APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...
Transcript of APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...
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APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR INDUCED BY THE
PRESENCE OF LARGE OPENINGS
A2.1 Shear variation at support in case of openings placed at the head of the floor
In case of openings placed near one head of the floor, design charts reports the shear
variation δ on the ith slab at support with respect to the corresponding case of floor without
openings:
δ = �V�,�� − V��� V��⁄ . (A2.1)
being V�,�� the shear force acting on the ith slab of the floor with openings and V�� the one acting on
the corresponding slab of the floor without openings.
For a given floor slenderness ratio L/H and a given thickness of concrete topping, the value of
parameter δ has been deduced by following a two-step procedure. First, a unique value of δ has
been calculated as the envelope of the results provided by numerical analyses at SLS, referred to
different reinforcement ratios. Numerical analyses have been indeed performed by considering
different reinforcement ratios for a same typology of hollow core floor, as a function of the applied
variable load (assumed equal to 5 kN/m2, 10 kN/m2 or 15kN/m2). By repeating this procedure for
each considered floor slenderness ratio L/H (25, 30, 35), a SLS design curve has been obtained
for a given value of concrete topping (0, 4, 8 cm). Subsequently, the so obtained SLS design curve
has been properly amplified, by shifting it upwards of a quantity ∆δ = |δ��� − δ|, being δmax the
maximum value assumed by the parameter δ during the loading history until the reaching of the
sectional resistant moment Mrd (or until the reaching of numerical failure, if the latter was
anticipated with respect to Mrd for the considered case of opening).
The verification of shear resistance can be then carried out by evaluating the applied shear force
through the following expression:
V�� = γ���G� + G�,�������� bl 2⁄ + ((γ�!G! + γ"Q) bl 2⁄ (1 + δ) ≤ V'� (A2.2)
being b the slab width, l its design span and G1,castings the dead weight of all the cast in situ
elements, such as joints and the concrete topping, if present. The parameter δ in Equation (A2.2)
can be instead obtained from design charts reported in the following pages.
It should be noticed that the parameter δ only affects 2nd phase loads, which are applied after floor
assembly (permanent loads G2 and variable ones Q), while it should not be applied to 1st phase
loads (that is to say the dead weight of HC slabs, as well as of cast in place joints and concrete
topping), since during mounting operations each slabs carry its own part of load, without
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redistribution effects. The so evaluated acting shear force Vsd should be then compared with the
shear resistance Vrd, calculated on the effective cross-section of the slab according to EN 1168
(2009).
In case of openings that affect only part of slab width (opening cases ftA and ftB), the
parameter δ referred to the slab interested by the opening assumes a negative value, so
representing a reduction of acting shear at support, due to the lower load applied on the slab itself.
In this case, the verification of shear resistance can be still performed by applying Equation (A2.2)
and considering a reduced value of both the acting shear force Vsd (due to the negative value of
δ parameter in the second addendum) and of the shear resistance Vrd (which should be indeed
calculated on the effective cross-section of the slab, reduced by the presence of the opening).
Table A2-1. Average percentage variation of shear force at support (by considering three values of concrete topping thickness, that is to say 0, 4 and 8 cm) with respect to floor without openings at SLS condition (“SLS”). Magnitude of the translation applied on the SLS reference curve (“shift”) by considering the maximum shear variation during the loading history and corresponding final value of the average percentage variation of shear force at support reported in ULS design charts (“MRd”) for different slab thickness (22, 32, 40 and 50 cm) and for a floor slenderness ratio L/H = 30.
left slab right slab left slab right slab left slab right slab left slab right slabSLS -29 8 -20 6 -22 10 -18 8MRd -27 12 -18 9 -18 13 -17 12
shift 2.7 4.6 1.9 3.0 4.2 3.2 1.7 3.4
SLS -19 -24 -11 -17 -11 -14 -9 -14MRd -13 -15 -7 -11 -5 -9 -7 -10
shift 6.3 9.0 4.2 6.5 5.9 5.8 1.9 3.7
SLS 37 35 39 36 41 39 42 39MRd 47 47 45 45 48 45 46 44
shift 10.3 12.5 5.7 8.9 7.5 6.0 4.2 5.1
SLS 76 77 77 77 81 81 83 82MRd 91 97 92 95 97 98 96 99
shift 15.1 19.6 14.7 17.6 15.3 16.9 14.0 16.7
ftA
ftB
ftC
ftD
foro L=30HShear average percentage variation with respect to the case of a floor without openings
H22 H32 H40 H50opening
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ftA ftB ftC ftD
ftA opening
Slab # 6 Slab # 7
H22
H32
H40
H50
Figure A2-1. Design charts ( - L/H relative to slabs 6 and 7, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with ftA opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321 7654321 7654321
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
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ftA ftB ftC ftD
ftB opening
Slab # 5 Slab # 6
H22
H32
H40
H50
Figure A2-2. Design charts ( - L/H relative to slabs 5 and 6, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with ftB opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321 7654321 7654321
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasatocappa 4 cmcappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
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ftA ftB ftC ftD
ftC opening
Slab # 4 Slab # 6
H22
H32
H40
H50
Figure A2-3. Design charts ( - L/H relative to slabs 4 and 6, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with ftC opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321 7654321 7654321
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.35
0.40
0.45
0.50
0.55
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
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ftA ftB ftC ftD
ftD opening
Slab # 3 Slab # 6
H22
H32
H40
H50
Figure A2-4. Design charts ( - L/H relative to slabs 3 and 6, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with ftD opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321 7654321 7654321
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
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ftA ftB ftC ftD
ftA opening
Slab # 6 Slab # 7
no topping
4 cm topping
8 cm topping
Figure A2-5. Design charts ( - L/H relative to slabs 6 and 7, obtained from numerical analyses carried out on floors with concrete topping of variable thickness (0, 4 and 8 cm) and with ftA opening, for different HC slabs thickness (H22, H32, H40 and H50 floors, where the number after H represents the slab thickness in cm).
7654321 7654321 7654321 7654321
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
H22H32H40H50
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
H22H32H40H50
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.05
0.10
0.15
0.20
0.25
25 30 35
δδδδ
L/H
H22H32H40H50
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ftA ftB ftC ftD
ftB opening
Slab # 5 Slab # 6
no topping
4 cm topping
8 cm topping
Figure A2-6. Design charts ( - L/H relative to slabs 5 and 6, obtained from numerical analyses carried out on floors with concrete topping of variable thickness (0, 4 and 8 cm) and with ftB opening, for different HC slabs thickness (H22, H32, H40 and H50 floors, where the number after H represents the slab thickness in cm).
7654321 7654321 7654321 7654321
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
H22H32H40H50
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ftA ftB ftC ftD
ftC opening
Slab # 4 Slab # 6
no topping
4 cm topping
8 cm topping
Figure A2-7. Design charts ( - L/H relative to slabs 4 and 6, obtained from numerical analyses carried out on floors with concrete topping of variable thickness (0, 4 and 8 cm) and with ftC opening, for different HC slabs thickness (H22, H32, H40 and H50 floors, where the number after H represents the slab thickness in cm).
7654321 7654321 7654321 7654321
0.00
0.10
0.20
0.30
0.40
0.50
0.60
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.10
0.20
0.30
0.40
0.50
0.60
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.10
0.20
0.30
0.40
0.50
0.60
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.10
0.20
0.30
0.40
0.50
0.60
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.10
0.20
0.30
0.40
0.50
0.60
25 30 35
δδδδ
L/H
H22H32H40H50
0.00
0.10
0.20
0.30
0.40
0.50
0.60
25 30 35
δδδδ
L/H
H22H32H40H50
10
ftA ftB ftC ftD
ftD opening
Slab # 3 Slab # 6
no topping
4 cm topping
8 cm topping
Figure A2-8. Design charts ( - L/H relative to slabs 3 and 6, obtained from numerical analyses carried out on floors with concrete topping of variable thickness (0, 4 and 8 cm) and with ftD opening, for different HC slabs thickness (H22, H32, H40 and H50 floors, where the number after H represents the slab thickness in cm).
7654321 7654321 7654321 7654321
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
H22H32H40H50
0.60
0.70
0.80
0.90
1.00
25 30 35
δδδδ
L/H
H22H32H40H50
11
A2.2 Openings placed near midspan
A2.2.1 Reduction of floor bearing capacity in presence of openings (with respect to the
corresponding case of floor without openings)
Table A2-2 reports the reduction of bearing capacity of a floor with openings with respect to
the corresponding case of floor without openings, defined through the parameter δ*:
δ∗[%] = -�M/��0,�� − M/��0,��� M/��0,��1 2100, (A2.3)
being Mfail,op and Mfail,int the values of bending moment corresponding to the numerical failure of the
floor (formed by the assembly of the considered 7 slabs) with and without openings. These values
have been conventionally evaluated as M = pl! 8⁄ , where p represents the applied load for slab
unit length acting in correspondence of numerical failure and l is net span of the floor.
It should be remarked that the reduction of bearing capacity reported in Table A2-2 is relative to
the whole floor, while it does not provide any “punctual” information on the state of stress acting on
each single slab cut by the opening or adjacent to the opening itself, differently from design charts
reported in § A2.2.3.
Table A2-2 – Reduction of bearing capacity δ* in case of a floor with an opening placed near midspan with respect to the corresponding floor without openings.
L/H = 25 L/H = 30 L/H = 35 L/H = 25 L/H = 30 L/H = 35
s = 0 cm -10.69 -16.32 -14.73 s = 0 cm -5.34 -11.42 -9.79
s = 4 cm -8.60 -16.11 -13.94 s = 4 cm -3.44 -10.07 -7.63
s = 8 cm -9.94 -9.70 -19.23 s = 8 cm -4.97 -5.54 -14.62
s = 0 cm -10.23 -14.05 -22.23 s = 0 cm -5.84 -9.36 -18.52
s = 4 cm -10.37 -13.08 -15.17 s = 4 cm -5.67 -7.76 -8.38
s = 8 cm -11.55 -14.11 -19.22 s = 8 cm -4.95 -11.79 -12.81
s = 0 cm -13.49 -34.07 -14.52 s = 0 cm -7.43 -22.15 -12.52
s = 4 cm -14.50 -31.00 -15.67 s = 4 cm -9.81 -20.67 -10.22
s = 8 cm -16.46 -27.72 -17.59 s = 8 cm -9.40 -19.51 -13.19
s = 0 cm -22.46 -24.91 -36.26 s = 0 cm -13.94 -18.18 -29.35
s = 4 cm -14.66 -25.26 -31.47 s = 4 cm -10.00 -13.34 -24.09
s = 8 cm -15.99 -24.14 -29.60 s = 8 cm -12.56 -16.93 -17.27
δ∗δ∗δ∗δ∗ =(M fail,op - M fail,int )/M fail,int [ %]
H22
H32
H40
H50
fmE opening fmF opening
H22
H32
H40
H50
12
A2.2.2 Increase of floor deformability near openings placed at midspan at SLS
In case of openings placed near midspan, design charts for SLS condition report the maximum
deflection variation φ of the ith slab with respect to the corresponding case of floor without
openings:
φ = -�f�,�� − f��� f��⁄ 2, (A2.4)
being fi,op and fint the maximum deflection of the ith slab of a floor with openings and of an intact
panel.
It should be remarked that the values of midspan deflection used for the evaluation of parameter φ
(f�,�� e f��, according to Equation A.4), have been subtracted from the initial deflection due to
prestressing, so representing the “net” deflection due to the application of slab dead weight
(including the concrete topping, if present) and of permanent and variable loads at SLS.
The values of parameter φ reported in design charts, for a given floor slenderness ratio L/H and a
given thickness of the concrete topping, have been obtained as the envelope of the results
provided by numerical analyses at SLS, referred to different reinforcement ratios (similarly to the
first step of the procedure already described for shear in § A2.1)
Deflection control at SLS can be then checked by calculating the total deflection of the ith slab of
the floor with openings (including the effects of prestressing) through the following expression:
f�,898 = f�,�� + f�,�� = �f�,�� + f:,���(1 + φ�) + f�,�� ≤ f0�� . (A2.5)
The total deflection of the ith slab of the floor with openings, f�,898, can be then evaluated by
summing up its own “net” deflection f�,��, deduced from that of the corresponding intact slab
suitably increased by parameter φ (through Equation A2.4), and the initial deflection of the intact
slab due to prestressing (with opposite sign). More in details, in Equation (A2.5) f�,�� represents
the midspan deflection of the intact slab due to the application of its dead weight and that of
concrete topping if present, while f:,�� represents the midspan deflection of the intact slab due to
permanent (G2) and variable loads at SLS. These deflections can be manually computed by
applying the usual expressions based on beam theory.
13
fmE fmF
fmE opening
Slab #5 Slab #6
H22
H32
H40
H50
Figure A2-9. Design charts ;- L/H relative to slabs 5 and 6, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with fmE opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLESLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLESLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLESLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLESLE
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
SLS SLS
SLS SLS
SLS
SLS SLS
SLS SLS
14
fmE fmF
fmF opening
Slab # 6 Slab # 7
H22
H32
H40
H50
Figure A2-10. Design charts <- L/H relative to slabs 6 and 7, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with fmF opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
25 30 35
ϕϕϕϕ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
SLS SLS
SLS SLS
SLS SLS
SLS SLS
15
A2.2.3 Maximum bending moment variation in case of openings placed at midspan at SLS
and ULS
In case of openings placed near midspan, design charts report the maximum bending moment
variation δ of the ith slab with respect to the corresponding case of floor without openings:
δ = -�M�,�� − M��� M��⁄ 2. (A2.6)
being M�,�� and M�� the maximum bending moments of the ith slab at midspan in case of a floor
with or without openings.
In this paragraph two different design charts are provided, respectively referred to SLS and ULS
limit states. Design charts for SLS have been here reported for sake of completeness and have
been used in the construction of ULS ones (by operating a suitable shifting upwards of the curves,
as already done for shear). However, resistance verifications should be carried out with reference
to ULS charts.
For a given floor slenderness ratio L/H and a given thickness of concrete topping, the value of
parameter δ reported in ULS charts has been deduced by following a two-step procedure. First, a
unique value of δ has been calculated as the envelope of the results provided by numerical
analyses at SLS, referred to different reinforcement ratios. Numerical analyses have been indeed
performed by considering different reinforcement ratios for a same typology of hollow core floor, as
a function of the applied variable load (assumed equal to 5 kN/m2, 10 kN/m2 or 15kN/m2). By
repeating this procedure for each considered floor slenderness ratio L/H (25, 30, 35), a SLS design
curve has been obtained for a given value of concrete topping (0, 4, 8 cm). Subsequently, the so
obtained SLS design curve has been properly amplified, by shifting it upwards of a quantity
∆δ = |δ��� − δ|, being δmax the maximum value assumed by the parameter δ for an applied load
corresponding to SLS or to the reaching of the cross-sectional resistant moment Mrd (or the
reaching of numerical failure, if the latter was anticipated with respect to Mrd for the considered
case of opening).
The verification of bending resistance can be then performed by evaluating the applied moment
through the following expression:
M�� =�=>?�?@=>A�A@=B"�0A
Cb(1 + δ) ≤ M'� , (A2.7)
where b is the slab width and l its net span, while parameter δ can be deduced from ULS design
charts. It should be observed that in this case the parameter δ has been obtained by considering
16
both the 1st and the 2nd phase loads (that is to say dead load G1 of slabs, joint and concrete
topping, if present; or permanent loads G2 and variable loads, respectively) applied on the floor.
The so evaluated acting moment Msd should be then compared with the resistant moment Mrd,
calculated on the effective cross-section of the slab.
In case of openings that affect only part of slab width (opening case fmF), the parameter
δ referred to the slab interested by the opening assumes a negative value, so representing a
reduction of acting moment at midspan, due to the lower load applied on the slab itself. In this
case, the verification of bending resistance can be still performed by applying Equation (A2.7) and
considering a reduced value of both the acting bending moment Msd (due to the negative value of
δ parameter) and of the resistant moment Mrd (which should be indeed calculated on the effective
cross-section of the slab, reduced by the presence of the opening).
The verification of intact slabs placed near the opening (that is to say slabs 4 and 7 for opening
fmE and slabs 5 and 7 for opening fmF), which are subjected to an increase of acting bending
moment at midspan, can be conservatively performed by adopting the values of δ provided by ULS
charts for slab 7 in case of opening fmF.
17
fmE fmF
fmE opening
SLS Slab # 5 Slab # 6
H22
H32
H40
H50
Figure A2-11. Design charts ;- L/H at SLS relative to slabs 5 and 6, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with fmE opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
SLS SLS
SLS SLS
SLS SLS
SLS SLS
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
18
fmE fmF
fmE opening
ULS Slab # 5 Slab # 6
H22
H32
H40
H50
Figure A2-12. Design charts ( - L/H at ULS relative to slabs 5 and 6, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with fmE opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasatocappa 4 cmcappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasatocappa 4 cmcappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
19
fmE fmF
fmF opening
SLS Slab # 6 Slab # 7
H22
H32
H40
H50
Figure A2-13 . Design charts <- L/H at SLS relative to slabs 6 and 7, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with fmF opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
SLE
SLS SLS
SLS SLS
SLS SLS
SLS SLS
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
20
fmE fmF
fmF opening
ULS Slab # 6 Slab # 7
H22
H32
H40
H50
Figure A2-14. Design charts ( - L/H at ULS relative to slabs 6 and 7, obtained from numerical analyses carried out on H22, H32, H40 and H50 floors (where the number after H represents the floor thickness in cm) with fmF opening, for different concrete topping thickness (respectively equal to 0, 4 and 8 cm).
7654321 7654321
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
25 30 35
δδδδ
L/H
rasato
cappa 4 cm
cappa 8 cm
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping
no topping 4cm topping 8cm topping