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We investigate the radial collapse of carbon nanotubes bundles using the density-functional based tight-binding method for a large number of chiralities in the small diameter range. We find that for tubes larger than about 0.6 nm the collapse pressure fits a law, but with collapse pressures considerably larger than previous estimates based on classical potentials. Furthermore, we show that the effect of chirality on the collapse pressure is small, and that previous reports of large chiral effects are probably due to the use of too small unit cells. Our results are in good agreement with available experimental data, and provide a better understanding of the process of collapse of nanotubes under pressure.