Effect of the reinforced leading or trailing edge on the aerodynamic performance of a two-dimensional (2D) perimeter-reinforced (PR) membrane wing at Re=2500 is investigated numerically, using a novel model of membrane wing and a fluid-structure interaction solution procedure based on the characteristic-based split (CBS) finite element method. For the PR membrane wing, it is found that the time-averaged lift will not be affected much when 20% of the membrane is reinforced at the leading edge. However, the same length of reinforcement at the trailing edge will deteriorate greatly the time-averaged aerodynamic performance. At lower angles of attack, the reinforced leading edge could trigger the shedding of the leading-edge vortices, which will increase the time-averaged lift but also make the response of the membrane much more irregular and uncontrollable. In addition, secondary vortex structures are observed at higher angles of attack when more than 60% area of the membrane wing is reinforced at the leading or trailing edge. The secondary vortices, which will push the shed leading-edge vortices away from the wing surface, are found to be the key reason for the significant drop of the lift. This paper could reveal more details of the vortex structures and structural responses of the PR membrane wing in laminar flow regime and provide a new membrane wing model for the micro air vehicles.