# Simplified Pushover Analysis for Rapid Assessment of Shear-Type Frames
## 2 Displacement Capacity of a 2D Frame
### Element Capacity for the Three LSs  
In those cases where a strong-beam/weak-column situation occurs, the global capacity depends only on the columns. A column is considered as a series system of flexural plastic hinges, column shear, and joints shear.  
The force-displacement capacity of a column exhibiting a flexural behavior is determined by considering, both, the rotations of the plastic hinges at the ends, which follow the above-described sectional moment-curvature diagram, and the elastic deformation of the column. The flexural capacity of the column is compared to its shear capacity and to the joint capacity to obtain the actual capacity of the column element.  
The analytical expressions for the force capacity \(v_{xLS}\) and the corresponding displacement \(u_{xLS}\) at the three limit states \(x=D,\ L,\ C\) are obtained from equilibrium and compatibility conditions as:  
\[v_{xLS}=min\Big{\{}\ \frac{2M_{xLS}-Nu_{xLS}}{H}\quad V_{R}\quad V_{jR}-\Big{(} A_{jt}+\kappa A_{jb}\Big{)}f_{y}\ \Big{\}} \tag{8}\]  
\[u_{xLS}=Q\bigg{[}\frac{2M_{xLS}}{K_{E}H}+\phi_{xLS}L_{p}\big{(}H-L_{p}\big{)} \bigg{]} \tag{9}\]  
where \(M_{xLS}\) is the flexural capacity at \(xLS\) in Table 1; \(N\) is the axial load; \(H\) is the column height; \(V_{R}\) is the column shear capacity; \(V_{jR}\) is the joint shear capacity; \(A_{jt}\) and \(A_{jb}\) are top and bottom rebars area passing through the joint or anchored in it, respectively, with \(\kappa=1\) for internal joints and \(\kappa=0\) for external joints; \(f_{y}\) is the steel yield tensile strength; and  
Figure 3: Cross-section with geometric parameters: \(d\) is the section depth, \(b\) is the section width, \(b_{x}\) is the concrete core width, \(b_{y}=d_{c}\) is the section depth at collapse, \(A_{s}\) is the tension steel area (equal to that in compression), \(A_{sw}\) is the stirrup area, and \(s\) is the stirrup spacing.  
\(\phi_{xLS}\) is the curvature capacity at \(xLS\) in Table 1. In the above equations, the p-delta factor \(Q\), the elastic stiffness \(K_{E}\), and the plastic hinge length \(L_{p}\) are given by:  
\[Q=\left(1+\frac{N}{K_{E}H}\right)^{-1} \tag{10}\]  
\[K_{E}=\frac{12E_{c}I}{\left(H-2L_{p}\right)^{3}} \tag{11}\]  
\[L_{p}=0.1H+0.17\frac{b+d}{2} \tag{12}\]  
where \(E_{c}\) is the concrete elastic modulus and \(I\) is the gross section moment of inertia.  
The column shear capacity \(V_{R}\) is computed as:  
\[V_{R}=0.9 dbf_{c}\omega_{w}\sqrt{\frac{\alpha_{c}v}{\omega_{w}}-1} \tag{13}\]  
where the term under square root should be within the interval \([1,\ \ 2.5]\) and \(\omega_{w}\). is the transverse reinforcement mechanical ratio:  
\[\omega_{w}=\frac{A_{sw}}{b\,s}\frac{f_{y}}{f_{c}} \tag{14}\]  
with \(A_{sw}\) the transverse reinforcement area, while \(\
u=0.5\) is the concrete compressive strength reduction factor and \(\alpha_{c}\) depends on the normalized axial load \(n_{S}\), as follows:  
\[\alpha_{c}=\begin{cases}\min(1+n_{S},1.25)&0.00\leq n_{S}<0.50\\ 2.5(1-n_{S})&0.50\leq n_{S}<1.00\end{cases} \tag{15}\]  
The joint shear capacity \(V_{JR}\) is computed as:  
\[V_{JR}=\eta d_{c}b_{j}f_{c}\sqrt{1-\frac{n_{S}}{\eta}} \tag{16}\]  
where \(d_{c}=b_{y}\) in Figure 3, \(b_{j}\) is the joint effective width, and \(\eta=\alpha_{j}(1-f_{c}/250)\) with \(f_{c}\) in MPa, with \(\alpha_{j}=0.6\) for internal joints and \(\alpha_{j}=0.48\) for external joints.