Free vibration analysis of functionally graded stiffened micro-cylinder based on the modified couple stress theory

Document Type : Article


Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran


In this paper, free vibration of the micro-cylinder made by functionally graded material that is stiffened in circumferential direction, has been investigated based on the modified couple stress and first order shear deformation theories. Modified couple stress theory (MCST) has been used to catch size effects in micro scales. By using first order shear deformation theory and Hamilton principle, general equations of motion and corresponding boundary conditions have been derived. Free vibration of the structure has been investigated by implementing simply supported boundary condition as a common case. The effects of different parameters such as dimensionless length scale parameter, distribution of FGM properties, number of stiffeners, thickness and length on the natural frequencies were calculated and compared with classical continuum theory. Results show that effects of the size are considerable and also using stiffeners lead to increase in natural frequencies which is because of increase in stiffness of the cylinder.


Main Subjects

1. Basdekas, N.L. and Chi, M. Response of oddly
sti ened circular cylindrical shell", Journal of Sound
and Vibration, 17, pp.187-206 (1971).
2. Zhou, X.P. Vibration and stability of ring-sti ened
thin-walled cylindrical shells conveying
uid", Acta
Mechanica Solida Sinica, 25, pp. 168-176 (2012).
3. Hoppmann, W.H. Some characteristics of the
vibrations of orthogonally sti ened cylindrical shells",
2608 S. Jabbarian and M.T. Ahmadian/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2598{2615
The Journal of the Acoustical Society of America,
30(1), pp. 77-82 (1958).
4. Mikulas, M.M. and McElman, J.A., On Free Vibrations
of Eccentrically Sti ened Cylindrical Shells and
Flat Plates, National Aeronautics and Space Administration
5. Sankar, B.V. An elasticity solution for functionally
graded beams", Composites Science and Technology,
61(5), pp. 689-696 (2001).
6. Aydogdu, M. and Taskin,V. Free vibration analysis
of functionally graded beams with simply supported
edges", Materials & Design, 28(5), pp. 1651-1656
7. Ying, J., Lu, C.F., and Chen, W.Q. Two-dimensional
elasticity solutions for functionally graded beams resting
on elastic foundations", Composite Structures,
84(3), pp. 209-219 (2008).
8. Xiang, H.J. and Yang, J. Free and forced vibration
of a laminated FGM Timoshenko beam of variable
thickness under heat conduction", Composites Part B:
Engineering, 39(2), pp. 292-303 (2008).
9. Kapuria, S., Bhattacharyya, M., and Kumar, A.N.
Bending and free vibration response of layered functionally
graded beams: a theoretical model and its experimental
validation", Composite Structures, 82(3),
pp. 390-402 (2008).
10. Prakash, T., Singha, M.K., and Ganapathi, M. Thermal
snapping of functionally graded materials plates",
Materials & Design, 30(10), pp. 4532-4536 (2009).
11. Koiter, W.T. Couple-stresses in the theory of elasticity
I and II", Proc. K Ned Akad Wet B, 67, pp. 17-44
12. Mindlin, R.D. and Tiersten, H.F. E ects of couplestresses
in linear elasticity", Arch. Ration. Mech.
Anal., 11(1), pp. 415-48 (1962).
13. Toupin, R.A. Elastic materials with couple-stresses",
Arch. Ration. Mech. Anal., 11(1), pp. 385-414 (1962).
14. Yang, F., Chong, A.C.M., Lam, D.C.C., and Tong,
P. Couple stress based strain gradient theory for
elasticity", Int. J. Solids Struct., 39(10), pp. 2731-2743
15. Zeighampour, H. and Tadi Beni, Y. Cylindrical thinshell
model based on modi ed strain gradient theory",
International Journal of Engineering Science, 78, pp.
27-47 (2014).
16. Zhou, X., Lin, W., and Peng, Q. Free vibration of
micro-and nano-shells based on modi ed couple stress
theory", Journal of Computational and Theoretical
Nanoscience, 9(6), pp. 814-818 (2012).
17. Hosseini-Hashemi, Sh., Sharifpour, F., and Ilkhani,
M.R. On the free vibrations of size-dependent closed
micro/nano-spherical shell based on the modi ed couple
stress theory", International Journal of Mechanical
Sciences, 115, pp. 501-515 (2016).
18. Park, S.K. and Gao, X.L. Bernoulli-Euler beam
model based on a modi ed couple stress theory",
Micromech Microeng, 16(11), pp. 2355-2359 (2006).
19. Kong, S., Zhou, S., Nie, Z., and Wang, K. The sizedependent
natural frequency of Bernoulli Euler microbeams",
Int. J. Eng. Sci., 46, pp. 427-437 (2008).
20. Simsek, M. Nonlinear static and free vibration analysis
of microbeams based on the nonlinear elastic
foundation using modi ed couple stress theory and
He's variational method", Composite Structures, 112,
pp. 264-272 (2014).
21. Akgoz, B. and Civalek,  O. Strain gradient elasticity
and modi ed couple stress models for buckling analysis
of axially loaded micro-scaled beams", International
Journal of Engineering Science, 49(11), pp. 1268-1280
22. Ghayesh, M.H., Farokhi, H., and Amabili, M. Nonlinear
dynamics of a microscale beam based on the
modi ed couple stress theory", Composites Part B:
Engineering, 50, pp. 318-324 (2013).
23. Wang, Y., Wen-Hui, L., and Ning, L. Nonlinear free
vibration of a microscale beam based on modi ed
couple stress theory", Physica E: Low-dimensional
Systems and Nanostructures, 47, pp. 80-85 (2013).
24. Jomehzadeh, E., Saidi, A.R., and Atashipour, S.R.
An analytical approach for stress analysis of functionally
graded annular sector plates", Materials & design,
30(9), pp. 3679-3685 (2009).
25. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H.,
and Rahaeifard, M. On the size-dependent behavior
of functionally graded micro-beams", Materials &
Design (1980-2015), 31(5), pp. 2324-2329 (2010).
26. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H.,
and Ahmadian, M.T. The modi ed couple stress
functionally graded Timoshenko beam formulation",
Materials & Design, 32(3), pp. 1435-1443 (2011).
27. Sadeghi, H., Baghani, M., and Naghdabadi, R. Strain
gradient elasticity solution for functionally graded
micro-cylinders", International Journal of Engineering
Science, 50(1), pp. 22-30 (2012).
28. Sahmani, S., Ansari, R., Gholami, R., and Darvizeh,
A. Dynamic stability analysis of functionally graded
higher-order shear deformable microshells based on the
modi ed couple stress elasticity theory", Composites
Part B: Engineering, 51, pp. 44-53 (2013).
29. Bedroud, M., Nazemnezhad, R., Hosseini Hashemi,
S., and Valixani, M. Buckling of FG circular/annular
Mindlin nanoplates with an internal ring support using
nonlocal elasticity", Applied Mathematical Modelling,
40(4), pp. 3185-3210 (2016).
30. Beni, Yaghoub Tadi, Mehralian, F. and Razavi, H.
Free vibration analysis of size-dependent shear deformable
functionally graded cylindrical shell on the
basis of modi ed couple stress theory", Composite
Structures, 120, pp. 65-78 (2015).