Static bending of micromorphic Timoshenko beams

This study develops a rigorous analytical framework for investigating the static bending behavior of micromorphic and nonlocal strain gradient Timoshenko beams, with particular emphasis on capturing size-dependent effects in micro- and nano-scale structural elements. The model is derived using a variational principle and it consists of a set of governing equations and boundary conditions that incorporate two distinct internal length-scales, one associated with nonlocal stress gradients and the other with strain gradient effects. The obtained system of two coupled differential equations governs the deflection and the rotation of the beam. Uncoupling both equations leads to sixth- and fifth-order differential equations for the deflection and the rotation, respectively. Exact solutions are obtained for standard boundary configurations, including simply-supported, clamped–clamped, and cantilever cases, under both point and distributed loads. The analytical model is shown to be theoretically equivalent to a class of two-length-scale nonlocal strain gradient theories, thereby offering a consistent and unified description of scale-dependent mechanics in microstructured beams. A distinct series-based solution is also constructed to verify the closed-form micromorphic results. Verification against established reference solutions demonstrates the accuracy and generality of the proposed model. A series of parametric studies is conducted to quantify the role of internal length-scales, revealing that the model successfully predicts both stiffening and softening trends, depending on the microstructural configuration. The derived exact solutions provide a reliable benchmark for assessing numerical schemes and serve as a foundation for further studies involving advanced materials with microstructural complexity.