This study presents *ab initio* calculations of the structural, elastic, and vibrational properties of single-wall carbon nanotubes with different radii and chiralities. The calculations are performed using pseudopotential-density-functional theory, allowing for large systems with a significant number of atoms per cell. The results show that the Young moduli of the nanotubes are similar to those of graphite and do not vary systematically with radius or chirality. The Poisson ratio retains graphitic values, except for a slight reduction for small radii, and exhibits a chirality dependence. The behavior of phonon branches, such as the breathing mode, twists, and high-frequency optic modes, is also studied, showing a small chirality dependence at the top of the band. The results are compared with predictions from the zone-folding approximation, which generally offers accurate results, even for small radii, except in the low-frequency vibrational region. The study highlights the importance of curvature effects and provides insights into the electronic and mechanical properties of carbon nanotubes.This study presents *ab initio* calculations of the structural, elastic, and vibrational properties of single-wall carbon nanotubes with different radii and chiralities. The calculations are performed using pseudopotential-density-functional theory, allowing for large systems with a significant number of atoms per cell. The results show that the Young moduli of the nanotubes are similar to those of graphite and do not vary systematically with radius or chirality. The Poisson ratio retains graphitic values, except for a slight reduction for small radii, and exhibits a chirality dependence. The behavior of phonon branches, such as the breathing mode, twists, and high-frequency optic modes, is also studied, showing a small chirality dependence at the top of the band. The results are compared with predictions from the zone-folding approximation, which generally offers accurate results, even for small radii, except in the low-frequency vibrational region. The study highlights the importance of curvature effects and provides insights into the electronic and mechanical properties of carbon nanotubes.