An analytical method is presented to determine the delamination buckling and growth of an axially loaded laminated beam-plate with through-the width delamination at an arbitrary location in loading direction. The influence of the delamination size, depth, and location on initial buckling and strain energy release rate is investigated with consideration of the transverse shear deformation. The buckling load is significantly affected by the location of the unsymmetric delamination in axial direction for the relatively thick delamination. The effects of the transverse shear deformation on the growth of delamination are investigated. The energy release rate with the transverse shear deformation is higher than that without transverse shear deformation.