# Simplified Pushover Analysis for Rapid Assessment of Shear-Type Frames
## 2 Displacement Capacity of a 2D Frame
### Moment-Curvature Relationship for the Three LSs  
The trilinear curve in Figure 2 describes the three limit states in a sectional moment-curvature diagram. The criteria adopted for defining the sectional LSs are: DLS (Damage Limit State), corresponds to tensile reinforcing bars yielding; LLS (Life safety Limit State), corresponds to concrete cover crushing; and CLS (Collapse prevention Limit State), corresponds to concrete core crushing. The coordinates of these points are given in terms of curvature \(\phi_{xLS}\) and moment \(M_{xLS}\) capacity, where \(x=D\), \(L\), \(C\) for DLS, LLS, and CLS. The corresponding equations for rectangular sections with symmetrical reinforcement, extended from [30], are shown in Table 1, with the notation in Figure 3, where \(d\) is the section depth, \(b\) is the section base, \(d_{c}=b_{y}\) is the section depth at collapse, \(A_{s}\) is the tension steel area, \(A_{sw}\) is the stirrup area, and \(s\) is the stirrup spacing. In addition, \(\varepsilon_{y}\) is the steel yield strain, \(\varepsilon_{cut}\) is the ultimate concrete strain, and \(f_{c}\) is the compressive concrete strength, while \(\omega_{s}\) is the longitudinal reinforcement mechanical ratio and \(n_{S}\) is the normalized axial load, given as:  
\[\omega_{s}=\frac{A_{s}f_{y}}{bdf_{c}}\qquad and\qquad n_{S}=\frac{N_{S}}{bdf_{c}} \tag{1}\]  
Figure 1: Mechanical idealization of an example two-story one-bay 2D frame with column-driven failure, where in each column three resisting mechanisms are considered: bending (\(M\)), shear (\(V\)), and joint shear (\(V_{j}\)). Moreover, \(\widetilde{\varphi}_{2}\) is the confining stress normalized with respect to the characteristic compressive concrete strength \(f_{fd}\), which can be computed as follows:  
\[\widetilde{\varphi}_{2}=\alpha_{n}\cdot\alpha_{s}\cdot\frac{1}{3}\omega_{w} \tag{2}\]  
where:  
\[\alpha_{n}=\Bigg{(}1-\frac{\sum b_{i}^{2}}{6b_{x}b_{y}}\Bigg{)}\approx 1-\frac{ 1}{3b_{x}b_{y}}\Bigg{(}\frac{b_{x}^{2}}{n_{ct,y}+1}+\frac{b_{y}^{2}}{n_{ct,x}+ 1}\Bigg{)} \tag{3}\]  
\[\alpha_{s}=\bigg{(}1-\frac{s}{2b_{x}}\bigg{)}\bigg{(}1-\frac{s}{2b_{y}}\bigg{)} \tag{4}\]  
\[\omega_{w}=\frac{A_{sw,x}b_{x}+A_{sw,y}b_{y}}{b_{x}b_{y}s}\cdot\frac{f_{y}}{f _{c}} \tag{5}\]  
where \(b_{i}\) is the distance between consecutive engaged bars, \(b_{x}\) and \(b_{y}\) are the sides of the confined concrete core in the two directions, \(n_{ct,x}\) and \(n_{ct,y}\) are the number of cross-ties in \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{3}{c}{**Damage Limitation Limit State (DLS)**} \\ \hline Moment capacity & \(M_{DLS}=\frac{\frac{\varepsilon_{s}^{2}}{\widetilde{\varepsilon}_{w}^{2}}}{[0.45 8n_{s}(0.8-n_{s})+2.4\ (\omega_{s}+0.015)]bd^{2}f_{c}}\) \\ \hline Curvature capacity & \(\phi_{DLS}=\frac{\varepsilon_{s}}{2d}\big{(}3+\sqrt{n_{s}}\big{)}\) \\ \hline \multicolumn{3}{c}{**Life Safety Limit State (LLS)**} \\ \hline Moment capacity & \(M_{LLS}=\Big{[}\omega_{s}+\frac{1}{2}n_{s}\big{(}1-\frac{n_{s}}{\widetilde{ \varepsilon}_{y}}\big{)}\Big{]}bd^{2}f_{c}\) \\ \hline Curvature capacity & \(\phi_{LLS}=\frac{\varepsilon_{m}}{d}\frac{0.8}{n_{s}}\eta_{f}\) \\ \hline \multicolumn{3}{c}{**Collapse Prevention Limit State (CLS)**} \\ \hline Moment capacity & \(M_{CLS}=\Big{[}\omega_{s}+\frac{1}{2}n_{s}\big{(}1-\frac{n_{s}}{\widetilde{ \varepsilon}_{y}}\big{)}\Big{]}bd^{2}_{c}f_{c}\) \\ \hline Curvature capacity & \(\phi_{CLS}=\phi_{LLS}\frac{d}{d_{c}}\ \Big{(}1+\frac{1}{4}\frac{\widetilde{ \varepsilon}_{y}}{\widetilde{\varepsilon}_{w}}\Big{)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Equations for moment and curvature capacity at three LSs for the case of rectangular sections with symmetric rebars. Figure 2: Moment-curvature relationship of a column section with the corresponding limit state criteria for Damage limitation Limit State (DLS, tension steel yielding), Life safety Limit State (LLS, concrete cover crushing), and Collapse prevention Limit State (CLS, concrete core crushing).