## 4 Results and Discussion
### Ultimate Energy Dissipation Capacity and Prediction of Failure  
Previous studies [24; 25; 26] have pointed out the dependence of the ultimate energy dissipation capacity of structural members subjected to low-cycle fatigue on the pattern of loading applied. An appropriate way [24; 25] to take into account this dependency consists of splitting the total energy dissipated by the damper into two parts: the energy dissipated on the so-called skeleton part and the energy dissipated on the Bauschinger part, as shown in Figure 17. The typical Q-\(\delta\) curve obtained by testing a steel component under arbitrarily applied cyclic loads until failure (Figure 17a) can be decomposed into the skeleton part (Figure 17b) and Bauschinger part (Figure 17c) as follows. The skeleton part is formed, in each domain of loading, by connecting sequentially the segments of the Q-\(\delta\) curve whose force \(Q\) exceeds the maximum force attained in previous cycles in the same domain of loading. The skeleton curves obtained in this way coincide with the Q-\(\delta\) obtained by subjecting the steel component to monotonic loadings [25]. The skeleton curves are idealized by the trilinear curve in Figure 17b (dashed and dotted lines), in turn defined by the following parameters: \(Q_{y}\) and \(\delta_{y}\) given by Equations (1) and (2); the elastic stiffness \(K_{e}\) (=\(Q_{y}/\delta_{y}\)); the force \(Q_{B}\) at the onset of the second plastic branch given by Equation (1); and the first and second plastic stiffness, \(K_{p1}\) and \(K_{p2}\). The displacement \(\delta_{B}\) associated with \(Q_{B}\) is readily obtained from \(K_{p1}\) and \(Q_{B}\). In Figure 17b, \(\delta_{0}\)\({}^{+}\) and \(\varepsilon\)\(\delta_{u}\)\({}^{-}\) denote the maximum deformation accumulated in the positive and negative skeleton parts. The portion of the total plastic strain energy dissipated by the damper on the skeleton part, in the positive and negative domains of loading, respectively \({}_{S}W_{u}\)\({}^{+}\) and \({}_{S}W_{u}\)\({}^{-}\), is readily obtained by integrating the areas enveloped by each skeleton curve in Figure 17b. The Bauschinger part is constituted by the segments of the Q-\(\delta\) curve that start at \(Q\) = 0 and end at the point corresponding to the maximum load attained in the preceding cycles in the same loading domain. The Bauschinger parts are graphically depicted in Figure 17c, and the area enveloped in each domain of loading, \({}_{B}W_{u}\)\({}^{+}\) and \({}_{B}W_{u}\)\({}^{-}\), represents the portion of the total energy dissipated by the steel element in the Bauschinger part. For convenience, \({}_{S}W_{u}\)\({}^{+}\), \({}_{S}W_{u}\)\({}^{-}\), \({}_{B}W_{u}\)\({}^{+}\), \({}_{B}W_{u}\)\({}^{-}\), \({}_{S}\delta_{u}\)\({}^{+}\) and \({}_{S}\delta_{u}\)\({}^{-}\), are expressed in a nondimensional form by:  
\[s\overline{\eta}\ ^{+}\ =\ \frac{sW_{u}^{+}}{Q_{y}\delta_{y}};_{S}\overline{\eta} ^{-}\ =\ \frac{sW_{u}^{-}}{Q_{y}\delta_{y}};_{B}\overline{\eta}^{+}\ =\ \frac{{}_{B}W_{u}^{+}}{Q_{y}\delta_{y}};_{B} \overline{\eta}^{-}\ =\ \frac{{}_{B}W_{u}^{-}}{Q_{y}\delta_{y}};_{\text{ }}\overline{\eta}^{+}\ =\ \frac{s\delta_{u}^{+}}{\delta_{y}};_{\text{ }}\overline{\eta}^{-}\ =\ \frac{s\delta_{u}^{-}}{\delta_{y}} \tag{4}\]  
and the total energy dissipated on the skeleton part, on the Bauschinger part, and the total energy dissipated by the steel component until failure can be expressed in nondimensional form by:  
\[{}_{S}\eta\ =\ \big{|}_{S}\overline{\eta}^{+}\big{|}+\big{|}_{S}\overline{\eta} ^{-}\big{|};_{B}\eta\ =\ \big{|}_{B}\overline{\eta}^{+}\big{|}+\big{|}_{B}\overline{\eta}^{-}\big{|};_{ \text{ }}\overline{\rho}\eta\ =\ \big{|}_{\text{ }\rho}\overline{\eta}^{+}\big{|}+\big{|}_{\text{ }\rho}\overline{\eta}^{-}\big{|};\eta\ =\ _{S}\eta+{}_{B}\eta. \tag{5}\]  
Past studies [12] showed that \({}_{B}\eta\) and \({}_{\text{ }\rho}\eta\) follow an approximately linear relationship:  
\[{}_{B}\eta\ =\ a_{\text{ }\rho}\eta+b^{\prime}, \tag{6}\]  
where \(a\) and \(b^{\prime}\) are two parameters determined from tests.