تحلیل ارتعاش آزاد و پایداری خمشی-پیچشی تیر جدار نازک ماهیچه‌ای ساخته‌شده از مواد مدرج تابعی بر بستر الاستیک

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مهندسی عمران، دانشکده مهندسی، دانشگاه کاشان، کاشان، ایران

2 گروه مهندسی عمران، دانشکده مهندسی، دانشگاه کاشان، ایران

چکیده

مواد مدرج تابعی، مصالحی نوین با ساختاری ناهمگن و ویژگی‌های منحصر به فرد هستند که در سال‌های اخیر نظر پژوهشگران زیادی را به خود جلب کرده‌اند. بنابراین در این مقاله برای اولین بار، ارتعاشات آزاد و پایداری تیر ماهیچه‌ای با مقطع جدار نازک ساخته‌شده از مواد مدرج تابعی محوری بر بستر الاستیک وینکلر مورد بررسی قرار می‌گیرد. فرض بر این است که خواص مکانیکی ماده (ضریب ارتجاعی و چگال) در راستای طول عضو به‌طور پیوسته و براساس توزیع توانی، متغیر و نسبت پواسون ثابت فرض شده است. معادلات حاکم بر حرکت و شرایط مرزی با استفاده از اصل همیلتون و روش انرژی به‌دست می‌آیند. تحلیل پایداری و ارتعاش آزاد بر پایه مدل ولاسو برای مقاطع جدار نازک باز و با اعمال اثرات محل اتصال بستر الاستیک و خروج از مرکزیت بار فشاری در معادلات انجام شده است. با استفاده از روش مربعات دیفرانسیلی، دستگاه معادلات گسسته و حل می‌گردد. سپس با اعمال شرایط مرزی روی پاسخ‌های به‌دست‌آمده و با استفاده از روش حل مقادیر ویژه، بار کمانشی و فرکانس ارتعاشی محاسبه می‌شوند. در پایان نتایج حاصل از این تحقیقدر خصوص تیر همگن با مقطع متغیر با نتایج دیگر تحقیقات جهت بررسی صحت و دقت محاسبات مقایسه گردیده و تاثیر پارامترهای مختلفی همچون طول عضو، شرایط مرزی، خروج از مرکزیت بار فشاری، تغییر ابعاد مقطع، توان ماده مدرج تابعی، سفتی بستر الاستیک و محل اتصال فنر وینکلر بر روی پایداری و ارتعاش آزاد مورد ارزیابی قرار گرفته است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Free vibration and flexural-torsional stability analyses of axially functionally graded tapered thin-walled beam resting on elastic foundation

نویسندگان [English]

  • Masoumeh Soltani 1
  • Ali Ahanian 2
1 Department of Civil Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran
2 Department of Civil Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran
چکیده [English]

The thin-walled beams are widely adopted in different structural components ranging from civil engineering to aeronautical applications due to their conspicuous characteristics. A slender thin-walled beam loaded initially in compression may buckle suddenly in flexural–torsional mode since its torsional strength is much smaller than bending resistance. In this paper, flexural-torsional stability and free vibration analyses of axially functionally graded tapered I-beam resting on Winkler elastic foundation are assessed. Considering the coupling between the flexural displacements and the twist angle, the motion equations are derived via Hamilton’s principle in association with Vlasov’s thin-walled beam theory. The differential quadrature method is applied to solve the system of differential equations and to acquire the critical buckling loads and natural frequencies. To validate the obtained results, at first, homogeneous tapered I-beam in the absence of elastic foundation was analyzed and compared with a finite element solution using ANSYS and other available benchmarks. Afterward, the numerical outcomes for axially graded non-prismatic I-beam resting on elastic foundation are reported in graphical form to find out the impacts of axial load position, beam’s length, end conditions, web and flanges tapering ratio, material gradient index, Winkler parameter and spring position on the non-dimensional buckling loads and vibration frequencies.
 

کلیدواژه‌ها [English]

  • Flexural-torsional buckling
  • Vibration frequency
  • Functionally graded materials
  • Winkler foundation
  • Differential quadrature method
[1] M. Yamanouchi, M. Koizumi, T. Hirai, I. Shiota, Proceedings of the   first   international symposium on functionally gradient materials, Japan, (1990).
[2] M. Koizumi, The concept of FGM, ceramic transactions, functionally gradient materials, 3(1) (1993) 3-10.
[3] S.B. Kim, M.Y. Kim, Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames, Eng Struct, 22 (2000) 446–58.
[4] C.N. Chen, Dynamic equilibrium of non-prismatic beams defined on an arbitrarily selected co-ordinate system. J Sound Vib, 230(2) (2000) 241–260.
[5] R.D. Ambrosini, J.D. Riera, R.F. Danesi, A modified Vlasov theory for dynamic analysis of thin-walled and variable open section beams, Eng. Struct. 22 (8) (2000) 890–900.
[6] J. Li, R. Shen, H. Hua, X. Jin, Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams, Int. J. Mech. Sci. 46 (2) (2004) 299–320.
[7] Jun L, Wanyou L, Rongying S, Hongxing H. Coupled bending and torsional vibration of non-symmetrical axially loaded thin-walled Bernoulli–Euler beam. Mechanics Research Communications 2004; 31:697–711.
[8] A.Y.T. Leung, Exact dynamic stiffness for axial-torsional buckling of structural frames, Thin-Walled Structures, 46 (2008) 1–10.
[9] F. Borbon, A. Mirasso, D. Ambrosini, Beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded, Computers and Structures, 89 (2011) 1406-1416.
[10] H.S. Shen, ZX. Wang, Assessment of Voigt and Mori–Tanaka models for vibration analysis of functionally graded plates. Composite Structure. 94(7) (2012) 2197-2208.
[11] A. Andrade, One Dimensional Models for the Spatial Behaviour of Tapered Thin-walled Bars with Open cross sections: Static, Dynamic and Buckling Analyses (Ph.D. Thesis), University of Coimbra, Portugal, 2012.
[12] K.K. Pradhan, S. Chakraverty, Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method, Composite. Part B, 51(2013) 175–184.
[13] K.K. Pradhan, S. Chakraverty, Effects of different shear deformation theories on free vibration of functionally graded beams, International Journal of Mechanical Sciences, 82 (2014) 149–160.
[14] M. Jabbarzadeh, M.K. Baghdar Delgosha, Thermal buckling analysis of FGM sector plates using differential quadrature method. Modares Mechanical Engineering, 13(2) (2013) 33-45.
[15] M. Jabbarzadeh, J.J. Eskandari, M. Khosravi, The analysis of thermal buckling of circular plates of variable thickness from functionally graded materials           . Modares Mechanical Engineerin Journal.12(5) (2013) 59-73. (In Persian)
[16] Y. Huang, L.E. Yang, Q.Z. Luo, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section,  Composites: Part B:, Engineering, 45(1) (2013) 1493-1498.
[17] Y. Zhao, Y. Huang, M. Guo, A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory, Composite Structures, 168 (2017) 277-284.
[18] F. Mohri, S.A Meftah, N. Damil, A large torsion beam finite element model for tapered thin-walled open cross-sections beams, Engineering Structures, 99 (2015) 132-148.
[19] P. Ruta, J. Szybinski, Lateral stability of bending non-prismatic thin-walled beams using orthogonal series, Procedia Engineering, 11 (2015) 694-701.
[20] J. Kuś, Lateral-torsional buckling steel beams with simultaneously tapered flanges and web, Steel and Composite Structures, 19(4) (2015) 897-916.
[21] K. Khorshidi, A. Bakhsheshi H. Ghadirian, The study of the effects of thermal environment on free vibration analysis of two dimensional functionally graded rectangular plates on Pasternak elastic, Journal of Solid and Fluid Mechanics, 6(3) (2016) 137-147.
[22] A. Paul, D. Das, Non-linear thermal post-buckling analysis of FGM Timoshenko beam under non-uniform temperature rise across thickness. Engineering science and technology, an international journal, 19(3) (2016) 1608-1625.
[23] S.T. Dennis, K.W. Jones, Flexural-torsional vibration of a tapered C-section beam, J. Sound Vib. 393 (2017) 401–414.
[24] T-T. Nguyen, N-I. Kim, J. Lee, Free vibration of thin-walled functionally graded open-section beams, Composite structures, 95 (2016) 105-116.
[25] T-T. Nguyen, N-I. Kim, J. Lee, Analysis of thin-walled open section beams with functionally graded materials, Composite structures, 138 (2016) 75-83.
[26] T-T. Nguyen, P.T. Thang, J. Lee, Lateral buckling analysis thin-walled functionally graded beams,” Composite structures, 160 (2017) 952-963.
[27] T-T. Nguyen, P.T. Thang, J. Lee, Flexural-torsional stability of thin-walled functionally graded open-section beams, Thin walled structures, 110 (2017) 88-96.
[28] W. Chen, H. Chang, Closed-form solutions for free vibration frequencies of functionally graded Euler-Bernoulli beams, Mechanics of Composite Materials, 53(1) (2017) 79-98.
[29] S.B. Beheshti-Aval, M. Lezgy-Nazargah, A coupled refined high-order global– local theory and finite element model for static electromechanical response of smart multilayered/ sandwich beams, Archive of Applied Mechanic, 82 (2012) 1709-1752.
[30] M. Lezgy-Nazargah, S.B. Beheshti-Aval, Coupled refined layerwise theory for dynamic free and forced responses of piezoelectric laminated composite and sandwich beams, Meccanica, 48(6) (2013) 1479–1500.
[31] M. Lezgy-Nazargah, Efficient coupled refined finite element for dynamic analysis of sandwich beams containing embedded shear-mode piezoelectric layers, Mechanics of Advanced Materials and Structures, 23(3) (2016) 337-352.
[32] M. Lezgy-Nazargah, A generalized layered global-local beam theory for elasto-plastic analysis of thin-walled members, Thin-Walled Structures, 115 (2017) 48-57.
[33] M. Soltani, B. Asgarian, Buckling Analysis of Axially Functionally Graded Beams with Variable Cross-Section, Modares Civil Engineering Journal, 18 (3) (2018) 87-99 (In Persian).
[34] M. R-Pajand, A.R. Masoodi, A. Alepaighambar, Lateral-torsional buckling of functionally graded tapered I-beams considering lateral bracing, Steel and Composite Structures, 28 (2018) 403-414.
[35] M. Soltani, B. Asgarian, F. Mohri, Improved finite element formulation for lateral stability analysis of axially functionally graded non-prismatic I-beams, International Journal of Structural Stability and dynamics, 19(9) (2019) 1950108.
[36] S. Rajasekaran, H.B Khaniki, H.B, Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part I/II: Theoretical formulations, Thin-Walled Structures, (2019).
[37] H. Li, B. Balachandran, Buckling and free oscillations of composite microresonators, Journal of Microelectromechanical Systems, 15(1) (2006) 42-51.
[38] M.A. Steinberg, Materials for aerospace, Scientific American 255(4) (1986) 59–64.
[39] C. Lyu, W. Chen, R. Xu, C.W. Lim, Semi-analytical elasticity solutions for bi-directional functionally graded beams, International Journal of Solids and Structures, 45(1) (2008) 258–275.
[40] A. Sears, R.C. Batra, Macroscopic properties of carbon nanotubes from molecular-mechanics simulations, Physical Review, 69(23) (2004) 235406.
[41] Winkler E. Die Lehre von Elastizitat und Festigkeit (“The theory of elasticity and stiffness”). H. Domenicus. Prague. (1867) (In German).
[42] V.Z. Vlasov, Thin-Walled Elastic Beams, Israel Program for Scientific Translations, Jerusalem (1961).
[43] Y.Y. Yung, D. Munz, Stress analysis in a two materials joint with a functionally graded material. In: Shiota, T., Miyamoto, M.Y. (Eds.), Functionally Graded Material, (1996) 41–46.
[44] Z. H. Jin and G. H. Paulino, Transient thermal stress analysis of an edge crack in a functionally graded material, International Journal of Fracture 107 (2001) 73–98.
[45] A. Shahba, R. Attarnejad, M. Tavanaie Marvi and S. Hajilar, Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites: Part B.  42(4) (2011) 801-808.
[46] F. Delale and F. Erdogan. The crack problem for a nonhomogeneous plane, ASME J Appl Mech, 50 (1983) 609–614.
[47] N. Kim II, S.S. Jeon, M.Y. Kim, An improved numerical method evaluating exact static element stiffness matrices of thin-walled beam-columns on elastic foundations, Computers and structures, 83(23-24) (2005) 2003-2022.
[48] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics, a review, Applied Mechanics Reviews 49: 1-28.
[49] Shu C. Differential Quadrature and Its Application in Engineering. Sprimger; 2000.
[50] Zong Z, Zhang Y. Advanced Differential Quadrature Methods. Chapman & Hall/CRC; 2009.
[51] M. Soltani, B. Asgarian, F. Mohri, Elastic instability and free vibration analyses of tapered thin-walled beams by power series method, Journal of constructional steel research, 96 (2014) 106-126.
[52] ANSYS, Version 5.4, Swanson Analysis System, Inc, 2007.