An effective computational scheme to calculate the complete set of independent elastic constants as well as other structural parameters including bulk modulus, shear modulus, Young's modulus, and Poisson's ratio for crystals is reported. The scheme is based on the stress-strain analysis approach with the appropriate selection of strain governed by symmetry consideration. The first principles Vienna ab initio simulation package (VASP) is used in stress calculations. Comprehensive tests were performed for alpha-SiO2 and spinel MgAl2O4 with different exchange-correlation potentials, and different sets of computational parameters to investigate the relative accuracies of the calculations. A wide range of oxides, nitrides, and carbonate crystals with different crystal symmetries were chosen to test the scheme under both LDA and GGA approximations at zero temperature and pressure. Some of these calculations for large complex crystals are believed to be attempted for the first time. The calculated elastic constants show quite good agreement with the existing experimental data for almost all the examined systems with the exception of the relatively soft material such as alpha-SiO2 and the C-14 parameter of some trigonal crystals expressed in the hexagonal form such as in alpha-Al2O3. Other structural properties derived from the elastic constants also show good agreements with the measured values.