•We describe a new mathematical formulation for the unidirectional QCSP with stability constraints (QCSPVS) based on trim stability.•We propose an efficient logic-based Benders approach to optimally solve the QCSPVS.•We present extensive computational experiments on benchmark instances aimed at validate the effectiveness of the proposed model and solution approach. This paper addresses the quay crane scheduling problem (QCSP) with vessel stability constraints. Vessel stability is essential to improve quay crane operations in container terminals, but it significantly complicates the basic QCSP and the corresponding solutions methods. We describe a novel mathematical formulation for the unidirectional QCSP with vessel stability, and we propose an exact algorithm based on logic-based Benders decomposition to solve the problem efficiently. The problem is decomposed into two subproblems, e.g., a task-assignment master problem without vessel stability constraints, and a time-allocation problem, aimed at determining the operation time of each task under the premise of the vessel stability requirements. The proposed algorithm is tested on benchmark instances derived from the literature, and the effectiveness of the proposed model and solution approach is demonstrated.