# Nonlinear Semi-Numeric and Finite Element Analysis of Three-Point Bending Tests of Notched Polymer Fiber-Reinforced Concrete Prisms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{sp}) was 125 mm. The measurement of the notched cross-sections showed that the tolerance for the prism dimensions could be assumed to be less than 1 mm (Table 1). Each prism’s notched cross-section width was measured four times, while the height was measured two times, and then the measurements were averaged.

## 3. Results

_{max}) and force (F

_{i}) values corresponding to the specified displacements (F

_{0.47}, F

_{1.32}, F

_{2.17}, and F

_{3.02}) are given in Table 4. The corresponding stress values (residual flexural tensile strength values (σ

_{i})) were calculated by considering the notched cross-section dimensions in Table 1 under the assumption of a linear stress distribution (as specified by the EN14651 standard [19]). The results’ average and standard deviation values are in the last four columns (F

_{aver.i}, F

_{sd.i}, σ

_{aver.i}, and σ

_{sd.i}).

_{c}, while the averaged results for the compressive strength are designated as f

_{c.aver}.

_{cm}(t)) was used:

_{cm.28}is the mean compressive strength of the concrete at age 28 days after pouring. Considering the age of the fiber-reinforced concrete cubes at testing (37 days), the mean compressive strength at an age of 28 days after pouring was estimated to be 44.17 MPa for cubes and 35.34 MPa for cylinders. A value of 1.25 was considered for the ratio of the compressive strength determined on cubes and cylinders (at a concrete age of 37 days after pouring, the estimated “cylinder” mean compressive strength was 36.26 MPa). The characteristic compressive strength determined on cylinders was assumed to be 27.34 MPa (8 MPa difference was considered for the difference between concrete’s characteristic and mean compressive strength). The used relations between the characteristic, mean, “cube” and “cylinder” compressive strength values were from the EN1992-1-1 standard [20]. It can be concluded that the chosen concrete composition (Table 2) resulted in a slightly higher compressive strength as anticipated by the mix-design.

## 4. Calculation

#### 4.1. Moment-Curvature Relation Calculation

_{cs}is the characteristic structural length, considered equal to the cross-section height for elements without conventional reinforcement. The crack width (w), according to the fib Model Code for Concrete Structures 2010 [17], is considered equal to the crack mouth opening displacement (CMOD), which is measured at the bottom of the concrete prism (at the notch start), while the crack tip opening displacement (CTOD), measured at the notch end, is the more precise measure for the crack width (w). The CMOD and CTOD measurement locations are shown in Figure 6.

_{c1}) at the maximum compressive stress (in our case, f

_{cm}), calculated by the following equation from the EN1992-1-1 standard [20]:

_{c1}is calculated in ‰, and f

_{cm}needs to be in MPa. The experimentally determined mean compressive strength of concrete was adjusted considering different ages of the concrete at compressive (37 days) and bending testing (49 days) according to Equation (2).

_{cm.28}) was also calculated from the mean compressive strength at the concrete age of 28 days with the following equation:

_{cm.28}is calculated in GPa, and f

_{cm.28}(the mean compressive strength of concrete at the age of 28 days needs to be in MPa). For the age of 49 days, the adjusted modulus of elasticity was calculated with:

_{ctm}) is linear considering the modulus of elasticity E

_{cm}. The strain at the tensile strength is equal to ε

_{P}. After reaching the tensile strength, the stress-strain diagram curve (line) continues in the direction of the coordinate (ε

_{Q}, 0.2∙f

_{ctm}) but only up to the strain ε

_{c}, which is calculated as the intersection point of the negative slope line of the stress-strain relation of plain concrete, after reaching the tensile strength, and of the line of the stress-strain relation of fiber-reinforced concrete between the coordinates (ε

_{SLS}, f

_{Fts}) and (ε

_{ULS,}f

_{Ftu}) (see Figure 7b,c). The strain ε

_{Q}can be calculated with:

_{F}is the fracture energy of plain concrete. G

_{F}can be calculated with:

_{F}is calculated in N/m (f

_{cm}needs to be in MPa). The strains ε

_{SLS}and ε

_{ULS}correspond to the CMOD values of 0.5 mm and 2.5 mm. However, as explained before, the CTOD values are considered for calculating the strains (according to the relation $\mathsf{\epsilon}=\frac{\mathrm{CTOD}}{{\mathrm{l}}_{\mathrm{cs}}}$). The part of the constitutive law in Figure 7b,c with tensile strains larger than ε

_{c}differs from the constitutive law of plain concrete. It indicates the specifics (differences) of the mechanism of fiber-reinforced concrete’s behavior compared with plain concrete. The fiber action is activated at larger strains (strain values at crack formation) and starts to dominate at larger strains (at larger crack widths).

_{i}in Table 4) at CMOD values of 2.5 mm and 0.5 mm is considered to calculate the residual tensile strength value (${\mathrm{f}}_{\mathrm{Ftu}}$) at the CMOD value of 2.5 mm (an explanation is given in [13]). The fib Model Code for Concrete Structures 2010 [17] also relates the flexural tensile strength to the (uniaxial) tensile strength. This relation results in an estimated tensile strength (based on the average maximum load in Table 4) of 2.89 MPa (for the considered cross-section height of 125 mm, the calculated ratio of the flexural tensile strength and the tensile strength is 1.57), which is close to the tensile strength result of the inverse analysis in Table 6. Interestingly, in [13], expressions for ${\mathrm{f}}_{\mathrm{Ftu}}$ and ${\mathrm{f}}_{\mathrm{Fts}}$ are presented, which are the basis of the fib Model Code for Concrete Structures 2010 [17] expressions; however, they are in a less-simplified form. The expressions are derived by considering a linear distribution of stresses in the cross-section (linear model) and the equilibrium conditions of forces and moments in the cross-section. The resulting residual strength values (0.77 MPa and 0.98 MPa, according to expressions from [13]) are relatively closer to the back-calculated values in Table 6.

_{c}, ε

_{SLS}, and ε

_{ULS}, the values 4.088 × 10

^{−4}, 4 × 10

^{−3}, and 2 × 10

^{−2}were considered. Note that for the moment-curvature relation (based on the simplified expressions for the tensile strength values from the fib Model Code for Concrete Structures 2010 [17]), the strains ε

_{SLS}and ε

_{ULS}were calculated considering the CMOD values for the crack width in Equation (3).

_{bot}) are additionally marked. The maximum load-bearing capacity is reached after the cross-section cracks and the load slightly increases. The strain at the bottom of the notched cross-section, corresponding to the maximum load, also marked in Figure 8, is greater than the strain corresponding to the tensile strength (ε

_{P}). The comparison of the bending moment versus curvature relations of the notched and the gross cross-section shows how the chosen notch depth ensures that at the load corresponding to crack formation in the notched cross-section, the gross cross-section curvature is still in the elastic region. The comparison of the bending moment versus curvature relations of the notched cross-section based on the constitutive law in Figure 7 and based on the constitutive law with tensile strength values according to the simplified expressions of the fib Model Code for Concrete Structures 2010 [17]) shows how the latter assumes an almost constant load-bearing capacity for large strains and how the latter results in a greater load-bearing capacity for strain values close to the strain ε

_{SLS}.

#### Load-Displacement Calculation

_{max})), the flexural deflection was calculated by integrating the curvature over the prism length (κ(x)) with the virtual bending moments resulting from a vertical virtual unit force at the midspan. For load steps corresponding to larger curvature values of the notched cross-section (κ > κ(M

_{max})), the plastic hinge approach was applied. Many studies state different values for the appropriate plastic hinge lengths, as shown in [25], where different studies’ plastic-hinge lengths range from half of the cross-section height to twice the cross-section height for steel fiber-reinforced concrete. The plastic hinge length (L

_{p}) applied for the deflection calculation in this study was equal to the height of the notched cross-section. After all, this value was already used (as the characteristic or structural characteristic length) for transforming the stress versus crack width relation to the stress versus strain relation in Figure 7.

_{max})), the notch effect was considered by linearly varying the curvature over the plastic hinge between the curvature of the gross cross-section (cross-section without the notch) and the curvature of the notched cross-section (Figure 10a). For load steps corresponding to larger curvatures of the notched cross-section (κ > κ(M

_{max})), it was considered that the curvature over the plastic hinge is constant and equal to the notched cross-section curvature (Figure 10b). On the interval 0 ≤ x ≤ (L − L

_{p})/2, the curvature distribution (κ(x)) for each load step was determined from the gross cross-section bending moment versus the curvature relation.

_{max})) was also calculated by integrating the curvatures and the virtual bending moments, a discontinuity in the load-displacement diagram would occur. Therefore, an approximation was used, where the contribution of the plastic hinge rotation to the displacement of a load step was accounted for by considering a plastic hinge rotation (φ

_{pl}) equal to:

_{1}is the curvature of the notched cross-section read from the positive slope part (with κ

_{1}< κ(M

_{max})) of the moment-curvature diagram in Figure 8. As shown in Figure 11, κ

_{1}corresponds to a bending moment equal to the bending moment of the current load step (M(κ

_{1}) = M(κ)).

_{el}) of the prism segment of the plastic hinge is already counted in by the integration of the curvature over the prism length (κ(x)) with virtual bending moments, the curvature κ

_{1}is subtracted from the total curvature of the notched cross-section κ in Equation (11). Finally, the flexural deflection (Δ

_{fl}) of a load step is calculated with:

_{max})), and $\mathsf{\kappa}$ is the curvature of the notched cross-section. Note that the curvature distributions (κ(x)) considered in Equations (12) and (13) both correspond to the curvature distribution in Figure 10a.

_{S}) equal to 0.83 (as, for example, per [26]). The shear displacement Δ

_{s}was calculated with:

_{p})/2 equal to the gross cross-section area and on the interval (L − L

_{p})/2 ≤ x ≤ L/2 equal to the notched cross-section area), V(x) is the shear force distribution, $\overline{\mathrm{V}\left(\mathrm{x}\right)}$ is the virtual shear force distribution, and ν is Poisson’s ratio for concrete (0.2). The total deflection (vertical displacement at the midspan) was calculated as the sum of the flexural (Δ

_{fl}) and shear displacement (Δ

_{s}). The results in the form of a load-displacement curve are given in Figure 12. The load-displacement results are also depicted based on the moment-curvature relation calculated by considering the tensile strength values determined directly by applying the expressions from the fib Model Code for Concrete Structures 2010 [17].

_{calculated}) and the displacement values defined as per the EN14651 standard [19] (δ

_{standard}), which correspond to specific CMOD values and CTOD values, are, together with the corresponding loads (F), given in Table 7 (based on the back-calculated tensile strength values) and Table 8 (based on the tensile strength values from expressions in the fib Model Code for Concrete Structures 2010 [17]) for comparison.

_{calculated}and δ

_{standard}values in Table 7 suggests that the presented calculation model (with the back-calculated tensile strength values) matches with the EN14651 standard [19] relations between, on one side, the CMOD and CTOD values and, on the other side, the vertical displacement values. The comparison of the δ

_{calculated}and δ

_{standard}values in Table 8, with the results based on the tensile strength values from the expressions in the fib Model Code for Concrete Structures 2010 [17]), shows much larger discrepancies, with the difference between the δ

_{calculated}and δ

_{standard}displacement values being larger than 0.6 mm for larger displacements. This is a consequence of the fib Model Code for Concrete Structures 2010’s [17] consideration of the CMOD values as the crack width values. The actual CMOD value is larger than the crack width (CTOD) value. Consequently, for the load stage denoted by CMOD = 2.5 mm in Table 8 for the M-κ (fib) model, the displacements correspond to about 20% larger CMOD values. If this is considered, the actual displacements of the M-κ (fib) calculation at the actual CMOD values of 0.5 mm and 2.5 mm are equal to 0.480 mm and 2.220 mm, which are much closer to the δ

_{standard}values.

#### 4.2. Finite Element Simulation of the Three-Point Bending Test

^{®}Mechanical, Release 2022 R2 [27]. Symmetry boundary conditions were applied, and only a quarter of the concrete prism was modeled. The geometry, coordinate system, loads, and support conditions can be seen in Figure 13.

_{Fo}is the base fracture energy dependent on the maximum aggregate size. In our case, the maximum aggregate size was 16 mm, and the base fracture energy was equal to 30 N/m. The parameter f

_{cmo}is equal was 10 MPa. It must be noted that the fracture energy impacts the calculation in Chapter 4.1 only in the sense of the strain at which the fiber action starts to dominate. In contrast, for the Menetrey-Willam material model, the fracture energy is the most important parameter for defining the exponential yield function in tension.

_{Fts}(as the lower value of f

_{Fts}and f

_{Ftu}) in Table 6 as the residual tensile stress in the Menetrey-Willam material model. In this sense, the finite element analysis results can be considered a lower-bound approximation of the structural behavior.

#### 4.3. Comparison of the Experimental, Moment-Curvature-Based Calculation, and Finite Element Analysis Results

_{Fts}) was considered in the material model. The chosen constant residual tensile strength value equal to f

_{Fts}is also the reason that for the corresponding displacement of 0.47 mm. The finite element results are quite close to the average experimental results. The load and displacement result values at the maximum load, the crack mouth opening displacement of 0.5 mm, and the crack mouth opening displacement of 2.5 mm are collected in Table 11.

## 5. Conclusions

- All fiber-reinforced prisms showed the same failure mode: a crack occurred at the notched cross-section. The fiber-reinforced prisms showed softening behavior (the residual load-bearing capacity was less than the load at (or shortly after) crack formation). However, the load started to increase again for larger displacements. This behavior can be described as post-crack hardening behavior.
- The compressive strength of the fiber-reinforced concrete cubes was slightly lower than the compressive strength of the plain concrete cubes. This can be attributed to the relatively high fiber dosage.
- The nonlinear semi-numeric computational procedure based on the moment-curvature relation with back-calculated tensile strength properties agreed with the averaged experimental results. The result matching was especially good at the maximum load and CMOD values of 0.5 mm and 2.5 mm. The calculated load-displacement curve matched (accurate to around 0.05 mm) the averaged experimental results at the maximum load and at CMOD values of 0.5 mm and 2.5 mm. This agreement can be attributed to the fact that the tensile strength properties for the calculation were determined by an inverse analysis, where the maximum load and loads at presumed CMOD values of 0.5 mm and 2.5 mm were matched with the experimental results. The load-displacement calculation proves that the back-calculated (by inverse analysis) mechanical properties (tensile strength and residual tensile strengths at CMOD values of 0.5 mm and 2.5 mm) are correct.
- The nonlinear semi-numeric computational procedure based on the moment-curvature relation with tensile strength properties calculated with simplified expressions from the fib Model Code for Concrete Structures 2010 [17] was in relatively lesser agreement with the averaged experimental results, whereas the calculated load-displacement curve closely aligned with the averaged experimental result at the maximum load, with an accuracy of approximately 0.02 mm and 0.19 kN, and at the CMOD value of 2.5 mm, with an accuracy of approximately 0.06 mm and 0.01 kN, the alignment of the load-displacement results demonstrated less than satisfactory alignment for the CMOD value of 0.5 mm, with an accuracy of approximately 0.02 mm and 1.75 kN, indicating a notable discrepancy between the experimental and computational outcomes. The relatively lesser agreement can be attributed to the fact that tensile strength properties were calculated with simplified expressions from the fib Model Code for Concrete Structures 2010 [17], which resulted in a notably higher residual tensile strength at the CMOD value of 0.5 mm and a lower residual tensile strength at the CMOD value of 2.5 mm compared with the back-calculated residual tensile strength values. Contrary to the experimental observations and contrary to the back-calculated residual tensile strength values, the residual tensile strength values calculated with simplified expressions from the fib Model Code for Concrete Structures 2010 [17] give an impression of post-crack softening behavior (with the difference between the residual tensile strength at the CMOD value of 0.5 mm being approximately 0.01 kN lower than the residual tensile strength at the CMOD value of 2.5 mm).
- The finite element analysis with the Menetrey-Willam material model, which used the back-calculated tensile strength and considered a constant residual tensile strength equal to the back-calculated residual tensile strength at the CMOD value of 0.5 mm, was in expectedly lesser agreement with the averaged experimental results compared with the computational procedure based on the moment-curvature relation with back-calculated tensile strength properties. In comparison with the computational procedure relying on the moment-curvature relation, wherein tensile strength properties were determined through simplified expressions drawn from the fib Model Code for Concrete Structures 2010 [17], the correlation between the finite element analysis load-displacement outcomes and the experimental results did not exhibit a degradation in quality: the results at the maximum load were accurate to 0.02 mm and 0.05 kN, at the CMOD value of 0.5 mm the results were accurate to 0.02 mm and 0.61 kN, and the results at the CMOD value of 2.5 mm were accurate to approximately 0.05 mm and 1.44 kN. Considering the straightforward Menetrey-Willam material model with its concise input parameters, performing finite element analyses with this material model and a carefully chosen constant residual tensile strength presents a practical way to analyze structural elements made of fiber-reinforced concrete. In this study, we picked the lowest value of the residual tensile strength within the CMOD range of 0.5 mm to 2.5 mm for the constant residual tensile strength. Consequently, the finite element analysis results present a lower limit of the investigated fiber-reinforced concrete prisms’ behavior within the CMOD range of 0.5 mm to 2.5 mm. Although the force results slightly overestimated the load-bearing capacity around a CMOD value of 0.5 mm, the calculated forces still came closer to the experimentally determined value (at a CMOD of 0.5 mm) compared with the results of the nonlinear semi-numeric computational procedure based on the moment-curvature relation where tensile strength properties were determined using simplified expressions from the fib Model Code for Concrete Structures 2010 [17].

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Fiber-reinforced prisms after bending tests (

**left**) and the notched region (

**right**) with the crack after the bending test (prism B23258/1).

**Figure 5.**Concrete cubes after the compression test: unreinforced cube BV1 (

**left**); fiber-reinforced cube ZV1 after the compression test (

**center**); and fiber-reinforced cube ZV1 after the compression test and removal of the cracked parts by hand (

**right**).

**Figure 7.**Constitutive law of fiber-reinforced concrete according to the fib Model Code for Concrete Structures 2010 [17] (the graphs have different scales for clarity): (

**a**) in compression; (

**b**) in tension for small strains of < 1‰; and (

**c**) in tension for large strains of > 1‰.

**Figure 8.**Calculated bending moment versus curvature relation for the gross cross-section, for the notched cross-section with tensile strength values determined from expressions in the fib Model Code for Concrete Structures 2010 [17], and for the notched cross-section with marked strains at the bottom edge (ε

_{bot}) of the notched cross-section (corresponding to specific curvature values).

**Figure 9.**Stress distributions for different load stages (CRACK–load of crack occurrence (ε

_{bot}= ε

_{P}), Mₘₐₓ—maximum bending moment (ε

_{bot}= 0.262∙10⁻³), SLS—load of CMOD = 0.5 mm (ε

_{bot}= ε

_{SLS}), and ULS—load of CMOD = 2.5 mm (ε

_{bot}= ε

_{ULS})).

**Figure 10.**Considered curvature distributions over the prism length prior to reaching the maximum load (

**a**) and after reaching the maximum load (

**b**), and definition of the angle of rotation at the notch after reaching the maximum load (

**c**).

**Figure 11.**Specific points of the bending moment versus curvature relation considered for the displacement calculation.

**Figure 12.**Calculated load-displacement relation (the dotted line presents the results based on tensile strength values from expressions of the fib Model Code for Concrete Structures 2010 [17]).

**Figure 13.**Geometry, coordinate system, loads, and support conditions of the finite element model (

**left**: view of the notched side of the model;

**right**: view of the end face of the model).

**Figure 15.**The deformed shape and visible mesh deformation with vertical displacements contour (displacements in mm) for the final load step with midspan displacement of 3.5 mm.

**Figure 16.**Deformed shape with (unaveraged) equivalent plastic strain contour (strain is unitless) for the final load step with midspan displacement of 3.5 mm.

**Figure 19.**Stress path (

**left**) and normal stress distributions (

**right**) for different load steps (CRACK—last load step prior to occurrence of equivalent plastic strain, Mₘₐₓ—load step with maximum bending moment (or load) at the notched cross-section, SLS—load step corresponding to CMOD = 0.5 mm, and ULS—load step corresponding to CMOD = 2.5 mm).

**Figure 20.**Averaged experimental values (Exp.), values calculated based on the back-calculated tensile strength (M-κ), and values calculated based on tensile strength values from expressions in the fib Model Code for Concrete Structures 2010 [17] (M-κ (fib)) and finite element analysis (FEA) load-displacement relations (with the shaded area representing the envelope of all experimental results).

B23258/1 | B23258/2 | B23258/3 | B23258/4 | B23258/5 | B23258/6 | |
---|---|---|---|---|---|---|

B [mm] | 150.33 | 149.57 | 149.62 | 149.21 | 149.87 | 150.96 |

H_{n} [mm] | 125.06 | 125.03 | 125.01 | 125 | 125.2 | 125.06 |

Component | Dosage [kg/m^{3}] |
---|---|

Aggregate (maximum grain size: 16 mm) | 711 |

Cement (CEM II/A-S 52.5 N) | 135 |

Total water | 71.4 |

Plasticizer | 0.4 |

Barchip 48 fibers | 5 |

Property | Value/Description |
---|---|

Tensile strength [MPa] | 640 |

Young’s modulus [GPa] | 12 |

Length [mm] | 48 |

Base material | Virgin polypropylene |

B23258/1 | B23258/2 | B23258/3 | B23258/4 | B23258/5 | B23258/6 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

i | F_{i}[kN] | σ_{i}[MPa] | F_{i}[kN] | σ_{i}[MPa] | F_{i}[kN] | σ_{i}[MPa] | F_{i}[kN] | σ_{i}[MPa] | F_{i}[kN] | σ_{i}[MPa] | F_{i}[kN] | σ_{i}[MPa] | F_{aver.i}[kN] | F_{sd.i}[kN] | σ_{aver.i}[MPa] | σ_{sd.i}[MPa] |

Max. | 14.63 | 4.67 | 16.57 | 5.31 | 13.53 | 4.34 | 13.06 | 4.20 | 12.43 | 3.97 | 14.64 | 4.65 | 14.14 | 1.345 | 4.52 | 0.43 |

0.47 | 7.40 | 2.36 | 8.56 | 2.75 | 6.27 | 2.01 | 4.53 | 1.46 | 4.88 | 1.56 | 7.59 | 2.41 | 6.54 | 1.462 | 2.09 | 0.47 |

1.32 | 8.80 | 2.81 | 10.11 | 3.24 | 8.32 | 2.67 | 6.61 | 2.13 | 6.48 | 2.07 | 10.02 | 3.18 | 8.39 | 1.449 | 2.68 | 0.46 |

2.17 | 8.96 | 2.86 | 10.22 | 3.28 | 8.15 | 2.61 | 6.07 | 1.95 | 6.42 | 2.05 | 10.71 | 3.40 | 8.42 | 1.749 | 2.69 | 0.55 |

3.02 | 8.62 | 2.75 | / | / | / | / | 5.71 | 1.84 | 6.96 | 2.22 | 11.49 | 3.65 | 8.20 | 2.166 | 2.61 | 0.68 |

Without Fibers | With Fibers | |||||
---|---|---|---|---|---|---|

BV1 | BV2 | BV3 | ZV1 | ZV2 | ZV3 | |

f_{c} | 48.53 | 48.44 | 47.64 | 45.38 | 44.76 | 45.87 |

f_{c.aver.} | 48.21 | 45.33 |

**Table 6.**Input parameters of the stress-strain diagram in Figure 7 (with tensile strength results).

Property | Unit | Value |
---|---|---|

f_{cm} | [MPa] | 37.10 |

E_{cm} | [GPa] | 32.60 |

ε_{c1} | [10^{−3}] | −2.15 |

G_{F} | [N/m] | 139.90 |

ε_{P} | [10^{−3}] | 0.15 |

ε_{C} | [10^{−3}] | 0.45 |

ε_{SLS} | [10^{−3}] | 3.33 |

ε_{ULS} | [10^{−3}] | 16.67 |

f_{ctm} | [MPa] | 2.82 |

f_{Fts} | [MPa] | 0.75 |

f_{Ftu} | [MPa] | 1.07 |

**Table 7.**Comparison of the calculated values (δ

_{calculated}), based on the back-calculated tensile strength values, and displacement values defined as per EN14651 standard [19] (δ

_{standard}), corresponding to specific CMOD values (with corresponding CTOD values, calculated curvature values (κ) of the notched cross-section, and load values (F)).

CMOD [mm] | CTOD [mm] | κ [1/m] | δ_{standard}[mm] | δ_{calculated}[mm] | F [kN] |
---|---|---|---|---|---|

0. | 0 | 2.27 × 10^{−3} | 0.04 | 0.04 | 12.31 |

2.11 × 10^{−2} | 1.76 × 10^{−2} | 0.0036 | 0.06 | 0.05 | 14.14 |

0.5 | 0.42 | 0.0301 | 0.47 | 0.47 | 6.54 |

1.5 | 1.25 | 0.0860 | 1.32 | 1.35 | 7.43 |

2.5 | 2.08 | 0.1416 | 2.17 | 2.22 | 8.42 |

3.5 | 2.92 | 0.1971 | 3.02 | 3.08 | 9.40 |

**Table 8.**Comparison of the calculated values (δ

_{calculated}), based on tensile strength values from expressions in the fib Model Code for Concrete Structures 2010 [17]), and displacement values defined as per EN14651 standard [19] (δ

_{standard}) corresponding to specific CMOD values (with corresponding CTOD values, calculated curvature values (κ) of the notched cross-section, and load values (F)).

CMOD [mm] | CTOD [mm] | κ [1/m] | δ_{standard}[mm] | δ_{calculated}[mm] | F [kN] |
---|---|---|---|---|---|

0 | 0 | 2.24 × 10^{−3} | 0.04 | 0.04 | 12.35 |

2.09 × 10^{−2} | 1.74 × 10^{−2} | 0.0036 | 0.06 | 0.05 | 14.33 |

0.5 | 0.42 | 0.0362 | 0.47 | 0.57 | 8.29 |

1.5 | 1.25 | 0.1032 | 1.32 | 1.62 | 8.38 |

2.5 | 2.08 | 0.1694 | 2.17 | 2.65 | 8.41 |

3.5 | 2.92 | 0.2354 | 3.02 | 3.68 | 8.41 |

Property | Unit | Value |
---|---|---|

Young’s modulus | MPa | 3.26 × 10^{4} |

Poisson’s ratio | / | 2.00 × 10^{−1} |

Bulk modulus | MPa | 1.81 × 10^{4} |

Shear modulus | MPa | 1.36 × 10^{4} |

Uniaxial compressive strength | MPa | 3.71 × 10^{1} |

Uniaxial tensile strength | MPa | 2.82 × 10^{0} |

Biaxial compressive strength | MPa | 4.31 × 10^{1} |

Plastic strain at uniaxial Compressive strength | / | 1.69 × 10^{−3} |

Plastic strain at transition from power law to exponential softening | / | 3.04 × 10^{−3} |

Relative stress at start of nonlinear hardening | / | 3.66 × 10^{−1} |

Residual relative stress at transition from power law to exponential softening | / | 5.11 × 10^{−1} |

Residual compressive relative stress | / | 2.00 × 10^{−1} |

Mode 1 area specific fracture energy | N/m | 7.51 × 10^{1} |

Residual tensile relative stress | / | 2.66 × 10^{−1} |

Dilatancy angle | ° | 9 |

**Table 10.**Finite element analysis results for CMOD, CTOD, vertical displacement (δ), and load (F) (the row values correspond to the same load step).

CMOD [mm] | CTOD [mm] | F [kN] | δ [mm] | CMOD/CTOD [/] |
---|---|---|---|---|

0.0032 | 0.0007 | 3.01 | 0.01 | 4.79 |

0.0417 | 0.0271 | 14.09 | 0.07 | 1.54 |

0.5036 | 0.4113 | 7.15 | 0.45 | 1.22 |

2.4978 | 2.0694 | 6.98 | 2.12 | 1.21 |

Average Experimental Load | Corresponding Experimental Displacement | M-κ Calculated Load | M-κ Calculated Displacement | M-κ (fib) Calculated Load | M-κ (fib) Calculated Displacement | FEA Load | FEA Displacement | |
---|---|---|---|---|---|---|---|---|

[kN] | [mm] | [kN] | [mm] | [kN] | [mm] | [kN] | [mm] | |

Max. load | 14.14 | 0.053 * | 14.14 | 0.051 | 14.33 | 0.051 | 14.09 | 0.067 |

CMOD = 0.5 mm | 6.54 | 0.465 | 6.54 | 0.473 | 8.29 | 0.480 | 7.15 | 0.452 |

CMOD = 2.5 mm | 8.42 | 2.165 | 8.42 | 2.216 | 8.41 | 2.220 | 6.98 | 2.117 |

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## Share and Cite

**MDPI and ACS Style**

Unuk, Ž.; Kuhta, M.
Nonlinear Semi-Numeric and Finite Element Analysis of Three-Point Bending Tests of Notched Polymer Fiber-Reinforced Concrete Prisms. *Appl. Sci.* **2024**, *14*, 1604.
https://doi.org/10.3390/app14041604

**AMA Style**

Unuk Ž, Kuhta M.
Nonlinear Semi-Numeric and Finite Element Analysis of Three-Point Bending Tests of Notched Polymer Fiber-Reinforced Concrete Prisms. *Applied Sciences*. 2024; 14(4):1604.
https://doi.org/10.3390/app14041604

**Chicago/Turabian Style**

Unuk, Žiga, and Milan Kuhta.
2024. "Nonlinear Semi-Numeric and Finite Element Analysis of Three-Point Bending Tests of Notched Polymer Fiber-Reinforced Concrete Prisms" *Applied Sciences* 14, no. 4: 1604.
https://doi.org/10.3390/app14041604