# Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}) to connect continuum mechanics, governed by a stress–strain constitutive relationship (σ − ε), and fracture mechanics, governed by a stress–crack opening relationship (σ − w). Hence, the aforementioned internal length is not a structural material property, but an artifact related to the length of the fracture process zone, the width of the band and the dimensions of the specimen. To properly perform this connection and prevent mesh dependency, several methods have related the internal length to physical parameters, such as maximum aggregate size for non-local approaches [17,18], or element size for local approaches [15,16]. In addition, several numerical crack models have been developed, such as the discontinuous numerical modeling of cracks using embedded discontinuities [19], the discrete strong discontinuity approach [20,21,22], dynamic fragmentation [23], the sequentially linear analysis method [24], or even a machine learning [25].

_{cs}) is defined as a parameter to convert stress–crack width (σ − w) curves into stress–strain (σ − ε) curves [26]. Although this structural length is used like the internal length (L

_{i}) mentioned above, it has a different meaning. While the first one, L

_{i}, is related to the distance between cracks and the depth of the neutral axis (macroscopic effects), the second one depends on the length of the fracture process zone (material properties) [27,28]. In all cases, these length parameters convey the idea that FRC is a continuous material, when in fact the main consequence of concrete post-cracking behavior is crack localization, which introduces a discontinuity in the material. Moreover, the L

_{i}parameter implies a structural dependence on the stress–strain curve (σ − ε), despite it should be totally independent of the type of structure considered.

## 2. Materials Hypothesis

#### 2.1. Non-Cracked Zone

#### 2.2. Cracked Zone

_{Fts}represents the serviceability residual strength, defined as the post-cracking strength for serviceability crack openings, and f

_{Ftu}represents the ultimate residual strength. f

_{Fts}and f

_{Ftu}are calculated through the residual values of flexural strength by using the following equations:

_{R}

_{1}is the residual flexural strength corresponding to a crack mouth opening displacement (CMOD) of 0.5 mm and f

_{R}

_{3}is the residual flexural strength corresponding to a CMOD of w

_{u}. These parameters are determined by performing a three-point bending test, on a notched beam, according to [36] (see Figure 3). w

_{u}is usually taken as 2.5 mm.

_{Ftu}in this linear model depends on the required ductility that is related to the allowed crack width. The ultimate crack width should not exceed 2.5 mm in any case. From Figure 2, the crack opening, w, in post-cracking constitutive law can be expressed as a function of the residual stress, σ:

_{F,FRC}. This area represents the theoretical energy required to open a unit area of crack surface, considering a linear softening.

## 3. Modeling of Crack Propagation

_{n}.

_{n}= y/h as the non-dimensional depth of the neutral axis; these parameters vary between 0 and 1. The non-dimensional crack opening is obtained by dividing it by the ultimate crack width, w* = w/w

_{u}.

_{b}and the stress at the top is σ

_{t}. Non-dimensional stresses are obtained by dividing by the serviceability residual strength, f

_{Fts}. So, we define σ

_{b}* = σ

_{b}/f

_{Ft}

_{s}and σ

_{t}* = σ

_{t}/f

_{Fts}.

_{b}< w

_{u}, and it is in case 2 when the crack mouth opening is bigger than the ultimate crack width w

_{b}> w

_{u}, see Figure 4.

_{0}. This value increases monotonically during the cracking process, so case 2 represents a decreasing curve [37] and for this reason, our study is focused on the development of case 1, where maximum load takes place.

_{b}(M,z) can be evaluated by the expression given by [34]. w

_{b}(σ

_{b}) is defined considering the softening law, Equation (3). Thus, Equation (9) can be expressed as:

_{b}*, σ

_{t}*, γ

_{n}, and M*, the only input data is β

_{H,FRC}. The crack depth, ξ, is used as a control parameter during the crack process. For each crack depth, only one equilibrium solution exists.

_{Ftu}and f

_{Fts}, (f

_{Ftu}/f

_{Fts}). Crack opening depends on the brittleness and on the α ratio previously defined. The maximum value for w

_{b}* in case 1 is 1.0. Once this value is surpassed, case 2 applies.

## 4. Model Response and Experimental Validation

_{H,FRC}influences the behavior of the FRC section. In Figure 5a,b, the x-axis represents the non-dimensional crack mouth opening, w

_{b}*, and the y-axis the non-dimensional bending moment during crack growth, M*. In Figure 5a, the ratio between f

_{Fts}and f

_{Ftu}, α, has a constant value of 0.8. The initial point of all curves is the cracking moment. If we consider an elastic material, this crack has a non-dimensional value of 0.167. As the brittleness number decreases, peak load increases. Thus for smaller values of β

_{H,FRC}the softening length development is bigger. Therefore, this is the main reason for the increase in peak load. Section behavior is analyzed through moment versus crack opening curves, instead of the moment versus curvature curves, normally used in reinforced concrete section design. These curves give a more physical approximation to the FRC sections’ flexural behavior.

_{Ftu}and f

_{Fts}, α, in the fiber reinforced concrete behavior. As in the previous case, the x-axis represents the non-dimensional crack mouth opening, w

_{b}*, and the y-axis represents the non-dimensional bending moment during crack growth, M*. The brittleness number β

_{H,FRC}has a constant value of 0.01 in the results shown. The peak load is not influenced by this parameter, as it is shown in the figure. As α increases, the nondimensional crack opening also increases. Therefore, when the slope of the softening curve decreases, so does the slope of the moment-opening curve after the peak.

_{H,FRC}= 0.01 and α = 0.4 in Figure 5b, or the maximum is reached in the interval limit, w

_{b}* = 1.00, see, for example, the curve for β

_{H,FRC}= 0.1, and α = 0.8 in Figure 5a. This last case takes place when w

_{u}* is reached at the bottom part of the crack, and the maximum is not within the interval [0, 1] of w

_{b}*.

^{3}, which is a usual range in fiber reinforced concrete elements. Figure 6 and Table 1 show the comparison, the x-axis represents the brittleness number, β

_{H,FRC}, and the y-axis the maximum non-dimensional bending moment during crack growth, M

_{max}*. The model follows the experimental trends of experimental results with good agreement. Dotted horizontal lines represent the theoretical limits to the non-dimensional moment M

_{max}* as will be explained in the next section.

_{max}* depending on the brittleness number, β

_{H,FRC}, is easy to obtain.

## 5. Size Effect on Flexural Strength for FRC

_{H,FRC}, and the y-axis represents the non-dimensional flexural strength, f

_{R}*, which is defined as the ratio between the flexural strength, f

_{R}, and f

_{Ft}

_{s}. The results obtained with the model are plotted with the expressions for the flexural strength evaluated by Uchida et al. [45] and Planas et al. [46]. These expressions were derived following a classical computational approach based on a cohesive constitutive law, in which secondary cracking is neglected [8]. Model results follow the same trend than the computational results, and they show the dependency of the flexural strength on the brittleness number, which represents the intrinsic size of the section.

_{R}* → 3 for β

_{H,FRC}→ 0 (plastic limit solution for cohesive cracks, M

_{max}* = 0.5) and f

_{R}* → 1 for β

_{H,FRC}→ ∞ (linear elastic solution, M

_{max}* = 0.166). Moreover, the plot represents the usual brittleness number ranges for FRC and plain concrete. For FRC members, the flexural strength can be 150% to 200% higher than the tensile strength (f

_{Fts}in FRC) while for plain concrete this range is around 10–25%.

_{H,FRC}, the FEM approach shows an asymptotic behavior that does not satisfy the plastic limit solution, f

_{R}* → 3, see Figure 7a. The existence of a non-negligible compressed area in the upper zone of the section stalls the crack growth, and therefore the plastic limit solution cannot be reached. Thus, for structural engineering purposes, it may be convenient to limit the value of f

_{R}* to 2.5.

_{H,FRC}, and the y-axis represents the value of σ

_{b}*, γ

_{n}, and ξ obtained with the proposed model.

_{H,FRC}increases, the crack depth and the depth of the neutral axis decrease. Crack depth shows an asymptotic trend of 0, and the neutral axis of 0.5, as is expected for the linear elastic solution. σ

_{b}* also decreases as β

_{H,FRC}increases, but its influence on the flexural strength gets smaller because the softening zone also shrinks, compared to the depth, as β

_{H,FRC}increases. As an example of how the stress profiles vary, Figure 9 plots the non-dimensional stress distributions for several values of β

_{H,FRC}, namely 1, 0.1, 0.01, and 0.001.

## 6. Practical Expression to Determine the Flexural Strength in FRC

_{R}*, can be expressed, fitting our model results, as:

_{Fts,d}and f

_{Ftu,d}.

_{R1}given in Table L.2: Residual Strength Classes for SRFC (Annex L) from Eurocode 2 draft [47] and Equation (22), it is possible to draw the maximum bending moment versus f

_{R1}. Figure 11 shows the variation of the curve depending on the ductility class. We considered two depths of the section (20 and 40 cm), for a characteristic compressive strength of concrete equal to 25 MPa. Elasticity modulus has been calculated according to Model Code 2010 [7].

_{H,FRC}, and α where equations are valid.

_{Ftu}/f

_{Fts}between 0 and 0.8 for conventional FRC, which correspond with ratios f

_{R}

_{3}/f

_{R}

_{1}in the range a–d according to Mode Code 2010 [7]. So, we conclude that the proposed expression covers most practical applications and can be used for structural design. In all cases, as mentioned in Section 5, for structural engineering purposes, it may be convenient to limit the value of f

_{R}* to 2.5.

## 7. Brittle–Ductile Transition in Flexural Failure for FRC Sections

_{cr}is the cracking moment of the concrete matrix and M

_{max}is described in Section 5 and can be evaluated from the model showed in this paper. Thus, Equation (24) can be expressed as:

_{t}is the tensile strength of the concrete matrix and f

_{R}is the flexural strength or modulus of rupture. So, the behavior of the FRC section can be described in a non-dimensional form as:

_{t}*, is equal to f

_{t}/f

_{Fts}. So, if f

_{t}* is bigger than f

_{R}*, fiber concrete section presents a brittle behavior, and, if f

_{t}* is lower than f

_{R}*, fiber concrete section presents a ductile behavior. Rearranging Equation (20) as a function of β

_{H,FRC}, and considering the limit condition in Equation (25), Equation (28) gives the value of the maximum brittleness number that has a ductile behavior for a given base concrete tensile strength. In other words, if a section has a brittleness number, β

_{H,FRC}, less than β

_{H,FRC,max}, for a given value of f

_{t}*, this section will be ductile.

_{t}, and the brittleness number, β

_{H,FRC}. In Figure 14, the x-axis represents the brittleness number, β

_{H,FRC}, and the y-axis the non-dimensional tensile strength of the base concrete, f

_{t}*. Two areas are drawn defining the FRC section behavior (ductile or brittle). The boundary curve between brittle–ductile behavior is given by Equation (28).

## 8. Conclusions

- The planar crack assumption can be considered as an alternative to Navier’s hypothesis to model the FRC cracked zone. Using this approach, we avoid using length parameters as l
_{cs}to evaluate strains from crack openings, as is commonly carried out in models based on stress–strain laws. - We propose a brittleness number, β
_{H,FRC}, analogous to the one of Hillerborg, as a characterization parameter of FRC structural sections. It is derived from a nondimensional analysis, which includes the beam size and FRC softening characteristics. - The model fits experimental results very well. Moreover, the model reproduces the asymptotic behavior expected from plastic limit solution for cohesive cracks—very short depths—to the linear elastic solution—overly large depths.
- We offer an expression to calculate the flexural strength of a fiber-reinforced concrete section based on the model results. It depends on the brittleness number, β
_{H,FRC}and on the serviceability residual stress, f_{Fts}. Its range of validity covers most of practical cases and thus, it can be profitably used for the structural design of FRC sections. - The model, also, allows studying the ductile–brittle transition in FRC sections. It depends only on two parameters, namely the related tensile strength of the base concrete, f
_{t}, and the brittleness number, β_{H,FRC}. - The planar crack model contributes to a better understanding of the nature of flexural behavior of FRC sections and gives a more physical approach to their failure behavior. In addition, the expressions derived from the model results can be used for structural engineering purposes, constituting a design toolset that avoids complex calculations through finite elements.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

b | beam width |

E_{c} | longitudinal elastic modulus of concrete |

f_{Fts} | serviceability residual stress of concrete |

f_{Ftu} | ultimate residual stress of concrete |

f_{R} | flexural strength |

f_{R}* | non-dimensional flexural strength |

f_{R1} | residual flexural tensile strength corresponding to CMOD = 0.5 mm |

f_{R3} | residual flexure tensile strength corresponding to CMOD = 2.5 mm |

f_{t} | tensile strength of concrete |

f_{t}* | non-dimensional tensile strength of concrete |

A_{F,FRC} | area under the softening crack law (energy per unit area to open a crack up to the ultimate crack opening) |

h | beam height |

l_{cs,FRC} | structural characteristic length of FRC |

L_{i} | specific internal length |

M | bending moment |

M* | nondimensional bending moment |

M_{cr} | critical bending moment |

M_{max} | maximum bending moment during crack growth |

M_{max}* | maximum nondimensional bending moment during crack growth |

w | crack opening |

w* | nondimensional crack opening |

w_{b} | crack mouth opening displacement (opening of the crack at the bottom edge of the section) |

w_{b}* | non-dimensional crack mouth opening displacement |

w_{u} | ultimate crack opening |

y_{n} | neutral axis depth |

z | crack depth |

z_{0} | crack depth at critical opening |

α | relation between f_{Ftu} and f_{Fts} |

β_{H,FRC} | brittleness number for FRC |

γ_{n} | non-dimensional neutral axis depth |

ε | strain |

ε_{b} | strain at bottom part of the beam |

ε_{t} | strain at the top part of the beam |

ξ | non-dimensional crack depth |

ξ_{max} | maximum nondimensional crack depth |

σ | stress |

σ_{b} | stress at the bottom part of the beam |

σ_{b}* | non-dimensional stress at the bottom part of the beam |

σ_{t} | stress at the top part of the beam |

σ_{t}* | non-dimensional stress at the top part of the beam |

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**Figure 1.**Material hypotheses. Cracked area is modeled according to the crack planar hypothesis and non-cracked area according to Navier’s hypothesis.

**Figure 2.**Simplified post-cracking constitutive law: stress–crack opening (softening post-cracking behavior). Model Code 2010 [7].

**Figure 5.**Influence of (

**a**) the brittleness number, β

_{H,FRC}; and (

**b**) the ratio between f

_{Ftu}and f

_{Fts}, α.

**Figure 7.**(

**a**) Dependency of the flexural strength on the brittleness number, β

_{H,FRC}; (

**b**) Asymptotic behavior.

**Figure 8.**Non-dimensional stress at the bottom part of the beam, σ

_{b}*, non-dimensional neutral axis depth, γ

_{n}, and non-dimensional crack depth, ξ, as a function of the brittleness number, β

_{H,FRC}.

**Figure 10.**Planar crack model results for the non-dimensional flexural strength, f

_{R}*, and corresponding crack-depth, ξ

_{max}, fitted by (

**a**) f

_{R}* given by Equation (20), and (

**b**) ξ

_{max}given by Equation (22), respectively.

**Figure 11.**Maximum bending moment versus residual strength f

_{R1}for two section’s depths: (

**a**) 20 cm; (

**b**) 40 cm.

**Table 1.**Experimental results from the bibliography compared in Figure 6 with numerical results from the planar crack model.

Reference | V_{f} (kg/m^{3}) | P_{max} (kN) | f_{R,1} (MPa) | f_{R,3} (MPa) | b (m) | h (m) | L (m) | E_{c} (MPa) | w_{u} (mm) | f_{Fts} (MPa) | f_{Ftu} (MPa) | A_{f,FRC} (N/mm) | β_{H} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

40 kg/m^{3} [38] | 40 | 16.00 | 7.13 | 5.69 | 0.10 | 0.20 | 1.20 | 35,728 | 2.50 | 3.21 | 1.42 | 5.79 | 0.0080 |

Malgorzata-0.57H [39] | 45 | 16.29 | 3.61 | 2.22 | 0.15 | 0.15 | 0.50 | 43,281 | 2.50 | 1.63 | 0.39 | 2.52 | 0.0034 |

Malgorzata-0.57CH [39] | 45 | 16.49 | 3.27 | 1.78 | 0.15 | 0.15 | 0.50 | 41,903 | 2.50 | 1.47 | 0.24 | 2.13 | 0.0035 |

Michels. 0.65% [40] | 51 | 31.40 | 7.73 | 6.93 | 0.15 | 0.15 | 0.60 | 30,000 | 2.50 | 3.48 | 1.92 | 6.75 | 0.0062 |

Michels. 0.52% [40] | 41 | 30.60 | 7.47 | 6.67 | 0.15 | 0.15 | 0.60 | 30,000 | 2.50 | 3.36 | 1.84 | 6.50 | 0.0061 |

Zhang (78.4 kg/m^{3}) [41] | 78 | 16.50 | 9.60 | 6.00 | 0.10 | 0.10 | 0.40 | 32,000 | 2.50 | 4.32 | 1.08 | 6.75 | 0.0081 |

Doo-S13 (157 kg/m^{3}) [42] | 157 | 27.90 | 12.15 | 9.00 | 0.10 | 0.10 | 0.30 | 50,876 | 2.50 | 5.47 | 2.07 | 9.42 | 0.0053 |

Doo-S16.3 (157 kg/m^{3}) [42] | 157 | 32.90 | 13.50 | 12.60 | 0.10 | 0.10 | 0.30 | 46,260 | 2.50 | 6.08 | 3.60 | 12.09 | 0.0043 |

Doo-S19.5 (157 kg/m^{3}) [42] | 157 | 37.90 | 14.85 | 15.75 | 0.10 | 0.10 | 0.30 | 46,126 | 2.50 | 6.68 | 4.91 | 14.48 | 0.0031 |

Barros (60 kg/m^{3}) [43] | 60 | 11.50 | 2.20 | 1.90 | 0.15 | 0.15 | 0.45 | 33,366 | 2.50 | 0.99 | 0.51 | 1.88 | 0.0017 |

Barros (45 kg/m^{3}) [43] | 45 | 7.50 | 1.40 | 1.20 | 0.15 | 0.15 | 0.45 | 33,935 | 2.50 | 0.63 | 0.32 | 1.19 | 0.0011 |

Ali (60 kg/m^{3}) [44] | 60 | 28.00 | 7.00 | 5.67 | 0.15 | 0.15 | 0.60 | 34,484 | 2.50 | 3.15 | 1.43 | 5.73 | 0.0060 |

Ali (40 kg/m^{3}) [44] | 40 | 28.00 | 7.33 | 6.33 | 0.15 | 0.15 | 0.60 | 35,808 | 2.50 | 3.30 | 1.70 | 6.25 | 0.0054 |

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**MDPI and ACS Style**

Carmona, J.R.; Cortés-Buitrago, R.; Rey-Rey, J.; Ruiz, G.
Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections. *Materials* **2022**, *15*, 5821.
https://doi.org/10.3390/ma15175821

**AMA Style**

Carmona JR, Cortés-Buitrago R, Rey-Rey J, Ruiz G.
Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections. *Materials*. 2022; 15(17):5821.
https://doi.org/10.3390/ma15175821

**Chicago/Turabian Style**

Carmona, Jacinto R., Raúl Cortés-Buitrago, Juan Rey-Rey, and Gonzalo Ruiz.
2022. "Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections" *Materials* 15, no. 17: 5821.
https://doi.org/10.3390/ma15175821