(Microsoft Word - \346\317\307\317 \346\322\355\344\31046- 55) Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, March, (2019) P.P. 46- 55 Buckling and Pre Stressed Dynamics Analysis of Laminated Compo- site Plate with Different Boundary Conditions Widad I. Majeed* Zainab Abdul Kareem Abed** *,**Department of Mechanical Engineering/ University of Baghdad *Email: wedad.ibrahim@coeng.uobaghdad.edu.iq **Email: zainabkareem2158@gmail.com (Received 8 February 2018; accepted 4 July 2018) https://doi.org/10.22153/kej.2019.07.002 Abstract Critical buckling and natural frequencies behavior of laminated composite thin plates subjected to in-plane uniform load is obtained using classical laminated plate theory (CLPT). Analytical investigation is presented using Ritz- method for eigenvalue problems of buckling load solutions for laminated symmetric and anti-symmetric, angle and cross ply composite plate with different elastic supports along its edges. Equation of motion of the plate was derived using prin- ciple of virtual work and solved using modified Fourier displacement function that satisfies general edge conditions. Various numerical investigation were studied to exhibit a convergence and accuracy of the present solution for consid- ering some design parameters such as edge conditions, aspect ratio, lamination angle, thickness ratio, orthotropic ratio, the results obtained gives good agreement with those published by other researchers. Keywords: Buckling load, different boundary conditions, composite laminated plate, Free vibration, Rayleigh-Ritz method. 1. Introduction The composite materials (C.M) reinforced by fiber are perfect for structural applications where high stiffness and strength to weight of ratios are necessary. Composite materials can be adapted to assemble the especial necessities of strength and stiffness by varying fiber orientations and lay-up. The capability to adapt a (C.M) to its work is very most important advantages of a (C.M) over a common material. In the past few decades the im- provement and study of (C.M) in the mechanical, civil and aerospace structures design has devel- oped. The structures might be exposed to dynamic loads in difficult environmental conditions, so it is necessary to know the characteristic of vibrations. The cause of failure of the structure components when the natural frequencies of structure and the forcing frequency close to each other's which is the resonance (when structure damping is consid- ered), may be occur large torsion \ translation de- flections and internal stresses. Many researchers have presented the stability of (C.M) subjected to buckling loads. [1]. Developed an exact solution on the base of the first order shear deformation theory (FSDT) to investigate the buckling behav- ior of symmetrical simply supported cross ply rectangular plates subjected to unidirectional line- arly varying in-plane loads. [2]. Used a semi- analytical attitude to the buckling analysis of symmetric laminated plates with general edge conditions. The multi-term extended Kantorovich method was used to decrease the partial differen- tial of the equations of buckling to a solution of ordinary differential equations. [3]. Using the fi- nite element method to study buckling and vibra- tion of composite laminated plates with variable fiber spacing. [4]. Extended a two variable re- fined theory to the free vibration analysis of ortho- Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 47 tropic plates using Navier's solution. [5]. Present- ed a study of buckling and post buckling behavior of simply supported composite plates subjected to non-uniform in-plane loading. The analytical solu- tions for laminated plate based on higher order shear deformation theory. The multiterm Galerkin method was used to solve the nonlinear partial differential equations governing post buckling behavior of plate. [6]. Investigated the natural fre- quencies and buckling of layered plates subjected to combined in-plane loadings with the aim of furnishing a few guidelines for the modeling and design of pre-stressed laminated panels. Different load conditions, stacking sequences, material properties and boundary conditions are consid- ered. [7]. Presented a study of bending and free vibration analysis based on simple first order shear deformation theory (FSDT). Were the ana- lytical solved by an exact method (Levy's meth- od). [8]. Presented an exponential shear defor- mation theory which extended for buckling and free vibration analysis. The theory takes into ac- count of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate. Were the analytical solved by an exact method (Navier's method). [9]. Studied buckling behavior and free vibration of composite plates subjected to in plane parabolic, linear and uniform distributed loads using (CLPT). Analytical investigation is shows using Ritz method for eigenvalues problems of buckling loads solution for laminated plate. The edges con- ditions take into account are (SSSS, CCCC, SCSC, SFSF and CFCF). [10].The vibration and buckling of laminated beams studied by using a shear deformation and refined theory. The dis- placement field is estimated by using the Ritz technique. The functions used in the Ritz tech- nique are chosen by way of either a hybrid poly- nomial-trigonometric series or a pure polynomial series. 2. Buckling and Pre Stressed Vibration Analysis of Laminated Plates The governing equation is derived by using CPLT, [8]: D�� ������ + �2D� + 4D�� ������ ��� + D ������ +I� ������ = N� ������ ... (1) Where, stress resultants are expressed in dis- placement form from below: �M�. M�. M��� = � �σ� ‚ σ�‚ σ���z dz �/ !�/ =∑ � �σ� ‚ σ�‚ σ���z dz#$%&#$'()� ... (2) I� = � ρ dz �/ !�/ ... (3) Integrating Eq. (2), (3) through thickness of the plate, the stress resultant is associated to the dis- placement (w) by the relatives: + M�M�M��, = - D�� D� D��D� D D �D�� D � D��. + k�k�k��, ... (4) D01 = 2Q4 015 � z dz67�7� ... (5) The twisting moments and bending, transversal shear forces can be written in terms of the dis- placement function as, [11]. M� = −D�� ������ − D� ������ ... (6) M� = −D ������ − D� ������ ... (7) M�� = −2D�� ����� �� ... (8) Q� = −D�� �9���9 − �D� + 4D�� �9���� �� ... (9) Q� = −D �9���9 − �D� + 4D�� �9���� �� ... (10) For a flexibly restricted rectangular plate shown in Fig.(1), the boundary conditions are : k�:w = Q� K�: ���� = −M� ….. at x=0 ... (11-12) k��w = −Q� K�� ���� = −M� ….. at x=a ... (13-14) k�:w = Q� K�: ���� = −M� ….. at y=0 ... (15-16) k��w = −Q� K�� ���� = −M� ….. at y=b ... (17-18) Where k�: k�� and k�: k�� are the transitional stiffness of spring,K�: K��and K�:, K�� are the rotations stiffness of spring. Eq.(11)-(18) express for different edge conditions, the classic homoge- neously edge conditions can be direct obtained by putting the constants of spring equalize to an very small or large number. By substituting Eq. (6-10) in Eq. (11-18), get the following equations: k�:w = −D�� �9���9 − �D� + 4D�� �9���� �� ... (19) k��w = D�� �9���9 + �D� + 4D�� �9���� �� ... (20) K�: ���� = D�� ������ + D� ������ ... (21) K�� ���� = D�� ������ + D� ������ ... (22) And similarly found other four equations in the y direction. Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 48 Fig. 1. Elastic restrained edges for a rectangular plate [12]. As mentioned by many plate and shell re- searches that exact solution for plate or shell with general boundary conditions is not available so we use Ritz method to get approximate solution from Hamilton's equation: δ ∭�U − W + T = 0 ... (23) Where U is the strain energy, W is potential energy due to the external forces and T is the ki- netic energy. δ the random variation. Where U = � � � �σ�ε� + σ�ε� + τ��γ���dx dy dzJ:K:7�67� + Elastic energy of springs at edges W = � � � LN� M���� N O dx dyJ:K: T = ρ � � � P������ wQ dx dy dzJ:K:7�67� ... (24) Substituting Eqs.(2- 4),(19-22) in Eq.(24),yield to: U − W = � � � LD�� M������ N + D M������ N +J:K: 4D�� M ����� ��N + 2D� ������ ������ O dx dy +� � Lk�:w +K:K�: M���� N O�): dy + � � Lk��w +K:K�� M���� N O�)J dy + � � Lk�:w +J:K�: M���� N O�): dx + � � Lk��w +J:K�� M���� N O�)K dx − � � � LN� M���� N OJ: dx dyK: ... (25) and T = � ω ∬ I� w� dx dy ... (26) 3. Admissible Functions With Ritz method the allowable functions play an important part. The products of beam functions are commonly selected like the displacement function and functions can be consequently like, [12]. w�x y = ∑ AUVX�x Y�y U V)� ... (27) WhereX�x , Y�y are the specific variables for beams that include a similar edge conditions in the (y, x) direction, correspondingly. The beam functions can be in general achieved like a linear collection of hyperbolic and trigono- metric functions, which involve some unknown variables which are exist from the edge condi- tions. Then, every edge conditions essentially conduct to a various beam functions. In actual uses, this is obviously disadvantageous, beside the damage of computing the specific functions for a different boundary beam. with a view to avert this difficulty, an developed the series of Fourier tech- nique has been suggested for beams with different boundaries at each ends in which the specific functions are written in the form of, [13]. w�x = ∑ aU cos λJUx + P�x aU): MλJU = UbJ N, 0c x c a ... (28) P (x) is the function in Eq. (28) considers an arbi- trarily continued function that, in any case of edge conditions, is constantly selected to satisfy the equations as following: Pddd�0 = Wddd�0 = α:. Pddd�a = Wddd�a =α�. ... (29-30) Pd�0 = Wd�0 = β:. and Pd�a = Wd�a =β�. ... (31-32) P (x) is here inserted to take care of the latent discontinuities of the function of displacement and its derivative at end points. Accurately, previously it is known that the smoothest a periodical func- tion, the quicker its Fourier extension conver- gence. Thus, adding of the P (x) will have two instantaneous interests: (1) the series of Fourier extension is presently agreed with any edge condi- tions, and (2) the solution of the series of Fourier its accurateness of convergences. Yet, P (x) have just been realized as a continu- ously function that satisfy Eq. (29) - (32), the function P (x) format is not a worry with regard to the convergences of the series solution. Therefore, it can be chosen in any required formula. Like a substantiation, supposes the P (x) is a the function of polynomial, P�x = ∑ CV PV M�JN .hV): ... (33) Where PV�x is the Legendre function of order n , CV is the coefficient of extension. It is clarifies that P (x) desires to be minimum a 4th polynomial to jointly satisfy Eq.(29) - (32). Sub- stituting Eq. (33) into Eq. (29) - (32) results yield to: Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 49 CiPiddd�0 + ChPhddd�0 = aiα:. ... (34) CiPiddd�1 + ChPhddd�1 = aiα�. ... (35) C�P�d�0 + C P d�0 + CiPid�0 + ChPhd�0 = aβ:. ... (36) C�P�d�1 + C P d�1 + CiPid�1 + ChPhd�1 = aβ�. ... (37) By using the above equations, the coefficients, C�. C . Ci and Ch are straight acquired in terms of the edge constants, α:. α�. β:. then β�. As the co- efficient C: doesn't really seem in Eq. (34) -(37), it can be a random number theoretically. For ex- ample, C: is presently chosen to content � P�x dx = 0J: ... (38) The last appearance for the P(x) can be shown as P(x) =ζJ�x lm4 ... (39) Where m4 = nα:. α�. β:. β�ol ... (40) and ζJ�x l = ⎩⎪⎨ ⎪⎧−�15xh − 60axi + 60a x − 8ah /360a�15xh − 30a x + 7ah /360a �6ax − 2a − 3x /6a�3x − a /6a ⎭⎪⎬ ⎪⎫ ... (41) The results in Eq. (39) - (41) are already de- rived from an additional simple but little common approach, [12]. So as to obtain the unknown of edge constants, α:. α�. β:. and β�. substitution of Eq. (28) and eq. (39) into the edge conditions Eq. (19)-(22) that results in m4 = ∑ HJ!�QJUaUaU): ... (42) Where HJ = ⎣⎢ ⎢⎢ ⎢⎢ ⎢⎡ 1 + �(��J9i�:�&& �(��J9i�:�&& !(��Ji�&& !(��J��(�&J9i�:�&& 1 + �(�&J9i�:�&& !(�&Ji�&& !(�&J� Ji J� ����&& + �J !�J J� Ji !�J ��&�&& + �J ⎦⎥ ⎥⎥ ⎥⎥ ⎥⎤ … (43) and QJU = ��−1 (���&& �−1 U (�&�&& −λJU �−1 UλJU � l ... (44) It must be reminded that a matrix Ha becomes single to a totally free Beam, [14], . By using of Eqs. (39) and (42), Eq. (28) becomes as: w�x = ∑ aUφUJ �x aU): ... (45) Where φUJ �x = cos λJUx + ζJ�x HJ!�QJU ... (46) Mathematically, Eq. (45) mention that every of the functions of the beam can be observed as a function in the functional space spanned by the base functions {φUJ �x : m = 0. 1. 2. … … … o. So, Eq.(27) can be consequently rewritten as: w�x. y = ∑ AUφUJ �x φVK�y aU.V): ... (47) Where: φVK�y = cos λKVy + ζK�y HK!�QKV ... (48) The terms for ζK�y . HKand QKV can be, corre- spondingly, obtained from Eqs. (41), (43) and (44) by easily changing the x- regarding parameters by the y- regarding. 4. Eigen Value Problem Orthotropic plate are consider, the material di- rections of width identify with the plate directions. Uniaxial in-plane compressive force N� along the both sides of edge (x) is subjected. To calculate the critical buckling load; the natural frequency ω is set to zero, to find the natural fre- quency without the action of buckling load; N� is set to zero, and to calculate the natural frequency under buckling load action; ; N� is left as a known after finding the critical buckling load N�� Previ- ously. Performing the required mathematical pro- cesses (differentiations and then integrations) of Eq.(25) and(26) and then putting the mechanical energy in the following equation: ������ = 0 ... (49) Eq. (48) gives homogenous equations as follow1; f�AUV. N�� = 0 for buckling problem f�AUV. ω = 0 for vibration problem f�AUV.N��. ω = 0 for vibration under bucklin... (50) Eq. (50) solving as an Eigen-value problem which is written as below: - a�.� ⋯ a�.�U∗V ⋮ ⋱ ⋮a�U∗V .� ⋯ a�U∗V .�U∗V . ¢ A��⋮AUV£ = 0 ... (51) Where ¤¥¦ are the coefficients of the nonzero Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 50 unknowns§¨©. Finding the determinant of the first term of Eq.(51) and equating it to zero will lead to get the natural frequencies ª and critical buckling load «¬­ . When M and N are more than 1, The «¬­and ª are determined by solving ei- genvalue problem. For different edge conditions and M &N more than 1, the solution becomes more difficult and needs computer programming to determine«¬­. In this study MATLAP R2015a is used to solve the eigenvalue problem to find the natural frequency under buckling action. For nu- merical study, ANSYS APDL programming is used. 5. Results and Discussion MATLAP (version15) programming is used to investigate and solved the buckling load and natu- ral frequency under buckling of composite lami- nated plate (CLP) with elastic edge condition. The plates are described by a four-letter symbol for example SCSC denotes a plate with simply sup- ported edge at y=b, y=0, clamped at x=a, x=0. To see the validly of the derived equations and per- formance of computer programming for buckling and vibration analysis of (CLP), orthotropic plate numerical results are compared with those found by Firas Hamzah Taya, 2014 [9]. And I. Shufrin, O. Rabinovitch, M. Eisenberger, 2008 [2].Table (1, 2) shows a good agreement results for different edge conditions. It show that the clamped plate along two or four edges can hold buckling load more than plate with simply supported boundary conditions, especially in table (1) for the case where the plate is FSFS. In the case where the plate is simply supports or mixed with free edges, it is weak to hold large load compared with clamped plates. While show results for laminated composite plate with different edge conditions, stacking sequence, aspect ratio and modulus ratio give good agreement when compared with J. N. Reddy, 2003[15] and I. Shufrin, O. Rabinovitch, M. Eisenberger, 2008[2]. As shown in Table (3, 4, 5). Table (6) present the results of anti-symmetric cross and angle ply with different aspect and modulus ratio and give a good agreement when compared with results obtained by J. N. Reddy, 2003. Table (7) shows the results of the natural frequency of laminated plate under buckling for different load ratio and compered with the results obtained by Firas Hamzah Taya, 2014, give very close results. It shows that the natural frequency is less than that found without loading because the stiffness reduction. In table (8,9,11) show the re- sults of the natural frequency under buckling for different load ratio and compared with the results obtained by numerical program ANSYS, also in table (10) show the natural frequency under load ratio (d=0.5) with effect of aspect , modulus ratio and Fig.2. shows the mode shapes of the case. 6. Conclusions Buckling of rectangular laminated plate with general elastic restraints along the edges is ob- tained by using modified Fourier function; also free vibration of this plate under in plane loading is investigated using Rayleigh–Ritz method. The results are compared with the results obtained by other researchers; the comparison showed good agreement between them. The effect of edge con- ditions, aspect ratio, lamination system, angle of lamination and load correction factor on buckling and vibration characteristics are studied. From the result it is concluded that, the buckling load de- creases rapidly with increasing aspect ratio till it is about 1.5, after that takes constancy or close val- ues for higher aspect ratio. The edge conditions affect the critical buckling load and fundamental natural frequency. Clamped edges conditions offer high stiffness, results in high critical buckling load and natural frequency. Clamped edges make the plate holds larger load than simply supported edg- es. The natural frequency changes reversely with buckling load ratio. Therefore, this investigation has actually showed that this function can be used to get buckling and vibration characteristics of laminated plate with various boundary conditions. Table 1, Non-dimensional buckling load�®4 = ®¯°±² ³²´µ⁄ , for [0 90 0] plates of different Boundary conditions, (³· ³²⁄ = ·¸, º·² = ¸. »³², ¼·² = ¸. ²½, ¾ = ± . References Type of boundary conditions SSSS CCCC SCSC FSFS FCFC Present work 11.550 40.38 35.900 8.049 32.544 Firas[9] 11.491 40.507 36.255 7.991 32.982 Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 51 Table 2, Non-dimensional buckling load (®4 = ®¯° ·²�· − ¼·²¼²· ³·⁄ ´µ), for [30 -30 30] plates of different boundary conditions, (³· ³² = ². ¿½,⁄ º·² = ¸. ¿À³²,¼·² = ¸. ²µ a=b). References Type of boundary conditions SSSS CCCC CSCS SCSC Present work 25.77 66.81 47.33 39.32 I. Shufrin[2] 26.67 65.26 49.18 40.93 Discrepancy% 3.4 2.3 3.9 4 Table 3, Non-dimensional buckling load�®4 = ®¯°±² Á²²Â²⁄ , for [0 90 90 0] plates (SSSS) of different aspect, modulus ratio, ( º·² = ¸. ½³², ¼·² = ¸. ²½ . References a/b ³· ³²⁄ =5 10 20 25 40 Present work 0.5 13.94 18.225 22 23.1 25 Reddy[16] 13.9 18.126 21.87 22.87 24.59 Present work 1 5.66 6.353 7 7.13 7.5 Reddy 5.65 6.347 6.96 7.12 7.4 Present work 1.5 5.238 5.28 5.317 5.326 5.34 Reddy 5.233 5.27 5.31 5.318 5.33 Table 4, Non-dimensional buckling load�®4 = ®¯°±² Á²²Â²⁄ , for ø Ä¸Å²Æ laminated plates (CCCF) of different aspect, modulus ratio, ( º·² = ¸. ½³², ¼·² = ¸. ²½ . ³· ³²⁄ References ¾ ±⁄ =1 1.5 2 3 Present work 6.7 3.48 2.47 I. Shufrin 6.4 3.3 2.34 Discrepancy% 4.7 5 5 10 Present work 8.08 3.96 2.6 I. Shufrin 7.84 3.78 2.48 Discrepancy% 2.9 4 4 Table 5, Non-dimensional buckling load�®4 = ®¯°±² Á²²Â²⁄ , for ø Ä¸Å²Æ laminated plates (CSCS) of different aspect, modulus ratio, ( º·² = ¸. ½³², ¼·² = ¸. ²½ . ³·³² References ¾ ±Ç =1 1.5 2 3 Present work 6.671 6.379 6.12 I. Shufrin 6.659 6.295 5.84 Discrepancy% 0.179 1.3 4.5 10 Present work 6.584 6.096 5.71 I. Shufrin 6.557 6.056 5.46 Discrepancy% 0.41 0.656 4.3 Table 6, Non-dimensional buckling load�®4 = ®¯° ±² ³²´µ⁄ , for anti-symmetric laminated plates (SSSS) with effect of different modulus ratio, ( º·² = ¸. ½³², ¼·² = ¸. ²½ . Ply Orientations References ³· ³²Ç =10 25 40 Ã0 90Åh Present work 11.174 23.523 35.874 Reddy 10.864 22.622 34.381 Discrepancy% 2.7 3.8 4.1 Ã45 − 45Åh Present work 18.2 42.81 67.38 Reddy 17.637 41.16 64.68 Discrepancy% 3 3.8 4 Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 52 Table 7, Dimensionless natural frequency�É4 = Éʾ²Ë̯ ³²⁄ ´⁄ �, for [0 90 0] plates under buckling of different load rati- os, (³· ³²⁄ = ·¸, º·² = ¸. »³², ¼·² = ¸. ²½, ¾ = ± . D References SSSS CCCC CSCS SFSF CFCF 0 Present work 10.656 22.255 21.065 8.892 20.106 Firas 10.649 22.323 21.119 8.886 20.143 0.25 Present work 9.228 19.27 18.243 7.701 17.413 Firas 9.223 19.33 18.29 7.443 17.444 0.5 Present work 7.535 15.736 14.895 6.287 14.217 Firas 7.53 15785 14.933 5.973 14.243 0.75 Present work 5.328 11.127 10.532 4.446 10.053 Firas 5.324 11.161 10.559 3.995 10.071 Table 8, Dimensionless natural frequency�É4 = Éʾ²Ë̯ ³²⁄ ´⁄ �, for [0 90 90 0] plates under buckling of different load ratios, (³· ³²⁄ = ¿¸, º·² = ¸. ½³², ¼·² = ¸. ²½, ¾ = ± . d References SSSS CCCC SCSC SFSF 0 Present work 18.817 41.216 38.668 6.916 Ansys 18.703 40.662 38.099 6.914 0.25 Present work 16.296 35.695 33.488 6.916 Ansys 16.196 35.434 33.169 6.902 0.5 Present work 13.306 29.145 27.343 5.812 Ansys 13.224 29.14 27.244 5.813 0.75 Present work 9.408 20.609 19.335 4.11 Ansys 9.351 20.777 19.394 4.115 Table 9, Dimensionless natural frequency�É4 = Éʾ²Ë̯ ³²⁄ ´⁄ �, for ÿ½ − ¿½Å¿ plates under buckling of different load ratios, (³· ³²⁄ = ·¸, º·² = ¸. ½³², ¼·² = ¸. ²½, ¾ = ± . D References SSSS CCCC SCSC SFSF 0 Present work 13.409 21.632 17.914 5.252 Ansys 13.111 21.165 17.533 4.671 0.25 Present work 11.613 18.734 15.514 4.548 Ansys 11.395 18.378 15.258 4.072 0.5 Present work 9.482 15.296 12.667 3.713 Ansys 9.341 15.05 12.529 3.35 0.75 Present work 6.704 10.816 8.957 2.626 Ansys 6.634 10.678 8.917 2.39 Table 10, Dimensionless natural frequency�É4 = Éʾ²Ë̯ ³²⁄ ´⁄ �, [30 -30 30] plates of different boundary conditions, (³· ³² = ². ¿½,⁄ º·² = ¸. ¿À³²,¼·² = ¸. ²µ a=b). D References SSSS CCCC SCSC 0 Present work 7.311 13.02 9.949 Ansys 7.237 13.04 9.913 0.25 Present work 6.332 11.289 8.619 Ansys 6.272 11.359 8.595 0.5 Present work 5.171 9.223 7.04 Ansys 5.124 9.334 7.027 0.75 Present work 3.657 6.526 4.98 Ansys 3.625 6.653 4.978 Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 53 Fig. 2. Mode shape for free vibration of (SSSS) for [30 -30 30] laminated square plate a-first b-second c-third d- fourth modes. Table11, Dimensionless natural frequency�É4 = Éʾ²Ë̯ ³²⁄ ´⁄ �, for [0 90 90 0] (SSSS) plates with effect of aspect and modulus ratios, (º·² = ¸. ½³², ¼·² = ¸. ²½ (natural frequency without load) d=0.5. References a/b ³· ³²⁄ =10 25 40 Present work 0.5 6.275 (8.875) 9.654 (13.652) 12.125 (17.147) Ansys 6.275 (8.875) 9.63 (13.62) 12.066 (17.064) Present work 1 7.426 (10.502) 10.774 (15.237) 13.306 (18.817) Ansys 7.411 (10.48) 10.731 (15.176) 13.22 (18.702) Present work 1.5 10.157 (14.364) 13.964 (19.748) 16.938 (23.954) Ansys 10.161 (14.367) 13.964 (19.747) 16.934 (23.948) Nomenclature Symbol Discretion Units A Length of a plate M B width of a plate M H Plate thickness M A vector of the expansion or Rayleigh–Ritz coef- ficients §¨© expansion or Rayleigh– Ritz coefficients ¤¨ expansion or Rayleigh– Ritz coefficient Í¥¦ flexural rigidity M,N numbers of expansion terms used in x- and y direction, respectively ÎÏ, ÎÐ, ÎÏÐ Moment result per unit length N.m/m ÑÏ . ÑÒ Transverse shear force result N ÓÏ:, ÓÏ� rotational stiffness at Rad.N/m Widad I. Majeed Al-Khwarizmi Engineering Journal, Vol. 15, No. 1, P.P. 46- 55 (2019) 54 x = 0 and a, respectively ÓÐ:, ÓÐ� rotational stiffness at y = 0 and b, respectively Rad.N/m ÔÏ:, ÔÏ� translational stiffness at x = 0 and a, respectively N/m ÔÐ:, ÔÐ� translational stiffness at y = 0 and b, respectively N/m P(x) a simple polynomial function x,y,z Cartesian coordinate system M Π Total potential energy of the System N.m U Strain energy of defor- mation N.m Õ¬ the elastic potential en- ergy N.m W(x) flexural displacement of a beam M W(x,y) flexural displacement of a plate M Ö�× , Ø�Ù beam characteristic function Ú:, Ú� Û ddd�¤ , Û ddd�0 Ü:, Ü� Û d�0 , Û d�¤ ÝÞ̈ �× admissible functions in x direction Ý©ß �Ù admissible functions in y direction S , C ,F Simply- clamped- free 7. References [1] Zhong, Hongzhi, and Chao Gu. "Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load." Composite structures Vol. 80 No.1 (2007): PP. 42-48. [2] Shufrin, I., O. Rabinovitch, and M. Eisen- berger. "Buckling of symmetrically laminated rectangular plates with general boundary con- ditions–A semi analytical ap- proach." Composite Structures Vol. 82 No.4 (2008) PP: 521-531. [3] Kuo, Shih-Yao, and Le-Chung Shiau. "Buck- ling and vibration of composite laminated plates with variable fiber spacing" Composite Structures Vol. 90 No.2 (2009) PP: 196-200. [4] Kim, Seung-Eock, Huu-Tai Thai, and Jaehong Lee. "Buckling analysis of plates using the two variable refined plate theory."Thin-Walled Structures Vol. 47 No. 4 (2009) PP: 455-462. [5] Kumar Panda, Sarat, and L. S. Ramachandra. "Buckling and postbuckling behavior of cross- ply composite plate subjected to nonuniform in-plane loads." Journal of engineering me- chanics Vol. 137 No. 9 (2011) PP: 589-597. [6] Carrera, E., et al. "Effects of in-plane loading on vibration of composite plates." Shock and Vibration Vol. 19 No. 4 (2012) PP: 619-634. [7] Thai, Huu-Tai, and Dong-Ho Choi. "A simple first-order shear deformation theory for lami- nated composite plates." Composite Struc- tures Vol. 106 (2013) PP: 754-763. [8] Sayyad, ATTESHAMUDDIN S., and Y. M. Ghugal. "Buckling and free vibration analysis of orthotropic plates by using exponential shear deformation theory." Latin American Journal of Solids and Structures Vol. 11 No. 8 (2014) PP: 1298-1314. [9] Firas Hamzah Taya., ‘‘Buckling and Vibration Analysis of Laminated Composite Plate’’, M.Sc. Thesis, University of Baghdad, Mech. Dep. 2014. [10] Mantari, J. L., and F. G. Canales. "Free vi- bration and buckling of laminated beams via hybrid Ritz solution for various penalized boundary conditions." Composite Struc- tures Vol. 152 (2016) PP: 306-315 [11] Khov, Henry, Wen L. Li, and Ronald F. Gib- son. "An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary condi- tions." Composite Structures Vol. 90 No. 4 (2009) PP: 474-481. [12] Li, Wen L. "Vibration analysis of rectangular plates with general elastic boundary sup- ports." Journal of Sound and Vibration Vol. 273 No. 3 (2004) PP: 619-635 [13] Li, Wen L. "Free vibrations of beams with general boundary conditions." Journal of Sound and Vibration Vol. 237 No. 4 (2000) PP: 709-725. [14] Li, W. L., and M. Daniels. "A Fourier series method for the vibrations of elastically re- strained plates arbitrarily loaded with springs and masses." Journal of Sound Vibration Vol. 252 (2002) PP: 768-781 [15] Reddy, Junuthula Narasimha. Mechanics of laminated composite plates and shells: theory and analysis. CRC press, 2004. )2019( 46- 55، صفحة 1د، العد15دجلة الخوارزمي الهندسية المجلم وداد مجيد ابراهيم 55 واالهتزاز لصفائح مركبة طبقية مع ظروف أسناد عامةعاج نبتحليل اال **زينب عبد الكريم عبد *ابراهيم مجيد وداد جامعة بغداد قسم الهندسة الميكانيكية/ كلية الهندسة/***، wedad.ibrahim@coeng.uobaghdad.edu.iq :البريد االلكتروني* zainabkareem2158@gmail.com :البريد االلكتروني** الخالصة الصفيحة بأستخدام نظرية ىمنتظمة ضمن مستو طبقات المعرضة الحمالالزاز واالنبعاج للصفائح الرقيقة المركبة المكونة من تتمت دراسة االه .في الجانب النظري التحليلي تم أشتقاق معادالت الحركة بأستخدام طريقة رتز للحصول على مجموعة معادالت متجانسة (CLPT) الصفائح الكالسيكسة الدوال المستخدمة وطبقات متعامدة وغير متعامدة الزوايا .من ةمكون ةلحل مسألة حمل االنبعاج لصفائح متناظره وغير منتاظر eigenvalueوحلها كمسألة ولعل االهم من ذلك، ان هذه الدراسة في هذا البحث يمكن ان تمثل بدوال مثلثية ودوال عشوائية مستمرة وذلك لضمان سالسة المطلوبة لعمل الدالة الرئيسية. مثلة عددية الثبات أبولة التي يمكن تطبيقها لشروط اسناد الحافات المختلفة. لقد تم دراسة عدة ققد طورت أسلوب عام الشتقاق مجموعة كاملة من الدوال الم مع األخذ بنظر االعتبار تغيير في بعض معايير التصميم مثل شروط الحدود نسبة االرتفاع وزاوية التصفيح ونسبة السماكة ونسبة وتقاربهادقة نتائج الحل ً ة النتائج مع باحثين اخرين واعطت تقاربحيث تم مقارن orthotropy ـال .جداً اً جيد ا