Progetto 3 Earthquake resistant design

Aχ

x y G I I H k k

G G x y x y

Element 2 4 4

m m kN m m m m kN/m kN/m

1 3.16 0.83 704.20 8.74 2.41 34.92 4.43 1.11E+05 6.38E+05

2 5.49 3.19 398.86 8.30 30.57 2.23 4.43 5.88E+05 1.03E+05

Walls 3 3.16 5.39 719.53 7.16 1.32 28.61 4.43 6.32E+04 5.23E+05

4 0.82 3.19 398.86 8.30 30.57 2.23 4.43 5.88E+05 1.03E+05

Table 14 : Parameters for the first floor

Aχ

x y G I I H k k

G G x y x y

Element 2 4 4

m m kN m m m m kN/m kN/m

1 3.16 0.40 262.60 4.21 0.27 16.75 2.89 4.70E+04 5.90E+05

2 5.49 2.74 333.03 8.61 33.73 2.34 2.89 1.20E+06 3.38E+05

Walls 3 3.16 5.47 518.63 8.31 2.07 33.08 2.89 3.05E+05 1.17E+06

4 0.82 2.74 331.01 8.56 33.53 2.30 2.89 1.20E+06 3.33E+05

Floor 3.16 2.84 52.72

Table 15 : Parameters for the second floor

Aχ

x y G I I H k k

G G x y x y

Element 2 4 4

m m kN m m m m kN/m kN/m

1 3.17 0.40 386.48 4.22 0.27 16.91 4.24 1.54E+04 3.32E+05

2 5.93 3.14 285.13 4.15 16.12 0.27 4.24 3.23E+05 1.52E+04

Walls 3 3.17 5.87 386.48 4.22 0.27 16.91 4.24 1.54E+04 3.32E+05

4 0.31 3.14 217.41 3.17 12.29 0.12 4.24 2.47E+05 6.83E+03

Floor 3.07 3.14 73.52

Table 16 : Parameters for the third floor

Aχ

x y G I I H k k

G G x y x y

Element 2 4 4

m m kN m m m m kN/m kN/m

1 3.17 0.31 293.17 3.27 0.13 13.10 4.15 7.76E+03 2.67E+05

2 6.00 3.12 246.02 3.43 13.30 0.15 4.15 2.77E+05 9.19E+03

Walls 3 3.17 5.92 293.17 3.27 0.13 13.10 4.15 7.76E+03 2.67E+05

4 0.31 3.12 227.38 3.17 12.29 0.12 4.15 2.56E+05 7.28E+03

Floor 3.14 3.12 80.80

Table 17 : Parameters for the fourth floor

Aχ

x y G I I H k k

G G x y x y

Element 2 4 4

m m kN m m m m kN/m kN/m

1 3.17 0.25 305.36 2.64 0.07 10.57 5.36 1.91E+03 1.37E+05

2 6.08 3.11 251.81 2.61 10.27 0.07 5.36 1.34E+05 1.89E+03

Walls 3 3.17 6.00 335.90 2.90 0.09 11.62 5.36 2.54E+03 1.51E+05

4 0.27 3.11 266.92 2.77 10.89 0.08 5.36 1.43E+05 2.25E+03

Floor 3.18 3.11 116.20 19

Table 18 : Center of gravity and stiffness of the floors

x x e y y e

G K x G K y

Level m m cm m m cm

Ground 3.16 3.15 -0.40 2.99 3.16 17.17

1 3.16 3.16 0.18 3.28 3.16 -11.90

2 3.28 3.48 19.68 3.14 3.20 6.28

3 3.20 3.26 5.72 3.12 3.17 5.95

4 3.13 3.09 -4.60 3.19 3.16 -2.35

2.4 Shear strength

Although towers are constructions mainly subjected to flexural actions, the verification against

shear is performed in order to identify possible weak storeys inside the tower. In the following,

the shear strength of the storeys of the tower is assessed through the simplified model proposed

in the Italian normative on cultural heritage.

Shear strength is evaluated for both directions x and y assuming that the walls of the building in

that direction reach the average tangential stress. The shear strength of the i-th floor in the two

directions is expressed in Equation(18):

,

,

= = (18)

, ,

, ,

, ,

Where:

• is the design value of shear resistance of the wall, expressed as:

0

= √1 + (19)

0 1.5 0

In which is the design average tangential stress, equal to the one from the table re-

0

duced by the confidence factor, and is the average vertical stress on walls of the i-th

0

floor, obtained by dividing the total vertical load by the area of the walls.

•

and are coefficients that account for the stiffness and resistance homogeneity of

, ,

the masonry walls in that direction.

2 2

∑ ∑

, ,

, ,

= 1 − 0.2√ −1 = 1 − 0.2√ −1 (20)

, ,

2 2

, ,

In which is the number of walls in direction for floor and is the are of the

, ,

j-th wall.

• is a coefficient linked to the type of failure expected mainly in the masonry walls of

the i-th floor; it is 1 in the case of collapse due to shear, while it can be assumed to be

equal to 0.8 in the case of collapse due to pressure-bending;

• is a coefficient linked to the resistance of the floor masonry bands in the walls arranged

.

in direction It is 1 in the case of resistant bands (breaking of vertical masonry walls),

while it can take a value between 0.8 and 1 in the case of weak bands, not able to block

the rotation at the ends of the masonry walls;

•

and are coefficients of irregularity in plan, expressed as:

, ,

, ,

= 1 + 2 = 1 + 2 (21)

, ,

, , –

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

In which and are the maximum distances between the centre of stiffness and

, ,

the outermost wall in the y and x directions, respectively;

•

is the ratio between the shear force at the floor and the total seismic force. In order

to determine this value, a modal shape is assumed. According to the Italian directive, a

linear modal shape has been assumed, equal to:

ℎ

Φ = (22)

ℎ

In which is the height of the floor and is the total height of the building. The factor

is then:

∑ Φ

=

= (23)

∑ Φ

In the following tables the results in x and y directions are reported.

Table 19 : Shear resistance in x direction

σ τ β Φ κ

A µ M F

0 di x x x SLV,x

Level 2 2 2

kN/m kN/m m - - - ton - kN

Ground 272.41 89.99 19.09 0.98 1.15 0.20 236.56 1.00 938.96

1 201.73 79.70 15.02 0.93 0.91 0.33 152.08 0.88 885.48

2 246.97 86.43 10.13 1.00 1.04 0.52 135.73 0.76 703.28

3 196.14 78.83 7.85 1.00 1.04 0.70 114.76 0.59 644.64

4 137.83 69.10 6.65 0.99 0.98 0.94 169.00 0.39 755.74

Table 19 : Shear resistance in y direction

σ τ β Φ κ

A µ M F

0 di y y y SLV,y

Level 2 2 2

kN/m kN/m m - - - ton - kN

Ground 272.41 89.99 19.91 1.00 1.00 0.20 236.56 1.00 1150.54

1 201.73 79.70 20.60 1.00 1.00 0.33 152.08 0.88 1184.89

2 246.97 86.43 8.78 0.97 1.12 0.52 135.73 0.76 551.76

3 196.14 78.83 7.91 0.99 1.04 0.70 114.76 0.59 646.46

4 137.83 69.10 6.46 0.99 0.97 0.94 169.00 0.39 748.16

As reported in NTC18, §5.4, the shear resistance in the two directions is the minimum of the

values found in every floor, therefore =644.64 kN and =551.76 kN.

, ,

2.4.1 Seismic mass of the building

The seismic mass to consider for the ultimate limit state is that associated to gravitational loads,

equal to: +∑

2

= = 808.12 ton (24)

Where are the permanent loads and the variable loads. it the gravity acceleration, equal

2

to 9.806 m/s . The participant mass e* is given by Equation(25):

2

Φ

(∑ )

e* = = 0.762 (25)

2

∑

Φ

21

2.4.2 Evaluation of the ordinate of the elastic spectrum

With reference to the condition that leads to the achievement of the SLV it is possible to obtain

the ordinate value of the elastic response spectrum as:

= (26)

, e*

Table 20 : Elastic response spectrum ordinate for the two directions

q F e* M S S

SLV e,SLV e,SLV

Direction 2

- kN - ton m/s g

x 2.3 644.64 0.7621 808.12 2.4074 0.2455

y 2.3 551.76 0.7621 808.12 2.0606 0.2101

Once the ordinate is known the return period associated can be obtained by linearly interpolating

the data with those coming from the spreadsheet “SpettriNTC”:

− −

1 1

= (27)

− −

2 1 2 1 ∗

Where is one of the quantities between , and . Making the hypothesis on the value

0

of each parameter is found and then the elastic spectrum ordinate is:

( )

= (28)

0

The procedure is iteratively made until the convergence on is reached, founding the corre-

,

sponding return period.

Table 21 : Parameters associated to the site for the 4 limit states

*C

T T a F S S

R g 0 e

yrs yrs g - - g

30 0.248 0.043 2.569 2.160 0.240

50 0.258 0.053 2.581 2.160 0.295

475 0.289 0.119 2.513 2.160 0.647

975 0.293 0.154 2.450 2.160 0.817

Being the elastic response spectrum ordinate associated to the tower lower than the one associated

to a return period of 30yrs, there is no need for interpolation and the values corresponding to T

R

=30yrs are assumed. Being T =0.515s, the ordinate of the response spectrum is equal to 0.240g.

1

Inverting Equation(28) we get the acceleration that leads to SLV state:

( )

1

= = 0.043g (29)

0

Using Equation(10) and (11) the seismic safety indices are obtained:

30 0.043

= = 0.06 = = 0.36 (30)

, ,

475 0.119

As can be seen from the indices, the vulnerability of the tower with respect to the expected rare

earthquake is very high, therefore an immediate intervention is needed. –

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

2.5 Flexural strength

This verification will be carried out by means of a comparison between the design moment acting

and the resisting moment at each section of the tower. The checks are performed in both main

directions of inertia and at each floor to identify the most critical section. For the computation of

the acting moment, a system of forces distributed along the entire height of the structure will be

considered, assuming a linear trend of the displacements. The resultant of the seismic action on

the i-th floor is equal to:

∑

= (31)

ℎ ∑

Where:

• is given by:

ℎ

( )

= 0.85 with = 2.8 (32)

ℎ 1

• ;

and are the weight of sections and

•

and are the heights, measured from the foundation, of the centers of mass of the

sections.