How to cite this paper
Lee, J., Lee, J., Choi, J & Le, B. (2025). Generalized stiffness of laterally functionally graded materials and implementation to dynamic beam element.Engineering Solid Mechanics, 13(1), 39-52.
Refrences
Akgoz, B., & Civalek, O. (2013). Free vibration analysis of axially functionally graded Bernoulli-Euler microbeams based on the modified couple stress theory. Composite Structures. 98, 314–322. https://doi.org/10.1016/j.compstruct.2012.11.020
Alshorbagy, A.E., Eltaher, M.A., & Mahmoud, F.F. (2011). Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling. 35, 412–425. https://doi.org/10.1016/j.apm.2010.07.006
Burden, R.L., Faires, D.J., & Burden, A.M. (2016). Numerical analysis, Boston, MA, USA: Cengage Learning.
Chopra, A.K. (2001). Dynamics of structures, Upper Saddle River, NJ, USA: Prentice-Hall, Inc.
Czechowski, L., & Kolakowski, Z. (2019). The study of buckling and post-buckling of a step-variable FGM box. Materials. 12(6), 918. https://doi.org/10.3390/ma12060918
Gere, J.M., & Timoshenko, S.P. (1980). Mechanics of materials, Boston, MA, USA: PWS Publishing Company.
Horibe, T.A., & Mori, K. (2015). Large deflections of tapered cantilever beams made of axially functionally graded materials. Mechanical Engineering Journal. 5(1), 1–10. https://doi.org/10.1299/mej.17-00268
Huynh, T., Luu, A., & Lee, J. (2017). Bending, buckling and free vibration analyses of functionally graded curved beams with variable curvatures using isogeometric approach. Meccanica. 52, 2527–2546. https://doi.org/10.1007/s11012-016-0603-z
Javania, M., Kianib, Y., & Eslamia, M.R. (2019). Free vibration of arbitrary thick FGM deep arches using unconstrained higher-order shear deformation theory. Thin-Walled Structures, 136, 258–266. https://doi.org/10.1016/j.tws.2018.12.020
Kang, Y.A., & Li, X.F. (2009). Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force. International Journal of Non-Linear Mechanics, 44(6), 696–703.
https://doi.org/10.1016/j.ijnonlinmec.2009.02.016
Lee, J.K., & Lee, B.K. (2019). In-plane free vibration of uniform circular arches made of axially functionally graded materials. International Journal of Structural Stability and Dynamics, 19(7), 1950084. https://doi.org/10.1142/S0219455419500846
Lee, J.K., & Lee, B.K. (2022a). Buckling optimization of axially functionally graded columns having constant volume. Engineering Optimization, 542(2), 269–285. https://doi.org/10.1080/0305215X.2020.1862824
Lee, J.K., & Lee, B.K. (2022b). Coupled flexural-torsional free vibration of an axially functionally graded circular curved beam. Mechanics of Composite Materials, 57(6), 833–846. https://doi.org/10.1007/s11029-022-10003-8
Liu, Y., & Shu, D.W. (2014). Free vibration analysis of exponential functionally graded beams with a single delamination. Composites Part: B-Engineering, 59, 166–172. https://doi.org/10.1016/j.compositesb.2013.10.026
Malekzadeh, P. (2009). Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties. Composite Structures, 91(1), 38–47. https://doi.org/10.1016/j.compstruct.2009.04.034
Malekzadeh, P., Atashi, M.M., & Karami, G. (2009). In-plane free vibration of functionally graded circular arches with temperature-dependent properties under thermal environment. Journal of Sound and Vibration. 326, 837–851.
https://doi.org/10.1016/j.jsv.2009.05.016
Noori, A.R., Aslan, T.A., & Temel, B. (2018). An efficient approach for in-plane free and forced vibrations of axially functionally graded parabolic arches with non-uniform cross section. Composite Structures. 200, 701–710.
https://doi.org/10.1016/j.compstruct.2018.05.077
Qatu, M.S., & Elsharkawy, A.A. (1993). Vibration of laminated composite arches with deep curvature and arbitrary boundaries. Computers & Structures. 47(2), 305–311. https://doi.org/10.1016/0045-7949(93)90381-M
Raki, M., Alipour, R., & Kamanbedast, A. (2012). Thermal buckling of thin rectangular FGM plate. World Applied Science Journal, 16(1), 52–62.
Sitar, M., Kosel, F., & Brojan, M. (2014). Large deflections of nonlinearly elastic functionally graded composite beam. Archives Civil and Mechanical Engineering, 14(4), 700–709. https://doi.org/10.1016/j.acme.2013.11.007
Trinh, T.H., Nguyen, D.K., Gan, B.S., & Alexandrov, A. (2016). Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation. Structural Engineering and Mechanics, 58(3), 515–532.
https://doi.org/10.12989/sem.2016.58.3.515
Yousefi, A., & Rastgoo, A. (2011). Free vibration of functionally graded spatial curved beams. Composite Structures. 93, 3048–3056. https://doi.org/10.1016/j.compstruct.2011.04.024
Zaczynska, M., & Kazmierczyk, F. (2020). Multi-mode buckling analysis of FGM channel section beams. Materials, 13(11), 2567. https://doi.org/10.3390/ma13112567
Zhao, F.Q., Wang, Z.M., & Zhang, R.P. (2012). Post-buckling analysis of FGM beam subjected to non-conservative forces and in-plane thermal loading. Applied Mechanics and Materials. 152/154, 474–479.
https://doi.org/10.4028/www.scientific.net/AMM.152-154.474
Alshorbagy, A.E., Eltaher, M.A., & Mahmoud, F.F. (2011). Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling. 35, 412–425. https://doi.org/10.1016/j.apm.2010.07.006
Burden, R.L., Faires, D.J., & Burden, A.M. (2016). Numerical analysis, Boston, MA, USA: Cengage Learning.
Chopra, A.K. (2001). Dynamics of structures, Upper Saddle River, NJ, USA: Prentice-Hall, Inc.
Czechowski, L., & Kolakowski, Z. (2019). The study of buckling and post-buckling of a step-variable FGM box. Materials. 12(6), 918. https://doi.org/10.3390/ma12060918
Gere, J.M., & Timoshenko, S.P. (1980). Mechanics of materials, Boston, MA, USA: PWS Publishing Company.
Horibe, T.A., & Mori, K. (2015). Large deflections of tapered cantilever beams made of axially functionally graded materials. Mechanical Engineering Journal. 5(1), 1–10. https://doi.org/10.1299/mej.17-00268
Huynh, T., Luu, A., & Lee, J. (2017). Bending, buckling and free vibration analyses of functionally graded curved beams with variable curvatures using isogeometric approach. Meccanica. 52, 2527–2546. https://doi.org/10.1007/s11012-016-0603-z
Javania, M., Kianib, Y., & Eslamia, M.R. (2019). Free vibration of arbitrary thick FGM deep arches using unconstrained higher-order shear deformation theory. Thin-Walled Structures, 136, 258–266. https://doi.org/10.1016/j.tws.2018.12.020
Kang, Y.A., & Li, X.F. (2009). Bending of functionally graded cantilever beam with power-law non-linearity subjected to an end force. International Journal of Non-Linear Mechanics, 44(6), 696–703.
https://doi.org/10.1016/j.ijnonlinmec.2009.02.016
Lee, J.K., & Lee, B.K. (2019). In-plane free vibration of uniform circular arches made of axially functionally graded materials. International Journal of Structural Stability and Dynamics, 19(7), 1950084. https://doi.org/10.1142/S0219455419500846
Lee, J.K., & Lee, B.K. (2022a). Buckling optimization of axially functionally graded columns having constant volume. Engineering Optimization, 542(2), 269–285. https://doi.org/10.1080/0305215X.2020.1862824
Lee, J.K., & Lee, B.K. (2022b). Coupled flexural-torsional free vibration of an axially functionally graded circular curved beam. Mechanics of Composite Materials, 57(6), 833–846. https://doi.org/10.1007/s11029-022-10003-8
Liu, Y., & Shu, D.W. (2014). Free vibration analysis of exponential functionally graded beams with a single delamination. Composites Part: B-Engineering, 59, 166–172. https://doi.org/10.1016/j.compositesb.2013.10.026
Malekzadeh, P. (2009). Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties. Composite Structures, 91(1), 38–47. https://doi.org/10.1016/j.compstruct.2009.04.034
Malekzadeh, P., Atashi, M.M., & Karami, G. (2009). In-plane free vibration of functionally graded circular arches with temperature-dependent properties under thermal environment. Journal of Sound and Vibration. 326, 837–851.
https://doi.org/10.1016/j.jsv.2009.05.016
Noori, A.R., Aslan, T.A., & Temel, B. (2018). An efficient approach for in-plane free and forced vibrations of axially functionally graded parabolic arches with non-uniform cross section. Composite Structures. 200, 701–710.
https://doi.org/10.1016/j.compstruct.2018.05.077
Qatu, M.S., & Elsharkawy, A.A. (1993). Vibration of laminated composite arches with deep curvature and arbitrary boundaries. Computers & Structures. 47(2), 305–311. https://doi.org/10.1016/0045-7949(93)90381-M
Raki, M., Alipour, R., & Kamanbedast, A. (2012). Thermal buckling of thin rectangular FGM plate. World Applied Science Journal, 16(1), 52–62.
Sitar, M., Kosel, F., & Brojan, M. (2014). Large deflections of nonlinearly elastic functionally graded composite beam. Archives Civil and Mechanical Engineering, 14(4), 700–709. https://doi.org/10.1016/j.acme.2013.11.007
Trinh, T.H., Nguyen, D.K., Gan, B.S., & Alexandrov, A. (2016). Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation. Structural Engineering and Mechanics, 58(3), 515–532.
https://doi.org/10.12989/sem.2016.58.3.515
Yousefi, A., & Rastgoo, A. (2011). Free vibration of functionally graded spatial curved beams. Composite Structures. 93, 3048–3056. https://doi.org/10.1016/j.compstruct.2011.04.024
Zaczynska, M., & Kazmierczyk, F. (2020). Multi-mode buckling analysis of FGM channel section beams. Materials, 13(11), 2567. https://doi.org/10.3390/ma13112567
Zhao, F.Q., Wang, Z.M., & Zhang, R.P. (2012). Post-buckling analysis of FGM beam subjected to non-conservative forces and in-plane thermal loading. Applied Mechanics and Materials. 152/154, 474–479.
https://doi.org/10.4028/www.scientific.net/AMM.152-154.474