In any nonlinear system as complex as an urban rail transit network or metrorail network, some incidence of perturbations of its state is inevitable. These perturbations, such as natural hazards, can highly affect the networks' resilience. Increasing the ability of metrorail networks to withstand such perturbations requires robustness and vulnerability assessments as key attributes of resilience and necessary steps toward developing reliable networks. Most models developed for this purpose associate a network's failures to binary representations of the failure of its components without incorporating weight factors. Since ridership is a primary factor to define the metrorail network performance, this paper proposes a general ridership pattern, considers different failure cases, and uses a novel methodology to quantitatively measure the weighted-network resilience attributes incorporating ridership throughout the Washington, DC Metrorail as a case study. The proposed methodology has clear relationships to adjacency and link-weight matrices and defines a new expression for the weighted global network efficiency based on the sum of weights on each geodesic path. Results show that the most vulnerable stations and links hold critical positions in the network topological structure and/or bear larger amounts of ridership. For the case study, the most vulnerable components include transfer stations located in the city center as well as stations and links on the northwest section of the Red Line. The methodology presented herein provides insights for enhancing critical components during the planning and operation of a metrorail by mitigating the risks associated with failure events.