## 2 Backbone Curve for RC Columns in ASCE 41-17  
In ASCE 41-17, an idealized nonlinear load-deformation relation for a structural component is provided to represent a generalized backbone curve, as shown in Figure 1. In Figure 1, \(V\) and \(V_{y}\) are the column shear force and the column yield shear strength, respectively, and \(\theta\) is the drift ratio; \(a\) and \(b\) are the parameters representing plastic drift ratios, for cases when a strength degradation begins and the shear force drops to the point \"D\", respectively.  
The generalized load-deformation relation consists of a linear portion and a nonlinear portion. The linear portion corresponds to the segment between points A and B, which can be determined using the effective stiffness of a structural component. In ASCE 41-17, equations are given to calculate  
Figure 1: Generalized load–deformation relation.  
the effective stiffness for various structural components. For RC columns, the proposed equations to calculate the effective stiffness are summarized in Table 1, where \(E_{c}\) is the modulus of elasticity of concrete, \(I_{g}\) is the moment of inertia of the concrete gross section, \(A_{w}\) is the summation of the net horizontal cross-sectional area for concrete in the direction of loading, and \(A_{g}\) is the gross sectional area of the column.  
The nonlinear portion is governed by two deformation parameters, that is, modeling parameters \(a\) and \(b\). For rectangular RC columns, parameter \(a\) can be calculated using Equation (1):  
\[a_{ASCE}=0.042-0.043\frac{P}{A_{g}f_{c}}+0.63\rho_{t}-0.023\frac{V_{y}}{V_{0}} \geq 0, \tag{1}\]  
where \(P\) is the axial compressive load applied on the column section, \(f_{c}\) is the concrete compressive strength, \(\rho_{t}\) is the ratio of the area of distributed transverse reinforcement to the concrete gross area perpendicular to that reinforcement, and \(V_{y}\) is the yield shear strength of the column. The column shear strength \(V_{0}\) is calculated from Equation (2) provided in ASCE 41-17 for RC columns:  
\[V_{0}=\left[\frac{A_{v}f_{yt}d}{s}+\left(\frac{6\sqrt{f_{c}}}{a_{v}/h}\sqrt{1 +\frac{P}{6A_{g}\sqrt{f_{c}}}}\right)0.8A_{g}\right], \tag{2}\]  
where \(a_{V}/h\) is the height-to-depth ratio of the column; \(k_{nl}\) is 1.0 for a displacement ductility demand (\(\mu\)) less than or equal to 2 and 0.7 for \(\mu\geq 6\), where \(\mu\) is calculated as \(\theta/\theta_{y}\) and \(\theta_{y}\) is the yield drift ratio as shown in Figure 2. For \(\mu\) between 2 and 6, \(k_{nl}\) is estimated using interpolation.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Flexural Rigidity** & **Shear Rigidity** & **Axial Rigidity** \\ \hline Columns with compression caused by design gravity loads & \(0.7E_{c}I_{g}\) & \(0.4E_{c}E_{aw}\) & \(E_{c}E_{g}A_{g}\) \\ \hline Columns with compression caused by design gravity loads or with tension & \(0.3E_{c}I_{g}\) & \(0.4E_{c}E_{aw}\) & \(0.4E_{c}E_{g}A_{g}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Effective stiffness equations proposed by ASCE 41-17.  
Figure 2: Measured modeling parameters from a measured first-cycle envelope.  
The load-deformation response representing a reduced resistance is predicted using modeling parameter \(b\), which is calculated using Equation (3):  
\[b_{ASCE}=\frac{0.5}{5+\frac{P}{0.8A_{g}f_{c}}\frac{1}{P_{t}}\frac{f_{c}}{f_{yt}}} -0.01\geq a_{ASCE}, \tag{3}\]  
where \(f_{yt}\) is the yield strength of the transverse reinforcement and modeling parameter \(b\) must be greater than or equal to modeling parameter \(a\).