Equilibrium and Static Deflection for Bending of a Nonlocal Nanobeam

This study presents art analytical study with numerical examples for bending of a nanobeam based on the nonlocal elastic stress theory. By solving the nonlocal constitutive relation directly, this paper reports new higher-order equilibrium conditions for bending of a nonlocal nanobeam with the corresponding higher-order boundary conditions. The paper also concludes that the state of equilibrium of a classical beam cannot be directly applied for a nonlocal nanobeam even if the relevant terms are replaced by the nonlocal counterparts, The new state of equilibrium is governed through the replacement of bending moment by a newly defined effective nonlocal bending moment. Subsequently, new analytical solutions for nanobeams with various boundary conditions are analyzed. The examples conclude that nanoscale size effect strengthens nanobeam stiffness and hence reduced static deflection which are against the previous results. The intriguing conclusions reported earlier that a nanobeam with certain boundary conditions and loading conditions is indifferent to nanoscale size effect is also addressed and new solutions presented.