The extension of a single-user-class macroscopic dynamic traffic assignment model to include multiple user classes is considered. The distinction between user classes is based on vehicle characteristics. Cars and trucks are two typical classes. To deal with various asymmetries that may occur, such as intra-user-class interaction and spatial and temporal asymmetries, the model is specified as a quasi-variational inequality problem. A nested modified projection method is successfully adopted to solve the assignment problem. The solution of the problem depends heavily on the choice of some very important input: the multiclass link travel time functions. Under mild restrictions there exists a solution, which need not be unique. A case study illustrates the model.