Buckling Analysis of FG Timoshenko Beam Based on Physical Neutral Surface Position

Abstract

The paper investigates the buckling analysis of functionally graded Timoshenko beams with different boundary conditions. The study focuses on the static and dynamic behavior of FG beams and plates. The critical buckling loads are investigated concerning the slenderness ratio, power law index, and boundary conditions. The method involves defining the effective properties of the FG beam using the rule of mixture, assuming the reference surface is the physical neutral surface. Numerical results are presented, showing the variation of the non-dimensional neutral surface position with the power law index for different material property ratios. The study concludes by discussing the influence of the slenderness ratio, power law index, and boundary conditions on the critical buckling load of FG Timoshenko beams.

Keywords

• Buckling analysis
• Critical load
• Functionally graded materials
• Physical neutral surface position
• Ritz method
• Timoshenko beam

References

1. [1] A. M. Zenkour, “A comprehensive analysis of functionally graded sandwich plates: Part 2-Buckling and free vibration,” Int J Solids Struct, vol. 42, no. 18–19, pp. 5243–5258, Sep. 2005, doi: 10.1016/j.ijsolstr.2005.02.016.
2. [2] Q. Li, V. P. Iu, and K. P. Kou, “Three-dimensional vibration analysis of functionally graded material sandwich plates,” J Sound Vib, vol. 311, no. 1–2, pp. 498–515, Mar. 2008, doi: 10.1016/j.jsv.2007.09.018.
3. [3] J. N. Reddy, Analysis of functionally graded plates, 2000.
4. [4] T. P. Vo, H. T. Thai, T. K. Nguyen, F. Inam, and J. Lee, “A quasi-3D theory for vibration and buckling of functionally graded sandwich beams,” Compos Struct, vol. 119, pp. 1–12, Jan. 2015, doi: 10.1016/j.compstruct.2014.08.006.
5. [5] H. T. Thai and D. H. Choi, “Finite element formulation of various four unknown shear deformation theories for functionally graded plates,” Finite Elements in Analysis and Design, vol. 75, pp. 50–61, 2013, doi: 10.1016/j.finel.2013.07.003.
6. [6] V. Taskin and P. A. Demirhan, “Static analysis of simply supported porous sandwich plates,” Structural Engineering and Mechanics, vol. 77, no. 4, pp. 549–557, Feb. 2021, doi: 10.12989/sem.2021.77.4.549.
7. [7] P. A. Demirhan and V. Taskin, “Bending and free vibration analysis of Levy-type porous functionally graded plate using state space approach,” Compos B Eng, vol. 160, pp. 661–676, Mar. 2019, doi: 10.1016/j.compositesb.2018.12.020.
8. [8] T. P. Vo, H. T. Thai, T. K. Nguyen, A. Maheri, and J. Lee, “Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory,” Eng Struct, vol. 64, pp. 12–22, Apr. 2014, doi: 10.1016/j.engstruct.2014.01.029.

9. [9] C. P. Sreeju Nair S B, “Functionally Graded Panels: A Review,” International Journal for Modern Trends in Science and Technology, no. 8, pp. 36–43, Aug. 2020, doi: 10.46501/ijmtst060808.
10. [10] C. Pany and G. V. Rao, “Calculation of non-linear fundamental frequency of a cantilever beam using non-linear stiffness,” Journal of Sound and Vibration, vol. 256, no. 4. Academic Press, pp. 787–790, Sep. 26, 2002. doi: 10.1006/jsvi.2001.4224.
11. [11] C. Pany and G. V. Rao, “Large amplitude free vibrations of a uniform spring-hinged beam,” J Sound Vib, vol. 271, no. 3–5, pp. 1163–1169, Apr. 2004, doi: 10.1016/S0022-460X(03)00572-8.
12. [12] C. Pany, “Large amplitude free vibrations analysis of prismatic and non-prismatic different tapered cantilever beams,” Pamukkale University Journal of Engineering Sciences, vol. 29, no. 4, pp. 370–376, 2023, doi: 10.5505/pajes.2022.02489.
13. [13] S. R. Li and R. C. Batra, “Relations between buckling loads of functionally graded timoshenko and homogeneous euler-bernoulli beams,” Compos Struct, vol. 95, pp. 5–9, Jan. 2013, doi: 10.1016/j.compstruct.2012.07.027.
14. [14] S. R. Li, D. F. Cao, and Z. Q. Wan, “Bending solutions of FGM Timoshenko beams from those of the homogenous Euler-Bernoulli beams,” Appl Math Model, vol. 37, no. 10–11, pp. 7077–7085, Jun. 2013, doi: 10.1016/j.apm.2013.02.047.
15. [15] V. L. Nguyen, M. T. Tran, V. L. Nguyen, and Q. H. Le, “Static behaviour of functionally graded plates resting on elastic foundations using neutral surface concept,” Archive of Mechanical Engineering, vol. 68, no. 1, pp. 5–22, 2021, doi: 10.24425/ame.2020.131706.
16. [16] T. K. Nguyen, T. P. Vo, and H. T. Thai, “Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory,” Compos B Eng, vol. 55, pp. 147–157, 2013, doi: 10.1016/j.compositesb.2013.06.011.
17. [17] S. Li, X. Wang, and Z. Wan, “Classical and homogenized expressions for buckling solutions of functionally graded material Levinson beams,” Acta Mechanica Solida Sinica, vol. 28, no. 5, pp. 592–604, Oct. 2015, doi: 10.1016/S0894-9166(15)30052-5.
18. [18] T. Morimoto, Y. Tanigawa, and R. Kawamura, “Thermal buckling of functionally graded rectangular plates subjected to partial heating,” Int J Mech Sci, vol. 48, no. 9, pp. 926–937, Sep. 2006, doi: 10.1016/j.ijmecsci.2006.03.015.
19. [19] S. Abrate, “Functionally graded plates behave like homogeneous plates,” Compos B Eng, vol. 39, no. 1, pp. 151–158, Jan. 2008, doi: 10.1016/j.compositesb.2007.02.026.
20. [20] D. G. Zhang and Y. H. Zhou, “A theoretical analysis of FGM thin plates based on physical neutral surface,” Comput Mater Sci, vol. 44, no. 2, pp. 716–720, Dec. 2008, doi: 10.1016/j.commatsci.2008.05.016.
21. [21] D. G. Zhang, “Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory,” Int J Mech Sci, vol. 68, pp. 92–104, Mar. 2013, doi: 10.1016/j.ijmecsci.2013.01.002.
22. [22] A. Fekrar, M. S. A. Houari, A. Tounsi, and S. R. Mahmoud, “A new five-unknown refined theory based on neutral surface position for bending analysis of exponential graded plates,” Meccanica, vol. 49, no. 4, pp. 795–810, 2014, doi: 10.1007/s11012-013-9827-3.
23. [23] D. G. Zhang, “Nonlinear bending analysis of FGM rectangular plates with various supported boundaries resting on two-parameter elastic foundations,” Archive of Applied Mechanics, vol. 84, no. 1, pp. 1–20, Jan. 2014, doi: 10.1007/s00419-013-0775-0.
24. [24] S. C. Han, W. T. Park, and W. Y. Jung, “A four-variable refined plate theory for dynamic stability analysis of S-FGM plates based on physical neutral surface,” Compos Struct, vol. 131, pp. 1081–1089, Nov. 2015, doi: 10.1016/j.compstruct.2015.06.025.
25. [25] H. Bellifa, K. H. Benrahou, L. Hadji, M. S. A. Houari, and A. Tounsi, “Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 38, no. 1, pp. 265–275, Jan. 2016, doi: 10.1007/s40430-015-0354-0.
26. [26] F. Ebrahimi, A. Jafari, and M. R. Barati, “Free Vibration Analysis of Smart Porous Plates Subjected to Various Physical Fields Considering Neutral Surface Position,” Arab J Sci Eng, vol. 42, no. 5, pp. 1865–1881, May 2017, doi: 10.1007/s13369-016-2348-3.
27. [27] V. L. Nguyen, M. T. Tran, V. L. Nguyen, and Q. H. Le, “Static behaviour of functionally graded plates resting on elastic foundations using neutral surface concept,” Archive of Mechanical Engineering, vol. 68, no. 1, pp. 5–22, 2021, doi: 10.24425/ame.2020.131706.
28. [28] A. Sadgui and A. Tati, “A novel trigonometric shear deformation theory for the buckling and free vibration analysis of functionally graded plates,” Mechanics of Advanced Materials and Structures, vol. 29, no. 27 pp. 6648-6663, 2021, doi: 10.1080/15376494.2021.1983679.
29. [29] L. S. Ma and D. W. Lee, “Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading,” European Journal of Mechanics, A/Solids, vol. 31, no. 1, pp. 13–20, Jan. 2012, doi: 10.1016/j.euromechsol.2011.06.016.
30. [30] L. O. Larbi, A. Kaci, M. S. A. Houari, and A. Tounsi, “An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams,” Mechanics Based Design of Structures and Machines, vol. 41, no. 4, pp. 421–433, Oct. 2013, doi: 10.1080/15397734.2013.763713.
31. [31] D. G. Zhang, “Thermal post-buckling and nonlinear vibration analysis of FGM beams based on physical neutral surface and high order shear deformation theory,” Meccanica, vol. 49, no. 2, pp. 283–293, Feb. 2014, doi: 10.1007/s11012-013-9793-9.
32. [32] K. S. Al-Basyouni, A. Tounsi, and S. R. Mahmoud, “Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position,” Compos Struct, vol. 125, pp. 621–630, Jul. 2015, doi: 10.1016/j.compstruct.2014.12.070.
33. [33] F. Ebrahimi and E. Salari, “A Semi-analytical Method for Vibrational and Buckling Analysis of Functionally Graded Nanobeams Considering the Physical Neutral Axis Position,” 2015.
34. [34] K. Zoubida, T. H. Daouadji, L. Hadji, A. Tounsi, and A. B. el Abbes, “A New Higher Order Shear Deformation Model of Functionally Graded Beams Based on Neutral Surface Position,” Transactions of the Indian Institute of Metals, vol. 69, no. 3, pp. 683–691, Apr. 2016, doi: 10.1007/s12666-015-0540-x.
35. [35] N. T. B. Phuong, T. M. Tu, H. T. Phuong, and N. van Long, “Bending analysis of functionally graded beam with porosities resting on elastic foundation based on neutral surface position,” Journal of Science and Technology in Civil Engineering (STCE) - NUCE, vol. 13, no. 1, pp. 33–45, Jan. 2019, doi: 10.31814/stce.nuce2019-13(1)-04.
36. [36] M. Derikvand, F. Farhatnia, and D. H. Hodges, “Functionally graded thick sandwich beams with porous core: Buckling analysis via differential transform method,” Mechanics Based Design of Structures and Machines, vol. 51, no. 7, pp. 3650-3677, 2021, doi: 10.1080/15397734.2021.1931309.
37. [37] P. Van Vinh, N. Q. Duoc, and N. D. Phuong, “A New Enhanced First-Order Beam Element Based on Neutral Surface Position for Bending Analysis of Functionally Graded Porous Beams,” Iranian Journal of Science and Technology - Transactions of Mechanical Engineering, vol. 46, no. 4, pp. 1141–1156, Dec. 2022, doi: 10.1007/s40997-022-00485-1.
38. [38] Y. Liu, S. Su, H. Huang, and Y. Liang, “Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane,” Compos B Eng, vol. 168, pp. 236–242, Jul. 2019, doi: 10.1016/j.compositesb.2018.12.063.