A reinforcing assembly for a tubular cross member of a rail and tram car bogie has a liner elongated along an axis, deformable radially, and insertable axially inside one end of the tubular cross member to line an inner cylindrical surface of the end; a tubular body insertable coaxially inside the liner; and a number of radial forcing devices, which are activated to force the tubular body, and hence the liner, radially outwards to lock the liner frictionally against the inner cylindrical surface of the cross member.