تحلیل خمشی صفحات مستطیلی ضخیم همسانگرد جانبی واقع بر بستر الاستیک دو پارامتری

نوع مقاله : مقاله کامل پژوهشی

نویسندگان

دانشکده مهندسی عمران، دانشگاه صنعتی نوشیروانی بابل

چکیده

در این پژوهش، حل دقیق خمش صفحات ضخیم همسانگرد جانبی واقع بر بستر الاستیک دو پارامتری با استفاده از توابع پتانسیل تغییر مکان ارائه می­شود. برای این منظور با استفاده از معادلات سه­بعدی الاستیسیته، دستگاه معادلات حاصل از شش معادله کرنش- جابه­جایی، شش معادله ساختاری و سه معادله تعادل در هم ادغام و بر حسب تغییرمکان­ها بیان شده است. سپس با استفاده از توابع پتانسیل جابه­جایی، پانزده معادله به دو معادله دیفرانسیل پاره­ای مرتبه دو و چهار مستقل از یکدیگر بر حسب دو تابع پتانسیل تغییر مکان تبدیل گردیدند. حل معادلات حاکم به روش جداسازی متغیرها و اعمال دقیق شرایط مرزی صورت پذیرفته و بر اساس آن پاسخ تغییرمکان­ها، تنش­ها، برش­ها و نیز لنگرها در صفحه بر روی بستر دوپارمتری به دست آمدند. نتایج در حالت خاص برای صفحه نازک همسانگرد و صفحه نسبتاً ضخیم همسانگرد با جواب­های موجود مقایسه گردیدند که نشان­دهنده صحت و درستی نتایج می­باشند. در ادامه رفتار خمشی چهار ماده همسانگرد جانبی منیزیم، اپوکسی گرافیت، اپوکسی شیشه­ای، الیاف کربنی و نیز و فولاد به عنوان یک ماده همسانگرد کامل با یکدیگر مقایسه شد. نتایج حاکی از آن است که منیزیم و الیاف کربنی به ترتیب بیش­ترین و کم­ترین میزان تغییرات تغییرمکان را در مقابل افزایش ضخامت از خود نشان می­دهند.

کلیدواژه‌ها


عنوان مقاله [English]

Bending Analysis of Transversely Isotropic Thick Rectangular Plates on Two-Parameter Elastic Foundation

نویسندگان [English]

  • Ghazaleh Samadi
  • Bahram Navayi Neya
  • Parvaneh Nateghi Babagi
Faculty of Civil Engineering, Babol Noshirvani University of Technology
چکیده [English]

The analysis of plates on elastic foundation has a wide range of applications in various fields of engineering, such as civil, mechanical, aerospace, nuclear and marine engineering. Due to soil behavior is dependent on numerous factors, providing full and comprehensive model for foundation is very complicated. Therefore, in order to express properties of the substrate are used the simpler models based on the elastic properties. The simplest of those models, which was presented by winkler in 1867, assumes that the soil medium containing a system of independent spring (Liew et al, 1996). To increase the accuracy of modeling two-parameter models arose that the effect of shear stresses in the substrate is also included. In these models a new parameter is proposed to establish a mechanical connection between the independent springs. An approximate solution for the bending of moderately thick rectangular plates using numerical differential quadrature method was proposed by Han and Liew (1997). Teo and Liew (2002), presented a solution for analysis of shear deformable rectangular plates on Pasternak foundations by using differential cubature method.
Displacement potential functions (DPF) method is one of the effective and efficient techniques that can be used to solve elasticity problems. The most advantage of DPF method is that the system of differential equations is uncoupled or at least simplified. This method has been used for 3D analysis of plates by a number of researchers such as Nematzadeh et al (2010) and Moslemi et al (2016) for the bending and buckling solution, respectively.

کلیدواژه‌ها [English]

  • thick rectangular plate
  • transversely isotropic
  • two-parameter elastic foundation
  • potential functions
اخوان ح، علی­بیگلو ا، "تحلیل کمانش ورق­های ضخیم مستطیلی برروی بستر الاستیک دو پارامتری، تحت بار صفحه­ای یکنواخت با چهار تکیه­گاه ساده"، پانزدهمین کنفرانس سالانه بین­المللی مهندسی مکانیک، دانشگاه صنعتی امیرکبیر، 1386، تهران، 25-27.
صفاری ح، حسینی خرمی س ا، "حل معادله دیفرانسیل حاکم بر تعادل صفحات با استفاده از موجک هار"، سومین کنگره ملی مهندسی عمران، 1386.
نوائی­نیا ب، "حل دقیق ارتعاش آزاد صفحات همسانگرد مستطیلی ضخیم بر روی تکیه­گاه­های ساده با استفاده از توابع پتانسیل تغییر مکان"، مهندسی عمران شریف، بهار 1393، 30-2 (1)، 33-41.
نوبختی ص، محمدی اقدم، م، "تحلیل استاتیکی ورق رایسنر روی بستر الاستیک دو پارامتره با شرایط مرزی گوناگون"، هفدهمین کنفرانس سالانه بین­المللی مهندسی مکانیک، 1388، اردیبهشت، دانشگاه تهران، 29-31.
Al-Hosani K, “A non-singular fundamental solution for boundary element analysis of thick plates on Winkler foundation under generalized loading”, Computers and Structures, 2001, 79 (31), 2767-2780.
Al-Hosani K, Fadhil S, El-Zafrany A, “Fundamental solution and boundary element analysis of thick plates on Winkler foundation”, Computers and Structures, 1999, 70, 325-336.
Ardeshir Behrestaghi A, Eskandari-Ghadi M, “Two layers half-space transversely isotropic medium under horizontal load on surface in frequency domain”, Journal of Civil and Surveying Engineering, 2009, 43 (1), 1-13.
Bergmann LA, Hall JK, Lueschen GGG, McFarland DM, “Dynamic Green’s function for Levy plates”, Sound Vibration, 1993, 162, 281-310.
Bezine G, “A new boundary element method for bending of plates on elastic foundations”, International Journal of Solids and Structures, 1988, 24 (6), 557-565.
Costa JA, Brebbia CA, “The boundary element method applied to plates on elastic foundation”, Engineering Analysis, 1985, 2, 174-183.
Ding H, Chen W, Zhang L, “Elasticity of transversely isotropic materials”, Springer Science and Business Media, 2006.
El-Zafrany A, Fadhil S, “A modified Kirchhoff theory for boundary element analysis of thin plates resting on two-parameter foundation”, Engineering Structures, 1996, 18 (2), 102-114.
Erattl N, Akoz AY, “The mixed finite element formulation for the thick plates on elastic foundations”, Computers and Structures, 1997, 64 (4), 515-529.
Eskandari-Ghadi M, “A complete solution of the wave equations for transversely isotropic media”, Journal of Elasticity, 2005, 81, 1-19.
Eskandari-Ghadi M, Amiri-Hezaveh A, “Wave propagations in exponentially graded transversely isotropic half-space with potential function method”, Mechanics of Materials, 2013.
Han JB, Liew KM, “Numerical differential quadrature method for Reissner/Mindlin plates on two parameter foundations”, International Journal of Mechanical Sciences, 1997, 39 (9), 977-989.
Hayashi K, “Theory of beams on elastic foundation”, Springer-verlag, German, 1921.
Henwood DJ, Whiteman JR, Yettram AL, “Fourier series solution for a rectangular thick plate with free edges on an elastic foundation”, International Journal for Numerical Methods in Engineering, 1982, 18 (12), 1801-1820.
Hetenyi M, “Beams on Elastic Foundation: theory with applications in the fields of civil and mechanical engineering”, University of Michigan, Michigan, 0-472-08445-3, 1971.
Huang MH, Thambiratnam DP, “Analysis of plates resting on elastic supported and elastic foundation by finite strip method”, Computers and Structures, 2001, 79, 2547-2557.
Huang ZY, Lu CF, Chen WQ, “Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations”, Composite Structures, 2008, 85, 95-104.
Kamal K, Durvasula S, “Bending of Circular Plate on Elastic Foundation”, Journal of Engineering Mechanics, 1983, 109 (5), 1293-1298.
Kerr A, “Elastic and Viscoelastic Foundation Models”, Journal of Applied Mechanics, 1964, 491-498.
Khojasteh A, Rahimian M, Eskandari-Ghadi M, “3D static analysis of a transversely isotropic half space”, Journal of Faculty of Technology, University of Technology, 2006, 40 (5), 611-624.
Lam KY, Wang CM, He XQ, “Canonical exact solutions for Levy-plates on two-parameter foundation using Green’s functions”, Engineering Structures, 2000, 22, 364-378.
Lekhnitskii SG, “Theory of elasticity of an anisotropic body”, Mir publishers Moscow, 1981.
Leon Sde, Paris F, “Analysis of thin plates on elastic foundations with boundary element method”, Engineering Analysis with Boundary Elements, 1989, 6 (4), 192-196.
Li Q, Soric J, “A locking-free meshless local Petrov Galerkin formulation for thick and thin plates”, Journal of Computational Physics, 2005, 208 (1), 66-79.
Liew KM, Han JB, Xiao ZM, Du H, “Differential quadrature method for Mindlin plates on Winkler foundations”, International Journal of Mechanical Sciences, 1996, 38 (4), 405-421.
Liu FL, “Rectangular thick plates on Winkler foundation: differential Quadrature element solution”, International Journal of Solids and Structures, 2000, 37, 1743-1763.
Matsunaga H, “An application of a two dimensional higher-order theory for the analyses of a thick elastic plate”, Computers and Structures, 1992, 45, 633-648.
Matsunaga H, “Buckling instabilities of thick elastic plates subjected to in-plane stress”, Journal of computer and Structures, 1997, 62, 205-214.
Matsunaga H, “Free vibration and stability of thick elastic plate subjected to in-plane forces”, International Journal of Solids and Structures, 1994, 31, 3113-3124.
Moslemi A, Navayi Neya B, Vaseghi Amiri J, “Benchmark solution for buckling of thick rectangular transversely isotropic plates under biaxial load”, International Journal of Mechanical Sciences, 2017, 131-132, 356-367.
Moslemi A, Navayi Neya B, Vaseghi Amiri J, “3-D Elasticity Buckling Solution for Simply Supported Thick Rectangular Plates using Displacement Potential Functions”, Applied Mathematical Modelling, 2016.
Nematzadeh M, Eskandari-Ghadi M, Navayi Neya B, “An Analytical Solution for Transversely Isotropic Simply Supported Thick Rectangular Plates using Displacement Potential Function”, Journal of Strain Analysis, 2010, 46,120-141.
Nobakhti S, Aghdam MM, “Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips”, European Journal of Mechanics and Solids, 2011, 30, 442-448.
Ozgan K, Daloglu A, T, “Effect of transverse shear strains on plates resting on elastic foundation using modified Vlasov model”, Thin-Walled Structures, 2008, 46, 1236-1250.
Pan B, Li R, Su Y, Wang B, Zhong T, “Analytical bending solutions of clamped rectangular thin plates resting on elastic foundations by the simplistic superposition method”, Applied Mathematics Letters, 2013, 26, 355-361.
Pasternak PL, “On a new method of analysis of an elastic foundation by means of two foundation constants”, Gps. Izd. Lit. Po Strait. I Arkh. (In Russian), 1954.
Qian LF, Batra RC, Chen LM, “Elastostatic deformations of a thick plate by using a higher-order shear and normal deformable plate theory and two meshless local Petrov-Galerkin (MLPG) Methods”, CMES, 2003, 4, 161-175.
Rahimian M, Eskandari-Ghadi M, Pak RYS, Khojasteh A, “Three dimensional dynamic analysis of a transversely isotropic half-space”, ASCE Journal of Engineering Mechanics, 2007, 133,134-1145.
Sadd MH, “Elasticity: theory, applications, and numeric", Academic Press, 2009.
Silva ARD, Silveira RAM, Goncalves PB, “Numerical methods for analysis of plates on tensionless elastic foundations”, International Journal of Solids and Structures, 2001, 38, 2083-2100.
Szilard R, “Theories and Applications of Plates Analysis: Classical, Numerical and Engineering Methods”, John Wiley & sons. Inc., 2004.
Teo TM, Liew KM, “Differential cubature method for analysis of shear deformable rectangular plates on Pasternak foundation”, International Journal of Mechanical Sciences, 2002, 44, 1179-1194.
Tian B, Li R, Zhong Y, “Integral transform solutions to the bending problems of moderately thick rectangular plates with all edges free resting on elastic foundations”, Applied Mathematical Modelling, 2015, 39 ,128-136.
Voyiadjis GZ, Kattan PI, “Thick rectangular plates on an elastic foundation”, Journal of Engineering Mechanices, 1986, 112, 1218-1240.
Wang CM, Lam KY, He XQ, “Solutions for Timoshenko Beams on Elastic Foundations Using Green’s Functions”, Mechanics of Structures and Machines, 1998, 26 (1), 101-113.
Wang J, Wang X, Huang M, “Fundamental solutions and boundary integral equations for Reissner’s plates on two parameter foundation”, International Journal of Solids and Structures, 1992, 29 (10), 1233-1239.
Wang W, Shi MX, “Thick plate theory based ongeneral solutions of elasticity”, Acta Mechanica, 1997, 123, 27-36.
Winkler E, “Die Lehre von der Elasticitaet und Festigkeit”, 1867.