3.3.1. Verification

Zhang et al. [41] conducted a cyclic fatigue performance test of a low yield strength steel shear plate damper and the failure mode is depicted in Figure 5b investigating obvious buckling. Using the method above, the finite elemental analysis (Figure 5a) is conducted and the results including the failure mode and the cyclic curve are shown in Figure 5c,d. It can be seen that simulated buckling deformation is consistent with the tested. The simulated curve agrees well with the test curve but a little deviation of the initial stiffness could be observed which might be caused by failing to accurately model the actual loading boundaries. In addition, this finite element model(FEM) accurately predicts the bearing capacity in each loading circle. Therefore, this modelling method could be used to predict the behavior of the HESs and the results could be used to evaluate their performance.

**Figure 5.** The validation of the finite element modeling method. (**a**) Finite element model, (**b**) tested failure mode, (**c**) simulated failure mode, (**d**) hysteresis curves.

Xu et al. [39] proposed a metal shear plate damper utilizing the low yield performance of BLY 160 materials for effective energy dissipation and conducted hysteric tests to evaluate its performance. This metal shear plate damper is composed of four components: shear panels, confined flanges, stiffening ribs and the roof/floor plates for connection (Figure 6b). To validate the simulating method used above, the above modelling method is employed to carry out a finite element analysis of the damper (Figure 6a). The failure mode, the buckling behavior and numerical simulated curves are shown in Figure 6c,d. It is found that the simulated curve has good coincidence with the test curve, but the load-displacement amplitude has some deviation. Some unpredicted slip occurred in the boundary during the experimental loading process, resulting in a deviation between the finite element loading displacement and the test. Therefore, the modelling method could simulate the hysteric behavior of the HESs and could be utilized to evaluate their performance.

**Figure 6.** The validation of the finite element modelling method. (**a**) Finite element model, (**b**) tested failure mode, (**c**) simulated failure mode, (**d**) hysteresis curves.

#### 3.3.2. The Simulation of the Double-Step Performance of the HESs

The detailed geometric parameters are listed in Table 1, Table 2 and Figure 7. The monotonic behavior of specimen HES1 is simulated and the results are shown in Figure 7 and its displacement threshold is set as 3mm. When the shear displacement is applied to 0.24 mm (the drift ratio = 0.12/100), the steel plate in the ED zone begins to yield and the yielding area gradually enlarges from the two ends of the steel plate to the middle part as shown in Figure 8a. With the increment to 3 mm (the drift ratio θ = 1.5/100), the steel plate totally comes into plastic and the shear strength of HES1 comes to the end of its first step, as shown in Figures 8b and 9a. With the development of deformation, the SL zone begin to work and when the horizontal displacement is up to 3.8 mm (the drift ratio = 1.9/100), the middle part of functional bolts yields, the shear stiffness lifting plate is almost in elastic with slight stress concentration at the bolt holes as shown in Figure 8c,d. Finally, the full section of the functional bolts come into plastic and the shear stiffness lifting plate locally yields, as shown in Figure 8e,f. The top and bottom boundaries utilized thick steel plates to simulate the connection to shear walls. When the HES is employed in steel composite shear wall system [42] and steel-damping-concrete composite wall sytems [43] the high strength bolt connection is available.

**Table 1.** Common geometric parameters for specimens.


**Table 2.** Specific dimensions for HES specimens.


**Figure 7.** The detailed dimensions of HES specimens. (**a**) 3D model, (**b**) energy dissipation (ED) zone/rectangular steel plates (HES1/HES4/HES5), (**c**) stiffness lifting (SL) zone/the shear stiffness lifting plate, (**d**) SL zone/the shear stiffness lifting plate, (**e**) the top view, (**f**) ED zone/X-shaped steel plates(HES2/HES3), (**g**) ED zone/Rectangular hole (HES6), (**h**) ED zone/Circular hole (HES7).

**Figure 8.** The validation of the finite element modelling method. (**a**) The stress distribution of ED zone (θ = 0.12%), (**b**) the stress distribution of ED zone (θ = 1.5%), (**c**) the stress distribution of SL zone (θ = 1.9%), (**d**) the stress distribution of functional bolts (θ = 1.9%), (**e**) the stress distribution of SL zone(θ = 5%), (**f**) the stress distribution of functional bolts (θ = 5%).

The double-step force-displacement curve of HES1 is depicted in Figure 9a and the typical double-step working mechanism of the HESs is shown in Figure 9b. The performance of the HES can be observed with four stages including the completely elastic stage, ED plastic stage, SL elastic stage and functional bolts plastic stage. In the completely elastic stage (the OA line in Figure 9a), steel plates in ED are in elastic and the initial shear stiffness is K1 = 525.19 kN/mm. When coming to the ED plastic stage (the AB line), the input seismic energy is mainly dissipated by the plastic deformation. From point B to point C (the SL elastic stage), the SL zone is almost in elastic and the HES restores the bearing capacity and the shear stiffness with K2 = 166.32 kN/mm. As the shear displacement increases, the bending deformation of the functional bolts gradually develops into the fourth stage (the CD line) and finally the come into yielding as depicted in Figure 9b.

**Figure 9.** The typical curve for the HES. (**a**) Monotonic force-displacement curve, (**b**) the failure mode.

#### 3.3.3. The Typical Hysteric Behavior of the HESs

The loading protocol is shown in Figure 10a which is divided into ten stages with two cycles in each stage. In this protocol, two types of calculating the loading displacement are adopted respectively considering the character of the ED zone and the SL zone. In the stages of only the ED zone working, the loading displacement is set according to the yield displacement of the steel plate and the latter loading amplitude is twice the amplitude of the previous loading displacement. In the stages of the ED zone and the SL zone working simultaneously, the loading displacement is set according to the yield displacement of the SL zone and the latter loading amplitude is 1.4 times of the previous loading amplitude. The simulated typical hysteresis curve is depicted in Figure 10b demonstrating the deforming characteristics and energy dissipation performance. Due to the in-plane shear resistance of the steel plate in the ED zone, the HES exhibits a character of the large initial stiffness and the high energy dissipation ability. It can be seen that the cyclic curves are close to rectangular shapes indicating the grea<sup>t</sup> energy dissipation ability. When the loading displacement is larger than the designed displacement threshold, the area of the hysteresis curve and the bearing capacity both gradually increase largely. The hysteretic curve exhibits double-step operating characters with both high energy dissipation ability and the shear stiffness re-lifting ability.

**Figure 10.** The typical hysteresis curve of the HESs. (**a**) The loading protocol, (**b**) the hysteretic curve.
