Size-dependent buckling loads of strain gradient Timoshenko beams

This paper presents both analytical and semi-analytical solutions for the size-dependent buckling loads of strain gradient Timoshenko beams (SGTB). Based on Hamilton’s principle, coupled governing equations for the deflection and rotation are established with consideration of micro-inertia and size effect. Then, a unified single-variable sixth-order governing equation for the SGTB is derived. The buckling problems are exactly and semi-analytically solved. Explicit expressions for the buckling loads are obtained for typical end supports. For arbitrary boundary conditions, two semi-analytical methods—the Rayleigh Quotient Method (RQM) and the Weight Function Method (WFM)—are developed to obtain closed-form approximate expressions for the buckling loads. Numerical comparisons demonstrate that the relative errors of the proposed approximate solutions are all below 0.5%, confirming their high accuracy. The study verifies that both Poisson’s ratio and size effects have significant influences on the critical loads of SGTBs. Finally, by fitting the experimental data, the intrinsic length of ZnO nanowires is estimated, providing valuable reference for future theoretical and numerical investigations.