Based on the analytical mechanics, the Euler-Lagrange equations of DNA chain with elastic rod model are derived in detail by the variation of the free energy functional, which depends on the curvature κ, torsion τ, twisting angle χ and its derivative with respect to the arc-length of central axis curve of rod. With the different shapes of rod, we obtain the equilibrium equations of DNA with circular and noncircular cross sections, which provide an approach to describe the physical behaviors of A-, B-, Z-DNA. The results show that the elastic rod model with circular cross section can accurately characterize the equilibrium configurations of A- and B-DNA, while the model with elliptical cross sections (k=0.141) is more suitable for Z-DNA. This study might be helpful to characterize the mechanical properties of DNA chains or design DNA-template devices in a wide range of applications.