Engineering Transactions, 67, 3, pp. 441–457, 2019

Bending of Beams with Symmetrically Varying Mechanical Properties Under Generalized Load – Shear effect

Krzysztof MAGNUCKI
Institute of Rail Vehicles TABOR

Institute of Rail Vehicles TABOR

The paper is devoted to simply supported beams with symmetrically varying mechanical properties in the depth direction. Generalized load of the beams includes the load types from uniformly distributed to point load (three-point bending). This load is analytically described with the use of a certain function including a dimensionless parameter. The value of the parameter is decisive for the load type. The individual nonlinear “polynomial” hypothesis is applied to deformation of a planar cross section. Based on the definitions of the bending moment and the shear transverse force the differential equation of equilibrium is obtained. The equation is analytically solved and the deflections are calculated for an exemplary beam family. The results of the study are specified in tables.
Keywords: FGM beams; bending; analytical modeling
Full Text: PDF
Copyright © The Author(s). This is an open-access article distributed under the terms of the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0).


Wang C.M., Reddy J.N., Lee K.H., Shear deformable beams and plates, Elsevier, Amsterdam, Lausanne, New York, Shannon, Singapore, Tokyo 2000.

Magnucki K., Stasiewicz P., Elastic buckling of a porous beam, Journal of Theoretical and Applied Mechanics, 42: 859–868, 2004.

Magnucka-Blandzi E., Axi-symmetrical deflection and buckling of circular porous-cellular plate, Thin-Walled Structures 46: 333–337, 2008.

Thai H-T., Vo T.P., Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. International Journal of Mechanical Sciences, 62(2012): 57–66.

Dehrouyeh-Semnani A.M., Bahrami A., On size-dependent Timoshenko beam element based on modified couple stress theory, International Journal of Engineering Science, 107: 134–148, 2016.

Magnucki K., Malinowski M., Magnucka-Blandzi E., Lewiński J., Three-point bending of a short beam with symmetrically varying mechanical properties, Composite Structures, 179: 552–557, 2017.

Paczos P., Wichniarek R., Magnucki K., Three-point bending of the sandwich beam with special structures of the core, Composite Structures, 201: 676–682, 2018 .

Sankar B.V., An elasticity solution for functionally graded beams. Composites Science and Technology, 61: 689–696, 2001.

Kadoli R., Akhtar K., Ganesan N., Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modelling, 32: 2509–2523, 2008.

Kapuria S., Bhattacharyyam M., Kumar A.N., Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation, Composite Structures, 82: 390–402, 2008.

Giunta G., Belouettar S., Carrera E., Analysis of FGM beams by means of classical and advanced theories, Mechanics of Advanced Materials and Structures, 17: 622–635, 2010.

Kahrobaiyan M.H., Rahaeifard M., Tajalli S.A., Ahmadian M.T., A strain gradient functionally graded Euler-Bernoulli beam formulation, International Journal of Engineering Science, 52: 65–76, 2012.

Li S-R., Cao D-F., Wan Z-Q., Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams, Applied Mathematical Modelling, 37: 7077–7085, 2013.

Zhang D-G., Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory, Composite Structures, 100(3): 121–126, 2013.

Rahaeifard M., Kahrobaiyan M.H., Ahmadian M.T., Firoozbakhsh K., Strain gradient formulation of functionally graded nonlinear beams, International Journal of Engineering Science, 65: 49–63, 2013.

Chen D., Yang J., Kitipornchai S., Elastic buckling and static bending of shear deformable functionally graded porous beam, Composite Structures, 133: 54–61, 2015

Li L., Hu Y., Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, 107: 77–97, 2016.

Nejad M.Z., Hadi A., Eringen’s non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams. International Journal of Engineering Science, 106: 1–9, 2016.

Sayyad A.S., Ghugal Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures, 171: 486–504, 2017.

Magnucki K., Witkowski D., Lewinski J., Bending and free vibrations of porous beams with symmetrically varying mechanical properties – Shear effect, Mechanics of Advanced Materials and Structures (published online: 16 May 2018).

Taati E., On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment, International Journal of Engineering Science, 128: 63–78, 2018.

Szyniszewski S.T., Smith B.H., Hajjar J.F., Schafer B.W., Arwade S.R., The mechanical properties and modelling of a sintered hollow sphere steel foam, Materials & Design, 54: 1083–1094, 2014.

DOI: 10.24423/EngTrans.987.20190509