{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,14]],"date-time":"2026-06-14T22:30:05Z","timestamp":1781476205889,"version":"3.54.1"},"reference-count":24,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,5,16]],"date-time":"2024-05-16T00:00:00Z","timestamp":1715817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSTC, Taiwan","award":["MOST 111-2221-E-164-001"],"award-info":[{"award-number":["MOST 111-2221-E-164-001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>A time-dependent third-order shear deformation theory (TSDT) approach on the displacements of thick functionally graded material (FGM) conical shells under dynamic thermal vibration is studied. Dynamic equations of motion with TSDT for thick FGM conical shells are applied directly with the partial derivative of variable R*\u03b8 in the curve coordinates (x, \u03b8, z) instead of y in the Cartesian coordinates (x, y, z) for thick FGM plates, where R* is the middle-surface radius at any point on conical shells. The generalized differential quadrature (GDQ) numerical method is used to solve the dynamic differential equations in equilibrium matrix forms under thermal loads. It is the novelty of the current study to identify the parametric effects of shear correction coefficient, environment temperature, TSDT model, and FGM power law index on the displacements and stresses in the thick conical shells only subjected to sinusoidal heating loads. The physical parts with values on the length-to-thickness ratio equals 5, and 10 FGMs can be used in an area of an airplane engine that usually operates near more than 1000 K of temperatures when the thermal stress is considered and affected. The important findings of the presented study are listed as follows. The values of normal stress are in decreasing tendencies with time in cases when the coefficient c1 equals 0.925925\/mm2 in TSDT and length-to-thickness ratio equals 5. The shear stress values in x plane z direction on the minor middle-surface radius (r) equals the major middle-surface radius (R) over 8 and length-to-thickness ratio equals to 5 can withstand T = 1000 K of pressure.<\/jats:p>","DOI":"10.3390\/ma17102403","type":"journal-article","created":{"date-parts":[[2024,5,16]],"date-time":"2024-05-16T11:26:17Z","timestamp":1715858777000},"page":"2403","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Thermal Vibration of Thick FGM Conical Shells by Using Third-Order Shear Deformation Theory"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6490-5528","authenticated-orcid":false,"given":"Chih-Chiang","family":"Hong","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Hsiuping University of Science and Technology, Taichung 412-406, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"105559","DOI":"10.1016\/j.ast.2019.105559","article-title":"A semi-analytical approach on the effect of external lateral pressure on free vibration of joined sandwich aerospace composite conical-conical shells","volume":"99","author":"Soureshjani","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"110906","DOI":"10.1016\/j.compstruct.2019.110906","article-title":"Nonlinear vibration response of higher-order shear deformable FG-CNTRC conical shells","volume":"222","author":"Ansari","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.matpr.2019.03.035","article-title":"Low velocity normal impact performance of functionally graded conical shell with simple power law","volume":"11","author":"Das","year":"2019","journal-title":"Mater. Today Proc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1016\/j.compositesb.2019.01.007","article-title":"Free vibration analysis of functionally graded rotating conical shells in thermal environment","volume":"163","author":"Shakouri","year":"2019","journal-title":"Compos. Part B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.compstruct.2018.12.047","article-title":"Review of research on the vibration and buckling of the FGM conical shells","volume":"211","author":"Sofiyev","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1016\/j.conbuildmat.2019.04.015","article-title":"The performance of steel conical shells reinforced with CFRP laminates subjected to uniform external pressure","volume":"214","author":"Taraghi","year":"2019","journal-title":"Constr. Build. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.tws.2019.04.024","article-title":"Nonlinear vibrations of FGM truncated conical shell under aerodynamics and in-plane force along meridian near internal resonances","volume":"142","author":"Yang","year":"2019","journal-title":"Thin-Walled Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.compstruct.2018.07.065","article-title":"Free vibration characteristic of laminated conical shells based on higher order shear deformation theory","volume":"204","author":"Javed","year":"2018","journal-title":"Compos. Struct."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"14302","DOI":"10.1016\/j.matpr.2018.03.012","article-title":"Free vibration of functionally graded conical shell","volume":"5","author":"Mouli","year":"2018","journal-title":"Mater. Today Proc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"116208","DOI":"10.1016\/j.mseb.2022.116208","article-title":"GDQ computation for thermal vibration of thick FGM plates by using third-order shear deformation theory","volume":"294","author":"Hong","year":"2023","journal-title":"Mater. Sci. Eng. B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2","DOI":"10.37256\/mp.1120221967","article-title":"Thermal vibration of thick FGM circular cylindrical shells by using TSDT","volume":"1","author":"Hong","year":"2022","journal-title":"Mater. Plus"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3657","DOI":"10.1016\/j.ijsolstr.2005.04.011","article-title":"Mechanical behavior of functionally graded material plates under transverse load, part I: Analysis","volume":"43","author":"Chi","year":"2006","journal-title":"Int. J. Solids Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1016\/S0168-874X(03)00073-8","article-title":"Transient analysis of laminated composite plates with embedded smart-material layers","volume":"40","author":"Lee","year":"2004","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1016\/j.ijnonlinmec.2004.11.003","article-title":"Non-linear response of laminated composite plates under thermomechanical loading","volume":"40","author":"Lee","year":"2005","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_15","unstructured":"Whitney, J.M. (1987). Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Company, Inc."},{"key":"ref_16","unstructured":"Reddy, J.N. (2002). Energy Principles and Variational Methods in Applied Mechanics, Wiley."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/BF01046487","article-title":"Nonlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature","volume":"5","author":"Bert","year":"1989","journal-title":"Comput. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/S0020-7683(96)00057-1","article-title":"Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analyses of beams and plates","volume":"34","author":"Shu","year":"1997","journal-title":"Int. J. Solids Struct."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"112701","DOI":"10.1016\/j.oceaneng.2022.112701","article-title":"Advanced dynamic thermal vibration of thick FGM plates-cylindrical shells","volume":"266","author":"Hong","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.ast.2018.12.011","article-title":"3D shell model for the thermo-mechanical analysis of FGM structures via imposed and calculated temperature profiles","volume":"85","author":"Brischetto","year":"2019","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1080\/014957399280841","article-title":"Thermal stresses in functionally graded materials","volume":"22","author":"Noda","year":"1999","journal-title":"J. Therm. Stress."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"112486","DOI":"10.1016\/j.compstruct.2020.112486","article-title":"A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition","volume":"249","author":"Malikan","year":"2020","journal-title":"Compos. Struct."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1016\/j.compstruct.2018.03.058","article-title":"New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment","volume":"192","author":"Dastjerdi","year":"2018","journal-title":"Compos. Struct."},{"key":"ref_24","unstructured":"Conte, S.D., and de Boor, C. (1980). Elementary Numerical Analysis an Algorithmic Approach, McGraw-Hill Book Company."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/17\/10\/2403\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:43:39Z","timestamp":1760107419000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/17\/10\/2403"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,16]]},"references-count":24,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["ma17102403"],"URL":"https:\/\/doi.org\/10.3390\/ma17102403","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,16]]}}}