Analysis of Simply Supported Composite Plates with Uniform Pressure using ANSYS and Maple

Second Progress Report11/28/2013

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 106 psi

Thickness (h) 0.250 inchPoisson's Ratio (ν) 0.3Edge Length (a) 24 inchApplied Surface Pressure (q)

10 psi

• wmax = α*q*a4/D

• D = E*h3/12*(1-ν2)

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 ANSYSwmax = 0.941085”

• % Error = 0.033%

Material Properties of Composite Laminate

Edge Length (a) 24 inch

Ply Thickness 0.040 inch

E1 2.25e7 psi

E2 1.75e6 psi

E3 1.75e6 psi

ν12 0.248

ν23 0.458

ν13 0.248

G12 6.38e5 psi

G23 4.64e5 psi

G13 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 PlateComposite Plate Results

[0 90 0 90]s Laminate

•From governing equations:wmax = 0.7146”

• From ANSYSwmax = 0.7182”

• % Error = -0.5%

Results of Composite Plate (ANSYS)Laminate Stack-up Deflection - ANSYS (in) Deflection - Maple (in) Percent Error

[0 90]s 5.682 5.666 -0.282

[0 90 0 90]s 0.7182 0.7146 -0.50

[0 90 0 90 0 90]s 0.2141 0.21196 -1.0

[0 90 0 90 0 90 0 90]s 0.091 0.0895 -1.7

[+/-30 0]s 1.404 1.4585 3.74

[+/-45 0]s 1.271 1.3822 8.04

[+/-60 0]s 1.405 1.4628 3.95

[+/-30 0 +/-30 0]s 0.1592 0.1777 10.4

[+/-45 0 +/-45 0]s 0.1452 0.1685 13.81

[+/-60 0 +/-60 0]s 0.1600 0.1796 10.9[+/-30]s 6.299 5.7712 -9.15

[+/45]s 5.796 5.680 -2.04[+/-60]s 6.299 5.7712 -9.15[+/-30 +/-30]s 0.5966 0.5546 -7.57

[+/-45 +/-45]s 0.4989 0.5614 11.13[+/-60 +/-60]s 0.5966 0.5546 -7.57[+/-30 +/-30 +/-30]s 0.1528 0.1719 11.11

[+/-45 +/-45 +/-45]s 0.1371 0.1611 14.9[+/-60 +/-60 +/-60]s 0.1528 0.1719 11.29

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:

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

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.