PSI - Issue 44

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

534

7

P1, P5 P2, P6 P3, P7 P4, P8 P9, P10

F θ, i , x [ kN ] or M zz , i , x [ kN / m ]

F θ, i , y [ kN ] or M zz , i , y [ kN / m ]

(c)

(a)

(b)

˜ η Cumulative percentage [%]

i

i

Fig. 6: (a) Modal contributions to F θ and M zz for earthquake component k = x and (b) for earthquake component k = y . (c) Cumulative percentage of modal response contributions ordered according to the decreasing impact on F θ and M zz .

(e**) The five curves of Figure 6c, with abscissa ˜ η , show the accumulated percentage contribution of modes and earthquake directions to the total response. The total response is the one calculated with all the modes, i.e. f T · b = F θ and f T · b = M zz in the last row of Table 2. (e***) Eq. (5) is used to define the new load pattern Q based on dominant modes / earthquake components and to compute the corresponding capacity curves. The factors α i , k are not recalculated and they remain the ones calculated with the non-reduced modal basis. The equivalent SDoF oscillator parameters are recalculated accounting for the ˜ η min dominant modes / components only. Capacity spectra and ADRS response spectra are also recalculated, as well as the target points, which are the vertices of the polygon plotted with the dotted line in 7a. Notice that the two polygons are close each other. Also the horizontal lines in 7b, indicating the torsion moment at the target point for the cases of entire modal basis and of dominant modes / components only, are close. One can conclude that the ˜ η min dominant mode / component contributions defined according to Table 3 su ffi ce for pushover analysis of the SPEAR building.

Table 3: Order of modes according to their impact (percentage) on the seismic response. The penultimate row indicates the number of dominant modes and component contributions ˜ η min .

Load case

P1, P5

P2, P6

P3, P7

P4, P8

P9, P10

i

k x x y y x x x

% i 89.3 2

k y x x y y y x

% i 50.6 2 39.9 3 11.2 2

k y y x y y x y

% i 87.0 1

k x y y x y x x

% i 70.1 2 20.5 1

k y x y y x x y

%

η 1 2 3 4 5 6 7

1 2 2 1 4 3 7

45.5 39.9

4.8 3.1 1.3 1.1 0.2 0.1

1 2 1 3 5 4

4.4 3.9 1.7 1.2 0.9 0.3

2 1 2 3 4 3

8.1 -3.8 2.9 0.8 0.5

3 1 2 3 5

8.4 6.9

-4.0 0.9 0.7 0.5

1 5 1 6

-3.1 1.8 0.4

2

2

2

2

3

˜ η min

Dominant modes

i = 1,2

i = 2,1

i = 2,3

i = 1,2

i = 2,1,3

P 3

P 9

P 4

P 2

— : all modes - - : dominant modes

M zz [ kNm ]

P 5

P 1

F y [ kN ]

P 10

P 8

P 6

P 7

F x [ kN ]

(a)

(b)

Time [ s ]

Fig. 7: Comparison between pushover and transient non-linear analysis results for the earthquake with spectrum of Figure 5a: (a) Forces F x and F y at the building basis and (b) Torsion moment M zz at the building basis.