THERMAL BUCKLING OF RECTANGULAR PLATES WITH DIFFERENT TEMPERATURE DISTRIBUTION USING STRAIN ENERGY METHOD Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 8, No. 2, PP 1 -11 (2012) Restrained Edges Effect on the Dynamics of Thermoelastic Plates under Different End Conditions Wael R. Abdul-Majeed* Muhsin J. Jweeg** Adnan N. Jameel*** Department of Mechatronics Engineering/Al-Khawarizmi College of Engineering/ University of Baghdad * ** College of Engineering/University of Al-Nahrain *** Department of Mechanical Engineering/College of Engineering/University of Baghdad (Received 22 March 2011; accepted 9 January 2012) Abstract Frequency equations for rectangular plate model with and without the thermoelastic effect for the cases are: all edges are simply supported, all edges are clamped and two opposite edges are clamped others are simply supported. These were obtained through direct method for simply supported ends using Hamilton’s principle with minimizing Ritz method to total energy (strain and kinetic) for the rest of the boundary conditions. The effect of restraining edges on the frequency and mode shape has been considered. Distributions temperatures have been considered as a uniform temperature the effect of developed thermal stresses due to restrictions of ends conditions on vibration characteristics of a plate with different will be investigated. it is noticed that the thermal stress will increase with increasing the heatnig temperature and that will cause the natural frequency to be decreased for all types of end conditions and for all modes of frequency. Keywords: Thermoelasticity, thin plate, ends condition, mode shape, natural frequency. 1. Introduction Thermoelasticity is concerned with questions of equilibrium of bodies treated as thermodynamic systems whose interaction with the environment is confined to mechanical work, external forces, and heat exchange. Because of constraints, a non -uniform temperature distribution in a component having a complex shape usually gives rise to thermal stresses. It is essential to know the magnitude and effect of these thermal stresses when carrying out on rigorous design of such components. The thermal stresses alone and in combination with the mechanical stresses produced by the external forces will be effect on dynamics properties of apart such as natural frequency and mode shape . Naji, et al. [1] studied the thermal stresses generated within a rapidly heated thin metal plate when a parabolic two-step heat conduction equation is used. The effect of different design parameters on the thermal and stress behavior of the plate is investigated. Al-Huniti, et al. [2] investigated the thermally induced vibration in a thin plate under a thermal excitation .The excitation is in the form of a suddenly applied laser pulse (thermal shock). The resulting transient variations of temperature are predicted using the wave heat conduction model (hyperbolic model), which accounts for the phase lag between the heat flux and the temperature gradient. The resulting heat conduction equation is solved semi analytically using the Laplace transformation and the Riemann sum approximation to calculate the temperature distribution within the plate. The equation of motion of the plate is solved numerically using the finite difference technique to calculate the transient variations in deflections. Norris and Photiadis [3] enabled direct calculation of thermoelastic damping in vibrating elastic solids. The mechanism for energy loss is thermal diffusion caused by inhomogeneous deformation, flexure in thin plates. The general result is combined with the Kirchhoff assumption to obtain a new equation for the flexural vibration of thin Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 2 plates incorporating thermoelastic loss as a damping term. The thermal relaxation loss is inhomogeneous and depends upon the local state of vibrating flexure, specifically, the principal curvatures at a given point on the plate. The influence of modal curvature on the thermoelastic damping is described through a modal participation factor. The effect of transverse thermal diffusion on plane wave propagation is also examined. It is shown that transverse diffusion effects are always small provided the plate thickness. Tran a, et al. [4] studied the thermally induced vibration and its control for thin isotropic and laminated composite plates. The structural intensity (SI) pattern of the plates which have different material orientations and boundary conditions was analyzed. The thermoelasticity simulation is performed using the finite element method. It shows that the structural energy flows are dependent on the material structures as well as the boundary conditions for a prescribed thermal source. The position to attach a damper for controlling the thermally induced vibration is investigated based on the virtual sources and sinks of the SI patterns. 2. Analytical Study The plate analyzed has usually been assumed to be composed of a single homogeneous and isotropic material with shape and dimensions as in Fig. (1) [5]. Fig. 1.Schematic Diagram of Thin Plate. 3. Boundary Conditions General closed – form solutions are given of a thermoelastic rectangular plate with various elementary boundary conditions on each of the four edges. Appendix A collect some important combinations of end boundary conditions. [Let the plate be placed in a coordinate system with the origin at it center and the edge width (a) be parallel to x – axis and and the edge width (b) be parallel to y as in Fig. (1) 4. Natural Frequency and Mode Shape of dynamic Thermoelastic plates Free, transverse vibrations of the thermoelastic structural with neglecting the effect of in plane vibrations are studied with different end boundary conditions under uniform temperature distribuation. 4.1. All Edges are Simply Supported The general governing differential equation of free vibration of thermoelastic plate is represented by [6]: …(1) Where )1(12 2 3   Eh D , and the quantities    2/ 2/ )( h h t dzTEN     2/ 2/ )( h h t zdzTEM  …(2) Which represents t he thermal stress resultants . T h e n t h e b o u n da r y c o n d i t i o n s f or t h e deflection w are represented in Appendix C 00   axx ww , 00   byy ww 0 2 2 2 0 2        x w x w axx , 0 2 2 2 0 2        y w y w byy The initial conditions assuming the plate initially at rest in the refrence position ,are given by 0)0,,()0,,(     yx t w yxw ax 0 , by 0 …(3) The displacement function ),,( tyxw is approximated by means of the expansion [7]. natural frequency (Hz) yx w N y w N x w N M whwD xyyx t             2 2 2 2 22 4 2 1    b Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 3 b yn a xm wt tyxwtyxw m n mn mn    sinsinsin sin),(),,( 1 1       …(4) And the displacement function ),( yxw is assumed from functions, that satisfies identically the boundary conditions; these functions are different due to the types of end conditions at x and y axis and this will be studied . The plate will have uniform temperature c TT  …(5) substitution of Eq.(5) in Eq. (2) we have ct EhTN  0tM …(6) So that for all edges are restrained   1 t yx N NN 0 xy N …(7) with all edges are restrained ,substituting the thermal forces in Eq. (7) and the deflection from Eq. (4) into the governing differential equation of free vibration of thermoelastic plate in Eq.(1) noting that 0Mt , one obtains the following frequency equation.   222 2 2224 )()( 1 )()( mn t h b n a m v N b n a m D             …(8) for natuaral frequancy of plate without thermal load 0 t N  2222 4 4 2 nrm ha D mnf     …(9) Then )( )1( 222 2 2 22 nrm avh Nt mnfmn       …(10) Substituting Eq. (7) into Eq. (10) , the natuaral frequancy as a function of uniform temperature c T can be presented as )( )1( 222 2 2 22 nrm avh ETc mnfmn       …(11) And for restrained edges at x=0,a and unrestrained at y=0,b thermal forces will be   1 t x N N 0 Yxy NN …(12) and the natuaral frequancy will be 2 2 22 )1( avh mNt mnfmn     …(13) and the function of the uniform temperature c T will be 2 22 22 )1( av mETc mnfmn      …(14) 4.2. All Edges are Clamped To derive the differential equation for lateral vibration of rectangular thermoelastic plate a kinetic energy of the plate in edition to the total strain energy of the plate and apply the Hamilton's principle to derive the equation of motion. The kinetic energy due to the velocity w only is represented as  A dxdywhT 2 2 1  …(15) the Hamilton's principle for the plate undergoing small deflection can be set as [8]: 0)( 2 1  dtT t t strain …(16) Then the lagrangian of the plate from the above equation can be written as   dxdywhdxdy y w x w v M dxdy y w x w N y w N x w Ndxdy y w x w DL RA t xyy A x A                                                                   2 2 2 2 2 2 22 2 2 2 2 2 1 1 2 2 1 2 1  …(17) For free vibration the solution is assumed tyYxXAtyxw ji m i n j ij sin)()(),,( 1 1     …(18) Substituting Eq. (18) by Eq. (19) and minimizing the resulting lagrangian with respect to ij A ,we get Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 4     ij a bm i n j a b t ij m k n j a b xyyx a b AdxdyYXhdxdyYXYX v M A dxdyYYXXNYXNYXN dxdyYXYYXXYXD                                    0 0 222 1 1 0 01 1 0 0 2222 2222 0 0 )( )1( )2)(( )(2)(  …(20) This is the general frequency equation. With uniform temperature c T and all edges are restrained with the aid of Eq. (2) for thermal forces and thermal moments into general frequency equation we have:           a b a a bb t dxdyYXh dxdyYXYX v N dxdyYXYYXXYXD 0 0 22 0 0 0 2222 0 2222 2 )()( )1( )(2)(   …(21) The frequency of plate without thermal effect has the form       a b a b ijf dxdyYXh dxdyYXYYXXYXD 0 0 22 0 0 2222 2 )(2)(   …(22) Then with substituting the mode shape of clamped ends i X and j Y from Appendix C   )1( 2 2 3 22 122 vha rNt ijfij       …(23) With ijf  for free vibration of clamped plate   4 4 3 4 2 24 12 2 ha rrD ijf      ...(24) Then ij  terms of uniform temperature will be as:   )1( 2 2 3 22 122 va rETc ijfij       …(25) Where 21 , and 3  are calculated from Appendix C For clamped edges restraind at x=0,a and unrestrained at y=0,b )1( 2 2 122 vha Nt ijfij      …(26) In terms of temperature )1( 2 2 122 va ETc ijfij      …(27) 4.3. Edges are Clamped at x=0,a and Simply Supported at y=0,b The general frequency equation of clamped edges Eq. (20) are suitable for edges clamped at x=0,a and simply supported at y=0,b. With uniform temperature c T and all edges restrained with the aid of Eq. (2) for thermal forces and thermal moments into general frequency equation and arranged with substituting the mode shape of two clamped ends and two simply supported ends i X and j Y from Appendix C into above equations the result will be   )1( 2 2 3 22 122 vha rNt ijfij       ...(28) With ijf  for free vibration suitable for edges clamped at x=0,a and simply supported at y=0,b .   4 4 3 4 2 24 12 2 ha rrD ijf      …(29) Then ij  in terms of uniform temperature will be:   )1( 2 2 3 22 122 va rETc ijfij       …(30) Where 21 ,  and 3  calculated from Appendix C For clamped edges restraind at x=0,a and simply supported unrestrained at y=0,b )1( 2 2 122 vha Nt ijfij      …(31) In terms of temperature Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 5 )1( 2 2 122 va ETc ijfij      …(32) 5. Results and Discussions The sample of calculations was made on Aluminum 1060-H18 rectangular plate which has the mechanical and thermal properties given in appendix A respectively. Rectangular plate with three aspect ratio a/b (r = 1.2). and a/h ( =120) and owing constant magnitude of a=0.12 m has been considered. The effects of the uniform increase of temperature of plates (thermoelastic behavior) on the natural frequency and mode shapes with different three types of ends conditions have been studied. Figures (2), (3) and (4) show the effect of temperature rising on natural frequencies analytical magnitudes till it reaches the thermal buckling temperature for plates with all edges restrained. The types are SSSS, CCCC and CSCS respectively It is observed that the lowest natural frequencies of all types reach zero when the temperatures get to the thermal buckling temperature; also the first five natural frequencies of plates decreas with increasing the temperature. Second and third natural frequencies of CSCS plate have the same magnitudes almost. Figures (5), (6) and (7) show the effect of temperature rising on natural frequencies analytical magnitudes till it reaches the thermal buckling temperature for plates with edges at x=0,a restrained the types are SSSS, CCCC and CSCS respectively The lowest natural frequencies of all types reach zero when the temperatures has the thermal buckling temperature. The first five natural frequencies of plates decrease with increasing the temperature. The fifth natural frequency of SSSS plate will become the fourth natural frequency and vice versa when the temperature has magnitude close to 6 C 0 . Also CCCC natural frequencies have the same behavior of SSSS type but they are switching at magnitude close to 3 C 0 . CSCS natural frequencies have the switching behavior between second and third natural frequencies at magnitude close to 1 C 0 . Fig. 2. Effect of Temperature on First Five Natural Frequencies Magnitude on SSSS Plate, All Edges are Restrained. 4s r =1.2 0 200 400 600 800 1000 1200 1400 1600 1800 2000 4.5813 4.5 4 3 2 1 0 T (C) natural frequancy (Hz) mode 1,1 mode 2,1 mode1,2 mode 2,2 mode 3,1 SSSS Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 6 Fig. 3. Effect of Temperature on First Five Natural Frequencies Magnitude on CCCC Plate with All Edges are Restrained. Fig. 4.Effect of Temperature on First Five Natural Frequencies Magnitude on CSCS Plate, All Edges is Restrained. SSSS r =1.2 0 200 400 600 800 1000 1200 1400 1600 1800 2000 11.17831086420 T (C) n a tu ra l f re q u a n cy ( H z) mode 1,1 mode 2,1 mode1,2 mode 2,2 mode 3,1 Fig. 5.Effect of Temperature on First Five Natural Frequencies Magnitude on SSSS Plate, Edges at y=0, b are Unrestrained. CSCS 0 500 100 150 200 250 5.47 5 4 3 2 1 0 T (C) natural frequency (Hz) mode 1,1 mode 1,2 Mode2,1 mode 2,2 mode 3,1 natural frequancy (Hz) 4C r =1.2 0 50 1000 1500 2000 2500 6.86.5 6 5 4 2 0 T (C) natural frequancy (Hz) mode 1,1 mode 2,1 mode1,2 mode 2,2 mode 3,1 Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 7 Fig. 6.Effect of temperature on First Five Natural Frequencies Magnitude on CCCC Plate, Edges at y=0, b are Unrestrained. Fig. 7. Effect of Temperature on First Five Natural Frequencies Magnitude on CSCS Plate, Edges at y=0, b are Unrestrained. 6. Conclusions The following are the main summarized conclusions of this paper: 1. Thermal stresses have a significant influence on the natural frequency for the free boundary conditions compared with clamped boundaries, so that the boundary condition is one of the important factors that influence the vibration and mode shapes. 2. The lowest natural frequencies of all types reach zero when the temperatures has the thermal buckling temperature 3. The first five natural frequencies of plates decreasing with increasing of the uniform temperature of the plates for all types of ends conditions 4. In the case of the two opposite edges which are unrestrained, there is a switching between the modes of natural frequency when the temperature increases for each type of ends conditions. Nomenclature Latin Symbols A Area (mm 2 ) a, b Plate side length (mm) D Flexural rigidity of an isotropic plate (N.mm) E Modulus of elasticity of isotropic material (N/mm^2) 2C2 r 0 500 100 150 200 250 8.9468 7 6 4 2 0 T (C) Natural frequency (Hz) mode 1,1 mode 1,2 mode 2,1 mode 2,2 CSCS mode 3,1 CCCC 0 50 1000 1500 2000 2500 6.86.6 5 4 2 0 T (C) Natural frequency (Hz) mode 1,1 mode 2,1 mode1,2 mode 2,2 mode 3,1 Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 8 h Plate thickness (mm) i ,j Integer Mt Thermal bending moment (N.m) m,n Integer Nx, Ny Edge forces per unit length (N/m) Nxy Shearing forces per unit length (N/m) Nt Thermal forces per unit length (N/m) r Dimensional aspect ratio a/b (m/m) T Temperature (C 0 ), Kinetic energy of the element (J) t Time (sec) x, y, z Cartesian coordinates Greek Symbols nm  , Coefficients  Poisson’s ratio  Mass density (Kg/mm^3) strain  Strain energy stored in complete plate (J) ijijf  , Angular frequency without and with thermal effect (rad/s)  Dimensional aspect ratio side / thickness (m/m)  Coefficient of thermal expansion (1/C 0 ) w Deflection (mm) Abbreviations Symbols CCCC Clamped-Clamped-Clamped-Clamped CSCS Clamped-Simply-Clamped-Simply SSSS Simply-Simply-Simply-Simply 7. Refrences [1] Malak Naji ,M. Al-Nimr and Naser S. Al- Huniti “THERMAL STRESSES IN A RAPIDLY HEATED PLATE USING THE PARABOLIC TWO-STEP HEAT CONDUCTION EQUATION “ Journal of Thermal Stresses, 24:399-410, 2001 Taylor & Francis [2] Naser S. Al-Huniti, M. A. Al-Nimr AND M. M. Meqdad “THERMALLY INDUCED VIBRATION IN A THIN PLATE UNDER THE WAVE HEAT CONDUCTION MODEL” Journal of Thermal Stresses, 26: 943–962, 2003 Taylor & Francis Inc [3] A. N. Norris and D. M. Photiadis “Thermoelastic Relaxation in Elastic Structures, WITH Applications to Thin Plates” arXiv: cond-mat/0405323 v2 20 Nov 2004 [4] T.Q.N. Tran a, H.P. Lee a,b, and, S.P. Lim a “Structural intensity analysis of thin laminated composite plates subjected to thermally induced vibration” Composite Structures. Article in press. [5] William L. Ko “Predictions of Thermal Buckling Strengths of Hypersonic Aircraft Sandwich Panels Using Minimum Potential Energy and Finite Element Methods “, NASA Technical Memorandum 4643, May 1995 [6] V. I. Kozolv "thermoelastic vibrations of arectangular plate" Pirk. Mekh. ,vol. 8, pp. 445-448, April 1972 [7] J. S. Rao, "DYNAMICS OF PLATES", Narosa Publishing House, 1999. [8] A. W. Leissa, "recent research in plate VIBRATIONS", Complicating Effects ,Shock & Vib. Digest Vol. 19, No. 3, 1987 . Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 9 Appendices Appendix A Some Combinations of End Boundary Conditions deflection Mid-plane deformation symbol clamped Restrained unrestrained supported restrained unrestrained free restrained unrestrained Appendix B Mechanical Properties of Aluminum 1060-H18 Thermal Properties of Aluminum 1060-H18 Heat Capacity 0.9 J/g °C Thermal Conductivity 233 W/m °C Coefficient of Thermal expansion 2.34e-5/°C Convection Coefficient 2.5 W/m² °C Appendix C For SSSS ends condition xX ii sin , yY jj sin 00   axx ww , 00   byy ww , 0 2 2 2 0 2        x w x w axx , 0 2 2 2 0 2        y w y w byy Density 2705 kg/m³ Hardness, Brinell 35 Ultimate Tensile Strength 27 MPa Tensile Yield Strength 20 MPa Elongation at Break 6 % Modulus of Elasticity 69 GPa Poisson's Ratio 0.3 Fatigue Strength 44.8 MPa Machinability 30 % Shear Modulus 26 GPa Shear Strength 75.8 MPa q Wael R. Abdul-Majeed Al-Khwarizmi Engineering Journal, Vol. 8, No.2, PP 1-11 (2012) 10 For CCCC ends condition )cosh(cossinhsin xxxxX iiiiii   )cosh/(cos)sinh(sin aaaa iiiii   )cosh(cossinhsin yyyyY jjjjjj   )cosh/(cos)sinh(sin bbbb jjjjj   0 0   axx ww , 0 0   byy ww , 00        x w x w axx , 0 0        y w y w byy For SCSC ends condition )cosh(cossinhsin xxxxX iiiiii   )cosh/(cos)sinh(sin aaaa iiiii   , yY jj sin 0 0   axx ww , 0 0   byy ww , 00        x w x w axx , 0 2 2 2 0 2        y w y w byy Where a i  and b j  are the roots of the above equations The roots of SSSS ends condition are a m i    , b n i    The roots of CCCC ends condition are For i=1 , j=2,3,4,…. For i=1 , j=1 )2(3.12 37.4 )5.0( 112 3 1       i For i=2,3,4,… j=1 )2()2( )5.0( )5.0( 33112 3 1       j i For i=2,3,4,. j=2,3,4,….\ )2()2( )5.0( )5.0( 33112 3 1       j i For i=2,3,4,. j=2,3,4,…. The roots of CSCS ends condition are For i=1 , j=1, 2, 3,.. 22 112 3 1 )2( )5.0(    j j i    For i=2,3,4,… j=1,2,3…. q q 3.151 73.4 2 31     )2(3.12 )5.0( 73.4 332 3 1       j 22 2 3 1 3.12 73.4    j j    (2012 )1 - 11 ، صفحت2، العذد 8 مجلت الخىارزمي الهىذسيت المجلذوائل رشيذعبذ المجيذ 11 حاثير الحافاث المحذدة مه الحركت على ديىاميكيت الصفائح المروت حراريا ححج ظروف وهاياث مخخلفت ***عذوان واجي جميل** محسه جبر جىيج* وائل رشيذ عبذ المجيذ جايؼت بغذاد/ كهٍت انهُذست انخىاسصيً/ قسى هُذست انًٍكاحشوَكس* جايؼت انُهشٌٍ/ كهٍت انهُذست ** جايؼت بغذاد/ كهٍت انهُذست/ قسى انهُذست انًٍكاٍَكٍت*** الخالصت كم , كم انُهاٌاث راث اسُاد بسٍط : صٍغ يؼادنت انخشدد انطبٍؼً نصفائح يسخطٍهت انشكم يغ وبذوٌ حاثٍش انًشوَت انحشاسٌت نحاالث انُهاٌاث انخانٍت انُهاٌاث يثبخت , و َهاٌخٍٍ يخقابهخٍٍ باسُاد بسٍط وَهاٌخٍٍ يثبخخٍٍ حى اٌجادها يٍ خالل انحم بانطشٌقت انًباششة نهُهاٌاث باسُاد بسٍط , وباسخخذاو يباديء حاثٍش حثبٍج انُهاٌاث افقٍا بىجىد دسجت حشاسة يُظًت انخىصٌغ ػهى انخشدداث انطبٍؼٍت . هايهخىٌ وانخخفٍض بطشٌقت سحض نهطاقت انكهٍت نباقً اَىاع انُهاٌاث وشكم انخشدد حى دساسخها كًا حى انخؼشف ػهى حاثٍش حىنذ االجهاداث انحشاسٌت انُاحجت يٍ حثبٍج انُهاٌاث افقٍا ػهى خىاص االهخضاصاث وحى يالحظت اٌ االجهاداث انحشاسٌت انًخىنذة حضداد يغ اصدٌاد دسجت حشاسة انخسخٍٍ وهزا ٌىدي انى َقصاٌ فً انخشدداث انطبٍؼٍت نكم اَىاع انُهاٌاث ونكم اشكال انخشدداث .انطبٍؼٍت