## 2 The Proposed Damper
### Predicting the Behavior of the Damper  
To impose yielding on the damper's main plates, the ultimate state of the shear and bending moment capacities reach values of about \(1.5V_{p}\) and \(1.2M_{p}\), respectively. To impose bending yielding before shear yielding of the damper's plates, the ultimate state of the shear and bending moment capacities reached values of about \(0.9V_{p}\) and \(1.2M_{p}\), respectively. For the main plate not connected to the cross_flexural plate, knowing \(M_{p}=\frac{lt^{2}}{4}F_{y}\) and \(V_{p}=\frac{F_{y}}{\sqrt{3}}bt\), the \(b/h\) ratio needs to be limited [34; 35; 36; 37; 38; 39; 40; 41] to \(b/h\leq 0.9\), where \(F_{y}\) is the yielding stress, and \(t\) is the thickness of the main plate. The ultimate strength and elastic stiffness of the proposed damper, \(F_{u}\), are obtained from Equations (1) and (2), respectively. \[F_{u}=F_{fu}+F_{wu}, \tag{1}\]  
\[K_{d}=K_{main\ plate}+K_{flexural\ cross\ plate}, \tag{2}\]  
where \(F_{fu}\) is the ultimate strength of the cross_flexural plate that is obtained from Equation (3), and \(F_{wu}\) is the ultimate strength of the main plate. The elastic stiffness of the cross_flexural plate, \(K_{f}\), is obtained using Equation (4). Since the cross_flexural plate is subjected to a weak axial moment, it always behaves in a flexural form. Therefore, it is assumed that the ultimate strength of the cross_flexural plate, \(F_{fu}\), can be obtained as  
\[F_{fu}=\frac{4M_{p}}{h}, \tag{3}\]  
\[K_{f}=\frac{24 EI_{f}}{h^{3}}, \tag{4}\]  
where \(M_{p}\) and \(I_{f}\) are the plastic moment and moment of inertia of the cross_flexural plate, respectively. Furthermore, \(E\) is the Young modulus, and \(h\) is the height of the plate, as shown in Figure 2, \(M_{p}=\frac{b/t_{f}^{\ 3}}{4}F_{y}\). For the main plate, it is under in_plane loading, according to [47]. Thus, considering the Poisson's ratio of steel of 0.3, the elastic buckling stress, \(\tau_{cr}\), can be expressed as  
\[\tau_{cr}=3.6E\bigg{(}1.33\frac{t^{2}}{h^{2}}+\frac{t^{2}}{b^{2}}\bigg{)}. \tag{5}\]  
Figure 1: Schematic views and placement of the damper in frame. Since the aspect ratio of the damper's plate is low, buckling mainly occurs in nonlinear zones after yielding. \[\tau_{cr}=\sqrt{(0.8\tau_{y})3.6E\bigg{(}1.33\frac{t^{2}}{\hbar^{2}}+\frac{t^{2}}{ b^{2}}\bigg{)}}=\sqrt{2.88E\bigg{(}1.33\frac{t^{2}}{\hbar^{2}}+\frac{t^{2}}{b^{2}} \bigg{)}\tau_{y}}\leq\frac{F_{y}}{\sqrt{3}}. \tag{6}\]  
For the damper with plates without cross_flexural plates, elastic buckling is minimum according to Equations (5) and (6). For the main plate, the ultimate strength, \(F_{wu}\), and elastic stiffness, \(K_{we}\), are calculated as  
\[F_{wu}=\sigma_{yx}bt, \tag{7}\]  
\[K_{we}=\frac{F_{wu}}{\Delta_{we}}, \tag{8}\]  
Figure 2: Dampers with different mechanisms. where \(\Delta_{ww}\) is the displacement due to buckling, \(\frac{\tau_{cr}}{G}d\), and yielding, \(\Delta_{wpb}\), as expressed in Equation (9). \[K_{d}=K_{main\ plate}+K_{flexural\ cross\ plate}, \tag{2}\]0  
where \(\Delta_{wpb}\) is the plastic displacement of the plate under a shear force that is determined as described below. The created stresses in the steel plate are developed along the angle \(\theta\). Accordingly, the amount of created stress can be expressed as  
\[K_{d}=K_{main\ plate}+K_{flexural\ cross\ plate}, \tag{2}\]1  
where \(\sigma_{ty}\) is equivalent yield stress. According to the [PERSON] yield criterion, plate yielding mainly occurs when the following equation is established:  
\[K_{d}=K_{main\ plate}+K_{flexural\ cross\ plate}, \tag{2}\]2  
For specimens without the cross_flexural plate or specimens with the cross_flexural plate unconnected to the main plate, the vertical length of the plate is free; therefore, \(\sigma_{z}=\sigma_{xx}=0\). It should be noted that, for the specimen with cross_flexural plates connected to the main plate, \(\sigma_{xx}\) is taken from Equation (10). Since plane stress is formed in the plate, \(\sigma_{z}=\sigma_{yz}=\sigma_{xz}=0\).