PSI - Issue 44

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

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actions, is applied to the structure and monotonically increased, while the profile shape is kept unchanged, until a control node (generally placed on the top of the building) reaches a target displacement or the structure collapses. The output of the analysis is the pushover curve, which is used for estimating the seismic capacity of the structure. The behavior of the entire multi-degree of freedom (MDOF) system is reduced to that of a single degree of freedom (SDOF) system, in order to compare seismic capacity to seismic demand, estimated by the acceleration-displacement response spectrum. This kind of analysis delivers realistic results only when the structural behavior is governed by a single mode that, moreover, remains unchanged during the simulation. The most complete method for evaluating seismic structural behavior is nonlinear dynamic analysis, also referred to as time history analysis. It consists in the direct solution of the semi-discrete di ff erential equations of motion for the entire MDOF system subjected to acceleration time histories representative of the expected earthquake in the site. The nonlinear material response is directly included in the analysis without the need for the behavior factor. A further step in time history analysis is represented by the incremental dynamic analysis (IDA)Vamvatsikos and Cornell (2002); Kohrangi et al. (2020), which consists in subjecting the structure to multiple ground motions scaled to di ff erent levels of intensity in order to obtain a curve relating the intensity measure to a damage measure conveniently chosen. The main shortcoming of nonlinear dynamic analysis is the high computational cost that makes this method not a ff ordable for complex large buildings.This is the reason why in engineering practice Pushover analysis is mainly used to assess structural seismic behavior despite of its limitations. Modifications to the nonlinear static analysis to achieve a better consideration of di ff erent aspects, such as the contribution of higher modes, torsional e ff ects, redistribution of inertia forces and irregular structures were also investigated Kassem et al. (2020); Chopra and Goel (2002). An alternative research path is represented by time history analysis of a reduced model (ROM) obtained by model order reduction Koutsovasilis and Beitelschmidt (2008), consisting in finding a low dimension subspace where the motion equations are projected. In this work a new ROM method for inelastic frames is proposed. A mechanical-based choice of the ROM basis is introduced. It includes the most influential linear mode shapes in terms of participant mass, their corresponding plastic mechanisms and single story plastic mechanisms. The additional plastic modes are obtained for each direction of the seismic action, once and for all, at the beginning of the analysis by means of static analyses with suitable horizontal loads.

2. Nonlinear dynamic analysis of 3D frames

In the nonlinear dynamic analysis, the structure is discretized spatially by means of finite elements Magisano et al. (2018). Then, the semi-discrete equations of motions are solved in time by direct integration. Newmark’s method will be adopted in the following. The spatial finite element discretization is based on a mixed beam finite element formulation based on fiber plasticity with elasto-perfectly plastic material model Magisano and Garcea (2020, 2021).

3. The reduced order model for a MDOF system

Reduction technique consists in approximating the space of displacement solution as a linear combination of modes. These are intended as particular configurations representative of the structural behavior both in elastic and plastic regime. When these modes are linearly independent, they become a basis which can be used to approximate the displacement field solution space. If we denote with u k the k-th mode, with k varying from 1 to m and m n , being m the dimension of the approximated solution space and n the dimension the whole displacement field, the following approximation ˜ u of u is assumed:

m k = 1

a k u k = u 0 + Ua ,

˜ u = u 0 +

(1)

where the vector a collects the coe ffi cients of the linear combination, the columns of matrix U are the m u k modes and u 0 is the initial solution for the dead loads. Our proposal di ff ers from others for a multi-scale strategy and for the choice of the reduced basis, i.e. the selection of the modes u k . We distinguish them into two subspaces: the first of dimension generated by linear elastic modes v k , well-describing the linear part of the displacement field, and the second of dimension m − generated by plastic collapse mechanisms w k , useful when the structure undergoes