Computational modeling of curved structures formed from woven composites presents unique difficulties. Woven composites can undergo large nonlinear deformation that is not due entirely to material properties but rather yarns rotating to align with the loading direction. Moreover, yarns go through significant reorientations during the forming process resulting in nonuniform yarn angles in the curved structure. Modeling such complex structures at the component level as a homogenized macroscale continuum is not always an acceptable simplification, and damage features of individual mesoscale constituents are lost. A damage analysis framework incorporating concurrent multiscale modeling is proposed to address these complexities. Experiments on carbon woven fabric/polycarbonate (PC) composite dome parts are performed and compared with the simulations. Accounting for yarn reorientations, mesoscale representative volume elements (RVEs) with distinct yarn angles, whose constituents have been explicitly modeled, are simultaneously localized to the Gauss points of macroscale elements of the dome through Direct FE² homogenization. In virtual compression tests, force-displacement curves rise as parts of the dome undergo large deformations. Damage evolution and eventual ​cruciform​ damage of the dome model dominated by dual brittle-ductile failure modes are forecasted on the principal meridians. High shear stresses generated by yarn rotation are demonstrated to cause the ductile damage. Experimental findings of dome parts with diverse ply counts validate the reliability of the framework, providing a multiscale modeling strategy for structural analysis of woven composite parts with complex geometries capable of concurrently revealing damage within mesoscale constituents.