{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T19:43:03Z","timestamp":1778096583843,"version":"3.51.4"},"reference-count":44,"publisher":"MDPI AG","issue":"15","license":[{"start":{"date-parts":[[2021,8,3]],"date-time":"2021-08-03T00:00:00Z","timestamp":1627948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>This is a review paper containing the governing equations and analytical solutions of the classical and shear deformation theories of functionally graded straight beams. The classical, first-order, and third-order shear deformation theories account for through-thickness variation of two-constituent functionally graded material, modified couple stress (i.e., strain gradient), and the von K\u00e1rm\u00e1n nonlinearity. Analytical solutions for bending of the linear theories, some of which are not readily available in the literature, are included to show the influence of the material variation, boundary conditions, and loads.<\/jats:p>","DOI":"10.3390\/app11157159","type":"journal-article","created":{"date-parts":[[2021,8,3]],"date-time":"2021-08-03T08:16:39Z","timestamp":1627978599000},"page":"7159","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Theories and Analysis of Functionally Graded Beams"],"prefix":"10.3390","volume":"11","author":[{"given":"J. N.","family":"Reddy","sequence":"first","affiliation":[{"name":"J. Mike Walker \u201966 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA"}]},{"given":"Eugenio","family":"Ruocco","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Campania \u201cLuigi Vanvitelli\", Via Roma 29, 81031 Aversa, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9797-2126","authenticated-orcid":false,"given":"Jose A.","family":"Loya","sequence":"additional","affiliation":[{"name":"Department of Continuum Mechanics and Structural Analysis, University Carlos III of Madrid, 28911 Madrid, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4662-2716","authenticated-orcid":false,"given":"Ana M. A.","family":"Neves","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,3]]},"reference":[{"key":"ref_1","unstructured":"Timoshenko, S.P. (1972). A Course of Easticity Theory. Part 2: Rods and Plates, Naukova Dumka. [2nd ed.]. (In Russian)."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1080\/14786442108636264","article-title":"On the correction for shear of the differential equation for transverse vibrations of prismatic bars","volume":"41","author":"Timoshenko","year":"1921","journal-title":"Philos. Mag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"106496","DOI":"10.1016\/j.ijmecsci.2021.106496","article-title":"Analytical solution for a 5-parameter displacement model","volume":"201","author":"Ruocco","year":"2021","journal-title":"Int. J. Mech. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1115\/1.3167719","article-title":"A simple higher-order theory for laminated composite plates","volume":"51","author":"Reddy","year":"1984","journal-title":"J. Appl. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1016\/0020-7683(84)90056-8","article-title":"A refined nonlinear theory of plates with transverse shear deformation","volume":"20","author":"Reddy","year":"1984","journal-title":"Int. J. Solids Struct."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Reddy, J.N. (2004). Mechanics of Laminated Plates and Shells, Theory and Analysis, CRC Press. [2nd ed.].","DOI":"10.1201\/b12409"},{"key":"ref_7","unstructured":"Reddy, J.N. (2017). Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons. [3rd ed.]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0022-460X(88)90244-1","article-title":"A higher order beam finite element for bending and vibration problems","volume":"126","author":"Heyliger","year":"1988","journal-title":"J. Sound Vib."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1115\/1.2777164","article-title":"Modelling and analysis of functionally graded materials and structures","volume":"60","author":"Birman","year":"2007","journal-title":"Appl. Mech. Rev."},{"key":"ref_10","first-page":"374","article-title":"Homogenization methods for multi-phase elastic composites: Comparison and benchmarks","volume":"30","author":"Klusemann","year":"2010","journal-title":"Tech. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4457","DOI":"10.1016\/S0020-7683(97)00253-9","article-title":"Nonlinear Transient Thermoelastic Analysis of Functionally Graded Ceramic-Metal Plates","volume":"35","author":"Praveen","year":"1998","journal-title":"J. Solids Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2197","DOI":"10.1016\/j.compstruct.2012.02.018","article-title":"Assessment of Voigt and Mori\u2013Tanaka models for vibration analysis of functionally graded plates","volume":"94","author":"Shen","year":"2012","journal-title":"Compos. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1080\/01495739808956165","article-title":"Thermomechanical Analysis of Functionally Graded Cylinders and Plates","volume":"26","author":"Reddy","year":"1998","journal-title":"J.Thermal Stress."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02327219","article-title":"Influence of couple-stresses on stress concentrations","volume":"3","author":"Mindlin","year":"1963","journal-title":"Exp. Mech."},{"key":"ref_15","first-page":"17","article-title":"Couple-stresses in the theory of elasticity: I and II","volume":"B67","author":"Koiter","year":"1964","journal-title":"K. Ned. Akad. Van Wet. (R. Neth. Acad. Arts Sci.)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF00253050","article-title":"Theories of elasticity with couple-stress","volume":"17","author":"Toupin","year":"1964","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/0020-7683(65)90006-5","article-title":"Second gradient of strain and surface-tension in linear elasticity","volume":"1","author":"Mindlin","year":"1965","journal-title":"Int. J. Solids Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2731","DOI":"10.1016\/S0020-7683(02)00152-X","article-title":"Couple stress based strain gradient theory for elasticity","volume":"39","author":"Yang","year":"2002","journal-title":"Int. J. Solids Struct."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1016\/S0022-5096(03)00053-X","article-title":"Experiments and theory in strain gradient elasticity","volume":"51","author":"Lam","year":"2003","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"873","DOI":"10.1016\/j.jmps.2012.10.008","article-title":"A model for a constrained, finitely deforming, elastic solid with rotation gradient dependent strain energy, and its specialization to von Karman plates and beams","volume":"61","author":"Srinivasa","year":"2013","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.ijnonlinmec.2014.06.003","article-title":"Nonlinear theories of beams and plates accounting for moderate rotations and material length scales","volume":"66","author":"Reddy","year":"2014","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2355","DOI":"10.1088\/0960-1317\/16\/11\/015","article-title":"Bernoulli-Euler beam model based on a modified couple stress theory","volume":"16","author":"Park","year":"2006","journal-title":"J. Micromech. Microeng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3379","DOI":"10.1016\/j.jmps.2008.09.007","article-title":"A microstructure-dependent Timoshenko beam model based on a modified couple stress theory","volume":"56","author":"Ma","year":"2008","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1615\/IntJMultCompEng.v8.i2.30","article-title":"A nonclassical Reddy-Levinson beam model based on a modified couple stress theory","volume":"8","author":"Ma","year":"2010","journal-title":"Int. J. Multiscale Comput."},{"key":"ref_25","first-page":"24","article-title":"A model for free vibration analysis of Timoshenko beams with couple stress","volume":"23","author":"Reddy","year":"2010","journal-title":"Acta Mech. Solida Sin."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1155\/2012\/307806","article-title":"Vibration of Timoshenko beams using non-classical elasticity theories","volume":"19","author":"Reddy","year":"2012","journal-title":"Shock Vib."},{"key":"ref_27","first-page":"1250056","article-title":"Free vibration and buckling analysis of beams with a modified couple-stress theory","volume":"4","author":"Reddy","year":"2012","journal-title":"Int. J. Appl. Mech."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2382","DOI":"10.1016\/j.jmps.2011.06.008","article-title":"Microstructure-dependent couple stress theories of functionally graded beams","volume":"59","author":"Reddy","year":"2011","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.compstruct.2012.10.003","article-title":"Nonlinear analysis of functionally graded microstructure-dependent beams","volume":"98","author":"Arbind","year":"2013","journal-title":"Compos. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1590\/S1679-78252014000300006","article-title":"Modified couple stress-based third-order theory for nonlinear analysis of functionally graded beams","volume":"11","author":"Arbind","year":"2014","journal-title":"Lat. Am. J. Solids Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1016\/S0304-3991(03)00077-9","article-title":"Mechanical characterization of micro\/nanoscale structures for MEMS\/NEMS applications using nanoindentation techniques","volume":"97","author":"Li","year":"2003","journal-title":"Ultramicroscopy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1021\/ac035048k","article-title":"Glucose biosensor based on the microcantilever","volume":"76","author":"Pei","year":"2004","journal-title":"Anal. Chem."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.ijengsci.2018.08.005","article-title":"The material length scale parameter used in couple stress theories is not a material constant","volume":"133","author":"Khorshidi","year":"2018","journal-title":"Int. J. Eng. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Reddy, J.N. (2022). Theories and Analyses of Beams and Circular Plates, CRC Press. in press.","DOI":"10.1201\/9781003240846"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Reddy, J.N. (2015). An Introduction to Nonlinear Finite Element Analysis, Oxford University Press. [2nd ed.].","DOI":"10.1093\/acprof:oso\/9780199641758.001.0001"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Mindlin, R.D., and Deresiewicz, H. (1953). Timoshenko\u2019s Shear Coefficient for Flexural Vibrations of Beams, Columbia University. Technical Report No. 10, ONR Project NR064-388.","DOI":"10.21236\/AD0013061"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1115\/1.3625046","article-title":"The shear coefficient in Timoshenko\u2019s beam theory","volume":"33","author":"Cowper","year":"1966","journal-title":"J. Appl. Mech."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0022-460X(81)90493-4","article-title":"A new rectangular beam theory","volume":"74","author":"Levinson","year":"1981","journal-title":"J. Sound Vib."},{"key":"ref_39","first-page":"137","article-title":"A consistent higher order beam theory","volume":"11","author":"Bickford","year":"1982","journal-title":"Dev. Theor. Appl. Mech."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"100166","DOI":"10.1016\/j.jcomc.2021.100166","article-title":"Theories and Analyses of Functionally Graded Circular Plates","volume":"5","author":"Reddy","year":"2021","journal-title":"Compos. Part C Open Access"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"103453","DOI":"10.1016\/j.ijnonlinmec.2020.103453","article-title":"Buckling analysis of elastic\u2013plastic nanoplates resting on a Winkler\u2013Pasternak foundation based on nonlocal third-order plate theory","volume":"121","author":"Ruocco","year":"2020","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1007\/s11012-020-01177-3","article-title":"On the application of fractional calculus for the formulation of viscoelastic Reddy beam","volume":"55","author":"Paola","year":"2020","journal-title":"Meccanica"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"111056","DOI":"10.1016\/j.engstruct.2020.111056","article-title":"An enhanced Hencky bar-chain model for bending, buckling and vibration analyses of Reddy beams","volume":"221","author":"Ruocco","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.ijengsci.2015.10.013","article-title":"Bending of Euler-Bernoulli beams using Eringen\u2019s integral formulation: A paradox resolved","volume":"99","author":"Zaera","year":"2016","journal-title":"Int. J. Eng. Sci."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/11\/15\/7159\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:39:35Z","timestamp":1760164775000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/11\/15\/7159"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,3]]},"references-count":44,"journal-issue":{"issue":"15","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["app11157159"],"URL":"https:\/\/doi.org\/10.3390\/app11157159","relation":{},"ISSN":["2076-3417"],"issn-type":[{"value":"2076-3417","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,3]]}}}