Bending and Free Vibration Analysis of Isotropic Plate Using Refined Plate Theory

Transcription

1 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 4 Bending nd Free Vibrtion Anlsis of Isotropic Plte Using Refined Plte Theor I.I. Sd, S.B. Chiklthnkr nd V.M. Nndedkr Abstrct--- In this pper, Trigonometric sher deformtion theor is pplied for bending nd free vibrtion nlsis of thick plte. In this theor in plne displcement field uses sinusoidl function in terms of thickness coordinte. It ccounts for relistic vrition of the trnsverse sher stress through the thickness nd stisfies the sher stress free surfce conditions t the top nd bottom surfces of the plte. The theor obvites the need of sher correction fctor like other higher order or equivlent sher deformtion theories. Simpl supported thick isotropic plte is considered for detil numericl stud. Nvier s solution technique is emploed for the nlticl solution. The results re obtined for displcements, stresses nd nturl bending mode frequencies nd compred with those of other refined theories nd ect solution from theor of elsticit. Kewords--- Nturl Frequencies, Sher Correction Fctor, Sher Deformtion, Trnsverse Sher Stress I. INTRODUCTION I n clssicl plte theor, it is ssumed tht line which is norml to the neutrl surfce before deformtion remin stright nd norml to the neutrl surfce fter deformtion. This ssumption results in under-estimtion of deflection nd over-estimtion of nturl frequencies nd buckling lods. The errors in deflection, stresses, nturl frequencies nd bulking lods re even higher for pltes mde out of dvnced composites like grphite epo; boron epo etc,. in the erl ds [1-7] the clssicl plte theor ws etended for the nlsis of the composite structures. Lter the importnce of sher effect in plte bending ws relized nd the higher order theories were developed which tke trnsverse stresses nd strins into ccount. Higher order theories im t improving the ccurc b incorporting trnsverse strins/stresses in the formultion. For higher order theories, primril two tpes of pproches re vilble. In the first one the stresses re treted s primr vrible nd the displcements re found. In the second pproch displcements re treted s primr vribles nd the stresses re found. I.I. Sd, P.G., Student, Deprtment of Mechnicl Engineering, Government College of Engineering, Aurngbd, Indi, E-mil: sdimrni@gmil.com S.B. Chiklthnkr, Associte Professor, Deprtment of Mechnicl Engineering, Government College of Engineering, Aurngbd, Indi. V.M. Nndedkr, Professor, Deprtment of Production Engineering, SGGS Institute of Engineering nd Technolog, Nnded, Indi First Order Sher Deformtion Theor (FSDT) [8-9] is the strting point in the development of plte theories in which trnsverse sher effects were included. In this theor, constnt sher strin is ssumed cross the thickness nd it predicts verge sher stress. It lso requires sher correction coefficient, ccurte evlution of which is problem specific [1]. This theor hs been widel used for sttic, free vibrtion nd trnsient nlsis becuse of its simplicit nd good globl predictions. The plte theories re further refined b ssuming prbolic (higher order prbols) sher strin vrition cross the thickness nd these re clled s Higher order Sher Deformtion Theories (HSDT). It is ver well known tht HSDT gives more ccurte results nd ver close to three dimensionl (-D) elsticit solution for sttic loding conditions nd free vibrtions. Higher order theories re bsed on relistic displcement models, which give rise to nonliner distribution of in-plne, norml nd trnsverse sher strins. The higher order theories re quite involved nd re more complicted s compred to the CPT nd FSDT. It is ver importnt to develop higher order theor, which is simple nd es to use. Good review on plte theories is given in references [11-17] In the present work, emphsis hs been lid on specific development of Trigonometric Sher Deformtion Theor (TSDT) for plte nlsis. And the effectiveness of this theor is shown b ppling it to sttic nd dnmic problems of isotropic pltes. The displcement models contin trigonometric terms in ddition to clssicl plte theor terms. This displcement models is different from the generlized model of LO nd Christensen [18-19] which contins onl polnomil terms. Use of trigonometric functions to describe the plte behvior in thickness direction ws first proposed b Stein [-1] nd ws used for lminted bem nd post buckling nlsis of pltes. Redd nd Phn hd crried out stbilit nd vibrtion of isotropic, orthotropic nd lminted pltes ccording to higher order sher deformtion theor []. However, the present TSDT model differs from tht of Stein. In Stein s the top nd bottom sher stress conditions re not stisfied. Higher order theories, which consider plte properties in smered mnner, fil to interlminr stresses t interfces ccurtel. Trnsverse stresses re ver importnt in the composite nlsis since interlminr stresses re primril responsible for delmintion. This filure is overcome b developing the theories, which considers the behvior of individul ler in the nlsis, nd these tpes of theories re clled s ler-b-ler theories. DOI: /BIJIEMS.49 ISSN Bonfring

2 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 41 The etensive use of dvnced composite mterils in the vrious high performnce structures led to the development of refined theories for nlsis of such structures in order to ddress the correct structurl behvior. The objective of this pper is to present trigonometric sher deformtion theor for isotropic thick pltes. It includes the effect of trnsverse sher. Results obtined for uniforml distributed loding cse nd re compred with those of refined theories like Ghugl nd Sd[], Krishnmurth[4], Redd[5], Mindlin[9], Kirchhoff[] nd ect elsticit theor [6] vilble in the literture. II. THEORETICAL FORMULATION A. The Plte under Considertion: The plte under considertion occupies in O - z Crtesin coordinte sstem the region: ; b; h / z h / (1) Figure 1: Geometr of Thick plte Where,, z re Crtesin coordintes, nd b re the edge lengths in the nd directions respectivel nd h is the thickness of the plte. The plte is mde up of homogeneous, linerl elstic isotropic mteril with the principl mteril es prllel to the nd es in the plne of plte. The plte mteril obes generlized Hooke's lw. For isotropic plte the mteril properties re E 1 =E =E=GP, G 1 =G =G 1 =G, μ 1 =μ 1 =μ=. Assumptions mde in the Theoreticl Formultion: 1. The displcements re smll nd, therefore, strins involved re infinitesiml.. The in plne displcement u in direction s well s displcement v in direction ech consists of three prts: ) Displcement component nlogous to the displcement in clssicl plte theor of bending. b) Displcement component due to sher deformtion, which is ssumed to be sinusoidl in nture with respect to thickness coordinte, such tht the mimum sher stress occurs t neutrl is.. The trnsverse displcement w in z direction is ssumed to be function of nd coordintes onl. 4. The plte cn be subjected to trnsverse s well s in plne lods. B. The Displcement Field Bsed on the before mentioned ssumptions, the displcement field of the Present Refined Plte Theor (RPT) cn be epressed s follows: w h π z u= -z + sin φ (,,t ) π h w h π z v= -z + sin ψ (,,t ) π h w=w,,t ( ) Here u nd v re the in plne displcement components in the nd directions respectivel nd w is the trnsverse displcement in the z direction. The trigonometric function in terms of thickness coordinte in both the displcements u nd v is ssocited with the trnsverse sher stress distribution through the thickness of plte nd the functions φ(, t, ) nd ψ ( t,, ) re the unknown functions ssocited with the sher slopes. C. Strin-Displcement Reltionships Norml nd sher strins re obtined within the frmework of liner theor of elsticit using the displcement field given b (). These reltionships re given s follows: Norml Strin: u w h π z ε = = -z + sin π h v w h π z ε = = -z + sin π h w ε z = z φ () () Sher Strins: u v w h π z φ γ = + = + sin + π h u w π z γ = + = cos φ z h v w π z γ z = + = cos ψ (4) z h Stress-Strin Reltionships: For linerl elstic isotropic mteril, stresses, σ nd σ re relted to strins γ, ε nd ε sher stresses re relted to sher strins b the following constitutive reltions: σ Q11 Q1 ε σ Q Q = ε nd 1 Q 66 γ Q γ z 44 z = Q γ 55 (5) ISSN Bonfring

3 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 4 where σ, σ nd re the in plne stresses nd z, re the trnsverse sher stresses. The ( ε, ε, ) γ γ γ re the sher re norml strin component, (, z, ) strin components nd Qij re the stiffness coefficients [7]. D. Governing Equtions nd Boundr Conditions Using the epressions for strins nd stresses through (5) nd using the principle of virtul work, vritionll consistent governing differentil equtions nd boundr conditions for the plte under considertion cn be obtined. The principle of virtul work when pplied to the plte leds to: z= h/ = b = z = - h/ = = b = σ δε + σ δε + σ z δε z + δγ + δγ + δγ (, ) q δ wdd z z dddz = z= h/ = b = u v w + ρ δu + δv + δ w dddz t t t z = - h/ = = b = w w ( N δ w + N δ w) dd (6) = Where smbol δ denotes the vrition opertor. Integrting (6) b prts nd collecting coefficients of δ w, δφ nd δψ following governing differentil equtions nd the ssocited boundr conditions re obtined. The governing differentil equtions obtined re s follows: w w w Q11 A + 4 ( Q1A+ 4Q66A) + Q A 4 φ φ Q11B ( Q1 B+ Q66B) q 4 = w QB ( Q1 B+ Q66B) ρ A t 4 w φ w ρa + ρb + ρb + ρh t t Y t t w w φ Q11B + Q ( 1B+ Q66B) Q 11C φ Q66C ( Q1C+ Q66C) + Q55Dφ w φ ρb + ρc (7) t t w w QB + ( Q1 B+ Q66B) φ ( Q1C+ Q66C) Q66C Q C w + Q44Dψ ρb + ρc t t The ssocited consistent boundr conditions obtined re s below: Along the edge nd = w φ ω Q11 A Q 11B + Q1A Q 1B w or is prescribed. w φ w Q11 A + Q 11B Q 1 A + Q 1B w w φ 4Q66A 4Q 66A + Q 66B φ w φ + Q66B + Q66B + ρa ρb t t or w is prescribed. w φ w Q11 B + Q 11C Q1B + Q 1C or φ is prescribed. w φ Q66B + Q66C + Q66C (8) or ψ is prescribed. Along the edge nd = b: w φ ω Q1A Q 1B + Q A Q B w or is prescribed. w φ w ψ Q1 A + Q 1B Q A + Q B w φ ψ 4Q66 A + Q 66B + Q66B w ψ + ρa ρa or w is prescribed. t t w φ Q66B + Q66C + Q66C or φ is prescribed. w φ ω QB 1 + QC 1 QB + QC or ψ is prescribed. (9) Thus, the vritionll consistent governing differentil equtions nd boundr conditions re obtined. The vlues of integrtion constnts A,B,C nd D re mentioned below. h/ h/ h h, C= ( ) 1 π h/ h/ h/ h/ A= z dz = f z dz = ( z) h df h B= z f(z)dz =,D= dz (1) π = dz h/ h/ ISSN Bonfring

4 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 4 The fleurl behvior of the plte is described b the solution stisfing these equtions nd the ssocited boundr conditions t ech edge nd corner of the plte. III. ILLUSTRATIVE EXAMPLES Emple 1: A simpl supported isotropic rectngulr plte subjected to uniforml distributed lod. The rectngulr plte occuping the region given b the (1) is considered, the plte is subjected to uniforml distributed trnsverse lod, q(,) on surfce z = - h/ cting in the downwrd z-direction s given below: mπ nπ q(, ) = qmn sin sin (11) m= 1 n= 1 b Where, qmn re the coefficients of Fourier epnsion of lod, which re given b, 16q q mn = for m=1,,5,..., nd n=1,,5,..., mnπ (1) q mn = for m=,4,6,..., nd n=,4,6,..., The plte mteril re considered s E=1 GP nd µ=., where E is the Young s modulus nd µ is the Poisson s rtio. The governing differentil equtions nd the ssocited boundr conditions for sttic fleure of rectngulr plte under considertion cn be obtined directl from (7) through (9). The following re the boundr conditions of the simpl supported isotropic plte on the edges = nd =. Nvier Solution The following is the solution form for w (, ), φ(, ), nd ψ (, ) stisfing the boundr conditions given b the (8) nd (9) perfectl for plte with ll the edges simpl supported: mπ nπ w(, )= wmn sin sin m=1 n=1 b mπ nπ (1) φ(, )= φmn cos sin m=1 n=1 b mπ nπ ψ(, )= ψmn sin cos m=1 n=1 b where w mn, φ mn, nd ψmn re coefficients, which cn be esil evluted fter substitution of (1) in the set of three governing differentil equtions (6) nd solving the resulting simultneous equtions. Hving obtined the vlues of w mn, φmn, nd ψmn one cn then clculte ll the displcement nd stress components within the plte. Displcement: Substituting the finl solution for w(, ), φ (, ), nd ψ(, ) in the displcement field, the finl displcements cn obtin. The displcements re obtined s follows. mπ mπ nπ u= zw + f ( z) φ qmn cos sin b nπ mπ nπ v= zw + f ( z) ψ qmn sin cos b b m π n π w= wqmn sin sin b (14) IV. NUMERICAL RESULTS AND DISCUSSION Results obtined for displcements, stresses will now be compred nd discussed with the corresponding results of Refined Plte Theor (RPT),Higher order sher deformtion theor (HSDT) of Redd, clssicl plte theor (CPT) of Kirchhoff, first order sher deformtion theor (FSDT) of Reissner nd n ect solution for nlsis of plte Srinivs. The trnsverse displcement, in-plne nd trnsverse stresses re presented in the following non-dimensionl form for the purpose of presenting the results in this pper 1 Ew σ, σ,, z w=, 4 ( σ, σ, ) =, (, z ) = qh AR q AR qar ( ) ( ) ( ) Further, it m be noted tht nd obtined b constitutive reltions re indicted b nd nd CR EE the re indicted b nd when the re obtined b using equilibrium equtions. Similr nottions re lso used for z. The percentge error in result of prticulr theor is shown in prenthesis in subsequent tbles with respect to the result of ect elsticit solution which is clculted s follows vlue b prticulr theor-vlue b ect elsticit solution % error= X 1 vlue b ect elsticit solution Results obtined for displcements nd stresses re compred nd discussed with the corresponding results of clssicl plte theor (CPT), first order sher deformtion theor (FSDT), higher order sher deformtion theories (HSDTs), of vrious uthors nd the ect elsticit solution of plte. For the purpose of comprison, results were specill generted ccording to the ect elsticit solution [6]. Emple 1: Tble 1 shows comprison of deflection nd stresses for the simpl supported homogenous rectngulr isotropic plte subjected to uniforml distributed lod for /b.5, /b=1. nd h/.1. The present theor gives more ccurte vlue of deflection thn tht is given b other refined theories s compred to ect vlue. The theor gives n ect vlue of in-plne norml stress σ nd σ s compred to the vlue of ect solution for /b=1.(refer figure nd ). The vlue of in-plne sher stress obtined b present theor is in ecellent greement with the vlues of other refined theories. Trnsverse sher stress when obtined b constitutive reltions using present theor is on higher side, however, use of equilibrium equtions ield more ccurte results in cse of present theor s shown in figure 4. EE CR ISSN Bonfring

5 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 44 Tble shows comprison of non-dimensionl nturl bending mode frequencies of simpl supported isotropic plte. It cn be observed from Tble tht the present theor ields ecellent vlues of frequencies for lmost ll modes of vibrtion. The minimum % error predicted b present theor is.1 %. V. CONCLUSION In this pper Trigonometric Sher deformtion Theor is presented for isotropic plte nlsis. This theor gives relistic vrition of trnsverse sher stress through the thickness of plte nd stisfies the sher stress free boundr condition on the top nd bottom plnes of the plte. This theor obvites the sher correction fctor. From the numericl results it is conclude tht the vlues of trnsverse deflection, norml stress nd sher stress obtined b this theor re in ecellent greement with those of the ect theor. Also it is notified tht bending frequencies obtined b this theor re ver close to the frequencies of ect theor nd others theories. REFERENCES [1] S.P. Timoshenko nd S.W. Krieger, Theor of Pltes nd Shell, McGrw Hill, New York, [] E. Reissner nd Y. Stvsk, Bending nd stretching of certin tpe of heterogeneous elotropic elstic pltes, Journl of Applied Mechnics, Vol. 8, Pp 4-48, [] G.R. Kirchhoff, Uber ds gleichgewicht und die bewegung einer elstischen Scheibe, Journl für die reine und ngewndte Mthemtik (Crelle's Journl), Vol.4, Pp 51-88, 185. [4] S.G. Lekhnitskii, Anisotropic pltes, Gorden nd Brech, New York, [5] J. M. Whitne nd A.W. Leiss, Anlsis of heterogeneous nisotropic pltes, Journl of Applied Mechnics, Vol. 6, Pp 61-66, [6] R. Szilrd, Theor nd nlsis of pltes-clssicl nd numericl methods, Prentice-Hll Inc., Englewood Cliffs, New Jerse, 4. [7] J.M. Whitne nd N.J. Pgno, Sher Deformtion in Heterogeneous Anisotropic plte, Americn Societ of Mechnicl Engineers. Journl Applied Mechnics, Vol. 7, Pp , 197. [8] E. Reissner, The Effect of trnsverse Sher Deformtion on the Bending Elstic Plte, Trnsctions of the Americn societ of Mechnicl Engineers, Journl of Applied Mechnics, Vol. 1, Pp , [9] R.D. Mindlin, Influence of Rottor Inerti nd Sher on Fleurl Motions of Isotropic, Elstic Pltes, ASME Journl of Applied Mechnics, Vol. 18, Pp.1-8, [1] J.M. Whitne, Sher Correction Fctors for Orthotropic Lmintes under Sttic Lod, ASME Journl of Applied Mechnics, Vol.4, Pp [11] C.W. Bert, A criticl evlution of new plte theories pplied to lminted composites, Composite Structure., Vol., pp 9-47, [1] A.K. Noor nd W.S. Burton, Assessment of SHEAR Deformtion Theories for Multilered Composite Pltes, Applied Mechnics Reviews, Vol. 4, Pp.1-1, [1] R. K. Kpni, nd S. Reciti, Recent dvnces in nlsis of lminted Bems nd Pltes: Prt-II, AIAA Journl., Vol.7, Pp , [14] J.N. Redd, On the generliztion of displcement-bsed lminte theories, Applied Mech. Rev., Vol. 4 S1-S, [15] Mllikrjun nd T. Knt, A criticl review nd some results recentl developed refine theories of fibre reinforced lminted composites nd sndwiches, Composite Structures, Vol. 6, Pp. 9-1, 199. [16] J. N. Redd nd D. H. Robbins Jr., Theories nd computtionl Models for composite lmintes, Applied Mech. Rev., 47, No 6, Pp , [17] A.K. Noor, W.S. Burton nd C.W. Bert, Computtionl models for sndwich pnels nd shells, Applied Mech. Rev., Vol.49, Pp , [18] K.H. Lo, R.M. Christensen, nd E.M. Wu, A High-Order Theor of Plte Deformtion, Prt-1: Homogeneous Pltes, ASME Journl of Applied Mechnics, Vol. 44, Pp , [19] K. H. Lo, R.M. Christensen nd E.M. Wu, A High-Order Theor of Plte Deformtion, Prt-: Lminted Pltes, ASME Journl of Applied Mechnics, Vol. 44, Pp , [] M. Stein, Nonliner Theor for Pltes nd Shells Including the Effects of Trnsverse Shering, AIAA Journl, Vol. 4, Pp , [1] M. Stein, D.C. Jegle, Effect of Trnsverse Shering on clindricl bending, vibrtion buckling of lminted pltes, AIAA Journl, Vol. 5, No. 1, Pp.1-19, [] J.N. Redd, nd N.D. Phn, Stbilit nd Vibrtion of isotropic, orthotropic nd lminted pltes ccording to higher order sher deformtion theor, Journl of Sound nd Vibrtion, Vol. 98, Pp , [] Y.M. Ghugl nd A.S. Sd, A sttic fleure of Thick Isotropic Pltes Using Trigonometric Sher Deformtion Theor, Journl of Solid Mechnics, Vol., Pp.79-9, 1. [4] A.V. Krishn Murt, Higher Order Theor for Vibrtions of Thick Pltes, AIAA Journl, Vol. 15, Pp , [5] J.N. Redd, A simple Higher-order Theor for lminted Composites Plte, Trnsction of the Americn Societ of Mechnicl Engineers, Journl Applied Mechnics, Vol. 51, Pp , [6] S. Srinivs, C.V. Jog Ro nd A.K. Ro, Bending, Vibrtion nd buckling simpl supported thick orthotropic rectngulr pltes nd lmintes, Interntionl Journl of Solids nd Structures Pp. 6: , 197. [7] R.M. Jones, Mechnics of composite Mteril, McGrw Hill Kogkush, Ltd., Imrn I.Sd. Aurngbd. D.O.B.-6//198 B.E.(Mech.),M.B.A.,PGDFT. E-mil-sdimrni@gmil.com ISSN Bonfring

6 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 45 Tble1: Comprison of deflection w t (=/, =b/,z=), in plne norml stresses σ t (=/, =b/,z=h/), σ t (=/, =b/,z=h/), in-plne sher stress (=, =,z=h/) nd trnsverse sher stress t (=, =b/,z=) in rectngulr isotropic plte subjected to uniforml distributed lod. /b h/ Theor w σ σ.5.1 Present Ghugl nd Sd[] Krishnmurth[4](HSDT) Redd[5] (HSDT) Mindlin[9] (FSDT) Kirchoff [] (CPT) Ect[6] Present Ghugl nd Sd[] Krishnmurth[4](HSDT) Redd[5] (HSDT) Mindlin[9] (FSDT) Kirchoff [] (CPT) Ect[6] Tble : Comprison of nturl frequencies ofω mn of simpl supported isotropic squre plte (m,n) Ect[6] Present Mindlin[9] Redd[5] Reissner CPT[] Ghugl nd Sd[] (1,1) (1,) (,) (1,) (,) (1,4) (,) (,4) (,4) (1,5) (,5) (4,4) (,5) CR EE mn mn h ρ ω = ω ; h/=.1 ;b/=1. ; E / E1.55; G1 / E1.69; G1 / E ; G / E ; μ1.4446; μ1.14 Q11 ISSN Bonfring

7 Bonfring Interntionl Journl of Industril Engineering nd Mngement Science, Vol., No., June 1 46 z/h Present Redd Ghugl FSDT CPT σ Figure : Comprison of in plne norml stresses σ t (=/, =b/,z=h/), in isotropic squre plte subjected to uniforml distributed lod(h/=.1). z/h Present Redd Ghugl FSDT CPT σ Figure : Comprison of in plne norml stresses σ t (=/, =b/,z=h/) in isotropic squre plte subjected to uniforml distributed lod(h/=.1) Present Redd Ghugl nd Sd FSDT CPT.1 z/h z Figure 4: Comprison of trnsverse sher stress t (=, =b/,z=) in isotropic squre plte subjected to uniforml distributed lod(h/=.1).. ISSN Bonfring