Refined Analysis of Shear Stress Distribution in Tapered Rods Accounting for Gradient Effects

In this article, we propose to apply the strain gradient elasticity theory for the refined stress analysis in the tapered rods with variable cross section. Corresponding statement of the higher-order boundary value problem with extended set of boundary conditions is derived based on the variational approach. Considering an example for the cylindrical/conical rod loaded by self-equilibrated body and end forces, we provide the comparison between the stress distributions that can be obtained within the classical 3D elasticity, classical 1D rod theory and the established 1D rod theory with the strain gradient effects. It is shown that the last one allows to obtain the smoothed solution for the shear stresses that can be fitted to 3D elasticity solution under appropriate choice of additional length scale parameter of gradient theory. In contrast, solution of classical rod theory cannot be fitted exactly to 3D elasticity solution and contains unavoidable discontinuities of shear stresses.