Empirical model for vertical-to-horizontal spectral amplitude ratio of seismic design spectra considering incident angle effect

The vertical design spectrum serves as a benchmark for assessing structural capacity under vertical seismic action. However, in most seismic design codes, vertical spectra are often oversimplified. In this study, a vertical-to-horizontal spectral amplitude ratio (VHSR) model is developed to derive a corresponding vertical design spectrum from its horizontal counterpart. The influence of the earthquake incidence angle on horizontal spectral ordinates is explicitly accounted for, revealing that this factor can introduce significant variability into empirical VHSR models. Specifically, the VHSR corresponding to the most unfavorable incidence angle can be up to twice that associated with the most favorable one. Accordingly, vertical spectra are determined not only based on the median VHSR but also by considering the maximum and minimum VHSR values across all possible incidence angles of horizontal excitation. The effects of earthquake magnitude, epicentral distance and local site conditions (site class) on VHSR are systematically examined. Results indicate that VHSR exhibits a strong dependence on site class. A site-specific VHSR model is proposed based on the derived VHSR–T relationships, demonstrating that VHSR decreases with increasing site softness for T ≥ 0.3 s, whereas an opposite trend is observed for 0.06 s ≤ T ≤ 0.3 s. Finally, it is shown that the record-to-record variability of VHSR follows a lognormal distribution, which can facilitate the development of probabilistic vertical design spectra from a consistent horizontal reference.