# Punching Shear Stress in Post-Tensioned Transfer Plate of Multi-Story Buildings
## 3 Theories and Current Design Approach in Punching Shear
### Factored Shear Stress in the Eccentric Shear Stress Model  
The eccentric shear stress model is presented in ACI 318-19 as a means for calculating punching shear acting on the slab-column connection. The model is based on unbalanced moment and direct shear force at the slab-column connection for which punching shear was determined using analysis software [20; 21]. In the eccentric shear stress model, \(\gamma_{v}M_{u}\), fraction of the unbalanced moment \(M_{u}\), is supported by the slab in shear, and the shear stress distribution is assumed to be linear. The fraction ratio of unbalanced moment, \(\gamma_{v}\), supported by slab in shear is given as follows:  
\[\begin{array}{l}\gamma_{vx}=1-\frac{1}{1+(2/3)\sqrt{l_{y}/l_{x}}}\\ \gamma_{vy}=1-\frac{1}{1+(2/3)\sqrt{l_{x}/l_{y}}}\end{array} \tag{2}\]  
where \(l_{x}\) and \(l_{y}\) are the lengths of each side in the direction of \(x\) and \(y\) axes of the critical section, respectively (Figure 4).  
The punching shear stress (\(v_{u}\)) due to unbalanced moment (\(M_{u}\)) and direct shear force (\(V_{u}\)) acting on the connection is calculated as follows:  
\[v_{u}=\frac{V_{u}}{A_{c}}+\frac{\gamma_{vx}M_{ux}\overline{y}}{J_{cx}}+\frac{ \gamma_{vy}M_{uy}\overline{x}}{J_{cy}} \tag{3}\]  
\[A_{c}=b_{0}d \tag{4}\]  
\[J_{cx}=\frac{d(c_{y}+d)^{3}}{6}+\frac{d^{3}(c_{y}+d)}{6}+\frac{d(c_{y}+d)^{2}( c_{x}+d)}{2}\ (\text{Interior column}) \tag{5}\]  
where \(A_{c}\) is the area of concrete section resisting transferred shear stress; \(\overline{x}\) and \(\overline{y}\) are the distances between the center of column and center of the critical section in each direction; \(J_{cx}\) and \(J_{cy}\) are the polar moments of inertia with respect to each principal direction; and \(c_{x}\) and \(c_{y}\) are the lengths of each side of the column section in each principle direction.  
For the exterior and corner columns, the effects of eccentric distances between the center of the column and critical section need to be also considered. The unbalanced moment in exterior and corner columns is calculated as follows:  
\[M_{ue}=M_{u}-V_{u}\cdot e \tag{6}\]  
where \(M_{ue}\) is the unbalanced moment considering eccentric distance; \(M_{u}\) is the unbalanced moment; \(V_{u}\) is the direct shear force; and \(e\) is the eccentric distance between the center of the column and center of the critical section.  
Given Equations (4) and (5) above, \(A_{c}\) and \(J_{c}\) are proportional to the effective thickness of the slab, \(d\). \(A_{c}\) is linearly related to \(d\). Whereas \(J_{c}\) in terms of \(d\) increases exponentially. Intuitively, punching shear stress for thick members, such as a transfer plate, is controlled by direct shear rather than unbalanced moment. Direct shear force and unbalanced moment can be calculated as shown in Figure 5.  
Figure 4: Definition of critical section. (**a**) Internal column, (**b**) Exterior column and (**c**) Corner column.  
Calculating direct shear force and the unbalanced moment is difficult in many cases. As a result, they are conservatively approximated by the axial force and bending moment acting on the top of the column below the plate [22]. However, if a shear wall exists within the critical section for the transfer plate, the magnitude of the direct shear force acting on the critical section will be significantly lower than the axial force at the top of the transfer column. Thus, the methodology to calculate punching shear stress of transfer plate using axial force and moment at the top of the transfer column could lead to a non-economic design exceeding that which may be construed as conservative.