Multi-objective optimal design of inerter-based vibration absorbers for earthquake protection of multi-storey building structures

In recent years different inerter - based vibration absorbers (IVAs) emerged for the earthquake protection of building structures coupling viscous and tuned - mass dampers with an inerter device . In the three most popular IVAs the inerter is functioning either as a motion amplifier [tuned - viscous - mass - damper (TVMD) configuration], mass amplifier [tuned - mass - damper - inerter (T MDI) configuration], or mass substitute [tuned - inerter - damper (TID) configuration]. Previous work has shown that through proper tuning , IVAs achieve enhanced earthquake - induced vibration suppression and/or weight reduction compared to conventional dampers/absorbers , but at the expense of increased control forces exerted from the IVA to the host building structure . These potentially large forces are typically not accounted for by current IVA tuning approaches. In this regard, a multi-objective IVA design approach is herein developed to identify the compromise between the competing objectives of (i) suppressing earthquake-induced vibrations in buildings, and (ii) avoiding development of excessive IVA (control) forces, while, simultaneously, assessing the appropriateness of different modeling assumptions for practical design of IVAs for earthquake engineering applications . The potential of the approach to pinpoint Pareto optimal IVA designs against the above objectives is illustrated for different IVA placements along the height of a benchmark 9-storey steel frame structure. Objective (i) is quantified according to current performanc e-based seismic design trends using first-passage reliability criteria associated with the probability of exceeding pre-specified thresholds of storey drifts and/or floor accelerations being the engineering demand parameters (EDPs) of interest . A variant, simpler, formulation is also considered using as performance quantification the sum of EDPs variances in accordance to traditional tuning methods for dynamic vibration absorbers. Objective (ii) is quantified through the variance of the IVA force. It is found that reduction of IVA control force of up to 3 times can be achieved with insignificant deterioration of building performance com pared to the extreme Pareto optimal IVA design targeting maximum vibration suppression , while TID and TMDI a chieve practically the same building performance and significantly outperform the TVMD. Moreover, it is shown that the simpler variant formulation may provide significantly suboptimal reliability performance . Lastly, it is verified that the efficacy of optimal IVA designs for stationary conditions is maintained for non-stationary stochastic excitation model capturing typical evolutionary features of earthquake excitations

Analytical reliability calculation of linear dynamical systems in higher dimensions

The recent application of reliability analysis to controller synthesis has created the need for a computationally efficient method for the estimation of the first excursion probabilities for linear dynamical systems in higher dimensions. Simulation methods cannot provide an adequate solution to this specific application, which involves numerical optimization of the system reliability with respect to the controller parameters, because the total computational time needed is still prohibitive. Instead, an analytical approach is presented in this paper. The problem reduces to the calculation of the conditional upcrossing rate at each surface of the failure boundary. The correlation between upcrossings of the failure surface for the different failure events may be addressed by the introduction of a multi-dimensional integral. An efficient algorithm is adopted for the numerical calculation of this integral. Also, the problem of approximation of the conditional upcrossing rate is discussed. For the latter there is no known theoretical solution. Three of the semi-empirical corrections that have been proposed previously for scalar processes are compared and it is shown that the correction should be based on the bandwidth characteristics of the system. Finally, examples that verify the validity of the analytical approximations for systems in higher dimensions are discussed

Stochastic System Design and Applications to Stochastically Robust Structural Control

The knowledge about a planned system in engineering design applications is never complete. Often, a probabilistic quantification of the uncertainty arising from this missing information is warranted in order to efficiently incorporate our partial knowledge about the system and its environment into their respective models. In this framework, the design objective is typically related to the expected value of a system performance measure, such as reliability or expected life-cycle cost. This system design process is called stochastic system design and the associated design optimization problem stochastic optimization. In this thesis general stochastic system design problems are discussed. Application of this design approach to the specific field of structural control is considered for developing a robust-to-uncertainties nonlinear controller synthesis methodology. Initially problems that involve relatively simple models are discussed. Analytical approximations, motivated by the simplicity of the models adopted, are discussed for evaluating the system performance and efficiently performing the stochastic optimization. Special focus is given in this setting on the design of control laws for linear structural systems with probabilistic model uncertainty, under stationary stochastic excitation. The analysis then shifts to complex systems, involving nonlinear models with high-dimensional uncertainties. To address this complexity in the model description stochastic simulation is suggested for evaluating the performance objectives. This simulation-based approach addresses adequately all important characteristics of the system but makes the associated design optimization challenging. A novel algorithm, called Stochastic Subset Optimization (SSO), is developed for efficiently exploring the sensitivity of the objective function to the design variables and iteratively identifying a subset of the original design space that has v i high plausibility of containing the optimal design variables. An efficient two-stage framework for the stochastic optimization is then discussed combining SSO with some other stochastic search algorithm. Topics related to the combination of the two different stages for overall enhanced efficiency of the optimization process are discussed. Applications to general structural design problems as well as structural control problems are finally considered. The design objectives in these problems are the reliability of the system and the life-cycle cost. For the latter case, instead of approximating the damages from future earthquakes in terms of the reliability of the structure, as typically performed in past research efforts, an accurate methodology is presented for estimating this cost; this methodology uses the nonlinear response of the structure under a given excitation to estimate the damages in a detailed, component level

Multi-objective optimal design of the tuned mass-damper-inerter (TMDI) for stochastically support excited building structures

The multi-objective optimal design is considered of tuned-mass-damper-inerter (TMDI) equipped linear building frames subject to seismic excitations modeled as stationary colored random processes. The TMDI couples the traditional tuned mass-damper (TMD) with an inerter. The latter is a two-terminal mechanical device developing a resisting force proportional to the relative acceleration of its terminals by the “inertance” constant. Previous work has shown that TMDI with large inertance can outperform the classical TMD for the same attached mass, if properly tuned/designed. This performance enhancement comes, though, at the expense of increased inerter forces that need to be accommodated by the host structure. A multi-objective design framework is considered to investigate the compromise between these two competing objectives. The first objective, representing the vibration suppression efficiency, is defined using first-passage reliability criteria, considering outcrossing of failure modes related to floor accelerations, inter-storey drifts, and attached mass displacement. A linear combination of the probabilities related to these modes is taken as objective function, following current performance-based seismic design practices. The second objective, representing the strengthening of the host structure required to accommodate the TMDI forces, corresponds to the stationary inerter force. A variant of the optimization problem is also considered by adopting as secondary objective the maximum force of either the inerter or the damper utilized in the TMDI configuration. In the illustrative example different topological configurations are examined for the TMDI. Results demonstrate that the proposed framework supports a comprehensive understanding of how the TMDI design establishes a compromise between the aforementioned objectives. Connection of the inerter at a lower floor than the one immediately below the TMDI mass can lead to significant reduction of both objectives

Reliability-based design of tuned mass-damper-inerter (TMDI) equipped multi-storey frame buildings under seismic excitation

The reliability based optimal design is considered of tuned mass-damper-inerter (TMDI) equipped linear building frames subject to seismic excitations modeled as stationary colored random processes. The TMDI is a recently introduced generalization of the classical linear tuned mass-damper (TMD) benefitting from the mass amplification property, the so-called inertance, of the inerter device to enhance the vibration suppression capabilities of the TMD. The frequency, damping ratio, and inertance TMDI properties are treated as design variables to minimize out-crossing rates of prespecified thresholds for building floor accelerations, inter-storey drifts, and TMDI mass displacement. Numerical data pertaining to a 10-storey frame structure equipped with a TMDI arranged in 12 different topologies are furnished indicating the enhanced performance of the TMDI over the classical TMD especially for relatively small additional attached mass

Robust reliability-based design of seismically excited tuned mass-damper-inerter (TMDI) equipped MDOF structures with uncertain properties

This paper considers a reliability-based approach for the optimal design of the tuned mass-damper-inerter (TMDI) in linear building frames with uncertain structural properties subject to seismic excitations defined as stationary colored random processes with uncertain parameters. The TMDI is a recently introduced generalization of the classical linear passive tuned mass-damper (TMD) comprising an additional mass attached to the primary structure whose oscillations are to be suppressed via a linear spring and dashpot in parallel. The TMDI benefits from the mass amplification property, the so-called inertance, of an inerter device that links the additional mass to a different floor from the one it is attached to which improves the vibration suppression capabilities of the TMD. Herein, the structural seismic performance is quantified through the probability of occurrence of different failure modes, related to the floor acceleration, the inter-storey drifts, and the attached mass displacement exceeding acceptable thresholds. The overall design objective is taken as a linear combination of these probabilities whereas the TMDI linear spring constant , viscous damping constant , and inertance properties are taken as the design variables. The parametric structural and excitation uncertainty is efficiently addressed through a two-stage approach combining a Taylor series approximation and Monte Carlo simulation. Numerical data for a 10-storey shear frame structure equipped with a TMDI with different values of attached mass and arranged in 8 different topologies are furnished indicating the enhanced performance of the TMDI over the classical TMD for relatively small attached masses. The reported numerical results evidence that the performance of optimally designed TMDIs is less affected by the parametric uncertainties as the total inertia TMDI properties (attached mass and inertance) increases, indicating that the inclusion of the inerter leads to more robust passive vibration control

Validation of hazard-compatible stochastic ground motion model modification techniques

An important consideration for the adoption of stochastic ground motion models in performance-based earthquake engineering applications is that the probability distribution of target intensity measures from the developed suites of time-histories is compatible with the prescribed hazard at the site and structure of interest. The authors have recently developed a computationally efficient framework to modify existing stochastic ground motion models to facilitate such a compatibility. For a given seismicity scenario, the framework identifies the modified stochastic ground motion model that can sufficiently match the prescribed hazard while maintaining similarity to regional physical ground motion model characteristics. This paper extends this effort through a validation study. Suites of recorded and stochastic ground motions, whose spectral acceleration statistics match the mean and variance of target spectra within a period range of interest, are utilized as input to perform response history analysis of inelastic single-degree-of-freedom case-study systems. The resultant engineering demand parameters distributions are then compared to assess the effect of the proposed modification

Modification of stochastic ground motion models for matching target intensity measures

Stochastic ground motion models produce synthetic time‐histories by modulating a white noise sequence through functions that address spectral and temporal properties of the excitation. The resultant ground motions can be then used in simulation‐based seismic risk assessment applications. This is established by relating the parameters of the aforementioned functions to earthquake and site characteristics through predictive relationships. An important concern related to the use of these models is the fact that through current approaches in selecting these predictive relationships, compatibility to the seismic hazard is not guaranteed. This work offers a computationally efficient framework for the modification of stochastic ground motion models to match target intensity measures (IMs) for a specific site and structure of interest. This is set as an optimization problem with a dual objective. The first objective minimizes the discrepancy between the target IMs and the predictions established through the stochastic ground motion model for a chosen earthquake scenario. The second objective constraints the deviation from the model characteristics suggested by existing predictive relationships, guaranteeing that the resultant ground motions not only match the target IMs but are also compatible with regional trends. A framework leveraging kriging surrogate modeling is formulated for performing the resultant multi‐objective optimization, and different computational aspects related to this optimization are discussed in detail. The illustrative implementation shows that the proposed framework can provide ground motions with high compatibility to target IMs with small only deviation from existing predictive relationships and discusses approaches for selecting a final compromise between these two competing objectives

Hazard-compatible modification of stochastic ground motion models

A computationally efficient framework is presented for modification of stochastic ground motion models to establish compatibility with the seismic hazard for specific seismicity scenarios and a given structure/site. The modification pertains to the probabilistic predictive models that relate the parameters of the ground motion model to seismicity/site characteristics. These predictive models are defined through a mean prediction and an associated variance, and both these properties are modified in the proposed framework. For a given seismicity scenario, defined for example by the moment magnitude and source-to-site distance, the conditional hazard is described through the mean and the dispersion of some structure-specific intensity measure(s). Therefore, for both the predictive models and the seismic hazard, a probabilistic description is considered, extending previous work of the authors that had examined description only through mean value characteristics. The proposed modification is defined as a bi-objective optimization. The first objective corresponds to comparison for a chosen seismicity scenario between the target hazard and the predictions established through the stochastic ground motion model. The second objective corresponds to comparison of the modified predictive relationships to the pre-existing ones that were developed considering regional data, and guarantees that the resultant ground motions will have features compatible with observed trends. The relative entropy is adopted to quantify both objectives, and a computational framework relying on kriging surrogate modeling is established for an efficient optimization. Computational discussions focus on the estimation of the various statistics of the stochastic ground motion model output needed for the entropy calculation

Optimal tuned mass-damper-inerter (TMDI) design for seismically excited MDOF structures with model uncertainties based on reliability criteria

The tuned mass-damper-inerter (TMDI) is a recently proposed linear passive dynamic vibration absorber for the seismic protection of buildings. It couples the classical tuned mass damper (TMD) with an inerter, a two-terminal device resisting the relative acceleration of its terminals, in judicial topologies, achieving mass-amplification and higher-modes-damping effects compared to the TMD. This paper considers an optimum TMDI design framework accommodating the above effects while accounting for parametric uncertainty to the host structure properties, modeled as a linear multi degree of freedom system, and to the seismic excitation, modeled as stationary colored noise. The inerter device constant, acting as a TMD mass amplifier, is treated as a design variable, whereas performance variables sensitive to high-frequency structural response dynamics are used to account for the TMDI influence to the higher structural modes. Reliability criteria are adopted for quantifying the structural performance, expressed through the probability of occurrence of different failure modes related to the trespassing of acceptable thresholds for the adopted performance variables: floor accelerations, interstory drifts, and attached mass displacement. The design objective function is taken as a linear combination of these probabilities following current performance-based seismic design trends. Analytical and simulation-based tools are adopted for the efficient estimation of the underlying stochastic integral defining the structural performance under uncertainty. A 10-story building under stationary Kanai-Tajimi stochastic excitation is considered to illustrate the design framework for various TMDI topologies and attached mass values. It is shown that the TMDI achieves enhanced structural performance and robustness to building and excitation uncertainties compared to same mass/weight TMDs