Progetto 2 Earthquake resistant design

A

Beam Location M max M min V max V min A + es

Ed Ed Ed Ed es 2 2

kNm kNm kN kN mm mm

0 0.00 -14.12 0.00 -49.80 0.00 89.30

1 l/2 11.91 0.00 16.37 0.00 75.32 0.00

l 0.00 -50.83 78.66 0.00 0.00 321.45

0 0.00 -77.85 0.00 -111.10 0.00 492.33

2 l/2 52.59 0.00 5.13 0.00 332.58 0.00

l 0.00 -96.67 118.81 0.00 0.00 611.35

0 0.00 -103.14 0.00 -123.93 0.00 652.26

3 l/2 58.25 0.00 0.00 -10.13 368.38 0.00

l 0.00 -60.57 106.26 0.00 0.00 383.05

0 2.21 0.00 0.00 -63.20 13.98 0.00

4 l/2 26.13 0.00 27.33 0.00 165.25 0.00

l 0.00 -71.09 116.36 0.00 0.00 449.58

0 0.00 -108.56 0.00 -153.66 0.00 686.54

5 l/2 68.95 0.00 2.58 0.00 436.04 0.00

l 0.00 -116.37 157.09 0.00 0.00 735.93

0 0.00 -97.31 0.00 -110.36 0.00 615.39

6 l/2 1.21 0.00 7.12 0.00 7.65 0.00

l 0.00 -98.73 117.92 0.00 0.00 624.38

7 0 0.00 -119.13 0.00 -158.93 0.00 753.39

–

36 EARTHQUAKE RESISTANT DESIGN A.Y. 2022-23 Sostegni

l/2 70.54 0.00 0.00 -4.27 446.10 0.00

l 0.00 -102.61 151.83 0.00 0.00 648.91

0 0.00 -60.00 0.00 -81.85 0.00 379.44

8 l/2 7.13 0.00 0.00 -19.19 45.09 0.00

l 0.00 -15.27 46.89 0.00 0.00 96.57

0 0.00 -15.15 0.00 -46.54 0.00 95.81

9 l/2 6.92 0.00 19.44 0.00 43.76 0.00

l 0.00 -60.61 82.10 0.00 0.00 383.30

0 0.00 -105.52 0.00 -153.94 0.00 667.32

10 l/2 72.49 0.00 2.87 0.00 458.43 0.00

l 0.00 -113.44 157.19 0.00 0.00 717.40

0 0.00 -43.03 0.00 -50.05 0.00 272.12

11 l/2 0.00 -9.24 4.22 0.00 0.00 58.43

l 0.00 -45.47 51.87 0.00 0.00 287.56

0 0.00 -110.47 0.00 -155.37 0.00 698.62

12 l/2 71.16 0.00 1.61 0.00 450.02 0.00

l 0.00 -111.37 155.88 0.00 0.00 704.31

0 0.00 -72.00 0.00 -116.73 0.00 455.33

13 l/2 25.60 0.00 0.00 -27.70 161.90 0.00

l 2.28 0.00 62.72 0.00 14.42 0.00

0 0.00 -60.82 0.00 -106.38 0.00 384.63

14 l/2 58.28 0.00 10.02 0.00 368.57 0.00

l 0.00 -102.81 123.81 0.00 0.00 650.18

0 0.00 -96.69 0.00 -118.84 0.00 611.47

15 l/2 52.65 0.00 0.00 -5.16 332.96 0.00

l 0.00 -77.73 111.08 0.00 0.00 491.57

0 0.00 -51.08 0.00 -78.85 0.00 323.03

16 l/2 11.91 0.00 0.00 -16.53 75.32 0.00

l 0.00 -13.86 49.58 0.00 0.00 87.65

Table 23 : Bending moment, shear and estimated steel area for beams in y-direction -

A

Beam Location M max M min V max V min A + es

Ed Ed Ed Ed es 2 2

kNm kNm kN kN mm mm

0 0.00 -6.41 0.00 -22.52 0.00 40.54

17 l/2 8.16 0.00 7.35 0.00 51.60 0.00

l 0.00 -19.83 32.27 0.00 0.00 125.41

0 0.00 -18.29 0.00 -31.00 0.00 115.67

18 l/2 7.91 0.00 0.00 -6.08 50.02 0.00

l 0.00 -8.43 23.78 0.00 0.00 53.31

0 0.00 -13.92 0.00 -17.66 0.00 88.03

19 l/2 7.71 0.00 0.00 -0.98 48.76 0.00

l 0.00 -12.98 17.18 0.00 0.00 82.09

0 0.00 -11.75 0.00 -15.85 0.00 74.31

20 l/2 5.77 0.00 2.17 0.00 36.49 0.00

l 0.00 -18.52 18.86 0.00 0.00 117.12

0 0.00 -29.17 0.00 -41.30 0.00 184.47

21 l/2 19.18 0.00 1.36 0.00 121.30 0.00

l 0.00 -31.81 42.38 0.00 0.00 201.17

0 0.00 -1.09 0.00 -13.13 0.00 6.89

22 l/2 9.96 0.00 4.32 0.00 62.99 0.00

l 0.00 -20.36 21.34 0.00 0.00 128.76 37

0 0.00 -8.32 0.00 -15.56 0.00 52.62

23 l/2 8.98 0.00 2.10 0.00 56.79 0.00

l 0.00 -15.77 19.02 0.00 0.00 99.73

0 0.00 -17.65 0.00 -20.75 0.00 111.62

24 l/2 11.59 0.00 0.00 -3.76 73.30 0.00

l 0.00 -0.76 13.69 0.00 0.00 4.81

0 0.84 0.00 0.00 -12.78 5.31 0.00

25 l/2 9.81 0.00 4.70 0.00 62.04 0.00

l 0.00 -21.40 21.72 0.00 0.00 135.34

0 0.00 -8.96 0.00 -16.24 0.00 56.66

26 l/2 9.83 0.00 1.56 0.00 62.17 0.00

l 0.00 -13.75 18.39 0.00 0.00 86.96

0 0.00 -19.64 0.00 -21.19 0.00 124.20

27 l/2 10.71 0.00 0.00 -4.17 67.73 0.00

l 0.00 -0.81 13.32 0.00 0.00 5.12

0 0.00 -31.06 0.00 -42.05 0.00 196.43

28 l/2 19.18 0.00 0.00 -1.03 121.30 0.00

l 0.00 -29.91 41.63 0.00 0.00 189.15

0 0.00 -17.89 0.00 -18.57 0.00 113.14

29 l/2 5.80 0.00 0.00 -1.91 36.68 0.00

l 0.00 -12.34 16.14 0.00 0.00 78.04

0 0.00 -12.31 0.00 -16.84 0.00 77.85

30 l/2 7.68 0.00 1.29 0.00 48.57 0.00

l 0.00 -14.66 18.00 0.00 0.00 92.71

0 0.00 -7.10 0.00 -22.88 0.00 44.90

31 l/2 7.98 0.00 6.97 0.00 50.47 0.00

l 0.00 -19.47 31.89 0.00 0.00 123.13

0 0.00 -18.65 0.00 -31.39 0.00 117.94

32 l/2 8.09 0.00 0.00 -6.47 51.16 0.00

l 0.00 -7.74 23.41 0.00 0.00 48.95

6.1 Minimum and maximum steel area for longitudinal reinforcement

Apart from the requested area of steel, there might be beams in which no steel reinforcement is

needed. Anyway, the NTC imposes to consider a minimum amount of steel reinforcement in ten-

sion zones, equal to: 2 2

= 0.26 = 187.85mm ≥ 0.0013 = 175.11mm (52)

,

On the other hand, the maximum steel reinforcement area is equal to:

2

= 0.04ℎ = 6000mm (53)

,

Section §7.4.6.2.1 of NTC imposes that in every section the longitudinal reinforcement is at least

composed by 2 bars of diameter greater or equal to Ø14, both for upper and lower reinforcement.

Moreover, for every bar layer, the following inequality must hold:

1.4 3.5

≤ ρ ≤ ρ + (54)

/ℎ ρ

Where is the geometrical rebar ratio, equal to of the tensioned reinforcement and

of the compressed one. Furthermore, the condition in Equation(55) must be respected everywhere.

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38 EARTHQUAKE RESISTANT DESIGN A.Y. 2022-23 Sostegni

ρ ≥ 0.5ρ (55)

6.2 Computation of the resisting bending moment

The resisting bending moment at the ultimate limit state condition is computed upon some hy-

pothesis on the collapse of the beam:

• Upper reinforcement is not yielded while the lower is;

• ε

Collapse of the structural element happens only when the ultimate limit strain in

the concrete fiber is reached.

Figure 24 : Scheme of the ultimate condition with strains and stresses

If any of these assumptions is violated, the equation needs to be rewritten. With reference to

Figure 24, some nondimensional quantities are defined: ′

′

ε

′ ′

α = = 1.79 ξ= δ = ω = ω = (56)

0

ε

The neutral axis depth is found by imposing equilibrium:

′ 2 ′ ′ ′

(α )ξ

+ = ⟹ 0.8ξ + ω − ω − α ω δ = 0 (57)

0 0

Once the neutral axis depth is computed, the strain in the upper reinforcement is:

′

δ −ξ

′

ε = ε (58)

ξ

If the strain is greater than zero, it means that the neutral axis falls before the depth of the upper

reinforcement, meaning that also upper reinforcement is tensioned.

the neutral axis is computed as:

ε = 1.86‰,

Then, if the strain is greater than

′

+

= (59)

0.8

The resisting bending moment is equal to: ′ ′ ′

( ( )

= 0.8 − 0.4) + ε − (60)

If the upper reinforcement is yielded: ′ ′

( ( )

= 0.8 − 0.4) + − (61)

39

In the following tables, values of the resisting bending moment and the chosen longitudinal rein-

forcement of the beams in Figure 19 are reported.

Table 24 : Positive resisting bending moments for beams in x direction

A + d A - d' x M + M +

s s Rd Ed

Beam Location Rebar + Rebar -

2 2

mm mm mm mm mm kNm kNm

0 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

1 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 11.91

l 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

2 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 52.59

l 2Ø16+1Ø12 515.22 449.67 2Ø16+2Ø12 628.32 50.00 49.82 86.69 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+2Ø12 628.32 50.00 49.82 86.69 0.00

3 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 58.25

l 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 2.21

4 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 26.13

l 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

5 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 68.95

l 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

6 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 1.21

l 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

7 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 70.54

l 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

8 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 7.13

l 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

9 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 6.92

l 2Ø16+1Ø12 515.22 449.67 2Ø16+1Ø12 515.22 50.33 50.01 86.70 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

10 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 72.49

l 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

11 l/2 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

l 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80 49.70 86.68 0.00

0 2Ø16+1Ø12 515.22 449.67 2Ø16+3Ø12 741.42 49.80