Nano Kirişlerin Değiştirilmiş Sonlu Elemanlar Taşima Matrisi Yöntemi ile Stabilite Analizi

Abstract

Bu çalşmada değiştirilmiş Sonlu elemanlar–taşıma matrisi yöntemi nano kirişlerin stabilite analizi için uyarlanmıştır. Çalımada önce nano kirişin stabilite denklemi yerel olmayan Euler kiriş teorisi yardımıyla oluşturulmuştur. Diferansiyel denklemin çözümü ile önce elemanın sonlu elamanlar matrisi yazılmış daha sonra yapılan dönüşümle Ricatti taşıma matrisi elde edilmiştir. Çalışmanın sonunda sunulan yöntemin uygunluğunu literatürden alınan bir örnek üzerinde gösterilmiştir. Sunulan yöntemde matris boyutları klasik sonlu elemanlar yöntemine göre kayda değer bir şekilde azalmakta ve dolayısıyla çözüm süreside kısalmaktadır. Sunulan yöntem özellikle çok açıklıklı ve değişken kesitli nano kirişlerin çözümünde kullanılabilir.

Keywords

• Nano kiriş
• Sonlu eleman
• Taşıma matrisi
• Euler kirişi

Stability Analysis of Nanobeams by Modified Finıie Element Transfer Matrix Method

Abstract

In this study, the modified finite element-transfer matrix method has been adapted for the stability analysis of nanobeams. In the study, the stability equation of the nanobeam was first established with the help of the nonlocal Euler beam theory. With the solution of the differential equation, first, the finite element matrix of the element was written and then the Ricatti transfer matrix was obtained by the transformation. The suitability of the presented method is shown on an example taken from the literature. In the presented method, matrix dimensions are significantly reduced compared to the classical finite element method and therefore the solution time is shortened. The presented method can be used especially for the solution of multi-span and variable cross-section nanobeams.

Keywords

• Nanobeam
• Finite element
• Transfer matrix
• Euler beam

References

1. [1]. Glabisz, W., Jarczewska, K., Hołubowski, R., Stability of nanobeams under nonconservative surface loading, Acta Mechanica, 2020, 231(9), 3703-3714.
2. [2]. Shariati, A., Mohammad-Sedighi, H., Żur, K. K., Habibi, M., Safa, M., On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams, Materials, 2020, 13(7), 1707.
3. [3]. Kumar, D., Heinrich, Ch., Waas, A.M., Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories. Journal of Applied Physics, 2008, 103, 073521.
4. [4]. Zhang, Y.Y., Wang, C.M., Challamel, N., Bending, buckling and vibration of micro/nanobeams by hybrid nonlocal beam model, Journal of Engineering Mechanics, 2010, 136(5), 562–574 .
5. [5]. Mohammadi, H., Mahzoon, M., Mohammadi, M., Mohammadi, M., Postbuckling instability of nonlinear nanobeam with geometric imperfection embedded in elastic foundation, Nonlinear Dynamics, 2014, 76, 2005–2016.
6. [6]. Wang, C.M., Zhang, H., Challamel, N., Duan, W.H.: On boundary conditions for buckling and vibration of nonlocal beams. European Journal of Mechanics-A/Solids, 2017, 61, 73–81.
7. [7]. Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of applied physics, 1983, 54(9), 4703-4710.
8. [8]. Eringen, A. C., Wegner, J. L., Nonlocal continuum field theories. Appl. Mech. Rev., 2003, 56(2), B20-B22.

9. [9]. Eringen, A. C., Nonlocal continuum mechanics based on distributions. International Journal of Engineering Science, 2006, 44(3-4), 141-147.
10. [10]. Civalek, Ö., Demir, Ç., Buckling and bending analyses of cantilever carbon nanotubes using the euler-bernoulli beam theory based on non-local continuum model. Asian Journal of Civil Engineering, 2011, 12(5):651–661.
11. [11]. Ebrahimi, F., Shaghaghi, G.R., Salari, E., Vibration analysis of sizedependent nano beams based on nonlocal timoshenko beam theory, Journal of Mechanical Engineering and Technology, 2014, 6(2).
12. [12]. Ebrahimi, F., Nasirzadeh, P., A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method, Journal of Theoretical and Applied Mechanics, 2015, 53(4):1041–1052.
13. [13]. Uzun, B., Numanoglu, H. M., Civalek, O., Free vibration analysis of BNNT with different cross-Sections via nonlocal FEM, Journal of Computational Applied Mechanics, 2018, 49(2):252–260
14. [14]. Uzun, B., Civalek, Ö., Nonlocal FEM Formulation for vibration analysis of nanowires on elastic matrix with different materials, Mathematical and Computational Applications, 2019, 24(2):38
15. [15]. Uzun, B., Civalek, Ö., Free vibration analysis Silicon nanowires surrounded by elastic matrix by nonlocal finite element method, Advances in nano research, 2019, 7(2):99
16. [16]. Numanoğlu, H. M., Akgöz, B., Civalek, Ö., On dynamic analysis of nanorods, International Journal of Engineering Science, 2018, 130:33–50
17. [17]. Yayli, M. Ö., An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach, Journal of Vibration and Control, 2018, 24(11):2230–2246
18. [18]. Murmu, T., Pradhan, S. C., Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity. Journal of Applied Physics, 2009, 106(10):104301
19. [19]. Farajpour, A., Danesh, M., Mohammadi, M., Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-dimensional Systems and Nanostructures, 2011, 44(3): 719–727
20. [20]. Karličić, D., Cajić, M., Adhikari, S., Dynamic stability of a nonlinear multiple-nanobeam system, Nonlinear Dynamics, 2018, 93(3), 1495-1517.
21. [21]. Behdad, S., Fakher, M., Hosseini-Hashemi, S., Dynamic stability and vibration of two-phase local/nonlocal VFGP nanobeams incorporating surface effects and different boundary conditions, Mechanics of Materials, 2021, 153, 103633.
22. [22]. Sourani, P., Hashemian, M., Pirmoradian, M., Toghraie, D., A comparison of the Bolotin and incremental harmonic balance methods in the dynamic stability analysis of an Euler–Bernoulli nanobeam based on the nonlocal strain gradient theory and surface effects, Mechanics of Materials, 2020, 145, 103403.
23. [23]. Hamed, M. A., Mohamed, N. A., Eltaher, M. A., Stability buckling and bending of nanobeams including cutouts, Engineering with Computers, 2020, 1-22.
24. [24]. Arda, M., Aydogdu, M., Dynamic stability of harmonically excited nanobeams including axial inertia, Journal of Vibration and Control, 2019, 25(4), 820-833.
25. [25]. Eltaher, M. A., Khater, M. E., Park, S., Abdel-Rahman, E., Yavuz, M., On the static stability of nonlocal nanobeams using higher-order beam theories, Advances in nano research, 2016, 4(1), 51.
26. [26]. Holzer, H., Analysis of Torsional Vibration, Springer, Berlin, 1921
27. [27]. Dokainish, M. A., A new approach for plate vibrations: combination of transfer matrix and finite-element technique, 1972, 526-530.
28. [28]. Rong, B., Rui, X., Wang, G., Modified finite element transfer matrix method for eigenvalue problem of flexible structures, Journal of applied mechanics, 2011, 78(2).
29. [29]. Rong, B., Rui, X., Tao, L., Perturbation finite element transfer matrix method for random eigenvalue problems of uncertain structures, Journal of applied mechanics, 2012, 79(2).
30. [30]. Ozturk, D., Bozdogan, K. A. N. A. T., Nuhoglu, A., Modified finite element-transfer matrix method for the static analysis of structures, Structural Engineering and Mechanics, 2012, 43(6), 761-769.
31. [31]. Bozdoğan, K. B., Maleki, F. K., An Application of the Modified Finite Element Transfer Matrix Method for a Heat Transfer Problem. Kırklareli Üniversitesi Mühendislik ve Fen Bilimleri Dergisi, 2019, 5(1), 15-28.
32. [32]. Wang, C.M., Zhang, Y.Y., Rames,S.S, ,Kitipornchai,S.,: Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory, Jornal of Phyisics D: Applied Physics,2006,39,3904-3909