Available online at ScienceDirect. Procedia Engineering 144 (2016 )

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1 Available online at ScienceDirect Procedia Engineering (06 ) t International onference on Vibration Problems, IOVP 05 Termal stresses, deformations and vibrations of lates and sells - A nonlinear aroac Abstract Paritos Biswas* Ex-Princial, Jalaiguri-7350,West Bengal, India Some reliminary discussion is resented on te nonlinear analysis of late and sell structures under mecanical and termal loading. As an illustrative examle non-linear couled artial differential equations derived in te von Karman sense and extended to termal loading wit te inclusion of curvature for a skew rectangular anel as been emloyed to analyze termal stresses for movable edge boundary condition. 06 Te Te Autors. Publised by Elsevier by Elsevier Ltd. Tis Ltd. is an oen access article under te BY-N-ND license (tt://creativecommons.org/licenses/by-nc-nd/.0/). Peer-review under resonsibility of te organizing committee of IOVP 05. Peer-review under resonsibility of te organizing committee of IOVP 05 Keywords: Nonlinear, Termal stresses, sallow sells, skew sae, Galerkin rocedure.. Introduction In structural mecanics and in many brances of engineering mecanics and aeronautics, te widesread use of lates and sells are made to deal wit termal stresses, deformations, buckling and vibrations of lates and sells for wic tere arises te need for reliance uon different metods of analysis. Analytical tecniques in dealing wit suc roblems ave serious limitations because of difficulties in deriving closed-form solutions of nonlinear differential equations. Tin lates and sells of regular olygonal and irregular saes made of isotroic, ortotroic and sandwic materials are often subjected to different kinds of mecanical and termal loading and as a result suc structural comonents are rone to deformations, bucking and vibrations for wic roer analysis are required to be made and of great interest to designers, engineers, scientists and researcers. * orresonding autor. Tel.: & address:.biswas93@gmail.com Te Autors. Publised by Elsevier Ltd. Tis is an oen access article under te BY-N-ND license (tt://creativecommons.org/licenses/by-nc-nd/.0/). Peer-review under resonsibility of te organizing committee of IOVP 05 doi:0.06/j.roeng

2 0 Paritos Biswas / Procedia Engineering ( 06 ) Te bulk of classical aroac in Alied Mecanics rests on te assumtion tat te enomenon involved as been described by te linear matematical model by many researcers. However, wit te advent of modern tecnology and systems exosed to oressive oerational conditions induce large deflections, i.e., deflections tat are of te same order as te late and sell tickness and small comared to in-lane dimensions of te structures. Tus, wen te deflections are no longer small in comarison to te tickness but small comared to te in-lane dimensions, te middle surface strains must be considered in deriving te differential equations of tin lates and sell structures. In tis way one gets te nonlinear artial differential equations in te classical nonlinear teory. For te analysis of large deflections of lates von Karman s couled nonlinear artial differential equations ave extensively been emloyed by many a earlier researcer. Tese equations wic are of te fourt order wit resect to te unknown deflection W and stress function F enables one to determine w and F wit extension to termal loading and wit te inclusion of curvature for sallow sells. Tese equations can also be exressed in terms of dislacement comonents u, v and w and can conveniently been used for te nonlinear analysis of different kind of lates and sells. von Karman s equations are generally difficult to deal wit because of its couled nonlinearity and as yet no general solutions of tese equations are known. However, aroximate and different numerical and comutational metods ave been adoted for te solution of suc large deflections analysis of lates and sells. Tere is a galaxy of outstanding researc workers of wom detailed references of works may be available in [-3] wo emloyed von Karman equations to te analysis nonlinear beavior of tin lates, bot isotroic and ortotroic under mecanical and oter kinds of loading and subsequently extended to sallow sells wit te inclusion of curvature and termal loading arameter Due to te matematical comlexities, many researcers ave resorted to te use of different comutational and numerical tecniques to deal wit roblems of nonlinear dynamic analysis of lates and sells under termal stresses. Since von Karman equations are in te couled form, H.M.Berger []roosed quasi-linear artial differential equations in te decouled form wit extension to termal stresses for lates and sallow sells. For nonlinear analysis of lates and sells under stationary and non-stationary temerature distribution and for eated ortotroic lates basic governing equations ave been roosed by Biswas [ 5- ] Te outstanding works of N.Kamiya [-9 ] of Jaan sould be mentioned for eated sandwic lates and sells using Berger s aroximation. For irregular-saed lates and for lates of olygonal saes, comlex variable teory and conformal maing tecnique could conveniently be emloyed by Biswas [ 7 ]. Now resented below te use of nonlinear couled artial differential equations derived in te von Karman sense and extended to termal loading wit te inclusion of curvature for a skew rectangular anel to analyze termal stresses for movable edge boundary condition.. Basic governing equations w w w w w t F E N T xy x y R y R x () E t D W MT F, xx w, yy F, xy w, xy F, yy w, xx Were () F Stress function, E Young s Modulus, Sell Tickness, w Normal Dislacement omonent R, R k = Princial Radii of urvature of te Rectangular Panel kx y

3 Paritos Biswas / Procedia Engineering (06 ) t o-efficient of Linear Termal Exansion, ab, Sell Dimension Lalacian Oerator, Poisson s Ratio,, a b T x, y, z x, y z( x, y) (3a) 0 were T is te temerature distribution witin te late or sell as stated by Nowacki [] and, /,, /,, /,,, / xy TT xy T T (3b) 0 T T x y T T x y (3c) T and T = temeratures at te lower and uer media resectively. N T M t E Tx, y, zdz Termal Stress oule () T T x, y, z dz termal stress resultant or termal moment (5) Eqs. () and () are nonlinear in w and couled. If we coose w as in () below satisfying clamed-edge boundary conditions (), Eq. () for te stress function becomes linear in form and easily solvable as sown in te analysis for clamed movable-edge boundary conditions. Te stress function tus obtained and inserted into equation () results into a cubic equation determining te central deformation after alying te tedious but simle and lengty matematical calculations. 3. Transformation to skew co-ordinates Geometry of te Sell-Panel is sown in Fig.. Fig.. Geometry of te Sell-Panel We consider a skew sell anel of lengt a along te -axis (taken orizontally) and erieral widt b along te -axis, te origin is being located in te mid-lane at te centre of te sell anel. is te skew angle measured between te direction and te tangential to te direction. In terms of te above geometry and co-ordinates systems,, reresents skew (oblique) co-ordinates and xy,

4 06 Paritos Biswas / Procedia Engineering ( 06 ) are te rectangular co-ordinates, wic are related as follows X sin, Y cos (6), x x y sin y cos sin cos (7) (8) Accordingly in terms of oblique (skew) co-ordinates equation () transform into F AB D (9) were, E w w w A cos E w sin w sin w B R cos E w R D te NT (0). Metod of solution Te deformation mode sae satisfying te boundary conditions for clamed edges w W 0, at a w W 0, at b () is cosen in te form ww (cos )( cos ) 0 () We assume z - indeendent lanner temerature distribution in te form [Ref.0, age-65] T(,, z) x, y Tsin sin (3) 0 suc tat T,, z vanises around te eriery of te sell anel T z,, 0 wen a; b)

5 Paritos Biswas / Procedia Engineering (06 ) Using Eq. (7) and (8) and after re-arranging terms one gets F cos cos cos cos coscos sinsin coscos cos cos () were i ( i =,, 8) are known coefficients given in te Aendix. Te articular integral F of Eq. () is obtained in te form F b cos b cos b cos b cos b cos cos b sin sin b cos cos c bsinsin bcos cos b sin sin (5) were, b i ( I =,,.. 0 ) are also known coefficients and given in te Aendix. Te comlementary function of Eq. () is taken in te form Fc L N M (6) were te constants LM,, N contribute directly to te in-lane stresses N, N and N resectively and will be determined by using te boundary conditions as stated below Since te above in-lane stresses can be exressed in terms of stress function F, te following set of relaxed boundary conditions for movable edges are enforced [ 0- ]. b F, d 0 F a b a a a F, d 0 F b a b, d 0 b, d 0 (7-0) a b Using te above boundary conditions (7-0) to te comlete stress function F F F, c one gets LM N 0 () Terefore, te stress function is comletely determined for movable edges in te form,, F F () Exressions for termal stresses are given by N F F F cos, sin, sin, (3) N F, () from wic matematical exressions for te inlane stresses N / E and N / E can be derived.

6 08 Paritos Biswas / Procedia Engineering ( 06 ) Numerical Results and Discussion Numerical comutations and detailed discussion for te non-dimensional termal stresses for a skew anel could be resented for cross-variation of different arameters involved in te analysis and is excluded for brevity of sace in te resentation. 6. onclusion Skew lates and skew sell anels under termal stresses find wide alications in aerosace industry since te nose of an aircraft structure is generally non-rectangular and skew-saed and exosed to severe termal environments. Te stress function so derived from Eq. () can conveniently be emloyed in Eq. () and alication of Galerkin Procedure or using oter suitable comutational metods will lead to a cubic equation to determine te normal dislacement function in terms of te arameters involved in te analysis. Suc exercise is too rigorous and may constitute a searate aer. Aendix A. Te big exressions for te constant coefficients i [ i =,,3,.., 8 ] and also for b i [ i=,,3,., 0] in Eqs. () and (5) resectively are exressed below Ew Ew sin Ew cos ab Ra cos Ra Ew ab cos 0 3 Ew ab cos a 3 cos b 6 a b5 cos 5 6 a b6 cos 5 6 a 7 cos 7 3 b Ew Ew 0 0 ab cos Rb cos Ew sin E T 0 t 6 Rab cos cos a b a b cos b cos b 6 a 8 cos 7 3 b b b cos a 9 cos b a 0 cos b

7 Paritos Biswas / Procedia Engineering (06 ) a a a a sin sin b b b b a a a a sin sin b b b b b b b b a a a a 3 sin 6 8sin a a a a sin 6 8sin b b b b a a a a 5 6 sin 8sin b b b b b b b b a a a a 6 6 sin 8sin For te sake of brevity te formidable-looking exressions for te inlane stresses N / E and N / E derived from Eqs. (3) and () are omitted. On request from any interested Reader to te Autor by .biswas93@gmail.com, detailed numerical comutations and interretation of results may be sent. Also detailed bibliograical references may also be sent on request. References [] R.B.Hetnarski (Princial Editor) Encycloedia of Termal Stresses, Sringer,0. [] Wiltold Nowacki, Termoelasticity, Addision Wesley, NY, 96. [3].Y.ia, Nonlinear Analysis of lates, Mc-Graw Hill, 978. [] H.M. Berger, A New Aroac to te Analysis of Large Deflection of Plates, Journal of Alied Mecanics,ASME,955,, [5] P.Biswas, Large Deflection of a Heated Ortotroic Plates,, Indian Journal of Pure and Alied Mats., 9(0), (97), [6] P.Biswas, Nonlinear analysis of lates and sells under stationary and non-stationary temerature distribution, Transactions of te t.smirt-onference,lyon,france,997,august 7- [7] P.Biswas, Large Termal Deflections of Elastic Plates by onformal Transformations, Proceedings of nd International onference on Numerical Metods in Termal Problems, Venice (Italy), 98,,6-7. [8] P.Biswas and P.Kaoor, Nonlinear Free Vibrations of Ortotroic ircular Plates at Elevated Temerature, Journal of te Indian Institute of Science,Bangalore, 65(B), (98), [9] P.Biswas, and P. Kaoor, Nonlinear Free Vibrations of Triangular Plates at Elevated Temerature, Journal of te Indian Institute of Science,Bangalore, 65(B), (98), [0] P.Biswas, Large Deflection of a Heated ylindrical Sell, International Journal of Mecanical Science, 0, (978), 7-0. [] P.Biswas, Large Deflection of Heated Ortotroic ylindrical Sallow Sell, Defence Science Journal, 30(), (980), [] N.Kamiya, Simlified Field Equations for Ortotroic Sandwic Plates and Sallow Sells Under Moderately Large Deflections, Trans. JSM, 976, () [3] N.Kamiya, Governing Equations for Large deflections of Sandwic Plates, AIAA Journal,, (976), [] N.Kamiya, Nonlinear Bending Analysis of an Isotroic Sandwic Plates by Berger Metod, Proc. Sym. Al. omuter Metods in Engineering,, (977), [5] N.Kamiya, An Aroximate Aroac Nonlinear Termal Bending of Sallow Sandwic Sells and Flat Plates, Journal of Termal Stresses,, (978), [6] N.Kamiya, Analysis of te Large Termal Bending of Sandwic Plates by a Modified Berger metod, Journal of Strain Analysis, 3(), (978), 7-. [7] N.Kamiya, Large Termal Bending and Termal Buckling, Acta Tecnica SAV,, (98), [8] N.Kamiya, An Aroximate aroac to Nonlinear Termal Bending of Sallow Sandwic Sells and Flat Plates, Journal of Termal Stresses,, (987),

8 030 Paritos Biswas / Procedia Engineering ( 06 ) [9] N.Kamiya, Sawaki,Y. and Nakamura,Y, Nonlinear Bending Analysis of Heated Sandwic Plates and Sells by te Boundary Element Metod, Researc Mecanics, 8, (983), [0] M. Satyamoorty and K. A. V. Pandalai, Nonlinear Flexural Vibrations of Ortotroic Skew Plates, Journal of Sound and Vibration, (), (97),5-0. [] M. Satyamoorty and K. A. V. Pandalai, Large Amlitude Flexural Vibrations of Simly Suorted Skew Plates, AIAA Journal,, (973), 79-8