Large column-free areas are a need of today’s industrial units. This need can be fulfilled by thin laminated shell structures made of composite material. Conoidal shell, being a doubly curved shell structure with a ruled surface, is comparatively easy to construct and is ideal for roofing purposes of such large areas. High specific strength of composite materials and curved form of shells provide greater stability to these shell roofs. The behaviour of these shell structures under loading largely depends on support conditions. Now, separation of layers, known as delamination, is considered to be a major critical defect of composite laminates, which occurs due to repeated cyclic stresses, impact and manufacturing defects. Delamination causes remarkable loss in mechanical toughness and ultimate failure. The most notable thing regarding delamination is that, in most cases, it is not visible from outside. Hence, this type of damage needs a wider attention from researchers. In this paper, the composite conoidal shells with delamination under static loading have been studied. A finite element formulation has been developed with Fortran code to observe the change in deflection in a conoidal shell with varying number of support conditions. An in-depth analysis is done based on number of supports, laminations, stacking sequences and percentage of damage; thereafter, some helpful observations have been highlighted so as to narrate the characteristics of this shell structure as a roofing unit.