Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access May 29, 2019

Nonlinear response of axially functionally graded Timoshenko beams on elastic foundation under harmonic excitation

  • Hareram Lohar , Anirban Mitra and Sarmila Sahoo EMAIL logo

Abstract

Forced vibration of non-uniform axially functionally graded (AFG) Timoshenko beam on elastic foundation is performed under harmonic excitation. A linear elastic foundation is considered with three different classical boundary conditions. AFG materials are an advanced class of materials that have potential for application in various engineering fields. In the present work, variation of material properties along the longitudinal axis of the beam are considered according to power-law forms. Five values of material gradation parameter provides different functional variation and their effect on the frequency response of the system is studied. The present approximate method is displacement based and Von-Karman type of geometric nonlinearity is considered with rotational component to incorporate transverse shear. Hamilton’s principle is used to derive nonlinear set of governing equation and Broyden method is implemented to solve the nonlinear equations numerically. The results are successfully validated with previously published article. Frequency vs. amplitude curve corresponding to different combinations of system parameters are presented and are capable of serving as benchmark results. A separate free vibration analysis is undertaken to include backbone curves with the frequency response curves in the non-dimensional plane.

References

[1] Goupee AJ and Vel SS 2006 Optimization of natural frequencies of bidirectional functionally graded beams Struct Multidisc Optim 32 473–48410.1007/s00158-006-0022-1Search in Google Scholar

[2] Simsek M 2016 Buckling of Timoshenko beams composed of two-dimensional functionally graded material 2D-FGM having different boundary conditions Compos Struct 149 304–31410.1016/j.compstruct.2016.04.034Search in Google Scholar

[3] Huynh TA, Lieu XQ and Lee J 2017 NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem Compos Struct 160 1178–119010.1016/j.compstruct.2016.10.076Search in Google Scholar

[4] Karamanlı A 2018 Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory Compos Struct 189 127–13610.1016/j.compstruct.2018.01.060Search in Google Scholar

[5] Khaniki HB and Rajasekaran S 2018 Mechanical analysis of nonuniform bi-directional functionally graded intelligent microbeams using modified couple stress theory Mater Res Express. 5055703 https://doiorg/101088/2053-1591/aabe62Search in Google Scholar

[6] Kadoli R, Akhtar K and Ganesan N 2008 Static analysis of functionally graded beams using higher order shear deformation theory Appl Math Model. 32 2509–252510.1016/j.apm.2007.09.015Search in Google Scholar

[7] Hemmatnezhad M, Ansari R and Rahimi GH 2013 Largeamplitude free vibrations of functionally graded beams by means of a finite element formulation Appl Math Model. 37 8495–850410.1016/j.apm.2013.03.055Search in Google Scholar

[8] Pradhan KK and Chakraverty S 2014 Effects of different shear deformation theories on free vibration of functionally graded beams Int J Mech Sci 82 149–16010.1016/j.ijmecsci.2014.03.014Search in Google Scholar

[9] Chen D, Yang J and Kitipornchai S 2015 Elastic buckling and static bending of shear deformable functionally graded porous beam Compos Struct 133 54–6110.1016/j.compstruct.2015.07.052Search in Google Scholar

[10] Ebrahimi F and Zia M 2015 Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities Acta Astronaut. 116 117–12510.1016/j.actaastro.2015.06.014Search in Google Scholar

[11] Paul A and Das D 2016 Free vibration analysis of pre-stressed FGM Timoshenko beams under large transverse deflection by a variational method Eng. Sci. Technol. Int J. 19 1003–101710.1016/j.jestch.2015.12.012Search in Google Scholar

[12] Aydogdu M 2008 Semi-inverse method for vibration and buckling of axially functionally graded beams J. Reinf. Plast. Compos 27(7) 683-69110.1177/0731684407081369Search in Google Scholar

[13] Shahba A, Attarnejad R and Hajilar S 2011 Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Shock Vib. 18 683–69610.1155/2011/591716Search in Google Scholar

[14] Alshorbagy AE, Eltaher MA and Mahmoud FF 2011 Free vibration characteristics of a functionally graded beam by finite element method Appl Math Model.35 412–42510.1016/j.apm.2010.07.006Search in Google Scholar

[15] Shahba A and Rajasekaran S 2012 Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials Appl Math Model.36 3094–311110.1016/j.apm.2011.09.073Search in Google Scholar

[16] Simsek M, Kocatürk T and Akbas SD 2012 Dynamic behavior of an axially functionally graded beam under action of a moving harmonic load Compos Struct 94 2358–236410.1016/j.compstruct.2012.03.020Search in Google Scholar

[17] Akgöz B and Civalek Ö 2013 Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory Compos Struct 98 314–32210.1016/j.compstruct.2012.11.020Search in Google Scholar

[18] Kumar S, Mitra A and Roy H 2015 Geometrically nonlinear free vibration analysis of axially functionally graded taper beams Eng. Sci. Technol. Int J. 18 579-59310.1016/j.jestch.2015.04.003Search in Google Scholar

[19] Ghayesh MH and Farokhi H 2018 Bending and vibration analyses of coupled axially functionally graded tapered beams Nonlinear Dyn 91 17–2810.1007/s11071-017-3783-8Search in Google Scholar

[20] Rajasekaran S and Khaniki HB 2018 Finite element static and dynamic analysis of axially functionally graded nonuniform smallscale beams based on nonlocal strain gradient theory Mech Adv Mater Struc 0 (0) 1–15Search in Google Scholar

[21] Sınıra S, Çevik M and Sınır BG 2018 Nonlinear free and forced vibration analyses of axially functionally graded Euler-Bernoulli beams with non-uniform cross-section Composites Part B 148 123–13110.1016/j.compositesb.2018.04.061Search in Google Scholar

[22] Shames IH and DymCL 2009 Energy and Finite Element Methods in Structural Mechanics New Age International Publishers DelhiSearch in Google Scholar

[23] Shahba A, Attarnejad R, Marvi MT and Hajilar S 2011 Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions Composites: Part B 42 801–80810.1016/j.compositesb.2011.01.017Search in Google Scholar

[24] Huang Y, Yang LE and Luo QZ 2013 Free vibration of axially functionally graded Timoshenko beamswith non-uniform crosssection Composites: Part B 45 1493–149810.1016/j.compositesb.2012.09.015Search in Google Scholar

[25] Huang Y, Zhang M and Rong H 2016 Buckling analysis of axially functionally graded and non-uniform beams based on Timoshenko theory Acta Mech Solida Sin. 29 (2) 200-20710.1016/S0894-9166(16)30108-2Search in Google Scholar

[26] Rajasekaran S 2013 Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods Appl Math Model.37 4440–446310.1016/j.apm.2012.09.024Search in Google Scholar

[27] Sarkar K and Ganguli R 2014 Closed-form solutions for axially functionally graded Timoshenko beams having uniform crosssection and fixed–fixed boundary condition Composites: Part B 58 361–37010.1016/j.compositesb.2013.10.077Search in Google Scholar

[28] Calim FF 2016 Transient analysis of axially functionally graded Timoshenko beamswith variable cross-section Composites Part B 98 472-48310.1016/j.compositesb.2016.05.040Search in Google Scholar

[29] Shafiei N, Kazemi M and Ghadiri M 2016 Comparison of modelling of the rotating tapered axially functionally graded Timoshenko and Euler–Bernoulli microbeams Physica E 83 74–8710.1016/j.physe.2016.04.011Search in Google Scholar

[30] Chen DQ, Sun DL, and Li XF 2017 Surface effects on resonance frequencies of axially functionally graded Timoshenko nanocantilevers with attached nanoparticle Compos Struct 173 116–12610.1016/j.compstruct.2017.04.006Search in Google Scholar

[31] Ghayesh MH 2018 Nonlinear vibrations of axially functionally graded Timoshenko tapered beams J Comput Nonlin Dyn. 13 041002-1-1010.1115/1.4039191Search in Google Scholar

[32] Huang Y, Wang T, Zhao Y and Wang P 2018 Effect of axially functionally graded material on whirling frequencies and critical speeds of a spinning Timoshenko beam Compos Struct 192 355–36710.1016/j.compstruct.2018.02.039Search in Google Scholar

[33] Mohanty SC, Dash RR and Rout T 2011 Parametric instability of a functionally graded Timoshenko beamonWinkler’s elastic foundation Nucl Eng Des. 241 2698–271510.1016/j.nucengdes.2011.05.040Search in Google Scholar

[34] Yan T, Kitipornchai S, Yang J and He XQ 2011 Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load Compos Struct 93 2992–300110.1016/j.compstruct.2011.05.003Search in Google Scholar

[35] Yas MH and Samadi N 2012 Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation Int J Pres Ves Pip. 98 119-12810.1016/j.ijpvp.2012.07.012Search in Google Scholar

[36] Esfahani SE, Kiani Y, and Eslami MR2013 Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations Int J Mech Sci 69 10–2010.1016/j.ijmecsci.2013.01.007Search in Google Scholar

[37] Komijani M, Esfahani SE, Reddy JN, Liu YP and Eslami MR 2014 Nonlinear thermal stability and vibration of pre/postbuckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation Compos Struct 112 292–30710.1016/j.compstruct.2014.01.041Search in Google Scholar

[38] Tossapanon P and Wattanasakulpong N 2016 Stability and free vibration of functionally graded sandwich beamsresting on twoparameter elastic foundation Compos Struct 142 215–22510.1016/j.compstruct.2016.01.085Search in Google Scholar

[39] Deng H, Chen KD, Cheng W and Zhao SG 2017 Vibration and buckling analysis of double-functionally graded Timoshenko beam system on Winkler-Pasternak elastic foundation Compos Struct 160 152–16810.1016/j.compstruct.2016.10.027Search in Google Scholar

[40] Arefi M and Zenkour AM 2017 Wave propagation analysis of a functionally graded magneto-electro-elastic nanobeam rest on Visco-Pasternak foundation Mech. Res. Commun. 79 51–6210.1016/j.mechrescom.2017.01.004Search in Google Scholar

[41] Arefi M and Zenkour AM 2017 Analysis of wave propagation in a functionally graded nanobeam resting on visco-Pasternak’s foundation Theoret. Applied Mech. Lett. 7 145–15110.1016/j.taml.2017.05.003Search in Google Scholar

[42] Huang Y and Luo QZ 2011 A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint Comput Math Appl. 61 2510–251710.1016/j.camwa.2011.02.037Search in Google Scholar

[43] Lohar H, Mitra A and Sahoo S 2016 Geometric nonlinear free vibration of axially functionally graded non-uniform beams supported on elastic foundation Curved and Layer. Struct. 3(1) 223– 23910.1515/cls-2016-0018Search in Google Scholar

[44] Lohar H, Mitra A and Sahoo S 2016 Natural frequency and mode shapes of exponential tapered AFG beams on elastic foundation International Frontier Science Letters 9 9-2510.18052/www.scipress.com/IFSL.9.9Search in Google Scholar

[45] Lohar H, Mitra A and Sahoo S 2018 Geometrically non-linear frequency response of axially functionally graded beams resting on elastic foundation under harmonic excitation International Journal of Manufacturing Materials and Mechanical Engineering 8(3) 23-4310.4018/IJMMME.2018070103Search in Google Scholar

[46] Calim FF 2016 Free and forced vibration analysis of axially functionally graded Timoshenko beams on two-parameter viscoelastic foundation Composites Part B 103 98-11210.1016/j.compositesb.2016.08.008Search in Google Scholar

[47] Press WH, Teukolsky SA, Vetterling WT and Flannery BP 2005 Numerical Recipes in Fortran 77 2nd ed Cambridge USA: Press SyndicateSearch in Google Scholar

[48] Nakamura T, Wang T and Sampath S 2000 Determination of properties of graded materials by inverse analysis and instrumented indentation Acta Mater. 48 1444-145010.1016/S1359-6454(00)00217-2Search in Google Scholar

[49] Ribeiro P 2004 Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods Comput Struct 82 1413–142310.1016/j.compstruc.2004.03.037Search in Google Scholar

Received: 2018-11-11
Accepted: 2018-12-27
Published Online: 2019-05-29
Published in Print: 2019-01-01

© 2019 Hareram Lohar et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Downloaded on 30.4.2024 from https://www.degruyter.com/document/doi/10.1515/cls-2019-0008/html
Scroll to top button