# Punching calculation (Punching)

The following checks should be carried out:

**At the column perimeter**

- v
_{Ed}≤ v_{Rd,max}

where:

- v
_{Ed}= β V_{Ed}/ (u_{0}d)- v
_{Ed}is the shear stress - β takes account of eccentricity as below
- V
_{Ed}is the reaction force in the column - u
_{0}is the lenght of column perimeter

- v
- v
_{Rd,max}= 0,5 ν f_{cd}- v
_{Rd,max}is the maximum punching shear

- v

- v
_{Rd,max}= min (0,5 ν f_{cd},1,6 v_{Rd,c}u_{1}/ u_{0})- v
_{Rd,max}is the maximum punching shear - ν = 0,6 (1 - f
_{ck}/ 250) (f_{ck}in MPa) - v
_{Rd,c}is the punching shear capacity without shear reinforcement as below - u
_{1}is the lenght of the control perimeter - u
_{0}is the lenght of column perimeter

- v

**At the control perimeter**

Shear reinforcement is not necessary if:

- v
_{Ed}≤ v_{Rd,c}

where:

- v
_{Ed}= β V_{Ed}/ (u_{1}d)- v
_{Ed}is the shear stress - β takes account of eccentricity as below
- V
_{Ed}is the reaction force in the column - u
_{1}is the lenght of the control perimeter

- v
- v
_{Rd,c}is the punching shear capacity without shear reinforcement as below

# Punching shear resistance without shear reinforcement

The design punching shear resistance (MPa) may be calculated as:

- v
_{Rd,c}= C_{Rd,c}k (100 ρ_{1}f_{ck})^{1/3}+ k_{1}σ_{cp}≥ (v_{min }+ k_{1}σ_{cp})

where:

- C
_{Rd,c}= 0,18 / γ_{c} - v
_{min}= 0,035 k^{3/2}fck^{1/2} - k
_{1}= 0,1 - f
_{ck}is in MPa - k = 1 + (200 /d)
^{0,5}≤ 2,0 d in mm - ρ
_{1}= (ρ_{1y}ρ_{1z})^{0,5}≤ 0,02- ρ
_{1y}and ρ_{1z}is the tension steel in y- and z- directions respectively calculated as mean values taking account a slab width equal to the column width plus 3d each side.

- ρ
σ

_{cp}= (σ_{cy}+σ_{cz}) / 2*σ

_{cy}and σ_{cz }are the normal concrete stresses in the critical section in y- and z- directions ( MPa, positive if compression)σ

_{cy}*= N*_{Ed,y }/ A_{cy}σ

_{cz}*= N*_{Ed,z }/ A_{cz}N

_{Ed,y }and_{ }N_{Ed,z }are the longitudinal forces across the full bay for internal columns and the longitudinal force across the control section for edge columns. The load may be from a load or a prestressing action.A

_{c}is the area of concrete according to the definition of N_{Ed}.

When the reinforcement area differs in the two directions the average value is used. A column is regarded as a corner column if the distance between slab edge and column edge in both x- and y-direction is < slab thickness. A column is regarded as an edge column if the distance between slab edge and column edge in x-direction is < slab thickness and the corresponding distance in y-direction > slab thickness. In all other cases the column is regarded as an interior column.

If the capacity is insufficient the following message will be displayed in the result comment:

**Not allowed!**

To increase the capacity shear reinforcement could be provided or the column could be increased to simulate a mushroom or the slab thickness could be increased locally over the column.

# Control perimeters

The punching check is performed at both the column perimeter and the control perimeter as shown below.

## Column perimeter u_{0}

**Interior column**

u_{0} = length of column periphery (mm)

**Edge column**

Rectangular: u_{0} = c_{2} + 3 d ≤ c_{1} + c_{2}, (mm),

Circular: u_{0} = π * D

**Corner column**

Rectangular: u_{0} = 3 d ≤ c_{1} + c_{2}, (mm),

Circular: u_{0} = 3/4 π * D

## Critical perimeter u_{1}

**Interior column**

**Edge column**

The distance s above cannot be larger than the slab thickness for an edge column.

**Corner column**

According to EN 1992-1-1 6.4.2 a hole in the slab situated closer than 6 d should be considered. This can be done by reducing the critical perimeter.

# Eccentricity factor

The eccentricity factor is calculated as below for interior columns but for edge and corner columns the simplified method is used.

**Rectangular internal column**

- β = 1 + 1,8 [(e
_{y}/ b_{z})^{2}+ (e_{z}/ b_{y})^{2}]^{0,5} **e**and_{y}**e**are the eccentricities_{z}**M**/_{Ed}**V**along_{Ed}**y**and**z**axes respectively,**b**and_{y}**b**are the dimensions of the column perimeter or the control perimeter respectively_{z}

**Circular internal column**

- β = 1 + 0,6 π e / (D + 4 d)
**e**is the max eccentricity**M**/_{Ed}**V**along_{Ed}**y**and**z**axes respectively**D**is the column diameter

**Simplified method**

These values only apply when the lateral stability does not depend on frame action between slab and columns and the length between adjacent spans do not differ more than 25%.

# Punching shear resistance with shear reinforcement

If the capacity is inadequate shear reinforcement can be provided as bent-down bars or links.

Where shear reinforcement is required the capacity should be calculated as:

- v
_{Rd,cs}= 0,75 v_{Rd,c}+ 1,5 (d / s_{r}) A_{sw}f_{ywd,ef}(1 / (u_{1}d)) sin(α) - as: v
_{Rd,cs}= v_{Ed}

Upper limit of shear resistance = *k _{max} v_{Rd,c}*

- If
*V*>_{Ed }*k*⇒ Inadequate capacity!_{max}v_{Rd,c }

The required area will be:

- A
_{sw}= (v_{Ed}- 0,75 v_{Rd,c}) s_{r}u_{1}/ (1,5 f_{ywd,ef}sin(α))

where:

- A
_{sw}is the area of one perimeter of shear reinforcement around the column (mm^{2}).

*k _{max}* = 1.6

Upper limit of shear resistance = 2* v _{Rd,c }u_{1 }d*

If *V _{Ed }* > 2

*v*⇒ Inadequate capacity!

_{Rd,c }u_{1 }d*A _{sw} *= (

*v*

_{Ed}_{ }- 0,25

*v*)

_{Rd,c}*s*sin(α)

_{r }u_{1 }/ (f_{ywd,ef}*)*with

*f*< 300 MPa

_{ywd,ef}## Radial spacing of perimeters of shear reinforcement sr

**Bent - down bars**

If a single line of bent-down bars is provided, then the ratio d / s_{r} may be given the value 0,67.

As shown below a single line is sufficient resulting in s_{r} = d / 0,67.

A_{sw,min} = (v_{Ed} - 0,75 v_{R,c}) u_{1} d / (1,005 f_{ywd,ef} sin(α))

**Links**

Max distance between link perimeters are 0,75 d as shown below resulting in s_{r} = 0,75 d and d / s_{r} = 1,33

- A
_{sw,min}= (v_{Ed}- 0,75 v_{Rd,c}) u_{1}d / (1,995 f_{ywd,ef}sin(α)) - f
_{ywd,ef}is the effective design strength of the punchimg shear reinforcement calculated as: - f
_{ywd,ef}= 250 + 0,25 d ≤ f_{ywd}(MPa) - α is angle between the shear reinforcement and the plane of the slab.

The control perimeter at which shear reinforcement is not required u_{out,er} should be calculated as:

u_{out,er} = β V_{Ed} / v_{Rd,c} d)

The outermost perimeter of shear reinforcement should be placed at a distance not greater than kd within u_{out,er} as seen below:

The factor k may be given the value k = 1,5.

k = 1,5 unless the perimeter at which reinforcement is no longer required is less than 3d from the face of the loaded area/column. In this case the reinforcement should be placed in the zone 0,3d and 1,5d from the face of the column.

k = 2,0

# Rules for punching reinforcement

Where punching shear reinforcement is required it should be placed between the loaded area/column and kd inside the control perimeter at which the shear reinforcement is no longer required (see above). It should be provided in at least two perimeters of links legs, see figure below. The spacing of the links leg perimeters should not exceed **0,75d**.

**Links**

The spacing of links legs around a perimeter should not exceed **1,5d** within the first control perimeter (**2d** from loaded area), and should not exceed **2d** for perimeters outside the first control perimeter where the part of the perimeter is assumed to contribute to the shear capacity, see figure below.

**Bent-down bars**

For bent-down bars as arranged in the figure below one perimeter of link legs may be considered sufficient.

**Minimum reinforcement**

Where shear reinforcement is required the area of a link leg (or equivalent) A_{sw,min} is given by:

- A
_{sw,min}= (1,5 sin(α) + cos(α)) / (s_{r}s_{t}) ≥ 0,08 f_{ck}^{0,5}/ f_{yk}

where:

- α is the angle between the shear reinforcement and the main steel (i.e. for vertical links α = 90 and sin(α) = 1)
- s
_{r}is the spacing of shear links in the radial direction - s
_{t}is the spacing of shear links in the tangential direction - f
_{ck}is in MPa

Bent-up bars passing through the loaded area or at a distance not exceeding **0,25d** from this area may be used as punching shear reinforcement, see figure above.

The distance between the face of the support, or the circumference of a loaded area, and the nearest shear reinforcement taken into account in the design should not exceed d/2. This distance should be taken at the level of the tensile reinforcement. If only a single line of bent-down bars is provided, their slope may be reduced to 30 deg.

# Layout for punching shear reinforcement

**Bent-down bars**

Bending angle 30 ≤ α ≤ 60 degrees

For bent-down bars as arranged in the figure below one perimeter may be considered sufficient.

The tangential spacing s_{t} ≤ 0,25 d + 0,5 c.

- c is the column width or diameter. For a rectangular column with sides c
_{1}and c_{2} - c = (c
_{1}+c_{2}) / 2

The same number of bars will always be used in both directions meaning that if e.g. 7 bars is required 8 will be chosen.

**Links**

Angle 45 ≤ α ≤ 90 degrees

- Links should be provided between the face of the column and the distance k
_{d}(see above) inside the outer perimeter. - There should be at least two perimeters of links.
- The calculated required area A
_{sw}is to be placed at each perimeter. - The radial spacing of the links s
_{r}should not exceed 0.75d. - The tangential spacing s
_{t}of the links should not exceed 1.5d within 2d of the column face. - The tangential spacing s
_{t}of the links should not exceed 2d for any perimeter. - The distance between the face of the column and the nearest shear reinforcement should be less than 0.5d.