Analytical and computational investigation of the flexural behavior of UHPFRC beams

Analytical and computational calibrations and subsequent parametric analyses were performed on the four UHPFRC beams under flexure to estimate their actual shear strength. The computational models were calibrated using different values for the CDP parameters, and a finite element mesh sensitivity study was conducted. These models predicted the experimental behavior satisfactorily, and the analytical model was also able to find the beams’ failure loads and proved to be a simple tool to estimate their behavior. The parametric analysis showed that the current beams did not require stirrups once the UHPFRC had high shear strength, and the maximum shear capacity was accurately determined.


INTRODUCTION
Ultra High Performance Fiber Reinforced Concrete (UHPFRC) was developed in the mid-1990s by Richard and Cheyrezy (1995).Because of its excellent mechanical performance, it has attracted the attention of researchers and engineers for practical applications in architectural and civil structures.One of the advantages of using this composite is that, according to Farhat et al. (2007), Graybeal and Tanesi (2007), and Yoo et al. (2014) have high compressive and tensile strengths of approximately 150 and 10 MPa, respectively.As a result, structures can have a lighter weight, which makes them look slenderer.In addition, lighter loads can be applied to foundations.Other remarkable properties include excellent durability, energy absorption capacity, and fatigue performance.
Although the great potential of UHPFRC for structural applications has been demonstrated through experimental research, few studies on finite element modeling (FEM) have been carried out to predict the flexural behavior of UHPFRC beams and to provide further insights into the interaction of this ductile material with reinforcements and damage mechanisms.According to Yin et al. (2019), a reliable computational model allows an effective improvement of structural performance and reduces not only costs but also the number of laboratory tests required.
In this sense, resorting to computational analysis tools is currently a relevant practice in civil engineering because the Finite Element Modeling (FEM) method is capable of predicting both beam failure modes and crack development under loading conditions.However, nonlinear models are inherently complex because of several factors, such as the constitutive models that are used to represent materials and the problem of mesh dependency, which commonly occurs in materials that exhibit softening behavior.These variables must be understood and adapted for UHPFRC.Some finite element (FE) software packages, namely, Abaqus, Diana, and Atena, are available with constitutive models for conventional concrete.The Concrete Damaged Plasticity (CDP) model implemented in ABAQUS is often used to predict conventional concrete behavior.However, the properties of UHPFRC are significantly different from those of conventional concrete; consequently, the parameters for CDP that establish the failure criterion, such as dilation angle (), eccentricity (є), fb0/fc0 (ratio of the biaxial and uniaxial compressive strengths), shape factor (  ) and viscosity (µ), may differ from commonly used parameters.Krahl et al. (2018) developed an experimental study to determine the parameters of the CDP for UHPFRC.These parameters were computationally tested in prisms using a four-point bending test, and good results were obtained in comparison with the experimental results.Slobbe et al. (2012) conducted a sensitivity analysis of mesh dependency and reported that the spatial discretization or topology of the finite element model can affect the numerical solution.According to the authors, mesh dependency can be subdivided into the sensitivity of crack propagation along continuous mesh lines (sensitivity to mesh polarization or mesh alignment) and the dependency of crack bandwidth on element size and orientation (sensitivity to mesh size).
Thus, further research is needed on the actual effects of not only the parameters for CDP but also of the mesh typology used in the computational models of UHPFRC beams, as there are no studies in the literature analyzing mesh sensitivity, finite element types, finite element approximation, or parameters of constitutive models.Table 1 shows the failure criteria for the CDP (implemented in the Abaqus software) used for UHPFRC in different studies.The major differences lie in the dilation angle and in the ratio of biaxial to uniaxial compressive strength f bc /f c in viscosity.The dilation angle governs the ratio of volumetric to diverting plastic strain (Abaqus, 2014), and is specific for each material.Chi et al. (2017) found a decreasing trend between the dilation angle and the properties of steel fibers.The f bc /f c ratio decreased with increasing compressive strength, as experimentally demonstrated by Speck (2008).The ratio of 1.16 was established on the basis of the findings of Kupfer et al. (1969), who reported values for the compressive strength of concrete of 18, 25, and 57 MPa.The viscosity parameter, on the other hand, is a mathematical resource to reduce the effect of permanent strain localization, which usually poses processing challenges to constitutive models that represent softening.
Therefore, a sensitivity analysis is required to apply the models to new materials.In this regard, a computational study was conducted based on experimental tests of four UHPFRC beams with different rates of flexural reinforcement, designed by Chen et al. (2018).Computational models were used to test the influence of CDP parameters, sizes and types of mesh elements, and mesh refinement.An analytical methodology was developed to determine the behavior of UHPFRC beams based on the model introduced by Fehling et al. (2014).Researchers, such as Al-Osta et al. (2017), Chen et al. (2018), Shafieifar et al. (2018), and Dogu and Menkulasi (2020), used analytical expressions to estimate the ultimate moment of UHPFRC beams.This paper presents a simplified methodology for calculating the moment-curvature and load-displacement diagrams based on the work of Yoo and Yoon (2015), which allows a more complete analysis of the behavior of beams under bending.

EXPERIMENTAL MODEL
This study used four UHPFRC beams, namely B1, B2, B3, and B4, which were previously tested by Chen et al. (2018).Figures 1 and Figure 2 and Table 2 show the geometric properties of the beams, as well as details of the reinforcements.Tables 3 and Table 4 show the mechanical properties of steel and concrete, respectively.

COMPUTATIONAL MODEL
Figure 3 shows the computational model designed according to the specifications for the geometry, loadings, and boundary conditions presented above.An embedded region constraint was assigned to the interaction between the reinforcement and concrete, simulating perfect adhesion between the steel and concrete.The interaction between the steel plates and the beam was modeled using a tie constraint.Thus, the possibility of displacement at the interface was disregarded.The steel plates were assumed to exhibit linear elastic behavior.The reference points (RPs) created to apply the loads and constraints on the supports were linked to the surfaces of the plates using coupling constraints.and Structures, 2024, 21(1), e518 5/18 strains (plasticity) that are typical of concrete.The plastic damage model was based on the model proposed by Lubliner et al. (1989) and Lee and Fenves (1998).According to Kmiecik and Kamiński (2011), this model is an adaptation of the Drucker-Prager model, which is also used to demonstrate the behavior of brittle materials.According to Abaqus (2014), this model is intended for the analysis of reinforced concrete structures under monotonic, cyclic, random or dynamic loadings.
Figure 4 shows the stress-strain curve calibrated by Krahl et al. (2018) that was used to represent the compression behavior of UHPFRC, with a fiber volume of 2%, according to the mechanical characteristics of concrete reported in Table 3.The authors used the model of Carreira and Chu (1985) to represent the stress-strain behavior of UHPFRC under compression loading.For the post-peak interval, two constants ( 1 and  2 ), proposed by Mansur et al. (1999), were added to the equation to simulate the effect of fiber addition, according to Equations 3 and 4, where the constants were only considered in the post-peak branch. 0 ,  0 ,  0 are peak stress, peak strain and initial elastic modulus, respectively, and they are known parameters. 1 and  2 are parameters based on experiments with the post-peak branch of the experimental results.Mansur et al. (1999) proposed equations for these variables that depend on fiber volume and peak stress of concrete ( 0 ).In the study developed by Krahl et al. (2018),  1 and  2 were calibrated for parameter , and total energy (  ) corresponds to the average area of the experimental stress-strain curves of concrete.For the present study,  1  2 ,  and   are 0.15, 0.30, 6.5 and 1.11 MPa, respectively.The tensile behavior of UHPFRC was determined using the constitutive law for stress-energy fracture.Fracture energy (  ) is calculated by Equation 5, according to Wille et al. (2014), where   is fiber volume ratio.Considering that the models used UHPFRC with 2% fiber volume, fracture energy was calculated to be 20,4 /.Tensile strength of UHPFRC was calculated with Equation 6, proposed by Schmidt (2008), where   is the average tensile strength of UHPFRC and   is its characteristic compressive strength.Simulations of the reinforcements were performed by considering the perfect constitutive elastoplastic model associated with the Von Mises yield criterion.Table 4 lists the elastic modulus and yield stress of each rebar.Poisson's ratio was assumed to be 0.3.The material underwent plastic strain, as shown in Table 4.

MOMENT-CURVATURE DIAGRAM
For an analytical estimation of the experimental behavior of the beams, the moment-curvature diagrams were first determined for each beam, followed by the calculation of the load-displacement curves.
The moment-curvature diagrams were calculated using the equilibrium equations of the UHPFRC cross-sections designed by Leutbecher and Fehling (2013) and Fehling et al. (2014).Figure 5 shows the distribution of stresses and strains for a rectangular cross-section under axial force () and bending moment ().The distribution of the tensile stresses on the concrete beams was simplified.There was triangular behavior for the linear-elastic branch and rectangular behavior for the branch where cracking occurred.This distribution is given by Equations 7 and 8, respectively, where ℎ stands for beam height;  is the depth of the neutral axis;  is beam thickness;   is the cracking stress of UHPFRC;  it is height that limits the branch between cracked and non-cracked concrete, as given by Equation 9.
Distribution of the resultant compressive force (  ) is determined according to Equation 10.Tensile strength in steel (  ) is given by Equation 11, where calculation of stress in the reinforcements is based on the perfect elastoplastic constitutive mode.Equations 7, 8, 10, and 11 were used to establish the balance of forces, according to Equation 12.In relation to the neutral axis (), the bending moment was calculated as shown in Figure 6 and Equation 13.One compressive strain value was assumed for concrete (  ), and the neutral axis () was made to range from 0 to h, until the forces were balanced (Equation 12).Subsequently, curvature (), strain in the reinforcements (  ) and tensile strain of concrete (  ) were calculated according to Equations 14, 15 and 16, respectively.Importantly, equations 14,15 and 16 represent the compatibility of the strains in the cross-section, as shown in Figure 5.According to Yoo and Yoon (2015), the moment-curvature curve can be converted to a load-displacement curve using Equations 17 and 18.

CALIBRATION
A mesh sensitivity study was conducted using Beam B1.Four discretization were used for the solid and rebar elements (60 mm, 50 mm, 40 mm, and 30 mm) to assess how the mesh affected the responses of the models.This assessment was carried out to obtain a mesh that had a low computational cost and adequately represented the experimental failure mode.
Two types of solid elements were tested: a mesh with hexahedral elements and a mesh with tetrahedral elements, as shown in Figures 7 and Figure 8.In both cases, quadratic approximations were used for the displacements, referred to as C3D8 (R)/C3D20 (R) and C3D4/C3D10.Elements T3D2 and T3D3 were used in the rebar elements, for example, the longitudinal reinforcements.Table 5 lists all types of finite elements used for the modeling.Table 1 shows that the parameters for the CDP used as a starting point for these analyses were the same as those experimentally calibrated by Krahl et al. (2018).However, a parametric analysis was performed while varying the parameters for CDP to assess how they could affect the response of the models, since other recommendations of these parameters for UHPFRC have been reported in the literature.
All models accurately captured the beginning of cracking as well as the post-cracking regime up to the peak load.Therefore, they are suitable for predicting the behavior of beams in a linear elastic regime.However, the maximum load and post-peak branch were better characterized by the models containing hexahedral finite elements with linear approximation and reduced integration (C3D8R) and sizes of 50 mm and 60 mm, and with reinforcements that were simulated with elements T3D2/T3D3, as shown in Figures 9g and 9h.In comparison, the other models presented a high computational cost and overestimated the beam tenacity after the application of the maximum load.

T3D2 T3D3
2 nodes 1 integration point 3 nodes 2 integration points Therefore, finite elements (C3D8R/T3D2) were chosen for subsequent analyses.When it comes to a model that considers the nonlinearity of materials, it is clear that the mesh size can have a significant impact on the behavior of the models.In their research, Genikomsou and Polak (2016) showed that even when using very similar mesh sizes (15 mm, 20 mm and 24 mm), the models behaved considerably different in the tests.Slobbe et al. (2012) reported that the spatial discretization or mesh topology of the finite element model influences the numerical solution.According to the author, mesh dependence can be subdivided into the sensitivity of crack propagation along the continuous mesh lines (mesh polarization sensitivity or mesh alignment) and the dependency of crack bandwidth on the size and orientation of the element (mesh size sensitivity).A study was conducted to assess the influence of CDP-related parameters for beam B1, such as the dilation angle (ψ), eccentricity (є), biaxial-to-uniaxial strength ratio (f bc /f c ), shape factor k c , and viscosity (µ).The results are shown in Figure 10.
Figure 10 shows that the variation in the parameters for the CDP did not significantly change the behavior of the beams, with the exception of eccentricity and viscosity (see Figure 10b and 10e).This is because the UHPFRC is a highly ductile material.Brittle materials, such as conventional concrete, are greatly influenced by the variation in these parameters, especially the dilation angle, as can be seen in the study developed by Genikomsou and Polak (2015).
Eccentricity defines the approach speed of the plasticity potential function asymptote (Labibzadeh, 2020) and is used to model the nonlinear behavior and damage progression in concrete structures (Benin et al., 2020).In this study, values greater than 0.1 showed post-peak hardening behavior, which does not corroborate the expected results.
Viscosity is a parameter used to solve the numerical convergence problems.Michal and Andrzej (2015) studied this parameter and noted that values between 0 to 10 -3 did not significantly affect the results.However, when adopting viscosity values above 10 -3 , the computational results found by the authors were overestimated; this is because the higher this parameter is, the greater the stress redistribution along the structural element will be.
The use of small values can improve the convergence rate and processing time without compromising results.Adopting a viscosity value µ = 10 −4 resulted in a good response when compared to the experimental one, in addition to a short processing time.Therefore, the parameters for the CDP adopted for all beams are the same as those determined by Krahl et al. (2018)

Figures 11a and 11b
show the moment-curvature and load-displacement diagrams of the beams, respectively.The results of the load-displacement diagram of the analytical and experimental models were very similar.Therefore, the analytical methodology satisfactorily predicted the behavior of UHPFRC beams.The models were processed with the parameters for the CDP, as shown in Table 1, and with elements C3D8R and T3D2 for concrete and reinforcement, respectively.All computational models satisfactorily represented the experimental behavior of the beams.The difference in the maximum load with the mesh size variation was less than 8%, as shown in Table 6.As can be seen in Figure 12, the models B1 and B4 showed a better post-failure behavior for mesh sizes of 50  and 60 ; by contrast, models B2 and B3 outperformed in the post-peak branch with mesh sizes of 30  and 40 .
Figure 13 shows the crack maps of the computational models for the 30 mm mesh together with the stress distribution in the reinforcement.All computational models predicted flexural failure and showed that cracking was more intense between load application points.The yield started in the longitudinal reinforcements under tensile stress immediately before the beams reached the failure load.

PARAMETRIC STUDY
Based on the calibration of computational models, a parametric study can be conducted to obtain more information on the final capacity of the beams.The new models evaluated the effects of the transverse (ρ w ) and longitudinal (ρ l ) reinforcement ratios on the shear strength of the beams.Overall, a parametric study was performed for 20 cases, as shown in Figure 14 and Table 7.  Figure 15 shows the results for models B1-I, B2-1, B3-I, and B4-I.These models are similar to those of Figure 2, but the difference lies in the fact that they do not have transversal reinforcement.When comparing the load-displacement diagrams in Figure 15 (without stirrups) with those in Figure 12 (with stirrups), the difference is insignificant; that is, the transverse reinforcement ratio does not change the behavior of the beams or their failure modes.Therefore, for these UHPFRC beams, there is no need to use shear reinforcement, which would consequently reduce the cost of labor and materials.
In this regard, through experimental research, Gomaa and Alnaggar (2019) showed that for UHPFRC with 2% fiber content, the failure mode of the beams changed from shear failure to bending failure.This finding confirms the high shear strength achieved by Ultra High-Performance Concrete.To assess the shear strength of the beams without transverse reinforcement, different values were used for the ratio and yield stress in the longitudinal reinforcements.Figures 16 -18 show that models B7 (f y = 600 MPa) and B8 (f y = 500, 550 and 600 MPa) exhibited shear failure, whereas the others exhibited flexural failure, with yield stress in the longitudinal reinforcement.It should be emphasized that models B5, B6, B7, and B8 are those without transverse reinforcement.Bahij et al. (2018) conducted experimental studies on UHPFRC beams with similar characteristics to those of the present study; they used steel rebars with high tensile strength (1320 MPa) and with flexural reinforcement ratios of approximately 2.0% and 3.0%.In the present study, the steel rebars of choice were those with yield stress ratios similar to those of the experimental models.When comparing the experimental beams (B1, B2, B3, and B4) with the model having the lowest shear strength B6 (f y = 600 MPa), it was found that these beams failed by 27.4%, 45.3%, 57.3%, and 67.2% of their shear strength capacity, respectively.
As concrete was crushed with a longitudinal reinforcement ratio of 9.03% for model B8 (f y = 600 MPa), a parametric study of the beams with stirrups was not carried out because a higher ratio would be needed for beam shear failure to occur; however, the limiting factor would still be the crushing of concrete, as can be seen in Table 8 and Figure 17.Graphically, at the maximum load frame, as shown in Figure 18, for the output variable LE, ultimate concrete strain, and S, the ultimate reinforcement stress.

CONCLUSION
This study reported the results of the analytical and computational calibrations of four UHPFRC beams.The calibration of the computational models used different parameters for the CDP, as well as different mesh sizes and types for the finite element analysis.The calibrated models satisfactorily predicted the experimental behavior.The parametric analysis showed that the beams did not require stirrups because the UHPFRC had high shear strength.Finite elements C3D8R / T3D2 has been shown to be effective for use in UHPFRC beams.The analytical model was able to satisfactorily predict the failure load of the beams and proved to be a simple tool for estimating the behavior of UHPFRC beams.

Figure 1 Figure 2
Figure 1 Beam geometry, reinforcement and loading

Figure 3
Figure 3 Geometry, loads and boundary conditions

Figure 4
Figure 4 Behavior of concrete under uniaxial compression

Figure 5
Figure 5 Stress and strain diagrams for cracked cross section

Figure 6
Figure 6 Flowchart to determine the moment-curvature diagram

Figure 7 Figure 8
Figure 7 Variation in the size of the hexahedral mesh for the elements C3D8 (R) and C3D20 (R)

Figure 10
Figure 10 Assessment of parameters for CDP

Figure 11
Figure 11 Diagrams for the behavior of the beams

18 Figure 12 Figure 13
Figure 12 Load curve versus displacement of the beams

Figure 14
Figure 14 Cross section of the beams

Figure 15
Figure 15 Load-displacement curve of the beams without stirrups

Figure 16
Figure 16Load-displacement curve of the beams (30 mm mesh)

Figure 17 Figure 18
Figure 17 Ultimate load diagram versus longitudinal reinforcement ratio of the beams

Table 5
Types of finite elements (see Table1).Paulo Sergio Mota dos Santos Junior et al.

Table 6
Experimental and computational results at failure

Table 8
Computational results of the parametric study of the beams Paulo Sergio Mota dos Santos Junior et al.Latin American Journal of Solids and Structures, 2024, 21(1), e518 15/18