Random hyper-network is an important hyper-network structure. Studying the structure and properties of random hyper-networks, which helps researchers to understand the influence of the hyper-network structure on its properties. Currently, studies related to the influence of the internal structure of the hyper-edge on robustness have not been carried out for research on the robustness of hyper-networks. In this paper, we construct three $k$-uniform random hyper-networks with different structures inside hyper-edges. The nodes inside hyper-edges are connected in the ways randomly connected, preferentially connected and completely connected. Meanwhile, we propose a capacity-load model that can describe the relationship between the internal structure and the robustness of the hyper-edge, based on the idea of capacity-load model. The robustness of the three hyper-networks was obtained by simulation experiments. The results show the variation of the internal structure of hyper-edge has a large influence on the robustness of the $k$-uniform random hyper-network. In addition, the larger number of ordinary edges $m_k$ inside the hyper-edges and the size of the hyper-network $k$, the more robust the $k$-uniform random hyper-network is.