# Bond Modelling for the Assessment of Transmission Length in Prestressed-Concrete Members

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## Abstract

**:**

## 1. Introduction

## 2. Roles of the Major Parameters Affecting the Transmission Length

## 3. Analytical Modelling of the Transmission Length

#### 3.1. General Calculation Procedure

_{bpd}is the bond stress along the considered element and A

_{sp}is the tendon cross-sectional area. This implies knowledge of the bond stress distribution along the transmission zone: the matter will be addressed in the next sections.

#### 3.2. Elastic Analysis Based on the Thick-walled Cylinders Theory

#### 3.3. Anisotropic Analysis for Cracked Concrete

## 4. Model Calibration and Results

#### 4.1. Global Behaviour: Transmission Length Assessment

_{t}”. This demonstrates the effectiveness of the analytical model in replicating the measured transmission length values.

#### 4.2. Local Behaviour: Radial Cracking and Bond Stress Development

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Statement of Data Availability

## Nomenclature

A_{c} | Cross-sectional area of concrete |

A_{sp} | Cross-sectional area of prestressing tendon |

b | Width of the concrete section |

c | Concrete cover thickness |

${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$ | Constants of integration for the solution of u |

E_{c} | Elastic modulus of concrete |

E_{ps} | Elastic modulus of prestressing steel |

e | Vertical eccentricity of the considered tendon with respect to the centre of gravity of the concrete section |

h | Height of the concrete section |

J_{x} | Moment of inertia of the concrete section |

L_{t} | Transmission length of the prestressing tendon |

L_{t, experimental} | Experimental value of the transmission length |

L_{t, theoretical} | Theoretical value of the transmission length |

l_{bp} | Basic anchorage length according to fib MC2010 |

P | Initial prestressing-force in the tendon |

r | Radial distance from the tendon centroid |

r_{jack} | Radius of the tendon after release |

r_{ps} | Radius of the unstressed tendon |

r_{tip} | Distance from the tendon centroid to the crack tip |

u | Radial displacement |

u_{c} | Radial displacement of the concrete |

u_{ps} | Radial displacement of the tendon outer surface |

y | Vertical distance from the centre of gravity of the concrete section |

$\Delta $z | Length of the single finite element in which the prestressing tendon is subdivided |

z | Longitudinal distance from the free-end of the PC member |

α_{p1} | Coefficient which takes into account the prestress release method, according to fib MC2010 |

α_{p2} | Coefficient which takes into account the action effect to be verified, according to fib MC2010 |

α_{p3} | Coefficient which takes into account the influence of bond situation, according to fib MC2010 |

α_{rel} | Coefficient which takes into account the prestress release method, according to the findings proposed by the authors |

${\mathsf{\epsilon}}_{1}$ | Concrete strain (assumed as 0.0003) corresponding to concrete tensile stress equal to 0.15 ${\mathrm{f}}_{\mathrm{t}}$, according to Han’s softening model |

${\mathsf{\epsilon}}_{\mathrm{c},\mathrm{ck}}$ | Cracking strain of concrete |

${\mathsf{\epsilon}}_{\mathrm{c},\mathrm{r}}$ | Concrete strain in the radial direction |

${\mathsf{\epsilon}}_{\mathrm{c},\mathsf{\theta}}$ | Concrete strain in the circumferential direction |

${\mathsf{\epsilon}}_{\mathrm{c},\mathrm{z}}$ | Concrete axial strain at the level of the tendon centroid |

${\mathsf{\epsilon}}_{\mathrm{u}}$ | Ultimate concrete strain (assumed as 0.002) corresponding to concrete tensile stress equal to zero, according to Han’s softening model |

η_{p1} | Coefficient which takes into account the type of tendon, according to fib MC2010 |

η_{p2} | Coefficient which takes into account the position of the tendon, according to fib MC2010 |

$\mathsf{\mu}$ | Overall friction coefficient between the tendon and the surrounding concrete, combining actual frictional and mechanical bond |

${\mathsf{\upsilon}}_{\mathrm{c}}$ | Poisson’s ratio of the concrete |

${\mathsf{\upsilon}}_{\mathrm{ps}}$ | Poisson’s ratio of the prestressing steel |

${\mathsf{\sigma}}_{\mathrm{bpd}}$ | Prestress transfer bond at the interface tendon-concrete |

${\mathsf{\sigma}}_{\mathrm{c},\mathrm{r}}$ | Concrete radial stress |

${\mathsf{\sigma}}_{\mathrm{c},\mathsf{\theta}}$ | Concrete circumferential stress |

${\mathsf{\sigma}}_{\mathrm{c},\mathrm{z}}$ | Concrete axial stress |

${\mathsf{\sigma}}_{\mathrm{ct}}$ | Tensile strength of concrete |

${\mathsf{\sigma}}_{\mathrm{ptd}}$ | Design tensile strength of the prestressing steel |

${\mathsf{\sigma}}_{\mathrm{r}}$ | Tendon radial stress |

${\mathsf{\sigma}}_{\mathrm{r}}$(r_{jack}) | Radial compressive stress at the interface between steel and concrete, arising from the Hoyer effect |

$\Delta {\mathsf{\sigma}}_{\mathrm{s}}$ | Increment in tendon stress resulting from the development of bond stress along the finite element |

${\mathsf{\sigma}}_{\mathrm{s}}$ | Tendon stress at the considered point along the length of the member |

${\mathsf{\sigma}}_{\mathrm{si}}$ | Jacking stress of the tendon at prestressing-force release |

φ | Nominal tendon diameter |

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**Figure 3.**Geometry of the idealised steel and concrete cylinders for the application of the thick-walled cylinders (TWC) theory.

**Figure 10.**Comparison between experimental and theoretical concrete strain build-up profiles ($\mu $ = 0.6) for specimen M12-H-C4-1; experimental results are derived from Oh et al., 2006.

**Figure 11.**Analysis of the transmission length at the cut end of specimens “FC350-2” from Russell and Burns, 1996 (

**a**) and “SS150-4” from Russell and Burns, 1997 (

**b**) with $\mu $ = 0.6.

**Figure 12.**Development of the radial cracking along the length of specimen M12-H-C4-1, $\mu $ = 0.6.

**Figure 13.**Bond stress distribution along the transmission length of specimen M12-H-C4-1, evaluated with the TWC model ($\mu $ = 0.6) and principal design codes.

**Table 1.**Detail of the dataset of experimental transmission length values for model calibration: test specimens and authors.

Reference Citation | No. of Experimental Tests |
---|---|

Mitchell et al. (1993) [15] | 14 |

Russell and Burns (1996) [24] | 20 |

Russell and Burns (1997) [28] | 12 |

Oh and Kim (2000) [11] | 36 |

Oh et al. (2006) [16] | 24 |

Martì-Vargas et al. (2007) [29] | 12 |

Dang et al. (2017) [30] | 12 |

Friction Coefficient | AVE | COV | RMSE |
---|---|---|---|

$\mu $ = 0.3 | 2.10 | 1.15 | 755.57 |

$\mu $ = 0.4 | 1.62 | 0.67 | 439.31 |

$\mu $ = 0.5 | 1.30 | 0.36 | 232.72 |

$\mu $ = 0.6 | 1.07 | 0.18 | 139.20 |

$\mu $ = 0.7 | 0.92 | 0.16 | 154.16 |

$\mu $ = 0.8 | 0.81 | 0.23 | 207.22 |

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**MDPI and ACS Style**

Fabris, N.; Faleschini, F.; Pellegrino, C. Bond Modelling for the Assessment of Transmission Length in Prestressed-Concrete Members. *CivilEng* **2020**, *1*, 75-92.
https://doi.org/10.3390/civileng1020006

**AMA Style**

Fabris N, Faleschini F, Pellegrino C. Bond Modelling for the Assessment of Transmission Length in Prestressed-Concrete Members. *CivilEng*. 2020; 1(2):75-92.
https://doi.org/10.3390/civileng1020006

**Chicago/Turabian Style**

Fabris, Nicola, Flora Faleschini, and Carlo Pellegrino. 2020. "Bond Modelling for the Assessment of Transmission Length in Prestressed-Concrete Members" *CivilEng* 1, no. 2: 75-92.
https://doi.org/10.3390/civileng1020006