Frequency Analysis of Bi-directional Porous Functionally Graded Beams with Variable cross section on Elastic Foundation using Reddy Third Order Shear Deformation Theory

Authors

1 Department of Mechanical Engineering,Shiraz Branch,Islamic Azad University,Shiraz,Iran

2 Mech. Eng., Shiraz Branch, Islamic Azad Univ., Shiraz, Iran

3 Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran

Abstract

In this paper, frequency analysis of Bi-directional Porous functionally graded beams with variable cross section which are resting on elastic foundation based on Reddy third order shear deformation theory is studied. Mechanical property gradients defined in accordance with two models of exponential and volume fraction power law. Governing equation which is obtained with the aid of third order shear deformation theory and by considering elastic foundation effect in conjunction with Hamilton’s principle. Due to intrinsic closed form solution of equations, differential equations solved with using Generalized Differential Quadrature Method by considering various end conditions. In order to validate the results comparisons are made with solutions which are available for other papers. This study reveals that the difference between the results of this paper and the results of others is negligible. Eventually the effects of geometrical parameters, power and exponential law indexes and elastic foundation coefficients on natural frequencies of Bi-directional FGM beams is studied. The results reveal that non dimensional frequencies increase with the rise of elastic foundation coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non dimensional frequencies.The results of this study can be used in optimal design, vibration control and detection of failure structure of functional graded structures

Keywords

References

[1]  Zahedinejad P (2016) Free vibration analysis of functionally graded beams resting on elastic foundation in thermal environment. Int J Str Stab and Dyn 16(7): 1550029-51.
[2]  Li SR, Wan ZQ, Zhang JH (2014) Free vibration of functionally graded beams based on both classical and first-order shear deformation beam theories. App Math and Mech 35(5): 591-606.
[3]  Ziou H, Guenfoud H, Guenfoud M (2016) Numerical modelling of a Timoshenko FGM beam using the finite element method. Int J Str Eng 7(3):   239-6.
[4] Karamanli A (2018) Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory. Comp Str 189: 127-36.
[5] ابراهیمی ممقانی ع، حسینی ر، شاهقلی م، سرپرست هـ (1397) تحلیل ارتعاشات عرضی آزاد تیرهای ناهمگن محوری نمایی با شرایط مرزی مختلف. نشریه علمی مکانیک سازه‌ها و شاره‌ها 133-125 :(3)8.
[6] Eisenberger M (1994) Vibration frequencies for beams on variable one-and two-parameter elastic foundations. J Sound Vib 176(5): 577-84.
[7] Catal S (2008) Solution of free vibration equations of beam on elastic soil by using differential transform method. App Math Mod  32(9): 1744-57.
[8] Malekzadeh P, Karami G (2008) A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations. App Math Mod 32(7): 1381-94.
[9] محمدی مهر م، قربانپور ع، روستاناوی ب (1394) تحلیل ارتعاشات آزاد پانل استوانه ای ساخته شده از مواد مدرج تابعی قرار گرفته بر روی بستر الاستیک پاسترناک با استفاده از تئوری برشی مرتبه اول. نشریه علمی مکانیک سازه­ها و شاره­ها 163-149 :(1)5.
[10] Akbas SD (2015) Free vibration and bending of functionally graded beams resting on elastic foundation. Research on Eng Str  Mat 1(1): 25-37.
[11] Wang ZH, Wang XH, Xu GD, Cheng S, Zeng T (2016) Free vibration of two-directional functionally graded beams. Comp Str 135: 191-198.
[12] ترشیزیان م­ر، اخوت پور ع ح (1394) تحلیل ارتعاشات تیر تیموشنکو ساخته شده از مواد تابعی دو بعدی. فصلنامه علمی پژوهشی دانشگاه آزاد مشهد 100-90 :8.
[13] Şimşek, M (2015) Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Comp Str 133: 968-978.
[14] Abrate S (1995) Vibration of non-uniform rods and beams. J  sound vib 185(4): 703-16.
[15] Shahba A, Attarnejad R, Marvi MT, Hajilar S (2011) Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Comp Part B: Eng 42(4): 801-808.
[16] Calim FF (2016) Transient analysis of axially functionally graded Timoshenko beams with variable cross-section. Comp Part B: Eng 98: 472-483.
[17] رحمانی ب، نغمه سنج م، حسینی م (1395) کنترل بهینه ارتعاشات عرضی تیر ساخته شده از مواد مدرج تابعی با سطح مقطع متغیر. مجله علمی پژوهشی مهندسی مکانیک تبریز 122-113 :(4)46.
[18] Shafiei N, Kazemi M (2017) Buckling analysis on the bi-dimensional functionally graded porous tapered nano beams. Aerspace Sci Tech 66: 1-11.
[19] Chen D, Yang J, Kitipornchai S (2016) Free and forced vibrations of shear deformable functionally graded porous beams. Int J of Mech Sci 108: 14-22.
[20] Fouda N, El-Midany T, Sadoun AM (2017) Bending, buckling and vibration of a functionally graded porous beam using finite elemen ts. J App and Comp Mech 3(4): 74-82.
[21] Şimşek M (2010) Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nuc Eng Desg 240(4): 697-705.
[22] Pradhan KK, Chakraverty S (2013) Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Comp Part B: Eng 51: 175-184.
[23] رحیمی ع ر، لیوانی م، نگهبان برون ع (1400)    تحلیل ارتعاشات آزاد تیر مدرج تابعی با وجود ترک عرضی. مجله علمی پژوهشی مهندسی مکانیک تبریز 281-277 :(1)51.
[24] Ghorbanpour Arani, Niknejad  S (2020) Dynamic Stability Analysis of Bi-Directional Functionally Graded Beam with Various Shear Deformation Theories under Harmonic Excitation and Thermal Environment. J of  Solid  Mech.
[25] Lei J, He Y, Li Z, Guo S, & Liu D (2019) Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory. Comp Str 209:  811-829.
[26] Bert CW, Malik M (1996)  Differential quadrature method in computational mechanics. a review. App Mech Rev 49(1): 1-28.
Journal of Solid and Fluid Mechanics
Volume 11, Issue 3
October 2021
Pages 97-118

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