In this study, transient response of the delaminated composite curved beam (DCCB) under radial moving force, traveling with the constant velocity along the beam, has been investigated. First order shear deformation theory (FSDT) is used, and general boundary conditions have been considered by employing artificial springs. The finite element method (FEM) is employed to solve these equations. In order to simulate the mechanical behavior of the delaminated region, dynamic contact between nodes has been applied on each pair of adjacent nodes, examining two critical issues: relative displacement between adjacent nodes, and the type of reaction force between them. Results for the free and forced vibration of the beam have been verified against those available in another literature and 3D model in ANSYS. The effects of different parameters such as the delamination’s length and location, fibers orientation, and material properties on the transient response of the DCCBs have been investigated. It has been concluded that the critical point of the beam, which has the greatest radial deformation, does not necessarily exist at the mid-point of the beam, and has different locations for different parameters. Also, the role of the dynamic contact is essential, particularly when delamination is in the lower layers.