This article is about the nonlinear theory of elastic rods (Cosserat-Timoshenko rods). This theory uses bending stiffness, shear stiffness and tensile-compression stiffness and there are not any limitations for movements and rotations. Equations of quadratic approximation were derived. The variational formulation of problem as the search for the stationary point of the Lagrangian functional for quadratic approximation was given. The coherence of equations of quadratic approximation with variational formulation was approved in this article. The border conditions of this problem for quadratic approximation were given. The examples of solution of the problems in nonlinear theory are given in this article