A time-domain semi-analytical framework for constructing input energy spectra from design spectra and its application to the Chinese seismic code

Constructing design spectra that accurately quantify seismic input energy is a fundamental task in energy-based seismic design. Existing time-domain empirical methods often rely on extensive or region-specific seismic data regressions, deep learning-based approaches lack physical interpretability, and frequency-domain theoretical methods face challenges in explicitly incorporating the cumulative effects of strong-motion duration ( t D ). To address these limitations, this study proposes a code-compatible, time-domain semi-analytical framework that derives elastic and plastic input energy equivalent velocity spectra ( V EI,E -spectra and V EI,P -spectra) directly from the design acceleration response spectra ( S a ) and t D. This study originally formulates harmonic time-domain steady-state solutions that decompose input energy into peak-response and time-accumulative components. Serving as the foundation, these solutions are extended to seismic scenarios via an inter-spectral mapping coefficient, C E ( t D , T g , ζ ), which links S a to input energy. Introducing a medium-to-long period attenuation index γ 1 ( t D , T g , ζ ) and categorizing t D into three types enable the proposed method to successfully establish V EI,E -spectra explicitly tailored for the Chinese Code for Seismic Design of Buildings. The V EI,E -spectra are further converted into V EI,P -spectra via regression-based conversion equations. Ground motion effects are already captured, which is why this conversion is governed solely by structural parameters—specifically the strength reduction factor ( R ), post-yield stiffness ratio ( λ ), and damping ratio ( ζ )—ultimately establishing V EI,P -spectra compatible with the Chinese code. Finally, the practicality and accuracy of the proposed V EI,E -spectra and V EI,P -spectra for the Chinese code are validated through four sets of case studies. This methodology provides a practical tool for rapidly estimating structural input energy directly from seismic codes.