The constrained-layer damping treatment is widely used to control the resonant response of vibrating structures, particularly in the automotive and aerospace industries. This technique requires design tools in order to select the best location, type and thickness of the damping and constraining-layer materials for optimum damping while minimizing the total weight of the damping treatments. In this study, a numerical tool is presented to analyze the vibration response of plates partially or fully covered with a constrained viscoelastic damping material. This tool is based on a finite element approach. The undamped plate structure is modeled using a discrete Kirchhoff theory (DKT) element. However, a sandwich plate element is specially derived to model the damped plate structure. This element is formulated in such a way that it is compatible with the DKT element. The natural frequencies and modal loss factors are derived from the modal strain energy method. The proposed approach is validated by comparing predictions with the results of various examples published in the literature. Hence, this approach is shown to predict accurately the damping characteristics of arbitrarily shaped sandwich plates, with different material properties and boundary conditions, fully or partially covered with constrained viscoelastic layers. The influence of the damping patch locations and the viscoelastic material's shear modulus on the vibration and damping characteristics is investigated. To damp the plate effectively using partial coverage, the damping patches must be distributed appropriately. To select the best damping patch locations, an indicator based on the energy dissipated through the viscoelastic layer is proposed.