Punching shear proof

According to EN 1992-1-1:2004/AC:2010 - SECTIONS 6.4 and 9.4.3

General

When a flat slab is exposed to a concentrated load larger than the capacity, the effect on the slab is referred to as punching shear. In these slabs, the shear force per unit length can become high close to the area of loading. If the capacity for shear punching in the slab is exceeded, a punching shear failure may occur within the discontinuity regions of the flat slab. This type of failure is a brittle failure mechanism, and may cause a global failure of the structure. Punching shear failure is a typical failure for slab-column connections.

Punching shear failure is a local failure mechanism, where diagonal tensile cracks form a failure surface around the loaded area of the slab. The failure occurs along a truncated cone shape in the structure.

A control perimeter at some distance from the loaded area defines the section for punching shear calculations. This control perimeter varies in the different methods for calculating punching shear.

Normal design practice is to always control for punching shear in cases where the structure functions as a flat slab.

 

Punching Shear Design in Eurocode 2

"Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings" (from now on referred to as EC2) defines rules for design for shear punching. The method in EC2 is based on experiments, and most of the formulas are therefore empirical. The next sections will briefly present the calculation method used in EC2. The formulas given in this section are gotten from Sections 6.4 and 9.4.3 in EC2, and are mainly given for cases with uniformly distributed loading. This section will only cover the effect of punching shear on slabs.

 

A calculation model for control of the punching shear capacity in the ultimate limit state is shown in Figures 2 and 3, where A_cont is the basic control area, u₁ is the basic control perimeter, A_load is the loaded area and r_cont is the further control perimeter. The shear capacity is to be controlled at the edge of the column and at the basic control perimeter u₁.

 

Critical Control Perimeter

The critical control perimeter u₁ can normally be evaluated at the distance 2 ⋅ d from the loaded area, and should be constructed to minimize the length of the perimeter. Figure 4 shows examples of typical critical control perimeters around loaded areas.

The effective thickness of the slab, d_eff, is assumed to be constant and is normally given by the expression in equation:

d_eff = d_y + d_z / 2

In equation above, d_y and d_z are the effective thicknesses for the reinforcement in two orthogonal directions.
If the loaded area is close to an opening in the slab, and the distance between the edge of the loaded area and the edge of the opening does not exceed 6d, the critical control perimeter must be reduced. This reduction is done by assuming the part of the control perimeter contained between two tangents drawn from the centre of the loaded area to the outline of the opening to be ineffective, as shown in Figure 5.

For columns situated near a corner or an edge, this must be taken into account when defining the critical control perimeter. Figure 6 shows the critical control sections for different situations where a column is situated near a corner or an edge. The section including continuation to the edge of the slab must be smaller than the one defined by Figure 6.

Figure 7 shows the location of the control perimeter for a column with an enlarged circular column head. For l_H < 2 ⋅ h_H, as shown in Figure 7, the punching shear capacity must only be controlled at a critical control section outside the column head. The distance from the center of the cross-section to the critical section, rcont, can be defined as given in equation for a circular column:

r_cont = 2 ⋅ d + l_H + 0,5 ⋅ c

In equation, l_H is the distance from the column edge to the edge of the column head and c is the diameter of a circular column.

For a rectangular column with a rectangular column head, l_H < 2h_H and edges with lengths l₁ and l₁, the value of r_cont can be given as the smallest of the values given in equations:

r_cont = 2 ⋅ d + 0,56 ⋅ (l₁ ⋅ l₂)^0,5
r_cont = 2 ⋅ d + 0,69 ⋅ l₁

Where:

l₁ = c₁ + 2 ⋅ l_H1
l₂ = c₂ + 2 ⋅ l_H2
and l₁ ≤ l₂

For l_H > 2h_H, as shown in Figure 8, the punching shear capacity must be controlled at a critical section outside the column head as well as inside the column head.

For circular columns the distance from the center of the cross-section to the control section within the column head can be assumed as given in equation:

r_cont,int = 2 ⋅ (d + h_H) + 0,5 ⋅ c

The distance from the center of the cross section to the control section outside the column head can be assumed as in equation:

r_cont,ext = l_H + 2 ⋅ d + 0,5 ⋅ c

 

Shear Force from Concentrated Loading

The controls given in equations below must be performed for a slab with concentrated loading.

ν_Ed ≤ ν_Rd,max
ν_Ed ≤ ν_Rd,c

In equations above, ν_Ed is the maximum shear stress from the concentrated loading, ν_Rd,c is the design value of the punching shear stress resistance along the control section for a slab without punching shear reinforcement and ν_Rd,max is the design value of the maximum punching shear stress resistance along the cross-section. If the condition in the second equation above is not fulfilled, shear reinforcement has to be added to the section according to Section 2.4.4.

For an eccentric support reaction force, V_Ed, the value of ν_Ed can be defined as given in equation below for a rectangular column:

ν_Ed = β ⋅ V_Ed / (u_i ⋅ d)

Where, u_i is the length of the considered control section, and β is given by equation:

β = 1 + k ⋅ (M_Ed / V_Ed) ⋅ (u₁ / W₁)

In equation above, u₁ is the length of the critical control perimeter, k is a coefficient that depends on the sizes of the column edges, as shown in table below, and W₁ corresponds to a shear distribution, as shown in Figure 9.

W₁ is given by equation:

where dl is a length increment of the perimeter and e is the distance from dl to the axis where the moment M_Ed acts.

Values of k for rectangular loaded areas:

c1 / c2	≤0.5	1.0	2.0	≥3.0
k	0.45	0.60	0.70	0.80

The expression for W₁, and therefore β, for rectangular columns differs from the expression for β for circular columns. The expressions also depend on whether there is eccentricity in one or two directions, and whether the column is an inner column, en edge column or a corner column. These expressions will not be given in detail here.

Where the adjacent spans do not differ in length by more than 25%, and the lateral stability does not depend on the frame action between the slab and the column, simplified expressions for β, as presented in Figure 10, may be used.

 

Punching Shear Capacity in Slabs Without Punching Shear Reinforcement

The punching shear capacity in slabs without punching shear reinforcement, ν_Ed, is to be controlled at the critical section, as described in Section 2.4.2, and can be calculated by equation:

ν_Rd,c = C_Rd,c ⋅ k ⋅ (100 ⋅ ρ_l ⋅ f_ck)^1/3 + k₁ ⋅ σ_cp ≥ (ν_min + k₁ ⋅ σ_cp)

The value given for C_Rd,c varies for different countries. It is set to be as in equation:

C_Rd,c = 0,18 / γ_c

The values for k, l, σ_cp and ν_min are given in following equations:

k₁ = 1 + (200 / d)^0,5 ≤ 2 ⋅ d    (d is in mm)

ρ_l = (ρ_lx ⋅ ρ_ly)^0,5 ≤ 0,02

σ_cp = (σ_cx ⋅ σ_cy) / 2 = (N_Ed,x / A_cx + N_Ed,y / A_cy) / 2

ν_min = 0,035 ⋅ k^3/2 ⋅ f_ck^1/2

ρ_lx and ρ_lz are the reinforcement ratios in x and y directions. σ_cx and σ_cy are the normal stresses in the concrete in x and y directions, resulting from the normal forces over the concrete areas A_cx and A_cy.
f_ck is the characteristic compressive strength of the concrete.

 

Punching Shear Capacity in Slabs With Punching Shear Reinforcement

In cases where punching shear reinforcement is shown to be necessary, the design value of the punching shear stress resistance for a slab with punching shear reinforcement, ν_Rd,cs, can be calculated by the expression given in the equation:

ν_Rd,cs = 0,75 ⋅ ν_Rd,c + 1,5 ⋅ (d / s_sr) ⋅ A_sw ⋅ f_ywd,ef ⋅ 1 / (u₁ ⋅ d) ⋅ sinα

In the equation above, A_sw is the area of one perimeter of shear reinforcement around the column, s_r is the radial spacing of perimeters of shear reinforcement, d is the mean value of the effective depth in orthogonal directions, α is the angle between the plane of the slab and the shear reinforcement and f_ywd,ef is the effective design strength of the punching shear reinforcement, given by the equation:

f_ywd,ef = 250 + 0,25 ⋅ d ≤ f_ywd

At the edge of a column, the punching shear capacity must be smaller than ν_Rd,max, as given in the equation:

ν_Ed = β ⋅ V_Ed / (u₀ ⋅ d) ≤ ν_Rd,max

In the equation above, β is as given in Section 2.4.2 and u₀ is the control perimeter at the edge of the column. The value of ν_Rd,max varies for different countries:

ν_Rd,max = min{ 0,4 ⋅ ν ⋅ f_cd   ;   1,6 ⋅ ν_Rd,c ⋅ u₁ / (β ⋅ u₀) }

The control section where shear reinforcement is not necessary is given by the equation:

u_out,ef = β ⋅ V_Ed / (ν_Rd,c ⋅ d) ≤ 1,6 ⋅ ν_Rd,c ⋅ u₁ / (β ⋅ u₀)

 

Punching Shear Reinforcement

If punching shear reinforcement is shown to be necessary, the punching shear reinforcement should be placed between the loaded area and the length k ⋅ d within the perimeter where shear reinforcement is not necessary, u_out,ef.
The value of k varies for different countries, it's recommended value is equal to 1.0.

Link legs should be provided in at least two perimeters, and the spacing of the link leg perimeters should not exceed s_r,max = 0,75 ⋅ d. The spacing of link legs around a perimeter, s_t,max, should not exceed 1,5 ⋅ d within the first perimeter and should not exceed 2 ⋅ d outside the first perimeter. The criteria for spacing of link legs are shown in figure 11.

Where punching shear reinforcement is provided, the area of a link leg is given by the equation:

A_sw,min ⋅ (1,5 ⋅ sinα + cosα) / (s_r ⋅ s_t) ≥ 0,008 ⋅ f_ck^0,5 / f_yk

Here, α is the angle between the shear reinforcement and the main reinforcement, s_r is the spacing of shear links in radial direction and s_t is the spacing of links in tangential direction.