Shear force capacity for hollowcores, EN 1168 4.3.3.2.2.2, Advanced method
Source: 
Date: 20221128
Version of PREStress: 6.7.028 (20221028)
PREStressfile is attached
.The file used for handcalculations will not be available for public use.
If there are parameters used in both PREStress and handcalculations there might be a postfix added (_{.PRE} for PREStress) or (_{.HC} for hand calculations)
Codes and standards used
Handcalculation  PREStress 

EN 1990 (200204xx)  SSEN 1990 (201412xx) 
EN 199211:2004/A1:2014 (201412xx)  SSEN 199211:2005/A1:2014 (20150116) 
EN 1168:2005+A3:2011  EN 1168:2005+A3:2011 
SS 21 25 53:2013  SS 21 25 53:2013 
Other documents

Prerequisites
Geometry and material
Hollowcore subjected to a uniform load.
 Total length of the element L_{tot} = 8000 mm, theoretical span L0 = 7800 mm.
 Exposure class XC1, Life class L50
Material  
Concrete  C40/50  
Prestressed reinforcement  Y1860S7  7strand wire ø_{p} = 12.9mm, cross section area = 100 mm^{2}, relaxation class 2 
Cross section
Handcalculation
The cross section is derived from among other things the following parameters:
 Schematic of a cross section
 H = 380 mm
 B = 1197 mm
 b = 1141 mm
 n_{cores.outer} = 2
 b_{t1} = 213
 b_{b1} = 213
 h_{1} = 295
 c_{1} = 45
 r_{t1} = 143
 r_{b1} = 130
 z_{rt1} = c_{1} + r_{t1} = 188 mm
 z_{rb1} = (H  c_{1}  h_{1}) + r_{b1} = 170 mm
 n_{cores.inner} = 2
 b_{t2} = 218 mm
 b_{b2} = 218 mm
 h_{2} = 295 mm
 c_{2} = 45 mm
 r_{t2} = 143 mm
 r_{b2} = 130 mm
 z_{rt2} = c_{2} + r_{t2} = 188 mm
 z_{rb2} = (H  c_{2}  h_{2}) + r_{b2} = 170 mm
 Lifting indentations
A formula to reduce the cross section due to lifting indentations is used
For the handcalculation a function of the width b_{w}(y) is shown in the figure above, both without the reinforcement (blue) and with the reinforcement (black, Idealized cross section). Two vectors with the width at each ycoordinate is used in the calculation. One for the concrete cross section b_{w0} and one for the idealized cross section b_{wρ}.
Cross section area
The area of the cross section is calculated as the width at each level x 1mm height
A_{c0.HC} = 211 689.6 mm^{2}
A_{cρ.HC} = 215 321.8 mm^{2}
Center of gravity
By integrating over the cross section a center of gravity is derived:
Y_{c0.HC} = 193.16 mm
Y_{cρ.HC} = 190.53 mm
Second moment of area (Moment of intertia)
By integrating over the cross section a center of gravity is derived:
I_{c0.HC} = 3867e6 mm^{4}
I_{cρ.HC} = 3954e6 mm^{4}
PREStress
Cross section area
Reinforcement is not considered in the section parameters shown when selecting the section parameters.
A_{c.PRE} = 214 300 mm^{2} (Reinforcement not considered)
 A_{c.PRE }/ A_{c0.HC }= 1.012
Center of gravity
Reinforcement is not considered in the section parameters shown when selecting the section parameters.
In the design the reinforcement is considered.
Y_{c0.PRE} = 191 mm
Second moment of area (Moment of intertia)
Reinforcement is not considered in the section parameters shown when selecting the section parameters.
In the design the reinforcement is considered.
I_{c0.PRE} = 3.931e9 mm^{4}
Design strengths
Strengths are calculated according to EN 199211, 3.1.6, 3.2.7, 3.3.6 and SS 21 25 53:2013
Concrete:
f_{ck} = 40 MPa
f_{cd} = α_{cc} * f_{ck}/γ_{c} = 1,0 * 40/1,5 = 26,67 MPa
f_{ctk0,05} = 2,456 MPa
f_{ctd} = α_{ct} * f_{ctk0,05}/γ_{c} = 1,0 * 2,456/1,5 = 1,64 MPa
f_{ctm} = 3,509 MPa
E_{cm} = 35,22 GPa
At release:
f_{ck,i} = 30 MPa
f_{ctk0,05,i} = 2,0 MPa
f_{ctm,i} = 2.9 MPa
E_{cm,i} = 33 GPa
Prestressing reinforcement:
7strand wire
ø_{p} = 12.9 mm
A_{pi} = 100 mm^{2}
f_{p0,1k} = 1640 MPa
f_{pd} = f_{p0,1k}/γ_{s} = 1640/1,15 = 1426 MPa
f_{puk} = 1860 MPa
f_{pud} = f_{pk}/γ_{s} = 1860 / 1,15 = 1617 MPa
E_{s} = 195 GPa
ε_{puk} = 3,15 * 10^{2}
Reinforcement
Coordinates  Handcalculation
No.  x [mm]  z [mm]  Prestress [MPa]  Diameter [mm]  Area [mm^{2}]  Material 

1  0  37  1000  12.9  100  Y1860S7 
2  0  37  1000  12.9  100  Y1860S7 
3  0  37  1000  12.9  100  Y1860S7 
4  0  37  1000  12.9  100  Y1860S7 
5  0  37  1000  12.9  100  Y1860S7 
6  0  37  1000  12.9  100  Y1860S7 
7  0  37  1000  12.9  100  Y1860S7 
8  0  37  1000  12.9  100  Y1860S7 
ycoordinates are not used, as an uneven placement (excentricity) is not considered in the hand calculation.
Coordinates  PREStress
No.  x (start) [mm]  x (end) [mm]  y [mm]  z [mm]  Prestress [MPa]  Diameter [mm]  Area [mm^{2}]  Material 

1  0  8000  49  37  1000  12.9  100  Y1860S7 
2  0  8000  271  37  1000  12.9  100  Y1860S7 
3  0  8000  361  37  1000  12.9  100  Y1860S7 
4  0  8000  554  37  1000  12.9  100  Y1860S7 
5  0  8000  644  37  1000  12.9  100  Y1860S7 
6  0  8000  837  37  1000  12.9  100  Y1860S7 
7  0  8000  927  37  1000  12.9  100  Y1860S7 
8  0  8000  1149  37  1000  12.9  100  Y1860S7 
Long term losses
Handcalculation
No losses (creep, shrinkage or relaxation) are considered for simplifying the calculations.
PREStress
No losses (creep, shrinkage or relaxation) are considered to be able to compare with the handcalculations.
Manufacturing
Sudden release EN 199211 8.10.2.2, is not considered.
Design loads
Permanent loads  

Deadload hollowcore Density of reinforced concrete (incl. reinforcement)  q_{G.HC} q_{G.PRE} ρ_{c}  5.12 kN/m 5.36 kN/m 24 kN/m^{3} + 1 kN/m^{3} = 25 kN/m^{3} 
Deadload joint concrete  q_{G.joint}  
Variable loads  
Live load  q_{d} ψ_{0}  5 kN/m^{2} 0.7 
q(x)HC = 1.35 * q_{G} + 1.5 * ψ_{0} * q_{d} * 1.2 m^{2}/m = 1.35 * 5.38 kN/m + 1.5 * 0.7 * 5kN/m^{2} * 1.2 m^{2}/m = 13.568 kN/m
q(x)PRE = 1.35 * q_{G} + 1.5 * ψ_{0} * q_{d} * 1.2 m^{2}/m = 1.35 * 5.36 kN/m + 1.5 * 0.7 * 5kN/m^{2} * 1.2 m^{2}/m = 13.531 kN/m
Reactions
Handcalculation
V_{A} = q_{G} * L_{tot} / 2 = 21.53 kN (Deadload only)
V_{B} = q_{G} * L_{tot} / 2 = 21.53 kN (Deadload only)
V_{A} = q(x) * L_{tot} / 2 = 54.27 kN (Design load)
V_{B} = q(x) * L_{tot} / 2 = 54.27 kN (Design load)
PREStress
V_{A} = 21.43 kN (Deadload only)
V_{B} = 21.43 kN (Deadload only)
V_{A} = 54.12 kN (Design load)
V_{B} = 54.12 kN (Design load)
Shear force
Handcalculation
PREStress
Design
Critical point on line of failure
Handcalculation
The vector consits of the ycoordinate for each xcoordinate. Plotting the vector gives the line of failure.
PREStress
PREStress gives the angle 35 degrees from the support and looks at some points along the line where the critical value might be.
It is important to select the Advanced (EN1168 4.3.3.2.2.2)option to get a comparable design.
Concrete compressive stress at height y and distance l_{x}
Handcalculation
This will give a matrix with the stress in each x and ypoint of the beam.
From this matrix the values along the critical line is extracted into a vector, with a value for each xvalue.
PREStress
No values are extracted graphically.
Concrete shear stress due to transmission of prestress at height y and distance l_{x}
Handcalculation
This will give a matrix with the shear stress in each x and ypoint of the beam.
From this matrix the values along the critical line is extracted into a vector, with a value for each xvalue.
PREStress
No values are extracted graphically.
Shear force capacity
Handcalculation
Using the input from previous calculations above we now put them into the final formula:
The result is a matrix with all possible Shear force capacities in the beam, but only the results along the critical line are relevant to look at.
By extracting those values we get a vector along the beam with the capacity for each xcoordinate:
Some values are not relevant, as they are not part of the design criterion according to EN 1168. Only values between the support and along the critical line are allowed.
So the final step is to find the lowest value of the shear capacity within the viable range.
V_{Rdc} = 181.7 kN
x = 235 mm from the end (or 135 mm from the support)
Concrete width at the critical point:
PREStress
The capacity in PREStress is shown as a range between coordinates.
Below is the capacity 188.26kN shown for range 0.10m0.19m