With the increasing need for analyzing graph data, graph systems have to efficiently deal with concurrent graph processing (CGP) jobs. However, existing platforms are inherently designed for a single job, they incur the high cost when CGP jobs are executed. In this work, we observed that existing systems do not allow CGP jobs to share graph structure data of each iteration, introducing redundant accesses to same graph. Moreover, all the graphs are real-world graphs with highly skewed power-law degree distributions. The gain from extending multiple external storage devices is diminishing rapidly, which needs reasonable schedulings to balance I/O pressure into each storage. Following this direction, we propose GraphScSh that handles CGP jobs efficiently on a single machine, which focuses on reducing I/O conflict and sharing graph structure data among CGP jobs. We apply a CGP balanced partition method to break graphs into multiple partitions that are stored in multiple external storage devices. Additionally, we present a CGP I/O scheduling method, so that I/O conflict can be reduced and graph data can be shared among multiple jobs. We have implemented GraphScSh in C++ and the experiment shows that GraphScSh outperforms existing out-of-core systems by up to 82%.