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NON-LINEAR AND DYNAMIC EXTENSION OF CONSISTENT LAMINATED SHELL ELEMENT AND
APPLICATION TO FRP CHIMNEYS
by
Albert Ernest Mikhail
Department of Civil and Environment Engineering
Faculty of Engineering Science
Submitted in partial fuKdlment
of the requirements for the degree of
Master of Engineering Science
Faculty of Graduate Studies
The University of Western Ontario
London, Ontario
danuary 1999
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ABSTRACT
Laminated composites are increasingly being used in many structural applications
where high resistance to corrosion and chernical attacks and/or high strength to weight
ratio is required. A Consistent Laminated shell element has been recently formulated by
Koziey and used in the linear analysis of laminated shef 1 structures. This element has a
major advantage of being free kom the locking phenornenon associated with
isoparametric shell elements. In this study, the formulation of the Consistent Laminated
shell element is extended to include both large displacement static and dynarnic analysis.
A detailed presentation of the nonlinear stiffriess matrix, unbalanced load vector, and the
consistent mass matrix as well as a simple approach for coding these matrices is included
in this study. Verification of the large displacement formulation reveals the excellent
performance of the element.
A nonlinear materid mode1 for FEW laminates is added to the large displacernent
static formulation of the Consistent Laminated shell element. The model is used to study
the nonlinear static behavior of FW chimneys subjected to both thermal and wind Ioads.
An assessrnent for the level of safety against strength and instability failure, provided by
a simplified design procedure previously developed by Awad, is then conducted using the
nonlinear Consistent Laminated shell element model. The results obtained provide an
indication about the adequacy of this simplified procedwe.
Keywords: Nonlinear, Dynamic, Large displacernents, Composite, Laminated Shell
Element, FRP chimneys, Cracks, Finite element, Themal load, Wind load.
TQ MY DEAR PARENTS
I wish to express my sincere appreciation to my s u p e ~ s o r Dr. A.A EI Darnatty
for his valuable guidance and encouragement throughout the course of this research and
the preparation of this thesis. But most of all, his energy and dedication to the research
which seemed to renew my interest and drive towards my goal. Without his continuous
support and brotherly advice, this study could not be completed.
The valuable support of Dr. B. I. Vickery and the financial support of the
University of Western Ontario are also sincerely appreciated. Without their support, my
involvement in this study would not have been possible.
Thanks also to my family, especially my parents, who always believed in me.
Their continuous encouragement and support made it possible for me to get to where 1 am
now.
TABLE OF CONTENTS
Page
CERTIFICATE OF EXAMINATION
ABSTRACT
ACKNOWLEDGMENTS
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
LIST OF SYMBOLS
CHAPTER 1 INTRODUCTION
1.1 Introduction
1.2 Laminated Shell and Plate Theories
1 2 . 1 Mechanics of Laminated Structures
1 -2.2 Classical Laminated Plate Theory
1-2.3 First-Order Laminated Plate Theory
1.2.4 Third-Order Laminated Plate Theory
1.2.5 Layer-Wise Theories
1.3 Shear Locking Phenomenon and Spurious Displacement Modes
1.4 Consistent Thick Shell Element
1.5 Fiber Reinforced Plastic chimneys
xiv
1.6 Objectives and Scope 20
CHAPTER 2 FORMULATION OF THE CONSISTENT LAMINATED SHELL
ELEMENT
2.1 Introduction 24
2.2 Consistent Laminated Shell Element Formulation 25
2.2.1 Coordinate S ystems and Geometry 25
2.2.2 Displacement Field 27
2.3 Strain-Displacement and Stress-Strain Relationships 32
2.4 Elernent Stiffhess Matrix 36
2.5 Kinematic Constraints 37
2.6 N m c a l example 40
CHAPTER 3 LARGE DISPLACEMENT STATIC AND DYNAMIC FORMULATION
OF THE CONSISTENT LAMiNATED S E L L ELEMENT
3.1 Introduction 46
3.2 Large Displacement Formulation 47
3 2.1 The Incremental Displacement Field 47
3 -2.2 Expressions for Nonlinear Stiffhess Matrix and
UnbaIanced Load Vector 49
3.2.3 Evaluation of the Initial Strains 55
3.2.4 Solution Technique for the Nonlinear Static Analysis 61
vii
3.3 Venfication of the large displacement formulation
3 -3.1 Simply Supported Isotropie Plate under Uniform Transverse Load
3 -3.2 Clamped Square Plate under Uniforrn Transverse Load
3 -3.3 : Anti-Symmetrical Cylindncal Panel under Central Point Load
3 -4 Dynamic formulation
3 -4.1 Derivation of The Consistent Mass Matrix
3 -4.2 Free Vibration Analysis Using the Consistent
Laminated SheH Element
3 -4.2.1 The Free Vibration Formulation
3 -4.2.2 Verification of the Free Vibration Formulation
3 -4.3 Nonlinear Time History Analysis
3 -4.3.1 Formulation of the Nonlinear Time History Analysis
3 -4.3 -2 Solution Technique for the Nonlinear Time History Analysis
3 -4.3 -3 Verification of the Dynarnic Large Displacement Formulation
3.5 Conclusion
CHAPTER 4 NON-LINEAR ANALYSIS OF FRP C H I M N E Y S USING THE
CONSISTENT LAMINATED S E L L ELEMENT
4.1 Introduction
4.2 Simple design approach by Awad
4.3 Thermal load formulation
4-3.1 Formulation details
4.3 -2 Verification example
4.4 Material Non-linearity formulation
4.5 Typical Behavior of FRP Chimneys under Wind and Thermal Load
4.5.1 Description of the Analysis
4-52 Discussion of the Results of the Anaiysis
4.5.2.1 Deflection Shapes Near the Base Due to Different
Thermal Loading Conditions
4.5.2.2 Stresses Resulting from Load Case Bhavl
4.5.2.3 Stresses Resulting fiorn Load Case Bhav2
4 5 2 . 4 Stresses Resulting from Load Case Bhav3
4.5.2.5 Stresses Resulting from Load Case Bhav4
4.5.2.6 Stresses Resulting fiom Load Case Bhav5 and Bhav6
4.5.2.7 Stresses Resuiting fiom Load Cases Bhav7 and Bhav8
4.6 Assessment for the Design Procedure Developed by Awad
4.6.1 Assessment for Strength
4.6.2 Assessment for hstability Failure
4.7 Conclusion
CHAPTER 5 CONCLUSIONS
5.1 Introduction
5.2 Summary and Conclusions
5.2.1 Nonlinear Extension of the Consistent Laminated Shell Element
5.2.2 Nonlinear Analysis of FRP Chirnneys
5.3 Recommendations for Future Research
REFERENCES
APPENDIX 1 Interpolation Functions for Consistent Laminated
ShelI Element
APPENDIX II Construction of orthogonal bais
APPENDIX III Derivation of Lamina through Thickness Interpolation
Functions M,,, M,,, M,, and M,, 151
APPENDIX IV Consistent Laminated ShelI Element S train-Disp lacernent
Matrix p'] 155
VITA 157
LIST OF FIGURES
CHAPTER ONE
1.1 A typical laminated plate
CHAPTER TWO
2.1 Consistent Laminated Shell Element Coordinate System and
Nodal Degrees of Freedom 42
2.2 Material Coordinate System of a Typical Laminated Shell 43
2.3 Dimensions and Material Properties for Laminated Fiber Reinfurced
Composite Plates, Koziey (1 993) 44
2.4 NonnaI Stress o, through Thickness of 9 Ply Laminate, Koziey (1 993) 45
2.5 Shear Stress T, through Thickness of 9 Ply Laminate, Koziey ( 1 993) 45
CHAPTER THREE
3.1 Typical Clamped Plate (2x2) 8 Elements
3.2 Load-Deflection Curve for an Isotropic Square Plate
3.3 Load-Deflection Curve for a Laminated Clamped Square Plate
3.4 Typical Cylindrical Panel
3.5 Load-Deflection Curve for a Cylindrical Panel
3.6 Load-Deflection Curve for a Cross-Ply (0/90) Cylindncal Panel
CHAPTER FOUR
4.1 A typical through Thickness Stress Distribution for an Angle-Ply Laminate
4.2 Fibers Configuration for Adjacent Angle-Ply Iayers
4.3 A Typical Finite Elernent Mesh used in the Analysis
4.4 Coordinate Systems used in the Analysis
4.5 A Section through the Wall Thickness of the Chimney used in the Analysis
4.6 Deflection Shape at the Base of the Chimney for
Di fferent Thermal Load Cases
4.7 Hoop Stress Distribution at the Bottom Sm of the Chimney for Bhavl
4.8 Meridional Stress Distribution at the Bottom 5rn of the Chimney for Bhavl
4.9 Hoop Stress Distribution at the Bottom 5m of the Chimney for BhavZ
4.10 Meridional Stress Distribution at the Bottom 5m of the Chirnney for Bhav2
4.1 1 Hoop Stress Distribution at the Bottom 5rn of the Chimney for Bhav3
4.12 Mendional Stress Distribution at the Bottom 5m of the Chirnney for Bhav3
4.13 Hoop Stress Distribution at the Bottom 5m of the Chimney for Bhav4
4.14 Mendional Stress Distribution at the Bottom Sm of the Chimney for Bhav4
4.15 Deflection Shape at the Base of the Chimney Due to Wind Load
4.16 Hoop Stress Distribution at the Bottom 5m of the Chimney
for Bhav5 & Bhav6
4.17 Meridional Stress Distribution at the Bottom Sm of the Chimney
for Bhav5 & Bhav6
xii
4.18 Hoop Stress Distribution at the Bottom of the Lee Side Ext. Surface of the
Chimney 134
4.19 Meridional Stress Distribution at the Bottom of the Lee Side Ext. Surface of the
Chimney 134
4.20 Hoop Stress Distribution at the Bottom of the Wind Side h t . Surface of the
Chimney 135
4.2 1 Meridional Stress Distribution at the Bottom of the Wind Side int. Surface
of the Chimney 135
4.22 Typical Design Curve for FRP Chimneys According to Awad (1 998) 136
4.23 Factor of Safety (FSJ for Chimneys Designed for R = 2 136
4.24 Factor of Safety (FS,) for Chimneys Designed for R = 3 137
4.25 Factor of Safety @.SI) for Chimneys Designed for R = 4 137
4.26 Typical Shear Stress (03 Distribution at the Base of a Chimney
under Wind Load 138
4.27 Buckling Modes at the Base of the Chimney for Different L/D Ranges 138
LIST OF TABLES
CHAPTER THREE
3.1 Dimensionless Fundamental Frequency Z = o , / p h ' / ~ z of SirnpIy
Supported Cross-Ply Square Plate. 76
CHAPTER FOUR
4.1 Non-Dimensional Central Denec tion (w *) of a Cross-Ply C ylindrical
Panel under Thermal Load
4.2 Surnmary of the Load Cases used in Behavior Study
4.3 Geometry of Chirnneys used in the Strength Assessment
4.4 Results of the Strength Assessment
4.5 Results ofthe Bucklinghalysis
4.6 Results of the Revised BuckIing Analysis
xiv
LIST OF SYMBOLS
Al1 symbols are defhed at their h t appearance. The principal symbols used are listed
belotv:
Displacement degrees of keedorn.
Rotations degrees of fieedom.
Cubic and Quadratic shape hctions, respectively.
Linear and Cubic through thickness shape functions, respectively.
Through thickness shape functions.
Elasticity matrix in local coordinate system.
2nd Piola-Kirchhoff stress tensor.
Linear part of the incremental Green-Lagrangian strain tensor.
Nonlinear part O f the incremental Green-Lagrangian strain tensor.
Initial thermal strain rnatrix of the L" layer in the local
coordinate system.
Elastic rnoduli in the material directions and the in-plane shear
modulus of orthotropic lamina.
Transverse shear moduli.
The sum of the linear and the initial strain stiffness matrices.
The initial stress stiffness rnatrix.
(F) The unbdanced load vector.
(RI The externd load vector.
h~ The mass density and the thickness of the L" layer, respectively.
[Ms 1 The consistent mass matrix.
CHAPTER ONE
INTRODUCTION
1-1 INTRODUCTION
Laminated plastic composites are finding increasing uses in many engineering
fields. The non-structural applications of these composites include appliances,
electronics, medical equipment and sporting goods. They have also been used for a
number of decades in the aerospace industry. In the past two decades, laminated
composites, especially fiber reinforced plastics (FRP), have shown a wide spread of
applications in the construction industry. These include the construction of FRP
chimneys, tanks, bridges and industrial buildings. FRP matenals are also widely used in
the retrofitting and strengthening of existing buildings.
The main advantages of laminated composites can be stated as:
- High corrosion and chemical resistance.
- High strength to weight and stifiess to weight ratios.
- Flexibility in tailoring the mechanical properties to suit the requirements of a certain
application.
- High fatigue strength.
- Ability to serve in a wide variety of thermal conditions.
This thesis includes two studies related to laminated structures. The first study
involves an extension of the formulation of a previously developed larninated finite
element mode1 to include non-linear static as well as dynarnic analysis. The mode1 is
based on a consistent shell element that was developed by Koziey and Mina (1997), and
then extended to a laminated formulation by Koziey (1993). This finite element mode1
which is referred to as the "Consistent Laminated Shell elernent" bas the advantage of
being free from the Locking phenomenon as well as providing accurate prediction for
stresses as will be discussed later. In the second study, this mode1 is used to investigate
the non-linear behavior of FRP chùnneys under both thermal and wind loads. A Iiterature
review of both studies is provided in this chapter.
1-2 LAMINATED S m L L AND PLATE THEORIIES
In light of the need to analyze laminated plate and shell structures, many
laminated shell elements have been developed in the last three decades. In general, shell
element models can be grouped under two wide categories (Liu and Surana 1993):
Displacement based formulations; and stress, mixed and hybrid formulations. The
Laminated Consistent shell element falls under the displacement based category.
Therefore, oniy a review of displacement based sheIl elements is provided in this chapter.
This section starts by providing a brief description of the rnechanics of laminated
structures. This is followed by a presentation of various theones used in the finite element
formulations of laminated structures, starting with the classicai laminated plate theory and
advancing up to the recent refined theories of larninated composites.
1-2-1 Mechanics of Larninated Structures
Consider the laminated plate s h o w in Fig, (1.1). The plate consists of a number
of layers having different orthotropic mechanical properties. At any point within the
plate, three normal strains E,, s, and s,, one in-plane shear strain y,, and two transverse
shear strains y, and y, are defined. The corresponding stresses are o,, ofl, a,, aKY, a,
and o,. The first three components represent normal stresses in the -Y, y and s directions,
respectively. Meanwhile, cr, is the in-plane shear stress and both a, and an are the
transverse (inter-laminar) shear stresses. Assuming a plane stress behavior of the plate,
the stress component o, can be neglected.
For fuily bonded layers, the kinematics between two layers at inter-laminar
locations can be described as, (see Jones 1975):
1) Compatibility of displacement components u, v and W.
2) Continuity of normal strains E, and E, as well as the in-plane shear strain y,,.
3) Discontinuity of transverse shear strains y, and y,.
The last condition allows a continuity of transverse shear stresses cr, and O,
(which is necessary for equilibrium) to be achieved for layers having different shear
modulus (G). A finite element mode1 that duplicates the true physical behavior of
laminated plate and shell structures should have the capability to sat ise the above three
conditions.
1-2-2 Classical Laminated PIate Theory
This simplified theory falls under the equivalent single layer laminate theories
which is characterized that one continuous funcrion represents the through thickness
displacement field. The formulation is based on the Kirchhoff hypothesis which includes
the following assurnptions:
1. Straight lines perpendicular to the mid-surface remah straight after deformation. This
implies that the transverse displacement is independent of the transverse coordinate
(Through thickness coordinate).
2. Straight lines perpendicular to the mid-surface remain perpendicular aRer
deformation. This implies that the transverse shear strains are equal to zero; Le. E, =
= O.
3. Straight lines perpendicular to the mid-surface maintain the same length &et-
deformation. This implies that the transverse normal strain is zero; i-e. E, = 0.
These assurnptions reduce the strain components of the strain vector to (E,, y,J.
The previous assurnptions lead to the following displacement field for a typical
sheIl element mode1 based on the CLPT:
where (zc,, v,, w,) are the displacement components at the mid-surface of the shell, in the
x, y and z directions, respectively;
(u, V, W) are the displacement components at any point of the shell in the x, y and z
directions, respectively;
x and y are two Iocal coordinates tangent to the mid-surface, whi1e z is the
coordinate perpendicular to the surface, measured from the mid-surface.
This assumed displacement field reduces the problem to the determination of the
mid-surface displacements of the shell. Once evaluated, the mid-surface displacements
c m be used to calculate the displacements at any point in the shell using Equations (1.1).
This method is used extensively to analyze thin isotropic plates. However, due to
the fact that transverse shear deformations are neglected, considerable errors cm arise
when this method is used in analyzing laminated composites. The transverse shear moduIi
of some of the modern advanced lamlnated composites tend to be very low compared to
the in-plane axial moduli. As such, transverse shear deformations can be important for
such materials.
1-2-3 First-Order Laminated Plate Theorv
This method is based on the First-Order shear deformation theory (FSDT), which
is aIso known as the Mindlin's theory. Again, this theory falIs under the classification of
equivalent single layer laminate theones. It employs the KirchhoE hypothesis descnbed
in subsection (1-2-2) with the exception that the second assumption "Straight iines
perpendicular to the mid-surface remain perpendicular after deformation" is waived. As
such, the transverse shear strains (y, and y d are included in the formulation.
The displacement field for shell elements formulated based on this theory are
govemed by the fol1owing relations:
where (u,, v,, w,, u, v and w ) are as defined in the previous section;
and 4,. are the rotations of a vector normal to the surface (transverse normal)
about the y and x axes, respectively.
This displacement field reduces a 3-D shell problern to the determination of the
displacements of the mid-surface of the shell as weli as the rotations of the normal vector.
Five strain cornponents (E,, E ~ , y,., yu, y d are considered for such analysis. n e in-plane
strains (E,, E, . - and y,,) vary linearly through the thickness and the transverse shear strains
(y, and yLJ are constant within the thickness of the laminate. This means that this
formulation predicts transverse shear stresses that are constant through the thickness of
the laminate. However, fiom basic theory of mechanics, one would expect that such
stresses have at Ieast a quadratic variation through the thichess (definitely quadratic for
isotropie stnictures and can have a higher order for Iaminated structures). This error is
compensated for by using a shear correction coefficient (K) that is multiplied by the
calculated transverse shear stresses.
This method provides good results for the global response of a larninated structure
(Deflection, buckling load and natural fiequency). However it fails to predict accuratel y
the through thickness stresses specially at areas of discontinuity such as the boundaries of
structures. Pipes and P agano (2970) have shown that near the boundaries of larninated
structures, the transverse stresses may increase significantly. This has been referred to as
the "boundary-layer effect". These areas can be the starting points for delarnination in the
structure. Therefore, a mode1 which can accurately predict transverse shear stresses is
needed to detect the delarnination failure. To overcome this problem, finite elements
based on higher order theories were developed. Reddy (1990) and Kapania and Raciti
(1989) provide a review of the different higher order theories used in the analysis of
laminated shell structures.
1-2-4 Third-Order Laminated Plate Theorv
This theory, which was developed by Reddy (1984), is descnbed in this section as
a sample of the higher-order theories. Again, the theory falls under the equivalent single
Iayer laminate category. It expands the displacements as cubic fünction of the through
thickness coordinate. Thus the only assumption included in this formulation is that
"Straight lines perpendicular to the mid-surface maintain the same lena@ afier
deformation". This irnplies that the transverse normal seah is equal to zero, Le. E, = 0.
The displacernent field for the finite element formulations based on the Third-
order Shear Deformation Theory (TSDT) is governed by the following relations:
where (uo, v,, w,, u, v, w, qix, and z) are as defined in the previous section;
h is the total thickness of the laminate.
This displacement field provides a quadratic distribution of the transverse shear
strains (y, and yJ and a linear distribution of the in-plane strains (E,, and y,!) throua
the thickness of the Iarninate. It also satisfies the conditions that the transverse shear
stresses at the top and bottom surfaces of the larninate are equal to zero. It can be seen
that Equation (1.3) includes the first derivative of the transverse displacement w,. As
such, a finite element formulation using this displacement field requires a Cl-continuity.
Such a requirement complicates the formulation of the element especially if a non-linear
formulation is attempted.
It is clear that this method provides a better representation for the transverse shear
stresses compared to the FSDT method. However it has the disadvantage, like al1 the
equivalent single layer laminate theones, of describing the entire laminate by a single
polynornial. This results in the continuity of the displacements as well as the first and
higher derivatives of the displacements at the interface between laminas. As such,
continuous transverse shear strains (y, and y& are forced between layers and a
discontinuity of transverse shear stresses (on and C Y ~ would occur at these locations if
the layers have different shear moduli (G). This behavior violates the conditions of
equilibnum as discussed in subsection (1-2-1). For thin laminates the error arïsing from
the discontinuity of transverse shear stresses is usually negligible. However for thick
Iarninates, such an error is significant and c m not be ignored. In general, finite element
formulations based on this theory can not predict delamination of layen. In order to be
able to predict such failure mode, a layer-wise approach was suggested by many authors.
1-2-5 Laver-Wise Theories
It is clear frorn the previous discussion that using one continuous fùnction for the
through thickness displacement fails to duplicate the tnie physical behavior of laminated
composites. In the layer-wise theories, the displacement fields are chosen in such a way
that continuity of displacements and discontinuity of the derivatives of these
displacements are provided between layers. As such, finite elements based on these
theones are not constrained to have continuous transverse shear strains and thus, the finite
element approximation can lead to continuous transverse shear stresses between different
layers.
According to Reddy (1997), the displacement based layer-wise theories can be
divided into the following two categories:
1) The partial layer-wise theories where a layer-wise approximation is only used for the
in-plane displacements (u and v). This formulation provides discrete transverse shear
strains (y, and y d through various Iayers.
2) The full layer-wise theories where a layer-wise approximation is used for al1 three
displacement components. This formulation provides discrete distribution for both the
transverse normal strain (EA and the transverse shear strains (y, and y a through
different layers. It should be noted that, unlike other laminated theories, the full layer-
wise theory includes the transverse normal strain (&a and the transverse normal stress
(c) in its formulation. The mechanics of laminated structures implies that a finite
element approximation should have the fkeedom of predicting discontinuous
transverse normal strains behveen Iayers. This can lead to a continuous distribution of
transverse normal stresses between layers having different properties and therefore
equilibrium can be satisfied. Finite elements based on the full layer-wise theory can
predict such a behavior.
A finite element formulation based on the full layer-wise theory, developed by
Reddy (1987), is presented in this subsection. In this formulation, a typical element is
subdivided into layer-like divisions through the thickness of the laminate. In each of
those subdivisions, the 2-D mesh (in the .Y-y coordinate) of the structure is repeated in the
thickness direction creating a 3-D discretisation. The number of subdivisions can be
equd, greater or Iess than the number of materid layers existing withui the thickness of
the laminate. However, if the number of subdivisions is Iess than the nurnber of layers,
each subdivision is represented as an equivalent single layer. The formulation of this type
of elements involves two levels of interpolation. A 2-D interpolation set for the
displacernent function (U, VI and W,) is first achieved using nodal degrees of freedorn.
Those interpolation sets c m have various forms depending on the type of the 2-D parent
element. The displacement functions (Ul, V, and WI) are then interpolated through the
thickness using the interpolation funciions (a1 and Y ') which can have a linear, quadratic
or a higher-order variation. Thus the displacement field of the laminate can be written as:
W ( X . y, Z) = C w1 ( x , y ) ~ (z) I=1
where ( U , VI, WI) are the displacement components at the node 1 calculated f?om the
interpolation of the displacements in the x-y plane;
a' are the gIobal interpolation functions for the discretization of the in-plane
displacements (U,, V,) through the thickness coordinate z;
Y ' are the global interpolation functions for the discretization of the transverse
displacement (WI) through the thickness coordinate z;
N is the number of nodes for the discretization of the in-plane displacements
through the thickness coordinate z;
M is the nunber of nodes for the discretization of the transverse dispiacement
through the thichess coordinate z.
The interpolation Functions a' and Y' are chosen in such a way that they are
continuous through subdivision boundaries. This satisfies the requirement that
displacements are continuous through the thickness of the laminate. On the other hand,
because a separate fùnction is used for each subdivision, the derivatives of the functions
can be discontinuous. This allows the transverse strains to be discontinuous at the
subdivision boundaries and therefore, a possibility of obtaining continuous transverse
stresses between layers having different properties. The use of separate interpolation
Functions for the transverse displacement permits the modeling of inextensibility of the
transverse normals. This c m be achieved by setting M=l and Y'=l.
In terms of the size of the problem, a full layer-wise finite element model is
similar to a 3-D displacement h i t e element modeI. As such, for structures with a large
number of layers, the use of such an element becomes unpractical. However, the full
layer-wise model has the advantage over 3-D formulations of requiring only a 2-D input
data (Le. a 2-D rnesh generation c m model the problem). According to Reddy (l997), this
type of element is computationally more efficient than a conventional 3-D finite element.
However, layer-wise theones have the disadvantage of suffering from the locking
phenomenon and spurious shear modes when used to mode1 thin structures (Reddy 1997).
In the next section, the locking phenomenon and spurious shear modes problems are
exp lained.
1-3 SHEAR LOCKING f HENOMENON AND SPURIOUS DISPLACEMENT
MODES
Shell elements (both single layer and larninated) based on the First-Order shear
deformation theory (FSDT) and the layer-wise theones suffered from a phenornenon
known as the Shear Locking phenomenon. This was encountered in the analysis of thin
plate and shell problems (generally when the length to thickness ratio exceeds 100). Very
stiff solutions resulted fkom these analyses with unpredicted distribution of transverse
shear stresses (spurious shear modes). For such thin shell structures, the transverse shear
strains are required to approach zero. The locking phenomenon results fiom the inability
of those elements to accurately mode1 a bending behavior of the shells under a state of
zero transverse shear strains.
A number of studies attempted to provide explanations and solutions for the
Iocking phenomenon. Arnong these solutions are the reduced integration and the selective
integration techniques. The reduced integration technique is based on choosing a lower
order integration scheme than what iç necessary for the numerical integration of the
stifiess matrix. This results in a singular shear stif iess matrix that yields zero transverse
shear stresses. However this method does not work properly for al1 types of boundary
conditions (eg. for clarnped plates). As for the seiective integration technique, it involves
applying full integration to the bending components of the stiffhess maû-ix and reduced
integration to the transverse shear stress components. Again this method solves only
special cases and its use can not be generaiized for al1 problems. In generai, the previous
methods often produce results that have an excess number of zero eigenvalues, which
produces spunous deformation modes.
Koziey and Mina (1997) have attempted to explain the cause of the locking
phenornenon and the associated spurious shear modes. According to their explmation. the
shear locking is a numericai problem resulting fiom the use of the sarne order of
interpolation for both the transverse displacement (w) and the normal vector rotations ( A
and 4,). In their study, Koziey and Mina (1 997) have proven that such formulation is
inconsistent and create spurious shear modes. They have proposed a solution for the
problern based on employing cubic interpolation functions for the displacements and
quadratic interpolation fùnctions for the rotations. This idea led to the development of the
Consistent Thick Shell element, which was then extended to a larninated formulation
known as the Consistent Larninated Shell element by Koziey (1993). A brief description
of the Consistent Thick Shell element showing the advantages of this element is provided
in the next subsection.
1-4 CONSISTENT THICK SHELL ELEMENT
This shell element, developed by Koziey and Mirza (1997), can be classified
under the category of elements based on high order theories. The formulation of the
element includes only the assurnption of plane stress conditions (Le. 0, = O). The element
has two major advantages:
1) The elernent's formulation is fiee fkom the spurîous shear modes and the associated
locking phenomenon. This feature is very important in analyzing thin plate and shell
prob lems.
2) The element's formulation includes special rotational degrees of fieedom that lead to
cubic variation of the through thickness displacements. As such, quadratic transverse
shear strain and shear stress distributions c m be predicted by the element. This
feature is very useful in anaiyzing thick plate and shell structures consisting of a
single layer.
The elernent's formulation includes 3 displacernent degrees of fieedom (u, v and
w ) directed dong the globai x, y and z axes, and 4 rotational degrees of fieedom (a, ,O, (
and y/). Both a and # are about a local axis y : and P and yare about a local axis xt The
two axes x' and y' are tangent to the mid-surface of the shell as s h o w in Fig. (2.1).
Rotations a and p are rnultiplied by a function varying linearly through the thickness of
the shell. Meanwhile, q5 and p are rnultiplied by a function that varies cubically through
the sarne thichess. The approximation of the degrees of fieedom within the mid-surface
of the shell was carefuIly chosen to provide cubic variation for the displacement degrees
of freedom and quadratic variation for the rotational degrees of freedom. This approach
leads to a consistent formulation that eliminates the qmrious shear modes and the
associated locking phenornenon as descnbed by Koziey and Mina (1997). As shown in
Fig. (2.1 ), the element has a triangular shape and Uicludes 13 nodes. Ten of these nodes (3
corner nodes, 6 one-third-side nodes and one central) are used to achieve the cubic
interpolations for the displacements (u, v and w) and six nodes (3 corner nodes and 3 mid-
side nodes) are employed to achieve the quadratic for the rotations (a, P, @ and y).
Thus, the displacement field for the Consistent Thick shell element is written as:
where (u, v and w) are the components of the displacement vector at any point within the
shell in the x, y and z directions, respectively;
v,, and N , are cubic and quadratic shape functions, respectively, varying within
the surface of the shell, (see Appendix i).
] is a matrix vector relating the local coordinates (x' and y? to the global
coordinates (x, y and z), (see Appendùc II).
M, and M2 are the lùiear and cubic through thickness shape fûnctions,
respectively, and are related to the thiclaess of the shell C H ) and the through
thickness coordinate (t) (normal to the surface) as follows:
This element presents an excellent tool in analyzing single layer plate and shell
problems. The element has been modified by Koziey (1993) to the analysis of larninated
shell structures. This Consistent Laminated shell element rnodel, which is extended in
this study to include non-linear static and dynamic behavior, is fully described in the next
chapter.
1-5 FIBER REDWORCED PLASTIC CHIMNEYS
The non-linear version of the Consistent Laminated shell element, which is
formulated in Chapter Three of this Thesis, is used to study the non-Iinear behavior of
Fiber Reinforced Plastic (Fm) chimneys. During the past two decades, a nurnber of FRP
chimneys have been constructed in North Arnerica and in various places around the
world. The spread of FRP as a construction matenal for chimneys is due to the superb
corrosion and acid resistance of such materials.
FRP chimneys are usually constructed Eom a large number of layers. The typical
thickness of a layer varies between 0.8 and 1 .O mm. Due to the ease of construction, the
angle-ply configuration (+ 8) is usuaily used in FRP stacks. For such a configuration. the
fibers of a certain layer are oriented by an angle 0 (with a specific direction), while the
fibers of the two adjacent layers are oriented by an angle -0 (with the sarne direction). A
common composite used in FRP chimneys consists of a Vinyl ester resin reinforced by E-
g las fibers. The fibers content usually varies between 60 - 70% (based on weight).
No national code currently exists for the design of FRP chimneys. However, as
stated by Pritchard (1996), an attempt is being undertaken by the International Cornmittee
on Industrial Chimneys CICIND (1988) to develop a code for such composite structures.
Studies conducted on FRP chimneys are very rare in the literature. Joseph M.
Plecnik et. el. (1984) have reported the design of a 170 fi. (5 1.8 m) high free-standing
FRP c h e y located in Washington, USA.
Recently Awad (1998) has conducted an extensive analyticai investigation for the
response of FRP chimneys to both temperature and wind loads. In his study, Awad
(1998) has shown that cracking of the resin (in the direction perpendicular to the fibers) is
expected to happen at localized parts in the bottom of ETW chirnneys due to thermal
loads. In the same study, Awad (1998) has developed a simplified computer code to be
used in the design of FRP chimneys subjected to both thermal and wind loads. This
simplified code, described in details in Chapter Four, uses the classical lamination theory
to obtain an equivalent orthotropic material of the chimney FRP section. The computer
code also includes a number of assumptions that need venfication. In the current study.
the non-linear version of the Larninated Consistent shell element is used to v e n e some
aspects of the design approach provided by this simplified computer code.
1-6 OBjECTIVES AND SCOPE
The objectives of the present study are as follows:
1) Extend the formulation of the Consistent Laminated Shell element to include large
deformation non-linear analysis.
2 ) Extend the formulation of the Consistent Larninated SheIl element to include
dynarnic analysis.
3) Use the Consistent Larninated Shell element to study the behavior of Fiber
Reinforced Plastic (Fm) chimneys under thermal and wind load taking large
deformation and material cracking into consideration.
4) Verie and check the range of applicability of the simplified computer code developed
by Awad (1 998).
Ln Chapter Two, a detailed description for the Consistent Larninated Shell
element, previously developed by Koziey (1993), is provided. The accuracy of the
element in predicting various stress components in laminated structures is demonstrated.
In Chapter Three, the formulation of the Consistent Laminated Shell element is
extended to include the effect of large deformations. The non-linear formulation is carrïed
out incrementally using the Total Lagrangian approach. Analyses of a number of plate
and shell problems are conducted using the non-linear version of the Consistent
Laminated Shell element and results of the analyses are compared to those available in
the literature. In the same Chapter, the formulation of the Consistent Lamùiated Shell
element is extended to account for dynarnic analysis. Both fiee vibration and non-linear
time history dynamic analyses are included. Again, the formulation is verified by
anaiyzing a nurnber of plate and shell problems and cornparhg the result of the analyses
with those available in the literature.
In Chapter Four, the large displacement formulation of the Consistent Larninated
Shell element is extended to include the matenal non-linearity resulting Eom cracking of
resin in a FRP structure. Thermal load vectors due to temperature variation is also
included in the non-linear model. The behavior of FRP chimneys under thermal and wind
loads is studied using the non-linear Consistent Laminated Shell element model.
Cracking and buckling of FRP chimneys are considered in the analysis. Finally, the non-
linear finite element mode1 is used to ven@ the simplified design code previously
developed by Awad (1998).
Fuially, in chapter Five, the conclusions drawn Eom this study and the
recommendations for the fùture research are presented.
Fin 1.1 : A tvpical laminated plate
CHAPTER TWO
F O ~ A T r O N OF THE
CONSISTENT LAMINATED SHELL ELEMENT
2-1 INTRODUCTION
The discussion provided in the previous chapter has demonstrated that iso-
pararnetric shell elements based on both the k t order and the Layer-Wise theones suffer
from the locking phenomenon. The use of these elements in analyzing thin plate and shell
problems should be considered with caution. In the same chapter, a consistent sub-
parametric shell elernent, which was developed by Koziey and Mina (1997), is presented.
This element has the major advantage of being fkee fkom the spunous shear modes that
cause the locking phenomenon. This feature is extremely usefirl when the element is used
to analyze thin plate and shelI problems.
The Consistent Thick Shell element has been extended by Koziey (1 993) to a
Iaminated formulation. Similar to its parent element, the Consistent Larninated Shell
element has the advantage of being free fiom the "lockuig phenornena" and the spurious
shear modes. Meanwhile, the element has the capability of predicting strain and stress
distributions that are consistent with the true physical behavior of laminated shell
structures. Since, one of the objectives of this thesis is to extend the formulation of the
consistent larninated shell element to include large disp lacement static and dparnic
analysis, it was decided that it is suitable to provide the reader with a brief description of
the consistent larninated shell element. For more details the reader is referred to Koziey
(1993).
2-2 CONSISTENT LAMINATED SHELL ELEMENT FORMUtATION
2-2-1 Coordinate Systems and Geometrv
Four coordinate systerns are used in the formulation of the Consistent Larninated
Shell Element. These coordinate systems are shown in Fig. (2.1) and are defined as
foIlows:
1) The global Cartesian coordinates as x, y and z.
2) The curvilinear coordinates r, s, t and t,, where r and s are tangent to the mid-surface
and t is perpendicular to the surface and ranges fkom -1 to +l for the bottom and the
top surface of the laminate, respectively (see Fig. 2.1). t, is a local coordinate for each
layer (lamina) and ranges fkom -1 to +1 for the bottom and the top of the lamina,
respectively. t, has the sarne orientation as the coordinate t. (See Fig. 2.1 ).
3) The local Cartesian coordinates x : y' and z ' ; This set of axes is used to define the
local strains and stresses. The local axis z' is always nomal to the surface, while x'
and y'are located in a plane tangent to the surface. The directions of these axes Vary
fiom one point to anorher on the sheli.
4) The material coordinates (1, 2 and 3) which are shown in Fig. (2.2). Axes 1 and 2 are
located in a plane tangent to the surface. For angle-ply laminated structures, a i s 1 is
directed along the fiber direction while axis 2 is perpendicular to the fiber direction.
Axis 3 is perpendicular to the surface (i.e. coinciding with the z'axis).
The element has 13 nodes (3 corner nodes, 3 mid-side nodes, 6 one-third-side
nodes and one central node) as shown in Fig. 2.1. In the elernent formulation, the
displacement degrees of freedom of the nodes are interpolated cubically, while the
rotational degrees of fieedom are interpolated quadratically within the plane of the
element. The cubic interpolation is achieved using cubic shape Functions rn and the
value of the variables at ten nodes (3 corner nodes, 6 one-third-side nodes and a central
node). Meanwhile, quadratic interpolation is achieved using the quadratic shape hct ions
N, and the value of the variable at six nodes (3 corner nodes and 3 mid-side nodes). The
shape hc t ions w, and N , are given in Appendix 1. The geomew of the element is
interpolated quadratically within the surface of the element and linearly within its
thickness. As such the global coordinates at any point within the element is calculated
using the following equation:
where N, are the quadratic interpolation functions. (see Appendix 1);
.r,, y,, and r, are the coordinates of the nh node (at the mid surface of the shell, i.e.
at t = 0);
V, is the unit vector perpendicular to the surface rnultiplied by the thickness of
the eIernent at the n* node.
Note that only 6 nodes (the comer and the rnid-side nodes) have an associated
non-zero value for the quadratic shape function N,. This means that only these 6 nodes
are used to interpolate the geornetry.
2-2-2 Disdacement Field
The displacements u, v and w at any point within the consistent laminated shell
element are approximated using the following degrees of fieedom:
1) The global displacements LI,, V, and W, of the laminate mid-surface directed dong
the global x, y and z axes, respectively. These degrees of freedom are interpolated
cubically within the mid-surface of the laminate using the corner, one-third-side and
the center nodes.
2) Two rotations, cr, and ,& about the local axes y'and x : respectiveIy. These rotational
degrees of freedom provide a Iinear variation of the through thickness displacements
and are interpolated quadratically withîn the mid-surface of the laminate using the
comer and mid-side nodes.
3) For each layer the following extra degrees of tieedom are used: Rotations 4,,?. vny.
A: 7 and displacements d # z , dry?, d@z , dv$. Here the subscript L denotes
the layer number and the superscnpts Top and B designate the top and the bottom of
the layer, respectively. Rotations 4 2 , #n: are about the local y'axis and rotations
Y?, are about the local x'axis. Displacements d # 2 , d@: are dong the local
x ' axis and displacements d , d ryz are dong the local y ' axis. These degrees O l
fieedom are interpolated cubically through the tfiicbess of each layer of the larninate
in order to obtain exact transverse shear stress distribution. Within the plane of the
larninate, these degrees of freedom are approximated quadratically similar to an and
P n -
As shown in Fig. (2.1), not al1 nodes have the same number of degrees of
keedom. Only the comer nodes have al1 types of degrees of fkeedom. Such nodes have
dq!$, dvn: , . . , d )). Assuming that the 13 nodes have the sarne number of degrees
of Ereedom as the corner nodes, the total number of degrees of fieedom per element
becomes (65+104*NL), where NL is the total number of layers. However, the active
number of degrees of fkeedorn is only (42+48*NL).
Following the description above, the global displacements u, v and w at any point
within the laminate can be wrîtten as:
where En and Nn are the cubic and quadratic shape hctions, respectively, and are
given in Appendix 1.
[en ] is a rnatrix which relates the direction cosines of the local axes .Y ' and y ' to
the global directions x, y and z. It is used in Equation (2.2) to transform the local
displacements resulting fiom the rotational degrees of fieedorn to the global
directions x, y and z. The rnatrix [fn] is calculated fkom the unit vectors Y,, and f&,
directed along the local x 'and y 'axes, respectively, i.e.
The procedure for evaluating the vectors En , as well as the vector En (which
is normal to both 6, and fZn ) was given by KoPey and Mirza (1997) and is presented in
Appendix II.
The shape function M, approximates the displacement field through the depth of
the shell due to constant rotations a and ,8 and is given by:
where H is the total thickness of the laminate. The through thickness shape funciions
M,,, MJL, M, and Ma, are selected in such a way that when multiplied by the local
'" , $g , y,". d@ and through thickness degrees of keedom (#y, vf"P, d ~ ,
d ) they provide cubic variation for the displacements through the thickness of the
laminate. The functions were given by Koziey (1993) as:
where hL is the thickness of the L" lamina The non-dimensional coordinate t, ranges
fiom -1 at the bottom to +l at the top of L& lamina as described earlier. It is related to
the coordinate t as follows:
A transformation matrix [d which includes the direction cosines I , between the
local and the global axes system, i.e.
is multiplied by Equation (2.2) to yield the following equation for the local displacements
u : v 'and w ' at any point within the element:
where the matrix [C,, 1 is given by:
In the above relation, a superscript n means that the direction cosine component is
evaluated at node o. Meanwhile, a direction cosine specified without a superscnpt means
that it is evaiuated at the point of interest (Le. at a general point withh the shell). The
procedure for the evduation of [@J as described by Koney and Mina (1997) is given in
Appendix II.
2-3 STRAIN-DISPLACEMENT AND STRESS-STMN RELATIONSHIPS
For small deflections, the local strains (É) are expressed by the following
equation:
Notice that the normal strain component is not considered in Equation (2.1 0)
as the shell element formulation is based on the plane stress assumption (Le. C J ~ , ~ = 0) and
such a strain wiIl not contribute in the total strain energy of the system.
Substituting Equation (2.8) into Equation (2.10) yields:
The strain matrix [B'j is given by Koziey (1993) and is also provided in
Appendix IV as it will be used in the nonlinear formulation presented in the next Chapter.
Using the assumption that stresses normal to the surface (cf;) are negligible, the
constitutive rnatrix [Dl'] in the material coordinates system (1, 2 and 3) is given by Johns
(1975) as:
where G,? GI3, G, = are the shear moduli in the 1-2, 1-3 and 2-3 material planes.
respectively.
Constants C, are defined as follows:
where E, , E?, E, is the elastic moduli in the principle matenal axis;
vu is the Poisson's ratio for transverse strain in the j" direction when stressed in
the ib direction.
For a laminated structure, the fiber orientation typically varies £iom layer to layer.
In order to analyze a lamuiated sheU structure, the fiber orientation for each layer must be
referenced to a certain cornmon axis. In the formulation of the larninated consistent sheil
element, the local axis x 'has been chosen as the reference axis. The fiber orientation with
respect to this local axis x'is defined by an angle @as show in Fig. (2.2).
The elasticity matrix is therefore referred to the local coordinate system using the
fo llowing transformation:
where the transformation matrix [TE ] relates the strains ( E,,,. , E ,.,. , E ,.,. , E,,=. . E,?. ) in . .
the local coordinate system to those in the material coordinate system ( 6 . E, , T , ~ ,
y,, , y?, ), and is given by:
2 4 ELEMENT STIFFNlESS MATRIX
- cos2 6 sin2 B C O S B S ~ 6 O O
sin' 8 COS' 8 -sinQcos0 O O
-2sinBcosB 2sinBcosB (cos2@-sin'@ O O
O O O cos8 -sin0
O O O s ine COSB -
The element stiffiiess mat& is calculated as:
(2-1 6 )
[k] = ,f[l?'lT [D' ] [B'](IV Y
The above equation is integrated numerically using the Gaussian-Quadrature
scheme. Since the above integral includes functions M,,, M,, Mx and M,, that are
discontinuous kom one layer L to the other (Within a layer L, these Functions are
descnbed by Equation (2.6) and are equal to zero outside this layer) Koziey (1993) has
performed the integration using the coordinates r, s and î, within each layer and then a
surnmation dong al1 the layers was performed. This procedure led to the following
equation:
h, where - H
relates differentials dt and dt, and is calculated fiom Equation (2.7);
[DL] is the elasticity matrix for layer L, and is evaiuated from Equation (2.15);
detl J'[ is the determinant of the Jacobian matnx [ J ' ] which relates the
derivatives with respect to the curvilinear coordinates system to those with respect
to the local coordinate systern, Le.:
2-5 KINEMATIC CONSTRAINTS
As discussed in subsection (1-2), the mechanics of larninated structures
necessitate the following conditions:
1) Transverse shear strains (y, and y 4 must have the fkeedom to be discontinuous
between layers. This discontinuity is essentiai in order for the mode1 to be able to
predict continuous transverse shear stresses between layers having different shear
moduli (G). This feature exists in the Consistent Laminated Shell element due to the
use of independent rotations at the interface between two adjacent layers. For
example, the rotation ($7 ) at the top of the f i t Iayer is not necessarily equal to the
rotation (4: ) at the botiom of the second layer. In general:
The evaiuation of the transverse shear strains includes derivatives of the fiinctions MIL
and M, with respect to the coordinate t,, which are multiplied by ( and y. Examining
Equations (2.5), it c m be shown that these derivatives have a non-zero value at the
interface between Iayers (at t, = i l ) . Therefore, the use of Equations (2.20) satisfies
the condition of discontinuous shear strains.
At the interface bebveen two layers, displacements have to be continuous. Studying
the displacement field in Equation (2.8), it can be seen that the first 2 tems are
continuous through the thickness of the laminate. Meanwhile, in the third terrn,
and Mx are equal to zero at the laminae interfaces. This means that the discontinuity
in q5 and y does not affect the continuity of the inter-laminar displacements.
Furthemore, the value of Ma at the top of a certain lamina is equal to the value of
M, at the bottom of the lamina located above. Thus, in the consistent laminated shell
element model, a continuity of displacements at the interfaces between larninae can be
satis fied by imposing the fo llowing constraints:
The above constraints are irnposed in the finite element code and they lead to the
reduction o f the active degrees of fieedom per element fiom 42+48*NL to
30+36*NL.
3) At the interface between two layers, in-plane strains (sr;: s,.;.' and y,;.) have to be
continuous. in view of Equation (2.8), the evaluation of these strains includes
denvatives of the shape fûnctions N and N with respect to x' and y' which are then
multiplied by the through thickness shape functions (Ml, Mx, Ma,, M, and Ma). M, is
a continuous function while M, and Mx are both equal to zero at the laminas
interface. This means that the discontinuity in 4 and yl does not affect the continuity
of the in-plane strains. Furthemore, the value of Ma at the top of a certain lamina is
equai to the value of Ma at the bottom of the lamina located above. Thus the
continuity of in-plane strains is achieved.
2-6 NUMERICAL EXAMPLE
Ln order to demonstrate the capability of the Consistent Laminated shell element
in predicting accurate stress distributions for laminated structures, the following example
is reproduced fkom Koziey (1993). This example involves the analysis of a simply
supported laminated thick plate subjected to a sinusoidal transverse load. The dimensions
of the plate and the mathematical expression for the sinusoidal load are shown in Fig.
(2.3). A through thickness cross section of the laminated plate is also given in the fi,we
showing that the plate consists of nine cross-ply layers arranged syrnmetrically about the
mid-surface. The span to thiclaiess ratio (LEI) of the plate is equal to 4 as shown in the
figure. For each layer, the material properties in the 1, 2,3 directions are dehed as:
E, = 25 x 106 psi (1 -724 x 10' MPa), E2 = 1 x 106 psi (6.895 x 103 MPa),
G,, = G,, = 0.5 x 106 psi (3.45 x 103 MPa),
- G, 4 . 2 x 106psi (1.38 x 10' MPa), v , ~ = v,, - vZ3 = 0.25
Due to symmetry, Only a quater of the plate is modeled using 3 2 Consistent
Laminated Shell elements (4x4 rnesh). The results were compared by Koziey (1993) to
the elasticity solution given by Pagano and Hatfield (1972). The normal stress (aL. at the
center of the plate and the transverse shear stresses (cd at point A (shown in Fig. 2.3),
are plotted in Figures (2.4) and (2.5), respectively. These two figures show the variation
of the stresses with the thickness of the plate. These stresses were compared by Koziey to
the closed form elasticity solution indicating an excellent agreement. It is worth noting
how the element predicts continuous shear stresses at the inter-lamina. faces for layers
having different shear moduli.
Fie. 2.1: Consistent Laminated Shell Elernent Coordinate System and Nodal Deqrees of
Freedom
Fie 2.2: Matenal Coordinate Svstem of a Tvpicai Laminated Shell
L = 10.0 in Sinusoidd transverse load q = 100 sin (d) sin(qdL) psi SimpIy supported
1 in = 25.4 mm 1 psi = 6895 N/m'
0" and 90" denote the fiber onentation with respect to the x ' axis
Fig. 2-3: Dimensions and Material Pro~erties for Larninated Fiber Reinforced
Composite Plates. Koziey (1 993)
q Koziey ( 1993)
Fig. 2.4: Normal Stress a, throueh niickness of 9 Plv Laminate. Koziev (1 993) -
Z N Elasticity Solution 0.5 * Koziey ( 1 993)
0.4 -
0.3 -
0.2 -
0.0 - Ksi
0.0 7
o.co0 0.100 -0.1 -
-0.2 -
-0.3 -
-0.4 -
-0.5 *
Fie. 2.5: Shear Stress r.., throueh Thickness of 9 Plv Laminate. Koziev (1993) -
CHAPTER THREE
LARGE DISPLACEMENT STATIC AND DYNAMIC
FORMULATION OF
CONSISTENT LAMINATED SHFlLL ELEMENT
3-1 INTRODUCTION
Due to their high flexibility, laminated structures can be subjected to large
defomations. Therefore, a large displacement analysis might be necessary to study those
structures. The excellent performance of the Consistent Laminated sheIl eiement in
performing linear analysis of laminated structures has encouraged the author to extend the
eiement's formulation to include large displacement static and dynamic analysis. This
chapter starts by presenting the large displacement nonlinear static development. The
nonlinear formulation is based on the total Lagrangian approach while the solution is
carried out incrementally using the Newton-Raphson method for iterations. A detailed
presentation of the nonlinear stifiess matnx and the unbalanced load vector is included.
To test the accuracy of the above development a number of larninated plate and shell
structures are modeled using the nonlinear version of the Consistent Laminated shell
element and the results obtained from these analyses are compared with those available in
the fiterature. This is followed by a presentation for the dynamic formulation of the
element. The consistent mass matrix of the element is denved and incorporated in both
fiee vibration arid the-history nonlinear analyses. Again, the dynamic development is
verified by modeling a nurnber of benchmark problems using the Consistent Laminated
shell elernent and comparing the results of the analyses to those available in the Literature.
3-2 LARGE DISPLACEMENT FORMULATION
In this section, the incremental form of the displacement field of the Consistent
Larninated shell element is fist presented. This is followed by a derivation of the
nonlinear stifhess matrut and the unbalanced load vector. These matrices are presented in
a form that can be easily coded using the components of the linear stifhess rnatnx
(developed by Koziey 1993). Finally, the solution technique used in the nonlinear
analysis is presented.
3-2-1 The Incremental Displacement Field
In tfiis study, an incrernental nonlinear formulation is conducted for the Consistent
Laminated shell element. For such a formulation, the displacement field has to be wntten
in an incremental form rather than a total one as presented in Chapter Two. In the
following nonlinear presentation, the superscript T identifies different confi,wations
corresponding to different load incrernents and a superscript k associated with a variable
indicates that this variable is evaiuated at the km iteration of the increment. Thus, the
displacement vector u:'~' at the km iteration of a certain configuration T is related to the
displacement vector at the previous iteration (k-1) as:
In view of Equation (2.2), the incremental global displacements Au, Av and Aw
at any point within the laminate can be wrïtten as:
where the shape functions vn and N,, the transformation rnatrix [cl, and the through
thickness interpolation functions M,, M,,, Ma, M,, and M, are as defined in Chapter
Two and are provided in an explicit fonn in Appendices 1, II and III.
Also in view of Equation (2.Q the incremental local displacements at any point
within the laminate can be written as:
Aw'
where Au' , Av' and Aw' are the local displacements dong the local x', y' and z' axes,
respectively; and the matrices [C,] and [ûJ are as defined in Chapter Two. In the above
equation, the incremental degrees of eeedorn follow the same definition provided for the
total degrees of fieedom in chapter Two.
It should be noted that, unlike the linear behavior, the large displacement analysis
would involve an update for the matrices [ B j and [C,] to reflect the variation of the
direction cosines Z, with the Ioading. This is done to assure that the local rotations (a, P, (
and y) are taken about axes which are always tangent to the surface.
3-2-2 Expressions for Nonlinear Stinnes Matrix and Unbalanced Load Vector
A detailed description of the total Lagrangian approach used to perfonn the
nonlinear formulation of a general finite element approximation is given by Bathe (1 996).
Based on the above approach, the equilibrium equation of a nonlinear structure is given
in a tensorial form as:
where Cg, is the constitutive relations of the matenal.
SQ is the znd Piola-Kirchhoff stress tensor. (See Bathe (1996) for a definition).
Aeu is the h e a r part of the incremental Green-Lagangian strain tensor.
A is the nonlinear part of the incremental Green-Lagrangian strain tensor.
The above equation can be expressed in a rnatrix form as:
where [KJ is the surn of the linear and the initial strain stiffhess matrices.
[KJ is the initial stress s t iaess matrix.
(F) is the unbalanced load vector.
{AU) is the incremental degrees of fieedom vector.
In view of the general total Lagrangian approach provided by Bathe (1996) and
using the assumption that the stresses normal to the surface O+. are neglected, the
nonlinear stifkess matrices and the unbalanced load vector for the Consistent Laminated
shell element, presented in the local axes system (x: y < z 3, are given by:
where M is the total number of degrees of fieedom per element, and is equal to
(6SH 04*NL).
The constitutive matrix [D 1 for laminated structures defined in the .r : y ' and r '
coordinate system is described by Equations (2.12) to (2.15). Matrices [SI and (Se)
include the local stress components and are given by:
The incremental stress vector {AS) can be calcdated using the constitutive matrix
[D 1 and the incremental strain vector as:
where the incremental strain vector (Ae) is calculated as:
El Damatty et. al. (1997) has presented a systematic approach which can be used
in the nonlinear formulation of a general shell element. The approach is based on relating
the components of the nonlinear stiffkess matrix and the unbalanced load vector to those
of the linear stiffness matrix. This approach was used by El Darnatty et. al. (1 997) in
deriving the large displacement formulation of the single layer Consistent shell element.
In view of this approach, the components of the [B,] and [B,] matrices can be obtained
using the following:
1) The linear strain matrix [B'],,, , given by Koziey (1 993), is written as the s u m of the
two matrices [BA, ] and [Biz ] :
Expressions of matrices [Bo, ] and [BA2] are given in Appendix IV.
2) The variables h and G included in the expressions of the matrices [Bi,] and [BA2]
represent derivatives of the shape function with respect to the local axes system. and
are given by the following tensonal equations:
In the above equation, n represent the node number; Fn, andrn , represent the
derivatives of the shape fuaction r, with respect to the r and s curvilinear
coordinates, respectively; J,; are the components of the inverse of the Jacobian
matnx [J'] given by Equation (2.19); M,, and dMqL are the through thickness shape
functions and their derivatives with respect to the curvilinear coordinate t ,
respectively; q takes the values (q = 1, 3, 4, 5 and 6 corresponding to different
through thichess shape functions).
3) The matrices [B,] and [B,] can be related to matrices [Bo,] , [Bo?] and the initial
strains in the manner outlined in El Damatty e t al. (1997) as follows:
First row of [Bo,]
Third row of [B,J
Fourth row of [Bo,]
Thirdrowof [Bo,]
Secondrow of [Bo,] [Bsl,x, =
Fifth row of [Be]
1 Fourth row of [Bol ]
cont' d . . . First row of [Bo, ] I 1 cont' d . . . Third row of [Bo? ] 1
/ cont' d . . . Fourth row of [Bo?]
j cont'd ... Third row of [B,,] cont'd ... Second row of [Bol]
I
1 cont' d . . . Fifth row of [Bo, J I
1 cont'd . .. Fourth row of [Bol ] I 1 cont' d . . . Fi& row of [Bo, ] I I I - c:~G~, C G - C G C:'G;~ - Cn2~,)5 C : ' G ~ ~ - C ~ I G , ~ ~ 11 -
(3.15)
z<> (row 1 of [Bo, 1 ) + v; (row 3 of [Ba 1) + w:. (row 4 of [Bot])
u',.. (row 3 of [ B o l ] ) + v:. (row 2 of [Bo, 1) + W.;. (row 5 of [B,? 1)
u->,(row 1 of [Bo , ] ) + v+;,.(row 3 of [BO2 1) + w."Jrow 4 of [B,])
+ u>(row 3 of [Bo , ] ) + v:.(row 2 of [ B o l ] ) + w:.(row 5 of [ B E ] )
u'.(row .- 1 of [ B o l ] ) + v'Jrow - 3 of [Bo2 ] ) + w:. (row 4 of [BO2])
+ u'Jrow 4 of [ B o l ] ) + v>(row 5 of [Bo,] )+ w>(row 9 of [ B S ] )
u;.(row .- 3 of [Bo, 1) + vl.(row .- 2 of [Bo,]) + wt.(row - 5 of [ B J ) + U.~~ . (~OW 4 of [ B o l ] ) + vf,..(row 5 of [Bol 1) + w'Jrow 9 of [ B , ] )
+ [Bol L x , + [Bo2 Ln .w
where ri :i , v :i , w :, (i = 1,2 and 3 corresponding to x ' , y 'and z : respectively) are the
derivatives of the local displacements u', v', w ' with respect to the local axes i (local
strains). The matrices [ B o l ] and [Bo?] are presented in appendix IV.
3-2-3 Evaluation of the Initial Strains
The matrix [B,] includes local initial strain terms evaluated at the local
coordinate system (x: y : 2'). In order to evaluate these terms, accurnulated local
displacements ut '('-", v' T"-l) and w"(*-" have to be evaluated at the beginning of kb
iteration of configuration T. These can be obtained fiom the global accumulated
nt-1) n k - 1 1 w r ( k - [ ) displacements ( u , v , ) and the transformation matrix [q. In evaluating the
accurnulated displacements resulting fiom various iterations, some of the degrees of
fieedom can not be added algebraically. These degrees of keedom are the incremental
, ~ 4 : and ~ r y : ) as well as the rotational degrees of fkeedom ( Acc, , APm A#F , A tyd
degrees of keedorn that include the local through thickness displacements ( ~ d # z ,
ady?, ~dq52 and adv:). This is due to the effect of large displacements, which
makes the direction cosines of the axes, around which the incrernental rotations are taken,
change with different configurations. To overcome this problem, the contribution of the
local rotational and the local displacement degrees of freedom to the global displacements
(IC, V, W) is evduated at each iteration.
The procedure to evaluate the global accumulated displacement at the beginning
of the kh iteration of configuration T, is described by the followùig steps:
- The accumulated effect of the global displacement degrees of keedom
( AU,, A V, , A W, ) c m be obtained by sumrning al1 the incrernental displacements
preceding the k& iterations of the configuration T.
The contribution of the hcremental rotations ( Aa, and A 4 ) to the total global
diçp lacements T( ' - 1 1 "'-1) w T ( k - l ) ,V , can be Uitroduced using the terms
T( k-1) ayn T( k-1) and m, , respectively. These terms are special rotations about the
global axes, which are multiplied by the shape bc t ions MI, to give the through
thickness displacements in the global x, y, z axes, respectively. These special rotations
are evaluated usùig the followiug equations:
- The contribution of the incremental rotations and displacements ( A4= , A y, , Ad#, and
T ( k - 1 ) Adyn ) aat the top and bottom of each layer, to the total global displacements u ,
n k - 1 ) and w '"-" can be introduced in a similar rnanner using the tems ( #cmT('-" and
T( k-1) ), ( 0 , y - l ) a,d &m dxn '('-" ) and ( &.'-l' and 6i. '-" ), respectively. These terms
when multiplied by the shape functions M,, , M,, , M,, and M,, , give the through
thickness displacements in the global x, y, z axes, respectively. These special rotations
are evaluated as follows :
Thus, the total global displacement at the beginning of the kh iterations of the
configuration T is aven by:
Using the transformation matrix [ûJ, the local displacements at the beginning of the same
iteration are given by:
Using the Jacobian matrix [J'] and Equation (3.36) the derivatives of the totaI locaI
displacements with respect to the local axes are evaluated as:
where u; = u' , ul = v r , U; = w' ,
and
The variable h,, and G& are as defined in Equations (3.13) and (3.14), respective1 y.
As mentioned earlier, although a Total Lagrangian formulation is used, it is
necessary to update the geometry of the structure to ensure that the rotations are always
considered around the local axes that are tangent to the surface. Therefore, the matrices
[ûJ and [Cm] are updated after each iteration.
3-24 Solution Technique for the Nonlinear Static Analvsis
The solution technique used in this study is similar to the one presented by El
Damatty et. al. (1997). It starts by the calculation of matrices [B:('-"1, [B,"*-"1,
{ s ~ } and [D 'J. These matrices are then substituted into Equations (3 -5 ) to
(3.7) to evaiuate the components of the s t iEess matrix [K:(~--"] , [K,"'-"] and the
unbalanced load vector {F*('-')} corresponding to the (k-l)h iteration of a certain
configuration T. Gaussian quadrature scheme is used to perform the numerical integration
of the previous equations. Seven integration points are used to integrate in the r-s plane,
while five integration points are used along t, for each layer. Al1 the t ems of the
expressions for [ ~ f ( ~ - " ] and [K;('-''] are evaluated at the integration points except those
having either a subscnpt or a superscnpt n which are evaluated at the nodal points.
Finally, Equation (3.4b) is used to solve for (AU). The solution then proceeds in an
iterative manner until convergence is reached. The convergence cnteria is based on the
ratio between the Euclidean n o m of the displacement vector occurring at a certain
iteration to the one evaiuated at the 1" iteration of the configuration T.
During each configuration, the external load {R? is kept constant while the
unbalanced load vector (F') and the matrices [K:] and [K:] are updated at each
iteration. This updating procedure is performed as follows :
At each node, the global coordinates ( x , , y, and 2 , ) and the vector VJn, are
updated using the incrernental displacement and rotational degrees of fieedom
calculated from the solution corresponding to the previous iteration (k-2)-
Those updated global coordinates and vectors are then used to update the matices
[8J, [C,] and [V,] defïned in Chapter Two.
Followiig the procedure explained in Section (3-2-3), the initial strains at the
current kh iteration are calculated and used to update the matrix [~f '*-"] given by
Equation (3.16).
Using Equations (3.10) and (3.1 L), the incremental stresses occumng at the
integration points, are calculated. Those stresses are then used to update the
matrices [Sn'-')] and ( s ~ ~ ' ~ ' ' ) } given by Equations (3.8) and (3.9), respectively.
The constitutive matrix [D 1, described by Equations (2.12) to (2.15), is used
together with the updated matrices [B,T"-"1, [s~(~-')] and (s"(~-')} and the
constant matrix TB, 1, to obtain the u~dated stiffiiess matrices TK?~-' ' I and
[K,"'-"] and the updated load vector {F~"-"} thr0ug.h the application of Equations
(3.5) to (3.7).
6 ) Equation (3.4b) is then used to obtain the incremental displacements for the current
iteration.
The solution then proceeds in an iterative rnanner until convergence is achieved.
At this equilibrium state, the total displacement {u*) for the current configuration T is
T-Ar given by the summation of the displacement {U ) at the previous configuration (T-AT)
and a11 the iterative solutions { A u condncted in this load step. This solution {uT is
then used to obtain the initial strains and stresses for the first iteration of the next
configuration (T+AT).
3-3 VERIFICATION OF THE LARGE DISPLACEMENT FORMULATION
3-3-1 S i m ~ l v SUD DO^^^^ Isotro~ic PIate under Uniform Transverse Load
In this example, the Consistent Laminated shell element is used to analyze an
isotropic structure. The problem consists of an isotropic square plate having simply
supported boundary conditions and subjected to a uniforrnly distributed transverse load.
The plate has the following elastic properties: Modulus of Elasticity E = 71 020.0 MPa
and Poison's ratio v = 0.3 16. The dimensions of the plate are s h o w in Fig. 3.1, where
a = b = 16 in (40.64 cm) and h = 0.5 in (12.7 mm). Due to double symmetry in both
geometry and loading, only one quarter of the plate is rnodeled using 8 (2x2 mesh)
Consistent Laminated shell elements. The central deflection w resulting firom the analysis
is plotted in Fig. (3.2) ushg the dimensionless parameters (qai/Ehi) and (w/h). Here q, a
and h are the load intensity, the length and the thichess of the plate, respectively. A
corresponding load-deflection cuve for the same problem was provided by Ostrowski
(1984) using thirty six non conforming plate bending elements. An excellent agreement
between the two sets of resuIts is indicated by Fig. (3.2).
3-3-2 Clamaed Sauare Plate under Uniform Transverse Load
The large displacement static analysis of a square plate clamped on four sides and
subjected to a uniform transverse load is considered in this example. The plate consists of
four laminas having the fibers orientation (0/90/90/0). For each lamina, the fiber
orientation is measured relative to the local x' axis (coinciding with the global x axis
direction shown in Fig. 3.1). The geometry and material properties of the plate are given
below (see Fig. (3.1 ) for description of notations).
a = b = 12 h(30.48 cm), h = 0.096 in. (2.44 mm)
E, = 1.8282 x 1 O" psi (12605.4 MPa), & = 1.83 15 x 1 O6 psi (12628.2 MPa),
G,, = G,, = G, = 0.3125 x 106 psi (2154.7 MPa),
v , ~ = v,, = v, = 0.2395
Due to double symmetry in both geornetry and loading, oniy a quarter of the plate
is rnodeled- The vertical deflection resulting korn the large displacement analysis of the
plate, using 8 (2x2 mesh) and 18 (3x3 mesh) Consistent Larninated shell elements, are
plotted against the load in Fig. (3.3). These c w e s when compared to the results of the
andysis pelformed by Putcha and Reddy (1986), using a Rehed mixed finite element,
ùidicate very good agreement especially with the (3x3) mesh.
3-3-3 Anti-Svmmetrical Cvlindrical Panel under Central Point Load
This example considers the large displacernent behavior of a cylindrical angle-ply
panel (45/-45) hinged at the straight edges and kee at the c w e d sides. For each lamina,
the fibers orientation is measured fiom the local x' axis which forms with the local y' axis
a plane tangent to the shell. The local y' axis is parallel to the global y axis show in Fig.
(3.4). The shell is subjected to a central point load P and has the following geometry and
material properties, (see Fig. (3.4) for description of notations):
Panel length b = 0.508 m (20.0 in), thickness h = 0.0124 m (0.496 in)
Shell radius R = 2.54 m (100 in), Shell angle 4 = 0.2 rad.
E, = 3299.3 MPa (4.785 x 1 Os psi), E2 = 1099.8 MPa (1.595 x 1 Os psi),
GL2 = G13 = 659.85 MPa (0.957 x 10' psi),
G, 4 4 - 2 8 MPa (0.64 x 10' psi), v , ~ = 0.25
The shell is modeled using 32 Consistent Laminated shell elements (4x4). The
vertical defiechon resulting £tom the large displacement anaiysis of the shell is plotted
against the central point load in Fig. (3.5). This curve is compared to the results obtained
by Yeorn and Lee (1989) uskg a (6x6) mesh of nine node shell elements. The
comparison indicates excellent agreement between the two sets of analysis even when a
coarser Consistent Larninated shell element mesh is used in the analysis.
3-4 DYNAMIC FORRlULATION
3-41 Derivation of The Consistent Mass Matrix
In order to extend the formulation of the Consistent Larninated shell element to
include dynarnic analysis, the mass matrix of the element has to be forrnulated. This is
achieved using the expression for the v h a l work done by the inertia force. Neglecting
the inertia force resulting fiorn the rotational degrees of fkeedom ( a,, , P, , 4 2 , vmy, #=:
and yn: ), the virtual work quantity (V,) at tirne T is given by:
where p,(L) and h, are the mass density and the thickness of the L" layer, respectively;
i iT , ijT and wT are the components of the accumulated acceleration of the mid-
surface of the laminate at time T, in the global x, y and z CO-ordinates respectively;
&Au), ~ ( A v ) and &Aw) are the components of the virtual incremental
displacements applied at the mid-surface of the laminate along the globaI x, y and
s CO-ordinates respectively.
Expressions for &(Au), ~ ( A v ) and ~ ( A w ) can be obtained fiom Equation (3.2) by
setting al1 rotational degrees of fieedom to be equal to zero.
Neglecting the rotational inertia, the components of the accumulated acceleration
at any point of the shell are given by:
where Ü:, vmT and wnT are the components of the total acceleration of the n' node at
tirne T which can be obtained by adding algebraically cornponents of the
accelerations resulting fiom al1 the time increments AT.
Substituting the expressions for &Au), ~ ( A v ) , ~ ( A w ) , ÜT 7 ,d ~r into
Equation (3.40), the inertia forces for a Consistent Laminated shell eIement are given by:
where NEL is the number of elements, M is the total number of degrees of freedom per
element (6S+l O4*NL);
{ Ü T } is a vector which contains the acceleration of the rnid-surface global degrees
of fkeedom, Le.
.- { i iT )Tra"={ü , v, w, O O .. Ü2 v1 O .. ü,) F3 q3 O ..}
[ M , ] , t f s , is the consistent mass matrix & M = (5+8NL).
A simple procedure for coding the components of the mass matrix M! is
provided below:
2 ) Indices i and j are given as follows:
In the equations above, k,, k2 can take values ranging fiom 1 to 13, while Z, and take
values ranging form 1 to (5+8NL). The numencal value 13 corresponds to the number
of nodes per element while (5+8NL) relates to the number of degrees of freedom per
node-
2) Any combination of (k,, kZ, Z, and 4) corresponds to certain values of i and j which
d e h e a component of the mass matrix MF which is given by the following relations:
where 6(Z, Z, - ) is the kroneker delta fùnction, Le. 6(1,4 ) = 1 for Z, = 4 and 6(1,[, ) = O
otherwise;
fl are the cubic interpolation bctions;
p, (L) is the density of the L" layer;
hL is the thickness of the L" layer.
3-4-2 Free Vibration Analvsis us in^ the Consistent Laminated Shell Element
3-4-2-1 The Free Vibration Formulation
The linear stifiess rnatrix [K,] of the Consistent Laminated shell element is
incorporated with the consistent mass matrix CMs] into a fiee vibration routine to
determine the natural fiequency (0,) of laminated shell structures and the corresponding
mode shape {q,). These are obtained by solving the following homogeneous equations:
The non-tnvial solution of the above homogeneous equation exists i l the
deteminant of the coefficients vanishes, Le.
Equation (3.47) is solved to obtain the natural fkequencies of a larninated shell
structure and these are back substituted into Equation (3.46) to obtain the corresponding
mode shapes.
3-4-2-2 Verification of the Free Vibration Formutation
To ver@ the fiee vibration formulation, a simply supported larninated square
plate is modeled using 8 (2x2 mesh) Consistent Laminated shell elements. The plate,
which was previously analyzed by Reddy & Khdeir (1989) using an element based on the
tirst order deformation theory, consists of a variable number of cross-ply layers (0/90/. . .)
with a span to total thickness ratio of 5. The mechanical properties of a layer of the plate
are al1 related to the modulus of elasticity in the cross fibers direction E,, as follows:
Meanwhile, the poisson's ratio of the layer v12 is assurned to be equal to 0.25. Free
vibration analysis is conducted for the plate and the results are presented using the
dimensionless parameter Z which is d e h e d by the following relation:
w, p and h are the natural fiindamental frequency, the mass density and the
thickness of the plate, respectively. Cornparison between Z predicted by the Consistent
Larninated shell element andysis and those aven by Reddy & Khdeir (1989) for various
number of layers are presented in Table (3.1) indicating an excellent agreement.
3-4-3 Nonlinear Time Histow Analvsis
3-4-3-1 Formulation of the Nonlinear Time Historv AnaIvsis
Neglecting the damping effect, the incremental equation of motion for nonlinear
dynarnic analysis is given by:
Using the Newark ' s method with implicit parameters (6 = 0.5 and a = 0.25) and
following the procedure described by Bathe (1 W 6 ) , the incremental solution (AU)
corresponding to the kU iteration of time step AT is obtained from the following equation:
where the effective stiffhess matnx [K'~('-"] and the vector {A"'-") are given b y :
Vector { A ~ ' is updated after each iteration using: (u"*-") the total
displacement degrees of Eeedorn of the preceding iteration (k-1) at the current tirne T and
{U '-"' 1 , ( U } , u ] which represent the total nodal displacement, total nodal
velocity and the total acceleration, respectively, evaluated at the equilibrium
configuration corresponding to the previous tirne increment (T - AT). Those are related to
the equilibrium solution corresponding to the preceding t h e increment (T - 2AT) as
follows:
3-4-3-2 Solution Techniaue for the Noniinear Time Histow Analvsis
The solution procedure for the non-linear dynamic analysis is camied out similar
to the procedure used in the nonlinear static analysis (see Section 3-2-4) with the
following added steps:
1) The consistent mass matrix is evaluated at the beginning of the analysis.
2) The vector {A"'-')} and the eEective sti&ess matrix [K'~('-"] are updated, at each
iteration, using Equations (3 -5 1) (3 SO), respectively.
3) The initial velocities and accelerations at each tirne step are evaluated using Equations
(3.52) and (3 S3), respectively.
34-3-3 Verification of the Dvnamic L a r ~ e Displacement Formulation
Cvlindrical SheM Panel Under Interna1 Uniform S t e ~ Pressure
A cross-ply (0/90) cylinder panel, is subjected to an interna1 uniform step pressure
q, = 5000 psi (34.475 m a ) . The cylinder is clamped on ail sides. The fiber orientation is
measured relative to the xf-axis and the directions of the local axes x' and y' foIIow the
description given in Section (3.3.3). The geometry and material properties of the plate are
given below, (see Fig. (3.4) for description of the notations).
b = 20 in (508 mm), thickness h = 1 in (25.4 mm)
R = 20 in (508 mm), 4 = d 2 rad, mass density p = 1 lb. s'lin4 (10.687*106 Kg/m3),
El = 7.5 x 106 psi (51 712.34 MPa), E, = 2 x IO6 psi (1 3789.96 MPa),
GI2 = GZt = G13 = 1.25 x IO6 psi (861 8-72 MPa), v,, = v,, = 0.25.
Due to double symmetry in geometry and loading, only one eighth of the cylinder
is modeled using 8 (2x2 mesh) Consistent Laminated shell elements. A time history large
displacement dynamic analysis was performed using a time step AT=0.001 sec. The
variation of the central defiection w, resulting fiom the analysis, is plotted against the
time T in Fig. (3.6). This cuve is compared to the results obtained by Reddy &
Chandrashekhara (1985) using the nine node isopararnetric shell elements. The
cornparison indicates excellent agreement, thus v e r i m g the high performance of the
Consistent Laminated shell element in perforxning the nonlinear dynarnic analysis of
larninated shell structures.
3-5 CONCLUSION
The formulation of the Consistent Laminated shell elernent has been extended in
this chapter to include both static and dynamic large displacements analysis. Expressions
for the noniinear stifhess matrix, the unbalanced load vector and the consistent mass
matrix, as well as a simple procedure for coding those matrices are presented in this
chapter. AIthough the nonlinear formulation is based on the total Lagrangian method, it
was necessary to update the geometry to ensure that local degrees of keedorn are taken
about (or dong) local axes always tangent to the mid-surface. The formulation has been
verified by performing large displacement static and dynamic analysis for a number of
plates and shells and comparing the results to those available in the literature. In al1 above
examples, the element has shown a very good performance.
Table (3.1): Dimençionless Fundamental Frequency Z = o J p h 2 / ~ ; of Simply
Supported Cross-Ply Square Plate.
Layers 1
2 Reddy & JShdeir (1 989)
0.35333 Current Results
0.349 1 1
Fig. 3 . f : T-mical Clamped Plate (2x2) 8 Elements
Fie. 3.2: Load-Deflection Curve for an Isotro~ic Square Plate
Present study 2 x 2 - -..---.-.- Present study 3x3
-- Reddy (1986)
0.0 0 .4 0.8 1 .2 1.6 2 .0
Pressure (psi) ( 1 psi=6.895 K P a )
Fig. 3-3: Load-Deflection Curve for a Laminated Clarn~ed Sauare PIate
Fie. 3.4: Tmical Cvhdrical Panel
Present study 4x4 0 Yeom&Lee(I989)
O - 7 4 6 8 10 1 I
Center deflection -w (mm)
Fie. 3.5: Load-Deflection Curve for a Cylindncal Panel
- Present study -- Reddy & Chandrashekhara ( 1985)
Time, T?t 1 O sec
Fie. 3.6: Load-Deflection Curve for a Cross-PLV (0/90) Cvlindrical Panel
CHAPTER FOUR
NON-LINEAR ANALYSIS OF FRB Cai[R/iN]EYS USING THE
CONSISTENT LAMINATED SHELL ELEMENT
4-1 INTRODUCTION
Industrial chimneys are usually subjected to a highly corrosive environment.
Therefore, corrosion resistant materials such as Fiber Reinforced Plastic matenals (FU)
are very suitable for such a structura1 application. Actually, the construction of chimneys
using FRP diminates the need to use any protective coatings or lining and thus reduces
the cost of the chimneys. Meanwhile, the recent advances in material engineering have
led to the development of FRP matenals that have hi& temperature resistance. Due to
these factors, FRP matenals are increasingly being used in the design and construction of
chimneys. However, the application of this material in the construction of chirnneys is
hindered by the lack of a code that regulates the design of FRP chimneys.
Awad (1998) recently conducted an extensive study where a simple approach for
designing FRP chimneys under both wuid and thermal loads was developed. This design
approach @riefly described later in this chapter) needs to be verified ushg a thorough
finite element analysis. Furthemore, this approach does not account for Iarge
deformation and possible local buckling. These factors c m be a concern for FRP
structures, which are usually relatively fIexibIe due to the low stifmess of FRP.
This chapter includes an extensive finite element anaiysis for FRP chimneys
conducted using the Consistent Laminated Shell element. In order to conduct a
comprehensive analysis, the large deformation formulation of the element (presented in
Chapter Three) is extended to include the matenal non-hear behavior as well as the
thermal effects. The shell element mode1 is then used to study the typical behavior of
FRP chimneys under wind and thermal loads. Finally the mode1 is used to evaluate and
verZy the simple design procedure developed by Awad (2998).
Before presenting the current study conducted using the Consistent Laminated
Shell element, it is appropriate to provide the reader with a brief description for the
design approach developed by Awad (1998). This is followed with a description for the
thermal and the material non-linear extension of the f i t e element formulation. A number
of analysis are then conducted using the f i t e element mode1 to determine typical
behavior of FRP chimneys under thermal and wind loads. Finally, an extensive
parameûic study is conducted to evaluate the safety against strength and instability failure
for chimneys designed using the approach developed by Awad (1998).
4-2 SIMPLE DESIGN APPROACH BY AWAD
Awâd (1998) developed a simplified computer code, which can be used to design
FRP chimneys. A purpose of this study is to verim the validity and determine the
limitation of this simplified approach. The simplified procedure is based on the
following:
The Classical Lamination Theory is used to obtain the elastic properties of an
orthotropic materid equivalent to the larninated FRP composite. The property of
interest is the axial modulus of Elasticity (EJ in the longitudinal direction of the
c himney .
The FRP chimney is then treated as a cantilever bearn having the rnodulus of
Elasticity (E,) calculated above.
The wind load response is then evaluated using an approach described by Davenport
(1 993).
Maximum strains obtained frorn the analysis are evaluated and then multiplied by the
tnie modulus of the laminate to obtaul cr,, and q,, which represent the maximum
stresses in the fiber and in the across fiber directions, respectively.
The thermal analysis is conducted using the linear version of the Larninated
Consistent Shell element. The analysis has indicated that localized cracks, in the
direction perpendicular to the fibers, near the base of the chimney are expected to
occur. However it is believed that these cracks are controllable if the inter-laminar
stresses are fairh low com~ared to the allowable inter-laminar strength. Thus- the
thexmal anaiysis proceeded by assuming that the modulus of Elasticity perpendicular
to the fibers (E-J is negligible. Under this assumption, the maximum stress in the
fibers direction (o,,) was evaluated as a function of the temperature variation.
6) This thermal stress (O,,) was added to the wind stress (CF,,) to obtain the maximum
total stress in the along fibers direction (qd.
7) The along fibers stress (cl,-) and the across fibers stress (q,) are used in the
quadratic faiiure cnteria developed by Tsai et. al. (1 971). In applying this failure
cnteria, a fixed factor of safety equal to 1.6 was adopted in the across fibers direction,
while a variable factor of safety (R) was applied for the along fibers direction. Each
value for the factor of safety (R) will correspond to a certain thickness for the
chunney.
It is clear that the philosophy behind the design approach suggested by Awad (1998)
is that localized cracks are allowed to happen due to thermal loads. Along fibers
stresses due to thermal loads are evaluated by virtually assuming that the whole
chimney has suffered from cracking. On the other hand, the wind design is controlled
by the first crack, Le. cracking is not allowed to happen due to wind loads.
A purpose of this study is to use the Consistent Larninated Shell element to
conduct an iterative nonlinear analysis that follows the propagation of cracks inside the
chimney (including also the effect of geometric non-linearity). The analysis can be used
to check the actual factor of safety for the along fibers direction stresses (O,-. and also
the safety of the structure against instability failure, when the simplified approach
developed by Awad (1 998) is used.
4-3 THERMAL LOAD FORMULATION
4-3-1 Formulation Detaiis
This section includes the layout of the thermal analysis applied to the non-linear
formulation of the Consistent Laminated SheIl element. A temperature change of AT at a
certain layer L in the shell introduces initial strains b:i 1, which are expressed relative to
the material axis system (1, 2 and 3), and are given as:
Where cr, and a, are the thermal expansion coefficients of the Layer L in the material
directions 1 and 2, respectively.
Using the transfomation matrix [TE] given in Equation (2.16) the initial strains
can be calculated in the local axes system (x: y 'and z ') as follows:
In general, the relation between a stress vector (a) and a strain vector {E] when
initial strains {EA} are included, is given by:
where [D 1 is the constitutive matrix given by Equations (2.12) to (2.1 5) .
In view of Equation (4.3), the inclusion of the initial thermal strains modifies Equation
(3.4a) to be written as:
where S; = [D']{E; } (4-5)
A cornparison between Equation (4.4) and Equation (3.4a) indicates that two extra
terrns are included in Equation (4.4). The fïrçt term J ( - s , ; ) G ( A ~ ) ~ v modifies the vo
initial stress stifiess matrix [ K;'"'' 1, that was previously defined by Equation (3.6), to
be given as:
The second added term: I ( s v ) G ( A ~ , ) ~ v , given on the right-hand side of the va
equation, leads to the thermal load vector vT} which is given as:
AI1 the components of Equations (4.6) and (4.7) were previously defined in chapter 3.
The incremental equilibrium equation, which accounts for both an extemal forces
(like wind loads) and initial thermal strains, is given as:
[ ~ L f ( ~ - l ) + Q k - I ' 1 (AU} = ( R } + {fi. ) - {F J
where [KJ is the sum of the linear and the initial strain stifhess matrices, aven by
Equation (3.6).
[KSI is the initial stress stifhess matrix, given by Equation (4.6).
{F) is the unbalanced Ioad vector, given by Equation (3-7).
Cf,) is the thermal load vector, given by Equation (4.7).
(AU) is the incremental degrees of fieedom vector.
(R} is the external load vector.
Equation (4.8) is solved in an iterative mamer using the same procedure outlined
in Section (3.4). The stresses at the end of each iteration are evaluated using Equation
(4.3).
This example is used to veri@ the thermal stress formulation. fhe problem
involves the thermal analysis of a cross-ply (0°/900/00/900) clarnped cy!indrical panel
subjected to a uniform temperature load. The geometry of the panel is shown in Fig.
(3.4). In this figure, the length of the two sides of the panel (a and b in Fig. 3.4) and its
radius R are related by the following ratios: a/b=l, R/b=l. Two values for the thickness H
are considered in this analysis. These values are chosen in such a way that a = l O and
100, respectively. Each layer of the panel has the following material properties:
E, = 181 GPa, & = 10.3 GPa, G,, = G,, = 7.17 GPa,
Gz = 6.21 GPa, vI2 = 0.25, a, = 0.02 x m / d ° C , a, = 22.5 x 104 rn/m/OC
Due to symnetry in loadùig and geometry, only one quarter of the panel is
modeled using 8 Consistent Laminated Shell elements (2x2 mesh). Linear analysis only is
considered in this problem and the results are compared to those results reported by
Chandrashekhara and Bhimaraddi (1 993) where a shear flexible fini te eiement mode1 was
used to solve the same problem. The displacement at the center of the panel (MI) is
presented using the dirnensionless parameter w* which is defined as: w* = ~w/a,~,b' ;
where T, is the value of the uniforni through thickness temperature change of the panel.
Values of w* obtained in the curent analysis together with those predicted by
Chandrashekhara and Bhimaraddi (1993) are presented in Table (4.1). It can be seen fkom
Table (4.1) that a very good agreement is shown between the two sets of results.
4-4 MATEMAL NON-LINEARITY FORMULATION
As mentioned earlier in this chapter, the recent study conducted by Awad (1998)
has shown that a moderate and practical temperature variation in a FRP chirnney resulrs
in across fibers thermal stresses exceeding the dtimate strength of the ma&. As such,
cracks are expected to occur in FRP chunneys as a result of the difference between the
curing, the ambient and the operational temperatures of the chimney. These hi& thermal
stresses were found to be independent of the thickness of the chimney and are Iocalized at
the extreme fibers at the bottom of the structure. Ln the same study, Awad (1998) adapted
a conservative approach by assuming a negligible value for the across fibers stiffhess
along the whole height of the chimney. This approach assumes that cracks happen in a11
layers of the chimney. Along fiben stresses as well as shear stresses evaluated based on
this approach were presented by Awad (1998).
The objective of the current study is to conduct an accurate analysis for FRP
chinineys subjected to both thermal and wind loads and to be able to follow the
progressive development of cracks inside the chimneys. To achieve this objective, a non-
linear material behavior is added to the formulation of the Consistent Laminated Shell
element. The approach is based on the same concept applied in finite element analysis of
cracked reinforced concrete sections (Chen 1982). The non-linear materid formulation is
adopted by applying the following steps:
1) Having evaluated the solution of a certain iteration, the local stresses (in the x : y 'and
z' coordinate system) are evaluated and then transfomed to the material coordinate
system (1,2 and 3) giving the stress components (o,, G,, alz, O,,, G~).
2) The across fibers normal stresses G,, at a certain point belonging to a layer L and
located at a distance z'measured from the mid-surface of the laminate, is divided to
an axial component q and a bending component O,, using the following steps:
a) The across fibers normal stress 0; , at a point located at a distance (-z 'j' relative to
the center of the laminate, is evaluated.
b) The axial stress (GA is evaiuated as follows: ou = f
2
c) The bending stress (o,,) is then aven as: a, = oz -0,
A typical through thickness stress distribution for an angle-ply laminate is shown in
Fig. (4.1). Notice that the expected jumps in stresses result frorn the variation of the
modulus of elasticity between one iayer and the adjacent one.
3) Since FRP materials usually have ultirnate axial strengths different Erom their ultimate
bending strengths, the state of cracking of the point has to be checked using the
fo llowing equation:
where a: and D: are the axial and bending ultimate strengths of the material.
respectively.
Depending on the value of R, two possibilities can arise:
a) R 5 1, no cracks is expected to develop at the considered point.
b) R > 1, across fibers crack is expected to develop at the considered point.
Another feature of FRP materials is that both o: and CT: have different values
depending whether the considered point is in a state of compression or tension. This
feature is accommodated when Equation (4.9) is applied.
During the fïrst iteration of the initial load increment, no matenal non-linearity is
considered and the displacements, strains and stresses are evaluated based on a hl ly
elastic response of the stnichire. At the end of this iteration (and for ail following
iteration), steps 1 to 3 are applied at al1 integration points to check the material state of
these points (either cracking or behaving elasticaLly). For a cracked integration point, a
constitutive matrix [Dl,, is calculated. This matrix is similar to the one described by
Equations (2.12) and (2.15) with the exception that the elastic modulus E2 and the shear
modulus G, have negligible values (approaching zero). The constitutive rnatrïx [D 'Js6 is
then multiplied by the accumulated total strains to give the accumulated total stresses at
this particular integration point. This yields zero values for both the axial stress
perpendicular to the fibers (02) and the shear stress (o,,) at a cracked integration point.
This integration point is then marked as cracked in al1 subsequent iterations. For
integration points which have not cracked, the constitutive matrix [D liU is calculated
using Equations (2.12) and (2.15). In this approach, no change of geometry is considered
to be associated with cracking. This is illustrated by considenng Fig. (4.2) which shows
cracks happening in one layer having the fibers onented by an angle -8. The two adjacent
layers are dso displayed in Fig. (4.2) showing the fibers oriented by an angle 8. It is
expected that if no inter-laminar shear failure occurs, the two adjacent layers, which have
a high stifiess in a direction perpendicular to the crack, wilI contrd the width of this
crack. As such, inter-laminar shear stresses have to be checked in this approach to be sure
that such a behavior occurs.
4-5 TYPICAL BEHAVIOR OF FRP CHIMTWYS UNDER WrIND AND
THERMAL LOAD
4-54 Descri~tion of the Analvsis
By including the geometric and material non-linearities as well as the effect of
initial thermal strains in the formulation of the Consistent Laminated SheIl elernent, the
h i t e element mode1 is used to study the typical behavior of FRP chimneys subjected to
wind and thermal loads. A FRP chimney having a height L=50 m, a diameter D=3.3 rn
and a total thickness H=30 mm is modeled using the Laminated Consistent Shell elernent.
The chimney is assumed to be fiee at its top and have sirnply supported boundq
conditions at its bottom end-
The fmite element mesh used to analyze the chirnney is shown in Fig. (4.3). A
total nurnber of 512 Larninated Consistent Shell elements are used to mode1 the entire
c b e y ; 32 tnangular elements are used around the circumference and 16 rows of
elements are applied along the height of the chimney. As shown in Fig. (4.3), a very fine
mesh is used at the bottom of the chimney where stress concentrations are anticipated.
This also allows the detection of possible local buckling effect at this region.
In order to follow the discussion of the results of the finite element analysis, the
two sets of local and matenal axes are first described. At a certain point on the shell of the
chimney, two sets of coordinates are shown in Fig. (4.4) together with the global
coordinates (x, y and 2). Fig. (4.4) includes a vertical and horizontal projection of the
walls of the chimney. From the figure, it can be seen that both the axes x' and y' are
tangent to the surface. The x ' axis is located in a horizontal plane and the y ' axis is parallel
to the global y axis (the vertical axis of the chirnney). Fig. (4.4) also shows the matenal
axes (1 and 2) for one of the angle-ply layers. Note that 8" is the angle between the fiber
direction (axis 1) and the x'axis.
Two different themal Ioad conditions and one wind load condition are considered
in the analyses. For al1 thermal loads, a curing temperature of 50°C is assumed for the
chimne y.
a) Load Condition (L,k -
This load case simulates a non-operating state of the chimney. The arnbient
temperature is assumed to be -30°C. The difference between the ambient and the curing
temperature leads to a constant through thickness distribution of the differential
temperature having a magnitude of -80°C.
h ) k -
This load case sirnulates an operational state of the chimney. The arnbient
(extemal) and the operating (intemal) temperatures are assurned to be equal to -30°C and
70°C, respectively. This leads to a linear profile of the through thickness distribution of
the differential temperature. The external and the intemal differential temperatures are
equal to -80°C and 20°C, respectively (relative to the curing temperature).
C) Load Condition a,): -
This load case consists of a wind load increased incrementally up to 1.5 times the
design wind load as suggested by the International cornmittee on industrial chimneys
CICIND (1988). According to this code the design wind load distribution -which is used
in the analysis- is described by the following equation:
where P o is the wind load per unit vertically projected area of the chimney; Po is a
variable related to the design wind veiocity and is equal to the wind load on the chimney
at height lm above the ground; and y is the height of the point above ground level. In
this load condition, Po is evaluated based on an assumed wind speed (at height 10m above
the ground) of 30m/s. A profile for a typical wind load distribution along the height of a
chimney is shown in Fig. (4.3).
A combination of load conditions 1 and 3 as well as load conditions 2 and 3 is
also considered in the analyses. Each load condition is conducted twice; the fint analysis
assumes a fülly elastic behavior of the chimney and the second one includes the effect of
cracking (material non-linearity) as described in Section (4-4). The two load
combinations are evaluated only with material non-linearity included. As such, a total
number of eight load cases are considered and are described in Table (4.2).
The chirnney's laminates consist of vinyl ester resïn reinforced by E-glas fibers.
The lamuiae are stacked in an angle-ply configuration using an angle 0 For fibers
orientation of +/- 55 (8 is the angle between the fibers direction and the .Y' axis as
descnbed above). The fibers content of each lamina is assurned to be 70% (based on
weight). Based on the above material configuration, and using the program Tnlam II (by
Osborne Composite Eng. Ltd. Ver 94), the values of the stifiess and ultimate strength of
each Iamina are given as:
El = 36.85 GPa, &= 11.16 GPa, GI2= G13 = 3-36 GPa,
vlz= 0.30, G,= 4.32 GPa, a, = 7.7 x 10" rn/d°C, a, = 43.4 x IOd m/d°C
O: = 552.775 MPa, O:=' = 442.22 MPa, oz1 = 1105.54 MPa, 0;; = 884.44 MPa,
a:=' = 16-74 MPa, a:' = 89.25 MPa, ce = 33.47 MPa, orb' = 178.45 M a ,
ru = 70.57 MPa
where ou is the ultimate strengths of the material; the superscripts 1 and 2 correspond to
matenal axes 1 and 2, respectively; the subscnpts T and C correspond to Tensile
and compression ultimate sQengths, respectively; the subscripts a and b
correspond to axial and bending ultimate strengths, respectively and ? is the
ultimate shear strength.
A total number of three layers is used in the analysis. The layers thickness are
provided in Fig. (4.5). Although in practice a larger number of layers is usually used (a
typical thickness of a layer is approximately 1 .O mm). A prelirninary analysis has been
conducted and has shown that using a less number of layers has a slight effect on the
results and is also a conservative approach. It was found that a larger number of layers
Uicrease the displacement slightly (less than 1%). As for the stresses, the effect on normal
stresses was negligible and the shear stresses decreased slightly (in the range of 1-2%).
4-5-2 Discussion of the Results of the Analvses
4-5-2-1 Deflection Shaoes near the Base Due to Different Thermal Loading
Conditions
The deflection shapes near the base of the chimney, resulting kom the first four
load cases (Bhavl to Bhav4), are presented in Fig. (4.6). Since al1 these load cases
include axisymmetric thermal loads and also due to the axisyrnmetric nature of the
geometry, an axisymrnetric deflection shape results fiom these analyses. Fig. (4.6) shows
radial deflections plotted dong one of the generations of the chimney. From this figure.
the following observations c m be noticed:
1) The generator of the chimney is subjected to an inward deflection in al1 the load
cases. This is anticipated since the mid-surface temperature change is negative for al1
these cases.
2) Aithough analyses (BhavZ and Bhav4) indicate the occurrence of cracking near the
base of the chimney, this matenal nonlinearity has no significant ef5ect on the
deflected shape. This is demonstrated by the fact that the deflected shape resulting
fkom Bhavl is ahost identical to the one resulting fiom Bhav2. The same
observation can be drawn when comparing Bhav3 and Bhav4.
3) Deflections resulting fiom the first two load cases (Bhavl and Bhav2) are larger than
those resulting fiom the last two load cases (Bhav3 and Bhav4). This is related to the
magnitude of the mid-surface temperature change associated with the load case. This
temperature change is equal to -80°C and -30°C for the fïrst two load cases (Bhavl
and Bhav2) and the 1st IWO load cases (Bhav3 and Bhav4), respectively. Tt seems that
the maximum inward deflection is proportional to the mid-surface temperature
variation.
4) At an elevation of approximately 1.25111 above the base of the chimneys, the radial
displacements become almost constant through the height of the chimney; i.e. no
bending effect above this elevation.
5) A localized bending is shown at the bottom of the chimneys associated with a double
curvature deflection to accommodate the restraint imposed at the base of the chimney.
This area is expected to be sensitive to local buckling.
4-5-2-2 Stresses Resultin~ from Load Case Bhavl
Thermal stresses resulling f?om constant through thickness temperature change
acting on FRP chimneys are itlustrated by studying the results of the load case Bhavl .
Note that this analysis is restricted to elastic behavior ody. The distribution of themal
stresses dong the bottom 5m of the chimney that resuIt fkom the analysis are plotted in
Figures (4.7) and (4.8). Again, this stress distribution is axisymrnetric since both the
geometty and thermal loading of the chimney are axisymmetnc. In both figures, the
stresses are presented at the exterior, the middle and the interior surfaces of the chirnney.
Note that the mid-surface stresses reflect an axial effect, while the difference between the
extenor and the interior stresses represent a bending effect.
Hoop stresses (or;.) and meridional stresses (G,,;.) are presented in Fig. (4.7) and
Fig. (4.8), respectively. The following observations can be drawn fiom those two figures:
1) Both the hoop stresses (or;.) and the meridionai stresses (o,.;,~) diminishes away korn
the base of the chimney. This is expected since thermal stresses develop mainly due
to the restrain to the displacements provided at the base of the chimney.
2) High tende axial hoop stresses (at the mid-surface) are shown at t?ie base of the
chirnney. These stresses result fiom the radial support provided by the base of the
chimney that restraints the shrinkage of the bottom part of the chimney.
3) The region located between a height of O.35m and lm shows a negative compression
Hoop stresses. This behavior is in agreement with the deflected shape shown in Fig.
(4.6a) which indicates deflections at this region that are less than those away fkom the
base. As such, negative hoop strains and associated negative hoop stresses occur in
this region.
4) By examining the meridional stress distributions for the extrerne surfaces, it c m be
seen a reversed bending moment effect (double curvature) happens at the bottom part
of the chimney. At the very bottom part of the chimney, this bending action causes
tensile and compressive stresses at the intenor and extenor surfaces of the chimney,
respectively. This effect is reversed at the region localized between elevations of O J m
and 1.0m f?om the base. Such a behavior is in agreement with the deflection shape
given in Fig. (4.6a) that shows a double curvature and a point of inflection at
approximately 0.7m fkom the base.
5) Negative meridional axial stresses (at the mid-surface) are shown near the base of the
chunney. It is anticipated that these are associated with the poisson's effect and the
hoop strains.
4-5-2-3 Stresses Resultin~ from Load Case Bhav2
The effect of cracking (material non-linearity) is examined by snidying the results
of the analysis Bhav2 and comparing thern to those obtained fiom Bhavl. As mentioned
earlier, both Ioad cases are similar with the exception that Bhavl is fully elastic, and
Bhav2 accounts for matenal non-linearity. The nonlinear analysis (BhavZ) indicates that
cracking due to the thermal loads occurs in the bottom 0.4m of the chirnney. These cracks
propagate al1 through the thickness of the chimney wall in the direction perpendicular to
the fibers. These cracks lead to a redistribution of stresses at the very bottom part of the
chimney (at the bottom 0.4m) as shown by comparing Fig. (4.7) to Fig. (4.9) as well as
Fig. (4.8) to Fig. (4.10).
A significant drop in the values of the maximum stresses is noticed once material
non-linearity is included. It should be noted that dthough cracks happened at the bonom
of the chimney, no sign of total failure or instability has been shown in the anaiysis.
4-5-2-4 Stresses Resultinp from Load Case Bhav3
Thermal stresses due to linear through thickness temperature change acting on
FFW chirnneys are illustrated by snidying the results of the load case Bhav3. This anaiysis
is restricted to eiastic behavior and involves temperature changes of -80°C and 20°C at
the extenor and interior surfaces, respectively. Hoop and meridional stresses resulting
from this analysis are plotted in Fig. (4.1 1) and Fig. (4.121, respectively. By plotting the
extenor, the middle and the interior surface stresses in these two figures, the following
observations can be drawn:
1) Away fkom the base of the chimney, the axial components of both stresses (hoop and
meridional) vanish. Such a behavior is expected since no restraint to axial motion is
provided in this region.
2 ) Away fkom the base of the chimney, a pure bending behavior occurs. This is in
agreement with the behavior of isotropie cylinders subjected to a similar temperature
change (Timoshenko et al. 1959)-
3) Axial tensile hoop stresses are shown at the very bottom of the chimney. Similar to
those observed in Bhavl, these stresses result fkom the restraint of the thennal
shrinkage of the chimney provided by the base support. However, these tensile hoop
stresses are much Iess than those observed for Bhavl. This is related to the smaller
magnitude of the ternperature variation at the mid-surface, in this case relative to
Bhavl. For this anaiysis a mid-surface temperature change of -30°C ( (20°C - 80°C) 2
)
is applied compared to -80°C for Bhavl.
4) A region bounded by heights 0.35m and 1.Om shows a small value of negative hoop
stresses. This behavior is in agreement with the deformation mode of the chimney
shown in Fig. (4.6~).
5 ) The variation of bending meridional stresses at the base follows the curvature of the
deflection given in Fig (4 .6~) . A smail peak is noticed at approximately an elevation
of O.8m. Through the effect of poisson's ratio, a similar behavior is s h o w for the
hoop bending stresses shown in Fig. (4.1 1).
4-5-2-5 Stresses Resaltinp - from Load Case Bhav4
The eMect of the materid non-linearity (cracking) on the behavior of FRP
chimneys subjected to iinearly varying through thickness thermal load is studied by
considering the results of the load case Bhav4. The analysis has shown that cracking due
to this load case is very Limited and localized at the external surface near the base of the
chirnney. The reasons that limit cracking of the chimney compared to the constant
through thickness thermal load case (Bhav2) are described as follows:
1) The mid-surface axial stresses resulting fiom the linear temperature distribution
(Bhav3) are much less than those due to the constant temperature distribution
(Bhavl). These stress values are directly proportional to the value of the temperature
change at the mid-surface of the shell. This is also manifested by the smaller
deformations occurring in Bhav3 and Bhav4 compared to those happening in Bhavi
and Bhav2.
2) The varying temperature load case (Bhav3) results in bending stresses that are larger
than those obtained f?om the constant temperature load case (Bhavl). These bending
stresses act through the whole length of the chimney. Since the ultimate capacity of
FRP materials in bending is double its capacity in axial, the bending stresses do not
cause significant cracking of the chimney.
Stresses resulting fiom the load case Bhav4 are presented in Figures (4.13) and
(4.14). Cornparison between the stresses shown in these figures and those given in
Figures (4.1 1 ) and (4.12) indicate that the linear analysis (Bhav3) and the non-linear
analysis (Bhav4) yield almost the same stresses.
4-5-2-6 Stresses Resulting from Load Cases Bhav5 and Bhav6
These load cases include the wind load described in sub-section (4-5-1) acting on
the chirnney. Load case (Bhav5) is a purely linear analysis, while load case (Bhav6)
involves the effect of matenal non-linearity (cracking). Results of the analysis Bhav6
indicates that no cracking happens as a result of the wind load. As such, the two load
cases yield exactly the same results. The deformation patterns resulting of those analyses
are plotted in Fig. (4.15) along two generators that belong to a plane parallel to the
direction of the wind load. This load is acting paraIlel to the x- axis shown in Fig. (4.4).
One of these generators (wind side) passes through point A shown in Fig. (4.4), while the
other one (lee side) passes through point B shown in the same fi,we.
The deforrnation patterns shown in Fig. (4.15) indicate a small bending effect at
the bottom of the chimney, especially at the lee side generator which is under
compression due to the wind load. Stresses resulting fiom these load cases are plotted in
Fig. (4.16) and Fig. (4.17). The following observations c m be drawn kom these figures:
1) Away fkorn the base, meridional axial stresses in both the lee and the wind sides are
constant through the thickness causing no overall bending (about the center of the
circular cross section). These axial stresses are compression at the lee side and tension
at the wind side.
2 ) A small meridionai bending effect is shown near the base (see Fig. 4.17) which agrees
wiîh the deformation patterns given in Fig. (4.15).
3) Hoop stresses away fiom the base are aimost negligible. Hoop sbesses develop near
the base as a result of the slight ovdling occumhg in this region. This is manifested
by the slight difference between the lee side and wind deformation in this area
presented in Fig. (4.15).
4-5-2-7 Stresses Resulting from Load Cases Bhav7 and Bhav8
The combined effect of thermal and wind loads is studied in these two load cases.
Load case Bhav7 is a combination of the wind load and the constant through thickness
temperature load. Meanwhile, load case Bhav8 combines the effect of the wind load and
the linearly varying through thickness temperature load. The two analyses inciude the
effect of material cracking. Results obtained Eoom boîh analyses (presented in Figures
4.18 to 4.2 1) indicate the following:
1) The maximum negative stresses occur at the extenor surface of the lee side of the
chimney. The hoop and mendional stress distributions aiong a generator, belonging to
this surface, are plotted in Fig. (4.18) and Fig. (4.19), respectively.
2) The maximum positive stresses occur at the interior surface of the wind side of the
chimney. The hoop and meridional stress distributions aiong a generator, belonging to
this surface, are plotted in Fig. (4.20) and Fig. (4.21), respectively.
3) The maximum positive stresses exceed the maximum negative stresses. A cornparison
between Figures (4.18), (4. N), (4.20) and (4.2 1) indicate that the absolute maximum
stress occurs near the base of the chimney at approximately an elevation of O.4m from
the base.
4) Stresses obtained frorn the load case Bhav7 exceed those obtained fiom the load case
Bhav8. As such, for stress calculation a constant through thickness temperature
variation, which simulates a non-operating state of the chimney, is the cntical one.
However, when resistance to local buckling is considered, which is discussed at a
following sub-section, linearly variable temperature change rnight be more critical.
In order to understand why the positive stresses at the interior surface of the wind
side exceed the negative ones occurring at the extenor surface of the lee side, stresses
obtained from individual thermal load cases (Bhav2 and Bhav4) are studied together with
the wind load case (Bhav5). Ln general, stresses encountered in the analyses Bhav7 and
Bhav8 result ftom a combination of thee effects:
1) Thermal stresses including both bending and axial components.
2) Axial stresses due to wind loads.
3) A relatively small bending component of the stresses associated with the wind loads.
At the intenor surface of the wind side, all the above stress cornponents are tensile
and their effects are added to give a maximum value for the stresses. MeanwhiIe, for the
extenor surface of the lee side, the two stress components are compressive while the
third is tensile. This explains the occurrence absolute of ma,uimum stresses at the interior
surface of the wind side of the chimney.
4-6 ASSESSMENT FOR THE DESIGN PROCEDURE DEWLOPED BY AWAD
The design procedure developed by Awad (1998) for FRP chirnneys subjected to
both wind and thermal loads has been bnefiy discussed in Section (4-2). The procedure is
simple and c m be easily applied by a design engineer since it uses a beam mode1 to
simulate the behavior of a FRP chimney. The effects of both the constant and the linearly
variable through thickness temperature distributions are considered in this simple
procedure. Meanwhile, the cornputer design tool, developed by Awad (1998), includes a
dynarnic analysis that considers the effects of various components of the wind load. In
general, the along wind component of the wind load can be accurately descnbed by the
equivalent static load described by Equation (4.10).
According to Awad (1998), it was found that for chimneys that have a relatively
high value for the length to diameter ratio (LJD), the design is govemed by the strength
capacity due to the effect of the along wind load. Meanwhile, chimneys that have iow
values for the ratio (Lm) are govemed by fatigue failure due to the vortex shedding
effect. A typical design curve, developed according to Awad (1998) procedure, is
presented in Fig. (4.22). In this figure, the ascending straight line represents the design
thicknesses governed by strength failure due to dong wind loads. Meanwhile, the
descendhg curve provides the design thicknesses that are controlled by the fatigue
strength due to across wind loads (vortex shedding effect).
As descnbed in Section (4-2), Awad's design procedure was based on the
constraint of not allowing any crack in the resin to happen due to wind loads (a factor of
safety of 1.6 was used in the across fibers direction). Meanwhile, a variable factor of
safety (R) is applied in the along fibers direction.
In addition to the geometry of the chùnney and the mechanical properties of the
laminate, a typical ascending straight line shown in Fig. (4.22) depends slightly on the
assurned value for the factor of safety (R). In fact, this line moves slightly up when (R) is
increased. The reason that the variation with (R) is not very significant is that the design
of the chimney is usually controlled by the condition that no cracking is allowed to
happen due to wind loads. (Le. design is controlled by the across fibers direction).
Beside al1 the above factors, the location of a typical descending design thickness
curve is greatly af5ected by the value assumed for the damping ratio of the material. This
is related to the fact that this descending curve is govemed by the across wind loads
(vortex shedding) that have a dynamic nature,
The Consistent Laminated shell element mode1 that includes the effect of
geometric non-linearity (large deformations) and cm simulate the propagation of cracks
through a laminated shell stnicture, which was developed in the current study, can be
used to asses the simplified procedure developed by Awad (1998). The procedure used to
assess Awad's design method is summarized in the fotlowing steps:
1) A FRP chimney having a certain length L and a diameter D is designed using the
cornputer code developed by Awad (1998). In the design, the FRP chimney consists
of an angle-ply laminate having Vinyl Ester resin and E-glas fibers. A 70% fibers
content and an angle of inclination 8 = 55" (measured with the horizontal plane) is
used for the angle ply laminate. The design of the chimney includes the effects of
both the wind and the thermal loads. A constant through thickness temperature
variation similar to the one described in the load condition (L,), given in Section
(4-5-1), is used in the design. Only the along wind direction component of the wind
Load is included in the design (Le. vortex shedding is not considered here). The wind
speed used in the design is assumed to be 30m/s at height of 10m and the load factors
of (1.35) and (1.5) are applied to the thermal and whd loads, respectively. A factor of
safety (R) is used for the along fibers direction in the failure critena described in
Section (4-2). Meanwhile, as mentioned earlier, no cracking is allowed in the acrass
fibers direction and a factor of safety of (1.6) is used in this direction. A certain
thickness for the chimney, that satisfies these design requirements, is predicted by the
simplified computer code (Awad 19%).
2) This chimney is modeled using a finite element mesh of Consistent Laminated shell
elements sirnilar to the one shown in Fig. (4.3). The thickness calculated in step (1) is
divided to three layers with the sarne proportionality shown in Fig. (4.5).
3) A non-linear incremental analysis (ïncluding both geometric and material non-
linearities) is conducted for the FRP chimney. The analysis includes the constant
through thiclaess thermal load used in step (1) and an equivalent static along wind
load descnbed by Equation (4.10). In this equation Po is calculated according to the
procedure provided by the International Committee for Indushial Chimneys CICIND
(1 988) and using a wind speed U,, of 30m/s, (similar to the one applied in step (1 )
above). A Load factor of (1.5) for the wind loads is also applied. In the non-linear
analysis, the hrst load increment includes oniy the temperature load. In the
subsequent increments the wind coefficient, used in Equation (4. IO), is incrementally
increased up to a value of 1 . 5 ~ Po. Note that cracks are allowed to happen and
propagate in the chùnney (in the direction perpendicular to the fibers) during the
incremental anaIy sis.
4) The maximum along fibers stresses (0,,3 resulting fiom the non-linear analysis are
evaluated. This is divided to an axial stress (03 and a bending stress (o,) using the
approach described by step (2) in Section (4-4).
5) A factor of safety (FS,) in the along fibers direction is then evaluated using the
following equation :
a: is the ultimate axial resistance in the along fibers direction and is substituted with
u 1 Ll 1 b ~ a or O,, , given in Section (4-5-l), depending whether oa is tension or
compression. Sirnilarly, o: is the ultimate bending resistance in the along fibers
direction and is substituted with O&' and O:', also given in Subsection (4-5-l),
depending on the sign of a, .
The maximum values for the transverse shear inter-laminar stresses a,,,, and G~,,
are also recorded. A factor of safety (FS,,) for inter-laminai- failure in the (1-3)
direction and a similar factor of safety (ES,) in the (2-3) direction are evaluated
using the following equations:
where P is the ultimate inter-larninar transverse shear strength of the material, given
previously in Subsection (4-5- 1).
6) FaiIure of the chimney would happen if any of these three situations occurs:
a) The factor of safety (ES,) is less than one; this Lead to strength ffaiure in the along
fibers direction.
b) Any of the transverse shear stresses a,,, and o,, exceed the ultimate inter-
laminar shear strength of the laminate(i.e. either (F-S, , ) or (FS,) are less than
unity) ; this results in delamination failure.
c) A local buckling happening at the base of the chimney and resulting in zero
stifhess of the entire structure; Ùiis results in instability failure.
The analysis has been conducted for a large number of chirnneys covering a
practical range of dimensions. Three different heights (L = 30, 40 and 50m) are
considered in the analysis and for each height five different values for the aspect ratio
(Lm) are considered (10, 12.5, 15, 17.5 and 20). This leads to a total number of 15
chirnneys. Each of the these chimneys is designed according to step (1) by assuming a
factor of safety (R) equal to 2, 3 and 4. Table (4.3) summarizes the properties of the
chimney designs used in this section. Results of these analyses are presented in the next
subsection.
4-64 Assessment for Strewth Failure
The factors of safety (F-SI), (F.S13) and (F-S,) resulting from the analyses are
presented in Table (4.4). The variation of the factor of safety (F-S,) with the aspect ratio
(Lm) is plotted in Figures (4.23) to (4.25) for R = 2, 3 and 4, respectively. Studying the
above plots and also the vaiues tabulated in Table (4.4), the following observation can be
no ted:
1) The factors of safety of transverse shear stresses (F-S,,) and (F-S,) are very hi&. The
minimum value of these factors is found to be equd to 24.56. This indicates that
inter-larninar failure is not a concem for FRP chimneys. It shodd be noted that these
analyses have been conducted using three layers for the FRP laminate. In reality, the
number of layers are usually much larger than that. However, the preliminary analysis
ùiat was conducted to assess the effect of the number of Iayers indicates that the inter-
laminar shear stresses are reduced with the increase of the number of layers. As such,
the values provided in Tabte (4.4) represent an upper limit for these stresses. Analyses
indicate that the shear stresses (O,,) and (oz) result mainly fi-om resolving the
transverse shear stress ( ~ 3 . A typical distribution for the shear stress (04 through
the thickness of the laminate resulting fkom one of the analyses is shown in Fig.
(4.26).
2) The values of the factor of safety (ES,) range between 5.77 and 8.33. This indicates
that chimneys, designed according to Awad (1998), with a factor of safety (R) equal
to 2 or higher, are fairly safe against strength failure. These values of the factor of
safety (FS,) seern to agree with common practice of FRP filament widening design
where a factor of safety of 5 is commonly targeted. Such a high factor of safety is
applied in order to account for some effects such as creep and degradation of strengîh
due to moisture absorption.
3) Although the factors of safety (ES ,) increase with the increase of the factors of safety
(R), used in the initial design, performed according to Awad (1988), this variation is
not signifiant and is not proportional with the increase in CR). The reason behind the
minor effect of (R), is that in Awad's design procedure the thickness of the chimney
is governed mainly by the condition of no crack imposed on the across fibers
directions (stress cd. As such, a variation in (R) does not have a major effect on the
selection of the appropriate thickness.
4) As a general trend, the factor of safety (ES ,) increases with the increase of the height
of the chimney-
The high values of the true factors of safety (ES,) relative to the assurned one (R)
can be interpreted by the following two reasons:
In Awad's procedure, the effect of the thermal stresses was accounted for by
assuming that the whole chimney has suffered fiom cracks. This is a conservative
assumption that overestimates the thermal stresses in the dong fibers direction.
The design procedure adopted by Awad (1998) is govemed by first cracking. This
condition limits the capacity of the laminate and significantIy reduces the stresses
acting in the along fibers direction.
It is evident that the first cracking condition, which is commonly used in the FRP
industry, does not make use of the high strength capacity of a FRP laminate in the fibers
direction. For FRP chimneys, where wind Ioads cause maidy meridional stresses, this
effect wi11 be more obvious when smaller values of the fibers inclination (8) are used. As
such, for angles of inclination (9 less than 55" (a value of 35" is cornmonIy used), higher
values for the factor of safety (ES,) would be expected.
Although, this study has been conducted using only the dong wind component of
the wind load, simulated as an equivalent static load, the same conclusions are anhcipated
to result fiom a dynamic analysis conducted using vortex shedding loads. However, this
c m not be confirmed until a dynamic analysis (which is beyond the scope of this study) is
conducted.
4-6-2 Assessrnent for Instabilitv Failure
The previous subsection covered an assessrnent for the two possible strength
failure modes (Fibers failure and delamination). The instability failure mode1 due to
possible local buckling of the FRP shell is discussed in this subsection. The following
steps are performed to evaluate the buckling capacity of FRP chimneys:
1) FRP chimneys modeled using the Consistent Laminated shell element are subjected to
both themal and wind loads. Thermal loads are applied as the first load hcrement in
the analysis. This is followed by adding incremental values for the wind loads. Both
constant and linearly variable through thickness temperature distribution are
considered separately for each chimney. The variation of the applied wind loads P&/
along the height of the chimney has the same distribution descnbed by Equation
(4.10). The magnitude of the load is incremented by multiplying the variable by an
incremental load factor (F).
2) The value of P(y) is gradually increased (by increasing the factor F) at various load
increments. The stifniess of the structure will correspondingly decrease reaching a
zero value at P=P, (P, = F, x Po). P, corresponds to the limit load of the structure.
Two possibilities can arise here:
a) P,, is smaller than the factored design wind load (Fm c 1.5). According to the limit
state design, this means that the chimney fails by instability due to the combined
effect of the themal and the wind loads.
b) P, is greater than the factored design wind load (Fm > 1.5). This implies that no
instability failure would occur. The factor F, = PJP0 is a measure for the factor of
safety of the structure against instability failure.
This study has been conducted for the chimneys designed using the factor of
safety R=2, which was considered in the previous subsection. These chimneys, which
have the least values for the thickness, are believed to be the most cntical to buckling.
Results of this buckling study are presented in Table (4.5). The fïrst four columns of the
table includes a description of the geometry of the analyzed chimneys. The last column
indicates the type of thermal load (either constant or linearly variable) which led to a
minimum value for P, Columns five and six provides the values of the critical buckling
loads P, and the critical load factor Fm respectively. The general observation that can be
made from the values provided in this table is that chimneys having aspect ratio (Lm)
less than 15 are very likely to buckle. Al1 of these chimneys (except for the 50m chimney
with L/D=12.5) have a critical buckling load which is less than the factored design wind
load.
In order to study the type of instability failure obtained in the analyses, the
displacement in the x axis direction corresponding to the critical load P, are plotted at the
bottom of two chimneys (see Fig. 4.27). The h s t chimney has a ratio (UD) less than 15
and the second one has a ratio (Lm) larger than 15. A clear sign of local buckling is
shown at the bottom of the first chirnney. This behavior is typical for al1 chimneys having
a ratio (Lm) less than 15. Meanwhile, the local buckling switch direction for the second
chimney. Again, this behavior is typical for chimneys having a ratio (Lm) exceeding 15
which have shown a very hi& lirnit load value (P,). It is anticipated that the failure of
these chimneys is due to this local buckling.
The buckling results obtained in this study depend largely on the magnitude of the
assumed thermal load. For the considered magnitude of thermal loads, it can be
concluded that chimneys having a (Lm) ratio greater than 15 are safe against buckling.
On the other hand, those having a (Lm) ratio less than 15, are shown to fail by local
buckling. However, according to Awad (1998) the designs of chimneys having (L/D)
ratios less than 15 are usually govemed by fatigue failure due to vortex shedding. This
effect is not accounted for when the thicknesses of the analyzed chimneys have been
selected.
Chimneys having a (L/D) ratio equal to 10 have been redesigned by considering
the fatigue failure due to vortex shedding (using Awad's cornputer code) and assurning a
damping ratio of 2% for the FRP materiai. Given the uncertainty in the damping ratio for
FRP materials, a 2% damping is considered a high value to be used by a design Engineer.
Actually, experirnental study conducted by Awad (1998), indicated that the damping of
E-glass Vinyl Ester composites can be as low as 0.7%. Since the thiclmess of the chimney
increases with the decrease of the assumed value of the damping ratio, the thickness
based on 2% damping can be considered as a lower bound for the thickness of the
chimneys. The new thicknesses of the three redesigned chimneys, that have (Lm = IO),
are provided in Table (4.6). Buckling analysis conducted on these three chimneys indicate
that load factors (Fm) exceeding a value of 2.8 is achieved for al1 of them. These results
are indicators that local buckling is not a major concern for chimneys designed according
to the procedure developed by Awad (1998).
4-7 CONCLUSION
This chapter includes an extension of the formulation of the Consistent Laminated
shell element to include a materid nonlinearity mode1 that accounts for the propagation
of cracks in a larninated FRP structure. Thennal loads due to constant and linearly
varying through thickness temperature changes are also incorporated into the model.
The model is used to study the typical behavior of FRP chimneys to both thermal
and wind load. The following conclusions cm be drawn fkom this study:
Both constant and linearly variable through thickness temperature change lead to hi&
stress concentrations near the base of the chimneys. These stresses are associated with
high bending effects especidly in the meridional direction.
Through thickness cracking in the across fibers direction and near the base of the
chimney is expected to happen due to a constant through thickness temperature
profile. This leads to a redistribution of stresses in this region. Meanwhile, it is found
that cracking is very limited in the case of linearly variable temperature change and
has no significant effect on the Final thermal stress distribution. Ln general, it can be
concluded that a constant through thickness temperature change, simulating a non-
operational state for the chimney, is more criticd for checking the strength failure of
FRP chimneys.
Wind loads result rnainly in axial saesses. A small component of bending stresses
occurs near the base of the chiwiev due to these loads- No cracks are shown to
happen due to wind loads. A cornparison between maximum stresses obtained from
different analyses indicates that the thermal ioads are equaliy important as the wind
loads as both leads to the same order of magnitude of stresses.
4) Due to the combined effects of wiad and themal loads, the absolute maximum stress
occurs near the base at the interior surface of the wind side of the chimney.
The mode1 is then used to assess the level of safety provided by the design
procedure previously developed by Awad (1998). Nonlinear analysis conducted on a
number of FRP chimneys, designed according to Awad (1998), indicate the following:
Very large factors of safety are achieved against inter-laminar shear failure. As such,
this type of fâilure is not a concem for FRP chimneys.
Factors of safety (ES,) for the along fibers stresses ranging between 5.77 and 8.33 are
predicted by the nonlinear analysis. Accordhg to Awad's procedure, these correspond
to a factor of safety (R) ranging between 2 and 4. The variation of (F-S,), with (R) is
shown to be not significant. The above results indicate that the design procedure
proposed by Awad (1998) is safe when static along wind loads are considered. It is
expected that Awad7s procedure for the design of the dynarnic effect of the across
wind loads (vortex shedding) to be similarly safe. However, this needs to be venfied
before a general conclusion cm be drawn.
FRP chimneys having a height to diameter ratio (Lm) larger than 15 are shown to be
safe against instability failure due to local buckling. Meanwhile, FRP chùnneys
having a height to diameter ratio (L/D) less than 15 are shown to fail due to instability
initiated by local buckling at the base of the chimneys. These chimneys are designed
according to Awad's procedure to sustain only dong wind loads. The buckling
capacity of these c h e y s is found to be more sensitive to linearly variable through
thickness temperature variations. However, the same chimneys, when designed
according to Awad's procedure to resist fatigue due to vortex shedding, have s h o w
to be safe against buckling. This buckiing study provides an indication that the design
procedure suggested by Awad (1998), including both strength and fatigue resistance,
provides safe design againçt instability failure.
Table (4.1): Non-Dimensional Central Deflection ( w * ) of a Cross-
Ply Cylindrical Panel under Thermal Load.
fi
L
10 1 O0
Load case
- - - -
Type of Analysis Lo ad Condit ions
(Mat enal No nlineari t y)
Percentage dinerence
(%) 1
1.5 0.75
ResuIts by Chandrashekhara
(tv* ) 9.9417 1.4100
I 1
1 Bhav6 1 L3 1 No nlinear
Results by present Study
(w*) 9.78876
1.4
Bhav 1 Bhav2 B hav3 Bhav4 Shav5
Table (4.2): Summary of the Load Cases used in Behavior Study.
L1 LI L2 L2 L3
Bhav7 Bhav8
Linear Nonlinear
Linear No nlinear
- Linear
L I +L3 L2+L3
No nlinear No nlinear
Table (4.3): Geometry of Chimneys used in the Strength Assessrnerit.
Table (4.4): Results of the Strength Assessment.
L = 30m
Dia- (ml
Dia. (ml
4 3.2
2.66 2.22
2
Thickness (mm)
WD
I O 12.5 15
17.5 20
L = 4 0 m
Thic kness (mm)
16 2 1 26 33 38
Design Io ad (Pa)
L = Som
10 12.5 15
17.5
Design 10 ad [Pa)
496.5 509.71 523.5 541.7 554.35
15 19
23.5 30
517.3 527.94 541 -4 555.2
569.93
Dia. (ml
5 4
3.33 2.86 2.5
3 1 11.5
3 1 10.5 2.4 1 13.5
20 10
12-5 15
17.5 20
494.6 506.03 518-94 536.5
479.03 491.9 505.85 521.3
2.4 2
1.71 1.5
5 4
3.33 2-86
514.88 1 4
2 1-71
Thickness (mm)
2 1 27 34 41 50
14.5 18.5 22.5 27
19.5 25 3 1 38
525.49 536.6
549.29 16.5 20
35 14 18
22.5 28 33
Design Io ad (Pa)
481.27 494.88 510.2 525.4 543-25 ,
3.2 2.66 2.22
1.5 3
2.4 2
1.71 1.5
563-11 513.57 524.16 535.3
548.06 560.73
24 1 O 13 16
19.5 23
2 4
3.2 2.66 2.22
2
537-8 476.66 490.29 504.33 515.11 532.9
46 18 24 30 36
42.5
549.4 492.65 504.08 516.98 532.75 545.66
2.5 5 4
3.33 2-86 2.5
Fcr = PcrfP,
2.22 1 28 1 18.0 1 3156 1 5.92 1 Constant
Thermal Io ad
3 1 O 10-0 330 2.4 13 12-5 450 2 16 15.0 2500
1.71 19.5 17.5 3050 1.5 23 20.0 3530 4 14 10-0 330
3 -2 18 12.5 449 2-66 22.5 15.0 2463
4 1 24 1 12.5 1 1700 1 3.47 1 Constant
0.64 0.86 4.67 5.57 6.30 0.67 0-89 4.76
2 33 20.0 3825 5 18 10-0 325
3.33 1 30 1 15.0 1 2515 1 4.99 1 Constant
Linear L inear
Constant Cons tant Constant
L inear Linear
Constant
2.86 1 36 1 17.5 1 3503 1 6.76 1 Constant
7.00 0.68
Constant Linear
Table (4.5): Results of the Buckling Analysis.
2.5 42.5 20.0 3700
Table (4.6): Results of the Revised ~ u c k l i n ~ Analysis.
6.94
Diameter (ml 3 4 5
Constant
Thickness t'pril)
33 45 58
L/D
@ 10.0 10.0 10-0
Buckting Thermal load Pcr
1450 1450 1600
la ad I
Fcr = P,PO
2.8 1 2.94 3.35
Linear Linear Linear
Typical stress Diagram
Fin 4- 1: A tvpical throunh Thickness Stress Distribution for an Angle-PLV Laminate
Fig 4.2: Fibers Configuration for Adjacent Angle-Ply Iavers
Wind Load distribution
Fig 4.3: A Typical Finite Element Mesh used in the halysis
Fig 4.4: Coordinate Systerns used in an the AnaIvsis
Vertical projection
Horizontai projection
Fig 4.5: A Section through the WalI Thickness of the Chimney used in the AnaIysis
a) Bhav 1 b) Bbav 2
-5 -4 -3 -2 -1 O
Radial displacement (mm)
-5 -4 -3 - 7 - 1 O
Radial d isplacenien t (nim )
C ) Bhav 3 c i ) Bhav 4
-5 -4 -3 -2 - 1 O
Radial displacement (mm) Radial displacernent (mm)
Fin 4.6: Deflection Shape at the Base of the Chirnnev for Different Thermal Load Cases
- - bhav 1 Exterior Surface - - - bhav 1 Intenor Surface
bhav 1 Mid-Surface
O 1 2 3 4 5
Height (m)
Fig 4.7: HOOD Stress Distribution at the Bottom Sm of the Chimnev for Bhav 1
- - bhav I Exterior Surface - - - bhav 1 lnterior Surface
bhav 1 Mid-Surface
Fig 4.8: Meridional Stress Distribution at the Bottom 5m of the Chimnev for Bhavl
- - bhav 2 Exterior Surface - - - bhav 2 Interior Surface
Fig 4.9: HOOD Stress Distribution at the Bottom 5m of the Chimnev for Bhav2
bhav 2 Exterior Surface bhav 2 Interior Surface
Height (m)
Fig 4.10: Meridional Stress Distribution at the Bottom 5m of the Chirnnev for Bhav2
- bhav 3 Exterior Surface - - - bhav 3 Interior Surface bhav 3 Mid-Surface
Fie; 4.1 1: Hoop Stress Distribution at the Bottom Sm of the Chimnev for Bhav3
- - bhav 3 Exterior Surface - - - bhav 3 Intenor Surface
bhav 3 Mid-Surface
= \ - > -,/ -2.5€+07 ; ' '
O 1 2 3 4 5
Height (m)
Fin 4.12: Meridional Stress Distribution at the Bottom Sm of the Chininev for Bhav3
- - bhav 4 Exterior Surface - - - bhav 4 Interior Surface
Height (m)
Fig 4.13: Hoop Stress Distribution at the Bottom 5m of the Chimnev for Bhav4
- - bhav4 Exlerior Surface - - - bhav 4 Interior Surface
Height (m)
Fie 4.14: Meridional Stress Distribution at the Bottom 5x11 of the Chimnev for Bhav4
a) Wind side b) Lee side
O 1 - 3 3 4 5
Displacement along x-axis (mm)
O I - 3 4 5 1
Displacement alorig s-asis ( r i i i i i )
Fin 4.15: Deflection S h a ~ e a t the Base of the Chimnev Due to Wind Load
Fip 4-14: WOOD Stress Distribution at the Bottom Sm of the Chimnev for Bhriv5 Lk Bhal.6
2E+07
1 5E+07
1E+07
- - bhav 5 Exterior Surface (Lee side) - - - bhav 5 lnterior Surface (Lee side)
bhav 5 Mid-Surface (Lee side) - - bhav 5 Exterior Surface (Wind side)
, , , bhav 5 lnterior Surface (Wind side) - bhav 5 Mid-Surface (Wind side)
\ \
- - bhav 5 Exterior Surface (Lee side) . - - - bhav 5 Interior Surface (Lee side) \ bhav 5 MidSurface (Lee side) - - bhav 5 Exterior Surface (Wind side) \ , , , bhav 5 Intefior Sutface (Wind side)
Fi2 4.17: Meridional Stress Distribution at the Bottom 5m of the Chimnev for Bhat-5 c9L Bhav6
- bhav 5 Mid-Surface {Wind side)
_-_----------- - - - - - _ _ -
I -1 E+07
'
-I -
l l l l l l l l l l ' l l i l l l l r l ' t l l l O 1 2 3 4 5
Height (m)
, , I t ~ , , l 1 1 1 1 1 1 l 1 1 , 1 ! 1 , 1 1
O 1 2 3 4 5
Height (rn)
Fip: 4.18: Hoop Stress Distribution at the Bottom of the Lee Side Ext. Surtace of the Chtmnet-
- - B h a ~ 7 !Vinci t I s t . ( [.cc ~IJL ' ) - - - Bhav S \Vinci E\I . (1.t.c iirit.,
Height (m)
Fig 4.19: Meridional Stress Distribution at the Bottom of the Lee Side Ext. Surface of the Clitnincv
- - Bhav 7 Wind [nt. ( Wirid srdr)
- - - B h a v 8 W i n d t n t . ( W i n d s i J c )
Fie 4-20: Hoop Stress Distribution at the Bottorn o f the Wind Side Int. Surface oK the Chimnev
- : 4 31 - $1 1 - 2 i 3 I
- - Bila\ 7 WÏnd lnt. ii hi - - Bha\ 8 Hïtid I i i i . ( \i'itid
\ :\ ' - - - - - - \ - - - + - - - _
- - - - \ - - \
\
- \ - - - \ - ---- - - - - - _ _ - - - - - - +_____
- - - - - - - - - - _ _ _ _ _ - - - -
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 1
O 1 2 3 4 5
Height (m
Fip 4-21: Meridional Stress Distribution at the Bottom of the Wind Side Int. Surface of thc Chinine\-
Fig 4.22: Twical Design Curve for FRP Chimnevs Accordine to Alvaci ( 19981
Fig: 4.23: Factor of Safetv (F-SI) for Chimneys Designed for R = 2 -
Fin 4.24: Factor of Safetv (F.S,) for Chimnevs Designed for R = 3 -
Firr 4.25: Factor of Safetv (FS,) - for Chimnevs Desiened for R = 4
S hear Stresses qz (Pa)
Fie; 4.26: Typical Shear Stress (O,) Distribution at the Base of a Chimncv undcr \Vinci L o d
-1.5 -1.0 -0.5 0.0 0.5
Displacement along x-auis (mm)
O 5 1 O 15
Displacement along s-mis (mm)
F ~ P 4.27: Buckling Modes at the Base of the Chimney for Di fferent WD Ranges
CHAPTER FIVE
CONCLUSIONS
5-1 INTRODUCTION
This Thesis includes bvo research studies related to larninated shelt stnictures. The
first study involves the extension of the formulation of the Consistent Lanlinated slit.11
element to include large displacement static and dynamic analysis. The secoiid study is an
application of the developed finite element mode1 involving nonlinear analysis of FRP
chimneys subjected to both thermal and wind loads.
The major conclusion drawn from these iwo studies are presented in the foIlo\ving
section.
5-2 SUMMARY AND CONCLUSIONS
5-2-1 Nonlinear Extension of the Consistent Laminated Shell Element
A literature review of the state-of-the-art finite element development related to the
analysis of larninated shell structures is presented in Chapter One. This review reveaIs tl-iat
a need exists to develop finite element models that are cornputationally eficient and can
also prcdict the physical behavior of laminated shell structures. Tlie Corisisteiil Laiiiiriatcd
shell eleinent, that was previously dcveloped by Koziey (1993) and is prcsciilccl i i i ('Iiaptcr
Two of this Thesis, satisfies these requirements. A major advantage of tliis elenicii~ is bciiig
free from the "locking phenomenon" associated with isoparanietric sheli clements.
Meanwhile, the element has the capability of predicting strain and stress distributiotis that
are consistent with the exact physical behavior of laminated sheI1 structures-
In chapter Three of the thesis, the formdation of the Consistent Laniinated shell
element is extended to account for large displacement static and dynamic analysis. Tlic
large displacement formulation is based on the total Lagrangian approach and the soltitio~i
is carrïed out incrementally using the Newton-Raphson method for iterations. Expressions
for the nonlinear stiffness matrix, the unbalanced load vector and the consistait mass
matrix, as well as a simpIe procedure for coding those matrices are presented. The c s ~ e n d d
formulation has been venfied by performing large displacement static and dynainic analysis
for a number of plate and shell problerns and cornparing the results to tliose availablr in the
Iiterature. In a11 the examples, the element has shown a very good perFornlance.
5-2-2 Nonlinear Analvsis of FRP Chimnevs
In Chapter Four, the large displacement formulation of the Consistent Larninatcd
shell elernent is extended to include thermal loads as well as a material nonlinearity mode1
that duplicates the propagation of cracks inside a FRP laminated structure. The extended
finite element mode1 is then used to study the nonlinear behavior of FRP ci-iirnneys. The
typical behavior of FRP chimneys under both constant and Iinearly variable tlirotigli
thickness temperature loads as well as under wind loads is exaniined usi~ig tlic liiiiic
element model. The following conclusions are drawn from this part of tlic siudy:
1 ) High stress concentrations near the base o f the chimneys results frotn both the conslant
and the linearly variable temperature loads. A meridional bending e fTcct is cissociarcd
with these stresses.
2) Cracks in the across fibers direction are more observed in FRP chimneys subjected io
constant through thickness temperature change than in the ones subjected to linearly
variable temperature change. For constant temperature loads, these cracks cciusc a
significant redistribution ofstresses near the base of the chimney. For the case of lincar
temperature change, the limited cracks have no significant effect on the stress
distribution. In general, it can be stated that constant temperature change. siniiilatiny a
non-operational state for the chimney, should be considered wlien the strengtli hi lure O f *
the chirnneys is checked.
3) Stresses due to thermal loads are found to be in the same order of niagnitude of those
resulting from wind loads. Both types of loading should be considered wlien the design
of such a composite structure is studied.
The investigation in Chapter Four proceeds by using the nonlinear finite clcitleiit
mode1 to assess the level of safety provided by a sirnplified design procedure for FRP
stacks previously developed by Awad (1998). Nonlinear analyses conducted Tor a nuniber
of F W chirnneys, designed according to Awad' s procedure, indicate the fo 110 w ing :
1) Very large factors of safety are achieved for inter-laminar shear failure. As sucli. tliis
type of failure is not a concern for FRP chimneys.
The factors of safety (F-S,) for the aiong fibers stresses resulting from the detailcd
nonlinear finite element analysis are larger than those assumed by Awad in lris design
procedure. In a11 the considered cases, the factors of safety ( F S , ) have cscccdcd ri valrit.
of 5. This indicates that the simplified procedure suggested by Awad ( 19W) providcs
reasonable values for the factors of safety against strength failure. It sIiould bc notcd-
that the level of safety provided by Awad's procedure against fütigiic fâilurc duc 10
vortex shedding is not considered in this study. It is anticipated that a conclusion siniiIar
to the one drawn for the strength safety would arise for the fatigue sakty. However, to
confim this point, nonlinear dynarnic analysis has to be performed for FRP cliinineys
subjected to across wind load resulting Eorn the vortex shedding effect.
Based on the assumed level of thermal stresses and the assumed material pi-opcrtics.
FRP chimneys having a height to diarneter ratio (L/D) Iarger than 1 5 are found to be no t
critical to instability failure due to local buckling. Meanwhile. under ttrc sanie
circumstances. FRP chimneys having a height to diameter ratio (L:D) less tlian 15 arc
shown to faiI due to instability initiated by Iocal buckIing at the base of the chiniiieys.
These chimneys are designed according to Awad's procedure to sustain only dong wind
loads. Linearly variable temperature variations are shown to be more critical in
initiating this instability failure than constant ones. The same cliirnneys. whcn d e s i p x i
according to Awad's procedure to resist fatigue due to vortex shedding. liave showr to
be safe against buckling. This buckling study provides an indication tliat the design
procedure suggested by Awad (1998)' including both strength and fatigue resistancc.
provides safe design against instability failure.
5-3 RECOMMENDATIONS FOR FUTURE RESEARCH
The following studies related to the Consistent Laminated sheli eicment. rire
recommended for future investigations:
i ) Extend the formulations of the Consistent Larninated shell element to include the ei-kct
of the normal stress component oz. .. (normal stress in the direction perpendicular to t k
surface). The inclusion of this stress component will allow delaniination l>iltirc in ttic
direction perpendicular to the surface to be studied.
2) Conduct an experimental verifkation for the material nonlinearity model included in
this study.
This study has laid down the analysis and provided a description for ihc nonlincar
behavior of FEU' chimneys. The developed model can be used to condtict csic~isi\.e
parametric studies in order to obtain a comprehensive understanding for the beliavior and
design of such type of composite structures. In that regard, the lollowing stiidies are
suggested:
1 ) Study the effect of varying the angle of inclination of the fibers and the fibers content
on the nonlinear behavior of FRP chimneys.
2) Assess the level of safety provided by Awad's procedure when fatigue hilure due to tlic
vortex shedding effect is considered. This can be achieved by conductiiig iioii 1 iiisar
dynamic analyses of FRP chimneys subjected to thermal loads combined npitli \.ortex
shedding dynamic wind loads.
3) Conduct a comprehensive buckling analysis by varying the assumed tcmperai~irc lad.
the angle of orientation of the fibers and the fibers content. This study \voiiid rcsiili iii
determining the range of applicability of Awad's procedure with respect to instabi l i ty
fai l ure.
REFERENCES
- Awad, A. (1998), 'Behavior of FRP chimneys under thermal and wind loads' MESc.
Thesis, University of Western Ontario, London, Canada.
- Bathe, K. J. (1996) 'Finite Element Procedures in Engineering Analysis' Prentice
Hall, Englewood Cliffs, NJ.
- Chandrashekhara, K. and Bhimaraddi, A. (1994): Thermal Stress Analysis of
Laminated Doubly Curved Shells Using a Shear Flexible Finite Element, Comp. &
Stnict., Vol. 52, NO. 5, pp. 1023-1030.
- Chen, W. F. (1982), 'Plasticity in reinforced concrete', McGraw-Hill book Company,
New York.
- CICIND Mode1 Code for Steel Chimneys (1988). By the International Cornmittee on
industrial chimneys.
- Davenport, A.G. (1993), 'The Response of Slender Structures to Wind', proceedings
of the NATO Advanced study Institute At Waldbronn, wind Climates in Cities,
Germany.
- El Darnatty, A. A., Korol, R. M. and Mirza, F. A. (1997)' 'Large displacement
extension of consistent shell element for static and dynamic analysis' Comp. &
Struct., Vol. 62, pp. 943-960.
Johns, R. M. (1975), 'Mechanics of composite materials' McGraw-Hi11 book
cornpany.
Kapania, Rakesh K. and Raciti, Stefano (1989), 'Recent advances in analysis of
laminated beams and plates, Part 1: Shear effects and buckling', PLLAA Joumal, Vol.
27, NO. 7, pp. 923-934.
Koziey, B. and Mirza, F. A. (1997), 'Consistent thick shell element' Comp. & Struct.,
Vol. 65, NO. 4, pp. 53 1-54 1.
Koziey, Bradley L. (1 993), 'Formulation and applications of Consistent shell and
beam elements', Phd- Thesis, McMaster University, Hamilton, Canada.
Liu, J. H. and Surana, K. S. (1995), 'Piecewise hierarchical P-version curved shell
element for geometrically non-hear behavior of laminated composite plates and
shells', Comp. & Stmct., Vol. 55, No. 1, pp. 47-66.
Ostrowski, P. (1984), 'Finite element modeling of single chord RHS gap K-joint',
Phd. Thesis, McMaster University, Hamilton, Canada.
Pagano, N. J. and Hatfield, S. J. (1972), 'Elastic behavior of multilayered
bidirectional composites', AIAA Journal, Vol. 10, pp. 93 1-933.
Pipes, R. B. and Pagano, N. J. (1970), 'Interlarninar seesses in composite laminates
under uniform axial extension', J. of Composite Matends, Vol. 4, pp. 538-548.
Plecnik, Joseph M., Whitman, Warren E., Baker, Treven E. and Pham, Mai (1 984)
'Design concepts for the tallest eee-standing fiber glass stack', Polymer Composites,
Vol. 5, NO. 3, pp. 186-190.
Pritchard, B.N. (1 996) 'fndusnial chimneys: A Review of the Current State of Art',
Proc. Instn. Civ. Engrs Structs & Bldgs, Vol. 1 16, pp. 69-8 1.
Putcha, N. S- and Reddy, J. N. (1986), 'A refined mïxed shear flexible f i t e element
for the nonlinear analysis of laminated plates' Comp. & Struct., Vol. 22, pp. 529-538.
Reddy, J. N. (1 984), 'A simple Higher-Order theory for iaminated composite plates',
J. of Applied Mechanics, Vol. 5 1, pp. 745-752.
Reddy, J. N- (1987), 'A generalization of two dimensiond theones of laminated
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180.
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Analysis' CRC Press, Boca Raton, Flonda.
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No. 2, pp. 79-90.
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2 768.
APPENDIX I
INTERPOLATION FUNCTXONS FOR CONSISTENT
LAMINATED SaELL ELEMENT
Ouadratic Interpolation Functions:
APPENDIX tI
CONSTRUCTION OF ORTHOGONAL BASIS
The Jacobian matrix can be written as
(II. 1 )
where the vectors R and S are tangent to the surface deked by t = constant. A normal
vector 5 to this surface can be obtained by applying the following cross product:
v = & * S -3 (11.2)
The other two vectors y, and of the orthogonal basis are obtained from the following
equations:
V = j * K 3 -7 - (11.3)
v =K, *y3 -1 (11.4)
where is the unit vector dong the global y-mis. It should be mentioned here that in case
that the vector is paraIIel to the global y-axis , the unit vector k along the z axis cm be
used instead o f i in Equation (2.3.
The normalisation of the vecton y,, and & then Ieads to the unit vecton y,, 3
and which c m be used in the transformation matrix as follows:
(II. 5 )
APPENDIX rn
DERIVATION OF LAMINA THROUGH TaLCKNESS INTERPOLATION
FUNCTIONS M,,, M,,, RI, AND M,,
Let rotation #L Vary quadratically through the thichess of the lamina and be
approximated by:
where Zis the dimensional CO-ordinate in the lamina thickness direction and a, b and c are
unlaiown constants. The displacement d@L due to rotation 4, is calculated as:
Substiniting the expression for q4, into the above equation and integrating gives:
(III .2)
where d is an additional unknown constant. The displacements and rotations at the top
and bottorn of the layer are taken as degrees of fieedom, Le.
where subscnpts T and B designate the top and bodom of the layer, respectively and h, is
the thickness of the lamina. Substituting the expressions for 4, and d$, into the above
equation gives the degrees of fkeedom in terms of the unknown constants as:
Since there are four equations and four unknowns, the above equation c m be
solved to give the unlaiown constants a, b, c and d in terms of the degrees of freedom as:
(III. 6 )
The above expressions for the constants are now substituted into equation (III. 1)
and equation (III.3), and the terms having like degrees of Greedorn are grouped. This
yields:
The relationship between the dimensional CO-ordinate z and the non-dimensionaI
CO-ordinate t, is given by:
which is used to re-write equation m.7) in terms of t, as:
(III. 9a)
Equation (III.9b) can afso be written as:
where interpolation functions Mx, Ma, Mx and Ma define the displacement variation
through the thickness of the lamina due to degrees of keedom #,', d hT, #:, d #/,
respec tively .
APPENDIX IV
CONSISTENT LAMINATED SEELL ELEMENT STIRAIN-
DISPLACEMENT MATRIX [B.]
where
J,' represents the components of the inverse of the Iacobian matrix as defined by Koziey
and Mirza (7997).
- N,, and ??, are the derivatives of the cubic shape function N, with respect to the r and s
resp ectiveiy .
2, are the direction cosines given in Appendix II.
C: are components of the matrix [Cn J given in equation (2.9).
MqL are the through thickness shape fùnction, and dM,, are the denvatives of these
hc t ion with respect to t.
To facilitate the formulation of the matrices [BJ and [B, 1, the matrix [B.] is divided to
the matrices [Bo, ] and [Bo, ] which are as follows: