A precise estimation of residual deformation of structures under bidirectional seismic excitation is essential, especially in the context of performance-based earthquake engineering. However, conducting bidirectional analysis is a challenging task even with the availability of modern software. In line with the recent investigations on estimating seismic responses to bidirectional loading per straightforward unidirectional analysis, this study aims to explore a simple approach to assess the residual response of a structure to bidirectional shaking by employing less-rigorous unidirectional analysis. The amplification in residual deformation of a bridge pier due to bidirectional excitation relative to the unidirectional counterpart has been computed under a set of simplified singe-pulse and multi-pulses (extracted from near-fault pulse-like motions) covering all incidence angles. For a given pulse, specific incidence angles are noted in which such amplification becomes minimum and maximum. These incidence angles are related to the orientations of the pulse where relative component intensity measured in terms of a physically meaningful ground motion parameter becomes maximum and minimum, respectively. As such, most preferred and least preferred orientations are identified corresponding to the smallest and the largest differences between the responses to bidirectional and unidirectional analyses. Hence, the unidirectional analysis in the most preferred orientation may be used for seismic design of bridge piers in future.