The potential use of individual carbon nanotubes as nano devices warrants detailed investigation of their mechanical behavior based on structural and geometrical configurations. The objective of this paper is to unravel the structural and chirality dependence of the axial Young's modulus of a carbon single-walled nanotube by analytical and numerical approaches. In this work, we employ the general homogenization composite shell model developed based on the asymptotic homogenization technique for analytical modeling of single-walled nanotubes. We derive the working formulae for the effective elastic properties of carbon single-walled nanotube of any chirality and predict the structural and chiral dependence of the effective axial Young's modulus of the nanotube. Also, a finite element analysis on the chirality dependence of the axial Young's modulus of the carbon nanotube is reported. The outcomes of our analyses are compared with available experimental and simulation results.