PSI - Issue 44

L. Navas-Sánchez et al. / Procedia Structural Integrity 44 (2023) 418–425 L. Navas-Sánchez et al. / Structural Integrity Procedia 00 (2022) 000 – 000

422

5

z

3(1

)

+

a q PGA

h

a a q S T z PGA = ( , )

(

) 0.5

− 

b

(1)

T

2

1 (1

)

+ −

a

a

T

1

MIT19-Nmodes: General Formulation. This formulation (Eq. (2, 3)) is based on a modal superposition approach, in which the results of the general modes i¸j are combined using the SRSS rule. It takes into account the particular spectral acceleration for each specific mode and the NSE damping factor ξ a . In order to apply this formulation to a building with torsional modes, the modal shapes (φ) for the specifi c points under study and modal participation factors ( Γ ) in both horizontal directions (X and Y independently) were taken from the 3D numerical model. φ ij is the j th component of the i th modal shape. This procedure cannot be followed without the 3D model for a building whose main modes are torsional. A different spectrum for each of the two directions was obtained. Therefore, it was worthwhile to study the actual direction of the NSE in order to combine the spectra and the type of loading to be considered (in-plane or out-of-plane loading). ( ) ( / ; ) ( , ) , i a a i i ij e a ij a a S T R T T S T    =  (2)

T

T

) (1 2

) ] , 2 −

[(2

0.4 0.5

 with

R

=

 a

+ −



= −

a

a

(3)

T

T

i

i

The amplification factor R considers the coupling between the i th vibration mode of the structure and the fundamental mode of the NSE through the coefficient β, herein considered 0.5. The values of all the values of φ and Γ cannot be reported for space limitation. MIT19-Simp-Nmodes: Simplified Formulation. This constitutes a simplified multimodal formulation based on Degli Abbati et al. (2018). In this case, the results of the different modes are also combined by using the SRSS rule. The non-linearity of the structure could be included by varyin g the period and the viscous damping of the building ξi corresponding to the period. Eqs. 4 shows its values for each mode considered, being η(ξ a ) the damping correction factor for the NS element ( η(ξ a )=[10/(5+ξ a )] 0.5 ).

        

0.5

−

11

a ( )

( )

PFA z

− −    i

 for T aT

i

a

i

0.5

−

1.6

+ 1 [11

( ) 1](1   

T aT /

)

i

a

a

i

0.5

−

a a ( , , )  z

11

a ( )

( )

a i S T ,

PFA z

  for aT T bT

  

=

(4)

i

i

i

a

i

0.5

−

11

a ( )

( )

PFA z

  

 for T bT

i

i

a

i

0.5

−

1.2

+ 1 [11

− ( ) 1]( /   

1)

T bT

−

i

a

a

i

where a and b are parameters defining the range of periods of maximum amplification of the spectrum ( a =0.8 and b =1.1), and PFA, the Peak Floor Acceleration, which can be determined by means of Eq. (5)

2 ( , ) ( ) 1 0.0004 z    + 

( ) PFA z S T =

(5)

i

e

i

i

i i

i

MIT19-Simp-MRF: The Simplified Formulation for MRF includes the parameters a, b and a p , which have been calibrated to take into account the elongation of the fundamental period due to system non linearity and the contribution of the higher modes. The values of parameters are a =0.8, b =1.4 and a p =5.0 for T 1 <0.5s; a =0.3, b =1.2 and