Acknowledgement
The research described in this paper, supported by the National Natural Science Foundation of China (No. 41877251).
References
- Ahmed Houari, M.S., Bessaim, A., Bernard, F., Tounsi, A. and Mahmoud, S.R. (2018), "Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter", Steel Compos. Struct., 28(1), 13-24. https://doi.org/10.12989/scs.2018.28.1.013.
- Affdl Halpin, J.C. and Kardos, J.L. (1976), "The Halpin-Tsai equations: A review", Polym. Eng. Sci., 16(5), 344-352. https://doi.org/10.1002/pen.760160512.
- Arefi, M. (2015), "Elastic solution of a curved beam made of functionally graded materials with different cross sections', Steel Compos. Struct., 18(3), 659-672. https://doi.org/10.12989/scs.2015.18.3.659.
- Arioui, O., Belakhdar, K., Kaci, A. and Tounsi, A. (2018), "Thermal buckling of FGM beams having parabolic thickness variation and temperature dependent materials", Steel Compos. Struct., 27(6), 777-788. https://doi.org/10.12989/scs.2018.27.6.777.
- Bambaeechee, M. (2019), "Free vibration of AFG beams with elastic end restraints", Steel Compos. Struct., 33(3), 403-432. https://doi.org/10.12989/scs.2019.33.3.403.
- Barka, M., Benrahou, K.H., Bakora, A. and Tounsi, A. (2016), "Thermal post-buckling behavior of imperfect temperaturedependent sandwich FGM plates resting on Pasternak elastic foundation", Steel Compos. Struct., 22(1), 91-112. https://doi.org/10.12989/scs.2016.22.1.091.
- Bouguenina, O., Belakhdar, K., Tounsi, A. and Bedia, E.A.A. (2015), "Numerical analysis of FGM plates with variable thickness subjected to thermal buckling", Steel Compos. Struct., 19(3), 679-695. https://doi.org/10.12989/scs.2015.19.3.679.
- Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "A new higherorder shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., 19(3), 521-546. https://doi.org/10.12989/scs.2015.19.3.521.
- Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: a review", Appl. Mech. Rev., 49, 1-27. https://doi.org/10.1115/1.3101882.
- Bouchafa, A., Bouiadjra, M.B., Houari, M.S.A. and Tounsi, A. (2015), "Thermal stresses and deflections of functionally graded sandwich plates using a new refined hyperbolic shear deformation theory", Steel Compos. Struct., 18(6), 1493-1515. https://doi.org/10.12989/scs.2015.18.6.1493.
- Celep, Z. (1980), "Stability of a beam on an elastic foundation subjected to a nonconservative load", J. Appl. Mech., 47(1), 116-120. https://doi.org/10.1115/1.3153587.
- Chen, C.S., Liu, F.H. and Chen, W.R. (2017), "vibration and stability of initially stressed sandwich plates with FGM face sheets in thermal environments", Steel Compos. Struct., 23(3), 251-261. https://doi.org/10.12989/scs.2017.23.3.251.
- Du, H., Liew, K.M. and Lim, M.K. (1996), "Generalized differential quadrature method for buckling analysis", J. Eng. Mech., 122(2), 95-100. https://doi.org/10.1061/(ASCE)0733-9399(1996)122:2.
- Ebrahimi, S., Zahrai, S.M. and Mirghaderi, S.R. (2019), "Numerical study on force transfer mechanism in through gusset plates of SCBFs with HSS columns & beams", Steel Compos. Struct., 31(6), 541-558. https://doi.org/10.12989/scs.2019.31.6.541.
- Finot, M. and Suresh, S. (1996), "Small and large deformation of thick and thin-film multilayers: effect of layer geometry, plasticity and compositional gradients", J. Mech. Phys. Solids, 44(5), 683-721. https://doi.org/10.1016/0022-5096(96)84548-0.
- Hadji, L., Daouadji, T.H., Tounsi, A. and Bedia, E.A. (2014), "A higher order shear deformation theory for static and free vibration of FGM beam", Steel Compos. Struct., 16(5), 507-519. https://doi.org/10.12989/scs.2014.16.5.507.
- Halpin, J.C. and Tsai, S.W. (1969), "Effects of environmental factors on composite materials", AFML-TR-67-423.
- Hauger, W. and Vetter, K. (1976), "Influence of an elastic foundation on the stability of a tangentially loaded column", J. Sound Vib., 47(2), 296-299. https://doi.org/10.1016/10.1016/0022-460x(76)90726-4.
- Karami, G., Malekzadeh, P. and Shahpari, S. (2003), "A DQEM for vibration of deformable non-uniform beams with general boundary conditions", Eng. Struct., 25, 1169-1178. https://doi.org/10.1016/S0141-0296(03)00065-8.
- Kitipornchai, S., Chen, D. and Yang, J. (2017), "Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets", Mater. Design, 116, 656-665. https://doi.org/10.1016/j.matdes.2016.12.061.
- Koizumi, M. (1993), "The concept of FGM", Ceram. Trans. Funct. Grad. Mater., 34, 3-10.
- Lee, S.Y., Yang, C.C. (1994), "Nonconservative instability of nonuniform beams resting on an elastic foundation", J. Sound Vib., 169, 433-444. https://doi.org/10.1006/jsvi.1994.1027.
- Lai, B., Richard, J.Y. and Xiong, M. (2019), "Experimental and analytical investigation of composite columns made of high strength steel and high strength concrete", Steel Compos. Struct., 33(1), 67-79. https://doi.org/10.12989/scs.2019.33.1.067.
- Leissa, A.W., McGee, O.G. and Huang, C.S. (1993), "Vibrations of sectorial plates having corner stress singularities", J. Appl. Mech. Transactions of the ASME, 60(1), 134-140. https://doi.org/10.1115/1.2900735.
- Liu, R. and Wang, L. (2015), "Thermal vibration of a singlewalled carbon nanotube predicted by semiquantum molecular dynamics", Physical Chemistry Chemical Physics, 7. https://doi.org/10.1039/C4CP05495D.
- Li, X., Zhou, X., Liu J. and Wang, X. (2019), "Shear behavior of short square tubed steel reinforced concrete columns with highstrength concrete", Steel Compos. Struct., 32(3), 411-422. https://doi.org/10.12989/scs.2019.32.3.411.
- Mahmoud, A.A., Awadalla, R. and Nassar, N.M. (2011), "Free vibration of non-uniform column using DQM", Mech. Res. Commun., 38, 443-448. https://doi.org/10.1016/j.mechrescom.2011.05.015.
- Marin, M. (2010), "Lagrange identity method for microstretch thermoelastic materials", J. Math. Anal. Appl., 363(1), 275-286'. https://doi.org/10.1016/j.jmaa.2009.08.045.
- Marin, M. (2010), "Some estimates on vibrations in thermoelasticity of dipolar bodies", J. Vib. Control, 16(1), 33-47. https://doi.org/10.1177/1077546309103419.
- Marin, M., Agarwal, R.P. and Mahmoud, S.R. (2013), "Nonsimple material problems addressed by the Lagrange's identity", Bound. Value Probl, 2013(1-14). https://doi.org/10.1186/1687-2770-2013-135.
- Marin, M. and Nicaise, S. (2016), "Existence and stability results for thermoelastic dipolar bodies with double porosity", Continuum Mech. Thermodyn., 28(6), 1645-1657. https://doi.org/10.1007/s00161-016-0503-4.
- Marin, M., Ellahi, R. and Adina, C. (2017), "On solutions of Saint-Venant's problem for elastic dipolar bodies with voids", Carpathian J. Math., 33(2), 219-232. https://doi.org/10.37193/CJM.2017.02.09
- Marin, M., Craciun, E.M. and Pop, N. (2016), "Considerations on mixed initial boundary value problems for micropolar porous bodies", Dyn. Syst. Appl., 25(1), 175-195.
- Martone, A., Faiella, G., Antonucci, V., Giordano, M. and Zarrelli, M. (2011), "The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix", Compos. Sci. Technol., 71(8), 1117-1123. https://doi.org/10.1016/j.compscitech.2011.04.002.
- Mirjavadi, S.S., Afshari, B.M., Shafiei, N., Hamouda, A.M.S. and Kazemi, M. (2017), "Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams", Steel Compos. Struct., 25(4), 415-426. https://doi.org/10.12989/scs.2017.25.4.415.
- Montazeri, A., Javadpour, J., Khavandi, A., Tcharkhtchi, A. and Mohajeri, A. (2010), "Mechanical properties of multi-walled carbon nanotube/epoxy composites", Mater. Des., 31, 4202-4208. https://doi.org/10.1016/j.matdes.2010.04.018.
- Moradi-Dastjerdi, R. and Momeni-Khabisi, H. (2016), "Dynamic analysis of functionally graded nanocomposite plates reinforced by wavy carbon nanotube", Steel Compos. Struct., 22(2). https://doi.org/10.12989/scs.2016.22.2.277.
- Moradi-Dastjerdi, R., Foroutan, M. and Pourasghar, A. (2013), "Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method", Mater. Des., 44, 256-266. https://doi.org/10.1016/j.matdes.2012.07.069
- Nguyen, D.K. and Tran, T.T. (2018), "Free vibration of tapered BFGM beams using an efficient shear deformable finite element model", Steel Compos. Struct., 29(3), 363-377. https://doi.org/10.12989/scs.2018.29.3.363.
- Nguyen, X.H., Le, D.D. and Nguyen, Q.H. (2019), "Static behavior of novel RCS through-column-type joint: Experimental and numerical study", Steel Compos. Struct., 32(1), 111-126. https://doi.org/10.12989/scs.2019.32.1.111.
- Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239.
- Pelletier Jacob, L. and Vel Senthil,S. (2006), "An exact solution for the steady state thermo elastic response of functionally graded orthotropic cylindrical shells", Int. J. Solid Struct., 43, 1131-1158. https://doi.org/10.1016/j.ijsolstr.2005.03.079.
- Quan, J.R. and Chan, C.T. (1989), "New insights in solving distributed system equation by the quadrature methods", Comput. Chem. Eng., 13, 779-788. https://doi.org/10.1016/0098-1354(89)85051-3.
- Sharma, A., Sharda, H.B. and Nath, Y. (2005a), "Stability and vibration of Mindlin sector plates: an analytical approach", AIAA J., 43(5), 1109-1116. https://doi.org/10.2514/1.4683.
- Sharma, A., Sharda, H.B. and Nath, Y. (2005b), "Stability and vibration of thick laminated composite sector plates", J. Sound Vib., 287(1-2), 1-23. https://doi.org/10.1016/j.jsv.2004.10.030.
- Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065.
- Shen H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026
- Shen, H.S. and Zhang, C.L. (2010), "Thermal buckling and postbuckling behavior of functionally graded carbon nanotubereinforced composite plates", Mater. Des., 31(7), 3403-3411. https://doi.org/10.1016/j.matdes.2010.01.048
- Shu, C. and Du, H. (1997a), "Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates", Int. J. Solids. Struct., 34, 819-835. https://doi.org/10.1016/S0020-7683(96)00057-1.
- Shu, C. and Du, H. (1997b), "A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates", Int. J. Solids. Struct., 34, 837-846. https://doi.org/10.1016/S0020-7683(96)00056-X.
- Shu, C. (2000), "Differential Quadrature and Its Application in Engineering", Springer, Berlin.
- Smith, T.E. and Herrmann, G. (1972), "Stability of a beam on an elastic foundation subjected to a follower force", J. Appl. Mech., 39, 628-629. https://doi.org/10.1115/1.3422743.
- Song, Y., Uy, B. and Wang, J. (2019), "Numerical analysis of stainless steel-concrete composite beam-to-column joints with bolted flush endplates", Steel Compos. Struct., 33(1), 143-162. https://doi.org/10.12989/scs.2019.33.1.143.
- Sundararajan, C. (1974), "Stability of columns on elastic foundations subjected to conservative or nonconservative forces", J. Sound Vib., 37(1), 79-85. https://doi.org/10.1016/S0022-460X(74)80059-3.
- Tahouneh, V. (2016), "Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates", Steel Compos. Struct., 20(3), 623-649. https://doi.org/10.12989/scs.2016.20.3.623.
- Tahouneh, V. (2017), "The effect of carbon nanotubes agglomeration on vibrational response of thick functionally graded sandwich plates", Steel Compos. Struct., 24(6), 711-726. https://doi.org/10.12989/scs.2017.24.6.711.
- Tornabene, F., Bacciocchi, M., Fantuzzi, N. and Reddy, J.N. (2018), "Multiscale Approach for Three-Phase CNT/Polymer/Fiber Laminated Nanocomposite Structures", Polymer Composites, In Press, DOI: 10.1002/pc.24520.
- Tornabene, F., Fantuzzi, N., Ubertini, F. and Viola, E. (2015), "Strong formulation finite element method based on differential quadrature: A survey", Appl. Mech. Rev., 67(2), 1-55. https://doi.org/10.1115/1.4028859.
- Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2019), "Refined shear deformation theories for laminated composite arches and beams with variable thickness: Natural frequency analysis", Eng. Anal. Bound. Elem., 100, 24-47. https://doi.org/10.1016/j.enganabound.2017.07.029.
- Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2017), "Foam core composite sandwich plates and shells with variable stiffness: Effect of the curvilinear fiber path on the modal response", J. Sandw. Struct. Mater., 21(1), 320-365. https://doi.org/10.1177/1099636217693623.
- Wagner, H.D., Lourie, O. and Feldman, Y. (1997), "Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix", Appl. Phys. Lett., 72(2), 188-190. https://doi.org/10.1063/1.120680.
- Wang, X. and Bert, C.W. (1993), "A new approach in applying differential quadrature to static and free vibrational analysis of beam and plates", J. Sound Vib., 162(3), 566-572. https://doi.org/10.1006/jsvi.1993.1143.
- Wang, J. and Sun, Q. (2019), "Seismic behavior of Q690 circular HCFTST columns under constant axial loading and reversed cyclic lateral loading", Steel Compos. Struct., 32(2), 199-212. https://doi.org/10.12989/scs.2019.32.2.199.
- Wattanasakulpong, and Ungbhakorn, V. (2013), "Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation", Comput. Mater. Sci., 71 201-208. https://doi.org/10.1016/j.commatsci.2013.01.028
- Wu, C.P. and Liu, Y.C. (2016), "A state space meshless method for the 3D analysis of FGM axisymmetric circular plates", Steel Compos. Struct., 22(1), 161-182. https://doi.org/10.12989/scs.2016.22.1.161.
- Xu, W., Wang, L. and Jiang, J. (2016), "Strain gradient finite element analysis on the vibration of double-layered graphene sheets", Int. J. Comput. Method., 13(3). https://doi.org/10.1142/S0219876216500110.
- Yaghoobi, H., Valipour, M.S., Fereidoon, A. and Khoshnevisrad, P. (2014), "Analytical study on post-buckling and nonlinear free vibration analysis of FG beams resting on nonlinear elastic foundation under thermo-mechanical loadings using VIM", Steel Compos. Struct., 17(5), 753-776. https://doi.org/10.12989/scs.2014.17.5.753.
- Yas, M. and Samadi, N. (2012), "Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation", Int. J. Pressure Vessel. Piping, 98, 119-128. https://doi.org/10.1016/j.ijpvp.2012.07.012
- Yeh, M.K., Tai, N.H. and Liu, J.H. (2006), "Mechanical behavior of phenolic-based composites reinforced with multi-walled carbon nanotubes", Carbon, 44(1), 1-9. https://doi.org/10.1016/j.carbon.2005.07.005.
- Yusheng, F. and Bert, C.W. (1992), "Application of quadrature method to flexural vibration analysis of a geometrically nonlinear beam", J. Nonlinear Dynam., 3, 13-18. https://doi.org/10.1007/BF00045468
- Zhang, Y. and Wang, L. (2018), "Thermally stimulated nonlinear vibration of rectangular single-layered black phosphorus", J. Appl. Phys., 124(13), 10.1063/1.5047584. https://doi.org/10.1063/1.5047584.
- Zhu, X.H. and Meng, Z.Y. (1995), "Operational principle fabrication and displacement characteristics of a functionally gradient piezoelectricceramic actuator", Sens. Actuators, 48(3), 169-176. https://doi.org/10.1016/0924-4247(95)00996-5.
Cited by
- Free vibration analysis of open-cell FG porous beams: analytical, numerical and ANN approaches vol.40, pp.2, 2021, https://doi.org/10.12989/scs.2021.40.2.157