Bending and free vibration analyses of ribbed plates with a hole based on the FSDT meshless method

•A meshless model for ribbed plate with straight ribs in any direction and a hole in any location is established.•A meshless method for free vibration and flexure analysis of the ribbed plates with a hole is proposed.•The proposed method allows the ribs to be placed anywhere on the plate, and remeshing of the plate under the finite element method can be avoided when rib location changes. A meshless method based on the first-order shear deformation (FSDT) theory and moving least-squares (MLS) approximation for bending and free vibration analyses of the ribbed plate with a hole is proposed. Firstly, a ribbed plate with a hole is regarded as a composite structure of a plate (with a hole) and ribs. Nodes are used to discretize the plate and ribs. Secondly, the displacement fields of the plate and ribs are established based on the FSDT and MLS, and their potential and kinetic energy functional are derived. The diffraction law is used to treat the discontinuity related to the hole on the plate. Thirdly, the total potential and kinetic energy of the ribbed plate is obtained by superposing the potential and kinetic energy of the plate and ribs and introducing the compatibility condition. Finally, the equations governing the elastic bending and free vibration of the ribbed plate with a hole are obtained according to the principle of Minimum Potential Energy and Hamilton’s Principle. The boundary conditions are enforced by the full transformation method. Thereafter, several ribbed plate with hole examples with different holes, rib arrangements, boundary conditions and load conditions are calculated. The results are compared with those given by ABAQUS and literature. The results show that the effectiveness and accuracy of the proposed method in analyzing the static and dynamic problems of the ribbed plate with a hole.