Nonlinear Vibration of a Beam Resting on a Nonlinear Viscoelastic Foundation Traversed by a Moving Mass: A Homotopy Analysis
Abstract
In this study, the dynamic response of an Euler-Bernoulli beam resting on the nonlinear viscoelastic foundation under the action of a moving mass by considering the stretching effect of the beam’s neutral axis is investigated. A Dirac-delta function is applied to model the location of the moving mass along the beam as well as its inertial effects. The Galerkin decomposition method is used to transform a partial dimensionless nonlinear differential equation of dynamic motion into an ordinary nonlinear differential equation. Subsequently, the well-known homotopy analysis method (HAM) is employed to obtain an approximate analytical solution of this equation. The validity and accuracy of the solution are examined numerically using the fourth-order Runge-Kutta method. Finally, several examples are provided to show the effects of parameters such as linear and nonlinear stiffness coefficients of a viscoelastic foundation, velocity of the moving mass as well as Coriolis force, centrifugal force and inertia force of the moving mass on the dynamic deflection of the beam.
Keywords:
Euler-Bernoulli beam, nonlinear viscoelastic foundation, moving mass, homotopy analysis methodReferences
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2. He W., Vertical dynamics of a single-span beam subjected to moving mass-suspended payload system with variable speeds, Journal of Sound and Vibration, 418: 36–54, 2018, https://doi.org/10.1016/j.jsv.2017.12.030
3. Parhi D.R., Jena S.P., Dynamic and experimental analysis on response of multi-cracked structures carrying transit mass, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 231(1): 25–35, 2017, https://doi.org/10.1177/1748006X16682605
4. Aldlemy M.S., Al-jumaili S.A.K., Al-Mamoori R.A.M., Ya T., Alebrahim R., Composite patch reinforcement of a cracked simply-supported beam traversed by moving mas, Journal of Mechanical Engineering and Sciences, 14(1): 6403–6415, 2020, https://doi.org/10.15282/jmes.14.1.2020.16.0501
5. Wang Y., Zhou A., Fu T., Zhang W., Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniformly distributed moving mass, International Journal of Mechanics and Materials in Design, 16(3): 519–540, 2020, https://doi.org/10.1007/s10999-019-09483-9
6. Rofooei F.R., Enshaeian A., Nikkhoo A., Dynamic response of geometrically nonlinear, elastic rectangular plates under a moving mass loading by inclusion of all inertial components, Journal of Sound and Vibration, 394: 497–514, 2017, https://doi.org/10.1016/j.jsv.2017.01.033
7. Liu X.-X., Ren X.-M., An IPEM for optimal control of uncertain beam-moving mass systems with saturation nonlinearity, Journal of Vibration and Control, 24(13): 2760–2781, 2017, https://doi.org/10.1177/1077546317693957
8. Sheng G.G, Wang X., The geometrically nonlinear dynamic responses of simply supported beams under moving loads, Applied Mathematical Modelling, 48: 183–195, 2017, https://doi.org/10.1016/j.apm.2017.03.064
9. Rodrigues C., Simões F.M.F., Pinto da Costa A., Froio D., Rizzi E., Finite element dynamic analysis of beams on nonlinear elastic foundations under a moving oscillator, European Journal of Mechanics-A/Solids, 68: 9–24, 2018, https://doi.org/10.1016/j.euromechsol.2017.10.005
10. Jahangiri A., Attari N.K.A., Nikkhoo A., Waezi Z., Nonlinear dynamic response of an Euler-Bernoulli beam under a moving mass–spring with large oscillations, Archive of Applied Mechanics, 90: 1135–1156, 2020, https://doi.org/10.1007/s00419-020-01656-9
11. Esen I., Dynamics of size-dependant Timoshenko micro beams subjected to moving loads, International Journal of Mechanical Sciences, 175: 105501, 2020, https://doi.org/10.1016/j.ijmecsci.2020.105501
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14. Green M.F., Cebon D., Dynamic interaction between heavy vehicles and highway bridges, Computers & Structures, 62(2): 253–264, 1997, https://doi.org/10.1016/S0045-7949%2896%2900198-8
15. Auciello N.M., On the transverse vibrations of non-uniform beams with axial loads and elastically restrained ends, International Journal of Mechanical Sciences, 43(1): 193–208, 2001, https://doi.org/10.1016/S0020-7403%2899%2900110-1
16. Chen Y.-H., Huang Y.-H., Shih C.-T., Response of an infinite Timoshenko beam on a viscoelastic foundation to a harmonic moving load, Journal of Sound and Vibration, 241(5): 809–824, 2001, https://doi.org/10.1006/jsvi.2000.3333
17. Azam S.E., Mofid M., Khoraskani R.A., Dynamic response of Timoshenko beam under moving mass, Scientia Iranica, 20(1): 50–56, 2013, https://doi.org/10.1016/j.scient.2012.11.003
18. Nikkhoo A., Farazandeh A., Ebrahimzadeh HassanabadiM., Mariani M., Simplified modeling of beam vibrations induced by a moving mass by regression analysis, Acta Mechanica, 226(7): 2147–2157, 2015, https://doi.org/10.1007/s00707-015-1309-3
19. Eftekhari S., A differential quadrature procedure for linear and nonlinear steady state vibrations of infinite beams traversed by a moving point load, Meccanica, 51(10): 2417–2434, 2016, https://doi.org/10.1007/s11012-016-0373-7
20. Kaya D., A review of the semi-analytic/numerical methods for higher order nonlinear partial equations, Contemporary Analysis and Applied Mathematics, 3(1): 133–152, 2015, https://doi.org/10.18532/caam.42606 https://doi.org/10.18532/caam.42606
21. Hemmi Y., Review on higher homotopies in the theory of H-spaces, Mathematical Journal of Okayama University, 60(1): 1–36, 2018, https://doi.org/10.18926/mjou/56008
22. Moutsinga C.R.B., Pindza E., Maré E., Homotopy perturbation transform method for pricing under pure diffusion models with affine coefficients, Journal of King Saud University-Science, 30(1): 1–13, 2018, https://doi.org/10.1016/j.jksus.2016.09.004
23. Samadani F., Moradweysi P., Ansari R., Hosseini K., Darvizeh A., Application of homotopy analysis method for the pull-In and nonlinear vibration analysis of nanobeams using a nonlocal Euler-Bernoulli beam model, Zeitschrift für Naturforschung A, 72(12): 1093–1104, 2017, https://doi.org/10.1515/zna-2017-0174
24. Vandewater L.A., Moss S.D., Non-linear dynamics of a vibration energy harvester by means of the homotopy analysis method, Journal of Intelligent Material Systems and Structures, 25(13): 1605–1613, 2014, https://doi.org/10.1177/1045389X135102
25. Turkyilmazoglu M., An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method, Filomat, 30(6): 1633–1650, 2016, https://doi.org/10.2298/FIL1606633T
26. Srivastava H.M., Kumar D., Singh J., An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45: 192–204, 2017, https://doi.org/10.1016/j.apm.2016.12.008
27. Hamarsheh M., Ismail A., Odibat Z., Optimal homotopy asymptotic method for solving fractional relaxation-oscillation equation, Journal of Interpolation and Approximation in Scientific Computing, 2015(2): 98–111, 2015, https://doi.org/10.5899/2015/jiasc-00081
28. Ganji S.S., Ganji D.D., Sfahani M.G., Karimpour S., Application of AFF and HPM to the systems of strongly nonlinear oscillation, Current Applied Physics, 10(5): 1317–1325, 2010, https://doi.org/10.1016/j.cap.2010.03.015
29. Temimi H., Ansari A.R., Siddiqui A.M., An approximate solution for the static beam problem and nonlinear integro-differential equations, Computers & Mathematics with Applications, 62(8): 3132–3139, 2011, https://doi.org/10.1016/j.camwa.2011.08.026
30. Rafiq A., Malik M.Y., Abbasi T., Solution of nonlinear pull-in behavior in electrostatic micro-actuators by using He’s homotopy perturbation method, Computers & Mathematics with Application, 59(8): 2723–2733, 2010, https://doi.org/10.1016/j.camwa.2010.01.040
31. Pirbodaghi T., Fesanghary M., Ahmadian M.T., Non-linear vibration analysis of laminated composite plates resting on non-linear elastic foundations, Journal of the Franklin Institute, 348(2): 353–368, 2011 https://doi.org/10.1016/j.jfranklin.2010.12.002
32. Rouabhia A., Chikh A., Bousahla A.A., Bourada F., Heireche H., Tounsi, A., Benrahou K.H., Tounsi A., Al-Zahrani M.M., Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory, Steel and Composite Structures, 37(6): 695–709, 2020, https://doi.org/10.12989/scs.2020.37.6.695
33. Bellal M., Hebali H., Heireche H., Bousahla A.A., Tounsi A., Bourada F., Mahmoud S.R., Bedia E.A.A., Tounsi A., Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model, Steel and Composite Structures, 34(5): 643–655, 2020, https://doi.org/10.12989/scs.2020.34.5.643
34. Guellil M., Saidi H., Bourada F., Bousahla A.A., Tounsi A., Al-Zahrani M.M.A., Hussain M., Mahmoud S.R., Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation, Steel and Composite Structures, 38(1): 1–15, 2021, https://doi.org/10.12989/scs.2021.38.1.001
35. Bendenia N., Zidour M., Bousahla A.A., Bourada F., Tounsi A., Benrahou K.H., Bedia E.A.A., Mahmoud S.R., Tounsi A., Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation, Computers and Concrete, 26(3): 213–226, 2020, https://doi.org/10.12989/cac.2020.26.3.213
36. Rabhi M., Benrahou, K.H., Kaci A., Houari M.S.A., Bourada F., Bousahla A.A., Tounsi A., Bedia E.A.A., Mahmoud S.R., Tounsi A., A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions, Geomechanics and Engineering, 22(2): 119–132, 2020, https://doi.org/10.12989/gae.2020.22.2.119
37. Bourada F., Bousahla A.A., Tounsi A., Bedia E.A.A., Mahmoud S.R., Benrahou K.H., Tounsi A., Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic-foundation, Computers and Concrete, 25(6): 485–495, 2020, https://doi.org/10.12989/cac.2020.25.6.485
38. Chikr S.C., Kaci A., Bousahla A.A., Bourada F., Tounsi A., Bedia E.A.A., Mahmoud S.R., Benrahou K.H., Tounsi A., A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin’s approach, Geomechanics and Engineering, 21(5): 471–487, 2020, https://doi.org/10.12989/gae.2020.21.5.471
39. Refrafi S., Bousahla A.A., Bouhadra, A., Menasria A., Bourada F., Tounsi A., Bedia E.A.A., Mahmoud S.R., Benrahou K.H., Tounsi A., Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations, Computers and Concrete, 25(4): 311–325, 2020, https://doi.org/10.12989/cac.2020.25.4.311
40. Kaddari, M., Kaci A., Bousahla A.A., Tounsi A., Bourada F., Tounsi A., Adda Bedia E.A., Mohammed A.A., A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: bending and free vibration analysis, Computers and Concrete, 25(1): 37–57, 2020, https://doi.org/10.12989/cac.2020.25.1.037
41. Addou F.Y., Meradjah M., , Bousahla A.A., Benachour A., Bourada F., Tounsi A., Mahmoud S.R., Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT, Computers and Concrete, 24(4): 347–367, 2019, https://doi.org/10.12989/cac.2019.24.4.347
42. Tounsi A., Dulaijan S.U.A., Al-Osta M.A., Chikh A., Al-Zahrani M.M., Sharif A., Tounsi A., A four variable trigonometric integral plate theory for hygro-thermo-mechanical bending analysis of AFG ceramic-metal plates resting on a two-parameter elastic foundation, Steel and Composite Structures, 34(4): 511–524, 2020, https://doi.org/10.12989/scs.2020.34.4.511
43. Senalp A.D., Arikoglu A., Ozkol I., Dogan V.Z., Dynamic response of a finite length Euler-Bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force, Journal of Mechanical Science and Technology, 24(10): 1957–1961, 2010, https://doi.org/10.1007/s12206-010-0704-x
44. Shahlaei-Far S., Nabarrete A., Balthazar J.M., Nonlinear vibrations of cantilever Timoshenko beams: a homotopy analysis, Latin American Journal of Solids and Structures, 13(10): 1866–1877, 2016, https://doi.org/10.1590/1679-78252766
45. Ganjefar S., Rezaei S., Pourseifi M., Self-adaptive vibration control of simply supported beam under a moving mass using self-recurrent wavelet neural networks via adaptive learning rates, Meccanica, 50(12): 2879–2898, 2015, https://doi.org/10.1007/s11012-015-0174-4
46. Qaisi M.I., A power series approach for the study of periodic motion, Journal of Sound and Vibration, 196(4): 401–406, 1996, https://doi.org/10.1006/jsvi.1996.0491
2. He W., Vertical dynamics of a single-span beam subjected to moving mass-suspended payload system with variable speeds, Journal of Sound and Vibration, 418: 36–54, 2018, https://doi.org/10.1016/j.jsv.2017.12.030
3. Parhi D.R., Jena S.P., Dynamic and experimental analysis on response of multi-cracked structures carrying transit mass, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 231(1): 25–35, 2017, https://doi.org/10.1177/1748006X16682605
4. Aldlemy M.S., Al-jumaili S.A.K., Al-Mamoori R.A.M., Ya T., Alebrahim R., Composite patch reinforcement of a cracked simply-supported beam traversed by moving mas, Journal of Mechanical Engineering and Sciences, 14(1): 6403–6415, 2020, https://doi.org/10.15282/jmes.14.1.2020.16.0501
5. Wang Y., Zhou A., Fu T., Zhang W., Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniformly distributed moving mass, International Journal of Mechanics and Materials in Design, 16(3): 519–540, 2020, https://doi.org/10.1007/s10999-019-09483-9
6. Rofooei F.R., Enshaeian A., Nikkhoo A., Dynamic response of geometrically nonlinear, elastic rectangular plates under a moving mass loading by inclusion of all inertial components, Journal of Sound and Vibration, 394: 497–514, 2017, https://doi.org/10.1016/j.jsv.2017.01.033
7. Liu X.-X., Ren X.-M., An IPEM for optimal control of uncertain beam-moving mass systems with saturation nonlinearity, Journal of Vibration and Control, 24(13): 2760–2781, 2017, https://doi.org/10.1177/1077546317693957
8. Sheng G.G, Wang X., The geometrically nonlinear dynamic responses of simply supported beams under moving loads, Applied Mathematical Modelling, 48: 183–195, 2017, https://doi.org/10.1016/j.apm.2017.03.064
9. Rodrigues C., Simões F.M.F., Pinto da Costa A., Froio D., Rizzi E., Finite element dynamic analysis of beams on nonlinear elastic foundations under a moving oscillator, European Journal of Mechanics-A/Solids, 68: 9–24, 2018, https://doi.org/10.1016/j.euromechsol.2017.10.005
10. Jahangiri A., Attari N.K.A., Nikkhoo A., Waezi Z., Nonlinear dynamic response of an Euler-Bernoulli beam under a moving mass–spring with large oscillations, Archive of Applied Mechanics, 90: 1135–1156, 2020, https://doi.org/10.1007/s00419-020-01656-9
11. Esen I., Dynamics of size-dependant Timoshenko micro beams subjected to moving loads, International Journal of Mechanical Sciences, 175: 105501, 2020, https://doi.org/10.1016/j.ijmecsci.2020.105501
12. Willis R., The effect produced by causing weights to travel over elastic bars, Report of Commissioners appointed to inquire into the application of iron to railway structures, Appendix, HM Stationery Office, London, UK, 1847.
13. Mackertich, S., Response of a beam to a moving mass, The Journal of the Acoustical Society of America, 92(3): 1766–1769, 1992, https://doi.org/10.1121/1.405276
14. Green M.F., Cebon D., Dynamic interaction between heavy vehicles and highway bridges, Computers & Structures, 62(2): 253–264, 1997, https://doi.org/10.1016/S0045-7949%2896%2900198-8
15. Auciello N.M., On the transverse vibrations of non-uniform beams with axial loads and elastically restrained ends, International Journal of Mechanical Sciences, 43(1): 193–208, 2001, https://doi.org/10.1016/S0020-7403%2899%2900110-1
16. Chen Y.-H., Huang Y.-H., Shih C.-T., Response of an infinite Timoshenko beam on a viscoelastic foundation to a harmonic moving load, Journal of Sound and Vibration, 241(5): 809–824, 2001, https://doi.org/10.1006/jsvi.2000.3333
17. Azam S.E., Mofid M., Khoraskani R.A., Dynamic response of Timoshenko beam under moving mass, Scientia Iranica, 20(1): 50–56, 2013, https://doi.org/10.1016/j.scient.2012.11.003
18. Nikkhoo A., Farazandeh A., Ebrahimzadeh HassanabadiM., Mariani M., Simplified modeling of beam vibrations induced by a moving mass by regression analysis, Acta Mechanica, 226(7): 2147–2157, 2015, https://doi.org/10.1007/s00707-015-1309-3
19. Eftekhari S., A differential quadrature procedure for linear and nonlinear steady state vibrations of infinite beams traversed by a moving point load, Meccanica, 51(10): 2417–2434, 2016, https://doi.org/10.1007/s11012-016-0373-7
20. Kaya D., A review of the semi-analytic/numerical methods for higher order nonlinear partial equations, Contemporary Analysis and Applied Mathematics, 3(1): 133–152, 2015, https://doi.org/10.18532/caam.42606 https://doi.org/10.18532/caam.42606
21. Hemmi Y., Review on higher homotopies in the theory of H-spaces, Mathematical Journal of Okayama University, 60(1): 1–36, 2018, https://doi.org/10.18926/mjou/56008
22. Moutsinga C.R.B., Pindza E., Maré E., Homotopy perturbation transform method for pricing under pure diffusion models with affine coefficients, Journal of King Saud University-Science, 30(1): 1–13, 2018, https://doi.org/10.1016/j.jksus.2016.09.004
23. Samadani F., Moradweysi P., Ansari R., Hosseini K., Darvizeh A., Application of homotopy analysis method for the pull-In and nonlinear vibration analysis of nanobeams using a nonlocal Euler-Bernoulli beam model, Zeitschrift für Naturforschung A, 72(12): 1093–1104, 2017, https://doi.org/10.1515/zna-2017-0174
24. Vandewater L.A., Moss S.D., Non-linear dynamics of a vibration energy harvester by means of the homotopy analysis method, Journal of Intelligent Material Systems and Structures, 25(13): 1605–1613, 2014, https://doi.org/10.1177/1045389X135102
25. Turkyilmazoglu M., An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method, Filomat, 30(6): 1633–1650, 2016, https://doi.org/10.2298/FIL1606633T
26. Srivastava H.M., Kumar D., Singh J., An efficient analytical technique for fractional model of vibration equation, Applied Mathematical Modelling, 45: 192–204, 2017, https://doi.org/10.1016/j.apm.2016.12.008
27. Hamarsheh M., Ismail A., Odibat Z., Optimal homotopy asymptotic method for solving fractional relaxation-oscillation equation, Journal of Interpolation and Approximation in Scientific Computing, 2015(2): 98–111, 2015, https://doi.org/10.5899/2015/jiasc-00081
28. Ganji S.S., Ganji D.D., Sfahani M.G., Karimpour S., Application of AFF and HPM to the systems of strongly nonlinear oscillation, Current Applied Physics, 10(5): 1317–1325, 2010, https://doi.org/10.1016/j.cap.2010.03.015
29. Temimi H., Ansari A.R., Siddiqui A.M., An approximate solution for the static beam problem and nonlinear integro-differential equations, Computers & Mathematics with Applications, 62(8): 3132–3139, 2011, https://doi.org/10.1016/j.camwa.2011.08.026
30. Rafiq A., Malik M.Y., Abbasi T., Solution of nonlinear pull-in behavior in electrostatic micro-actuators by using He’s homotopy perturbation method, Computers & Mathematics with Application, 59(8): 2723–2733, 2010, https://doi.org/10.1016/j.camwa.2010.01.040
31. Pirbodaghi T., Fesanghary M., Ahmadian M.T., Non-linear vibration analysis of laminated composite plates resting on non-linear elastic foundations, Journal of the Franklin Institute, 348(2): 353–368, 2011 https://doi.org/10.1016/j.jfranklin.2010.12.002
32. Rouabhia A., Chikh A., Bousahla A.A., Bourada F., Heireche H., Tounsi, A., Benrahou K.H., Tounsi A., Al-Zahrani M.M., Physical stability response of a SLGS resting on viscoelastic medium using nonlocal integral first-order theory, Steel and Composite Structures, 37(6): 695–709, 2020, https://doi.org/10.12989/scs.2020.37.6.695
33. Bellal M., Hebali H., Heireche H., Bousahla A.A., Tounsi A., Bourada F., Mahmoud S.R., Bedia E.A.A., Tounsi A., Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model, Steel and Composite Structures, 34(5): 643–655, 2020, https://doi.org/10.12989/scs.2020.34.5.643
34. Guellil M., Saidi H., Bourada F., Bousahla A.A., Tounsi A., Al-Zahrani M.M.A., Hussain M., Mahmoud S.R., Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation, Steel and Composite Structures, 38(1): 1–15, 2021, https://doi.org/10.12989/scs.2021.38.1.001
35. Bendenia N., Zidour M., Bousahla A.A., Bourada F., Tounsi A., Benrahou K.H., Bedia E.A.A., Mahmoud S.R., Tounsi A., Deflections, stresses and free vibration studies of FG-CNT reinforced sandwich plates resting on Pasternak elastic foundation, Computers and Concrete, 26(3): 213–226, 2020, https://doi.org/10.12989/cac.2020.26.3.213
36. Rabhi M., Benrahou, K.H., Kaci A., Houari M.S.A., Bourada F., Bousahla A.A., Tounsi A., Bedia E.A.A., Mahmoud S.R., Tounsi A., A new innovative 3-unknowns HSDT for buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions, Geomechanics and Engineering, 22(2): 119–132, 2020, https://doi.org/10.12989/gae.2020.22.2.119
37. Bourada F., Bousahla A.A., Tounsi A., Bedia E.A.A., Mahmoud S.R., Benrahou K.H., Tounsi A., Stability and dynamic analyses of SW-CNT reinforced concrete beam resting on elastic-foundation, Computers and Concrete, 25(6): 485–495, 2020, https://doi.org/10.12989/cac.2020.25.6.485
38. Chikr S.C., Kaci A., Bousahla A.A., Bourada F., Tounsi A., Bedia E.A.A., Mahmoud S.R., Benrahou K.H., Tounsi A., A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin’s approach, Geomechanics and Engineering, 21(5): 471–487, 2020, https://doi.org/10.12989/gae.2020.21.5.471
39. Refrafi S., Bousahla A.A., Bouhadra, A., Menasria A., Bourada F., Tounsi A., Bedia E.A.A., Mahmoud S.R., Benrahou K.H., Tounsi A., Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations, Computers and Concrete, 25(4): 311–325, 2020, https://doi.org/10.12989/cac.2020.25.4.311
40. Kaddari, M., Kaci A., Bousahla A.A., Tounsi A., Bourada F., Tounsi A., Adda Bedia E.A., Mohammed A.A., A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: bending and free vibration analysis, Computers and Concrete, 25(1): 37–57, 2020, https://doi.org/10.12989/cac.2020.25.1.037
41. Addou F.Y., Meradjah M., , Bousahla A.A., Benachour A., Bourada F., Tounsi A., Mahmoud S.R., Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT, Computers and Concrete, 24(4): 347–367, 2019, https://doi.org/10.12989/cac.2019.24.4.347
42. Tounsi A., Dulaijan S.U.A., Al-Osta M.A., Chikh A., Al-Zahrani M.M., Sharif A., Tounsi A., A four variable trigonometric integral plate theory for hygro-thermo-mechanical bending analysis of AFG ceramic-metal plates resting on a two-parameter elastic foundation, Steel and Composite Structures, 34(4): 511–524, 2020, https://doi.org/10.12989/scs.2020.34.4.511
43. Senalp A.D., Arikoglu A., Ozkol I., Dogan V.Z., Dynamic response of a finite length Euler-Bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force, Journal of Mechanical Science and Technology, 24(10): 1957–1961, 2010, https://doi.org/10.1007/s12206-010-0704-x
44. Shahlaei-Far S., Nabarrete A., Balthazar J.M., Nonlinear vibrations of cantilever Timoshenko beams: a homotopy analysis, Latin American Journal of Solids and Structures, 13(10): 1866–1877, 2016, https://doi.org/10.1590/1679-78252766
45. Ganjefar S., Rezaei S., Pourseifi M., Self-adaptive vibration control of simply supported beam under a moving mass using self-recurrent wavelet neural networks via adaptive learning rates, Meccanica, 50(12): 2879–2898, 2015, https://doi.org/10.1007/s11012-015-0174-4
46. Qaisi M.I., A power series approach for the study of periodic motion, Journal of Sound and Vibration, 196(4): 401–406, 1996, https://doi.org/10.1006/jsvi.1996.0491
