BEHAVIOR OF FRP CHIMNEYS UNDER THERMAL WIND LOADS · 2.3.4 Phenolic resins 2-3.5 Polyimides resins...
Transcript of BEHAVIOR OF FRP CHIMNEYS UNDER THERMAL WIND LOADS · 2.3.4 Phenolic resins 2-3.5 Polyimides resins...
BEHAVIOR OF FRP CHIMNEYS UNDER THERMAL AND WIND LOADS
by Ahmed Shawky Awad
Faculty of Engineering Science Department of Civil and Environment Engineering
Submitted in partial fu~1Iment of the reqairements for the degree of
Master of Engineering Science
Faculty of Graduate Studies The University of Western Ontario
London, Ontario December 1998
BAhmed Shwaky Awad 1998
uisiins and Acquisitions et Bib iographi Services services bibliographques Y*
The author has granted a non- exciusive licence dowing the National Library of Canada to reproduce, loan, distrïiute or seil copies of this thesis m microform, paper or electronic formats.
The author retains ownership of the copyright in this thesis. Neither the thesis nor substantial extracts fiom it may be printed or otherwise reproduced without the author's pdss ion .
L'auteur a accordé une licence non exclusive permettant à la BWothèque nationale du Canada de reproduire7 prêter, dis6ri'buer ou vendre des copies de cette thèse sous la forme de mic&che/nim, de reproduction sur papier ou sur format électronique.
L'auteur conserve la propriété du droit â'aukur qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son
Due to their high corrosion and chemical resistance, nber reinforced plastic (FRP)
materials are increasingly being used in the construction of industrial chimneys.
However, no national code currently exists to guide the design of such type of composite
structures. This thesis attempts to investigate the structural behavior of FRP chimneys
under both thermal and wind Ioads. The study also Uicludes a s w e y to identify the
appropnate type of composite for chimneys applications and an experimental study for
evaluating the damping of such composite.
The thermal study is conducted using an in-house developed h i t e element mode1
which is used to predict values for thermal stresses that can be used in the deign of FRP
chimneys.
A cornputer code that incorporates the classical lamination theory together with a
procedure previously developed by Davenport for estimating wind loads, is developed
and used to study the wind behavior of FRP chimneys. An extensive parametric study for
both the dong and the vortex shedding respomes of FRe chimneys is conducted using the
developed code. Appropriate thicknessa for FRP chimneys that satisfy the strength and
the fatigue limits of the material are presented in a graphical form.
Finaily, dynarrric testing of samples of the materiai, commonly used in the
construction of FRP chirnneys, is conducted and reveals a relatively low damping ratio
for such materials.
Keywords: FRP materials, chimneys, themial stresses, wind loads, damping, design.
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to my supenisor Dr. A.A El Darnatty
not only for his guidance, criticisrn and encouragement through the research and
preparation of this thesis but also for his unseifish desire to give the best possible
background to his student. Without his continuous support and his Enendiy advice, this
study could not be completed.
The valuable advice of Drs. B. J. Vickery and A. Davenpoa and the tremendous
cooperation of al1 the staff in The Boundary Layer Wind Tunnel are also sincerely
appreciated.
Fhally, sincere thanks to my lamily, their nipport helped me to succeed where I
thought 1 never could.
TABLE OF CONTENTS
CERTIFICATE OF EXAMNATION
ABSTRACT
ACKNOWLEDGMENTS
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
NOMENCLATURE
CHAPTER 1 NTRODUCTION
1.1 General Review
1.1 .1 Industrial Chimneys
1.1.2 Composite Materials in Industrial Chimneys
1.2 Objectives and Scope
xvi
CHAPTER 2 FIBER REINFORCED PLASTIC MA-S iN CONSTRUCTlON
2.1 Introduction 7
2.2 Fibers 8
2.3 The M m (polymers) 10
2.3.1 Polyester Resins 1 1
2.3.2 Vinyl Esters 12
2.3.3 Epoxy resins 13
2.3.4 Phenolic resins
2-3.5 Polyimides resins
2.4 Fiber-Matrix Composite
2.5 Environmental E ffect on Glass Fiber Reinforced Plastics
2.5.1 Moisture Absorption
2.5.2 Thermal Instability
2.5.3 Chernical Attack
2.6 Fatigue in Fiber ReUiforced Plastics
2.6.1 Temperature and Environment al E ffec t on FRP Fatigue S trength
2.7 Long Tenn Perfomance of FRP Matenals
2.8 Applications of FRP Matenals in the Construction Industry
2.9 Recornmended FibenResin in indumial Chimey s
CHAPTER 3 THERMAL STRESS ANALYSIS OF FRP CHIMNEYS USiNG
CONSISTENT LAMINATED SHELL ELEMENT
3.1 Introduction
3.3 Description of The CLS Element and Thermal Loading
3.2.1 Geometry and Disp lacement Interpolations
3.2.2 Stress-Strain Relations and the Thermal Load Vector
3 -3 Verification of the Mode1
3.3.1 Iso tropic Plate Subjected To Linearly Varying Temperature Change
3.3.2 Anti-symmetric Angle Ply Laminate Plate Subjected to Linearly
Varying Temperahne Change
3.3.3 Isotropic Cylinder Subjected to Linearly V-g Temperature Change
3.3.4 Anti-symmetric Cross Ply Cylindrical panel Subjected to Uniform
Temperature
3.4 Thermal Stress Anaiysis of FRP Chimneys
3.4.1 The Effect of the Larninate Thickness
3.4.2 Effect of the Diameter of the Chimney
3.4.3 The Effect of the Height of the Chimney
3.4.4 The Effect of the Number of Layen and Fiber ûrientation
3.4.5 Summary of the Results of the Parametric Study
3.5 Practical Considerations for Atternpting Design Rocedure of FRP Chimneys
3.6 Thermal St~ess Values to be Used in Practical Design of FRP Chimneys
3.7 Conclusions
CHAPTER 4 COMPUTER AIDED-DESIGN CODE TO EVALUATE W N D RESPONSES OF FBER REINFORCED PLASTIC CHIMNEYS
4.1 Introduction
4.2 Laminate Equivalent Elastic Properties Ushg Classical Lamination Theory
4.2.1 Stress Strain Relations for Larnînated FRP Material
42.2 Extemion and Bending Stiffiiesses for Laminated FRP Materials
4.2.3 Properties of An EquivaIent ûrthotropic Material
4.3 Beam Bending Behavior of Chimneys
4.4 Evaluation o f Dynarnic Characteristics Using Stodola Method
4.5 The Wind Loads
4.5.1 Along-Wind Response
4.5.2 Across-Wind Response
4.5.3 Vortex Shedding Response
4.5.4 Wind Loads As An EquivaIent Static Loads
4.5.5 Wind Load Cases Considered in the Study
4.6 The Stresses CalcuIation
4.7 The Failure Criteria
4.8 Fatigue Calculation
4.9 Veri fication of the Mode1
4.10 Parametric Study
4.10.1 Fibers Orientation
4.10.2 Damping and Mass Density of FRP
4.10.3 Effect ofTapenng
4.1 1 Design Thicknesses For FRP Chirnneys
4.12 Conciusions
C W T E R 5 DAME'ING OF FRP MATERiALS
5.1 Introduction 1 1 1
5.2 Review of Damping Evduation of Fiber Reinforced Plastic Materials 112
5.3 Measures And Techniques For Determining Materiai Damping 114
5.3.1 Logarithmic Decrement Technique 115
5.3 2 Half Power Band-Width Method 115
5.4 Experimental Evaiuation of the Damphg Roperties of G l a s Reinforced
Vmyl Ester Matenal
5.4.1 Experiment Set-up and Procedure
5.4.2 Damping Results and Discussion
5.5 Correction for Aerodynamic Damping
5.6 Conclusions
CHAPTER 6 CONCLUSIONS AND RECOMMENDA'TTONS
6.1 Introduction
6.2 Suitable FRP Material For Chimneys' Construction
6.3 Thermal Stresses Induced in FRP Chimneys
6.4 Effect of Wind Loads on FRP Chimneys
6.5 Experimental Evaiuation of Damping Ratio of Vinyl Ester Glass Reinforced
Composite
6.5 Recommendations For Further Research
APPENDIX A
REFERENCES
VITA
LIST OF FIGURES
CHAPTER TWliEE
Coordinate systerns and nodal degrees of Eeedom of CLS element.
Variation of outer and mid-surface longitudinal and c ircum ferential stresses
at fkee end ofcylinder due to linearly varying temperature change.
Cross-piy cylinâricd panel.
Cross section and vertical projection of laminated Cylindrical FRP chimney. 52
Thermal stresses of 5 layers angle-ply (+/- 55') FRP chimney versus the
laminate thickness at a section away frorn the boundaries of the c h e y .
Thermal stresses of 5 layers angle-ply (+/- 550 ) FRP chimney versus the
laminate thickness at the base section of the chimney.
The effect of the diameter on the thermal stresses induced at the base of a
FRP chimney.
The hoop and axial stresses at the inside face dong the height of a FRP
chimney subjec ted to linearly varyîng temperature.
The maximum longitudinal and transverse stresses at the outside face
of the laminate vs. the angle of orientation.
3.10 The maximum longitudinal and transverse stresses at the outside
face of the Iaminate vs. the angle of orientation.
3.1 1 The maximum longitudinal srcesses at the inner and outer face of
the laminate vs. the angle of orientation afler degraduig the a m s s
fibers stiflhess of the layers.
3. L 2 Ln-plan shear stress t,, , transverse shear stresses t,, , t, of 1 O
layer laminate at the bottom of the chimney after degrading the
across fibers stifiess of the layers.
3.13 The longitudinal thennal stresses of 35" angle-ply FRP chimney
for different temperature fields (degraded across fibers modulus E,
= E,/1000)
3.14 The longitudinal thermal stresses of 55" angle-ply FRP c himney
for different temperature fields (degraded across fibers modulus &
= E,/1000)
CHAPTER FOUR
Vertical and horizontal cross sections of FRP chimney.
Vertical projection of the laminate showing the set of axes.
The geometry of the laminate.
Normalized longitudinal extension modulus versus fiber orientation angle
First naturai fiequnicy of chimoey 1 with the fiber orientation angle.
Along and across-wind tip deflection versus angle of orientation angle
Nomalized across-wind tip deflection vmus the mass density for
1, II and III.
Normaiized tip deflections vernis damping ratio for 1, II and III.
The estimated across-wind mponse versus the struchiral damping for
chimney with height H= 40m, bottom diameter &=3.0m for 0.0,0.3
and 0.6 tapering ratio.
4.10 Estimated thicknesses of FRP chimneys versus the aspect ratio
for factor of safety = 2.0,5 = 0.70%.
Estimated thicknesses of FRP chimneys versus the aspect ratio
for factor ofsafety = 3.07< = 0.70%.
Estimated thicknesses o f FRP chimneys versus the aspect ratio
for factor of safety = 4.0, < = 0.70%.
Estimated thicknesses o f FRP chimneys vernis the aspect ratio
for factor of safety = 5.0,< = 0.70%.
Estimated thicknesses of FRP chimneys versus the aspect ratio
for factor of safety = 2.0, Ç = 0.85%.
Estimated thicknesses of F W chimneys vernis the aspect ratio
for factor of safety = 3.0, = 0.85%.
Estimated thicknesses of FRP chimneys versus the aspect ratio
for factor of safety = 4.0,c = 0.85%.
Estimated thicknesses of FRP chunneys versus the aspect ratio
for factor of safety = 5.0, < = 0.85%.
Estimated thicknesses of FFW chimneys vasus the aspect ratio
for factor of safety = 2.0, c = 1.0%.
Estixnated thicknesses of FRP chimneys versus the aspect ratio
for factor of safety = 3 . 4 < = 1 .O%.
Estimated thicknesses of FRP chimneys versus the aspect ratio
for fractr of safety = 4.0, = 1 .O%.
Estimated thîcknesses of FRP chîmneys versus the aspect ratio
for factor of safety = 5.0, & = 1.0%.
xii
4.22 Tip defiection nonnalized to diarneter of FRP chimneys venus the
aspect ratio for factor of safety = 2.0 and 5 = 0.7%. 110
4.23 Estimated thicknesses of FRP chimneys vernis the aspect ratio
for factor of safety = 5.4 5 = 0.7%, 8 = f3S0. 110
CHAPTER EwE
A photo showing various components of the shaker system.
Schematic diagram of the Shake Table and the Data Acquisition System
A photo showing the epoxy glue and steel plate used in mounting the
specimen.
A photo of a typical specimen mounted to the slide table.
Typical experimental fkequency-response curve and the fitted response
of single degree of fieedom system.
The damping of the fkst mode vernis the Eequency nom the resonant test.
The damping of both first and second mode versus the nequency nom
the resonant test.
The damping ratio of the nrst mode verrus the maximum bending strain
amplitude h m the decay test.
The damping ratio versus the maximum baiding seain amplitude in the
longitudinal direction for specimen (2 in diameter)
LIST OF TABLES
CHAPTER TWO
2.1 Typical properties of some selected fibers. 9
2.2 Mechanicd properties of Thennoset resins. 14
2.3 Effcct of moisture on mer reinforced plastics. 17
2.4 Cornparison of HDT of some unreinforced matrix mins and fiber
reinforced composites. 18
2.5 The mechanical properties of orthotropic layer with 70% fiber content 28
CHAPTER THREE
Results of the analysis of an isotropie plate subjected to linearly
varying temperature.
Results of the analysis of an anti-symmetric angle-ply plate
subjected to linearly varying temperature.
Results of the analysis of anti-symmetric cross-ply (0°/900/00/900)
cylindncal panel subjected to uniforrn temperature (AB= 1, A/H =
100, IU8 = 1).
The in-plane and transverse shear stresses associated with the
longitudinal stresses in Fig.3.12 for an angle-ply laminate f 3S0.
The in-plane and transverse shear stresses associated with the
longitudinal stresses in Fig3.13 for an angle-ply laminate k 55'.
CHAPTER FOUR
4.1 The dimensions, the lay-ups and the tip deflections of FRP chimneys 85
CHAPTERFIVE
5.1 The measured nahuai fi.equencia and damping values nom the resonant
and fiee decay tests
NOMENCLATURE
Al1 symbols are defined at their first appearance. The principal symbols used are listed
below :
ai, pi
P'1
1% 1
Rotations degrees of fieedom.
Elasticity rnatrix in local coordinate system.
Initial thermal strain matrix of the L~ layer in the local
coordinate system.
Thermal load vector.
Reduced and transfonned reduced stifiesses of the Imina.
Laminate extensional, coupling and bending stiffhess matrices.
Longitudinal , transverse modului and in-plane shear modulus
of orthotropic lamina.
Equivalent modulus of the laminate in X and y directions,
equivalent in-plane shear moduius and equivalent Poisson's ratio.
Natural kquency and mode shape of the i" mode of vibration.
Vortex shedding f?equency and Strouhal number.
Structural damping and aerodynamic darnping.
Wind speed at the top and the criticai wind speed.
htensity of turbulence at the top and the drag coefficient.
Specifïc damping capacity, loss factor and logarithmic
decmnent,
CHAPTER 1
INTRODUCTION
1.1 Generai Review
1.1.1 Industrial Chimneys
The cornmon impression about industriai chimneys is that they are simple vertical
structures. The tnith is that they are complex structures in their behavior and design
requirements. in fact the design of industrial c h e y s is rather sophisticated and requires
knowledge of structure dynamics, fluid rnechanics, materiai science, chemistry and heat
transfer. Aithough the casualties due to the collapse of chimneys are very few, the
consequences of chimneys' deterioration are usually disastrous in ternis of losses of the
output fiom the served equipment or industry.
Early constnicted induskiai chimneys were relatively shon (rarely exceeding
5ûm) and were usually comtmcted fiom bnckwork. With the increase of the labor cost
and the availability of high quality welded steel, steel chimneys became more economic
than brickwork for relatively short chimneys. With the increase concem about air
pollution, the heights of the chimneys became usuaily detennined by the need to disperse
the flue gases over a wide area. As such, chimneys haMng a height ranging between 80 m
and 200 m (sometimes up to 300 m) became widely used. For such long chimneys, steel
is not econornicai as a material for construction and remforced concrete is the alternative.
Concrete chimneys are usually provided with bnckwork or a steel liner to protect the
concrete shell (windshield) h m hot and aggressive flue gases.
Industrial chimneys face a variety of environmental hazards during their life time.
These include high wind loads, across-wind oscillations, earthquakes, chernical effects
and thermal loads. Any of these hazards cm govem the design of chimneys. Besides acid
corrosion. across-wind oscillation caused by vortex shedding is the most comrnon cause
of failure in steel chimneys (Pritchard, 1996). Ail ta11 structures are subjected to the
vortex shedding phenomenon. Meanwhile, the low structural damping of steel chirnneys
makes them more vuinerable than other structures. On the other hand, concrete chimneys
possess sufficient mass and structural damping to suppress the across-wind oscillations.
This phenomenon has been researched For more than 50 yean. Till the late the L9807s,
chimneys' design codes did not provide a simple and reliable method which assists
chimneys' designers in addressing this phenomenon. Recently, chimney codes (e.g.
C I C N ( 1988), AC1 ( 1 995)) have provided approximate means for predicting excessive
across-w ind amplitudes.
1.1.2 Com~osite Materiab in ludustrial Chimnevs
The science of composites repraents a way of material optimizaûon. In other
words, the materiais properties are optimized by combination. It is common that two or
more components c m be combined to form a composite matenal. That combination
makes best use of the more favorable properties of the components while negating the
effects of some of their less desirable properties. Although the practical applications of
composite materials go back to perhaps the Second World War, they have found wide
spread use in aerospace and marine industry only in the past two decades.
nie moa commonly used composite is the fiber reinforced plastic (FRP). In such
a composite, fibers act as a reinforcement for a polymer matrix. The fibers may be
aligned continuously or randomly in the matrix material. The fiben can be also aligned
unidirectionally or in an inter-woven arrangement. The composite is usually stacked in
multilayer fashion to form the basic structure which is called the laminate. As such,
strength, stiflhess and any other property could be tailored to provide the optimum
structural performance by changing the type of fibers or matrix and also by aitering the
fibers orientation and the stacking sequence of the layers.
Composite materials have been shown to have many advantages over
conventional structural matenals. Composite materials provide high strength low weight
structures, sutam moderate and hi& temperature as well as have high corrosion
resistance for wide ranges of acids and bases. Despite of ail of these advantages, the
potentid use of composites in structural applications is stiil creeping. The reasons which
slow down their use in construction could be the lack of design experience, the
unavailability of design codes and the incomplete understanding of the behavior of FRP
matenals under long term exposure to various environmental effects.
Since FRP materials have an excellent corrosion resistance, considerable interest
has been shown to use thern in the construction of industrial chimneys during the past few
years. btially, some polymeric composites which exhibit excellent acid raistance have
been used as liners. This is foliowed by constructing chuiineys totally tiom FRP matenals
(Plecnik, 1984). The poor performance of polymers at hi& temperature is one of the
early difficulties faced by the materiai producers and the designers. The rapid
development of the FRP industry provideci polymen which cm sustain higher
temperatures and consequently encouraged the use of' FRP in the construction of
chimneys. Till now, no design code exists for FRP chimneys but there is an attempt by
the International Cornmittee on Industrial Chimneys CICIND to develop a mode1 code
for such structures.
1.2 Obiectives and S c o ~ e
The objectives of the present study are as follows:
I . Searching various types of fibers and polymeric matrices which have
mechanical and enviromeatal properties suiting the FRP chunneys
applications.
2. Studying the behavior of FRP chimneys under thermal loads using detailed
h i t e element analysis.
3. Studying the behavior of FRP chimneys under wind loads, developing a
simplifieci cornputer code to be useci in the design of FRP chimneys under both
wind and thermal loads-
4. Detemining the viscous damping of glass reinforced vinyl ester composite
using dynamic testing.
Chapter Two includes a presentation for the mechanical properties of various
types of fibers and polyrneric matrices used in FRP composites. The infiuence of the
environmental effects on the mechanical properties of FRP as a composite including
moisture absorption, thermal instability and chemicai attack are then discussed. This is
followed by a discussion of the fatigue performance of FRP materials and how it is
affected by the environmental conditions. Finally, an overview of the applications of FRP
materials in the construction industry and suggestions for the favorable types of fibers
and polyrnenc matrices ihat suit FRP chimneys application are presented.
Chapter Three starts by providing a brief description of a Consistent Laminated
Shell Element which was developed by Koney (1993) and is used in this suidy for
conducting thermal analysis of FRP chimneys. The extension of the Consistent
Laminated Shell Elernent formulation to Uiclude thermal stress analysis is then presented.
This thermal formulation is venfied using a number of benchmark problems available in
the literature. This mode1 is then used to conduct a parametric study to assess the effect of
various parameters which might influence thermal stresses hduced in FRP chimneys.
Finally, stress values resuiting fiam themial loads acting on FRP chimneys are evaluated
and suggested for consideration in the design of FRP chimneys.
Chapter Four includes a development of computer code to be used in studying the
behavior of FRP chimneys under wind loads as well as in designing such type of
structures. The computer code is based on the Classicai Lamination Theory to evaluate an
equivalent elastic modulus for FRP chimneys, a dynarnic procedure @avenport, 1993) to
evaluate the wind response and a quadratic interaction failure critenon. Fatigue stresses
due to vortex shedding are aiso accounted for in the design. A parametric study
examining the sensitivity of the along and across-wind rrsponses of FRP chimneys to the
orientation angle of the Iayen, the mass density, the damping ratio of the composite and
the tapenng ratio is conducted using the developed code. Finally, design thicknesses for
FRP chimneys covering a certain range of dimensions are introduced.
Chapter Five includes description of the experllnentai program conducted to
evaluate the damping capacity of glas reinforced vïnyl ester composite. The experiments
are conducted using a shake table facility. The variation of the viscous damping of the
material with the fkequency and the strain amplitude is examined.
In chapter six, conclusions that are drawn fiom this study as well as
recommendations for M e r research are presented.
CEAPTER 2
EXBER REINFORCED PLASTIC MATEXIALS IN CONSTRUCTION
2.1 Introduction
A composite material is a combination of two or more different materiais. The
purpose of combination is to optimize the materials properties. The constituent materials
of a composite maintain their separate identities microscopically. Meanwhile,
combination of the materials produces propdes and characteristics different from those
of the constituents. Among composite materials, mer reinforced plastics (FRP) represent
a very attractive material, which were used for a number of decades in the aerospace
industry and very recently in civil engineering applications. One of the main advantages
of FRP materials is their high sûength to weight ratio. FRP matenals have two major
constituents; the matrix which forms a continuous media and the fibers which act as a
reinforcement that are added to the matrix to improve the matrix properties. The fibers'
surface is usually chemically treated or coated with a very thin layer to enhance the
bondhg with the ma&, protect the fiers nom moi- or chemicds reacting with the
rnatrix at high temperature.
Fiber reinforced plastic materîals have usually a polymeric matrix. Plastics have
iow density, good chernical resistance and can be easily fabricated Memwhile, the lack
of thermal stability and the relatively poor mechanical properties of plastics can be
remedied by adding fiers to the plastic matrix.
in this chapter, a survey of different types of fibers and polymers is presented.
This is foilowed by a presentation for the environmental and mechanical properties of the
fiber-matrix composite. Finally, a survey of the applications of FRP materials in
construction as well as a recommendation for the suitable FR. to be used in the
construction of chimneys are presented.
Fibers are the main load-carrying component in a fiber reinforced composite
material. The effectiveness of a fiber reinforcement depends on the type, length, volume
fraction and orientation of the fibers in the matrix. The proper selection of these fiber
parameters is very important as they influence the density, strength, modulus, fatigue
performance, themal properties and the cost of the fiber reinforced composite.
The reinforcements used with FRP materials are either E-glass, S-glass, carbon or
ararnid fiben. E-glass is selected in the majority of structural applications due to its low
cost. Carbon and aramici fibers are mainly used in the aerospace and marine applications
because of their higher modulus compared to E-glas fibers. Table 2-1 shows the
mechanical properties of some of the commonly used fiers at room temperature.
Table 2-1 Typical properties of some selected fibers. (Composite Engineering Handbook, 1997)
Density Tensile Temile Failure CoefK of thermal (g/cm3) strength modulus strain expansion
Fiber (MPa) (GPa) (%) (1 O?C) E-glas 2.54 3450 72.4 4.8 5.0 S-gla~s 2.49 4300 86.9 5.0 2.9 Carbon
High strength 1.70-1.80 3100-4000 210-250 1.3-1.6 -0.6 (long.) Inter. Modulus 1 -78- 1 -8 1 5300-5650 290-300 1.80 1 0.0 (radial) High modulus 1.80- 1 -90 22 10-2760 340-390 0.75
Ultra-high 1 -90-2.0 1 520- 1 860 480-520 0.40 rnodulus
Aramid 1.39-1.47 3000-3620 70-1 79 1.9-4.4 -2.0 (long.) ( K e v l a r - 4 9 ) ( r a d i a l ) * Kcivar-49 is the most commonly used aramid fiber in the advanccd composite industry.
It should be mentioned that these types of fiben have different maximum working
temperatures. Depending on the type of glass, the tende strength of glas fibers starts to
decrease between 220 and 260°C, reaching only 50% of its room temperature strength at a
temperature range of 480-560°C. On the other hand, carbon fiben usually start to oxidize
between 300400°C. Aramid fibers have maximum working temperature of 90°C. Carbon
and aramid fibers are characterized by having a negative thermal expansion coefficient in
the longitudinal direction which can be used to produce a composite having zero themal
expansion.
Most of the fibers are mdactured in the fom of Long continuous filaments and
then combined in various fashions to produce strands, tows, rovings, yarns or mats. Short
fibers are obtained by cuning the continuous fiers into lengths ranges fiom 3 to 50 mm.
Fibers have generdly a linearly elastic tende stress-strain response hl1 they fail in a
brittle mamer.
2 3 The Matrix (~olvmers)
The composite matrix is required to hlfill several functions. The matrix binds the
fiben and holds them in the desired direction, acts as a stress transFemng media to the
fibers and protects the f ibm from the mechanical damage, chernical and moisture attack.
The matrix has a minor role in the longitudinal strength and modulus of a unidirectional
continuous fiber composite. However, the matrix properties influences the transverse
strength and modulus as well as shear strength and modulus OF a unidirectional fiber
composite.
The matrix cm either be a thermoset or a thennoplastic polyrner. Thermoset
polymers include epoxies, vinyl esters and polyesters. Phenolics bismaleimides and
polyimides are also thennoset polymers which are used for high temperature applications.
Thermoset polyrners are charactenzed by having low viscosity (i.e. liquid-like polymen)
which are suitable for long continuous fibers. Thennoplastics such as polypropylenes and
nylons have hi& viscosity even at hi& temperature. Therefore, themioplastic polymea
are used more commonly with short fiers because of the difficulty of processing high
viscosity resin with continuous fibers. Thennosets are more thermdy stable and
chemically resistant than thermoplastics. Therefore, thermosets are more suitable for FRP
chimneys applications. A bnef discussion about the properties of the previously
mentioued thermosets will be introduced in the foIlowing ab-sections.
2.3.1 Polvester Resins
Polyester resins are mwively used in numerous FRP applications, e.g. the
construction of pipes and tanks. Polyester resins have a relatively low cost and
meanwhile, have adequate mechanical properties as well as reasonable environmental
durability. Polyester resins can be classified as orthophthalic, isophthalic, Chlorendics
and Bisphenol A fumarates. Orthophthaiic resin is among the least expensive polyesters.
However, this type of resin has relatively poor corrosion resistance. The applications of
that resin is limited to some structural applications where neither corrosion resistance nor
high temperature resistance is required. Isophthalic polyesters cost approximately 10%
higher than the orthophthalics. Meanwhile, this type of resin has improved corrosion
resistance, better water resistance, superior mechanical properties and higher heat
distortion temperature(I3DT); HDT is the sofiening temperature of' the polymer at which
Young's modulus of the material starts to &op. Chlorendics and Bisphenol-A fumarates
are two special polyesters which are fomulated for use in applications requinng supenor
corrosion resistance to that provided by isophthalic polyester. These two types of resins
have a very high rpsistance to concentrated acids. However, their resistance to alkaline
enWonments is poor. Bisphenol-A fumarates polyester resins are used for high durability
and for hi& performance applications where their relatively hi& cost can be justified.
Chlorendic polyesters have high fhe resistance but their strength and toughness properties
are lower than the isophthaiic resins.
2.3.2 Vinvl Esters
Vinyl esters have a higher failure strain as well as better impact darnage resistance
and Fatigue properties than typical polyesters. Vinyl esters have replaceci polyesters and
epoxies as well in many applications. They can be cured at room temperature without
postcuring and still have KDT 90°C (this is a big advantage compared to epoxies). The
vinyl esters cm be classified as Fire retardant, Novolac and high elongation vinyl esters.
Fire retardant version of vinyl esters contains brominated Bisphenol-A epoxy and are
suitable for chemical resistant structures. This type of resin has a high tensile failure
strain (typically 6%). Commercial names of this type of resin which are available in
North Amenca include Derakane 510 series, CoRezyn VE 8400 series, Dion VER
9300NP and Hetron FR9911992. Novolac vinyl ester resin is particularly suited to
applications requiring both high serving temperatures and solvent resistance. The tensile
failure straïn of this resin is relatively low (typically 3%). Commercial names of this type
of resin, which are available in North America include Derakane 5 ION, CoRezyn VE
8730 senes, Dion VER Y480NP, Hetron FR980 and Corin Vibrin E-085 series. High
elongation vinyl esters cm reach up to 10% failure tensile strain. Commercial names of
this resin hclude Derakane 8084 and CoRezyn 85-DA-5000.
Vinyl esters are high performance resins compared to isophthalic polyesters. They
have a superior resisrance to wata and chemical attack, higher stiaess at elevated
temperatures and greater toughness. The fdure shah of these resins is dso higher than
orthophthalic and isophthalic polymer resins (typically twice). Due to their excellent
chernical resistance, low maintenance requirement, design flexibility and ease of
installation, vinyl ester resin based composites have demonstrated low price-to-
performance characteristics compared to steel and its ailoys in many corrosive industrial
applications. As such, vinyl ester resins relliforced with fiber glass have been widely used
in pipes, ducts, Bue stacks and storage tanks.
2.3.3 E ~ o x v resins
Epoxy resins are widely used in aerospace applications. In general, epoxies have
higher values of fracture toughness compaml to polyesters and vinyl esters which usually
result in superior fatigue performance. Epoxies have hi& resistance to water absorption,
high mechanical properties and high working temperature. They are much expensive than
polyesters and vinyl esters and confineci to special applications requiring good
mechanical properties, specially high shear strength and high working temperature.
23.4 Phenolic resins
Phenolic resins have low thermal expansion coefficient as weil as excellent
electrical M a t i o n properties, creep &stance, hardness and flammability
characteristics. Pheno lic resins are convenient for performance under heat with retention
of p r o p d w under fk conditions. There are two basic types of phenolic resins: resole
and novolacs. Resole phenolics have mechanical properties comparable with those of
orthophthalic polyesters with extra thermal stability and fïre resistance. ïhe Iow failure
strain of phenolic resins, which leads to composite having poor mechanical properties,
has limited the application of this type of resin.
2.3.5 Polyimides resins
Polyimide resins have hi& thermal stability which results in service temperatures
of about 300°C (among the highest of currently available resins). A study done by Buyny
(1990) has shown that composite laminates having polyimide as a resin suffer &tom
microcrackings upon thermal cycling. These lead ta a significant reduction in the
mechanical properties and the t h m a l stability of the laminates.
Table 2-2 shows a typical range for the mechanicd properties of thermoset resins,
(Neil, 1994). As stated by Neil (1994), this information is just indicative and the actual
properties of the polymer depend on the exact system used and the curing schedule.
Fiber aiignment in ma& can be unidirectional, two directional, three directional
or random (discontinuous fibers). Contmuous fibers are used in filament-winding,
pulmided or larninated structures in which the fibers can be oriented precisely. Two
directional fiber alignent is used with larninated composite and three directional is
usually used when delamination is anticipated to be a problem. The unidirectional
arrangement provides the most effective use of the fiben when the load is acting in the
fiber direction. For such anangement, the strength and the modulus in the transverse
direction of that lamina are very low compared to those in the longitudinal direction (such
composite is highly anisotropic). Randomly oriented fibers give equal properties in al1
directions on plane of the lamina (ahost isotropie). in a two dimensional alignment,
Fibers are woven in both 0" and 90' directions which brings the lamina properties in the
two directions to be identical if the f i e r content is the same in the two directions.
A fiber reinforced plastic structure is usually a multiple layered structure. Each
Iayer is called a lamina and the whole composite is called the laminate. nie typical
thickness of a lamina varies between 0.8 to l.Omm. The order in which various laminae
(having different fibers orientation) are stacked in the laminate is engineered to obtain the
desired global properties. A laminate denoted by (0/+30/-30/-30/+30/0) is a symrnenic
laminate consisting of six lamina whose angle of f i e r orientation are 0°,+300,-30°,-
30°,+300 and 0°, respectively. A laminate which is symmetric about its mid plan is
prefemd because it does not exhibit extension-bending coupling (as will be discussed in
details in chapter 4).
The possibility of combining different fiber orientations in different Iayers gives a
tremendous design flexibiiity for the laminated composites that is not available with any
other structural matenal. As such, the mechanical properties and the thermal
charactenstics of a laminate can be tailored and suited to the desired application.
2.5 Environmental Effect on Glas Fiber Reinforced Plastics
The mechanicd properties of polymeric composites depend on their constituents
and their interaction with the environmental conditions such as moisture and temperature.
Two major environmental problems are usually associated with polyrners; moisture
absorption and thermal instability.
2.5.1 Maisture Absomtion
in environmental conditions, polymeric composites absorb water. This leads to
change in the mechanical properties of these composites. The absorption rate depends on
the matrix type, exposure time, operating temperature, geometry of the composite and
relative humidity. The moisture absorption generally reduces the HDT of polyrnen. As
reponed by Delasi (1987), a 10% increase of the moisture content of five different types
of epoxy r e s h has led to 50% reduction in the HDT of these resins. Both the strength
and modulus of a composite are dkcted by its moisture content but the modulus is less
sensitive. in general, the moisture absorption of glass-epoxy composite is less than glas-
polyester composite. The effect of moisture absorption has to be taken into account in the
design of poiymeric composites especiaily if the composite is highly stresseci (compared
to its ultimate strength) or if the operating temperature is close to the HDT of the min. A
usehl survey is presented by Bulder (1991) on the effects of moisture on mechanical
properties of glass and carbon reinforced plastics and is shown in Table 2-3. As seen fiom
Table 2.3, the glass/epoxy absorbs less moisture thau glasdpolyester composite. So that
the change in the properties is greater with polyester as a matrix than with epoxy.
Carbonkpoxy composite absorbs the lest arnount of moisture and consequently it is the
less affected by water than glasskpoxy composite.
Matrix (weight %) ( YO ) (_%) 103cycle(%) 10'cycle(%) Glasdpo lyest er 4 -10 -15 -35 - 1 O Glasdepoxy 2 - 10 - 10 -20 O ,
Carbon/polyester - - -5 - - Carbon/epoxy 1.5 +1 -2 O O Glass-carbon/epoxy 1 < 2 1 O / -3 1 -
2.5.2 Thermal Instabilitv
As mentioned before, most of the polymeric matrices possess a certain HDT after
which a significant loss of the composite strength and modulus usually occur. The
reinforcing fibers have a much higher thermal stability cornpared to the matrix. As such,
the presence of the fibers in the matrix causes signincant improvement in the HDT of the
composite. By studying three ciiffirent types of matrix resin reinforced by E-glas and K
g l w fibers, Ghosh (1995) has reported that glas fiber reinforcement has trernendously
improved the HDT of the m a h . R d t s of this study are presented in Table 2-4. As c m
be noted fiom Table 2.4, the HDT of the polyester resin changed h m 79OC to 1 70°C due
to adding 33% (by weight) of N-glas fibea to the resin. A significant improvement of
the HDT of the epoxy and the phenolic resins is also noted nom the table. The British
Standard Specification For GRP Pipes (BS 5480, 1991) recommends an upper limit for
the working temperature of polymeric composite to be 20°C less than the HDT of the
composite to ensure that the composite possesses its ambient mechanical properties.
A typical environment can have hot andlor wet conditions. The stiffhess and
strength of a composite in such environment may be considerably reduced in cornparison
to its mbient properties due to the combined effect of temperature and moisture
(hygrothermai). The composite matrix is more sensitive than Abers to the hygrothermal
effect. For that, the composite properties that are dominated by the matrix are much more
affected. The hygrothermal conditions generate the most severe degradation of the plastic
composite properties; mainly the transverse normal and in-plane shear strength and
stiffhess properties. On the other hand, longitudinal properties of the plastic composites
are very slightly alfected as they are dominated by the fiber properties.
Table 2-4 Cornparison of HDT of some unreùiforced m a h resins and fiber reinforced composites (Ghosh, 1995)
Reinforcing fibers 1 Isophthalic 1 Phenolic used (6 layers) polyester Epoxy (Resol)
HDT CC) 1 %mers 1 HDT (OC) 1 %mers HDT("C) 1 %fibers Unreinforced resin 79 O 94 O 130 O Composites
N-glas 170 33 187 33 208 50 E-dass 194 36 200 36 210 50
2.53 Chernical Attack
Chernicals existing in the enviromnent surrounding a polymeric composite cm
attack and penetrate the resin matrïx. This can result in damage of the fibedresin interface
as well as exposure of the fibers. The corrosion of reinforced plastics is dependent upon
the type of resin used. Glass/polyester has good resistance at low operating temperature to
most chernicals except strong bases and strong oxidants. Glass/epoxy shows better
resistance at low temperature to al1 chernicals except strong oxidants with decrease in the
resistance at high temperature. Glasdvinyl ester has an excellent chernical resistance to
alkalis and acids and is suitable for tough industrial applications. Above certain chemical
concentration and working temperature, the use of vinyl esters under strong oxidants or
aggressive solvents attack is not recommended The polymeric resins can be ranked frorn
low to hi& according to their chemical resistance as ortho-po 1 yester, iso-po 1 yester,
Bisphenol-polyester and vinyl ester. For denning the appropriate resin for specific
c hemical attack under a certain service temperature, the designer should consult the
technical product inibnnation.
2.6 Fatigue in Fiber Reinforced PIastics
Due to the cyclic nature of the wind loads acting on chimneys, fatigue is a major
consideration when the design of a FRP chimney is attempted. The damage modes of
fiber reinforced composites under fatigue loading are similar to those due to static
loading. These damage modes include mairk cracking, interfacial debonding (sepration
of the fibea from the matrix), delarnination (separation of the adjacent layers) and fiber
breakage. Fatigue of FRP is characterized by three stages of damage accumulation
(Mallick, 1997). The initial stage C O ~ ~ S ~ S of pnrnary matnx cracking perpendicular to the
fiber direction and diseibuted along the length of the fibers. Another cracks paralle1 to the
fibers develop and mers failure then initiates in the region of stress concentration created
by the primary cracks. This is usually fiollowed by delarnination in the intenor of the
laminate and excitation of al1 the damage modes till one of the lamina fils.
The fatigue behavior of fiber reinforced composite is influenced by large number
of parameten; type and frequency of loading, stress level as well as the parameters
affecting the mechanicd properties of the fibers and the matrix (e.g. type of fiber and
resin, fiber orientation, fiber content, serving temperature and moisture content).
Generally, the increase of the fiber content increases the fatigue strength of the
composite. The angle of orientation of the fibers measured fiom the direction of the
applied load (0) has a significant role in the fatigue strength. For example, for the angle-
ply laminate (Hl), a rapid reduction of the fatigue strength is associated with increasing
the angle of orientation of the layers. Such an increase in angle makes the mechanicd
properties of the composite more dependent on the ma& which has poorer fatigue
properties compared to the fibers, (Curtis, 1989).
The %ers alignment in the matrix contributes to the fatigue behavior of FRP
laminate. Unidirectional laminate, stressecl in the &ers direction, has higher fatigue
strength than multidirectional and woven fàbncs. For muitidirectional laminates, the
fatigue strength as ratio of its static strength is less than that of the unidirectional
laminate, (Kim, 1989). In general, the woven fabncs have lower stifiess, strength and
fatigue strength than the unidirectional and nonwoven cross-ply laminates. This is due to
the stress concentration at fiber tow crossover points in the fabric, which become sites for
fatigue damage in the resin and fiber resin-interface, @ais, 1975).
Unidirectional glasdepoxy laminate has a ratio between the fatigue strength (at 10
million cycles) and the ultirnate static strength equal to one third, (Curtis, 1989). This
value is comparab!e with the 40% and 20% ratio provided by mild steel and aluminum,
respectively. On the other hand unidirectional carbonkpoxy laminate exhibit superior
fatigue strength (80% of the ultirnate static strength at 10 million cycles). As such, carbon
fibers are preferred in aerospace applications which usually need high fatigue strength
and low weight matenals.
in a study done by Echtenneyer (1991), the fatigue performance of various
composites having the same type and content of fibers with different resin types were
studied. The following resins were used in the tested composites: ortho-polyester (Noprol
41-90), iso-polyester (Noprol 72-80), iso-NPG-polyester (Noprol 20-80), flexible vinyl
ester (Noprol 92-20) and rubber modined vinyl ester (Noprol 92-40). The results of the
study have shown that generally for a high amplitudes of fatigue stresses, vinyl ester
resins and iso-NPG polyester have longer fatigue Life for the same stress level compared
to iso- and ortho-polyesters. For low amplitudes of fatigue stresses, all the laminates
exhibit a sirnilar Me time. In another study, Forsdyke (1 988) has shown that Phenolic
matrix gives higher fatigue strength than polyesters.
2.6.1 Tem~erature and Envir~omental Effect on F'RP Fati~ue Strewth
Generdly, the effect of temperature on the fatigue strength of FRP materials is
similar to what is stated for the static properties. For instance, the glass/thermoset
composites have higher fatigue strength at low temperature than at high temperature.
Unfortunately, there are not enough experimental fatigue data conducted at high
temperature for various kinds of laminates having different resins.
The fatigue behavior under wet envimnrnents depends on the sensitivity of the
matrix and the fiber-matrix interface io moistue absorption. in general, composites
having strong fiber-matrix interface show linle sensitivity to moishm content when
subjected to fatigue loads at room temperature. in a study done by Jones (1984), it was
shown that simple exposure to humid air does not affect the fatigue response of E-glas
fibedepoxy composites. Meanwhile, in the same study, it was s h o w that immersion in
boihg water significantly reduce the fatigue strength in the low cycle region. The fatigue
strength of 0°/9û" glasslepoxy laminate reduced nom 90% (normaiized to the static tende
strength) at dry or exposed to 65% relative humidity to 40% after boiling in water in the
low cycle range of S-N curve, (Jones, 1984). As mentioned before, hi& temperatures
accelerate the rnoistrire absorption and consequcntly reduce the mechanical and fatigue
property of the composite. The cornbineci effm of temperature and moistlne on fatigue
strength is similar to its effect on the static strength of the FRP composite. However, a
reliable data for hot/wet fatigue strength of FRP materials are not available.
2.7 L o n ~ Term Performance of FFW Materials
Polymencs as viscoelastic materiais undergo elastic and the-dependent (viscous
or creep) deformation. Creep W n of FRP material depends on the stress level,
temperature, lay-up of the laminate and the mechanical properties of its constituents, Le.
fiber and matrix. In generai, the creep response in laminate dominated by fibes is less
significant than that dominated by matrix. Therefore, unidirectional laminate stressed in
the fiber direction exhibits very low creep strain compared to that with angle-ply Iay-up
(Carlile. 1989). The creep rate of different laminates consisting of various types of fibers
acting as a remorcement for vinyl ester resin have been studied by Yeung (1 987) and
compared with the creep rate of steel. It was reported that carbon fibers composite
exhibits the less creep rate, E glas fibers composite has creep rate comparable with steel
while aramid composite (Keivar 49) has the highest creep rate.
The failure of a materid under sustained constant load is known as stress rupture.
Considering the moisture effect on the stress rupture of E-glass/polyester Bisphenol A
(lay-up, rnatlwoven rovinghat), Munscheck (1987) has pointed out that the tende
strength of the composite has been reduced under wet condition. The moa important
hding obtained hm this shidy is that E-glass/polyesta c m withstand 50% of its
ultimate strength when subjected to a load for 100,000 hours at a temperature equa130°C.
This finding provides a limit for the safe working strength of glass/polyester exposed to
long tem stresses at 30°C. On the other han& Ho fer (1975) has show that the moisture
has improved the stress rupture of graphitekpoxy (lay-up, [0/45/-45/0/90], and [O]) at
177°C. The combined effect of moisture and temperature on stress rupture is much more
complicated. The stress rupture of unidirectional glass/polyester imrnened in water was
investigated at temperatures 30°C, 4S°C and 60°C by Pritchard (1 988). It was surprising
to h d that tirne of failure was longer for the 45°C case than the 30°C and the 60°C cases.
2.8 A~~l icat ions of FRP Materials in the Construction Industnt
As mentioned before, fiber reinforced plastic materials (FRP) are being widely
used in a nurnber of structural applications replacing conventional materials like steel
and concrete. The reasons which make FRP an attractive material for structural
applications can be stated as:
1 ) FRP materials possess high corrosion resistance which increases the life expectancy
of FRP saucnires compared to conventional steel and reinforced concrete structures
and also reduces the repair and maintenance cost of these structures.
2) FRP materials have very low weightlstrength ratio as weii as a relatively low
weight/stifiess ratio. This light weight reduces the foundation costs as well as rnakes
transporthg and assembling the structure cornponents much easier.
FRP materials have been applied in the foilowing structurai applications:
1) FRP bars are used as a reinforcement for concrete members. For instance, the Taylor
Bridge (Manitoba, Canada) has been constnicted recently using carbon fiber reinforced
plastic ( C m ) as prestressing and shear reinforcement of four main girden as well as
reinforcement for a part of the deck slab of the bridge. Glass fiber reinforced plastic
(GFRP) bars were also used to reinforce a portion of the barrier wall of the bridge.
2) FRP structural shapes such as wide flange sections, angles, channels, hollow rounded
and rectangular sections are used as the main sûuctural elements in the construction of
bridges and industrial buildings. These sections are usualy fabricated nom E-glus
fibers with polyester or vinyl ester resins. A large nurnber of composite plastic bridges
aiready exist in many countries ail over the world. For exarnple, Fiberline Bridge-
Kolding (Denmark), one of the largest GFRP bridges in Europe, the lightweight of the
composite allowed the bridge to be easily erected in only 18 night-time hours. Pa10
Alto (California) bridge, 10- long x 56m wide polyrner composite bridge,
demonstrates the feasibility of using E-glas fiber reinforced polymer in short span
bridges.
3) FRP laminates are used in up-grading and retrofitting of existing structures.
4) FRP laminates are used as main structural components in the construction of pipes,
tanks, stacks and large roofs (e.g. Haj terminal in Saudi Arabia, the Pontiac Silver
Dome in Detroit and Denver international airport).
The major disadvantages of FRP materials for construction applications are:
1) The cost of FRP materials is a major obstacle to the spread use in structurai
application. The cost of the cheapest FRP composite @-glasdpolyester) is twice that
of iow carbon steel.
2) The FRP materials have relatively low stifiess. For that, most structural applications
are not govemed b y the strength but by the stifiess of the material.
3) Some of the FRP composites can sustain the individual effects of environment
(moisture, temperature or chanicals), but when combined together, the properties of
the composite could be severely degraded. Due to the lack of relevant long tenn
behavior of FRP matenals, researchers rely on extrapolation nom laboratory data. This
gives rise to a suspicious long term behavior and uncertain.
4) The lack of practical codes for the variety of structurai applications hinden the use of
E3.P matenals with confidence, as the required experience does not exist.
2.9 Recommended FibedResin in Industrial Chimnevs
ïhe aim of the previous discussion was to present the mechanical propemes and
environmentai resistance of fiber reinforceci plastic polymers. The final objective is to be
able to choose the constihients of an FRP composite which suit the industrial chimneys
applications. From the previous discussion and based on the serviceability conditions of
industrial chimneys which include high thermal effect, chernical environment and cyclic
loading due to wind actions, the following constituents of FRP are comidered to be
appropriate for chimneys:
a) Fibers
For the level of stresses expected in an industrial chimney, E-glas fibers are the most
suitable type of fibers h m the cost point of view. However, the surrounding matnx
(resin) has to assure the necessary chemical and abtasion resistance to protect the fiben
from any chemical substances.
b) Matrix
In view of the mechanical properties of the available polymeric resins, given in table 2-2,
one may conclude the following:
Polyester resins (the least expensive resins) have almost the same modulus and
strength ranges as vinyl esters, epoxies and Phenolic. These resins have reasonable
failure strain. moderate chernical resistance and low continuous service temperature
(up to 130°C). As such, the use of polyester resins should be confined to low
temperature applications which does not involve strong chemical environment in the
form of strong bases and strong oxidants.
Vinyl ester resins have slightly higha continuous service temperature (up to 150°C)
than polyesters, better fatigue strength, wider failure çtrain range and much better
chernical resistance. For that, vinyl esters are suitable for the consmiction of indusnial
chimneys having moderately service temperature.
Epoxy resins, while having better continuous service temperature than vinyl esters (up
to 180 OC), are not as good in chemical resistance and have a higher cost.
Phenolic and Polyimide resins have higher lange of service temperature (120 -300 O C )
wîth good chemical resistance. Both of hem have very low failure strain. PhenoIic
mins are low tensile strength resins having a low cost. Polyirnide resins have very
high tensile strength and a high cost.
As a conclusion, vinyl ester polymers reinforced by E-glas fibers are suitable to
be used in the construction of industrial chimneys as long as the service temperature of
the chirnney is less than the continuous service temperature of the vinyl ester polymer.
For the case of chimneys having high service temperature and chernical attack is not a
conceni, epoxy polymers reinforced by E-glas fiben cm be used for such applications.
Typical mechanical properties of vinyl ester E-glas composite layer having 70% fibers
content (by weight) which will be used in the rest of this thesis, are shown in Table 2-5.
Table 2-5 The mechanical properties of E-glasdvinyl ester layer with 70% fiber content
~~~~~
Poisson's Ratios v,', = 0.3, VI3 = 0.3 ,v,? = 0.29 Thermal Expansion Coefficients
Longitudinal a, = 7 . 7 ~ IO4 m/d0C Transverse m = 43.4~ 1 O4 m/m /OC
Modulus Longitudinal Transverse In-plane shear
GPa E, = 36.85 E1 = 1 1.16 G,,=3.36
Ultimate Strength Longitudinal (tende) Transverse (tende) In-planeshear
MPa O, = 552.77 O, = 16.74 O,? = 70.57
CHAPTER 3
THERMAL STRESS ANALYSIS OF FRP CHIMNEYS USING CONSISTENT
LAMINATED SHELL ELEMENT
3.1 Introduction
Due to their high corrosive resistance, fiber reinforced plastic materials are
increasingiy being used in the construction of industrial chimneys. The design of a
chimney is governeci mainly by wind loads and thermal stresses resulting frorn the
differences among the ambient, the operathg and the curing temperanires.
Although, a large number of theoreticai studies investigated thermal stresses
induced in various shells of revolution (exarnple Padovan (1 W6), Fettahlioglu and Wang
(1988) and Lin and Boyd (1971)). it appears that no attempt has been made to study
themiai stresses in FRP chimneys. It is clear that a lack of knowledge about the expected
thermal stresses in FRP ctiimneys affects both structural engineers and FRP
manufachuers attempting the design and construction of FRP chimneys.
This chapter includes a theoretical hite elment development which is then
employed in performing thermal stress analysis of FRP chimneys. The theoretical
development is based on a consistent Iaminated subparametric shell element which was
initiaily formulaîed by Koziey (1993) and has the advantage of being k e fkom spurious
(locking phenornena) shear modes associatecl with isopafametric sheU elements. The
formulation of the element is extended in this study to include thermal stress analysis of
larninated structures.
The finite element development is verified using results of thermal stress analysis
for a number of plate and shell problems which are available in the literature. The finite
element mode1 is then used to perform an extensive parametric study in order to identify
the main parameters affecting the themal stresses induced in FRP chimneys. Practical
considerations which should be accounted for when perfonning thermal stress anaiysis of
FRP chimneys are discussed. Finally, a chart predicting thermal stresses induced in EXP
chimneys as a function of the parameters defining the through thickness temperature
distribution is presented.
3.2 Description of the CLS Element and Thermal Loading
The stress andysis of FRP chimneys subjected to temperature variation cm be
performed using a lamhdted sheil eiement . Due to their simp licity , degenerated shell
elernents, which are based on the Mincilin plate theory (Mindlin, 195 l), provide a suitable
numerical tool for nich an application. Degenerated sheil elements were first introduced
by Ahmed et ai. (1970) through the nine-node isoparamehic element. However,
degenerated isoparametric shell elements have shown to predict very stiff solutions
(locking) when used to mode1 thin plate and sheil structures. A large number of attempts
have been done to overcome this looking phenornenon which achially results kom the
existence of spurious shear modes in the formulation of isopafametric degenerated shel1
elements. However, none of these attempts have been successfbi in a general application.
A consistent subpararnetric laminated shell elernent was developed by Koziey ( 1993).
The main advantage of this element is being k e Eom the spurious shear modes; i.e. does
not exhibit locking when used to mode1 thin shell structures. This has been achieved by
using a consistent formulation which includes cubic approximations for displacements
and quadratic interpolation for rotations as will be seen later.
3.2.1 Geometrv and Dis~ïacement Intemolations
Different coordinate systems, used Ui the formulation of the consistent shell
element, are s h o w in Fig.3.1. These coordinate systems are given as:
1 ) Global set of axes (x,y,z)
2) Local set of axes (x',yt,z'), x' and y' are tangent to the surface, while 2' is
perpendicular to the surface
3) Curvilinear set of axes (r,s,t), where t is perpendicular to the suface
The location of a point within the element in the global coordinate system is determined
by the coordinates of the corner and the mid-side nodes (x,,x,zJ and vector V,, at each
node as
Ni are quadratic interpolation hmctions and V,i is the unit vector perpendicdar to the
d a c e (at node i) multiplied by the thickness of the sheil and t is the through thickness
coordinate varying nom - 1 at the bottom to +l at the top of the shell.
The mid-surface displacements u, v, w are interpoiated cubically within the
element using displacement degrees of fieedom at the corner nodes, one-third side nodes
and the center node. Quadratic approximation of the rotations a (about y') and P (about
x') are achieved using rotational degrees of keedom at the corner and mid-side nodes. It
should be noted that these rotational degrees of fkeedom provide a linear displacement
through the thickness. The degrees of freedorn associated with various nodes of the
element are shown in Fig.3.1.
Based on the above, the global displacements u, v and w in terms of the nodal degrees of
Freedom are
H t where M, = - , H is the thiclmess of the shell, Ni and Ni are cubic and quadratic
7
interpolation func tions, respectively. [y ] = [vli - V2i ] , where the unit vecton
- V,i and Tzi are directed dong the local axis x' and y' axes, respectively. The procedure
for calculating the above vecton is desm'bed by Koziey (1993). It should be noted that
another set of degrees of &dom, which provide cubic variation of the displacement
through the thickness of the shell (and consequently exact distnibution of the transverse
shear stresses), were used in the formulation of the element. Such degrees of hedom are
important when the analysis of a thick &el1 stmchire is considered. In the current
application, which is directed towards the analysis of thin structures, these degrees of
fkedom were deactivaid A fûll description of the element as well as the stiffiiess maau<
formulation are provided by Koziey (1993).
3.2.2 Stress-Strain Relations and the Thermal Load Vector
In this sub-section, the load vector of the laminated consistent shell element
resulting nom thermal variation is developed.
Within a lamina, the stress-strah relationship in the local coordinate system
x',y',zf is
where E', and do are the initial local strain and stress vectors, respectively. The matrix
[Df] is related to the constitutive matrix for orthotropic matenal [Dl using the following
transformation:
P'I = [TJT Pl [TJ (3.4)
The constitutive matrix [Dl, given by Jones (1975), is defined in the materiai axes system
1-2-3 (axis 1 is pardel to the fiber direction, while axes 2 and 3 are perpendicuiar to the
fibers, both axes 1 and 2 are tangent to the mid-dace of the shell). The matrix [TJ
represents the transformation matrut relating the local axes system (x',y' and z') to the
matend axes system (12 and 3). An expression for [Thl is given by Cook et al. (1989). It
should be noted that the transformation angle 0 (orientation angle), which is included in
the ma& [T J, is the angie between the £%a direction (axis 1) and the local axis x'. In
practice, practitioners usually use the angle betwem the fiers and the vertical axes of the
chimney as a reference angle. However, in this study, it was decided to define 0 as the
angle between the x' axis (i.e. horizontal plane) and the fibers direction. This was done
for convenience to be consistent with the way consistent shell element is developed.
A temperature change AT vdl induce initial thenaal strains {&} (in the material axes 1-
2) which are given by :
where a,, and a ?, are the thermal expansion coefficients of the Lh Iayer in the direction
of axes 1 and 2, respectively. The transformation m a h [Tc] is applied to {&"a} to
obtain the local initial strains {EL } expresseci relative to the local axes X'J' and z' i.e.
by d e m g the potential energy of the system n (subjected only to thermal stresses) as
and substituthg Eq.3.3 into Eq.3.7 (putthg {sa } = {O} ), the following expression for
the load vector {f) due to temperature change is obtained.
where the strain ma& P'J relates the nodal degrees of fkedom to the local strains and
is defined by Koziey (1993), H and hL are the total thickness of the sheii and the thickness
of the L' layer, respectively, detlJl is the determinant of the Jacobian matrix and is also
given by Koziey (1993), and t, is the through thiclmess variable varying from -1 at the
bottom to +1 at the top of the L~ layer, and is related to t using the following relation:
The integration of Eq.3.8 is performed numericaiiy using Gaussian-Quadrature
scheme dong r, s and t,.
Using the load vector ( f ) resuiting fiom temperature variation AT and the
stifl'hess matrix F] (given by Koziey, 1993), displacements and consequently
strains { E ' ) resulting fkom such temperature change are obtained. Substituthg {o . } and
{si i into Eq.3.3, the thermal stresses {a'} cm be evaiuated.
3 3 Verification of the Mode1
in order to ver@ the accuracy of the above f i t e element development, thermal
analysis of a number of plate and sheU problems is pdormed using the consistent
laminated shell eIernent.
3.3.1 Isotrooic Plate Subiected To Linearlv VpRrinp Tem~eratnre Chawe
A simply supported isotmpic plate is anajyzed under a linearly varying through
thickness temperature distribution The temperature variation at any point w i t h the plate
is expressed as: T(x,y,z) = TL s/H, where TL is the value of the temperature at the top and
bonom fibibas of the plate; z is the coordinate normal to the plate and measured from the
rnid-surface; and H is the thickness of the plate. The boundary condition of the plate are
such a way that the x-displacements are prevented at the two edges perpendicular to the
x-axis, the y-displacements are prevented at the two edges perpendicular to the y-axis,
and the z-displaciments are prevented dong the four edges of the plate. Results of the
analysis are presented using the dimensionless parameter w, which is defuied as:
wL = H w/a, TL A'
where: w, is the centrai deflection of the plate, a, is the coefficient of thermal expansion,
and A is the length of the plate dong the x-axis.
The analyses are conducted For different A/B and W A ratios; where B is the length of the
plate dong the y-axis. Values of the dimensioniess parameter w, resulting from these
analyses together with those predicted by Timoshenko et al. (1959) are presented in Table
3.1 showing an excellent agreement. It should be noted that the displacements resulting
fkom the thermal analysis of isotropic plate are independent of the modulus of elasticity
of the plate. The Poisson's ratio of the plate considered in this exarnple is assurned to be
II Table 3.1. Results of the analysis of an isotropic plate subjected to lineariy varying tem~erahm
w, for various A/B ratios
Source A/B=I .O A/B=2.0
Timoshenko (1959)
Timoshaiko (1959)
Timoshenko (1959)
33.2 Anti-svrnrnetric Annle Plv Laminated Plate Subiected to Linear Tem~erature
An angle ply (#) square plate is considered for thermal stress analysis using the
consistent shell element. The plate has the same boundary conditions and is subjected to
the same through thickness temperature distribution described in the previous exarnple.
The mechanical properties of the orthotropic lamina dong the 1-2 directions (1 is the
fiben direction and 2 is an axis perpendicular to the fibers in the plane of the plate) are
givenas: E, = 53.8 GPa, E, = 17.9 GPa,G,, =G,,=G, =8.62 GPa, v,,= v,, = v,=0.25,
a, = 6 . 3 ~ IO4 m,m/ O C , a = 20% IO6 dm/ OC. The aspect ratio of the plate is chosen in
nich a way that: ARI = 100; where A is the length of the plate and H is the thickness.
The analyses are conducted for various angles of orientation 8 and considering 1
and 4 layers, respectively. Results of the analyses are also presented using the
dimensionless parameter w, and are given in Table 3.2 together with those predicted by a
finite element soIution conducted by Wu et al. (1980). An excellent agreement between
the results of two sets of analyses is shown.
Table 3.2. Results of the analysis of an anti-symmetric angle-ply plate subjected to linearly varyîng temperature
W*
Source 2 layers 4 layers
3.3.3 Isotro~ic Cvlinder Subiected to Linearlv var vin^ Tem~erature Chan~e
A closed fonn solution for the thennai stress anaiysis of free standing isotropic
c y linder subjected to through th ichas linearly varying temperature is given b y
Timoshenko et al. (1959). According to this solution, the longitudinal stress o, and the
circumferential stress a, at the outer and inner faces of the cylinder evaluated at a point
away from the boundary (ei ther the restrained or the fkee end) are given by the lollowing
relation:
T, and T, are the temperatures at the inside and outside faces of the shell, respectively. In
the above equation, the stresses at the outer Face are t e d e if T,>T,. An isotropic free
standing cylinder having a modulus of elasticity E =36.85 GPa , a Poisson's ratio v = 0.3,
a coefficient of thermal expansion a = 7.7~10' m/m/ OC and a diameter D = 3 . h is
modeled using the consistent laminated shell element. The cylinder is subjected to the
through thickness temperature distribution shown in Fig.3.2. The above parameters are
substituted into Eq.3.10 to obtain the stresses at a cross section away from the boundaries.
According to Eq.3.10 such a section is subjected to pure fircdereatial and longitudinal
bending stresses (Le. stresses at the mid-suffice e q d zero) which are equal to 2.027 MPa
and -2.027 MPa at the inner and outer faces, respectively. Results of the nnite element
analysis together with those predicted by Timoshenko e t al. (1959) are presented in
Fig.32 by plotting the stresses a,, G , and <r, dong the le@ of the cylinder (in this
figure y = O corresponds to the k e end), where: a, is the outer circumferential bending
stress, a ,, is the outer longitudind bending stress and a, is the mid-plane
circumfetential stress. Fig.3.2 shows that the longitudinal bending stress G ,, vanishes at
the fkee end and that both oh, c, approach the exact value (2.027 MPa) away fiom the
boundaries. It is also clear fkom the figure that a full agreement between the finite
element and the closed form solution is achieved.
3.3.4 Anti-svrnmetric Cross Ph Cvlindrical ~ a n e l Subiected to Uniform
Tem~erature
A four-layer cross-ply (0°/900/00/900) cylindrical panel has been considered for
thermal stress analysis. The panel is subjected to unifonn temperature field and its
geometry is deked by the following ratios: A/B=I, A/H=100 and RI B = 1 , where the
variables A, B and R are shown in Fig.3.3 and H is the thickness of the shell. Two types
of boundary conditions are considered in the analysis; BC, has the four edges of the panel
clamped while BC2 has the circular edges klly clamped and the straight edges (dong y-
ais) satisQ the following boundary conditions (see Fig.3.3 for axes description):
i) The x and z displacements as well as the rotation about the x-axis are restrained
ii) Al1 other motions are ailowed
The layers have the following properties defineci in the directions of the material axes ( 1-
2): E, = 181 GPa, & = 10.3 GPa,GI,=G,,=7.17GPa. G, = 6 1 1 GPa, v , ~ = v , , =v,
4-25, a, = 0.02x104 dm/ OC, - = 22.5x104 m/m/ O C . Results of the analyses are
prwnted using the dimensionles parameter w, ,defïned in nib-section 3.3.1, and are
given in Table 3.3 together with those predicted by a f i t e element analysis conducted by
Chandrashekhara et ai. (1993). It could be concluded fiom the results shown in table 3.3
that the consistent laminated shell element provides a very good agreement with the
analyses conducted by Chandrashelchara et ai. (1 993).
Table 3.3. Results of the anaiysis of anti-symmeûic cross-ply (0°/900/00/900) a= 1)
Chandrashekhara (1 993)
3.4 Thermal Stress Analvsis of FRP Chimnevs
Having verified the accuracy of the consistent laminated shell element when
extended to thermal stress analysis, the finite element mode1 is then used to study the
effect of various parameten affecting thermal stresses induced in FRP chimneys. FRP
chimneys are usually constmcted fiom angle-ply laminates with orientation angles varies
between 8 = k 3 5 O and +5S0. The description of the laminate is following what is
presented in sub-section 2.4. Figure 3.4 shows a typical cross section of cylindrical
chimney and a vertical projection of the laminate. It is also s h o w in the figure, the
material axes 1-2, the local axes x' and y' and the orientation angle 8.
Thermal stresses induced in ERP chimneys depend on the curing temperature of
the composite. Unsaturated polyester and vinyl ester resins, which are cornrnonly used in
the construction of FRP chimneys, are usudIy cured at a temperature ranging between
50-1 50 OC, (Mallick, 1997). in order to snidy thermal stress resulting nom the difference
between the operating and the curing ternperatures, a number of FRP chimneys
constructed ffom a Der41 1-45 matrix reinforced by E-giass fibea are modefed using the
consistent laminated shell elernent. For a 70% fiber content (based on weight), the
mechanical properties for a laminate dong the material axes are given by: E,= 36.85 GPa,
E2 = 11.16 GPa, G,, = G,, = 3.36 GPa, G, = 4.32 GPa, a, = 7.7~10-6 mlmPC, a2 =
4 3 . 4 ~ 10-6 mlmPC, where a, and a, are the coefficient of thermal expansion in the Bbers
direction and perpendicular to the fibers, respectively.
The curing temperature (reference temperature of the composite) is assurned to be
equal to 100°C and the chimneys are analyzed at interior (operating) temperature and
extenor (arnbient) temperature equal to 70°C and -3 O°C, respectively . This leads to
temperature change (relative the curing temperature) at the interior (AT,J and the exterior
(AT.3 surfaces of the chimneys equd to -30°C and -130°C. respectively. The above
mechanicd properties and temperature variation are used to perforrn a parametric study
hvestigating the effect of various parameters (thickness of the shell, number of laminate
layen, orientation angle of' the fibers, diameter and height of the chimney) on the themal
messes induced in FRP chimneys during their operating stage. in al1 analyses, the
boundary conditions are assumed to be full £kation at the bottom of the chimney and fkee
displacements and rotations at the top.
3.4.1 The Effect of the Laminate Thickness
A FRP chimney having a diameter D = 3 . h and a height L = 40.0 m is
considered for thermal stress analysis. The larninate of the chimney consists of 5 angle-
ply layers (5S0/-55" 155"/-55"/55"). Notice that these angles are measured relative to the
axis x', x' is an axis tangent to the surface and located in a horizontal plane. The andysis
is carried out by varying the laminate thickness in the range of 1 Omm to l3Ornm. The
temperature distribution is assumed to be linear with values of -30°C and -130°C at the
interior and exterior surfaces, respectively (as described in the previous section). The
thermal stresses that resulted h m the analysis are plotted in Figs.3.5 and 3.6 for a
location away nom the boundary and for a point located at the base of the chimney,
respectively. Fig.3.5 indicates that the thickness has no effect on the thermal stresses at
sections located away nom the boundary. This is due to the fact that by increasing the
thickness of the shell, both the initial thermal strains (extemal load) and the stifiess of
the shell increase and thus the same values of final thermal stresses are obtained. Fig.3.6
shows that up to a thickness of 3 0 m , an increase in the thickness leads to a
corresponding increase in the themial stresses at the base of the chimney. The same figure
shows that beyond a thickness value of 30mm, stresses become almost constant. This
behavior was also reported for laminated plates by Thangaratnam et al. (1987).
In summary, it can be concluded that beyond a certain thickness value, an increase
of the thickness of FRP chimney bas no effect on the induced thermal stresses.
3.4.2 EfKect of the Diameter of the Chimnev
A chimney having a height equal to L = 40m, a thickness H = 65mm and
consisting of 5 layers symmetric angle-ply laminate (8 = t5s0 ) is considered for thermal
stress analysis in order to asses the efTect of the diameter of the chimney. The temperature
variation follows the linear dishibution previously described when studying the effect of
the thickness. The parametric study is perfomed by varying the diameter of the chimney
in the range between 1.5m and 6m. The variation of the themal stresses induced ai the
base of the chimney versus the diameter is presented in Fig.3.7. It could be concluded
from the figure that the change in the diameter has no significant effect on the thermal
stresses. in Fig.3.8 both the hoop thermal stresses (q) and the axial (meridional) thermal
stresses (a,) are ploaed along the height of one of the analyzed chimneys. As might be
expecied, both the hoop and the axial thermal stresses have rapid fluctuations near the
boundaries (for both the fixed and the fiee botindaries). In general. the thermal stress
distributions show hi& stress values occmhg very close to the boundaries and are
localized in a narrow region.
3.4.3 The Effect of the Heieht of the Chimnev
The effect of the height of the chimney on the induced thermal stresses is studied
by fixing both the diameter and the thickness of the FRP chimney and varying its height.
Analyses indicated that the maximum values of thermal stress (occurring near the fixed
bottom of the chimney) are independent of the height of the chimney. The stress
distniutions along the height are Clpcaily as shown in Fig.3.8; the change of the height
only aects the length of the region having constant stress distriiution.
3.4.4 The Effect of the Number of Lavers and Fiber Orientation
In this section, the effects of varying both the number of layers (keeping the total
thickness constant) and the orientation of the fibers on the thermal stresses induced in
FRP chimneys are studied. The parametric study is conducted by considering a FRP
chimney having a height L = 4ûm, a diameter D = 3m and a total thickness H = 65mm.
This thickness was chosen by considering 2 , 4 5, 6 and 10 layers larninate, respectively.
The laminates 2, 4, 6 and 10 consist of anti-syrnrnetric angle-ply layers (28) and the 5
layer laminate is a symmetric angle-ply laminate. For each larninate configuration, the
angle of orientation 8 has been varied between 0' and 90'. In Fig.3.9, the variations of the
maximum longitudinal stresses a, and transverse stresses q (occurring near the base) for
the outside face of the c b e y are plotted versus the angie of orientation 8 for different
laminate configurations. Fig.3.10 shows similar graphs plotted for the inside face. Both
figures indicate that the number of layers has no significant effect on both the
longitudinal and the transverse stresses. At the inside face of the shell, the increase of the
fiber orientation 8, increases the longitudinal stresses reaching maximum values at 0 =
90" and decreases the transverse stresses which reach minimum values at O = 90"- For the
outside face, the increase of the angle ply 8 leads to a slight decrease in the stresses which
is then followed by a significant increase of the stresses with the angle 8 (at 0 = 37.5" for
the case of 0,).
3.4.5 Summary of the Results of the Parametric Studv
From the above conducted parametric study, it can be concluded that the height,
the diameter, the thickness and the number of layen used to achieve the thickness have
almost no effect on the maximum thermal stresses induced in FRP chimneys. Such
stresses are usually very localized in a nanow region near the base of the chimney. The
main parameten affecting the values of the stresses are the temperature profile, the angle
of the orientation of the fibers, the coefficient of thermal expansion and the modulus of
elasticity along the fibea direction. For practical FRP chimneys consisting of glass fibers
and vinyl ester resin, the last two parameten depend rnainly on the percentage of the
fibers content.
The practical range for the angle of inclination 0 is between 3S0 and 55". Examining
the stress values shown in Figs.3.6 and 3.7 (these figures represent results for chimney
having 0 = 5 5 O ) , it cm be concluded that the maximum value for the stresses a, (along
the fiben direction) and q (perpendicdar to the fibers direction) are approxirnately 100
MPa and 80 MPa, respectively. Typical d t h a t e strength dong the fiben a,, and
perpendicular to the fibers ou have approximately the following values o,, =il00 MPa
and sZu = 33.5 MPa (for 70% E-glas content based on weight). Cornparison between the
induced stresses and the ultimate strength indicates that although large factor of safety is
achieved along the libers direction, the cross &ers direction is unsafe. As such, one
would expect that cracks localized at the bottom part of the chimneys paralle1 to the fiber
direction would occur (independent of the value of the thkkness) due to thermal stresses.
3.5 Practical Considerations for Attern~tinp Desipu Procedure of FRP Chimnevs
From the above discussions, it is clear that the temperature distribution assumed
in the analysis results in across thermal stresses which are approxirnately 2.5 times the
allowable stresses in that direction. As such, it is aixnost impossible to avoid cracking in
the across fiber direction. Moreover, if the design is govemed by preventing such cracks,
the fiber reinforcement would be redundant. Knowing that cracks will occur, it has been
decided to anaiyze the FRP chimneys under thermal loads by assuming that the stiffhess
in the direction perpendicular to the fibers alrnost vanishes (Le. & is very mall). R i s
assumption is made for al1 layers along the height of the chimney. The author believes
that this assumption is conservative because, in practice, cracks will not occur in al1
layen and not necessary along the whole height of the chimney. The safety of an FRP
chimney analyzed under such an assumption can be checked by assuring that the stresses
dong the fiben do not exceed the ultimate strength divided by a suitable factor of safety
and also that the interlarninar shear stresses are also well below the ultimate shear
snength. By assuring that the interlaminar shear stresses are safe and using an angle-ply
configuration, it is expected that the cracks in one layer will be very much controlled by
the stifiess of the two adjacent layers along the fibers direction. Figure 3.1 I shows the
variation of the longitudinal stresses o, with the angle ply 0 for a typicd FRP chimney
using the temperature distriiution descrihi above (after degrading the across fibers
stiflhess). It should be noted that the anaiysis has been perfomed for a practical range of
8 varying between 3S0 and 60". From Fig.3.11, it c m be concluded that the maximum
stresses o, do not exceed value of 180 MPa This value leads to a factor of safety of
approximately six when compared to the ultimate strength. in order to check safety
against shear failure, the in-plane shear stress r,, as well as the transverse shear stresses
r,, and r,, resulted from the same analyses are plotted in Fig.3.12 versus the angle of
orientation 8. The typical values For the ultimate shear strength in-plane and transverse
are given by r,, = 70.6 MPa , r,, = 70.6 MPa and 7, = 18.85 MPa Cornparison between
the induced shear stresses and the ultimate ones reveals that factor of safety of
approximately 3.5, 15 and 5.6 are achieved for the in-plane and the transverse shear
stresses, respectively.
3.6 Thermal Stress Vaiues to be Used in Practical Desien of FRP Chirnnevs
As mentioned before, the thermal stresses induced fiom temperature variation in
FRP chimney depend on the followuig factors:
1. Fibers content
2. Angle of inclination of the fibm
3. Temperature pro file
4. Type of fibers and resin
Restrainîng the design to FRP chimneys constnicted nom vinyl ester resin
reinforced by 70% (based on weight) E-glas fibers, for a certain angle of inclination 8 of
the fibers, the themal stresses depend only on the temperature profile. This profile is
govemed by two parameters which are:
1. The variation of mid-dace tempcratrne with respect to the curing temperature Tm.
2. The Merence between the temperahire at the inside and the outside faces (AT); AT =
Using the approach described in sub-section 3.5. analyses have been conducted to
determine the maximum stresses o, as function of Tm and AT for huo angle
configurations, 8 = f 35' and k 55'. respectively. Figures 3.13 and 3.14 show the
variation of the maximum dong fiber stresses CF, vems the temperature variation AT for
different values of' Tm for 0 = f 35" and i 5S0, respectively. These graphs can be used to
estimate the stresses induced in a FRP chimney, having the above-described properties
under various temperanite variations. Cornparison between the two graphs indicates that
in general higher themal stresses are introduced when the fiben become more vertically
inclined (i.e. 0 = f 55' leads to higher thermal stresses than 0 = + 35'. The shear stresses
associated with various temperature profiles are shown in Tables 3.4 and 3.5 for 0 = + 35O and f 5S0, respectively. It should be noted that the shear stresses Vary linearly with the
parameter Tm and independently of AT. The designer of FRP chimney has to assure that a
sufficient factor of salety is achieved against shear failure.
Table 3.4. The in-plane and transverse shear stresses associated with the longitudinal
Table 3.5. The in-plane and transverse shear stresses associated with the longitudinal stresses in Fig.3.14 for an angle.
Middle surface 1 Maximum in-plane s-
Maximum transverse shear stresses
In this chapter, the formulation of the consistent laminated shell element is
extended to include thermal stress analysis. A number of plate and shell structures are
rnodeled for themial stress anaiysis and the redts are compared with those available in
the literature. in al1 examples, the elernent gives adequate predictions for thermal stresses.
The developed h i t e element formulation is then used to study the effect of various
parameters which might influence the thermal stresses induced in angle-ply laminated
fiber reinforced plastic chimneys. Redts of the parametric midy indicate that the
thickness, the diameter, the height and the number of laminae have no significant effect
on the induced thermal stresses. Analyses indicate that the thermal stresses depend
rnainly on the through thiclmess temperature distnàution (relative to the curing
temperature), the angle of orientation of the fiers, the coefficient of tfiennal expansion
and the moduîus of elastici@ dong the fîôers direction. The last two parameters depend
on the mer content in the matrix. The thermal stress analyms of typical FRP chimneys
shows high stress concentration near the boundaries with in-plane across fiber stresses
exceeding the typical ultimate strength in this direction. As such, cracks are expected to
occur in FRP chimneys as a result of through thickness temperature variations. However,
it is believed that these cracks will be controlled if the interlarninar shear stresses are less
than the ultimate shear strength divided by an appropriate factor of safety.
The analysis then proceeds by assuming a negligible value for the modulus of
elasticity in the direction perpendicular to the fibers. Results of this 1 s t set of analysis
indicate that for the practical range of the early mentioned influential parameters, the
dong fiber direction stresses as well as the shear stresses of cracked chimneys are within
acceptable values. Finally charts predicting the dong fiber thermal stresses induced in
typical cracked FRP chimneys (but lirnited to 70% nber content and angles of inclination
9 = f 35 O and f 55') as a function of the through thickness temperature distribution are
presented These stress values cm be considered when the design of a FRP chimney is
atternpted.
0 u , % W , c C P i
A sR 0 4 v, W'
Fig.3.1 Coordinate systems and nodal de- of M o m of CLS element.
i Elasticity
A
0 present B
1 I 6
O 1 2 3 Y (ml
Fig.3 -2 Variation of outer and mid-dace longitudinal and cucumferentid stresses at fÎee end of cylinder due to ünearly varying temperature change.
Fig.3.3 Cross-ply cylindrical panel
Fig3.4 Cross section of FRP chimney and vertical projection of the laminate.
Fig.3.5 Thermal stresses of 5 layers angle-ply (+/O 55' ) FRP chimney versus the laminate thickness at a section away fiom the boundaries of the chimney.
0.02 0-04 0-06 0.08 O. 1 O 0.12 0.14
thickness (m)
a" O -
bF -25 -
-50 -
-75 -
Fig3.6 Thermal stresses of 5 layers angle-ply (+/- 5 9 ) FRP chimney versus the laminate thickness at the base section of the chimney.
-, 02
inside Face
-100 -
Fig.3.7 The eKect of the diameter on the thexmai stresses induced at the base of a FRP chimney .
120
80 7,
40 -
-30 -25 -20 -15 -10 -5 O
Hoop stresses O, (MPa)
0 i,
-40 -
-80 -70 -60 -50 -40 -30 -20 -10 O
kWai stresses O, (MPa)
O2 ...----.A. ....... A ...... ..A-----..-, ........ -........A- - . . * * * - - . * - . . d, inside face
Fig.3.8 The hoop and axial streses at the inside face dong the height of a FRP chimney subjected to linearly varyi.ng temperature.
-80 - 01 -1 20 I I I 1
2 3 4 5 6
Diameter (m)
Fig.3.9 The maximum longitudinal and transverse stresses at the outside face of the larninate vs. the angle of orientation.
+/- 8 Fig.3.10 The maximum longitudinal and transverse stmses at the outside fafe of the laminate vs the angle of orientation.
AT,=-30 OC - E y E l n 000
AT,=- 1 30 O C
O, mid surface stresses 10 layers
- ..... 2 layers ---* -- 4 layers O, (inside fàce)
- A
I - 10 layers
Fig.3.11 The maximum longitudinal stresses at the inner and outer Face of the laminate vs the angle of orientation after degrading the across fibers stitfness of the layers.
Fig.3.12 In-plan shear stress s,, , transverse shear stresses r,, , r, of 10 layer lamiBate at the bottom of the chimney after degxading the across fibers dffhess of the layers.
1 . . . - . - - inside face 1
Fig.3.13 The longitudinal thermal stresses of 3 5' angle-ply FRP chimney for Merent temperature fields (degraded across fibers modulus E, = E,!1000)
AT,, T r\ inside face 10 layers antisymrnetric angle-ply laminate +l-55"
Fig.3.14 The longitudinal themai stresses of 55' angle-ply RIP chimney for different temperature fields (degraded across fibers modulus E, = E ,11000)
CEAPTER 4 COMPUTER A[DED-DESIGN CODE TO EVALUATE WIND RESPONSES OF
FlBER REINFORCED PLASTIC CEfIMNEYS
4.1 Introduction
The design of industrial chimneys is usuaily govemed by the stresses and
displacements induced by the wind loads. Meanwhile, a proper design should also
account for various phenomenae associated with wind loads acting on slender structures
such as vortex shedding and ovalling.
One of the engineering problerns that interest the researchea and the designers of
industrial chimneys is understanding the complete behavior of the vortices in the
downstream of cylinden created by the oncoming flow. Strouhal stated the relationship
between the fkequency of the vortices, the wind speed and the diameter of the cylinder
more than a cenhny ago. Many efforts have been made in the past to estimate the
magnitude of the fluctuating forces acting on cyünders and associated with the turbulent
wind in the wake of the structure (Van Koten (1969), Scurton (1963), Vickery (1997),
Davenport (1993)). The vortex shedding phenornenon is stilI an open area of research. As
stated by Vickery (1997). the difficulty in predicting the across-wind behavior of
chimneys is that the current available wind hmneis are not capable of achievkg high
Reynolds number associated with prototype chimneys.
The purpose of this chapter is to investigate anaiytically the response of FRP
chimneys under wind loads. A simple and efficient computer code, to be used in
achieving this task, is developed and desdeci in this chapter. The developed computer
code can be used to perforrn static and dynamic analysis of tapered cantilever like
laminated structures (e.g. FRP fiee standing chimneys) subjected to wind loads. The
developed computer code is based on the following:
1) Classical lamination theory to obtain apparent elastic properties of the laminate based
on the mechanical properties of each lamina in its local axes.
2) The Stodola method to evaluate the natural fiequemies and the correspondhg mode
shapes of a FRP tapered chimney.
3) The wind loads acting on the chimney (dong, across and vortex shedding) treated as
an equivalent static loads according to the CICIND code for steel chimneys (1988), or
as a combination of static and dynamic loads based on a procedure developed by
Davenport ( 1993).
4) Tsai and Wu (1971) failure criterion used to constnict a failure envelope representing
the limit bearing capacity of each lamina.
5) Fatigue stresses due to vortex shedding evduated and encountered in the design by
applying the fatigue damage indicator defined by the EUROCOMP Design Code of
FRP (1997).
in this chapter, a bnef description for the above theones and procedures and how
they are incorporated in the development of a computer design-aided code for EXP
chimneys subjected to wind loads, is presented. A flow chart showing the interaction
between different parts of the cornputer code is givm. A verification for the developed
code using results of detailed finite element is presented and a parametnc study on the
parameters affecthg the vortex shedding respome is performed. Finally, design
thicknesses for different aspect ratios and factors of safety are provided for FRP chirnneys
subjected to both wind and themai loading.
4.2 Laminate Eauivalent Elastic Pro~erties Usin? Classical Lamination Theorv
A typical cross section of a tapered FRP chimney is given in Fig.4.1. As shown in
the figure, the FRP chirnney is constxucted from a number of curved larninae. in each
lamina, fibers have a specific orientation forrning an orthotropic layer. The principle
material axes for each lamina are those parallel and perpendicular to fiber direction as
shown in a vertical projection of a typical laminate given in Fig.4.2. In this figure, axes 1
and 2 are tangent to the surface and are parallel and perpendicular to the fiben,
respectively. Axis 3 is perpendicular to the surface, Le. perpendicular to both axes 1 and
2. In the sarne figure, another set of local orthogonl axes X, y,Z are shown; both K and
y are tangent to the surface of the chimney, X lies in a horizontal plane and y has an
inclination a (a is the tapering angle of the chimney) with the vertical, Z is perpendicular
to both X and 7, Le. perpendicular to the surface of the chimney (coinciding with the
material axis 3). The direction of the local axes (Z-7) relative to the matenal axes (1 J ) is
shown in Fig.42, where 0 is the angle between axes 1 and X.
The laminate is constructed by stacking a number of laminae having fibers
onentated in different directions to give desirable stifniess and strength in vanous global
directions. The material properties of each lamina in the directions parallet and
perpendicular to the fibers (material directions) are usually available from the
manufacturer. Starting h m the stress-strain relations for each orthotropic lamina, the
classical lamination theory can be used to evaiuate the elastic pmperties for an equivalent
orthotropic section. The aeps used to obtain the equivaient orthotropic properties are
described in the following sub-sections.
4.2.1 Stress Strain Relations for Laminated FRP Material
The classicd lamination theory is based on the assumption that transverse shear
stresses r,, and t, as well as the nomial stresses perpendicular to the surface a,, are
neglected. Based on the above assumptions,
material axes (1,2,3) for an orthotropic lamina
T . P
the stress strain relations in the principal
are given by:
where Qij are the reduced stiffhesses and are defineci in terms of engineering constants as
Qll = El I - u12uT,
E, and Et are the longitudinal and transverse Young's modulus of the lamina (paralle1 and
perpendicular to the fibers, respectively) , G,, is the in-plane shear modulus, v , ~ is the
major Poisson's ratio of the lamina defined as the ratio between the transverse strain - to
the longitudinal strain E, when the lamina is stresseci in the longitudinal direction only,
v2, is the minor Poisson's ratio and can be dehed the same way.
Stresses and strains in the local axes systern (%y) are related to the material axes
system ( 1 -2) b y the following transformation:
where the transformation matru< [TI is defined as
where 0 is the angle of orientation between the material lamina axes and local laminate
axes as shown in FigAl.
Usùig Eqs. 4.1 to 4.5, the stress strain relations in the X, y,Z coordinates are given by
where
- Q,, = Q, ,cos4B + 2(Q,, + 2Q,)sin20cos'0 +Qn sinie
O,, = (QI, + Q, - 4~,)s in~Ocos~e +Q,2(~U14e +COS%)
an = Q,,sin4B + 2(Q,, + 2Q,)sin20cos20 +Q, cos% - QI, = (Q,, - QI2 - 2Qs6)sin0cos30 + (QI,-Q, + 2Q,)sin30cos0
= (Q, , - QI, - 2Q,)slli38cos0 + (Q,, -Qn + 2Q,)sin0cos3~ - Q, = (QI, + Qu - 2Q12 - 2 ~ ~ ) s i n ~ 9 c o s % + Q,(sin% + cos%)
It is noticed that the coefficients of the transforrned reduced stifhess matrix 4, given by
Eq.4.7 have no zero cross-diagonal terms in contrast with the reduced stifiess matnx Q,,
given by Eq.4.1. This means that coupling exists between shear strain and normal stresses
as well as between shear stresses and normal strains.
4.2.2 Extension and Bendinp Stiffiiesses for Laminated FRlP Materials
The classical lamination theory assumes perfectly bonded layen, Le., no slip is
allowed between the layea. For a thin laminate, the transverse shear deformations y, and
y, are neglected. As such, a straight line perpendicular to the middle surface will remain
straight and perpendicular to the middle surface when the laminate extends or bends,
(Kirchhoff-Love hypothcsis). Also, by neglecting the normal stmh si, the middle
d a c e strains and curvature c m be expresseci as
where u,, vo and wo are the middle W a c e displacement in Z, Y andf directions,
respectively. K ~ , K , and K are the middle surface curvatures respectively. Based on
the above, the strain components through the laminate thickness can be obtained From the
following relation:
By substituting Eq.4.9 into Eq.4.6, the stresses in the kh layer cm be expressed in ternis
of the rniddle surface strains and curvature as following
Due to the variation of 4, within various layen, jump in the stresses at the interlarninar
locations is expected to occur in spite of the continuity of the strains through the
thickness of the laminate.
The resulting laminate forces and moments can be obtained by integrating the
stresses through the laminate thickness which yields the following
O %
O - #E E; &+k, Y:, K-
where NI, N , and N , are the membrane forces per unit length, M, , Mi and M , are
the bending moments per unit length of the laminate, N is the total number of layers and
Z, andZk-, are the distance betwem the top and the bottom faces of the km layer from the
middle surface as shown in Fig.4.3. Since the middle surface strains and curvatures do
not vary with the coordinate, Z , Eq.4.11 and Eq.4.12 can be rewritten as
Where
N
a, B, and D, are the extensional, coupling and bending stiffnesses. The prnence of Bi,
terms produces couphg between the bending and the extension of the laminate. As such ,
a laminate which has non zero BQ terms will bend a d o r twist if it is extended. For more
details, the reader is referred to Jones, W. (1 975) and Eckold, G. ( 1 994).
The purpose of this section is to evaluate the elastic properties for an equivalent
single orthotropic layer to replace the laminated FRP in perfonning the analysis. As
mentioned before, when the laminate is constnicted by stacking a number of orthotropic
iaminae in an arbitras, sequence of orientations, the laminate stiffhess matrices will be
hlly populated. The presence of the coupling rnatrix p] produces coupling between
bending and extension. Meanwhile, the presence of A,, and A, in the matrinix A produces
coupling between the normal stresses and shear strains ( D,, and DI, do the sarne effect).
For laminates which are symmetric about their mid-plane (Le. for each lamina above the
rnid-plane, there is an identicai one at equal distance below the mid-plane), the
components of the coupling matrix [BI vanish. On the other hanci, for laminates which are
anti-symmetric about their mid-plane (Le. for each lamina above the mid-plane having a
positive angle0 , there is another lamina at qua1 distance below the mid-plane with same
thickness and have a negative angle û), the t e m A,, A,, Dl, and D2, vanish. For the
case of laminates constnicted fiom large number of angle-ply layers M. the cornponents
of the matrïx p] fùlly vanish and the ternis A,, A,. D,, and D, become very close to
zero and thus can be negiected. The chimneys considered in this study are chosen to meet
such criteria
Substituthg Bi,, A,,, A , D,, and D2, with zeros into Eqs. 4.13 and 4.14, yields:
(NI = [Al @) and (MI = [Dl {KI
w hem:
Eq.4.16 is inverted to give
(4.1 7)
(4.18)
For an equivalent orthotropic materials, the extension stress-strain relations are given by:
(4.19)
where
and h is the thickness of the orthotropic matenal.
Also for orthotropic materials, the bending-sîrain relations are given by:
(KI = F I {Ml
where
1 [F7=p
Equations (4.17), (4.19) and (4.20) can be used to obtain the properties of an equivalent
orthotropic materid having the same extensional stifiess of the laminate. This leads to
the following equivalent larninate extension properties:
(Extension)
where q, are the components of the ma& [A]". Meanwhile, Equations 4.18, 4.21 and
4.22 can be used to obtain the propdes of an equivalent orthotropic matenal having the
sarne bending stifhess of the larninate. This leads to the following equivalent laminate
bending properties:
dl2 v* =-- dl,
(Bending)
4.3 Beam Bendin~ Behavior of Chimnevs
The behavior of a Eee standing chimney subjected to wind loads can be simulated
as beam bending about axis A-A shown in Fig.4.l (the load is acting dong the x axis
shown in the figure). Considering a horizontal cross section of the chimney, such global
bending will cause mainly elongation (or shortening) straîns which vary fiom point to
another on the circUZIlference of the section (the through thichess local bending saain
can be neglected as the thickness of the chimney is much smaller than its radius). As
such, the main parameter which govem the strains induced in the equivdent beam mode1
is E,, (equivalent extension rnodulus of elasticity dong the y axis shown in Fig.4.1). E,
can be obtained by evaluating El fom Eq.4.23, then applying the following
transformation:
E, = E~ COS' a
where a is the angle between axes y and y. Note that this bearn mode1 can be used to
evaluate the displacements and the strains of the chirnney, stresses should be evaluated
fiorn the calculated strains and the tnie modulus o f each lamina as will be seen later.
4.4 Evaluation of Dvnamic Characteristics Usine Stodola Method
Due to the dynarnic nature of wind loads, the natural fkequencies and the
associated mode shapes of chimneys are required in order to estirnate their dynarnic
response. From the practical point of view, it is enough to consider the First few modes of
vibration in evaiuathg the dong-wind response because of the small contribution of the
higher modes to the variance of the dynamic response.
Ln this study, tapered chimneys having lineiffly varying thicknesses are modeled
as cantilever beams with varying moment of inertia Aluiuugh, closed form solution for
the natural Ekequencies and mode shapes of constant mass and constant inertia cantilevea
exisîs, the variation of the structure thickness and diameter introduce extra complication.
As such, it was decided to use the Stodola method (maîrix iteration) to detennine the
nahuai hquencies and mode shapes of the chimneys. Considering a FRP chunney, the
elastic properties for the equivalent orthotropic chimney can be obtained using the
procedm described in section 4.2. As the chimney is rnodeled as a beam, subjected
mainly to transverse wind loads, the quantity of interest is E, (extension) which is
calculated ushg Eq.4.25.
Stodola method is probably one of the best iterative methods in evaiuating natural
kquencies and mode shapes. The method starts by assumhg a trial shape for the fim
mode. This is followed by evaluation of the deflected shape resulting fkom the inertia
loads associated with the fint trial shape. This deflected shape, after normalization to a
certain desired amplitude, is used as a new mal shape (which is more accurate than the
first trial). The process continues iteratively dl1 the deflected shape in two consecutive
cycles are identical. For more detaiis about the Stodola method, the reader is referred to
Berg (1989). An expression for the naniral fkequencies ai is provided by Berg (1989) as
follows
where m(y) is the mass at elevation y,qi (y) is the finai computed deflected shape before
normalizing, L is the height of the chimney and yi (y) is the last normalized trial shape.
It should be noted that ,in rnanipulating the above integral, the height of chimney is
divided to equaily spaced intervals and the integrations are evaluated numericaliy using
the Simpson's method.
For any trial hct ion , the process will converge to the fimdamental mode as long
as it is not forced to converge to another. For proceedmg to the higher modes, the Iower
mode cornponents have to be swept out From the trial shape used in the iterations. For
exarnple, the third mode trial shape should be swept out frorn the first and the second
mode shapes which are already defined. Because the integration is perfonned
numerically, the trial shape will never be completely swept out From the lower modes
depending on the accuracy of the integration. As such, the sweeping process should be
done in each cycle of the iteration to ensure the convergence to the desired mode.
The natural frequencies and the corresponding mode shapes evaluated using the
Stodola method are incorporated into the dynarnic analysis as descnbed in the following
sub-sec tions.
4.5 The Wind Loads
The wind load acting on a typical chimney has two basic components; the mean
component which is rnainly static and the fluctuahg component which has a dynamic
nature. The Bucniating part is divided to an irreguiar and slowly varying component
known as the background component, and an oscillatory component having a definite
frequency and known as the resonant component The dynarnic whd response of a
c himney is contro lled b y various €omis of aerodynamic parameters; the turbulent
fluctuations in the oncoming flow which cause the dong and across-wind response; and
the vortices shedding in the wake of the structure which mate an across-wind response
and aerodynamic damping forces. Herein, the method used for calculating the wind load
response of FRP chimney is based on a procedure developed by Davenport (1993) to
evaluate the wind load response of a generai slender structure.
As mentioned above, the total response of the structure to wind loads is a
combination of the mean, the background and the resonant response. The peak
generalized response f (may be bending moment, shearing force or deflection) is
presented in the form
r =r+gr (4.27)
where T is the mean response, 7 is the root mean square of the fluctuating response
(including the effect of both the background and the resonant response) and g is the peak
factor with typical values between 3 and 4. The root mean square (ms) of the fluctuating
component is defined as
where is the mis background response (slowly varybg component) and ?,, is the rms
resonant response associated with the jm natural mode. The peak factor is dehed as
where T is the sample period and v is the effective cycling rate and is given as
v = f ,$ / ,/$ + i ~ , ~ (4.30)
4 is the j' natural fiequency of the structure-
In the next sub-sections, the procedure developed by Davenport (1993) to evaluate
various components of the wind responses of slender structures is bnefly d e s d e d .
Consider a structure subjected to a lateral force F(y) (force per unit length) which
has a mean component F(y) and a fluctuating component Ff(y) . F(y) is the ms of the
fluctuating component. The mean response Fof a certain function (r can be deflection,
bending moment or shear force) is given by
- ? = [ F ( Y ) * ~ , ( Y ~ (4.3 1 )
where iiy) is the influence line of this specific function defining the response due to a
unit lateral load acting at a height y. An expression for the mean wind force F(y) is
presented in appendix A as hc t ion of the height of the chimney L, the top diameier of
the chirnney DL, the drag coefficient CD, the reference velocity pressure at the top of the
structure Q, the variation of the wind speed 4, (y) and the diameter 4, (y) dong the
height of the chimney.
The background response which is slowiy varying response at kequencies below
the naturd kequencies of the structure, can be considered a s a quasi-static response. The
mean square background response can be written as:
where R(y,,y,J is the correlation of the Eluctuating forces at heights y, and y?. F(y,) and
F(y3 are the mean forces, i,(y,) and i&) are the influence Lines of the response r, and
[(y,) and I(y3 are the intensiry of the turbulence at height y, and yz, respectively.
RecogniPng that the correlation firnction R(y,,yJ depends only on the separation (Le.
depends on Ay = =,-y3 and providing a transition formula between large scale and local
scale correlations, Davenport (1993) simpüfied the double integration in the expression of
Eq.4.32 to be a single integration over the height. The hpl if ied expression of the rms
background response is given by Eq.A.3 in appendix A.
The resonant response in the vicinity of the natural fkquencies of the stnicture
can be evaiuated using modal andysis. Such procedure gives satisfactory results provided
that the modal f?equencies are well separated and the structure is lightly damped. Based
on this method, the resonant modal contribution of the jm mode to the root mean square
response P (given in Eq.4.28) can be written as:
- 2 - 1
rRi = ZRI-.R,- (4.33)
w here
- zRJ2 =
- 2, is the generalized modal CO-ordinate of j' mode and is approximated by;
SG5 (f,) is the generaiized modal force spectnim (defined by Eq.A.6 in appendix A), M, is
the modal mass, 4% is the structurai darnping ratio, 6 , is aerodynamic damping ratio
(given by Eq.A.7 in appendix A). R, is the value of the response r due to unit 2,. R, can
be defined by the influence üne of the response r and the mode shape of j" mode as:
R, = ,' 1 i, (Y)-p, (~ ) -m( Y)~Y (4.35)
where m(y) is the m a s per unit Length at height y, pJ(y) is the j" mode shape. Using
Eqs.4.33 to 4.35, a general expression for the resonant modal response $ of the j' mode
is given by Eq.A.5 shown in appendix A. Having evduated the mean, the background and
the resonaut responses, Eq.4.27 to Eq.4.30 can be appüed to obtain the peak dong-wind
response of the chimney.
It should be noted that the aerodynamic damping associated with the along-wind
response expressed by Eq.A.7 in appendix A has always a positive and well defined
value. On the other hand, the aerodynamic damping associated with the vortex response,
which is discussed later, reaches a maximum at resonance with a negative vaiue reducing
the overall effective damping of the structure in laterai osciIlations.
4.5.2 Across-Wind Resaonse
The across-wind turbulent response due to lateral turbulence can be related to the
along-wind response. Knowing that the across wind forces are ody fluctuating
components having zero mean values, Davenport (1993) showed that typicdly the across-
wind background response is 0.4 t h e s the dong-wind background response. Also,
Davenport (1993) proved that if the difference between the dong-wind and across-wind
aerodynamic damping is negligible, the across-wind resonant response is equal to f i
times the dong-wind resonant response. However, when a slender structure is excited
monantly by the vortex shedding (causing lateral movement), the lateral turbulence has
Little influence on the maximum laterd response and therefore can be ignored.
4.5.3 Vortex Sheddin~ Res~onse
The response of a slender structure to vortices which shed behind the structure in
a smooth or turbulent flow is not completely undentood yet. This difficulty of predicting
the vortex response is due to the limited full scale data and the difficulties in achieving
Reynolds Number associated with large chimneys in the wind tunnels testing (Vickery,
1997). The response of FRP c b e y s to vortex shedding is evaluated based on a mode1
developed by Davenport (1 993).
The frequency of the eddies f , shedding behind a cylinder when a flow passes the cylinder
is given by
where S, is Strouhal number, U and D are the mean wind speed and the diameter of the
cylinder, respectively. When the fiequency of the eddies matches any of the natural
fkequencies of the structure, it will not only dnve the cylinder in a resonant lateral
response but dso causes a rapid change of the aerodynamic damping from positive to
negative leading to signincant amplification of the lateral motion. This explains why the
structure continues to resonate (locking) even if the mean wind speed fluctuates by
e5%-90% about the critical wind speed. At this range of wind speed the fkequencies of
the vortex eddies are not following Strouhai behavior (Le. Eq.4.36) but are controlled by
the motion of the structure. Expressions for the vortex generaiized force specmim
(across-wind forces) for the j' mode of vibration of the cylinder (when the vortex
fkquency f , is close to the f?equency of the j' mode) and the maximum negaîive
aerodynarnic damping ratio are given in appendix A. Suggested values of mis lifi
coefficient, Strouhai nurnber and correlation length of lift forces, which are part of these
expressions, are given by Vickery (1997) and are dso presented in appendix A. Using the
generalized vortex force spectnun of the j' mode and the maximum negative
aerodynarnic damping ratio (Eq.A.8 and Eq.A.12 in appendix A), the mean square
resonant vortex response can be obtained h m Eq.A.5 in appendix A.
In Eq.A.8, the integration is evaluated numerically and the ratio ( f,' / St ) as a part
of this equation ( f,' / St = U, / U, ; U, is the wind speed at the top, U, is the j' critical
wind speed which creates eddies matching one of the natural frequencies of the structure
4) is scanned nom 0.7 to 1.4 to obtain the maximum value of the integral. It shouid be
mentioned that the vortices can excite the h t mode and even the second mode if the
critical velocity associated with the second mode U, is below the design wind speed.
4.5.4 Wind Load -4s Eauivaient Static Loads
Beside using the dynamic approach given by Davenport (1993), it was decided to
use the code for steel chimneys developed by international Committee on industrial
Chirnneys CICIND (1988) to evaluate an equivalent static wind load for FRP chimney. In
this code, a gust factor is employed to account for the innuence of the fluctuating part of
dong-wind loads. The mean wind Ioad is scaied up dong the height of the chimneys
using this gust factor to obtain an equivalent static load. As reported by Vickery ( 1995),
this procedure is not totally consistent. On the other han& the American Concrete
institute's Code for concrete chimneys (1995) applies the gust factor to the mean base
bending moment not to the mean distriiuted loads. Neithm approach is totally correct and
both can lead to high error for some structures as guyed towen (Vickery, 1995).
Conceming the across-wind response multing fiom vortex shedding, the CICMD
code for steel chimneys provides an approximate method for calculating upper bound
limits for the across-wind amplitudes. The procedure is based on wind tunnel tests as well
as full-scale observations. The code recommends considering the response of the second
as well as the first modes of vibration for slmder chimneys having iow first critical wind
speed (associated with the fint mode). It is also stated in the code that if critical wind
speed exceeds 1.2 the design wiod speed at the top of the chimney, no signifiant
oscillations would be expected in the lateral direction of the chimney.
4.5.5 Wind Load Cases Considered in This Studv
Based on the dynamic approach suggested by Davenport (1993). the following
three load cases are considered in the analysis of FRP chimneys conducted in this study
(each wind load case is evaluated at a different wind speed):
1) The peak dong-whd load is combined with the associated peak across-wind load
resulting fiom the lateral fluctuations OC the oncoming flow.
2) The vortex shedding load resulting nom exciting the nrst mode of vibration is
combined with the peak dong and across-wind loads associatcd with the first
criticai wind speed.
3) The vortex shedding load that resulted fiom exciting the second mode ( if the
second critical wind speed is less than the design wind speed) is combined with the
peak dong and across-wind loads associated with the second critical wind speed.
The wind load as defined by the code for steel chimneys developed by
International Cornmittee on Indusnial Chimneys (CICIND, 1988) is also calculated for
cornparison with the dynamic method discussed above.
4.6 The Stresses Calculation
Having evaluated the bending moment diagram using the dynarnic procedure
given by Davenport (1993), the maximum normal strain ( E, ) acting on the kh layer of
any horizontal cross-section of a FRP chimney cm be evaluated as follows:
% is maximum normal strain acting on the kh lamina in the direction of the global axis
y; M( y ) is the maximum bending moment resulting from the three load cases described
in section 4.5.5 at level y, N(y) is the nomal force due to the own weight oîchimney and
acting at a section having an elevation y from the base, and Et, A(y), I(y) and 4 are the
outer radius, the cross sectional area, the moment of iuertia of the cross-section and the
thiclmess of the id> lamina, respectively. E, is the equivdcnt extension Young's modulus
of the laminate as obtained h m Eq.425. The in-plane strain zW in the direction of the
local axis Y and the strains in the local lamina axes 1-2 can be obtained using the
followuig relations:
Consequently, the stresses defined in the local axes 1-2 and acting on the k" lamina are
given by:
where QI,, Qll, Q,,, and Q, are the reduced &esses of the lamina and are defined in
4.7 The Failure Criterion
Different types of lailure critena (Jones, 1975) were developed by various
raearchers to determine the carrying capacity of a lamina under various load
combinations A failure criterion for a Iaminated structure can be based on either stresses
or strains. Using any of the failme aiteria, it is possible to construct failure envelopes
representing the b i t bearing capacity of a lamina This means that if a given loading
condition is within the envelope, the material will not fail. FRP materials have almost no
ductility. This means that once the failure envelope is reached, the material cannot sustain
any more load and will fd in a brittle manner. A failure criterion which has shown an
excellent agreement with experimental d t s is the quadratic failure criterion developed
by Tsai and Wu (1971). This failure d e r i o n accounts for interaction of stress
cornponents in determining the capacity of a biaxial stress field. The general Form of the
criterion in terms of the ultimate strength is
where O,, o,, are the dong fibers tensile and compressive ultimate strengths,
respectively. 4, o, are the tensile and compressive dtimate strengths in the direction
perpendicular to the fibers and T,, is the in-plane shear strength. In practical design of
FRP structures, the above defined ultimate strengths have to be divided by an appropriate
factor of safety.
4.8 Fati~ue - Calculation
The vortex shedding response is not oniy excessive lateral amplitudes experienced
by the chimneys but also it is a fatigue concem. The fatigue behavior of FRP materials is
more complicated that other structurai materials because of the different possible damage
mechanisms experienced by this type of plastic composite. The damage in composites
involves a widespread number of microstnicturai mechanisms, manix cracking,
interfacial debonding, delamination and fiber breakage. Damage mechanisms are
generally related to the matrix properties. The current understanding of the influence of
various environmentai effects on the fatigue strength of composite materials (as discussed
in chapter 2) is a long way h m providing a simple and accunite technique for life fatigue
prediction.
A number of atternpts have been done to develop a relation between the along
fibers fatigue strength and the number of cycles based on the knowledge of the static
strength of the composite (Mandell ( 198 1 ), Jones ( 1984)). The relationship between the
along fibers Fatigue strength S and the number of cycles can be written as:
S = o,, (rn.logN + b) (4.4 1 )
where o,, is the ultimate static strength in the fibers direction, rn and b are constants.
Such a relation represents a straiaight line when drawn on a semi-log scale; rn is the dope
of this line. Based on the intermediate range of the experimental results conducted by
Mandell ( 198 1) and Jones (1 984), values of -0.12 and 1.0 are assumed in this study For rn
and 6, respectively.
The fatigue strength of FRP materials in the cross fiben direction is not
sufficiently covered in the literaîure. Based on test clah, the code of design for reinforced
plastic pipes developed by the American Society of Mechanical Engineering ASME
RTP-lb (1997), defines limiting strain values which assure nomcracking of the matrix of
a composite when subjected to long term cyclic loading. These limiting lateral strain
values are equal to 0.0015 and 0.008 for tende and compressive stresses, respectively. in
this study, it was decided to use these limit values (for laterd strain) to check for the
across-wind fatigue induced by vortex sheddhg. In view of the approach described by the
EUROCOMP (1997). and using the avaiiable information about fatigue strength of FRP
materials, the following equation is used to check the fatigue damage under combined
date of stresses.
where N, and N,, are the number of cycles to cause longitudinal and in-plane shear
stresses failure, respectively. These are calculated using Eq.4.41 by substituting o,,, equal
to the ultimate along fiber normal strength and the in-plane shear strength (both divided
by a suitable factor of safety), respectively, while S is equal to the induced factored
stresses o, and a,,, respectively, resulting fkom the vortex shedding analysis. N is
calculated as defked by the CICIND (1988) for 20 years design life as:
N=0.4 I O ~ A ' ~ - " ~ f
3SU, where A=- , U, is the critical wind speed, U, is the design wind speed at the top
UL
and f is the resonant frequency. ~2~ is the maximum factored transverse strain due to
vortex shedding and E, is the limiting strain in the laterd direction which is equd to
0.00 15 divided by factor of safety e q d to 1.6.
A flow chart, nimmaripng various steps incorporateci into the cornputer code for
designing a FRP chimney, is presented in appendix A.
4 9 Verification of the Model
A sophisticated hite element mode4 based on the laminateci consistent shell
element (Koaey, 1993) which was descn'bed in chapter two, is used to veTify the simpler
approach developed in this chapter. Three different chimneys are modeled and analyzed
under static load conditions using both the laminated shell element and the simple
computer code. The chimneys' laminates conskt of angle-ply larnhae al1 having 50 % E-
glas as reinforcement and Der 41 1-45 as resin. The layers have the following mechanical
properties defined in the directions of the material axes (1-2): E, = 23.46 GPa, E2 = 6.95
GPa, G,, = G,, = 2.2 GPa, G, = 2.65 GPa, v,, = v,, = vu = 0.32. The dimensions of the
chimneys and the stacking sequence of the layers are shown in Table 4.1. The three
chimneys are subjected to wind speed equal to 30m/sec at elevation 1Om above the
ground and are assumed to have 1.0% viscous damping ratio. The equivalent static load
based on the CICIND (1 988) is applied to both the finite element mode1 and the computer
code developed in this chapter. The deflections at the top of the chimneys, axial and hoop
stresses resulting h m both analyses are presented in Table 4.1. Cornparison between the
results of the analyses Uidicates an excellent agreement and shows that the simple
approach adopted in the chapter accurately predicts the response of iarninated FRP
chimneys.
Table 4.1 The dimensions, the lay-ups and the tip deflections of FRP chimneys Deflection Stresses (MPa)
L D t lay-UP m e n t FE. present F.E. (m) (ml (mm) =x =, =x G y
3 0 1.5 3 0 OOWOO\OO 0.41 0.401 3.1 32.55 1 2.95 31.1
4.1 0 Pararnetric Studv
The geometry of a FRP chimney, the properties of its laminate and the wind
characteristics have a direct influence on the structurai response of the chimney. The
main material and geometry parameters are: the elastic properties of the basic materials
(fibers, matrix), fibers content ratio, fibers orientation, laminate stacking sequence
(symmetric, anti-symmetric, u~f~ymmetric), mass daisity of the composite, material
damping, tapering ratio, and the aspect ratio (height/diameter) of the chimney. The
intensity of the turbulence, Strouhal number, Reynolds number, rms of lift coefficient and
the mean wind speed are the main wind parameters which affect the along and across-
wind responses.
Using the cornputer code developed in this chapter, a parameûic study is done for
investigating the effect of some of the previously mentioned parameters on the along and
across-wind responses of FRP chimneys.
4.10.1 Fibers Orientation
The fiber orientation of the individual lamina plays a significant role in defining
the apparent elastic properties of the laminate. In order to shidy the effect of the fiber
orientation, 3 cylindricai c b e y s are considered. The chimneys (1, II, IiI) have 40, 60
and 80m height, 3.0, 4.5 and 6.5m diameter, and 75, 105 and 140m.m wail thickness,
respectively. The &ers type is E-glas-roving (50 % of the weight of the composite) used
as reinforcement for Der 4 t 1-45 resin. AU chimneys consists of angle-ply laminate (f0)
and the thickness of each lamina is taken equal to 1.ûm.m. The elastic properties of each
lamina in the material axes are as follows: longitudinal moduius E, = 23.46 GPa, lateral
modulus E, = 6.95 GPa, in-plan shear modulus G,, = 2.20 GPa and Poisson's ratio v,, =
0.32.
Fig.4.4 shows typical relation between the apparent longitudinal flexural modulus
E, that resulted nom the developed cornputer code, normalized with respect to the lateral
modulus E, of the basic laniina, and the angle-ply f 0 (for chimney 1). Lt is clear from the
figure that the longitudinal modulus is strongly dependent on the laminae orientation. The
maximum modulus is achieved at 8 = 90°, Le. fibers are ail oriented parallei to the y
direction. The minimum value of the apparent longitudinal modulus occurs at the vicinity
of 0 = 30'. It should be mentioned that due to the contribution of the shear modulus, the
minimum longitudinai apparent modulus E,, cm be larger or srnaller than the lateral
lamina modulus & and also the maximum can be larger or maller than the longitudinal
lamina modulus E,. Fig.4.4 suggests that in order to benefit fiom the presence of fibers in
enhancing the longitudinal stifiess of the chimney, fibers have to be oriented by an angle
0 >5S0. in Fig.4.5, the first natural frequency of chimney 1 are plotted versus the angle of
orientation 0. As expected, the natitrai fresuency has the same trend as the variation of
the apparent longitudinal rnodulus.
The dong and across-wind tip deflection for chimney 1, II and III are evaiuated
using a wind speed U,, = 30.0 m/s (at 10m above the gromd), a drag coefficient CD = 0.7,
an intensity of hrrbdeace 1, = 0.14 and a damping ratio<,= 0.80 %. The maximum tip
defiedon is plotted vernis the angle of orientation 0 in Fig.4.6. 11 should be noted that
the maximum dong-wind response corresponds to the design wind speed (U = 3 0 . M ~ ) .
Meanwhile, the maximum across-wind response occurs at the vicinity of the critical wind
speed which is evaluated using Eq.4.36 with fc equai to either the first or second natural
kquency of the chunney. It is noted that the angle of orientation has a strong effect on
the longitudinal tip deflection. On the other hand, the lateral tip deflection is found io be
unaffected by v w n g the angle of orientation. The across-wind response has little
sensitivity to the change of the longitudinal modulus due to the associated change of the
natural kquency of the chimney. in fact, the change of the natural nequency alters the
critical wind speed and the response is ahnost the same.
It is also noted that the maximum dong-wind response occurs in the vicinity of an
angle of orientation 8 = 30". It could be concluded that the angle of orientation is an
excellent design tool for tailoring the laminate to give the optimum structural
performance.
4.10.2 Damoin~ and Mass Densitv of FRP
The mass density of FRP materials depends on the percentage of the fibers in the
maîrix as well as the mass density of both the fibers and the ma&. However, the
variation of the mass density of the fibas and the ma& is mal1 and the mass density of
ERP composite mainly alters with the percentage of fiber content. The typical density of
FRP varies between 1400 kg/m3 for low fiber ratio to 1900 kg/m3 for high f ier ratio.
The across-wind tip deflection (normalized to the diameter) for c h h e y s 1, II and
UI (Tor 0.8 % damping ratio and angle-ply k45) are plotted in Fig.4.7 venus the mass
density of the composite. It should be noted that the mass density was varied by changing
the percentage of the fibers which consequently changes the stif'hess of the composite.
As seen in Fig.4.7. a sharp increase in the tip deflection occurs with the decrease of the
mass density. In view of Eq.A. 12, a decrease in the m a s density of the chirnney leads to
an increase in value of the negative aerodynamic damping associated with vortex
shedding and consequently a decrease in the total damping of the structure. This sharp
increase in the response may be shifted to the nght or lefl depending on the damping ratio
and the average diameter over the upper third. In general, FRP materials have light
weight compared to other structural materiais. By examinhg Eq.A. 12 given in appendix
A, it cm be stated that due to their Light weight, FRP chunneys are expected to experience
relatively higher negative aerodynamic damping in the vicinity of the critical wind speed
compared to steel and concrete chimneys. The negative aerodynamic damping should be
cornpensated by sufficient materiai damping or extemal damping devices to prevent
excessive oscillations.
Chimneys 1, iII (both have angle-ply 8 = k45 and mass density 1580 kg/m3) are
analyzed using variable structural damping ratios. Fig.4.8 shows the variation of both
dong and across-wind response with the damping d o . It is ciear that the dong-wind
response is aot sensitive to the damping ratio. This is due to the fact that the dong-wind
response is govemed rnainly by the static components (mean and background) and the
resonant component (a£Fécted by the damping ratio) has a Littie effect On the other hand,
the damping ratio has a significant contribution to the across-wind response which is a
resonant response. It is noted that the variation of damping ratio has the same effect as the
variation of the mass density since both of them contribute directly to the total damping
of the structure (see Eq.A. 12).
The damping of FRP materials depends on a large number of parameters; the fiber
orientation, stacking sequence of the layers, amplitude and frequency of vibration and the
manufacturing process. As shown in Figs.4.7 and 4.8, the location of the critical response
zone varies significantly with the two main parameters which influence the total damping
of the structure (structural damping, average mass over top third). With the uncenainty
about the propa damping ratio for FRP matends, the designer of a FRP chirnney should
be consewative in estimating the damping ratio in order to ensure the stability of the
chimney against vortex shedduig. Otherwise, by overestimating the damping ratio, the
chimney rnight become located in the critical region of the across-wind response.
in Figs.4.7 and 4.8, the across-wind response of chimneys 1, iI and iii are
calculated using the CICIND code (1988) for Steel Chimneys and are plotted versus the
mass density and the dampîng ratio. respectively. It is clear nom these figures that the
CICTND code provides a consemative response for chimney 1 and slightly unconservative
for chimneys II and DI.
4.103 Effect of Ta~ering
The tapering ratio is another parameter contniuting in the across-wind response
through varying both the spectrum of the lif€ forces and the value of the aerodynarnic
damping. Tapering spreads the fhquency of the excitation forces dong the height of the
chimney and consequently reduces the vortex shedding response. To investigate the effect
of tapering on the maximum response of FRP chimney, the across-wind response due to
vortex shedding has been calculated for chimney 1 for tapenng ratios equal to 0.0,0.3 and
0.6 respectively. The tapenng ratio T is defined by the following relation: T= (D, - DJD,
where D,, and D, are the bottom and the top diameters of the chimney, respectively. The
tapenng has been achieved by fixing the bottom diameter Db and reducing the top
diameter Dr Fig.4.9 shows the vortex response plotted versus the damping ratio of the
structure 5, . As expected, the vortex response is reduced with the increase of the tapering
ratio of the chimney. It is noted fÎom Fig.4.9 that the region in which the response rapidly
increases is shifted to lower damping ratios when chimneys are provided with tapering.
As discussed before that rapid change in the response occurs when the total damping ratio
approaches a zero value. As seen in Fig.4.9, providing 0.6 tapering ratio reduces the
needed mctural damping by about 0.3%. This damping vdue is significant when
compared to values of the structurai damping for steel and FRP materials.
The responses of the same three chimneys, based on the CICIM) (1988), are also
s h o w in Fig.4.9. It can be seen h m the figure that the values predicted by the CICIM)
are very conservative for the tapering ratios 0.0 and 0.3. Meanwhile, the sarne graph
shows that for a tapering ratio T = 0.6, both the dynamic andysis and the CICID predict
very close behavior.
The tapering ratio seerns to be a very good tool for reducing the across-wind
response for FRP chimneys. Tapenng reduces the aerodynamic damping forces and
disperses the fi-equencies of the eddies dong the height. This dispersion makes the
spectrum of the lift forces flatter and reduces the dynamic effect of the vortex forces.
4.1 1 Desim Thicknesses For FRP Chimnevs
in this section, a number of FRP cylindrical chimneys are designed, i.e. an
adequate thickness of each chimney is evaluated to sustain both wind and thermal loads.
The wind loads are considered as dacribed in section 4.5 with wind speed U,,=30 &sec2,
drag coefficient C,=0.7 and intensity of turbulence I,=O. 14. For each chimney, three
different designs are attempted by assuming that the viscous damping ratio 6, is equd to
0.70 %, 00.5 % and 1 .O %, respectively. The chimneys' laminates conskt of Der 41 1-45
resin reinforced by 70% (based on weight) E-glas fibers. The layers have the folîowing
properties defined in the directions of the materiai axes (1 -2): E, = 36.85 GPa, & = 1 1.16
GPa, G,, = G,, = 3.36 GPa, G, = 4.32 GPa, v,, = v,, = 0.3 and v, = 0.29. Al1 laminates
consist of angle-ply (H) layen; 0 is measured with the tangentid axis located in a
horizontal plane as shown in Fig.42. The values of uitimate strengths of the layers
dehed in the material axes are as follows: longitudinal tende o,, = 552.77 MPa,
longitudinal compressive a, = 44220 MPa, tramverse tende sZt = 16.74 MPa,
transverse compressive CL, = 89.28 MPa and in-plane shear strength q2 = 70.57 MPa.
Most of the study is conducted assuming an inclination angle 0 = fis0. However, for the
sake of comparison, one set of anaiysis for an angle of inclination 8 = f3S0 is conducted.
Three different heights L are considered in the design; L = 30,40 and S b , respectively.
For each height, the diameter D is varied in such a way that a range of 10 to 20 is covered
for the aspect ratio UD.
The thermal stresses induced nom temperature change are estimated based on
tindings of chapter three. The curing tempemure Tc, the operating (inside) ternperature
Ti=, and the ambient (outside) temperature T,,, are assumed equd to 80°C, 70°C and
-30°C, respectively. Based on this temperature distribution, and using Fig.3.14, a
localized region at the bottom of the chimneys is expected to suffer nom lateral cracks
with maximum almg fiber compression and tensile thermal stresses equal to 1 13 MPa
and 76 MPa, respectively, for the case of angie-ply 0 = Also, using Fig.3.13, the
chimneys expenence only compressive stresses (near the base only) with stress values are
equai to 57.5 MPa and 12.6 MPa at the inside and outside face of the chimney,
respectively, for the case of angle-ply 0 = fis0.
The design of FRP chimneys is based on the tollowing aspects:
1) As a resuit of the thermal loads, cracking is expected to occur in the direction
perpendicular to fibers (especially at the bottom of the chimney). The maximum dong
nbers stress a,, resulting h m the tanperature variation c m be obtained nom Fig.3.13
or Fig.3.14. It should be noted that this stress value is independent of the thickness of
the shell as discussed in chapter three.
2) A desired factor of safety FS is selected for the dong fibers stresses. A factor of safety
of 1.6 is chosen for the across fibers direction (sllnilar to the value used in the ASME
RTP- 1 b-( 1997)). As such, the ultimate tensile strength o,,, compression strength o,,
in the along fibm direction and the in-plane shear strength a,, are given as:
w here O,,, a,, and a,, are the ultimate tensile, compressive and in-plane shear strengths as
defined earlier.
3) A certain thickness t is assumed for the chimney.
4) The dong and the across &en stresses resuiting fiom the three load cases descriid in
subsection 4.5.5 are evaluated. The maximum values obtained h m the three Ioad
cases are denoted as a,, and a,. Meanwhile, fatigue messes a,, a,, a,, (and the
correspondhg main E,.) resulting eoom the across wùid loads induced by vonex
shedding are evaluated.
5) Stresses due to the own weight of the chimney a,,, a,, a,, are dso evaluated.
6) Load factors of 1.10, 1 S O and 1.35 are used for the gravity, the wind and the thermal
loaàs, respectively. As such, the applied stresses are cdcuiated as:
a, = 1.1 a,,+ 1.5a,,+ 1 . 3 5 ~ , ~
q = 1.1 a,+ 1.5 q,
o,,= 1.1 a,,+ 1.5 o,, + 1.35 a,,
where O,, a, and a,? are the along fibm , transverse &ers and the in-plane shear stresses,
respec tively .
7) Equation 4.40 is used to check the adequacy of the chosen thickness by substituting O,,
q, q*, b l t . 9 Clcu q m > u, a,, respectively*
8) Meanwhile, Eq.4.42 is used to check the safety of the chimney against fatigue failure.
9) In case that the leît hand side of Eq.4.40 or Eq.4.42 is Iarger than 1.0, a larger
thickness has to be chosen and steps 4 to 7 are repeated till the lefi hand side of the
two equations is less than unity.
nie above design steps are conducted to various FRP chimneys (dl having 8 =
f5S0) covering the dimension range mentioned early in this subsection. For each
chimney, the adequate thickness of the sheil is evaiuated based on a factor of safety FS
equal to 2, 3, 4 and 5, respectively. The design is also repeated assuming a viscous
darnping ratio S, equd to 0.7 %, 0.8 % and 1.0 %, respectively. The calculated
thicknesses are plotted versus the aspect ratio UD in Figs.4.10 to 4.21. Each of these
figures shows two thickness values; a thickness evaluated by considering only the along-
wind loads and another thichess evaluated based on both the dong-wind and the vortex
shedding loads (Le. includes both static and dynamic loads and consider fatigue failure).
Both design thicknesses account for the thexmal and the gravity loads.
AU analyses indicate that, when considering loads due to vortex shedding, the
design is govemed by the fatigue failure rather than strength. AU figures show a typicd
behavior reflecting a linear increase of the thickness with the increase of the aspect ratio
(Le. with the decrease of the diameter, keeping the height constant) when only along wind
loads are considered. Such behavior is expected, since for static behavior, a decrease in
the diameter kads to a linear mcrease in normal stresses. Such an increase in stresses has
to be compensated by a magnification for the thickness in order to keep the same stress
level.
On the other hand, Figs.4.10 to 4.21 show that when both along wind and vortex
shedding loading are considered, an increase in the aspect ratio is associated with a
decrease of the required thickness. This behavior can be interpreted by considering
Eq.A. 12 (appendix A), which shows that an increase in the diameter (smaller aspect ratio
UD) leads to an increase in the negative aerodynamic damping associated with the vortex
shedding. Also, accorcling to EqA. 12, such an amplification of the negative aerodynamic
darnping can be reduced by increasing the m a s of the chimney and consequently
increasing the thickness.
The plotted figures indicate that except for a srnail range of high aspect ratios of
the design presented in Figs.4. 17, 4.20 and 4-21, the design of the chirnneys is govemed
by fatigue failure.
The effect of the viscous damping ratio can be assessed by comparing the
thicknesses of the chimneys designeci using the s m e factor of siûety (FS) and having
difkrent damping ratio t, (e.g. cornparison between Figs.4.10, 4.14 and 4.18). Such
cornparisons show that the thicknesses based on the dong wind T o n s e are not affected
by the variation of the damping. On the other hand, an increase in the damping ratio &
significantly decreases the required thickness when vortex shedding is considered.
The tip deflections 6 remlting Born the analysis of the chimneys designed for a
Factor of safety FS equal to 2 and having 0.70% viscous damping ratio are presented in
Fig.4.22. tt is expected that this case exhibits the maximum deflection as it has the least
factor of safety and damping ratio. As s h o w in the graph, the ratio of the tip deflection to
the diameter does not exceed 0.24D which is Iess than 0.3D; the limiting deflection for
the serviceability requirements specified by the CICIND (1988) for steel chimneys.
As mentioned eariy, one set of analysis for an angle of orientation 8 = c 3 S 0 is
conducted for cornparison with 0 = S 5 O . The design thicknesses for factor of safety FS =
5 and 0.7% damping ratio are shown in Fig.4.23. As expected, the design thicknesses are
increased for al1 the range of the aspect ratio. By changing the fibers orientation from 5 5 O
to 3S0, the stresses in the lateral direction of the layers are increased. The lateral direction
of the layers has low ultimate strength. For that, the required thickness to satisQ Eqs.4.40
and 4.42 is increased compared to the 55Oorientation angle laminate. It is noted aiso that
the fatigue stresses governs the design of the chimney ody up to aspect ratio 15 which is
l e s ihan the corresponding value in Fig.4.13, up to 20.
A factor of safety 5 is cornmonly used in the design of FRP materials for the
longitudind direction. As cm be excluded h m Figs. 4.13.4.17 and 421, an aspect ratio
between 15 and 20 gives the minimum design thicknesses when the vortex shedding
response is considered in the design of FRP chimney. Within this range of aspect ratios,
the chimney will not expenence excessive lateral oscillations due vortex shedding or high
fatigue stresses if the design thickness is chosen appropnately. In some cases the designer
can not optimize the aspect ratio to reduce the fatigue stresses produced by the vortex
shedding. Therefore, a choice between satisfj4ng the Fatigue strength requirements
(increase the thickness of the chimney) or reducing the vortex shedding response by
adding darnping to the system has to be made. This will ultimately depend on the most
cost efficient solution.
4.12 Conclusions
The response of FRP chimneys to wind loads depends on a large number of
parameters. These include the wind characteristics, the laminate properties and the
geomehy of the chimney. From the pafametric study conducted in this investigation to
assess the effect of various parameters on the wind responses of FRP chimneys, one cm
conclude the following:
The fiber orientation d e h e s most of the laminate properties such as stiffhess, strength
and damping ratio. To achieve a considerable improvement in the longitudinal
stifiess of the chlliuiey, fibers have to be oriented by an angle 8 255' (0 is measured
with a horizontal direction). It should be mentioned that an angle of orientation 5 5 O
produces an intermediate level of thermal stresses as shown in chapter 3.
The across-wind Ioad respome of FRP chimneys is very sensitive to the cornbined
effect of the composite mass density and the damping ratio. Since FRP are very Iight
materials and do not have a well defined damping ratio, a conservative approach mut
be used in estimating the across-wind response of such chimneys.
Tapering ratio is a very efficient way of reducing the vortex shedding response.
The CICIND code for steel chimneys (1988), when applied to FRP chimneys Ieads to
overly consefyative results in some cases and slightly unconservative in other cases.
When vortex shedding response is considered, the design of FRP chimneys is
govemed by the fatigue stresses for almost al1 the range of aspect ratio and heights
considered in this shidy.
The suggested optimum aspect ratio which produces minimum thickness when the
vortex shedding response is included in the design varies between 15 and 20 (
considering both wind and thermal loads).
t A Fig.4.1 Vertical and horizontd cross sections of FRP chhmq.
- X
Fig.4.2 Vertical projection of the laminate showing the set of axes.
etry of the laminate.
N
Orientation angle (+/-O)
1
Fig.4.4 Normalized longitudinal extension modulus versus fiber orientation angle
Fig.4.3 The geom(
20 40 60
Orientation angle (+/-O)
Fig.4.5 First natural fhquency of chimney 1 with the fiber orientation angle.
1 across-wind -
0.7 -
along-w ind . . -. . . + .
III
r *
1 -9. ' --.m.-
. . . * . - - - - - * - - - - * . * - I I * * P . . * - - . . . - -. . r - -. -. * . - 9 -
1 *: œ
* * . - - * - --. - 0 - - - - - - - -
O - .
- - * - a - - - *-*----..--*.....
Angle of orientation(+/-8)
Fig.4.6 Along and across-wind tip deflection versus angle of orientation.
III 5,=0.8 O h II
- - - - - - - - - --------• - * - - - - - - - - -------.---._.-_*_--.---
Fig.4.7 Nomdized across-wind tip deflection versus the mass density for 1, II and m.
0.0 10
Damping ratio (Q
Fig.4.8 N o d i z e d tip deflections versus damping ratio for chimneys 1, II and 111.
Damping ratio (Q
Fig.4.9 The estimated across-wind response versus the structural damping for chimney with height H= 40m, bottom diameter @=3.Om for 0.0,03 and 0.6 tapering ratios.
120 -
Factor of safety = 2.0 30m 5 = 0.7 % - - 40m alang-w ind
*..-.. Som -c- 30m -- 40m along-wind --A-. and vortex
---- O-, k 1 I 1 1
HfD Fig.4.10 Estimated thicknesses of FRP chimney s vernis the aspect ratio
for factor of safety = 2.0,5 = 0.70%.
A--. Factor of safety = 3.0 5 = 0.7 %
- 30m - - 40m along-w ind
4 30m -r- 40m along-w ind - - & - - Som and vortex
Fig.4. i 1 Estimated thicknesses of FRP chimneys vernis the aspect ratio for factor of dety = 3.0, & = 0.70%.
Factor of safety = 4.0
m Fig.4.12 Estimated thicknesses of FRP chimneys vernis the aspect ratio
for factor of safety = 4.0,c = 0.70%.
A- - . * - - - - -. Factor of safety = 5.0 .-, 30m 4 = 0.7 % - - -. dom along-wind
---.-- 50m + 30m
-Y -c - 40m along-w ind
\ - - & - * . 50m and vonex '.
\ 'S.
8 10 42 14 16 18 20 22 WD
Fig.4.13 Estimated thicknesses of FRP chimneys vernis the aspect ratio for factor of safety = 5.0,1; = 0.70%.
A-. . Factor of safety = 2.0 - <= 0.85 % 30m
-. 4orn along-wind m. . . m . - - S o m
-\ - 30m along-wind '. \
-- 40m and vortex m... .*A--
\ 50m
Fig.4.14 EstMated thicknesses of FRP chimneys versus the aspect ratio for factor of safety = 2.45 = 0.85%.
- - Factor of safety = 3.0 - & = 0.85 % 30m -. 40m dong-w ind
-. - - - - - - 501x1
.W.-
.&*. 50m and vortex
Fig.4.15 Estmiated thicknesses of FRP chinmeys vernis the aspect ratio for factor of d e t y = 3.0, < = 0.85%.
Factor of safety = 4.0 < = 0.85 % - 30 m
-. 40 m along-wind .m.-.- 50 m + 30m -- along-wind * * & * - Som and vortex
m Fig.4.16 Estimated thicknesses of FRP chimoeys versus the aspect ratio
for factor of de ty = 4.0,< = 0.85%
A - * - - * - - Factor of de ty = 5.0 30m & = 0.85 % - -
-. 40m along-wind * - * * * - Som - 30m - - 40m along-w ind
-\ \ * * A - - Som and vortex \ \ k
8 10 12 14 16 18 20 22 EUD
Fig.4.17 Estimated tbicknesses of FRP chimneys vernis the aspect ratio for factor of safety = 5.0, < = 0.85%
&-. . Factor of safety = 2.0 - 30m <= 1.0% -. 40m along-wind
....*. Som Y*
r, - 30m 4- 40rn along-wind .. \ -*&.- Som and vortex
m Fi& 1 8 Estunated thicknesses of FRP chimneys versus the aspect ratio
for factor of safety = 2.0, < = 1 .O%
Factor of safety = 3.0 A- . - <= i.0 % - 30rn
-. 40m along-w ind -*...* - . 50m 4 30m
k 4- 40m along-w ind -\ .\ --).- 50m and vonex
\ =A - \
FigA. 19 Estimated thicknesses of FRP chimneys v e n u the aspect ratio for factor of s a f i = 3.0,< = 1 .OYO
A-** Factor of safety = 4.0
- 9 . - 0 - c;= 1.0% - 30 m
Y. - - 40 m dong-wind .... 0 . 50 m
Lm Fig.4.20 Estimated thicknesses of FRP chimneys versus the aspect ratio
for factor of safety = 4.0.6 = 1 .O%.
Factor of safety = 5.0 - 30m along-w ind
aiong-wind and vonex
8 10 12 14 16 18 20 22 L/D
Fig.421 Estimated thicknesses of FRP chimneys vernis the aspect ratio for factor of safety = 5.0, < = 1 .O%
Factor of safty = 2.0 <=û.70 %
Lm Fig.4.22 Tip deflection normaüzed to diameter of FRP chimneys versus the aspect ratio
for factor of safety = 2.0 and 5 = 0.7%.
180 -, Factor of safety = 5.0
A- < = 0.7 % _.-- a r . -
* : - * . . . - -
along-w ind
4- 40m along-w ind --A-. Som and vortex
8 10 12 14 16 18 20 22
m Fig.4.23 Estimateci thicknesses of FRP chllnneys versus the aspect ratio for factor
of sâfety = 5.0, & = 0.7%, 0 = +/- 35'.
CHAPTER 5
DAMPING OF FRP MATERIALS
5.1 Introduction
For structural applications which are requked to withstand a harsh dynarnic
environment, damping is a very important parameter. The damping capacity of the
structure plays a vital role by limiting the resonant response and forcing the transient
response to die out quickly. One of such applications are fiber reinforced plastic
chimneys which are constantly subjected to dynamic forces in the form of wind loads.
Material damping is the ability of the material to dissipate energy by converting
the mechanical energy to heat. Composite plastic materials have multiple sources of
energy dissipation, such as the viscoelastic Cesponse of the ma&, thennoelastic
conversion of mechanical energy into heat, fiction at fiber-matrix interface and energy
dissipated berneen layers due to delamination.
The damping of FRP materials depends on many parameters such as: the rnatrix
property, fibers content, fibm onentaiion, fkquency, strain amplitude and method of
manufacturing. Although a number of studies exists in the iiterature for evaluating the
damphg of FRP mater&, no &ta exkt for typical materiai used in the construction of
FRP chimneys. The purpose of this chapter is to evaluate experimentally the material
damping of glass reinforced Vinyl ester materials which are typicaiiy used in the design
of FRP stacks.
This chapter starts by presenh'ng a bnef review of the research existing in the
literature and pertaùiing to the damping of FRP matends in general. This is followed by a
description of two techniques which are used to evaiuate the material damping
experimentally. Finally, the experiments conducted in this study are described and the
obtained results are presented.
5.2 Review of Dam~inp Evaluation of Fiber Reinforced Plastic Materials
For more than three decades, f ibs reinforced plastic materials have been
investigated for dynamic properties and damping capacity. In general, results of the
studies indicate that for FRP materiais tested at low strain levels, the material damping is
independent of the snain amplitude but does depend on the fiber content, fiber
orientation, temperature, moisture, fkequency of load and matrix properties.
In the late sixties, Schultz (1968) published remarkable resuits of damping ratios
of unidirectionai (UD) glass-epoxy cantilever beam using the decrement and the
bandwidth techniques. In this study, it was observed that the damping capacity mainly
depends on the fkequency of Ioading and the angle of orientation of the fiers. In generd,
it was found that the damping capacity increases with the increase in kquency.
Meanwhile, by varyhg the angle of orientation, it was found that the maximum damping
is achieved at an angle of 45". Damping values for unidirectional and cross-ply E-glass
fiber reinforced epoxy beams under flexurai vibration were reported by Gibson (1976) for
a wide range of frequency (20-500 Hz). It was found that the damping values
considerably increase with the increase of the load frequency and are independent of
strain amplitude (up to 0.002 strain for cross-ply laminate). Mymon, Biley and Rehfield
(1 978) conducted an experimental investigation studying the effoct of temperature,
moisture content and angle of orientation on the damping capacity of a variety of graphite
epoxy laminates. It was found that the angle-ply [+4S0] laminate exhibits higher damping
than [O0], and [0°J+450J900,/-450,] laminates for both dry (2S°C) and hot-wet (93°C)
conditions. The same shidy showed that the hot-wet environment increases the damping
for the [O0] laminate by about 29 %. Meanwhile, a remarkable decrease for the damping
(about 28%) was observed for the other two laminates due to the effect of the hot-wet
environment.
The effect of the frequency of the loading on the damping values of FRP materials
was studied by Robert (1982) showhg in grnerai an inmase of 10%-20% in the darnping
ratio for tenfold increase in the fkquencies. It should be mentioned that d l of these
studies dealt with linear viscoelastic damping, at low strain, well bonded and undamaged
composite. A complete Litazture review about theoretical and experimental studies
conducted for evaiuating the damping capacity of FRP materiais is done by Gibson ( 1979
and 1977) and Vantomme (1995).
5.3 Measnres And Techniaues For Determioinp Material Dam~ing
Material damping is of€en characterized by the specific darnping capacity
(SDC), loss factor q, darnping ratio 5 and logarithrnic decrement d. Specific Damping
capacity y is defined as the ratio of the energy dissipated during one cycle of loading to
the maximum strain energy stored in the specirnen during this cycle. Loss factor q is
equal to the tangent of the phase angle 6 which represents the phase shift between the
response and the hannonic excitation. The damping ratio t; is defined using the following
C relation: -, where c is the damping coefficient and c, is the criticai damping coefficient
c m
defined as c, = 2.rn.o; m is the m a s and o is the naturd frequency. The logarithmic
decrement d characterizes the decay of the fke vibration response of a single degree of
teedom system and is defined by the n a d logarithm of the ratio of two successive
maximum amplitudes. The relations between the above defined damping parameters are
given as:
qj =2nq=41r(;=Zntan6=2d (5- 1)
Numerous testîng techniques can be used to determine the above damping
properties. niese include: the forced oscillation technique which is based on resonance
testing (half power band width, resonant dwell), the modally tmed impulse technique, the
logarithmic decranent (sometimes called the fke decay technique) and the off-resonance
impedance technique. A brief description of the logarithmic decrement technique and the
half power band-width methoci, which are used in this study, are presented in the next
sub-sections.
5.3.1 Lo~arithmic Decrement Techniaue
n i e logarithmic decrement technique represents the classical way for estimating
the damping ratio of a material. The technique is based on free vibration testing of the
specimen. The test is conducted by exciting one ofthe natural modes of the specimen and
then measuring the decay amplitudes after mnoval of the driving force. The decay of the
measured time history cuve can be used to atimate the modal damping coefficient for
that particular mode using the following expression:
where 4, and &, are the response amplitudes at the nh and n + m' cycles. It should be
mentioned that the expression given by Eq.5.2 is based on the assumption that the
damping ratio is very maIl i.e.5 cc 1 .O%. As such, the fiee decay method is most suited
to the determination of damping values for lightly damped systems (typically less than
0.0 1 ).
53.2 Half Power Band-Width Method
This technique is the most wideiy used method in damping testing. For structures
with well separated modes, single de- of fieedom modeliug of each nanual mode
when excited resonantiy gives very accurate results. In this methoâ, the steady state
amplitudes correspondhg to discrete fkquency values of forced harmonic excitations,
covering a wide range around the natural frequency of interest, are measured. For a given
fiequency response curve, the damping ratio can be calculated fkom
where and f, are the fbquencies at which the amplitudes of response are 1 I f i tirnes
the maximum amplitude. For tightly damped structures, fitting the modal peaks of
continuous structure to the steady state response of single degree of keedorn system is
more convenient than applying Eq.5.3 to estimate the modal damping. in the current
study, the measured response of the specimens ovet the fkequency range in the
neighborhood of the modal fkequency of interest has been fitted to the following equation
(which represents the steady state response of single degree of fieedom system excited by
a harmonic load).
where y( E T ) is the measured amplitude of the response, p,, is the amplitude of the applicd
hamonic force, k is the stiffiiess of the specimen, m is the dnving fiequency, o is the
naturai fiequency of the specirnen, 5 is damping ratio of the specimen and = -. In the k
tests, the response of the specimen due to varying hquency of harmonic load is
measured. in view of Eq.5.4 and using the above measured response, a curve fitting
technique can be used to estimate o , & and y .
5.4 Exaerimental Evaluation of the Damaine Prmerties of Glass Reinforced
Vinvl Ester Com~osite
Damping is a very important parameter controlling the dynarnic response of
chimneys in general. As discussed in ctiapter 2, glass reinforced vinyl ester represents the
favorable composite to be used in the construction of FRP chirnneys. Due to the Iack of
damping values of this specific composite, it was decided in this study to conduct some
dynarnic tests in order to evaluate the damping capacity of this type of polymeric
composite.
Four cyiindrical specimens having diameten equal to 2", 3", 4" and 6" and
thicknesses equai to O. 19", O. 19", 0.2" and 0.24", respectively, are used in the dynamic
testing. Al1 specimens conskt of filament winding angle-ply g las reinforced vinyl ester
laminates. The specimens have an equivalent axial modulus of elasticity E = 1.3.10' psi
(8.97 GPa) and an axial tensile strength o = 9000 psi (62 MPa). The specimens are
stacked as follows: 0.0 1 " chemical barrier reinforced with Nexus Veil having 10% fiber
content, 0.1" Anti-wicking barrîer of two chopped sîrand mats 1 10 oz with 25% fiber
content, structurai layers of a continuous nlament winding with fis0 (angle-ply)
orientation angle measured h m the longitudinai axis of the specimen with 70% fiber
content, and W y 0.01" exterior protection resh coating. In order to cover a wide range
of ~ u e n c i e s , various lengths of each s p e c h (Le. hawig different naturai
fkequencies) are used in the testing. The specimens are donated by Reinforced PIarrc
Systern hc..
5.4.1 Ex~eriment Set-u~ and Procedure
The damping testes are perforrned using a uni-directional shake table recently
constnicted at ïïze University of Watern Ontario. The shake table system consists of an
electro-magnetic shaker connected to a 4'x4' slide table, an amplifia and a PC based data
acquisition system. The output sipds which excite the shaker are generated and
controlled by enomous speed data acquisition board.
Figure 5.1 represents a photo showhg various components of the shaker system.
The schematic illustration of the shake table system is shown in Fig.5.2. For more details
about the shake table and the data acquisition system, the reader is referred to ECDamaty
( 1998).
The dynamic tests are conducted by mounting the specimens to the slide table.
The specimens' clamping is designed such that there is no extraneous loss mechanisrn
neither from any created damage in the matend nor through fiction losses at the clamped
end. As such, the specimens have been carefully glued to steel plates using epoxy glue
and then mounted on the table using four corner steel bolts comected to the steel plates as
s h o w in the photo provided in Fig.5.3.
The response of the specimen is monitored by mounting high sensitive charge
signal accelerometers at various locations dong the specimen height. The signals are
conditioned (nltered and amplifïed) using high accuracy charge signal amplifier. The
signals are then stored to the hard disk of the PC through the data acquisition system.
Figure 5.4 shows a photo of a typicd specimen mounted to the slide table.
The haif power band-width technique is adopted to evaluate the damping of the
specimens. The following steps are applied to identify the darnping ratio of each
specimen:
1) The specimen is driven by a harmonic excitation having a specific fiequency.
2) The steady state response (acceleration) of the specimen (usually at the top of the
specimen) is measured and stored.
3) Step (1) and (2) are repeated for a fkequency range in the vicinity of a naturai
fkequency of the specimen.
4) The fiequency response cuve, the relation between the steady state acceleration and
the kequency, is plotted for each naiural mode of excitation.
5) The fkequency response c w e is fitted to the response of a single degree of fkedorn
system, Le. to Eq.5.4, to give the estimated damping value.
Steps (1) to (5) are repeated for the fkst and second modes of vibration of each specimen.
Low amplitudes of excitation are chosen to minimize the effect of aerodynamic darnping
and also to Limit the specimens' strain to the level at which damping of the composite is
independent of the amplitude. Fig.5.5 shows a typical fkquency response curve as
measured h m a test, togetha with the response of an quivalent single degree of
needom system. It shodd be mentioned that in order to represent accurately the
fkequency response curve (specially around the natural fkquency), a very small step of
kquency variation has been usd
Logarithmic decrement tests have bem conducted as well for the fiat mode of
each specimen by sirnply pulling the top of the specimen and measuring the decay
response after removing the applied force. Since exciting only the Fbndarnental mode of
the specimen manually is possible, the acquired sigals for the decay test have been
filtered to eliminate the contributions of the higher modes to the response. Exciting the
second mode of vibration manually for the decay test was not possible because the
specimens are relatively stiff. For that, the fint mode of vibration has been only tested
using the decay test. These decay tests are conducted for cornparison with the resonant
tests results and also to check the dependency of the damping ratios on the strain
ampli tude.
5.4.2 D a m ~ i n ~ Results and Discussion
Resonant tests have been conducted for the first two modes of vibration of the
specimens dacribed in section 5.4. Table 5.1 shows the measured natural Eequencies and
dampuig values for various tested specimens. in Fig.5.6, The damping values
corresponding to the first mode are plotteci vernis the fundamental Eequency. The
damping ratios are fitted with a second order polynomial bction. However, the results
of the c w e fitting shows an almost Linear behavior. It is clear fiom the figure that the
variation of the damping ratio with the fkquency is almost negligible for the considered
rage of kequencies. Figure 5.7 shows the variation of the damping ratios of both the first
and the second mode with the modal fkequencies. It is clear firom the figure that the
results of the second mode show more scattered damping values about the fitting curve
compared to those corraponding to the fint mode.
The damping ratios corresponding to the fiindamental mode of the specirnens and
based on decrement decay tests are presented in Table 5.1 as well as Fig.5.8. in general,
most of the tests results show a good agreement between the decrement and the resonant
tests. It has been noted that typically the damping values obtained from the decay test are
slightly larger than those obtained nom the resonant test. Meanwhile, the dependency of
the damping ratios on the fkquency is much stronger for the decay test results compared
to those obtained using the resonant tests (specially for kquencies higher than 40 Hz).
The average damping ratios obtained h m all the conducted tests are equai to 0.6551 %
for the resonant tests and 0.75 14 % for the decay tests.
During the tests, the strains at the base of the specimens have not been measured.
However, these cm be easily caiculated using the values of the measured acceleration at
the top of the specimen. As mentioned earîier the decay tests are conducted by pulling the
specimen at its top point and measuring the free decay acceleration. Ignoring the
contribution of the higher modes is a reasonable approximation since the initial imposed
deflection shape is very close to the k t mode shape and consequently the expected
behavior is mostly according to the first mode. Assuming that the specimen is vibrating
with only its fundamental mode, the base moment M(t) can be evaluated by considenng
the moment of the inertia forces about the base, i.e.
where Y(t) is the measured tip acceleration of the specimen, m(x) is the m a s per unit
length, m, is the mass of the acc~ierumeter at level i, $(x) is the fhdarnentai mode shape
of cantilever beam normalized to be equal to unity at the top of the specimen, x, is the
distance between the base and the iLh acceleforneter and x is the vertical coordinate
measured from the base of the specirnen. Having evaluated the base moment M(t) using
Eq.5.5, the longitudinal saains ~ ( t ) at the base of the specimen are given by:
where R is the outer radius of the specimen, E is the &ai longitudllial modulus and 1 is
the moment of inertia of the section.
Figure 5.9 shows the variation of the damping ratios vernis the maximum
amplitude of the bending strain obtabed nom a decay test (for a specimen having a
diameter D = 2 in and length L = 1.45 m). Figure 5.9 indicates that the increase of the
damping ratio with the strain level is fairly srnall. The small increase in the damping ratio
can be related to an added aerodynamic darnping and not to permanent damage in the
composite. The later usually results in a significant and rapid increase in darnping.
The maximum O ff-axes longitudinal strain amplitude show in Fig.5.9 is 0.0009 1 .
This corresponds to strains equal to 0.000299 and 0.00061 in the fibers and the across
fibers directions, respectively. This level of strain is much lower than the maximum strain
level(0.002) at which the damping ratio is independent ofthe strain amplitude as reported
by Gibson (1976).
It should be mentioned that the maximum level of strain expected for FRP
chirnneys subjected to wind loads, varies between 0.0003-0.0005 (see chapter 4). These
values are Iess than the threshold value desmibed earlier by Gibson (1976). As such, the
values of damping obtained nom the experirnentai work conducted in this study can be
used in the design of FRP chùnneys if giass resorced vinyl ester angle ply laminates (0
= f i 5 O with the longitudinal axis) are used in the construction or the chirnneys. It is
obvious that the evaluated damping ratios are luniteci to a construction involving an
angle-ply 0 = S S O (measured with the longitudinal axis of the specimen). However, by
contacting many FRP manufacturers in Canada, it has been infonned that due to the ease
of Fabrication, this value of angle-ply is the most commody used in practice.
5.5 Correction for Aerodvnamic Damnine
If a structure vibrates in a Buid environment, the motion is retarded by the fluid
drag. Due to the interaction between the structure and the surroundhg Buid, some energy
transfers to the Buid through the work done by the drag forces. This source of energy
dissipation is known as the aerodynamic damping. The drag forces FD acting on structure
vibrating in still air is given as
where, CD is the drag coefficient, D is the diameter of the structure, p, is the air density
and y is velocity of the structure. For a continuous structure vibrating in a single mode,
the displacernent is w&en as Y(x,t) = y(t) +(x), where y(t) is the modal amplitude and
$(x) is the mode shape. The equivalent viscous damping factor for a single mode of
vibration in still air (mode shape is always positive dong the height such as the
iündamental mode of fiee standing structure) can be written as
where T is the penod of oscillation, m is the m a s per unit length and L is the length of
the structure. The drag coefficient is not constant as the stmcture vibrates in the Buid, it is
in fact a function of Reynolds number which in hmis is a fimction of the relative velocity
(Le. the structure velocity assuming that the air is still). The drag coefficient of a cùcular
cyhder in steady flow can be approximated as a function of Reynolds nimiber (Blevins,
1986) as
CD = b, + bJRe
w here
b, = 1.3 and 4 = 10 for the following range of Reynolds nurnber: 1 < Re c 1 04, and v is
the kinematic viscosity of the air.
The aerodynamic darnping associateci with the tested specimens has been
calculated using Eqs.S.8 to 5.10. These values have been subtracted fiom the measured
darnping values to obtain the tnie material damping and are plotted in Figs.5.6 and 5.7.
Figures 5.6 and 5.7 show that the aerodynamic damping did not change the general trend
of the results and in general can be neglected for both the first and second modes of
vibration. The maximum value of the aemdynarnic damping is only 2.7% of the total
measured darnping. It should be noted that no correction for aerodynamic damping are
needed for the values obtained nom the decay tests presented in Fig.5.8. This is due to the
fact that the ploned values are obtained by extending the ntting curve of the rneasured
data to intersect with the vertical axes (which basicaüy corresponds to zero amplitudes).
On the other hand, the damping ratios which are plotted versus the strain amplitudes in
Fig.5.9 need to be corrected for aerodynamic damping. Figure 5.9 shows the values of the
cdcuiated aerodynsunic damping ratio as well as those evaiuated by subtracting the
aerodynamic damping nom the measirrwi one. It can be easily concluded fiom the graph
that values of aerodynamic damping corresponding to the shains adopted in the tests are
negligi'ble.
5.6 Conclusions
Experimental testing has been conducted to evaluate the darnping values of FRP
laminates commonly used in the construction of FRP chimneys. Such laminates consist of
angle-ply (0 = f 5 5 O with the longitudinal axis of the specimens) glass reinforced vinyl
ester composite. Both resonant and logarithmic decrement tests have been conducted on a
number of cylindrical specirnens. The damping results h m the decay test exhibited
slightly higher darnping ratios. For the range of frequency tested the damping value has
shown slight increase with the increase of the frequency. For the range of the applied
snain , results indicate that the damping values are strain-independent. The added
damping fkom the surrounding air has b m i calculated and found to be negligible. The
average darnping values fiom al1 conducted tests are equal to 0.66 % for the resonant tests
and 0.75 % for the decay tests.
Fig.5.1 A photo showing various components of the shaker system.
Conditioned Signais
f
AT-MIO- lm- 10
Charge Signal Conditioning
Amplifier Shaker Table
# The Dampmg Ratio
aculations
* 2692-A-OS4
1 1 Pentiumpc 1
Accckromctcrs Output Sqpah
Fig.S.2 Schematic diagram of the Shake Table and the Data Acquisition System.
Fig.5.3 A photo showing the epoxy glue and steel plate used in mounting the specimen.
Fig.5.4 A photo of a typical specimen mounted to the slide table.
f (Hz) Fig.5.5 Typical experimental fiequency-response curve and the fitted response of single degree of freedom system.
Measured (mode 1 ) - 2 order fitting - t
<-d - L,
8 8 a Y Y
- 8 O O
8
a
Fig.5.6 The damping of the nrst mode versus the fkquency h m the resomnt test.
First and second mode Mode 1 a - 2 nd order
0 Mode2
L - 5 U t
Fig.5.7 The damping of both first and second mode versus the fkquency fiom the resonant test.
Fig.5.8 The damping ratio of the fïrst mode versus the hdamental hquency fiom the decay test.
Strain amplitude .1 o5 Fig.5.9 The damping ratio versus the maximum bending strain amplitude in the longitudinal direction for specimen (2 in diameter).
CaAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 Introduction
This thesis includes an extensive investigation about the application of FRP
materials in the construction of industrial chunneys. An attempt to answer the following
questions has been provided in the thesis:
1) What type of FRP materials suit the chimneys application and what are the mechanical
and envuonmental properties of this materials?
2) What level of thermal stresses is expected during the operation of a FRP chimney?
3) How to assess the wind response of FRP stacks?
4) What is a typical value for damping ratio that c m be used in designing FRP chimneys?
The nrst three questions are addressed using an analyticai approach, while an
experimental study is conducted to a t l ~ ~ ~ e r the fourth question.
Conclusions that can be drawn nom the whole study are summarized in the
following sub-sections.
6.2 Suitable FRP Material For Chimnevs' Construction
Knowing that the serviceability conditions of industrial chunneys include hi&
thermal eff- an aggressive chernical environment and a loading haWig a cyclic nature,
the constituents of FRP should be carefidly chosen in order to provide a durable structure.
Following the discussion in Chapter 2, Wiyl ester polymer reinforced by E-glas fibers
are suitable to be used in the construction of industrial chimneys as long as the service
temperature of the chimney is less than the continuous service temperature of the vinyl
ester polymer. For the case of chimneys having hi& service temperature, epoxy polymea
reinforced by E-glas f ibm can be the alternative though a higher cost is expected.
6.3 Thermal Stresses Induced in II'RP Chimnevs
in this study, the formulation of a consistent laminated shell element has been
extended to include themal stress analysis. The thermal formulation has been checked by
modeling and analyzing a number of benchmark problems and comparing the results of
the analyses with those availabie in the titerature. An excellent agreement has been shown
in all the analyzed examples. The effect of various parameten which might influence the
thermal stresses induced in angle-ply laminated mer reinforced plastic chimneys have
been studied using the developed model. Results of the pararneûic study indicate that the
thickness, the diameter, and the height of the chimney as well as the nurnber of laminae
bave no signincant effect on the induced thmal stresses. Analyses indicate that the
thermal stresses mainly depend on the through thickness temperature distribution (relative
to the curing temperature), the angle of orientation of the nbers, the coefficient of thermal
expansion and the modulus of elasticity dong the fibers direction. The last two
parameters depend mainly on the fiber contait in the mat*. The thermal stress analysis
of typical FRP chimneys shows high stress concentration near the boundaries with in-
plane across fiber stresses usually exceedmg the ultimate strength of the matrix. As such,
cracks are expected to occur in FRP chimneys as a result of a through thickness
temperature variation. Xowever, it is beiieved that these cracks can be controlled if the
interlaminar shear stresses are less than the ultimate interlaminar shear strength divided
by an appropriate factor of safety. The andysis then pmceeds by wuming a negligible
value for the modulus of elasticity in the direction perpmdicular to the fiben to simulate
a cracked chimney. Results of this last set of analysis indicate that for typical FRP
chimney, the along fibers stresses as well as the shear stresses of cracked chimneys are
within acceptable values. Finally, charts predicting the dong fibers thermal stresses
induced in typicd cracked FFtP chimneys as a fhction of the through thickness
temperature distribution are presented. These stress values can be considered when the
design of a FRP chimney is attempted.
6.4 Effect of Wind Loads on FRP Chimnevs
The response of FRP chimneys to wind loads depends on the wind characteristics,
the laminate properties and the geometry of the chimney. From the paramenic study
conducted to assess the effect of various pisrazeters on the wind responses of FRP
chimneys, one can conclude the foiiowing:
The fibers orientation defines most of the lammate properties such as stiffiiess and
strength. To achieve a considefable improvement in the longitudinal stifniess of the
chimney and consequently reduce the wind response, fibers have to be onented by an
angle 0 S 5 " (O is measured with a horizontal direction).
The across-wind load response of FRP chimneys is vey sensitive to the combined
effect of the composite mass density and the damping ratio. Since FRP are very light
materials and do not have a weil dehed damping ratio, a consmative approach must
be used in estimahg the across-wind response of such chimneys.
Tapering ratio is very efficient way of reducing the vortex shedding.
The CICIND code for steel chimneys (1988), when applied to FRP chimneys leads to
overly conservative results in some cases and slightly unconsetvative in other cases.
When vortex shedding is considad, the design of FRP chimneys is show to be
governed by the fatigue strength for the range of aspect ratio and height considered in
this shidy.
The optimum aspect ratio (height to diameter ratio) which produces minimum
thickness of FRP chimneys subjected to both wind and thermal loads varies between
15 and 20.
6.5 Ex~erimeutai Evaluation of D a r n ~ i n ~ Ratio of Vinvl Ester Glass Reinforced
Com~osite
Dynamic testing has been performed to evaluate the damping ratio of E-
glassNinyl ester fiber reinforceci plastic materiai. Both the resonant and the logarithmic
decrement techniques are used in this study. Based on the d t s of the damping tests, the
following conclusions cm be drawn:
The damping ratios obtained f hn the decay tests are shown to be slightiy higher than
those obtained fkom the resonant tests,
For the range of fkquencies applied in the tests, the decay tests show damping values
which are more f?equency-dependent compared to those obtained from the resonant
tests. However, the variation of the damping ratio with the fixquency is usually small.
For the range of strains applied in the tests (maximum expected strains in reai
chimneys are within this range), the damping ratios are show to be strain-
independent.
The average damping value obtained fkom the whole experimental study are equal to
0.66% and 0.75% for the resonant and the decay tests, respectively.
6.5 Recommendations For Further Research
As mentioned previously in the thesis, this investigation represents the fint
extensive study conducted on FRP chimneys. Future research is needed, as extension to
this study, to obtain a full understanding about the behavior of FRP chimneys. The
following points are suggested as a fùture direction for research to be conducted on FRP
chimneys.
1) The uneven distniution of wind loads around the top part of cylindrical chimneys
might lead to ovalling of the chimneys in these Locations. This phenomenon, which
was s h o w to happen for steel chimneys, was not considered in this study. An
investigation for such effect is needed.
2) The fatigue strength for E-giassNinyl esta angle-ply composite which was shown in
this study to be corivenient for chimneys applications, is not well defined in the
fiterature speciaily for variable angles of orientation of the fibers. As such,
expenmental testing for evaluating the fatigue strength of E-glassNiny1 ester
composite is highly recommended.
3) The pararnetric studies conducted in this thesis to wess the behavior of FRP
chimneys under thermal and wind loads assume constant thickness through the height
of the chimney. Practical design of FRP chimneys includes often variation of the
thickness through the height of the chimney. As such, it is recommended to
investigate the effect of varying the thickness of the chimneys on the induced thermal
stresses and also on the wind responses.
4) The local buckling of thin shells is very much important when assessing the stability
of such type of structures. FRP chimneys are very susceptible to local buckling
specially due to the highly localized thermal stresses at the base of the shell.
Therefore, buckliag of FRP chimney has to be investigated.
APPENDIX A (Davenport, 1993)
Mean drag force
F(Y) = (q,D,HC,) +,'(Y) +,(Y)
where
- L is the height of the chimney.
- CD is the drag coefficient.
- DL is the diameter at the top of the chimney.
- UL is the mean wind speed at the top of the structure.
- q, is the reference velocity pressure at the top q, = I / PU ,' , p is the air density
1,
- ( , (y) is function defines the wïnd speed pmfùe ( U(y) = (, (y).U, ), 4, (y) = (t) and
1, is intensity of the turbulence at the top of the chimney.
- #, (y) is function d e W the variation of the diameter of the chimney dong the height
( WY) = O,(Y)-D, ).
The mean drag remonse
The rms of backeroand remonse
where Lu is the scale of turbulence, Lu=30-60m.
The expression between parenthesis in Eq.3 is a reduction factor to accommodate the
correlation of the forces with the height of the structure.
The rms of resonant remonse
The mean square resonant response of the j' mode is;
where 4 is the naturai frequency of the jm mode,+, (y) is the variation of the mass dong
the height, 5, and 5, are the structural and the aerodynamic damping and p(y) is the
mode shape.
- Along-wind generaiized force spectnim is;
where C is the decay constant =6-10
- Along-wind aemdynamic damping
where rn, is the mass at the top of the chimney.
Vortex shedding
The generalized force spectrum of vortex shedding is;
fiSGf, (fi ) =
where CL is the mis of the left coefficient. h is a coefficient defines the correlation of
the wake forces at fiequencies near F, and approximately is equal 1 as suggested by
Vickery (1997). f ' = FD, 1 U, is the reduced frequency. The mis of the lift coefficient
is believed to be strongly dependent on the scde and intensity of turbulence, based on
full-scale measurernents Vic kery (1 983). The suggested value for m i s Ii R
coefficient EL, Vickery (1997), is
e, = (0.1 5 + 055 i') - (0.09 + 055 i') e-'20'*" (A.9)
where i' = 1 @/L) and L =100(y/10)'" the scale of turbulence. There is a reduction to
the ms Ieît coefficient with the aspect ratio and to accommodate the rapid decrease of
the left coefficient nea. the tip of the chimney. Strouhal number S, is surface roughness.
Reynolds number, turbulent and aspect ratio dependent. The suggested value for
S trouhal number S, is
S(1)=0.14 + 0.05 h(h/4) for 4< h > 25 (A- 10)
and constant above A=25, where h is the aspect ratio (Lm).
In the across-wind vibration at a fkquncy near the vortex shedding fkquency, the
aemdynamic damping is expresseci by;
where K ( u ' ~ ) is the aerodynamic damping coefficient. With the associated uncertainty
of the aerodynamic coefficient and with the dramatic change nom positive to negative
in the vicinity of the aitical wind speed, it was suggested by Vickery that the maximum
negative aerodynamic is
D' and rn' are the average diarneter
1 (A. 12)
and average m a s over the upper third of the
chimney. It should be noted that the non-linear ternis were neglected from the general
expression of the aerodynamic damping given by Vickery. This assumption is valid if the
vibrations have relatively small amplitudes.
Set a laminate configuration
Obtain the lamina properties E,, E, G,,, v , , h and 8 Determine lamina reduced stifhess QG h m Eq.4.2
Calculate the lamina transformed reduced stiffhess from Eq.4.7 I
r Calculate the extensional, couplhg and bending stifkess matrices (A, B, D) for the larninate from Eq.4.15
Calculate the equivalent elastic properties of the larninate trom the inverse of the matrices A, D kom Eq.4.23,4.24 - - --
Calculate the dynamic properties of the chimney, naturai fkequencies and the mode shapes by usmg the equivalent bending rnodulus E,
Calculate the maximum wind response fiom the three cases of toading For each section dong the height
I Calculate the maximum strains for each tamina h m Eq.4.37 in x-y axes
Transforrn the maximum strains to 1-2 axes fiom Eq.4.3 8
1 Calcdate the maximum stresses in 1-2 in each Iamina fiom Eq.4.39 1
r
Apply the failure criteria for each lamina fkom Eq.4.40 I
l Check the fatigue stresses EqA-42 1 1
Compare the deflection with the maximum pamisïble deflection
Ftow chart descri-bes the design sequence of FRP chimney.
REFERENCES
Ahmeci, B.M. Irons and Zienkiewicz, O.C. (1970): Analysis of Thick and Thin Shell Shucîures By Curved Finite Elements, International J m a l for Numerical Methoàs in Engineering, vol. 2, pp. 41 9-45 1.
Berg, G.V. (1989): Elements of Structurai Dynarnics, Prentice Hall, Englewood Cliffs, New Jersy.
Blevins, R.D. (1986): Flow Induced Vibration, Reinhold, pp. 220-223.
Bulder, B.H. and Bach, P.W. (1991): Literature Survey on the Effects of Moimire on the Mechanical Roperties of Glas and carbon plastic Laminate, ECN-C-91-033, ECN, Petten.
Buyny, R.A. (1 990): Predicîing the Durability of High Temperature Composite Matenals, SPE Conference proceeding, Dallas, TX.
Carlile, D.R., Leach, D.C. and Zahlan, N. (1989): Mechanical Properties of The Carbon FiberlPEEK Composite APC-BAS-4 For Structural Applications, Advances in Themioplastic Matrix Composite Materials, ASTM STP 1044, pp. 199.
Chandrashekhara, K. and Bhimaraddi, A. (1993): Thermal Stress Analysis of Laminated Doubly Curved Shells Using a Shear Flexible Finite Element, Cornputers & Sfrucctwes, vol. 52, No. 5, pp. 1023-1030.
Cook, R-D, Malkus, D.S. and Plesha, ME. (1989): Concepts and Applications of Finite Element Analysis, 3rd edition, Jhon Wiley& Sons Inc., New York, NY.
Curtis, P.T. (1989): The Fatigue Behavior of Fibrous Composite Materials, J. of Strain Analysis vol. 24 No. 4, pp. 225-243.
Davenport, A.G. (1993): The Response of Slender Structures to Wind, proceedings of the NATO Advanced study W h i t e At Waldbronn, wind Climates in Citia, Gemany.
Davis, IL. (1975): The Fatigue Resistance of Reinforced Plastics, Mater. Des. Eng., pp. 87-
Echtermeyer, AT. (1991): Significance of damage caused by fatigue on mechanical properties of composite laminates, Roc. International conference on composite materials 8, Hawaii.
Eckold, G. (1994), Design and Manufacture of Composite Structures, McGraw-Hill Book Company, New York, NY.
El-Damaty, M.A. ( 1998): Data Acquisition System for the Dynarnic Shaker, BLWT, The University of Western Ontario.
Fettahlioglu, O.A. and Wang, P.C. (1988): Asymptotic Solutions for Thermal Stress and Ceformation in ûrthotropic Nonhomogeneous Shells of Revolution, J. Tlienn. Stresses, vol. 1 1, pp. 305-324.
Fondyke, K.L. (1988): Phenolic FRP today, Roc. British Plastics Federation conference, Blackpool.
Ghosh, P.G. and BOS, N. (1995): FRP Composites Based on Different Types of Glass Fibers and Matnx Resins: A Comparative Study, J. Polymers & Science, vol. 58, pp. 2 177-2 184,
Gibson, R.F. and Plunkett, R. ( 1976): Dynamic Mechanical Behavior of Fiber-Reinforced Composites: Measurement and Analysis, J. Comp. Materials, vo i. l O, pp. 325-34 1 .
Gibson, R.F. and Plunkett, R. (1977): Dynamic Stifiess and Damping of Fiber Reinforced Composite Xaterials, Shock and Vib. Digest, vol. 9(2), pp. 9-1 7.
Gibson, R.F. and Wilson, D.G. (1979): Dynamic Mechanical Properties of Fiber Reinforced Composite Materials, Shock and Vib. Digest, vol. 1 l(1 O), pp. 3-1 1.
Hofer, K.E., Larsen, D. And Hurnphreys, V.E (1975): Development of Engineering data on the mec hanical and physicai properties of Advanced Composite Materials, AFML-TR- 74-266, Air Force Materials Laboratory, Wright-Patterson AFB, OH.
Jones, C.J., Dickson, R., Adam, T. and Harris, B. (1984):Environmental Fatigue of Reinforced Plastics, Composites vol. 14, pp. 288.
Jones, R.M ( 1975): Mechanics of Composite Materials. McGraw Hill Book Company, New York, W.
Kim, R.Y. (1989): Fatigue Behavior, in Composite Design (S. W. Tsai, Ed.), Technomic, Lancaster, PA.
Koziey, B.L. (1993): Formulation and Applications of Consistent Shell and Beam Elements, Phd Thesis, McMaster University, Hamilton, Canada
Lin, T.D. and Boyd, D. E (1971): Thermal Stresses in Multilayer Anisotropic Shells J. Engng. Mech. Div., Roc. ASCE 97, pp. 829-845.
Mallick, P X (1997): Composites Engineahg Handbook, Marcel Dekker hc., New York
Mailick, PX. (1993): Fiber Reinforced Composites, 2& ed., Marcel Dekker, New York. pp. 248.
Mandell, J.F. & othen (1 98 1 ): Tensile Fatigue Performance of Glass Fiber Dominated Composites, Comp. Tech. Rev., pp. 96- 102.
Maymone, G., Bnley, R.P. and Rehfïeld, L.W. (1978): Muence of Moisnire Absorption and Elevated Temperature on the Dynamic Behavior of Re in Matrix Composites, ASTM S'Il? 658, pp. 221-223.
Mindlin3.D. (1951): Influence of Rotary M a and Shear Deformation on Flexural Motions of Isotropic Elastic Plates. ASME Journal of app. Mech., vol. 18, pp. 3 1-38.
Munscheck, H. (1987): Pnifbericht nr B-8404, KV, Aachen (E-CR Glass Update I ( 199 1) Owens Corning Fibergias, Battice)
Neil, L.H. and Rayner, M.M. (1994): Design Data For Reinforced Plastics, Chapman & Hall, New York.
Padovan, J. (1976): Thermoelasticity of Cylindrical Anisotropic Generaily Laminated Cylinders, J. Appl. Mech., vol. 43, pp. 124- 130.
Plecnikn SM, Hm, T.L, Howward, J., Baker, TE, Pham, M. (1983): Fibergiass Concepts for the Tallest Free-Standing Fiberglass Stack, Polymer Composites, vol. 3, pp. 186- 189.
Pritchard, D. and Speake, S.D. (1988): Effect of Temperature On Stress Rupture Times in Glasdpo lyester Laminates, Composites, vol. 1 9, pp. 29-3 5.
Pritchard, G. and Speake, S.D. (1987): The Use of Water Absorption Kinetic Data to Predict Laminate Property Change, Composites vol. 18, pp. 227-232.
Pritchard, B.N. (1996): Industrial chimneys: A Review of the Current State of Art, froc. Instn. CIY. Engrs Stmcts % Bldgs, vol. 1 16, pp. 69-8 1.
Robert, P. (1982): Damping Mechanisms in Fiber Reinforced Laminates, Proceedings of the IUTAM Symposium on Mechanics of Composite Materials, pp. 93-104.
Schmaltz, AB., Tsai, S.W. (1968): Dynamic Moduli and Damping Ratios in Fiber- Relliforced Composites, J. Composite Matenah, vol. 2(3), pp. 368-379.
Scurton, C. (1963): On the Wmd Exciteci Oscillations of Stacks, Towers and Masts, Int. Conference of Wind Effects on Buildings and Sûuctuns, NRL. Teddhgton.
Sims, GD. and Gladman, D.G. (1982): A Framework for specifling the Fatigue Performance of Fiber Reinforced Plastic, Report DMA(A) 59, National Physical Laboratory, Teddmgton.
Thangaratriam, RX., Palaninathan and Ramachandran ( 1 987): Thermal Stress Anal y sis of Laminated Plates and Shells, Cornputers & Stmciwes, vol. 30, No. 6, pp. 1403-141 1
Timoshenko, S. and Woinowsky-Krieger, W. (1959): Theory of Plates and Shells, McGraw-Hill, New York.
Tsai, S.W. and Wu, E.M. (1971): A General Theory of Strength for Anisotropic Materials, J. Composite Matenals, January, pp. 58-80.
Van Koten, H. (1 969): Vortex Excitation of Slender Structures, Proceeding of Conference on Tower-Shaped Stnichires, The Hague, Int. Assn. Shell Stmctures.
Vantomme, J., De Visscher, J., Sol, K. and De Wilde, W.P. (1995): Determination and Parameüic Study of Matenal Darnping in Fiber Reidiorced Plastics: A Review, Europeun Journal Mech. Eng. M, vo l.40(4), pp. 203-2 1 3.
Vickery, B.J. and Basu, R.I. (1983): Across-Wind Vibration of Structures of Circular Cross-Section, Part 1 and II, I. W.E. and LA., vol. 12, pp. 49-97, 1 983.
Vickery, B.I. (1995): The Response OF Chimneys and Tower-like Structures to Wind Loading, Ninth international Conference on Wind Engineering, New Delhi, pp. 205-233, 1995.
VikeryJ3.J. (1997): Wind Loads & Design Cnteria for Chimneys, 8' U.S National Conference on Wind Engineering, Johns Hopkins, Baltimore, Md..
Whitney, I. & others ( 1 982): Experimental Mechanics of Fiber Reinforced Composite Materials, Society for Experirnental Stress Analysis Monograph No.4.
Wu, H. and Tauchert, T.R. (1980): Thermoelastic Analysis of Laminated Plates. 2: Arti symmetric Cross-ply and Angle-ply Laminates, J. hm. Stresses, vol. 3, pp. 365-378.
Yeung, Y.C. and Paker, B.E (1987): Composite Tension Memben For Structural Applications, Composite Structures, vol.4, pp. 1309-13 19.
Related codes
Amencan Concrete Institute: AC1 1 307 (1 995)
Australia Standard AS1 1702 SAA Loading Code, Part 2: Wind Loads
ASME RTP- 1 b ( 1 997)
BS 5480 (1991): Specification For GRP Pipes, Joints And Finings For Use For Water Supply and Sewerage, BSI. Milton Keynes.
CICIND Mode1 Code for Steel Chimneys (1988).
EUROCOMP Design Code of FRP, (1997).