PSI - Issue 44

Domenico Magisano et al. / Procedia Structural Integrity 44 (2023) 456–463 Magisano et al. / Structural Integrity Procedia 00 (2022) 000–000

458

3

significant plastic deformations: By projecting the dynamic equilibrium onto this basis, a reduced order system is obtained whose unknowns are the coe ffi cient of the linear approximation: m ¨ a + d ˙ a + s ( a ) = p ( t ) . (2) Matrices m and d are the projection of the mass matrix and of the damping matrix respectively, while p is the projection of the load vector. The vector s in Eq. 2 represents the projection of the internal force vector obtained from the original size displacement vector ˜ u . This means that our approach can be framed as a multi-scale strategy: the internal forces are computed on the full model, while equilibrium in imposed on the reduced space. The advantages of the proposal is that the iterative solution of Eq. 2 requires only the factorization of a small sized iteration matrix: (3) where k is the projected tangential sti ff ness matrix. In this way, a significant part of the computational cost for these global operations, hard to parallelize in a computer code, is avoided, while the internal forces accounting for the material nonlinearity is accurately evaluated at the full model scale with element-by-element operations easy to implement in parallelized form. g = 4 ∆ t 2 m + 2 ∆ t d + k

4. Definition of the approximation space for the reduced order model

4.1. Linear elastic modes subspace

The first set of modes is obtained by solving the generalized eigenvalue problem that allows to determine linear mode shapes and corresponding frequencies. Mode shapes are listed according to the descending order of their per centage of participation mass factor. Then, the subspace is defined choosing a number of modes such that the total percentage of participant mass involved is higher than 90%.

4.2. Collapse mechanisms subspace

Plastic mechanisms are obtained evaluating the structural limit load solving the nonlinear static problem under a suitable reference load vector: • Collapse mechanisms associated to modal forces: for each linear mode included in the first subspace, the corresponding plastic mechanism is calculated applying a load vector representing inertial forces associated to that mode; • Story collapse mechanisms: Each single story mechanism is evaluated subjecting the structure to a horizontal load profile shaped to produce a non-null shear force at the considered story, in order to cause its local collapse. Load vector results in a story force at the considered story and an equal and opposite one for the story immediately above and below.

5. Numerical results: mildly irregular 3D frame

0 . 2

a g / g

0.199 2.420 0.374

¨ u g ( g )

F 0 T ∗ c

0

soil category B topographical category 1 ξ 5

− 0 . 2

0

10

20

5

15

t ( s )

Table 1: Target spectrum parameters from EC8 guidelines Eurocode (1996)

Fig. 1: Ground motion acceleration time history.