A new composite finite element of a three-layer partial-interaction sandwich composite beam with interlayer interfacial slip was derived for static, dynamic and buckling analysis. The partial-interaction effects of the sandwich beam were considered by deriving the energy principle based on the Timoshenko beam theory. Then a high-order interpolation function with internal degrees of freedom was used for determining the nodal displacement, section rotary angle and interfacial slips of the sandwich beam, which could solve the frequent "slip locking" phenomenon in numerical analysis. The explicit stiffness matrix, mass matrix and geometric stiffness matrix of the sandwich beam were obtained through variational principle. The accuracy of the proposed composite finite element for the corresponding sandwich structure was verified through the numerical program which was developed based on the MATLAB software. The static, dynamic and buckling analysis of the three-layer sandwich beam with different cross-sections were then carried out, under the circumstances of various loads and different boundary conditions. The variation laws of the calculation results and their errors of sandwich composite beams with different span-depth ratios and different interfacial shear stiffnesses were also analyzed. The proposed composite finite element with explicit expressions has high calculation efficiency and is easy to be applied into other finite element programs or commercial software subroutines.