Consistent Euler – Bernoulli beam theories in statics for classical and explicit gradient elasticities


Gradient elasticity has developed into an important area of continuum mechanics with numerous applications in engineering mechanics, structural analysis, experimental and computational mechanics. With the advancement in micro-and nano-electromechanical systems over the past few decades, these activities have been significantly intensified. Most practical applications involving the bending of beams, such as static bending, often take the engineering or elementary approach, popularly denoted as the Euler – Bernoulli beam theory.

In engineering and elementary mechanics, this bending theory is governed by two cardinal assumptions. The first one is about the constitutive law, and it assumes that the material is isotropic elastic. The second one concerns the deformation geometry, postulating that the beam cross-sections would remain plane, rigid and perpendicular to the beam axis without any changes in the shape of the cross-sections. This assumption suggests that the cross-sections of the beam only undergo rigid body motion rather than deformation.

Generally, most beam bending approaches are based on semi-inverse methods that assume a part of deformation from the beginning. Unfortunately, due to these fundamental assumptions, the Euler – Bernoulli beam theory suffers from well-known inconsistencies that, if not well controlled, could compromise its practical applications. The reason is the isotropic elasticity law and the assumed geometry, which implies vanishing shear strain and hence vanishing shear stress. This is always incorrect as it contradicts the equilibrium equations. It is worth noting that this theory is the most simple for modeling the real bending of beams, especially for structural and experimental engineering mechanics. Therefore, providing a framework for formulating this well-established theory by eradicating the inconsistencies is highly desirable.

To this note, Dr. Stergios Alexandros Sideris and Professor Charalampos Tsakmakis from TU Darmstadt aimed at removing the inconsistencies in the classical Euler – Bernoulli beam theory. Their work assumed anisotropic elastic response of the material subject to internal constraints – a rather obvious assumption given that the aforementioned assumptions imply different deformation behaviors in the cross-sections and along the axis of the beam. The present work mainly addressed the deflection distribution and bending rigidity. In addition, consistent formulations of the bending problems for gradient elasticity models to determine and analyze the stress distributions were presented. The validity of the engineering mechanics results was verified. Their research work is currently published in the journal, Composite Structures.

The authors showed that all the engineering mechanics results obtained within the inconsistent isotropic elasticity remained valid. Similarly, known results for deflection curves predicted using the Kelvin gradient elasticity model also remained valid. Nevertheless, the distributions of Cauchy stress components were derived in a consistent approach, and the obtained results were presented based on parameter-dependent limiting processes and the effects of the gradient stiffening. Furthermore, the relationship between the stress components and the section forces and that between the boundary condition of the 3D beam theory and the corresponding loading and boundary conditions of the 1D theory were revealed and discussed.

In a nutshell, Dr. Sideris and Professor Tsakmakis developed consistent Euler – Bernoulli beam theories in classical and explicitly gradient elasticity, assuming that the material response transverse anisotropic elasticity. The associated gradient elasticity results reflected the robustness and feasibility of a Kelvin gradient elasticity model and is useful in validating the response predictions via classical elasticity. Notably, this approach could be extended to modeling of bending of Euler – Bernoulli beam in gradient explicit elasticity. In a statement to Advances in Engineering, the authors said their findings provided an appropriate framework for rendering the well-established Euler – Bernoulli beam theory as well-formulated and suitable for practical applications.


Sideris, S., & Tsakmakis, C. (2022). Consistent Euler – Bernoulli beam theories in statics for classical and explicit gradient elasticitiesComposite Structures, 282, 115026.

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