Exact solutions for the static bending of nonlocal higher-order shear beams under various boundary conditions

This paper presents some exact solutions, for the static bending behavior of higher-order shear elastic nanobeams using the nonlocal differential constitutive relation of Eringen, and under various boundary conditions. The nonlocal higher-order shear beam referred as a Bickford-Reddy’s beam model, assumes a cubic interpolation field for the displacement along the cross section, associated with a parabolic shear strain measure. The governing equations and boundary conditions are derived using the principle of virtual displacements. The nonlocality is applied to the generalized higher-order shear constitutive law, which is formulated in term of shear force, bending moment and higher-order moment. Subsequently, it is shown that the governing equations can be reduced to a single linear sixth-order differential equation for the transverse deflection, which can be solved using exact methods, for general boundary conditions. The present approach generalizes the results derived by Reddy [Citation1] for the static bending of a simply supported nonlocal higher-order shear beams under uniform distributed loading. Furthermore, we conduct an in-depth investigation to quantify the effects of the nonlocal parameter and the length-to-thickness ratio of the nanobeam on its bending response, for various boundary conditions, including the simply-supported, clamped and free boundary conditions at each end. Comparative studies demonstrate that our results not only cover the case of nonlocal higher-order shear beam theories, but can degenerate asymptotically into nonlocal Euler–Bernoulli beam theory, and nonlocal Timoshenko beam theory, which are better documented for the aforementioned boundary conditions.