This paper proposes a novel traffic assignment model considering uncertainties in both demand and supply sides of a road network. These uncertainties are mainly due to adverse weather conditions with different rainfall intensities on the road network. A generalized link travel time function is proposed to capture these effects. The proposed model allows the risk-averse travelers to consider both an average and uncertainty of the random travel time on each path in their path choice decisions, together with the impacts of weather forecasts. Elastic travel demand is considered explicitly in the model responding to random traffic condition in the network. In addition, the model also considers travelers' perception errors using a logit-based stochastic user equilibrium framework formulated as fixed point problem. A heuristic solution algorithm is proposed for solving the fixed point problem. Numerical examples are presented to illustrate the applications of the proposed model and efficiency of the solution algorithm.