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Analysis of Simply Supported Composite Plates with Uniform Pressure using ANSYS and Maple Second Progress Report 11/28/2013

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Thin Plate Theory Three Assumptions for Thin Plate Theory There is no deformation in the middle plane of the plate. This plane remains neutral during bending. Points of the plate lying initially on a normal-to-the- middle plane of the plate remain on the normal-to- the-middle surface of the plate after bending The normal stress in the direction transverse to the plate can be disregarded

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Material Properties and Governing Equations Modulus of Elasticity (E) 10 x 10 6 psi Thickness (h)0.250 inch Poisson's Ratio (ν)0.3 Edge Length (a)24 inch Applied Surface Pressure (q) 10 psi w max = α*q*a 4 /D D = E*h 3 /12*(1-ν 2 )

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ANSYS Model with Mesh Side 2 Side 1 Side 4 Side 3 Origin Due to Symmetry only a quarter of the plate needs to be modeled The mesh size has an edge length of 0.75 Side 1 and Side 2 are constrained against translation in the z-direction. Side 2 and Side 3 is constrained against rotating in the x-direction Side 1 and Side 4 is constrained against rotation in the y-direction The origin is constrained against motion in the x- and y-directions A pressure of 10 psi is applied to the area Side 2 Origin Side 4 Side 1 Side 3

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Results of Aluminum Plate From governing equations: wmax = 0.941399 From ANSYS wmax = 0.941085 % Error = 0.033%

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Material Properties of Composite Laminate Edge Length (a)24 inch Ply Thickness0.040 inch E1E1 2.25e7 psi E2E2 1.75e6 psi E3E3 ν 12 0.248 ν 23 0.458 ν 13 0.248 G 12 6.38e5 psi G 23 4.64e5 psi G 13 6.38e5 psi Applied Surface Pressure (q) 10 psi

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Governing Equations Analysis and Performance of Fiber Composites: Agarwal & Nroutman ABD Matrix:

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Governing Equations (cont.) Mechanics of Composite: Jones For Cross-ply Laminates the [D] matrix simplifies and the governing equation reduces to:

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Governing Equations (cont.) For symmetric angle laminates, the ABD matrix is fully defined. The boundary conditions for a symmetric angle laminate are:

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Governing Equations (cont.) Using the Rayleigh-Ritz Method based on the total minimum potential energy will provide an approximation of the deflection of the plate

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ANSYS Model with Mesh Side 2 Side 1 Side 4 Side 3 Origin Due to Symmetry only a quarter of the plate needs to be modeled The mesh size has an edge length of 0.75 Side 1 and Side 2 are constrained against translation in the z-direction. Side 2 and Side 3 is constrained against rotating in the x-direction Side 1 and Side 4 is constrained against rotation in the y-direction The origin is constrained against motion in the x- and y-directions A pressure of 10 psi is applied to the area Side 2 Origin Side 4 Side 1 Side 3

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Results of Composite Plate Composite Plate Results [0 90 0 90]s Laminate From governing equations: wmax = 0.7146 From ANSYS wmax = 0.7182 % Error = -0.5%

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Results of Composite Plate (ANSYS) Laminate Stack-upDeflection - ANSYS (in)Deflection - Maple (in)Percent Error [0 90] s 5.6825.666-0.282 [0 90 0 90] s 0.71820.7146-0.50 [0 90 0 90 0 90] s 0.21410.21196 [0 90 0 90 0 90 0 90] s 0.0910.0895-1.7 [+/-30 0] s 1.4041.45853.74 [+/-45 0] s 1.2711.38228.04 [+/-60 0] s 1.4051.46283.95 [+/-30 0 +/-30 0] s 0.15920.177710.4 [+/-45 0 +/-45 0] s 0.14520.168513.81 [+/-60 0 +/-60 0] s 0.16000.179610.9 [+/-30]s6.2995.7712-9.15 [+/45]s5.7965.680-2.04 [+/-60]s6.2995.7712-9.15 [+/-30 +/-30]s0.59660.5546-7.57 [+/-45 +/-45]s0.49890.561411.13 [+/-60 +/-60]s0.59660.5546-7.57 [+/-30 +/-30 +/-30]s0.15280.171911.11 [+/-45 +/-45 +/-45]s0.13710.161114.9 [+/-60 +/-60 +/-60]s0.15280.171911.29

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Failure Criterion

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Failure Criterion (cont.) The Tsai-Wu Failure Criterion is based on the following equations:

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Failure Criterion (cont.) Maximum Stress Criterion for bi-axial loading of composite plate:

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Failure Criterion (cont.) Tsai-Wu Criterion for bi-axial loading of composite plate:

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Conclusions The composite plate that had the smallest deflection was the 12 ply [+/-45 +/-45 +/-45]s laminate. The thinnest plate that had the smallest deflection was the 8 ply [+/-30 +/-30]s and [+/-60 +/-60]s laminates The larger percent error for the results occurred for the symmetric angle ply trials. This is because of the nature of the Rayleigh-Ritz Method. When the composite has symmetric angle plies there is a full [D] matrix. The full [D] matrix does not allow for a separation of variables method to be used to calculate the deflection because not all of the boundary conditions can be satisfied. The Rayleigh-Ritz Method approximates the deflection by using a Fourier expansion for the total potential energy. The calculated percent error seems to be within reason for the analysis that was done for this project. The Rayleigh-Ritz Method does not provide an exact solution when compared to the method for a specially orthotropic plate. The most reasonable plate arrangement that would be suitable for replacing the aluminum plate is the 8 ply orientations of [+/-30 +/-30]s, [+/-45 +/-45]s, [+/-60 +/-60]s. These three ply combinations can withstand a significant stress in the 1-direction, 2-direction, and 12-direction (shear) in comparison to other composite plates. These 8 ply plates will also be marginally thicker than the 0.25" aluminum plate, but provide a significant decrease in overall weight.

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1 CM 197 Mechanics of Materials Chap 10: Strength of Materials Strains Professor Joe Greene CSU, CHICO Reference: Statics and Strength of Materials, 2.

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