A model of internally structured emulsion droplets is presented that accounts for the traction forces generated by interfacial tension and the von Mises yield criterion of the internal supporting network. For symmetric droplets, the method calculates the total stress acting on a droplet locally, allowing droplet stability and location of failure to be predicted. It is not regions of high interfacial curvature that prompt droplet reconfiguration, rather regions transitioning from high to low curvature. The model enables the design of emulsion droplet response and reconfigurability to external triggers such as changes in surface tension (surfactant concentration) and temperature.