PSI - Issue 44

Olivier LHERMINIER et al. / Procedia Structural Integrity 44 (2023) 528–535 LHERMINIER Olivier / Structural Integrity Procedia 00 (2022) 000–000

531

4

600

P 3

Load case P 1 : = ° max( ) = max( ) = ,

Load case P 2 : = ° max( ) = max(−( + )/√2) = −(

P 4

P 2

400

,2 + ,2 )/√2 , 2

200

F y [ kN ]

P 5

P 1

0

1 = [ ,1 , ,1 ] = [ , , ,1 ] , 1

−200

P 8

P 6

2 = [ ,2 , ,2 ]

−400

P 7

−600

(b)

−600 −400 −200

0

200 400 600

(a)

F x [ kN ]

Fig. 2: (a) Example of elliptical envelope of forces F x , F y at the basis of a structure ( n r = 2) and maximization of F θ = F x cos θ + F y sin θ for θ = 0 ◦ and θ = 225 ◦ . (b) Elliptical envelope of forces F x , F y at the basis of the SPEAR structure (Section 3) and response vectors corresponding to the maximization of F θ for eight θ -values ( θ = 0 ◦ , 45 ◦ , 90 ◦ , 135 ◦ , − 180 ◦ , − 135 ◦ , − 90 ◦ , − 45 ◦ ): load cases P1, ..., P8.

2.4. Procedure for the definition of the pushover load pattern

In this Section, a procedure is proposed for the definition of a multi-modal load pattern for pushover analysis under multi-component earthquakes, based on all the n modes calculated for the analyzed structure : (a) Chose the n r seismic responses F 1 , F 2 , ... (forces, moments, drifts...) composing the response vector f ; (b) Calculate the response matrix X f ; (c) Define a scalar response F to be maximized when the pushover load pattern is applied on the structure ; (d) Calculate the weighting factors α i , k using Eq. (4). They allow expressing f of step (c) and any other simultaneous seismic response as a weighted sum of the response spectrum vectors F i , k ;

(e) Use α i , k of step (d) to define the nodal force field Q by Eq. (2); (f) Use Eq. (1) to define the load pattern with increasing amplitude q pushover

suited for the pushover analysis.

2.5. Procedure for the definition of dominant modes and of the associated load pattern for pushover analysis

In several practical situations, instead of using ”all” the modes, it is interesting to define the load pattern for the pushover analysis by using only few dominant modes , i.e. the modes which give the most important contributions to the response F = f T · b . Only the components having an important impact on the response f will be retained in the dominant mode definition. In the case of dominant modes analysis, steps (a), (b), (c), (d) and (f) of section 2.4 remain unchanged, while step (e) has to be replaced by the following list of three items: (e ∗ ) Based on the α i , k factors calculated at step (d), compute the products F b , i , k = α i , k F T i , k · b (contributions of each mode i for each earthquake component k to the response F = f T · b of step (c)). Order these products from the one with the maximum absolute value to the one with the smallest one. The index describing this order is called η = 1 , ... , n ec n . One has η = η ( i , k ). (e ∗∗ ) Compute the cumulative sum of the products F b ,η = α η F T η · b according to the order defined at item (e*) and find the number of these products ˜ η min < n ec n such that the di ff erence between their accumulated contribution and the total response F = f T · b is less than 10% as suggested in ASCE-4-16 (2017) at chapter 4.3.1(b). (e ∗∗∗ ) Compute the nodal force field Q based on dominant modes using the following updated version of Eq. (2):

˜ η min η = 1

α η M · A max ,η

(5)

Q =