In this paper, we propose a set of two-stage robust optimization models to design reliable p-median facility location networks subject to disruptions. A customized column-and- constraint generation approach is implemented and shown to be more effective than Benders cutting plane method. Numerical experiments are performed on real data and management insights on system design are presented. Our study also demonstrates the strong modeling capability of two-stage robust optimization scheme by including two practical issues, i.e., facility capacities and demand losses due to disruptions, which receive little attention in reliable network design research. Results show the significant influence of the demand loss factor on the network configuration.