PSI - Issue 44

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

532

5

3. Numerical application: SPEAR building

SPEAR Workshop – An event to honour the memory of Jean Donea – Ispra, 4-5 April 2005

j P 1 − P 8

j P 9 , P 10

C 6

(c)

(a)

(b)

Figure 1. The SPEAR structure Eight out of the nine columns have a square 250 by 250 mm cross-section; the ninth one, column C6 in Figure 1, has a cross-section of 250 by 750 mm, which makes it much stiffer and stronger than the others along the Y direction, as defined in Figure 1, which is the strong direction for the whole structure. As the structure is regular in elevation and has the same reinforcement in the beams and columns of each storey and its resisting elements in both directions are all of the same kind (frames), the structure belongs to a special class of multi-storey buildings, the so-called regularly asymmetric multi-story structures, meaning that the centre of mass (CM), the centre of stiffness (CR) and the centre of strength (CP) of each storey are located along three vertical lines separated by the distances e r and e s . The centre of stiffness (CR) (based on column secant-to-yield stiffness) is eccentric with respect to the mass centre (CM) by 1.3 m in the X direction (~13% of plan dimen sion) and by 1.0 m in the Y direction (~9.5% of plan dimension). The reference system used in the PsD test and the location of the CM of the structure at the first and second floor are shown in Figure 1. The origin of the reference system is in the centreline of column C3. The coordinates of the CM of the first two storeys with respect to this reference system are (-1.58m, -0.85m); at the third storey the coordinates of the CM vary slightly, becoming (-1.65m; -0.94m). Given the doubly non-symmetrical plan configuration of the specimen, the PsD test needed to be bi-directional, with the input applied along two orthogonal directions. As a consequence, three degrees of freedom (DoFs) per storey were taken into account: two translations and one rotation along the vertical axis, as opposed to the single de gree of freedom per storey that is usually taken into account in conventional unidirec tional PsD testing. The accelerograms used as input were the Montenegro ’79 Herceg-Novi records for the longitudinal and transverse component, artificially fitted to the EC8 spectrum, as shown in Figure 2, scaled to different levels of PGA and applied to the structure ac- E ff ect. mass X(%) E ff ct. mass Y(%) 81.20 3.33 0.00

Fig. 3: SPEAR building: (a) experimental mock-up; (b) plan-view with center of mass (CM) and center of sti ff ness (CR), figures extracted from Negro and Fardis (2005); (c) FE m del with areas of application of added masses and with control points.

In this chapter, the E-DVA method is applied f llowing the tep-by-step p ocedure 2.4 to the SPEAR building. For a larger spectrum of possible applications of th E-DVA approach, th re der is referred to the following papers published by the authors Erlicher and Huguet (2016); Lherminier et al. (2018); Huguet et al. (2018, 2019). The building analyzed here is the RC framed non-symmetric structure of SPEAR experimental program of Negro and Fardis (2005), which is submitted to an earthquake with two horizontal components. The water barrels added on the slabs (see Figure 3a) simulate a live load and they are also implemented in the FE model (see Figure 3c). The FE model of Figure 3c is implemented into Code Aster FE software from EDF (1989–2022).

Table 1: Modal frequencies, damping ratios and e ff ective mass percentages of the FE model and associated pseudo-acceleration

2 )

2 )

S a , x ( m / s

S

a , y ( m / s

Mode

Frequency (Hz)

Damping ξ (%)

E ff ect. mass Z(%)

1.07 1.21 1.51 3.14 5.36

6.64 6.80 7.44

1 2 3 4 8

2.450 2.608 2.651 2.252 1.890

2.493 2.483 2.524 2.045 1.680

5.72 1.35 8.35 0.00

70.37

0.00 0.01 0.00

9.79

12.01 19.38

0.3

140

0.03

36.61 95.14

Total(74)

-

-

99.17

99.14

-

-

74 + 1 (rigid resp.)

34.8

1.22

0.83

0.87

4.86

1.463

1.463

(a) The modal basis of Table 1, the response-spectrum analysis and Eq. (3) with CQC coe ffi cients are used to determine the response matrix of the six generalized forces and moments f = [ F x , F y , F z , M xx , M yy , M zz ] T at the basis of the structure, for the average x- and y-spectra. (b) The response matrix X f (2 × 2) defining the elliptical envelope of the seismic e ff orts at the building basis F x − F y is obtained (in the case of torsion moment, X f degenerates into a 1 × 1 matrix).

Table 2: Definition of ten load cases for pushover analysis

Load case

P1 (P5)

P2 (P6)

P3 (P7)

P4 (P8)

Load case

P9 (P10) + M zz (-)

Response to be maximized

F θ

F θ

F θ

F θ

Response to be maximized

θ ( ◦ )

0 (-180)

45 (-135) + 344.8 (-) + 287.0 (-) + 137.1 (-)

90 (-90) + 23.1 (-) + 414.4 (-) + 903.2 (-)

135 (-45) -330.8 ( + ) + 269.9 (-) + 1101.9 (-)

-

-

F x (kN) F y (kN)

+ 456.4 (-) + 21.0 (-) -630.2 ( + )

F x (kN) F y (kN)

-213.4 ( + ) + 277.7 (-) + 1347.6 (-)

M zz (kNm)

M zz (kNm) | M zz | (kNm)

F θ (kN)

456.9

448.6

415.1

426.9

1347.6

(c) The first eight responses to be maximized are chosen as follows: F = F θ = F x cos θ + F y sin θ , with eight di ff erent values for θ : θ = 0 ◦ , 45 ◦ , 90 ◦ , 135 ◦ , − 180 ◦ , − 135 ◦ , − 90 ◦ and − 45 ◦ (named P1, ..., P8). Any other choice for θ is possible, see e.g. Lherminier et al. (2018). Notice that F θ = f T · b , with b = [cos θ, sin θ ] T . For each direction, the maximization condition is applied, in order to obtain eight response vectors f = [ F x , F y ] T whose orientation into the plane of horizontal forces is θ t = atan( F y / F x ); see Figure 2b. Then, the same principle is applied for the maximization of f = M zz (P9) and of f = − M zz (P10) (see Table 2).