[1] Euler L., 1744, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, Lausanne & Geneva: Marcum-Michaelem Bousquet, .
[2] Bresse J.A.C., 1859, Cours de mécanique appliquée: professé a l’École Imperiale des Ponts et Chaussées. Résistance des matériaux et stabilité des constructions. Gauthier-Villars.
[3] Timoshenko S.P., 1921, LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(245): 744–6. doi: 10.1080/14786442108636264.
[4] Timoshenko S.P., 1922, X. On the transverse vibrations of bars of uniform cross-section, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 43(253): 125–31. doi: 10.1080/14786442208633855.
[5] Challamel N., Elishakoff I., 2019, A brief history of first-order shear-deformable beam and plate models, Mechanics Research Communications, 102: 103389. doi: 10.1016/j.mechrescom.2019.06.005.
[6] Elishakoff I., 2020, Who developed the so-called Timoshenko beam theory?, Mathematics and Mechanics of Solids, 25(1): 97–116. doi: 10.1177/1081286519856931.
[7] Elishakoff I., 2019, Handbook on Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories. WORLD SCIENTIFIC.
[8] Thomas J., Abbas B.A.H., 1975, Finite element model for dynamic analysis of Timoshenko beam, Journal of Sound and Vibration, 41(3): 291–9. doi: 10.1016/S0022-460X(75)80176-3.
[9] Sarma B.S., Varadan T.K., 1985, Ritz finite element approach to nonlinear vibrations of a Timoshenko beam, Communications in Applied Numerical Methods, 1(1): 23–32. doi: 10.1002/cnm.1630010106.
[10] Zhou D., Cheung Y.K., 2001, Vibrations of tapered timoshenko beams in terms of static timoshenko beam functions, Journal of Applied Mechanics, Transactions ASME, 68(4): 596–602. doi: 10.1115/1.1357164.
[11] Ruta P., 2006, The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem, Journal of Sound and Vibration, 296(1–2): 243–63. doi: 10.1016/j.jsv.2006.02.011.
[12] Park Y.H., Hong S.Y., 2006, Vibrational energy flow analysis of corrected flexural waves in Timoshenko beam - Part I: Theory of an energetic model, Shock and Vibration, 13(3): 137–65. doi: 10.1155/2006/308715.
[13] Li X.F., 2008, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams, Journal of Sound and Vibration, 318(4–5): 1210–29. doi: 10.1016/j.jsv.2008.04.056.
[14] Pradhan K.K., Chakraverty S., 2013, Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method, Composites Part B: Engineering, 51: 175–84. doi: 10.1016/j.compositesb.2013.02.027.
[15] Gul U., Aydogdu M., Karacam F., 2019, Dynamics of a functionally graded Timoshenko beam considering new spectrums, Composite Structures, 207: 273–91. doi: 10.1016/j.compstruct.2018.09.021.
[16] Şimşek M., 2010, Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load, Composite Structures, 92(10): 2532–46. doi: 10.1016/j.compstruct.2010.02.008.
[17] Attarnejad R., Jandaghi Semnani S., Shahba A., 2010, Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams, Finite Elements in Analysis and Design, 46(10): 916–29. doi: 10.1016/j.finel.2010.06.005.
[18] Quintana V., Grossi R., 2010, Eigenfrequencies of generally restrained Timoshenko beams, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-Body Dynamics, 224(1): 117–25. doi: 10.1243/14644193JMBD189.
[19] Asghari M., Rahaeifard M., Kahrobaiyan M.H., Ahmadian M.T., 2011, The modified couple stress functionally graded Timoshenko beam formulation, Materials and Design, 32(3): 1435–43. doi: 10.1016/j.matdes.2010.08.046.
[20] Yang W., He D., 2017, Free vibration and buckling analyses of a size-dependent axially functionally graded beam incorporating transverse shear deformation, Results in Physics, 7: 3251–63. doi: 10.1016/j.rinp.2017.08.028.
[21] Shahba A., Attarnejad R., Marvi M.T., 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: Engineering, 42(4): 801–8. doi: 10.1016/j.compositesb.2011.01.017.
[22] Rajasekaran S., 2013, Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods, Applied Mathematical Modelling, 37(6): 4440–63. doi: 10.1016/j.apm.2012.09.024.
[23] Huang Y., Yang L.E., Luo Q.Z., 2013, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites Part B: Engineering, 45(1): 1493–8. doi: 10.1016/j.compositesb.2012.09.015.
[24] Sarkar K., Ganguli R., 2014, Closed-form solutions for axially functionally graded Timoshenko beams having uniform cross-section and fixed-fixed boundary condition, Composites Part B: Engineering, 58: 361–70. doi: 10.1016/j.compositesb.2013.10.077.
[25] Tang A.Y., Wu J.X., Li X.F., Lee K.Y., 2014, Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams, International Journal of Mechanical Sciences, 89: 1–11. doi: 10.1016/j.ijmecsci.2014.08.017.
[26] Akbaş Ş.D., 2014, Free Vibration of Axially Functionally Graded Beams in Thermal Environment, International Journal Of Engineering & Applied Sciences, 6(3): 37–37. doi: 10.24107/ijeas.251224.
[27] Bambill D. V., Rossit C.A., Felix D.H., 2015, Free vibrations of stepped axially functionally graded Timoshenko beams, Meccanica, 50(4): 1073–87. doi: 10.1007/s11012-014-0053-4.
[28] Calim F.F., 2016, Free and forced vibration analysis of axially functionally graded Timoshenko beams on two-parameter viscoelastic foundation, Composites Part B: Engineering, 103: 98–112. doi: 10.1016/j.compositesb.2016.08.008.
[29] Calim F.F., 2016, Transient analysis of axially functionally graded Timoshenko beams with variable cross-section, Composites Part B: Engineering, 98: 472–83. doi: 10.1016/j.compositesb.2016.05.040.
[30] Deng H., Cheng W., 2016, Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams, Composite Structures, 141: 253–63. doi: 10.1016/j.compstruct.2016.01.051.
[31] 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): 141–55. doi: 10.1007/s00707-016-1705-3.
[32] Ghayesh M.H., 2018, Nonlinear Vibrations of Axially Functionally Graded Timoshenko Tapered Beams, Journal of Computational and Nonlinear Dynamics, 13(4). doi: 10.1115/1.4039191.
[33] Ghayesh M.H., 2018, Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams, Applied Mathematical Modelling, 59: 583–96. doi: 10.1016/j.apm.2018.02.017.
[34] Ghayesh M.H., 2019, Resonant dynamics of axially functionally graded imperfect tapered Timoshenko beams, JVC/Journal of Vibration and Control, 25(2): 336–50. doi: 10.1177/1077546318777591.
[35] Cao D., Gao Y., 2019, Free vibration of non-uniform axially functionally graded beams using the asymptotic development method, Applied Mathematics and Mechanics, 40(1): 85–96. doi: 10.1007/s10483-019-2402-9.
[36] Yuan J., Mu Z., Elishakoff I., 2020, Novel Modification to the Timoshenko–Ehrenfest Theory for Inhomogeneous and Nonuniform Beams, AIAA Journal, 58(2): 939–48. doi: 10.2514/1.J056885.
[37] Elishakoff I., Tonzani G.M., Marzani A., 2018, Effect of boundary conditions in three alternative models of Timoshenko–Ehrenfest beams on Winkler elastic foundation, Acta Mechanica, 229(4): 1649–86. doi: 10.1007/s00707-017-2034-x.
[38] Elishakoff I., Tonzani G.M., Zaza N., Marzani A., 2018, Contrasting three alternative versions of Timoshenko-Ehrenfest theory for beam on Winkler elastic foundation – simply supported beam, ZAMM Zeitschrift Fur Angewandte Mathematik Und Mechanik, 98(8): 1334–68. doi: 10.1002/zamm.201700019.
[39] Tonzani G.M., Elishakoff I., 2020, Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation, Mathematics and Mechanics of Solids, . doi: 10.1177/1081286520947775.
[40] Hosseini M., Hadi A., Malekshahi A., Shishesaz M., 2018, A review of size-dependent elasticity for nanostructures, Journal of Computational Applied Mechanics, 49(1): 197–211. doi: 10.22059/JCAMECH.2018.259334.289.
[41] Hadi A., Nejad M.Z., Hosseini M., 2018, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, 128: 12–23. doi: 10.1016/J.IJENGSCI.2018.03.004.
[42] Hosseini M., Khoram M.M., Hosseini M., Shishesaz M., 2019, A concise review of nano-plates, Journal of Computational Applied Mechanics, 50(2): 420–9. doi: 10.22059/JCAMECH.2019.293625.459.
[43] Shariati M., Azizi B., Hosseini M., Shishesaz M., 2021, On the calibration of size parameters related to non-classical continuum theories using molecular dynamics simulations, International Journal of Engineering Science, 168: 103544. doi: 10.1016/J.IJENGSCI.2021.103544.
[44] Shariati M., Shishesaz M., Sahbafar H., Pourabdy M., 2021, A review on stress-driven nonlocal elasticity theory, Journal of Computational Applied Mechanics, 52(3): 535–52. doi: 10.22059/jcamech.2021.331410.653.
[45] Hosseini M., Shishesaz M., Tahan K.N., Hadi A., 2016, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, 109: 29–53. doi: 10.1016/J.IJENGSCI.2016.09.002.
[46] Shishesaz M., Hosseini M., Naderan Tahan K., Hadi A., 2017, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica 2017 228:12, 228(12): 4141–68. doi: 10.1007/S00707-017-1939-8.
[47] Hadi A., Nejad M.Z., Rastgoo A., Hosseini M., 2018, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, 26(6): 663–72. doi: 10.12989/scs.2018.26.6.663.
[48] Hosseini M., Shishesaz M., Hadi A., 2019, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, 134: 508–23. doi: 10.1016/J.TWS.2018.10.030.
[49] Mohammadi M., Hosseini M., Shishesaz M., Hadi A., Rastgoo A., 2019, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics - A/Solids, 77: 103793. doi: 10.1016/J.EUROMECHSOL.2019.05.008.
[50] Haghshenas Gorgani H., Mahdavi Adeli M., Hosseini M., 2019, Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches, Microsystem Technologies, 25(8): 3165–73. doi: 10.1007/S00542-018-4216-4/FIGURES/7.
[51] Shishesaz M., Hosseini M., 2019, Mechanical Behavior of Functionally Graded Nano-Cylinders Under Radial Pressure Based on Strain Gradient Theory, Journal of Mechanics, 35(4): 441–54. doi: 10.1017/JMECH.2018.10.
[52] Khoram M.M., Hosseini M., Hadi A., Shishehsaz M., 2020, Bending Analysis of Bidirectional FGM Timoshenko Nanobeam Subjected to Mechanical and Magnetic Forces and Resting on Winkler–Pasternak Foundation, Https://Doi.Org/10.1142/S1758825120500933, 12(8). doi: 10.1142/S1758825120500933.
[53] Arda M., 2021, Axial dynamics of functionally graded Rayleigh-Bishop nanorods, Microsystem Technologies, 27(1): 269–82. doi: 10.1007/s00542-020-04950-2.
[54] Aydogdu M., Arda M., Filiz S., 2018, Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter, Advances in Nano Research, 6(3): 257–78. doi: 10.12989/anr.2018.6.3.257.
[55] Leissa A.W., Qatu M.S., 2011, Vibrations of Continuous Systems. New York: McGraw-Hill Education.
[56] Arda M., Aydogdu M., 2020, Vibration analysis of carbon nanotube mass sensors considering both inertia and stiffness of the detected mass, Mechanics Based Design of Structures and Machines, 0(0): 1–17. doi: 10.1080/15397734.2020.1728548.
[57] Wright E.M., Kantorovich L. V., Krylov V.I., Benster C.D., 1960, Approximate Methods of Higher Analysis, The Mathematical Gazette, 44(348): 145. doi: 10.2307/3612589.