Probabilistic capacity models and seismic fragility estimates for RC columns subject to corrosion

doi:10.1016/j.ress.2006.12.015

Reliability Engineering & System Safety

Volume 93, Issue 3, March 2008, Pages 383-393

https://doi.org/10.1016/j.ress.2006.12.015 Get rights and content

Abstract

In this paper, probabilistic drift and shear force capacity models are developed for corroding reinforced concrete (RC) columns. The developments represent a merger between a probabilistic model for chloride-induced corrosion, a time-dependent corrosion rate, and previously developed probabilistic models for drift and shear force capacity of pristine (undamaged) RC columns. Fragility estimates are obtained for an example corroding column by applying the developed models at given shear and drift demands. Model uncertainties in both the capacity and corrosion models are considered in the fragility estimation, in addition to uncertainties in environmental conditions, material properties, and structural geometry. Sensitivity analyses of the corroding RC column are carried out to identify the parameters to which the reliability of the example column is most sensitive. The developed models consider different combinations of chloride exposure condition, environmental oxygen availability, water-to-cement ratios, and curing conditions. They are applicable to both existing and new RC columns and may be employed for the prediction of service-life and life-cycle cost analysis of RC structures.

Introduction

Corrosion of reinforcement is detrimental to the serviceability and capacity of reinforced concrete (RC) structures. Corrosion is a long-term process that effectively weakens structural elements and increases their vulnerability to extreme loads (vehicles) and natural hazards. A structure that is originally designed to meet code specifications may not have the same margin of safety once the structure has undergone significant corrosion. Of particular significance to the developments in this paper is the inevitable presence of uncertainties; the prediction of onset and progression of corrosion can only be made in a probabilistic manner. For example, the corrosion process is highly influenced by the material and environmental factors. To this end, a key objective in this paper is to put forward structural capacity models that include the effects of deterioration, with appropriate consideration of uncertainties.

In recent decades, significant research efforts have been devoted to the quantification and inclusion of corrosion in design, construction and maintenance of RC structures. Tuutti [1] and Liu and Weyers [2] suggested deterministic corrosion models for RC reinforcement, while Thoft-Christensen et al. [3], [4] and DuraCrete [5] presented probabilistic models for the deterioration process. Stewart and Rosowsky [6], Vu and Stewart [7], Enright and Frangopol [8], [9] developed probabilistic corrosion models for bridge slabs, beams and girders by extending commonly employed RC capacity models. This paper further extends these developments by constructing deterioration models that incorporate both probabilistic models for the drift and shear capacity of RC columns and probabilistic models for the deterioration process.

Gardoni et al. [10], [11] and Choe et al. [12] developed probabilistic capacity models for pristine (not corroded) RC columns with circular cross section that properly account for all the prevailing uncertainties, including model error arising from potential inaccuracies in the model form and potentially missing variables, as well as measurement errors and statistical uncertainty. However, these models may not be appropriate once the structure begins deteriorating due to corrosion.

In this study, we incorporate information about material deterioration—employing a probabilistic corrosion model—into the probabilistic capacity models for RC columns developed by Gardoni et al. [10], [11] and Choe et al. [12]. The corrosion model is based on a probabilistic model for chloride induced corrosion [5] and a time-dependent corrosion rate function [7]. The uncertainties in parameters and model inexactness (or model error) in both the material deterioration model and the structural capacity model are considered.

The developed probabilistic models are used to assess the time-varying reliability of an example column subject to different environmental and material conditions. The reliability (or the lack of) is expressed in terms of a fragility function where fragility is defined as the conditional probability of failure of the column for given demand variables. Sensitivity analysis is carried out to identify to which parameter(s) the reliability of the example column is most sensitive. The models developed in this paper may be used for the prediction of service-life of existing and new structures and life-cycle cost analysis for RC structures.

Section snippets

Probabilistic model for reinforcement corrosion

There are three deterioration phases in the corrosion process of reinforcement steel [2]. In the first phase, termed the diffusion phase, chloride ions diffuse to the surface of steel to initiate corrosion. The second phase, termed the corrosion propagation phase, comprise the time from initiation of corrosion to initiation of cracks in the concrete cover. The third phase, termed the deterioration phase, is the process that takes place after the initiation of cracks.

To determine the time to the

Probabilistic capacity models for corroding RC columns

The probabilistic capacity models presented herein for corroding RC columns represent a merger between the work by Gardoni et al. [10] and Choe et al. [12] and the probabilistic diffusion model described above. A “model” in this study means a mathematical expression that relates one or more quantities of interest, e.g., capacities of structural components, to a set of measurable variables x=(x₁, x₂, …), e.g., material properties, member dimensions, and imposed boundary conditions. In its

Example column

The numerical studies in this paper are carried out with the OpenSees software (http://opensees.berkeley.edu) for an example column. OpenSees is a comprehensive, open-source, object-oriented finite element software that has previously been extended with reliability and response sensitivity capabilities [17]. In this study we employ OpenSees to obtain fragility estimates for the example column based on the probabilistic models presented above. The column is modeled by fiber-discretized cross

Fragility estimates

The probabilistic capacity models developed above enable estimating the fragility of corroding RC columns during their service-life. In particular, shear and drift fragilities are considered in this paper. The fragility of a structural component is defined as the conditional probability of failure, given demand variable values. According to the conventional notation in structural reliability theory, e.g., Ditlevsen and Madsen [19], a limit state function g_k(x, Θ_k) is defined such that the event

Predictive estimates of fragility

A “predictive” estimate of the fragility at time t, $\tilde{F} (t, x_{s})$ , incorporates the uncertainties in the model parameters Θ_k and T_corr, by characterizing Θ_k and T_corr as random variables and taking the expectation of F(t, x_s, Θ_k, T_corr) over the posterior distribution of Θ_k and T_corr, i.e., $\tilde{F} (t, x_{s}, T_{corr}) = \int F (x_{s}, Θ_{k}, T_{corr}) f (Θ_{k}) d Θ_{k}$ $\tilde{F} (t, x_{s}) = \int F (t, x_{s}, T_{corr}) f (T_{corr}) d T_{corr} = \int {\int F (t, x_{s}, Θ_{k}, T_{corr}) f (Θ_{k}) d Θ_{k}} \times f (T_{corr}) d T_{corr}$ where f(Θ_k) and f(T_corr) are the probability density functions of Θ_k and T_corr.

These

Sensitivity and importance measures

Sensitivity analysis is employed to determine to which parameter(s) the reliability of RC columns is most sensitive. Such results may provide physical insight and guidance in further data gathering and model development. To compute the sensitivity measures, we decompose x into a vector of constant parameters x_c and a vector of random variables x_p, so that x can be written as x=(x_c, x_p). Also, let f(x_p, Θ_f) be the probability density function of the basic variables x_p, where Θ_f is a set of

Conclusions

This paper presents probabilistic capacity models for corroding reinforced concrete columns to estimate the fragility of deteriorated structural components. The models are applicable to existing and new columns that are subject to current or future deterioration and may be employed in service-life and life-cycle cost analyses. Fragility estimates for an example corroding column subjected to shear force and drift demands are developed using the models developed herein. The fragilities show that

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Probabilistic capacity models and seismic fragility estimates for RC columns subject to corrosion

Abstract

Introduction

Section snippets

Probabilistic model for reinforcement corrosion

Probabilistic capacity models for corroding RC columns

Example column

Fragility estimates

Predictive estimates of fragility

Sensitivity and importance measures

Conclusions

Structural Safety

Engineering Structures

Modeling bond strength of corroded reinforcement without stirrups Cement and Concrete Research

Cement and Concrete Research

Structural Safety

Corrosion of Steel in Concrete Swedish

Modeling the time-to-corrosion cracking in chloride contaminated reinforced concrete structures

ACI Materials Journal

Assessment of the reliability of concrete slab bridges reliability and optimization of structural systems