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The paper presents a new analytical model to study the evolution of radial cracking around a corroding steel reinforcement bar embedded in concrete. The concrete cover for the corroding rebar is modelled as a thick-walled cylinder subject to axisymmetrical displacement constraint at the internal boundary generated by expansive corrosion products. A bilinear softening curve reflecting realistic concrete property, together with the crack band theory for concrete fracture, is applied to model the residual tensile stress in the cracked concrete. A governing equation for directly solving the crack width in cover concrete is established for the proposed analytical model. Closed-form solutions for crack width are then obtained at various stages during the evolution of cracking in cover concrete. The propagation of crack front with corrosion progress is studied, and the time to cracking on concrete cover surface is predicted. Mechanical parameters of the model including residual tensile strength, reduced tensile stiffness, and radial pressure at the bond interface are investigated during the evolution of cover concrete cracking. Finally, the analytical predictions are examined by comparing with the published experimental data, and mechanical parameters are analysed with the progress of reinforcement corrosion and through the concrete cover.

The serviceability and durability of concrete structures may be seriously affected by the corrosion of steel reinforcement in structures that are exposed to aggressive environments, such as motorway bridges, car parks, and marine structures. Reinforcement corrosion consumes original steel rebar, generates much lighter rust products, and creates expansive layer at the interface between the reinforcement and the surrounding concrete cover. As corrosion progresses, the expansive displacement at the interface generated by accumulating rust products causes tensile stress in the hoop direction within the concrete cover, leading to radial splitting cracks in the concrete. The cracking and eventually spalling of the concrete cover significantly affect the bond strength between the rebar and the surrounding concrete cover and consequently influence the service ability and resistance of reinforced concrete structures [

Many investigations have been undertaken during the last two decades regarding the influence of reinforcement corrosion and concrete cracking on the performance of reinforced concrete structures. Al-Sulaimani et al. [

The paper presents a new approach for studying the evolution of cover concrete cracking due to reinforcement corrosion, based on the thick-walled cylinder model for the concrete cover and the cohesive crack model for the cracked concrete. A governing equation for directly solving crack width within cover concrete is established with considering the realistic bilinear softening curve for the cracked concrete and the estimated number of cracks in the concrete cover. The closed-form solutions to crack width are then obtained for various cases that may occur during the evolution of cover concrete cracking. The propagations of the cracked front and critical crack front are investigated, and the time to concrete cover cracking is predicted. Mechanical parameters, such as residual tensile strength, reduced tensile stiffness, and radial pressure at the bond interface, are also studied with the progress of rebar corrosion. Finally, the developed analytical model is examined through its ability to reproduce reported experimental measurements and theoretically provides the evolution of concrete cracking and the deterioration of tensile stiffness and strength of the cracked concrete over the time of reinforcement corrosion.

The thick-walled cylinder model for concrete cover, initially proposed by Tepfers [

In the thick-walled cylinder model for cover concrete cracking induced by reinforcement corrosion, as shown in Figures

Thick-walled cylinder model for cover concrete cracking evolution due to reinforcement corrosion.

Idealisation of cover concrete

Geometry of cylinder model subject to internal displacement

To accommodate the volume increase due to steel corrosion, the prescribed displacement at the interface between the steel rebar and the surrounding concrete over time

The prescribed displacement

In the case when the prescribed displacement is given, the mass of rust products can be calculated by

Based on the assumption that the steel rebar has uniform corrosion at the surface, the thick-walled cylinder model for cover concrete cracking can be considered as an axis symmetrical problem. The thick-walled cylinder model could be further treated as a plane stress problem because the normal tension-softening stress in the direction of longitudinal axis could be ignored [

Concrete cracking could be modelled as a process of tensile softening if the cracking is considered as cohesive and the crack width does not exceed a limited value [

Bilinear softening curve for cohesive cracking in concrete.

From the crack band theory for the fracture of concrete [

From the results for a thick-walled cylinder subject to internal pressure given by Timoshenko and Goodier [

It is well known that, for an anisotropic thick-walled cylinder subject to axis symmetrical actions, radial strain

The stress equilibrium equation for the thick-walled cylinder is

From the cohesive crack model, the residual tensile stress in hoop direction for the cracked concrete can be obtained from

Cracks initiate in cover concrete when the tensile hoop stress at the internal boundary reaches tensile strength and then propagate through the concrete cover until reaching the free cover surface. Depending on the crack width at the internal boundary

Since the cover concrete remains intact and elastic before the tensile hoop stress reaches the tensile strength of concrete, the classical elastic solution of radial displacement

The thick-walled cylinder is now divided into two zones, an intact outer ring (

In the cracked inner ring, where the crack width at the internal boundary does not exceed the critical value, the displacement condition at internal boundary (

In this case, the thick-walled cylinder is divided into three zones, as shown in Figure

After the crack front reaches the external surface, the concrete cover becomes completely cracked. Depending on the crack widths at the internal and external boundaries, three cases are considered, crack width within the concrete cover does not exceed the critical value (

A single cracked zone within the concrete cover exists in this case, and the crack width at the internal boundary does not exceed the critical value when the cover surface is cracked. To determine the two constant coefficients

The critical crack front divides the thick-walled cylinder into two zones: a cracked outer ring, where crack width does not exceed the critical value (

A single cracked zone is considered for the thick-walled cylinder in this case, and the crack width over the concrete cover now exceeds the critical value. The boundary conditions for this case are given by

To validate the proposed approach, the published experimental data such as the time to cracking on the concrete cover surface and the concrete crack width with reinforcement corrosion progress are adopted. Liu and Weyers [

Predicted and observed times to cracking on concrete cover surface.

Specimen number | Steel rebar diameter (mm) | Cover thickness (mm) | Corrosion rate (^{2}) | Predicted time (year) | Observed time* (year) |

S1 | 16 | 48 | 2.33 | 1.83 | 1.84 |

S2 | 16 | 70 | 1.79 | 3.44 | 3.54 |

S3 | 16 | 27 | 3.75 | 0.40 | 0.72 |

S4 | 12 | 52 | 1.80 | 2.20 | 2.38 |

The predicted results for the crack width on concrete cover surface with the progress of reinforcement corrosion are now compared with the experimental measurements presented by Andrade et al. [

Comparison of predicted crack width over time with published experimental results.

The specimen S1 shown in Table

Radial displacement at rebar surface with time

Crack widths at rebar surface (

The plot in Figure

Propagation of cracked front

Figure

Hoop stress

Hoop stiffness reduction factor

The bursting pressure

Radial pressure

Figures

Selected times during cover concrete cracking evolution.

Time | Concrete crack evaluation |
---|---|

0.014 year | Cracking initiation at internal boundary |

0.97 year | Critical cracks developed at internal boundary |

1.83 years | Cracks reaching at external surface |

1.84 years | Cracks just occurred at external surface |

11.2 years | Cracks reaching |

42.9 years | Cracks reaching ultimate cohesive width over concrete cover |

Crack width

Hoop stress

Stiffness reduction factor

Radial stress

A new method for theoretically analysing the evolution of cracking in concrete cover subject to expansive internal displacement caused by steel rebar corrosion is presented based on the thick-walled cylinder model for the concrete cover and the tensile softening model for the cracked concrete. The governing equation for directly solving the crack width in the cracked concrete is established and a general closed-form solution is obtained for the proposed boundary value problem. The formulas for calculating actual crack width as well as other mechanical parameters of the cracked concrete, including residual strength, residual stiffness, and radial stress, are derived for various stages during the cracking evolution in the cover concrete. The predicted results for the time to cracking for various concrete cover dimensions and reinforcement corrosion rates and for the crack width over time are examined and demonstrated to be in good agreement with the published experimental measurements.

The time taken for cracked front to propagate from the internal boundary of the concrete cover to the cover surface is substantially long, and the existing models for estimating the time to cracking on the cover surface by ignoring the crack propagation through the concrete cover may be improper. The time to cracking is a function of cover dimensions, concrete material properties, and reinforcement corrosion rate. The crack width of the concrete cover depends on concrete material properties and the expansive displacement developed at the internal boundary due to reinforcement corrosion. The residual stiffness in hoop direction reduces significantly when concrete is cracked and decays faster than the hoop residual strength. The radial pressure at the interface between the steel rebar and the concrete cover reaches peak value well before the cracks occur at the cover surface, drops suddenly when concrete becomes completely cracked through the cover, and decays fast from the bond interface over the concrete cover. The time taken for cracks to reach the ultimate cohesive width and for hoop residual strength and stiffness to vanish is relatively long, comparing with the time to cracking.