Size effects in Kirchhoff flexible rods

The Kirchhoff equations for flexible rods are extended to be able to describe size effects. Thus, the extended equations are applicable not only for macroscopic rods but also for ultrathin rods whose thickness is close to the material length. It is found that extensional size effects exist in flexible rods, namely, the external forces exerted on both ends of rods will increase as the rods are getting thinner if an identical deformation is remained.