Engineering Transactions, 69, 1, pp. 43–61, 2021

On Computational Solution of the Dynamic and Static Behaviour of a Coupled Thermoelastic Timoshenko Beam

Theddeus Tochukwu AKANO
University of Lagos

Akintoye O. OYELADE
University of Lagos

The Timoshenko beam theory caters for transverse shear deformations, which are more pronounced in short beams. Previous works were examined, and Hamilton’s principle was used in deriving the governing equation. This research considers two dimensions (2-D): heat and displacement response. A more comprehensive mathematical expression that incorporates this 2-D model on the vibration of a coupled Timoshenko thermoelastic beam and axial deformation effect is formulated. The significance of this model will be expressed through its finite element method (FEM) formulation. The results compared favourably with those of previous works. It was re-established that the amplitude of deflections, as well as cross-sectional rotations, increases considerably as the aspect ratio of the beam decreases. In this way, for larger aspect ratios, the response of the beam is like the quasi-static heating condition. This is expected since the increase in the aspect ratio of the beam reduces its structural stiffness and consequently its natural frequencies. So, the amplitude and temporal period of its vibrations become greater. The beam under the applied thermal loading experiences thermally-induced vibrations. Also, the dynamic solution is substantially influenced by the coupling between strain and temperature fields. The results also reveal that the aspect ratio of the beam could have a significant impact on the vibratory response of the beam. Specifically, it is proportional to the amplitude and temporal period of the thermally-induced vibrations of the beam.
Keywords: thermally-induced; vibrations; Timoshenko beam; finite element method
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Zhang F., Wang G., A study of bending-bending-torsional coupled vibrations of axially-loaded Euler-Bernoulli beams including warping effects, [In:] Computational Structural Engineering, Yuan Y., Cui J., Mang H.A. (Eds), pp. 733–741, Springer Netherlands: Dordrecht, 2009, doi: 10.1007/978-90-481-2822-8_80.

Manolis G.D., Beskos D.E., Thermally induced vibrations of beam structures, Computer Methods in Applied Mechanics and Engineering, 21(3): 337–355, 1980, doi: 10.1016/0045-7825(80)90101-2.

Manoach E., Ribeiro P., Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams, International Journal of Mechanical Sciences , 46(11): 1589–1606, 2004, doi: 10.1016/j.ijmecsci.2004.10.006.

Giunta G., De Pietro G., Nasser H., Belouettar S., Carrera E., Petrolo M., A thermal stress finite element analysis of beam structures by hierarchical modelling, Composites Part B: Engineering, 95: 179–195, 2016, doi: 10.1016/j.compositesb.2016.03.075.

Wen J.J., Yi L.Y., Analyze on a coupled beam vibration system by FEM ( I ), Theory, Advanced Materials Research, 933: 281–284, 2014, doi: 10.4028/

Zhang W., Zhang L.-L., Li Q., Lu Z.-R., Vibration analysis of a coupled beam system carrying any number of sprung masses, Advances in Structural Engineering, 15(2): 217–230, 2012, doi: 10.1260/1369-4332.15.2.217.

Trinh L.C., Vo T.P., Thai H.-T., Nguyen T.-K., An analytical method for the vibration and buckling of functionally graded beams under mechanical and thermal loads, Composites Part B: Engineering, 100: 152–163, 2016, doi: 10.1016/j.compositesb.2016.06.067.

Boley B.A., Approximate analyses of thermally induced vibrations of beams and plates, Journal of Applied Mechanics, 39(1): 212–216, 1972, doi: 10.1115/1.3422615.

Kidawa-Kukla J., Vibration of a beam induced by harmonic motion of a heat source, Journal of Sound and Vibration, 205(2): 213–222, 1997, doi: 10.1006/jsvi.1997.0980.

Kidawa-Kukla J., Application of the Green functions to the problem of the thermally induced vibration of a beam, Journal of Sound and Vibration, 262(4): 865–876, 2003, doi: 10.1016/S0022-460X(02)01133-1.

Li P., Du S. J., Shen S. L., Wang Y. H., Zhao H.H., Timoshenko beam solution for the response of existing tunnels because of tunneling underneath, International Journal for Numerical and Analytical Methods in Geomechanics, 40(5): 766–784, 2016, doi: 10.1002/nag.2426.

Kahrobaiyan M.H., Zanaty M., Henein S., An analytical model for beam flexure modules based on the Timoshenko beam theory, [In:] Proceedings of ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Volume 5A: 41st Mechanisms and Robotics Conference, pp. 1–8, 2017, doi: 10.1115/detc2017-67512.

Bitar I., Kotronis P., Benkemoun N., Grange S., A generalized Timoshenko beam with embedded rotation discontinuity, Finite Elements in Analysis and Design, 150: 34–50, 2018, doi: 10.1016/j.finel.2018.07.002.

Li X.Y., Li P.D., Kang G.Z., Pan D.Z., Axisymmetric thermo-elasticity field in a functionally graded circular plate of transversely isotropic material, Mathematics and Mechanics of Solids., 18(5): 464–475, 2013, doi: 10.1177/1081286512442437.

Kumar R., Response of thermoelastic beam due to thermal source in modified couple stress theory, Computational Methods in Science and Technology, 22(2): 95–101, 2016, doi: 10.12921/cmst.2016.22.02.004.

Nayfeh A.H., Faris W., Dynamic behavior of circular structural elements under thermal loads, [In:] Proceedings of 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 9 pages, 2003, doi: 10.2514/6.2003-1618.

Zenkour A.M., Two-dimensional coupled solution for thermoelastic beams via generalized dual-phase-lags model, Mathematical Modelling and Analysis, 21(3): 319–335, 2016, doi: 10.3846/13926292.2016.1157835.

Blandino J.R., Thornton E.A., Thermally induced vibration of an internally heated beam, Journal of Vibration and Acoustics, 123(1): 67–75, 2001, doi: 10.1115/1.1320446.

Du C., Li Y., Nonlinear resonance behavior of functionally graded cylindrical shells in thermal environments, Composite Structures, 102: 164–174, 2013, doi: 10.1016/j.compstruct.2013.02.028.

Benaarbia A., Chrysochoos A., Robert G., Kinetics of stored and dissipated energies associated with cyclic loadings of dry polyamide 6.6 specimens, Polymer Testing, 34: 155–167, 2014, doi: 10.1016/j.polymertesting.2014.01.009.

Yasumura K.Y. et al., Quality factors in micron- and submicron-thick cantilevers, Journal of Microelectromechanical Systems, 9(1): 117–125, 2000, doi: 10.1109/84.825786.

Khisaeva Z.F., Ostoja-Starzewski M., Thermoelastic damping in nanomechanical resonators with finite wave speeds, Journal of Thermal Stresses, 29(3): 201–216, 2006, doi: 10.1080/01495730500257490.

Lifshitz R., Roukes M. L., Thermoelastic damping in micro-and nanomechanical systems, Physical Review B, 61(8): 5600, 2000, doi: 10.1103/PhysRevB.61.5600.

Smoltczyk U., Geotechnical Engineering Handbook,Volume 2: Procedures, Ernst & Sohn, 2003.

Tilmans H.A.C., Elwenspoek M., Fluitman J.H.J., Micro resonant force gauges, Sensors and Actuators A: Physical, 30(1–2): 35–53, 1992, doi: 10.1016/0924-4247(92)80194-8.

Lothe J., Aspects of the theories of dislocation mobility and internal friction, Physical Review, 117(3): 704-708, 1960, doi: 10.1103/PhysRev.117.704.

Zhang C., Xu G., Jiang Q., Analysis of the air-damping effect on a micromachined beam resonator, Mathematics and Mechanics of Solids, 8(3): 315–325, 2003, doi: 10.1177/1081286503008003006.

Houston B.H., Photiadis D.M., Marcus M.H., Bucaro J.A., Liu X., Vignola J.F., Thermoelastic loss in microscale oscillators, Applied Physics Letters, 80(7): 1300–1302, 2002, doi: 10.1063/1.1449534.

Yang J., Ono T., Esashi M., Energy dissipation in submicrometer thick single-crystal silicon cantilevers, Journal of Microelectromechanical Systems, 11(6): 775–783, 2002, doi: 10.1109/JMEMS.2002.805208.

Duwel A., Gorman J., Weinstein M., Borenstein J., Ward P., Experimental study of thermoelastic damping in MEMS gyros, Sensors and Actuators A: Physical, 103(1–2): 70–75, 2003, doi: 10.1016/S0924-4247(02)00318-7.

Thornton E.A., Chini G.P., Gulik D.W., Thermally induced vibrations of a self-shadowed split-blanket solar array, Journal of Spacecraft and Rockets, 32(2): 302–311, 1995, doi: 10.2514/3.26610.

Li X.-Y., Chen W.Q., Wang H.-Y., General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application, European Journal of Mechanics – A/Solids, 29(3): 317–326, 2010, doi: 10.1016/j.euromechsol.2009.11.007.

Zhu X., Zhong S., Sun D., Ye A., Deng F., Investigation of phononic band gap structures considering interface effects, Physica B: Condensed Matter, 450: 121–127, 2014, doi: 10.1016/j.physb.2014.06.012.

Massalas C.V., Kalpakidis V.K., Coupled thermoelastic vibrations of a Timoshenko beam, International Journal of Engineering Science, 22(4): 459–465, 1984, doi: 10.1016/0020-7225(84)90081-8.

Guo X.-X, Wang Z.-M, Wang Y., Zhou Y.-F, Analysis of the coupled thermoelastic vibration for axially moving beam, Journal of Sound and Vibration, 325(3): 597–608, 2009, doi: 10.1016/j.jsv.2009.03.026.

Manoach E., Warminska A., Warminski J., Dynamics of beams under coupled thermo-mechanical loading, Applied Mechanics and Materials, 849: 57–64, 2016, doi: 10.4028/

Sharma V., Kumar S., Influence of microstructure, heterogeneity and internal friction on SH waves propagation in a viscoelastic layer overlying a couple stress substrate, Structural Engineering and Mechanics, 57(4): 703–716, 2016, doi: 10.12989/SEM.2016.57.4.703.

Bower A.F., Applied Mechanics of Solids, CRC Press, Boca Raton, Florida, USA, 2009.

Fakhrabadi M.M.S., Yang J., Comprehensive nonlinear electromechanical analysis of nanobeams under DC/AC voltages based on consistent couple-stress theory, Composite Structures, 132: 1206–1218, 2015, doi: 10.1016/j.compstruct.2015.07.046.

Reddy J.N., Romanoff J., Loya J.A., Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory, European Journal of Mechanics – A/Solids, 56: 92–104, 2016, doi: 10.1016/j.euromechsol.2015.11.001.

DOI: 10.24423/EngTrans.1149.20210126