PSI - Issue 44

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

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(d) The α -vectors corresponding to the ten chosen maximization conditions are computed using Eq. (4). In the case of torsion, F x and F y are computed simultaneously with the maximum and minimum M zz , allowing to calculate θ t = atan ( F y / F x ). Five α -vectors are shown in Figure 4. The cases P5, P6, P7, P8 and P10 are symmetric to P1, P2, P3, P4 and P9, respectively, and α -factors have opposite signs.

: α i , x : α i , y

P1: θ = 0 ◦

P3: θ = 90 ◦

P2: θ = 45 ◦

P4: θ = 135 ◦

P9: torsion

1

1

1

1

1

Mode i

Mode i

Mode i

Mode i

Mode i

Fig. 4: Graphical representation of the α -vectors for load cases P1-P4 and P9 ( α i , z = 0 since there is no vertical earthquake component).

3.1. Load pattern based on the modal basis with all modes

The load fields Q are calculated for the 10 considered load cases (P1,...,P10) with Eq.(2) (step (e)) and the corre sponding incremental loads are applied to the FE model (step (f)).10 capacity curves are then determined, the abscissa being the displacement of control point [ u ] j , t along direction t , and the ordinate being the force F t at the building basis along the same direction. Control points “ j ” are indicated in Figure 3c. Capacity curves are normalized to get capacity spectra using the parameters of the 10 equivalent SDOF oscillators. Capacity spectra are shown in Figure 5. Calculated response spectra are shown for both initial damping ratio ξ ∗ (dotted line) and final elastic plus hysteretic damping ratio ˜ ξ ∗ (solid line). The straight line associated with linear behaviour and the target points are also indicated.

P2 θ t = 39 . 77 ◦

P3 θ t = 86 . 80 ◦

P4 θ t = 140 . 79 ◦

P9 θ t = 127 . 54 ◦

P1 θ t = 2 . 64 ◦

2 ]

S a , t [ m / s

S d , t [ m ]

S d , t [ m ]

S d , t [ m ]

S d , t [ m ]

S d , t [ m ]

Fig. 5: Capacity and response spectra in the ADRS plane for load cases P1-P4 and P9.

Target points are determined at the intersection of the capacity spectrum with the ADRS spectrum; see Erlicher et al. (2020). Symmetry is lost due to non-linear e ff ects and results for all ten cases have to be explicitly given.

3.2. Load pattern based on dominant modes

The procedure for the selection of the dominant modes / earthquake components presented in Section 2.5 is applied here to the SPEAR building. Ten pushover analyses analogous to the ones of previous section are performed, using a load pattern Q defined according to Eq. (5). In detail: (e*) With the α i , k factors calculated at step (d), we compute for both k = x and k = y the products F θ, i , k = α i , k ( F x , i , k cos θ + F y , i , k sin θ ) for the load cases P1,...,P8 and M zz , i , k = α i , k M zz , i , k for load cases P9 and P10. Figure 6a shows F θ, i , x and M zz , i , x . Figure 6b shows F θ, i , y and M zz , i , y . Notice that abscissa in Figures 6a and 6b is the index i of modes, according to increasing frequency (Table 1). Once the responses F θ, i , k and M zz , i , k are known for all i and k , then for each load case they are ordered from the one with the maximum absolute value to the smallest one. The index η is introduced to describe this new order.