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

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

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

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 )

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

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

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

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

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

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

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

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

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%

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

Failure Criterion

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

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