Prediction of flow and sediment transport in open channels and rivers, in particular the evolution of the bed profile and cross-sectional shape, is of critical importance in many river engineering applications due to the possible effects on the river eco-system, structural integrity of hydraulic structures, navigability, etc. This paper presents a one dimensional (1D) model RubarBE that has been developed to simulate non-equilibrium sediment transport and river morphological changes under unsteady flow. The model has the capability of computing mixed flow regimes, i.e., combined subcritical and supercritical flows. Sediment size distribution is represented by the median diameter and the standard deviation instead of the usual modelling with multiple particles size classes. Equations of water and sediment transport are solved in a coupled procedure that includes the changes in bed elevation and bed material composition. The bed elevation changes in a cross-section are calculated using an innovative method that links bed deformation to the boundary shear stress distribution around the wetted perimeter. The model handles irregular cross-sections and the availability of multiple geometry updating methods enables the user to select the method that represents adequately the channel morphology evolution. The model is validated using hypothetical simple cases characterized by the evolution of the bed to an equilibrium state and a laboratory experiment of dam-break wave that permits to simulate very unsteady situations. The application of the model to real rivers shows the suitability of the model to deal with field problems and to obtain realistic results for operational objectives. The Ha!Ha! dam break event reveals very strong bed changes (erosion up to 20 m deep) that are reasonably well reproduced if the appropriate type of cross sectional shape updating method is selected, which proves the influence of the shape of cross section on bed longitudinal profile. The Rivière aux Herbes case shows that during one extreme event, the accumulation of sediment may raise peak water level strongly but the erosion process should also been taken into account. Finally, the reservoir flushing of the Arc River is simulated; in this case the calibration of the model is possible because of the detailed field measurements.