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This paper studies the capacity restraint transit assignment problem with elastic demand. We analyze the urban transit network characteristics, such as the links have finite capacities, equilibrium delay only arises when capacity is reached; and then a variational inequality model of the stochastic user equilibrium transit assignment with elastic demand under capacity constrain is proposed. The proposed model can simultaneously predict how passengers would choose their optimal routes and estimate passenger flows in a congested transit networks. Based on the penalty function method, an algorithm for solving the proposed model is presented. Finally, the algorithm is illustrated with a numerical example. The results show that the algorithm is quite satisfactory.