{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T15:30:41Z","timestamp":1771947041861,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T00:00:00Z","timestamp":1558051200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Foundation for Science and Technology Development at Nguyen Tat Thanh University","award":["2019-01\/H\u0110KHCN-NTT"],"award-info":[{"award-number":["2019-01\/H\u0110KHCN-NTT"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique for triangular shell element (MITC3), called ES-MITC3, for free vibration analysis of functionally graded shells is investigated in this work. In the formulation of the ES-MITC3, the stiffness matrices are obtained by using the strain-smoothing technique over the smoothing domains that are formed by two adjacent MITC3 triangular shell elements sharing an edge. The strain-smoothing technique can improve significantly the accuracy and convergence of the original MITC3. The material properties of functionally graded shells are assumed to vary through the thickness direction by a power\u2013rule distribution of volume fractions of the constituents. The numerical examples demonstrated that the present ES-MITC3method is free of shear locking and achieves the high accuracy compared to the reference solutions in the literature.<\/jats:p>","DOI":"10.3390\/sym11050684","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T11:06:46Z","timestamp":1558091206000},"page":"684","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method"],"prefix":"10.3390","volume":"11","author":[{"given":"Tien Dat","family":"Pham","sequence":"first","affiliation":[{"name":"Faculty of Mechanical Engineering, Le Quy Don Technical University, Ha Noi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6316-3120","authenticated-orcid":false,"given":"Quoc Hoa","family":"Pham","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Le Quy Don Technical University, Ha Noi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Van Duc","family":"Phan","sequence":"additional","affiliation":[{"name":"Center of Excellence for Automation and Precision Mechanical Engineering, Nguyen Tat Thanh University, Ho Chi Minh 700000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9949-0702","authenticated-orcid":false,"given":"Hoang Nam","family":"Nguyen","sequence":"additional","affiliation":[{"name":"Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical &amp; Electronics Engineering, Ton Duc Thang University, Ho Chi Minh 700000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Van Thom","family":"Do","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Le Quy Don Technical University, Ha Noi 100000, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7409","DOI":"10.1016\/S0020-7683(01)00048-8","article-title":"Nonlinear analysis of functionally graded plates and shallow shells","volume":"38","author":"Woo","year":"2001","journal-title":"Int. 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