About the Link between the Relative and Absolute Motion Equations under a Differentiated Seismic Impact

Abstract

When a structure is subjected to a differentiated seismic impact, each of its support points is characterized by different functions of displacements, velocities and accelerations as it moves with the ground. The main purpose of this article is to show how a multidimensional kinematic impact can be converted to a form suitable for carrying out standard earthquake engineering analysis, i.e., how it is expressed in terms of translation seismic forces and seismic intensity. A description of differentiated impact in the form of a field of kinematic parameters is presented. A definition of seismic impact intensity in terms of intensity at one of the support points (anchor point) and acceleration field alteration functions (according to Y.P. Nazarov's formulation) is given. A definition of translational motion is given, and the equations of motion in absolute and relative coordinates are obtained. In the equations of motion written in absolute generalized coordinates, seismic load is expressed in the form of forces defined as the product of the support elements system stiffness matrix by the vector of support displacements. It is shown that in the equations of motion written in relative coordinates, inertial seismic forces depend, as in case of using the traditional integral description of a seismic impact, on the seismic intensity at the anchor point. The theoretical formulas are illustrated using the example of a two-dimensional frame, for each support point of which individual functions of vertical, horizontal and angular displacements are specified.