Dynamic Response of Bi-Directional Functionally Graded Materials (BDFGMs) Beams Rested on Visco-Pasternak Foundation Under Periodic Axial Force

Document Type : Research Paper

Authors

Mechanical Engineering Faculty, University of Kashan, Kashan, Iran

10.22034/jsm.2020.678358

Abstract

Since the temperature or stress distribution in some advanced machines such as modern aerospace shuttles and craft develops in two or three directions, the need for a new type of FGMs is felt whose properties vary in two or three directions. On the other hand, dynamic buckling behavior of structures is a complicated phenomenon which should be investigated through the response of equations of motion. In this paper, dynamic response of beams composed of bi-directional functionally graded materials (BDFGMs) rested on visco-Pasternak foundation under periodic axial force is investigated. Material properties of BDFGMs beam vary continuously in both the thickness and longitudinal directions based on the two types of analytical functions (e.g. exponential and power law distributions). Hamilton's principle is employed to derive the equations of motion of BDFGMs beam according to the Euler-Bernoulli and Timoshenko beam theories. Then, the generalized differential quadrature (GDQ) method in conjunction with the Bolotin method is used to solve the differential equations of motion under different boundary conditions. It is observed that a good agreement between the present work and the literature result. Various parametric investigations are performed for the effects of the gradient index, length-to-thickness ratio and viscoelastic foundation coefficients on the dynamic stability region of BDFGMs beam. The results show that the influence of gradient index of material properties along the thickness direction is greater than gradient index along the longitudinal direction on the dynamic stability of BDFGMs beam for both exponential and power law distributions.

Keywords

[1] Bolotin V.V., 1964, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, CA.
[2] Simitses G.J., 1987, Instability of dynamically-loaded structures, Applied Mechanics Reviews 40(10): 1403-1408.
[3] Simitses G.J., 1990, Dynamic Stability of Suddenly Loaded Structures, New York, Springer.
[4] Iwatsubo T., Sugiyama Y., Ogino S., 1974, Simple and combination resonances of columns under periodic axial loads, Journal of Sound and Vibration 33: 211-221.
[5] Abbas B.A.H., Thomas J., 1978, Dynamic stability of Timoshenko beams resting on an elastic foundation, Journal of Sound and Vibration 60: 33-44.
[6] Aristizabal-Ochoa J.D., 1993, Statics stability and vibration of non-prismatic beams and columns, Journal of Sound and Vibration 162(3): 441-455.
[7] Briseghella L., Majorana C.E., Pellegrino C., 1998, Dynamic stability of elastic structures: a finite element approach, Computers & Structures 69: 11-25.
[8] Ozturk H., Sabuncu M., 2005, Stability analysis of a cantilever composite beam on elastic support, Composites Science and Technology 65: 1982-1995.
[9] Shastry B.P., Rao G.V., 1986, Dynamic stability of columns with two symmetrically placed intermediate supports, Journal of Sound and Vibration 104(3): 524-527.
[10] Li X.F., 2008, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernaulli beams, Journal of Sound and Vibration 318(4-5): 1210-1229.
[11] Ke L.L., Wang Y.S., 2011, Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composites Structures 93(2): 342-350.
[12] Mohanty S.C., Dash R.R., Rout T., 2012, Static and dynamic stability analysis of a functionally graded Timoshenko beam, International Journal of Structural Stability and Dynamics 12(4): 1250025-1250033.
[13] Fu Y., Wang J., Mao Y., 2012, Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment, Applied Mathematical Modelling 36(9): 4324-4340.
[14] Zamanzadeh M., Rezazadeh G., Jafarsadeghi-poornaki I., Shabani R., 2013, Static and dynamic stability modeling of a capacitive FGM micro-beam in presence of temperature changes, Applied Mathematical Modelling 37(10-11): 6964-6978.
[15] Ke L.L., Yang J., Kitipornchai S. 2013, Dynamic stability of functionally graded carbon nanotube-reinforced composite beams, Mechanics of Advanced Materials and Structures 20(1): 28-37.
[16] Ghorbanpour Arani A., Hashemian M., Kolahchi R., 2013, Nonlocal Timoshenko beam model for dynamic stability of double-walled boron nitride nanotubes conveying nanoflow, Proceedings of the Institution of Mechanical Engineers, Part N, Journal of Nanomaterials, Nanoengineering and Nanosystems 229(1): 2-16.
[17] Ghiasian S.E., Kiani Y., Eslami M.R., 2015, Nonlinear thermal dynamic buckling of FGM beams, European Journal of Mechanics - A/Solids 54: 232-242.
[18] Xu Y., Qian Y., Chen J., Song G., 2015, Stochastic dynamic characteristics of FGM beams with random material properties, Composites Structures 133: 585-594.
[19] Shegokara N.L., Lal A., 2016, Stochastic dynamic instability response of piezoelectric functionally graded beams supported by elastic foundation, Advances in Aircraft and Spacecraft Science 3(4): 471-502.
[20] Saffari S., Hashemian M., Toghraie D., 2017, Dynamic stability of functionally graded nanobeam based on nonlocal Timoshenko theory considering surface effects, Physica B 520: 97-105.
[21] Ghorbanpour Arani A., Cheraghbak A., Kolahchi R., 2016, Dynamic buckling of FGM viscoelastic nano-plates resting on orthotropic elastic medium based on sinusoidal shear deformation theory, Structural Engineering and Mechanics 60(3): 489-505.
[22] Ghorbanpour Arani A., Kolahchi R., Zarei M.S., 2015, Visco-surface-nonlocal piezoelasticity effects on nonlinear dynamic stability of graphene sheets integrated with ZnO sensors and actuators using refined zigzag theory, Composites Structures 132: 506-526.
[23] Ghorbanpour Arani A., Jalaei M.H., 2016, Nonlocal dynamic response of embedded single-layered graphene sheet via analytical approach, Journal of Engineering Mathematics 98(1): 129-144.
[24] Yao J.C., 1963, Dynamic stability of cylindrical shells under static and periodic axial and radial loads, AIAA Journal of Air Transportation 1(6): 1391-1396.
[25] Nagai K., Yamaki N., 1978, Dynamic stability of circular cylindrical shells under periodic compressive forces, Journal of Sound and Vibration 58(3): 425-441.
[26] Ghorbanpour Arani A., Mortazavi S.A., Khoddami Maraghi Z., 2015, Dynamic stability of nanocomposite viscoelastic cylindrical shells coating with a piezomagnetic layer conveying pulsating fluid flow, Science and Engineering of Composite Materials 24(3): 401-414.
[27] Arefi M., Zenkour A.M., 2017, Analysis of wave propagation in a functionally graded nanobeam resting on visco-Pasternak’s foundation, Theoretical and Applied Mechanics Letters 7(3): 145-151.
[28] Arefi M., 2016, Considering the surface effect and nonlocal elasticity in wave propagation of a nano functionally graded piezoelectric rod excited to two dimensional electric potential and applied voltage, Applied Mathematics and Mechanics 37(3): 289-302.
[29] Arefi M., Zenkour A.M., 2017, Employing the coupled stress components and surface elasticity for nonlocal solution of wave propagation of a functionally graded piezoelectric Love nanorod model, Journal of Intelligent Material Systems and Structures 28(17): 2403-2413.
[30] Arefi M., Soltan Arani A.H., 2018, Higher order shear deformation bending results of a magnetoelectrothermoelastic functionally graded nanobeam in thermal, mechanical, electrical, and magnetic environments, Mechanics Based Design of Structures and Machines 46(6): 669-692.
[31] Zenkour A.M., Arefi M., 2017, Nonlocal transient electrothermomechanical vibration and bending analysis of a functionally graded piezoelectric single-layered nanosheet rest on visco-Pasternak foundation, Journal of Thermal Stresses 40(2): 167-184.
[32] Ghorbanpour Arani A., Haghparast E., BabaAkbar-Zarei H., 2017, Vibration analysis of functionally graded nanocomposite plate moving in two directions, Steel and Composite Structures 23(5): 529-541.
[33] Ghorbanpour Arani A., GS Jafari G.S., 2015, Nonlinear vibration analysis of laminated composite Mindlin micro/nano-plates resting on orthotropic Pasternak medium using DQM, Applied Mathematics and Mechanics 36(8): 1033-1044.
[34] Arefi M., Mohammad-Rezaei Bidgoli E., Dimitri R., Tornabene F., 2018, Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets, Aerospace Science and Technology 81: 108-117.
[35] Arefi M., Pourjamshidian M., Ghorbanpour Arani A., 2017, Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electro-thermal environment, Applied Physics A 123(5): 323.
[36] Arefi M., Zenkour A.M., 2017, Size-dependent vibration and bending analyses of the piezomagnetic three-layer nanobeams, Applied Physics A 123(3): 202.
[37] Arefi M., Zenkour A.M., 2017, Electro-magneto-elastic analysis of a three-layer curved beam, Smart Structures and Systems 19(6): 695-703.
[38] Arefi M., Zenkour A.M., 2016, Employing sinusoidal shear deformation plate theory for transient analysis of three layers sandwich nanoplate integrated with piezo-magnetic face-sheets, Smart Materials and Structures 25: 115040.
[39] Arefi M., Zenkour A.M., 2017, Thermo-electro-mechanical bending behavior of sandwich nanoplate integrated with piezoelectric face-sheets based on trigonometric plate theory, Composite Structures 162: 108-122.
[40] Arefi M., Zenkour A.M., 2017, Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates, Mechanics Research Communications 84: 27-42.
[41] Arefi M., Zenkour A.M., 2017, Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation, Physica B : Condensed Matter 521: 188-197.
[42] Arefi M., Zamani M.H., Kiani M., 2017, Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation, Journal of Intelligent Material Systems and Structures 29(5): 774-786.
[43] Karamanli A., 2017, Elastostatic analysis of two-directional functionally graded beams using various beam theories and Symmetric Smoothed Particle Hydrodynamics method, Composites Structures 160: 653-669.
[44] Hao D., Wei C., 2016, Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams, Composites Structures 141: 253-263.
[45] Ghorbanpour Arani A., Haghparast E., BabaAkbar-Zarei H., 2016, Nonlocal vibration of axially moving graphene sheet resting on orthotropic visco-Pasternak foundation under longitudinal magnetic field, Physica B 495: 35-49.
[46] Mohammadimehr M., Rousta Navi B., Ghorbanpour Arani A., 2017, Dynamic stability of modified strain gradient theory sinusoidal viscoelastic piezoelectric polymeric functionally graded single-walled carbon nanotubes reinforced nanocomposite plate considering surface stress and agglomeration effects under hydro-thermo-electro-magneto-mechanical loadings, Mechanics of Advanced Materials and Structures 24(16): 1325-1342.
[47] Shu C., 2000, Differential Quadrature and its Application in Engineering, New York, Springer.
[48] Shu C., Du H., 1997, Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates, Journal of Sound and Vibration 34: 819-835.
[49] Kim Y.W., 2005, Temperature dependent vibration analysis of functionally graded rectangular plates, Journal of Sound and Vibration 284(3): 531-549.
[50] Ebrahimi F., Ghasemi F., Salari E., 2016, Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities, Meccanica 51(1): 223-249.
[51] Nguyen D.K., Nguyen Q.H., Tran T.T., Bui V.T., 2017, Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica 228(1): 141-155.

Articles in Press, Corrected Proof
Available Online from 15 December 2020