Under the same assumptions as those needed for the convergence of the covariant components of the displacement fiel we establish the convergence of the contravariant components of the scaled linearized strain tensor (T-ij(epsilon)). More precisely, the scaled strains T-alpha beta (epsilon) and T-i3(epsilon)/epsilon for ''membrane'' shells, and T-alpha beta(epsilon)/epsilon and T-i3(epsilon)/epsilon(2) for ''flexural'' shells, have a limit, which is explicitly computed, when the thickness 2 epsilon approaches zero.