This paper presents ab initio calculations of the structural, elastic, and vibrational properties of single-wall carbon nanotubes (SWNTs) with different radii and chiralities. The calculations are based on pseudopotential-density-functional theory, allowing for the study of systems with a large number of atoms per cell. The results show that the Young modulus of SWNTs is very similar to that of graphite and does not systematically vary with radius or chirality. The Poisson ratio retains graphitic values, except for a possible slight reduction for small radii, and shows chirality dependence. The vibrational properties, including the breathing mode, twistons, and high-frequency optic modes, are studied, with the latter showing a small chirality dependence at the top of the band. The results are compared with the predictions of the simple zone-folding approximation, which offers accurate results, even for relatively small radii, except for the known deficiencies in the low-frequency vibrational regions. The structural properties of the nanotubes show that the average carbon bond-length is within 1% of the graphitic value. The bond lengths and angles vary with tube radius, with the longer bond being perpendicular to the tube axis in (n,n) tubes. The structural distortions affect the electronic structure of these metallic tubes, with the Fermi level position being influenced by the reduced symmetry of the nanotubes. The elastic properties show that the strain energy per atom follows the classical elasticity theory behavior, with a value of C/r², where C is a constant dependent on the Young modulus and thickness of the wall. The Young modulus of SWNTs is found to be around 56 eV/atom, with small variations between tubes of different radii and chirality. The Poisson ratio is found to be around 0.14 for (n,n) tubes, with a slight increase for other chiralities. The vibrational properties show that the breathing mode and acoustic bands are more stable, with the breathing mode frequency showing a 1/r dependence. The results confirm that the effect of curvature on the elastic properties is small, and the zone-folding approach provides accurate results for the vibrational properties of the nanotubes. The study highlights the importance of ab initio calculations in understanding the properties of carbon nanotubes, and the limitations of simpler theoretical approaches.This paper presents ab initio calculations of the structural, elastic, and vibrational properties of single-wall carbon nanotubes (SWNTs) with different radii and chiralities. The calculations are based on pseudopotential-density-functional theory, allowing for the study of systems with a large number of atoms per cell. The results show that the Young modulus of SWNTs is very similar to that of graphite and does not systematically vary with radius or chirality. The Poisson ratio retains graphitic values, except for a possible slight reduction for small radii, and shows chirality dependence. The vibrational properties, including the breathing mode, twistons, and high-frequency optic modes, are studied, with the latter showing a small chirality dependence at the top of the band. The results are compared with the predictions of the simple zone-folding approximation, which offers accurate results, even for relatively small radii, except for the known deficiencies in the low-frequency vibrational regions. The structural properties of the nanotubes show that the average carbon bond-length is within 1% of the graphitic value. The bond lengths and angles vary with tube radius, with the longer bond being perpendicular to the tube axis in (n,n) tubes. The structural distortions affect the electronic structure of these metallic tubes, with the Fermi level position being influenced by the reduced symmetry of the nanotubes. The elastic properties show that the strain energy per atom follows the classical elasticity theory behavior, with a value of C/r², where C is a constant dependent on the Young modulus and thickness of the wall. The Young modulus of SWNTs is found to be around 56 eV/atom, with small variations between tubes of different radii and chirality. The Poisson ratio is found to be around 0.14 for (n,n) tubes, with a slight increase for other chiralities. The vibrational properties show that the breathing mode and acoustic bands are more stable, with the breathing mode frequency showing a 1/r dependence. The results confirm that the effect of curvature on the elastic properties is small, and the zone-folding approach provides accurate results for the vibrational properties of the nanotubes. The study highlights the importance of ab initio calculations in understanding the properties of carbon nanotubes, and the limitations of simpler theoretical approaches.