Methods of enforcing earthquake base motions in seismic analysis of structures

It is important to design structures, systems, and components subjected to earthquake base motions to ensure their structural integrity and secure functionality. This paper presents methods to enforce base excitations for designing structures, qualifying equipment, and assessing seismic fragility against such base motions in seismic analysis. The concept of the large force method is extended from that of the large penalty technique applied in static analysis accounting for displacement boundary conditions. An action exerted on the artificial base block is equivalently enforced by applying the corresponding base motion using the large penalty technique. Depending on the way to solve the equations of motion, the large force method may be expressed as an indirect-enforcing or direct-enforcing method. The indirect-enforcing method is derived by decomposing whole system motion into free structural motion and constrained base motion, with the structural motion divided into two parts to simplify the input of base motion. By contrast, upon applying the large penalty factor, the direct-enforcing method is achieved by directly solving the equations of motion without the breakdown of the system motion. Modal analysis for the direct-enforcing method is conducted to investigate the characteristics of frequency and mode properties in the extreme low/high frequency domains using the Jacobi method and the frequency shift technique. To compare the response results, the Newmark direct integration scheme is applied in the large force method for analyzing structural systems under harmonic and random base excitations. The compatibility of base time histories taken from an earthquake motion database is discussed to avoid errors in using the large force method. Information presented in this paper is useful in adopting a proper way to enforce base motions and interpreting the response in vibration analysis.