Introducing electric vehicles (EVs) into urban transportation network brings higher requirement on travel time reliability and charging reliability. Specifically, it is believed that travel time reliability is a key factor influencing travelers' route choice. Meanwhile, due to the limited cruising range, EV drivers need to better learn about the required energy for the whole trip to make decisions about whether charging or not and where to charge (i.e., charging reliability). Since EV energy consumption is highly related to travel speed, network uncertainty affects travel time and charging demand estimation significantly. Considering the network uncertainty resulted from link degradation, which influences the distribution of travel demand on transportation network and the energy demand on power network, this paper aims to develop a reliability-based network equilibrium framework for accommodating degradable road conditions with the addition of EVs. First, based on the link travel time distribution, the mean and variance of route travel time and monetary expenses related to energy consumption are deduced, respectively. And the charging time distribution of EVs with charging demand is also estimated. Then, a nested structure is considered to deal with the difference of route choice behavior derived by the different uncertainty degrees between the routes with and without degradable links. Given the expected generalized travel cost and a psychological safety margin, a traffic assignment model with the addition of EVs is formulated. Subsequently, a heuristic solution algorithm is developed to solve the proposed model. Finally, the effects of travelers' risk attitude, network degradation degree, and EV penetration rate on network performance are illustrated through an example network. The numerical results show that the difference of travelers' risk attitudes does have impact on the route choice, and the widespread adoption of EVs can cut down the total system travel cost effectively when the transportation network is more reliable.