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This method according to EN 1992-1-1, clause 6.4.3(6) can be used for structures where the lateral stability does not depend on frame action between the slabs and the columns, and where the adjacent spans do not differ in length by more than 25%. The value of β factor is the same for all control perimeters and depends on position of the column, where for
The value of β factor for simplified method and for different position of column can be set or edit for each NA in Manager of national annex > EN 1992-1-1 > ULS > Punching, see picture below
This method of calculation is described in to EN 1992-1-1, clause 6.4.3(6) and the calculation of β factor depends on two factors: (i) on the position of the column and (ii) on the direction of eccentricity (whether it is oriented inward or outward of the slab; this applies to columns on the edge or at the corner of a slab). The value of β factor is calculated for each control perimeter as minimum value from β factor in first critical and in current critical perimeter. It means the this value can be different for each control perimeter.
The following formula are used for calculation of β factor
This formula is used too for biaxial unbalanced bending moment too, because approximate expression 6.43 in EN 1992-1-1 does not take into account opening around and interruption of control perimeter for dimension of column greater than 3d
The β factor in first critical perimeter is used for calculation the control perimeter at which shear reinforcement is not required (value uout,eff), see clause 6.4.5(4) in EN 1992-1-1.
the eccentricity is toward the interior of the slab ( MEdy > 0)
the eccentricity is toward the exterior of the slab ( MEdy ≤ 0)
the eccentricity is toward the interior of the slab ( MEdx > 0)
the eccentricity is toward the exterior of the slab ( MEdx ≤ 0)
the eccentricity is toward the interior of the slab (MEdx> 0 and MEdy >0 )
the eccentricity is toward the interior of the slab
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kx |
is a coefficient dependent on the ratio between the column dimensions in direction of x axis of LCS for punching. The value is calculated by interpolation from the table 6.1 in EN 1992-1-1, for ratio cx/cy |
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ky |
is a coefficient dependent on the ratio between the column dimensions in direction of y axis of LCS for punching. The value is calculated by interpolation from the table 6.1 in EN 1992-1-1, for ratio cy/cx |
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ke |
is a coefficient dependent on the ratio between the column dimensions in direction of eccentricity of bending moment
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MEd,x |
is unbalanced transferred bending moment between the slab and the columns(supports) in direction of x axis |
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MEd,y |
is unbalanced transferred bending moment between the slab and the column(supports) in direction of y axis |
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MEd |
is unbalanced transferred bending moment between the slab and the column (supports)recalculated to direction of eccentricities
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Wi |
is the modulus of i-th control perimeter in direction of eccentricity (value α) recalculated to the centre of gravity of the column (support) |
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Wx,i |
is the modulus of i-th control perimeter in direction of x axis recalculated to the centre of gravity of the column (support) |
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Wy,i |
is the modulus of i-th control perimeter in direction of y axis recalculated to the centre of gravity of the column (support) |
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ui |
is the length of i-th control perimeter |
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u* |
reduced control perimeter, see picture 6.20 in EN 1992-1-1 |
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VEd,i |
is shear force for i-th control perimeter |
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α |
Is angle between x-axis and direction of eccentricities
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where:
| A |
Direction of eccentricity |
| B |
the centre of gravity of the column (support) |
This method of calculation is based on the formula below, where value β is calculated from the biggest value of the shear stress caused by shear force and unbalanced transferred bending moments between the slab and the columns(supports). The value of β factor is calculated for each control perimeter as minimum value from β factor in first critical and in current critical perimeter. It means the this value can be different for each control perimeter.
The β factor in first critical perimeter is used for calculation the control perimeter at which shear reinforcement is not required (value uout,eff), see clause 6.4.5(4) in EN 1992-1-1.
where:
| A |
the centre of gravity of the i-th control perimeter |
| B |
the centre of gravity of the column (support) |
