The reinforced concrete (RC) is one of the most used construction materials. During the hydration process the pH becomes highly alkaline inside the concrete matrix due to the formation of the products of the cement hydration. This alkaline pH produces a passive film of iron oxide or hydroxide on the surface of the steel bars that protect the bars themselves from the corrosion. Therefore, in RC element the reinforcements are not corroded until this layer stays. During its service life a RC structure may be exposed to aggressive species that migrate inside the concrete from their exposed surface and reduce the alkaline protective pH. When the concentration of these aggressive species reaches a threshold value and consequently the pH in the matrix concrete is sufficiently low, the oxidation of the steel bars begins. This process is referred to as “depassivation” (Val and Trapper 2008; Martin-Perez et al. 2001; Saetta et al. 1995; Izquierdo et al. 2004) of the reinforcement bars. The reduction of the pH in the concrete matrix is produced by two main mechanisms: the CO_{2} diffusion from the atmosphere and the diffusion of active ions, like chloride ions (Martin-Perez et al. 2001; Saetta et al. 1995). The diffusion of chloride ions is the principal cause of the reinforcement bar corrosion in marine environment or cold climates.

Chloride ions are sometimes present inside the hydration water used to mix the concrete. In that case these ions slow down or prevent the formation of the protective passive layer on the steel surface. In the other cases the chloride ions migrate into the contrete matrix from the external enviroment. The time at which the critical chloride concentration is reached on the passive layer is called “time to initiation”, according to the Tuutti model (Tuutti 1982) (see Fig. 1). This time devides the initiation phase from the propagation phase. During the initiation phase the chloride ions diffuse in the concrete matrix, but the steel corrosion is absent. In the propagation phase the corrosion started and proceed to a critical value of steel loss that causes failure of the structural element or all the structure. The propagation phase is considerably shorter than the initiation phase. Therefore, the prediction of the time to corrosion initiation is an important aspect in reliability analysis of RC structures because it indicates the beginning of the strenght reduction of the structural elements (Nogueira and Leonel 2013; Marano et al. 2008). For this reason the estimation of the time to corrosion initiation has been investigated by several researchers.

The chloride penetration in the concrete matrix occurs through either permeation and absorbition of a chloride solution or diffusion of free chloride ions into the saturated concrete (Val and Trapper 2008; Martin-Perez et al. 2001). These mechanisms may also occur sequentially. The absorption runs out in short time and after that the diffusion begins due to the saturation of the concrete pores (Martin-Perez et al. 2001). Each transport process is modelled by a different mathematical law depending on the forces that are at stake. In the convection process the motion of the solution inside the concrete pores (and then the motion of the chloride ions) is caused by the moisture/humidity gradient. The gradient of chloride concentration between the external surface and the inside of saturated concrete matrix causes the chloride ions motion in the diffusion process (Val and Trapper 2008; Nogueira and Leonel 2013; Basheer et al. 2002). This paper proposes the investigation of the chloride ingress in piers partially submerged in seawater, but fully saturated: the part of the pier out of the water is subject to continuous wetting cycles, therefore it is constantly wet and saturated. In these conditions the diffusion phenomenon is prevalent. The diffusion process of chloride ions in the concrete matrix is not irreversible in case of inversion of the concentration gradient and is history independent, so it is a normal diffusion process (Nogueira and Leonel 2013). For that reason the Fick’s Second Law is used to model the time-variant chloride ions diffusion. The validity of the Fick’s second law to model the diffusion of chloride ions in concrete is determined on an empirical basis (Chatterji 1995).

Different elements influence the chloride penetration into RC elements as combination of air or water pressure, humidity and concetration differences or temperature differences of solutions (Izquierdo et al. 2004; Nogueira and Leonel 2013; Basheer et al. 2002). Some studies investigated the influence of the different factors on the chloride threshold value that lead to the depassivation of the rebars (Izquierdo et al. 2004; Azad 1998). Most of those studies are based on the simple plane monodirectional diffusion model, called “slab model”. In this model the concrete matrix is assumed to be an isotropic homogeneous semi-infinite medium with higher chloride concentration on its external plane surface than the inside concentration. In this model the concentration gradient causes the diffusion of chloride ions inside the concrete matrix along the direction perpendicular to the external surface. The Fick’s Second Law (Chatterji 1995) describing the diffusion process in this slab model depend only on one spatial variable *x* and the time *t* (Collepardi et al. 1972):

$$ \left( {\frac{\partial C}{\partial t}} \right) = D\frac{{\partial^{2} C}}{{\partial x^{2} }} $$

(1)

The spatial variable *x* is the distance of the inside point, where the concentration is estimated, from the external surface, while the constant *D* is the diffusion coefficient. During the diffusion process the chloride concentration *C* changes inside the concrete matrix and the concentration gradient decreases.

The parameters affecting the chloride diffusion inside the concrete matrix are initial chloride concentration in the concrete mix, type of cement used in the mix, water-cement ratio (*w*/*c* ratio), curing conditions, external temperature and external concentration of the chloride ions and variation of the external concentration in time. The effect of the type of cement, the *w*/*c* ratio and the curing conditions on the chloride diffusion are all included in the value of the diffusion coefficient *D* (in m^{2}/s) (Azad 1998). In the most simple studies the chloride diffusion coefficient is kept constant during the diffusion process (Collepardi et al. 1972). Some scholars presented works in which the diffusion coefficient is estimated as a function of different parameters, like the *w*/*c* ratio (Lin 1990). Different laws to estimate the diffusion coefficient of ordinary portland concrete (OPC) Type I were compared in (Vu and Stewart 2000). The diffusion coefficients estimated according to these laws resulted to be similar for *w*/*c* ratio in the range 0.3–0.5. Furthermore the diffusion coefficient is highly dependent from the temperature, but the *w*/*c* ratio influences the value of this coefficient in the same way at different temperatures, as showed in (Kirkpatrick et al. 2002). Other studies demonstrated the dependence of diffusion coefficient on the age the temperature and the humidity at the same time (Val and Trapper 2008; Martin-Perez et al. 2001). It is important to notice that in literature the diffusion coefficient is always evaluated by assuming the occurrence of diffusion in a semi-infinite medium with plane external surface. This assumption is questionable in case the estimation of the diffusion coefficient is supported by experimental data: both laboratory samples and several real RC structural member from which samples are taken are not semi-infinite plane elements.

Although the deterministic approach to investigate the chloride diffusion in RC structural members is the most adopted, some studies proposed a probabilistic assessment of the time to corrosion initiation in RC elements. The time-dependent probability of steel corrosion initiation in partially saturated RC members after exposure to chloride ions is calculated in (Val and Trapper 2008) by taking into account both the diffusion and convection phenomena. In (Nogueira and Leonel 2013) and (Saassouh and Lounis 2012) a probabilistic model of chlorides diffusion is coupled to reliability algorithms to determine the probability of failure of an RC structure. Both numerical approach, like Monte Carlo simulation, (Nogueira and Leonel 2013; Saassouh and Lounis 2012) and analytical approaches, like FORM (Nogueira and Leonel 2013; Saassouh and Lounis 2012) and SORM (Saassouh and Lounis 2012), were tested by scholars to estimate the limit state of corrosion in RC structures. The uncertainties affecting the diffusion coefficient D have a large scatter in the first life years of an RC structure, while after that time the overall uncertainty of the diffusion coefficient results reduced (Saassouh and Lounis 2012). The time and space invariant diffusion coefficient is an questionable assumption, but it is often supported by measures on the real concrete elements (Saassouh and Lounis 2012). It is clear that some uncertainty could be reduced by increasing the knowledge of the corrosion mechanism or testing the structure: in this case the uncertainty should be modelled as epistemic instead of aleatory (Saassouh and Lounis 2012; Do et al. 2005). The different kind of uncertainty could be handled by random, fuzzy or fuzzy random variables (Sobhani and Ramezanianpour 2011). In (Do et al. 2005) the uncertainty of the parameters on which the time to corrosion initiation depends is treated with the fuzzy logic. The defuzzification of the time to corrosion initiation could be considered a deterministic value to use in the maintenance planning of the structure. The fuzzy logic is an effective tool also in case the model defined to assess the reliability of RC structures subject to pitting reinforcement corrosion depends on both probabilistic and nonprobabilistic parameters (Marano et al. 2008). In complex models for the life cycle prediction and service life design of RC structures the uncertainties of the model parameters and the degradation mechanisms are not negligible. The interdependence among both parameters and mechanisms is also critical for the life cycle prediction of RC structure, therefore also in the assessment of the time to corrosion initiation. In (Sobhani and Ramezanianpour 2011) the high complexity of this problem depending on several parameters modelled as fuzzy random variables and interdependent mechanisms was managed by proposing an algorithm with a soft computing core.

In previous deterministic studies (Val and Trapper 2008; Tuutti 1982; Basheer et al. 2002) the time to corrosion initiation is evaluated with the Fick’s second law formulated for the slab model (Eq. (1)). The solution of this equation is (Crank 1975)

$$ C\left( {x,t} \right) = C_{0} \left[ {1 - erf\left( {\frac{x}{{2\sqrt {Dt} }}} \right)} \right] $$

(2)

where \( erf( \bullet ) \) is the error function, *C*
_{0} is the chloride concentration on the external surface (expressed in kg/m^{3} or in %), *D* is the apparent chloride diffusion coefficient and *t* is the duration of the exposure to chlorides (Tang and Joost 2007). Assuming D as time-invariant, the complement of the error function \( (1 - erf\left( \bullet \right)) \) describes the evolution of the chloride diffusion front in a monodimensional model in dependence to the exposure time *t* (Tang and Joost 2007; Tuutti 1982) and the distance from the external surface *x*.

Generally, the solution of the Fick’s second law depends on the geometry of the element where the diffusion occurs, as well on the shape of the source (Crank 1975). Therefore, the time to corrosion initiation due to chloride ions diffusion depends on the geometry of the RC element. Some scholars proposed solutions of the Fick’s second law for aggressive species in the two-dimensional (2-D) models (Val and Trapper 2008; Martin-Perez et al. 2001; Frier and Sørensen 2007) and one-dimensional not plane (1-D) model of RC elements (Arora et al. 1997). First Martin-Perez et al. (2001) and after Val et al. (Val and Trapper 2008) used a 2-D ingress of chloride ions into partially saturated RC members with rectangular cross-section. In these studies the chloride penetration is due to both the diffusion and the convection, while in (Frier and Sørensen 2007) only the chloride diffusion is estimated. In these 2-D chloride diffusion models it is assumed that the chloride ions penetrate at the same into the concrete section time along two directions perpendicular to the exposed external surfaces. This kind of model is applicable to RC elements with rectangular cross-section. Val and Trapper (2008) proved that the total chloride content increases faster near the reinforcement in a corner of a 2-D cross-section than in the proximity of the rebar in a 1-D model. For this reason the life-time of a rectangural RC element is shorter than the life-time of a RC element with geometry approximable to a plane semi-infinite element.

The piers cross-section of bridges or quays could be not only rectangular, but also circular. In (Arora et al. 1997) the chloride diffusion process was investigated for columns with a circular cross-section. In that article the diffusion in a model of one steel bar covered by saturated concrete and exposed to a chloride solution was analysed. Sensitivity analyses for different initial and boundary conditions were made. Finally, the analytical results were compared with experimental data taken on circular columns with radii smaller than ones used in real structures. Although that article proposed the solution of the Fick’s second law for element with circular cross section, the results were estimated for RC members with unrealistic dimensions.

This paper proposes the analytical solution of Fick’s second law for RC members with circular cross section calculated for RC cylindrical columns with realistic dimensions. These results are compared with the results of the chloride diffusion in the slab model to highlight the difference between the time to corrosion initiation of the RC elements estimated with the model of a cylindrical column and the time estimated with the semi-infinite plane model. This comparison leads to indications about the limit of use the simple “slab model” to estimate the time to corrosion initiation in case of RC members with circular cross-section.