Modeling and analysis of out-of-plane behavior of curved nanobeams based on nonlocal elasticity

The purpose of this study is to analytically investigate the out-of-plane static behavior of curved nanobeams with non-uniform distributed loads. Nonlocal constitutive equations are implemented into the governing equilibrium equations in cylindrical coordinates. Use of these equations enables revealing the scale effect on static response of curved nanobeams. Shear deformation is considered in the formulation of out-of-plane static behavior of nanobeams. The exact analytical solution to the theoretical model is derived by using initial value method. The fundamental matrix required for the initial value method is obtained analytically. Then, the displacement, slopes and stress resultants are found analytically using the fundamental matrix. Illustrative examples for such nano structures with non-uniform distributed loads are given in subsequent sections. Results showed that the scale effect plays crucial role at nano scale which agree with the results given in the literature. The effects of small scale parameter, slenderness ratio and opening angle on the displacements ratio, rotation angles ratio and stress resultants ratio are also investigated in details. Considering scale effect and shear deformation in the exact solutions of the governing equilibrium equations presented here, the results can be considered as a reference for nonlocal theories of curved nanobeams.