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Punching

The punching analysis is based on the chapter 6.4 of EN 1992-1-1. Openings are considered exactly according to the input (the condition, that the distance of the opening should be shorter than 6d, isn't checked and consideration of distant openings depends on the designer's decision). Control perimeters are found according to the chapter 6.4.2(1) using the distance between perimeters equal to 2d.

Shear stress in control perimeter

The maximum shear stress in control perimeter is given by the expression

Where is:

β

  • The coefficient

VEd

  • The design value of shear force

d

  • The mean effective depth of the slab

ui

  • The length of the control perimeter being considered

The mean effective depth of the slab is calculated using formula

Where is:

dy

  • The effective depth of the slab in direction y

dz

  • The effective depth of the slab in direction z

The coefficient β may be entered manually, selected according to the figure 6.21N or calculated using expression (6.39):

Where is:

u1

  • The length of the control perimeter being considered

k

  • The coefficient dependent on the ratio between the column dimensions c1 and c2. The value is a function of proportions of unbalanced moment transmitted by uneven shear and by bending and torsion. Values are based on the table 6.1. The factor is calculated in the direction of bending moment.

W1

  • The modulus that corresponds to the shear distribution in the figure 6.19

The program provides also an option to use general formula for calculation of the factor β, (the option "Calculate β according to 6.4.3(3-5) - in axes directions" in the part "Analysis").

Where is:

u1

  • The length of the control perimeter being considered

kx, ky

  • The coefficients dependent on the ratio between the column dimensions c1 and c2. The values are functions of proportions of unbalanced moment transmitted by uneven shear and by bending and torsion. Values are based on the table 6.1. The factor is calculated in the direction of bending moment.

MEd,x, MEd,y

  • Bending moments in directions x and y

ex,1, ey,1

  • The eccentricities of centre of gravity of control perimeter relative to the centre of gravity of column

Wx,i, Wy,i

  • The moduli in directions x and y, recalculated relatively to the centre of gravity of control perimeter

For the factor β, it is checked whether the value does not exceed 2.0. The stress from the bending moment is higher than the stress from the shear force in such cases. Such a detail should not be verified according to the punching theory.

The modulus W1 is calculated according to the expression (6.40) using the numerical integration:

Where is:

dl

  • The length increment of the perimeter

e

  • The distance of dl from the axis about which the moment MEd acts

Maximum punching shear resistance

The following verification is done for any control perimeter:

Where is:

vEd

  • The design value of shear stress

vRd,max

  • The design value of the maximum punching shear resistance

The design value of the maximum punching shear resistance is calculated according to the chapter 6.4.5(3):

where

Punching shear resistance of a slab without punching shear reinforcement

The punching shear reinforcement isn't necessary if the following condition is fulfilled:

Where is:

vEd

  • The design value of shear stress

vRd,c

  • The design value of the punching shear resistance of a slab without punching shear reinforcement

The design value of the punching shear resistance of a slab without punching shear reinforcement is calculated in accordance with the chapter 6.4.4(1):

where

and

Where is:

ρly, ρlz

  • The reinforcement ratios for bonded tension reinforcement in slab parallel to axis y and z

The compressive stress in the concrete from axial load σcp is given by the expression

The normal concrete stresses in the critical section σcy and σcz are given by expressions:

Where is:

NEd,y , NEd,z

  • The longitudinal forces across the full bay (internal columns) or the longitudinal force across the control section (edge columns). The force may be from a load or prestressing.

Acy, Acz

  • Corresponding areas of of concrete

The value vmin is calculated according to the chapter 6.2.2(1):

Punching shear resistance with shear reinforcement

If the shear reinforcement is required, the procedure according to 6.4.5(1) is used:

Where is:

Asw

  • The area of shear reinforcement in the perimeter

sr

  • The radial spacing of perimeters of shear reinforcement

fywd,eff

  • The effective design strength of the punching shear reinforcement

d

  • The mean of the effective depths in the orthogonal directions

α

  • The angle between the shear reinforcement and the plane of the slab

The previous expression requires constant value of sr between individual perimeters of shear reinforcement and also constant area Asw in all these perimeters. The expression was modified to allow input of different values of sr and Asw:

Where is:

Asw,x

  • The real area of shear reinforcement between verified and previous control perimeter (the zone with the width equal to 2d). The area of reinforcement between control perimeter and the column is used for perimeters with the distance shorter than 2d in the foundation slabs. The considered area may be extended to 2d when using the setting "Always consider reinforcement in the range 0 to 2d". This setting is not recommended.

The area Asw,x alternates the following expression in the formula (6.52)

This expression describes the reinforcement area in the strip with the width 2d for constant Asw and sr.

The effective design strength of the punching shear reinforcement is given by the expression

The length of the control perimeter where the shear reinforcement is not required is deinfed in the chapter 6.4.5(4):

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