# Non-Linear Analysis of Flat Slabs Prestressed with Unbonded Tendons Submitted to Punching Shear

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Analysis

#### 2.1. Solution Methods

^{3}.

^{−2}, 10

^{−2}, and 10

^{−4}, for displacement, force, and residual energy, respectively.

#### 2.2. Concrete

_{c}is the crack width where the stress is zero, and σ is the normal stress in the crack. The constant values are c

_{1}= 3 and c

_{2}= 6.93. The parameter G

_{f}represents the fracture energy required for the crack to be stress-free. The standard value adopted by the program was based on [26], as shown in Equation (2).

_{c}

_{1}= 0 and σ

_{c}

_{2}= f’

_{c}to the region of tension with compression, with a linear decrease in force.

_{t}. In the state of tensile stress with compression, the tensile force is reduced by Equation (5).

_{c}

_{1}and σ

_{c}

_{2}are the principal stresses in concrete, f′

_{c}is the uniaxial cylinder strength, r

_{ec}is the reduction factor of the compressive strength in principal direction two due to the tensile stress in principal direction one, and r

_{et}is the reduction factor of the tensile strength in principal direction two due to the compressive stress in principal direction one.

_{ts}f

_{t}(Figure 4). The recommended default value for C

_{ts}= 0.4 was considered, as indicated by the model code [14].

#### 2.3. Steel

## 3. Numerical Models

^{2}and a thickness of 160 mm. This experimental program used to calibrate the numerical models. In this section, the modeling of slabs M1 (reinforced concrete) and M4 (prestressed concrete) will be presented. Both models were without shear reinforcement.

#### 3.1. M1 Slab in Reinforced Concrete

_{c}is the strength of concrete, f

_{ct}is the tensile strength of concrete, E

_{c}is the modulus of elasticity of concrete, Ø

_{s}is the steel diameter, f

_{sy}is the yield stress of steel, f

_{sr}is the failure strength of steel, and E

_{s}is the modulus of elasticity of steel.

#### 3.2. M4 Slab in Prestressed Concrete

- Application of incremental displacement on the column until the slab reaches a reaction of 80 kN.
- Application of a standard force on each of the prestressing tendons by the command Prestressing for reinf line, under boundary conditions.
- At the end of prestressing, the master-slave node connections between the end of the tendon and the node closest to the anchor plate are activated. Finally, incremental displacement is applied to the column until the slab reaches failure.

_{FEA}/V

_{exp}ratio was 1.016, which is sufficiently accurate.

## 4. Parametric Study

- Series A: three models varying the spacing between the central prestressing tendons;
- Series T: seven models varying the thickness of the slab;
- Series C: fifteen models varying the column rectangularity.

^{3}.

#### Comparison of Results

_{c,máx}), and the V/V

_{M}

_{4}ratio were analyzed.

_{c,máx}). For the models with greater spacing, as shown in the table, a reduction in the maximum strain is observed, which indicates a more fragile rupture, justified by the anticipation of the rupture of models.

_{c,máx}) and the V/V

_{M}

_{4}ratio of the Series T are compared and are presented in Table 4. Based on the presented data, it is noticeable how alterations in the thickness of the slab interfere in its capacity of resistance and in the rigidity of the models. The configurations showed contributions from 16% to 138% for the ultimate load and, for some, a displacement reduction of up to 20%. Regarding the strain of the concrete, a behavior very similar to that of the reference slab E-16 is observed, with a reduction of −5.77% for the E-30 model.

_{c,máx}) and the V

_{C}/V

_{M}

_{4}ratio of the 16 slabs were analyzed.

## 5. Normative Comparison

_{FEA}) is arranged with the estimated ultimate load predicted in the normative codes (V

_{R}). For the calculation of the breaking load values, all partial safety factors provided for in the design codes were considered as equal to 1.

#### 5.1. Comparison of Series A Results

_{FEA}/V

_{R}, ratio, along with their respective mean values, standard deviation (SD.), and coefficient of variation (C.V.). Their respective breaking load evolution is shown in Figure 18.

#### 5.2. Comparison of Series T Results

_{FEA}/V

_{R}ratio was also used to compare the normative codes and the numerical results obtained from the 8 Series T models, as shown in Figure 19.

#### 5.3. Comparison of Series C Results

_{FEA}/V

_{R}ratio are in Table 7, and the evolution of the failure estimate is shown in Figure 20.

_{1}, which, for the calculation check, considers that the maximum measurement of the column does not exceed 3d. With this, it was observed that 10 of the 16 models presented very conservative results, even at the highest level of the fib model code [14], which means that the equations require adjustments.

_{R,FEM}/V

_{R,NBR}ratio did not exceed 1.34, which indicates estimates always in favor of safety. Regarding Eurocode 2 [12], it can be noted that the results differ from the Brazilian standard due to the fact that its greater conservatism regarding the scale factor results in it being set at 2.0. The ACI [11] showed satisfactory results, with an average of 1.26; however, the dispersion percentage of 13.26% should be noted as well as the more satisfactory results in the models with columns of larger dimensions.

## 6. Conclusions and Suggestions for Future Investigations

- Series A: the increase in spacing between tendons promoted a reduction in punching resistance and a decrease in concrete strain. The panorama of cracking of the slab with smaller spacing showed a greater concentration of cracks and strains, accompanied by an increase in resistance.
- Series T: the thickness of the slab causes great interference in the resistance and rigidity capacity of the models. The models revealed that strains in thicker slabs tend to concentrate, before failure, in a more central point and on the upper face of the slab.
- Series C: the rectangularity of the column has a great influence on the resistance capacity of prestressed flat slabs. Larger strains are achieved by slabs with a smaller column contact area. The larger support area for the prestressing tendons contributes to increased resistance.

- Series A: in general, the rules worked in favor of safety. Note that the design codes did not show a significant parameter capable of interfering with the results when changing the spacing between the tendons. It was identified that the fib model code [14] presents the best estimates even in the LoA III approximation, where the code considers that this level is not suitable for estimating cracked elements. ACI [11] was the most conservative in the estimates.
- Series T: it was observed that the fib model code [14] has the best trend of results as the thickness of the slab is increased; additionally, there is little difference in the results in its two levels of approximation. In general, the other standards tend to show the same behavior, except for ACI [11], which is characterized by more conservative behavior in estimates.
- Series C: the predictions of Eurocode 2 [12] and NBR [13] follow the same trend of behavior, differing due to the scale factor k. Considering the coefficient of variation and the average, the best results were due to NBR [13], followed by ACI [11]. In the case of the ACI [11], better predictions were noted with the increase in the rectangularity of the column. For the fib model code [14], it is observed that, in levels II and III, there is a stability of the results from the C-40 Series, which is justified due to the discontinuity of the critical perimeter, limited to 3d.

- It is proposed to use the constitutive model adopted in this work to simulate more experimental tests of slabs and beams, as well as other types of reinforced and prestressed concrete structures, such as columns, corbels, and shells.
- The simulation of isolated connections between columns and slabs is adequate for the localized punching analysis, but it does not cover the existing membrane forces in the global model (panels of slabs). With the facilities offered by computational tools, it would be interesting to model a slab supported by several pillars to study this phenomenon.
- Regarding prestressing, little information was found in the literature review and in the standards regarding the differences between the types of prestressing (bonded or unbonded). It is recommended to carry out a study to compare these two alternatives in behavior from flat slabs to punching. An experimental study of prestressed slabs is also suggested to confirm the numerical results achieved in this research.
- It is proposed to perform modeling of concrete slabs with shear reinforcement, varying the number of layers and radial spacing between elements.
- It is proposed to analyze punching situations for slabs with applied moments (eccentric punching).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{R}), differentiating by the subindexes with the abbreviations of each normative code, and the other parameters, with the same uniform nomenclature according to the following equations for dimensioning prestressed flat slabs.

_{g}is the maximum aggregate size in mm and ψ is the slab rotation. In the case of prestressed slabs, rotation can be calculated using Equations (A7) and (A8).

_{s}is the distance between the geometric center of the column and the radius at which the bending moment is zero; f

_{yd}is the design value of the bending reinforcement yield stress; E

_{s}: modulus of elasticity of bending reinforcement; m

_{sd}is the average bending moment per unit width in the control strip; b

_{s}= 1.5r

_{s}, or internal slab–column connections, where it can be considered V

_{Ed}/8 (V

_{Ed}is the shear force acting on the slab); m

_{Rd}: average design flexural strength per unit length in a loaded range, according to Equation (A9).

_{p}is the moment of decompression in the width of the band b

_{s}, which can be determined by the isostatic moment of the prestressing according to the equation:

_{p}is the normal force due to the sum of the prestressing of each strand that intersects the control strip (b

_{s}); e

_{p}is the eccentricity of the normal force in relation to the geometric center of the section at the point where the prestressing tendons intersect the control perimeter (b

_{1}).

_{cp}: plastic strength of concrete, given by ${f}_{\mathit{cp}}{\left(\frac{30\mathrm{MPa}}{{f}_{c}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\le {f}_{c}$ and ${\chi}_{\mathit{pl}}$: height of the concrete’s compressed region in the section, assuming a constant plastic stress distribution.

_{1}: critical control perimeter, according to each standard;

_{c}is the compressive strength of concrete;

_{p}: increase in vertical force due to prestressing in the b

_{s}control range for the fib model code [14]. For ACI [11], EC2 [12], and NBR [13], it is calculated in the critical control perimeter b

_{1}.

_{s}: constant, equal to 40 for internal columns;

^{3};

_{cp}: average compressive stress caused by prestressing;

_{c}: the value of 1.0 was assigned to determine the maximum value of the resistant load;

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**Figure 3.**Biaxial failure function for concrete [18].

**Figure 4.**Tension Stiffening [18].

**Figure 6.**(

**a**) Numerical model M1 and boundary conditions, (

**b**) Characteristics of the experimental model, and (

**c**) Position of extensometers on the reinforcement.

**Figure 9.**(

**a**) Numerical model M4 and boundary conditions, (

**b**) Characteristics of the experimental model, (

**c**) Position of extensometers on the reinforcement.

**Table 1.**Mechanical properties of materials, model M4 [1].

Concrete Properties | Prestressing Tendon Properties | Steel Properties | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

f_{c}(MPa) | f_{ct}(MPa) | E_{c}(MPa) | Ø_{p}(mm) | A_{p}(mm ^{2}) | V_{p} (1%)(kN) | E_{p}(MPa) | Ø_{s}(mm) | f_{sy}(MPa) | f_{sr}(MPa) | E_{s}(MPa) |

51.92 | 3.87 | 30.46 | 12.7 | 99.9 | 182 | 208,000 | 12.5 | 651.4 | 792.3 | 201,541 |

8.0 | 601.8 | 711.9 | 206,900 |

_{c}is the compressive strength of concrete, f

_{ct}is the tensile strength of concrete, Ø

_{p}is the equivalent diameter of a prestressing tendons, A

_{p}is the tendon’s area, V

_{p}is the vertical component of tendon force acting on a specified section, E

_{p}is the modulus of elasticity of tendons.

**Table 2.**Prestressing force on tendons (kN) [1].

Direction x | Direction y | |||||||
---|---|---|---|---|---|---|---|---|

Tendon | 5 | 6 | 7 | 8 | 5 | 6 | 7 | 8 |

Prestressing | 97.09 | 94.38 | 121.10 | 95.24 | 93.53 | 97.59 | 85.80 | 98.24 |

Re-stress | 128.77 | 129.61 | 133.21 | 130.61 | 128.65 | 132.29 | 127.48 | 131.94 |

Rupture | 136.03 | 137.89 | 140.07 | 134.10 | 137.15 | 141.03 | 134.88 | 135.52 |

Models | V (kN) | $\frac{\mathit{V}}{{\mathit{V}}_{\mathit{M}\mathbf{4}}}$ | Displacement (mm) | ε_{c,máx}(‰) |
---|---|---|---|---|

A-5 | 788.12 | 1.00 | 8.89 | −2.08 |

A-10 | 785.06 | 1.00 | 8.79 | −2.18 |

A-15 | 753.17 | 0.96 | 7.85 | −2.04 |

A-20 | 690.83 | 0.88 | 6.31 | −1.77 |

_{M}

_{4}is the failure load of slab M4 obtained experimentally by [1].

Models | V (kN) | $\frac{\mathit{V}}{{\mathit{V}}_{\mathit{M}\mathbf{4}}}$ | Displacement (mm) | ε_{c,máx}(‰) |
---|---|---|---|---|

T-16 | 785.06 | 1.00 | 8.79 | −2.08 |

T-18 | 909.46 | 1.16 | 7.04 | −2.04 |

T-20 | 1059.23 | 1.35 | 6.22 | −2.00 |

T-22 | 1187.16 | 1.51 | 5.33 | −1.90 |

T-24 | 1408.22 | 1.79 | 5.62 | −2.04 |

T-26 | 1534.49 | 1.95 | 4.81 | −1.95 |

T-28 | 1669.18 | 2.13 | 4.19 | −1.86 |

T-30 | 1870.16 | 2.38 | 4.22 | −1.96 |

Models | V (kN) | $\frac{\mathit{V}}{{\mathit{V}}_{\mathit{M}\mathbf{4}}}$ | Displacement (mm) | ε_{c,máx}(‰) |
---|---|---|---|---|

C-18 | 785.06 | 1.00 | 8.79 | −2.08 |

C-20 | 812.31 | 1.04 | 9.54 | −2.28 |

C-25 | 846.78 | 1.08 | 10.02 | −1.93 |

C-30 | 847.04 | 1.08 | 9.61 | −1.55 |

C-35 | 833.47 | 1.06 | 8.51 | −1.10 |

C-40 | 869.48 | 1.11 | 9.09 | −0.98 |

C-45 | 932.41 | 1.19 | 10.26 | −0.85 |

C-50 | 977.14 | 1.24 | 10.89 | −0.76 |

C-55 | 917.22 | 1.17 | 8.65 | −0.46 |

C-60 | 1019.94 | 1.30 | 11.16 | −0.50 |

C-65 | 1047.08 | 1.33 | 10.85 | −0.25 |

C-70 | 1064.57 | 1.36 | 10.63 | −0.16 |

C-75 | 1014.59 | 1.30 | 9.19 | −0.02 |

C-80 | 1047.25 | 1.33 | 9.87 | 0.11 |

C-85 | 1129.02 | 1.44 | 10.59 | 0.27 |

C-90 | 1200.37 | 1.53 | 12.11 | 0.28 |

Models | V_{FEA} (kN) | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{ACI}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{ECI}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{NBR}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{MC}\mathit{II}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{MC}\mathit{III}}}$ |
---|---|---|---|---|---|---|

A-5 | 788.12 | 1.53 | 1.39 | 1.28 | 1.23 | 1.21 |

A-10 | 785.06 | 1.55 | 1.38 | 1.28 | 1.22 | 1.21 |

A-15 | 753.17 | 1.51 | 1.37 | 1.27 | 1.17 | 1.16 |

A-20 | 690.83 | 1.39 | 1.26 | 1.16 | 1.07 | 1.06 |

Mea. | 1.50 | 1.35 | 1.25 | 1.17 | 1.16 | |

SD. | 0.06 | 0.05 | 0.05 | 0.06 | 0.06 | |

C.V. (%) | 4.26 | 3.93 | 4.01 | 5.19 | 5.19 |

Models | V_{FEA} (kN) | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{ACI}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{ECI}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{NBR}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{MC}\mathit{II}}}$ | $\frac{{\mathit{V}}_{\mathit{FEA}}}{{\mathit{V}}_{\mathit{R},\mathit{MC}\mathit{III}}}$ |
---|---|---|---|---|---|---|

C-18 | 785.06 | 1.55 | 1.38 | 1.28 | 1.22 | 1.21 |

C-20 | 812.31 | 1.56 | 1.40 | 1.29 | 1.24 | 1.23 |

C-25 | 846.78 | 1.51 | 1.41 | 1.30 | 1.27 | 1.25 |

C-30 | 847.04 | 1.41 | 1.35 | 1.25 | 1.24 | 1.22 |

C-35 | 833.47 | 1.30 | 1.29 | 1.19 | 1.20 | 1.18 |

C-40 | 869.48 | 1.27 | 1.29 | 1.19 | 1.23 | 1.20 |

C-45 | 932.41 | 1.28 | 1.35 | 1.24 | 1.32 | 1.30 |

C-50 | 977.14 | 1.27 | 1.36 | 1.26 | 1.38 | 1.35 |

C-55 | 917.22 | 1.14 | 1.24 | 1.15 | 1.31 | 1.28 |

C-60 | 1019.94 | 1.20 | 1.34 | 1.23 | 1.45 | 1.42 |

C-65 | 1047.08 | 1.18 | 1.34 | 1.23 | 1.49 | 1.46 |

C-70 | 1064.57 | 1.15 | 1.32 | 1.21 | 1.52 | 1.48 |

C-75 | 1014.59 | 1.05 | 1.22 | 1.13 | 1.45 | 1.42 |

C-80 | 1047.25 | 1.04 | 1.22 | 1.13 | 1.50 | 1.46 |

C-85 | 1129.02 | 1.08 | 1.29 | 1.19 | 1.62 | 1.58 |

C-90 | 1200.37 | 1.11 | 1.33 | 1.23 | 1.72 | 1.68 |

Mea. | 1.26 | 1.32 | 1.22 | 1.39 | 1.36 | |

SD. | 0.17 | 0.06 | 0.05 | 0.15 | 0.14 | |

C.V. (%) | 13.26 | 4.17 | 4.18 | 10.94 | 10.66 |

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## Share and Cite

**MDPI and ACS Style**

Brigo, H.; Ashihara, L.J.; Marques, M.G.; Liberati, E.A.P. Non-Linear Analysis of Flat Slabs Prestressed with Unbonded Tendons Submitted to Punching Shear. *Buildings* **2023**, *13*, 923.
https://doi.org/10.3390/buildings13040923

**AMA Style**

Brigo H, Ashihara LJ, Marques MG, Liberati EAP. Non-Linear Analysis of Flat Slabs Prestressed with Unbonded Tendons Submitted to Punching Shear. *Buildings*. 2023; 13(4):923.
https://doi.org/10.3390/buildings13040923

**Chicago/Turabian Style**

Brigo, Heraldo, Luana J. Ashihara, Marília G. Marques, and Elyson A. P. Liberati. 2023. "Non-Linear Analysis of Flat Slabs Prestressed with Unbonded Tendons Submitted to Punching Shear" *Buildings* 13, no. 4: 923.
https://doi.org/10.3390/buildings13040923