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This paper proposes an analytical solution and isogeometric analysis numerical approach for buckling analysis of size-dependent beams based on a reformulated strain gradient elasticity theory (RSGET). The superiority of this method is that it has only one material parameter for couple stress and another material parameter for strain gradient effects. Using the RSGET and the principle of minimum potential energy, both non-classical Euler–Bernoulli and Timoshenko beam buckling models are developed. Moreover, the obtained governing equations are solved by an exact solution and isogeometric analysis approach, which conforms to the requirements of higher continuity in gradient elasticity theory. Numerical results are compared with exact solutions to reveal the accuracy of the current isogeometric analysis approach. The influences of length–scale parameter, length-to-thickness ratio, beam thickness and boundary conditions are investigated. Moreover, the difference between the buckling responses obtained by the Timoshenko and Euler–Bernoulli theories shows that the Euler–Bernoulli theory is suitable for slender beams.

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This paper presents a novel non-classical Timoshenko–Ehrenfest beam model based on a reformulated strain gradient elasticity theory. The strain gradient effect, couple stress effect, and velocity gradient effect for vibration are included in the new model by only one material length scale parameter for each. The variational formulation and Hamilton’s principle are applied to derive the governing equations and boundary conditions. Both an analytical solution and an isogeometric analysis approach are proposed for static bending and free vibration of the microbeam. A non-uniform rational B-splines (NURBS) isogeometric analysis with high-order continuity can effectively fulfill the higher derivatives of the displacement variables in the reformulated gradient beam model. Convergence studies and comparisons to the corresponding analytical solutions verify the model’s performance and accuracy. Finally, different boundary conditions, material length scale parameters, and beam thicknesses are investigated in order to certify the applicability of the proposed approach.