# Punching Shear Stress in Post-Tensioned Transfer Plate of Multi-Story Buildings
## 4 Analytical Model
### Punching Shear Stress on the Post-Tensioned Transfer Plate  
The two-way slab is designed to have sufficient resistance to punching shear stress at the critical section of the column. The critical section is defined at the effective depth of the slab, \(d\), in accordance with ACI 318-19, Section 22.6.4. Here, the effective thickness of the prestressed concrete member considers both the prestressing tendon as well as the rebar and is calculated as the average distance from the extreme compressive fiber to the center of the tendon and from the center of the rebar. Moreover, for the shear design of prestressed concrete, the effective thickness does not exceed 0.8\(h\). Where, \(h\) is the total height of the member. Hence, the effective thickness of the transfer plate is determined to be 0.8\(h\), since the clear span of the transfer plate is relatively short compared to its thickness, and the tendon is located at the center of the plate due to its curvature limit.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Effective ground acceleration (\(S\)) & 0.22 g \\ Site classification & \(S_{D}\) \\ Design effective ground acceleration & 0.66 \\ at 1 s period (\(S_{D1}\)) & 0.6235 \\ Design effective ground acceleration & 4 \\ at short period (\(S_{DS}\)) & 1.2 \\ Response modification factor (\(R\)) & 1.2 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Seismic load parameters summary.  
The transverse shear stress of the thick-shell element was calculated based on shear strain, which is calculated directly from shear deformation and the following equation [16].  
\[\begin{array}{l}\gamma_{13}=\frac{du_{1}}{dx_{3}}+\frac{du_{3}}{dx_{1}}\\ \gamma_{23}=\frac{du_{2}}{dx_{3}}+\frac{du_{3}}{dx_{2}}\end{array}\text{ and }\begin{array}{l}\tau_{13}=G_{13}\cdot\gamma_{13}\\ \tau_{23}=G_{23}\cdot\gamma_{23}\end{array} \tag{13}\]  
where \(\gamma_{13}\) and \(\gamma_{23}\) are transverse shear strains in each direction; \(u_{1}\), \(u_{2}\), and \(u_{3}\) are the displacements in each direction; \(x_{1}\), \(x_{2}\), and \(x_{3}\) are the coordinates of in the material's 1, 2 and 3 directions, and \(G_{13}\) and \(G_{23}\) are the shear moduli.  
For the complete model, transverse shear stress (\(\tau_{13}\), \(\tau_{23}\)) of the shell element at the point with constant spacing along the critical section was monitored as punching shear stress. \(\tau_{13}\) for the side of local axis 1 and \(\tau_{23}\) for the side of local axis 2 of the critical section were also monitored (Figure 8a).  
For the separated model, punching shear stress is calculated with Equation (3). As mentioned in Section 3.1, direct shear force (\(V_{u}\)) and unbalanced moment (\(M_{ux}\) and \(M_{uy}\)) are approximated by the axial force and bending moment acting on the top of the column below the transfer plate. Since the column size is assumed to be 1 m by 1 m, the fraction ratio of the unbalanced moment, \(\gamma_{v}\), is calculated as 0.4 for corner column and interior column for both \(x\) and \(y\) directions. For the exterior column, the fraction ratios of the unbalanced moment in \(x\) and \(y\) direction are calculated as 0.37 and 0.44, respectively.