*6.3. Time-Dependent Seismic Fragilities of Aged Bridge System*

The joint impacts of the multi-deterioration of multiple components on system vulnerabilities are quantified by developing time-dependent fragility curves of the overall bridge. Figure 15 shows the aging bridge system fragility curves at three different points in time (e.g., 0, 25 and 50 years) for moderate and complete states. It can be clearly seen that the seismic fragility curves at the system level increased steadily with age from 0 to 50 years. However, the seismic fragility of the individual components tended to show a reduction in vulnerability when the bridge component continued to be corroded due to the individual consideration of multi-deterioration for a single component. Furthermore, the seismic fragility of the bridge expansion bearing along the transverse direction can decrease at the initial time (e.g., 0, 15 and 25 years) due to the added stiffness of the bearing NR pads induced by thermal oxidation. The results indicated the importance of considering the joint

effects of multi-deterioration mechanisms for multiple bridge components. Furthermore, it was found that the multi-deterioration of the aging MSC concrete girder bridge had a potentially negative influence on the overall seismic fragility at the system level. ing the joint effects of multi-deterioration mechanisms for multiple bridge components. Furthermore, it was found that the multi-deterioration of the aging MSC concrete girder bridge had a potentially negative influence on the overall seismic fragility at the system level.

*Materials* **2022**, *15*, x FOR PEER REVIEW 19 of 24

**Bridge Slight Moderate Extensive Complete Component Med. Disp. Med. Disp. Med. Disp. Med. Disp.**  RC columns 1.29 0.59 2.10 0.51 3.52 0.64 5.24 0.65 Fixed bearing—longitudinal 28.9 0.60 104.2 0.55 136.1 0.59 186.6 0.65 Fixed bearing—transverse 28.8 0.79 90.9 0.68 142.2 0.73 195.0 0.66 Expansion bearing—longitudinal 28.9 0.60 104.2 0.55 136.1 0.59 186.6 0.65 Expansion bearing—transverse 28.8 0.79 90.9 0.68 142.2 0.73 195.0 0.66

[47,48]. The JPSDMs can be written as Equation (48), where

*6.3. Time-Dependent Seismic Fragilities of Aged Bridge System* 

calculated by using Monte Carlo analysis.

**Table 6.** Capacity limit state for different bridge components for an MSC concrete girder bridge.

The assessment of bridge system vulnerability is performed by assuming the bridge as a series system, as presented by Nielson [47,48]. The demands of the bridge components under seismic loading are considered dependent and then the correlation coefficient between the peak responses can be estimated by constructing a joint probability density function (JPDF) for component demands. The generalized formula for the aged bridge system fragility can be derived using joint probabilistic seismic demand models (JPSDMs)

the median values (in units of g PGA) and logarithmic standard deviations of the system fragility at different points in time, respectively. Solutions to Equation (48) can be directly

The joint impacts of the multi-deterioration of multiple components on system vulnerabilities are quantified by developing time-dependent fragility curves of the overall bridge. Figure 15 shows the aging bridge system fragility curves at three different points in time (e.g., 0, 25 and 50 years) for moderate and complete states. It can be clearly seen that the seismic fragility curves at the system level increased steadily with age from 0 to 50 years. However, the seismic fragility of the individual components tended to show a reduction in vulnerability when the bridge component continued to be corroded due to the individual consideration of multi-deterioration for a single component. Furthermore, the seismic fragility of the bridge expansion bearing along the transverse direction can decrease at the initial time (e.g., 0, 15 and 25 years) due to the added stiffness of the bearing NR pads induced by thermal oxidation. The results indicated the importance of consider-

γ

*sys* ( )*t* and

ζ

*sys* ( )*t* are

**Figure 15.** Time-dependent system seismic fragility curves for an MSC concrete girder bridge for (**a**) a moderate damage state and (**b**) a complete damage state under multi-deterioration of multiple components. **Figure 15.** Time-dependent system seismic fragility curves for an MSC concrete girder bridge for (**a**) a moderate damage state and (**b**) a complete damage state under multi-deterioration of multiple components.

#### *6.4. Time-Dependent Seismic Fragilities Considering Prestress Losses and Cracking*

A reference model for the corrosion-induced presetress loss was developed [20,21] by considering the effects of concrete cracking. Corrosion-induced prestress loss can be modeled as the difference between the effective prestress in an uncorroded strand and that in the corroded strand [49]. Here, it was suggested that the strain compatibility and force equilibrium equations in concrete bridge-girders could also be used to evaluate the effective prestress in the corroded strand. If the pre-stressing force in the pre-tensioned concrete bridge-girders is released, then the strand pre-stress would transfer to the concrete via the bonding stress at the stand-concrete interface. Then, the effective prestress in the corroded strand can be calculated using Equation (49) [49], where *T<sup>p</sup>* represents the tension force of the corroded strand and *Ap*(*η*) denotes the residual cross-sectional area of the corroded strand.

During the corrosion process, RC components of bridges also suffer from the prestress and the expansive pressure. When the tensile stress induced by the expansive pressure exceeds the concrete tensile strength, the concrete is considered to be cracking. The concrete cover can contain a cracked inner region and an uncracked outer region. Here, the outer wires can be considered to be first corroded when the strand suffers from corrosion. The corrosion loss of a strand can be expressed as Equation (50) [50], where *R*<sup>0</sup> and *R<sup>ρ</sup>* are the radiuses of wire before and after corrosion, respectively, and *A<sup>p</sup>* is the strand crosssectional area. Meanwhile, it is assumed that the smeared cracks in the cracked region are distributed uniformly, and then a reduction factor can be used to reflect the residual tangential stiffness in the cracked concrete. Consequently, by combining stress equilibrium equations with the strain compatibilities, the crack width on the concrete surface can be written as Equation (51) [50], where *R<sup>t</sup>* is the radius of wire with corrosion products; *R<sup>c</sup>* = *R*<sup>0</sup> + *C*, where *C* is the concrete cover; *ν<sup>c</sup>* = √ *ν*1*ν*2, where *ν*<sup>1</sup> and *ν*<sup>2</sup> are the Poisson ratios of concrete in the radial and tangential directions, respectively; *a* is a reduction factor; *ft* is the concrete tensile strength under the biaxial stress state; and *E<sup>c</sup>* is the elastic modulus of concrete.

Subsequently, the combined effects of the prestress losses and cracking on the seismic fragilities of bridge systems can be quantified by using Equation (48), which also includes the effects of the multi-deterioration of multiple components. Figure 16 shows the aging bridge system fragility curves at three different points in time (e.g., 0, 25 and 50 years) for moderate and complete states. It can be derived from Figures 15 and 16 that the system fragility when considering the combined effects of prestress losses and cracking was more conservative than that without considering these effects. The results indicated

that the combined effects of the prestress losses and cracking should not be neglected when performing the seismic fragility of aging bridge systems. *Materials* **2022**, *15*, x FOR PEER REVIEW 21 of 24

Finally, it is stressed that other time-variant approaches, such as the empirical exper-

1. Deals with multi-deterioration mechanisms among multiple compo-

iment and trial-and-error [51–53], for the corrosion process of aging bridge systems are very expensive and very unrealistic when it comes to obtaining a series of multi-deterioration mechanisms among multiple components of aging bridge systems. Moreover, the empirical experiment and trial-and-error method are very resource-consuming when dealing with the time-dependent system fragility of such complex aging bridges when undergoing the earthquake excitation test. Therefore, when we would like to efficiently develop the overall time-variant deterioration process of aging bridge systems, the finite element modeling combined with the theoretical modeling of the corrosion process is the best choice for performing the time-dependent overall seismic fragility curves of aging bridge systems. The applicability of different approaches can be summarized in Table 7. **Table 7.** Applicability of different approaches. **Different Approaches Applicability**  Finally, it is stressed that other time-variant approaches, such as the empirical experiment and trial-and-error [51–53], for the corrosion process of aging bridge systems are very expensive and very unrealistic when it comes to obtaining a series of multi-deterioration mechanisms among multiple components of aging bridge systems. Moreover, the empirical experiment and trial-and-error method are very resource-consuming when dealing with the time-dependent system fragility of such complex aging bridges when undergoing the earthquake excitation test. Therefore, when we would like to efficiently develop the overall time-variant deterioration process of aging bridge systems, the finite element modeling combined with the theoretical modeling of the corrosion process is the best choice for performing the time-dependent overall seismic fragility curves of aging bridge systems. The applicability of different approaches can be summarized in Table 7.

nents (e.g., RC columns, bearing systems); **Table 7.** Applicability of different approaches.


## **7. Conclusions**

This paper provides a probabilistic method for identifying the time-dependent fragility of an aging bridge system under earthquake events by considering the impacts of multideterioration of multiple bridge components, as well as the combined effects of the prestress

losses and cracking. A typical MSC concrete girder bridge is used to evaluate the seismic performance by considering the uncertainty models in structural material and corrosion parameters. Chlorine corrosion from deicing salts, which are widely utilized across northern China, can be considered the cause of the degradation and aging of bridge systems. Here, the multi-deterioration mechanisms include the corrosion deterioration of an RC column due to loss of both the steel area (i.e., longitudinal bars, stirrups) and concrete covers, and the deterioration of elastomeric bearing assemblies due to the reduction in both the shear stiffness of bearing NR pads and steel dowel area. The multi-deterioration mechanisms of RC components of the bridge systems affect the lateral force resistance and seismic responses. In addition, the models for corrosion-induced prestress loss and cracking are introduced to develop time-dependent seismic fragilities of bridge systems. The overall seismic demands found using PSDMs showed a steady increase as the growing service life when joint effects of multi-deterioration of bridge multiple components (e.g., RC columns and fixed bearings) were considered. However, there was a decrease in the demand (e.g., bearing deformation) on other multiple components (e.g., RC columns and expansion transverse bearings).

Time-variant PSDMs and the components' capacities were constructed for calculating the seismic fragility of the MSC concrete bridges, which considered the variability in attributes of bridges, ground motion and degraded parameters. The seismic fragility curves at the system level demonstrated a steady increase over time due to corrosion aging. However, individual components, such as expansion bearings in the transverse direction, revealed a decreased vulnerability at the initial time due to the added bearing pad shear stiffness. Subsequently, the dominant effect of the loss of steel dowel areas led to an increase in vulnerability. Moreover, the system seismic fragility considering the combined effects of prestress losses and cracking was more conservative. That meant the effects of both prestress losses and cracking should not be ignored when investigating the seismic fragility of the aging bridge systems. All these system fragility curves presented a more authentic estimation of the seismic vulnerability of aging bridges, and they offered a more accurate basis for life-cycle cost analysis.

**Author Contributions:** Data curation, W.Z.; Funding acquisition, X.L.; Investigation, P.S. and M.L.; Resources, P.S.; Visualization, W.Z.; Writing—original draft, X.L.; Writing—review & editing, M.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant numbers 51801149 and 11802224.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No data were used to support this study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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