Finite element mesh quality, which affects analysis accuracy and computational cost, is usually determined based on the designer's experience. In this study, we propose a GCN (Graph Convolutional network)-based method for evaluating the quality of 3D meshes composed of shell elements. In the proposed method, a three-layer network is constructed using GENConv with graph adjacency matrix and feature matrix as input. The feature matrix is created based on geometric shapes such as coordinates and the mesh qualities such as aspect ratio. A dual graph is constructed by converting finite elements into graph nodes and their adjacent parts into graph edges, and the created features are given to the graph nodes for learning. The proposed method is applied to a dataset of automotive side-member FE models and the effectiveness of the constructed network and the introduced mesh features is confirmed. We also visualize the attribution of mesh features and important factors in the obtained results by calculating the integrated gradients of the network. This allows us to select the important mesh features both positively and negatively and explains the basis for our predictions.