Internet of Things (IoT) is an important component of the 5(th) generation mobile network, which is being deployed worldwide. To achieve the goal of "connecting everything", a great amount of devices are connected. New scenarios and applications are developed for IoT, some of which require intensive computations and low response delay. This makes edge computing and cloud computing two main technologies to back the Internet of Things. Edge-cloud computing was proposed to utilize the advantages of both these two computing forms. In this paper, we construct a secure edge-cloud computing (SECC) framework. Sensor nodes and applications can interact with the framework through unified interfaces. We implement the edge server as a collection of services, including Orchestration, Data Processing and Storage, Communication Management, Authentication and Authorization, and Situation Awareness. Through a daisy-chain approach, our framework can be secured for heterogeneous security needs of different parts of the system. We also build a prototype of the proposed framework and demonstrate the efficacy and security of the SECC framework through experiments and comprehensive analysis.