## 2 Designing Flat Slabs against Punching Shear Failures with Shear Reinforcement
### General Considerations
It should be noted here that limiting the shear stresses directly at the perimeter of the load application area \(u_{0}\) to a value equal to the web-crushing limit of beams \(v_{\rm Rd,max}\) is, according to DIN EN 1992-1-1+NA(D) [4; 5; 6; 7], obsolete since a previous comparison with test results clearly showed a dependency on the concrete compressive strength and the specific column perimeter \(u_{0}/d\)[2; 26; 27]. In addition, limiting the shear stresses at the face of the column to \(v_{\rm Rd,max}\) means that it is not possible to consider a potential increase in the punching shear resistance due to improved punching shear elements, such as double-headed studs or lattice girders [28; 29; 30]. As in DIN EN 1992-1-1 [4; 5], the punching shear capacity without shear reinforcement is calculated according to Equation (3). For interior column-slab connections and \(u_{0}/d\) ratios \(<4.0\), a reduced empirical pre-factor \(C_{\rm Rd,c}\) is given by  
\[C_{\rm Rd,c}=\frac{0.18}{\gamma_{\rm C}}\cdot\left(0.1\cdot\frac{u_{0}}{d}+0. 6\right)\quad\text{ for }\frac{u_{0}}{d}<4.0 \tag{6}\]  
To avoid compression reinforcement, DIN EN 1992-1-1+NA(D) [4; 5; 6; 7] limits the flexural reinforcement ratio to \(\rho_{1}\leq\min\left(0.02;\ 0.5\cdot f_{\rm cd}/f_{\rm yd}\right)\). For effective depths \(>600\) mm, the minimum punching shear capacity is reduced in comparison to DIN EN 1992-1-1 [4; 5] and is equal to the minimum shear capacity of the German bridge-building provisions \"DIN Fachbericht 102\" [31]:  
\[v_{\rm min}=(0.0525/\gamma_{\rm C})\cdot k^{3/2}\cdot f_{\rm ck}^{1/2}\quad \text{ for }d\leq 600\text{ mm} \tag{7}\]  
\[v_{\rm min}=(0.0375/\gamma_{\rm C})\cdot k^{3/2}\cdot f_{\rm ck}^{1/2}\quad \text{ for }d>800\text{ mm} \tag{8}\]  
For effective depths between 600 mm and 800 mm, the pre-factor \(C_{\rm Rd,c}\) may be interpolated linearly. Limiting the shear stresses to \(\
u_{\text{Rd,max}}\) (i.e., web-crushing limit) at the column perimeter was not adopted in Germany. The maximum punching shear capacity was defined as 1.4 times the punching shear capacity without shear reinforcement [2; 3; 26; 27]:  
\[v_{\text{Ed}}\leq\
u_{\text{Rd,max}}=1.4\;\
u_{\text{Rd,c}},\text{ for stirrups as punching shear reinforcement } \tag{9}\]  
Furthermore, DIN EN 1992-1-1+NA(D) [4; 5; 6; 7] demands that the amount of punching shear reinforcement is increased by a factor of \(\kappa_{\text{sw,1}}=2.5\) for the first row (at a distance of \(0.3d\) to \(0.5d\)) and by a factor of \(\kappa_{\text{sw,2}}=1.4\) for the second row. From the third row onwards, a factor of \(\kappa_{\text{sw}}=1.0\) is assumed. The distance between adjacent rows needs to be adjusted to a maximum of \(0.75d\). In principle, as many rows of punching shear reinforcement are to be arranged as it is required until the verification of the punching shear capacity of a linearly supported slab (one-way shear) is fulfilled along an outer control perimeter at the distance of \(1.5d\) from the outermost punching shear reinforcement row. The required length of the outer circular section \(u_{\text{out}}\) is determined using  
\[v_{\text{Ed}}=\frac{\beta\cdot V_{\text{Ed}}}{u_{1}\cdot d}\leq v_{\text{Rd,c}} \tag{2}\]0  
with the \(v_{\text{Rd,c}}\) corresponding to the punching shear capacity without shear reinforcement (Equation (3)). Note that for the verification of the punching shear capacity outside the punching shear reinforced area according to NA(D) [6; 7], the term \(v_{\text{Rd,c}}\) in Equation (3) shall have an empirical pre-factor \(C_{\text{Rd,c}}=0.15/\gamma_{\text{C}}\) (with \(\gamma_{\text{C}}=1.5\)) and a factor \(k_{1}\) equal to \(0.10\).