**3. Multi-Degradation Mechanisms of RC Columns Due to Corrosion**

**3. Multi-Degradation Mechanisms of RC Columns due to Corrosion**  All equations used in Section 2 can be found in Table 1. All equations used in Section 2 can be found in Table 1.





#### *3.1. The Diffusion Process and Corrosion Initiation Time*

Chloride-induced corrosion is considered to be one of the major causes of the aging of reinforced concrete bridges. Corrosion of reinforcing steel in an RC column is initiated by the ingress of chloride ions through the concrete cover to the bar's surface. The main source of chloride ions is considered to be the deicing salts used in winter. The diffusion process of chloride ions that result in corrosion can be described by Fick's second law [25]. The formulation can be seen as Equation (1) in Table 2, where *C* is the chloride ion concentration, *D* is the chloride diffusion coefficient of concrete, *x* is the depth of the concrete cover and *t* is the time step in years. We assumed that the initial condition (initial chloride ion content), boundary condition (surface chloride content) and material property (chloride diffusion coefficient) were equal to zero, the mean-invariant *C<sup>s</sup>* and the mean-invariant *D*, respectively. The closed solution of the chloride ion content is shown as Equation (2) [32], where *er f*(·) is the Gaussian error function. The corrosion parameters involved in the diffusion process were assumed to be random variables whose distribution type, mean *u* and coefficient of variation COV are listed in Table 2 [5,19].

**Table 2.** Probability distribution and the random parameters involved in the diffusion process.


The diffusion process of chloride ions can be described in probabilistic terms by using the Monte Carlo method. It was assumed that the depth of concrete cover *x* was 40 mm and the service life *t* was 50 years; the resulting probability density functions (PDFs) of chloride ion concentrations that were obtained using a Monte Carlo simulation with a sample size of 50,000 are shown in Figure 2. With reference to Fick's second law of diffusion and Equation (2), the corrosion initiation time *T<sup>I</sup>* was evaluated using Equation (3) [5], where *T<sup>I</sup>* is the corrosion initiation time in years, *C<sup>S</sup>* is the chloride concentration of concrete surface and *C<sup>C</sup>* is the critical chloride concentration that can dissolve the protective passive film around the reinforcement steels. Therefore, a Monte Carlo simulation was performed with a sample size of 50,000 for random variables to evaluate the probability distribute of the corrosion initiation time *T<sup>I</sup>* , as shown in Figure 3. A lognormal distribution with a mean *µ* = 1.0374 years and standard deviation *σ* = 0.2829 was found to be a good estimation of the simulated distribution for the corrosion initiation time *T<sup>I</sup>* . This distribution had an influence on the probabilistic modeling of the steel corrosion in the bridge RC columns. sample size of 50,000 are shown in Figure 2. With reference to Fick's second law of diffusion and Equation (2), the corrosion initiation time *TI* was evaluated using Equation (3) [5], where *TI* is the corrosion initiation time in years, *CS* is the chloride concentration of concrete surface and *CC* is the critical chloride concentration that can dissolve the protective passive film around the reinforcement steels. Therefore, a Monte Carlo simulation was performed with a sample size of 50,000 for random variables to evaluate the probability distribute of the corrosion initiation time *TI* , as shown in Figure 3. A lognormal distribution with a mean μ = 1.0374 years and standard deviation σ = 0.2829 was found to be a good estimation of the simulated distribution for the corrosion initiation time *TI* . This distribution had an influence on the probabilistic modeling of the steel corrosion in the bridge RC columns.

The diffusion process of chloride ions can be described in probabilistic terms by using the Monte Carlo method. It was assumed that the depth of concrete cover *x* was 40 mm and the service life *t* was 50 years; the resulting probability density functions (PDFs) of chloride ion concentrations that were obtained using a Monte Carlo simulation with a

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process of chloride ions that result in corrosion can be described by Fick's second law [25]. The formulation can be seen as Equation (1) in Table 2, where *C* is the chloride ion concentration, *D* is the chloride diffusion coefficient of concrete, *x* is the depth of the concrete cover and *t* is the time step in years. We assumed that the initial condition (initial chloride ion content), boundary condition (surface chloride content) and material property (chloride diffusion coefficient) were equal to zero, the mean-invariant *Cs* and the mean-invariant *D* , respectively. The closed solution of the chloride ion content is shown as Equation (2) [32], where *erf* ( )⋅ is the Gaussian error function. The corrosion parameters involved in the diffusion process were assumed to be random variables whose distribution type, mean *u* and coefficient of variation COV are listed in Table 2 [5,19].

**Table 2.** Probability distribution and the random parameters involved in the diffusion process.

**Random Variable Unit Distribution Type Mean COV**  Concrete cover, *x* mm Lognormal 40.00 0.20 Diffusion coefficient, *D* cm2/year Lognormal 1.29 0.10 Surface chloride concentration, *Cs* wt%/cem Lognormal 0.10 0.10 Critical chloride concentration, *CC* wt%/cem Lognormal 0.040 0.10

Corrosion rate, *corr r* mm/year Lognormal 0.127 0.3

**Figure 2.** PDFs of the chloride concentration *C xt* ( , ) . **Figure 2.** PDFs of the chloride concentration *C*(*x*, *t*).

**Figure 3.** PDFs of the corrosion initiation time *T*1. **Figure 3.** PDFs of the corrosion initiation time *T*<sup>1</sup> .

#### *3.2. Reduction in the Cross-Sectional Area of Longitudinal Bars and Stirrups 3.2. Reduction in the Cross-Sectional Area of Longitudinal Bars and Stirrups*

The corrosion of rebars in the existing RC column was initiated by reducing the crosssection. However, some recent studies [33,34] showed that the corrosion of stirrups, which causes a reduction in confinement behavior, is more severe than that of the longitudinal bars in an RC column. Furthermore, the corrosion can degrade the shear resistance of the column by potentially shifting the failure modes from a ductile failure mode to a shear failure mode. Therefore, the effect of both the loss of the cross-sectional area and the ductile degradation of stirrups on the seismic behavior of bridge systems should be considered in this study. Corrosion from deicing salts leads to a reduction in the effective area The corrosion of rebars in the existing RC column was initiated by reducing the crosssection. However, some recent studies [33,34] showed that the corrosion of stirrups, which causes a reduction in confinement behavior, is more severe than that of the longitudinal bars in an RC column. Furthermore, the corrosion can degrade the shear resistance of the column by potentially shifting the failure modes from a ductile failure mode to a shear failure mode. Therefore, the effect of both the loss of the cross-sectional area and the ductile degradation of stirrups on the seismic behavior of bridge systems should be considered in this study. Corrosion from deicing salts leads to a reduction in the effective area of both

of both the longitudinal and transverse rebars. First, the corrosion penetration *R* is introduced as Equation (4) [19], where *corr r* is the corrosion rate and represents the implicit

agation time. The time-variant loss of longitudinal reinforcement and stirrup diameter in the BLG model can be evaluated using Equation (5) and Equation (6), respectively [19].

the diameters of a longitudinal bar and stirrup at the end of *<sup>I</sup> t T*− years. The corrosion

Equation (11), where ( ) 0 *AS* is the initial area of longitudinal reinforcement and ( ) *<sup>S</sup>*

is the loss rate of a longitudinal bar. The time-dependent area of a corroding stirrup can be expressed in Equation (12), where ( ) 0 *<sup>s</sup> a* is the initial area of a stirrup and ( ) *<sup>k</sup>*

the loss rate of a stirrup. It is acknowledged that the corrosion rate is considered a constant on average and it is assumed to be lognormally distributed, as listed in Table 1. Moreover, the diameters of the longitudinal rebar and stirrup are assumed to be lognormally distributed, whose mean *u* and coefficient of variation COV are listed in Table 3. On the basis of the above assumptions, the reduction in the diameter of the rebar due to corrosion could

time-variant mean of the longitudinal bar and the stirrups, as well as the corresponding standard deviations, are shown in Figure 4. The scattering region around the mean represents the stochastic time-dependent loss of both the longitudinal bars' and the stirrups'

Then, the time-dependent area of a corroding longitudinal bar can be expressed as

φ

( ) 0 are the initial diam-

*<sup>i</sup>* are the diameters of a

φ

is introduced as Equation (9) and Equation (10) [35].

φ

φ

( )*t* can be ex-

φ

( )*t* are

η*t*

λ*t* is

( ) 0 = 12.70 mm). The

pressed as Equation (7) and Equation (8), where Φ( ) 0 and

η

penetration index of rebars

eters of the longitudinal bar and stirrup, respectively; Φ*i* and

be evaluated for a lifetime of 50 years (Φ( ) 0 = 28.58 mm and

Combining Equation (4), Equation (5) and Equation (6), Φ( )*t* and

corroding longitudinal bar and stirrup at time *t i* = , respectively; Φ( )*t* and

the longitudinal and transverse rebars. First, the corrosion penetration *R* is introduced as Equation (4) [19], where *rcorr* is the corrosion rate and represents the implicit function of the time-dependent area due to corrosion, while *t* − *T<sup>I</sup>* is the corrosion propagation time. The time-variant loss of longitudinal reinforcement and stirrup diameter in the BLG model can be evaluated using Equation (5) and Equation (6), respectively [19]. Combining Equation (4), Equation (5) and Equation (6), Φ(*t*) and *φ*(*t*) can be expressed as Equation (7) and Equation (8), where Φ(0) and *φ*(0) are the initial diameters of the longitudinal bar and stirrup, respectively; Φ*<sup>i</sup>* and *φ<sup>i</sup>* are the diameters of a corroding longitudinal bar and stirrup at time *t* = *i*, respectively; Φ(*t*) and *φ*(*t*) are the diameters of a longitudinal bar and stirrup at the end of *t* − *T<sup>I</sup>* years. The corrosion penetration index of rebars *η* is introduced as Equations (9) and (10) [35].

Then, the time-dependent area of a corroding longitudinal bar can be expressed as Equation (11), where *AS*(0) is the initial area of longitudinal reinforcement and *ηS*(*t*) is the loss rate of a longitudinal bar. The time-dependent area of a corroding stirrup can be expressed in Equation (12), where *as*(0) is the initial area of a stirrup and *λ<sup>k</sup>* (*t*) is the loss rate of a stirrup. It is acknowledged that the corrosion rate is considered a constant on average and it is assumed to be lognormally distributed, as listed in Table 1. Moreover, the diameters of the longitudinal rebar and stirrup are assumed to be lognormally distributed, whose mean *u* and coefficient of variation COV are listed in Table 3. On the basis of the above assumptions, the reduction in the diameter of the rebar due to corrosion could be evaluated for a lifetime of 50 years (Φ(0) = 28.58 mm and *φ*(0) = 12.70 mm). The timevariant mean of the longitudinal bar and the stirrups, as well as the corresponding standard deviations, are shown in Figure 4. The scattering region around the mean represents the stochastic time-dependent loss of both the longitudinal bars' and the stirrups' crosssectional areas because of the effect of uncertainties in degradation parameters. This figure illustrates that the effects of stirrup corrosion were more significant than that of longitudinal reinforcement corrosion because the smaller diameter of a transverse bar led to higher levels of corrosion of the stirrup. Thus, the area loss of steel corrosion in RC columns could be modeled as the reduction in the cross-sectional area of both longitudinal reinforcing and stirrups in the BLG model with the fiber section.


**Table 3.** Probability distribution of random parameters for bridge modeling.

<sup>1</sup> Mean and standard deviation of the lognormal distribution, lower bound and upper bound of the uniform distribution.

#### *3.3. Reduction in Strength and Ductility of Corroded Longitudinal Bars and Stirrups*

The corrosion of stirrups is more serious than that of longitudinal reinforcement in RC columns and it leads to a reduction in confinement behavior. The corrosion can degrade the shear-resistant capacity of RC columns by potentially changing the ductile failure to brittle failure or even shear failure [34]. According to some studies [36–38], the reduction in both the strength and the ductility in longitudinal reinforcement can be considered an explicit function of the cross-sectional loss. The cross-sectional loss rate of the longitudinal bar is considered a function of the corrosion penetration index *η* in Equation (13) [19]. Therefore, the time-dependent ultimate strain of the longitudinal reinforcement in Steel 01

material can be expressed as Equation (14) [19], where *εsu*(0) is the initial nominal value of the ultimate strain of the longitudinal rebar. The time-variant ultimate strain is shown in Figure 5. to higher levels of corrosion of the stirrup. Thus, the area loss of steel corrosion in RC columns could be modeled as the reduction in the cross-sectional area of both longitudinal reinforcing and stirrups in the BLG model with the fiber section.

cross-sectional areas because of the effect of uncertainties in degradation parameters. This figure illustrates that the effects of stirrup corrosion were more significant than that of longitudinal reinforcement corrosion because the smaller diameter of a transverse bar led

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**Figure 4.** Time-evolving longitudinal bar and stirrup diameters for Φ ( ) 0 = 28.58 mm and φ ( ) 0 **Figure 4.** Time-evolving longitudinal bar and stirrup diameters for Φ(0) = 28.58 mm and *φ*(0) = 12.70 mm. and λ*<sup>k</sup>* , which depend on the reduction in the diameter in both the corroding longitudi-

**Distribution Parameters A 1 B 1** 

η*S*

= 12.70 mm.

nal rebar and hooping.

uniform distribution. **Figure 5.** Ultimate strain of a longitudinal bar over time. **Figure 5.** Ultimate strain of a longitudinal bar over time.

*3.3. Reduction in Strength and Ductility of corroded Longitudinal Bars and Stirrups*  The corrosion of stirrups is more serious than that of longitudinal reinforcement in RC columns and it leads to a reduction in confinement behavior. The corrosion can degrade the shear-resistant capacity of RC columns by potentially changing the ductile failure to brittle failure or even shear failure [34]. According to some studies [36–38], the reduction in both the strength and the ductility in longitudinal reinforcement can be considered an explicit function of the cross-sectional loss. The cross-sectional loss rate of the longitudinal bar is considered a function of the corrosion penetration index η in Equation (13) [19]. Therefore, the time-dependent ultimate strain of the longitudinal reinforcement in Steel 01 material can be expressed as Equation (14) [19], where ( ) 0 *su* ε is the *3.4. Reduction in Strength and Ductility in Corroded Concrete*  For the corrosion initiation phase, the confinement behavior of the concrete can be enhanced due to the expansion of the corrosion products. However, the propagation of longitudinal cracks and the cracking of concrete cover inversely result in a reduction in the confinement with the accumulation of rust products. According to the unconfined and confined stress–strain rules (i.e., BLG model, Figure 6), the residual strength of the unconfined concrete can be evaluated using Equation (17) [40], where *K* is a coefficient related to the rebar roughness and the diameter (a value *K* = 0.1 is used for medium-diameter ribbed rebars), ( ) 0 *<sup>c</sup>* ε is the initial strain of the unconfined concrete at the peak compres-Furthermore, the residual yield strength of the longitudinal rebar depends on the cross-sectional loss rate *ηS*(*t*), which is expressed as Equation (15) [39], where *fsy*(0) is the initial nominal yield strength of the longitudinal bar. Similarly, the time-dependent yielding strength of the hooping in the BLG model depends on the cross-sectional loss rate *λk* (*t*). The yielding strength is expressed as Equation (16) [39], where *fyh*(0) is the initial yielding strength of the stirrups. The yielding strengths of both the longitudinal bar and stirrup were assumed to be lognormally distributed, whose mean *u* and the coefficient of variation COV are listed in Table 1. The time-dependent yielding strengths *fsy*(*t*) and *fyh*(*t*) can be respectively evaluated using the cross-sectional loss rate *η<sup>S</sup>* and *λ<sup>k</sup>* , which depend on the reduction in the diameter in both the corroding longitudinal rebar and hooping.

#### initial nominal value of the ultimate strain of the longitudinal rebar. The time-variant ulsive stress (0) *pc f* and ε *<sup>t</sup>* ( )*t* is the average tensile strain in the cracked concrete at time *3.4. Reduction in Strength and Ductility in Corroded Concrete*

timate strain is shown in Figure 5. Furthermore, the residual yield strength of the longitudinal rebar depends on the cross-sectional loss rate η*<sup>S</sup>* ( )*t* , which is expressed as Equation (15) **[39]**, where ( ) 0 *sy f t* . The average tensile strain can be evaluated using Equation (18) [40], where *bars n* is the number of longitudinal rebars, *<sup>i</sup> r* is the width of the cross-sectional area in a pristine state and *w* is the crack width for each longitudinal rebar. The relationship between the crack width and cross-sectional loss of the longitudinal steel bar can be expressed as Equation (19) [41], where *wk* = 0.0575 mm−1, Δ*As* is the cross-sectional loss of a longitudinal For the corrosion initiation phase, the confinement behavior of the concrete can be enhanced due to the expansion of the corrosion products. However, the propagation of longitudinal cracks and the cracking of concrete cover inversely result in a reduction in the confinement with the accumulation of rust products. According to the unconfined and confined stress–strain rules (i.e., BLG model, Figure 6), the residual strength of the unconfined concrete can be evaluated using Equation (17) [40], where *K* is a coefficient

steel bar for cracking propagation and Δ*As*0 is the cross-sectional loss of the longitudinal

( ) 0 is the initial diameter of the longitudinal rebar, *x* is the depth of the concrete

is the pit concentration factor. For uniform corrosion, one has

*S S A* . The cross-sectional loss rate of the longitudinal

*<sup>S</sup>* can be evaluated using Equation (20) [40], where

α

= 2, and

η

*t A* and Δ*As*<sup>0</sup> <sup>=</sup> () () 0 0

rebar for the cracking initiation ( ) 0

<sup>=</sup> () ( ) 0 *S S*

cover and

α

η

φ

related to the rebar roughness and the diameter (a value *K* = 0.1 is used for mediumdiameter ribbed rebars), *εc*(0) is the initial strain of the unconfined concrete at the peak compressive stress *fpc*(0) and *εt*(*t*) is the average tensile strain in the cracked concrete at time *t*. The average tensile strain can be evaluated using Equation (18) [40], where *nbars* is the number of longitudinal rebars, *r<sup>i</sup>* is the width of the cross-sectional area in a pristine state and *w* is the crack width for each longitudinal rebar. The relationship between the crack width and cross-sectional loss of the longitudinal steel bar can be expressed as Equation (19) [41], where *k<sup>w</sup>* = 0.0575 mm−<sup>1</sup> , ∆*A<sup>s</sup>* is the cross-sectional loss of a longitudinal steel bar for cracking propagation and ∆*As*<sup>0</sup> is the cross-sectional loss of the longitudinal reinforcement for cracking initiation. Then, one obtains the following expressions: ∆*A<sup>s</sup>* = *ηS*(*t*)*AS*(0) and ∆*As*<sup>0</sup> = *ηS*(0)*AS*(0). The cross-sectional loss rate of the longitudinal rebar for the cracking initiation *ηS*(0) can be evaluated using Equation (20) [40], where *φ*(0) is the initial diameter of the longitudinal rebar, *x* is the depth of the concrete cover and *α* is the pit concentration factor. For uniform corrosion, one has *α* = 2, and for the localized corrosion, one obtains 4 < *α* < 8. The initial compressive strength of unconfined concrete is assumed to be lognormally distributed, whose mean *u* and the coefficient of variation COV are listed in Table 1. The residual strength *fpc*(*t*) can be evaluated using the cross-sectional losses ∆*A<sup>s</sup>* and ∆*As*0, which are dependent on the swedged steel bars, as shown in Figure 7. unconfined concrete is assumed to be lognormally distributed, whose mean *u* and the coefficient of variation COV are listed in Table 1. The residual strength ( ) *pc f t* can be evaluated using the cross-sectional losses Δ*As* and Δ*As*<sup>0</sup> , which are dependent on the swedged steel bars, as shown in Figure 7. Since a failure criterion for confined concrete is not provided in the BLG model, the ultimate compressive strain proposed by Scott et al. [42] was assigned to the concrete core fibers. The time-dependent ultimate compressive strain of the confined concrete related to the first stirrup facture can be estimated with the failure criterion. Furthermore, the strain is expressed as Equation (21), where ( ) *<sup>s</sup>* ρ *t* is the volume–stirrup ratio at time *t* and *yh*<sup>0</sup> *f* is the initial yield strength of a transverse stirrup. The ultimate compressive strain of the confined concrete ( ) *cu* ε *t* can be calculated using the stirrup ratio ( ) *<sup>s</sup>* ρ *t* , which has a direct dependency on the percentage loss of the total swedged reinforcement in the cross-section of an RC column, as shown in Figure 8. The result indicated that the ultimate compressive strain of the confined concrete is also time-dependent because the progressive reduction of stirrup area was closer to the concrete surface, which can be more susceptible to corrosion. Therefore, the material degradation of the RC column was properly simulated by using the BLG model and Steel 01. for the localized corrosion, one obtains 4 < < 8. The initial compressive strength of unconfined concrete is assumed to be lognormally distributed, whose mean *u* and the coefficient of variation COV are listed in Table 1. The residual strength ( ) *pc f t* can be evaluated using the cross-sectional losses Δ*As* and Δ*As*<sup>0</sup> , which are dependent on the swedged steel bars, as shown in Figure 7. Since a failure criterion for confined concrete is not provided in the BLG model, the ultimate compressive strain proposed by Scott et al. [42] was assigned to the concrete core fibers. The time-dependent ultimate compressive strain of the confined concrete related to the first stirrup facture can be estimated with the failure criterion. Furthermore, the strain is expressed as Equation (21), where ( ) *<sup>s</sup>* ρ *t* is the volume–stirrup ratio at time *t* and *yh*<sup>0</sup> *f* is the initial yield strength of a transverse stirrup. The ultimate compressive strain of the confined concrete ( ) *cu* ε *t* can be calculated using the stirrup ratio ( ) *<sup>s</sup>* ρ *t* , which has a direct dependency on the percentage loss of the total swedged reinforcement in the cross-section of an RC column, as shown in Figure 8. The result indicated that the ultimate compressive strain of the confined concrete is also time-dependent because the progressive reduction of stirrup area was closer to the concrete surface, which can be more susceptible to corrosion. Therefore, the material degradation of the RC column was

α

α

< 8. The initial compressive strength of

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for the localized corrosion, one obtains 4 <

properly simulated by using the BLG model and Steel 01.

**Figure 6.** Unconfined and confined stress–strain law obtained with the BLG model. **Figure 6.** Unconfined and confined stress–strain law obtained with the BLG model. **Figure 6.** Unconfined and confined stress–strain law obtained with the BLG model.

**Figure 7.** Time variance of the compression strength of unconfined concrete. **Figure 7.** Time variance of the compression strength of unconfined concrete. **Figure 7.** Time variance of the compression strength of unconfined concrete.

Since a failure criterion for confined concrete is not provided in the BLG model, the ultimate compressive strain proposed by Scott et al. [42] was assigned to the concrete core

Equation (24)

Equation (30)

Equation (31)

fibers. The time-dependent ultimate compressive strain of the confined concrete related to the first stirrup facture can be estimated with the failure criterion. Furthermore, the strain is expressed as Equation (21), where *ρs*(*t*) is the volume–stirrup ratio at time *t* and *fyh*<sup>0</sup> is the initial yield strength of a transverse stirrup. The ultimate compressive strain of the confined concrete *εcu*(*t*) can be calculated using the stirrup ratio *ρs*(*t*), which has a direct dependency on the percentage loss of the total swedged reinforcement in the cross-section of an RC column, as shown in Figure 8. The result indicated that the ultimate compressive strain of the confined concrete is also time-dependent because the progressive reduction of stirrup area was closer to the concrete surface, which can be more susceptible to corrosion. Therefore, the material degradation of the RC column was properly simulated by using the BLG model and Steel 01. *Materials* **2022**, *15*, x FOR PEER REVIEW 11 of 24

**Figure 8.** Time-dependence of the ultimate compressive strain of confined concrete. **Figure 8.** Time-dependence of the ultimate compressive strain of confined concrete.

#### **4. Degradation of Elastomeric bridge bearings due to Corrosion and Thermal 4. Degradation of Elastomeric Bridge Bearings Due to Corrosion and Thermal Oxidation**

**Oxidation**  All equations for Section 3 can be found in Table 4.

All equations for Section 3 can be found in Table 4. **Table 4.** Equation list for Section 3. **Equation Number Equation Expression Equation Number Equation Expression**  Equation (22) <sup>0</sup> / 1 *BB kt s sr* = + Equation (32) () = 1 + ∆௦ ( ) ( ) 2 2 ( )/ = 0 ( ( )) / *mh h m lh h* τ <sup>−</sup> − − Elastomeric bridge bearings are widely used in concrete girders, where they allow for force transmission from the superstructure to the substructure. These types of bearing systems consist of an elastomeric nature rubber (NR) pad and steel dowels, but they often form "walk-out" bearings during seismic events due to the effects of aging and deterioration [43]. Corrosion of these assemblies, caused by chloride-laden water from deicing salts, may potentially result in a larger deformation of the bearing systems under seismic loading. For the steel dowels of the elastomeric bridge bearings, corrosion deterioration can lead to a reduction in both the cross-section and the shear strength. Moreover, the elastomeric NR pads suffer an increase in both the shear modulus and shear stiffness due to corrosion and thermal oxidation [28–30].

Equation (23) 3 2 *<sup>s</sup>* 1 2 34 *ka a a a* = + ++ ξξξ Equation (33) (0 ( ( *m h hmlh lhml* ≤ ≤ ≤ ≤− −≤ ≤ ) ) ) 222 1 23 222 0 01 2 3 ( 3) ( 3) *B G s s B G* λλλ λλλ = ++− = ++− Equation (34) () = (1 + ∆௦) *E* Additionally, each member of this type of bearing system makes a contribution to the transfer of forces. For example, an elastomeric NR pad transfers a lateral load by developing a frictional force, while steel dowels provide resistance via the action of the beam. The assemblies of the elastomeric bridge bearings consist of the fixed and expansion types, which depend on the dimensions of the slot in the NR pad [31], as shown in Figure 9. The sliding behavior of the elastomeric NR pad and the yield of steel dowels in pristine bridge bearings can be modeled by using the proper composite action.

Equation (25) 0 0 // 1 *GG BB kt s s sr* = =+ Equation (35) 1 1 ( ) / *<sup>r</sup> R T T rtt e* <sup>−</sup> = Equation (26) Δ− − *G G G GGG ss s* =( ) / = / 1 00 0 Equation (36) Δ*Gt k t s s* ( )== Equation (27) \* *h T* =α β exp( / ) Equation (37) () = ()/ Equation (28) 0 0 ( )/ ( )/ 1 *Gt G Bt B G* = = +Δ *<sup>s</sup>* ( *<sup>m</sup>* =0 or *<sup>l</sup>* ) Equation (38) ( ) ( )/ 3 *F t fA t y yd* <sup>=</sup> As above-mentioned, aging and thermal oxidation may lead to an increase in the shear modulus in the elastomeric NR pads, where these deterioration mechanisms can be modeled with a changed value of the constitutive parameter in Steel 01 materials. Normally, the value of the shear modulus in the elastomeric bridge bearings recommended by AASHTO [44] is assumed to be a mean constant in the analytical modeling of bridges. However, a battery of accelerated exposure tests proposed by Itoh et al. [28] illustrated that the shear modulus of the NR pad is not a constant and can be changed over time due to

> Elastomeric bridge bearings are widely used in concrete girders, where they allow for force transmission from the superstructure to the substructure. These types of bearing systems consist of an elastomeric nature rubber (NR) pad and steel dowels, but they often form "walk-out" bearings during seismic events due to the effects of aging and deterioration [43]. Corrosion of these assemblies, caused by chloride-laden water from deicing salts,

= ଵ<sup>ଶ</sup> + ଶ+ଷ

Equation (29) <sup>0</sup> *Gt G* ( )/ =1 ( <sup>0</sup> *h mlh* ≤ ≤− ) Equation (39) ( ) ( )/ 3 *F t fA t u ud* =

*dG t dm* ( )/ 0 = ( *m h* = or *l h* − )

() 

thermal oxidation. The derivation process of the time-dependent shear modulus for an NR pad can be expressed as follows.


**Table 4.** Equation list for Section 3.

**Figure 9.** Fixed and expansion elastomeric bearing types, depending on the dimensions of the slot. **Figure 9.** Fixed and expansion elastomeric bearing types, depending on the dimensions of the slot.

As above-mentioned, aging and thermal oxidation may lead to an increase in the shear modulus in the elastomeric NR pads, where these deterioration mechanisms can be modeled with a changed value of the constitutive parameter in Steel 01 materials. Normally, the value of the shear modulus in the elastomeric bridge bearings recommended by AASHTO [44] is assumed to be a mean constant in the analytical modeling of bridges. However, a battery of accelerated exposure tests proposed by Itoh et al. [28] illustrated that the shear modulus of the NR pad is not a constant and can be changed over time due to thermal oxidation. The derivation process of the time-dependent shear modulus for an NR pad can be expressed as follows. First, an aging model of NR bearings proposed by Itoh and Gu [29,30] showed the relation between the variation of strain energy and the aging time under accelerated exposure test conditions (i.e., the simulated thermal oxidation). This aging model can be provided by Equation (22), where *Bs*/*B*<sup>0</sup> represents the relative strain energy versus its initial state at the NR block surface, *k<sup>s</sup>* is the strain-dependent coefficient for the strain energy and *t<sup>r</sup>* is the aging time at the test temperature in hours. Here, 60 ◦C was taken as the reference temperature of the NR, and the empirical formula for *k<sup>s</sup>* was calculated using Equation (23), where *a*<sup>1</sup> = 0.54, *a*<sup>2</sup> = −4.19, *a*<sup>3</sup> = 8.16 and *a*<sup>4</sup> = 9.59 at 60 ◦C, and *ξ* is the nominal strain.

First, an aging model of NR bearings proposed by Itoh and Gu [29,30] showed the relation between the variation of strain energy and the aging time under accelerated exposure test conditions (i.e., the simulated thermal oxidation). This aging model can be provided by Equation (22), where <sup>0</sup> / *Bs B* represents the relative strain energy versus its initial state at the NR block surface, *<sup>s</sup> k* is the strain-dependent coefficient for the strain energy and *<sup>r</sup> t* is the aging time at the test temperature in hours. Here, 60 °C was taken as Second, a relationship between the strain energy and the shear modulus can be expressed using a one-parameter neo-Hoolean material model [45], as shown in Equation (24), where *B* is the strain energy; *G* is the shear modulus; and *λ*1, *λ*<sup>2</sup> and *λ*<sup>3</sup> are the stretches due to the uniaxial tension. If this material is incompressible, then *λ* 2 1 *λ* 2 2 *λ* 2 <sup>3</sup> = 1. Combining Equations (22) and (23), a relative change in the shear modulus of the NR bearing due to thermal oxidation can be calculated using Equation (25), where *Gs*/*G*<sup>0</sup> is the relative shear modulus versus its initial state for the NR bearing at time *t* = *t<sup>r</sup>* . Thus, the normalized shear

the reference temperature of the NR, and the empirical formula for *<sup>s</sup> k* was calculated

Second, a relationship between the strain energy and the shear modulus can be expressed using a one-parameter neo-Hoolean material model [45], as shown in Equation

the stretches due to the uniaxial tension. If this material is incompressible, then

 1 23 =1. Combining Equations (22) and (23), a relative change in the shear modulus of the NR bearing due to thermal oxidation can be calculated using Equation (25), where <sup>0</sup> / *G G <sup>s</sup>* is the relative shear modulus versus its initial state for the NR bearing at time

λ1, λ2 and λ<sup>3</sup> are

(24), where *B* is the strain energy; *G* is the shear modulus; and

ξ

222 λλλ

is the nominal strain.

modulus variation ∆*G<sup>s</sup>* can be obtained using Equation (26). Finally, the shear modulus variation in the outer region from the NR bearing's surface to the critical depth *h* should be expressed by using an equation. The relationship between the critical depth and the temperature can be obtained using Equation (27) [29], where *T* ∗ is the absolute temperature and *α* and *β* are the coefficients determined by the thermal oxidation test. It is assumed that the shear modulus variation *G*(*t*)/*G*<sup>0</sup> might be a function of the position *m*. The boundary conditions can be derived using Equations (28) to (30), where *G*(*t*) and *G*<sup>0</sup> are the shear modulus at time *t* and the initial state, respectively; ∆*G<sup>s</sup>* is the normalized shear modulus variation; and *l* is the width of the NR bearing. The normalized shear modulus *G*(*t*)/*G*<sup>0</sup> can be also assumed to be a square relation of the position *m*, and the function is given as Equation (31). Combining the boundary conditions, the normalized shear modulus *G*(*t*)/*G*<sup>0</sup> can be written as Equations (32) and (33). The time-dependent shear modulus in a bearing NR pad can be calculated using Equation (34), where *G*(*t*) is the shear modulus at real time *t*; *τ* is coefficient correlated with the position *m*, the critical depth *h* and the width *l* of the bridge NR bearing. Since thermal oxidation is commonly assumed to be a first-order chemical reaction for NR materials, the relationship between the deterioration time under the service condition and the time in the accelerated exposure tests can be expressed as Equation (35) [29], where *t<sup>r</sup>* is the test time, *t* is the aging time, *T<sup>r</sup>* is the absolute temperature in the thermal oxidation test, *T* is the absolute temperature under the service condition, *R* is the gaseous constant (8.314 J/mol K) and *E* is the activation energy of the rubber (94,900 J/mol). Consequently, the normalized shear modulus variation ∆*G<sup>s</sup>* can be written as Equation (36).

The initial shear modulus of the elastomeric bearing pad is assumed to be uniformly distributed, whose upper bound *a* and lower bound *b* are listed in Table 1, and the timedependent shear modulus *G*(*t*) is plotted in Figure 10. However, the time-variant shear stiffness of the NR pad is considered a critical element in the analytical modeling of elastomeric bearing pads, and the aging model of the shear stiffness can be expressed as Equation (37), where *G*(*t*) is the shear modulus of the NR pad at the real time *t*, *A<sup>p</sup>* is the area of the elastomeric bearing *NR* pad and *S<sup>p</sup>* is the thickness of the bearing pad. For the assemblies of the fixed and the expansion steel dowels, corrosion deterioration may lead to a loss of the cross-sectional area of this component type. Then, this deterioration effect is modeled by considering the variation of parameters in hysteretic material, which includes a reduction in the yield strength and ultimate lateral strength. The time-dependent yield strength and the ultimate lateral strength can be calculated using Equations (38) and (39) [46], where *f<sup>y</sup>* and *f<sup>u</sup>* are the tensile strength and the ultimate shear strength of the steel dowels, respectively, and *A<sup>d</sup>* (*t*) is the time-dependent cross-sectional area of the steel dowels. The initial area of the steel dowels and dowel tensile strength, as well as the lateral strength of the bearing dowel, are assumed to be uniformly distributed and whose upper bound *a* and lower bound *b* are listed in Table 1, respectively. Thus, the yield strength *Fy*(*t*) and the ultimate lateral strength *Fu*(*t*) can be evaluated using the cross-sectional loss of dowel bars *A<sup>d</sup>* (*t*), as shown in Figure 11. The results indicated that a reduction in the dowels' area has a significant impact on the strength over time due to corrosion.

**Figure 10.** Time evolution of the shear modulus in a bearing NR pad. **Figure 10.** Time evolution of the shear modulus in a bearing NR pad. **Figure 10.** Time evolution of the shear modulus in a bearing NR pad.

**Figure 11.** Time evolution of the (**a**) yield strength and (**b**) ultimate lateral strength for swedged steel dowels **Figure 11.** Time evolution of the (**a**) yield strength and (**b**) ultimate lateral strength for swedged steel dowels **Figure 11.** Time evolution of the (**a**) yield strength and (**b**) ultimate lateral strength for swedged steel dowels.

#### **5. Impact of Corrosion on the Seismic Response of Bridge Components 5. Impact of Corrosion on the Seismic Response of Bridge Components 5. Impact of Corrosion on the Seismic Response of Bridge Components**

All equations for Section 4 can be found in Table 5. All equations for Section 4 can be found in Table 5. All equations for Section 4 can be found in Table 5.

To analyze the impact of corrosion multi-deterioration mechanisms on the seismic fragility of the MSC concrete girder bridge, the seismic responses of the MSC bridge components, which considers the effect of the time-dependent aging, should be presented using dynamic simulations. For the geometry of this bridge type considered in a pristine state, the first two fundamental modes along longitudinal and transverse directions were 0.49 s and 0.37 s, respectively. Moreover, the foundations for the MSC bridge were assumed to be on medium hard soil represented by site class II (shear wave velocity between 500 m/s and 250 m/s), and the target response spectrum of 0.53 s was calculated by considering the site soil conditions. Then, twenty real earthquake records, whose response spectra had the greatest similarity to the target response spectrum, were selected to capture the uncertainty of ground motions. Thus, the dynamic responses of the bridge were illustrated through nonlinear time history analysis with the above twenty seismic inputs. The impact of corrosion multi-deterioration mechanisms on the evolving dynamic response of the bridge is presented using probabilistic seismic demand models (PSDMs) of aging components (Equation (42)). The influence of multi-deterioration mechanisms of a single component and the joint effects of both column and bearing on the seismic demands are compared in the following sections. To analyze the impact of corrosion multi-deterioration mechanisms on the seismic fragility of the MSC concrete girder bridge, the seismic responses of the MSC bridge components, which considers the effect of the time-dependent aging, should be presented using dynamic simulations. For the geometry of this bridge type considered in a pristine state, the first two fundamental modes along longitudinal and transverse directions were 0.49 s and 0.37 s, respectively. Moreover, the foundations for the MSC bridge were assumed to be on medium hard soil represented by site class II (shear wave velocity between 500 m/s and 250 m/s), and the target response spectrum of 0.53 s was calculated by considering the site soil conditions. Then, twenty real earthquake records, whose response spectra had the greatest similarity to the target response spectrum, were selected to capture the uncertainty of ground motions. Thus, the dynamic responses of the bridge were illustrated through nonlinear time history analysis with the above twenty seismic inputs. The impact of corrosion multi-deterioration mechanisms on the evolving dynamic response of the bridge is presented using probabilistic seismic demand models (PSDMs) of aging components (Equation (42)). The influence of multi-deterioration mechanisms of a single component and the joint effects of both column and bearing on the seismic demands are compared in the following sections. To analyze the impact of corrosion multi-deterioration mechanisms on the seismic fragility of the MSC concrete girder bridge, the seismic responses of the MSC bridge components, which considers the effect of the time-dependent aging, should be presented using dynamic simulations. For the geometry of this bridge type considered in a pristinestate, the first two fundamental modes along longitudinal and transverse directions were0.49 s and 0.37 s, respectively. Moreover, the foundations for the MSC bridge were assumed to be on medium hard soil represented by site class II (shear wave velocity between 500 m/s and 250 m/s), and the target response spectrum of 0.53 s was calculated by considering the site soil conditions. Then, twenty real earthquake records, whose response spectra had the greatest similarity to the target response spectrum, were selected to capture the uncertainty of ground motions. Thus, the dynamic responses of the bridge were illustrated through nonlinear time history analysis with the above twenty seismic inputs. The impact of corrosion multi-deterioration mechanisms on the evolving dynamic response of the bridge is presented using probabilistic seismic demand models (PSDMs) of aging components (Equation (42)). The influence of multi-deterioration mechanisms of a single component and the joint effects of both column and bearing on the seismic demands are compared in the following sections.


**Table 5.** Equation list for Sections 4 and 5.

#### *5.1. The Impact of Deterioration on the Seismic Demand of RC Columns*

As elaborated on earlier, the aging of RC columns is modeled via a relationship between stress and strain, which includes multiple deterioration mechanisms. There is a great shift in the demands of the aging RC columns due to the loss of both the steel area (i.e., longitudinal bars, stirrups) and concrete cover when the bridge experiences seismic loading. Then, the seismic demand placed on the column can be expressed using Equation (40), where *u<sup>θ</sup>* is the curvature ductility demand ratio, *θ<sup>m</sup>* is the maximum curvature demand of a column under seismic loading and *θ<sup>y</sup>* is the yield curvature in the column. The peak curvature ductility demands at 0 years and 50 years are shown in Figure 12. It can be seen that if only column multi-degradation was considered, it had a great impact on the column demands, which were significantly increased with aging from 0 to 50 years. Additionally, when the joint effects of both the column and bearing multi-deterioration were taken into account, the column demands at a specific time can be higher than when the multideterioration of an individual column component is considered. It is worth noting that the only bearing multi-deterioration had a minimal impact on the column demands. The results indicated the significance of considering the multi-deterioration mechanisms of the multiple components.

#### *5.2. The Impact of Deterioration on the Seismic Demand of Bridge Bearings*

Similarly, Figure 13 shows the influences of the multi-deterioration of the multiple components and a single component on the seismic response of the fixed bearing along the longitudinal direction. When only the multi-degradation of the fixed bearing was considered, there was a significant influence on the bearing deformation, which constantly increased from 0 to 50 years. If only column multi-degradation was considered, there was a small impact on the bearing deformation. Both the joint considerations of the column and bearing degradations did not have a higher effect on the bearing demand along the longitudinal direction than when only bearing deterioration is considered. However, Figure 14 shows the dependency between the multi-deterioration mechanisms of the column and expansion bearings in the transverse directions.


Column ductility demand ln(uθ

)

*Materials* **2022**, *15*, x FOR PEER REVIEW 16 of 24

Only column multi-deterioration(50 years) Both column and bearing multi-deterioration(50 years) Only bearing multi-deterioration(50 years)

Pristine(0 years)

**Figure 12.** Mean value of the demand placed on an RC column through PSDM. **Figure 12.** Mean value of the demand placed on an RC column through PSDM. ure 14 shows the dependency between the multi-deterioration mechanisms of the column and expansion bearings in the transverse directions.

Pristine(0 years)

3.5 4 **Figure 13.** Mean values of the demand placed on a fixed bearing in the longitudinal direction using a PSDM. **Figure 13.** Mean values of the demand placed on a fixed bearing in the longitudinal direction using a PSDM. *Materials* **2022**, *15*, x FOR PEER REVIEW 17 of 24

**Figure 14.** Mean values of the demand placed on an expansion bearing in the transverse direction using a PSDM. **Figure 14.** Mean values of the demand placed on an expansion bearing in the transverse direction using a PSDM.

If only column multi-deterioration is considered, the demands on the expansion transverse bearing at 50 years were lower than the demands on the pristine expansion bearing due to the domination of the corroded RC column. When the multi-deterioration If only column multi-deterioration is considered, the demands on the expansion transverse bearing at 50 years were lower than the demands on the pristine expansion

of the expansion transverse bearing was considered, the demands on the expansion bear-

stiffness of the NR pad induced by thermal oxidation. The multi-deteriorations of both the column and the expansion transverse bearing had a greater impact on the bearing deformation than when only column deterioration was considered. However, the deterioration mechanisms of multiple components had a lower influence on the bearing deformation than when only bearing degradation was taken into account. The reason for this was that the individual consideration of the corrosion degradation of the expansion bearing system had a greater role on the seismic responses after 50 years of exposure to deicing salts.

The impact of multi-deterioration mechanisms on the seismic performance of aged bridge components can be evaluated by developing time-dependent bridge fragility curves at a system level. Moreover, the increase in the probability of a damaged state exceedance with continuous degradation of the bridge along its service life can be quantified using such fragility curves. A methodology for the development of time-dependent fragility curves is presented in the following section using a typical case study of an aged MSC concrete girder bridge. The uncertainties in bridge modeling attributes, ground motion and deterioration parameters were considered in probabilistic terms for this case.

Time-dependent fragility curves represent the probability of damage exceedance of structures under earthquake excitation at different points in time throughout the service life. Such time-evolving fragility curves show the impact of multi-deterioration mechanisms on the seismic performance of aging structures at the system level. The generalized time-dependent seismic fragility function can be expressed as Equation (41), where *P t <sup>f</sup>* ( ) is the probability of damage state exceedance of a specific MSC bridge at an aging

time *t* . *D*( )*t* and *C t*( ) are the seismic demand and the capacity of an aged bridge at time *t* , respectively. *IM* is the intensity of the ground motion. Before developing the

**6. Impact of Corrosion on the Seismic Fragility of an Aged Bridge System** 

All equations for Section 5 can be found in Table 5.

*6.1. Time-Dependent Probabilistic Seismic Demand Models (PSDMs)* 

bearing due to the domination of the corroded RC column. When the multi-deterioration of the expansion transverse bearing was considered, the demands on the expansion bearing at 50 years were higher than that of the pristine demands. The reason for this was that the increase in strength reduction of steel dowels was higher than the increase in the shear stiffness of the NR pad induced by thermal oxidation. The multi-deteriorations of both the column and the expansion transverse bearing had a greater impact on the bearing deformation than when only column deterioration was considered. However, the deterioration mechanisms of multiple components had a lower influence on the bearing deformation than when only bearing degradation was taken into account. The reason for this was that the individual consideration of the corrosion degradation of the expansion bearing system had a greater role on the seismic responses after 50 years of exposure to deicing salts.

#### **6. Impact of Corrosion on the Seismic Fragility of an Aged Bridge System**

All equations for Section 5 can be found in Table 5.

The impact of multi-deterioration mechanisms on the seismic performance of aged bridge components can be evaluated by developing time-dependent bridge fragility curves at a system level. Moreover, the increase in the probability of a damaged state exceedance with continuous degradation of the bridge along its service life can be quantified using such fragility curves. A methodology for the development of time-dependent fragility curves is presented in the following section using a typical case study of an aged MSC concrete girder bridge. The uncertainties in bridge modeling attributes, ground motion and deterioration parameters were considered in probabilistic terms for this case.

#### *6.1. Time-Dependent Probabilistic Seismic Demand Models (PSDMs)*

Time-dependent fragility curves represent the probability of damage exceedance of structures under earthquake excitation at different points in time throughout the service life. Such time-evolving fragility curves show the impact of multi-deterioration mechanisms on the seismic performance of aging structures at the system level. The generalized timedependent seismic fragility function can be expressed as Equation (41), where *P<sup>f</sup>* (*t*) is the probability of damage state exceedance of a specific MSC bridge at an aging time *t*. *D*(*t*) and *C*(*t*) are the seismic demand and the capacity of an aged bridge at time *t*, respectively. *IM* is the intensity of the ground motion. Before developing the time-dependent seismic fragility curves, the relationship between the time-varying seismic demand and capacity should be established by using probabilistic seismic demand models (PSDMs). Such PSDMs for bridge critical components are constructed using nonlinear time history analysis to capture the impact of multi-deterioration mechanisms on the dynamic responses. Thus, to consider the uncertainties in the ground motion and structural parameters, a total of 20 real ground motions from PEER were used in the analysis and an equal number of random bridge samples in pristine states were generated through Latin hypercube sampling. The peak ground acceleration (PGA) of the 20 random samples was modulated ranging from 0.095 g to 1.05 g. In addition, to generate 20 random aging bridge samples, the uncertainties in the corrosion parameters that affect the multi-deterioration mechanisms of the column and bearing components were also taken into account in the finite element modeling.

To develop probabilistic seismic demand models for bridge components at different points in time (e.g., 0, 25 and 50 years), the 20 random bridge samples for each time instant at a specific intensity level of the ground motion were performed using nonlinear time history analysis. Then, probabilistic seismic demand models, which reflect the relationship between the peak demands of aged bridge components and ground motion intensity, were developed using linear regression. The demands considered in this research included RC columns and fixed and expansion elastomeric bearings. Consequently, the time-dependent probabilistic seismic demand models can be expressed as Equation (42) [31], where *D*ˆ <sup>i</sup>(t) is the time-dependent median value of the seismic demand for bridge component *i*, *ai*(*t*) and *bi*(*t*) are the linear regression parameters, and *IM* is the intensity of ground motion. A logarithmic seismic demand for the specific component ln *Di*(*t*) was assumed to be normally distributed, and then ln *Di*(*t*) and *βDm*,*i*(*t*) were the median value and dispersion, respectively. *βDm*,*i*(*t*) can be expressed as Equation (43), where *dm*(*t*) is the *m*th peak seismic demand for bridge component *i* and *IM<sup>m</sup>* is the *m*th PGA. Then, the time-evolving PSDMs can also be expressed by using Equation (44), where *λi*(*t*) is the natural logarithm of the seismic demand related to the median value of PGA and *λi*(*t*) = (ln *di*(*t*) − ln *ai*(*t*))/*bi*(*t*); *ξi*(*t*) is the lognormal standard deviation and *ξi*(*t*) = *βDm*,*i*(*t*)/*bi*(*t*).

#### *6.2. Time-Dependent Seismic Fragility of an Aged Bridge System*

In addition to probabilistic seismic demand models, the structural capacities of different bridge components should also be estimated during time-dependent seismic fragility analysis. The limit state capacities of different components considered in this study are lognormal and presented in Table 6 [31]. When the seismic demands and the limit state capacities for different bridge components are assumed to be lognormally distributed, the timedependent seismic fragility at the component level can be obtained using Equation (45), where *γi*(*t*) and *ζi*(*t*) are median values (in units of g PGA) and logarithmic standard deviations of the *i*th component fragility, respectively. *γi*(*t*) and *ζi*(*t*) can be expressed

as Equations (46) and (47), where ∧ *Ci*(*t*) and *βC*,*i*(*t*) are the median and dispersion of the *i*th component capacity, respectively, and *βDm*,*i*(*t*) is the dispersion of the *i*th component demand.

**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) [47,48]. The JPSDMs can be written as Equation (48), where *γsys*(*t*) and *ζsys*(*t*) are 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 calculated by using Monte Carlo analysis.
