## 2 Simplified Trilinear Non-Dimensional Pushover Curve
### Theory Remarks  
The simplified performance-based methodology herein investigated is based on a simplified trilinear non-dimensional pushover curve, derived as the envelope of three branches [27]: (1) elastic branch; (2) maximum bearing capacity branch, obtained through the calibrated Merchant-Rankine formula; and (3) mechanism equilibrium curve. The user can reproduce a pushover or a non-dimensional pushover curve, as depicted in Figure 1, with target points identified through mathematical relationships. The identification of the most-prone collapse mechanism can be attained through rigid-plastic analysis. By considering column sections and the distribution of static forces, this method yields the determination of the first-order collapse multipliers \(\alpha_{0}\) and, regarding the second-order effects, the slopes \(\gamma_{s}\) of the collapse mechanism equilibrium curve for each type of potential collapse mechanism [19; 27] (Figure 2). The mechanism that will be more prone to develop is the one corresponding to the curve characterised by the lower values of \(\alpha\), in the range of displacements compatible with the local ductility resources. Below are the main equations of the proposed model, whose complete discussion can be found in [25; 26; 27; 28; 29]. In Figure 1, the most prone collapse mechanisms affecting seismic-resistant structures are reported. The equations of the three branches in the \(\alpha-\delta\) plane (horizontal force multiplier--top sway displacement) are as follows:  
\[\mathit{Elastic\ branch}\ \rightarrow\alpha=\frac{1}{\delta_{1}}\delta \tag{1}\]  
where \(\delta_{1}\) is the top sway displacement corresponding to the design horizontal forces. The horizontal branch can be expressed by the following:  
\[\mathit{Horizontal\ branch}\ \rightarrow\alpha=\alpha_{max}=\frac{\alpha_{0}}{1+ \Psi\alpha_{0}\gamma_{s}\delta_{1}} \tag{2}\]  
where:  
\[\Psi=a+b\xi\qquad\qquad\qquad\xi=\sum\frac{El_{b}}{L_{b}}/\sum\frac{El_{c}}{L_{ c}} \tag{3}\]  
\(E\) is [PERSON]'s modulus, \(I_{b}\) and \(I_{c}\) are the inertia moduli of the beams and columns, respectively, while \(L_{b}\) and \(L_{c}\) represent the lengths of the beams and columns, respectively. For the coefficient \(\Psi\), the following relation is provided:  
\[\Psi=0.28488-0.14042\xi \tag{4}\]  
Figure 1: Trilinear approximation of the non-dimensional pushover curve. Figure 2: Collapse mechanism of MR-Frames. Finally, in accordance with the rigid-plastic analysis, the final branch is given by the following:  
\[Softening\ branch\ \rightarrow\alpha=\alpha_{0}-\gamma_{s}\big{(}\delta-\delta_{y} \big{)} \tag{5}\]  
where \(\delta_{y}\) is the top sway displacement corresponding to the formation of the first plastic hinge. The specific performance points (A, B, C, D, see Figure 1) associated with pre-defined limit states [30], identifying a target performance level, can be derived as follows:  
* Point A--Fully Operational : \[\alpha_{A}=\frac{1}{\delta_{1}}\delta_{A}\] (6) where \(\delta_{A}\) is the displacement corresponding to the minimum between the displacement in the service conditions, and the formation of the first plastic hinge. * Point B--Operational : \[\alpha_{B}=\alpha_{max}\delta_{B}=\alpha_{max}\delta_{y}\] (7) where \(\alpha_{max}\) is the maximum multiplier [25]. * Point C--Life Safety : \[\alpha_{C}=\alpha_{max}\delta_{C}=\delta_{mecc}=\frac{\alpha_{0}-\alpha_{max}}{ \gamma_{s}}+\delta_{y}\] (8)
* Point D--Near Collapse : \[\mathit{Horizontal\ branch}\ \rightarrow\alpha=\alpha_{max}=\frac{\alpha_{0}}{1+ \Psi\alpha_{0}\gamma_{s}\delta_{1}} \tag{2}\]0 (9) where \(\theta_{p,u}\) is the plastic hinge rotation capacity, equal 8.0 \(\vartheta_{y}\) according to EN 1988-1-3 [31], \(\vartheta_{y}\) represents the chord rotation at yielding and it is defined with reference to the property of the members [25], \(H_{0}\) is the collapse mechanism height. Finally, \(\theta_{p,mec}\) is the plastic hinge rotation demand to attain the collapse mechanism.