A Finite Element Study of the Deflection of Simply Supported

Composite Plates Subject to Uniform Load

by

Mer Arnel Manahan

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 Adviser

Rensselaer Polytechnic Institute Hartford, Connecticut

December 2011

ii

Copyright 2011

By

Mer Arnel Manahan

All Rights Reserved

iii

CONTENTS LIST OF TABLES............................................................................................................. vLIST OF FIGURES .......................................................................................................... viLIST OF SYMBOLS....................................................................................................... viiACKNOWLEDGMENT ................................................................................................ viiiABSTRACT ..................................................................................................................... ix1. Introduction.................................................................................................................. 12. Methodology................................................................................................................ 3

2.1 Classical Lamination Theory ............................................................................. 32.1.1 Kirchhoffs Hypothesis .......................................................................... 4

2.2 Governing Equations for the Response of a Laminated Plate............................ 62.3 Plate Modeling ................................................................................................. 11

3. Results........................................................................................................................ 143.1 Introduction...................................................................................................... 143.2 MATLAB Validation....................................................................................... 15

3.2.1 Deflection Function.............................................................................. 163.3 4-Ply Deflection Results .................................................................................. 18

3.3.1 Optimization of 4-Ply Composite Plate and Effect of D16 and D26...... 193.3.2 MATLAB Inaccuracy When Calculating Symmetric Angle-Ply

Laminate............................................................................................... 203.4 20-Ply Deflection Results ................................................................................ 21

4. Conclusion ................................................................................................................. 255. References.................................................................................................................. 27Appendix A: ANSYS Analysis code for 4-Ply Composite Plate .................................... 28Appendix B: MATLAB Validation code for 4-Ply Composite Plate.............................. 30Appendix C: ANSYS Analysis code for 20-Ply Composite Plate .................................. 32Appendix D: MATLAB Validation code for 10-Ply Composite Plate............................ 34

iv

Appendix E: Derivation of Deflection Solution for a Uniform Load.............................. 37

v

LIST OF TABLES

Table 1: Material properties for VM 824 ......................................................................... 12Table 2: Summary of Nineteen Cases Investigated......................................................... 14Table 3: ANSYS and MATLAB Results for 4-Ply Composite Plates ............................ 18Table 4: ANSYS and MATLAB results for 20-Ply Composite Plates............................ 21

vi

LIST OF FIGURES

Figure 1: Cross-section of cross-ply laminate2.................................................................. 1Figure 2: Plate Dimension1................................................................................................ 4Figure 3: Kirchhoffs straight line2.................................................................................... 4Figure 4: Showing Geometry, Nomenclature and Loading2 ............................................. 6Figure 5: Resultants at the x = a/2 boundary2.................................................................... 6Figure 6: Forces in the Z-direction2................................................................................... 7Figure 7: Moments on all sides about the x-axis2.............................................................. 8Figure 8: SHELL181 Geometry4 ..................................................................................... 11Figure 9: Visual Representation of Boundary Conditions............................................... 13Figure 10: Fourier Expansion for a Uniform Load8 ........................................................ 17Figure 11: Deflection of 4-Ply [0 90]S Plate.................................................................... 19Figure 12: ANSYS Deflection Results for 4-Ply Laminate............................................. 20Figure 13: ANSYS Deflection Results for 20-Ply Plates ................................................ 23Figure 14: % Error between MATLAB and ANSYS as 60 Ply Changes Location ....... 24

vii

LIST OF SYMBOLS

N = Normal force resultant (N/m)

M = Bending moment resultant (N-m/m)

Q = Transverse shear force resultant (N/m)

E = Elastic Modulus (N/m2)

G = Shear Modulus (N/m2)

= Poissons Ratio q = Load applied (N/m2)

S = Compliance matrix (N/m2)

Q = Stiffness matrix (N/m2)

A = Extensional stiffness matrix (N/m)

B = Coupling stiffness matrix (N)

D = Bending stiffness matrix (N-m)

= Fiber orientation angle (o) e0,0, 0 = reference surface extensional strains u0 = displacement in the x-direction (m)

v0 = displacement in the y-direction (m)

w0 = displacement in the z-direction (m)

t = Thickness (m)

viii

ACKNOWLEDGMENT

To my parents Arnel and Susan Manahan: Thank you for your support not only in my

pursuit of higher academic achievements, but in life in general. Thank you to all my

professors at SUNY Binghamton and RPI for their time and effort spent to help me grow

academically. A special thanks to Professor Ernesto Gutierrez-Miravete for his patience,

understanding and guidance throughout the masters project.

Give thanks to the LORD, for he is good; his love endures forever

- 1 Chronicles 16:14

ix

ABSTRACT

This paper discusses the response of simply supported, symmetrically laminated

composite plates subjected to a uniformly distributed load. ANSYS is used to both

model and perform FEA analysis on the plate of a 4-ply composite at first. The response

of the plate to the uniformly distributed load was analyzed and validated through

calculations using MATLAB in coherence with the classical lamination theory. Once the

FEA had been validated, an investigation on the affects of the fiber orientation to the

displacement value was conducted for different cases of symmetric 4-ply layup. The

same analysis was completed and validated for a 20-ply plate and in addition will reveal

the effect of stacking sequence on bending stiffness.

1

1. Introduction

Fiber-reinforced composites (FRC) have an extensive array of applications. They

range from structural to recreational use. The aerospace and automotive industries look

to composites to improve fuel economy due to its high strength to weight ratio. The

sports industry looks to composites to improve sports equipment technologies. The fact

that composites offer increased strength without sacrificing additional weight is what

gives composites the advantage from most structural and recreational materials1.

FRCs are often manufactured in laminates. A laminate consists of individual

lamina or plies as seen in Figure 1. These plies consist of different combinations of fiber

materials embedded in a matrix material, usually of a polymer resin.

Figure 1: Cross-section of cross-ply laminate2

When designing for the use of composites, there are many aspects to investigate as there

are many variables that affect a laminates response to load. Fiber and matrix material,

fiber orientation, layer stacking sequence are some of the variables affecting the

response of a laminate. The virtually limitless combinations of ply materials, ply

2

orientations and ply-stacking sequences increase the design flexibility of composite

structures. This paper will investigate the response of composite plates to uniformly

distributed load.

3

2. Methodology

This project develops FEA models of a simply supported, symmetric, composite

laminate plate under a uniformly distributed load. The first step is to develop an FEA

model of a composite plate that consists of 4 layers. The model will be subjected to a

uniformly distributed load. Deflection results will be analyzed and validated with hand

calculations using MATLAB. The hand calculations will apply classical lamination

theory to arrive at a deflection value of the plate. Once the hand calculations validate the

4-ply FEA model, an investigation of the affects of fiber orientations on deflection

results will be conducted. Deflection results will then be discussed, and a case of a 20-

ply laminate, symmetric plate will be modeled and analyzed. This model will also be

validated through hand calculations with investigations on how additional plies and

layup sequence affects the deflection results.

2.1 Classical Lamination Theory

The analysis of laminate plates involves the Classical Lamination Theory which is

almost identical to the classical plate theory except for anisotropy. Both require that the

laminate be thin where the span of a and b is greater than 10 times the thickness t and is

visualized through Figure 2 below. They also require a small displacement in the

transverse direction w, where w is significantly smaller than t. They both share the

assumptions made by Kirchhoffs Hypothesis.2

4

Figure 2: Plate Dimension1

2.1.1 Kirchhoffs Hypothesis

Kirchhoffs Hypothesis assumes that normals remain straight, normals remain

unstretched, and normals remain normal. Figure 3(a) below shows the cross-section of a

laminate plate. The midplane is located on the x-axis, and line AA is perpendicular to

the midplane.

Figure 3: Kirchhoffs straight line2

5

Kirchhoff proposed that the straight line AA in Figure 3 above, will remain straight,

will not stretch and will remain perpendicular to the midplane under various loading

conditions. Figure 3(a) shows an undeformed laminate plate. The laminate plate,

depending on loading conditions may bend or twist, but the line AA will remain straight

and perpendicular to the midplane. Line AA will translate and rotate accordingly as

seen in Figure 3(b). The rotations and translations can be described by the following

equations where u, v and w are the displacements in the x, y and z directions

respectively. uo, vo and wo are the translations at any point on the AA line in the x, y,

and z directions respectively and y

wo

and

xwo

are the rotations about the x and y axes

respectively:

xyxwzyxuzyxu

oo

),(),(),,( (1)

yyxwzyxvzyxv

oo

),(),(),,( (2)

),(),,( yxwzyxw o (3)

Another important assumption is that the line AA does not change in length. Line

segment tt is the same length in both Figure 3(a) and Figure 3(b). This indicates that

there is zero strain in the z-direction.

In addition to Kirchhoffs Hypothesis, perfectly bonded layers are assumed. This

proposes that there is no flaw or gap between the layers, and no slipping between the

layers. No slipping between the layers means that the layers will remain parallel with

one another and hence remain perpendicular to the normal AA.

6

2.2 Governing Equations for the Response of a Laminated Plate

The governing equations consist of the behavior of the plate internally as well as

the behavior of the boundary conditions. The governing equations will be derived below

using the Newtonian approach where summing the forces and moments on the plate is

used to develop the differential equations. Figure 4 below shows a composite plate in the

x-y-z orientations, a loading of q(x,y) and a width a and a length b with the reference

point at the center of the plate.

Figure 4: Showing Geometry, Nomenclature and Loading2

Figure 5: Resultants at the x = a/2 boundary2

7

Figure 5 above shows the resultant forces N and moments M acting on the x = a/2 side of

the plate. The results are then required from the x = - a/2 and y = +- b/2 sides. Once

these resultant forces are drawn, the sum of the forces in the x and y directions can be

found. This results in two of the three governing equations.

0

y

Nxyx

Nx (4)

0

y

Nyx

Nxy (5)

The third governing equation is derived by the summation of the shear force resultants Q

acting on the interior of the plate. In Figure 6 below, q(x,y) is the applied load and the

resultant forces are shown.

Figure 6: Forces in the Z-direction2

8

Summation of these forces results in:

0

qy

Qyx

Qx (6)

The moment equilibrium is now taken about the x-axis as seen in Figure 7 below. The

sum of the moments results in

xMxy

yMyQy

(7)

Summing up the moments about the y-axis results in:

yMxy

xMxQx

(8)

Substituting equations (7) and (8) in equation (6), results in the third governing equation.

02 222

2

2

q

yMy

yxMxy

xMx (9)

Figure 7: Moments on all sides about the x-axis2

9

Since the purpose of this paper is to investigate a composite laminate plates deflection

in response to a uniformly distributed load, it is beneficial to express the governing

equations in terms of displacements. Fortunately, the stress resultants can be expressed

in terms of strains and curvatures by

0

0

0

0

0

0

662616111116

262212111112

161211111111

662616662616

262212262112

161211161211

xy

y

x

xy

y

x

ee

DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA

MxyMyMxNxyNyNx

(10)

The strains and curvatures can be expressed in terms of displacements.

yyxu

xyxvyx

yyxvyxe

xyxuyxe

xy

y

x

),(),(),(

),(),(

),(),(

000

00

00

yxyxwyx

yyxwyx

xyxwyx

xy

y

x

),(2),(

),(),(

),(),(

020

2

020

2

020

(11)

Likewise, the stress resultants can be expressed in terms of displacements3. Substituting

equation (11) into equation (10), we get equation (12)-(17):

yxwB

ywB

xwB

xv

yuA

yvA

xuANx

ooooooo

2

162

2

122

2

11161211 2 (12)

yxwB

ywB

xwB

xv

yuA

yvA

xuANy

ooooooo

2

262

2

222

2

12262212 2 (13)

yxwB

ywB

xwB

xv

yuA

yvA

xuANxy

ooooooo

2

662

2

262

2

16662616 2 (14)

10

yxwDB

ywD

xwD

xv

yuB

yvB

xuBMx

ooooooo

2

162

2

122

2

11161211 (15)

yxwD

ywD

xwD

xv

yuB

yvB

xuBMy

ooooooo

2

262

2

222

2

12262212 2 (16)

yxwD

ywD

xwD

xv

yuB

yvB

xuBMxy

ooooooo

2

662

2

262

2

16662616 2 (17)

Substituting expressions in equations (12)-(17) into the governing equations (4), (5) and

(9), we get the equilibrium equations that govern the response of a laminated plate in

terms of displacements.3

0)2(3

)(2

3

3

262

3

66122

3

163

3

11

2

2

26

2

66122

2

162

2

66

2

162

2

11

ywB

yxwBB

yxwB

xwB

yvA

yxvAA

xvA

yuA

yxuA

xuA

oooo

oooooo

(18)

03)2(

2)(

3

3

262

3

66123

3

16

2

2

22

2

262

2

662

2

26

2

66122

2

16

ywB

yxwBB

xwB

yvA

yxvA

xvA

yuA

yxuAA

xuA

ooo

oooooo

(19)

03)2(

)2(3

4)2(24

3

3

223

3

262

3

6612

3

3

163

3

262

3

66122

3

163

3

11

4

4

223

4

2622

4

66123

4

164

4

11

yvB

yxvB

yxvBB

xvB

yuB

yxuBB

yxuB

xuB

ywD

yxwD

yxwDD

yxwD

xwD

ooo

ooooo

ooooo

(20)

11

Equations (18)-(20) are the equations in terms of displacements that govern the response

of a composite plate. In this project, we will limit the investigation to a simply

supported, symmetrically laminated composite plate.

2.3 Plate Modeling

Verification Manual 82 (VM82) ANSYS code was used for the purposes of this paper.

VM82 is a solved solution to a simply supported, composite plate subjected to a

uniformly distributed load. VM82 was utilized to validate the MATLAB code and

modified to run several different cases of laminate plates discussed in a later section.

VM82 uses several elements to perform the analysis on the plate. The SHELL181 was

chosen as the element to use for this paper because the input parameters of a composite

plate can be easily modified due to it being a shell element. SHELL181 was used as the

element type because it is suitable for thin to moderately thick shell structures and

commonly used for laminated composite shells. SHELL181 is a 4 node element with six

degrees of freedom at each node.4 Figure 8 below shows its geometry.

Figure 8: SHELL181 Geometry4

12

VM82 analyzes a [0 90]S , 5m by 5m composite laminate plate subjected to a uniformly

distributed load in the transverse direction with the material properties defined in Table

1. The ANSYS code can be found in APPENDIX A.

Table 1: Material properties for VM 824

The code in APPENDIX A shows that nine material property inputs were entered to run

the analysis, however Table 1 only consists of five.

VM82 assumes that Ey = Ez, xzxy , Gxy = Gxz, and yz = )(x

yxyyx E

E . The plate

is modeled using four key points at (0,0), (5,0), (5,5) and (0,5) xy coordinates. An area is

then made from these key points where the mesh is applied. Considering that the origin

is at (0,0), the boundary conditions are such that the side at x = 5, the plate is constrained

in the X and Z directions and at y = 5 it is constrained in the Y and Z directions. This

means that there will be no deflection and moments along these sides. A pressure of 1.0

N/m2 is then applied along the surface area of the plate. Figure 9 below provides a visual

representation of the boundary constraints applied in ANSYS. Figure 9 is one of the four

quadrants that the plate is divided into. Since the original plate is simply supported on all

sides, only two sides of the quadrant need to be constrained. This reduces the complexity

of the model.

13

Figure 9: Visual Representation of Boundary Conditions

14

3. Results

3.1 Introduction

This section discusses the results found during the FEA analysis and validation of

the composite plate subjected to a uniformly distributed load. Several cases were

analyzed and grouped by the amount of layers existent for each case. One group consists

of symmetric plates with only four layers. The other group consists of symmetric plates

with twenty layers. The group with four layer plate analyses ranged from a symmetric

plate to a symmetric, angle ply plates in which the fiber orientations of the outer layers

were changed to varying angles ranging from 0 to 90. The group of twenty layer plates

ranged from various stack up sequences of symmetric plates to symmetric, angle ply

plates also. Overall, there were nineteen cases performed. All of these cases were

modeled and analyzed in ANSYS and were also attempted to be validated in MATLAB.

The layup sequences of the different cases analyzed are tabulated in Table 2 below.

Table 2: Summary of Nineteen Cases Investigated

Layup(4PlyGroup) Layup(20PlyGroup)[090]s [6009090090900900]s[1090]s [0609090090900900]s[2090]s [006090090900900]s[3090]s [009060090900900]s[4090]s [0090906090900900]s[4590]s [009090060900900]s[5090]s [009090090600900]s[6090]s [0090900909060900]s[7090]s [009090090900600]s[8090]s [0090900909009060]s[9090]s

15

The first step of this project in the investigation of a composite plates response to

a uniformly distributed load begins with the validation of the FEA analysis through hand

calculations involving the Classical Lamination Theory as described in Section 2.1 of

this paper. Section 2.1.1 describes the assumptions applied by Kirchhoffs Hypothesis

and Section 2.2 shows the derivations of the equations that govern the response of a

composite plate in terms of displacements. The MATLAB code that applies the classical

lamination theory to a 4-ply symmetric laminate plate is found in APPENDIX B.

3.2 MATLAB Validation

The MATLAB code starts off by stating the material properties for each layer. In

this case, each layer has the same material properties listed in Table 1. In this section, the

code only enters in five material constants, E1, E2, G12, v12 and v21 (calculated), but

ANSYS requires nine material property inputs. This is because of the plane stress

assumption. In applications such as the plate analyzed in this project, three of the six

components of stress are generally much smaller than the other three. In the example of

a plate, the stresses acting in the plane of the plate are much larger than the stresses

perpendicular to that plane. In all calculations then, the stress components perpendicular

to the plane of the structure can be set to zero.2

The code then computes the values of the stiffness matrix [ ijQ ] for each ply. It does

this by first computing the compliance matrix [ ijS ] for each ply. This matrix is then

transformed and inverted to acquire the values of [ ijQ ]. Next, the code calculates the z

distances of each plate from the midplane which will provide the distances to calculate

the bending stiffness matrix [D]. This bending stiffness matrix is used to compute the

16

deflection of the plate. Since the composite that is analyzed by VM82 is a symmetric,

simply supported plate with the boundary conditions set to limit deflection in the x and y

directions, the governing equations (18)-(20) reduce to:

),()2(2 44

2222

4

66124

4

11 yxqywD

yxwDD

xwD

ooo

(21)

This reduction is due to the fact that for symmetric, orthotropic plates, D16 and D26 are

zero. The boundary conditions described cause the displacements uo and yo to be zero,

therefore those components with uo and yo go to zero.

3.2.1 Deflection Function

The deflection function for a symmetric composite rectangular plate, simply

supported on all sides under a uniformly distributed load is expressed by equation (22)5

1 422

226612

411

6

0

1 ])()())(2(2)([

sinsin16),(

nm

bnD

bn

amDD

amDmn

byn

axmq

yxw

(22)5

For m,n = 1,3,5

The method in APPENDIX E to derive equation (22) utilized the Navier solution for a

simply supported rectangular plate6. For any kind of loading given by )(xyqq , load

q(x,y) is represented by:

)sin()sin(),(11 b

yma

xmqyxq mnnm

(23)7

Then the differential equation (21) and boundary conditions can be satisfied by:

17

)sin()sin(),(11 b

yma

xmwyxw mnnm

(24)7

The derivation in APPENDIX E shows that when substituting equation (23) and (24)

into equation (21), we get:

])()(2(2[ 4222

66124

114

4

nRDmnRDDmDqa

w mn (25)7

Where R = a/b, the plate aspect ratio

APPENDIX E continues to solve for qmn, which for a uniform load, q(x,y) = qo, a

constant, the Fourier coefficients are7:

mnq

q omn 216 for m,n = 1,3,5 (26)

6,7,8

Figure 10 below shows that the Fourier expansion equation (26) represents a uniform

load distributed along the entire surface of the plate.

Figure 10: Fourier Expansion for a Uniform Load8

18

Equation (25) and (26) are now plugged into equation (24) to get the deflection equation

for a simply supported orthotropic laminated plate under a uniformly distributed load

expressed in equation (22).

3.3 4-Ply Deflection Results

The following results pertain to the analyses run through MATLAB and the FEA

through ANSYS. Table 3 below shows the results of not only analyses run for a [0 90]S

but for a symmetric angle-ply plate in which the inner layers are kept at 90 while the

fiber orientation of the outer layers are increased ranging from 0 to 90. The bending

stiffness values for D16 and D26 shown in Table 3 below were computed in the

MATLAB code in APPENDIX B. The code outputs the [D] matrix where the D16 and

D26 values were acquired.

Table 3: ANSYS and MATLAB Results for 4-Ply Composite Plates

LayupANSYSDeflection(m)

MatlabDeflection(m) %Error D16(Nm) D26(Nm)

[090]s 0.0681 0.0680 0.1468 0.0000 0.0000[1090]s 0.0883 0.0626 29.1053 288.1000 12.0000[2090]s 0.1078 0.0523 51.4842 493.4000 70.4000[3090]s 0.1164 0.0440 62.1993 565.8000 193.8000[4090]s 0.1160 0.0399 65.6034 505.3907 358.4758[4590]s 0.1154 0.0393 65.9445 438.5965 438.5965[5090]s 0.1155 0.0398 65.5411 358.4758 505.3907[6090]s 0.1165 0.0436 62.5751 193.8000 565.8000[7090]s 0.1100 0.0514 53.2727 70.4000 493.4000[8090]s 0.0889 0.0605 31.9460 12.0000 288.1000[9090]s 0.0651 0.0650 0.1536 0.0000 0.0000

Figure 11 below is the graphical output of the ANSYS FEA result for a [0 90]S plate.

The graphic as well as the result in Table 3 show that the MATLAB code is validated

19

through the FEA result. The MATLAB code resulted in a 0.0680m deflection while the

FEA resulted in 0.0681m deflection, a 0.14% error.

Figure 11: Deflection of 4-Ply [0 90]S Plate

3.3.1 Optimization of 4-Ply Composite Plate and Effect of D16 and D26

After the validation of the MATLAB code for a [0 90]S, an investigation on the

effects of the fiber orientation of the outer layers of the plate was conducted. This

investigation will use the same material properties in Table 1 and will keep the inner

layers of the plate arranged at 90 while the outer layers are changed ranging from 10 to

90. For this investigation the MATLAB code, as well as the ANSYS VM82 code was

modified for each layup. Table 2 provides the results for the deflections acquired from

MATLAB and ANSYS, as well as the value of the D16 and D26 of the bending stiffness

matrix. Figure 12 below is a graph showing how changing the orientation angle of outer

20

layers a 4-layer composite plate affects its deflection response. As the angles of the outer

layers are increased, the deflection of the plate increases as well until the angle of 60.

This angle resulted in the maximum deflection of the plate. After this angle, the behavior

of the plate starts to be that of an isotropic material until it the layup was [90 90]S.

Figure 12: ANSYS Deflection Results for 4-Ply Laminate

3.3.2 MATLAB Inaccuracy When Calculating Symmetric Angle-Ply Laminate

As can be seen in the results of Table 2, the MATLAB code only agreed with the

ANSYS results when the outer layers were either 0 or 90. This is because of the elastic

coupling that exists when D16 and D26 of the bending stiffness matrix [D] are non-zeros.

Equation (21) becomes:

),(44

)2(2

3

4

263

4

16

4

4

2222

4

66124

4

11

yxqyx

wDyx

wD

ywD

yxwDD

xwD

oo

ooo

(27)

Because of the separation of variables involving D16 and D26, the proposed solution to

equation (22) above and derived in APPENDIX E does not satisfy the governing

21

differential equation (27). Thus, the variables are not actually separable. Moreover, the

deflection expansion equation (24) also does not satisfy the boundary conditions5. This

problem has been solved using the Rayleigh-Ritz method5. However, this paper does not

cover this method. Instead, an investigation on how additional plies and stacking

sequence affects deflection results and the values of D16 and D26 was conducted.

3.4 20-Ply Deflection Results

Campbell states in Reference 1 that the values of D16 and D26 become small when a large

number of plies are stacked and become insignificant for thicknesses of more than

sixteen plies. This section investigates how the values of D16 and D26 become

insignificant when the number of plies is increased to more than sixteen plies. FEA

analysis and MATLAB calculations were conducted for 20-ply plates. These plates

consist of the same material properties as those in Table 1 and same loading is applied.

Thickness of each layer however was reduced from 0.025m to 0.0025m to keep the

requirement of the classical lamination theory where thickness is ten times smaller than

the length or width of the plate. The results of the analyses are tabulated in Table 4

Table 4: ANSYS and MATLAB results for 20-Ply Composite Plates

LayupANSYSDeflection(m)

MATLABDeflection(m) %Error D16(Nm) D26(Nm)

[6009090090900900]s 0.5829 0.4698 19.4030 7.5042 21.9060[0609090090900900]s 0.5733 0.4842 15.5416 6.0089 15.1954[006090090900900]s 0.5657 0.4970 12.1442 4.6797 13.6609[009060090900900]s 0.5606 0.5093 9.1509 3.5167 10.2659[0090906090900900]s 0.5576 0.5208 6.5997 2.5199 7.3559[009090060900900]s 0.5547 0.5300 4.4529 1.6891 4.9309[009090090600900]s 0.5533 0.5379 2.7833 1.0246 2.9909[0090900909060900]s 0.5525 0.5440 1.5385 0.5261 1.5358[009090090900600]s 0.5519 0.5481 0.6885 0.1938 0.5658[0090900909009060]s 0.5517 0.5501 0.2900 0.0277 0.0808

22

The analyses for the 20-ply plates were conducted to see how additional plies along with

layup sequencing would affect the deflection results. There are many possible stacking

sequences when a laminate plate involves twenty plies. The stacking sequence chosen

and displayed in the layup column of Table 4 was to keep the layup symmetric and

orthotropic. This section also investigated the effects on the values of D16 and D26 when

the plate is made of twenty plies. Comparing Table 3 with Table 4, there are noticeable

differences in the value of D16 and D26, as well as the % error between the ANSYS and

the MATLAB deflection results. The increase to 20-plies certainly decreased the value

of D16 and D26 when compared to the values of D16 and D26 of the 4-ply plates in Table

3. Sequentially moving the 60 ply closer to the midplane caused a decrease in the

deflection results as can be seen in Figure 13 below. When the layup had the 60 ply in

layers straddling the midplane, the laminate plate was at its stiffest point. Figure 14

below shows the decrease in % error between the ANSYS and MATLAB deflection

results. The % error decreased as the 60 ply moved closer to the midplane due to the

decreasing value of D16 and D26 which eventually got closer to zero.

23

Figure 13: ANSYS Deflection Results for 20-Ply Plates

24

Figure 14: % Error between MATLAB and ANSYS as 60 Ply Changes Location

25

4. Conclusion

An investigation on the response of a symmetric composite laminate plate was

conducted. ANSYS was utilized to model and analyze the response of a simply

supported symmetric plate subjected to a uniform load. VM82 was used for the purposes

of this project because it is a solved simply supported, symmetric plate problem. It was

used to validate the hand calculations performed using MATLAB that invoked the

classical lamination theory and modified accordingly to analyze the different cases in

Table 2. The MATLAB calculations along with the ANSYS analyses were performed to

investigate how fiber orientation, lay-up sequencing and total number of plies affect the

deflection results.

The response of a symmetric 4-ply plate with a layup of [0 90]S was performed using

VM82 to validate the MATLAB code. The effect of fiber orientation of the outer plies of

the 4-ply plate was then investigated. The results are tabulated in Table 3 and the

following conclusions are observed:

1. The minimum deflection of the 4-ply plate is when the lay-up is [90 90]S

2. The maximum deflection occurs when the outer layers of are oriented at 60.

3. The large % error between the MATLAB and ANSYS deflection results is

due to values of D16 and D26 being non-zeros for plies whose angles are those

other than 0 or 90, thus changing the governing differential equation.

26

The response of a symmetric 20-ply plate with an original layup up of

[0 0 90 90 0 90 90 0 90 0]S was conducted to investigate how an increase in the number

of plies affects the deflection results. Table 4, Figure 13 and Figure 14 tabulate and show

the results and thus the following conclusions are observed:

1. The % error between the MATLAB and ANSYS deflection results has

decreased when sixteen plies were added to the original four. This is due to

the decrease of D16 and D26 values in Table 4. These values are significantly

smaller than those in Table 3 when there were only four plies to the plate.

2. As the 60 ply was re-located from the outer plates to the inner plates, the

stiffness of the plate increased. The min deflection in Table 4 is when the 60

ply was in layers ten and eleven, straddling the midplane as shown in Figure

13.

When designing for composite materials with a specific purpose, there are several

variables an engineer should investigate. Fiber and matrix material, fiber orientation,

layer stacking sequence and so on affect the response of a laminate. The virtually

limitless combinations of ply materials, ply orientations and ply-stacking sequences

increase the design flexibility of composite structures.

27

5. References

1. Campbell, F.C. Structural Composite Materials ASM International. Copyright

2010

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

Updated Edition. 2009.

3. MANE-6180 Mechanics of Composite Materials Prof. R. Naik, Lecture Notes

2010.

4. ANSYS Element Reference. pp 1171-1186. Release 12.0. April 2009

5. Jones, Robert M. Mechanics of Composite Materials. First Edition. McGraw-Hill

Book Company. Copyright 1975

6. Timoshenko P., Stephen, S. Woinowsky-Kreiger. Theory of Plates and Shells.

Second Edition. McGraw-Hill, Inc. Copyright 1959

7. Gibson, F. Ronald. Principles of Composite Material Mechanics. Second Edition.

Taylor and Francis Group, LLC. Copyright 2007

8. Szilard, R. Theory and Analysis of Plates: Classical and Numerical Methods.

Prentice Hall 1974, pp 58-59.

28

Appendix A: ANSYS Analysis code for 4-Ply Composite Plate

/TITLE, 4-Ply SIMPLY SUPPORTED LAMINATED PLATE UNDER PRESSURE C*** USING SHELL181 /PREP7 SMRT,OFF ANTYPE,STATIC ET,1,SHELL181 ! 4 NODE LAYERED SHELL KEYOPT,1,3,2 ! FULL INTEGRATION KEYOPT,1,8,1 ! WRITE LAYER RESULTS SECTYPE,1,SHELL SECDATA,0.025,1,0 ! LAYER 1: 0.025 THK, THETA 0 SECDATA,0.025,1,90 ! LAYER 2: 0.025 THK, THETA 90 SECDATA,0.025,1,90 ! LAYER 3: 0.025 THK, THETA 90 SECDATA,0.025,1,0 ! LAYER 4: 0.025 THK, THETA 0 MP,EX,1,25E6 ! ORTHOTROPIC MATERIAL PROPERTIES MP,EY,1,1E6 MP,EZ,1,1E6 ! EZ=EY ASSUMED MP,GXY,1,5E5 MP,GYZ,1,2E5 MP,GXZ,1,5E5 MP,PRXY,1,0.25 ! MAJOR POISSONS RATIO MP,PRYZ,1,0.01 ! MAJOR POISSONS RATIO MP,PRXZ,1,0.25 ! MAJOR POISSONS RATIO K,1 ! CORNER KEYPOINTS OF QUADRANT (AREA) K,2,5 K,3,5,5 K,4,,5 A,1,2,3,4 ESIZE,,20 ! 20x20 MESH USING QUARTER SYMMETRY AMESH,1 NSEL,S,LOC,X,0 DSYM,SYMM,X NSEL,S,LOC,Y,0 DSYM,SYMM,Y NSEL,S,LOC,X,5 ! APPLY FREELY SUPPORTED B.C. D,ALL,UZ,,,,,UY NSEL,S,LOC,Y,5 D,ALL,UZ,,,,,UX NSEL,ALL /USER, 1 /VIEW, 1, 0.429637382252 , 0.886870752090E-01, 0.898635811919 /ANG, 1, -0.384698175032 /REPLO EPLOT FLST,2,1,5,ORDE,1 FITEM,2,1 /GO !* SFA,P51X,2,PRES,1 FINISH /SOL /STATUS,SOLU SOLVE

29

FINISH /POST1 !* /EFACET,1 PLNSOL, U,Z, 0,1.0

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Appendix B: MATLAB Validation code for 4-Ply Composite Plate

% This Matlab script calculates the laminate bending stiffness and then % calculates the deflection of a simply supported plate. % clear; close all; % % Ply lay-up for a four ply laminate [(0/90)]s % theta(1)=60; E1(1)=25e6; E2(1)=1e6; G12(1)=500000; v12(1)=0.25; t(1)=0.025; theta(2)=90; E1(2)=25e6; E2(2)=1e6; G12(2)=500000; v12(2)=0.25; t(2)=0.025; theta(3)=90; E1(3)=25e6; E2(3)=1e6; G12(3)=500000; v12(3)=0.25; t(3)=0.025; theta(4)=60; E1(4)=25e6; E2(4)=1e6; G12(4)=500000; v12(4)=0.25; t(4)=0.025; % % % Define cell arrays to store the transformed stiffness matrices of % %each % individual ply % Qbar=cell(length(theta),1); % % Loop through all plies, calculate Qbar for each ply % for i=1:length(theta) v21(i)=v12(i)*(E2(i)/E1(i)); S=[1/E1(i) -v21(i)/E2(i) 0; -v12(i)/E1(i) 1/E2(i) 0; 0 0 1/G12(i)]; c=cosd(theta(i)); s=sind(theta(i)); T=[c^2 s^2 2*c*s; s^2 c^2 -2*c*s; -c*s c*s (c^2 - s^2)]; Sbar=T'*S*T; Qbar{i,1}=inv(Sbar); end % % Calulate the z distances for each ply interface % total_thick=sum(t); z(1)=-total_thick/2; for k=2:length(theta)+1 z(k)=z(k-1)+t(k-1); end % % Calculate the D matrix % D=[0 0 0; 0 0 0; 0 0 0]; for i=1:length(theta) k=i+1; Dply=(1/3)*Qbar{i,1}*(z(k)^3-z(k-1)^3); D=D+Dply; end % % Applied Transverse Loading q0=1;

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% % Plate dimensions a=10; b=10; % %************************************************************************* m=[1 3 5 7]; n=[1 3 5 7]; % % The deflection function expressed in Fourier Expansion for i=1:length(m) for j=1:length(n) den=D(1,1)*(m(i)/a)^4 + 2*(D(1,2) + 2*D(3,3))*((m(i)*n(j))/(a*b))^2 + D(2,2)*(n(j)/b)^4; wm(i,j)=16*q0/((pi^6)*m(i)*n(j)*den); dw(i,j)=wm(i,j)*sin(m(i)*pi/2)*sin(n(j)*pi/2); end end % for k=1:4 wc(k)=sum(sum(dw(1:k,1:k))); end % Outputs the Deflection value wc(k) % Outputs the Bending Stiffness Matrix to get D16 and D26 values D

32

Appendix C: ANSYS Analysis code for 20-Ply Composite Plate

/TITLE, 20-Ply SIMPLY SUPPORTED LAMINATED PLATE UNDER PRESSURE with Angle Ply at Layer 1 & 20 C*** USING SHELL181 /PREP7 SMRT,OFF ANTYPE,STATIC ET,1,SHELL181 ! 4 NODE LAYERED SHELL SECTYPE,1,SHELL SECDATA,0.0025,1,0 ! LAYER 1: 0.025 THK, THETA 0 SECDATA,0.0025,1,0 ! LAYER 2: 0.025 THK, THETA 90 SECDATA,0.0025,1,90 ! LAYER 3: 0.025 THK, THETA 90 SECDATA,0.0025,1,90 ! LAYER 4: 0.025 THK, THETA 45 SECDATA,0.0025,1,0 ! LAYER 5: 0.025 THK, THETA 0 SECDATA,0.0025,1,90 ! LAYER 6: 0.025 THK, THETA 0 SECDATA,0.0025,1,90 ! LAYER 7: 0.025 THK, THETA 45 SECDATA,0.0025,1,0 ! LAYER 8: 0.025 THK, THETA 90 SECDATA,0.0025,1,90 ! LAYER 9: 0.025 THK, THETA 90 SECDATA,0.0025,1,60 ! LAYER 10: 0.025 THK, THETA 0 SECDATA,0.0025,1,60 ! LAYER 6: 0.025 THK, THETA 0 SECDATA,0.0025,1,90 ! LAYER 7: 0.025 THK, THETA 45 SECDATA,0.0025,1,0 ! LAYER 8: 0.025 THK, THETA 90 SECDATA,0.0025,1,90 ! LAYER 9: 0.025 THK, THETA 90 SECDATA,0.0025,1,90 ! LAYER 10: 0.025 THK, THETA 0 SECDATA,0.0025,1,0 ! LAYER 6: 0.025 THK, THETA 0 SECDATA,0.0025,1,90 ! LAYER 7: 0.025 THK, THETA 45 SECDATA,0.0025,1,90 ! LAYER 8: 0.025 THK, THETA 90 SECDATA,0.0025,1,0 ! LAYER 9: 0.025 THK, THETA 90 SECDATA,0.0025,1,0 ! LAYER 10: 0.025 THK, THETA 0 MP,EX,1,25E6 ! ORTHOTROPIC MATERIAL PROPERTIES MP,EY,1,1E6 MP,EZ,1,1E6 ! EZ=EY ASSUMED MP,GXY,1,5E5 MP,GYZ,1,2E5 MP,GXZ,1,5E5 MP,PRXY,1,0.25 ! MAJOR POISSONS RATIO MP,PRYZ,1,0.01 ! MAJOR POISSONS RATIO MP,PRXZ,1,0.25 ! MAJOR POISSONS RATIO K,1 ! CORNER KEYPOINTS OF QUADRANT (AREA) K,2,5 K,3,5,5 K,4,,5 A,1,2,3,4 ESIZE,,6 ! 20X20 MESH USING QUARTER SYMMETRY AMESH,1 NSEL,S,LOC,X,0 DSYM,SYMM,X NSEL,S,LOC,Y,0 DSYM,SYMM,Y NSEL,S,LOC,X,5 ! APPLY FREELY SUPPORTED B.C.

33

D,ALL,UZ,,,,,UY NSEL,S,LOC,Y,5 D,ALL,UZ,,,,,UX NSEL,ALL FLST,2,1,5,ORDE,1 FITEM,2,1 /GO !* SFA,P51X,2,PRES,1 FINISH /SOL /STATUS,SOLU SOLVE FINISH /POST1 !* /EFACET,1 PLNSOL, U,Z, 0,1.0

34

Appendix D: MATLAB Validation code for 10-Ply Composite Plate

% This Matlab script calculates the laminate bending stiffness and then % calculates the deflection of a simply supported plate. % clear; close all; % % Ply lay-up for a four ply laminate [(0/90)]s % theta(1)=0; E1(1)=25e6; E2(1)=1e6; G12(1)=500000; v12(1)=0.25; t(1)=0.0025; theta(2)=0; E1(2)=25e6; E2(2)=1e6; G12(2)=500000; v12(2)=0.25; t(2)=0.0025; theta(3)=90; E1(3)=25e6; E2(3)=1e6; G12(3)=500000; v12(3)=0.25; t(3)=0.0025; theta(4)=90; E1(4)=25e6; E2(4)=1e6; G12(4)=500000; v12(4)=0.25; t(4)=0.0025; theta(5)=0; E1(5)=25e6; E2(5)=1e6; G12(5)=500000; v12(5)=0.25; t(5)=0.0025; theta(6)=90; E1(6)=25e6; E2(6)=1e6; G12(6)=500000; v12(6)=0.25; t(6)=0.0025; theta(7)=90; E1(7)=25e6; E2(7)=1e6; G12(7)=500000; v12(7)=0.25; t(7)=0.0025; theta(8)=0; E1(8)=25e6; E2(8)=1e6; G12(8)=500000; v12(8)=0.25; t(8)=0.0025; theta(9)=90; E1(9)=25e6; E2(9)=1e6; G12(9)=500000; v12(9)=0.25; t(9)=0.0025; theta(10)=0; E1(10)=25e6; E2(10)=1e6; G12(10)=500000; v12(10)=0.25; t(10)=0.0025; % MIDPLANE theta(11)=0; E1(11)=25e6; E2(11)=1e6; G12(11)=500000; v12(11)=0.25; t(11)=0.0025; theta(12)=90; E1(12)=25e6; E2(12)=1e6; G12(12)=500000; v12(12)=0.25; t(12)=0.0025; theta(13)=0; E1(13)=25e6; E2(13)=1e6; G12(13)=500000; v12(13)=0.25; t(13)=0.0025; theta(14)=90; E1(14)=25e6; E2(14)=1e6; G12(14)=500000; v12(14)=0.25; t(14)=0.0025; theta(15)=90; E1(15)=25e6; E2(15)=1e6; G12(15)=500000; v12(15)=0.25; t(15)=0.0025; theta(16)=0; E1(16)=25e6; E2(16)=1e6; G12(16)=500000; v12(16)=0.25; t(16)=0.0025; theta(17)=90; E1(17)=25e6; E2(17)=1e6; G12(17)=500000; v12(17)=0.25; t(17)=0.0025; theta(18)=90; E1(18)=25e6; E2(18)=1e6; G12(18)=500000; v12(18)=0.25; t(18)=0.0025; theta(19)=0; E1(19)=25e6; E2(19)=1e6; G12(19)=500000; v12(19)=0.25; t(19)=0.0025; theta(20)=0; E1(20)=25e6; E2(20)=1e6; G12(20)=500000; v12(20)=0.25; t(20)=0.0025; % %

35

% Define cell arrays to store the transformed stiffness matrices of each % individual ply % Qbar=cell(length(theta),1); % % Loop through all plies, calculate Qbar for each ply % for i=1:length(theta) v21(i)=v12(i)*(E2(i)/E1(i)); S=[1/E1(i) -v21(i)/E2(i) 0; -v12(i)/E1(i) 1/E2(i) 0; 0 0 1/G12(i)]; c=cosd(theta(i)); s=sind(theta(i)); T=[c^2 s^2 2*c*s; s^2 c^2 -2*c*s; -c*s c*s (c^2 - s^2)]; Sbar=T'*S*T; Qbar{i,1}=inv(Sbar); end % % Calulate the z distances for each ply interface % total_thick=sum(t); z(1)=-total_thick/2; for k=2:length(theta)+1 z(k)=z(k-1)+t(k-1); end % % Calculate the D matrix % D=[0 0 0; 0 0 0; 0 0 0]; for i=1:length(theta) k=i+1; Dply=(1/3)*Qbar{i,1}*(z(k)^3-z(k-1)^3); D=D+Dply; end % % Applied Transverse Loading q0=1; % % Plate dimensions a=10; b=10; % %************************************************************************* m=[1 3 5 7 11]; n=[1 3 5 7 11]; % The deflection function expressed in Fourier Expansion for i=1:length(m) for j=1:length(n) den=D(1,1)*(m(i)/a)^4 + 2*(D(1,2) + 2*D(3,3))*((m(i)*n(j))/(a*b))^2 + D(2,2)*(n(j)/b)^4; wm(i,j)=16*q0/((pi^6)*m(i)*n(j)*den); dw(i,j)=wm(i,j)*sin(m(i)*pi/2)*sin(n(j)*pi/2); end end % for k=1:4 wc(k)=sum(sum(dw(1:k,1:k)));

36

end %Outputs the deflection wc(k) % Outputs the Bending Stiffness Matrix to get D16 and D26 values D

37

Appendix E: Derivation of Deflection Solution for a Uniform Load

38

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