To construct an asymptotically “accurate” model of plane multi-layer structures (rods), we use the following assumptions: (1) each layer is described by using a proper system of coordinates and a system of basic functions for the components of displacements (V.V. Bolotin and Yu.N. Novichkov); (2) displacements of one-layer structure are represented in terms of infinite power series with respect to the transversal coordinate and, consequently, equations governing stress-strain states of a structure may be obtained on the basis of the Lagrange variational principle (Kh.M. Mushtari and I.G. Teregulov); (3) the asymptotic analysis of the obtained infinite system of differential equations may be performed by employing the procedure of separation of the stress-strain state into the main stress state and the solutions corresponding to boundary layers. Based on these assumptions, we construct the theory for determining the main stress states with the accuracy of e2 (where e=H/l is the small parameter) for multi-layer rods of arbitrary properties of layers. We consider a particular case of soft filler and compare the equations obtained with the model by V.V. Bolotin and Yu.N. Novichkov. The equations of our model may be integrated. We compared the results obtained with the exact solution.