Document Type : Article

**Authors**

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

**Abstract**

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.

**Keywords**

**Main Subjects**

1. Basdekas, N.L. and Chi, M. \Response of oddly

stiened circular cylindrical shell", Journal of Sound

and Vibration, 17, pp.187-206 (1971).

2. Zhou, X.P. \Vibration and stability of ring-stiened

thin-walled cylindrical shells conveying

uid", Acta

Mechanica Solida Sinica, 25, pp. 168-176 (2012).

3. Hoppmann, W.H. \Some characteristics of the

exural

vibrations of orthogonally stiened 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 Stiened Cylindrical Shells and

Flat Plates, National Aeronautics and Space Administration

(1965).

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

(2007).

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

(1969).

12. Mindlin, R.D. and Tiersten, H.F. \Eects 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

(2002).

15. Zeighampour, H. and Tadi Beni, Y. \Cylindrical thinshell

model based on modied 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 modied 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 modied 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 modied 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 modied 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 modied couple stress models for buckling analysis

of axially loaded micro-scaled beams", International

Journal of Engineering Science, 49(11), pp. 1268-1280

(2011).

22. Ghayesh, M.H., Farokhi, H., and Amabili, M. \Nonlinear

dynamics of a microscale beam based on the

modied 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 modied

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 modied 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

modied 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 modied couple stress theory", Composite

Structures, 120, pp. 65-78 (2015).

Transactions on Mechanical Engineering (B)

September and October 2018Pages 2598-2615