A finite element (FE) method was proposed to calculate the corrosion penetration depth (_{crit}) on steel reinforcement necessary for the first visible crack to appear on the concrete cover. The FE analysis was carried out using the commercial software from ANSYS. The obtained FE method is a function of free concrete cover depth (_{crit} value. This influence can only be analysed three-dimensionally. The proposed FE method is validated with experimental results from literature. This approach is a novelty in considering the longitudinal direction in the analysis to account for the extension of the anodic cell. Corrosion type strongly depends on the

_{crit}) en las armaduras de acero, necesaria para originar la aparición de la primera grieta visible en el recubrimiento de hormigón. El análisis por FE se ha llevado a cabo utilizando el programa comercial ANSYS. El método de FE obtenido se ha desarrollado en función del espesor del recubrimiento de hormigón libre (_{crit}. Esta influencia sólo puede ser analizada tridimensionalmente. El modelo de FE propuesto se validó utilizando datos de la bibliografía. La originalidad del procedimiento propuesto radica en la consideración de la dirección longitudinal en el análisis, para tener en cuenta la extensión de la zona anódica. El tipo de corrosión tiene una gran dependencia de la relación entre

Reinforced concrete (RC) is one of the most widely used materials in the construction sector because of its versatility, cost and durability. The highly alkaline environment (pH above 12.5) of a good quality concrete leads to the formation of a passive layer on the embedded steel rebar, protecting it from corrosion and assuring a low steel corrosion rate of less than 1.16 μm/year (González

Steel corrosion and the loss of rebar cross-section take place in the presence of oxygen and water through an electrochemical process (Ahmad,

Depending on the location of the anode and the cathode, steel reinforcement corrosion may occur in micro-cells, where anodic and cathodic half-cell reactions occur in the same place, or in macro-cells where corroded zones (anode) are distinguished from uncorroded zones (cathode) (González _{crit}) on steel necessary to induce concrete cover cracking, they have considered uniform corrosion conditions (micro-cells), and only a few cases have assessed non-uniform corrosion (macro-cells) (Torres-Acosta and Sagües,

Analytical efforts have been made to propose a model that can predict either the time to corrosion-induced concrete cover cracking of a corroded structure or the residual flexural strength of RC beams subjected to rebar corrosion (Bažant,

The aim of this paper is to develop a numerical method by FE methodology to calculate the _{crit} on reinforcing steel in concrete and its validation through experimental results from the literature. In this way it is proposed to analyse the parameters which define the mechanical properties of the concrete based on the stress-strain response curve as the concrete is damaged by stress.

A model is proposed to describe the problem, which consists of a steel rebar with an initial radius (_{0}) embedded in concrete, the distance from the steel rebar centre to the nearest free concrete surface is defined by the _{c} parameter (_{c}=_{0}), where _{0} as defined above, _{c}, _{0} are showed in _{t}) to the maximum tensile strength of the concrete ( _{t}) to start the cracking process at the steel/concrete interface (Bhargava

Model’s parameters for the estimation of stresses during corrosion process. _{c} is the distance from steel rebar to the nearest free concrete surface, _{0} is the initial radius of steel rebar, and

The reliability of the common thick-walled concrete cylinder approach, under the action of internal pressure originated by the expansion of iron oxide of the corroded steel rebar, is only approximate (Sánchez-Deza _{0}, is smaller than the diameter of the corroded steel rebar, 2_{0}+

In order to describe the steel/concrete interface, contact elements may be used, which are considered as layers without thickness that adhere to the surfaces and allow them to come into contact. One side of the contact element is referred to as the ‘

When using contact elements, the main difficulty is that the contact surface between the two bodies is unknown in advance. Under ideal conditions, an initial value of

where _{g} is the radial pressure at the steel/concrete interface; _{s} is the Poisson’s coefficient of steel; _{s} is the modulus of elasticity for reinforcement steel; _{c} is the Poisson’s coefficient of concrete; _{c} is the modulus of elasticity for concrete; _{0} and _{c} have been defined above. It is possible to demonstrate that _{g} parameter can be calculated using the expression (Wang,

where all parameters have been defined previously.

In order to calculate _{g} parameters using Eq. (

A 3D model for the FE method has been utilized to represent the interaction of the steel/oxide/concrete system, on which the compressibility of rust products is not considered, and the expansion of corrosion products, denoted by _{y}) and the ultimate strength ( _{u}). And finally, (iii) the use of contact elements for friction modelling associated to iron oxide layers often involves great converge problems in the fitting process due to their highly non-linear behaviour. Friction and stiffness (also called ‘penalty stiffness’) are dependent upon the type of iron oxide layer and the permissible penetration is obtained from the FE method analysis. The most important challenge in modelling with contact elements is to determine when the two surfaces come into contact, and this depends on several factors such as the separation between them or the mesh size used.

Stress-strain curve for concrete. Considering _{0}=0.0021, _{0} is the unit strain in concrete at the compressive strength of concrete (

Tri-linear stress-strain curve for steel. Where _{u} is the unit strain in reinforcing steel at the _{u} ultimate strength, and _{y} is the unit strain in reinforcing steel at the _{y} yield strength.

It is assumed that the reinforcing steel and the surrounding concrete cover always remain in contact, but with the possibility of sliding, it is necessary to provide a friction coefficient, contact stiffness, and permissible penetration parameters. Steel _{crit} (see

where _{m}, cm^{3}·mol^{-1}) of an iron oxide is defined as the ratio between the molar mass and the density of a given oxide phase.

Molar volume (_{m}) and molar volume expansion ratio coefficient (_{m(oxide)}/_{m(Fe)}) for oxides of iron generated in a marine environment (Sánchez-Deza

Iron and Iron Oxide Phase | Molar Volume (V_{m}), (cm^{3}·mol^{−1}) |
Molar Volume Expansion Ratio Coefficient ( |
---|---|---|

Iron, Fe | 7.09 | − |

Akaganeite, |
23.32 | 3.29 |

Goethite, |
20.82 | 2.94 |

Lepidocrocite, |
22.42 | 3.16 |

Hematite, _{2}O_{3} |
30.27 | 2.13 |

Maghemite, _{2}O_{3} |
32.63 | 2.30 |

Magnetite, Fe_{3}O_{4} |
44.52 | 2.09 |

where _{m(oxide)} and _{m(Fe)} are the molar volumes of oxide and iron, respectively. The

The smeared crack model proposed by Pantazopoulou and Papoulia (_{c} is obtained using the formula:

where _{t} is the strain in polar coordinates; and _{c} has been defined above.

The procedure to carry out the non-linear FE method analysis is as follows, the mechanical properties of the concrete are defined by: (a) their stress-strain curves, (b) the steel diameter (_{0}), (c) the _{crit}, and Eq. (_{crit} and

The strategy followed in the present analysis to avoid convergence problems is similar to the proposal by Mohyeddin

To validate the present FE method results from Torres-Acosta and Sagües (^{−2}.

_{crit} by FE method modelling of a reinforcement with a cylindrical shape, due to their symmetry and simplification of the calculation, and that it is equally applicable for beams of rectangular section.

Comparison between tangential stress (_{t}, MPa) results obtained for different boundary conditions (cylinder and beam). _{0} is the radius of the steel reinforcement (mm), and _{c}=28836 MPa), Poisson’s coefficient of concrete (_{c}=0.24), steel reinforcement radius (_{0}=4.75 mm), modulus of elasticity of reinforcing steel (_{s}=210000 MPa), and Poisson’s coefficient of reinforcing steel (_{s}=0.3).

During the second step, when the first visible crack appears on the concrete cover surface, the numerical FE method and the experimental results from Torres-Acosta and Sagües (_{t}) on concrete due to localized corrosion of an anodic zone having a length

Variation of pressure (_{c}) originated by expansion of iron oxides and the ratio of corrosion penetration depth on steel and initial radius of steel rebar (_{crit}/_{0}) at the steel/concrete interface. “First Crack” indicates that cracking of concrete initiates at the steel/concrete interface.

Tangential stresses (_{t}) in concrete due to localized corrosion. Parameters used in the simulation: _{c}=0.24.

For a given ratio of _{g} using Eq. (^{2}=0.9200. The obtained model is described by the proposed expression:

where

Comparison between results obtained using the present FE method (_{crit} using Eq. (_{crit} Experimental) obtained from Torres-Acosta and Sagües (

The influence of mechanical properties of concrete and geometrical conditions of steel rebar on _{crit} defined by Eq. (_{c}) and tensile strength ( _{t}) (Mohyeddin _{crit} is not relevant, possibly because of the difference in stiffness between steel and concrete, which is on the order of 7 to 8 times as average. This fact is of great practical importance, since Eq. (

Influence of the mechanical properties of concrete on _{rcrit} for a

Influence of the mechanical properties of concrete on _{crit} for a

_{crit} for a _{c}
_{0} appears to be as power of two in Eq. (

Influence of the size of the steel reinforcement diameter, _{crit} for

Finally the influence of _{crit} was analysed, taking _{crit} is shown. From the analysis of _{crit}), and for values between the range 0.5<

Influence of _{crit} for different

Based on the results of this experimental investigation under marine environment, the following conclusions are drawn:

Using contact elements methodology it is possible to model initiation and propagation stages of the concrete cover cracking due to corrosion of steel reinforcements.

A FE method is proposed to determine the corrugated steel corrosion penetration depth at the point when the first visible crack appears. The FE method is based on geometry, free concrete cover depth (

A good correlation was found between experimental results from the literature and the developed FE method. Corrosion type (uniform or localized) strongly depends on the

The authors wish to express their gratitude to the National System of Researchers of CONACYT-Mexico (SNI) for financial support. D.M. Bastidas acknowledges funding from The University of Akron.