# Numerical simulation of rebar/concrete interface debonding of FRP strengthened RC beams under fatigue load

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## Abstract

The aim of this paper is to simulate the rebar/concrete interface debonding of FRP strengthened RC beams under fatigue load and also, to ascertain the influence of design parameters such as the elastic modulus, thickness and length of the FRP plate on the debonding performance. In order to simplify the simulation, some basic equilibrium equations are formulated and then the stresses of the rebar and FRP plate are numerically solved, and stress intensity factor is avoided in the simulation by fundamentals of fracture mechanics because of its complexity around the crack tip of bi-material interface. With the combination of finite element method and difference approximation, authors program the degradation model of coefficient of friction, debond criterion, propagation law and loop of load process into a commercial finite element code to investigate the fatigue debonding. The relationships between the debond length as well as other fatigue parameters and number of cyclic load are obtained and discussed.

## Keywords

FRP plate RC beam Finite element method Difference approximation Interface fracture## Nomenclature

*A*Interfacial area per unit length

*a*Radius of rebar

*b*Width of concrete beam

*b*_{p}Width of FRP plate

*C*_{c}Volume fraction of concrete of ERR calculation zone

*C*_{s}Volume fraction of rebar of ERR calculation zone

*c*Parameter of Paris law

*d*Concrete cover measured to center of rebar closest to tensile face of concrete

*E*Elastic modulus of ERR calculation zone when treated as an isotropic material

*E*_{c}Tensile Young’s modulus of concrete

*E*_{p}Young’s modulus of FRP plate

*E*_{s}Young’s modulus of rebar

*F*External cyclic load

*G*Energy release rate

*h*Height of beam

- \(\tilde{h}\)
Height of ERR calculation zone

*h*_{d}Step length in difference approximation

*kd*Distance from FRP plate end to the nearest support

*L*Length of beam

*l*Current debond length

*l*_{0}Initial debond length

*m*Parameter of Paris law

*N*Fatigue cyclic number

- \(\overline{N}\)
Elapsed cycle when μ(0) attains the value (μ

_{0}+ μ_{f})/2*N*_{int}Number of integration points

*n*Parameter describing the degradation velocity of coefficient of friction

*q*_{0}Residual fiber compressive stress in the radial direction

*r*Distance between neutral axis of concrete and

*x*axis*t*Thickness of FRP plate

*U*Strain energy of ERR calculation zone

*U*_{e}Strain energy of one element

*vol*_{i}Volume of integration point

*i*- ɛ
_{0} Strain of compression concrete when the stress gets σ

_{0}- ɛ
_{c} Strain of concrete

- ɛ
_{cu} Ultimate strain of compression concrete

- ɛ
_{p} Strain of FRP plate

- ɛ
_{s} Strain of rebar

- {ɛ
^{el}} Elastic strain vector

- μ(
*N*,*z*) Coefficient of friction on the debonded interface

- μ
_{0} Original value of the coefficient of friction before reduction

- μ
_{f} Steady state or final value of the coefficient of friction where there is no further degradation on μ

- σ
_{c} Stress of concrete

- σ
_{c0} Yield stress of compression concrete

- σ
_{p} Stress of FRP plate

- σ
_{s} Stress of rebar

- {σ}
^{T} Elastic stress vector

- τ
Frictional stress

## Notes

### Acknowledgements

This research project is funded by the National Nature Science Foundation of China (No. 50378001) and the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education institutions of MOE, P.R. China.

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