NONLINEAR FINITE ELEMENT ANALYSIS OF CONCRETE COLUMNS

CONFINED BY FIBRE- REINFORCED POLYMERS

SABREENA NASRIN

STUDENT NO: 1009042348

MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURAL)

DEPARTMENT OF CIVIL ENGINEERING

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

DHAKA-1000, BANGLADESH

JULY, 2013

NONLINEAR FINITE ELEMENT ANALYSIS OF CONCRETE COLUMNS

CONFINED BY FIBRE- REINFORCED POLYMERS

SUBMITTED BY

SABREENA NASRIN

STUDENT NO: 1009042348

A Thesis submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Civil Engineering (Structural)

DEPARTMENT OF CIVIL ENGINEERING

BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY

DHAKA-1000, BANGLADESH

JULY, 2013

DEDICATED

TO

My Beloved Parents

Late Dr. Nazrul Islam Miah

Mrs. Meherunnesa Khan

Engr. Humayun Kabir

&

Mrs. Nurunnahar Begum

CERTIFICATE OF APPROVAL

The thesis titled “Nonlinear Finite Element Analysis of Concrete Columns Confined by Fibre-Reinforced Polymers” submitted by Sabreena Nasrin, Student Number 1009042348F, Session: October, 2009 has been accepted as satisfactory in partial fulfillment of the requirement for the degree of Master of Science in Civil Engineering (Structural) on 29th July, 2013.

BOARD OF EXAMINERS

1. Dr. Mahbuba Begum Chairman Associate Professor (Supervisor) Department of Civil Engineering BUET, Dhaka-1000

2. Dr. Md. Mujibur Rahman Member Professor and Head (Ex-Officio) Department of Civil Engineering BUET, Dhaka-1000

3. Dr. Sk. Sekender Ali Member Professor Department of Civil Engineering BUET, Dhaka-1000

4. Dr. A. M. M. Taufiqul Anwar Member Professor Department of Civil Engineering BUET, Dhaka-1000

5. Dr. Md. Mozammel Hoque Member Associate Professor (External) Department of Civil Engineering

DUET, Gazipur

1  

CHAPTER 1

INTRODUCTION 1.1 General

In recent years, considerable attention has been focused on the use of fibre-reinforced

polymer (FRP) composite materials for structural rehabilitation and strengthening purpose.

Highly aggressive environmental conditions have a significant effect on the durability and

structural integrity of steel reinforced concrete piles, piers and columns. Corrosion of steel

rods is a potential cause for the structural damage of these reinforced concrete columns.

Dealing with the problem of steel reinforcement corrosion has usually meant improving the

quality of the concrete itself, but this approach has had only limited success. A traditional

way of repair of damaged concrete columns is wrapping a sheet of steel around the column.

While the strength of repaired columns can be increased for a short-term, the steel wrapping

suffers from the same problem as the steel rebar, corrosion and poor durability. It also suffers

from labor-intensive construction problem due to its weight.

In a new approach, FRPs are now being used as alternatives for steel wrappings in repair,

rehabilitation and strengthening of reinforced concrete columns. If correctly applied, the use

of FRP composites for strengthening reinforced concrete (RC) structures can result in

significant enhancements to durability, and decreased maintenance costs, as well as in

improved serviceability, ultimate strength, and ductility. Moreover, the FRP composites can

generally be applied while the structure is in use, with negligible changes in the member

dimensions. Other advantages include high strength and stiffness-to-weight ratios, a high

degree of chemical inertness, controllable thermal expansion, damping characteristics, and

electromagnetic neutrality. In addition to repair, FRP confined concrete columns have been

developed in new construction and rebuilding of concrete piers/piles in engineering

structures.

Extensive experimental studies have been conducted by several research groups on the

behavior of confined concrete columns (Benmokrane and Rahman, 1998; Saadatmanesh and

Ehsani, 1998; Meir and Betti 1997; El-Badry 1996). However, most of these studies are

confined to circular shaped columns. Experimental studies related to rectangular and square

columns are limited (Bousias et.al. 2004). Despite of the availability of a large amount of

experimental data for predicting the behavior of FRP confined concrete circular columns, a

2  

complete 3-D finite element model for understanding the influence of geometric shapes,

aspect ratios and FRP stiffness is somewhat lacking. As a contribution to fill this need an

attempt has been taken to develop a complete 3-D finite element model to investigate the

effect of aspect ratios, corner radius and thickness of FRP wrap on the behavior of FRP

wrapped concrete columns. This study also aims to evaluate the effect of FRP-concrete

interface on the behavior of FRP confined concrete.

1.2 Objectives of the Study

The objectives of the study are

1 To perform a nonlinear 3D finite element analysis on concrete columns of different

shapes confined with FRP wrap.

2 To validate the numerical model with respect to the experimental database available in

the literature.

3 To study the effect of selected parameters such as aspect ratio (a/b), the corner radius (R)

and the thickness of FRP wrap (tf) on the strength and ductility of FRP confined concrete

columns under concentric axial loading only.

1.3 Scope

The numerical simulation of concentrically loaded FRP confined concrete column has been

performed using ABAQUS, a finite element software package. A 3D finite element model

incorporating the nonlinear material behavior of concrete has been developed. The interface

between concrete and FRP has been modeled using contact pair algorithm in ABAQUS. A

perfect bond and a cohesion based surface interaction model have been assumed to define the

contact behavior of the concrete-FRP interface. The nonlinear load displacement response up

to failure of the confined columns has been traced using Riks solution strategy.

The performance of the developed model has been studied by simulating test columns

confined with FRP available in the published literature. These columns had various geometric

shapes as well as various FRP configurations. Finally the effect of the selected parameters

like cross-section shape factor, corner radius and the thickness of the FRP wrap on the

strength and ductility of FRP confined concrete columns have been investigated.

3  

1.4 Organization of the Study

The thesis has been organized in six chapters. Chapter 1 includes the background of the

work along with the objectives and scope of current study. A brief review on the available

literatures regarding the characteristics and available types of composites as well as different

rehabilitation schemed for various structural components has been reported in chapter 2.

Moreover, this chapter presents various analytical models proposed by different research

groups for predicting the behavior of concrete rectangular and square columns confined with

Fibre reinforced polymers

.

Chapter 3 includes the properties of reference columns and the characteristics of the finite

element. The performance of the FE model has been studied in chapter 4 by comparing the

numerically obtained graphs with available experimental graphs.

Chapter 5 incorporates the parametric study which includes the effects of aspect ratio, corner

sharpness and confinement effectiveness of FRP-strengthened concrete columns. Finally, the

summary and conclusions of the work along with the recommendations for future research

have been included in chapter 6.

4  

CHAPTER 2

LITERATURE REVIEW  

2.1 General

Recent evaluation of civil engineering infrastructure has demonstrated that most of it will

need major repairs in the near future. The strength and stability of these structural members,

bridges, water retaining structures, sewerage treatment plants, wharfs, etc. are provided by

concrete. Therefore it is very important to protect concrete and any deterioration or damage

to concrete must be repaired promptly in order not to compromise the integrity of structures

built with concrete. Concrete rehabilitation particularly in critical infrastructures is as

important as any other maintenance activity and must be carried out in a timely manner.

Repairs performed at early stage would save extremely expensive remediation that may

become necessary at latter stages. Concrete can be deteriorated for many reasons such as-

Accidental Loadings

Chemical Reactions

Construction Errors

Corrosion of Embedded Metals

Design Errors

Abrasion and Cavitations

Freezing and Thawing

Settlement and Movement

Shrinkage

Temperature Changes

Weathering etc.

The strengthening and retrofitting of existing concrete structures to resist higher design loads,

correct deterioration-related damage or increased ductility has traditionally been

accomplished using conventional materials and construction techniques. Externally bonded

steel plates, steel or concrete jackets and external post tensioning are just some of the many

techniques available. However, to repair and extend the life of damaged structures externally

bonded fibre reinforced polymers (FRP) have been proved to be the most effective alternative

to the conventional ones. Despite a high material cost, some advantages like high strength to

weight ratio, high corrosion resistance, easy handling and installation processes are

5  

establishing them as the most convenient option over the traditional strengthening materials

for rehabilitation of corroded RC structures, seismic damaged structures and so on. (Nasrin et

al., 2010). The composition and the type of this new composite material are presented in this

chapter. The material’s mechanical behavior is also included here. This chapter mainly

focuses on the repairing techniques by FRP laminates for shear and flexural strengthening of

corroded RC structures, strengthening of concrete beam-column joints and strengthening of

rectangular concrete columns in accordance with the numerical and experimental

investigations. The behavior of FRP confined concrete columns along with the design

guidelines are also reported in the literatures.

2.2 Fibre-Reinforced Polymers

Fibre-reinforced polymer (FRP) composites consist of continuous carbon (C), glass (G) or

aramid (A) fibres bonded together in a matrix of epoxy, vinylester or polyester. The fibres are

the basic load carrying component in FRP whereas the plastic, the matrix material, transfers

shear. FRP products commonly used for structural rehabilitation can take the form of strips,

sheets and laminates as shown in Figure 2.1.

                      

Figure 2.1 FRP products for structural rehabilitation, (a) FRP strips and (b) FRP sheets (Rizkalla et al. 2003).

Use of FRP has now become a common alternative over steel to repair, retrofit and strengthen buildings and bridges. FRP materials may offer a number of advantages over steel plates which include,

1. High specific stiffness (E/ρ).

2. High specific strength (σult /ρ)

3. High corrosion resistance

4. Ease of handling and installation

Moreover, its resistance to high temperature and extreme mechanical and environmental

conditions has made it a material of choice for seismic rehabilitation. Some of the

(a) (b)

6  

disadvantages of using FRP materials include their high cost, low impact resistance and high

electric conductivity.

2.3 Properties and Behavior of FRP

2.3.1 Tensile Behavior

The tensile strength and stiffness of FRP material is dependent on several factors. As the

fibres of FRP are the main load-carrying constituents, so the type of fibres, the orientation of

fibres and the quantity of fibres govern the tensile behavior mostly. When this FRP is loaded

under direct tension it does not exhibit any plastic behavior (yielding) before rupture. Most of

the time, FRP shows a linearly elastic stress-strain relationship until failure. Table 2.1 present

the tensile properties of commercially available FRP system.

Table 2.1 The tensile properties of some of the commercially available FRP systems

Fibre type Elastic modulus Ultimate Strength Rupture

strain, min

103 ksi GPa ksi MPa %

Carbon

General Purpose 32-34 220-240 300-550 2050-3790 1.2

High Strength 32-34 220-240 550-700 3790-4820 1.4

Ultra- High Strength 32-34 220-240 700-900 4820-6200 1.5

High modulus 50-75 340-520 250-450 1720-3100 0.5

Ultra- High modulus 75-100 520-690 200-350 1380-2400 0.2

Glass

E-glass 10-10.5 69-72 270-390 1860-2680 4.5

S-glass 12.5-13 86-90 500-700 3440-4140 5.4

Aramid

General Purpose 10-12 69-83 500-600 3440-4140 2.5

High performance 16-18 110-124 500-600 3440-4140 1.6

(Italian National Research Council, 2004)

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8  

2.4 Applications of FRP in Structural Rehabilitation

In the last ten to fifteen years, FRP materials have emerged as promising alternative repair

materials for reinforced concrete structures and they are rapidly becoming materials of choice

for strengthening and rehabilitation of concrete infrastructure. There are currently three main

applications for the use of FRPs as external reinforcement of reinforced concrete structures

such as

Flexural strengthening (FRP materials are bonded to the tension face of a beam)

Shear strengthening (FRP materials are bonded to the side faces of a beam) and

Confining reinforcement (columns are wrapped in the circumferential direction with

FRP sheets)

2.4.1 Beam Strengthening with FRP Laminates

Flexural strengthening of reinforced concrete beams using FRP composites is generally done

by bonding of FRP sheets at the tension side of the beam. The bonded sheets work as tension

reinforcement and in turn increase the flexural capacity of the beam considerably.Bonding of

FRP plates and laminates to RC beams has now become a popular strengthening technique

which was first introduced by Meier’s group (Meier 1997) at the Swiss Federal Laboratories

for Materials Testing and Research. Since then, extensive experimental and analytical studies

(Colalillo and Sheikh 2009; Saxena et al. 2008; Choi et al. 2008; Nitereka and Neal 1999;

Brena et al. 2003; Bonacci and Maalej 2000) have been carried out all over the world on

flexural strengthening of concrete beams. These studies have concluded that introduction of

FRP can significantly enhance the flexural strength of a reinforced concrete beam.

Considerable research has been conducted to establish a better understanding of these

laminated system behavior .Several types premature failure modes such as tensile failure of

the bonded plate, concrete failure in the compressive zone, and sudden or continuous peeling

off of the laminate have also been observed. According to ACI code 2005 the following

failure modes should be investigated for an FRP strengthened section

Yielding of the steel in tension followed by rupture of the FRP laminate.

Yielding of the steel in tension followed by concrete crushing.

Debonding of the FRP from the concrete substrate.

Shear / tension delamination of the concrete cover (cover delamination);and

Crushing of the concrete in compressive before yielding of the reinforcing steel.

9  

Some factors like the composite ratio Ac/As, the percentage of conventional tensile steel

reinforcement ρ; and the bond achieved between the FRP and the concrete influence the

degree of strength enhancement attained. It is reported that bonding very thin FRP plates to

the tension face of the beams can introduce a significant amount of enhancement in the

flexural strength of lightly reinforced beam, while more heavily reinforced beams requires an

increased amount of FRP, or a comparable composite ratio to achieve comparable strength

enhancement (Ross et.al., 1999). High composite ratio plays an important part in the

strengthening effect of light to moderately reinforced beams. By CFRP application,

approximately 10 to 35% higher load carrying capacity can be obtained along with a 10 to

32% decrease in the beam deflections at ultimate failure (Bonnaci et al.,2000) .

In addition to the strength enhancement, the FRP strengthening scheme with anchoring

system improves the ductility of the retrofitted beam by confining the concrete. Various

analytical models (Saadatmanesh et al. 1996, Niterika and Neale, 1999) have been proposed

to predict the ultimate moment capacities of reinforced concrete beams strengthened with

externally bonded composite laminates. In general, these models ignore the nonlinear stress–

strain behavior of the concrete and the contribution of tension concrete. Applications based

on such models are limited to structures with fairly simple geometries and loading conditions.

In addition to flexural strengthening, many experiments are now being carried out on shear

strengthening with FRP composites. The results show that significant increases in shear

capacity are possible with this FRP repair technique. The failure modes and degree of

strength enhancement, however, are strongly dependent on the details of the bonding scheme

and anchorage method. Shear strengthening using external FRP may be provided at locations

of expected plastic hinges or stress reversal and for enhancing post yield flexural behavior of

members in moment frames resisting seismic loads only by completely wrapping the section.

However, since the FRP materials behave differently than steel, the contribution of FRP

materials need to be included carefully in the design equations on the basis of detailed

experimental evaluation.

The bond behavior and load transfer behavior between concrete beam and FRP laminates is

an important tool to predict the failure behavior and stress distribution of retrofitted beams.

Experimental studies (Brena et al. 2003; Hamad et al. 2004; Saxena et al. 2008; and Choi

et al. 2008) indicated that debonding of the bottom strip from the concrete surface is the most

10  

common mode of failure for concrete beams strengthened by externally bonded FRP sheets.

The debonding results in the loss of the composite action between the concrete and FRP

laminates. The effective stress transfer between FRP and concrete is essential to develop the

composite action. The local debonding initiates when high interfacial shear and normal

stresses exceed the concrete strength (Kotynia et al. 2008). Additional U-jacket strips or

sheets can be provided in the debonding initiation region to delay the FRP debonding

resulting in increased efficiency of the FRP retrofitting scheme. More experimental and

analytical studies should be carried out to find a more reliable relation between bond behavior

of FRP laminates and concrete to make sure that the FRP fitted structure does not fail

prematurely.

2.4.2 Column Strengthening

Reinforced concrete columns are considered to be the most important part of a typical

reinforced concrete structure as they are the major load carrying element of the building.

Minimum cross section size and lack of steel reinforcement in under designed columns leads

to a weak column—strong beam construction. To avoid a soft story collapse of a building due

to seismic action, columns should be adequately designed.

During an earthquake, plastic hinges are most likely to form in columns in weak column—

strong beam construction which may result in a sudden story collapse of the whole structure.

So it is very necessary to strengthen the columns so that plastic hinges are formed in the

beams since it allows more effective energy dissipation. It is reported that, closely spaced

transverse reinforcement used in the plastic hinge zone of concrete bridge columns will help

in increasing the compressive strength as well as increase the ultimate compressive strain in

the core concrete (Mirmiran and Shahawy 1997). Therefore, a significant amount of increase

in compressive strain will result in increasing the ductility of concrete columns. Researchers

have shown that an increase in the thickness of CFRP and AFRP jacket proportionally

increases the shear strength of the upgraded column or pier (Fujisaki et al. 1997; Masukawa

et al. 1997). 

2.4.2.1 Experimental investigations

Unidirectional FRP sheets can be wrapped around the concrete columns as an external

reinforcement and confinement. Several investigations (Benzoni et al., 1996; Masukawa

et al., 1997; Seible et al., 1997; Lavergne and Labossiere, 1997; Saadatmanesh et al., 1997;

11  

Seible et al., 1999; Mirmiran and Shahawy 1997; Fukuyama et al., 1999; Pantelides et al.

2000b; Bousias et al. 2004 and Harajli et al. 2006) have been conducted to study the

effectiveness of FRP in restrengthening of circular, square and rectangular reinforced

concrete columns. Most of the research works were done for identifying the behavior of FRP

confined concrete circular columns.

Saafi et.al. (1999) confirmed that for circular columns external confinement of concrete by

FRP tubes can significantly enhance the strength, ductility, and energy absorption capacity of

concrete.

Experiments regarding behavior of rectangular columns confined with FRP laminates are

limited. Haralji et al. (2006) reported that for square column sections without longitudinal

reinforcement (plain concrete) the increase in axial strength was found to be 154%, 213%,

and 230% for one, two, or three layers of CFRP wraps, respectively.

Rochette and Labossie`re (2000) performed experimental research for identifying the

influence of FRP thickness and corner radius of rectangular columns. They reported that for a

given number of wraps around a section (or a given transverse reinforcement ratio), the

confinement effect is directly related to the shape of the section and the section corners

should always be rounded off sufficiently to prevent premature failure by punching of the

fibres in the wrap. To investigate the influence of aspect ratio Chaallal, O. et al. (2003)

performed an experiment having different cross sectional properties and material properties

of rectangular columns. The gain in performance of axial strength and ductility due to the

wrapping was found greater for the 3 ksi concrete wrapped columns than for the

corresponding 6 ksi concrete columns. The maximum gain achieved for the 3 ksi concrete

wrapped columns was approximately 90% as compared to only 30% for the 6 ksi columns.

Figure 2.3 shows a picture of FRP applications on concrete column for retrofitting.

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13  

the concrete shortly after the concrete has reached its ultimate compressive strength. To better

understand the FRP confinement of the concrete a proper stress-strain model has to be

developed. Exclusive work has been done in understanding this behavior.

2.5.1. Circular Columns

The confinement action exerted by the FRP on the concrete core is of the passive type, that is,

it arises as a result of the lateral expansion of concrete under axial load. As the axial stress

increases, the corresponding lateral strain increases and the confining device develops a

tensile hoop stress balanced by a uniform radial pressure which reacts against the concrete

lateral expansion (De Lorenzis & Tepfers, 2003.). When an FRP confined cylinder is subject

to axial compression, the concrete expands laterally and this expansion is restrained by the

FRP. The confining action of the FRP composite for circular concrete columns is shown in

Figure 2.4.

For circular columns, the concrete is subject to uniform confinement, and the maximum

confining pressure provided by FRP composite is related to the amount and strength of FRP

and the diameter of the confined concrete core. The maximum value of the confinement

pressure that the FRP can exert is attained when the circumferential strain in the FRP reaches

its ultimate strain and the fibres rupture leading to brittle failure of the cylinder. This

confining pressure is given by Equation 2.1:

(2.1)

Figure 2.4 Confinement action of FRP composite in circular sections

(Benzaid and Mesbah, 2013)

ffrp ffrp

tfrp tfrp

14  

Where fl is the lateral confining pressure, Efrp is the elastic modulus of the FRP composite, εfu

is the ultimate FRP tensile strain, ffrp is the ultimate tensile strength of the FRP composite, tfrp

is the total thickness of the FRP, d is the diameter of the concrete cylinder, and ρfrp is the FRP

volumetric ratio given by the following Equation 2.2 for fully wrapped circular cross section:

= / (2.2)

2.5.2. Rectangular Columns

A square column with rounded corners is shown in Figure 2.4. To improve the effectiveness

of FRP confinement, corner rounding is generally recommended. Due to the presence of

internal steel reinforcement, the corner radius R is generally limited to small values. Existing

studies on steel confined concrete (Park and Paulay, 1975; Mander, et al.1988; Cusson and

Paultre, 1995). have led to the simple proposition that the concrete in a square section is

confined by the transverse reinforcement through arching actions, and only the concrete

contained by the four second-degree parabolas as shown in Figure 2.5 (b) is fully confined

while the confinement to the rest is negligible. These parabolas intersect the edges at 45°.

While there are differences between steel and FRP in providing confinement, the observation

that only part of the section is well confined is obviously also valid in the case of FRP

confinement. Youssef et al. (2007) showed that confining square concrete members with FRP

materials tends to produce confining stress concentrated around the corners of such members,

as shown in Figure 2.5(a). The reduced effectiveness of an FRP jacket for a square section

than for a circular section has been confirmed by experimental results (Rochette &

Labossière, 2000). Despite this reduced effectiveness, an FRP-confined square concrete

column generally also fails by FRP rupture (Benzaid et. al., 2008). For finding the confining

pressure for rectangular columns in Equation (2.1), d is replaced by the diagonal length of the

square section. For a square section with rounded corners, d can be written as:

√2 2 √2 1 (2.3)

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enerally cle

tween the tw

er corner rad

hown in Figu

g occurred f

mn confined w

tion of FRP Mesba

15

ete specimen

height of th

d catastroph

ncrete core

ing and wav

failure of the

o failed by f

gh the stress

ncrete prism

e of the com

hstand the l

nement thus

ng various

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first. It was,

(b)with

composite inah, 2013)

ns is general

he specimen

hic, accomp

in the form

ving of the t

e FRP tube a

fracture of th

s-strain curv

s occurred w

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oad, which

triggers a su

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ng of the ag

rpendicular t

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e ultimate co

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n square sec

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n However,

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as reported b

he CFRP co

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without adva

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correspond

udden failure

loading. The

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to the fibre

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omposite at o

an increase i

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Chaallal et al

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ss

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re

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In

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For stron

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2.7 Desi

The Ame

of FRP a

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calculate

factored

Vertical

also limi

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calculate

ACI 318

If the me

FRP jack

Figure

unded off co

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ign Guideli

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as confining r

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(2002).

ember is sub

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rete Institute

reinforceme

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e (ACI 2002

ent for streng

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FRP-strength

CI 318 (2002

duction fact

combined co

sed on the cr

imens (a) cir(Chaallal

16

fined weakly

kage in the w

age was obse

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l reduction f

rain limitatio

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i et al., 1999

b)

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nly 50–150 m

most the full

ommendation

te columns.

ed with an F

) substitutin

factor is set t

ons in the FR

be achieved

should be co

ession streng

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the effective

9) and (b) rec

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mm in length

l height of th

ns for the us

FRP jacket

ng for f’c th

to ψf = 0.95.

RP jacket ca

with an FR

omputed wit

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be

by

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17  

0.004 0.75 (2.4)

At load levels near ultimate, damage to the concrete in the form of significant cracking in the

radial direction occurs. The FRP jacket contains the damage and maintains the structural

integrity of the column. At service load levels, this type of damage should be avoided. In this

way, the FRP jacket will only act during overloads that are temporary in nature. To ensure

that radial cracking will not occur under service loads, the stress in the concrete is limited to

0.65f ′c. In addition, the stress in the steel should remain below 0.60fy to avoid plastic

deformation under sustained or cyclic loads. By maintaining the specified stress in the

concrete at service, the stress in the FRP jacket will be negligible. (Nanni, A. 2001). These

guidelines are only for the circular FRP-wrapped columns under concentric axial load

because test data on square and rectangular, slender, and eccentrically-loaded columns are

comparatively scarce.

Guidelines in Canada (CSA and ISIS) and Europe (FIB) provide design equations for

strengthening rectangular columns retrofitted with externally-bonded confining composite

wrap. The enhancement of confined concrete strength depends on the passive confinement

due to the lateral pressure generated by the lateral FRP fibres. The design and construction

guide for strengthening concrete structures with externally bonded FRP systems reported by

the ACI Committee 440 (2002) is aware of the enhanced concrete strength reported by

researchers, but it still considers it as marginal and no recommendations have yet been

provided, given the many unknowns related to this type of application. It should be

mentioned that most of the research on confinement of rectangular concrete columns was

presented after the ACI 440.2R guidelines were published.

2.7.1 CSA-S806-022 (2002)

According to the Canadian Standards CSA-S806-02 the load carrying capacity of a confined

column can be calculated as follows

(2.5)

where ke is a resistance factor (= 0.80 for columns with transverse steel ties), φc and φs are the

resistance factors for concrete and steel (φc = 0.6 and φs = 0.85), α1 is the ratio of average

18  

compression stress to the concrete strength, that is, α1 = (0.85 – 0.0015f ′ c ) ≥ 0.67, Ag and Ast

are the gross concrete area and the area of steel bars respectively.

The CSA guidelines limit the applicability of the design equations to columns with small

aspect ratios and rounded corners. The maximum aspect ratio is limited to 1.5 (that is, b/h ≤

1.5). Also, the corner radius is to be greater than or equal to 20 mm (0.8 in.) (r ≥ 20 mm [0.8

in.]). For rectangular columns meeting these conditions, the confined concrete strength f ′cc

can be calculated using Equation (2.6) to (2.8).

0.85 (2.6)

6.7 -0.17 (2.7)

(2.8)

Where, kc is a confinement coefficient equal to 0.25 for rectangular columns, D is the

diameter of an equivalent circular column; tj is the thickness of the FRP jacket, fFj is the

stress in the FRP jacket (= minimum [0.004Ej, φF fFu]), φF is the resistance reduction factor

for FRP and fFu is the ultimate FRP tensile strength.

2.7.2 ISIS Canada (2001)

According to the design guidelines provided by ISIS Canada the confined concrete strength

can be calculatedusing Equation (2.9) ,

1 (2.9)

where αpr is a performance coefficient (=1); Ww is the volumetric strength ratio. To ensure an

effective confinement, the ISIS guidelines limit the applicability of the design equations to

quasi-square columns with rounded corners because the maximum aspect ratio is limited to

1.1 (b/h ≤ 1.1). Also, the corner radius should be greater than or equal to b/6 and not less than

35 mm (1.4 in.) [(r ≥ b/6] and [r ≥ 35 mm (1.4 in.)]. The guidelines, however, do not specify

any limiting values on the confining pressure as was the case for circular columns.

19  

2.7.3 FIB Guidelines (2001)

In its technical report “Externally Bonded FRP Reinforcement for RC Structures,” the

International Federation of Structural Concrete (FIB) provides equations for the design of

rectangular columns confined with FRP wrap. The ultimate confined concrete strength is

calculated using Equation (2.10) to (2.13).

0.2 3 (2.10)

Where can be calculated from Equation 2.11

(2.11)

(2.12)

(2.13)

where ke is the effectiveness coefficient representing the ratio of the effectively confined area

of the cross section to the total cross-sectional area and εju is the effective ultimate

circumferential strain of the FRP jacket. The guidelines state that in view of the limited

proper values of εju, the value chosen should be justified by experimental evidence.

2.8 Summary  From the review of literature presented in this chapter it is clear that extensive experimental

investigations have been performed on strengthening of concrete circular columns using fibre

reinforced polymers. The performance of FRP confined concrete circular columns is now

relatively well understood from the experimental point of view. But information about

behavior of confined rectangular columns is limited. Since laboratory experiments are

expensive and time-consuming, reliable analytical procedures should be developed for

predicting the structural response of concrete columns confined by fibre-reinforced polymers.

To fully simulate their behavior up to failure, numerical models which are capable of

predicting the complexities of material nonlinearity, concrete post cracking tension softening,

as well as interaction between the concrete and FRP surface, is required. Therefore, an

attempt has been made in current study to address these issues and thereby to develop a full

scale 3D finite element model for FRP confined concrete columns under axial loading.

20  

CHAPTER 3

FINITE ELEMENT MODELING  

3.1 General

Due to relatively high cost of large-scale experimental research, a means of modeling FRP-

confined concrete columns using computer aided program is needed to broaden the current

knowledge about the complete behavior and influence of the geometric properties. In this

study an attempt has been made to develop a complete Finite Element model that can be

applied for a variety of geometries of FRP confined concrete columns subjected to uniaxial

loading and provide accurate simulations of the compressive behavior. The model therefore is

to be capable of simulating numerically the compressive behavior of concrete columns

confined by Fibre-Reinforced Polymers. The model is developed using the

ABAQUS/Standard finite element software code.

A concrete damage plasticity model which is capable of predicting both compressive and

tensile failures is used to model the concrete material behavior. The FRP–concrete interface

in the confined concrete column is modeled using the contact pair algorithm in ABAQUS.

Both cohesion and perfect bond formulation having simple master–slave contact are used at

the interface of the FRP laminate and concrete infill. Nonlinear material behavior as well as

the geometric nonlinearities is accounted for in the numerical model. A static Riks solution

strategy is used to trace a stable post-peak response of the composite system up to failure.

Experimental results of 11 specimens, representing FRP-confined concrete columns are used

to validate the numerical results. To validate the model, simulations are conducted for axially

loaded rectangular test specimens reported in the literature, varying in cross section from

152×152 mm to 108×165 mm, including a variety of corner radius and concrete compressive

strength (25 MPa to 42 MPa). For circular columns the diameter of the columns are 152 mm.

The thickness of the FRP sheets is also varied here.

Detailed descriptions of the test specimens are provided in the following section. This is

followed by a description of the finite element model geometry used to simulate the various

tests, the material model parameters, as well as the loading program.

21  

3.2 Properties of Reference Test Specimens

3.2.1 Geometric and Material Properties of Square Columns

 

The column sets tested by Rochette and Labossie`re (2003) includes five square specimens

named S5C5, S25C3, S25C4, S25C5 and S38C3 are modeled for finite element analysis. The

lists of these specimens, along with their geometric properties, are given in Table 3.1 and

shown in Figure 3.1. These specimens had square cross sections of 152 mm X 152 mm with a

height of 500 mm. The corner radiuses of the specimens were varied from 5 mm to 38 mm

where 5 mm represented the sharpest square column. The material properties of these test

specimens are presented in Table 3.2. These specimens were wrapped with two to five plies

of carbon fibre. In all cases, the principal fibres were oriented perpendicular to the column

axis, in a so-called 0º orientation. The mechanical properties of these test specimens are

presented in Table 3.2. To provide confinement, composite sheets were wrapped around the

column models in a continuous manner. Once the appropriate number of laps had been

placed, the outermost confining sheet was extended by an additional overlapping length, in

order to provide a sufficient anchorage and prevent slip between layers. An overlap length of

100 mm was applied and was found to be sufficient. After placement of the external 0º layer,

a 25 mm wide strip was added at each end of the specimens. This additional local

confinement prevents local damages and ensures that compressive failure occurs in the

central portion of the model. The specimens were subjected to a monotonic uniaxial

compression loading up to failure. The load was applied at a strain rate of 10 µε/s with a

hydraulic press. Prior to the test, a thin sulfide layer was put on both ends of the column to

ensure that contact areas were flat and parallel. These specimens were modeled to investigate

the confinement efficiency and influence of the corner radius for a constant FRP laminate

thickness.

3.2.2 Geometric and Material Properties of Rectangular Columns

 

Four rectangular columns SC-1L3-0.7, SC-2L3-0.7, SC-3L3-0.7 and SC-4L3-0.7 constitute

the column set of Chaallal et al.(1999) having different aspect ratio (a/b=0.7) are also

modeled to validate the numerical results. These specimens had rectangular cross sections of

165 mm X 108 mm with a height of 305 mm which is shown in Figure 3.1. The compressive

strength of concrete was around 21 MPa. For the specimens receiving carbon lamination, the

required layers of the standard CFRP system were applied. The standard system consists of a

22  

bidirectional weave with an average of 6.7 yarns per inch in each direction and per layer.

Details of the material properties of the CFRP are presented in Table 3.2. For each specimen,

the corners were rounded with a corner radius equal to 25.4 mm to improve their behavior

and to avoid premature failure of CFRP material due to shearing at sharp corners.  All

specimens were tested using a 550 kip (2,446 kN) MTS compression machine and an

automatic data acquisition system. Specimens were tested to failure under a monotonically

increased concentric load and a displacement control mode. Failure was usually caused by

sudden rupture of the composite wrap. After failure, the confined concrete was found to be

disintegrated in about one third of the total volume of the specimen. Experimental

observations suggest that the micro-cracking occurs in a more diffuse manner than in

unconfined concrete. Despite all measures, it was impossible to precisely identify the exact

location where failure initiated in the confining laminate (Chaallal et al., 2003)

3.2.3 Geometric and Material Properties of Circular Columns

To test the performance of circular concrete columns confined with FRP tubes, two circular

columns named C1 and C2 of Saafi et.al. (1999) with different thickness of FRP laminates

are also modeled under compression. All specimens consisted of short columns with a length-

to-diameter ratio of 2.85. Each specimen measured 152.4 mm in diameter and 435 mm in

length. The geometric properties are summarized in Table 3.3. The mechanical properties of

the FRP tubes are summarized in Table 3.4. The FRP tubes used in that study were made of

carbon-fibre filament winding-reinforced polymers, all consisting of 60 percent fibre and 40

percent polyester resin. The fibres oriented in the circumferential direction of the cylinders.

The concrete consisted of ASTM Type I Portland cement, river sand aggregate with a

fineness modulus of 2.6 and a crushed limestone aggregate with a maximum size of 10 mm.

The water-cement ratio (w/c) was 0.5 by mass. The average 28-day compressive strength of

the concrete was 38 MPa, and the modulus of elasticity was 30 GPa. Concrete encased with

carbon FRP tubes of thicknesses of 0.11 and 0.23 mm were designated as C1 and C2. The

confined cylinders, as well as unconfined samples, were tested using a 300-kip testing

machine. The load was applied to the specimen through a pad having the same area as the

concrete core. Failure of the composite specimens was initiated by fracture of the fibre tube.

23  

Figure 3.1 Geometric properties of square, rectangular and circular columns

b = 152 mm 

R 

a=152 mm 

500 mm

R 

b =165.10 mm

108 mm

305 mm 435 mm 

D=152.4

24  

Table 3.1 Geometric properties of square and rectangular columns

Reference

Column

Designation

Columns’ Dimensions (mm) Fibre-Reinforced

Polymers

(CFRP)

a (shorter

side)

b(longer

side)

H

Corner

Radius (R)

No. of

Layers

Thickness

(mm) (mm) (mm) (mm) (mm)

Rochette and

Labossie`re(2000)

S25C3 152 152 500 25 3 0.9

S25C4 4 1.2

S25C5 5 1.5

S38C3 38 3 0.9

S5C5 5 5 1.5

Chaallal, O. et

al.(2003)

SC-1L6-0.7 108.00 165.1 305 25.4 1 0.5

SC-2L6-0.7 2 1.0

SC-3L6-0.7

SC-4L6-0.7

3

4

1.5

2.0

Table 3.2 Material properties of square and rectangular columns

 

Reference

Column

Designation

Concrete Properties Fibre-Reinforced Polymers

(CFRP)

ρ

(%)

f�c

(MPa)

w

(g/cm3)

εult

%

Ej

(GPa)

ffu

(MPa)

Rochette, and

Labossie`re,(2000)

S25C3 2.26 42.00 1.80 1.5 82.7 1265

S25C4 3.02 43.90

S25C5 3.79 43.90

S38C3 2.25 42.00

S5C5 3.93 43.90

Chaallal, O. et al.(2003) SC-1L6-0.7 0.37 25.10 - 0.28 231 3650

SC-2L6-0.7 0.75 0.50

SC-3L6-0.7

SC-4L6-0.7

1.12

1.5

0.60

0.50

25  

Table 3.3 Geometric properties of circular columns

Reference

Column

Designation

Columns’ Dimensions

Fibre-Reinforced

Polymers

(CFRP)

D

H

No. of

Layers

Thickness

(mm) (mm) (mm)

Saafi et.al. (1999) C1 152.4 435 1 0.11

C2 2 0.23

Table 3.4 Material properties of circular columns

Reference

Column

Designation

Concrete properties Fibre-Reinforced

Polymers

(CFRP)

f’c

Ej

ffu

(MPa) (GPa) (MPa)

Saafi et.al. (1999) C1 35 367 3300

C2 390 3550

3.3 Cha

3.3.1 Geo

 

In this st

like cross

stiffness

The mod

section is

3.3.1.1 E

As report

the simu

capture t

whereas

used to s

Typically

brick ele

translatio

Figure

L

X 

racteristics

ometric Pro

tudy FRP co

s-sectional s

factor on

del used in th

s shown.

Element selec

ted in chapte

ultaneous oc

this behavio

eight-node

simulate the

y, the numb

ement is ca

onal degrees

e 3.2 (a) 3-D

Y 

Z 

s of the Fin

operties and

onfined conc

shape factor

n confineme

he analysis i

ction

er 2, the FR

ccurrence of

or eight node

finite strain

FRP sheets

er of nodes

alled C3D8

of freedom

D view of the

(a)

nite Elemen

d Finite Elem

crete column

(a/b), the co

ent efficien

is shown in

RP-confined c

f rupture of

e brick elem

n reduced in

s and lamina

in an eleme

8R and the

are consider

e column me

26

nt Model

ment Model

ns are mode

orner sharpn

ncy of exp

Figure 3.2(a

concrete col

f FRP lamin

ments (C3D8

ntegration co

ates, Details

ent is clearl

8-node she

red in each n

sh and (b) C

a

ls

eled to study

ness factor (a

perimental

a) and in the

lumns reach

nates and c

8R) are use

ontinuum sh

s are shown

y identified

ell element

node for both

Cross section

(b)

y the effect o

a/R) and the

FRP confin

e Figure 3.2

their ultima

crushing of

d to model

hell element

in Figure 3

in its name

is called S

h elements.

n (with CFRP

b

of paramete

e confinemen

ned column

2 (b) the cros

ate capacity

concrete. T

the concret

s (SC8R) ar

.3(a) and (b

e. The 8-nod

SC8R. Thre

For modelin

P laminate)

R

rs

nt

n.

ss

at

To

te,

re

b).

de

ee

ng

)

the circu

provide f

A body

symmetry

3.3(c) sh

point on

coordinat

Figur

ular columns

for the mode

of revolutio

y axis) and

ows a typica

this cross-se

tes coincide

re 3.3 Finite

(a) 

s axisymmet

eling of bodi

on is genera

is readily d

al reference

ection are de

with the glo

elements us

(c

tric element

ies of revolu

ated by revo

described in

cross-sectio

enoted by r a

obal Cartesia

                       

sed in the mo

c) Axisymme

(c) 

27

ts (CAX4R)

ution under a

olving a pla

cylindrical

on at �=0. T

and z, respec

an X- and Y-c

    

odel, (a) 8-no

etric solid el

) were used.

axially symm

ane cross-se

polar coord

The radial a

ctively. At �

coordinates.

ode solid (b

lement

(b)

. Axisymme

metric loadin

ection about

dinates r, z, a

and axial coo

�=0. , the ra

b) 8- shell el

etric elemen

ng condition

t an axis (th

and �. Figur

ordinates of

adial and axi

 

lement and

nts

ns.

he

re

f a

al

28  

3.3.1.2 Mesh description

The mesh configuration for the full FRP confined concrete column model is shown in Figure

3.2. A sensitivity analysis was performed using 5×5×5 mm, 25×25×25 mm and 50×50×50

mm rectangular block to optimize the mesh in order to produce proper representation of the

rupture of FRP sheets. Since, the rupture of the FRP sheets always started at the corners

(Rochette and Labossie`re, 2000) a finer mesh was defined at the corners of the rectangular

and square columns. The mesh size of other portions didn’t have any significant influence on

the compressive behavior of confined columns. The element sizes of the concrete and FRP

are selected to be approximately 50×50×50 mm rectangular block as it can properly simulate

the behavior and minimize the computational time.

3.3.1.3 Modeling of concrete-FRP interface

One of the most challenging aspects of this study was to model successfully the concrete –

FRP interaction at their interfaces with a contact algorithm. Contact conditions are a special

class of discontinuous constraint in numerical analysis. They allow forces to be transmitted

from one surface to another only when they are in contact. When the surfaces separate, no

constraint is applied. ABAQUS provides two algorithms for modeling contact: a general

contact algorithm and a contact-pair algorithm. The general contact algorithm is more

powerful and allows in simpler cases where as a contact pair algorithm is needed for

specialized contact features such as in the current problem.

Two different models were used to represent the interface between concrete and CFRP. In the

first model the interface was modeled as a perfect bond while in the second it was modeled

using a cohesive zone model. In perfect bond model contact pair algorithm is used between

concrete –FRP interface. First, two surfaces were defined geometrically. The surface of the

FRP laminates was defined as slave surface whereas the concrete surface was defined as

master surface. As long as the two surfaces were in contact, they transmitted shear and

normal force across the interface.

In cohesive based interface model simple traction-separation law is used in between master –

slave interfaces. Figure 3.4 shows a graphic interpretation of a simple bilinear traction–

separation law written in terms of the effective traction τ and effective opening displacement

δ.

29  

Figure 3.4 Bilinear traction separation constitutive law

The interface is modeled as a rich zone of small thickness and the initial stiffness K0 is

defined as: 1

3.1

where, ti is the resin thickness, tc is the concrete thickness, and Gi and Gc are the shear

modulus of resin and concrete respectively.

The values used for this study were ti = 1 mm, tc = 5 mm, Gi = 0.665 GPa, and Gc = 10.8 GPa.

From Figure 3.4, it is obvious that the relationship between the traction stress and effective

opening displacement is defined by the stiffness, K0, the local strength of the material, τmax, a

characteristic opening displacement at fracture, δf, and the energy needed for opening the

crack, Gcr, which is equal to the area under the traction– displacement curve. Equati` on.

3.2 provides an upper limit for the maximum shear stress, τmax, giving τmax = 3 MPa in this

case:

1.5 (3.2)

where

2.25 / 1.25

3.3

and bf is CFRP plate width, bc is concrete width and fct is concrete tensile strength.

Gcr

δ

τ

τmax

δf δ0

K0

Eff

ectiv

e tr

actio

n, τ

Effective opening displacement, δ

30  

The initiation of damage was assumed to occur when a quadratic traction function involving

the nominal stress ratios reached the value one. This criterion can be represented by

1 3.4

where σn is the cohesive tensile and τs and τt are shear stresses of the interface, and n, s, and t

refer to the direction of the stress components.

Interface damage evolution was expressed in terms of energy release. The description of this

model is available in the Abaqus material library. The dependence of the fracture energy was

defined based on the Benzaggah–Kenane fracture criterion. Benzaggah–Kenane fracture

criterion is particularly useful when the critical fracture energies during deformation purely

along the first and the second shear directions are the same;

i.e., Gsc= Gt

c. It is given by:

3.5

where Gζ = GS + Gt , Gξ= Gn + Gs and η are the material parameter. Gn, Gs and Gt refer to

the work done by the traction and its conjugate separation in the normal, the first and the

second shear directions, respectively. (Obaidat et.al, 2009)

Axial stress vs. axial strain responses of confined columns found from both models ensured

same ultimate capacity. Actually the failure of the FRP-confined concrete columns is

governed by the rupture of the FRP laminates at corners. Debonding of FRP sheets is not an

important criterion of failure in FRP confined concrete columns. So, the cohesion model

didn’t affect the ultimate capacity at all. Figure 3.5 (a) and (b) clearly illustrates that for

concentric loading there is no significant influence of cohesion model. However, it may

affect the ultimate capacity for eccentric loading condition. Hence perfect bond model is used

for further numerical modeling as it minimizes the computational time.

31  

(a) (b)

Figure 3.5 Axial stress vs. axial strain responses of column S25C5 (a) using cohesive zone model (b) using perfect bond model.

3.3.1.4. Load application & boundary condition

In the experiments the specimens were subjected to a monotonic uniaxial compression

loading up to failure. The load was applied with a hydraulic press. Prior to the test, a thin

sulfide layer was put on both ends of the column to ensure that contact areas were flat and

parallel. Uniaxial compressive load is applied in the model just like the experimental way

shown in Figure 3.6. As full cylinders and prisms have been modeled so fixed support is

applied at bottom end and displacement controlled loading is applied on the top. The top

surface is made rigid to ensure uniform transfer of the applied loading to the adjacent

concrete and FRP nodes.

Figure 3.6 Load application and boundary condition

FEM FEM

TEST TEST

Axial Strain % Axial Strain %

Axi

al S

tress

(MPa

)

Displacement control loading

Fixed Support

0

20

40

60

80

0 0.5 1 1.5

0

10

20

30

40

50

60

70

0 0.5 1 1.5

32  

3.3.2 Material Properties

3.3.2.1 Concrete

Concrete is modeled using concrete damaged plasticity model provided by ABAQUS

software. The concrete damaged plasticity model is primarily intended to provide a general

capability for the analysis of concrete structures under cyclic and/or dynamic loading. The

model is also suitable for the analysis of other quasi-brittle materials, such as rock, mortar

and ceramics; but it is the behavior of concrete that is used in the remainder of this section to

motivate different aspects of the constitutive theory. Under low confining pressures, concrete

behaves in a brittle manner; the main failure mechanisms are cracking in tension and crushing

in compression. The brittle behavior of concrete disappears when the confining pressure is

sufficiently large to prevent crack propagation. In these circumstances failure is driven by the

consolidation and collapse of the concrete micro porous microstructure, leading to a

macroscopic response that resembles that of a ductile material with work hardening.

The model is capable of taking into consideration the degradation of elastic stiffness (or

“damage”) induced by reversible cycles as well as high temperatures both in tension and

compression. The concrete damage plasticity model uses a non-associated plastic flow rule in

combination with isotropic damage elasticity. The Drucker–Prager hyperbolic function is

used to define the plastic flow potential. The dilation angle defines the plastic strain direction

with respect to the deviatoric stress axis in the meridian plane. The volumetric expansion of

concrete can be controlled by varying the dilation angle.

The model uses the yield function of Lubliner et al. (1989), with modifications to account for

a different evolution of strength under tension and compression using multiple hardening

variables. The two hardening variables used to trace the evolution of the yield surface are the

effective plastic strains in compression and in tension, εc~pl and εt

~pl, respectively. The start of

compressive yield in a numerical analysis using this model occurs when εc~pl > 0, whereas

when εt~pl > 0 and the principal plastic strain is positive, it indicates the onset of tensile

cracking.

Uniaxial tension and compression stress behavior  

The uniaxial tensile and compressive responses (Figures 3.3(a) and 3.3(b), respectively) of

concrete used in this model are somewhat simplified to capture the main features of the

response. Under uniaxial compression, the stress–strain response (as shown in Figure 3.7(b))

33  

is assumed to be linear up to the initial yield stress, which is assumed to be 0.30fcu in the

current study. The plastic region is characterized by stress hardening, followed by strain

softening after reaching the ultimate strength, fcu. The uniaxial compression hardening curve

is defined in terms of the inelastic strain, εc~in, which is calculated using Equation (3.6). The

damage plasticity model automatically calculates the compressive plastic strains, εc~pl,

Equation (3.7), using a damage parameter, dc, that represents the degradation of the elastic

stiffness of the material in compression.

~ 3.6

~ ~ 1 3.7

Figure 3.7(a) shows the uniaxial tensile behavior of concrete used in the damage plasticity

model. The stress–strain curve in tension is assumed to be linearly elastic until the failure

stress, ftu , is reached. After this point strain softening represents the response of the cracked

concrete that is expressed by a stress versus cracking displacement curve. The values of the

plastic displacements calculated by the damage model are equal to the cracking

displacements since the tensile damage parame ter, dt , is zero for current study.

34  

Figure 3.7 Response of concrete to uniaxial loading in (a) tension and (b)compression.

A general form of serpentine curve, as given by the following equations (Carriera and Chu,

1985) is used to represent the complete stress-strain relationship of unconfined concrete

1 3.8

1

1 3.9

(a)

(b)

 

35  

Where, β is a material parameter which depends on the shape of the stress- strain diagram.

The value of β = 3 is used in this thesis which is proposed by Tulin and Grestle (1964).A

stress-strain relationship curve of concrete for different values of ε′c is plotted using the

above equations and this curve is shown in Figure 3.8 (a). Figure 3.8 (b) shows axial stress

versus plastic strain curve for compression hardening of concrete.

 

Figure 3.8 Stress-strain relationship curve of concrete for compression hardening

(a) stress versus total strain (b) stress versus plastic strain

(a)

(b)

0

5

10

15

20

25

30

35

40

45

0 1000 2000 3000 4000 5000 6000

Stre

ss, M

Pa

Strain , µε

0

5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000 5000

Stre

ss, M

Pa

Plastic Strain (µε)

36  

A same form of serpentine curve is used shown in Figure 3.9 (a) and (b) for the average

stress-strain diagram and stress- inelastic strain diagram of reinforced concrete in tension.

Figure 3.9 Stress-strain relationship curve of concrete for tension stiffening

(a) stress versus total strain (b) stress versus inelastic strain

3.3.2.2 FRP laminate

The FRP laminate is modeled as an isotropic homogeneous material as shown in Figure 3.10

using a linear elastic stress-strain curve with a poison’s ratio of 0.30. The density used in

modeling is 1.8 g/cc (Rochette and Labossie`re, 2000).

Stre

ss, M

Pa

(a)

(b)

Stre

ss, M

Pa

0

0.5

1

1.5

2

2.5

3

0 0.0001 0.0002 0.0003 0.0004 0.0005

Strain, ε

0

0.5

1

1.5

2

2.5

3

0 0.0001 0.0002 0.0003 0.0004 0.0005Inelastic strain

37  

Figure.3.10 Typical Elastic stress-strain curve of CFRP

Most of the FRP confined rectangular columns failed due to rupture of the FRP laminate at

the corners when the FRP sheets reached their hoop strength (Chaallal et al., 2003; Rochette

and Labossière, 2000). Hoop strength of carbon sheet generally ranges from 0.41 to 0.61 of

the tensile ultimate strength as reported by Rousakis et al. To simulate the failure behavior

hoop stress is provided 0.5 of the tensile strength.

3.3.3 Solution Strategy

The solution strategy is based on the Riks method. In simple cases linear eigenvalue analysis

may be sufficient for design evaluation; but if there is concern about material nonlinearity,

geometric nonlinearity prior to buckling, or unstable postbuckling response, a load-deflection

(Riks) analysis must be performed to investigate the problem further. The Riks method:

• Generally is used to predict unstable, geometrically nonlinear collapse of a structure.

• Can include nonlinear materials and boundary conditions.

• Often follows an eigenvalue buckling analysis to provide complete information about

a structure's collapse and

• Can be used to speed convergence of ill-conditioned or snap-through problems that do

not exhibit instability.

The Riks method uses the load magnitude as an additional unknown; it solves simultaneously

for loads and displacements. Therefore, another quantity must be used to measure the

progress of the solution; Abaqus/Standard uses the “arc length,” s, along the static

equilibrium path in load-displacement space. This approach provides solutions regardless of

whether the response is stable or unstable shown in Figure 3.11.

Tens

ile

Stre

ss, M

Pa

Strain, %

εult= 1.5%

ftf=1265 MPa

If the Rik

the step

whose m

prescribe

specified

The load

defined b

Where,

proportio

Abaqus/S

incremen

The Riks

initial inc

step, the

∆

ks step is a c

and are not

magnitude is

ed loads are

d.

ding during a

by

P0 is the “

onality facto

Standard pri

nt.

s procedure u

crement in a

initial load p

∆

continuation

redefined a

s defined in

e ramped fr

a Riks step i

“dead load,

or.” The loa

ints out the

uses only a

arc length a

proportional

Figure 3.1

of a previou

are treated a

n the Riks

rom the ini

s always pro

,” Pref is th

ad proportio

e current va

1% extrapol

long the sta

ity factor, ∆

38

11 Riks meth

us history, an

s “dead” loa

step is refe

itial (dead l

oportional. T

he reference

onality facto

alue of the

lation of the

atic equilibri

, is comp

hod.

ny loads tha

ads with con

erred to as

load) value

The current l

e load vec

or is found

load propo

e strain incre

um path, ∆

puted as

t exist at the

nstant magn

a “referenc

to the refe

load magnitu

tor, and

as part of

ortionality fa

ement. After

and after

e beginning o

nitude. A loa

ce” load. A

erence value

ude, ,

(3.10

is the “loa

the solution

actor at eac

r providing a

r defining th

(3.11

of

ad

All

es

is

0)

ad

n.

ch

an

he

1)

39  

Where lperiod is a user-specified total arc length scale factor (typically set equal to 1). This

value of ∆ is used during the first iteration of a Riks step. For subsequent iterations and

increments the value of λ is computed automatically, so there is no control over the load

magnitude. The value of λ is part of the solution. Minimum and maximum arc length

increments, ∆ and ∆ , can be used to control the automatic incrementation.

   

40  

CHAPTER 4

PERFORMANCE OF FINITE ELEMENT MODELS  

4.1 General

The finite element models developed in chapter 3 are validated using simulations of 11 FRP-

confined concrete columns reported in literature (Chaallal et al.,2003 ; Rochette and

Labossière, 2000 and Shaafi et al., 1999). The tests were performed on a wide variety of

concrete columns confined with fibre-reinforced polymers with different geometric properties

and material properties. The descriptions of the geometric and material properties of these

columns have been reported in chapter 3. From the finite element analysis of each of these

test columns, the predicted axial stress versus axial strain and transverse strain response are

obtained and compared with the corresponding experimental results. Moreover the finite

element model is also used to study the effect of corner radius, confinement effectiveness and

shape factor on the strength of confined concrete columns.

4.2 Performance of FEM Models

4.2.1 Ultimate Capacity and Strain

A finite element model with FRP wrapping was developed to predict the compressive

behavior of confined column under uniaxial loading. The ultimate capacities obtained from

numerical models are compared with those obtained from the experiments in Table 4.1. The

maximum axial stresses are found to be very close to those observed in the experiments. The

mean value of the experimental-to-numerical stress ratio is 1.01 with a standard deviation of

0.03.

The axial strain values at the ultimate strain for numerical models, along with the ratios of the

experimental-to-numerical failure strains are shown in Table 4.1. The numerically predicted

ultimate axial strains are found to be higher compared to the experimental values with an

average experimental-to-numerical ratio of 0.96 with a standard deviation of 0.09.

Table 4.1 contains all the results for concrete columns confined with carbon sheets. As

expected, the ratio increases with the confinement effectiveness. It also confirms that each

41  

additional layer for a given section shape provides a significant increase in compressive

strength and for any constant number of confining layers, an increase of the corner radius has

positive consequences on the axial strength.

Table 4.1 Performance of Numerical Models

Specimen

Designation

Axial Stresses fexp./fnum Axial strain at ultimate

point

εexp/εnum

Experimental Numerical

Experimental

Numerical

fz,max fz,max εmax εmax

(MPa) (MPa)

% %

SC-1L3-0.7 29.2 29.0 1.01 0.38 0.35 1.09

SC-2L3-0.7 34.3 33.7 1.01 0.50 0.49 1.01

SC-3L3-0.7 41.2 40.5 1.01 0.60 0.62 0.97

SC-4L3-0.7 47.6 46.8 1.02 0.60 0.71 0.85

S5C5 43.9 46.8 0.94 1.02 1.58 0.65

S25C3 41.6 43.1 0.97 0.94 0.94 1.00

S25C4 50.9 47.5 1.07 1.35 1.25 1.08

S25C5 47.9 47.9 1.00 0.90 1.1 0.82

S38C3 47.5 45.5 1.04 1.08 1.13 0.96

C1 55.0 56.9 0.97 1.00 1.13 0.88

C2 68.0 69.4 0.98 1.25 0.93 1.34

Mean* 1.01 0.96

Standard deviation* 0.03 0.09

*Excluding the value of S5C5 as it was not confined properly during experiment.

42  

4.2.2 Axial stress versus Axial Strain Response

Figures 4.1 to 4.11 show the numerically and experimentally obtained axial strains plotted

against average longitudinal strain of the concentrically loaded five square specimens, four

rectangular specimens and two circular specimens. In all graphs shown in the figures, the

axial stress was calculated by dividing the axial load by the concrete cross sections, assuming

that the composite wrapping has a negligible stiffness in the longitudinal direction.

4.2.2.1 Rectangular columns

Figures 4.1 to 4.4 show the comparison of axial stress versus axial strain response of

rectangular columns having 25mm corner radius and different confinement effectiveness. In

the initial stages of loading, up to a value close to the concrete f′c, the relationship follows a

curve typical of unconfined concrete specimens in compression. It is then followed by a

plastic zone in which maximum measured strains are much more important than for the

unconfined concrete.

In general, the initial portions of the numerical stress versus strain curves match very well

with the experimental ones, though a slight underestimation of axial stiffness is observed in

the initial curves for specimens SC-3L3-0.7 and SC-4L3-0.7. The axial stress versus axial

strain responses of the SC-1L3-0.7 and SC-2L3-0.7 specimens are in good agreement with

the experiment in both peak and post peak region.

It can be observed that the number of layers had little effect on the initial slope. However, as

the number of layers increased, the inflection point moved up to a higher stress level. The

slope of the second branch of the stress-strain curves increased with the number of CFRP

layers, while the first branch was generally not affected.

43  

Figure 4.1 Numerical and experimental axial stress versus axial strain response for column SC-1L3-0.7

Figure 4.2 Numerical and experimental axial stress versus axial strain response for column SC-2L3-0.7

Axial Strain (%)

Axi

al S

tres

s (M

Pa)

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4

TEST

FEM

EXPERIMENT

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6

TEST

FEM

EXPERIMENT

44  

Figure 4.3 Numerical and experimental axial stress versus axial strain response for column SC-3L3-0.7

Figure 4.4 Numerical and experimental axial stress versus axial strain response for column SC-4L3-0.7

Axi

al S

tres

s (M

Pa)

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

Axial Strain (%)

0

5

10

15

20

25

30

35

40

45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

TEST

FEM

EXPERIMENT

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8

TEST

FEM

EXPERIMENT

45  

4.2.2.2 Square columns

In square columns the maximum axial strain has reached an average value of 1.2%. For square

specimens, at low strain levels in the wraps, a small strength increase is initially produced, but

at higher strains the concrete decay in the center of the prism faces is too rapid and the strain

increase in the wraps is not sufficient to compensate for it, resulting in strain softening. It

occurs when the confining material has higher strength and higher deformation capability. In

experiment each concrete column had an overlap length of 100 mm and 25 mm wide CFRP

strip was added at each end of the specimens which prevented local damages and ensured

compressive failure at the centre of the concrete core. This local confinement is not modeled

in the FE model. May be for this reason a softening branch is found in the post peak region.

Experimental prism data reported by some researchers exhibit this same strain softening after

the first peak Figure 6 of Mirmiran et al. (2000). So the maximum axial strength was

measured at first peak. For high degree of confinement this softening branch seems to

disappear.

For models S25C3, S25C4 and S25C5 the maximum strength were found 43.1 MPa, 47.5 MPa

and 47.9 MPa respectively. It confirms that additional layer of FRP laminate increase the

capacity of the columns. The ultimate strength were measured 43.1 MPa and 45.5 MPa for

columns S25C3 and S38C3 which indicate that for a constant thickness of FRP more rounding

off of corners increase the capacity of square columns.

Figure 4.5 Numerical and experimental axial stress versus axial strain response for column S5C5.

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

0

10

20

30

40

50

60

0 0.5 1 1.5

FEM

TESTEXPERIMENT

46  

Figure 4.6 Numerical and experimental axial stress versus axial strain response for column S25C3.

Figure 4.7 Numerical and experimental axial stress versus axial strain response for column S25C4

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

FEM

TESTEXPERIMENT

0

10

20

30

40

50

60

70

0 0.5 1 1.5

FEM

TESTEXPERIMENT

47  

Figure 4.8 Numerical and experimental axial stress versus axial strain response for column S25C5

Figure 4.9 Numerical and experimental axial stress versus axial strain response for column S38C3

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

Axial Strain (%)

Axi

al S

tres

s (M

Pa)

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2

FEM

TESTEXPERIMENT

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4

FEM

TESTEXPERIMENT

48  

4.2.2.3 Circular columns

Carbon fibre-confined circular concrete columns exhibited strength, ductility, and energy

absorption capacity superior to that of unconfined concrete Figures 4.10 and 4.11 show the

axial stress plotted as a function of axial strain carbon fibre reinforced polymer encased

columns.

Figures 4.10 to 4.11 show that the curves are bilinear in nature with a small transition zone.

In the first linear zone, concrete primarily takes the axial load; the slope of the confined

concrete is the same as the slope for the unconfined concrete. The stress-strain curves show

that confinement with FRP tubes does not have much effect on the elastic modulus of the

concrete specimens. At stress levels near to ultimate stress of unconfined concrete, a

transition zone to the second portion of the bilinear curve starts. This region signified that

concrete had cracked, and the FRP tube started to show its full confining characteristics. This

phenomenon can be explained in terms of the composite action between the FRP tube and the

concrete core. At the earlier stage of loading, the Poisson’s ratio of the concrete is lower than

that of the composite tube, thus, the FRP tube has no confining effect on the concrete core.

As the longitudinal strain increases, the lateral expansion of unconfined concrete gradually

becomes greater than that of the FRP tube. A radial pressure develops at the FRP-concrete

interface; the concrete is stressed triaxially and the tube uniaxially. The intersection point

between the two linear branches on the stress-strain curve denotes the initial failure of the

confined concrete core.

The observed increase in axial stress over the unconfined specimen ranged from 51 to 137

percent for the concrete-filled carbon FRP tube. For the carbon FRP tube-confined concrete,

the increase in the ultimate axial strain ranged from 660 to 1100 percent. The relatively low

axial strain reported for Specimen C1 reflects both lower strain and thickness of the tubes.

49  

Figure 4.10 Numerical and experimental axial stress versus axial strain response for column C1

Figure 4.11 Numerical and experimental axial stress versus axial strain response for column C2

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1 1.2

FEM

TESTEXPERIMENT

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4

FEM

TESTEXPERIMENT

50  

4.3 Summary

The performance of finite element model in predicting the behavior of a variety of FRP

confined concrete columns under concentric loading in summarized as follows

The FE model developed in the study was observed to predict the experimental

capacity quite accurately.

In the post-peak region, a softening branch was achieved in square columns due to

high deformation capacity of FRP laminate but the model predicted the post-peak

response for both rectangular and circular columns very well.

The static, Riks solution strategy used in the finite element models made it possible

to trace the full behavior of confined columns without any numerical difficulties.

The interaction between the concrete surface and FRP surface is successfully modeled

using contact pair algorithm with perfect bond formulation.

 

 

 

 

 

 

 

 

 

51  

CHAPTER 5

PARAMETRIC STUDY

5.1 General

Most of the building columns and bridge piers are made up of rectangular columns. These

columns are often in need of strengthening and retrofitting. The use of externally bonded FRP

composites for repair can be a cost-effective alternative for restoring or upgrading the

performance of existing concrete columns. However, majority of the CFRP confining

procedures and models were developed for circular columns and cannot be used in the case of

rectangular columns. The fibre-reinforced polymer (FRP)-jacketed rectangular concrete

prisms in axial compression reveals a number of differences with the circular counterparts

and a set of important new phenomena. For example, while an FRP jacket can lead to an

increase in compression capacity; this increase may be significantly less for rectangular

column than that associated with a circular geometry. As a second example, while the

performance of jacketed circular columns is a strong function of jacket thickness, the

rectangular counterpart is a strong function of geometry. Furthermore, jacketed circular

cross-sections lead to a hardening stress-strain response whereas rectangular specimens often

exhibit strain-softening after the peak strength (Rochette and Labossière, 2000).

The effective confined cross section of rectangular columns depends on the aspect ratio and

the diameter of the rounded corners, as well as the lateral confining pressure. Therefore to

extend the range of applications of CFRP wrapping in strengthening and to enhance the

limited data on rectangular columns retrofitted by FRP wrapping, a parametric analysis is

required using a validated analytical model. The results and observations presented in this

chapter are useful to practicing engineers who have to predict the enhanced compressive

strength of concrete columns retrofitted with externally bonded FRP wrap.

5.2 Design of Parametric Study

The finite element model generated in this research will be used to conduct a detailed

parametric study on the behavior of FRP wrapped concrete rectangular columns. The variable

and fixed geometric as well as material properties used in the parametric study with their

range of values are included in the following sections.

52  

5.2.1 Variable Parameters

For designing the parametric study, the geometric properties of FRP-confined concrete

rectangular column that can significantly affect their behavior under uniaxial compression are

identified as potential variables. Among these, the aspect ratio (a/b) of column cross section,

corner sharpness factor (a/R) and thickness of FRP are identified as the most important

geometric variables. The geometric and material properties of the columns designed for

parametric study are included in Table 5.1. The definition of each parameter, along with its

selected range for this study, is presented in turn below.

5.2.1.1 Aspect ratio (a/b) of the column cross section

The level of concrete confinement is significantly affected by column geometry. Three cross

sections were considered in the analysis as shown in Figure 5.1: a square cross section with

dimensions 300 mm x 300 mm and two rectangular cross sections with dimensions 150 mm x

300 mm and 210 mm x 300 mm. The variation of side dimension is presented by introducing

a term aspect ratio (a/b, where a and b are, respectively, the shorter and longer sides of the

cross section). The cross-sectional aspect ratio for the selected columns varied from 0.5 to 1

with an intermediate value of 0.7.

Figure 5.1 Cross sections of rectangular and square columns used in the parametric study

5.2.1.2 Corner radius(R)

The failure behavior of FRP wrapped concrete columns can be greatly affected by the corner

radius of the column to be retrofitted. To evaluate these effects three different corner radii

i.e., R=10 mm, 25 mm and 50 mm were used for each a/b ratio selected in the parametric

study as shown in Figure 5.2. These values were selected on the basis of the most commonly

encountered radii in rehabilitation practices for reinforced concrete columns. The 10 mm

radius corresponds practically to the sharp edge whereas, the 25 mm and 50 mm are typical

150 mm 210 mm 300 mm

300m

m

R=50mm R=50mm R=50mm

53  

of edge rounding that are encountered in rectangular columns to which FRP wrapping is

applied.

 

 

 

Figure 5.2 Variation of corner radius for square columns

 

5.2.1.3 Thickness of FRP wrap (tf)

The ultimate strength and the ductility of the CFRP confined concrete increase with

increasing number of confining layers. The no. of FRP laminate increases the thickness of

FRP wrap which in turn increases the confinement stiffness. The thickness of a single FRP

layer is assumed to be 0.5 mm. The thickness varied from 0.5mm, 1mm and 2mm which are

used practically. The effect of the selected layers of FRP wrap is studied for fixed values of

a/b and R.

5.2.2 Fixed Parameters

For all column specimens the longer dimension (b) was fixed at 300mm. The length of the

columns was also fixed at 1500 mm. Only the shorter dimension (a) was varied to obtain the

required aspect ratios. The compressive strength of concrete for these columns was fixed at

21 MPa (3 ksi) which is most common value for a deteriorated structure. The equations

proposed by Carriera and Chu (1985) were used to define the concrete stress-strain curves for

the finite element analyses of these columns. Both curves for response of concrete to uniaxial

loading in compression and tension are shown in figure 5.3 (a) and (b) respectively. The

material properties of FRP were also kept constant. Carbon fibre reinforced polymer wrap

with a tensile strength of 3.65 GPA (530 ksi) and tensile modulus of elasticity of 230 GPa

(33500 ksi) were used in current study. The ultimate tensile elongation for the wrap was

defined as 1.4% (Chaallal et al., 2003).

R=50mm R=10mm R=25mm

300mm 300mm 300mm

300m

m

54  

Figure 5.3 Response of concrete to uniaxial loading in (a) compression and

(b) tension

5.3 Results and Discussion

This section presents the influence of each parameter on the behavior of FRP concrete

columns having various geometric properties in comparison to the concrete column in its

unconfined state which are not confined by FRP wrap. The specimens are named as PCX-

YTZ where X, Y and Z represent aspect ratio, thickness of FRP wrap in mm and corner radius

in mm respectively. All of the numerical models failed due to rupture of the FRP laminates at

(a)

(b)

0

5

10

15

20

25

0 1000 2000 3000 4000 5000 6000

Stre

ss, M

Pa

Strain , µε

0

0.5

1

1.5

2

2.5

3

0 0.0001 0.0002 0.0003 0.0004 0.0005Strain, ε

Stre

ss,M

Pa

55  

corners. The output parameters that have been extracted from the analysis are: the axial

stress, f′cc, and average axial strain, εau. The average axial strain is calculated by dividing the

total displacement in the axial direction by the length of the column. The summary of the

results is shown in Table 5.2. The axial stress versus axial strain and the axial stress versus

transverse strain curves are then generated from the numerical analyses for each parametric

column. The effects of the selected geometric parameters on the enhancement of ultimate

load carrying capacity and ductility of the concrete columns are also investigated in this

study.

56  

Table 5.1 Geometric properties of parametric columns

Column

Specimen

Short

dimension

Long

Dimension

Height

Corner

Radius

No. of

FRP

layer

Variables used in

parametric study

(a) (b) (H)

(R)

(a/b) tf

(mm) (mm) (mm) (mm) (mm)

PC1-0T50 300 300 1500 50 0 1 0

PC1-0.5T50 300 300 1500 50 1 1 0.5

PC1-1T50 300 300 1500 50 2 1 1

PC1-2T50 300 300 1500 50 4 1 2

PC0.7-0T50 210 300 1500 50 0 0.7 0

PC0.7-0.5T50 210 300 1500 50 1 0.7 0.5

PC0.7-1T50 210 300 1500 50 2 0.7 1

PC0.7-2T50 210 300 1500 50 4 0.7 2

PC0.5-0T50 150 300 1500 50 0 0.5 0

PC0.5-0.5T50 150 300 1500 50 1 0.5 0.5

PC0.5-1T50 150 300 1500 50 2 0.5 1

PC0.5-2T50 150 300 1500 50 4 0.5 2

PC1-1T10 300 300 1500 10 2 1 1

PC1-1T25 300 300 1500 10 2 1 1

PC0.5-2T10 150 150 1500 10 4 0.5 2

PC0.5-2T25 150 150 1500 10 4 0.5 2

57  

Table 5.2 Results of parametric study

Column Specimen Maximum Axial Stress

Maximum Axial Strain

Maximum Transverse

Strain

Ductility Ratios

f′cc εau εtu

(MPa) (%) (%) Apost/Apeak Atot/Aep

PC1-0T50 20.9 0.28 0.11 - -

PC1-0.5T50 22.0 0.31 0.55 0.41 0.96

PC1-1T50 23.0 0.52 0.75 1.62 0.98

PC1-2T50 30.2 0.67 0.90 3.01 1.07

PC0.7-0T50 20.6 0.30 0.08 - -

PC0.7-0.5T50 21.9 0.42 0.47 0.98 0.92

PC0.7-1T50 23.4 0.55 0.69 1.00 0.96

PC0.7-2T50 31.4 0.70 0.77 3.47 1.13

PC0.5-0T50 20.4 0.24 0.05 - -

PC0.5-0.5T50 21.8 0.38 0.11 0.73 0.48

PC0.5-1T50 23.3 0.66 0.86 1.60 1.02

PC0.5-2T50 33.0 0.87 1.06 5.00 1.15

PC1-1T10 22.1 0.38 0.44 0.88 0.93

PC1-1T25 22.7 0.43 0.48 1.27 0.94

PC0.5-2T10 23.6 0.82 1.01 2.50 0.95

PC0.5-2T25 24.3 0.84 1.02 2.12 1.01

58  

5.3.1 Effect of Aspect Ratio (a/b) of the Column Cross Section

Three aspect ratios were considered: a/b=0.5, a/b=0.70, and a/b=1.0 to evaluate the influence

of different cross sectional aspect ratios on gain in compressive strength. Table 5.3 along with

the Figures 5.4(a), (b) and (c) show the effect of the aspect ratios on the stress versus strain

response. The results are presented by organizing the parametric columns in three sets (Set I,

II and III-I as shown in Table 5.3 (a)). Each set of columns have a specific combination of

corner radius and thickness of the FRP wrap. Within each set the shorter dimension (a) is

varied only.

5.3.1.1 Axial stress versus strain response

As observed from Figures 5.4 (a), (b) and (c), a significant increase in ultimate compressive

strength is anticipated with the decrease in aspect ratio from 1 to 0.5. However, the effect of

aspect ratio has negligible effect on initial stiffness of the column prior to peak axial stress.

As the aspect ratio decreases the capacity increases as observed in the stress strain response

of all three sets of columns (Figure 5.4 (a), (b) and (c)). The stiffness of the columns also

increases with the decrease in the aspect ratio after reaching the peak. This is due to the fact

that after concrete crushing only confinement effect of FRP wrapping exists. The maximum

axial strain and maximum transverse strain is significantly affected by the aspect ratio. From

the Figure 5.4 it is observed that both the axial and transverse stain increase with the decrease

in aspect ratio for all the sets.

59  

(R=50 mm and tf = 0.5 mm, L/a = variable)

(R=50 mm and tf = 1 mm, L/a = variable)

Figure 5.4 Effect of aspect ratio on the strain-strain responses of FRP wrapped columns (continued)

Transverse Strain (%)

(b) Set II columns

Transverse Strain (%)

(a) Set I columns

0

5

10

15

20

25

‐0.6 ‐0.4 ‐0.2 0 0.2 0.4 0.6

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

a/b=1

a/b=0.7

a/b=0.5

0

5

10

15

20

25

30

‐1 ‐0.8 ‐0.6 ‐0.4 ‐0.2 0 0.2 0.4 0.6 0.8

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

a/b=1

a/b=0.7

a/b=0.5

60  

(R=50 mm and tf = 2 mm, L/a = variable)

Figure 5.4 Effect of aspect ratio on the strain-strain responses of FRP wrapped columns

Transverse Strain (%)

(c) Set III-I columns

0

5

10

15

20

25

30

35

‐1.5 ‐1 ‐0.5 0 0.5 1

Axi

al S

tres

s (M

Pa)

Axial Strain (%)

a/b = 1

a/b=0.7

a/b=0.5

61  

Table 5.3(a) Effect of aspect ratio (a/b) with variable slenderness ratio (L/a)

Column Set

Column Designation

Aspect Ratio

Corner Radius

Thickness of FRP Wrap

Ultimate Axial Stress

Axial Strain

Transverse Strain

% increase in axial

stress with respect to

unconfined column

Slenderness Ratio

a/b R tf f′cc εau εtu L/a

(mm) (mm) (MPa) % %

Set I PC1-0.5T50 1 50 0.5 22.0 0.31 0.55 5 5

PC0.7-0.5T50 0.7 50 0.5 21.9 0.42 0.47 6 7

PC0.5-0.5T50 0.5 50 0.5 21.8 0.38 0.11 7 10

Set II PC1-1T50 1 50 1 23.0 0.52 0.75 10 5

PC0.7-1T50 0.7 50 1 23.4 0.55 0.69 14 7

PC0.5-1T50 0.5 50 1 23.3 0.66 0.86 14 10

Set III-I

PC1-2T50 1 50 2 30.2 0.67 0.90 45 5

PC0.7-2T50 0.7 50 2 31.4 0.70 0.77 52 7

PC0.5-2T50 0.5 50 2 33.0 0.87 1.06 62 10

Set III-II PC1-2T50 1 50 2 30.2 0.67 0.90 45 5

PC0.7-2T50 0.7 50 2 25.6 0.62 0.84 24 5

PC0.5-2T50 0.5 50 2 23.3 0.77 0.97 13 5

62  

5.3.1.2 Effect of aspect ratio on ultimate capacity

The specimens with a/b=0.50 confirmed in all cases the highest compressive strength as

shown in Figure 5.5. Small variation is observed between the maximum strength of the

square columns (a/b=1) and that of the corresponding rectangular columns with a/b=0.70.

For column set III-I, the columns having an aspect ratio of a/b= 0.5 showed considerable

increase in strength as compared to that with a/b=1. For column set I, there is no significant

effect of aspect ratio on gain in compressive strength. Figure 5.5 clearly explains that the

maximum effect of aspect ratio is achieved for column Set III-I having 4 layers of

confinement. An increase of 52%, 62% and 45% is found for the columns having a/b=0.7,

a/b=0.5 and a/b=1 respectively. The results show that the ultimate strength gain for the

columns with an aspect ratio of a/b=0.5 ranges between 7% and 62%. It can be seen that

rectangular short columns achieved higher ultimate strength than square columns.

Figure 5.5 Effect of aspect ratio (a/b)

But usually decrease in the aspect ratio should result in a decrease in the confinement.

However, the unexpected behavior with a/b = 0.5 occurred due the fact that the slenderness

ratio (L/a) of these specimens was variable. For a/b=0.5, slenderness ratio (L/a) was 10

which was higher than square columns having slenderness ratio 5.

Therefore, to identify the effect of aspect ratio another set (Set III-II) of analysis is conducted

with columns having a fixed slenderness ratio (L/a=5) with cross sections 300 x 450 mm and

300 x 600 mm. The details of the results and axial stress vs. axial strain responses of this set

R=50mm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2

f'cc/

f'co

a/b ratio

tf=0.5

tf=1

tf=2

63  

of columns are shown in Table 5.3(b) and Figure 5.6 respectively. The results shows that for

column set III-II, the columns having an aspect ratio of a/b= 0.5 showed considerable

decrease in strength as compared to that with a/b=1 for a fixed slenderness ratio.

Table 5.3(b) Effect of aspect ratio (a/b) with fixed slenderness ratio

Column Set

Column Designation

Aspect Ratio

Corner Radius

Thickness of FRP Wrap

Ultimate Axial Stress

Axial Strain

Transverse Strain

% increase in axial stress with

respect to unconfined column

Slenderness Ratio

a/b R tf f′cc εau εtu L/a

(mm) (mm) (MPa) % %

Set III-II PC1-2T50 1 50 2 30.2 0.67 0.90 45 5

PC0.7-2T50 0.7 50 2 25.6 0.62 0.84 24 5

PC0.5-2T50 0.5 50 2 23.3 0.77 0.97 13 5

(R=50 mm and tf = 2 mm, L/a = 5)

Figure 5.6 Effect of aspect ratio on the strain-strain responses of FRP wrapped columns with fixed slenderness ratio

Axi

al S

tres

s (M

Pa)

Axial Strain (%) Transverse Strain (%)

Set III-II columns

0

5

10

15

20

25

30

35

‐1.5 ‐1 ‐0.5 0 0.5 1

a/b=1

a/b=0.67

a/b=0.5

a/b=0.7

64  

5.3.2 Effect of Corner Radius (R)

Figure 5.6 (a) and (b) show the effect of corner radius (R) on the stress strain curve of the

confined columns. The effect of corner radius is studied for three different corner radii (10

mm, 25 mm and 50 mm). Again for each corner radius the thickness of FRP layer is varied

between 1 mm and 2 mm. Two sets (Set IV and Set V) of analysis have been conducted. The

geometric properties and the results of the analyses are presented in Table 5.4 and Figure 5.6.

5.3.2.1 Axial stress versus strain response

Figure 5.7(a) illustrates the stress-strain curve obtained for three concrete square models

confined with 2 layers of FRP laminates compared to an unconfined column. Each square

column had a different corner radius. The right portion of the curve shows the axial stress vs.

axial strain response whether the left side represents the response of transverse strain with the

axial stress. The Figure clearly shows that the confinement effectiveness increases gradually

with the rounding of the corners. In addition, the curves illustrate the three types of behavior

in the post-peak region of the load deflection response. For a corner radius R = 10 mm,

softening behavior is characterized by a rapid decrease of the curve after it reaches a

maximum. For R = 25 mm, less softening behavior is obtained. Finally, a stiffening behavior

is observed for sections with a small side-to-radius ratio, including the one with R = 50 mm.

The rectangular columns having aspect ratio 0.5 shown in Figure 5.7(b) also confirmed

similar behavior for different corner radius.

It can be seen that the post-peak drop of the curve is always followed by an increase in

stiffness. As the confinement effectiveness increases with the most important rounding of the

corner, the curves show that it has an actual effect only on the strains and ductility

characteristics of the sections.

65  

(a/b = 1 and tf = 1 mm)

(a/b = 0.5 and tf = 2 mm)

 

Figure 5.7 Effect of corner radius on confinement effectiveness

   

Axial Strain (%) A

xial

Str

ess (

Mpa

) Transverse Strain (%)

Transverse Strain (%)

Axi

al S

tres

s (M

pa)

(a) Set IV columns

(b) Set V columns

0

5

10

15

20

25

‐1 ‐0.8 ‐0.6 ‐0.4 ‐0.2 0 0.2 0.4 0.6

R=50 mmR=10 mmR=25mmunconfined

0

5

10

15

20

25

30

35

‐1.5 ‐1 ‐0.5 0 0.5 1 1.5Axial Strain (%)

UNCONFINEDR= 50 mmR=25 mmR=10mm

66  

5.3.2.2 Effect of corner radius (R) on ultimate capacity

The ultimate capacity of the columns of Set IV and Set V are shown in Table 5.4 along with

the percent increase in the capacity with respect to the unconfined column. For an increase in

corner radius (R) from 10 mm to 50 mm, the increase in ultimate capacity ranges from 6% to

62%. This increase in load carrying capacity is significant in rehabilitation practice. It is seen

that for column Set IV, rounding off the corner from 10 mm to 25 mm and 50 mm can

increase the capacity 3% and 8% respectively. However, for column Set V, the increase in

capacity is found 3% and 46% for rounding off the corner radius from 10 mm to 25 mm and

50 mm respectively. So it can be said that the corner radius plays a significant role in gaining

strength of the FRP confined columns especially for columns with lower aspect ratio and

higher thickness of the FRP wrap.

 

Table 5.4 Effect of corner radius (R)

Column Set

Column Designation

Aspect Ratio

Corner Radius

Thickness of FRP Wrap

Ultimate Axial Stress

Axial Strain

Transverse Strain

% increase in axial stress with respect to unconfined column

a/b R tf f′cc εau εtu

(mm) (mm) (MPa) % %

Set IV PC1-1T10 1 10 1 22.1 0.38 0.44 6

PC1-1T25 1 25 1 22.7 0.43 0.48 9

PC1-1T50 1 50 1 23.0 0.52 0.75 14

Set V PC0.5-2T10 0.5 10 2 23.6 0.82 1.01 16

PC0.5-2T25 0.5 25 2 24.3 0.84 1.02 19

PC0.5-2T50 0.5 50 2 33.0 0.87 1.06 62

 

 

 

67  

5.3.3 Effect of Thickness of FRP Wrap (tf)

The effect of the thickness of the FRP wrap on the behavior of the column models is shown

in Figure 5.8 (a) to (c) presenting the axial stress-axial strain responses and axial stress-

transverse strain responses. Three thicknesses were considered: tf =0.5 mm, tf = 1mm and tf

=2 mm to evaluate the influence of thickness on gain in compressive strength. The results are

presented by organizing the parametric columns in three sets (Set VI, VII and VIII as shown

in Table 5.5). Each set of columns have a specific combination of corner radius and aspect

ratio. Within each set the thickness of FRP is varied only.

5.3.3.1 Axial stress versus strain response

The Figure 5.8 gives the average axial stress versus the axial strain responses and axial stress

versus the transverse strain responses for the specimens with zero, one, two, three, and four

layers. It can be observed that the thickness of layers had little effect on the initial slope for

all curves. However, as the thickness increased, the inflection point moved up to a higher

stress level. The slope of the second branch of the stress-strain curves increased with the

number of CFRP layers, while the first branch was generally not affected. This behavior was

also observed for circular columns (Mirmiran and Shahawy, 1997; Picher et al. 1996 and

Nanni and Bradford, 1995). The Figure clearly shows that confinement with CFRP can

significantly enhance the performance of concrete, both its strength and its ductility, under

axial load.

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5

‐1

‐1

5.8 Effect of

‐0.8

‐0.8

Transve

FRP thickne

‐0.6 ‐0.

‐0.6 ‐

Transv

erse Strain (%

(a/b = 1 an

(a/b = 0.7 a

ess on the cocolumns

.4 ‐0.2

0.4 ‐0.2

verse Strain

(b)

(a

%)

Axi

al S

tres

s (M

pa)

nd R= 50 m

 

 

and R= 50 m

 

ompressive bs (continued

0

5

10

15

20

25

30

35

0 0

‐5

0

5

10

15

20

25

30

35

0

Axi

al S

tres

s (M

pa)

n %

) Set VII col

a) Set VI co

mm)

mm)

behavior of Fd)

.2 0.4

0.2

Axial Stra

lumns

olumns

Axial Strain

FRP confine

0.6 0

unconfinedtf=1tf =0.5tf=2

0.4 0

ain %

unconfitf=0.5tf =1tf=2

n (%)  

ed rectangula

0.8

0.6 0.8

fined

 

ar

69  

(a/b = 0.5 and R= 50 mm)

 

 

 

Figure 5.8 Effect of FRP thickness on the compressive behavior of FRP confined rectangular columns

 

 

 

Figure 5.9 Effect of layer thickness on confinement effectiveness.

(b) Set VIII columns

0

5

10

15

20

25

30

35

‐1.5 ‐1 ‐0.5 0 0.5 1

Axi

al S

tres

s (M

Pa)

Transverse Strain % Axial Strain %

unconfinedtf=0.5tf =1tf=2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.5 1 1.5 2 2.5

Nor

mal

ized

axi

al st

ress

(f

'cc/

f'co)

Thickness of CFRP (mm)

a/b=1

a/b=.7

a/b=0.5

R= 50 mm

70  

5.3.3.2 Effect of FRP thickness (tf) on ultimate capacity

Figure 5.9 shows that the thickness of FRP wrap has significant effect on ultimate capacity of

confined columns. The ultimate capacity of the columns of Set VI, Set VII and Set VIII are

shown in Table 5.5 along with the percent increase in the capacity with respect to the

unconfined column. The increase in ultimate capacity of confined columns ranges from 5% to

62%. However, the percent increase in capacity is not proportional to the jacket thickness.

For example, for Set VII, the increase in compressive strength when the number of layers

increased from two to four is 38%, while the increase in compressive strength when the

number of layers increased from one to two is 8%. For Set VI consisting square columns, the

increase in compressive strength when the number of layers increased from two to four is

35%, while the increase in compressive strength when the number of layers increased from

one to two is 5%. An increase of 48% in compressive strength is found when the number of

layers increased from 2 to four for Set VIII consisting rectangular columns having aspect

ratio 0.5.  

 

Table 5.5 Effect of the FRP thickness (tf)

Column Set

Column Designation

Aspect Ratio

Corner Radius

Thickness of FRP Wrap

Ultimate Axial Stress

Axial Strain

Transverse Strain

% increase in axial stress with respect to unconfined column

a/b R tf f′cc εau εtu

(mm) (mm) (MPa) % %

Set VI PC1-0.5T50 1 50 0.5 22.0 0.31 0.55 5

PC1-1T50 1 50 1 23.0 0.52 0.75 10

PC1-2T50 1 50 2 30.2 0.67 0.90 45

Set VII PC0.7-0.5T50 0.7 50 0.5 21.9 0.42 0.47 6

PC0.7-1T50 0.7 50 1 23.4 0.55 0.69 14

PC0.7-2T50 0.7 50 2 31.4 0.70 0.77 52

Set VIII PC0.5-0.5T50 0.5 50 0.5 21.8 0.38 0.11 7

PC0.5-1T50 0.5 50 1 23.3 0.66 0.86 14

PC0.5-2T50 0.5 50 2 33.0 0.87 1.06 62

71  

5.3.4 Proposed Equation

Simple polynomial equations are developed using a statistical software SPSS v.17 to predict

the confined concrete strength and ultimate axial strain of FRP-confined concrete rectangular

columns having different aspect ratios, corner radius and thickness of FRP laminates. The

trend line of the numerical data can be closely approximated using the following equations:

13.36 5.65 0.002 1.15 (5.1)

3292 2500 0.10 1625 (5.2)

Where, tf is the thickness of FRP layers in mm, R is the corner radius in mm and (a/b) is the

aspect ratio of rectangular columns. The numerical results of peak compressive strength and

ultimate axial strain and the predicted compressive strength and strain are presented in Table

5.6.

72  

Table 5.6 Comparison between numerical values and predicted values by equation

Column

designation f′cc

Predicted

ultimate axial

stress from

Equation 5.1

Error (∆ %) ε'cc

Predicted

ultimate axial

strain from

Equation 5.2

Error (∆ %)

(MPa) (MPa)

(µε) (µε)

PC1-0.5T50 22.0 22.3 1.3 3100 3167 2.1

PC1-1T50 23.0 25.2 8.7 5200 4417 17.7

PC1-2T50 30.2 30.8 1.9 6700 6917 3.1

PC0.7-0.5T50 21.9 21.6 1.4 4200 4235 0.8

PC0.7-1T50 23.4 24.4 4.1 5500 5485 0.3

PC0.7-2T50 31.4 30.1 4.3 7000 7985 12.3

PC0.5-0.5T50 21.8 21.3 2.3 3800 4589 17.2

PC0.5-1T50 23.3 24.2 3.7 6600 5839 13.0

PC0.5-2T50 33.0 29.8 10.7 8700 8339 4.3

PC1-1T10 22.1 20.4 8.3 3800 4177 9.0

PC1-1T25 22.7 21.4 6.1 4300 4230 1.7

PC0.5-2T10 23.6 25.0 5.6 8200 8099 1.2

PC0.5-2T25 24.3 26.1 6.9 8400 8151 3.0

Figures 5.10 and 5.11 are the plots of the predicted values vs. numerical values of peak axial

stress and ultimate axial strain. The trend line of this figure shows a very good correlation

between the predicted and numerical values.

73  

Figure 5.10 Predicted values vs. numerical values of ultimate axial stress

 

 

 

Figure 5.11 Predicted values vs. numerical values of ultimate axial strain

 

y = 0.8278x + 4.257R² = 0.8446

15

20

25

30

35

15 20 25 30 35

Pred

icte

d ax

ial s

tres

s by

Equ

atio

n (M

Pa)

Axial stress measured from numerical model (MPa)

y = 0.928x + 421.9R² = 0.928

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 2000 4000 6000 8000 10000

Pred

icte

d ax

ial s

trai

n fr

om E

quat

ion

(µε)

Axial strain measured from numerical model (µε)

5.3.5 Eva

Ductility

strain. Th

element

areas und

before th

differenc

material.

as follow

Figure

A

pr

re

A

st

an

th

ad

el

aluation of D

y can be cha

his surface p

under the ap

der the stress

he initial str

ce of behavi

These two r

ws:

e 5.12 Area u

Apost/Apeak, wh

rovides info

eached.

Atot/Aep, wher

tress-strain c

nd the plasti

he shaded ar

dding the s

lement show

Ductility

aracterized b

provides info

pplied loadi

s-strain curv

ress peak to

ior between

ratios are ca

used to calcu

here parame

rmation on t

re paramete

curve and th

ic plateau. I

reas (Apost an

haded and

ws an almost

by the area

ormation ab

ing. Two fo

ve, are presen

o the area a

the actual

lculated usin

ulate ductility

eters Apost an

the strain res

ers Atot and

he total area

n Figure 5.1

nd Apeak) an

gray areas.

elastic-perfe

under the c

out strain en

ormulations,

nted. The fir

after this po

material an

ng areas illu

y ratios.( Ro

nd Apeak are d

serve still av

Aep are, res

bounded by

12, the first

nd the secon

When this

fectly plastic

curve of the

nergy accum

based on d

rst compares

oint; the oth

nd that of an

strated in Fi

ochette, P. an

defined in th

vailable after

spectively, th

y a slope of

parameter c

nd parameter

ratio is clo

behavior.

axial stress

mulated in th

ifferent ratio

s the area be

her ratio is

n elastic-per

gure 5.12 an

nd Labossie

he Figure 5.

r the initial p

he area und

f constant el

corresponds

r is the tota

ose to 1.0,

s versus axi

he structure o

os of specif

low the curv

based on th

rfectly plast

nd are define

`re, P.2000)

12. This rati

peak has bee

der the actu

astic stiffnes

to the sum o

l obtained b

the structur

al

or

fic

ve

he

tic

ed

io

en

al

ss

of

by

ral

75  

For heavily confined specimens, it is difficult to identify the value of the peak stress level

shown in Figure 5.12. The various portions of the stress-strain curve are thus determined as

follows. The longitudinal strain at failure of an unconfined specimen is first identified; its

value usually corresponds to the strain level in the concrete when the axial stress reaches f′c.

The value of fpeak for a confined specimen is then defined as that reached when the strain εco

is attained. The value of fpeak delimits the pre peak and post peak surfaces under the stress-

strain curve.

For all of the confined column models, the values of the two ratios defined above are given in

the right-hand side of Table 5.2. The value of Apost/Apeak is particularly useful to identify the

specimens that reached the most important axial deformations, even in the case when plastic

deformations continued to increase at a stress level lower than f′c.

The ratio, Atot/Aep, allows identification of the specimens exhibiting a behavior similar to that

of a strain hardening material. Only the highly confined specimens present a value above 1.0

for this ratio. Considering the definition of ductility, which assumes that plastic deformations

must occur without any substantial loss of strength, this ratio should be more appropriate to

qualify the ductility of the specimens with square or rectangular sections confined with

composite materials. The specimens with a value of Atot/Aep of 1.0, or higher, can be

considered ductile. Although the rectangular specimens with corners rounded to 25 mm with

four layers of CFRP wrapping showed improved ductility properties, the radius has to be

increased to 50 mm for 2 layers of CFRP wrapping for same cross section. This clearly

indicates that the stiffness of the confining material is of major importance in improving the

ductility of the structural element.

For confined columns to achieve improved behavior under axial compression, it is suggested

that the confinement characteristics should be selected in such a way that the ratio Atot/Aep is

always greater than 1.0. In addition, the ratio Apost/Apeak should also exceed a value of

Apost/Apeak >3.0 (Rochette, P. and Labossie`re, P., 2000). If the same rule is applied to the

carbon-wrapped rectangular and square columns modeled here, it would appear that only

configurations with corners rounded to 50 mm within at least four plies having thickness

2mm provide satisfy the ductility requirements.

76  

5.4 Summary

A comprehensive parametric analysis was performed to study the behavior of FRP confined

concrete columns subjected to axial compression. Three geometric parameters were varied

and their influences were demonstrated with respect to the ultimate axial stress and over all

column stress-strain responses. The important findings of the study presented in this chapter

are summarized below.

The parameters controlling the geometric confinement efficiency are the cross-section

aspect ratio (a/b), the corner radius (R), and the thickness of the FRP wrap (tf).

The confinement provided by the CFRP improves both the load-carrying capacity and

the ductility of the rectangular columns.

The strength of the confined rectangular columns increases with the decrease in the

aspect ratio or the increase in the rectangularity of the cross section. On the other hand

considering only enhancement of load carrying capability, the capacity of confined

rectangular columns increases with the decrease in rectangularity. So, further study is

required with a wide range of aspect ratio to take any concrete decision about effect of

aspect ratio.

The thickness of FRP wrap has significant effect on ultimate capacity of confined

columns. The increase in ultimate capacity of confined columns ranges from 5% to

62%. The maximum increase in axial strength is found 62% for confined column

having aspect ratio 0.5 and 2 mm of FRP wrapping.

The axial capacity of confined concrete increases with the increases in corner radius

of the rectangular columns. For an increase in corner radius (R) from 10 mm to 50

mm, the enhancement in ultimate capacity can be achieved 6% to 62%.

77  

 

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 General

Nonlinear 3D finite element models have been developed using ABAQUS finite element

software to investigate the compressive behavior of FRP confined concrete rectangular

columns. A static, Riks solution strategy was implemented in the numerical model to trace a

stable post peak response in the load-deformation curve. The concrete material in the

confined column was modeled using the damage plasticity model available in ABAQUS. The

FRP-concrete interface was simulated using two contact pair algorithms: perfect bond model

and cohesion model. To investigate the performance of this FEM model, simulations were

conducted for FRP confined concrete column test, reported in the literature. The confined

rectangular columns with normal strength concrete were varied in cross-sectional size from

152 mm x 152 mm to 108 mm x 165 mm, including a variety of confinement stiffness and

corner radius. The loading condition was limited to only concentric loading. The model was

found able to predict the peak load and post-peak behavior quite accurately. A parametric study was conducted using the numerical model to investigate the effect of a

full range of parameters on concentrically loaded FRP confined concrete column. The

parameters that were varied include the aspect ratio (a/b), corner radius (R) and thickness of

FRP wrap (tf). The cross sections were selected 150 mm x 300 mm, 210 mm x 300 mm and

300 mm x 300 mm. The length was selected 1500 mm. The effects of the selected parameters

on the behavior of FRP confined concrete columns were studied with respect to axial stress

versus strain curve. An equation on the basis of this model was proposed to identify the peak

axial stress. The conclusions of the parametric study and the performance of the finite

element model are listed below.

6.2 Conclusions

6.2.1 Performance of the Finite Element Model

In general the finite element models for FRP confined concrete columns developed in this

study were able to simulate the full behavioral histories of a variety of confined rectangular

and circular columns with a very good accuracy. The interaction between FRP-concrete

interfaces was successfully modeled using contact pair algorithm with perfect bond

78  

interaction at the FRP concrete interfaces. The numerical model also provided good

representations of ultimate axial stress and ultimate axial strain. The average experimental-to-

numerical ratio of the ultimate axial stress was obtained 1.01 with a standard deviation of

0.03. The numerically predicted ultimate axial strains are found to be higher compared to the

experimental values with an average experimental-to-numerical ratio of 0.96 with a standard

deviation of 0.09. These values demonstrate a very good correlation between the

experimental and numerical results.

6.2.2 Parametric Study

The parametric study was conducted using the validated finite element to investigate the

effects of the aspect ratio, corner radius and the thickness of the FRP wraps on rectangular

concrete columns. The following conclusions can be drawn from the parametric study.

6.2.2.1 Effect of aspect ratio (a/b)

The aspect ratio of the cross section had a significant influence on the increase in

compressive strength. The aspect ratio was varied from 0.5 to 1 in this study. It was found

that the confined-to-unconfined concrete strength ratio changes considerably with the

variation of the aspect ratio in the range of 0.5 to 1. The gain in confined compressive

strength ranges from 5% to 62%. The results show that decrease in aspect ratio causes a

reduction in the ultimate capacity for a fixed slenderness ratio.

6.2.2.2 Effect of corner radius (R)

The corner radius of rectangular columns influences significantly the strength and the

ductility of the columns. Rounding off the corners from 10 mm to 25 mm and 50 mm can

significantly increase the axial stress. The gain in confined compressive strength of column

due to the increase in the corner radius enhances the ultimate capacity from 6% to 62% with

respect to unconfined column. It should be noted that increasing the corner radius is not

always feasible due to the existence of reinforcing steel bars at the corners of rectangular

columns. Therefore, a corner radius of 25.4 mm (1 in.) is suitable in practical cases to ensure

an effective confinement.

6.2.2.3 Effect of FRP thickness (tf)

It is evident that in all cases the presence of external CFRP jackets increased the mechanical

properties of concrete columns in different amount according to the number of composite

layers. Increasing the amount of CFRP sheets can significantly increase the compressive

79  

strength of the confined column. But it is not proportional to the jacket thickness. For square

columns, an increase in compressive strength was found 5% and 35% when the thickness

increased from 0.5 mm to 1 mm and 1 mm to 2 mm respectively. On the other hand for

rectangular columns having aspect ratio 0.5 the increase in compressive strength was found

7% and 48% for increasing the thickness from 0.5 mm to 1 mm and 1 mm to 2 mm

respectively.

6.3 Recommendations for Future Research

In this study the finite element model developed herein was verified for concentric loading

only. However, eccentric loading may cause buckling or bending in a short column.

Therefore, in future research eccentric loading should be incorporated in the finite element

model.

This study only considers normal strength concrete. A limited number of tests have been

performed to date on high strength concrete. To include the effect of high strength concrete

on confined concrete stress large-scale experimental investigations are required. Also, the

effect of lateral reinforcement is not included in this study

The concrete columns were assumed to be in its original state before applying FRP

confinement. But in real case, damages are present. Therefore, in future research the

deteriorated state of the concrete columns should be integrated in the numerical model.

The aspect ratios considered in this study were 0.5, 0.7 and 1. For better understanding the

effect of aspect ratios, more research should be carried out including a wide range of aspect

ratios, dimension effect and the effect of shape factor. Further work is required to expand the

current work and integrate it with the effects of the FRP jacket stiffness on the strength and

ductility of the FRP confined concrete columns.

80  

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iii  

 

iv  

DECLARATION

Except for the contents where specific references have been made to the work of others, the

studies embodied in this dissertation are the outcome of the research conducted by the author.

It is here by declared that, this thesis or any part of it has not been submitted elsewhere for

award of any degree or diploma or other qualification (except for publication).

Sabreena Nasrin

v  

ACKNOWLEDGEMENT

“Praise be to Allah, The Cherisher and Sustainer of the world.”

The author would like to sincerely thank the Almighty, the most Gracious, and the most

Merciful. The author prays to Almighty Allah for being in good health and condition, for the

successful completion of the study.

The author expresses, with due respect her deepest gratitude to her supervisor Dr. Mahbuba

Begum, Associate Professor, Department of Civil Engineering, BUET, whose guidance and

valuable directives have made the research project possible. Her guidance on research

methods, incredible editing skills and encouragement in all stages of this research work has

made this task a significant degree less difficult than it could have been. Her valuable

comments and insights helped to improve my work enormously. The author’s thanks to her

are endless.

The author would like to show appreciation to the Head, Department of Civil Engineering,

BUET for providing all the computation facilities to materialize this work.

The author would like to thank her family, specially her parents for their undying support,

continuous inspiration and kind co-operation.

The author acknowledges with gratitude the help granted by her husband Mr. Nurullah Bin

Humayun who enriched this research work by providing his precious opinions. His

unwavering support in all of her endeavors continues to provide her with the confidence that

she can complete the tasks ahead.

Finally, the author admits the priceless supports of her friends and colleagues specially Ms.

Zasiah Tafheem, Mr. Nazmus Sakib and Mr. A.K.M Abir.

vi  

ABSTRACT

The use of fibre-reinforced polymer (FRP) is an efficient and technically sound method for

strengthening the damaged structures or upgrading the inadequately designed members or

retrofitting of seismically damaged reinforced concrete structures. Although there is a large

amount of experimental data available on the compressive behavior of fibre-reinforced

polymer (FRP) confined concrete columns, a full understanding of the behavior of FRP

confined rectangular concrete columns is somewhat lacking. This study aims to generate a 3D

finite element model for FRP confined concrete columns under axial loading to overcome the

deficiencies in the available experimental database.

The nonlinear finite element analysis on FRP confined plain concrete was performed using

ABAQUS/Standard (HKS 2009) finite element code. Both material and geometric

nonlinearities were included in the model. A damage plasticity model was used to simulate

the behavior of confined concrete. The interface between FRP and concrete was simulated

using contact pair algorithm. Two different types of formulation: cohesion based surface

interaction and friction type perfect bond interactions were defined at the FRP-concrete

interface. A static Riks formulation was implemented to trace the stable load-displacement

history of FRP confined concrete up to failure. The load was applied through displacement

control technique. The numerical model was successfully applied to simulate the behavior of

eleven columns from three experimental programs including square, rectangular and circular

columns. The model reliably reproduced the peak axial stress, axial deformation at the peak

stress, the post-peak behavior and the failure mode observed in the tests.

A parametric study was conducted to investigate the influence of several geometric

parameters such as aspect ratio, corner radius and the thickness of the FRP wrap on strength

and ductility of FRP confined rectangular columns. The maximum effect of confinement was

achieved for square columns. Decreasing the aspect ratio from 1 to 0.7 and 0.5 reduces the

ultimate capacity of the confined column by 20% and 30% respectively with respect to the

square column. Moreover, the axial capacity and ductility of the rectangular columns were

found to increase significantly with the increase in corner radius and thickness of the FRP

laminates. Finally, a simple form of polynomial equation was proposed to predict the

confined compressive strength and the ultimate axial strain of concrete.

vii  

TABLE OF CONTENTS

DECLARATION IV

ACKNOWLEDGEMENT V

ABSTRACT VI

TABLE OF CONTENTS VII-IX

LIST OF FIGURES X-XI

LIST OF TABLES XII

NOTATIONS XIII-XIV

CHAPTER 1

INTRODUCTION

1.1. General 1

1.2 Objectives of the Study 2

1.3 Scope 2

1.4 Organization of the Study 3

CHAPTER 2

LITERATURE REVIEW

2.1 General 4

2.2 Fiber-Reinforced Polymers 5

2.3 Properties and Behavior of FRP 6

2.3.1 Tensile Behavior 6

2.3.2 Compressive Behavior 7

2.4 Applications of FRP in Structural Rehabilitation 8

2.4.1 Beam Strengthening with FRP Laminates 8

2.4.2 Column Strengthening 10

2.4.2.1 Experimental investigations 10

2.4.2.2 Numerical investigations 12

2.5. Behavior of FRP Confined Concrete Columns 12

2.5.1 Circular Columns 13

2.5.2 Rectangular Columns 14

2.6 Failure Mechanism 15

2.7 Design Guidelines 16

2.7.1 CSA-S806-022 (2002) 17

viii  

2.7.2 ISIS Canada (2001) 18

2.7.3 FIB Guidelines (2001) 19

CHAPTER 3

FINITE ELEMENT MODELING 20

3.1 General 20

3.2 Properties of Reference Test Specimens 21

3.2.1 Geometric and Material Properties of Square Columns 21

3.2.2 Geometric and Material Properties of Rectangular Columns 21

3.2.3 Geometric and Material Properties of Circular Columns 22 3.3 Characteristics of the Finite Element Model 26

3.3.1 Geometric Properties and Finite Element Models 26

3.3.1.1 Element selection 26

3.3.1.2 Mesh description 28

3.3.1.3 Modeling of concrete-FRP interface 28

3.3.1.4. Load application & boundary condition 31

3.3.2 Material Properties 32

3.3.2.1 Concrete 32

3.3.2.2 FRP laminate 36

3.3.3 Solution Strategy 37

CHAPTER 4

PERFORMANCE OF FINITE ELEMENT MODELS

4.1 General 40

4.2 Performance of FEM Models 40

4.2.1 Ultimate Capacity and Strain 40

4.2.2 Axial Stress versus Axial Strain Response 42

4.2.2.1 Rectangular columns 42

4.2.2.2 Square columns 45

4.2.2.3 Circular columns 48

4.3 Summary 50

ix  

CHAPTER 5

PARAMETRIC STUDY

5.1 General 51

5.2 Design of Parametric Study 51

5.2.1 Variable Parameters 52

5.2.1.1 Aspect ratio (a/b) of the column cross section 52

5.2.1.2 Corner radius(R) 52

5.2.1.3 Thickness of FRP wrap (tf) 53

5.2.2 Fixed Parameters 53

5.3 Results and Discussion 54

5.3.1 Effect of Aspect Ratio (a/b) of the Column Cross Section 58

5.3.1.1 Axial stress versus strain response 58

5.3.1.2 Effect of aspect ratio on ultimate capacity 61

5.3.2 Effect of Corner Radius (R) 64

5.3.2.1 Axial stress versus strain response 64

5.3.2.2 Effect of corner radius (R) on ultimate capacity 66

5.3.3 Effect of Thickness of FRP Wrap (tf) 67

5.3.3.1 Axial stress versus strain response 67

5.3.3.2 Effect of FRP thickness (tf) on ultimate capacity 70

5.3.4 Proposed Equation 71

5.3.5 Evaluation of Ductility 74

5.4 Summary 76

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 General 77

6.2 Conclusions 77

6.2.1 Performance of the Finite Element Model 77

6.2.2 Parametric Study 78

6.2.2.1 Effect of aspect ratio (a/b) 78

6.2.2.2Effect of corner radius (R) 78

6.2.2.3 Effect of FRP thickness (tf) 78

6.3 Recommendations for Future Research 79

REFERENCES 80

x  

LIST OF FIGURES

Figure 2.1 FRP products for structural rehabilitation, (a) FRP strips and (b) FRP sheets

(Rizkalla et al. 2003). 5 

Figure 2.2 The typical tensile strengths, and stress-strain relationship of FRP and steel

reinforcements (https://www.build-on-prince.com) 7 

Figure 2.3 Applications of FRP for column retrofitting (Obaidat, et al., 2010) 12 

Figure 2.4 Confinement action of FRP composite in circular sections (Benzaid and Mesbah,

2013) 13 

Figure 2.5 Confinement action of FRP composite in square sections (Benzaid and Mesbah,

2013) 15 

Figure 2.6 Typical failed specimens (a) circular (Saafi et al., 1999) and (b) rectangular

(Chaallal et al., 2003) 16 

Figure 3.1 Geometric properties of square, rectangular and circular columns 23 

Figure 3.2 (a) 3-D view of the column mesh and (b) Cross section (with CFRP laminate) 26 

Figure 3.3 Finite elements used in the model, (a) 8-node solid (b) 8- shell element and 27 

(c) Axisymmetric solid element 27 

Figure 3.4 Bilinear traction separation constitutive law 29 

Figure 3.5 Axial stress vs. axial strain responses of column S25C5 (a) using cohesive zone

model (b) using perfect bond model. 31 

Figure 3.6 Load application and boundary condition 31 

Figure 3.7 Response of concrete to uniaxial loading in (a) tension and (b)compression. 34 

Figure 3.8 Stress-strain relationship curve of concrete for compression hardening 35 

(a) stress versus total strain (b) stress versus plastic strain 35 

Figure 3.9 Stress-strain relationship curve of concrete for tension stiffening 36 

(a)stress versus total strain (b) stress versus inelastic strain 36 

Figure.3.10 Typical Elastic stress-strain curve of CFRP 37 

Figure 3.11 Riks method. 38 

Figure 4.1 Numerical and experimental axial stress versus axial strain response for column

SC-1L3-0.7 43 

Figure 4.2 Numerical and experimental axial stress versus axial strain response for column

SC-2L3-0.7 43 

Figure 4.3 Numerical and experimental axial stress versus axial strain response for column

SC-3L3-0.7 44 

xi  

Figure 4.4 Numerical and experimental axial stress versus axial strain response for column

SC-4L3-0.7 44 

Figure 4.5 Numerical and experimental axial stress versus axial strain response for column

S5C5. 45 

Figure 4.6 Numerical and experimental axial stress versus axial strain response for column

S25C3. 46 

Figure 4.7 Numerical and experimental axial stress versus axial strain response for column

S25C4 46 

Figure 4.8 Numerical and experimental axial stress versus axial strain response for column

S25C5 47 

Figure 4.9 Numerical and experimental axial stress versus axial strain response for column

S38C3 47 

Figure 4.10 Numerical and experimental axial stress versus axial strain response for column

C1 49

Figure 4.11 Numerical and experimental axial stress versus axial strain response for column

C2 49 

Figure 5.1 Cross sections of rectangular and square columns used in the parametric study 52 

Figure 5.2 Variation of corner radius for square columns 53 

Figure 5.3 Response of concrete to uniaxial loading in (a) compression and(b) tension 54 

Figure 5.4 Effect of aspect ratio on the strain-strain responses of FRP wrapped columns 59 

Figure 5.4 Effect of aspect ratio on the strain-strain responses of FRP wrapped columns 60 

Figure 5.5 Effect of aspect ratio (a/b) 62 

Figure 5.6 Effect of aspect ratio on the strain-strain responses of FRP wrapped columns with

fixed slenderness ratio 63 

Figure 5.7 Effect of corner radius on confinement effectiveness 65 

Figure 5.8 Effect of FRP thickness on the compressive behavior of FRP confined rectangular

columns (continued) 68 

Figure 5.8 Effect of FRP thickness on the compressive behavior of FRP confined rectangular

columns 69

Figure 5.9 Effect of layer thickness on confinement effectiveness 69

Figure 5.10 Predicted values vs. numerical values of ultimate axial stress 73 

Figure 5.11 Predicted values vs. numerical values of ultimate axial strain 73 

Figure 5.12 Area used to calculate ductility ratios.( Rochette, P. and Labossie`re, P.2000) 74 

xii  

LIST OF TABLES

Table 2.1 The tensile properties of some of the commercially available FRP systems 6

Table 3.1 Geometric properties of square and rectangular columns 24

Table 3.2 Material properties of square and rectangular columns 24

Table 3.3 Geometric properties of circular columns 25

Table 3.4 Material properties of circular columns 25

Table 4.1 Performance of numerical models 41

Table 5.1 Geometric properties of parametric columns 56

Table 5.2 Results of parametric study 57

Table 5.3(a) Effect of aspect ratio (a/b) with variable slenderness ratio (L/a) 61

Table 5.3(b) Effect of aspect ratio (a/b) with fixed slenderness ratio 63 

Table 5.4 Effect of corner radius (R) 66

Table 5.5 Effect of the FRP thickness (tf) 70

Table 5.6 Comparison between numerical values and predicted values by equation 72

xiii  

NOTATIONS

a Short Dimension of column cross section, mm

b Long Dimension of column cross section, mm

H Height of column cross section, mm

d Diameter of concrete cylinder, mm

R Corner radius of concrete section

Ec Modulus of elasticity of concrete, GPa

Efrp Modulus of elasticity of FRP, GPa

εnom Nominal strain, mm/mm

fc' Specified compressive stress of concrete, MPa

ffrp Tensile strength of FRP, MPa

fl Lateral confining pressure, MPa

ρfrp Volumetric ratio of FRP

ψf Reduction factor

φ Strength reduction factor

Prmax capacity of a confined column, MPa

kc Confinement coefficient for rectangular columns

tf Thickness of the FRP jacket, mm

αpr Performance coefficient

Ww Volumetric strength ratio

α1 Ratio of average compression stress to the concrete strength

Effective strain in concrete, mm/mm

εfu Ultimate strain in FRP, mm/mm

ke Resistance Factor

φc Resistance factors for concrete

φs Resistance factors for steel εju Effective ultimate circumferential strain of the FRP jacket

ke Effectiveness coefficient

ti Resin thickness

tc Concrete thickness

Gi Shear modulus of resin

Gc Shear modulus of concrete

K0 Initial stiffness of interface

xiv  

τmax Maximum shear stress, MPa

fct Concrete tensile strength, MPa

bc Concrete width, mm

bf CFRP plate width, mm

δf Opening displacement at fracture, mm

Gcr Energy needed for opening crack

σn Cohesive tensile

τs Shear stresses of the interface, MPa

τt Shear stresses of the interface, MPa

η Material parameter

εc~in Inelastic strain, mm/mm

εc~pl Plastic strain, mm/mm

dc Damage parameter

Gn Work done by the traction in the normal direction

Gs Work done by the traction in the first shear direction

Gt Work done by the traction in the second shear direction

ρ Transverse reinforcement ratio

f‘cc Compressive stress of confined concrete, MPa