Abstract: The meshless local radial point interpolation method (LRPIM) for the free vibration analysis of a plate of moderate thickness under several boundary conditions is presented. The discretized system equations are obtained using a locally weighted residual method. It uses a radial basis function coupled with a polynomial basis function as a trail function, and uses the quartic spline function as a test function of the weighted residual method. The shape functions obtained in the trail function have the Kronecker delta function property, and the essential boundary conditions can be easily imposed. The present method does not need any grids，and all integrals can be easily evaluated over regularly shaped domains and its boundary. The present method can also avoid the shear locking when it is used to solve the problem for the plate of moderate thickness. Examples show that results obtained by the presented method are found to agree well with the exiting solutions in the literature and with the results obtained by the finite element method, and the presented method has a number of advantages, such as the high efficiency and the quite good accuracy and high rate of convergence.