We consider a linearly elastic loaded shell with a uniformly elliptic middle surface, clamped along a portion of its lateral boundary. Under some regularity assumptions on the middle surface, we prove that the space of the inextensional displacements is reduced to zero, by using unique continuation results. Consequently, when the thickness of the shell goes to zero, the limit of the average, with respect to the thickness, of the three-dimensional displacement vector, solves the `generalized membrane' shell model, under some asymptotic assumptions on the loads.