Textile composites are widely used in architecture and civil engineering. The periodic structure and multiscale nature of this composite motivate the adoption of homogenization frameworks in investigating their mechanical properties. Existing studies mainly focus on the in-plane (membrane) behaviors of textile composites, although the out-of-plane properties can also be important in certain applications. In this contribution, we present a comprehensive study of the mechanical properties of a textile composite using computational homogenization for thin sheets within the frameowrk of Kirchhoff-Love shell theory. A representative volume element (RVE) model is developed based on the virtual fiber method (VFM), in which yarns are modelled as bundles of one-dimensional truss elements. This approach avoids the complexity associated with geometric modeling and meshing. Different stiffness contrasts between fibers and matrix are chosen to represent various fiber materials. For multiple loading scenarios, the mechanical response is computed and the corresponding tangent stiffness matrices are extracted throughout the loading history. The evolution of geometric nonlinearity and the effective stiffness as a function of the applied loadings is then analyzed and compared for models with different stiffness ratios. Finally, the effects of different geometric imperfections on the composite mechanical properties are investigated.