Progetto 1 Earthquake resistant design

G G

2

Ton Ton·m m m

First 35.7097 433.7319 5.525 2.000

Second 34.7381 421.9310 5.525 2.000 –

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

Third 34.7381 421.9310 5.525 2.000

Roof 29.7831 361.7464 5.525 2.000

In the following, the procedure to create the frame is illustrated:

1. Creation of the material. Properties > Material > Material Properties. Select the Italian

Building Code NTC18 and then concrete class C25/30 from the menu;

2. Creation of the nodes. Node/Element > Nodes > Create Nodes. Coordinates of the nodes

of the plant are inserted and then they are copied at different heights. The first row of

nodes is composed of nodes at (0,0), (2.25,0), (6.95,0), (11.05,0) m. Then the row is re-

peated in direction Y with a step of 4 m. Once the nodes of the ground level are created,

they are repeated at heights 3.275, 6.275, 9.275, 12.275m. At the end it is obtained the

model reported in Figure 4;

3. Creation of the elements. Node/Element > Elements > Create Elements. Beam, columns

and walls are created taking in account shear deformability but no warping effects;

4. Creation of the sections. Properties > Section > Section Properties. Different sections

are created in order to represent beams, columns and walls. In particular, L-shaped wall

and corner column cross-sections needed an offset in order to be in the position of the

actual structure;

5. Creation of the rigid diaphragms. Structure > Building > Control Data > Story > Auto

Generate Story Data;

6. Imposing boundary conditions. Boundary > Supports > Define Supports. The 8 nodes of

the ground floor are restrained by perfects clamps;

7. Creation of the masses. Load > Static Loads > Loads to Masses > Floor Diaphragm

Masses. For each floor values reported in Table 7 are inserted.

Figure 4 : The basic model represented in the software

Before running the modal analysis, the number of modes should be considered by using the com-

mands Eigenvalues > Analysis, in which the number of modes to be extracted is set to 12, as in

the experimental study.

Once obtained the results, the quality of the model is evaluated looking at the relative error in

terms of modal periods. The relative error is obtained with the following:

−

ε= (2)

To obtain a better understanding of the results it is useful to consider two types of errors: 9

1. Mean relative error on the 12 modes;

2. Mean relative error on the number of modes which represents at least the 90% of the

participating mass of the structure. In this case the first 7 modes will be considered.

In Table 8 are reported the relative errors for each mode; the assumptions made on the first model

lead to an error of 7.20% and of 8.36% for the first 7 modes.

Table 8 : Relative errors of model 1.1

Mode T T Error

model lab

s s %

1 0.533 0.4854 9.80

2 0.2739 0.2538 7.92

3 0.2678 0.2320 15.42

4 0.1659 0.1570 5.68

5 0.0903 0.0861 4.93

6 0.0786 0.0775 1.39

7 0.074 0.0653 13.37

8 0.0612 0.0604 1.35

9 0.0394 0.0411 -4.22

10 0.034 0.0317 7.10

11 0.0253 0.0289 -12.46

12 0.0212 0.0218 -2.76

2.2.2 Model 1.2

Referring to Equation (3), it is evident that natural periods are inversely proportional to the stiff-

ness of the structure. For this reason, in order to reduce the relative error with the experimental

data, it is necessary to decrease the stiffness of the model.

= 2π√/ (3)

For this purpose, the panel zone effect was considered: in MIDAS it is possible to define a

value between 0 and 1 to take in account the rigidity of the nodes; if the value is set 1, then the

stiffness of the node is taken at full; instead, if the value is null the stiffness is not taken into

account. Since the goal is to reduce the relative error, the value set for the panel zone effect will

be in between 0 and 1.

By Boundary > Misc > Panel Zone Effect, a value 0.62 was set. The new results are reported

in Table 9. The mean relative error in this case is equal to 4.37% and the error on the first 7 modes

is 3.37%. As expected, including the panel zone effect has improved the model.

. Table 9 : Relative errors of model 1.2

Mode T T Error

model lab

s s %

1 0.5052 0.4854 4.07

2 0.2434 0.2538 -4.10

3 0.2341 0.2320 0.90

4 0.1581 0.1570 0.71

5 0.0864 0.0861 0.40

6 0.0728 0.0775 -6.09

7 0.0655 0.0653 0.35

8 0.0589 0.0604 -2.46

9 0.0375 0.0411 -8.84

10 0.0313 0.0317 -1.40

11 0.0246 0.0289 -14.88

12 0.0202 0.0218 -8.26 –

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

2.2.3 Model 1.3

In the real frame the slab is connected uniformly with the beams of the floor and therefore there

is collaboration between the two members. If this fact is taken in account two approaches may be

used: 1. Beam cross-section is modified with the adding of a flange of height equal to that of

the slab and width to be determined;

2. A plate element is used, joined by nodes with the beams.

In this model the first option is considered. In order to determine the flange width is made refer-

ence to the book by Paulay and Priestley (1992). The authors suggest to respect the quantities in

Figure 5. Figure 5 : Effective flange width dimensioning, Paulay et. al (1992)

The L-shape cross section is used for all the beams except for the central vertical beam. The

effective width of each flange is reported in Table 10, in which the numbering of the beams makes

reference to Figure 6. Table 10 : Effective flange width of the beams

Beam b eff

m

1 0.42

2 0.42

3 0.50

4 0.45

5 0.34

6 0.42

Implementing the new cross-sections in the basic model and launching the modal analysis the

errors reported in Table 11 were obtained. The mean relative error for all the 12 modes is 5.21%

and for the first 7 ones is 4.91%. 11

Beam 2

Beam 4

Beam 5

Beam 5 Beam 4 Beam 2

Beam 3 Beam 1

Beam 6 Beam 3 Beam 1

Beam 5 Beam 2

Beam 4 Beam 2

Beam 4

Beam 5 Figure 6 : Numbering of beams

Table 11 : Relative errors of model 1.3

Mode T T Error

model lab

s s %

1 0.5083 0.4854 4.71

2 0.2607 0.2538 2.72

3 0.2545 0.2320 9.69

4 0.1603 0.1570 2.11

5 0.0886 0.0861 2.95

6 0.0758 0.0775 -2.22

7 0.0718 0.0653 10.00

8 0.0609 0.0604 0.85

9 0.0387 0.0411 -5.92

10 0.0336 0.0317 5.84

11 0.0252 0.0289 -12.81

12 0.0212 0.0218 -2.76

2.2.4 Model 1.4

As done previously with the basic model, it is considered the panel zone effect on the structure

with the effective flange width. After many trials the value of 0.62 was chosen as input in the

program. With this setting the value of the error is 4.78% and the error on the first 7 modes is

3.35%. Table 12 : Relative errors of model 1.4

Mode T T Error

model lab

s s %

1 0.4982 0.4854 2.63

2 0.256 0.2538 0.86

3 0.2531 0.2320 9.09

4 0.1567 0.1570 -0.18

5 0.0863 0.0861 0.28

6 0.0751 0.0775 -3.12

7 0.0701 0.0653 7.39

8 0.0589 0.0604 -2.46

9 0.0382 0.0411 -7.14

10 0.0325 0.0317 2.38

11 0.0247 0.0289 -14.54

12 0.0202 0.0218 -7.34

2.2.5 Model 1.5

Deformability of the slab can be considered by removing the hypothesis of rigid diaphragm. From

Node/Element > Elements > Create Elements > Plate, a plate member of thickness 15cm is

–

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

created. Moreover, the concentrated mass is removed and mass of structural elements is consid-

ered distributed and automatically computed by the code. In order to do this the following com-

mands are used:

1. Definition of the conversion of the weight load. Load > Static Load > Structure

Loads/Masses > Load to Masses;

2. Auto-generation of the structural masses. Structures > Structure Type > Convert Self-

weight into Masses.

Results are reported in Table 13: a relative error of the 12 modes equal to 24.39% is found and an

error of the first 7 modes equal to 10.54%. The plate added has made worse the error between the

model and the experimental test.

Table 13 : Relative errors of model 1.5

Mode T T Error

model lab

s s %

1 0.4922 0.4854 1.39

2 0.2996 0.2538 18.04

3 0.266 0.2320 14.65

4 0.1581 0.1570 0.71

5 0.0921 0.0861 7.02

6 0.0904 0.0775 16.62

7 0.0753 0.0653 15.36

8 0.0657 0.0604 8.80

9 0.0504 0.0411 22.52

10 0.0498 0.0317 56.87

11 0.0419 0.0289 44.97

12 0.0405 0.0218 85.77

2.2.6 Choice of the model

Looking at Table 14 the following observations can be made:

1. The rigid diaphragm hypothesis is correct, as can be seen comparing the error in the last

model, without the hypothesis, and the rest of the models;

2. The addition of the panel zone effect to the model leads to a best approximation of ex-

perimental data both in model 1 and 3. This can be explained by the fact that in the real

structure, due to effects such as cracking of the concrete, the stiffness of the nodes is

lower than the theoretical one;

3. Looking at the relative error on the first 7 modes it is evident that the relative error on the

smaller normal modes is greater than the one on the first normal modes. This fact can be

accepted considering that the participating mass associated to the smaller normal modes

is very small and therefore we can accept these greater errors, since they do not affect so

heavily the dynamic response of the frame.

Table 14 : Comparison of the mean relative error in the 5 models

Model Error on the first 12 modes Error on the first 7 modes

% %

1.1 7.20 8.36

1.2 4.37 3.37

1.3 5.21 4.91

1.4 4.78 3.35

1.5 24.39 10.54

In conclusion, comparing all the models created, the better ones are the second and the fourth

model, which differ of a few points percent. Therefore, the second model created will be the

reference model for the next developments. 13

3 HIGH INTENSITY VIBRATIONS

This kind of analysis aims to study the response to seismic load excitation by using an accelero-

gram and/or a response spectrum, considering a life-safety limit state (ULS) and extracting nu-

merical values in terms of base shear and maximum displacements.

The real structure was subjected to 5 tests with different value of PGA, by performing Pseudo-

Dynamic Tests in the laboratory.

Table 15 : Pseudo dynamic tests performed on the dual frame structure

Test Type Design PGA am- PGA Length

plification factor g s

D01 Free oscillations

D02 Pseudo Dynamic 0.05 0.028 8

D03 Pseudo Dynamic 0.1 0.056 15

D04 Pseudo Dynamic 0.15 0.085 4

D05 Pseudo Dynamic 1.00 0.56 15

D06 Pseudo Dynamic 1.50 0.85 6

This kind of simulations is done by an experimental-numerical hybrid approach, in which in-

ertia forces and the viscous damping are analytically modelled and non-linear reactions of the

structure are directly measured. Fixing an initial time-step of 0.005s the equation of motion is

in