Asymptotic behavior of the equivalent Timoshenko linear beam model used in the analysis of tower buildings

We study the asymptotic properties of a continuous Timoshenko linear beam model immersed in a three-dimensional space and used in the analysis of tower buildings. Assume that the bending and axial behaviors are coupled on the one hand, while the shear and torsional behaviors are coupled on the other hand. If the displacement vector is totally damped and the rotational vector is undamped, or if the rotational vector is totally damped and the displacement vector is undamped, then the system is proved to be exponentially stable, under some assumptions on the physical parameters involved in the problem. We also consider the case when the bending and axial behaviors and/or the shear and torsional behaviors are uncoupled. In this situation, the stability properties are different if the rotational or the displacement vector is damped or not: the system can be exponentially stable, polynomially stable or even unstable.