2.6.1. Slabs without shear reinforcement
The Model Code (MC 2010) is like SIA262 2003 based on the CSCT. The punching strength depends on the slab rotation, which results from the applied load and the stiffness of the slab defined by the flexural strength. Since the punching strength depends on the applied load, the equation has to be solved so that VRc = V, as it was described previously for SIA262 2003 in
Section 2.5. Another specialty of the MC 2010 is that different levels of approximation exist. Level I approximation enables a fast pre-dimensioning, Level II approximation is recommended for the typical design of new structure, Level III approximation is recommended either for special design cases or for the analysis of existing structures, and Level IV approximation is recommended for special design cases or for a more detailed assessment of existing structures (Tassinari 2011). In this research, Level II and Level III approximation is used for the prediction of the tested specimen presented herein and Level II approximation is used for the comparison to tests from literature. For slabs without shear reinforcement, the punching strength is defined as:
, ∙ ∙ ∙ (2.29)
where b0 is a control perimeter set at d/2 of the border of the support region with circular corners, dv is the shear-resisting effective depth of the slab, fc is the compressive strength of
concrete in MPa, and kψ is defined as:
1
1.5 0.9 ∙ ∙ ∙ 0.6 (2.30) where d is the effective depth in mm, ψ is the rotation of the slab, and kdg is a factor accounting
for the influence of aggregate size defined as:
32
16 0.75 (2.31)
where dg is the maximum aggregate size in mm.
For a Level II calculation, the rotation of the slab can be estimated by:
1.5 ∙ ∙ ∙ . (2.32)
where rs distance to the point where the radial bending moment is zero, d is the effective depth, Es is the Young’s modulus of the flexural reinforcement, mR is the flexural strength, and ms the
For inner columns, ms can be assumed as:
8 (2.33)
where V is the applied shear force.
In the case of a level III, the factor 1.5 can be decreased to 1.2 due to the more accurate prediction of the average bending moment ms. Thus, the rotation for a Level III calculation can
be estimated by:
1.2 ∙ ∙ ∙ . (2.34)
where rs distance to the point where the radial bending moment is zero, d is the effective depth, Es is the Young’s modulus of the flexural reinforcement, mR is the flexural strength, and ms is
the is the average moment per unit length in the support strip determined by a linear-elastic finite element analysis.
Figure 2.9: Control perimeter according to MC 2010
2.6.2. Slabs with shear reinforcement
While most punching provisions of MC 2010 are similar to SIA262 2003, the provision regarding failure within the shear-reinforced area is different. While SIA262 2003 completely neglects the contribution of concrete, MC 2010 takes the summation of the shear forces transferred by the concrete and the shear reinforcement. Both values, the shear contribution of the concrete and that of the shear reinforcement depend on the rotation of the slab accounting for the activation of the shear reinforcement and the reduction in the concrete contribution with increasing rotation. More information about the mechanical model behind this approach can be found in the next subchapter explaining the CSCT in detail.
The contribution of the shear reinforcement can be calculated as the sum of the multiplication of the cross-sectional area of the shear reinforcement within an area between a distance of 0.35dv
and dv from the column face (Figure 2.10) and the stresses in the shear reinforcement.
Therefore, the contribution of the shear reinforcement is defined as:
∙ (2.35)
where ΣAsw is the cross-sectional area of all the shear reinforcement intersected by the potential
failure surface (conical surface with angle 45°) within a distance of 0.35dv to dv from the column
face and σsw are the stresses in the shear reinforcement defined by the rotation of the slab and the
bond conditions of the shear reinforcement. The stresses in the shear reinforcement are given by:
6 ∙ 1 2
∙
øw (2.36)
where Esw is the Young’s modulus of the shear reinforcement, ψ is the rotation of the slab, fct is
the tensile strength of the concrete defining the maximum bond stress, fyw is the yield strength of
the shear reinforcement, d is the effective depth, and øw is the diameter of the vertical branch of
the shear reinforcement.
(a) (b)
Figure 2.10: Shear reinforcement considered by MC 2010
The concrete contribution can be calculated according to the provisions for punching of slabs without shear reinforcement (Equation 2.29). Finally, the punching strength for failure within the shear-reinforced area can be obtained by the summation of the contributions of the concrete and the shear reinforcement.
(2.37)
Similar to other codes, MC 2010 uses the same formulation as for punching without shear reinforcement for the calculation of the punching strength for failure outside the shear- reinforced area. However, for failure outside the shear reinforced area, the control perimeter is
set at the distance of 0.5dv,out from the outermost shear reinforcement perimeter. Thus, the
punching strength is defined as:
∙ ∙ ∙ , (2.38)
where kψ is defined according to Equation 2.30, fc is the concrete compressive strength, bout is a
control perimeter set at a distance of 0.5dv,out from the outermost perimeter of shear
reinforcement, and dv,out is the distance between the flexural reinforcement and the bottom end
of the vertical branch of the shear reinforcement.
(a) (b)
Figure 2.11: Control perimeter for punching shear verification outside the shear- reinforced area according to MC 2010
With respect to the failure of the concrete strut, MC 2010 uses a similar approach as other codes by increasing the punching strength of slabs without punching shear reinforcement by a factor. It has to be noted that the predicted punching strength VR is a function of the applied shear force V, which is included in the calculation of the rotation. Therefore, the maximum punching
strength is not the proportionally increased punching strength estimated by the formulation for slabs without shear reinforcement. The maximum punching strength is defined as:
∙ ∙ ∙ ∙ ∙ ∙ (2.39)
where factor ksys is proposed as 2.4 for slabs with stirrups and 2.8 for slabs with double headed
studs. All other parameters correspond to the formulation of punching of slabs without shear reinforcement.