(Microsoft Word - \337\307\344\355 \346\307\310\312\345\307\34134-44) Vibration Analysis of Cross-Ply Plates Under Initial Stress Using Refined Theory Ibtehal Abbas Sadiq* Kani Hussein Bawa** *,**Department of Mechanical Engineering / University of Baghdad *Email:ebtialabas@yahoo.com **Email:kanihussein1991@gmail.com (Received 2 January 2018; accepted 22 May 2018) https://doi.org/10.22153/kej.2018.05.002 Abstract Natural frequency under initial stresses for simply supported cross-ply composite laminated plates (E glass- fiber) are obtained using Refind theory (RPT). This theory accounts for parabolic distribution of the transverse shear strain through the plate thickness and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The governing equations for Eigen value problem under initial stress are derived using Hamilton’s principle and solved using Navier solution for simply supported cross-ply symmetric and antisymmetric laminated plates. The effect of many design factors such as modulus ratio, thickness ratio and number of laminates on the Natural frequency and buckling stresses of orthotropic plates are studied. The results are compared with other researcher. Keyword: Composite laminated plate, buckling analysis, free vibration analysis, Refined plate theory. 1. Introduction Laminated composite plates have very importance in the engineering applications because of their useful features so a many variety of laminated theories for static and dynamic behavior have been developed such as approximate, experimental and exact methods. [1] presented static analysis using higher-order refined theory of angle ply plate and sandewich plates hitherto. No requirement to use shear correction factors (SCF), because the transverse- shear strains vary parabolicaly from side to side which lead to vanish the shear-stresses on the upper and bottom surface of the plate. From principle of potential energy, the equations of equilibrium are derived and solved by using Navier-type method. Correctness of the theoretical preparations and the solution method confirmed by comparing the results with other theory described in the literature. [2] Presented buckling analysis of SS plate exposed to in-plane loading using refined plate theory of orthotropic and isotropic plates. The governing equations G.E which derivative from the principle of virtul-displacements, and solved by using the Navier method. This theory is simple, comparable to the(FSDT) theory and there no exists a need for using SCF. [3] Studied a two-variable Refind theory (RPT) of lamineted composite plates. The theory contents the zero traction B.C on the upper and bottom faces of the plate without wanting to use SCF. The equations of motion are derivative using Hamilton’s principle (H.P) and solved using Naveir method of angle-ply and cross-ply antisymmetric laminate. This theory is simple and accurate in solving the buckling behaviors and static bending of laminated composite plates. [4] Studied free vibration of laminated composite plates using two variable Refined plate theory (RPT) and using Hamilton’s principle to derive the equations of motion, and these equations solved using Navier solutions of cross-ply and angle-ply antisymmetric laminates. This theory is accurate and effective in obtain the natural frequencies N.F of laminated composite plates. [5] Studied the buckling analysis using Refind theory for orthotropic plates. No requirement to use SCF in this theory and the Governing equations solved Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 14, No.4, December, (2018) P.P. 34- 44 Ibtehal Abbas Sadiq Al-Khwarizmi Engineering Journal, Vol. 14, No. 4, P.P. 34- 44 (2018) 35 using Levy-type method. It considering the effect of some design limitations such as boundary conditions, orthotropy ratio, thickness ratio and loading condition on the critical-buckling load. [6] Presented free-vibration investigation of functionaly arranged material (FGM) sandwich rectangular plates by the four variable refined theory (RPT) which not requirement to use SCF. the equation of motion achieved using Hamilton’s principle for the (FGM) sandwich plates and these equations solved by using the Navier type. This theory simple and accurate in resolving the free- vibration behavior of the functionaly arranged material sandwich plates when its results comparing with other theories such as classical laminated theory(CLP), first order theory (FSDT). [7] Presented free vibration analysis of simply supported plate which made of functionaly arranged materials using four variable Refind theory. No requirement to use shear correction factors, because the transverse-shear strains vary parabolicaly from side to side the thickness which lead to disappear the shear stresses on the upper and bottom faces of the plate. From the principle of virtual displacements, the governing equations for the (FGM) rectangular plates are derived and solved by using Navier-type method. The natural frequencies are found using the Ritz method in the case of FG clamped plates. The strength of this present theory which gave accurate free vibration of FG plate shown by comparing the present results with others theories and also the influence of vying rises, aspect ratios, and thick ratio on the free- vibration of the FG plates is showed. [8] Presented free vibration analysis of rectangular plate with two opposite edges simply supported (SS) and the other two edges having arbitrary boundary conditions using 'refined plate theory'. From the principle of virtual displacements, the governing equations are derived and solved by using the Levy-type method. No need to use shear correction factors in this theory, it considering the effect of some design parameters such as boundary conditions, modulus ratio, and aspect ratio on the natural frequency. In present work, the equation of motion of Refined plate theory are programming to find the Critical buckling and fundamental natural frequency for cross-ply plate for different thickness ratio, symmetric and antisymmetric and orthotropy ratio, while to obtain vibration characteristic of plate under initial stress, we derive equation of motion depending on Refined plate theory for simply supported plates using Navier solution. 2. Theoretical Analysis 2.1 Displacement Field In present work, a rectangular plate of total thickness (h) of (n) orthotropic layers with the coordinate system as shown in Fig (1) are considered the displacement of Refined plate theory (RPT) which satisfies equilibrium conditions at the top and bottom faces of the plate without using shear correction factor is developed. The transverse displacement W contains three components; bending ��� , extension �� and shear ��� which these components are functions of coordinates x, y, and time t only. Similarly, the displacements u in x-direction and v in y-direction have bending, extension and shear components [3]. U= + �� + �� V= � + ��� + ��� W�x‚y‚z‚t� = w��x‚y‚t� + w���x‚y‚t� + w���x‚y‚t� The shear components �� and ��� , ��� lead to the parabolic variations of shear strains � !, �"! and to shear stresses # !, #"! through the thickness of the plate in such a way that shear stresses # !, #"! are zero at the bottom and top surfaces of the plate. ��= $ %&' − )* +!�,-. /012/ ���= $ %&' − )* +!�,-. /012/" The following displacement field assumptions [3]: 3�4‚5‚$‚6� = u�x‚y‚t� − z +89:;8< , + z %&' −)* +=�,-. 89>?8< @�4‚5‚$‚6� = v�x‚y‚t� − z +89:;8B , + z %&' −)* +=�,-. 89>?8B C�4‚5‚$‚6� = �� �4‚5‚6� + ��� �4‚5‚6� +��� �4‚5‚6� …(1) For small strain, the strain-displacement relations take the form: D = /E/ D" = /F/" D " = &- +/E/" + /F/ , = &- � " D ! = &- +/E/! + /0/ , = &- � ! D"! = &- +/F/! + /0/" , = &- �"! …(2) By substituting eq. (1) into eq. (2) to give: D = /E/ − $ /G0HI/ G + $ [&' − )* +!�,-] /G012/ G D" = /F/" − $ /G0HI/"G + $ [&' − )* +!�,-] /G012/"G Ibtehal Abbas Sadiq Al-Khwarizmi Engineering Journal, Vol. 14, No. 4, P.P. 34- 44 (2018) 36 � " = /E/" + /F/ − 2$ /G0HI/ /" + 2$ [&' −)* +!�,-] /G012/ /" �"! = /0M/" + [)' − 5 =G�G] /012/" � ! = /0M/ + [)' − 5 =G�G] /012/" …(3) The strain field is: O D D"� " P = Q D RD" R� "R S + $ Q T �T"�T "� S + U Q T � T"�T "� S V� !�"! W = X� !��"!� Y + Z X � !� �"!� Y … (4) Where: Q D RD" R� "R S = ⎩⎪⎨ ⎪⎧ /E/ /F/" /E/" + /F/ ⎭⎪⎬ ⎪⎫ Q T �T"�T "� S = ⎩⎪⎨ ⎪⎧ − /G0HI/ G− /G0HI/"G−2 /G0HI/ /" ⎭⎪⎬ ⎪⎫ Q T � T"�T "� S = ⎩⎪⎨ ⎪⎧ − /G012/ G− /G012/"G−2 /G012/ /" ⎭⎪⎬ ⎪⎫ , X� !��"!� Y = Q /0M/ /0M/" S , X � !� �"!� Y = Q /012/ /012/" S f= − &' $ + )* $ +!�,- ‚ Z = )' − 5 +!�,- … (5) Fig. 1. coordinate system of laminated plates. 2.2 Principle of Virtual Work Using Hamilton’s principles, the equations of motion of the refined plate theory will be derived. Reddy, 2004 0 = c �d3 + d@ − de�fR g6 … (6) • The virtual strain energy d3 is: d3 = [c +h [# dD i + σBdD"i + σ? �8?8< , δu� + +−I-u� +I* 89� :;8< − I) 89� >?8< , � 89� :;8< + +I'u� − I) 89� :;8< +I� 89� >?8< , � 89� >?8< + +I&v� − I- 89� :;8B + I' 89� >?8B , δv� ++−I-v� + I* 89� :;8B − I) 89� >?8B , � 89� :;8B + +I'v� −I) 89� :;8B + I� 89� >?8B , � 89� >?8B + �w� � + w� �� +w� ���δw� � + �w� � + w� �� + w� ���δw� � + �w� � +w� �� + w� ���δw� ��� dx dy …(12) Where: x�& ‚�- ‚�* ‚�' ‚�) ‚ �� z =c �2Gn2G �1‚ $‚ $-‚ U�$�‚ $U�$�‚ [U�$�]- � g$ 2.3 Equation of Motion The Euler-Lagrange is obtained by substituting equation(8 - 12) into equation (6), then setting the coefficient of (d , d�, d�� ‚d��� ‚ d���) of Eq.(6) to zero separately, this give five equations of motion as follows: δu ∶ 8� 8< + 8� ¡8B = I&u¢ δv ∶ 8� ¡8< + 8�¡8B = I&v¢ δw�� : 8G¤ :888< 8B + 8G¤¡>8BG + 8¦ §>8< + 8¦¡§>8B +N�w� = I&�w¢ � + w¢ �� + w¢ ��� − I� 8G8¥G �8G9>?8?8BG � δw� ∶ 8¦ §�8< + 8¦¡§�8B + N�w� = I&�w¢ � + w¢ �� +w¢ ��� …(13) Where: N�w� = N