Structural Optimization of a Simply Supported
Composite Plate
by
Kevin S. Van Keuren
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Troy, New York
August, 2011
© Copyright 2011
by
Kevin S. Van Keuren
All Rights Reserved
ii
CONTENTS
Structural Optimization of a Simply Supported Composite Plate ...................................... i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
NOMENCLATURE ......................................................................................................... vi
ABSTRACT ...................................................................................................................... 1
1. Introduction and Background ...................................................................................... 2
2. Methodology ................................................................................................................ 4
2.1
Kirchhoff’s Assumptions ................................................................................... 4
2.2
Classical Lamination Theory ............................................................................. 5
2.3
2.2.1
Transformed Moduli for unidirectional ply rotated an angle, θ, relative
to plate reference axis............................................................................. 7
2.2.2
Modal Analysis .................................................................................... 14
Finite Element Model ....................................................................................... 14
3. Results........................................................................................................................ 16
3.1
3.2
Finite Element Model Validation ..................................................................... 17
3.1.1
Test Case 1: [0]s Plate – Analytical Classical Laminated Theory ....... 17
3.1.2
Test Case 2: [+45/-45]s Plate – Analytical Classical Laminated Theory
.............................................................................................................. 20
3.1.3
Test Case 2: [0/90]s Plate – Analytical Classical Laminated Theory .. 23
3.1.4
ANSYS Mesh Sensitivity Study .......................................................... 25
3.1.5
Validation Model ANSYS Results ...................................................... 26
Finite Element Results ..................................................................................... 27
4. CONCLUSION.......................................................................................................... 36
5. REFERENCES .......................................................................................................... 37
6. Appendix A - Maple version 12 files ........................................................................ 38
iii
LIST OF TABLES
Table 1 Graphite-Epoxy Composite Material Properties ................................................ 16
Table 2 Full Size Plate Geometry .................................................................................... 16
Table 3 Applied Pressure Load........................................................................................ 16
Table 4 ANSYS Results for Displacement and Max Stress ............................................ 28
Table 5 Modal Analysis Results ...................................................................................... 33
iv
LIST OF FIGURES
Figure 1 Example Laminated Plate.................................................................................... 2
Figure 2 Unidirectionally Reinforced Lamina ................................................................... 3
Figure 3 Plate Deflection According to Kirchhoff Assumptions ...................................... 5
Figure 4 Ply Axis vs. Plate Axis, θ=0 ................................................................................ 8
Figure 5 Fiber Axis Oriented at Angle θ Relative to Plate Axis ....................................... 8
Figure 6 Geometry of an N-Layered Laminate ............................................................... 13
Figure 7 ANSYS SOLID95 Element Geometry .............................................................. 15
Figure 8 Finite Element Model of Single Ply Dimensions .............................................. 17
Figure 9 ANSYS Model – 4 layer plate ¼ Symmetry Model ......................................... 28
Figure 10 Plot of Maximum Displacement versus Angle, θ............................................ 29
Figure 11 Von Mises Stress Plot [0]s Laminate............................................................... 30
Figure 12 Von Mises Stress Plot [15/-15]s Laminate ...................................................... 30
Figure 13 Von Mises Stress Plot [30/-30]s Laminate ...................................................... 31
Figure 14 Von Mises Stress Plot [45/-45]s Laminate ...................................................... 31
Figure 15 Von Mises Stress Plot [60/-60]s Laminate ...................................................... 32
Figure 16 Von Mises Stress Plot [75/-75]s Laminate ...................................................... 32
Figure 17 Von Mises Stress Plot [90]s Laminate............................................................. 33
Figure 18 Mode Frequency vs. Orientation Angle .......................................................... 34
Figure 19 Plot of Mode Frequency for Various Orientations .......................................... 34
Figure 20 Mode Shapes 1 thru 5 for [0]s Laminated Plate .............................................. 35
v
NOMENCLATURE
D
Flexural Rigidity
E
Modulus of Elasticity
G
Shear Modulus
N
Resultant Force, N/m
M
Resultant Moment, N-m/m
P
Pressure, Pa
p0
Uniform Pressure, Pa
Qij
Reduced Stiffness coefficients
𝑄̅ ij
Transformed Reduced Stiffness Coefficients
t
Plate Thickness
π
Pi
ε
Strain
γ
Shear Strain
κ
Midplane curvature
θ
Theta, Fiber Orientation, degree (°)
σ
Normal Stress
τ
Shear Stress
ν
Poisson’s ration
ω
Frequency, Hz
ωn
Natural Frequency, Hz
vi
ABSTRACT
This paper analyzes the response of a rectangular flat plate subjected to a uniform
pressure as the ply orientation is varied from 0° to 90°. The plate is composed of 4
layers of graphite-epoxy composite. The orientation of the layers is assumed to be
symmetric about the neutral axis of the laminate. The orientation of the outer layers is
set at +θ, the inner layers set at –θ. The orientation is varied from θ=0° to θ=90°. For
this analysis, 0° is assumed to be aligned with the x-axis which also coincides with the
long dimension of the rectangular plate.
The finite element method is used to evaluate the displacement and frequency
response of the system. A finite element model was created using the ANSYS finite
element package.
This information is then used to optimize the flat plate for maximum stiffness,
maximum flexibility and also find the modes of vibration.
1
1. Introduction and Background
Thin plates are classified as initially flat structural members that are bounded by two
parallel planes, known as faces, and geometrically by either straight or curved
boundaries. The thickness is defined as the distance between the two parallel faces and
is assumed to be much smaller than any of the other dimensions of the plate. For small
deflections, the loads are perpendicular to the faces.
A plate has similar load carrying characteristics as a network made up of an infinite
number of beams or cables. When deflected, the plat resists transverse loads thru the
development of shear forces along with bending and twisting moments. The plate is
considerably stiffer than a beam of comparable span and thickness because the loads are
also generally carried in both directions and the twisting rigidity is quite significant.
Plates combine light weight and efficiency along with a relatively high-load carrying
capacity, economy and effectiveness.
The advantages of plates make them a good choice for many structural applications.
Thin plates are used in bridges, architectural structures, dams, pavements, aircraft,
missiles and space applications. The use of composite materials with thin plates allows
designers to exploit the inherent anisotropy and tailor the plate for specific needs. A
composite plate could be designed to achieve high stiffness or maximum deflection. The
frequency response can also be tailored to resist high or low frequencies as desired.
Figure 1 Example Laminated Plate
2
Figure 2 Unidirectionally Reinforced Lamina
3
2. Methodology
For analytical analysis of a plate it is convenient to use the assumption of 2-D
behavior of the plate when loaded transversely.
This assumption is known as
Kirchhoff’s Assumption and is explained below. The reduction of the 3-D plate to 2
dimensions greatly simplifies the analytical calculation of the plate’s behavior while
maintaining a high degree of accuracy when compared to experimental data.
2.1 Kirchhoff’s Assumptions
This analysis assumes small deflection, thin plate bending.
A thin plate is
defined as one in which the thickness, t, is much smaller than the edge length of the
plate, b. This assumption is required by Kirchhoff’s assumptions for the small deflection
theory of bending.
𝑡
1
≪
𝑏 20
1
Kirchhoff assumptions for small deflection theory of bending:
1. Deflection of midplane is small compared to the thickness of plate (w<<t)
Therefore the slope of the deflected surface is small (∂w/∂x<<1) and the
square of the slope is negligible ([∂w/∂x]2=0)
2. Midplane remains unstrained subsequent to bending
3. Plane sections initially normal to the midplane remain plane and normal to
the midplane after bending, thus out of plane shear strains are zero (γxz=
γyz=0). Deflection is associated principally with bending strains. Out of plane
normal strain (εz) is also omitted.
4. Out of plane normal stress (σz) is neglected. This assumption is unreliable
near concentrated transverse loads.
4
Figure 3 Plate Deflection According to Kirchhoff Assumptions
2.2 Classical Lamination Theory
A laminated plate composed of composite materials is very similar to a plate made
from orthotropic materials, except each layer may have mechanical properties that are in
a direction and angle, θ, relative to the geometric axis of the plate.
Orthotropic materials are those in which the properties differ on three mutually
perpendicular planes of symmetry. The governing equation for rectangular plates is
modified to account for the different properties in the x- and y- directions. To modify
these equations for orthotropic materials we need to reformulate the governing equation:
∇4 𝑤 =
𝑝
𝐷
For an orthotropic material the properties differ in mutually perpendicular
directions. It is assumed that the properties vary in the x and y directions. Taking this
into account the governing equation becomes:
5
𝜕 4𝑤
𝜕 4𝑤
𝜕 4𝑤
∇ 𝑤 = 𝐷𝑥 4 + 2𝐻 4 2 + 𝐷𝑦
=𝑝
𝜕𝑥
𝜕𝑥 𝜕𝑥
𝜕𝑥𝑦 4
4
2
Where the flexural rigidities Dx , Dy , Dxy , Dyx and the tortional rigidity Ds are
defined as:
𝐷𝑥 =
𝐸𝑥
ℎ3
1 − 𝜈𝑥 𝜈𝑦 12
3
𝐷𝑦 =
𝐸𝑦
ℎ3
1 − 𝜈𝑥 𝜈𝑦 12
4
𝐷𝑥𝑦
𝐸𝑥 𝜈𝑦 ℎ3
=
1 − 𝜈𝑥 𝜈𝑦 12
5
𝐷𝑦𝑥 =
𝐸𝑦 𝜈𝑥 ℎ3
1 − 𝜈𝑥 𝜈𝑦 12
6
𝐷𝑠 =
𝐺ℎ3
12
7
The shear modulus, G, is the same for both isotropic and orthotropic materials:
𝐺=
√𝐸𝑥 𝐸𝑦
2(1 + √𝜈𝑥 𝜈𝑦 )
8
And finally, H:
6
𝐻 = 𝐷𝑥𝑦 + 2𝐷𝑠
9
The equation for displacement of a rectangular orthotropic plate is therefore:
∞
∞
16𝑝0
𝑤 = 6 ∑ ∑{
𝜋
𝑚=1 𝑛=1
𝑚𝜋𝑥
𝑛𝜋𝑦
sin 𝑎 sin
𝑏
}
𝑚 4
𝑚 2 𝑛 2
𝑛 4
𝑚𝑛 [𝐷𝑥 ( 𝑎 ) + 2𝐻 ( 𝑎 ) ( ) + 𝐷𝑦 ( ) ]
𝑏
𝑏
(𝑚, 𝑛 = 1,3,5, … )
10
The equations above apply well to an orthotropic material that is consistent
throughout its entire thickness with the material properties oriented with the geometric
axis of the plate. Composite materials, by definition, have material properties that can
vary from layer to layer. To address this, a set of equations are developed to describe the
behavior of an orthotropic layer that can be oriented at an angle, θ, relative to the
geometric axis of the plate. This angle can also vary layer to layer which means for the
same orthotropic material, the stresses can vary by layer according to the layer
orientation angle, θ. This directional behavior allows the laminate to be custom tailored
to particular design parameters and achieve a specific behavior.
2.2.1
Transformed Moduli for unidirectional ply rotated an angle, θ, relative to
plate reference axis
The challenge now is to determine the rigidities for the composite materials oriented
an angle ,θ, from the x-axis.
Each ply of the laminate is oriented a specified angle relative to the x-axis. The xaxis is defined as parallel to the longest dimension of the plate. The material of each ply
is assumed to be unidirectional and the longitudinal axis of the ply may be rotated
relative to the plate axis as shown in the following figures.
7
The unidirectional material is orthogonal and has material constants that are
determined relative to the 1-2 axis. The 1-2 axis is oriented with axis 1 being parallel to
the longitudinal direction of the fibers. Axis 2 is perpendicular to axis 1.
When a ply is oriented at an angle θ relative to the x-y axis system, the material
constants in the 1-2 axis system must be transformed to the x-y system to provide Ext ,
Eyt and G12. These values will be used to develop a transformed flexural rigidity, Dt.
Figure 4 Ply Axis vs. Plate Axis, θ=0
Figure 5 Fiber Axis Oriented at Angle θ Relative to Plate Axis
The stress-strain law for a single ply of unidirectional laminate is as follows:
8
1
𝐸1
𝜀1
−𝜈12
{ 𝜀2 } =
𝐸1
𝛾
12
[
−𝜈21
𝐸2
1
𝐸2
0
0
0
𝜎1
0 { 𝜎2 }
𝜏12
1
𝐺12 ]
11
E1 , E2 = Young’s modulii in the principle directions of the unidirectional ply
respectively
ν12 = Poisson’s ratio governing the contraction in the 2 direction for a tension
applied in the 1 direction.
ν21 = Poisson’s ratio governing the contraction in the 1 direction for a tension
applied in the 2 direction.
𝜈12 𝜈21
=
𝐸1
𝐸2
12
G12 = Shear Modulus in the plane of the laminate
For the unidirectional material being considered it is noteworthy that E1 is much
larger than either E2 or G12. This is because E1 is a “fiber dominated” property while E2
and G12 are “matrix dominated” properties. The above statement is given for a single
ply, however, since a unidirectional material is considered the equations would Apply to
a full laminate of unidirectional plies all oriented in the same direction.
For analysis, it is often more convenient to express the stress-strain law in the
following form which is the inverse of the equation presented earlier:
{𝜎} = [𝑄]{𝜀}
13
9
𝜎1
𝜀1
𝑄11 (0) 𝑄12 (0)
0
0 ] { 𝜀2 }
{ 𝜎2 } = [𝑄12 (0) 𝑄22 (0)
𝜏12
0
0
𝑄66 (0) 𝛾12
14
Where Qij are defined as the reduced stiffness coefficients and given by:
𝑄11 (0) = 𝐸1 ⁄(1 − 𝜈12 𝜈21 )
𝑄12 (0) = 𝜈21 𝐸1 ⁄(1 − 𝜈12 𝜈21 )
𝑄22 (0) = 𝐸2 ⁄(1 − 𝜈12 𝜈21 )
𝑄66 (0) = 𝐺12
The above equations describe the behavior of a unidirectional laminate in which the
material axis is parallel to the principal axis of the laminate. Now we consider the
behavior of an off-axis laminate. An off-axis laminate is one in which the fibers are
oriented to make an angle θ with the reference axis fixed in the laminate. We will
consider the reference axis to be the x-y coordinate system and the material axis to be
the 1-2 system fixed to the fibers. The angle θ is measured from the x axis to the 1 axis
and is positive in the counter-clockwise direction. Calculations are generally made using
the x-y reference or structural axis so it is required to transform the stress-strain law
from the material axis to the structural axis. If the stresses in the structural axes are
defined as σx, σy and τxy then these are related to the stresses in the material axes by the
transformation equation:
𝜎𝑥
𝜎1
𝜎
{ 𝑦 } = [𝑇] { 𝜎2 }
𝜏𝑥𝑦
𝜏12
15
Where the transformation matrix [T] is defined as:
10
𝑐𝑜𝑠 2 𝜃
[𝑇] = [ 𝑠𝑖𝑛2 𝜃
−sin 𝜃 cos 𝜃
𝑠𝑖𝑛2 𝜃
𝑐𝑜𝑠 2 𝜃
sin 𝜃 cos 𝜃
2 sin 𝜃 cos 𝜃
−2 sin 𝜃 cos 𝜃 ]
𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃
16
Using the transformation matrix we can create an equation that relates the stresses
and strains in the material principal directions to the x-y axis system. If we transform
the reduced stiffness matrix by an angle, θ, relative to the x-axis we obtain the
Transformed Reduced Stiffness Matrix:
[𝑄̅ ] = [𝑇]−1 [𝑄][𝑇]−𝑇
17
̅̅̅̅̅
𝑄11 (𝜃) ̅̅̅̅̅
𝑄12 (𝜃) ̅̅̅̅̅
𝑄16 (𝜃)
̅
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
[𝑄 ]𝜃 = [𝑄12 (𝜃) 𝑄22 (𝜃) 𝑄26 (𝜃)]
̅̅̅̅̅
̅̅̅̅̅
̅̅̅̅̅
𝑄
16 (𝜃) 𝑄26 (𝜃) 𝑄66 (𝜃)
18
Where
𝑄̅11 = 𝑄11 𝑐𝑜𝑠 4 𝜃 + 2(𝑄12 + 2𝑄66 )𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠 2 𝜃 + 𝑄22 𝑠𝑖𝑛4 𝜃
𝑄̅12 = (𝑄11 + 𝑄22 − 4𝑄66 )𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠 2 𝜃 + 𝑄12 (𝑠𝑖𝑛4 𝜃𝑐𝑜𝑠 4 𝜃)
𝑄̅22 = 𝑄11 𝑠𝑖𝑛4 𝜃 + 2(𝑄12 + 2𝑄66 )𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠 2 𝜃 + 𝑄22 𝑠𝑖𝑛4 𝜃
𝑄̅16 = (𝑄11 − 𝑄12 − 2𝑄66 )𝑠𝑖𝑛𝜃𝑐𝑜𝑠 3 𝜃 + (𝑄12 − 𝑄22 + 2𝑄66 )𝑠𝑖𝑛3 𝜃𝑐𝑜𝑠𝜃
𝑄̅26 = (𝑄11 − 𝑄12 − 2𝑄66 )𝑠𝑖𝑛3 𝜃𝑐𝑜𝑠𝜃 + (𝑄12 − 𝑄22 + 2𝑄66 )𝑠𝑖𝑛𝜃𝑐𝑜𝑠 3 𝜃
𝑄̅66 = (𝑄11 + 𝑄22 − 2𝑄12 − 2𝑄66 )𝑠𝑖𝑛2 𝜃𝑐𝑜𝑠 2 𝜃 + 𝑄66 (𝑠𝑖𝑛4 𝜃 + 𝑐𝑜𝑠 4 𝜃)
If the values from Equation 33 for σ1, σ2 ,τxy, ε1, ε2, and γ12 are substituted in we
obtain the equations for the stress-strain law as it relates to the structural axes.
̅̅̅̅̅
𝜎𝑥
𝑄11 (𝜃)
𝜎
{ 𝑦 } = [̅̅̅̅̅
𝑄12 (𝜃)
𝜏𝑥𝑦
̅̅̅̅̅
𝑄16 (𝜃)
̅̅̅̅̅
𝑄12 (𝜃)
̅̅̅̅̅
𝑄22 (𝜃)
̅̅̅̅̅
𝑄26 (𝜃)
̅̅̅̅̅
𝑄16 (𝜃) 𝜀𝑥
̅̅̅̅̅
𝑄26 (𝜃)] { 𝜀𝑦 }
̅̅̅̅̅
𝑄 (𝜃) 𝛾𝑥𝑦
66
19
11
̅̅̅𝑖𝑗 is the reduced stiffness matrix for plane stress oriented an angle, θ,
Where [𝑄]
relative to the plate axes. The above equations give us the stress or strains in a single
layer of the laminate for a given orientation relative to the geometric axis of the plate.
To determine the loads and moments according to the applied strains and curvatures
we need to develop another matrix. This matrix is referred to as the ABD matrix. It
links the reduced stiffness matrix to the loads and moments in the laminate to the applied
strains and curvatures.
𝑁𝑥
𝐴11
𝑁𝑦
𝐴12
𝑁𝑥𝑦
𝐴
= 16
𝑀𝑥
𝐵11
𝑀𝑦
𝐵12
𝑀
[ 𝑥𝑦 ] [𝐵16
𝐴12
𝐴22
𝐴26
𝐵12
𝐵22
𝐵26
𝐴16 𝐵11 𝐵12
𝐴26 𝐵12 𝐵22
𝐴66 𝐵16 𝐵26
𝐵16 𝐷11 𝐷12
𝐵26 𝐷12 𝐷22
𝐵66 𝐷16 𝐷26
𝐵16 𝜀°𝑥
𝐵26 𝜀°𝑦
𝐵66 𝜀°𝑥𝑦
𝐷16 𝜅°𝑥
𝐷26 𝜅°𝑦
𝐷66 ] [𝜅°𝑥𝑦 ]
20
Where the A, B and D matrices are defined as:
𝑁
𝐴𝑖𝑗 = ∑(𝑄̅𝑖𝑗 )𝑘 (𝑧𝑘 − 𝑧𝑘−1 )
𝑘=1
21
𝑁
1
𝐵𝑖𝑗 = ∑(𝑄̅𝑖𝑗 )𝑘 (𝑧𝑘 2 − 𝑧𝑘−1 2 )
2
𝑘=1
22
𝑁
1
𝐷𝑖𝑗 = ∑(𝑄̅𝑖𝑗 )𝑘 (𝑧𝑘 3 − 𝑧𝑘−1 3 )
3
𝑘=1
23
12
The orthotropic layers are oriented such that the complete laminate is symmetric
across the middle surface of the plate. Due to this symmetry, the [B] matrix that relates
the bending-extension coupling is effectively zero, [B]ij=0. The z dimensions are defined
from the middle surface of the laminate as shown in Figure 6.
The equation for the deflection of a rectangular orthotropic laminated plate is
similar to the Classic Thin Plate, Equation 10, described earlier. The difference is that
the bending stiffness [D]ij, come from the reduced stiffness matrix [𝑄̅ ]ij according to
Equation 22.
Figure 6 Geometry of an N-Layered Laminate
(From Mechanics of Composite Materials, p 197)
The equation of a rectangular orthotropic laminated plate subjected to a uniform
transverse load is thus:
∞
∞
16𝑝0
𝑤 = 6 ∑ ∑{
𝜋
𝑚=1 𝑛=1
𝑚𝜋𝑥
𝑛𝜋𝑦
sin 𝑎 sin
𝑏
}
𝑚 4
𝑚 2 𝑛 2
𝑛 4
𝑚𝑛 [𝐷11 ( 𝑎 ) + 2(𝐷12 + 2𝐷66 ) ( 𝑎 ) ( ) + 𝐷22 ( ) ]
𝑏
𝑏
13
(𝑚, 𝑛 = 1,3,5, … )
24
Once deflections are known the stresses and strains can be calculated by substituting
into the stress-strain relation equations.
2.2.2
Modal Analysis
As described in Jones, the vibration frequencies and mode shapes for a simply
supported, rectangular plate composed of orthotropic layers can be described by the
differential equation:
𝐷11 𝜕𝑤,𝑤𝑤𝑤𝑤 + 2(𝐷12 + 2𝐷66 )𝜕𝑤,𝑥𝑥𝑦𝑦 + 𝐷22 𝜕𝑤,𝑦𝑦𝑦𝑦 + 𝜌𝜕𝑤,𝑡𝑡
25
For a laminate composed of orthotropic layers arranged symmetrically about the
midplane, the natural frequencies, ω, are given by the following equation:
𝜋4
𝑚 4
𝑚 2 𝑛 2
𝑛 4
𝜔 =
[𝐷11 ( ) + 2(𝐷12 + 2𝐷66 ) ( ) ( ) + 𝐷22 ( ) ]
𝜌
𝑎
𝑎
𝑏
𝑏
2
26
The various modes of vibration are defined by the values chosen for m and n.
2.3 Finite Element Model
ANSYS 11.0 was used to perform the static structural and modal analysis of the
laminated rectangular plate. The SHELL181 element technology, Figure 7, was chosen
for the plate model because it is a 4 node solid element that is suitable for modeling thin
to moderately thick structures. This element is can also be used for layered applications
where a laminated composite shell or sandwich is modeled. The element is limited to
14
only 250 layers. The accuracy of the modeled shells is governed by the first order shear
deformation theory (Mindlin-Reissner shell theory).
Figure 7 ANSYS SHELL181 Element Geometry
15
3. Results
To validate the accuracy of the finite element model a case was solved using both
analytic and finite element methods. The model is a six layer symmetric laminate with a
[90]s layup. This case simulates a true orthographic material as the principle directions
of each ply are aligned with the coordinate axes of the plate. The problem is solved
analytically and then using the ANSYS finite element code. The results are compared to
evaluate the accuracy of the computer model. For each case evaluated in this paper the
material used is a unidirectional graphite-epoxy composite with material properties
shown in the following table.
Table 1 Graphite-Epoxy Composite Material Properties
E1
155.0 GPa
ν23
0.458
G23
3.20 GPa
E2
12.10 GPa
ν13
0.248
G13
4.40 GPa
E3
12.10 GPa
ν12
0.248
G12
4.40 GPa
ρ
1.75x10-3 g/mm3
The plate geometry used for this analysis is given below in Table 2.
Table 2 Full Size Plate Geometry
Length (a)
6 mm
Width (b)
3 mm
Number of layers (plies)
4
Single ply thickness
.15 mm
Total Plate Thickness
.60 mm
The load applied to the plate for this analysis is defined in Table 3.
Table 3 Applied Pressure Load
Pressure, p0
1000 Pa
(1.0 x10-6 GPa)
16
The model used for this analysis is composed of 4 plies with the shape and
dimensions shown in Figure 3. The model is one quarter of the actual plate geometry.
Symmetric boundary conditions allow the use of this model to take advantage of a
smaller model and reduced element counts leading to reduced computation time.
Figure 8 Finite Element Model of Single Ply Dimensions
3.1 Finite Element Model Validation
To validate the finite element model, two test cases are solved both analytically and
using the finite element code. A comparison of the results obtained from both methods
is used to validate the accuracy of the finite element model.
3.1.1
Test Case 1: [0]s Plate – Analytical Classical Laminated Theory
The test case is chosen to be a four layer laminate in which all four layers have the
material oriented parallel to the x-axis, θ=0. This corresponds to a laminate defined as
[0]s. This creates a plate with orthotropic properties that vary only in the x and y
directions.
17
There are no off axis layers so the following relation can be used to calculate the
stress-strain behavior of the plate:
̅̅̅𝜃=0
[𝑄] = [𝑄]
27
{𝜎} = [𝑄]{𝜀}
28
𝜎1
𝜀1
𝑄11 (0) 𝑄12 (0)
0
0 ] { 𝜀2 }
{ 𝜎2 } = [𝑄12 (0) 𝑄22 (0)
𝜏12
0
0
𝑄66 (0) 𝛾12
29
Plugging in the given values for the material properties gives us the particular
relationship for this case. If we take into account that the material is aligned with the
geometric axis of the plate we find the stress-strain relation is:
𝜎𝑥
155.75𝑥109
{ 𝜎𝑦 }
= [ 3.02𝑥109
𝜏𝑥𝑦 𝜃=0°
0
3.02𝑥109
12.16𝑥109
0
𝜀𝑥
0
] { 𝜀𝑦 }
0
9 𝛾𝑥𝑦
4.40𝑥10
30
𝜀1 = 𝜀𝑥
𝜀2 = 𝜀𝑦
𝜏12 = 𝜏𝑥𝑦
31
Due to the principal axis of the material being aligned with the geometric axis of the
plate, the [Q] matrix is equal to the [𝑄̅ ] matrix.
18
To validate the model a unidirectional strain is imposed upon the laminate in the xdirection. The stresses are calculated using the above equation and compared to the
ANSYS result. The applied strain loading is given to be:
𝜀𝑥 = 0.001
𝜀𝑦 = 0
𝜏𝑥𝑦 = 0
32
𝜎𝑥
155.75𝑥109
𝜎
{ 𝑦}
= [ 3.02𝑥109
𝜏𝑥𝑦 𝜃=0°
0
3.02𝑥109
12.16𝑥109
0
. 001
0
]{ 0 }
0
0
4.40𝑥109
33
𝜎𝑥
155.75𝑥106
{ 𝜎𝑦 }
= { 3.02𝑥106 } 𝑃𝑎
𝜏𝑥𝑦 𝜃=0°
0
34
The forces and moments created by the applied strain are calculated using the ABD
matrices as shown previously.
[𝐴]𝜃=0°
93.450
= [ 1.812
0
1.812
7.296
0
0
0 ]
2.640
35
0
[𝐵]𝜃=0° = [0
0
0 0
0 0]
0 0
36
2.803 0.054
0
[𝐷]𝜃=0° = [0.054 0.219
0 ]
0
0
0.079
19
37
𝑁𝑥
𝐴11
𝑁𝑦
𝐴12
𝑁𝑥𝑦
𝐴
= 16
𝑀𝑥
𝐵11
𝑀𝑦
𝐵12
[𝑀𝑥𝑦 ]𝜃=0° [𝐵16
𝐴12
𝐴22
𝐴26
𝐵12
𝐵22
𝐵26
𝐵16 𝜀°𝑥
𝐵26 𝜀°𝑦
𝐵66 𝜀°𝑥𝑦
𝐷16 𝜅°𝑥
𝐷26 𝜅°𝑦
𝐷66 ] [𝜅°𝑥𝑦 ]
𝐴16 𝐵11 𝐵12
𝐴26 𝐵12 𝐵22
𝐴66 𝐵16 𝐵26
𝐵16 𝐷11 𝐷12
𝐵26 𝐷12 𝐷22
𝐵66 𝐷16 𝐷26
38
𝑁𝑥
93.45𝑥106
𝑁𝑦
1.812𝑥106
𝑁𝑥𝑦
0
=
𝑀𝑥
0
0
𝑀𝑦
[
]
0
[𝑀𝑥𝑦 ]𝜃=0°
39
3.1.2
Test Case 2: [+45/-45]s Plate – Analytical Classical Laminated Theory
The test case is chosen to be a symmetric four layer laminate in which the four
layers have the material oriented at a +/-45 degree angle relative to the x-axis. This
corresponds to a laminate defined as [45/-45/-45/45].
This creates a plate with
orthotropic properties that vary only in the x and y directions.
Since the laminate is composed of off-axis plies, the following relation is used to
calculate the stress-strain behavior of the plate:
{𝜎} = [𝑄̅ ]𝜃 {𝜀}
40
Using the equations for [𝑄̅ ] defined earlier, the resulting matrix is:
20
[𝑄̅ ]𝜃=45°
47.888𝑥109
= [39.088𝑥109
35.898𝑥109
39.088𝑥109
47.888𝑥109
35.898𝑥109
35.898𝑥109
35.898𝑥109 ]
40.468𝑥109
41
[𝑄̅ ]𝜃=−45°
47.888𝑥109
= [ 39.088𝑥109
−35.898𝑥109
39.088𝑥109
47.888𝑥109
−35.898𝑥109
−35.898𝑥109
−35.898𝑥109 ]
40.468𝑥109
42
To validate the model a unidirectional strain is imposed upon the laminate in the xdirection. The stresses are calculated using the above equation and compared to the
ANSYS result. The applied strain loading is given to be:
𝜀𝑥 = .001
𝜀𝑦 = 0
𝜏𝑥𝑦 = 0
43
𝜎1
47.888𝑥109
{ 𝜎2 } = [39.088𝑥109
𝜏12 45°
35.898𝑥109
39.088𝑥109
47.888𝑥109
35.898𝑥109
35.898𝑥109 . 001
35.898𝑥109 ] { 0 }
0
40.468𝑥109
44
𝜎1
47.888𝑥106
𝜎
{ 2 } = {39.088𝑥106 }
𝜏12 45°
35.898𝑥106
45
𝜎1
47.888𝑥109
{ 𝜎2 }
= [ 39.088𝑥109
𝜏12 −45°
−35.898𝑥109
39.088𝑥109
47.888𝑥109
−35.898𝑥109
21
−35.898𝑥109 1000 𝑥 10−6
}
−35.898𝑥109 ] {
0
40.468𝑥109
0
46
𝜎1
47.888𝑥106
{ 𝜎2 }
= { 39.088𝑥106 }
𝜏12 −45°
−35.898𝑥106
47
The forces and moments created by the applied strain are calculated using the ABD
matrices as shown previously.
[𝐴]𝜃=+/−45°
28.733𝑥109
= [23.453𝑥109
0
23.453𝑥109
28.733𝑥109
0
0
]
0
9
24.281𝑥10
48
0 0 0
[𝐵]𝜃=+/−45° = [0 0 0]
0 0 0
49
[𝐷]𝜃=+/−45°
0.862𝑥109
= [0.704𝑥109
0.485𝑥109
0.704𝑥109
0.862𝑥109
0.485𝑥109
0.485𝑥109
0.485𝑥109 ]
0.728𝑥109
50
𝑁𝑥
𝐴11
𝑁𝑦
𝐴12
𝑁𝑥𝑦
𝐴
= 16
𝑀𝑥
𝐵11
𝑀𝑦
𝐵12
[𝑀𝑥𝑦 ]𝜃=+/−45° [𝐵16
𝐴12
𝐴22
𝐴26
𝐵12
𝐵22
𝐵26
𝐴16 𝐵11 𝐵12
𝐴26 𝐵12 𝐵22
𝐴66 𝐵16 𝐵26
𝐵16 𝐷11 𝐷12
𝐵26 𝐷12 𝐷22
𝐵66 𝐷16 𝐷26
𝐵16 𝜀°𝑥
𝐵26 𝜀°𝑦
𝐵66 𝜀°𝑥𝑦
𝐷16 𝜅°𝑥
𝐷26 𝜅°𝑦
𝐷66 ] [𝜅°𝑥𝑦 ]
51
22
𝑁𝑥
28.733𝑥106
𝑁𝑦
23.453𝑥106
𝑁𝑥𝑦
0
=
𝑀𝑥
0
0
𝑀𝑦
[
]
0
[𝑀𝑥𝑦 ]𝜃=+/−45°
3.1.3
Test Case 2: [0/90]s Plate – Analytical Classical Laminated Theory
The test case is chosen to be a symmetric four layer laminate in which the four
layers have the material oriented at a 0 and 90 degree angle relative to the x-axis. This
corresponds to a laminate defined as [0/90]s. This creates a plate with orthotropic
properties that vary only in the x and y directions.
Since the laminate is composed of off-axis plies, the following relation is used to
calculate the stress-strain behavior of the plate:
{𝜎} = [𝑄̅ ]𝜃 {𝜀}
52
Using the equations for [𝑄̅ ] defined earlier, the resulting matrix is:
[𝑄̅ ]𝜃=0°
155.75𝑥109
= [ 3.02𝑥109
0
3.02𝑥109
12.16𝑥109
0
0
]
0
9
4.40𝑥10
53
12.16𝑥109
̅
[𝑄 ]𝜃=90° = [ 3.02𝑥109
0
3.02𝑥109
155.75𝑥109
0
0
]
0
9
4.40𝑥10
54
To validate the model a unidirectional strain is imposed upon the laminate in the xdirection. The stresses are calculated using the above equation and compared to the
ANSYS result. The applied strain loading is given to be:
𝜀𝑥 = .001
23
𝜀𝑦 = 0
𝜏𝑥𝑦 = 0
55
𝜎1
155.75𝑥109
{ 𝜎2 } = [ 3.02𝑥109
𝜏12 0°
0
3.02𝑥109
12.16𝑥109
0
0
1000 𝑥 10−6
]{
}
0
0
9
4.40𝑥10
0
56
𝜎1
155.75𝑥106
{ 𝜎2 } = { 3.02𝑥106 }
𝜏12 0°
0
57
𝜎1
12.16𝑥109
{ 𝜎2 }
= [ 3.02𝑥109
𝜏12 𝜃=90°
0
3.02𝑥109
155.75𝑥109
0
0
1000 𝑥 10−6
]{
}
0
0
9
4.40𝑥10
0
58
𝜎1
12.16𝑥106
𝜎
{ 2 } = { 3.02𝑥106 }
𝜏12 90°
0
59
The forces and moments created by the applied strain are calculated using the ABD
matrices as shown previously.
[𝐴]𝜃=0°,90°
50.373𝑥109
= [ 1.812𝑥109
0
1.812𝑥109
50.373𝑥109
0
0
]
0
2.64𝑥109
60
24
0 0
[𝐵]𝜃=0°,90° = [0 0
0 0
0
0]
0
61
[𝐷]𝜃=0°,90°
2.480𝑥109
= [5.436𝑥107
0
5.436𝑥107
5.419𝑥108
0
0
]
0
7
7.92𝑥10
62
𝑁𝑥
𝐴11
𝑁𝑦
𝐴12
𝑁𝑥𝑦
𝐴
= 16
𝑀𝑥
𝐵11
𝑀𝑦
𝐵12
[𝑀𝑥𝑦 ]𝜃=0°,90° [𝐵16
𝐴12
𝐴22
𝐴26
𝐵12
𝐵22
𝐵26
𝐴16 𝐵11 𝐵12
𝐴26 𝐵12 𝐵22
𝐴66 𝐵16 𝐵26
𝐵16 𝐷11 𝐷12
𝐵26 𝐷12 𝐷22
𝐵66 𝐷16 𝐷26
𝐵16 𝜀°𝑥
𝐵26 𝜀°𝑦
𝐵66 𝜀°𝑥𝑦
𝐷16 𝜅°𝑥
𝐷26 𝜅°𝑦
𝐷66 ] [𝜅°𝑥𝑦 ]
63
𝑁𝑥
50.373𝑥106
𝑁𝑦
1.812𝑥106
𝑁𝑥𝑦
0
=
𝑀𝑥
0
0
𝑀𝑦
[
]
0
[𝑀𝑥𝑦 ]𝜃=0°,90°
64
3.1.4
ANSYS Mesh Sensitivity Study
A study was performed to determine the optimum mesh for this analysis. The edge
length of the element
25
3.1.5
Validation Model ANSYS Results
To validate the ANSYS model created to solve the engineering problem proposed
by this project, a strain in the x-direction is applied. The resulting values are compared
to the analytical calculations shown above.
The following strain is applied to the model along one edge that is perpendicular to
the x-axis:
𝜀 = .001
ANSYS allows the application of initial displacements to simulate initial strains. To
do this we need to calculate the equivalent displacement that would create the above
strain in the plate. Strain is calculated as follows:
𝜀=
𝛥𝐿
𝐿
Plugging in the desired strain value and the known length of the plate we obtain:
𝜀 = .001
𝐿 = 6𝑚𝑚
𝛥𝐿 = .006
𝑈𝑥 = .006
An initial displacement Ux=.006 is applied to the edge oriented perpendicular to the
x-axis.
3.1.5.1 Test Case 1: [0]s Plate – ANSYS
3.1.5.2 Test Case 2: [+/-45]s Plate – ANSYS
26
3.1.5.3 Test Case 3: [0,90]s Plate – ANSYS
3.1.5.4 Comparison of Analytical and ANSYS Results
Laminate
[0]s
Theta Value
0°
45°
[-45/+45]s
-45°
0°
[0/90]s
90°
Analytical
ANSYS
σx
155.75x106
155.748 x106
σy
3.02X106
3.015 x106
σxy
0
σx
47.888x106
σy
39.088x106
σxy
35.898x106
σx
47.888x106
σy
39.088x106
σxy
-35.898x106
σx
155.75x106
155.748 x106
σy
3.02X106
3.015 x106
σxy
0
σx
12.16 X106
σy
3.02 X106
σxy
0
% Error
0
0
3.2 Finite Element Results
A quarter symmetry model was constructed as shown in Figure 9. The model
contains 4 layers. Each layer has a defined material direction that is defined as an angle,
θ, relative to the x-axis. The angle was varied from θ=0° to θ=90° in 15° steps. For
each value of θ, a static and modal analysis was performed.
27
Figure 9 ANSYS Model – 4 layer plate ¼ Symmetry Model
The displacement and max stress calculated in each run has been tabulated below.
Table 4 ANSYS Results for Displacement and Max Stress
Theta
Displacement, mm (Max)
0
15
30
45
60
75
90
28
Stress, MPa (Max)
Displacement vs. Ply angle
4.00E-06
Displacement
3.50E-06
3.00E-06
2.50E-06
2.00E-06
1.50E-06
1.00E-06
5.00E-07
0.00E+00
0
15
30
45
60
75
90
Ply Angle, θ
Figure 10 Plot of Maximum Displacement versus Angle, θ
The deformation and stress distribution for each value of θ is shown in Figures 9
thru 15.
29
Figure 11 Von Mises Stress Plot [0]s Laminate
Figure 12 Von Mises Stress Plot [15/-15]s Laminate
30
Figure 13 Von Mises Stress Plot [30/-30]s Laminate
Figure 14 Von Mises Stress Plot [45/-45]s Laminate
31
Figure 15 Von Mises Stress Plot [60/-60]s Laminate
Figure 16 Von Mises Stress Plot [75/-75]s Laminate
32
Figure 17 Von Mises Stress Plot [90]s Laminate
The first 5 results of the Modal analysis for the plate geometry are tabulated below in
Table 4. The first 5 mode shapes for the θ=0° case are shown in Figure 16.
Table 5 Modal Analysis Results
Mode
Orientation, θ
0
15
30
45
1
2
3
4
5
33
60
75
90
Mode Frequency vs. Orientation Angle (θ)
700
Frequency (Hz)
600
500
Mode 1
400
Mode 2
300
Mode 3
200
Mode 4
Mode 5
100
0
0
15
30
45
60
75
90
Ply Angle, θ
Figure 18 Mode Frequency vs. Orientation Angle
Vibration Modes For Various Orientation Angles
700.00
Frequency (Hz)
600.00
90
500.00
75
60
400.00
45
30
300.00
15
200.00
0
100.00
0.00
1
2
3
Mode
4
5
Figure 19 Plot of Mode Frequency for Various Orientations
34
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Figure 20 Mode Shapes 1 thru 5 for [0]s Laminated Plate
35
4. CONCLUSION
36
5. REFERENCES
Bak, Michael. MANE6200 Course Notes. Hartford, CT: 2008. Print.
Schmalberger, Brian. Optimization of an Orthotropic Composite Beam. Hartford, CT:
2010. Print.
Ali, Arshad, and Zeeshan Azmat. Analysis of Composite Structure Under Thermal Load:
Using ANSYS. 1. LAP Lambert Academic Publishing, 2010. Print.
Ventsel, Eduard, and Theodor Krauthammer. Thin plates and shells: theory, analysis,
and applications. CRC, 2001. Print.
C., B., and Alan A. Composite materials for aircraft structures. American Institute of
Aeronautics & Astronautics, 1986. Print.
Jones, Robert M. Mechanics of Composite Materials. 2nd. New York, NY: Taylor &
Francis Group, LLC, 1999. Print.
37
6. Appendix A - Maple version 12 files
38