For a better understanding of the elastic behaviour of the crystalline aggregate of cold-rolled steel having preferred orientations, the lattice strains of ideally oriented crystallite groups under uniaxial loading and residual microstresses after unloading have been determined by x-ray measurements. It is necessary to understand the elastic behaviour of a single crystal in a polycrystalline aggregate in order to understand the problems with the X-ray elastic constants and furthermore the origin of the residual microstresses after plastic deformation. The elastic and plastic anisotrophies of the crystallites of the textured steel are observed in the curves of the applied stresses versus the lattice strains of the crystallite groups. Internal microstresses of one of the crystallite groups of the textured steel under various uniaxial loadings and the residual microstresses after unloading could have been determined by the X-ray stress measurements. A model is proposed to present a better understanding of the elastic behaviour of the crystalline aggregate. From the analysis by the model it is seen that the elastic behaviours of the crystalline aggregate of the cold-rolled steel for the rolling and transverse directions are different. The difference may explain the dependence of Young`s modulus on the relative direction of the sample cut to the rolling direction of cold-rolled materials.