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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the considered seismic response is small, for instance less than 10% for each of the three earthquake directions, see ASCE-4-16 (2017), then these higher frequency modes can be neglected and the frequencies are truncated at i = n m . Otherwise, higher modes are combined into the so-called residual rigid response vectors of displacements.

2.2. Proposal: load pattern as a linear combination of modes for multi-component earthquakes (E-DVA approach).

The method proposed in Erlicher et al. (2020) defines the load pattern by a linear combination of modal load patterns, without removing any sign; the contribution from each mode is fully taken with no modification. Such procedure, called Enhanced Direct Vectorial Addition (E-DVA), is based on the linear combination formula:

n ec k = 1

n ec k = 1

n i = 1

n i = 1

α i , k Q

α i , k M · A max , i , k = M · A = K · U

(2)

Q =

=

i , k

where 1 ≤ n ec ≤ 3 is the number of ground motion components (x and / or y and / or z) taken into account for the pushover analysis, A = n ec k = 1 n i = 1 α i , k A max , i , k and U = n ec k = 1 n i = 1 α i , k U max , i , k , with the number of given modes n ; α i , k are the so-called weighting or combination factors. Since n ec can be greater than 1, this method is multi-component. The load pattern of Eq. (2) has been first defined by Erlicher et al. (2014) with reference to the linear equivalent static method. Then, Erlicher and co-workers proposed the use of the same multi-modal and multi-component load pattern for pushover analyses Erlicher and Huguet (2016)-Lherminier et al. (2018). In Erlicher et al. (2020), this pushover approach is presented in more detail. The weighting factors α are defined as those creating a load pattern Q which originates a response vector f . The response vector f accounts for n r values of interest quantities (forcesat the basis of the building or at a particular floor, the displacement of a point, the stress in a fiber of a beam. . . ). The most classical example is the response vector f = [ F x , F y , F z , M xx , M yy , M zz ] T accounting for the 6 generalized forces and moments at the basis of the building. Under linear elastic behavior, f is proportional to the applied load pattern Q so it can also be expressed as a weighted sum of the modal-spectrum response vectors f i , k . The response envelope for a seismic response vector f = [ F 1 ,..., F r ,..., F n r ] T is defined, from Menun and Der Kiureghian (2000), as follows: f T · X − 1 f · f ≤ 1 with X f = R T f · ˜ H · R f (3) Among all the α such that R T f · α = f , it is possible to find an optimal one by solving the following optimization problem: for the given f , find α that minimises α T · ˜ H − 1 · α + λ T α · R T f · α − f where λ α is the vector of the n r Lagrange multipliers. This condition leads to α factors defining a combination of modes with the ”smallest” modal contributions. The previous optimization problem has a unique solution, see Erlicher et al. (2014), which reads : α = ˜ H · R f · X − 1 f · f (4) The vector α from Eq. (4) is such that α T · ˜ H − 1 · α = f T · X − 1 f · f and this implies: α T · ˜ H − 1 · α ≤ 1. The ellipsoid defined by Eq. (4) can be indicated as the locus of the probable modal weighting factors α which is the locus of probable response vectors f . In this section, a procedure is proposed for the choice and calculation of a particular response vector f , from which the associated unique vector α can be then determined by using Eq. (4). A pushover analysis is carried out by plotting a displacement of a “control point” as a function of a force at the building basis. It is interesting to choose a response vector f collecting the horizontal forces at the building basis ( F x , F y ) or the horizontal displacements ( u j , x , u j , y ) of the control point j . The corresponding response matrix X f can be calculated by Eq. (3) and is associated with an elliptical response envelope : each of the points on the ellipsoid boundary is related to the maximization of the response vector along a particular direction. For instance, among all the response vectors f = [ F x , F y ] T fulfilling the inequality (3), it is interesting to find the one on the ellipsoid boundary having the largest (maximum and positive) value of F x . This particular response vector / point f 1 = [ F x , max , F y , 1 ] T is shown in Figure 2a (Load case P 1 ). 2.3. Choice and calculation of a response vector f suitable for the α -factor definition