Browse > Article
http://dx.doi.org/10.12989/scs.2019.33.1.133

Static stability analysis of axially functionally graded tapered micro columns with different boundary conditions  

Akgoz, Bekir (Department of Civil Engineering, Faculty of Engineering, Akdeniz University)
Publication Information
Steel and Composite Structures / v.33, no.1, 2019 , pp. 133-142 More about this Journal
Abstract
In the present study, microstructure-dependent static stability analysis of inhomogeneous tapered micro-columns is performed. It is considered that the micro column is made of functionally graded materials and has a variable cross-section. The material and geometrical properties of micro column vary continuously throughout the axial direction. Euler-Bernoulli beam and modified couple stress theories are used to model the nonhomogeneous micro column with variable cross section. Rayleigh-Ritz solution method is implemented to obtain the critical buckling loads for various parameters. A detailed parametric study is performed to examine the influences of taper ratio, material gradation, length scale parameter, and boundary conditions. The validity of the present results is demonstrated by comparing them with some related results available in the literature. It can be emphasized that the size-dependency on the critical buckling loads is more prominent for bigger length scale parameter-to-thickness ratio and changes in the material gradation and taper ratio affect significantly the values of critical buckling loads.
Keywords
critical buckling load; size dependency; variable cross section; axially functionally graded materials; Rayleigh-Ritz method;
Citations & Related Records
Times Cited By KSCI : 22  (Citation Analysis)
연도 인용수 순위
1 Ghayesh, M.H. (2019), "Resonant dynamics of axially functionally graded imperfect tapered Timoshenko beams", J. Vib. Control, 25(2), 336-350. https://doi.org/10.1177/1077546318777591   DOI
2 Ghayesh, M.H. and Farokhi, H. (2018a), "Bending and vibration analyses of coupled axially functionally graded tapered beams", Nonlinear Dyn., 91(1), 17-28. https://doi.org/10.1007/s11071-017-3783-8   DOI
3 Ghayesh, M.H. and Farokhi, H. (2018b), "Mechanics of tapered axially functionally graded shallow arches", Compos. Struct., 188, 233-241. https://doi.org/10.1016/j.compstruct.2017.11.017   DOI
4 Ghayesh, M.H., Farokhi, H., Gholipour, A. and Tavallaeinejad, M. (2017), "Nonlinear bending and forced vibrations of axially functionally graded tapered microbeams", Int. J. Eng. Sci., 120, 51-62. https://doi.org/10.1016/j.ijengsci.2017.03.010   DOI
5 Gusso, A., Viana, R.L., Mathias, A.C. and Caldas, I.L. (2019), "Nonlinear dynamics and chaos in micro/ nanoelectromechanical beam resonators actuated by two-sided electrodes", Chaos Soliton Fract., 122, 6-16. https://doi.org/10.1016/j.chaos.2019.03.004   DOI
6 Anandrao, K.S., Gupta, R.K., Ramchandran, P. and Rao, G.V. (2010), "Thermal post-buckling analysis of uniform slender functionally graded material beams", Struct. Eng. Mech., Int. J., 36(5), 545-560. https://doi.org/10.12989/sem.2010.36.5.545   DOI
7 Arefi, M. (2018), "Nonlocal free vibration analysis of a doubly curved piezoelectric nano shell", Steel Compos. Struct., Int. J., 27(4), 479-493. https://doi.org/10.12989/scs.2018.27.4.479
8 Atmane, H.A., Tounsi, A., Ziane, N. and Mechab, I. (2011), "Mathematical solution for free vibration of sigmoid functionally graded beams with varying cross-section", Steel Compos. Struct., Int. J., 11(6), 489-504. https://doi.org/10.12989/scs.2011.11.6.489   DOI
9 Atmane, H.A., Tounsi, A., Bernard, F. and Mahmoud, S.R. (2015), "A computational shear displacement model for vibrational analysis of functionally graded beams with porosities", Steel Compos. Struct., Int. J., 19(2), 369-384. https://doi.org/10.12989/scs.2015.19.2.369   DOI
10 Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., Int. J., 30(6), 603-615. https://doi.org/10.12989/scs.2019.30.6.603
11 Aydogdu, M. (2008), "Semi-inverse method for vibration and buckling of axially functionally graded beams", J. Reinforced Plast. Comp., 27(7), 683-691. https://doi.org/10.1177/0731684407081369   DOI
12 Belkorissat, I., Houari, M.S.A., Tounsi, A., Bedia, E.A.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., Int. J., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063   DOI
13 Hamzehkolaei, N.S., Malekzadeh, P. and Vaseghi, J. (2011), "Thermal effect on axisymmetric bending of functionally graded circular and annular plates using DQM", Steel Compos. Struct., Int. J., 11(4), 341-358. https://doi.org/10.12989/scs.2011.11.4.341   DOI
14 Hadi, A., Nejad, M.Z., Rastgoo, A. and Hosseini, M. (2018), "Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory", Steel Compos. Struct., Int. J., 26(6), 663-672. https://doi.org/10.12989/scs.2018.26.6.663
15 Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235   DOI
16 Ghayesh, M.H. (2018d), "Vibration analysis of shear-deformable AFG imperfect beams", Compos. Struct., 200, 910-920. https://doi.org/10.1016/j.compstruct.2018.03.091   DOI
17 He, X.J., Wu, Q., Wang, Y., Song, M.X. and Yin, J.H. (2009), "Numerical simulation and analysis of electrically actuated microbeam-based MEMS capacitive switch", Microsyst. Technol., 15(2), 301-307. https://doi.org/10.1007/s00542-008-0702-4   DOI
18 Houari, M.S.A., 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., Int. J., 28(1), 13-24. https://doi.org/10.12989/scs.2018.28.1.013
19 Jahangiri, R., Jahangiri, H. and Khezerloo, H. (2015), "FGM micro-gripper under electrostatic and intermolecular Van-der Waals forces using modified couple stress theory", Steel Compos. Struct., Int. J., 18(6), 1541-1555. https://doi.org/10.12989/scs.2015.18.6.1541   DOI
20 Payam, A.F. and Fathipour, M. (2009), "Modeling and dynamic analysis of atomic force microscope based on Euler-Bernoulli beam theory", Dig. J. Nanomater. Bios., 4(3), 565-578.
21 Pradhan, K.K. and Chakraverty, S. (2013), "Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method", Compos. Part B-Eng., 51, 175-184. https://doi.org/10.1016/j.compositesb.2013.02.027   DOI
22 Rahmani, O., Deyhim, S. and Hosseini, S.A.H. (2018a), "Size dependent bending analysis of micro/nano sandwich structures based on a nonlocal high order theory", Steel Compos. Struct., Int. J., 27(3), 371-388. https://doi.org/10.12989/scs.2018.27.3.371
23 Rahmani, O., Hosseini, S.A.H., Ghoytasi, I. and Golmohammadi, H. (2018b), "Free vibration of deep curved FG nano-beam based on modified couple stress theory", Steel Compos. Struct., Int. J., 26(5), 607-620. https://doi.org/10.12989/scs.2018.26.5.607
24 Rezaiee-Pajand, M., Masoodi, A.R. and Alepaighambar, A. (2018), "Lateral-torsional buckling of functionally graded tapered Ibeams considering lateral bracing", Steel Compos. Struct., Int. J., 28(4), 403-414. https://doi.org/10.12989/scs.2018.28.4.403
25 Sahraee, S. and Saidi, A.R. (2009), "Free vibration and buckling analysis of functionally graded deep beam-columns on twoparameter elastic foundations using the differential quadrature method", Proc. Inst. Mech. Eng. C-J. Mech. Eng. Sci., 223(6), 1273-1284. https://doi.org/10.1243/09544062JMES1349   DOI
26 Saidi, H., Houari, M.S.A., Tounsi, A. and Bedia, E.A. (2013), "Thermo-mechanical bending response with stretching effect of functionally graded sandwich plates using a novel shear deformation theory", Steel Compos. Struct., Int. J., 15(2), 221-245. https://doi.org/10.12989/scs.2013.15.2.221   DOI
27 Karami, B., Janghorban, M. and Tounsi, A. (2018), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., Int. J., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201
28 Bennai, R., Atmane, H.A. and Tounsi, A. (2015), "A new higherorder shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., Int. J., 19(3), 521-546. https://doi.org/10.12989/scs.2015.19.3.521   DOI
29 Boeing (2019), https://www.boeing.com/commercial/787/bydesign/#/advanced-composite-use
30 Jia, X.L., Ke, L.L., Zhong, X.L., Sun, Y., Yang, J. and Kitipornchai, S. (2018), "Thermal-mechanical-electrical buckling behavior of functionally graded micro-beams based on modified couple stress theory", Compos. Struct., 202, 625-634. https://doi.org/10.1016/j.compstruct.2018.03.025   DOI
31 Khaniki, H.B. and Rajasekaran, S. (2018), "Mechanical analysis of non-uniform bi-directional functionally graded intelligent microbeams using modified couple stress theory", Mater. Res. Express., 5(5), 055703. https://doi.org/10.1088/2053-1591/aabe62   DOI
32 Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803   DOI
33 Salamat-Talab, M., Nateghi, A. and Torabi, J. (2012), "Static and dynamic analysis of third-order shear deformation FG micro beam based on modified couple stress theory", Int. J. Mech. Sci., 57(1), 63-73. https://doi.org/10.1016/j.ijmecsci.2012.02.004   DOI
34 Ebrahimi, F. and Mahmoodi, F. (2019), "A modified couple stress theory for buckling analysis of higher order inhomogeneous microbeams with porosities", Proc. Inst. Mech. Eng. C-J. Mech. Eng. Sci., 233(8), 2855-2866. https://doi.org/10.1177/0954406218791642   DOI
35 Ehyaei, J. and Akbarizadeh, M.R. (2017), "Vibration analysis of micro composite thin beam based on modified couple stress", Struct. Eng. Mech., Int. J., 64(4), 403-411.
36 Eringen, A.C. (1967), "Theory of micropolarplates", Z. Angew. Math. Phys., 18, 12-30. https://doi.org/10.1007/BF01593891   DOI
37 Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5   DOI
38 Fleck, N.A. and Hutchinson, J.W. (1993), "A phenomenological theory for strain gradient effects in plasticity", J. Mech. Phys. Solids, 41, 1825-1857. https://doi.org/10.1016/0022-5096(93)90072-N   DOI
39 Fleck, N.A. and Hutchinson, J.W. (2001), "A reformulation of strain gradient plasticity", J. Mech. Phys. Solids, 49, 2245-2271. https://doi.org/10.1016/S0022-5096(01)00049-7   DOI
40 Ghayesh, M.H. (2018a), "Mechanics of tapered AFG sheardeformable microbeams", Microsyst. Technol., 24(4), 1743-1754. https://doi.org/10.1007/s00542-018-3764-y   DOI
41 Ghayesh, M.H. (2018b), "Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams", Appl. Math. Model., 59, 583-596. https://doi.org/10.1016/j.apm.2018.02.017   DOI
42 Koiter, W.T. (1964), "Couple stresses in the theory of elasticity", I and II. Proc. K. Ned. Akad. Wet. B, 67, 17-44.
43 Khorshidi, M.A., Shariati, M. and Emam, S.A. (2016), "Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory", Int. J. Mech. Sci., 110, 160-169. https://doi.org/10.1016/j.ijmecsci.2016.03.006   DOI
44 Kiani, Y. and Eslami, M.R. (2010), "Thermal buckling analysis of functionally graded material beams", Int. J. Mech. Mater. Des., 6(3), 229-238. https://doi.org/10.1007/s10999-010-9132-4   DOI
45 Kocaturk, T. and Akbas, S.D. (2013), "Thermal post-buckling analysis of functionally graded beams with temperaturedependent physical properties", Steel Compos. Struct., Int. J., 15(5), 481-505. https://doi.org/10.12989/scs.2013.15.5.481   DOI
46 Shafiei, N., Kazemi, M. and Ghadiri, M. (2016b), "Nonlinear vibration of axially functionally graded tapered microbeams", Int. J. Eng. Sci., 102, 12-26. https://doi.org/10.1016/j.ijengsci.2016.02.007   DOI
47 Shafiei, N. and Kazemi, M. (2017), "Buckling analysis on the bidimensional functionally graded porous tapered nano-/microscale beams", Aerosp. Sci. Technol., 66, 1-11. https://doi.org/10.1016/j.ast.2017.02.019   DOI
48 Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", Int. J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004   DOI
49 Shafiei, N., Kazemi, M. and Ghadiri, M. (2016a), "Comparison of modeling of the rotating tapered axially functionally graded Timoshenko and Euler-Bernoulli microbeams", Physica E, 83, 74-87. https://doi.org/10.1016/j.physe.2016.04.011   DOI
50 Shafiei, N., Mousavi, A. and Ghadiri, M. (2016c), "On sizedependent nonlinear vibration of porous and imperfect functionally graded tapered microbeams", Int. J. Eng. Sci., 106, 42-56. https://doi.org/10.1016/j.ijengsci.2016.05.007   DOI
51 Lei, J., He, Y.M., Guo, S., Li, Z.K. and Liu, D.B. (2016), "Sizedependent vibration of nickel cantilever microbeams: Experiment and gradient elasticity", Aip. Adv., 6(10), 105202. https://doi.org/10.1063/1.4964660   DOI
52 Ghayesh, M.H. (2018c), "Nonlinear vibrations of axially functionally graded Timoshenko tapered beams", J. Comput. Nonlinear Dyn., 13(4), 041002. https://doi.org/10.1115/1.4039191   DOI
53 Kong, S., Zhou, S., Nie, Z. and Wang, K. (2009), "Static and dynamic analysis of micro beams based on strain gradient elasticity theory", Int. J. Eng. Sci., 47, 487-498. https://doi.org/10.1016/j.ijengsci.2008.08.008   DOI
54 Lam, D.C.C. and Chong, A.C.M. (2001), "Model and experiments on strain gradient hardening in metallic glass", Mat. Sci. Eng. AStruct., 318(1-2), 313-319. https://doi.org/10.1016/S0921-5093(01)01329-6   DOI
55 Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X   DOI
56 Lee, H.L. and Chang, W.J. (2009), "Effects of damping on the vibration frequency of atomic force microscope cantilevers using the Timoshenko beam model", Jpn. J. Appl. Phys., 48(6), 065005. https://doi.org/10.1143/JJAP.48.065005   DOI
57 Li, B., Tang, X.S., Xie, H.M. and Xin, Z. (2004), "Strain analysis in MEMS/NEMS structures and devices by using focused ion beam system", Sensor Actuator A-Phys., 111(1), 57-62. https://doi.org/10.1016/j.sna.2003.07.014   DOI
58 Shafiei, N., Kazemi, M. and Fatahi, L. (2017), "Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method", Mech. Adv. Mater. Struct., 24(3), 240-252. https://doi.org/10.1080/15376494.2015.1128025   DOI
59 Shafiei, N., Ghadiri, M. and Mahinzare, M. (2019), "Flapwise bending vibration analysis of rotary tapered functionally graded nanobeam in thermal environment", Mech. Adv. Mater. Struct., 26(2), 139-155. https://doi.org/10.1080/15376494.2017.1365982   DOI
60 She, G.L., Yuan, F.G. and Ren, Y.R. (2017), "Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory", Appl. Math. Model., 47, 340-357. https://doi.org/10.1016/j.apm.2017.03.014   DOI
61 Li, S.R., Zhang, J.H. and Zhao, Y.G. (2006), "Thermal postbuckling of functionally graded material Timoshenko beams", Appl. Math. Mech., 27(6), 803-810. https://doi.org/10.1007/s10483-006-0611-y   DOI
62 Ma, H.M., Gao, X.L. and Reddy, J.N. (2010), "A nonclassical Reddy-Levinson beam model based on a modified couple stress theory", Int. J. Multiscale Comput. Eng., 8(2), 167-180. https://doi.org/10.1615/IntJMultCompEng.v8.i2.30   DOI
63 She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133, 368. https://doi.org/10.1140/epjp/i2018-12196-5   DOI
64 Simsek, M. (2011), "Forced vibration of an embedded singlewalled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory", Steel Compos. Struct., Int. J., 11(1), 59-76. https://doi.org/10.12989/scs.2011.11.1.059   DOI
65 Li, Z.K., He, Y.M., Lei, J., Guo, S., Liu, D.B. and Wang, L. (2018), "A standard experimental method for determining the material length scale based on modified couple stress theory", Int. J. Mech. Sci., 141, 198-205. https://doi.org/10.1016/j.ijmecsci.2018.03.035   DOI
66 Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001   DOI
67 Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructuredependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solids., 56(12), 3379-3391. https://doi.org/10.1016/j.jmps.2008.09.007   DOI
68 Mahmoud, S.R. and Tounsi, A. (2017), "A new shear deformation plate theory with stretching effect for buckling analysis of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 24(5), 569-578. https://doi.org/10.12989/scs.2017.24.5.569
69 Marques, L., da Silva, L.S. and Rebelo, C. (2014), "Rayleigh-Ritz procedure for determination of the critical load of tapered columns", Steel Compos. Struct., Int. J., 16(1), 47-60. https://doi.org/10.12989/scs.2014.16.1.047
70 Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Comp. Mater. Sci., 61, 257-265. https://doi.org/10.1016/j.commatsci.2012.04.001   DOI
71 Simsek, M. and Reddy, J.N. (2013), "A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory", Compos. Struct., 101, 47-58. https://doi.org/10.1016/j.compstruct.2013.01.017   DOI
72 Tagrara, S.H., Benachour, A., Bouiadjra, M.B. and Tounsi, A. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., Int. J., 19(5), 1259-1277. https://doi.org/10.12989/scs.2015.19.5.1259   DOI
73 Thanh, C.L., Tran, L.V., Vu-Huu, T. and Abdel-Wahab, M. (2019), "The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis", Comput. Method. Appl. Mech., 350, 337-361. https://doi.org/10.1016/j.cma.2019.02.028   DOI
74 Toupin, R.A. (1964), "Theories of elasticity with couple-stresses", Arch. Ration. Mech. Anal., 17, 85-112. https://doi.org/10.1007/BF00253050   DOI
75 Vardoulakis, I. and Sulem, J. (1995), Bifurcation Analysis in Geomechanics, CRC Press.
76 Wang, C.M., Wang, C.Y. and Reddy, J.N. (2005), Exact Solutions for Buckling of Structural Members, CRC Press.
77 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., Int. J., 29(3), 363-377. https://doi.org/10.12989/scs.2018.29.3.363
78 Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couplestresses in linear elasticity", Arch. Ration. Mech. Anal., 11, 415-448. https://doi.org/10.1007/BF00253946   DOI
79 Nateghi, A., Salamat-talab, M., Rezapour, J. and Daneshian, B. (2012), "Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory", Appl. Math. Model., 36(10), 4971-4987. https://doi.org/10.1016/j.apm.2011.12.035   DOI
80 Nazemnezhad, R. and Kamali, K. (2018), "Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory", Steel Compos. Struct., Int. J., 28(6), 749-758. https://doi.org/10.12989/scs.2018.28.6.749
81 Nguyen, N.T., Kim, N.I. and Lee, J. (2014), "Analytical solutions for bending of transversely or axially FG nonlocal beams", Steel Compos. Struct., Int. J., 17(5), 639-663. https://doi.org/10.12989/scs.2014.17.5.641
82 Nguyen, K.T., Thai, T.H. and Vo, T.P. (2015), "A refined higherorder shear deformation theory for bending, vibration and buckling analysis of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 91-120. https://doi.org/10.12989/scs.2015.18.1.091   DOI
83 Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng., 16(11), 2355-2359. https://doi.org/10.1088/0960-1317/16/11/015   DOI
84 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., Int. J., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239   DOI
85 Akgoz, B. and Civalek, O. (2013a), "Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity", Struct. Eng. Mech., Int. J., 48(2), 195-205. https://doi.org/10.12989/sem.2013.48.2.195   DOI
86 Aifantis, E.C. (1999), "Gradient deformation models at nano, micro, and macroscales", J. Eng. Mater. Technol., 121, 189-202. https://doi.org/10.1115/1.2812366   DOI
87 Ak, C., Yildiz, A., Akdagli, A. and Bicer, M.B. (2017), "Computing the pull-in voltage of fixed-fixed micro-actuators by artificial neural network", Microsyst. Technol., 23(8), 3537-3546. https://doi.org/10.1007/s00542-016-3128-4   DOI
88 Akgoz, B. and Civalek, O. (2011), "Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories", J. Comput. Theor. Nanostruct., 8(9), 1821-1827. https://doi.org/10.1166/jctn.2011.1888   DOI
89 Akgoz, B. and Civalek, O. (2013b), "Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory", Compos. Struct., 98, 314-322. https://doi.org/10.1016/j.compstruct.2012.11.020   DOI
90 Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070   DOI
91 Amar, L.H.H., Kaci, A., Yeghnem, R. and Tounsi, A. (2018), "A new four-unknown refined theory based on modified couple stress theory for size-dependent bending and vibration analysis of functionally graded micro-plate", Steel Compos. Struct., Int. J., 26(1), 89-102. https://doi.org/10.12989/scs.2018.26.1.089
92 Wattanasakulpong, N., Prusty, B.G. and Kelly, D.W. (2011), "Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams", Int. J. Mech. Sci., 53(9), 734-743. https://doi.org/10.1016/j.ijmecsci.2011.06.005   DOI
93 Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X   DOI
94 Younis, M.I., Abdel-Rahman, E.M. and Nayfeh, A. (2003), "A reduced-order model for electrically actuated microbeam-based MEMS", J. Microelectromech. Sci, 12(5), 672-680. https://doi.org/10.1109/JMEMS.2003.818069   DOI