Sachpazis verification of the ultimate punching shear resistance to ec2 1992 1-1-2004 with na cen

VERIFICATION OF THE ULTIMATE
PUNCHING SHEAR RESISTANCE
EUROCODE 2
GEODOMISI Ltd. - Dr. Costas Sachpazis
Civil & Geotechnical Engineering Consulting Company for
Structural Engineering, Soil Mechanics, Rock Mechanics, Foundation Engineering & Retaining Structures.
Tel.: (+30) 210 5238127, 210 5711263 - Fax.:+30 210 5711461 - Mobile: (+30) 6936425722 & (+44) 7585939944,
www.geodomisi.com - costas@sachpazis.info

Structural Engineering, Soil Mechanics, Rock Mechanics,
Foundation Engineering & Retaining Structures.
Tel.: (+30) 210 5238127, 210 5711263 - Fax.:+30 210 5711461 -
Mobile: (+30) 6936425722 & (+44) 7585939944,
Project: VERIFICATION OF THE ULTIMATE PUNCHING
SHEAR RESISTANCE to EUROCODE2 1992-1-1:2004
with NA=CEN.
Job Ref.
www.geodomisi.com
Section
Civil & Geotechnical Engineering
Sheet no./rev. 1
Calc.
Dr. C. Sachpazis
Date
16/01/2016
Chk'd by
Date App'd by Date
INDEX
DESCRIPTION
CHECKS
1.- Zone adjacent to the support or load (Persistent situations) 3
2.- Zone with punching shear reinforcement (Persistent situations) 5
3.- External zone to the punching shear reinforecement (Persistent
situations) 7
4.- Punching shear reinforcement (EN 1992-1-1:2004/AC:2008, 9.4.3(2)) 9
5.- Clear distance between two isolated consecutive bars 9
6.- Distance between the support's face and the first punching shear
reinforcement 10
7.- Distance between transverse consecutive reinforcement perimeters 10
8.- Distance between two consecutive reinforcements in peripheral direction 11
9.- Distance between the support's external face and the outermost bar
inclined at 45° 11

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DESCRIPTION
Calculation of the punching shear perimeters
CHECKS
1.- Zone adjacent to the support or load (Persistent situations)
The worst case design forces occur for load combination 1.35·SW+1.35·DL1+1.5·LL1.
The following criteria must be satisfied:
Where:
vEd: Design value of the sheer stress along the control section considered.
vRd,max: Design value of the maximum punching shear resistance along the
control section considered.
The design value of the sheer stress along the control section considered is obtained
from the following expression (EN 1992
Where:
VEd: Design value of the applied shear force.
Ed Rd,maxv v≤
Ed
Ed
0
V
v
u d
β ⋅
=
⋅
Dr. Costas Sachpazis
Fax.:+30 210 5711461 -
SHEAR RESISTANCE to EUROCODE2 1992-
with NA=CEN.
Section
Calc.
Dr. C. Sachpazis
Date
16/01/2016
Chk'd by
Calculation of the punching shear perimeters
Perimeter of the s
u0:
Critical perimeter
u1:
xG:
yG:
W1x:
W1y:
Punching shear reinforcement perimeter
uout,ef:
xG:
yG:
Wout,ef,x:
Wout,ef,y:
Zone adjacent to the support or load (Persistent situations)
: Design value of the sheer stress along the control section considered.
: Design value of the maximum punching shear resistance along the
of the sheer stress along the control section considered is obtained
from the following expression (EN 1992-1-1:2004/AC:2008, 6.4.5):
: Design value of the applied shear force.
VERIFICATION OF THE ULTIMATE PUNCHING
-1-1:2004
Job Ref.
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Sheet no./rev. 1
Date App'd by Date
Page 3 - 11
Perimeter of the support (C1)
1600 mm
Critical perimeter
4362 mm
0 mm
0 mm
19163.6 cm²
19163.6 cm²
Punching shear reinforcement perimeter
5031 mm
0 mm
0 mm
51487.8 cm²
51487.8 cm²
2.43 MPa ≤≤≤≤ 4.50 MPa
vEd : 2.43 MPa
vRd,max : 4.50 MPa
of the sheer stress along the control section considered is obtained
vEd : 2.43 MPa
VEd : 756.75 kN

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Dr. C. Sachpazis
Date
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Page 4 - 11
ββββ: Coefficient which takes into account the effects of load eccentricity. (EN
1992-1-1:2004/AC:2008, 6.4.3). ββββ : 1.13
kx: Coefficient which depends on the relationship between the dimensions
cy (dimension in direction of the y-axis) and cx (dimension in direction of the
x-axis) of the column (EN 1992-1-1:2004/AC:2008, Table 6.1). kx : 0.60
ky: Coefficient which depends on the relationship between the dimensions cx
(dimension in direction of the x-axis) and cy (dimension in direction of the
y-axis) of the column (EN 1992-1-1:2004/AC:2008, Table 6.1). ky : 0.60
MEdx: Design moment around the x-axis, regarding the center of gravity of
the critical perimeter u1. MEdx : 58.65 kN·m
MEdy: Design moment around the y-axis, regarding the center of gravity of
the critical perimeter u1. MEdy : 12.30 kN·m
MEdOx: Design moment around the x-axis, regarding the center of gravity of
the column. MEdOx : 58.65 kN·m
MEdOy: Design moment around the y-axis, regarding the center of gravity of
the column. MEdOy : 12.30 kN·m
u1: Critical punching shear perimeter (EN 1992-1-1:2004/AC:2008, 6.4.2). u1 : 4362 mm
W1x : 19163.6 cm²
dl: Differential element of the critical perimeter length.
ey: Distance from dl to the axis where the moment MEdx acts about.
W1y : 19163.6 cm²
ex: Distance from dl to the axis where the moment MEdy acts about.
u0: Verification critical punching shear perimeter of the area adjacent to the
support or load (EN 1992-1-1:2004/AC:2008, 6.4.5). u0 : 1600 mm
d: Nominal depth of the slab. d : 220 mm
The design value of the maximum punching shear resistance along the control section
considered is obtained from the following expression (EN 1992-1-1:2004/AC:2008,
6.4.5):
vRd,max : 4.50 MPa
νννν : 0.54
Where:
fck: Concrete compressive strength. fck : 25.00 MPa
EdyEdx 1 1
x y
Ed 1x Ed 1y
MM u u
1 k k
V W V W
β = + ⋅ ⋅ + ⋅ ⋅
1u
1x y
0
W e dl= ⋅∫
1u
1y x
0
W e dl= ⋅∫
Rd,max cdv 0.5 f= ⋅ ν ⋅
ckf
0.6 1
250
 
ν = ⋅ − 
 

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fcd: Design value of the concrete compression force in the direction of
longitudinal member axis.
2.- Zone with punching shear reinforcement (Persistent situations)
Where:
vRd,cs: Design value of the punching shear resistance of a slab with punching
shear reinforcement along the control section cons
Where:
1992-1-1:2004/AC:2008, 6.4.3).
cy (dimension in direction of the y
the x-axis) of the column (EN 1992
ky: Coefficient which depends on the relationship between the dimensions
cx (dimension in direction of the x
the y-axis) of the column (EN 1992
MEdx: Design moment around the x
the critical perimeter u1.
MEdy: Design moment around the y
the critical perimeter u1.
MEdOx: Design moment around the x
of the column.
MEdOy: Design moment around the y
of the column.
u1: Critical punching shear perimeter (EN 1992
Ed Rd,csv v≤
Ed
Ed
1
V
v
u d
β ⋅
=
⋅
Edx 1 1
x y
Ed 1x Ed 1y
M u u
1 k k
V W V W
β = + ⋅ ⋅ + ⋅ ⋅
1u
1x y
0
W e dl= ⋅∫
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Section
Calc.
Dr. C. Sachpazis
Date
16/01/2016
Chk'd by
: Design value of the concrete compression force in the direction of the
longitudinal member axis.
Zone with punching shear reinforcement (Persistent situations)
: Design value of the punching shear resistance of a slab with punching
shear reinforcement along the control section considered.
: Coefficient which takes into account the effects of load eccentricity. (EN
1:2004/AC:2008, 6.4.3).
: Coefficient which depends on the relationship between the dimensions
(dimension in direction of the y-axis) and cx (dimension in direction of
axis) of the column (EN 1992-1-1:2004/AC:2008, Table 6.1).
(dimension in direction of the x-axis) and cy (dimension in direction of
axis) of the column (EN 1992-1-1:2004/AC:2008, Table 6.1).
: Design moment around the x-axis, regarding the center of gravity of
.
: Design moment around the y-axis, regarding the center of gravity of
.
: Design moment around the x-axis, regarding the center of gravity
: Design moment around the y-axis, regarding the center of gravi
: Critical punching shear perimeter (EN 1992-1-1:2004/AC:2008, 6.4.2).
: Differential element of the critical perimeter length.
Edy1 1
x y
Ed 1x Ed 1y
Mu u
1 k k
V W V W
β = + ⋅ ⋅ + ⋅ ⋅
-1-1:2004
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the
fcd : 16.67 MPa
0.89 MPa ≤≤≤≤ 4.90 MPa
vEd : 0.89 MPa
: Design value of the punching shear resistance of a slab with punching
vRd,cs : 4.90 MPa
vEd : 0.89 MPa
VEd : 756.75 kN
ββββ : 1.13
(dimension in direction of
kx : 0.60
on of
ky : 0.60
axis, regarding the center of gravity of
MEdx : 58.65 kN·m
ng the center of gravity of
MEdy : 12.30 kN·m
axis, regarding the center of gravity
MEdOx : 58.65 kN·m
ty
MEdOy : 12.30 kN·m
1:2004/AC:2008, 6.4.2). u1 : 4362 mm
W1x : 19163.6 cm²

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Calc.
Dr. C. Sachpazis
Date
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Page 6 - 11
W1y : 19163.6 cm²
The design value of the punching shear resistance of a slab with punching shear
reinforcement along the control section considered is obtained from the following
expression (EN 1992-1-1:2004/AC:2008, 6.4.5):
vRd,cs : 4.90 MPa
Where:
vRd,c : 0.95 MPa
with a minimum value of:
vRd,c,min : 0.68 MPa
Where:
γγγγc: Concrete resistance reduction coefficient. γγγγc : 1.50
k: Coefficient which depends on the nominal depth of 'd'. k : 1.95
ρρρρl: Geometric steel area of the main tensile longitudinal
reinforcement. ρρρρl : 0.0132
Where:
ρρρρlx: Ratio in X-direction. ρρρρlx : 0.0132
ρρρρly: Ratio in Y-direction. ρρρρly : 0.0132
σσσσcp: Average axial stress on the critical verification surface (positive
compression), with a maximum value of σcp,max. σσσσcp : 2.00 MPa
σσσσcp,max : 3.33 MPa
fcd: Design value of the concrete compression force in the
direction of the longitudinal member axis. fcd : 16.67 MPa
1u
1y x
0
W e dl= ⋅∫
sw
ywd,ef
r
Rd,cs Rd,c
1
A
f sin
s
v 0.75 v 1.5
u
 
⋅ ⋅ α 
 = ⋅ + ⋅
∑
( )
1 / 3
Rd,c l ck cp
c
0.18
v k 100 f 0.1= ⋅ ⋅ ⋅ ρ ⋅ + ⋅ σ
γ
3 / 2 1 / 2
Rd,c,min ck cpv 0.035 k f 0.1= ⋅ ⋅ + ⋅ σ
200
k 1 2
d
 
= + ≤  
 
l lx ly 0.02ρ = ρ ⋅ ρ ≤
cp,max cd0.20 fσ = ⋅

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Asw: Total area of punching shear reinforcement within a permieter
concentric with the support or loaded area.
sr: Radial distance between two concentric perimeters of rei
αααα: Angle between the shear reinforcement and the plane of the slab.
Reference
111
111
fywd,ef: Effective design strength of the punching shear
fywd: Design yield strength of the shear reinforcement.
(EN 1992-1-1:2004/AC:2008, 6.2.3(3))
u1: Critical punching shear perimeter (EN 1992
3.- External zone to the punching shear reinforecement (Persistent situations)
Where:
vRd,c: Design value of the punching shear resistance of a slab without punching
shear reinforcement along the control section considered.
Where:
1992-1-1:2004/AC:2008, 6.4.3).
ywd,ef ywdf 250 0.25 d f= + ⋅ ≤
ywd ywkf 0.8 f= ⋅
Ed Rd,cv v≤
Ed
Ed
out,ef
V
v
u d
β ⋅
=
⋅
Edx out,ef out,ef
x y
Ed out,ef,x Ed out,ef,y
M u u
1 k k
V W V W
β = + ⋅ ⋅ + ⋅ ⋅
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Section
Calc.
Dr. C. Sachpazis
Date
16/01/2016
Chk'd by
: Total area of punching shear reinforcement within a permieter
concentric with the support or loaded area.
: Radial distance between two concentric perimeters of reinforcement.
: Angle between the shear reinforcement and the plane of the slab.
Asw
(mm²)
sr
(mm)
α
(degrees)
2262 80 45.0
2262 80 45.0
: Effective design strength of the punching shear reinforcement.
: Design yield strength of the shear reinforcement.
1:2004/AC:2008, 6.2.3(3))
perimeter (EN 1992-1-1:2004/AC:2008, 6.4.2).
External zone to the punching shear reinforecement (Persistent situations)
must be satisfied:
: Design value of the punching shear resistance of a slab without punching
nt along the control section considered.
1:2004/AC:2008, 6.4.3).
EdyEdx out,ef out,ef
x y
Ed out,ef,x Ed out,ef,y
MM u u
1 k k
V W V W
β = + ⋅ ⋅ + ⋅ ⋅
-1-1:2004
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Sheet no./rev. 1
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nforcement.
Asw/sr
(cm²/m)
282.8
282.8
fywd,ef : 305.00 MPa
fywd : 320.00 MPa
fywk : 400.00 MPa
u1 : 4362 mm
External zone to the punching shear reinforecement (Persistent situations)
0.72 MPa ≤≤≤≤ 0.95 MPa
vEd : 0.72 MPa
: Design value of the punching shear resistance of a slab without punching
vRd,c : 0.95 MPa
vEd : 0.72 MPa
VEd : 756.75 kN
ββββ : 1.05

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Date
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Page 8 - 11
cy (dimension in direction of the y-axis) and cx (dimension in direction of
the x-axis) of the column (EN 1992-1-1:2004/AC:2008, Table 6.1). kx : 0.60
ky: Coefficient which depends on the relationship between the dimensions
cx (dimension in direction of the x-axis) and cy (dimension in direction of
the y-axis) of the column (EN 1992-1-1:2004/AC:2008, Table 6.1). ky : 0.60
MEdx: Design moment around the x-axis, regarding the center of gravity of
the critical perimeter uout,ef. MEdx : 58.65 kN·m
MEdy: Design moment around the y-axis, regarding the center of gravity of
the critical perimeter uout,ef. MEdy : 12.30 kN·m
MEdOx: Design moment around the x-axis, regarding the center of gravity
of the column. MEdOx : 58.65 kN·m
MEdOy: Design moment around the y-axis, regarding the center of gravity
of the column. MEdOy : 12.30 kN·m
uout,ef: Critical punching shear perimeter outside the reinforced zone (EN
1992-1-1:2004/AC:2008, 6.4.5). uout,ef : 5031 mm
Wout,ef,x : 51487.8 cm²
Wout,ef,y : 51487.8 cm²
The design value of the punching shear resistance of a slab without punching shear
reinformcement along the control section considered is obtained from the following
expression (EN 1992-1-1:2004/AC:2008, 6.4.4):
vRd,c : 0.95 MPa
with a minimum value of:
vRd,c,min : 0.68 MPa
Where:
γγγγc: Concrete resistance reduction coefficient. γγγγc : 1.50
k: Coefficient which depends on the nominal depth of 'd'. k : 1.95
ρρρρl: Geometric steel area of the main tensile longitudinal reinforcement. ρρρρl : 0.0132
out,efu
out,ef,x y
0
W e dl= ⋅∫
out,efu
out,ef,y x
0
W e dl= ⋅∫
( )
1 / 3
Rd,c l ck cp
c
0.18
v k 100 f 0.1= ⋅ ⋅ ⋅ ρ ⋅ + ⋅ σ
γ
3 / 2 1 / 2
Rd,c,min ck cpv 0.035 k f 0.1= ⋅ ⋅ + ⋅ σ
200
k 1 2
d
 
= + ≤  
 

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Where:
ρρρρlx: Ratio in X-
ρρρρly: Ratio in Y-
σσσσcp: Average axial stress on the critical verification surface (positive
compression), with a maximum value of
fcd: Design value of the concrete compression for
the longitudinal member axis.
4.- Punching shear reinforcement (EN 1992
Where shear reinforcement is required the area of a link leg (or equivalent), A
(9.11).
Reference
Asw
(mm²)
111 113
111 113
where:
Asw: the area of a link leg (or equivalent).
αααα: is the angle between the shear reinforcement and the main steel (i.e. f
α = 90° and sin α = 1).
sr: is the spacing of shear links in the radial direction.
st: is the spacing of shear links in the tangential direction.
fck: is in MPa
5.- Clear distance between two isolated consecutive bars
The horizontal and vertical clear spacing d
smin (EN 1992-1-1:2004/AC:2008, 8.2(2)):
l lx ly 0.02ρ = ρ ⋅ ρ ≤
cp,max cd0.20 fσ = ⋅
( ) ( ),min 1,5 sin cos / 0,08 / 9.11sw r t ck ykA s s f fα α⋅ ⋅ + ⋅ ≥ ⋅
( ) ( )
( )
,min
,min
1,5 sin cos /
0,08 /
w sw r t
w ck yk
A s s
f f
ρ α α
ρ
= ⋅ ⋅ + ⋅
= ⋅
l mind s≥
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Dr. C. Sachpazis
Date
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Chk'd by
-direction.
-direction.
: Average axial stress on the critical verification surface (positive
compression), with a maximum value of σcp,max.
: Design value of the concrete compression force in the direction of
the longitudinal member axis.
Punching shear reinforcement (EN 1992-1-1:2004/AC:2008, 9.4.3(2))
Where shear reinforcement is required the area of a link leg (or equivalent), Asw,min
(mm²)
sr
(mm)
st
(mm)
α
(degrees)
ρw ρw,min
80 33 45.0 0.0757 0.0010
80 33 45.0 0.0757 0.0010
: the area of a link leg (or equivalent).
: is the angle between the shear reinforcement and the main steel (i.e. for vertical links
: is the spacing of shear links in the radial direction.
: is the spacing of shear links in the tangential direction.
between two isolated consecutive bars
The horizontal and vertical clear spacing dl between two consecutive bars should be greater than or equal to
1:2004/AC:2008, 8.2(2)):
0.02
( ) ( )1,5 sin cos / 0,08 / 9.11sw r t ck ykA s s f f⋅ ⋅ + ⋅ ≥ ⋅
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ρρρρlx : 0.0132
ρρρρly : 0.0132
σσσσcp : 2.00 MPa
σσσσcp,max : 3.33 MPa
ce in the direction of
fcd : 16.67 MPa
1:2004/AC:2008, 9.4.3(2))
sw,min, is given by Expression
w,min ρw ≥ ρw,min
0.0010
0.0010
or vertical links
fck : 25.00 MPa
between two consecutive bars should be greater than or equal to
21 mm ≥≥≥≥ 20 mm

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Where:
smin: Maximum value of s1, s2
Where:
dg: Maximum size of aggregate.
Ømax: Diameter of the thickest bar of the transverse reinforcement.
111
111
6.- Distance between the support's face and the first punching shear reinforcement
The distance between the face of the support or loaded area and the first punching shear r
not be greater than smax (EN 1992-
Where:
d: Nominal depth of the slab.
7.- Distance between transverse consecutive reinforcemen
The distance dl between consecutive transverse reinforcement perimeters should be, at most, equal to s
1992-1-1:2004/AC:2008, 9.4.3):
Where:
1 maxs = ∅
2 gs 5 d= +
3s 20mm=
l maxd s≤
maxs 0.5 d= ⋅
l maxd s≤
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Dr. C. Sachpazis
Date
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2, s3.
: Maximum size of aggregate.
: Diameter of the thickest bar of the transverse reinforcement.
dl
(mm)
smin
(mm)
Ømax
(mm)
21 20 12
21 20 12
Distance between the support's face and the first punching shear reinforcement
The distance between the face of the support or loaded area and the first punching shear r
-1-1:2004/AC:2008, 9.4.3):
: Nominal depth of the slab.
Distance between transverse consecutive reinforcement perimeters
between consecutive transverse reinforcement perimeters should be, at most, equal to s
-1-1:2004
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smin : 20 mm
s1 : 12 mm
s2 : 17 mm
s3 : 20 mm
dg : 12 mm
Ømax : 12 mm
Distance between the support's face and the first punching shear reinforcement
The distance between the face of the support or loaded area and the first punching shear reinforcement should
80 mm ≤≤≤≤ 110 mm
smax : 110 mm
d : 220 mm
between consecutive transverse reinforcement perimeters should be, at most, equal to smax (EN
80 mm ≤≤≤≤ 165 mm

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8.- Distance between two consecutive reinforcements in peripheral direction
The distance dl between two consecutive perimeter reinforcements should not be greater than s
1-1:2004/AC:2008, 9.4.3):
Where:
9.- Distance between the support's external face and the outermost bar inclined at 45°
This combination does not proceed since the reinforcement is locate
support.
maxs 0.75 d= ⋅
l maxd s≤
= ⋅maxs 1.5 d
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Dr. C. Sachpazis
Date
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Distance between two consecutive reinforcements in peripheral direction
between two consecutive perimeter reinforcements should not be greater than s
Distance between the support's external face and the outermost bar inclined at 45°
This combination does not proceed since the reinforcement is located between the external faces of the
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smax : 165 mm
d : 220 mm
Distance between two consecutive reinforcements in peripheral direction
between two consecutive perimeter reinforcements should not be greater than smax (EN 1992-
33 mm ≤≤≤≤ 330 mm
smax : 330 mm
d : 220 mm
Distance between the support's external face and the outermost bar inclined at 45°
d between the external faces of the
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Sachpazis verification of the ultimate punching shear resistance to ec2 1992 1-1-2004 with na cen

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Sachpazis verification of the ultimate punching shear resistance to ec2 1992 1-1-2004 with na cen