Use of the equivalent system concept to calculate the natural frequency of buildings considering the effect of axial loading on columns with adjustable restraint

Resumo

Column stiffness is a fundamental parameter for the dynamic analysis of buildings. One of the mostimportant applications of structural dynamics theory is the use of the equivalent system concept, which allowstransforming systems with infinite degrees of freedom into one containing only a single degree of freedom. Withthis approach, multi-story buildings can be modeled as single-degree-of-freedom systems, allowing the stiffnessand mass of the columns to be included in the analysis. In the approach developed in this work, a standard columnis assumed to compose the floors of a building, with its properties being calculated based on the generalizedcoordinate assumed for its boundary conditions. In this context, the column's stiffness is calculated to take intoaccount its geometry, the influence of concentrated and distributed axial loading, and a certain degree of rotationalrestriction at the base, considered by the application of a rotational spring located at this position. The simulationassumes an arbitrary number of columns per floor and a certain number of floors in the building. A threedimensionalsystem is therefore represented by an equivalent system obtained by combining springs and masses,first in parallel, representing the grouping of columns on the floors, and then in series, representing the associationof the floors within the building. With this strategy, it was possible to determine the equivalent stiffness of eachfloor as the sum of the column stiffnesses, and that of the floors as the inverse of the sum of the inverses of thefloor stiffnesses at the different levels. The equivalent mass is obtained by eigenvalues in relation to the seriessystem. Thus, the natural frequency of the original model could be calculated using the concept of an equivalentsystem calculated under the influence of the loading, making this a non-linear approach.