We consider an -parametrized collection of cylinders of cross section , where R², and of fixed length . By Korn’s inequality, there exists a positive constant K such that |symu|²d³x ≥ K |u|²d³x provided that u H¹(;R³) satisfies a condition that rules out infinitesimal rotations. We show that K/² converges to a strictly positive limit, and we characterize this limit in terms of certain parameters that depend on the geometry of and on .