A general eigen-value buckling solution is developed for the buckling of long thick pipes subjected to internal and external hydrostatic pressure. The principle of stationary potential energy is used to formulate the conditions of equilibrium, neutral stability conditions, and associated boundary conditions using polar coordinates. The formulation accounts for shear deformation effects and is suited for composite pipe systems with thick cores. It involves destabilizing terms: one is due to the external hydrostatic pressure and incorporates the follower effects, and the other, is due to the pre-bucking stresses undergoing the nonlinear components of strains. The formulation adopts a work conjugate triplet consisting the Cauchy stress tensor, the Green Lagrange strain tensor, and constant constitutive relations. A Fourier series expansion of the displacement fields is adopted to transform the 2D problem into a series of independent 1D problems, thus keeping the computational effort to a minimum while preserving the accuracy of the solution. Two numerical solutions were developed and implemented under MATLAB; the first one is based on the finite difference technique and the second one is based on the finite element solution. Both solutions were shown to converge to the same solution, the finite difference from below, while the finite element converges from above. The finite element solution is then applied to predict the buckling capacity of sandwich pipes consisting of two steel pipes with a soft core. A comprehensive verification study is conducted and the validity of the formulation was established through comparison with other solutions. A parametric study is then conducted to investigate the effect of hydrostatic internal pressure, core material, core thickness, and internal and external pipe thicknesses, on the external buckling pressure of sandwich pipes.