## 2 Designing Flat Slabs against Punching Shear Failures with Shear Reinforcement
### General Considerations
As referred above, the term \(f_{\text{ck}}\) refers to the characteristic compressive strength of concrete (for cylinders). The term \(k\) is given by \(1+\left(200/d\right)^{1/2}\leq 2.0\) and refers to a size effect factor for the effective depth \(d\); the parameter \(v_{\text{min}}=0.035\cdot k^{3/2}\cdot f_{\text{ck}}^{1/2}\) refers to the minimum punching shear capacity, which is especially critical for small flexural reinforcement amounts in combination with a high concrete compressive strength. The term \(k_{1}\) refers to the empirical factor that defines the amount of normal stress to be considered (=0.10 is recommended in DIN EN 1992-1-1 [4; 5]) and \(\sigma_{\text{cp}}\) refers to the normal concrete stress in the critical section (e.g., due to pre-stressing). The punching shear design of prestressed interior column-slab connections is discussed in detail in [25]. The punching shear resistance inside the shear-reinforced zone is calculated by using a strut-and-tie model with a compression strut inclination of 33\({}^{\circ}\) in combination with a constant concrete contribution:  
\[v_{\text{Rd,cs}}=0.75\cdot v_{\text{Rd,c}}+1.5\cdot\left(\frac{d}{s_{\text{ r}}}\right)\cdot A_{\text{sw}}\cdot f_{\text{ywd,ef}}\cdot\left(\frac{1}{u_{1} \cdot d}\right)\cdot\sin\alpha\leq\ k_{\text{max}}\cdot v_{\text{Rd,c}} \tag{4}\]  
with \(v_{\text{Rd,c}}\) being the punching shear resistance without shear reinforcement (calculated with Equation (3)), \(k_{\text{max}}\) being a factor that limits the punching shear capacity (=1.5 recommended for stirrups as punching shear reinforcements), \(A_{\text{sw}}\) being the amount of shear  
Figure 2: Typical critical control sections around load introduction areas according to DIN EN 1992-1-1 [4; 5]. reinforcement provided in each row and \(s_{\rm r}\) being the radial spacing between adjacent rows of shear reinforcement. The term \(f_{\rm ywd,ef}\) refers to the effective design strength of the punching shear reinforcement, which considers the worse anchorage of the punching shear reinforcement for small slab thicknesses and is defined as \(250+0.25d\ \leq\ f_{\rm ywd}\). The term \(\alpha\) refers to the angle between the punching shear reinforcement and the plane of the slab. The amount of punching shear reinforcement that is calculated must be provided in several rows. The precise number of rows required is determined by the shear strength without shear reinforcement \(v_{\rm Rd,c}\) (calculated with Equation (3)) checked along the outer perimeter, which is located at a distance of \(1.5d\) from the outermost row. At least two rows of shear reinforcement are needed. The first row must be placed at a distance of \(0.30d\leq s_{0}\leq 0.50d\) from the boundary of the loaded area. The radial spacing between additional rows of shear reinforcement \(s_{\rm r}\) should be limited to \(0.75d\). For inclined bars, the effective cross-section must be determined using \(A_{\rm sw}=A_{\rm s}\cdot\sin\alpha\). Note that the punching shear capacity with punching shear reinforcement is limited by the maximum capacity \(v_{\rm Rd,max}\) (Equation (1)), which must be checked along the column perimeter \(u_{0}\). Complementary Design Provisions for Punching Shear with Shear Reinforcement in the German National Annex NA(D)  
The design value of the applied shear stresses along the critical perimeter at a distance \(2.0d\) from the boundary of the loaded area can be calculated in accordance with DIN EN 1992-1-1+NA(D) [4; 5; 6; 7] using  
\[v_{\rm Ed}=\frac{\beta\cdot V_{Ed}}{u_{1}\cdot d} \tag{5}\]  
For interior column-slab connections, the load increase factor \(\beta\) is slightly reduced in comparison to DIN EN 1992-1-1 [4; 5] to \(1.1\).