{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T06:03:40Z","timestamp":1762063420287,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Ministry of Science, Innovation and Universities","award":["DIN-2018-009940","PID2019-110928RB-C31","RTC-2017-6736-3","SBPLY\/19\/180501\/000220"],"award-info":[{"award-number":["DIN-2018-009940","PID2019-110928RB-C31","RTC-2017-6736-3","SBPLY\/19\/180501\/000220"]}]},{"DOI":"10.13039\/501100011698","name":"Junta de Comunidades de Castilla-La Mancha Spain","doi-asserted-by":"publisher","award":["DIN-2018-009940","PID2019-110928RB-C31","RTC-2017-6736-3","SBPLY\/19\/180501\/000220"],"award-info":[{"award-number":["DIN-2018-009940","PID2019-110928RB-C31","RTC-2017-6736-3","SBPLY\/19\/180501\/000220"]}],"id":[{"id":"10.13039\/501100011698","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This article describes a model based on concepts of Fracture Mechanics to evaluate the flexural strength of fiber-reinforced concrete (FRC) sections. The model covers the need by structural engineers to have tools that allow, in a simple way, the designing of FRC sections and avoiding complex calculations through finite elements. It consists of an analytical method that models FRC post-cracking behavior with a cohesive linear softening law (\u03c3 \u2212 w). We use a compatibility equation based on the planar crack hypothesis, i.e., the assumption that the crack surfaces remain plane throughout the fracture process, which was recently proven true using digital image correlation. Non-cracked concrete bulk follows a stress\u2013strain law (\u03c3 \u2212 \u03b5) combined with the Bernoulli\u2013Navier assumption. We define a brittleness number derived from non-dimensional analyses, which includes the beam size and the softening characteristics. We show that this parameter is key to determining the FRC flexural strength, characterizing fiber-reinforced concrete, and reproducing the size-effect of sections in flexure. Moreover, we propose an expression to calculate the flexural strength of FRC as a function of the cited brittleness number. The model also gives the ratio between the residual strength in service conditions and the flexural strength. Model results show a good agreement with tests in the scientific literature. Finally, we also analyze the brittle\u2013ductile transition in FRC sections.<\/jats:p>","DOI":"10.3390\/ma15175821","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T23:48:58Z","timestamp":1661384938000},"page":"5821","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Planar Crack Approach to Evaluate the Flexural Strength of Fiber-Reinforced Concrete Sections"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8896-6998","authenticated-orcid":false,"given":"Jacinto R.","family":"Carmona","sequence":"first","affiliation":[{"name":"Escuela T\u00e9cnica Superior de Arquitectura de Madrid (ETSAM), Universidad Polit\u00e9cnica de Madrid (UPM), 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ra\u00fal","family":"Cort\u00e9s-Buitrago","sequence":"additional","affiliation":[{"name":"Escuela T\u00e9cnica Superior de Arquitectura de Madrid (ETSAM), Universidad Polit\u00e9cnica de Madrid (UPM), 28040 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5314-3433","authenticated-orcid":false,"given":"Juan","family":"Rey-Rey","sequence":"additional","affiliation":[{"name":"Escuela T\u00e9cnica Superior de Arquitectura de A Coru\u00f1a, Universidad de A Coru\u00f1a (UDC), 15071 A Coru\u00f1a, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0352-0701","authenticated-orcid":false,"given":"Gonzalo","family":"Ruiz","sequence":"additional","affiliation":[{"name":"E.T.S.I. Caminos, Canales y Puertos, Universidad de Castilla-La Mancha (UCLM), 13001 Ciudad Real, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1617\/s11527-009-9529-4","article-title":"Fibre reinforced concrete: New design perspectives","volume":"42","author":"Plizzari","year":"2009","journal-title":"Mater. Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1590\/1807-1929\/agriambi.v24n3p187-193","article-title":"Development of mortar for laying and coating with pineapple fibers","volume":"24","author":"Azevedo","year":"2020","journal-title":"Rev. Bras. Eng. 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