Energy-based seismic design is an innovative structural design approach that systematically considers the energy-related demands of ground motion to analyze and design structures. An energy-based paradigm in seismic design is vitally relevant for structural systems in near-field areas. This research investigates the seismic behavior of three multi-span continuous concrete box-girder bridges (MSCC-BG) subjected to 328 ground motions from pulse-like and no-pulse databases using the OpenSees framework. It examines twenty-eight energy-related, residual, and displacement-based demands and thirty-six intensity measures (IMs) related to horizontal and vertical earthquake components. The study discusses the correlation between these demands and various IMs, identifying the most affected demands in each category, particularly under pulse-like earthquakes. Additionally, a multivariable probabilistic seismic demand model (PSDM) is developed for critical demands such as column hysteretic energy (EhColumn) and column residual drift ratio (RDRCol) using Lasso and Step-wise regression. The resulting model enhances prediction accuracy compared to single-IM PSDMs, while acceleration-related IMs play a more significant role. The research emphasizes the necessity of incorporating horizontal and vertical ground motion-related IMs in empirical equations to predict seismic demands precisely.