Dynamics of the Rocking Frame with Vertical Restrainers

Abstract

This paper investigates the rocking response and stability analysis of an array of slender columns caped with a rigid beam which are vertically restrained with elastic prestressed tendons that pass through the centerline of the columns while anchored at the foundation and the cap-beam. Following a variational formulation, the nonlinear equation of motion is derived in which the stiffness and the prestressing force of the tendons are treated separately. In this way, the postuplift stiffness of the vertically restrained rocking frame can be anywhere from negative to positive depending on the axial stiffness of the vertical tendons. The paper shows that the tendons are effective in suppressing the response of rocking frames with small columns subjected to long-period excitations. As the size of the columns, the frequency of the excitations, or the weight of the cap-beam increases, the vertical tendons become immaterial given that most of the seismic resistance of tall rocking frames originates primarily from the mobilization of the rotational inertia of their columns. The paper concludes with the presentation and validation of an equivalent rigid-linear system so that the rocking response of vertically restrained rocking frames can be computed with popular open-source or commercially available software simply by employing existing elastic-mutilinear elements.