We have examined some basic properties of damped Ly$\alpha$ systems (DLAs) by semi-analytic modelling. We assume that DLA hosts are disk galaxies whose mass function is generated by a Press-Schechter formalism at redshift 3. The disk undergoes star formation and chemical evolution. We select modelled DLAs according to their observational criterion by Monte Carlo simulation using random lines of sight and disk inclinations. The DLA ages are set to be 1 to 3 Gyr. By best-fitting the predicted metallicity distribution to the observed ones, we get the effective yield for DLAs of about $0.25Z_{\odot}$. On the basis of this constraint, we further compared our model predictions with observations at redshift 3 in the following properties: number density, gas content, HI frequency distribution, star formation rate density, and the relationship between metallicity and HI column density. We found that the predicted number density at redshift 3 agrees well with the observed value, but the gas content $\Omega_{DLA}$ is about 3 times larger than observed since our model predicts more DLA systems with higher column density. The frequency distribution at higher HI column density is quite consistent with observations while some difference exists at the lower HI end. The predicted star formation rate density contributed by DLAs is consistent with the most recent observations. In addition, we have investigated the connection between DLAs and Lyman Break galaxies (LBGs) by comparing their UV luminosity functions, which shows that DLAs host galaxies are much fainter than LBGs. However, there is a discrepancy between the model predictions and observations in the correlation between metallicity and HI column density for DLAs. Further investigations are needed into the mode of star formation in high redshift environments.