitious matrix with fine grains (
) and a high dosage of discontinuous short fibers, usually made of steel (
by volume) (
Brühwiler 2016). This material often is combined with steel reinforcement bars to form reinforced UHPFRC (R-UHPFRC). The number of implementations in structural engineering to rehabilitate and strengthen existing structures and to design and build new structures is increasing rapidly in Switzerland (
MCS EPFL 2020) and around the world (
Azmee and Shafiq 2018;
Graybeal et al. 2020;
Yoo and Yoon 2016).
Most of the research on stress distribution in R-UHPFRC members has focused on the ultimate resistance (
Adel et al. 2019;
Farhat et al. 2007;
Habel et al. 2007;
Qiu et al. 2020) or monotonic loading (
Huang et al. 2019;
Yang et al. 2019), without taking into account unloading of the member. The structural response under service loading with an analytical method of taking into account change of stiffness due to loading–unloading was discussed by Gao et al. (
2020), but without detailed analysis of stress distribution in the cross section or composite behavior between steel reinforcement bars and UHPFRC (
Wang et al. 2020).
As far as the ultimate resistance of members determines structural safety, the serviceability state prevails during service duration of a structure. Understanding the behavior of a structure under loading–unloading conditions, for example, due to live loads, and taking into account intrinsic scatter of material properties in a structural member (
Oesterlee et al. 2009) is necessary to eliminate discrepancies between modeled and measured responses (
Brühwiler et al. 2019).
The principle of inverse analysis is based on modeling of an experiment, from which the material properties are indirectly retrieved. Using this method, the direct tensile test, which unequivocally is difficult to conduct, can be replaced by relatively simple bending tests. In this way, the inherent variation of UHPFRC properties in different elements (
Oesterlee et al. 2009) can be quantified. Several inverse analysis methods are available, using simplified closed-form solutions (
AFGC 2013;
López et al. 2015,
2016;
Qian and Li 2008) as well as numerical (
Baby et al. 2013;
de Oliveira e Sousa and Gettu 2006;
Denarié et al. 2017), analytical (
Baril et al. 2016), and finite-element methods (
Mezquida-Alcaraz et al. 2019;
Tailhan et al. 2004), taking into account scatter of material properties in members (
Rossi et al. 2015). Inverse analysis also can be done for larger members (
Pan et al. 2016;
Shen et al. 2020). The present work used analytical and numerical finite-element modeling (FEM) methods.
The tensile strength of UHPFRC depends on fiber geometry, content, orientation, and the pull-out shear stress of fibers from the cementitious matrix (
Naaman 2018;
Oesterlee et al. 2009). Fiber content and orientation in a structural element can be determined using tomography or X-ray scanning (
Barnett et al. 2010); however, these methods are impractical on-site. In the case of UHPFRC with steel fibers, their magnetic conductivity can be used to determine content and orientation in non-destructive way, and further link it with tensile strength (
Nunes et al. 2016) resulting in magnetic nondestructive testing (NDT).
This study had two objectives: (1) to deduce the stress distribution in R-UHPFRC bent members under loading–unloading action, and (2) to observe the scatter of material performance in small and large UHPFRC elements. The following experiments were conducted: (1) magnetic NDT and four-point bending tests on full-scale R-UHPFRC beams; (2) standardized four-point bending tests on companion plates; and (3) magnetic NDT and four-point bending tests on thicker plates for calibration of the NDT method. Analytical and FEM inverse analysis methods were applied to determine material properties from bending tests of full-scale members and companion plate specimens. Obtained results were compared against results of magnetic NDT. The stress distribution in cross sections of R-UHPFRC beams under loading–unloading can be modeled precisely using the identified material properties. Finally, the results of modeling were validated with strain values of reinforcement bars measured during experiments.
This paper is structured as follows. Section “Methods” presents all the methods: (1) setups for testing of UHPFRC plates and R-UHPFRC beams, (2) analytical and numerical methods of inverse analysis, (3) constitutive model of UHPFRC under loading–unloading in tension, (4) numerical model for calculation of stress profiles, and (5) principles and calibration of magnetic NDT. In the section “UHPFRC Properties,” properties of UHPFRC obtained using different methods are compared and discussed. In the section “Ultimate Resistance of Members,” the importance of the variation of obtained material properties is quantified and compared with experiments using the bending resistance of R-UHPFRC beams. In the section “Stress Distribution in Members under Service Conditions,” previously obtained material properties are used to calculate stress profiles in R-UHPFRC under loading–unloading; the results are validated with experiments.
UHPFRC Properties
Comparison of FEM and Analytical Methods for Inverse Analysis of Plates
For some of the plates tested in four-point bending, finite-element modeling was used to verify agreement between the two analytical inverse analysis methods. Representative specimens for each group were chosen. Specimens with critical cross sections close to the midspan were favored because the two analytical methods should be more precise at this location, and thus the comparison more reliable. The comparison of material properties obtained for 9 of plates total 48 from different castings at 28 and 90 days using 3 methods is presented in Table
3.
The elastic limit stress obtained with FEM was between the values obtained with the analytical methods (except in Plate 6). This also was found for strain hardening deformation (except in Plates 3 and 7). Similar ultimate tensile strength was obtained with all methods; the variation was less than 15%.
The discrepancy between results obtained with each method illustrates the difficulty of fitting Point A in the analytical inverse analysis procedure. FEM should give the most precise results because the critical crack was modeled exactly where it appeared. Furthermore, the full sectional stress distribution was obtained, instead of a simplified distribution. Because the results obtained with FEM were between results of analytical analysis, it can be stated that the two methods approached the solution from two sides. The importance of these discrepancies is discussed subsequently.
Limit of Elasticity and Modulus of Elasticity
The Young’s modulus
and elastic limit stress
obtained with inverse analysis of all tested plates are presented in Table
4. The average values for each type of beam tested to failure also is given. The mean values (
) and standard deviations (
) were computed for six plates in each test series after 28 and 90 days to quantify the scatter of results.
Comparison of the results shows that no change of properties occurred between 28 and 90 days age. Only the values obtained with Method 2 for Group 2 were the two mean values outside the interval, indicating a scatter that was larger than expected assuming normal distribution of properties.
The elastic limit stress
for plates was smaller than that for beams, confirming the beneficial influence of reinforcement on material properties (
Oesterlee 2010). In contrast, the
obtained was higher for plates than for beams. This may be explained by the neglected shear deformation in the calculation of deflection in inverse analysis.
Inverse analysis Method 2 for plates have in average 36% higher elastic limit stress than did Method I; the case of Type 3 casting after 28 days, this parameter almost doubled. This may be have been due to lack of rapid loss of stiffness or to regaining it at a later stage due to the fiber orientation and content stratification in the specimen. The moduli of elasticity found with the two methods were similar, and the average scatter was less than 1%.
The elasticity limit of Type 3 beams was lower than that of other types. This probably was caused by early age shrinkage cracking due to (1) lower matrix tensile resistance because of age of premix (); and (2) addition of omega stirrups, which changed the restrain level of the setting mix. Only in this group of beams were localized microcracks detected after spraying with alcohol before loading. Such defects are not considered in inverse analysis; a lower apparent elasticity limit and thus elastic limit stress are obtained. This group of beams was cast 1 year after the other specimens, with the same material.
Tensile Strength
The tensile strength
and hardening strain
obtained with inverse analysis of plates are presented in Table
5 together with values obtained for respective beams. The average value for each type of beam tested to failure is given. The mean values (
) and standard deviations (
) are given for each series of tests after 28 and 90 days to quantify the scatter of results.
The mean
for beams obtained with magnetic NDT and on the basis of
retrieved previously for the current UHPFRC is presented in Table
5 as well. The average value for each beam was taken because the influence of fiber nonuniformity is negligible for the overall resistance of the beam (
Pimentel and Nunes 2016) in R-UHPFRC members. Still, it determines the failure crack location (
Sawicki and Brühwiler 2019).
The estimated hardening strain was about 50% lower using the analytical inverse analysis Method 2 than using Method I. In the case of beams, larger values were obtained with the analytical method than with the FEM method.
Similar values for plates and beams were obtained with all methods except Type 3 beams. As mentioned previously, due to the early age cracking, the apparent material strength was lower in the beams from this group.
Ultimate Resistance of Members
To quantify the influence of variation of material properties obtained with different methods, the computed ultimate resistance of the beams was compared with the testing results. The simplified method from the Swiss UHPFRC standard SIA 2052 (
Swiss Society of Engineers and Architects 2017) was used (Fig.
8), where
is compressive stress resulting from strain distribution, and
,
, and
are resultant forces from compressive and tensile action of the UHPFRC and tensile action of the reinforcement bar, respectively.
The method assumes that the plain sections remain plain and that both reinforcement and UHPFRC in tension are fully activated. The tensile stress block of UHPFRC is taken as 90% of height below the neutral axis to consider the fact that part of the material is in the elastic state. Then the neutral axis can be found to comply with force balance in the cross section. For the sake of comparison, the mean values of resistance were used here. As mentioned previously, strain at the bottom of the beam was assumed to be equal to
by analogy to SIA 2052 (
Swiss Society of Engineers and Architects 2017), and supported by FEM simulations. For values based on magnetic NDT, the
and
mean values obtained from plates using Method 1 for the respective types of beams were adopted. To quantify the composite behavior of reinforcement and UHPFRC, the ratio
of the sectional tensile force shared between them is presented in Table
6.
The bending resistance based on material testing for Groups 1 and 2 consistently was below the experimental value of ultimate resistance, but within a 10% margin, thus showing good agreement. For Group 3, the ultimate resistance was overestimated by 20% due to the previously mentioned early age shrinkage cracking of the matrix. Importantly, the ultimate resistance values determined using the value obtained by NDT were closer than those based on values from material testing. This was because the fiber orientation and content variation can be grasped correctly by the NDT method. The ultimate resistance based on the inverse analysis of beams is shown for the sake of comparison to quantify the error of the model. Finally, the smaller rebars () contributed as much as the UHPFRC to the tensile sectional force, and the contribution of UHPFRC decreased with increase of rebar diameter to 34 mm.
Stress Distribution in Members under Service Conditions
Two additional beams, one of Type 2 with
rebar and one of Type 3 with
rebar, were tested to investigate flexural stiffness and stress distribution in the cross section under service conditions. They were instrumented with strain gauges on the rebars prior to casting. Multiple loading–unloading cycles were imposed to simulate structural response under possible service conditions, up to about 50% of ultimate resistance (S). Figs.
9 and
10 present measured force vs strain and calculated stress distribution for Type 2 beam (
) respecitvely, while Figs.
11 and
12 present the same information for Type 3 beam (
).
The results of member modeling are presented in Tables
7 and
8. The structural response of the beams was calculated using the material properties obtained from plates with the two methods of analytical inverse analysis, and with the inverse FEM analysis of beams. Good agreement of the modeled and the measured reinforcement bar strains were obtained, validating the method. Because the beams used for validation were not those used to obtain the material properties, it is demonstrated that the method can be applied to structural members.
The residual strain after loading–unloading leads to increased stress in reinforcement bars under a given force. For beams of Type 2, stress in the reinforcement under loading of 20 kN doubled, from about 31 to 60 MPa, when preloaded to a force level of 135 or 250 kN, respectively (Table
7). In the case of Type 3 beams, stress in the reinforcement under a force of 5 kN increased from about 35 to 85 MPa when previously loaded with 60 or 102 kN, respectively (Table
8).
The ratio of tensile sectional force carried by the reinforcement bar and the UHPFRC describes the level of cooperation between them. As stress increases and the UHPFRC enters the strain-hardening domain, the load bearing contribution of the rebar increases. After unloading at , the rebar contribution is more pronounced than during loading to due to difference in loading ( and ) and unloading () secant values of the UHPFRC. The variation of the ratio is due to modification of the cross-sectional properties due to the strain hardening and unloading constitutive laws of UHPFRC. This mechanism greatly reduces the stress variation in the rebar during loading–unloading cycles, which is particularly important in the case of fatigue.
Loss of member stiffness was quantified with the
-ratio of bending inertia of the cross section showing UHPFRC strain-hardening to initial elastic inertia. The moment of inertia was calculated separately for
and
in each cycle. For
, a composite cross section with three moduli of elasticity was assumed (
AFGC 2013): (1) UHPFRC in the elastic state with
, (2) UHPFRC in the strain-hardening state with
, and (3) the reinforcement bar with
. The moment of inertia was calculated with respect to the neutral axis position
at
. For calculation of member inertia at unloading, the secant
was calculated for each computational layer separately. Then, the inertia of the composite cross section about
at
was obtained. The inertia at
was higher than at
, which is reflected by the slopes of the curves in Figs.
9 and
11.
The distribution of strain and stress in the UHPFRC for each load step is presented in Figs.
10 and
12. The response at
depended on the stress distribution at
. Interestingly, the UHPFRC in the bottom part of the member, which usually is in tension, may even have higher compressive stresses than the upper part when the beam is unloaded, depending on geometrical dimensions and loading history. In the case of the presented T-shaped beams, the bottom part was contributing as much as 90% of total compressive sectional force [Fig.
10(f)]. Similar behavior, but with smaller compressive stress activated because of a different cementitious material used, was observed by Wang et al. (
2020). With increasing load, the neutral axis position moves up due to strain-hardening and increase of
. At unloading, the axis moves even higher, compared with the respective
, due to the UHPFRC response, and especially when compressive stress is activated in the bottom part of section.
The aforementioned mechanism determines the structural response and should be taken into account when calculating the stress state of a member under service conditions. During loading, the range of elastic limit stress in the cross section should be found and the modified composite section should be taken into account for stress calculations. When the structure is unloaded, a more complex method should be applied, with calculation of . However, because stiffness during unloading is higher than that at primary loading, neglecting the modified moment of inertia at unloading is acceptable for the sake of simplification, leading to a higher computed deflection range, and thus to a conservative solution. Nevertheless, modified inertia at unloading should be taken into account during monitoring of deflection of R-UHPFRC structures under service loading, as well as calculation of stress ranges under fatigue actions.
The method was validated for both beam types, and good agreement was obtained between measured and calculated strain using material properties obtained from inverse analysis of beams. The agreement with properties based on plate testing was lower, with an average error of 20%. It is not obvious which method of inverse analysis of plates gives better results for the beam in the service state.
This paper analyzed test results of R-UHPFRC members and UHPFRC plates subjected to four-point bending. Using magnetic nondestructive testing and inverse analysis principles based on analytical and finite-element models, the UHPFRC material properties were determined. The results were compared, and the importance of their variation was quantified for R-UHPFRC beam under loading–unloading in the service state and at ultimate resistance.
This research showed that:
•
The analytical inverse analysis methods also can be used for structural elements, such as full-scale beams, including elements with reinforcement bars; correctness of the results was confirmed by finite-element modeling.
•
UHPFRC in the tensile zone of R-UHPFRC member enters into compression if it previously was loaded beyond the elasticity limit. This phenomenon leads to significantly increased tensile strain in the rebar in the unloaded state, and thus influences the global response of the structural member. This increase is notable in particular at high loading levels, and should be taken into account during design and verification of structures.
•
Magnetic NDT allows determining the UHPFRC tensile strength when the average fiber pull-out stress of the UHPFRC mix is known. Better estimation of ultimate bending resistance of structural members is obtained than that based on material testing because fiber distribution in the element explicitly is taken into account. Therefore, this technique can be used to check the quality of elements.
•
Magnetic NDT allows for determining only the tensile resistance, and therefore must be combined with material testing using small specimens to obtain the full set of material properties. This method gives results comparable to those of UHPFRC characterization with inverse analysis of a prototype element, and thus can be considered as an alternative to testing of a prototype element.
Furthermore, the knowledge gap regarding the behavior of UHPFRC in tension–compression regimes was identified. It is recommended that this research topic should be investigated, which in turn would allow improving the quality of modeling of R-UHPFRC members under loading–unloading in the serviceability domain.