نویسندگان
گروه مهندسی مکانیک، واحد کرج - دانشگاه آزاد اسلامی، کرج، ایران
چکیده
این مقاله به تحلیل کمانش تیرهای گوه‌ای با پهنای ثابت و ضخامت متغیر و ساخته شده از مواد مدرج تابعی دو بعدی پرداخته است. فرض بر آنست که تیر از ترکیب فلز با سرامیک ساخته شده باشد، بطوریکه کسر حجمی هر یک از آنها در راستای طول تیر و همچنین در راستای ضخامت آن بر اساس توابع توانی تغییر کند. همچنین فرض می‌شود که تیر بطور همزمان تحت اثر نیروهای محوری متمرکز و گسترده قرار دارد. معادلات حاکم و شرایط مرزی با استفاده از اصل انرژی پتانسیل کمینه استخراج گردیده و سپس با استفاده از روش تربیع دیفرانسیلی برای تیری با شرایط مرزی یک سر گیردار حل شده‌اند. پس از تایید همگرایی و صحت تحلیل ارائه شده، تأثیر مشخصات گوناگون بر روی مقدار بار بحرانی تیر در هر دو حالت تیر تحت بار متمرکز و تیر تحت بار گسترده بر واحد طول بررسی شده‌اند که از آن جمله می‌توان به مشخصات هندسی تیر، ایندکس‌های توانی در تغییرات خواص در هر دو راستای طول و ضخامت و همچنین چگونگی تغییرات ضخامت تیر اشاره کرد. تحلیل کمانش تیر در صورت اعمال همزمان بارهای نقطه‌ای و گسترده نیز مورد بررسی قرار گرفته است. در تحلیل کمانش تیر تحت بار گسترده سه الگوی توزیع برای بار در نظر گرفته شده‌اند که شامل توزیع خطی، درجه دو و نمایی است. نتایج این تحقیق نشان می‌دهند که در بین الگوهای بررسی شده برای توزیع بار گسترده بیشترین مقدار بار بحرانی متعلق به توزیع خطی است و کمترین مقدار بار بحرانی متعلق به توزیع نمایی بار است.
کلیدواژه‌ها
عنوان مقاله [English]
Buckling Analysis of FGM Timoshenko Beam with Variable Thickness under Concentrated and Distributed Axial loads Using DQM
نویسندگان [English]
- M. Mohieddin Ghomshei
- Sh. Namazi
چکیده [English]
In this article, mechanical buckling analysis of tapered beams having constant width and variable thickness, made of two-dimensional functionally graded materials is studied. The beam is assumed to be made of metal and ceramic, where their volume fractions vary in both longitudinal and thickness directions based on the power law. The beam is generally subjected to combined concentrated and distributed axial loads. The set of governing equations are derived using the Principle of Minimum total Potential Energy (PMPE), and are solved numerically using Differential Quadrature Method (DQM) for clamped-free boundary conditions. Convergence and accuracy of the presented solution are confirmed for both cases of concentrated and distributed axial loads. The effects of different parameters on the critical buckling load of the beam for both load cases are studied including geometrical parameters, gradation indices in longitudinal and thickness directions, and variation of thickness. Also buckling analysis of the beam under a combination of concentrated load and distributed axial loads of linear, quadratic and exponential types are investigated. Numerical results show that the highest values of the critical buckling load belong to the linear distributed load, and the lowest value is owned by exponential load.
کلیدواژه‌ها [English]
- Buckling
- Distributed load
- Tapered beam
- Two-dimensional functionally graded materials
2. Gauss , R.C., and Antman S.S., “Large Thermal Buckling of nonuniform Beams and Plates”, International journal of solids and structures, Vol. 20, pp. 979-1000, 1984.
3. Huang, Y., Zhang, M.,and Rong, H., “Buckling Analysis of Axially Functionally Graded and Non-Uniform Beams Based on Timoshenko Theory”, Acta Mechanica Solida Scinica, Vol. 29, pp. 200-207, 2016.
4. Rajasekaran, S., Khaniki, H. B., “Bending, Buckling and Vibration of Small-Scale Tapered Beams”, International Journal of Engineering Science,Vol. 120, pp.172-188, 2017.
5. Ozbasaran, H., and Yilmaz, T., “Shape Optimization of tapered I-beams with Lateral-Torsional Buckling, Deflection and Stress Constraints”, Journal of Constructional Steel Research, Vol. 143, pp. 119-130, 2018.
6. Tankova, T., Martins, J. P., da Silva, L. S., Marques, L., Craveiro, H. D., and Santiago, A., “Experimental Lateral-Torsional Buckling Behaviour of Web Tapered I-Section Steel Beams”, Engineering Structures, Vol.168, pp. 355-370, 2018.
7. Wang, C., Thevendran, V., Teo, K., and Kitipornchai, S., “Optimal design of tapered beams for maximum buckling strength”, Engineering Structures, Vol. 8, pp. 276-284, 1986.
8. Li, Q., “Exact Solutions for Buckling of Non-Uniform Columns Under Axial Concentrated and Distributed Loading”, European Journal of Mechanics-A/Solids, Vol. 20, pp. 485-500, 2001.
9. Darbandi, S., Firouz-Abadi R., and Haddadpour, H., “Buckling of Variable Section Columns Under Axial Loading”, Journal of Engineering Mechanics, Vol. 136. pp. 472-476, 2010.
10. Robinson, M. T. A., and Adali, S., “Buckling of Nonuniform Carbon Nanotubes Under Concentrated and Distributed Axial Loads”, Mechanical Sciences, Vol. 8, pp. 299-311, 2017.
11. Karamanli, A., and Aydogdu, M., “Buckling of Laminated Composite and Sandwich Beams Due to Axially Varying in-Plane Loads”, Composite structures, Vol. 210, pp. 391-408, 2019.
12. Melaibari, A., Abo-bakr, R. M., Mohamed, S., and Eltaher, M., “Static Stability of Higher Order Functionally Graded Beam Under Variable Axial Load”, Alexandria Eng,neering Journal, Vol. 26, pp. 48-65, 2020.
13. Şimşek, M., “Buckling of Timoshenko Beams Composed of Two-Dimensional Functionally Graded Material (2D-FGM) Having Different Boundary Conditions”, Composite Structures, Vol. 149, pp. 304-314, 2016.
14. Chen, X., Zhang, X., Lu, Y.,and Li, Y., “Static and Dynamic Analysis of the Postbuckling of Bi-Directional Functionally Graded Material Microbeams”, International Journal of Mechanical Sciences, Vol. 151 pp. 424-443, 2019.
15. Rajasekaran, S., and Khaniki, H. B., “Bi-Directional Functionally Graded Thin-Walled Non-Prismatic Euler Beams of Generic Open/Closed Cross Section Part I: Theoretical Formulations”, Thin-Walled Structures, Vol. 141, pp. 627-645, 2019.
16. Rajasekaran, S., and Khaniki, H. B., “Bi-Directional Functionally Graded Thin-Walled Non-Prismatic Euler Beams of Generic Open/Closed Cross Section Part II: Static, Stability and Free Vibration Studies”, Thin-Walled Structures, Vol. 141, pp. 646-674, 2019.
17. Timoshenko, S. P., “On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 41, pp. 744-746, 1921.
18. Sadd, M. H., Elasticity: Theory, Applications, and Numerics, Academic Press, London, 2009.
19. Kaneko, T., “On Timoshenko's Correction for Shear in Vibrating Beams”, Journal of Physics D: Applied Physics, Vol. 8 pp. 1927-1936, 1975.
20. Ventsel, E., Krauthammer, T., and Carrera, E., Thin Plates and Shells: Theory, Analysis, and Applications, CRC press, New York, 2020.
21. Hwang, C. Y., “The Buckling Problem of A Beam on An Elastic Foundation Under Distributed Axial Loads”, Thesis, Kansas State University, 1965.
22. Bert, C. W., and Malik, M., “Differential Quadrature Method in Computational Mechanics: A Review”, Applied Mechanics Reviews, Vol. 49, pp. 1-28, 1996.
23. Afshari H., Differential Quadrarure Method in the Solution of the Mechanical Engineering Problems, Pouyesh Andisheh Publications,1398 (In Persian).
24. Beer, F., Johnston, E, and DeWolf, J., Mechanics of Materials, 5th SI Edition, Stress, Ed. 1, pp. 1-12, 1999.
25. Soltani, M., and Asgarian, M. B., “Finite Element Formulation for Linear Stability Analysis of Axially Functionally Graded Nonprismatic Timoshenko Beam”, International Journal of Structural Stability and Dynamics, Vol. 19, pp. 195-202, 2019.
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