Three-Dimensional Elasticity Solution for Laminated Cross-Ply Panels Under Localized Dynamic Moment

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1 Tansaction B: Mechanical Engineeing Vol. 6, No. 3, pp. 9{39 c Shaif Univesity of Technology, June 009 Thee-Dimensional Elasticity Solution fo Laminated Coss-Ply Panels Unde Localized Dynamic Moment Abstact. A. Alibeigloo and M. Shakei Thee-dimensional elasticity solutions have been pesented fo a thick laminated coss-ply cicula cylindical panel. The cylindical panel is unde a dynamic localized patch moment, is simplysuppoted at all edges and has nite length. Odinay dieential equations with vaiable coecients ae fomulated using an expansion of Fouie seies applied to the displacement eld in cicumfeential and axial diection. The esulting odinay dieential equations ae solved by the Galekin nite element method. Numeical esults ae pesented fo [0/90/0], [90/0/90], [0/90/0/90/0], [0/90/0/90/0/90/0] and [0/90/0/90/0/90/0/90/0] stacking sequences. Keywods: Dynamic Patch moment Elasticity Panel Multi-layeed Coss-ply. INTRODUCTION Modeling of composite cylindical shells and panels is usually based on one of the following thee types of theoy: The Classical Laminated Theoy (CLT), shea defomation theoies, o the thee-dimensional theoy of elasticity. Because of the high atio of inplane Young's modulus to tansvese shea modulus, ignoance of the shell tansvese shea defomation in CLT can lead to seious eos, even fo thin cylindical shells/panels. These theoies also fail to accuately pedict the stesses and displacements in the vicinity of discontinuous loads and fo thick panels. To incease the ange of validity of classical shell theoies, the eects of shea defomation and tansvese nomal stesses ae incopoated in shea defomation theoies. Howeve, to establish the validity and limitations of these appoximate theoies, thee-dimensional elasticity solutions ae necessay. Vaious appoaches that have been used fo the analysis of composite shells ae eviewed in []. The plane stain defomation of tansvesely loaded,. Depatment of Mechanical Engineeing, Bu Ali Sina Univesity, Hamedan, P.O. Box , Ian.. Depatment of Mechanical Engineeing, Ami Kabi Univesity of Technology, Tehan, Ian. *. Coesponding autho. a beigloo@yahoo.com Received Febuay 007 eceived in evised fom 6 Septembe 007 accepted 7 Novembe 007 simply-suppoted coss-ply cylindical panels using a stess-function appoach was consideed and the esults wee povided in the fom of maximum displacements and stesses fo vaious stacking sequences in []. The moe geneal case of nite length cossply cylindical shells subjected to tansvese loading, vaying sinusoidally in both the axial and cicumfeential diections, was analyzed late by employing the Fobenius powe seies method [3,4]. Elasticity solutions have also been obtained fo simply-suppoted angle-ply cylindical shells undegoing: () Bending in the cicumfeential diection, alone, i.e. the genealized plane stain poblem [5] and () Axisymmetic defomation [6]. An exact closed-fom solution has been obtained in the st case which, as the second case, has been analyzed using Taylo seies. All of the above wok is esticted to smoothly vaying sinusoidally distibuted tansvese loads. The poblem of a simply-suppoted nite coss-ply cylindical shell pinched by diametically opposite localized adial foces was analyzed in [7]. It was shown that this analysis is a sevee test case fo judging the accuacy of the two-dimensional shell theoy fo the case of smooth tansvese loading. An exact theedimensional elasticity solution fo an innitely long, thick, tansvesely isotopic cicula cylindical shell unde a static patch load was investigated in [8]. In this pape, the bounday value poblem is educed to Bessel's dieential equation and the esult shows

2 30 A. Alibeigloo and M. Shakei that the tansvese shea stess distibution ove the shell thickness is not paabolic at o nea the vicinity of a highly discontinuous load. Elasticity solutions fo laminated othotopic cylindical shells subjected to a localized static moment wee studied in [9]. It was shown that a classical appoach fails to captue the localized defomations and stesses coectly, even fo quite thin shells. It has also been pointed out that localized moment loading is moe sevee and igoous than the cases of smooth and localized adial loading fo judging the ange of applicability of a two-dimensional laminate theoy. Shuvalov and Soldatos [0] used successive appoximation methods fo the thee-dimensional analysis of adially inhomogeneous tubes with abitay cylindical anisotopy. Chen et al. [] conducted a 3D coupled vibation analysis of uid-lled othotopic cylindical shells of functionally gaded mateials. The elasticity method was developed to study the bending and fee vibation of simply-suppoted angle-ply laminated cylindical panels in cylindical bending []. The dynamic stability analysis of thin, laminated panels unde static and peiodic axial foces is pesented, using the meshfee kp-ritz method [3]. The mesh-fee kenel paticle estimate is employed to appoximate the D tansvese displacement eld. Eects of lamination schemes such as the instability egions wee examined in detail. A eview of the published liteatue shows that the elasticity solution of laminated coss-ply cylindical panels unde a dynamic patch moment has not yet been investigated. Recently, the autho's have studied the esponse of multi-layeed anisotopic cylindical panels and shallow panels unde dynamic adial nomal loading with a constant distibution, using the theoy of elasticity [4,5]. In the pesent investigation, a theedimensional elasticity solution fo a thick laminated coss-ply cicula cylindical panel, simply-suppoted along fou edges and subjected to a dynamic patch moment, has been pesented. FORMULATION A laminated cylindical panel composed of N homogeneous othotopic thick layes is consideed (Figue ). The mateial axes coincide with the geometic axes, and z. The fou edges of the panel ae simplysuppoted and the constitutive equations of each laye ae stated as: 6 4 z z z = 6 4 C C C C C C C 3 C 3 C C C C " z " " z z : () Figue. Geomety and the coodinate system of laminated panel. The govening dieential equations of motion in the thee-dimensional theoy of elasticity + z @ + U : () Stain-displacement elations ae expessed as: " z = U " = z @ : (3) Combining Equations to 3, the govening equations, in tems of displacements fo each laye of a cylindical panel, become: C U + C66 k + C k U C3 = U

3 Thee-Dimensional Elasticity Solution Coss-Ply Panels 3 C k 66 C44 U + C U C U + @ U + C k 3 + C @! U C k 3 + Ck 3 C k = + C k 33 C @ U + C33 U z Ck = : (4) The simply-suppoted bounday conditions ae taken as: U = = z = 0 at = 0 U = z = z = 0 at z = 0 L: (5) Fo a laminate consisting of N laminae, the continuity conditions to be enfoced at any abitay inteio kth inteface can be witten as: ( ) k = ( ) k+ ( ) k = ( ) k+ ( z ) k = ( z ) k+ (6a) (U ) k = (U ) k+ (U ) k = (U ) k+ (U z ) k = (U z )k + (6b) whee suxes k and k + epesent the coesponding stesses and displacements at the kth and (k + )th laminae. Since the panel is unde the longitudinal patch moment M, applied in the fom of a linea vaiation of adial stess ( ) distibuted ove a small ectangula patch (L L p) Z (L+L p) and ( p) (+ p) on the oute suface, the bounday conditions on the oute and inne sufaces of the panel ae taken as: 6M = R b p L 3 (L Z) z = = 0 p at = R b (7a) = z = = 0 at = R a : (7b) SOLUTION The unknown displacements that satisfy the bounday conditions (Equations 5) ae expanded into a Fouie seies as follows: X X u = sin m sin P n z:u ( t) m= n= X X u = cos m sin P n z:u ( t) m= n= X X u z = sin m cos P n z:u z ( t) (8) m= n= whee: p n = n L m = m (m n = : : : ): Afte substituting Equation 8 into Equation 4, the patial dieential equations educe to odinay dieential equations with vaiable coecients. By consideing linea shape functions fo thee eld vaiable, U, U and U z, as: U = bn i N j c Ui U U = bn i N j c j Ui U j U z = bn i N j c Uzi (9) U zj and then by applying the fomal Galekin nite element method to the st govening odinay dieential equation as: Z j i 6 4 CP k nu x + C k Pn (U m U ) +C3P +C66 k ( mp n U m U x ) +C k U P nu # U N i d = 0 (0) 3 7 5

4 3 A. Alibeigloo and M. Shakei the following esult is obained: A 4 u i + B 4 u j + C 4 u i + D 4 u j + E 4 u zi + F 4 u zj + G 4 u zi + G 4 u zj = C55 i () whee A 4 B 4,.... and G 4 ae constants (Appendix A) and the indices i and j depend on the numbe of nodes in a adial diection. By integating the second and thid odinay dieential equations, two equations simila to Equation ae obtained. By changing N i to N j and then epeating the above pocedue, thee othe equations ae obtained. The esult is witten in the following dynamic nite element equilibium equation fo each non-bounday element: [M] k f Xgk + [K] k fxg k = ff (t)g k () whee: fxg k = fu i U i U zi U j U j U zj g : By efeing to the st equation of Equations 9 and using Equations and 3 and by expessing the petinent deivatives in backwad and fowad nite dieences fo kth and (k + )th layes, U ki, U ki and U zki can be obtained in tem of the displacement values of neighboing nodes as follows: U k ki = U k+ ki+ = A:U k ki + B:U k+ ki+ + C:U k ki + D:U k+ ki+ + E:U k zki + F:U k+ zki+ U k ki = U k+ ki+ = A0 :U k ki + B 0 :U k+ ki+ + C 0 :U k ki + D 0 :U k+ ki+ + E 0 :U k zki + F 0 :U k+ zki+ U k zki = U k+ zki+ = A00 :U k ki + B 00 :U k+ ki+ + C 00 :U k ki + D 00 :U k+ ki+ + E 00 :U k zki + F 00 :U k+ zki+ (3) whee A B F 00 ae constant coecients (Appendix B). Substituting Equation 3 into Equation, the dynamic nite element equilibium equations fo the two neighboing elements at the inteface of kth and (k + )th layes, become: [M] K f Xg + [K]k fxg k = f0g [M] k+ f Xgk+ + [K] k+ fxg k+ = f0g: (4a) (4b) Applying the bounday conditions (Equations 7a and 7b) fo the st and last nodes on the inne and oute sufaces, the displacement values fo these nodes will be as follows: U = C 0 U + D 0 U + E 0 U z U = C 0 0U + D 0 0U + E 0 0U z U z = C 00 0U + D 00 0U + E 00 0U z (5) U MI = G 0 U MI + H 0 U MI + I 0 U zmi + F 0 :M (t) U MI = G 0 0U MI + H 0 0U MI + I 0 0U zmi + F 0 0:M (t) U zmi = G 00 0U MI + H 00 0U MI + I 00 0U zmi + F 00 0:M (t) (6) whee C 0, D 0 and F 00 0 ae constants (Appendix C). Fom Equations, 5 and 6 the dynamic equations fo the st and last elements become: [M] f Xg + [K] fxg = ffg [M] MI f XgMI + [K] MI fxg MI = ffg MI : (7a) (7b) Assembling Equations, 4a, 4b, 7a and 7b the geneal dynamic nite element equilibium equations ae obtained as: [M] f Xg + [K] fxg = ffg: (8) The Newmak diect integation method with a suitable time step is used, and then Equation 8 is solved. RESULTS AND DISCUSSION Stacking sequences that wee mentioned in the abstact fo coss-ply laminated cylindical panels ae consideed. The extenal dynamic moment in the fom of a Fouie seies is fo example: M = M 0 ( e t ) = whee: X X m= n= q mn = 9M 0L m 3 n p R b L 3 p sin nlp L q mn ( e t ) sin m sin p n z (9) n cos nlp L sin mp sin cos nlp L : m

5 Thee-Dimensional Elasticity Solution Coss-Ply Panels 33 (m n = : : : ) and is the time constant, which in this pape is chosen as 300. The mateial popeties ae consideed as follows: E L E T = 40 G LT E T = 0:5 G T T E T = 0: LT = T T = 0:5 = 600(kg/m 3 ): Numeical esults ae obtained in the fom of maximum non-dimensional displacements and stesses which ae as: U = E LR 0U M 0 (U z U ) = 0E LR 0 M 0 (U z U ) = R H S = R 0 H ( i ij ) = ( i ij ) R3 0 M 0 fo i j = z: The extenal load at time T = 0:3 msec eaches its peak value of M 0 and then emains constant at this value. The time histoy of the adial displacement of thee-, ve-, seven- and nine-layeed panels ae shown in Figues a to d. As expected, the panels vibate in a adial diection due to initial conditions, (U _U U ). By inceasing the numbe of layes, the amplitude of vibation deceases and the natual fequencies incease. These esults ae acceptable, as the stiness of the panel is inceased by inceasing the numbe of layes. The vaiation of adial, cicumfeential and axial displacements, (U U U z ), with time at = R 0 and =, ae shown in Figues 3a to 3c fo a half length of the panel. As expected, by inceasing the loading duation up to 0.35 msec, the displacements ae st inceased and then deceased due to the vibation chaacteistic of the poblem. The main advantage of an elasticity solution, compaed with othe appoximate solutions, is that all in the elasticity solution (bounday conditions as well as inte-lamina continuity conditions) can be satised, wheeas this is not the case fo othe methods. Figue. Time histoy of displacements at z L = 0:48, = R b and S = 5.

6 34 A. Alibeigloo and M. Shakei It is seen that at the simply-suppoted edge whee z = 0, u and u become zeo, while the panel is fee to move in an axial diection, (U z 6= 0). At mid-length of the panel (z = L ), the adial and cicumfeential displacements ae zeo because of symmety. The inte-lamina continuity conditions ae also satised fo all thee components of displacement. To show the satisfaction of bounday and inte lamina continuity conditions fo stesses, the distibution of stess components acoss the thickness of the panel at T = 0:3 msec is shown in Figues 4a to 4d. It is seen that the bounday conditions at the inne and oute sufaces and the continuity conditions between the layes ae satised. Figue 4d also shows the Figue 3. Vaiation of displacements with time vesus z at = 6, = R 0, T = 0:3 msec [90=0=90]. Figue 4. Distibution of stesses acoss the thickness at T = 0:3 msec, S = 0, = 6, L z = 0:48 [90=0=90].

7 Thee-Dimensional Elasticity Solution Coss-Ply Panels 35 Figue 5. Vaiation of stesses with time vesus z at = 6, = R 0 and S = 5 [90=0=90]. paabolic distibution of the in-plane shea stess fo each laye. The pesented elasticity solution is applicable fo both thin and thick panels. Stess distibutions along the z-diection ae vaied with time (see Figues 5a to 5d). These distibutions satisfy the bounday conditions at z = 0 L. The stesses incease with inceasing the time up to T = 0:35 msec, and then begin to decease, due to the natue of extenal loading. In Figues 6a to 6d, the inuence of a stacking sequence in a thee-layeed panel on stess distibution along the axial diection is shown. The axial stess distibution (Figue 6a) in the stacking sequence of [0=90=0] in mid- suface is smalle than the one fo the [90=0=90] sequence lay-up. This esult is evident, because whee = R 0, the bes ae along the cicumfeential diection fo the [0=90=0] layeed panel, but they ae along the axial diection fo the [90=0=90] one. The existence of bes in a cicumfeential diection in the middle laye fo the [0=90=0] stacking sequence causes the cicumfeential stess at = R 0 to be highe than that of the [90=0=90] panel (see Figue 6b). Using simila easoning and noting that E = E 3, the dieence in adial stesses fo a two stacking sequence is small (see Figue 6a). Finally, the distibution of a tansvese shea stess ( ) fo two stacking sequences is shown in Figue 6d. CONCLUSION The main objective of this pape was to intoduce a pocedue to obtain the stess and displacement eld fo a panel in thee dimensions while the panel is subjected to a dynamic localized patched moment. Fo this pupose, the thee-dimensional theoy of elasticity was used to deive the govening dieential equations of motion in tems of displacements, and then the Galekin nite element method was used to obtain displacements and stesses vesus time. Numeical esults obtained fo vaious stacking aangements show that

8 36 A. Alibeigloo and M. Shakei Figue 6. Inuence of stacking sequence in stesses distibution vesus z at T = 0:3 msec., = 6, = R 0 and S = 5. the magnitude of stesses and displacements depends on the stacking sequence and the numbe of layes. When the extenal moment eaches peak value (Equation 9), the panel will be unde static loading. Since the damping coecient is not consideed, the panel vibates with constant amplitude. Meanwhile, the adial stess behaviou of the panel depends stongly on the S = R0 H. Although this pocedue is not an exact solution, the authos believe that it is the only solution (elasticity solution) which satises all bounday and inte-lamina conditions an almost accuate method fo showing the coect pattens of displacement and stess components vesus time and thickness of panel. NOMENCLATURE C ij (i j = 6)constants elated to elastic stiness E E G G 3 3 elastic mateial constants ff (t)g foce matix of (3MI 6 3N) size H thickness of panel h i thickness of ith laye h thickness incement [K] global stiness matix of L L p [M] M I N M 0 (3MI 6 3N) (3MI 6 3N) size axial dimension of panel axial length of patch global mass matix of (3MI 6 3N)(3MI 6 3N) size numbe of elements numbe of layes peak of mechanical load

9 Thee-Dimensional Elasticity Solution Coss-Ply Panels 37 R 0 i j R a R b t U U U z U i U i U zi U j U j U j U zj U i Ui Uzi U j Uj Uzj U k ki U k ki U k zki UkI+ k+ U ki+ k+, UzkI+ k+ z z i (i = z) mid adius adius of i and j nodes in one element inne suface adius oute suface adius time vaiable displacements in and z diections, espectively displacements of i node displacements of j node acceleation of i node acceleation of j node displacements at kith node of kth laye displacements at ki+th node of (k + )th laye constant elated to time shea stains nomal stesses " i (i = z) nomal stains z z shea stesses cicumfeential length of patch p REFERENCES of applied moment cicumfeential dimension. Noo, A.K. and Buton, W.S. \Assessment of computational models fo multi-layeed composite shells", Appl. Mech. Rev., Tans. ASME, 43(4), pp (990).. Ren, J.G. \Exact solutions fo laminated cylindical shells in cylindical bending" J. of Composites Science and Technology, 9, pp (987). 3. Ren, J.G. \Analysis of simply-suppoted laminated cicula cylindical shell oofs", Composites Stuctues,, pp (989). 4. Vaadan, T.K. and Bhaska, K. \Bending of laminated othotopic cylindical shells - An elasticity appoach", Composite Stuctues, 7(), pp (99). 5. Bhaska, K. and Vaadan, T.K. \Exact elasticity solution fo laminated anisotopic cylindical shells", J. of Applied Mechanics, 60, pp (993). 6. Bhaska, K. and Vaadan, T.K. \A benchmak elasticity solution fo an axisymmetically loaded angle-ply cylindical shell", Composites Engineeing, 3(), pp (993). 7. Bhaska, K. and Vaadan, T.K. \Benchmak elasticity solution fo locally loaded laminated othotopic cylindical shells", AIAA J., 3, pp (994). 8. Chandashekhaa, K. and Nanjunda, R. \Analysis of a long thick othotopic cicula cylindical shell panel", J. of Engineeing Mechanics, (6), pp (996). 9. Bhaska, K. and Ganapathysaan, N. \Elasticity solution fo laminated othotopic cylindical shells subjected to localized longitudinal and cicumfeential moments", J. of Pessue Vessel Technology, 5, pp (003). 0. Shuvalov, A.L. and Soldatos, K.P. \On the successive appoximation method fo thee-dimensional analysis of adially inhomogeneous tubes with abitay cylindical anisotopy", J. of Sound and Vibation, 59, pp (003).. Chen, W.Q., Bian, Z.G. and Ding, H.J. \Theedimensional vibation analysis of uid-lled othotopic FGM cylindical shells", Intenational Jounal of Mechanical Science, 46, pp (004).. Chen, W.Q. and Kang, Y.L. \State-space appoach fo statics and dynamics of angle-ply laminated cylindical panels", Intenational Jounal of Mechanical Science, 47, pp (005). 3. Liew, K.M., Lee, Y.Y., Ng, T.Y. and Zhao, X. \Dynamic stability analysis of composite laminated cylindical panels via the mesh-fee kp-ritz method", Intenational Jounal of Mechanical Science, 49, pp (007). 4. Alibeigloo, A., Shakei, M. and Eslami, M.R. \Elasticity solution fo thick laminated anisotopic cylindical panels unde dynamic load", J. of Mechanical Engineeing Science (ImechE), 6, Pat C, pp (00). 5. Shakei, M., Eslami, M.R. and Alibeigloo, A. \Elasticity solution fo multi-layeed shallow cylindical panels subjected to dynamic loading", Steel and Composite Stuctues, An Intenational Jounal, (3), pp (00). APPENDIX A "C k + C55 k A 4 = P n ( j i ) j ln j i + ( j i )( i 3 j ) B 4 = P n C k + C k 55 ( j i ) ( j i ) i j ln j i + (Ck 3 + C k 55) C k + C66 k C 4 = P n m ( j i ) j ln j i + ( j i ) ( i 3 j ) (Ck 3 + C k 55)

10 38 A. Alibeigloo and M. Shakei C k + C66 k D 4 = P n m ( j i ) ( j i ) i j ln j i G 4 = 6 ( j i ) E 4 = 3 Ck P n( j i ) j ln j i + ( j i ) C k 55 ( j i ) i ln j i F 4 = 6 Ck P n( j i ) ( j + i ) ln j i ( j i ) APPENDIX B A = H B C 3 det A 0 C = D B C 3 det A 0 E = E 3B C det A 0 A 0 = H A C 3 det A 0 mc k 66 ( j i ) + j i mc k 66 ( j i ) B = I B C 3 C 0 = D (A C 3 A 3 C ) D 0 = F (A C 3 A 3 C ) E 0 = E 3A C det A 0 D = F B C 3 F = G 3B C B 0 = I A C 3 F 0 = G 3A C + Ck 55 ( j i ) j ln j i : whee: A = Ck 3 C3 k+ + kp h i + R a i= i= h Ck 33 + C33 k+ C3 k+ C3 k B = m C kp = P n C3 k + C3 k+ h i + R a H = Ck 33 B = Ck+ 44 C44 k + kp h i + R a i= D = Ck 44 A 3 = P n C k 55 C k+ 55 I = Ck+ 33 A C44 k C44 k+ = m kp h i + R a h Ck 44 + C44 k+ F = Ck+ 44 E 3 = Ck 55 C 3 = h Ck 55 + C55 k+ i= det A 0 = A (B C 3 B 3 C ) A (B C 3 B 3 C ) + A 3 (B C B C ) G 3 = Ck+ 55 h : APPENDIX C C 0 = S T C33 h: det BB D S T 0 = h: det BB E 0 = S T h: det BB C0 0 = R T C 33 h: det BB D0 0 = R 3T R T h: det BB E0 R T 0 = h: det BB A 00 = H B A 3 det A 0 B 00 = I A 3 B C 00 0 = S R 3 C 33 h: det BB D00 S R 3 0 = h: det BB C 00 = D B A 3 det A 0 D 00 = F B A 3 E 00 = E 3(A B A B ) F 00 = G 3 (A B A B ) det A 0 : E0 00 = S R S R h: det BB G 0 = L N 3 C33 M det AA H 0 = L N 3 h det AA I L N 0 = hdetaa F 0 = L N 3 det AA G0 0 = R N 3 C M 33 h: det AA

11 Thee-Dimensional Elasticity Solution Coss-Ply Panels 39 H 0 0 = N 3N 5 N R 3 h: det AA I0 0 = R N h det AA F 0 0 = R N 3 det AA H 00 0 = L R 3 h det AA F 00 0 = L R 3 det AA whee: G00 0 = L R 3 C M 33 h: det AA I00 0 = L N 5 L R h: det AA N = P n C M 3 N 3 = N 5 = CM 3 R + CM 33 L = m C M 3 R b L = R + R = C 3 R C 33 R = m R a R 3 = P n S = mc 3 R a S = R a T = P n C 3 T = det AA = N 5 L N 3 R L N 3 R 3 L N det BB = R S T R S T R 3 S T :