Natural Frequency and Dynamic Analyses of Functionally Graded Saturated Porous Beam Resting on Viscoelastic 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, natural frequencies and dynamic response of thick beams made of saturated porous materials resting on viscoelastic foundation are investigated for the first time. The beam is modeled using higher-order beam theory. Kelvin-voight model is used to model the viscoelastic foundation. Distribution of porosity along the thickness is considered in two different patterns, which are symmetric nonlinear and nonlinear asymmetric distributions. The relationship between stress and strain is based on the Biot constitutive law. Lagrange equations are used to express the motion equations. Finite element and Newark methods are used to solve the governing equations. The effect of different boundary conditions and various parameters such as porosity and Skempton coefficients, slenderness ratio as well as stiffness and damping coefficients of viscoelastic foundation on natural frequency and transient response of beam have been studied. Results show that in a drained condition, beam has smallest fundamental frequency and by increasing the Skempton coefficient, the fundamental frequency of the beam increases.


Main Subjects

[1] Hromadová L., 2009, Thermal Pressurization of Pore Fluid During Earthquake Slip, Comenius University, Bratislava.
[2] Biot M.A., 1955, Theory of elasticity and consolidation for a porous anisotropic solid, Journal of Applied Physics 26(2): 182-185.
[3] Theodorakopoulos D.D., Beskos D.E., 1994, Flexural vibrations of poroelastic plates, Acta Mechanica 103(1-4): 191-203.
[4] Leclaire P., Horoshenkov K.V., Swift M.J., Hothersall D.C., 2001, The vibrational response of a clamped rectangular porous plate, Journal of Sound and Vibration 247(1): 19-31.
[5] Magnucki K., Stasiewicz P., 2004, Elastic buckling of a porous beam, Journal of Theoretical and Applied Mechanics 42(4): 859-868.
[6] Magnucka-Blandzi E., 2008, Axi-symmetrical deflection and buckling of circular porous-cellular plate, Thin-Walled Structures 46(3): 333-337.
[7] Magnucka-Blandzi E., Magnucki K., 2007, Effective design of a sandwich beam with a metal foam core, Thin-Walled Structures 45(4): 432-438.
[8] Debowski D., Magnucki K., 2006, Dynamic stability of a porous rectangular plate, PAMM 6(1): 215-216.
[9] Mojahedin A., Joubaneh E.F., Jabbari M., 2014, Thermal and mechanical stability of a circular porous plate with piezoelectric actuators, Acta Mechanica 225(12): 3437-3452.
[10] Jabbari M., Mojahedin A., Khorshidvand A.R., Eslami M.R., 2013, Buckling analysis of a functionally graded thin circular plate made of saturated porous materials, Journal of Engineering Mechanics 140(2): 287-295.
[11] Jabbari M., Mojahedin A., Haghi M., 2014, Buckling analysis of thin circular FG plates made of saturated porous-soft ferromagnetic materials in transverse magnetic field, Thin-Walled Structures 85: 50-56.
[12] Ebrahimi F., Mokhtari M., 2015, Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method, Journal of the Brazilian Society of Mechanical Sciences and Engineering 37(4):1435-1444.
[13] Ebrahimi F., Jafari A., 2018, A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities, Mechanics of Advanced Materials and Structures 25(3): 212-224.
[14] Ebrahimi F., Ghasemi F., Salari E., 2016, Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities, Meccanica 51(1): 223-249.
[15] Chen D., Yang J., Kitipornchai S., 2015, Elastic buckling and static bending of shear deformable functionally graded porous beam, Composite Structures 133: 54-61.
[16] Chen D., Yang J., Kitipornchai S., 2016, Free and forced vibrations of shear deformable functionally graded porous beams, International Journal of Mechanical Sciences 108: 14-22.
[17] Chen D., Kitipornchai S., Yang J., 2016, Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core, Thin-Walled Structures 107: 39-48.
[18] Arshid E., Khorshidvand A.R., 2018, Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method, Thin-Walled Structures 125(1): 220-233.
[19] Galeban M.R., Mojahedin A., Taghavi Y., Jabbari M., 2016, Free vibration of functionally graded thin beams made of saturated porous materials, Steel and Composite Structures 21(5): 999-1016.
[20] Grygorowicz M., Magnucki K., Malinowski M., 2015, Elastic buckling of a sandwich beam with variable mechanical properties of the core, Thin-Walled Structures 87: 127-132.
[21] Ghorbanpour Arani A., Khani M., Khoddami Maraghi Z., 2018, Dynamic analysis of a rectangular porous plate resting on an elastic foundation using high-order shear deformation theory, Journal of Vibration and Control 241(6): 3698-3713.
[22] Ghorbanpour Arani A., Khoddami Maraghi Z., Khani M., Alinaghian I., 2017, Free vibration of embedded porous plate using third-order shear deformation and poroelasticity theories, Journal of Engineering 2017: 1474916.
[23] Ghorbanpour Arani A., Zamani M.H., 2018, Nonlocal free vibration analysis of FG-porous shear and normal deformable sandwich nanoplate with piezoelectric face sheets resting on silica aerogel foundation, Arabian Journal for Science and Engineering 43(9): 4675-4688.
[24] Magnucka-Blandzi E., 2009, Dynamic stability of a metal foam circular plate, Journal of Theoretical and Applied Mechanics 47: 421-433.
[25] Mojahedin A., Jabbari M., Khorshidvand A.R., Eslami M.R., 2016, Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory, Thin-Walled Structures 99: 83-90.
[26] Ebrahimi F., Habibi S., 2016, Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate, Steel and Composite Structures 20(1): 205-225.
[27] Zimmerman R.W., 2000, Couplingin poroelasticity and thermoelasticity, International Journal of Rock Mechanics and Mining Sciences 37: 79-87.
[28] Detournay E., Cheng A.H.D., 1993, Fundamentals of poroelasticity, Analysis and Design Methods 1993:113-171.
[29] Peng X.Q., Lam K.Y., Liu G.R., 1998, Active vibration control of composite beams with piezoelectrics: a finite element model with third order theory, Journal of Sound and Vibration 209(4): 635-650.
[30] Meyers M., Chawla K., 1999, Mechanical Behavior of Materials, Prentice Hall, Inc, Section.
[31] Majzoobi GH., 2013, Concepted and Applications of Finite Element Method, Bu Ali Sina University of Technology Publication Center.
[32] Shakeri M., Akhlaghi M., Hoseini S. M., 2006, Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder, Composite Structures 76(1-2): 174-181.