To construct a tire component while avoiding problems resulting from both the action of fastening force and deterioration of adhesion of basic components in constructing a carcass structure.

An apparatus for constructing a tire for vehicular wheels includes a construction drum (10) having at least one first outer peripheral surface (14) and at least a pair of auxiliary support member (16) engaged in the construction drum (10) in a removable way, and the auxiliary support members (16) are individually adjacent to each end part of the construction drum (10) in the axial direction and has a second outer peripheral surface (16a) continuously extending to the first outer peripheral surface (14) so as to support at least one component (3, 19 and 20) of a tire (2). Each of the second outer peripheral surfaces (16a) has at least one non-adhesive circumferential part (25', 25'' and 25''') and at least one adhesive circumferential part (26' and 26'').