Buckling and static analyses of functionally graded saturated porous thick beam resting on elastic foundation based on higher order beam theory

Document Type : Research Paper


1 Department of Mechanical Engineering, Shahid Beheshti University, Tehran, Iran

2 Department of Mechanical Engineering, Islamic Azad University, Tehran North Branch, Tehran, Iran


In this paper, static response and buckling analysis of functionally graded saturated porous beam resting on Winkler elastic foundation is investigated. The beam is modeled using higher-order shear deformation theory in conjunction with Biot constitutive law which has not been surveyed so far. Three different patterns are considered for porosity distribution along the thickness of the beam: 1) poro/nonlinear non-symmetric distribution, 2) poro/nonlinear symmetric distribution and 3) poro/monotonous distribution. To obtain the governing equations, geometric stiffness matrix concept and finite element method is used. The effect of various parameters such as: 1) Stiffness of elastic foundation 2) Slender ratio 3) Porosity coefficient 4) Skempton coefficient 5) Porosity distributions and 6) Different boundary conditions has been investigated to draw practical conclusions.


[1] Biot, M.A., “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid”, Journal of Applied Physics, Vol. 26, pp. 182-185, (1955).
[2] Magnucki, K., and Stasiewicz, P., “Elastic Buckling of a Porous Beam”, J. Theor Appl. Mech, Vol. 42, pp. 859-868, (2004).
[3] Magnucka-Blandzi, E., “Axi-symmetrical Deflection and Buckling of Circular Porous Cellular Plate”, Thin-Walled Structures, Vol. 46, pp. 333–337, (2008).
[4] Jasion, P., Magnucka-Blandzi, E., Szyc, W., and Magnucki, K., “Global and Local Buckling of Sandwich Circular and Beam-rectangular Plates with Metal Foam Core”, Thin-Walled Struct, Vol. 61, pp. 154-161, (2012).
[5] Jabbari, M., Mojahedin, A., Khorshidvand, A.R., and Eslami, M.R., “Buckling Analysis of Functionally Graded Thin Circular Plate made of Saturated Porous Materials”, ASCE's J. Eng. Mech. Vol. 140, pp. 287-295, (2013).
[6] Mojahedin, A., Jabbari, M., Khorshidvand, A.R., and Eslami, M.R., “Buckling Analysis of Functionally Graded Circular Plates Made of Saturated Porous Materials Based on Higher order Shear Deformation Theory”, Thin-Walled Struct, Vol. 99, pp. 83–90, (2016).
[7] Farzaneh Joubaneh, E., Mojahedin, A., Khorshidvand, A.R., and Jabbari, M., “Thermal Buckling Analysis of Porous Circular Plate with Piezoelectric Sensor-actuator Layers under Uniform Thermal Load”, J. Sandwich Struct. Mater, Vol. 17, pp. 3-25, (2015).
[8] Almitani, K.H., “Buckling Behaviors of Symmetric and Antisymmetric Functionally Graded Beams”, Journal of Applied and Computational Mechanics, Vol.  pp. 115-124, (2018).
[9] Tornabene, F., Fantuzzi, N., and Bacciocchi, M., “Refined Shear Deformation Theories for Laminated Composite Arches and Beams with Variable Thickness: Natural Frequency Analysis”, Engineering Analysis with Boundary Elements, Vol. 100, pp. 24-47, (2019).
[10] Fouda, N., El-Midany, T., and Sadoun, A.M., “Bending, Buckling and Vibration of a Functionally Graded Porous Beam using Finite Elements”, Journal of Applied and Computational Mechanics, Vol. 3, pp. 274-282, (2017).
[11] Chen, D., Yang, J., and Kitipornchai, S., "Elastic Buckling and Static Bending of Shear Deformable Functionally Graded Porous Beam", Composite Struct, Vol. 133, pp. 54-61, (2015).
 [12] Chen, D., Yang, J., and Kitipornchai, S., "Free and Forced Vibrations of Shear Deformable Functionally Graded Porous Beams", International Journal of Mechanical Sciences, Vol. 108, pp. 14-22, (2016).
[13] Galeban, M.R., Mojahedin, A., Taghavi, Y., and Jabbari, M., "Free Vibration of Functionally Graded Thin Beams Made of Saturated Porous Materials”, Steel and Composite Structures, Vol. 21, pp. 999-1016, (2016).
 [14] Sahmani, S., Aghdam, M.M., and Rabczuk, T., "Nonlinear Bending of Functionally Graded Porous Micro/nano-beams Reinforced with Graphene Platelets Based upon Nonlocal Strain Gradient Theory", Composite Struct, Vol. 186, pp. 68-78, (2018).
[15] Arani, A.G., Khoddami Maraghi, Z., Khani, M., and Alinaghian, I., "Free Vibration of Embedded Porous Plate using Third-order Shear Deformation and Poroelasticity Theories", Journal of Engineering. Vol. 2017, pp. 1-13, (2017).
 [16] Tang, H., Li, L., and Hu, Y., "Buckling Analysis of Two-directionally Porous Beam", Aerospace Science and Technology, Vol. 78, pp. 471-479, (2018).

[17] Handbook of Geophysical Exploration: Seismic Exploration Chapter 7, "Biot's Theory for Porous Media", Vol. 31, pp. 219-293, (2001).


[18] Zimmerman, R.W., "Coupling in Poroelasticity and Thermoelasticity", International Journal of Rock Mechanics and Mining Sciences, Vol. 37, pp. 79-87, (2000).
 [19] Detournay, E., and Cheng, A.H.D., "Fundamentals of Poroelasticity", In Omprehensive          Rock Engineering: Principles, Practices and Projects, Vol. 2, ed. J.A. Hudson, pp. 113-171, Pergamon Press, Oxford, UK, (1993). 
[20] Peng, X.Q., Lam, K.Y., and Liu, G.R., "Active Vibration Control of Composite Beams with Piezoelectrics: A Finite Element Model with Third order Theory", Journal of Sound and Vibration, Vol. 209, pp. 635-650, (1998).
[21] Daloglu, A.T., and Vallabhan, C.G., "Values of k for Slab on Winkler Foundation", Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, pp. 463-471, (2000).
[22] Ferreira, A.J., "MATLAB Codes for Finite Element Analysis", Springer Science & Business Media, Solids and Structures, Vol. 157, (2008).