## 2 Fundamental Equations of the Trilinear Model  
The simplified trilinear model needs only the elastic structural analysis and the rigid-plastic analysis [28; 29; 30; 31] not requiring any static or dynamic non-linear analyses. Therefore, the user can quickly obtain the capacity curve through the intersection of three linear branches (Figure 1), computed using simple relationships proposed in the following. Referring to the proposed model, the first branch of the curve can be approximated by the elastic response curve; the horizontal one is defined using the maximum load-bearing capacity given by the calibrated Merchant\(-\)Rankine formula, while, the softening branch is given by the collapse mechanism equilibrium curve of the structure, influenced by the second-order effects [32; 33; 34; 35; 36]. The definition of the third branch is linked to the concept of the mechanism equilibrium curve. In particular, the equation of the third branch can be obtained by equating the virtual internal work of the dissipative zones with the virtual external work of the structure, accounting for second-order effects. The equations of the three identified branches in the \(\alpha\)\(-\)\(\delta\) plane (horizontal force multiplier--top sway displacement) are reported below:  
* Elastic response curve:  
Figure 1: Trilinear approximation of the capacity curve. \[\alpha\ =\ \frac{1}{\delta_{1}}\delta \tag{1}\]
* Maximum load-bearing capacity curve according to the modified Merchant\(-\)Rankine formulation [23; 37; 38] \[\alpha\ =\ \alpha_{max}\ =\ \frac{\alpha_{0}}{1\ +\ \Psi\alpha_{0}\gamma_{s} \delta_{1}}\] (2) where: \[\Psi\ =\ \alpha\ +\ b\xi\] (3) \[\xi\ =\ \frac{\sum\frac{Et_{k}}{L_{b}}}{\sum\frac{Et_{c}}{L_{c}}}\] (4)  
The use of Equation (3) is proposed by assuming for the coefficient \(\Psi\) with the following relation:  
\[\Psi\ =\ 0.28488\ -\ 0.14042\ \xi \tag{5}\]
* The mechanism equilibrium curve according to rigid-plastic analysis [37]: \[\alpha\ =\ \alpha_{0}\ -\ \gamma_{s}\big{(}\delta\ -\ \delta_{y}\big{)}\] (6)  
Characteristic performance points (points A, B, C, D of Figure 2) have been identified on the trilinear model. The points are associated with specific limit states [35], provided by codes, identifying a target performance level [34; 39; 40; 41]. * Point A--\"Fully Operational\" \[\alpha_{A}\ =\ \frac{1}{\delta_{1}}\delta_{A}\] (7) where \(\delta_{A}\) is the displacement corresponding to the minimum between the displacement in the service conditions and \(\delta_{y}\) is the displacement corresponding to the formation of the first plastic hinge. * Point B--\"Operational\" \[\alpha_{B}\ =\ \alpha_{max}\ ;\ \delta_{B}\ =\ \alpha_{max}\delta_{y}\] (8) where \(\alpha_{max}\) is the maximum multiplier defined according to the Equation (2). * Point C--\"Life Safety\"  
Figure 2: Characteristic performance points. \[\alpha\ =\ \frac{1}{\delta_{1}}\delta \tag{1}\]0  
This point derives from the intersection of the horizontal branch of equation \(\alpha=\alpha_{max}\) with the softening branch, representative of the collapse mechanism equilibrium curve, of equation \(\alpha=\alpha_{0}-\gamma_{S}(\delta-\delta_{y})\). * Point D--\"Near Collapse\"  
\[\alpha\ =\ \frac{1}{\delta_{1}}\delta \tag{1}\]1  
The chord rotation at yielding \(\phi_{y}\) is defined as a property of the member. For ductile elements and brittle elements involved into soft-storey mechanisms, are given by:  
\[\alpha\ =\ \frac{1}{\delta_{1}}\delta \tag{1}\]2  
while for brittle elements involved in partial and global mechanisms:  
\[\alpha\ =\ \frac{1}{\delta_{1}}\delta \tag{1}\]3  
For the evaluation of plastic rotations occurring in the critical members, an analytical formulation is proposed, based on a \"shear\(-\)type\" single\(-\)storey portal with different plastic moments at the top and base of the columns.