This paper is part of a more general mixed analytical/numerical strategy aiming at computing residual stresses of metallic strips after coiling process. Multiphase transitions and transformation induced plasticity occur during coil cooling. Thus, each layer of coil is subjected to an overall eigenstrain that can be sufficient to generate macroscopic plastic deformations. For each layer, a solution of the problem of an elastic-plastic hollow cylinder undergoing an arbitrary eigenstrain is derived. Mathematical developments relies on the linear inhomogeneous Navier equation by dealing with plasticity through the introduction of a deviatoric unknown plastic strain. An analytical solution is obtained in the form of series expansion, for any trial plastic strain. Then, an energetic principle enables to determine the plastic strain chosen as a solution of the problem. Practically, a numerical optimization procedure is performed directly on coefficients of the plastic strain series expansion.