*3.3. Resilience Assessment*

Resilience is another important structural performance indicator defining the capability of a civil infrastructure system of maintaining its post-hazard functionality. Generally, resilience includes four gradients, namely, rapidity, robustness, redundancy, and resourcefulness. In this paper, the resilience of the bridges under the considered seismic hazards was assessed, where the functionality is deemed to be resumed when the traffic becomes normal after earthquake.

It is worth mentioning that various functionality levels may be considered for different periods, e.g., emergency response and post-earthquake recovery stages. In the former stage, the main focus should be on the capability of transferring the resources to the disaster area. In the latter phase, the functionality of the bridge can be defined with different service statuses, e.g., "closed", "limited use", and "open". The resilience of the bridge can be evaluated through its recovery pattern. As illustrated graphically in Figure 5, one of the most widely adopted approaches to quantify resilience is [54,55] expressed as follows:

$$R\_{Resi} = \frac{1}{\Delta t\_r} \int\_{t\_0}^{t\_0 + \Delta t\_r} Q(t)dt \tag{16}$$

where *Q*(*t*) is the functionality of a bridge at time *t* (e.g., days); *t*<sup>0</sup> is the initial time at the investigated point; and ∆*t<sup>r</sup>* is the investigated time interval (e.g., days or months).

**Figure 5.** Schematic representation of resilience.

The shape of the recovery pattern curve is related to the repair and recovery efforts. Generally, bridge functionality can be assessed by defining the damage state to a value between 0 and 1.0, where a value of 0 means collapse of the bridge. Considering various levels of damage state, the bridge functionality can be expressed as:

$$Q = \sum\_{i=1}^{5} FR\_i \cdot P\_{DS\_i \mid IM} \tag{17}$$

where *FR<sup>i</sup>* is the functionality ratio (Func) associated with damage state *i*. For a typical bridge, possible scenarios include immediate access (Func ≥ 0.9), weight restriction (0.6 ≤ Func < 0.9), one lane open only (0.4 ≤ Func < 0.6), emergency access only (0.1 ≤ Func < 0.4), and bridge closure (Func < 0.1). Similar concepts have also been adopted by Padgett and DesRoches [56] and Decò et al. [57]. Once repair actions are initiated, the functionality of the bridge starts to recover and the performance restoration curve starts to rise. The Applied Technology Council (ATC-13) report proposed an approach to quantify the change of functionality *Q<sup>j</sup>* (*t*) during the recovery phase [58], expressed as follows:

$$Q\_{j}(t) = \frac{1}{\sigma\_{j}\sqrt{2\pi}} \int\_{-\infty}^{t} \exp[-\frac{(\tau - \mu\_{j} \cdot A\_{t})^{2}}{2\sigma\_{j}^{2}}]d\tau \tag{18}$$

where *µ<sup>j</sup>* and *σ<sup>j</sup>* are the mean and standard deviation of the recovery time for the *j*th damage state, as listed in Table 2; and *A<sup>t</sup>* is an amplification factor considering the increase of mean recovery time due to underwater repair work. The functionality of the considered bridge on a daily basis during the recovery phase can be calculated with the above-mentioned parameters and methodology. The probabilities of the bridge staying in different damage states can be used to obtain the expected recovery functionality *Q*(*t*), and the resilience can be assessed by Equation (18).


**Table 2.** Parameters associated with bridge restoration functionality [58].

#### **4. Case Study 4. Case Study**

#### *4.1. Description of Prototype Bridges 4.1. Description of Prototype Bridges*

A continuous RC bridge with two equal spans (20 + 20 m) supported by a middle pier was developed to investigate the life-cycle loss and resilience performances of the novel SMA-washer-based SCR bridge system. Figure 6a presents the bridge's geometric layout. Sliding bearings were placed on each abutment, and fixed bearings were placed on the bent cap above the middle pier. A sufficient separation space between the bridge deck and the abutment was assumed [59–61]. The concrete used for the box girder has a compressive strength of 50 MPa, and that for the abutment, pier, and rocking pile caps have a compressive strength of 40 MPa. The longitudinal reinforcement and stirrup utilized in the RC pier have diameters of 32 mm and 16 mm, respectively, with yield strengths of 440 MPa and 300 MPa, respectively. A continuous RC bridge with two equal spans (20 + 20 m) supported by a middle pier was developed to investigate the life-cycle loss and resilience performances of the novel SMA-washer-based SCR bridge system. Figure 6a presents the bridge's geometric layout. Sliding bearings were placed on each abutment, and fixed bearings were placed on the bent cap above the middle pier. A sufficient separation space between the bridge deck and the abutment was assumed [59–61]. The concrete used for the box girder has a compressive strength of 50 MPa, and that for the abutment, pier, and rocking pile caps have a compressive strength of 40 MPa. The longitudinal reinforcement and stirrup utilized in the RC pier have diameters of 32 mm and 16 mm, respectively, with yield strengths of 440 MPa and 300 MPa, respectively.

**Figure 6.** Geometric layout of two-span continuous concrete-girder bridge (Unit: mm): (**a**) general layout, (**b**) detailed dimensions. **Figure 6.** Geometric layout of two-span continuous concrete-girder bridge (Unit: mm): (**a**) general layout, (**b**) detailed dimensions.

A total of 55 longitudinal steel bars are equally distributed around the pier perimeter, corresponding to a 1.74% reinforcement ratio. Spiral stirrups are 100 mm apart. The RC pier has a 65-mm-thick concrete cover layer. The rocking pier employed six SMA washer sets, each of which has eight washers, 4 in parallel × 2 in series. Figure 6b shows the key characteristics of each unique SMA washer, where a maximum deformation of 32 mm and a maximum compressive resistance of roughly 250 kN are provided by every single washer. Each SMA washer spring set was preloaded with 960 kN, corresponding to a precompression deformation of 22 mm, to guarantee that the rocking pier does not uplift under normal service loads or small earthquakes. In other words, each SMA washer spring set has remaining deformability of 42 mm prior to the fully compressed status. When the SMA washer sets were totally flattened, a "locking" mechanism was produced, and no further rocking was allowed beyond this allowable angle. After locking, damage to the RC pier is expected.

For comparison, an extra conventional bridge was evaluated with a fixed-base RC pier that is 10 m tall, i.e., equal to the height of the rocking pier from the bottom surface of the bent cap to the rocking interface. The other design of the bridge remains the same.
