Vibration Analysis of Cross-Ply Plates Under Initial Stress Using Refined Theory

Natural frequency under initial stresses for simply supported cross-ply composite laminated plates (E glassfiber) 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.


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 transverseshear 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 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 freevibration 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 freevibration 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.

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 ‫ݓ‬ , extension ‫ݓ‬ and shear ‫ݓ‬ ௦ 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].
The shear components ‫ݑ‬ ௦ and ‫ݒ‬ ௦ , ‫ݓ‬ ௦ lead to the parabolic variations of shear strains ߛ ௫௭ , ߛ ௬௭ and to shear stresses ߪ ௫௭ , ߪ ௬௭ through the thickness of the plate in such a way that shear stresses ߪ ௫௭ , ߪ ௬௭ are zero at the bottom and top surfaces of the plate.
The following displacement field assumptions [3]: ) For small strain, the strain-displacement relations take the form: By substituting eq. ( 1) into eq.( 2) to give: The strain field is: Where:

Principle of Virtual Work
Using Hamilton's principles, the equations of motion of the refined plate theory will be derived.Reddy, 2004 • The virtual strain energy ߜܷ is: 4) into Eq.( 7) The virtual strains are known in terms of virtual displacement eq.( 4) and then Substituting the virtual strain into Eq.( 8) and integrating by parts to relative virtual displacement ‫,ݑߜ(‬ ‫,ݒߜ‬ ‫ݓߜ‬ ‫ݓߜ‚‬ ‚ ‫ݓߜ‬ ௦ ) in range of any differentiation, then we get: The virtual work done ߜV is:

Equation of Motion
The Euler-Lagrange is obtained by substituting equation (8 -12) into equation ( 6), then setting the coefficient of ‫,ݑߜ(‬ ‫,ݒߜ‬ ‫ݓߜ‬ ‫ݓߜ‚‬ ‚ ‫ݓߜ‬ ௦ ) of Eq.( 6) to zero separately, this give five equations of motion as follows: ) Where: The result forces are given by: Reddy [9].
The plane stress reduced stiffness ܳ is: From the constitutive relation of ݇ ௧ layer lamina, the transformed stress-strain relation are: The force results are related to the strains by relations:

Navier's Solution
To solve equations of motion (15-16), Navier's generalized displacements are used which satisfy the boundary conditions of the problem as shown in Fig. 2, therefore Simply supported boundary conditions are satisfied by assuming the following form of displacements: Reddy [ The following stiffnesses are zero if the Navier solution exists,

Vibration Analysis
Developing mass matrix and stiffness matrix from solution of homogeneous equations, when mechanical loading is equal to zero for free vibration, then eigenvalue equation is derived and the natural frequencies of vibration for simply supported plate are obtained.

Buckling
The applied loads for buckling analysis, are supposed to be in-plan forces

Free Vibration Analysis Under Initial Stress
The natural frequency is investigated with action of buckling, a ratio critical load (d) is applied.The effect of the load ratio (d) is studied to present the behavior of the plate and its frequency.

Vibration and Buckling Results
The fundamental natural frequency and critical buckling for cross-ply plate with different design parameters for simply supported boundary condition, is analyzed and solved used MATLAB programming.We derive equation of motion depending on Refined plate theory using Navier solution to obtain vibration characteristic of plate under initial stress.To examine the validity of the derived equation and performance of computer programming for vibration and buckling stress of cross-ply laminated simply supported plate, a comparison with others researchers for different layers, thickness ratio (a/h) and orthotropy ratio (E1/E2).The non-dimensional natural frequency of antisymmetric cross-ply two, four, six and ten layer of thick plate (a/h=5) as a function of orthotropy ratio (E1/E2) shown in table (1) for the mechanical properties ‫ܩ[‬ ଵଶ = ‫ܩ‬ ଵଷ = 0.6 E ଶ ‚ ‫ܩ‬ ଶଷ = 0.5 E ଶ ‚ ‫ݒ‬ ଵଶ = 0.25] while Table (2) shows the Non-dimensional fundamental frequencies of antisymmetric square laminated plate for various values of thickness ratio and modulus ratio (E1/E2=40).The natural frequency shows good agreement with other researchers.The present theory is also close agreement with other theory for critical buckling as shown in table (3) which compared with [2] for Non-dimensional uniaxial buckling load of simply supported antisymmetric layer for (a/h=10), while Table (4) and Table (5) are compared with [9] which show the Nondimensional uniaxial and biaxial buckling load of simply supported antisymmetric cross-ply for two and eight layers for various thickness ratio (a/h) and as a function of modulus ratios (E1/E2).

Vibration of Plate Under Initial Stress Results
Vibration analysis of present work is used but adding initial in-plane stress to investigate the validity of Refined theory for such case.Table (6) shows Non-dimensional natural frequency under various uniaxial loads ratio(d) for simply supported cross-ply [0/90/0] square plate with orthotropy ratio (E1/E2=10).Table (7) shows Nondimensional natural frequency under various uniaxial loads ratio with various thickness ratio (a/h) for simply supported cross-ply [0/90/0] square plate with orthotropy ratio (E1/E2=40).The fundamental frequency decrease when increasing the value of compressive stress until the lowest natural frequency vanished when inplane stress reaches the critical buckling stress, which proved by other researchers, as shown in Fig ( 3) and Fig ( 4).

Conclusion
Natural frequency and buckling stress of simply supported cross-ply square plate subject to initial axial stress have been obtained by using Refind plate theory.It is observed good results for natural frequency and critical buckling for uniaxial and biaxial load as compared with other researchers.The following conclusions may be drawn from the present analysis: 1. Refined plate theory for analyzing natural frequency and buckling stresses of cross-ply square plate has been presented.It is observed that the natural frequency and buckling load increasing as the number of layer and thickness ratio increases.

Table 5 , Non-dimensional biaxial buckling load of antisymmetric cross-ply laminates Table 6, Dimensionless natural frequency of a laminated plate under buckling different ratio (d).
2. The buckling stresses can be calculated through the stability equation as Eigen value problems.Another method to obtain the critical stress of cross-ply plate subject to axial and uniaxial inplane stresses is to compute natural frequency by increasing the absolute value of compressive stress until the lowest natural frequency vanishes.