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*

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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.

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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).