The classical strength of materials for beams is represented through the first two terms of the asymptotic expansion of the solution of Navier's equations. The method of asymptotic expansions with respect to the inverse of the slenderness of the beam permits us to obtain an approximate solution of Saint-Venant's problem. For the elasticity of the second order, the displacement field is obtained as the sum of a series, the general term of which at the nth order is the solution of a differential recursive system. We presently propose a general way of solving this kind of system. The exact solution is given explicitly in the case of a slender field (beam).