{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T12:57:34Z","timestamp":1777640254513,"version":"3.51.4"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T00:00:00Z","timestamp":1726185600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Italian Ministry of University and Research (MUR)","award":["PRIN2020 #20209F3A37"],"award-info":[{"award-number":["PRIN2020 #20209F3A37"]}]},{"name":"Italian Ministry of University and Research (MUR)","award":["PRIN2022 P20229YAYL_001"],"award-info":[{"award-number":["PRIN2022 P20229YAYL_001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The dynamic stiffness method is developed to analyze the natural vibration characteristics of functionally graded beams, where material properties change continuously across the beam thickness following a symmetric law distribution. The governing equations of motion and associated natural boundary conditions for free vibration analysis are derived using Hamilton\u2019s principle and closed-form exact solutions are obtained for both Euler\u2013Bernoulli and Timoshenko beam models. The dynamic stiffness matrix, which governs the relationship between force and displacements at the beam ends, is determined. Using the Wittrick\u2013Williams algorithm, the dynamic stiffness matrix is employed to compute natural frequencies and mode shapes. The proposed procedure is validated by comparing the obtained frequencies with those given by approximated well-known formulas. Finally, a parametric investigation is conducted by varying the geometry of the structure and the characteristic mechanical parameters of the functionally graded material.<\/jats:p>","DOI":"10.3390\/sym16091206","type":"journal-article","created":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T05:31:44Z","timestamp":1726205504000},"page":"1206","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Closed-Form Exact Solution for Free Vibration Analysis of Symmetric Functionally Graded Beams"],"prefix":"10.3390","volume":"16","author":[{"given":"Lorenzo","family":"Ledda","sequence":"first","affiliation":[{"name":"Department of Civil Engineering and Architecture, University of Catania, 95123 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2929-5044","authenticated-orcid":false,"given":"Annalisa","family":"Greco","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering and Architecture, University of Catania, 95123 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0673-5409","authenticated-orcid":false,"given":"Ilaria","family":"Fiore","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering and Architecture, University of Catania, 95123 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8063-8359","authenticated-orcid":false,"given":"Ivo","family":"Cali\u00f2","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering and Architecture, University of Catania, 95123 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,13]]},"reference":[{"key":"ref_1","unstructured":"Mahamood, R.M., Akinlabi, E.T., Shukla, M., and Pityana, S. 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