Summary: In this paper, the size effect on beam structures with centrosymmetric anisotropy is studied within strain gradient elasticity theory. Applying dimension reduction to the three dimensional anisotropic gradient elasticity, the third-order shear deformable (TSD) beam is analysed. A variational approach is used to determine the equilibrium equations of TSD beam together with consistent (classical and non-classical) boundary conditions. The TSD beam theory which is suitable for deep beam structures can be replaced by (less complicated) Euler-Bernoulli beam model for thin beam structures. The anisotropic Euler-Bernoulli beam model is also formulated within the framework of strain gradient theory. This anisotropic beam theory can be used to study size effects for any types of centrosymmetric anisotropy. To address the more practical cases of composite structures, the formulation is simplified for orthotropic and transversely isotropic materials. Finally, the analytical solutions are provided for bending of simply supported (TSD and Euler-Bernoulli) beams as well as clamped Euler-Bernoulli beams. The effect of the crystal orientation with respect to the beam geometry is investigated in these examples.
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