Quintela Estévez, PeregrinaSánchez Rúa, María Teresa2024-02-062024-02-062011Quintela, P., Sánchez, M.T. (2011). Three-point bending tests. Part I: Mathematical study and asymptotic analysis. Mathematical Methods in the Applied Sciences, 34(10), pp. 1211-12350170-4214http://hdl.handle.net/10347/32448This is the peer reviewed version of the following article: Quintela, P., Sánchez, M.T. (2011). Three-point bending tests. Part I: Mathematical study and asymptotic analysis. Mathematical Methods in the Applied Sciences, 34(10), pp. 1211-1235, which has been published in final form at https://doi.org/10.1002/mma.1434. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.The goal of this work is to study the static behaviour of a three-dimensional elastic beam when is subjected to a three-point bending test. In the rst part, under suitable compatibility conditions on the applied forces and on the geometry of the beam, we will prove the existence of a unique solution for the associated contact elastic problem; these conditions of compatibility on the data come from the absence of a Dirichlet condition on the beam boundary. In the second part, we will study the asymptotic behaviour of this problem; in particular, we will deduce the one-dimensional models associated to the displacement components, and we will give the existence and uniqueness of solution for them. Moreover, we will give an expression for the normal axial stress in the beam which is related to the modulus of rupture of brittle materials. In the nal part of the work, we will deal with the regularity of the solution for the bending problem and we will prove some properties of the coincidence setengFor non-commercial and non-promotional research and private study purposes individual users may view, print, download and copy self-archived articles, as well as text and data. The authors' moral rights are not compromised. Where content in the article is identified as belonging to a third party, it is the obligation of the user to ensure that any reuse complies with the copyright policies of the owner of that content. Self-archived content may not be re-published verbatim in whole or in part, whether or not for commercial purposes, in print or online. This restriction does not apply to use of quotations with appropriate citation, or text and data mining provided that the mining output is restricted to short excerpts of text and data and excludes images, unless further consent is obtained from Wiley.Modulus of ruptureContact conditionsAssymptotic analysisjournal article10.1002/mma.14341099-1476open access