Unmanned aerial vehicles (UAVs) can be used as aerial wireless base stations when cellular networks go down. Prior studies on UAV-based wireless coverage typically consider downlink scenarios from an aerial base station to ground users. In this paper, we consider an uplink scenario under disaster situations (such as earthquakes or floods), when cellular networks are down. We formulate the problem of optimal UAV placement, where the objective is to determine the placement of a single UAV such that the sum of time durations of uplink transmissions is maximized. We prove that the constraint sets of problem can be represented by the intersection of half spheres and the region formed by this intersection is a convex set in terms of two variables. This proof enables us to transform our problem to an optimization problem with two variables. We also prove that the objective function of the transformed problem is a concave function under a restriction on the minimum altitude of the UAV and propose a gradient projection-based algorithm to find the optimal location of the UAV. We validate the analysis by simulations and demonstrate the effectiveness of the proposed algorithm under different cases.