*2.1. Working Mechanism and Design Objectives*

According to post-earthquake field investigation reports, many bridge systems that followed the displacement-based seismic design philosophy in earthquake-prone areas suffered catastrophic damage or even collapsed. To enhance the resilience of newly designed bridges in earthquake regions, an SMA-washer-based SCR bridge pier was proposed in the previous work by the authors and co-workers [38], as shown in Figure 1. The rocking control capability of the novel bridge system is enabled by its SMA-washer-based SCR piers. The SCR pier is mainly composed of three parts: the upper pile cap, the lower pile cap, and the SMA washer springs (also known as disc springs), which are the kernel components providing self-centering capability and energy dissipation for the bridge system. These washer springs can be stacked either in parallel or in series (or in combination), which makes them flexible in terms of load resistance and available deformability. More technical details of the SMA washer springs can be found elsewhere [39].

**Figure 1.** SMA-washer-based SCR bridge pier: (**a**) illustration of the SMA washer springs, (**b**) illustration of the bridge system, (**c**) working principle of the SCR bridge pier.

In the new pier, each SMA washer spring set consists of several SMA washers, steel bars embedded in the lower pile cap, plastic tubes cast in the concrete of the upper pile cap, and several nuts and shims. As illustrated in Figure 1c, the seismic behavior of such a pier with increasing lateral load could be divided into three stages: (1) decompression stage (I) (where the gap is just about to open), (2) post-decompression stage (II) (III), and (3) locking stage (IV). In the first stage, the behavior of the pier behaves similar to a fixed

pier, i.e., the lateral deformation of the pier relies on the elastic deformation of the pier due to the application of appropriate preload of the SMA washer springs. In the subsequent stage, the lifting force exceeds the decompression force provided by SMA washer spring sets, and the pier starts rocking. When the allowable deformation of washer sets is consumed, the pier is "locked". Further lateral displacement may rely on the nonlinear deformation of the pier. In light of the above, three basic design goals could be set: (1) the pier does not uplift during small earthquakes; (2) a maximum drift ratio of the bridge is less than the "lock rotation" at moderate (E1) earthquake level; and (3) collapse is prevented at large (E2) earthquake level. deformation of the pier. In light of the above, three basic design goals could be set: (1) the pier does not uplift during small earthquakes; (2) a maximum drift ratio of the bridge is less than the "lock rotation" at moderate (E1) earthquake level; and (3) collapse is prevented at large (E2) earthquake level. *2.2. Experimental Verification of SMA-Washer-Based SCR Pier* 

#### *2.2. Experimental Verification of SMA-Washer-Based SCR Pier* Figure 2 schematically depicts the test arrangement for the pier specimen. The spec-

*Materials* **2022**, *15*, x FOR PEER REVIEW 4 of 24

Figure 2 schematically depicts the test arrangement for the pier specimen. The specimen was held down via four anchor bars passing through the designated slots in the pier base. In order to consider the dead weight of the bridge's superstructure, a PT tendon was used to apply the axial force. A double-action electro-hydraulic servo actuator was used to provide the lateral load to the loading head, and the lever arm, or the distance between the loading head's centroid and the rocking interface, was 1625 mm. The RC pier's diameter and height were 0.3 and 1.05 m, respectively, and other relevant dimensions are shown in Figure 2, with more details given in Fang et al. [38]. The typical test result (hysteretic response) is illustrated in Figure 3. The specimen displayed stable flag-shaped hysteretic curves under cyclic load, with no noticeable decrease in strength and stiffness responses and negligible residual drift. Numerical simulation and system-level analysis of a novel bridge system incorporating the new bridge pier are discussed in detail in Section 4. imen was held down via four anchor bars passing through the designated slots in the pier base. In order to consider the dead weight of the bridge's superstructure, a PT tendon was used to apply the axial force. A double-action electro-hydraulic servo actuator was used to provide the lateral load to the loading head, and the lever arm, or the distance between the loading head's centroid and the rocking interface, was 1625 mm. The RC pier's diameter and height were 0.3 and 1.05 m, respectively, and other relevant dimensions are shown in Figure 2, with more details given in Fang et al.[38]. The typical test result (hysteretic response) is illustrated in Figure 3. The specimen displayed stable flag-shaped hysteretic curves under cyclic load, with no noticeable decrease in strength and stiffness responses and negligible residual drift. Numerical simulation and system-level analysis of a novel bridge system incorporating the new bridge pier are discussed in detail in Section

4.

**Figure 2. Figure 2.** Test of the SCR pier: ( Test of the SCR pier: ( **a**) drawing of the SCR pier specimen, ( **a**) drawing of the SCR pier specimen, ( **b b** ) photo of the test setup. ) photo of the test setup.

**Figure 3.** Shear force–drift ratio hysteretic curves of the test and numerical simulation results.

#### **3. Methodology of Performance-Based Assessment**

The main steps of the analysis framework are shown in Figure 4. Seismic fragility analysis and loss assessment are two key steps: the former gives the probabilities of exceeding certain component or system damage levels and the latter enables translation from the damage level to economic loss quantities.

**Figure 4.** Flowchart of the performance-based life-cycle assessment of bridge systems under seismic hazard.

### *3.1. Seismic Fragility Analysis*

Structural seismic fragility assessment was carried out first according to the flowchart shown in Figure 4. Fragility analysis is a frequently used technique in the seismic risk

assessment in order to calculate the conditional probability of a structure's or component's demand reaching or beyond its corresponding capacity [40]. Analytical fragility curves were derived using the probabilistic seismic demand model (PSDM) based on nonlinear time history analysis series. A PSDM is typically developed using two methods: incremental dynamic analysis (IDA) [41] or a cloud technique [42]. The former method requires scaling all the ground motions to specific intensity measurement (*IM*) and conducting a nonlinear time history analysis at each level. In the later procedure, a collection of un-scaled ground motion data is used in the nonlinear time history analysis. Both methods are dependent on *IM*s, and extensive research has been carried out on the selection of suitable *IMs*, considering, for example, Peak ground acceleration (PGA), Peak ground velocity (PGV), and response spectrum type at a specific period. The optimal selection of *IM* may vary with different characteristics of structures [43].

The probabilistic seismic demand model (PSDM) is the probability distribution of structural demand conditioned on specified *IM* and based on the cloud technique. The probability that a structure's seismic demand (D) exceeds its capacity (C) may be written as follows:

$$P[D \ge \mathbb{C} | IM] = P[\frac{\mathcal{D}}{\mathbb{C}} \ge 1] \tag{1}$$

Equation (1) might be rewritten as a lognormal cumulative probability density function provided that C and D have a two-parameter lognormal distribution:

$$P[D \ge \mathcal{C}|IM] = \Phi(\frac{\ln(\mathcal{S}\_d) - \ln(\mathcal{S}\_c)}{\sqrt{\mathcal{P}\_{d|IM}^2 + \mathcal{P}\_c^2}}) \tag{2}$$

where *S<sup>c</sup>* signifies the median structural capacity estimate and *β<sup>c</sup>* denotes the standard deviation. Lognormal median estimate and standard deviation of structural demand in terms of an *IM* are represented by *S<sup>d</sup>* and *βd*|*IM*, respectively. Regression analysis was used to determine the relationship between *IM* and *S<sup>d</sup>* . The median value of seismic demand, according to Cornell's power exponent model, may be stated as:

$$S\_d = aIM^b \text{ or } \ln(S\_d) = \ln(a) + bln(IM) \tag{3}$$

where *a* and *b* represent the regression parameters obtained from the response analysis. *βd*|*IM* can be characterized as:

$$\beta\_{d|IM} \cong \sqrt{\frac{\sum\_{i=1}^{n} \left(\ln(d\_i) - \ln(S\_d)\right)^2}{N - 2}}\tag{4}$$

where *d<sup>i</sup>* represents the structural demand, also known as the seismic response of components, and *i*th represents the earthquake-model sample that corresponds to it. The following steps need to be taken to obtain *βd*|*IM*:

Given the fragility curves of the components, the fragility curve of a bridge system can be developed according to the first-order reliability theory (explained in Equations (5)–(7)). Such a theory assesses structural performance as an overall system by accounting for the relationship between the vulnerable components. Equation (5) provides the upper and lower bounds of the system fragility functions. The lower bound assumes complete correlation among components, while the upper bound refers to the components with no correlation.

$$\max\_{i=1}^{n} [P(F\_k)] \le P(F\_{sys}) \le 1 - \prod\_{i=1}^{n} (1 - P[F\_{component,i}]) \tag{5}$$

where *n* is the total number of components that might fail, *P*(*F<sup>k</sup>* ) is the probability that the component in concern will fail, *P*[*Fcomponent, i*] and *P*[*Fsys*] are the failure probabilities of the *ith* component and system, respectively, and Π is the product operator.

If a bridge is supposed to operate as a serial system, with each component executing an essential function separately, any component failure will result in system failure at the same level. As a result, the most significant damage state at the component level is as follows:

$$DS\_{\text{sys}} = \max(DS\_{\text{Pier}}, DS\_{\text{Bearing}}) \tag{6}$$

When a bridge is supposed to be a parallel system, however, it will attain a specific damage state once all of its components have reached that condition. As a result, the system damage state *DSsys* is determined by the component with the minimum damage state:

$$DS\_{sys} = \min(DS\_{Pier}, DS\_{Bearing})\tag{7}$$

The intersection of component probability and its lower and upper limits, as shown in Equation (8), yields the failure probability of a parallel system:

$$\prod\_{i=1}^{n} P(F\_i) \le P(F\_{\text{sys}}) \le \min[P(F\_i)] \tag{8}$$

Independent components are represented by the lower bound, whereas the upper limit represents entirely correlated components. These boundaries are often quite broad, showing the importance of component correlation. In fact, a bridge is neither a parallel nor a serial system, and component responses are often coupled to some degree. The first-order constraints in Equations (5)–(7), which assume total correlation or perfect independence between components, cannot accurately estimate the bridge system's failure probability. According to the work of Kim et al. [44], the bearing damage due to the load moving to other components of a multi-span simply supported bridge has a substantial impact on the bridge's overall seismic behavior. The global damage state is hence located in-between the limits set by Equations (5)–(7).

A composite *DS* based on component *DSs*, proposed by Zhang and Huo [45], was employed in this work. Piers and isolation devices were given a weighted ratio of 0.75 and 0.25, respectively, based on their proportional value for load carrying and maintenance cost. This ratio highlights that piers are more important than isolation devices and, as a result, should be given greater weight. However, since either excessive bearing displacement or pier collapse damage (*DS* = 4) might cause a single span or the whole bridge to collapse, a serial mechanism for the collapse damage was used. The following equation summarizes the resulting composite *DSsys* for system behavior:

$$DS\_{sys} = \text{int}(0.75 \cdot DS\_{Pier} + 0.25 \cdot DS\_{Bearing}) \qquad DS\_{Pier} \, DS\_{Bearing} < 4$$

$$\begin{array}{c} DS\_{sys} = 4\\ DS\_{Pier} \, \text{or } DS\_{Baring} = 4 \end{array} \tag{9}$$

#### *3.2. Life-Cycle Loss Assessment*

The fragility analysis could be followed by a life-cycle loss assessment, a framework that was initially proposed by the Pacific Earthquake Engineering Research (PEER) Center. Life-cycle loss assessment is an effective tool to evaluate the long-term benefit of the newly proposed bridge system. Direct loss (mostly repair loss) and indirect losses (e.g., running cost and property loss) are important qualities in the life-cycle loss assessment.

The selected seismic events should cover both frequent low-magnitude events with a high probability of occurrence and the high-magnitude earthquakes with a low probability of occurrence. Six hazard events with return periods of 225 years (E1), 475 years (E2), 975 years (E3), 1500 years (E4), 2475 years (E5), and 5000 years (E6) were considered [46]. The relationships between earthquake intensity measurement (*IM*) and the frequency of occurrence for the location of the bridge can be obtained from the USGS national seismic hazard map [47].

The obtained fragility curves were then used to quantify the damage probability of the bridge system. Under a given hazard event, the seismic loss can be calculated by summing up the consequences weighted with the damage probability. Equation (10) gives the expression of the expected annual loss under a specific hazard [48].

$$\mathcal{R} = \sum\_{LS} \mathbb{C}\_{LS} P\_{LS|IM} \tag{10}$$

where *CLS* and *PLS|IM* are the consequences at a specific limit state of the bridge and the conditional probability of the bridge at a limit state for a given *IM*, respectively. Direct and indirect losses are the two types of consequences considered in the present study, and the consequences were evaluated in terms of monetary values.

#### 3.2.1. Direct Loss

It is assumed that the necessary repair cost at a certain limit state, *i*, is proportional to the cost needed to rebuild the bridge, as expressed as [49]:

$$\mathbf{C}\_{REP,i} = \mathbf{R}\_{rcr} \cdot \mathbf{c}\_{reb} \cdot \mathbf{W} \cdot \mathbf{L} \tag{11}$$

The total repair cost of a bridge, *CREP*,*<sup>i</sup>* , at damage state *i* can be obtained by multiplying the rebuilding cost per square meter (unit: \$/m<sup>2</sup> ) *creb* by the width *W* and length *L* of the bridge (unit: m), with an extra consideration of the repair cost ratio *Rrcr* at damage state *i*. As suggested by Mander [50], the repair cost ratios at the slight, moderate, extensive, and collapse levels can be taken as 0.1, 0.3, 0.75, and 1.0, respectively.

#### 3.2.2. Indirect Loss

Societal and economic issues often occur following a seismic hazard, and these consequences result in indirect loss which can be even higher than the direct loss (i.e., repair cost) for highway bridges [51]. The indirect loss after an earthquake is somehow related to structural damage which, for example, affects the traffic flow in the route as the drivers are forced to detour during the closure of the bridge. In this study, the running cost, *CRUN*, and the monetary value converted from the time loss for users (i.e., vehicle drivers) through the detour, *CTL*, were considered as the indirect loss. *CRUN* under a given limit state *i* can be expressed as [49]:

$$\mathcal{C}\_{RUN,i} = \left[\mathcal{c}\_{Run,car}(1 - \frac{T\_0}{100}) + \mathcal{c}\_{Run,track}\frac{T\_0}{100}\right] \cdot D\_{l} \cdot ADT \cdot d\_{l} \tag{12}$$

where *cRun*,*car* and *cRun*,*truck* are the average running costs for cars and trucks per kilometer (\$/km), respectively; *T*<sup>0</sup> is the average daily truck traffic, defined as the total volume of vehicle traffic of a highway or road for a year divided by 365 days; *D<sup>l</sup>* is the detour length (km); *ADT* is the average daily traffic to detour, which is the average detour distance of all vehicles influenced by the bridge damage. ADT is generally determined by the bridge damage level; *d<sup>i</sup>* is the duration of the downtime associated with the damage levels, where 7, 30, 120, and 400 days are typically adopted corresponding to the slight, moderate, extensive, and complete damage states, respectively [52]. The monetary value of detour-induced time loss, *CTL*, can be calculated from:

$$\mathcal{L}\_{\rm TL,i} = \left[c\_{\rm AW} o\_{\rm crr} (1 - \frac{T\_0}{100}) + (c\_{\rm ATC} o\_{\rm truck} + c\_{\rm goods}) \frac{T\_0}{100} \right] \cdot \left[\frac{D\_l \cdot ADT}{\mathcal{S}} + ADT \mathcal{E} \cdot \left(\frac{l}{\mathcal{S}\_D} - \frac{l}{\mathcal{S}\_0} \right) \right] d\_l \tag{13}$$

where *cAW* and *cATC* are the average wage plus compensation per hour (\$/h) for car and truck drivers, respectively; *ocar* and *otruck* are the average vehicle occupancies for cars and trucks, respectively; *cgoods* is the time value of the goods transported in a cargo (\$/h); *S* is the average detour speed (km/h); *l* is the route segment containing the bridge (km); *S*<sup>0</sup> and *S<sup>D</sup>* represent the average speed on the intact link and damaged link (km/h), respectively.

#### 3.2.3. Long-Term Loss

By substituting Equations (11)–(13) into Equation (10), the annual loss of the bridge under a specific hazard event can be calculated. By assuming a Poisson distribution for the occurrence of an earthquake during an investigated time interval (0, *t*int), the total life-cycle loss of the bridge can be expressed as [53]:

$$LCL\_i(t\_{\rm int}) = \sum\_{i=1}^{N(t\_{\rm int})} L\_i(t\_k) \cdot e^{-\tau t\_k} \tag{14}$$

where *L<sup>i</sup>* (*tk* ) is the expected annual loss at time *t<sup>k</sup>* , and *τ* is the monetary discount rate. The total expected lifetime failure loss of the bridge during the time interval, *t*int, can be expressed as:

$$E[LCL\_i(t\_{\rm int})] = \frac{\lambda\_f \cdot E(L\_i)}{\tau} \cdot (1 - e^{-\tau t\_{\rm int}}) \tag{15}$$

where *λ<sup>f</sup>* denotes the mean rate of the Poisson model. The values of all necessary parameters mentioned above are summarized in Table 1.

**Table 1.** Parameters associated with consequences.

