PSI - Issue 84

Alessandro Lipari et al. / Procedia Structural Integrity 84 (2026) 1087–1094

1091

= √ 2 , + 2 ,

(5)

The effective depth d and the reinforcement ratio ρ to be used in Eq. (1) depend on the ratio of the design shear forces v Ed,y / v Ed,x , with the x -axis implicitly taken to be along the slab longitudinal axis. If 0.5 < v Ed,y / v Ed,x < 2: = 0.5( + ) (6) = 4 + 4 (7) However, Lipari (2025) recommends extending the range of v Ed,y / v Ed,x to 0.25–4, thus avoiding treating significantly skewed slabs as if they were straight. Indeed, beyond these thresholds, the calculations can be effectively carried out disregarding the effects of the slab skewness – for instance, when v Ed,y / v Ed,x < 0.25, the slab can be designed using d = d x and ρ = ρ x . Eq. (7) implicitly assumes that the reinforcement remains in an elastic state, which is generally conservative (Lipari, 2025). In the case of line or distributed loads and linear supports, the principal shear force is typically perpendicular to the support line. The case study of two bridge decks predominantly subjected to concentrated loads showed that α v is still approximately perpendicular to the support line (Lipari, 2025). EC2 allows shear forces to be averaged “over a width not larger than 2 d on both sides from the peak of the shear force, provided that the moment equilibrium after redistribution is fulfilled”. However, it is not stated whether the widths at each side of the peak should be the same or can differ as long as each width does not exceed 2 d . If other internal forces are required for the calculation of the shear resistance, they may be also averaged over the same width. In practice, the process of averaging shear forces in planar members is not straightforward, as principal shear forces vary in magnitude and direction across the slab. Instead, a control section is often used, which is taken parallel to the support line at a certain distance from the support edge (Pacoste et al., 2012, International Federation for Concrete Structures, 2023). The control section does not necessarily coincide with the plane on which the principal shear force action occurs, i.e., perpendicular to α v , but these are expected to be close in most cases. When significant concentrated loads are applied between d and 2 d from the face of the support ( d ≤ a q ≤ 2 d ), a control section located at a distance d from the face of the support should be verified. The contribution of such concentrated loads to the design shear force may be multiplied by 0.5 a q / d . No indications on the control section location are given for loads with a q > 2 d , but in such cases it may be assumed that the control section is at a distance d from either the face of the support or the load – whichever is more conservative. Finally, it is noted that, when using Eq. (2), the applied bending moment m Ed is that concurrent with the principal shear force v Ed and needs to be resolved along its direction α v , which can be done by equilibrium: = , 2 + , 2 + 2 , (8) 4.1. Application to skewed slabs Flexural reinforcement in slabs is typically placed according to one of the following arrangements (Fig. 3): • Arrgt. 1: orthogonal grid with longitudinal reinforcement parallel to free edges (like in specimen “O”). • Arrgt. 2: orthogonal grid with longitudinal reinforcement perpendicular to supported edges (not used in the experiments). • Arrgt. 3: non-orthogonal grid with longitudinal reinforcement parallel to free edges and transverse reinforcement parallel to supported edges (skewed reinforcement, like in specimens “N”). For Arrgts. 1 and 2, the verification can be carried out as described in the previous section. In the case of Arrgt. 2, it is convenient to rotate the local x - y axes so that the x -axis lies along the longitudinal reinforcement. Arrgt. 3 is not taken into account in EC2 and the following equations may be used (Lipari, 2025): ≈ 2 + 2 (9)