SCIA Engineer Beta Factors
Short Description
Punching shear desing in SCIA...
Description
Beta Be ta factor factor for f or punching pun ching EN 1992-1-1
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Table of contents Content 1
Abst ract ..........................................................................................................................................1
2
Setting of calculation .................................................................................................................... 2 2.1
Concrete solver ....................................................................................................................... 2
2.2
Punching data ......................................................................................................................... 3
3
Output for calc ulat ion for Beta fact or .........................................................................................4 3.1
Member check ......................................................................................................................... 5
3.2
Single check ............................................................................................................................ 7
4
Theoretical background ............................................................................................................... 9 4.1
Simplified method (EN 1992-1-1,6.4.3(6)) ............................................................................. 9
4.2
General method (EN 1992-1-1,6.4.3(3-5)) ............................................................................ 10
4.3
General method (general formula) ...................................................................................... 13
5
Comparis on of calc ulat ion Beta fact or .....................................................................................15 5.1
Internal column ..................................................................................................................... 15
5.2
Edge column (parallel x) ...................................................................................................... 16
5.3
Corner column ....................................................................................................................... 17
6
Ab breviati on ................................................................................................................................18
7
Li terature ...................................................................................................................................... 19
3
1
Abstract
The maximum punching shear stress in EN 1992-1-1, clause 6.4.3(3) is calculated according to formula below.
∙ ∙ ∙
It follows that maximum punching shear stress depends on:
design ultimate shear force on the perimeter (VEd)
length of control perimeter (ui)
the mean effective depth of the slab at control perimeter (d)
shear factor taking into account of moment transfer (). The calculation of this factor is based on the principle, that the moment transferred between the slab and the column must be equal to the moment produced by the shear v distributed around the perimeter ui
The version SCIA Engineer 2012 offers the following possibilities for calculation of factor :
Simplified method according to EN 1992-1-1, clause 6.4.3(6), default method
General method according to EN 1992-1-1, clause 6.4.3(3-5)
General method according to general formula
User input of factor (only if punching data are defined)
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2
Setti ng of calculatio n The type of method for calculation of factor can be set in:
Concrete solver > ULS > Punching for punching check in node without Punching data
Punching data for punching check in node with punching data
2.1 Concrete solv er Type of calculation of factor can be set in Concrete solver > ULS > Punching via combo box Type of calculation Beta factor , where is available three methods: o
Simplified (EN 1992-1-1,6.4.3(6)) - default method
o
General (EN 1992-1-1,6.4.3(3-5))
General ( general formula) o This setting will be taken into account for check of punching in the node without defined punching data.
2
2.2 Punchin g data Type of calculation of factor can be set in combo box Type of calculation Beta factor , where is available four possibilities: o
Simplified (EN 1992-1-1,6.4.3(6))
o
General (EN 1992-1-1,6.4.3(3-5))
o
General ( general formula)
o
User input
The default method is loaded form Concrete solver and this setting will be taken into account for check of punching in the node with defined punching data
Note
If Type of calculation Beta factor = User input , then new property User input of beta factor is active, where user can define own value of factor. Default value is 1,0.
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3
Output for calculation for Beta factor In the numerical output are new values, which are used in calculation factor according to generals methods. The following new values can be presented in numerical output :
MEd,x
For all methods
is unbalanced transferred
bending moment between the slab and the columns
(supports) in of x axis of LCS for punching MEd,y
is unbalanced transferred
bending moment between the slab and the columns
(supports) in of y axis of LCS for punching
k x,k y
For general ( general formu la) method
is a coefficient dependent on the ratio between the column dimensions in direction of x(y) axis of LCS for punching
ex(y),i
is eccentricity of
i-th control perimeter related to centroid of cross-section in
direction of x(y) axis of LCS for punching Wx(y),i
is modulus of i-th control perimeter in direction of x(y) axis of LCS for punching corresponds to a distribution of shear and recalculated to centre of gravity of the control perimeter
k x,k y
For General (EN 1992-1-1,6.4.3(3-5)) method
is a coefficient dependent on the ratio between the column dimensions in direction of x(y) axis of LCS for punching
ex(y),i
is eccentricity of
i-th control perimeter related to centroid of cross-section in
direction of x(y) axis of LCS for punching Wx(y),i
is modulus of i-th control perimeter in direction of x(y) axis of LCS for punching corresponds to a distribution of shear and recalculated to centre of gravity of the node , where is column (support linked to the slab)
ke
Is the coefficient dependent on the ratio between the column dimensions in direction of eccentricity of bending moments MEd,x and MEd,y.
W,i
is modulus of i-th control perimeter in direction of eccentricities of bending moments MEd,x and MEd,y.
4
Note
The orientation of axis’s in LCS for punching is always transformed to the first quadrant of GCS of the project and the LCS for punching can be presented in single check, see picture below
3.1 Member check The numerical output can be presented after clicking on action button Preview in service Punching check (tree Concrete > Punching ) or after adding the item Punching check to the document Numerical output in service Punching check
Numerical output in document
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There are presented following tables in member check
The factor and other values, which are used for calculation factor can be presented in table Check punching shear resist ance and design shear reinforc ement.
6
Note
Existing table can be edit via icon Table composer
or via double clicking on the header
of the table.
The new table can be created via icon Table manager
3.2 Singl e check The dialog for detailed check of one node (single check) will be opened after clicking on action button Single check in service Punching check (tree Concrete > Punching ). The numerical output can be
presented in tab-sheet Design of reinforcement or in tab-sheet Document setup
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Note
The numerical output in single check can be presented directly in the dialog (item Preview ) or can be set to the main document (item Document )
There is a special table in Results in critical section , where are presented ( depending on the selected method of calculation factor ) detailed values, which are used for calculation of factor
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4
Theoretic al backgr ound
4.1 Simp lified meth od (EN 1992-1-1,6.4.3(6)) 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 factor is the same for all control perimeters and depends on position of the column, where for
corner column = int
edge column (parallel x or parallel y) = edge
internal column = int
cor
edge
in t
The value of 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
9
4.2 General metho d (EN 1992-1-1,6.4.3(3-5)) This method of calculation is described in to EN 1992-1-1, clause 6.4.3(6) and the calculation of factor depends on position of column and on it, if the eccentricity toward the interior or exterior of the slab for edge and corner column. The value of factor is calculated for each control perimeter as minimum value from factor in first critical and in current critical perimeter. It means the this value can be different for each control perimeter. Internal c olumn
The following formula are used for calculation of factor
1 , ∙ ∙
Note
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 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.
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Edge (Parallel x)
the eccentricity is toward the interior of the slab ( MEdy > 0)
∗ , ∙ ∙ ,, 1 , ∙ ∙
the eccentricity is toward the exterior of the slab ( MEdy ≤ 0)
Edge (Parallel y)
the eccentricity is toward the interior of the slab ( MEdx > 0)
the eccentricity is toward the exterior of the slab ( MEdx ≤ 0)
∗ , ∙ ∙ ,, 1 , ∙ ∙
Corner
the eccentricity is toward the interior of the slab (MEdx 0 and MEdy 0 )
the eccentricity is toward the interior of the slab
∗ 1 , ∙ ∙
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
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
11
ke
is a coefficient dependent on the ratio between the column dimensions in direction of eccentricity of bending moment
, ,, , MEd,x
is unbalanced transferred bending moment between the slab and the columns(supports) in direction of x axis
MEd,y
is unbalanced transferred bending moment between the slab and the column(supports) in direction of y axis
MEd
is unbalanced transferred bending moment between the slab and the column (supports)recalculated to direction of eccentricities
, , 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)
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)
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)
ui
is the length of i-th control perimeter
u*
reduced control perimeter, see picture 6.20 in EN 1992-1-1
VEd,i
is shear force for i-th control perimeter
α
Is angle between x-axis and direction of eccentricities
tan,/,
12
u*
Direction of eccentricity
MEd,x α
u
y
y
min(1,5·d;0,5cy)
MEd,y
c
cx x
the centre of gravity of the column (support)
4.3 General metho d (general formula) This method of calculation is based on the formula below, where value 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 factor is calculated for each control perimeter as minimum value from factor in first critical and in current critical perimeter. It means the this value can be different for each control perimeter.
. . , , , , , 1 , ∙ ∙ , ∙ , , Note
The 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.
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
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
MEd,x
is unbalanced transferred bending moment between the slab and the columns(supports) in direction of x axis
MEd,y
is unbalanced transferred bending moment between the slab and the column(supports) in direction of y axis
13
ex(y),i
is eccentricity of i-th control perimeter related to centroid of cross-section in direction of x(y) axis. Distance between centre of gravity of control perimeter and centre of gravity of the column (support)
Wx,i
is the modulus of i-th control perimeter in direction of x axis recalculated to the centre of gravity of the control perimeter
Wy,i
is the modulus of i-th control perimeter in direction of y axis recalculated to the centre of gravity of gravity of the control perimeter
ui
is the length of i-th control perimeter
VEd,i
is shear force for i-th control perimeter
the centre of gravity of the
MEd,x
i-th control perimeter
ei
ui
y
c
y
cx x
14
MEd,y the centre of gravity of the column (support)
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Comparison of calculation Beta factor
5.1 Internal col umn b= 300 mm; h = 450 mm; d = 160 mm, VEd(REd) = 100kN MEd,x [kNm/m]
0
30
-30
0
0
30
30
-30
-30
MEd,y [kNm/m]
0
0
0
30
-30
30
-30
30
-30
Method
factor
Simplified EN 1992-1-1
1.15
General EN 1992-1-1
1.0
1.45
1.45
1.53
1.53
1.71
1.71
1.71
1.71
General (general fo rmula)
1.0
1.45
1.45
1.53
1.53
1.98
1.98
1.98
1.98
EN 1992-1-1, equati on 6.43
-
-
-
-
-
1.76
1.76
1.76
1.76
1 1 2 9 9 1 N E l a r e n e G
) a l u m r o f l a r e n e g ( l a r e n e G
15
5.2 Edge colum n (parallel x) b= 300 mm; h = 450 mm; d = 160 mm; ay=300mm; VEd(REd) = 100kN MEd,x [kNm/m]
0
30
-30
0
0
30
30
-30
-30
MEd,y [kNm/m]
0
0
0
30
-30
30
-30
30
-30
Method
factor
Simplified EN 1992-1-1
1.4
General EN 1992-1-1
1.0
1.39
1.39
1.60
1.59
1.99
1.72
1.99
1.72
General (general for mula)
1.2
1.67
1.67
1.91
1.36
2.30
1.74
2.30
1.74
1 1 2 9 9 1 N E l a r e n e G
) a l u m r o f l a r e n e g ( l a r e n e G
* The factor for method =General(general formula) is calculated for biggest value of shear stress along the control perimeter, therefore in some cases (the eccentricity perpendicular to the slab edge is toward the interior of the slab) there is quite big difference of results in comparison with method = General EN 1992-1-1..
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5.3 Corner colu mn b= 300 mm; h = 450 mm; d = 160 mm; ax=300mm; ay=300mm; VEd(REd) = 100kN MEd,x [kNm/m]
0
30
-30
0
0
30
30
-30
-30
MEd,y [kNm/m]
0
0
0
30
-30
30
-30
30
-30
Method
factor
Simplified EN 1992-1-1
1.5
General EN 1992-1-1
1.0
1.39
1.39
1.56
1.56
2.11
1.66
1.66
1.73
General (general for mula)
1.95
2.54
1.61
2.58
1.61
3.17
2.20
2.24
1.26
1 1 2 9 9 1 N E l a r e n e G
) a l u m r o f l a r e n e g ( l a r e n e G
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Abbreviation
Ab breviati on
Exp lanati on
SEN
Software SCIA Engineer
PNL
Physical nonlinear calculation
GNL
Geometrical nonlinear calculation
LCS
Local coordinate system
GCS
Global coordinate system
REDES
The module in SEN for inputting user reinforcement to 1D member via
SLS
template Serviceability limit state
ULS
Ultimate limit state
FEM
Finite element on 2D member, which is created by meshing of 2D member
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Literature [1]
EN 1992-1-1: 2004 Eurocode 2 : design of concrete structures – Part 1: General rules and rules for building Designer’s Quide to EN 1992-1-1 and EN 1992-1-2 Eurocode 2: Design of concrete
[2]
structures. General rules and rules for building and structural fire design. R.S. Narayanan & A.Beeby
[3]
Železobetónové nosne sústavy, Navrhovanie podľa európskych noriem, I.Harvan
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