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In this paper, a procedure has been presented to develop fragility curves of structures equipped with optimal variable damping or stiffness semi-active tuned mass dampers (SATMDs). To determine proper variable damping or stiffness of semi-active devices in each time step, instantaneous optimal control algorithm with clipped control concept has been used. Optimal SATMDs have been designed based on minimization of maximum inter-story drift of nonlinear structure which genetic algorithm(GA) has been used to solve the optimization problem. For numerical analysis, a nonlinear eight-story shear building with bilinear hysteresis material behavior has been used. Fragility curves for the structure equipped with optimal variable damping and stiffness SATMDs have been developed for different performance levels and compared with that of uncontrolled structure as well as structure controlled using passive TMD. Numerical analysis has shown that for most range of intensity measure optimal SATMDs have been effective in enhancement of the seismic fragility of the nonlinear structures which the improvement has been more than passive TMDs. Also, it has been found that, although variable stiffness SATMD shows to be more reliable in lower mass ratios, however in higher mass ratios variable stiffness and damping SATMDs performs similarly to improve reliability of the structure.

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In this paper, a procedure has been introduced to the multi-objective optimal design of semi-active tuned mass dampers (SATMDs) with variable stiffness for nonlinear structures considering soil-structure interaction under multiple earthquakes. Three bi-objective optimization problems have been defined by considering the mean of maximum inter-story drift as safety criterion of structural components, absolute acceleration as the criterion of occupants’ convenience, and safety of non-structural acceleration sensitive components, as well as SATMD relative displacement as the cost criterion of the control device. The parameters of the weighting matrices of the instantaneous optimal control algorithm and the maximum and minimum level of variable stiffness of the semi-active device have been considered as design variables. An improved version of the non-dominated sorting genetic algorithm (NSGA-II), has been employed to solve the optimization problems and figure out the set of Pareto optimal solutions. SATMDs with different mass ratios have been designed for an eight-story shear type building with bilinear elasto-plastic stiffness model where the soil-structure interaction has been incorporated by Cone model with three degrees of freedom for the soil. Results show the capability and simplicity of the proposed procedure to design SATMDs considering multiple performance criteria. It is observed that this procedure can offer a wide range of optimal solutions throughout the Pareto front which can be chosen by the designer based on desired performance and application of the structure.

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The semi-active bracing system locks or unlocks the stand-by braces in an on-off mode utilizing a variable stiffness device (VSD). In this paper, the optimal design of a semi-active bracing mechanism and evaluating its performance in mitigating structural vibration under seismic loading have been studied. The optimal stiffness values of the semi-active braces have been determined by solving two optimization problems including minimizing the maximum acceleration and also minimizing the maximum inter-story drift by imposing a constraint on the maximum acceleration. The genetic algorithm (GA) has been applied to solve the optimization problems. To illustrate the design procedure, an eight-story linear shear frame under earthquake record has been considered and the optimal semi-active braces have been designed. In addition, to assess the performance of optimal bracing system under other records which are different from design record in terms of intensity and frequency content, the structure equipped with optimally designed semi-active braces has been tested under several ground motion records. The results show that the optimal semi-active bracing system has simultaneously reduced different responses of the structure although the acceleration reduction has mainly been less compared to the drift reduction.