Computer Methods in Applied Mechanics and Engineering
Block Krylov subspace methods for the computation of structural response to turbulent wind
Introduction
In typical civil engineering structures such as buildings or bridges, wind velocity fluctuations are responsible for small oscillations and relatively small strains, so that no significant geometrical nor mechanical non-linearities arise. The dynamic response to wind turbulence is thus computed upon linearization of structural behavior. Consequently, linearization of the wind-structure interaction forces is normally pursued, based on the hypothesis that the mean wind velocity is much larger than those associated with turbulence and structural motion.
Under such premises two typical situations arise; the first involves “normal” structures, whose dynamic behavior is rather simple and thus governed by a single vibration mode for each component of horizontal motion. Code provisions explicitly address this case, by prescribing simple criteria, based on equivalent static forces, in which the dynamic effect is condensed into a single coefficient; this accounts, in an approximate way, for the one-degree-of-freedom (or single-mode) response to wind turbulence, modeled as a zero-mean stationary stochastic process. A different situation arises for more complex and/or flexible structural systems, such as tall buildings, in which the dynamic structural behavior under stochastic turbulence loading is fully simulated, though in a linearized setting, so that more refined techniques are necessary.
In this context, we address the development of a numerical procedure for computing the dynamic response of slender structures under wind loading; in terms of fluid–structure interaction the analysis is limited to the so-called “quasi-steady” range, in which the experimental characterization can be performed on aerodynamic (i.e., rigid) structural models [1]. Aeroelastic effects, such as galloping or vortex induced vibration in lock-in condition, are not considered here; the two aspects, however, are presently under consideration in parallel research activity.
The proposed methodology encompasses two different approaches: the first makes use of the whole set of time history recordings, coming from Boundary Layer Wind Tunnel (BLWT) testing, as forcing terms to be applied to the structure in a deterministic context. The second follows a stochastic approach, based on a probabilistic description of the wind turbulence field, on a suitable fluid–structure interaction model and on a characterization of the building aerodynamic properties; this can be taken, in standard situations, from the literature or a code, while for important structures will be derived from BLWT testing.
It is worth noticing that the stochastic analysis approach is often converted in the time domain through the explicit generation of velocity time histories, which have to satisfy well determined correlation characteristics [2], [3]. Then, Monte Carlo simulation techniques are adopted to recreate a statistical basis for the probabilistic evaluation of the structural response [4]. If the system under examination can be assumed to be linear, however, the stochastic procedure in the frequency domain can represent a simpler and more effective way to face the problem. Indeed, the Direct Frequency Domain (DFD) approach may often be preferred to more traditional time-domain integration techniques for several reasons. First, DFD analysis shows superior capabilities in the field of modeling complex structural systems; typical situations are related to soil-structure interaction, to the use of materials or structural components characterized by different damping properties or to the adoption of special devices, such as Tuned Mass Dampers. Moreover, direct analysis procedures are almost unaffected by the introduction of linearized fluid-structural interaction effects leading to nonsymmetric aerodynamic stiffness and damping matrices. Even though these effects show limited influence on the behavior of most buildings, generality is here sought in view of extremely demanding applications related to very important and sensitive structures.
From the strictly computational viewpoint, the DFD approach has to be dealt with by the use of specifically suited numerical tools, to reduce the requirements in terms of computing resources. One of the most critical aspects of DFD lies in the solution of the linear system, which is sought for all frequency steps. Actually, the mathematical problems stemming from the frequency domain analysis of a structural system arise in many fields of applied engineering as well, thus promoting the continuous development of effective strategies for their solution. Such strategies are based on a class of iterative solvers, namely Krylov subspace projection methods [5], for which all the solutions of the linear system corresponding to different frequency values can be projected onto the same subspace.
The application of these methods to the frequency analysis of mechanical systems has been addressed, by this research group, in a series of papers [6], [7], [8]. The purpose of these works has been rather general; attention was mainly devoted to the study of the capabilities of different solution strategies for solving problems of increasing size. The target application, in this respect, was the analysis of large 3D finite-element discretizations, such as the ones stemming from soil-structure interaction modeling. “Single-input” problems have been addressed in the quoted activity, in which a single time variation controls all external dynamic forces, as it happens in one-component seismic analysis or in the computation of foundation impedances.
Here, on the contrary, the research is focused on a specific field of application of DFD, i.e. the response of slender structures to turbulent wind loading. As it will be clear in the following, “slenderness”, is not intended here in strict aerodynamic terms, but refers to structural systems for which a sectional approach is possible, i.e. fluid–structure interaction can be treated in terms of forces and moments instead of local pressure values. In such setting, the procedure here proposed is targeted to the analysis of “medium size” structural models, as the ones resulting from the dynamic analysis of tall buildings or slender bridges, under “multiple-input” dynamic loading, the latter stemming from the spatial correlation of the turbulent velocity field.
Section snippets
Deterministic approach
The dynamic equilibrium equation of the n-dofs discretized system at time t is given bywith the usual notational meaning. When sufficient experimental data from wind tunnel testing are available, it is possible to obtain a set of time histories simulating the real effect of the wind loads on the structure. This is achieved by integrating the pressure field measured on the surface of the scale model; for an elongated body this can deliver the real scale loads as a couple
Frequency domain algebraic formulation
Given a linear structural system, the solution for the frequency problem, both in the deterministic and in the stochastic case, is computed by first considering the systemand then substituting into Eq. (4), (9). Here, in general, M, and are assumed to be nonsingular: M is always symmetric, K∗ is symmetric only in the deterministic case and the same has been here assumed for C, while , s being in the most of cases much smaller than the number of global degrees
Krylov subspace methods
Given the standard shifted linear system (12), iterative methods based on Krylov subspace techniques [5], [17] approximate the system solution by projecting the problem onto the subspace of where m is the subspace dimension and v1 is a generating vector. An important property of (15) is its shift-invariance, meaning that the same projection subspace can be used for all the values of the parameter λ, provided that the starting vector v1 does not
Description of test cases
The first structure (case A) is a mid-rise residential building, the “Lighthouse Tower”, located in the urban area of S. Benigno (Genova, Italy). The project was developed by the architect Giovanni Pellegrino, while the structural design was carried out by the M&L Genova Progetti Company in Genova. The building is composed of 22 stories, for a total height of 68.5 m above-ground, and the plan dimensions of the standard floor are 38.18 × 19.17 m. The horizontal loads resisting system is composed by
Conclusions
We presented a numerical procedure for the computation of the three-dimensional dynamic response of slender-like structures to turbulent wind excitation under the assumption of linear behavior. The procedure combines a stochastic and a deterministic approach exploiting Wind Tunnel tests data. On numerical grounds the proposed methodology is based on a frequency domain approach, which shows superior capabilities in terms of structural behavior modeling, with particular reference to dissipative
Acknowledgements
The authors wish to thank Prof. Alberto Zasso and his co-workers Stefano Giappino and Lorenzo Rosa of the Mechanics Department of Politecnico di Milano for their support in terms of experimental data and stimulating discussions. The help from M&L Genova Progetti Company (Genova, Italy) and the AI Engineering Company (Torino, Italy), who supplied structural data of the example buildings, is also gratefully acknowledged.
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