Reliability‐based approach to the robustness of corroded reinforced concrete structures

Currently, decisions on the maintenance and repair of infrastructural assets, structures in particular, are mostly based on the results of inspections and the resulting condition index, neglecting system robustness and therefore not making optimal use of the limited funds available. This paper presents a definition and a measure of structural robustness in the context of deteriorating structures which are compatible with asset management systems for optimal maintenance and repair planning. The proposed index is used to define the robustness of existing reinforced concrete (RC) structures to rebar corrosion. Structural performance and the corresponding reliability index are assessed using combined advanced reliability and structural analysis techniques. Structural analysis explicitly includes deterioration mechanisms resulting from corrosion, such as reinforcement area reduction, concrete cracking, and bond deterioration. The first‐order reliability method, combined with a response surface algorithm, is used to compute the reliability index for a wide range of different corrosion levels, resulting in a fragility curve. Finally, structural robustness is computed and discussed based on the results obtained. A robustness comparison of different structures can then be used to determine structural types more tolerant to corrosion and these results used for planning maintenance and repairs.

Currently, decisions on the maintenance and repair of infrastructural assets, structures in particular, are mostly based on the results of inspections and the resulting condition index, neglecting system robustness and therefore not making optimal use of the limited funds available. This paper presents a definition and a measure of structural robustness in the context of deteriorating structures which are compatible with asset management systems for optimal maintenance and repair planning. The proposed index is used to define the robustness of existing reinforced concrete (RC) structures to rebar corrosion. Structural performance and the corresponding reliability index are assessed using combined advanced reliability and structural analysis techniques. Structural analysis explicitly includes deterioration mechanisms resulting from corrosion, such as reinforcement area reduction, concrete cracking, and bond deterioration. The first-order reliability method, combined with a response surface algorithm, is used to compute the reliability index for a wide range of different corrosion levels, resulting in a fragility curve. Finally, structural robustness is computed and discussed based on the results obtained. A robustness comparison of different structures can then be used to determine structural types more tolerant to corrosion and these results used for planning maintenance and repairs. Maintaining the safety and serviceability of existing structures and bridges by making better use of available resources is one of major challenges of transportation agencies in most developed countries since the number of structures reaching the end of their design life is growing every year. 1 Strategies that give priority to the poorest condition are clearly insufficient as they do not take advantage of structural robustness and tolerance to damage. Currently, decisions on maintenance and repair are reactive and mostly based on the results of visual inspections and the resulting condition index. The condition index is a convenient indicator of the deterioration of a structure, but provides little information regarding structural safety, as neither the initial (intact) safety margin nor the impact of deterioration on safety is considered. Experience has shown that different structures can, for similar deterioration levels, present significantly different safety margin reductions and safety levels, with a dramatic effect on the need for repairs and the optimal allocation of funds in a network. This paper presents a framework for assessing the robustness of structures suffering deterioration. Considering the fact that a detailed safety assessment of every existing structure is impossible due to financial limitations and the uncertainty regarding the real deterioration, the robustness concept proposed here can serve as an approximate measure of the mean loss in safety irrespective of the deterioration level for a given bridge type. The proposed robustness framework can then be combined with the bridge deterioration information to obtain a better indication of current and future reductions in safety due to deterioration and, therefore, to define an optimum maintenance policy. For instance, the robustness indicator may help the decisionmaker to take a wise decision regarding the maintenance operations to be carried out on two bridges with an equal or similar condition rating.
Although a robustness analysis is also complex, the robustness of similar structures is believed to be relatively uniform, allowing a classification of structures in a network based on the detailed analysis of a limited number of structural typologies. This classification can be used in conjunction with the observed or predicted condition state to define the need for or urgency of maintenance considering, explicitly, the structural properties of a particular structure. This allows a clear distinction between structures that, although exhibiting similar deterioration levels for specific main components, have very different safety levels as a result of different geometries or critical failure paths or other factors.
The focus is on reinforced concrete (RC) structures (due to the representativeness of this structural type worldwide) suffering reinforcement corrosion as this is one of the major causes of structural deterioration.

| STRUCTURAL ROBUSTNESS
Research on robustness has focused on extreme events such as terrorist attacks. However, the concept can also be very useful in the context of structural aging and deterioration, especially in the asset management field. The robustness of some structural types can be crucial to the planning and design of future infrastructures so that they require fewer repair and maintenance actions during their service lives.
When it comes to the corrosion of RC structures, although the mechanisms responsible for rebar corrosion are relatively well known, 2 the prediction of future deterioration is associated with a very large uncertainty. For this reason, the deterioration of RC structures can be analyzed in a robustness framework, considering corrosion as unpredictable and assuming levels within a wide range. This approach is useful for both new and existing structures as it indicates, on the one hand, the structural designs less susceptible to corrosion and, on the other hand, the existing structures for which higher complete life repair costs can be expected.
Although robustness is a desirable property, there is still no consensual definition of robustness or a framework to assess it. 3 Significant work has been done, in particular under COST 1 Action TU-0601 "Robustness of Structures," but no unanimous methodology has yet been found.
Some authors suggest robustness to be a structural property, 4-7 while for others, robustness also depends on the surrounding environment. 8 In this latter case, robustness is a much broader concept, because it takes into account the indirect consequences of failure, which depend on several aspects, for example, social and economic. An in-depth discussion of the robustness concept can be found in Reference 7. In this paper, the perspective of robustness being a structural property is adopted in order to characterize the damage tolerance of existing structures to deterioration. The proposal of Reference 7 is considered because it is sufficiently generic to be applied to most structural types and damage scenarios and can be used in a probabilistic or deterministic framework. Robustness is defined as a structural property that measures the degree of structural performance remaining after damage occurrence. This relation can take many different forms depending on the limit state (from serviceability to ultimate) that is adopted in the structural evaluation. Damage can vary from simple degradation to a more serious damage scenario involving a local failure.
In order to assess robustness, it is fundamental to define a measure of structural performance f and a damage D causing a decrease in performance. The next step is to define the performance function of the damaged structure f(D) for the complete damage spectrum. The maximum value of damage in the spectrum corresponds to the maximum expected loss of performance during the service life. This is important when comparing the robustness of different structural types, where the performance profile can be highly different as a function of the damage level or, alternatively, the service life. In the final step, both the damage and performance indicator are normalized and the robustness indicator R D is computed as follows: For null robustness structures, a low level of damage produces a total loss of structural performance, and vice versa.
The proposed index R D is a generalization of the proposals of References 4 and 6 and the damage-based measure R D,int proposed by Reference 3, but with some advantages that appear to solve some of the limitations found in the robustness measures referred to. The proposal of Reference 4 is not suitable for dealing with continuous damage, which is the case with reinforcement corrosion. This problem appears to be solved by the proposal in Reference 6. Although this index considers continuous values for the damage variable, it results in different values for the robustness index depending on the damage level. These problems have been solved by the proposed index R D by considering normalized and continuous values for both structural performance and damage. Additionally, since the whole damage domain is integrated, robustness is given by a unique value independently of the damage level. Thus, the resulting robustness may be similar for different structures even if one degrades continuously and the reaction of the other is brittle. However, this can be overcome by the use of a probabilistic approach to measure the structural performance.
This paper analyses the robustness of RC structures subjected to corrosion. The damage inflicted on the structure is considered to be the corrosion level of the reinforcement measured in terms of rebar weight loss percentage. The difficulties in defining a probabilistic model for a hazard, in this case corrosion, lead to the analysis for a range of different corrosion levels. This strategy has been used in seismic engineering, for instance, where fragility curves resulting from exposing structures to different earthquake intensities have been used to characterize structural performance in seismic events. However, the concept can be extended to a wide range of other hazards, such as structural deterioration and, in particular, to reinforcement corrosion.
In this paper, structural performance is measured by the reliability index as this is a consistent measure of structural safety which takes uncertainty into account.

| Corrosion process
When RC is exposed to the environment, steel bar corrosion, and iron oxides formation are likely to occur due to the energetic potential of the iron-carbon alloy. The iron oxides resulting from the corrosion reaction do not have mechanical properties comparable with those of steel and undergo a volume increase that can reach seven times the original steel volume. The final result is the occurrence of several deteriorating mechanisms that lead to a deterioration in the structural capacity.
During the lifetime of a RC structure, two periods of corrosion can be distinguished. 9 The initiation period, that is, the stage where reinforcement is protected by a thin oxide layer. During this period, corrosion takes place at a negligible rate and no deterioration effects are expected. The second phase, the propagation period, starts when the concrete cover is contaminated and the passive oxide layer is destroyed. This results in an increased rate of corrosion and a deterioration in the structure's condition.
Steel depassivation occurs mainly due to concrete carbonation and chlorides contamination, which are typical of industrial and maritime environments, respectively. In the first case, corrosion is likely to occur uniformly along the length of the steel bars, while in the second case, corrosion tends to be more localized and pronounced (also known as pitting corrosion). In both cases, several deterioration mechanisms are expected to aggravate the structure's condition: reduction in effective reinforcement area, reduction in ductility of steel bars, cracking, and spalling of concrete and bond degradation between steel bars and surrounding concrete. The influence of these mechanisms on the structural behavior depends on several factors, such as type of corrosion, reinforcement ratio, concrete strength, loading, and cross section geometry. 10 In general, the reduction in the effective area and ductility of the steel bars is of more concern in cases of localized or pitting corrosion, 11,12 whereas concrete cracking and spalling and debonding effects play a greater deteriorating role in cases of general corrosion. [13][14][15][16] The reduction in the ductility of the steel bars is partly due to a chemical transformation of the steel which occurs during the corrosion process and is known as hydrogen embrittlement. 17,18 This is partly due to a localization phenomenon resulting from nonuniform corrosion 19 and can explain the reason why the reduction in the ductility of the steel bars has been especially considered in cases of pitting corrosion. In these cases, however, concrete cracking and spalling and debonding of reinforcement are, in general, not critical, as steel bars can be anchored in less corroded and uncracked zones. 20 However, if corrosion attacks the entire length of a bar, spalling of the concrete cover is likely to occur and loss of bond between steel bars and concrete, compromising the composite behavior of the two materials, can be expected. Val et al 21 concluded that the effects of localized and generalized corrosion can be potentially more hazardous for the bending ultimate and serviceability limit states of highway bridges, respectively. However, it should be noted that the authors assumed perfect anchorage of reinforcement in the abutments. Even if hooks are provided at rebar ends, anchorage can be greatly impaired by the existence of lapped rebars. 22 Additionally, it should be noted that the corrosion rate usually increases in zones of reinforcement concentration or where it is bent. According to References 13 and 14, reinforcement debonding is the main cause of impaired flexural behavior if corrosion is found to be generalized and uniform. This paper addresses generalized and uniform corrosion; localized and pitting corrosion are beyond the scope of this paper. From this point onwards, and for the sake of simplicity, only the effects of concrete cracking and spalling, debonding of steel bars, and reduction in effective reinforcement area will be considered. The impaired ductility of reinforcement and reduction in steel strength, including the spatial variability of corrosion, are not considered here, although it is recognized, and as suggested by References 11, 12 and 23, that these are factors of paramount importance in cases of localized corrosion, which is not considered here.

| Methodology
As discussed in the previous section, to model the effects of generalized corrosion adequately, it is necessary to take into account some undesirable consequences of the oxidation process of rebars, including reduction in reinforcement net area and expansion due to the accumulation of corrosion products. This latter phenomenon leads to damage, cracking, and splitting of the surrounding concrete and degradation of the steel-concrete bond responsible for the stress transfer between the two materials.
In order to model all these effects, an advanced finite element methodology was used coupled with advanced constitutive models for modeling materials. The method's capability to reproduce the behavior of corroded RC was demonstrated by comparing numerical results with results obtained experimentally. 24 The methodology employed considers a two-step analysis.
In the first step, a finite element analysis of the structure cross section is carried out, simulating the formation, and accumulation of corrosion products as an expansion of the steel bars. In this phase, steel bars are modeled using a linear elastic law and are coupled to the concrete through an interface model that regulates the shear stress transfer between the two materials. For the sake of simplicity, corrosion is considered to attack uniformly around the bar perimeter, although it is known that corrosion is more pronounced on the part of the steel bar closest to the external surface of the concrete. For concrete, an isotropic continuum damage model was used enriched with kinematics provided by the strong discontinuities theory. 25 The combination of these two approaches for modeling concrete behavior allows the simulation of crack development caused by the corrosion and expansion of the rebars.
In the second step, the results obtained during the cross section analysis are then used to build a two-dimension (2D) structural model of the corroded structure used to assess the impaired structural capacity. The RC is modeled by means of a composite material constituted by a matrix, representing the concrete, mixed with long fibers that represent the steel bars, as proposed by Reference 26. This is the main difference from the modeling strategy proposed by Reference 24, which used a mesoscopic approach for the 2D longitudinal model, using different finite elements for concrete, reinforcing bars, and interface. In the homogenized model used here, a unique composite finite element was enriched to reproduce the composite behavior of all the components. An advantage of this is that the homogenized model requires far fewer computational resources, due to the smaller size of the numerical model. This is an important aspect in this case, since many different analyses are required to perform the fragility curves. Additionally, the homogenized model seems to reproduce better the global structural behavior, since the interface between concrete and steel bars is implicitly considered. In the mesoscopic approach, the bond effect is reproduced using interface elements. In this manner, results can be affected by the mesh size, usually resulting in a less stiff global behavior.

| Cross section analysis
This section depicts the results obtained in the first step of the corrosion analysis methodology, obtained for a rectangular section (0.20 × 0.40 m 2 ) with the mean value properties of a C30/37 concrete and 2φ10 + 2φ20 steel rebars (grade S400) placed at the top and bottom faces, respectively. Corrosion was simulated considering a volumetric expansion of the steel bars, with similar penetration rates on both bars. The resulting iron oxides, as suggested by Reference 27, were considered to be incompressible and to occupy twice the initial iron volume. Figure 1 shows the effect of corrosion at cross section level. Figure 1a shows a damage map d on concrete due to the expansion of steel bars for a corrosion penetration depth X = 0.5 mm, which corresponds to an area percentage loss X P 1 = 10% and X P 2 = 20% for the bottom and top reinforcement, respectively. Damage d = 1 crack (e) crack (d) means the concrete had lost all its strength and cracking is imminent. Figure 1b shows the corresponding isodisplacement lines, the concentration of which indicates crack development as shown in Figure 1c. Figure 2 shows the width evolution of cracks (a)-(e) as corrosion increases. Cracks (a) and (e) are those reaching the range of visible cracks (0.1-0.2 mm) for X P 1 and X P 2 equal to 1 and 2%, respectively, and thus consistent with experimental results. 10 Figure 2 shows that, for corrosion X P 1 > 5% and X P 2 > 10%, crack widths increase linearly and no additional cracks were detected. This allows the definition of the effective concrete cross section as shown in Figure 1d. For the sake of simplicity, the corrosion of transverse reinforcement was neglected, 21 although it is recognized that, on the one hand, there is a positive confinement effect and, on the other hand, an additional negative contribution to the cross section deterioration.

| Structural analysis
The results obtained from the cross section analysis were used to build a 2D structural model of the corroded structure. A simply supported 5-m span beam was used to illustrate the proposed methodology. The RC was modeled by means of a composite material constituted by a matrix, representing the concrete, mixed with long fibers representing the steel bars, as proposed by Reference 26. Three types of composite materials had to be considered (see Figure 3): concrete cover (unreinforced plain concrete), concrete in the beam web (reinforced transversely) and concrete surrounding flexural bars (reinforced longitudinally). As for the cross section analysis, and in order to be able to model crack development, in the longitudinal model, the finite elements were also enriched with the strong discontinuities kinematics 25 and the isotropic continuum damage model was adopted for the concrete. 28 For the embedded fibers, the objective was to model the reinforcement behavior and the debonding effect resulting from corrosion simultaneously. In order to achieve such a goal, the slipping-fiber model proposed in Reference 26 was adopted, which considers slipping-fiber ε f strain as the sum of the fiber's mechanical deformation and the deformation of interface (see Figure 4).
Assuming a two-component serial system constituted by the fiber and the interface, the corresponding slippingfiber stress σ f is identical to the stress in each component. In both cases, the stress-strain relation can be obtained via a one-dimensional elastoplastic hardening-softening model. The resulting constitutive behavior for the slipping-fiber is also an elastoplastic model with the following characteristics:   Note that when E i ! ∞ and σ d y < σ i y , the system provides the mechanical behavior of the fiber only, reproducing a perfect adhesion between concrete and reinforcing bars.
For the slipping-fiber model characterization, pull-out tests can be performed in order to assess the parameters required. In this paper, for the uncorroded state, perfect adhesion between steel bars and concrete is considered and rigid-plastic behavior is adopted for the interface. This hypothesis may be considered acceptable since it is considered that the anchorage lengths are respected and only ultimate limit states related to structural capacity are being analyzed. For corroded states, the bond limit stress σ i y was considered to be lower than the reinforcement yield stress σ d y and to be dependent on the corrosion level X P . This means that perfect adhesion it is not valid for the corroded states, as suggested by several researchers. [14][15][16]29,30 The steel yield strength was considered to be unaffected by corrosion, although reductions have been documented, especially in cases of localized corrosion. In order to characterize bond strength reduction as a function of corrosion level, the M-pull model proposed by Reference 31 was adopted. This is an empirical model based on several authors' experimental tests (see Figure 5), so the following results must be considered with care. The M-pull model gives the normalized bond strength reduction depending on the corrosion level: For the sake of simplicity, bond degradation was considered to be uniform around the steel bar perimeter.
In order to build the 2D structural longitudinal model of the deteriorated structure, it was necessary to import the results obtained from the cross section corrosion analysis (see Figure 6a). As mentioned, special attention was given to the crack pattern, that is, when a crack crossed two cross section faces, the smaller section part was considered to be disconnected from the section core (see Figure 6b) and then, for simplicity, considered with damage d = 1 (see Figure 6c). In this case, and as observed in the previous section, for advanced corrosion stages, concrete corners at both the top and bottom of the beam tended to split away from the section core. The next step was to divide the cross section into thin horizontal slices and compute the average damage d for each slice (see Figure 6d). Finally, the damage values for each slice, as shown in Figure 6e, were projected onto the 2D longitudinal structural model (see Figure 6f ) defining the deteriorated structure.

| RELIABILITY ANALYSIS
As previously mentioned, the reliability index β is the structural performance indicator chosen to assess robustness since it is a consistent measure of safety. However, the reliability of an existing corroding structure is a time-dependent problem, which can be expressed by the following equation: where P f (t) instantaneous probability of failure at time t; X(t), random variables vector; G[X(t)], limit state function; and f X (t) joint probability density function of random variables. The instantaneous probability of failure can be integrated over an interval of time [0; t], resulting in the probability of failure over that time period P f (0, t). The random variables X(t) depend on time and, therefore, so does P f (t). The time t at which the limit state function G[X(t)] becomes zero is denoted time-to-failure and Equation 5 corresponds to a first-passage probability assessed with the out-crossing theory. 32 Time-integrated approaches for solving Equation 5 are much simpler, as lifetime maximum distributions for loads are used as presented in Equation 6: where R(t) is the resistance and S max (t) the maximum load effect for the time period [0; t]. However, as resistance also depends on time, decreasing with deterioration, it is extremely unlikely that the maximum load effect coincides with the time of minimum resistance. By dividing the structure's lifetime into n limited time periods for which resistance can be considered as time-invariant, it is possible to approach the first-passage problem by using Equation 7: where R i is the resistance at time interval [t i − 1 ; t i ], considered to be constant, and S max,i is the maximum load effect within the same period. Despite the independence of S max,i between time periods, the subset of events presented in Corrosion Level X P (%) Normalized bond strength Thus, if relatively short time periods are considered, taking into account the corrosion rate, the probability of failure, given a certain level of corrosion, can be considered to be approximately independent of time. The corresponding reliability index β is therefore used here as the performance indicator independent of time and Equation 1 results in Negative reliability indices might occur in the case of severe deterioration, meaning the structure is very likely to fail. Such a high risk will significantly decrease the robustness index, indicating the high potential consequences of deterioration. In order to compute the reliability index, the response surface method (RSM) is used to obtain an explicit approach for the structural response and allow the use of the first-order reliability method (FORM). 34,35 To illustrate the proposed methodology, the simply supported beam analyzed in the previous section is used and considered to support a 0.075-m deep × 1.25-m wide concrete deck for pedestrians.
The number of random variables considered in this study had to be restricted to the most fundamental because of the demanding reliability analysis, the sophisticated numerical models, and the limited computational resources. Table 1 shows the distributions and parameters of the six random variables considered as uncorrelated. The statistical properties of concrete 36,37 and reinforcing bars have been considered. Live load is the result of a concentration of pedestrians and is modeled by an exponential distribution with a 98% quartile of 7.0 kN/m 2 for the maximum distribution over a reference period of 50 years. This results in an exponential rate parameter λ = 1.1 and a mean value of 0.90 kN/m 2 for an annual occurrence rate. Thus, for the probability of failure to be computed with respect to a period of 1 year, and for the usual corrosion rates, the resistance of the deteriorating structure can be considered as constant. The width of the deck (1.25 m) was considered in the surface loads.
The limit state function G is defined as the resistance R minus the acting load S due to self-weight and live load. The resistance is considered as the maximum uniform load that could be applied to the structure until its failure in bending either defined by yielding of the steel bars or crushing of the concrete. The load effect S can be obtained from Equation 9: where W is the deck width (1.25 m) and A beam c and d slab c are the beam cross section and slab depth, respectively. The resistance can be computed with Equation 10: where R(f c , f y , X p ) is the resistance obtained from the corrosion analysis methodology described previously, explicitly approached by a response surface defined for each design point d P .
5 | DISCUSSION Figure 7 shows the evolution of the reliability index β(X P ) and the respective failure probability P f (X P ) for the corrosion level X P . The reliability of the intact structure is 3.5, decreasing significantly as corrosion increases, especially in the first 15% of reinforcement area lost. For corrosion levels ranging from   Figure 7 also shows two additional fragility curves: β(X P ) * , where the debonding effect has been neglected, and β(X P ) ** , where only reinforcement area reduction has been considered. Comparing β(X P ), β(X P ) * , and β(X P ) ** shows that the reduction in the safety margin due to the loss of reinforcement area is almost linear until corrosion reaches about 80%. From this stage onwards, the effective reinforcement area is below the minimum required to avoid structural failure immediately after initiation of flexural cracks. The cracking effect is more significant for corrosion >80%, as from this stage onwards, flexural strength is provided by the plain concrete section, which in this case has deteriorated as shown in Figure 1d. Figure 8 shows the normalized performance obtained from the ratio between the reliability of the corroded structure and the intact one as a function of the normalized damage, in this case considered as the corrosion level of the bottom reinforcement. The maximum damage is limited to 50% as for existing structures such a level of deterioration would trigger a repair action and considering more advanced corrosion levels is clearly unrealistic.

| Robustness assessment
Robustness computed according to Equation 8 results in R = 28%, showing that tolerance to generalized corrosion is relatively low and a reduced level of safety should always be a concern. The mean normalized performance reduction is therefore 72%. This is a result of the lack of redundancy in a simply supported beam, but is also due to the absence of a second layer of bottom reinforcement less affected by corrosion.
Computing the robustness of the remaining cases presented in Figure 7 results in R* = 75% and R ** = 82% if debonding and debonding including cracking have been neglected, respectively. Establishing the difference between the computed robustness indicators Δf provides the relative importance of each deteriorating mechanism for the lack in robustness. The result is that the debonding effect is the main cause of structural deterioration, producing a mean safety reduction Δf 1 = 47%, followed by reinforcement area lost and then cracking, causing a mean performance reduction of Δf 3 = 18% and Δf 2 = 7%, respectively (see Figure 8). Figure 9 shows the beam's time-dependent probability of failure P f (0, t) with respect to the time period [0; t] and considering a corrosion progression of 1% annually. The initiation period has been neglected and time t = 0 refers to the onset of corrosion. The upper and lower bounds of the probability of failure turned out to be very narrow and overlapped in Figure 9 as weak dependency was found between different time periods. Figure 9 also shows the timedependent probability of failure for a similar beam but considered fully protected against corrosion P f (0, t) *** . A comparison between the unprotected and protected beams shows the impact of corrosion on the time-dependent safety level, indicating the former to be a case for concern, requiring premature intervention. The probability of failure is approximately the same within the periods of 5 and 50 years for the unprotected and protected beam, respectively. Figure 9 also shows the time-dependent probability of failure when neglecting the debonding effect P f (0, t) * and considering only the effect of reinforcement area reduction P f (0, t) ** . As mentioned above, debonding is the major cause of impaired robustness and thus has a major impact on the timedependent probability of failure.

| Decision-making based on robustness
Similarly, a longer time between periodic inspections could be adopted depending on robustness. Figure 10 shows the time-dependent probability of failure for the same cases of Figure 9, given the observed corrosion level at the time of inspection and within a period of 3 years P f (3y|X P ). For illustration purposes, the mean time between periodic inspections was considered here as 3 years. As observed, the probability of failure within the time between FIGURE 7 Reliability index β and failure probability P f as a function of corrosion level X P. FIGURE 8 Normalized performance as a function of normalized damage considering: β(X P )/β(0), all three deteriorating effects; β(X P )/β(0)*, neglecting debonding effect; β(X P )/β(0) * * , neglecting cracking, and debonding effects.
inspections is constant for the beam protected against corrosion due to full robustness. For the unprotected beam, the probability of failure increases with the degree of corrosion. Therefore, a reduction in the time between inspections is required over the beam's lifetime.

| CONCLUSIONS
A probabilistic framework for the evaluation of the structural robustness of structures subjected to continuous damage is presented in this work. Within this framework, damage is defined in terms of an unpredictable continuous variable, making this robustness index particularly suitable for structural management systems and allowing the analysis and comparison of different structural types, with the final objective being to define those requiring more and earlier maintenance. The proposed robustness index can be used to estimate the need to repair a structure when damage is identified, since it provides an estimate of current and future structural safety. However, the inclusion of this index in existing management systems will require a calibration process, including a large pool of typical structures, where the condition index and the robustness index are related to the remaining time before a safety threshold is reached.
The results obtained showed that the proposed index is able to characterize the robustness of a structure, from a structural viewpoint, with a single indicator, independently of the structural safety level of the undamaged structure. The robustness of the example presented resulted in a figure of 28% which shows the structure's tolerance to generalized reinforcement corrosion. The mean performance loss was 72%, of which 47, 18, and 7% are caused by bond deterioration, reinforcement area reduction, and concrete cover cracking, respectively. The comparison with a similar, but fully robust, beam (due to corrosion protection) shows that the unprotected beam is thus less robust, requires maintenance sooner, and shorter periods between periodic inspections. For the sake of simplicity in introducing the concept of the robustness index, the example presented in this paper corresponds to a single simply supported beam. However, it is known that in statically indeterminate structures, damage effects include a redistribution of internal forces. How this redistribution affects the final value of robustness owing to the redistribution of stresses and activation of alternative loading paths will be the subject of future research.