Analytical and Numerical Study on the Buckling of Homogeneous Beams Coated by a Functionally Graded Porous Layer with Different Boundary Conditions
Author
Abstract
In this paper, static buckling of homogeneous beams coated by a functionally graded porous layer with different boundary conditions is investigated based on the Timoshenko beam theory. The principle of virtual work has been used to obtain the governing equations. Two different methods, namely analyticalsolution and numerical solution are used to solve the governing equations and extract the buckling force. The governing equations are coupled as a series of ordinary differential equations. In the analytical solution, these equations are first uncoupled using a series of mathematical operations, and are then solved. The obtained solution has a series of parameters and unknown constants. Using the boundary conditions at the boundaries of the beam, a homogeneous system of equations is extracted, from which the axial buckling force is obtained. In the numerical solution, the generalized differential quadrature method is used to solve the static equations. Finally, the numerical results are presented and the effects of various parameters such as thickness to beam length ratio, porous layer thickness, porosity parameter, etc. on the buckling of the beam are investigated. Comparison of the results obtained from the two analytical and numerical solution methods confirms the accuracy and validity of both methods.
Keywords
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30. Nguyen, T-K., Truong-Phong Nguyen T., Vo, TP., and Thai, H-T., “Vibration and Buckling Analysis of Functionally Graded Sandwich Beams by a New Higher-Order Shear Deformation Theory”, Composite Structures, Vol. 76, pp. 273-85, 2015.
31. Kahya, V., and Turan, M.,“Finite Element Model for Vibration and Buckling of Functionally Graded Beams Based on the First-Order Shear Deformation Theory”, Composites Part B, Vol. 109, pp. 118-115, 2017.
2. Rafiee, M., Yang, J., and Kitipornchai, S.” Large Amplitude Vibration of Carbon Nanotube Reinforced Functionally Graded Composite Beams with Piezoelectric Layers”, Composite Structures, Vol. 96, pp. 716-725, 2013.
3. Smith, B., Szyniszewski, S., Hajjar, J., Schafer, B., Arwade, S., “Steel Foam for Structures: a Review of Applications, Manufacturing and Material Properties”, Journal of Constructional Steel Research, Vol. 71, pp. 1-10, 2012.
4. Ashby, MF., Evans, T., Fleck, NA., Hutchinson, J., Wadley, H. and Gibson, L., Metal Foams :a Design Guide, Elsevier. 2000.
5. Badiche, X., Forest, S., Guibert, T., Bienvenu, Y., Bartout, J-D., Ienny, P., Croset, M., and Bernet, H., “Mechanical Properties and Non-Homogeneous Deformation of Open-Cell Nickel Foams :Application of the Mechanics of Cellular Solids and of Porous Materials”, Materials Science and Engineering: A, Vol. 289, No. 1, pp. 276-88, 2000.
6. Banhart, J., “Manufacture, Characterization and Application of Cellular Metals and Metal Foams”, Progress in Materials Science, Vol. 46, No. 6, pp. 559-632, 2001.
7. Lopatnikov, SL., Gama, BA., Haque, MJ., Krauthauser, C., Gillespie, JW., Guden, M., and Hall, IW., “Dynamics of Metal Foam Deformation During Taylor Cylinder–Hopkinson Bar Impact Experiment”, Composite Structures, Vol. 61, No. 1, pp. 61-71, 2003.
8. Pinnoji, PK., Mahajan, P., Bourdet, N., Deck, C., and Willinger, R., “Impact Dynamics of Metal Foam Shells for Motorcycle Helmets: Experiments and Numerical Modeling”, International Journal of Impact Engineering, Vol. 37, No. 3, pp. 274-284, 2010.
9. Lefebvre, L-P., Banhart, J., and Dunand, D., “Porous Metals And Metallic Foams: Current Status and Recent Developments”, Advanced Engineering Materials, Vol. 10, No. 9, pp. 775-787, 2008.
10. Ahmad, Z., and Thambiratnam, DP., “Dynamic Computer Simulation and Energy Absorption of Foam-Filled Conical Tubes under Axial Impact Loading”, Composite Structures, Vol. 87, No. 3, pp. 186-97, 2009.
11. Mojahedin, A., Jabbari, M., Khorshidvand, AR., and Eslami, MR., “Buckling Analysis of Functionally Graded Circular Plates Made of Saturated Porous Materials Based on higher Order Shear Deformation Theory”, Thin-Walled Structures, Vol. 992, pp. 83-90, 2016.
12. Kitipornchai, S., Chen, D., and Yang, J., “Free Vibration and Elastic Buckling of Functionally Graded Porous Beams Reinforced by Graphene Platelets”, Materials & Design, Vol. 116, pp. 656-665, 2017.
13. Yang, J., Chen, D., and Kitipornchai, S., “Buckling and Free Vibration Analyses of Functionally Graded Graphene Reinforced Porous Nanocomposite Plates Based on Chebyshev-Ritz Method”, Composite Structures, Vol. 193, pp. 281-294, 2018.
14. Dong, YH., He, LW., Wang, L., Li, Y H., and Yang, J. “Buckling of Spinning Functionally Graded Graphene Reinforced Porous Nanocomposite Cylindrical Shells: An Analytical Study”, Aerospace Science and Technology, Vol. 82-83, pp. 466-478, 2018.
15. Zhenhuan, Zh., Yiwen, N., Zhenzhen, T., Shengbo, Zh., Jiabin, S.,and Xinsheng, X., “Accurate Nonlinear Buckling Analysis of Functionally Graded Porous Graphene Platelet Reinforced Composite Cylindrical Shells” International Journal of Mechanical Sciences, Vol. 151, pp. 537-50, 2019.
16. Liu, Y., Su, Sh., Huang, H., and Liang, Y. “Thermal-Mechanical Coupling Buckling Analysis of Porous Functionally Graded Sandwich Beams Based on Physical Neutral Plane”, Composites Part B: Engineering, Vol. 168, pp. 236-242, 2019.
17. Yas, MH., and Rahimi, S. “Thermal Buckling Analysis of Porous Functionally Graded Nanocomposite Beams Reinforced by Graphene Platelets Using Generalized Differential Quadrature Method”, Aerospace Science and Technology, Vol. 107, p. 106261, 2020.
18. Trinh, MCh., Mukhopadhyay, T., and Kim, S-E., “A Semi-Analytical Stochastic Buckling Quantification of Porous Functionally Graded Plates”, Aerospace Science and Technology, Vol. 105, p. 105928, 2020.
19. Rafiei Anamagh, M., and Bediz, B., “Free Vibration and Buckling Behavior of Functionally Graded Porous Plates Reinforced By Graphene Platelets Using Spectral Chebyshev Approach”, Composite Structures, Vol. 253, p. 112765, 2020.
20. Hung, DX., Tu, TM., Long, NV., and Anh, PH., “Nonlinear Buckling and Postbuckling of FG Porous Variable Thickness Toroidal Shell Segments Surrounded by Elastic Foundation Subjected to Compressive Loads”, Aerospace Science and Technology, Vol. 107, p. 106253, 2020.
21. Foroutan, K., Shaterzadeh, A., and Ahmadi, H., “Nonlinear Static and Dynamic Hygrothermal Buckling Analysis Of Imperfect Functionally Graded Porous Cylindrical Shells”, Applied Mathematical Modelling, Vol. 77, pp. 539-53, 2020.
22. Nguyen, QH., Nguyen, LB., Nguyen, HB., and Nguyen-Xuan, H., “A Three-Variable High Order Shear Deformation Theory for Isogeometric Free Vibration, Buckling and Instability Analysis of FG Porous Plates Reinforced by Graphene Platelets”, Composite Structures, Vol. 245, p. 112321, 2020.
23. Li, Zh. “Exploration of the Encased Nanocomposites Functionally Graded Porous Arches: Nonlinear Analysis and Stability Behavior”, Applied Mathematical Modelling, Vol. 82, pp. 1-16, 2020.
24. Li, Zh., Chen, Y., Zheng, J., and Sun, Q., “Thermal-Elastic Buckling of the Arch-Shaped Structures with FGP Aluminum Reinforced by Composite Graphene Platelets”, Thin-Walled Structures, Vol. 157, p. 107142, 2020.
25. Zhu, B., Chen, X-Ch., Guo, Y., and Li, Y-H., “Static and Dynamic Characteristics of the Post-Buckling of Fluid-Conveying Porous Functionally Graded Pipes with Geometric Imperfections”, International Journal of Mechanical Sciences, Vol. 189, p. 105947, 2021.
26. Cuong-Le, Th., Nguyen, KhD., Nguyen-Trong, N., Khatir, S., Nguyen-Xuan, H., and Abdel-Wahab, M., “A Three-Dimensional Solution for Free Vibration and Buckling of Annular Plate, Conical, Cylinder and Cylindrical Shell of FG Porous-Cellular Materials Using IGA”, Composite Structures, Vol. 259, pp. 113216, 2021.
27. Talebizadehsardari, P., Salehipour, H., Shahgholian-Ghahfarokhi, D., Shahsavar, A., and Karimi, M., “Free Vibration Analysis of the Macro-Micro-Nano Plates and Shells Made of a Material with Functionally Graded Porosity: A Closed-Form Solution”, Mechanics Based Design of Structures and Machines, under publication, 2020.
28. Shu, C., “Differential Quadrature and Its Application in Engineering”, Berlin: Springer. 2000.
29. Vo, TP., Thai, HT., Nguyen, TK., Inam, F., and Lee, J., “A Quasi-3D Theory for Vibration and Buckling of Functionally Graded Sandwich Beams”, Composite Structures, Vol. 119, pp. 1-12, 2015.
30. Nguyen, T-K., Truong-Phong Nguyen T., Vo, TP., and Thai, H-T., “Vibration and Buckling Analysis of Functionally Graded Sandwich Beams by a New Higher-Order Shear Deformation Theory”, Composite Structures, Vol. 76, pp. 273-85, 2015.
31. Kahya, V., and Turan, M.,“Finite Element Model for Vibration and Buckling of Functionally Graded Beams Based on the First-Order Shear Deformation Theory”, Composites Part B, Vol. 109, pp. 118-115, 2017.
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