APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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1 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 i th 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 i th 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/m 2 , 10 kN/m 2 or 15kN/m 2 ). 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 δ max the maximum value assumed by the parameter δ during the loading history until the reaching of the sectional resistant moment M rd (or until the reaching of numerical failure, if the latter was anticipated with respect to M rd 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 G +G , bl 2 ⁄ +((γ G Q) bl 2 ⁄ (1+δ)≤V (A2.2) being b the slab width, l its design span and G 1,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 2 nd phase loads, which are applied after floor assembly (permanent loads G 2 and variable ones Q), while it should not be applied to 1 st 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

Transcript of APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

Page 1: 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

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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

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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

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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.

Page 13: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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

Page 14: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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

Page 15: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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

Page 16: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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.

Page 17: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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

Page 18: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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

Page 19: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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

Page 20: APPENDIX 2 - STRESS VARIATION IN A HOLLOW CORE FLOOR ...

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