Most traffic assignment tasks in practice are performed by using deterministic network (DN) models, which assume that the link travel time is uniquely determined by a link performance function. In reality, link travel time, at a given link volume, is a random variable. Such stochastic network (SN) models are not widely used because the traffic assignment algorithms are much more computationally complex and difficult to understand by practitioners. In this paper, we derive an equivalent link disutility (ELD) function, for the case of risk averse drivers in a SN, without assuming any distribution of link travel time. We further derive a simpler form of the ELD function in a SN which can be easily implemented in deterministic user equilibrium traffic assignment algorithms like a DN. By comparing our two derived ELD functions, the bound of the coefficient of the simpler ELD functions is obtained, so that drivers will make the same risk averse route choice decisions. A method to estimate the coefficient of the simpler ELD function has been proposed and demonstrated with questionnaire survey data gathered in El Paso, Texas. The results of user equilibrium traffic assignments in a test network using the Bureau of Public Roads (BPR) function and the simpler ELD function are then compared. Our simpler ELD function provides a mean for practitioners to use deterministic user equilibrium traffic assignment algorithms to solve the traffic assignment problem in a SN for risk averse drivers during the peak hour commute.