Bulletin of Earthquake Engineering

, Volume 17, Issue 7, pp 4247–4267 | Cite as

Seismic fragility analysis of deteriorating RC bridge columns with time-variant capacity index

  • Hu Cheng
  • Hong-Nan LiEmail author
  • Y. B. Yang
  • Dong-Sheng Wang
Original Research


This paper presents a methodology to perform seismic fragility analysis of deteriorating reinforced concrete (RC) bridge columns induced by corrosion based on time-variant capacity estimates. The typically simplified uniform reduction model of cross-section of a corroded reinforcing steel bar is assumed, and the strain penetration and slippage of reinforcements anchored to the foundation are considered in the performance assessment of corroded bridge columns. To be conveniently used in engineering practice, the drift ratio is taken as the capacity index to quantify various damage states, and the plastic hinge model is adopted and verified to calculate the tip displacement of corroded columns due to its simple form and wide use in elastic–plastic analysis. Seismic fragility curves are then developed with reference to both the pristine (or original) construction state and current deteriorated state to illustrate the differences of potential damage probabilities calculated by time-invariant and time-variant capacity indexes. Further, significance test is performed. The results highlight that some of the empirical plastic hinge length models can also be used to predict the ultimate displacements of corroded columns by introducing the corroded constitutive models of materials; the seismic performance of a deteriorating RC bridge column will be significantly underestimated by using pristine capacity index as the measurement criterion. As a matter of fact, in engineering practice, one of the main concerned issues should be the current structural failure probability when evaluating the performance of a deteriorated RC structure. Hence, it is suggested that the time-variant capacity index should be adopted to estimate the seismic performance of a deteriorating RC structure during its life cycle.


Seismic fragility RC bridge column Deteriorated structural capacity Chloride-induced corrosion Capacity index Significance test 

List of symbols

\( A_{g} \)

Column section area

\( A_{s0} \)

Area of undamaged steel bar

\( b_{0} \)

Width of undamaged concrete cross-section without corrosion cracks

\( c \)

Concrete cover

\( D \)

Column diameter for circular section or short size for rectangle section

\( d_{s} \)

Original diameter of longitudinal reinforcement

\( E_{s} \)

Young’s modulus

\( f_{c} \)

Peak compressive strength of concrete

\( f_{c}^{ * } \)

Reduced concrete strength due to cracking

\( f_{y} \)

Yield strength of longitudinal reinforcement

\( f_{u} \)

Ultimate strength of longitudinal reinforcement

\( H \)

Length from the point of maximum moment to the point of inflection

\( L_{p} \)

Plastic hinge length

\( n_{0} \)

Axial load ratio

\( n_{bars} \)

Number of steel bars

\( P \)

Axial load

\( P_{f} \)

Failure probability for a specific damage state

\( p \)

Corrosion penetration depth

\( S_{a} (T_{1} ) \)

Spectral acceleration at the first-mode period of vibration

\( S_{c} \)

Structural capacity

\( S_{d} \)

Seismic demand

\( w \)

Mean crack opening for each steel bar

\( w_{cr} \)

Critical width of crack corresponding to delamination and spalling of concrete cover

\( \Delta_{tip} \)

Tip displacement of cantilever column

\( \Delta_{y} \)

Yield displacement of cantilever column

\( \Delta_{p} \)

Plastic displacement of cantilever column

\( \delta \)

Dimensionless corrosion penetration index

\( \delta_{s} \)

Damage function of steel bar

\( \delta_{s0} \)

Amount of steel damage necessary for cracking initiation

\( \varepsilon_{c0} \)

Concrete strain at the peak compressive strength

\( \varepsilon_{ \bot } \)

Average (smeared) tensile strain in the cracked concrete at right angles to the direction of the applied compression

\( \varepsilon_{y} \)

Yield strain of longitudinal reinforcement

\( \varepsilon_{sh} \)

Steel strain at the starting point of hardening of longitudinal reinforcement

\( \varepsilon_{u} \)

Ultimate strain of longitudinal reinforcement

\( \kappa \)

Coefficient related to steel bar roughness and diameter

\( \lambda \)

Aspect ratio

\( \eta_{s} \)

Corrosion level of longitudinal reinforcement in terms of cross-sectional areas loss

\( \bar{\eta }_{s} \)

Average corrosion level of all longitudinal reinforcements for each specimen

\( \eta_{s,cr} \)

Critical corrosion level representing disappearing of yield plateau region

\( \phi \)

Section curvature at bottom of cantilever column

\( \phi_{y} \)

Yield curvature at bottom of cantilever column

\( \theta \)

Drift ratio of cantilever column

\( \theta_{y} \)

Yield drift ratio of cantilever column

\( \theta_{p} \)

Plastic drift ratio of cantilever column

\( \rho_{l} \)

Longitudinal reinforcement ratio

\( \rho_{s} \)

Transverse reinforcement ratio



This research is supported by the National Key Research and Development Program of China (2016YFC0701108) and the National Natural Science Foundation of China under Grant (NNSFC51778206). The first author would like to thank the China Scholarship Council for providing financial support for him to study in Politecnico di Milano as a visiting Ph.D. student.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Hu Cheng
    • 1
    • 2
  • Hong-Nan Li
    • 2
    • 3
    Email author
  • Y. B. Yang
    • 4
  • Dong-Sheng Wang
    • 5
  1. 1.School of Environment and Civil EngineeringJiangnan UniversityWuxiChina
  2. 2.State Key Laboratory of Coastal and Offshore EngineeringDalian University of TechnologyDalianChina
  3. 3.School of Civil EngineeringShenyang Jianzhu UniversityShenyangChina
  4. 4.School of Civil EngineeringChongqing UniversityChongqingChina
  5. 5.School of Civil and Transportation EngineeringHebei University of TechnologyTianjinChina

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