Analysis of Simply Supported Aluminum and Composite Plates with Uniform Loading to

Determine Equivalent Plate Ply Stack-Up

by

Kenneth Carroll

An Engineering Project Submitted to the

Graduate Faculty of Rensselaer Polytechnic Institute

In Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING IN MECHANICAL ENGINEERING

Approved:

________________________________________________

Ernesto Gutierrez-Miravete, Engineering Project Advisor

Rensselaer Polytechnic Institute

Hartford, Connecticut

December 2013

CONTENTS

List of Tables............................................................................................................

List of Figures..........................................................................................................

List of Symbols.........................................................................................................

Acknowledgement....................................................................................................

Abstract....................................................................................................................

Introduction...............................................................................................................

Methodology.............................................................................................................

Aluminum Plate Analysis...............................................................................

Thin Plate Theory................................................................................

Material Properties..............................................................................

Equation for Thin Plate Theory..........................................................

Composite Plate Analysis.............................................................................

Composite Thin Plate Theory............................................................

Material Properties of Composite Material........................................

Equations for Composite Thin Plate Theory.....................................

Maximum Stress Failure Criterion.....................................................

Tsai-Wu Failure Criterion..................................................................

Plate Model...................................................................................................

ANSYS Model for Aluminum Plate.....................................................

ANSYS Model for Composite Plate....................................................

Results.....................................................................................................................

Aluminum Plate.............................................................................................

Composite Plate...........................................................................................

Composite Failure Criterion Results............................................................

ANSYS Results - Aluminum Plate................................................................

ANSYS Results - Composite Plate...............................................................

Error Analysis...............................................................................................

Conclusion...............................................................................................................

References..............................................................................................................

Appendix A: Thin Plate Theory Analysis................................................................

Appendix B: Table from Thin Plate Theory by Timoshenko...................................

Appendix C: Composite Thin Plate Theory.............................................................

Appendix D: ANSYS Code for Aluminum Plate.......................................................

Appendix E: ANSYS Code for Composite Plate......................................................

LIST OF TABLES

Table 1: Material Properties of Aluminum

LIST OF FIGURES

LIST OF SYMBOLS

t - thickness

σ - stress

τ - shear stress (N/m2)

ε - strain (m/m)

ν - Poisson's Ratio

E - Modulus of Elasticity

G - Shear Modulus

γ - Engineering Shear Strain

C - Stiffness Matrix

S - Compliance Matrix

Q - Reduced Stiffness Matrix

[T] - Transformation matrix m = cos(θ) n = sin(θ)

Q - Transformed Reduced Stiffness

θ - ply angle

M - Bending Moment Resultant

N - Force Resultant

uo - displacement in x-direction

vo - displacement in y-direction

wo - displacement in z-direction

εo - Extensional Strain of Reference Surface

κox, κ

oy - Curvature of Reference Surface

γoxy - Surface In-plane Shear Strain

κoxy - Reference Surface Twisting Curvature

A - Extensional Stiffness Matrix

B - Coupling Stiffness Matrix

D - Bending Stiffness Matrix

N+x - Normal Force Resultant

N+XY - Shear Force Resultant

M+x - Bending Moment Resultant

M+xy - Twisting Moment Resultant

Q+x - Transverse Shear Force Resultant

q - Applied Distributed Force

x - x-direction

y - y-direction

z - z-direction

a - length in x-direction

b - length in y-direction

wmax - maximum deflection

ACKNOWLEDGMENT

ABSTRACT

INTRODUCTION

2. Methodology

2.1 Thin Plate Theory of Aluminum Plate

The analysis of thin plates with small deflection makes the following three assumptions

when the deflection, w, is small in comparison to the thickness of the plate1:

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 stresses in the direction transverse to the plate can be disregarded.

These three assumptions for thin plate theory is based off of Kirchhoff-Love Plate

Theory. Thin plate theory relies on different boundary conditions to constrain the plate.

The assumptions that are made for thin plates with small deflections means that the

material of the plate will not be stretched. With these three assumptions and the

boundary conditions, the deflection of the plate, w, can be calculated.

2.1.1 Material Properties of Aluminum

For this series of experiments the following material properties for Aluminum were used:

Table 1: Material Properties of Aluminum

Modulus of Elasticity (E) 10 x 106 psi

Thickness (h) 0.250 inch

Poisson's Ratio (ν) 0.3

Edge Length (a) 24 inch

Applied Surface Pressure (q) 10 psi

2.1.2 Equation for Thin Plate Theory

For this analysis it was assumed that the thin plate is square and will have a uniformly

applied surface pressure of 10 psi. The square plate will be simply supported along

each edge.

There are a series of equations that can be used for analyzing a simply supported

rectangular plate. With the simply supported boundary conditions for the square plate

with a uniform load, the best equation for the maximum deflection of the plate is from

Theory of Plates and Shells (Timoshenko & Woinowsky-Krieger) Article 30: Alternate

Solution for Simply Supported and Uniformly Loaded Rectangular Plates. The

1 Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 1

derivation of the maximum deflection equation from Article 30 can be found in Appendix

A. The derived equation for the maximum deflection of the plate can be expressed by:

wmax = α

(141)2

where D =

(3)3

α is a factor that is dependent on the ratio of the edge length of the plate. Appendix B

shows Table 8 from Theory of Plates and Shells for the numerical factors for a uniformly

loaded and simply supported rectangular plate. Equation (141) is the governing

equation for calculating the maximum deflection of the aluminum plate using thin plate

theory.

2.2 Thin Plate Theory for Composite Plate

2.2.1 Material Properties of Composite

2.2.2 Equations for Composite Thin Plate Theory

2.2.3 Maximum Stress Failure Criterion

2.2.4 Tsai-Wu Failure Criterion

2.3 ANSYS Model for Aluminum Plate

2 Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 117

3 3 Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 5

The aluminum plate was modeled in ANSYS using a SHELL63 element. The SHELL63

element was used for the aluminum plate analysis because the element has both

bending and membrane capabilities and is used for a linear elastic analysis. Each node

of the element has six degrees of freedom, three translational degrees of freedom and

three rotational degrees of freedom.

The SHELL63 element was used to create an area by dimensions in the active

workspace. Due to the symmetry of the aluminum plate, only a quarter of the plate has

to be created for the analysis. The 12 inch square was oriented in the x-y plane. A

thickness of 0.250 inch was entered for the plate in the z-direction. The edge length for

the generate mesh was 0.75 inch. Figure 1 below shows the generated plate in ANSYS

with meshing.

Figure 1: SHELL63 Element with Mesh (Edge Length = 0.75")

The aluminum plate had a series of boundary conditions applied in order to measure the

maximum deflection. The keypoint at the origin of the active workspace is where the

maximum deflection should be measured. Side 1 is constrained in the Uz direction to

prevent translation in the z-direction and in ROTy to prevent rotation in the y-direction.

Side 2 is constrained in the Uz direction to prevent translation in the z-direction and in

ROTx to prevent rotation in the x-direction. Side 3 is constrained in ROTx to prevent

rotation in the x-direction. Side 4 is constrained in ROTy to prevent rotation in the y-

direction. The keypoint at the origin is constrained in Ux and Uy directions. These

boundary conditions create the simply supported edges along Side 1 and Side 2. The

sides are prevented from freely rotating in both the x- and y-directions. Figure 2 shows

Side 1

Side 2

Side 3

Side 4

Origin

the model in ANSYS with the applied boundary conditions and a pressure of 10 psi

acting on the plate in the negative z-direction.

Figure 2: SHELL63 Element with Boundary Conditions and Pressure Applied

2.4 ANSYS Model for Composite Plate

3. Results

3.1 Aluminum Thin Plate Results

Using the equations listed in Section 2.1.2:

D =

= 14,308.608 lb*inch

For b/a = 1.0, α = 0.00406

wmax = 0.00406 *

= 0.941399 inch

3.2 Composite Thin Plate Results

3.3 Composite Failure Criterion Results

3.4 ANSYS Results - Aluminum Plate

Figure 3: Aluminum Plate Deformed + Undeformed

Deflection in negative z-direction = 0.941085"

Figure 4&5: Aluminum Plate Nodal Solution - Z-Component of Displacement

3.5 ANSYS Results - Composite Plate

3.6 Error Analysis

The maximum deflection using ANSYS to analyze the simply supported aluminum plate

is 0.941085". The maximum deflection using the governing equations for the simply

supported aluminum plate is 0.941399".

% Error =

* 100

% Error = -

* 100 = 0.0334%

4. Conclusions

DEADLINES

Deliverable Due: Date

Tentative Project Proposal 9/28/2013

Project Proposal Draft 9/30/2013

Development of Aluminum Model in ANSYS

10/7/2013

Organization of Thin Plate Analysis 10/7/2013

First Progress Report 10/7/2013

Development of Composite Model in ANSYS

10/21/2013

Organization of Composite Ply Thin Plate Analysis in Excel

10/21/2013

Second Progress Report 10/28/2013

Final Draft 11/11/2013

Preliminary Final Report 11/25/2013

Final Report 12/13/2013

Final Presentation 12/13/2013

REFERENCES

1. Hyer, Michael W. Stress Analysis of Fiber-Reinforced Composite Materials

Update Edition, 2009

2. Timoshenko, S. and Woinowsky-Krieger, S. Theory of Plates and Shells

2nd Edition, 1959

3. Notes from MANE 6180 Mechanics of Composite Materials R. Naik 2013

4. Manahan, Mer Arnel A Finitie Element Study of the Deflection of Simply Supported

Composite Plates Subject to Uniform Load. RPI Hartford Master's Project December

2011

5. Kirchoff-Love Plate Theory Wikipedia

http://en.wikipedia.org/wiki/Kirchhoff%E2%80%93Love_plate_theory

APPENDIX A

APPENDIX B

Timoshenko & Woinowsky-Krieger Theory of Plates and Shells page 120

APPENDIX C

APPENDIX D

APPENDIX E