Stress Analysis of a Fiber Reinforced - Polymer Matrix
Orthotropic Plate with an Elliptical Hole
by
Matthew J. Tauris
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, Thesis Advisor
Rensselaer Polytechnic Institute
Hartford, Connecticut
April, 2009
CONTENTS
LIST OF SYMBOLS ......................................................................................................... 4
LIST OF TABLES ............................................................................................................. 6
LIST OF FIGURES ........................................................................................................... 7
ABSTRACT ...................................................................................................................... 9
1. INTRODUCTION ..................................................................................................... 10
2. THEORETICAL BACKGROUND........................................................................... 14
2.1
Stress-Strain Relationships for Linear Elastic Solids ...................................... 14
2.2
Mechanics of Composites ................................................................................ 17
2.3
Stress Concentrations ....................................................................................... 21
3. APPROACH & METHODOLOGY .......................................................................... 24
4. RESULTS & DISCUSSION ..................................................................................... 31
4.1
Circular Notch in an Isotropic Plate ................................................................. 31
4.2
Elliptical Notches in an Isotropic Plate ............................................................ 31
4.3
4.4
4.2.1
Numerical Results ................................................................................ 31
4.2.2
Theoretical SCFs for Notched Isotropic Plates .................................... 34
Elliptical Notches in an Orthotropic Plate ....................................................... 34
4.3.1
Effect of the Ellipse Axis Ratio ........................................................... 34
4.3.2
SCFs as a Function of Measure of Orthotropy..................................... 37
Mesh Refinement of Elliptical Shaped Notches .............................................. 38
4.4.1
Meshing Parameters ............................................................................. 38
4.4.2
Numerical Results ................................................................................ 39
5. CONCLUSION & RECOMMENDATIONS ............................................................ 41
6. REFERENCES .......................................................................................................... 43
2
7. APPENDIX................................................................................................................ 45
7.1
Calculations of the Material Properties for the CFRP Material ....................... 45
7.2
Stress Concentration Calculation for an Orthotropic Plate .............................. 45
7.3
ANSYS Log File Input Command Templates ................................................. 47
7.4
Supplementary ANSYS Contour Plots for an Orthotropic Plate ..................... 51
3
LIST OF SYMBOLS
a = semi-major axis of ellipse, in.
Ac = net cross-sectional area of the composite, in.2
Af = net cross-sectional area of the fiber, in.2
Am = net cross-sectional area of the matrix, in.2
b = semi-minor axis of ellipse, in.
Cijkl = stiffness constant, psi
CFRP = carbon fiber reinforced polymer
E11 = elastic tensile modulus in 1-direction, psi
E22 = elastic tensile modulus in 2-direction, psi
E33 = elastic tensile modulus in 3-direction, psi
Ef = elastic tensile modulus of fiber, psi
EL = longitudinal modulus, psi
Em = elastic modulus of matrix, psi
ET = transverse modulus, psi
F = tensile force, lb.-f
G12 = elastic shear modulus (1-2 plane), psi
G13 = elastic shear modulus (1-3 plane), psi
G23 = elastic shear modulus (2-3 plane), psi
Gf = elastic shear modulus of fiber, psi
Gm = elastic shear modulus of matrix, psi
Kt = stress concentration factor
mx = coefficient of mutual influence (x direction)
my = coefficient of mutual influence (y direction)
p = pressure load, psi
Sijkl = compliance constant, psi-1
SCF = stress concentration factor
Uij = internal strain energy, in.·lb
Vf = volume fraction of the fiber
Vm = volume fraction area of the matrix
μ = complex parameter
4
ν12 = major Poisson’s ratio (1-2 plane)
ν13 = major Poisson’s ratio (1-3 plane)
ν21 = minor Poisson’s ratio (1-2 plane)
ν23 = major Poisson’s ratio (2-3 plane)
σ0 = nominal stress, psi
σapp = applied normal stress, psi
σc = normal stress in the composite, psi
σf = normal stress in the fiber, psi
σm = normal stress in the matrix, psi
σmax = maximum normal stress, psi
σij = stress component, psi
σr = normal stress in radial direction, psi
σx = normal stress in radial direction, psi
σθ = normal stress in tangential direction, psi
5
LIST OF TABLES
Table 1 – Mechanical properties of high tenacity carbon fiber and epoxy resin ............. 19
Table 2 – Theoretical mechanical properties of a unidirectional CFRP laminate ........... 19
Table 3 – Comparison of SCFs for elliptical shaped notches in isotropic and ................ 36
Table 4 – Comparison of numerically and calculated SCF for an isotropic plate ........... 40
6
LIST OF FIGURES
Figure 1 – Schematic of a cross-ply laminate formed by stacking .................................. 10
Figure 2 – Three orthogonal planes of symmetry ............................................................ 16
Figure 3 – Schematic of a unidirectional FRP lamina ..................................................... 17
Figure 4 – Circular hole in a plate resulting in a stress concentration ............................. 21
Figure 5 – Elliptical hole in a plate resulting in a stress concentration ........................... 22
Figure 6 – Loading of the orthotropic plate ..................................................................... 24
Figure 7 – Dimensions of the unidirectional FRP plate with an elliptical notch. ............ 25
Figure 8 – Dimensions of the unidirectional FRP plate with a circular notch. ............... 25
Figure 9 – ANSYS elements PLANE42 and PLANE82 ................................................. 26
Figure 10 – Quarter plate with an elliptical notch created in ANSYS ............................ 27
Figure 11 - Plate boundary conditions ............................................................................. 28
Figure 12 – Isotropic and orthotropic solutions by mesh refinement for
a
b
= 2 ............... 28
Figure 13 – Line divisions and scaling defining mesh size ............................................. 29
Figure 14 – Line divisions and scaling defining mesh size on the ellipse ....................... 29
Figure 15 – Stress distribution for and isotropic plate with a circular hole ..................... 31
Figure 16 – Maximum SCF for isotropic plates as a function ellipse axis ratio. ............ 32
Figure 17 – Stress in an isotropic plate using PLANE 42, ( ba = 2) ................................. 32
Figure 18 – Stress in an isotropic plate using PLANE 82, ( ba = 2) .................................. 33
Figure 19 – SCF for isotropic plates with elliptical notches. .......................................... 34
Figure 20 - Maximum SCF for orthotropic plates as a function ellipse axis ratio .......... 35
Figure 21 – Stress in an orthotropic plate using PLANE 42, ( ba = 2) .............................. 36
Figure 22 – Stress in an orthotropic plate using PLANE 82, ( ba = 2) .............................. 37
Figure 23 – Maximum SCFs as the orthotropic properties are varied ............................. 37
Figure 24 – Line divisions and scaling defining the refined mesh parameters. .............. 38
Figure 25 – Line divisions and defining the refined mesh parameters on the ellipse. .... 39
Figure 26 – Comparison of SCFs after mesh refinement for an isotropic plate .............. 39
Figure 27 – Comparison of SCFs after mesh refinement for an orthotropic plate .......... 40
Figure 28 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 2) .............. 51
7
Figure 29 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 3) .............. 51
Figure 30 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 4) .............. 52
Figure 31 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 5) .............. 52
Figure 32 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 6) .............. 53
Figure 33 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 7) .............. 53
Figure 34 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 8) .............. 54
Figure 35 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 9) .............. 54
Figure 36 – Stress in orthotropic plate using refined PLANE 82 mesh ( ba = 10) ............ 55
8
ABSTRACT
The aim of this paper is to analyze the effect of elliptical shaped notches on anisotropic
plates. More specifically, stress concentration factors shall be developed for a thin
orthotropic plate under plane stress with a single elliptical shaped notch. The model shall
simulate a unidirectional plate under plane stress. Loading is to be uniaxial applied in
the direction of the fiber. Stress analysis is to be accomplished utilizing the finite
element method. The analysis will briefly examine the effect circular and elliptical
notches on isotropic plates for which established stress concentration factors exist.
Stresses for a circular notch in an orthotropic plate will be characterized followed by
elliptical notches with varying semi-minor radii. The effect of the ratio between semimajor and semi-minor radii on the stress concentration will be examined. In addition,
the project will investigate how this stress concentration factor varies with respect to
changes in the degree of orthotropy of the material.
9
1. INTRODUCTION
Fiber reinforced polymer (FRP) materials play a critical role part in modern engineering
design. Although the major structural application areas of FRP materials are contained
with the aerospace, automotive and marine industries, their use has spread to numerous
commercial and industrial and infrastructure applications [1-4]. One reason for the
popularity of FRP material is undoubtedly due to their versatility in design. By altering
fiber orientation, laminate thickness and stacking sequence (Figure 1) the designer may
tailor the material’s properties as required specifically for the application. Complex
shapes that may have been required to be cast or forged can be now produced with less
energy. Near net-shape manufacturing processes can also be applied in a way that
finishing operations such as machining and/or grinding can be reduced or eliminated.
Part integration eliminates the need for multiple piece assemblies, which reduce the
number of parts, fasteners and sub-assembly operations.
fiber direction
of lamina 1
1
fiber direction
of lamina 2
2
fiber direction
of lamina 3
3
Figure 1 – Schematic of a cross-ply laminate formed by stacking
of uni-directional fiber reinforced lamina
In addition to offering these examples of design freedom, FRP composites offer other
significant advantages over their metallic counterparts.
FRP materials exhibit
significantly higher strength to weight ratios than steel, titanium, aluminum and nickel
alloys. For example, high strength unidirectional carbon fiber with an epoxy matrix has
comparable tensile strength to Inconel 718 nickel alloy yet has less than ¼ the density.
These materials also have demonstrated to possess better fatigue resistance than
10
aluminum alloys in aerospace applications.
High internal damping and corrosion
resistance are also some favorable characteristics of FRP materials [4].
In mechanical design, the need for discontinuities in part geometry is likely to arise. A
discontinuity may include holes, steps, fillets and other types of notches. In other cases
the discontinuity may not be intended in the design but is the result of a flaw in the part
such as cracks, casting voids or tool marks. Discontinuities are responsible for nonuniformity in the stress distribution in locations near the changes in geometry.
Understanding how discontinuities in part geometry affect the level of stress is critical to
the design of a structure. Stress raisers can include but are not limited to abrupt changes
in geometry, cracks, lightening holes and fastener holes. Isotropic materials have a
significant amount of theoretical stress concentration factors (SCF) that have been
developed for various types of geometry and loading. These SCFs are established and
commonly used in practice. Because an FRP material can have different properties
depending on the direction of measurement, theoretical SCFs corresponding to isotropic
material do not apply. In addition, anisotropic material behavior can result in more
severe stress concentrations than those that result from an identical geometric
discontinuity in an isotropic material [5-9]. The effect of notched geometry in materials
exhibiting anisotropic behavior depends on the type of loading and the characterization
of material mechanical properties. Therefore even for simple geometry and loading
conditions, SCFs may not exist.
Due to the versatility and subsequent popularity of fiber reinforced polymer matrix
materials in structural design, researchers in both academia and industry have realized
the extreme value in the development and understanding of the mechanics of composite
materials. Because stress concentrations due to cutouts are a relatively common root
cause of failure, the notched behavior of anisotropic plates and shells is a widely
researched topic.
11
Rezaeepazhand and Jafari [10] investigated several types of cutout shapes in isotropic
and anisotropic plates under a uni-axial load. In this case a classical method that
provides a general solution for a circular opening in an anisotropic plate was extended to
various shapes utilizing complex variable mapping. The effect of material properties,
fiber orientation and bluntness or curvature of the shape’s corners on the resulting stress
distribution was analyzed. Hufenbach et al. [11] has also used complex variables in
conjunction with stress functions to calculate the distribution of stress and displacement
around an elliptical notch in an anisotropic plate.
Although the analytical results provided good agreement with experimental findings,
Hwu [12] found that using the complex variable method, the accuracy of the
approximate solution varied for different shaped openings. By incorporating conformal
mapping combined with Stroh formalism, a single unified expression was developed to
determine stress distribution along the edge of several types of openings [12]. The
expression’s parameters are modified as required to accommodate different shapes. The
results of this work show that for notched plates subjected to unidirectional loads and
pure bending, the effect of anisotropy on the opening’s stress concentration is
completely determined by the fundamental elasticity matrices developed by Stroh.
Although the single equation obtained for any shape opening is more convenient than
previous methods it is still a fairly complicated approach.
Wu and Mu [13] have attempted utilizing geometrical scale factors (hole diameter-toplate width) to assist in the prediction of stress concentration factors in orthotropic finite
width plates from values derived by the infinite plate width assumption. Scale factors
were characterized for isotropic and orthotropic plates exposed to uni-axial and biaxial
stress. It is reported by this work that for common orthotropic materials the scale factors
are similar to those of isotropic materials when under a uni-axial loading. For a hole
diameter-to-plate width ratio of 0.5 or less, the orthotropic scale factors may be replaced
with those of an isotropic plate.
12
Xu et al. [5-7] have performed significant work in this area. Series solutions for the
stress distributions were obtained for anisotropic plates through a Faber series expansion
in conjunction with the complex potential method, conformal mapping and the least
squares collocation. The Yamada-Sun failure criterion was used to predict the strength of
anisotropic plates with multiple holes.
The analyses investigated a great deal of
parameters, which include hole spacing, hole size, position of the holes and ply
orientation. The response to tensile, compressive and shear loading were included.
Similar work has also been accomplished by Rao, Ukadgaonker et al [8,9]. As a result
of this work it is reported that an increase of plies in the in ±45° direction can
significantly reduce the effect of stress raisers from notched geometry [5-9].
The effect of several types of reinforcements in response to shear loading was analyzed
by Guo [14]. Carbon fiber/epoxy plates were modeled and with 45 flanged edges,
bonded metallic and composite rings and combinations of these types were analyzed
utilizing the finite element method. The numerical results demonstrated good agreement
with experimental data. In this analysis double steel ring reinforcements have shown
have shown to reduce the stresses around the cutouts most effectively. This type of
reinforcement also had the greatest effect on a composite plate’s buckling load factor.
This paper will investigate how geometric discontinuities affect the strength of
orthotropic laminates.
Specifically, orthotropic laminate plates with elliptical holes
exposed to a uni-axial load will be examined under linear elastic conditions.
The
analysis will determine numerical values for the stress concentration factor due to
elliptical notches. The ratio between the semi-major and semi-minor axes of the ellipse
will be varied.
The effect of the material’s measure of orthotropy on stress a
concentration will also be analyzed.
13
2. THEORETICAL BACKGROUND
2.1 Stress-Strain Relationships for Linear Elastic Solids
The generalized for Hooke’s law may be expressed as [15-17]
 ij  Cijkl kl
(1)
Where Cijkl are known as the elastic constants. If this tensor is expanded it results in 9
equations each with 9 unknowns, a total of 81 unknown elastic constants. The symmetry
of the stress tensor, σij allows the reduction of unknowns to 36. The system of equations
is written as follows
 11  C1111 11  C1122 22  C1133 33  C1123 2 23  C1113 2 13  C1112 2 12
.............................................................................................................
.............................................................................................................
 23  C2311 11  C2322 22  C 2333 33  C 2323 2 23  C2313 2 13  C 2312 2 12
(2)
............................................................................................................
............................................................................................................
To express equations (2) in a more convenient form the contracted notation often used.
Here the constants contain only two subscripts, which represent only the row and
column location of the constant,
 11  C11 11  C12 22  C13 33  C14 2 23  C15 2 13  C16 2 12
..............................................................................................
..............................................................................................
 23  C41 11  C 42 22  C 43 33  C 44 2 23  C45 2 13  C 46 2 12
(3)
..............................................................................................
..............................................................................................
Utilizing strain energy methods, the number of independent elastic constant may be
further reduced. The strain energy Uij is related to stress and strain by the following
relationship [15],
U ij
 ij
  ij
(4)
Taking the second derivative of results in another symmetric property,
Cij  C ji
14
Therefore, the generalized form of Hooke’s law has 21 independent elastic constants that
must be determined. In matrix form the stress-strain relations may be expressed as,
 11  C11 C12 C13 C14
  
C 22 C 23 C 24
 22  
 33  
C33 C34
 
C 44
 23  
 13   Symmetric
  
 12  
C15
C 25
C35
C 45
C55
C16   11 
C 26   22 
C36   33 
 
C 46   23 
C56   13 
 
C66   12 
(5)
An isotropic material has identical properties that are independent of the direction of
measurement. The resulting stiffness matrix is greatly simplified.
0
C11 C12 C12

C11 C 23 0


C11 0

C 44

 Symmetric


0
0
0
0
C 44
0 
0 
0 

0 
0 

C 44 
(6)
In this case only 2 independent elastic constants are required because of the following
relationship [15-17].
C44 
C 11C12
2
(7)
Orthotropic materials possess three mutually perpendicular planes of symmetry as
illustrated in Figure 2. Within a plane of symmetry mechanical properties are constant.
The stiffness matrix for general orthotropic material may be expressed as,
C11 C12 C13 0

C 22 C 23 0


C33 0

C 44

 Symmetric


0
0
0
0
C55
0 
0 
0 

0 
0 

C66 
(8)
There are 9 independent constants that must be specified [4]. Under the condition of
plane stress it is assumed that stresses cannot develop in one of the principle directions.
15
2-3 plane
1-3 plane
1-2 plane
Figure 2 – Three orthogonal planes of symmetry
Therefore the number of independent constants for an orthotropic material may be
further reduced. If it is assumed that the stresses exist only in the 1-2 plane,
 33   23   13   33   23   13  0
the constitutive relationship may then be written as,
 11 
 
 22 
 0 
 
 0 
 0 
 
 12 
0
0  11 
C11 C12 C13 0

C22 C23 0
0
0   22 


C33 0
0
0   0 

 
C44 0
0  0 

 Symmetric
C55 0   0 

 
C66  12 

(9)
In this case the stiffness matrix is commonly re-written as [4,17,18],
 C11
C
 12
 0
C12
C 22
0
0 
0 
C 66 
(10)
There are only 4 independent constants that must be specified for an orthotropic material
under plane stress conditions.
16
2.2 Mechanics of Composites
As mentioned above, general orthotropic materials can be characterized by 9 constants,
E11, E22, E33, G12, G13, G23, ν12, ν13, and ν23. Unidirectional fiber composites are a special
class of orthotropic materials in which the fiber reinforcements are orientated in a single
direction. For a composite lamina in which the fibers are oriented in the 1-2 plane, we
may assume that the properties in directions 2 and 3 are equal or E22 = E33, G12 = G13 and
ν12 = ν13 [4,15,16].
Figure 3 – Schematic of a unidirectional FRP lamina
Consider a rectangular unidirectional composite plate with its fibers parallel to the
longitudinal axis of the plate as in Figure 3. If a tensile load is applied in an orientation
direction parallel to the fibers, the force exerted on the composite material is [4],
 c Ac   f A f   m Am
(11)
Where σc is the normal stress in the composite, σf is the normal stress in the fiber, σm is
the normal stress in the matrix. Ac, Af and Am are the net cross-sectional areas of the
composite, fiber and matrix respectively. The stress present in the composite material is,
c   f
Af
Ac
m
Am
  f V f   mVm
Ac
(12)
where the volume fractions are defined as,
Vf 
Af
Ac
and Vm 
Am
Ac
(13)
Here Vf is the volume fraction of the fiber constituent and Vm is the volume fraction area
of the matrix constituent. If there are no other constituents within the composite material
is obvious that,
17
Vm  1  V f
(14)
Assuming a perfect bond between the fibers and matrix, the strains in both materials are
identical. Substituting equation (14) in to equation (12) and dividing by the normal
strain, εc results in,
EL  E f V f  Em (1  V f )  E11
(15)
Equation (13) is often referred to as the rule of mixtures [4]. Here, EL is longitudinal
modulus of the composite material. The transverse modulus, ET may be defined as
[4,16,17],
ET 
E f Em
EmV f  E f (1  V f )
 E22
(16)
Recalling the constants required to characterize an orthotropic material,
E11  EL
E22  E33  ET
G12  G13 
G f Gm
(17)
GmV f  G f (1  V f )
 12   13   f V f   m (1  V f )
(18)
To identify the final two constants, the minor Poisson’s ratio is required. This is defined
as the ratio of the strain in the 1-direction due to the applied strain in 2-direction.
Alternatively it may be determined by [4],
 21   12
E22
E11
(19)
Equation (19) allows calculation of the remaining major Poisson’s ratio [4].
 23   12
(1   21 )
(1   12 )
(20)
and subsequently the final shear modulus,
G23 
E22
2(1   23 )
(21)
The ratio,
E22
E11
(22)
18
is often referred to as the measure of orthotropy [15].
To calculate realistic material properties for a unidirectional composite plate assume a
fiber volume fraction, Vf of .60, which is representative of manufactured CFRP laminate
sheets. Some typical mechanical properties of CFRP material constituents are listed in
Table 1.
Table 1 – Mechanical properties of high tenacity carbon fiber and epoxy resin [18]
HTA Carbon Fiber Mechanical Properties
RTM Liquid Epoxy Resin Mechanical Properties
Ef – Tensile
Modulus
(longitudinal),
106 psi
Gf – Shear
Modulus,
106 psi
υf – Poisson’s
Ratio
Em – Young’s
Modulus,
106 psi
Gm – Shear
Modulus,
106 psi
υm – Poisson’s
Ratio
34.08
14.20
.20
.4525
.1638
.38
Utilizing equations (15-21) and the values listed in Table 1, the general mechanical
properties for a carbon fiber reinforced unidirectional laminate are obtained. These
values are listed in Table 2. Refer to the Appendix for detailed calculations.
Table 2 – Theoretical mechanical properties of a unidirectional CFRP laminate
Elastic Tensile Moduli
Elastic Shear Moduli
Poisson’s Ratio
E11,
106 psi
E22= E33,
106 psi
G12= G13,
106 psi
G23,
106 psi
υ12= υ13
(major ratio)
υ23
(major ratio)
υ21 = υ31
(minor ratio)
20.63
1.109
.4027
.4018
.272
.368
.0146
The strain-stress relationships may now be expressed for a an orthotropic lamina under
plane stress [4]
 xx 
 xx
Exx
 yy   yx
  yx
 xx
Exx
 yy
E yy

 mx xy
 yy
E yy
(23)
 my xy
 xy  mx xx  my yy 
(24)
 xy
(25)
Gxy
19
Where mx and my are referred to as the coefficients of mutual influence. They are given
by [4],

 1
1
1
2
1
1 

mx  (sin 2 )  12 

 (cos 2  )
 12 

E
E
2
G
E
E
E
2
G
22
12
11
22
12  
 11
 11
(26)

 1
1
1
2
1
1 

m y  (sin 2 )  12 

 (sin 2  )
 12 

E
E
2
G
E
E
E
2
G
11
22
12
11
11
22
12



(27)
These coefficients represent the effects of coupling between normal and shear strains in
the anisotropic material. For θ = 0° or θ = 90° these coefficients vanish. This is logical
because specially orthotropic materials do not exhibit extension-shear coupling. The
constitutive equations may then be rewritten,
 xx  11 
 xx
E11
 21
 yy   22   12
 xy  12 
 xx
E11
 yy
(28)
E22

 yy
(29)
E22
 xy
(30)
G12
These may be arranged in matrix form,
 xx   S11
  
 yy    S 21
   0
 xy  
S12
S 22
0
0   xx 
 
0   yy 
S66   xy 
(31)
where Sij are the components of the compliance matrix.
This is the strain-stress
relationship for an orthotropic material under plane stress. If equation is inverted the
stress-strain relationship is defined.
 xx   C11 C12
  
 yy   C21 C22
   0
0
 xy  
0   xx 
 
0   yy 
S66   xy 
(32)
This is stiffness matrix, which is identical to equation (10). The components of the
stiffness matrix have the following values [15-17],
C11 
E11
(33)
(1   12 21 )
20
C22 
E22
(34)
(1   12 21 )
C12  C12 
 12 E22
 21E11

(1   12 21 ) (1   12 21 )
C66  G12
(35)
(36)
2.3 Stress Concentrations
Many engineering texts define stress concentrations for simple geometry and loading. In
most cases the SCF Kt is defined as ratio of maximum stress to nominal stress in a net
cross section of the part.
Kt 
 max
0
(37)
Consider a large plate with a central hole of radius a, with a uni-axial load F applied as
shown in Figure 4. If is assumed that the hole is sufficiently far enough from the edges
of the plate the stresses as a function of radial position, r and angular position, θ are
expressed as [15],
r 
 

a2   
a4
a2 
1  2   1  3 4  4 2  cos 2
2 r  2
r
r 

a2   
a4 
1  2   1  3 4  cos 2
2 r  2
r 
Figure 4 – Circular hole in a plate resulting in a stress concentration
21
(38)
(39)
The stress concentration factor for an elliptical shaped notch also in an isotropic plate for
the geometry and loading conditions shown in Figure 5 have been derived and are given
by [19],
2
 2a 
 2a 
 2a 
K t  C1  C2    C3    C4  
D
D
D
3
a
C1  1  2 
b
C2  .351  .021
(40)
(41)
a
a
 2.483 
b
b
C3  3.621  5.183
C4  2.27  5.204
(42)
a
a
 4.494 
b
b
(43)
a
a
 4.011 
b
b
(44)
Where 2a and 2b are semi-major and semi-minor axes of the ellipse, respectively in a
plate of width D. It should be noted that formulae used to calculate the coefficients are
valid over the range of axis ratios .5  ba  10.
notch
base
Figure 5 – Elliptical hole in a plate resulting in a stress concentration
Other sources [15, 20] use the following relationship to define the maximum SCF in an
infinite isotropic plate with an elliptical hole.
22
a
K t  1  2 
b
It is apparent that as the ratio
(45)
a
b
approaches unity, the solution becomes identical to the
circular case of equation (39) for r = a, and at an angle of θ = 90°.
Hufenbach et al. [11] proposed an analytical method for determining the stress field
around an elliptical hole in an anisotropic plate loaded under plane stress conditions.
The solution’s complex parameters, μk are derived from the components of the
compliance matrix,
2
 2S  S 66 
2S  S66
S
  22
 k    12
  12
2S11
S11
 2S11 
(k = 1, 2, 3 4)
(46)
which have the form,
 k   k  i k
(47)
If the anisotropic problem is reduced to a unidirectional FRP plate with an elliptical
shaped hole, under a uniaxial load, with a fiber orientation parallel to the loading
direction and the hole is oriented as shown in Figure 5, the maximum SCF is defined by,
 a 1   2 

K t  1 
 b 1 2 
(48)
The location of this maximum stress is at the base of the notch [11].
23
3. APPROACH & METHODOLOGY
The effect of a small elliptical shaped notch in a thin, unidirectional CFRP composite
plate is analyzed utilizing the finite element method. Figure 6 indicates the loading of
the plate shall be uniaxial and applied in the direction parallel to the orientation of the
carbon fibers. It was assumed that the thin composite plate was under the condition of
plane stress. Figure 7 and Figure 8 represent the plate and notch geometry that were
investigated. It should be noted that the ratio of plate’s width to the diameter of the
circular hole or the semi-major axis of the ellipse is large enough to prevent edge effects
from affecting the stress level in the vicinity of the notch. Therefore the infinite plate
assumption is valid for this analysis.
2
1
p = 1000 psi
p = 1000 psi
direction of fibers
Figure 6 – Loading of the orthotropic plate
24
Figure 7 – Dimensions of the unidirectional FRP composite plate with an elliptical notch.
(Note: dimensions are in inches, drawing is not to scale)
Figure 8 – Dimensions of the unidirectional FRP composite plate with a circular notch.
(Note: dimensions are in inches, drawing is not to scale)
25
This project utilized the ANSYS 11.0 software package to determine the normal stress
distribution in the direction parallel to the plate’s fibers. Included are the results from
the utilization of 2 types of elements. The first type of element that was used is called
PLANE42. It is 4-noded, quadrilateral shaped element that is typically used for twodimensional analysis of solids under plane stress or plane strain conditions, flat plates in
bending or structures possessing axisymmetry.
Each of the four nodes has two
translational degrees of freedom in the x and y directions. The geometry of this element
is shown in Figure 9.
Figure 9 – ANSYS elements PLANE42 and PLANE82
The second type of element that was used is called PLANE82. It is simply a higher
order version of the PLANE42 element possessing 8 nodes. Refer to Figure 9 for a
graphical representation of this element. Again, this is very popular element for the
stress analysis of thin plates and shells often approximated as being under condition of
plane stress. Because the PLANE82 element has mid–nodes, it can tolerate irregular
geometry without a significant loss of accuracy. It is expected that element is better
suited to handle the shape of the ellipse once the axis ratio is increased, creating a very
narrow hole comparable to a crack.
Symmetry of the geometry and loading allowed a quarter plate to be used in the analysis.
Due to the relatively simple shape of the notched plates, ANSYS’s modeling features
26
were used to generate the geometry. To accomplish this, a square area was created with
a circular area created with its center located at the square’s lower left corner. The
elliptical shape was generated by scaling the geometry of a circle in the x-direction. The
ellipse was then subtracted from the square area resulting in the notched plate shown in
Figure 10. The required boundary conditions and loading are such that the plate is to be
in a state of plane stress due to a uni-axial load. Symmetry on the two lines adjacent to
the ellipse and a 1000 psi pressure load as shown in Figure 11 satisfies this condition.
The elastic properties for E11, E22, G12 and ν12 were taken from the values in Table 1.
Figure 10 – Quarter plate with an elliptical notch created in ANSYS
(Note: image adjusted to show detail of ellipse – full quarter plate not shown)
The isotropic case was first analyzed.
Meshing was accomplished within ANSYS
software. The required level of mesh refinement was determined by a convergence test
for
a
b
 2 . The number of subdivisions on selected lines was successively increased
generating a finer mesh during each step. Figure 12 shows the number of divisions on
the ellipse that begin to converge to a common solution. It should be noted the element
size near the free edge and loaded edge of the plate have little effect on the stress
gradients and the mesh may be constructed much coarser. Therefore, element size was
adjusted locally (in the vicinity of the notch) to improve accuracy of the stress gradients.
On the symmetry edges (adjacent to the notch) a scaling factor was used such that mesh
would become finer as approaching the ellipse. Figure 13 and Figure 14 illustrate the
initial element sizing parameters that were used in all cases.
27
Free Edge
Outward pressure
Load
Symmetry
Boundary
Conditions
Figure 11 – Plate boundary conditions
Maximum Stress, psi (x-direction)
12000
10000
8000
Isotropic PLANE42
6000
Orthotropic PLANE42
Orthotropic PLANE82
4000
2000
0
0
10
20
30
40
50
60
No. of Element Divisions on Ellipse
Figure 12 – Isotropic and orthotropic solutions by mesh refinement for
28
a
b
=2
Line 3
35 divisions
Line 10
50 divisions
.25 scale ratio
Line 2
35 divisions
Line 9
50 divisions
.25 scale ratio
Figure 13 – Line divisions and scaling defining mesh size
Line 5
40 divisions
Figure 14 – Line divisions defining mesh size on the ellipse
(Note: image adjusted to show detail of ellipse – full quarter plate not shown)
To establish a baseline for comparison, a preliminary analysis was accomplished with
notched isotropic plates.
Following the isotropic analysis, the element’s material
29
properties were modified to simulate the orthotropic behavior of a unidirectional fiber
reinforced composite material. The ratio of the semi-major axis to the semi-minor axis
of the elliptical notch was also varied. In addition, the measure of orthotropy of the plate
was varied for both the circular and elliptical notches.
To initiate the analyses, the ANSYS input file from the geometry used for the
convergence test.
This input file already defined boundary conditions, meshing
parameters and loading. From this input file, the following separate “template” input
files were created:

Isotropic plate quarter plate with elliptical notch with constant tensile and shear
moduli and Poisson’s ratio

Orthotropic quarter plate with elliptical notch and constant E11, E22, G12 and ν12

Orthotropic quarter plate with an elliptical notch and set up for constant E11 and
variable E22, G12 and ν12

Orthotropic quarter plate with a circular notch set up constant E11 and variable
E22, G12 and ν12
Each one of these input files was then altered as required to complete the finite element
analysis. This offered a simple way to modify element type, ellipse axis ratios, and
material properties rapidly for each case. The details of the template input files are
available in the Appendix.
The results of the analysis characterized the maximum stress level in the notched plates
in the x or 1-direction. Stress concentration factors were assigned for each case as,
Kt 
 max
 max

 app 1000 psi
(49)
The finite element results were compared with established analytic solutions for stress
concentration factors.
30
4. RESULTS & DISCUSSION
4.1 Circular Notch in an Isotropic Plate
The initial ANSYS run was performed on a thin isotropic plate containing a centric
circular shaped notch. Figure 15 shows σx in the vicinity of the hole for this classical
example. The primary purpose for this was to establish a baseline for the numerical
results. The circular hole in this case resulted in Kt = 3.005, which is within .16% of the
value calculated in Equation (39) with θ = 90° and r = a.
Figure 15 – Stress distribution for and isotropic plate with a circular hole. Results generated
using ANSYS PLANE82 elements and meshing parameters in Figures 13 & 14.
4.2 Elliptical Notches in an Isotropic Plate
4.2.1
Numerical Results
Analysis of elliptical notch geometry was then accomplished.
The ANSYS nodal
solutions for isotropic plates with varying ellipiticity are summarized in Figure 16.
Figure 17 is a magnified view if the computed stress in the x-direction for axis ratio
a
b
 2.
31
Stress Concentration Factor
(Maximum Stress/Applied Stress)
9
8
7
6
5
PLANE42
4
PLANE82
3
2
1
0
0
2
4
6
8
10
12
Ellipse Axis Ratio (Semi-Major/Semi-Minor)
Figure 16 – Maximum SCF for isotropic plates as a function ellipse axis ratio.
Figure 17 – Stress concentration for an isotropic plate with an elliptical (
a
b
= 2) notch. Results
generated using ANSYS PLANE42 elements and meshing parameters in Figures 13 & 14.
32
Figure 18 – Stress concentration for an isotropic plate with an elliptical (
a
b
= 2) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 13 & 14.
In general it can be stated that the SCF becomes more severe as the axis ratio ( ba ) of the
ellipse becomes more pronounced. At an axis ratio value of 10 the elliptical notch
approaches the geometry of a crack, void or some other flaw-type discontinuity. The
results displayed in Figure 16 indicate that a notch of this shape could produce a
localized stress approximately 8 times the stress due to the applied load. It is noted from
this plot that the PLANE42 element results shows a local maximum where the
PLANE82 solution approaches an asymptotic value.
During a majority of these software runs ( ba  5 ) the use of quad-node elements
generated an “element shape warning” by ANSYS. Poor element shapes can affect the
accuracy of the solution and should be reviewed for criticality [21].
Further
investigation into the warnings revealed that this message is displayed when one more
elements in the mesh possess edges that are approaching parallel directions. ANSYS’s
threshold in this case is a 165° angle between adjacent edges. Shape warnings were also
issued during the use of PLANE82 element as well, although the number of elements
33
affected was slightly less significant. The elements generating this warning were found
to be located in the vicinity of the highest stress level.
4.2.2
Theoretical SCFs for Notched Isotropic Plates
The analysis has included the calculation of theoretical stress concentration factors for
circular and elliptical notches in isotropic plates. Equation (39) yields a maximum stress
concentration factor Kt = 3 the plate shown in Figure 8. The location of the maximum
stress is located at a position of θ = 90° from longitudinal axis of the plate. Utilizing
equations (39-45) the maximum stress concentration factors for the plate shown in
Figure 5 are plotted in Figure 19 for comparison with the numerical results. The circular
stress concentration ( ba  1) and some of the elliptical geometry demonstrate good
agreement with one another. Although it is apparent that as the ellipiticity increases to
values of
a
b
 3 , the finite element results rapidly deviate from the calculated solution.
Stress Concentration Factor
(Maximum Stress/Applied Stress)
25
20
PLANE42
15
PLANE82
Roark's
10
Dieter
5
0
0
2
4
6
8
10
12
Ellipes Axis Ratio (Semi-major/Semi-minor)
Figure 19 – SCF for isotropic plates with elliptical notches. Numerical solutions are plotted
in comparison to theoretical results calculated from Ref [18].
4.3 Elliptical Notches in an Orthotropic Plate
4.3.1
Effect of the Ellipse Axis Ratio
Analysis of the elliptical shaped notches in an orthotropic material was accomplished
utilizing both 4-node and 8-node elements. The results as a function of ellipiticity have
been plotted in Figure 20. Figure 21 and Figure 22 are the graphical outputs from
34
ANSYS. Reviewing the data listed in Table 3, generally, notched orthotropic plates
display greater stress concentrations than isotropic plates with identical shaped notches.
Due to the orientation of orthotropic properties with respect to the loading direction, the
location of maximum stress is identical to an isotropic plate with an elliptical hole.
An observation is that the use of PLANE42 elements results in a concave downward that
slightly increase then begins to decrease as the ellipiticity of the plate’s opening is
increased. This behavior is somewhat similar to the corresponding solution in the
isotropic analysis. Shape warnings were also issued while meshing with both element
types. Additionally, the PLANE42 solution shows very little variation as the shape of
the ellipse is altered. It should also be noted that at an axis ratio of 8, the PLANE82
solution has a value that is uncharacteristic with respect to the trend of the curve. This
data point produces a sharp spike in the curve where the remaining points generate a
relatively smoothly increasing curve.
Stress Concentration Factor
(Maximum Stress/Applied Stress)
14.000
12.000
10.000
8.000
PLANE42
PLANE82
6.000
4.000
2.000
0.000
0
2
4
6
8
10
12
Ellipse Axis Ratio (Semi-Major/Semi-Minor)
Figure 20 – Maximum SCF for orthotropic plates as a function ellipse axis ratio
35
Table 3 – Comparison of SCFs for elliptical shaped notches in isotropic and
orthotropic plates
Axis Ratio
 
major
minor
PLANE42
PLANE82
PLANE42
PLANE82
2
3
4
5
6
7
8
9
4.268
5.399
5.871
6.146
6.363
6.372
6.584
6.493
4.602
5.711
6.469
6.943
7.403
7.592
7.909
8.099
5.022
6.649
6.026
6.694
6.262
6.016
6.296
5.866
7.461
8.240
8.987
9.290
9.961
9.854
12.304
10.516
10
6.265
8.143
5.661
10.111
Isotropic SCF
Orthotropic SCF
Figure 21 – Stress concentration for an orthotropic plate with an elliptical ( ba = 2) notch. Results
generated using ANSYS PLANE42 elements and meshing parameters in Figures 13 & 14.
36
Figure 22 – Stress concentration for an orthotropic plate with an elliptical ( ba = 2) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 13 & 14.
4.3.2
SCFs as a Function of Measure of Orthotropy
The effect of the ratio
E2 2
E1 1
was examined for both circular and elliptical shaped notches.
The results in Figure 23 were obtained for orthotropic plate geometry defined in Figure 5
and Figure 6. Only PLANE82 type elements were used. From this solution it is quite
apparent that as the material becomes less orthotropic, the effect of stress concentration
factors becomes less pronounced.
Stress Concentration Factor
(Maximum Stress/Applied Stress)
8
7
6
5
Elliptical SCF
4
Circular SCF
3
2
1
0
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Degree of Orthotropy (E2 2 /E11 )
Figure 23 - Maximum SCFs as the orthotropic properties are varied in
plates with elliptical ( ba = 2) and circular holes.
37
4.4 Mesh Refinement of Elliptical Shaped Notches
4.4.1
Meshing Parameters
Due to the large discrepancy between the numerical and theoretical solutions at ratios of
a
b
 2 in conjunction with some of the shape warning error messages generated by
ANSYS, it is warranted to investigate how the changing the meshing parameters affects
the results. The mesh size is particularly important in the ellipse geometry where the
vertical ends begin to form very sharp transitions to symmetry edge of the plate parallel
to the y-axis. The mesh sizing was modified as shown in Figure 24 and Figure 25. This
produced a significantly finer mesh in the vicinity of the notch. Additionally regions far
from the notch with low stress gradients were increased in size to save computation time.
Line 3
.250”element edge length
Line 2
.250”element edge length
Line 10
.010”element edge length
.25 scale ratio
Line 9
.010”element edge length
.25 scale ratio
Figure 24 – Line divisions and scaling defining the refined mesh parameters.
38
Line 5
.005” element
edge length
Figure 25 – Line divisions and scaling defining the refined mesh parameters on the ellipse.
(Note: image adjusted to show detail of ellipse – full quarter plate not shown)
4.4.2
Numerical Results
The isotropic results of the refined PLANE82 mesh in comparison to previously
obtained results have been plotted in Figure 26. Although as you approach higher values
of the ellipse axis ratio the ANSYS solution continues to diverge slightly from the
results calculated from equations (40-45), the solution is in close agreement with the
theoretical results for at a majority of the data points. Table 4 lists the data used to
generate Figure 26. Shape warnings only occurred in instances and affected only a
single element.
Stress Concentration Factor
(Maximum Stress/Applied Stress)
25
20
PLANE42
15
PLANE82
PLANE82 Refined
Roark's
10
Dieter
5
0
0
2
4
6
8
10
12
Ellipes Axis Ratio (Semi-major/Semi-minor)
Figure 26 – Comparison of SCFs after mesh refinement for an isotropic plate with an elliptical
notch.
39
Table 4 – Comparison of numerically generated and calculated SCF for an isotropic plate
with an elliptical notch.
Isotropic SCF
Axis Ratio

major
minor

PLANE42
PLANE82
Dieter
Roark's
PLANE82
Refined
1
2.998
3.005
3
2.882
3.128
2
4.268
4.602
5
4.666
5.036
3
5.399
5.711
7
6.512
7.008
4
5.871
6.469
9
8.359
8.914
5
6.146
6.943
11
10.207
10.756
6
6.363
7.403
13
12.056
12.335
7
6.372
7.592
15
13.905
13.653
8
6.584
7.909
17
15.754
14.708
9
6.493
8.099
19
17.603
15.733
10
6.265
8.143
21
19.453
16.349
The orthotropic results were also dramatically changed by refining the meshing
parameters. The plot in Figure 27 illustrates that the mesh size can greatly affect the
solution. The SCF has increased by an average multiplier of 2.2 between the two the
initial and refined solutions. This reiterates how critical a role selecting element type
and mesh size is to obtaining a valid solution. Utilizing equations (46-48) the solution
proposed by Hufenbacher et al. is also plotted in figure 27. There is a significant
discrepancy between this solution and the refined ANSYS output. The use of the
solution in [11] has generated SCFs similar to isotropic plates with elliptical holes. It is
felt that equations (46-48) were interpreted improperly.
Stress Concentration Factor
(Maximum Stress/Applied Stress)
30
25
20
PLANE42
PLANE82
15
PLANE82 Refined
Hufenbach
10
5
0
0
2
4
6
8
10
12
Ellipes Axis Ratio (Semi-major/Semi-minor)
Figure 27 – Comparison of SCFs after mesh refinement for an orthotropic
plate with an elliptical hole.
40
5. CONCLUSION & RECOMMENDATIONS
Fiber reinforced polymers and other types of composite materials have been
implemented into every conceivable structural application in aerospace, automotive,
industrial and infrastructure industries. Composites offer extremely favorable strengthto-weight ratios in comparison to their metal alloy counterparts. In addition to this
composite materials offer design flexibility because the mechanical properties (and the
manner in which they vary) may be specified based on an application. The widespread
use of these materials facilitates the need to understand how notches and other
discontinuities affect the local stress level of the component in the vicinity of the
discontinuity.
Although significant work has been accomplished, a general need for SCF for
orthotropic (and other anisotropic) material still exists. Analytical methods formulated
by researchers have provided reliable solutions for various types of notch geometry. The
drawback associated with these methods is ease of use. Even the single unified equation
developed by Hwu is somewhat cumbersome for practical utilization in industry. This
report has demonstrated that modern finite element analysis software package can assist
with this challenge.
In summary, the results of this paper have shown the following

Orthotropic plates with elliptical shaped notch exhibit higher SCF than isotropic
plates with identical geometry. This is in general agreement with past research in
this subject area.

As the case for isotropic plates, increased ellipiticity of the notch will cause a
more severe SCF. Narrow elliptical shapes that mimic cracks begin to produce
significant SCFs that could easily surpass the lamination strength of a FRP plate.

The SCF becomes larger as the degree of orthotropy increases.

Element selection is critical and will depend on geometry. As a minimum,
features possessing curvature should use higher order elements with mid-nodes.

Mesh size in the vicinity of irregularly shaped discontinuities is critical and if
possible known solutions should be used to establish baselines from comparison.
41
The advancement and availability of high performance computers enables almost every
mechanical designer or structural analyst to perform fairly complicated stress analyses.
A majority of modern finite element packages possess the ability to model materials
exhibiting anisotropic mechanical properties. Based on the results of this investigation,
the finite element method should be applied with discretion to orthotropic (or other types
of anisotropic) structures to ensure reasonably accurate results for stress concentration
factors are obtained.
Due to the vast number of configurations of FRP materials there is almost an unlimited
potential for future work in this area. Although the fact that the ANSYS results for an
orthotropic plate did not agree with the solution derived by Hufenbacher et al. for a
majority of ellipse axis ratios warrants further investigation, even for the simplified case
of a uni-directional composite plate with a elliptical hole under a uni-axial tensile load.
Because the material and loading are relatively simple it is recommended that the work
of report is verified with experimental evidence.
42
6. REFERENCES
[1] Romano, F., Fiori, J., Mercurio, U., May 2009, “Structural Design and Test
Capability of a CFRP Aileron,” Composite Structures, 88(3), pp. 333-341.
[2] Tsai, S. W., December 2005, “Three decades of composites activities at US Air
Force Materials Laboratory,” Composites Science and Technology, 65(15) pp. 22952299.
[3] Yang, D., Park, S., Neale, K. W., May 2009, “Flexural Behaviour of Reinforced
Concrete Beams Strengthened with Prestressed Carbon Composites,” Composite
Structures, 88(4), pp. 497-508.
[4] Mallick, P., 1993, Fiber Reinforced Composites (2nd Edition), New York, NY, Chap.
1-3.
[5] Xu, X., W., Man, H., C., Yue, T., M., May 2000, Strength Prediction of Composite
Laminates with Multiple Elliptical Holes International Journal of Solids and Structures,
37(21), pp. 2887-2900.
[6] Xu, X. W., Sun L., Fan., October 1995, Stress concentration of finite composite
laminates with elliptical hole, Computers & Structures, 57(1), pp. 29-34.
[7] Xu X., Sun L., Fan., October 1995, Stress Concentration of Finite Composite
Laminates Weakened by Multiple Elliptical Holes, International Journal of Solids and
Structures, Volume 32(20), pp. 3001-3014.
[8] Ukadgaonker, V. G., Rao, D. K. N., July 2000, “A General Solution for Stresses
Around Holes in Symmetric Laminates Under In-plane Loading,” Composite Structures,
49(3), pp. 339-354.
[9] Ukadgaonker, V. G., Kakhandki, V., September 2005, “Stress Analysis for an
Orthotropic Plate With an Irregular Shaped Hole for Different In-Plane Loading
Conditions—Part 1,” Composite Structures, 70(3), pp. 255-274.
[10] Rezaeepazhand, J., Jafari, M., December 2005, “Stress Analysis of Perforated
Composite Plates,” Composite Structures, 71(3), pp. 463-468.
[11] Hufenbach, W., Schaefer, M., Herrmann, A. S., 1990, “Calculation of the Stress and
Displacement Field of Anisotropic Plates with an Elliptical Hole (In German),”
Ingenieur-Archiv, 60(8), pp. 507-515.
[12] Hwu, C., September 1990, “Anisotropic Plates with Various Openings Under
Uniform Loading or Pure Bending,” ASME Journal of Applied Mechanics,
57(3), pp.700 - 706.
43
[13] Wu, H., Mu, B., March 2003, “On stress concentrations for isotropic/orthotropic
plates and cylinders with a circular hole,” Composites Part B: Engineering, 34(2), pp.
127-134.
[14] Guo, S. J., September 2007, “Stress Concentration and Buckling Behaviour of
Shear Loaded Composite Panels with Reinforced Cutouts,” Composite Structures, 80(1),
pp. 1-9.
[15] Dieter, G. E., 1986, Mechanical Metallurgy, McGraw-Hill, New York, NY,
Chap.1-2, 7, 11-14.
[16] Baker, A., Dutton, S., Kelly, D., 2004, Composite Materials for Aircraft Structures
(2nd Edition), American Institute of Aeronautics and Astronautics, Chap. 6.
[17] Staab, G. H., 1999, Laminar Composites, Elsevier, Chap. 2.
[18] Lee H. G., Hwang, H. Y., Lee, D. G., August 2006, “Effect of Wear Debris on the
Tribological Characteristics of Carbon Fiber Epoxy Composites,” Wear, 261(3), pp.
453-459.
[19] Young, W. C., Budynas, R.G., 2002, Roark's Formulas for Stress and Strain
(7th Edition), McGraw-Hill, New York, NY, Chap. 17.
[20] Budynas, R.G., Nisbett, J. K., 2008, Shigley’s Mechanical Engineering Design
(8th Edition), McGraw-Hill, New York, NY, Chap. 3.
[21] Stolarski, T., Nakasone, Y., and Yoshimoto, S., 2006, Engineering, Analysis With
ANSYS Software, Elsevier Butterwoth-Heinemann, Oxford, UK, Chap. 1-4.
[22] Segerlind, L. J., 1984, Applied Finite Element Analysis, John Wiley & Sons, New
York, NY, Chap. 1-6, 12, 17-28.
[23] Whitworth, H. A., Mahase, H., September 1999, “Failure of Orthotropic Plates
Containing a Circular Opening,” Composite Structures, 46(1), pp. 53-57.
44
7. APPENDIX
7.1 Calculations of the Material Properties for the CFRP Material
The following values have been calculated using the values of Table 1 and equations
(15-21).
Tensile Moduli longitudinal and transverse directions
E11  EC  E f V f  Em( 1  V f ) 3.408E 7  .6  4.525E 5  .4  2.063E7 psi
E22  E33 ET 
E f Em
EmV f  E f (1  V f )

3.408E7  4.525E5
 1.109E6 psi
4.525E5  .6  3.408E7  .4
Shear modulus in 1-2 plane
G12  G21
G f Gm
GmV f  G f (1  V f )

1.420 E 7  1.638E
 4.027E5 psi
1.638E 7  .6  1.420 E 7  .4
Major Poisson’s ratios for 1-2 plane and 1-3 plane
 12   13   f V f   m (1  V f )  .2  .6  .38  .4  .272
Minor Poisson’s ratios for 2-3 plane
 21   31   12
E22
1.109 E 6
 .272 
 .0146
E11
2.063E 7
Major Poisson’s ratios for 2-3 plane
 23   12
(1   21 )
(1  .0146)
 .272
 .368
(1   12 )
(1  .272)
Shear modulus in 2-3 plane
G23 
E22
1.109 E 6

 4.018E5 psi
2(1   23 ) 2(1  .368)
7.2 Stress Concentration Calculation for Orthotropic Plate with an
Elliptical Hole
The following values have been calculated using equations (46-48)
S11 
1
1

 .04847
E11 20.63
45
S 22 
1
1

 .9017
E 22 1.109
S12 
 12  .272

 .01318
E11
20.63
S 66 
1
1

 2.483
G12 .4027
2
 2S  S 66 
2S  S 66
S
  22
 k   12
  12
2S11
S11
 2S11 
2(.01318)  2.483
 

2(.04847)
2
 2(.01318)  2.483 
.9017

 
2(.04847)

 .04847
k    25.342  24.972
1  i.608  i1
2  i.608  i1
3  i7.093  i 2
4  i7.093  i 2
46
7.3 ANSYS Log File Input Command Templates
Input commands for an isotropic plate quarter plate with an elliptical notch with constant
tensile and shear moduli and Poisson’s ratio
/Title, Isotropic Plate with an Elliptical Hole
*SET,a,2.953
*SET,r,.1968
*SET,p,1000
/PREP7
ET,1,PLANE82
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,2.0613e7
MPDATA,PRXY,1,,.272
RECTNG,0,a,0,a,
CYL4,0,0,0,0,r,360
FLST,2,1,5,ORDE,1
FITEM,2,2
ARSCALE,P51X, , ,.100,1,1, ,1,1
ASBA,1,2
LESIZE,9,,,50,0.25,,,,0
LESIZE,10,,,50,0.25,,,,0
LESIZE,5,,,40,,,,,,0
!element type
!create plate
!geometry
!scaling factor to create ellipse
!specifying element
!divisons and scaling
!factors if required
SMRT,1
MSHAPE,0,2D
MSHKEY,0
AMESH,3
DL,9,3,SYMM
DL,10,3,SYMM
SFL,2,PRES,-p,
/REPLOT
47
Input commands for an orthotropic quarter plate with elliptical notch and constant E11,
E22, G12 and ν12
/Title, Orthotropic Plate with an Elliptical Hole
*SET,a,2.953
*SET,r,.1968
*SET,p,1000
*SET,E1,2.0613e7
!properties
*SET,E2,1.109249e6
*SET,G12,4.027e5
*SET,nu21,0.0146
/PREP7
ET,1,PLANE82
!element type
MP,EX,1,E1
MP,EY,1,E2
MP,NUXY,1,NU21
MP,GXY,1,G12
RECTNG,0,a,0,a,
CYL4,0,0,0,0,r,360
FLST,2,1,5,ORDE,1
FITEM,2,2
ARSCALE,P51X, , ,.500,1,1, ,1,1
ASBA,1,2
LESIZE,9,,,50,0.25,,,,0
LESIZE,10,,,50,0.25,,,,0
LESIZE,5,,,40,,,,,,0
!create plate
!geometry
!scaling factor to create ellipse
!specifying element
!divisions and scaling
!factors if required
SMRT,1
MSHAPE,0,2D
MSHKEY,0
AMESH,3
DL,9,3,SYMM
DL,10,3,SYMM
SFL,2,PRES,-p,
48
Input commands for an orthotropic quarter plate with an elliptical notch and set up for
constant E11 and variable E22, G12 and ν12
/Title, Orthotropic Plate with an Elliptical Hole
*SET,a,2.953
*SET,r,.1968
*SET,p,1000
*SET,E1,2.0613e7
*SET,E2,7.5e6
*SET,G12,(E2/E1)*8.11e6
!properites are
*SET,nu21,(E2/E1)*.272
!fcn of E2/E1
/PREP7
ET,1,PLANE82
MP,EX,1,E1
MP,EY,1,E2
MP,NUXY,1,NU21
MP,GXY,1,G12
RECTNG,0,a,0,a,
CYL4,0,0,0,0,r,360
FLST,2,1,5,ORDE,1
FITEM,2,2
ARSCALE,P51X, , ,.500,1,1, ,1,1
ASBA,1,2
LESIZE,9,,,50,0.25,,,,0
LESIZE,10,,,50,0.25,,,,0
LESIZE,5,,,40,,,,,,0
!create plate
!geometry
!scaling factor to create ellipse
!specifying element
!divisions and scaling
!factors if required
SMRT,1
MSHAPE,0,2D
MSHKEY,0
AMESH,3
DL,9,3,SYMM
DL,10,3,SYMM
SFL,2,PRES,-p,
49
Input commands for an orthotropic quarter plate with a circular notch set up constant E11
and variable E22, G12 and ν12
/Title, Orthotropic Plate with a Circular Hole
*SET,a,2.953
*SET,r,.1968
*SET,p,1000
*SET,E1,2.0613e7
*SET,E2,5e6
*SET,G12,(E2/E1)*8.11e6
!properties are
*SET,nu21,(E2/E1)*.272
!fcn of E2/E1
/PREP7
ET,1,PLANE82
MP,EX,1,E1
MP,EY,1,E2
MP,NUXY,1,NU21
MP,GXY,1,G12
RECTNG,0,a,0,a,
CYL4,0,0,0,0,r,360
!create plate
!geometry
ASBA,1,2
LESIZE,9,,,50,0.25,,,,0
LESIZE,10,,,50,0.25,,,,0
LESIZE,5,,,40,,,,,,0
!specifying element
!divisions and scaling
!factors if required
SMRT,1
MSHAPE,0,2D
MSHKEY,0
AMESH,3
DL,9,3,SYMM
DL,10,3,SYMM
SFL,2,PRES,-p,
50
7.4 Supplementary ANSYS Contour Plots for an Orthotropic Plate
with an Elliptical Notch
Figure 28 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 2) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
Figure 29 – Stress concentration for an orthotropic plate with an elliptical ( ba = 3) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
51
Figure 30 – Stress concentration for an orthotropic plate with an elliptical ( ba = 4) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
Figure 31 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 5) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
52
Figure 32 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 6) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
Figure 33 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 7) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
53
Figure 34 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 8) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
Figure 35 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 9) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
54
Figure 36 – Stress concentration for an orthotropic plate with an elliptical (
a
b
= 10) notch. Results
generated using ANSYS PLANE82 elements and meshing parameters in Figures 24 & 25.
55