Robust optimal design of tuned mass dampers for tall buildings with uncertain parameters

Abstract

In this paper a newly proposed approach to robust optimization is adopted in order to study the influence of uncertain structural parameters on the design of a passive control system The control devices are Tuned Mass Dampers placed on a tall building subjected to wind load. The uncertain structural variables are the floors’ mass and stiffness. The spatial variability of the mass over the floor’s surface is considered by ideally dividing each floor into a certain number of partitions in which the mass varies as a random variable. The robust optimization procedure is based on an enhanced Monte Carlo simulation technique and the genetic algorithm. The Latin Hypercube Sampling is employed to reduce the number of samples. The optimization of stiffnesses and dampings of the devices is carried out for each sample set of structural mass and stiffness matrices, allowing the evaluation of the probability distributions of the optimal parameters and the objective function. The robust designs are those having corresponding values of the objective function lying in the neighborhood of the expected value of the distribution The robust optimization is applied to a tall building controlled by a Tuned Mass Damper and subjected to wind loads obtained from wind tunnel tests. Several numerical analyses are carried out considering different numbers of surface partitions. Both the separate and joint influence of the mass and stiffness uncertainty is considered. The results show that the selected designs are robust, as they guarantee a small variability of structural performance caused by uncertainties.

Appendix A Latin Hypercube Sampling

The Latin Hypercube sampling is a well-known technique used to efficiently sample variables from their multivariate distributions.

Let N denote the number of realizations and p the number of random variables [m ₁,m ₂,...,m _p]. The sample space is then p-dimensional. To generate N samples from p variables with probability density function $f\left (m \right )$, the procedure is as follows. The range of each variable is subdivided into N non overlapping intervals of equal probability 1/N. From each interval one value is selected at random according to the probability density of the interval. The current practice is to choose samples directly from the cumulative distribution function:

$$ m_{j,i} =F_{j}^{-1} \left( {\frac{i-0.5}{N}} \right) $$

(19)

where m _j,i is the i-th sample of the j-th random variable m _j , $F_{j}^{-1}$ is the inverse cumulative distribution of the variable m _j . In order to account for the correlations between the random variables, the samples of each random variable are permuted to match the target covariance matrix.