*3.1. Constitutive Models of ECC Material*

In order to study the seismic response of the ECC-reinforced structures, a constitutive model [45,46], which is capable of considering the computational efficiency and accuracy, is essential for establishing finite element (FE) model, e.g., OpenSees model. The envelop curves of the constitutive model is composed of two parts, of which one is related to the tension state (i.e., *O-A-F-I*) and the other part is associated with the compression state (i.e., *O-J-P*), as shown in Figure 3. The envelop curve in tension is multilinear curves which are expressed as Equation (1), and the envelop curve in compression is also the multilinear curves as written in Equation (2).

**Figure 3.** Envelop curves of the ECC constitutive model [46].

Segment *A*-*F* in the tension region is a strain hardening stage, which indicates that the ECC material is more resilient than the normal concrete. The three segments on the envelope curve within the tensile region (i.e., segments *O*-*A*-*F*-*I*) are given in

$$F\_{t} = \begin{cases} E\varepsilon & 0 \le \varepsilon \le \varepsilon\_{t0} \\ \sigma\_{t0} + \left(\sigma\_{tp} - \sigma\_{t0}\right) \frac{\left(\varepsilon - \varepsilon\_{t0}\right)}{\left(\varepsilon\_{tp} - \varepsilon\_{t0}\right)} & \varepsilon\_{t0} \le \varepsilon < \varepsilon\_{tp} \\ \sigma\_{tp} \frac{\left(\varepsilon\_{tn} - \varepsilon\right)}{\left(\varepsilon\_{tn} - \varepsilon\_{tp}\right)} & \varepsilon\_{tp} \le \varepsilon < \varepsilon\_{tn} \\ 0 & \varepsilon\_{tn} \le \varepsilon \end{cases} \tag{1}$$

where *E* is Young's modulus. The first microcrack will occur if the strain *ε* is larger than the cracking strain *εt*<sup>0</sup> at point *A*. The loading path further follows the segment *A*-*F* until the strain reaches the peak tensile strain *εtp* at point *F*, where the stress simultaneously reaches the peak tensile stress *σtp*. However, a soft stiffness (i.e., segment *F*-*I*) will occur if the tensile strain exceeds *εtp* but is smaller than the ultimate tensile strain *εtu* at point *I*, where the corresponding stress becomes zero. It is worth noting that the loading path will continuously move forward along the positive abscissa axis if the strain is larger than *εtu*.

The envelope curve consists of multilinear curves in the compressive region (i.e., segments *O*-*J*-*P*) which can be expressed as

$$F\_{\mathcal{C}} = \begin{cases} E\varepsilon & \varepsilon\_{lp} \le \varepsilon < 0\\ \sigma\_{cp} \frac{(\varepsilon\_{\rm cf} - \varepsilon)}{\left(\varepsilon\_{\rm cf} - \varepsilon\_{cp}\right)} & \varepsilon\_{cp} \le \varepsilon < \varepsilon\_{\rm cf} \\\ 0 & \varepsilon\_{\rm cf} \le \varepsilon \end{cases} \tag{2}$$

where *σcp* and *εcp* are the peak compressive stress and corresponding strain, respectively. *εcu* is the ultimate compressive strain. It is noted that loading path will further move forward along the negative abscissa axis if the strain is smaller than *εcu*.

The loading, the unloading and the reloading rules of the ECC model in the tension region (see Figure 4a) are given by

$$F\_{l} = \begin{cases} E\varepsilon & 0 \le \varepsilon\_{tm} < \varepsilon\_{t0} \\ \sigma\_{tl}^{\prime} \left(\frac{\varepsilon - \varepsilon\_{tl}}{\varepsilon\_{tm}^{\prime} - \varepsilon\_{tl}}\right)^{a\_{l}} & \varepsilon\_{t0} \le \varepsilon\_{tm} < \varepsilon\_{tp}, \dot{\varepsilon} < 0 \\ \sigma\_{tl}^{\prime} + \left(\sigma\_{tm} - \sigma\_{tl}^{\prime}\right) \left(\frac{\varepsilon - \varepsilon\_{tl}^{\prime}}{\varepsilon\_{tm}^{\prime} - \varepsilon\_{tl}^{\prime}}\right) & \varepsilon\_{l0} \le \varepsilon\_{tm} < \varepsilon\_{tp}, \dot{\varepsilon} \ge 0 \\ \sigma\_{tm} \left(\frac{\varepsilon - \varepsilon\_{tl}}{\varepsilon\_{tm} - \varepsilon\_{tl}}\right) & \varepsilon\_{tp} \le \varepsilon\_{tm} < \varepsilon\_{tn} \end{cases} (5)$$

$$\left( \begin{array}{c} \sigma\_{tm} \left( \frac{\varepsilon - \varepsilon\_{tl}}{\varepsilon\_{tm} - \varepsilon\_{tl}} \right) \end{array} \right) \qquad \begin{array}{c} \varepsilon = \dots \\\\ \varepsilon\_{tp} \le \varepsilon\_{tm} < \varepsilon\_{tu} \end{array} \tag{6}$$

where *α<sup>t</sup>* is a constant that is larger than or equal to 1. It can be calibrated using the experimental data. The segment *B*-*C*-*E* is the initial unloading path expressed by the Equation (4). *ε tm* is the maximum strain reached in history on the envelop curve where the unloading is triggered. *ε tl* is the strain corresponding to the stress vanishing on the initial unloading path (i.e., the strain associated with the point *E* or *H*). The value *ε tl* = *β<sup>t</sup>* ·*εtm*, where *β<sup>t</sup>* is a constant. Segment *C*-*D* is a typically partial reloading path, where the stress at point *C* is not zero, expressed by Equation (5), which ensures the extension of the segment *C*-*D* passing through the historically reached maximum strain point on the envelop curve (i.e., point *B*). The *ε tr* and *ε tu* are strains at points *C* and *D*, respectively. The unloading path starting from *D* is controlled by Equation (4). The subscript on the parameter *ε* 0 *tm* denotes that *ε* 0 *tm* should be set as *ε tm* or *ε tr* when they are used to define the initial unloading path or partially reloading path, respectively. The same specification applies to the parameters *ε* 0 *tl* and *ε* 0 *tu*.

**Figure 4.** Unloading and reloading rules of the ECC model (**a**) Tension region; (**b**) Compression region [46].

The unloading and reloading paths after point *F* (i.e., *F*-*G*) are a linear curve given by Equation (6). The loading, the unloading and the reloading paths of the ECC model in the compression region (see Figure 4b) are expressed as:

$$F\_{\mathfrak{c}} = \begin{cases} E\varepsilon & \varepsilon\_{\mathfrak{c}\mathfrak{m}} < \mathfrak{e}\_{\mathfrak{c}\mathfrak{m}} < 0 \\\ \sigma\_{\mathfrak{c}\mathfrak{m}}^{\prime} \left(\frac{\varepsilon-\varepsilon\_{\mathfrak{c}\mathfrak{l}}}{\varepsilon\_{\mathfrak{c}\mathfrak{m}}^{\prime}-\varepsilon\_{\mathfrak{c}\mathfrak{l}}}\right)^{\mathfrak{a}\_{\mathfrak{c}}} & \varepsilon\_{\mathfrak{c}\mathfrak{m}} \le \varepsilon\_{\mathfrak{c}\mathfrak{m}} < \mathfrak{e}\_{\mathfrak{c}\mathfrak{p}}, \dot{\varepsilon} > 0 \\\ \sigma\_{\mathfrak{c}\mathfrak{m}}^{\prime} + \left(\sigma\_{\mathfrak{c}\mathfrak{m}} - \sigma\_{\mathfrak{c}\mathfrak{m}}^{\prime}\right) \left(\frac{\varepsilon-\varepsilon\_{\mathfrak{c}\mathfrak{c}}^{\prime}}{\varepsilon\_{\mathfrak{c}\mathfrak{m}}^{\prime}-\varepsilon\_{\mathfrak{c}\mathfrak{c}}^{\prime}}\right) & \varepsilon\_{\mathfrak{c}\mathfrak{m}} \le \mathfrak{e}\_{\mathfrak{c}\mathfrak{m}} < \mathfrak{e}\_{\mathfrak{c}\mathfrak{m}}, \dot{\varepsilon} \le 0 \end{cases} (8)$$

The segment *K*-*N*, as presented in Figure 4b, is formulated by Equation (8). *εcm* is the strain on the envelope curve where unloading is triggered (i.e., the strain at point *K*) and the *ε cl* is the strain on the initial unloading path corresponding to zero stress (i.e., the strain at point *L*). The value *ε cl* = *βc*·*εcm*, where *β<sup>c</sup>* is a constant. Segment *N*-*M* is formulated by Equation (9). *εcr* and *εcu* are strains of the points *M* and *N*, respectively. The unloading path from *N* is given by Equation (8). The parameter *ε* 0 *cm* should be set to *εcm* or *εcr* when they are employed to define the initial unloading path or partially reloading path.

## *3.2. ECC-Reinforced Column and Numerical Verification*

A 1/5 scaled ECC-reinforced column [47,48], as shown in Figure 5a, is used to verify the effectiveness of the ECC constitutive model proposed in Section 3.1. To provide lateral load on the top of the cantilever column, a rigid ECC transverse beam is monolithically casted with the cantilever base. This loading configuration is chosen to promote a flexural deformation mode in the specimen and to investigate the effect of ECC material properties on the expected plastic hinge region in particular. Longitudinal steel reinforcement was bent at a 90º angle at the bottom of the transverse beam and further extended to provide sufficient anchorage. A total of four reinforcing steels with a diameter of 10 mm are arranged at the four corners of the cross section of the pier, resulting in a reinforcement ratio of 3.14%. The compressive strength of the ECC material is 80.0 MPa at a strain of 1.2%. The tensile strength is 6.0 MPa at a strain of 6.0%. The Young's modulus of the ECC is 16,000.0 MPa. The Passion's ratio of the ECC material is 0.15. The yielding strength of the reinforcing steel is 410.0 MPa at a strain of 0.02% and the ultimate uniaxial strength is 640.0 MPa at a strain of 14.0%. The lateral load protocol for the quasi-static test is shown in Figure 5b.

**Figure 5.** Specimen and loading protocol. (**a**) ECC-reinforced column specimen (Unit: mm); (**b**) Loading protocol for quasi-static test.

The FE model of the ECC-reinforced column is established in OpenSees (Version 2.4.1). A total of 5 displacement-based beam-column elements are used to model the ECC column. The *ECC02 material model* that has been developed and implemented into OpenSees is employed to capture the response of the column. The analysis results regarding the horizontal force against drift ratio together with the test results are displayed in Figure 6. It indicates that the numerical simulation results agree well with the test results. It confirms that the ECC constitutive model proposed in this study is sufficiently accurate for further dynamic analysis.

**Figure 6.** Numerical simulation results against test results.

#### *3.3. Constitutive Models of SMA Washer*

The chemical composition of the SMA washer is 55.87% nickel and 44.13% titanium alloys by weight and it is supposed to exhibit super elasticity at room temperature (austenite finish temperature is 4.5 ◦C). The constitutive model of the SMA washer has been developed in [40], which has a flag-shaped envelop curves, including loading stage (i.e., O-A-B-E) and unloading stage (i.e., E-B-C-E-O), as shown in Figure 7, where *F*<sup>2</sup> and *δ*<sup>2</sup> represent the "yield" force and the corresponding deformation, respectively; *F*<sup>3</sup> and *δ*<sup>3</sup> are the force and deformation when the SMA washer is fully flattened. Similarly, *F*4, *δ*4, *F*1, and *δ*<sup>1</sup> represent the characteristic forces and deformations during the reverse plateau. Once the SMA washer reaches its maximum compressive deformation, the axial stiffness increases abruptly (i.e., BE).

**Figure 7.** Idealized constitutive model for SMA washer.

The loading stag on the envelop curve is composed of three segments, which are expressed as:

$$F = \begin{cases} \begin{array}{c} E\_{w}\delta \\ F\_{2} + k\_{1}E\_{w}(\delta - \delta\_{2}) \\ F\_{3} + k\_{2}E\_{w}(\delta - \delta\_{3}) \end{array} & \delta\_{2} \le \delta < \varepsilon\_{3} \\\ \begin{array}{c} F\_{3} + k\_{2}E\_{w}(\delta - \delta\_{3}) \end{array} & \varepsilon\_{3} \le \delta \end{cases} \tag{10}$$

where *E<sup>w</sup>* is the elastic stiffness of the single SMA washer; *k*<sup>1</sup> and *k*<sup>2</sup> are constants which can be determined from test.

The unloading stage on the envelop curve consists of four segments, which are given as follows:

$$F = \begin{cases} E\_w \delta & 0 \le \delta < \delta\_1 \\\ F\_4 - k\_1 E\_w (\delta\_4 - \delta) & \delta\_1 \le \delta < \delta\_4 \\\ F\_3 - E\_w (\delta\_3 - \delta) & \delta\_4 \le \delta < \delta\_3 \\\ F\_3 + k\_2 E\_w (\delta - \delta\_3) & \delta \ge \delta\_3 \end{cases} \tag{11}$$

The uploading and reloading stiffness away from the envelop curve is *Ew*.

The key parameters for a single SMA washer are shown in Figure 7.

The diagram of a typical SMA washer is presented in Figure 8, where the outer and inner diameters of the SMA washer are D and d, respectively. The thickness and the height of the SMA washer are h and t, respectively. These parameters of a SMA washer can be appropriately adjusted according to the seismic design objectives of rocking bridges. Once all the parameters are designated, the maximum compressive deformation and the restoring force provided by a SMA washer can be calculated.

**Figure 8.** Configuration of a typical SMA washer.

#### **4. Validation of the Numerical Simulation Method for Capturing Rocking Behavior**

A 1/4 scale RC rocking pier specimen equipped with SMA washer sets [29] is employed to validate the numerical simulation method used in this study for capturing the overall rocking behavior. The configuration of the rocking pier is shown in Figure 9. The test specimen includes one bent cap, the main body of a pier, two separated pile caps, four SMA washer sets, and a tendon. The diameter of the RC pier is 0.3 m. The strength of the normal concrete was 39.0 MPa at the test day. A total of 12 longitudinal HRB400 rebars with a diameter of 16 mm were uniformly arranged along the perimeter, resulting in a reinforcement ratio of 0.3%. The average yielding strength of the longitudinal rebar was 400.4 MPa. The diameter of the transverse reinforcement was 10 mm and the space between two adjacent stirrups was 75 mm. A total of four SMA washer sets were installed on the top of the upper pile cap and each SMA washer set composed of the loading protocol is shown in Figure 10. Other information regarding this rocking pier can be find elsewhere [29].

**Figure 9.** Configuration of a rocking pier specimen and loading protocol.

**Figure 10.** FE model of the rocking pier specimen.

To validate the numerical simulation method proposed in this study for capturing the overall rocking behavior, the experimental results of the rocking pier specimen are used to make comparison with the numerical analysis results. The RC pier was divided into seven displacement-based fiber beam-column elements, of which the normal concrete fiber was assigned with *Concrete02 material model* and the reinforcement fiber was assigned with *Steel02 material model*. The connection between the pier and the bent cap or the pile cap is modeled by rigid elements. The interface between upper and lower pile caps was simulated by eight zero-length spring elements, of which the tension strength is ignored. Each SMA washer set was modeled by a zero-length spring element, of which the assigned material was the *Self-centering material model*. The FE model is shown in Figure 10. It is worth noting that the nonlinear material properties of the normal concrete and the reinforcement can be considered by the *Concrete02 material model* and *Steel02 material model*, respectively. The geometric nonlinearity of the RC pier can also be accounted for by the co-rotational geometric transformations [39]. The numerical simulation results and the test ones are presented in Figure 11, which shows that the analysis results match well with the test regarding the amplitude of the lateral load and the trend of the hysteretic curve. It confirms that the numerical simulation method proposed in this study can be used to capture the overall rocking behavior of the rocking bridge system.

**Figure 11.** Comparison between numerical simulation and test results.
