## 2 Designing Flat Slabs against Punching Shear Failures with Shear Reinforcement  
### General Considerations  
As mentioned above, the main design provisions for punching shear with and without shear reinforcement are given in the European structural code DIN EN 1992-1-1 [4; 5], i.e., the \"base\" Eurocode 2. As explained, for example, in [19; 20], such provisions are grounded on the design concept given in CEB/FIP Model Code 1990 [21], which is mostly based on empirical models (i.e., those that were fitted to experimental evidence). Yet, in Germany, the punching shear design provisions partly deviate from the original provisions in DIN EN 1992-1-1 [4; 5], mainly to guarantee that the safety levels recommended in DIN EN 1990 [22] and DIN 1055-100 [23] can be reached (e.g., [2; 3]). As a result, complementary design provisions were added to the National Annex NA(D) [6; 7]. For the sake of this investigation, the design provisions for punching shear with and without shear reinforcement in the combined code DIN EN 1992-1-1+NA(D) [4; 5; 6; 7] were adopted. Design Provisions for Punching Shear with Shear Reinforcement according to DIN EN 1992-1-1 (Eurocode 2)  
For RC flat slabs without shear reinforcement, the punching shear resistance is evaluated along different control perimeters. For slabs without shear reinforcement, DIN EN 1992-1-1 [4; 5] also demands that the shear stress \(v_{\text{Ed}}\) at the periphery of the loaded area \(u_{0}\) does not exceed the web-crushing limit (i.e., crushing of the compressive strut) for beams \(v_{\text{Rd,max}}\):  
\[v_{\text{Ed}}=\frac{\beta\cdot V_{\text{Ed}}}{u_{0}\cdot d}\leq v_{\text{Rd, max}}\text{ with }v_{\text{Rd,max}}=0.4\cdot v\cdot f_{\text{cd}} \tag{1}\]  
with \(d\) being the distance from the extreme compression fibre to the centroid of the longitudinal tensile reinforcement (also known as the effective depth) and \(v\) being a factor accounting for the strength reduction of concrete compression struts in cracked concrete due to lateral tension stresses and given by \(v=0.6\cdot(1-f_{\text{ck}}/250)\), with \(f_{\text{ck}}\) referring to the characteristic compressive strength of concrete (for cylinders). The term \(\beta\) refers to a load increase factor to account for load eccentricities (i.e., the non-uniform shear distribution along the column perimeter). The second control perimeter \(u_{1}\) applies at a distance of 2.0\(d\) from the periphery of the loaded area (Figure 2). The shear stress \(v_{\text{Ed}}\) along this perimeter should not exceed the punching shear resistance without shear reinforcement \(v_{\text{Rd,c}}\):  
\[v_{\text{Ed}}=\frac{\beta\cdot V_{\text{Ed}}}{u_{1}\cdot d}\leq v_{\text{Rd,c}} \tag{2}\]  
Note that DIN EN 1992-1-1 [4; 5] provides three methods to allow for the effect of combined shear and an unbalanced moment [24]: (1) the assumption of a fully plastic distribution of shear stresses along the critical perimeter, (2) a reduction in the basic control perimeter, or (3) the use of constant shear force multipliers. For non-sway systems, and where the adjacent spans do not differ in length by more than 25%, the load increase factors may be taken as \(\beta\) equal to 1.5, 1.4 and 1.15 for the corner, edge and interior columns, respectively. Two other methods are described in [24]. The punching shear resistance without punching shear reinforcement is determined by using  
\[v_{\text{Rd,c}}=C_{\text{Rd,c}}\cdot k\cdot\left(100\cdot\rho_{1}\cdot f_{\text {ck}}\right)^{1/3}+k_{1}\cdot\sigma_{\text{cp}}\geq v_{\text{min}}+k_{1}\cdot \sigma_{\text{cp}} \tag{3}\]  
with \(C_{\text{Rd,c}}\) being the empirical factor for flat slabs (=0.18/\(\gamma_{\text{C}}\) recommended in DIN EN 1992-1-1 [4; 5]), with \(\gamma_{\text{C}}\) being the partial factor for concrete (=1.5 recommended in DIN EN 1992-1-1 [4; 5]); \(\rho_{1}\) referring to the mean flexural reinforcement ratio considering a slab width equal to the column width plus 3\(d\) on each side and given by \(\left(\rho_{1\text{x}}\rho_{\text{ly}}\right)^{1/2}\leq 0.02\).