Dimensionality reduction of the 3D inverted pendulum cylindrical oscillator and applications on sustainable seismic design of bridges

Abstract

The simplest 3D extension of Housner's planar rocking model is a rocking (wobbling) cylinder allowed to uplift and roll on its circumference, but constrained not to roll out of its initial position. The model is useful for the description of bridges that use rocking as a seismic isolation technique, in an effort to save material by reducing the design moment and the size of the foundations. This paper shows that describing wobbling motion in terms of displacements rather than rotations is more useful. It unveils that a remarkable property of planar rocking bodies extends to 3D motion: A small and a large wobbling cylinder of the same slenderness will sustain roughly equal top displacement, as long as they are not close to overturning. This allows for using the response of an infinitely large wobbling cylinder of slenderness alpha as a proxy to compute the response of all cylinders having the same slenderness, irrespectively of their size. Thus, the dimensionality of the problem is reduced by one. Moreover, this paper shows that the median wobbling response to sets of ground motions can be described as an approximate function of only two non-dimensional parameters, namely (gtan alpha/PGA,u/PGD)or (gtan alpha/PGA,uPGA/PGV2) where u is the top displacement of the wobbling body.