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• A unified DPIM-COM approach for time-invariant and time-variant RBDO is proposed. • DPIM together with Heaviside function is devised to compute failure probability directly. • DPIM-COM can attack time-invariant and time-variant RBDO with MPPs effectively. • DPIM combines with extreme value distribution to assess time-variant probabilistic constraint. Reliability-based design optimization (RBDO) can fulfill performance-based design of static and dynamic structures considering various uncertainties. For time-invariant and time-variant RBDO problems with multiple most probable points (MPPs, also termed as design points), the existing approaches encounter some difficulties in searching for all MPPs and performing time-variant RBDO which requires complicated time-variant reliability analysis. In this paper, the direct probability integral method (DPIM) combining with the change of probability measure (COM), i.e., DPIM-COM approach, is proposed to attack time-invariant and time-variant RBDO problems with multiple MPPs in a unified framework. Firstly, the DPIM in conjunction with Heaviside function is developed to estimate time-invariant and time-variant probabilistic constraints directly and explicitly. Then, an efficient formula for calculating the sensitivity of probabilistic constraints with respect to random design variables is derived based on the DPIM and COM strategy, where the representative points for reliability computation keep unchanged and only the corresponding assigned probabilities need to be updated. Accordingly, the corresponding structural responses of representative points at the current design variables can be reused to evaluate the sensitivity of probabilistic constraints, which saves much computational cost, especially for time-variant RBDO problems. Finally, three numerical examples, including the ten-story building with tuned mass damper under deterministic earthquake ground motion and near-fault stochastic impulsive ground motions, demonstrate the effectiveness and versatility of the proposed DPIM-COM approach for addressing time-invariant and time-variant RBDO problems with multiple MPPs. [ABSTRACT FROM AUTHOR]