The Time Capacitated Arc Routing Problem (TCARP) extends the classical Capacitated Arc Routing Problem by considering time-based capacities instead of traditional loading capacities. In the TCARP, the costs associated with traversing and servicing arcs, as well as the vehicle’s capacity, are measured in time units. The increasing use of electric vehicles and unmanned aerial vehicles, which use batteries of limited duration, illustrates the importance of time-capacitated routing problems. In this paper, we consider the TCARP with stochastic demands, i.e.: the actual demands on each edge are random variables which specific values are only revealed once the vehicle traverses the arc. This variability affects the service times, which also become random variables. The main goal then is to find a routing plan that minimizes the expected total time required to service all customers. Since a maximum time capacity applies on each route, a penalty time-based cost arises whenever a route cannot be completed within that limit. In this paper, a strategic oscillation simheuristic algorithm is proposed to solve this stochastic problem. The performance of our algorithm is tested in a series of numerical experiments that extend the classical deterministic instances into stochastic ones.