NUMERICAL AND ANALYTICAL MODELING OF CONCRETE …

The Pennsylvania State University

The Graduate School

Civil & Environmental Engineering

NUMERICAL AND ANALYTICAL MODELING OF CONCRETE

CONFINED WITH FRP WRAPS.

A Thesis in

Civil Engineering

by

Omkar Pravin Tipnis

2015 Omkar Pravin Tipnis

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

August 2015

ii

The thesis of Omkar P. Tipnis will be reviewed and approved* by the following:

Maria Lopez De Murphy

Associate Professor of Civil Engineering

Thesis Adviser

Ali Memari

Professor of Civil Engineering

Gordon P. Warn

Associate Professor of Civil Engineering

Peggy Johnson

Professor of Civil Engineering

Department head, Civil Engineering

iii

Abstract

This thesis is intended at studying and comparing empirical models that have been proposed for

the modeling of the stress-strain response of a FRP confined concrete subjected to axial load. An attempt

has been made to model the experimental set up for the compression test of a concrete cylinder confined

with FRP sheet in AbaqusCAE. The results so obtained have been compared and analyzed against the

experimental test results & the results obtained from a chosen mathematical model (Modified Lam &

Teng). An attempt was made to create a new material model in Opensees that follows the chosen

mathematical model. However, this was not achieved due to the reasons that will be explained in the later

sections.

Reinforced concrete confined with steel is typically designed by considering the Manders model

(Mander et al., 1988), which assumes a constant confining pressure. This is true with the case of steel as it

is a ductile material and one assumes the steel to be yielded. However with the case of FRP jackets, this is

not true. FRP is a linear elastic and brittle material and does not yield, which makes the Manders model

inaccurate for its analysis. Many models have been proposed which take into account the increasing

confining pressure due to the FRP wrap. A comparative study of the constitutive models proposed for

FRP confined reinforced concrete has been done in this study.

Finally after a series of numerical interpretations of different specimens and their comparison

with the experimental data, the utility and accuracy of the new modified Lam & Teng’s model was

validated. The validation process included comparison and analytical data obtained via finite element

simulation in Abaqus, empirical model results and the experimental data.

Contents

List of Figures v

Chapter 1 Introduction 1

1.1 Introduction 1

1.2 Scope of the study 2

Chapter 2 Literature Review 4

2.1 Introduction 4

2.2 Mechanism for Concrete Confinement by Transverse Reinforcement 4

2.3 Modeling of Concrete in Compression 6

2.3.1 Modified Hognestad Model: 6

2.3.2 Kent and Park model 7

2.4 Stress-Strain Response of FRP-Confined Concrete 9

2.4.1 First Zone 9

2.4.2 Transition Point 10

2.4.3 Second Zone 10

2.4.4 Failure Mode 10

2.4.5 Post Failure 12

2.5 Proposed models 15

2.5.1 Samaan and Mirmiran Model (1998) 16

2.5.1 Mander’s Model (1984) 18

2.5.3 Lam & Teng Model (2003) 24

2.5.4 Modified Lam & Teng (Liu et al., 2013) 26

2.5.5 Drucker-Prager Plasticity Model 30

2.6 Conclusions 34

Chapter 3 Experimental Database & Preliminary results. 35

3.1 Introduction 35

3.2 Preliminary study 39

3.3 Test Database 39

3.4 Abaqus modeling 40

3.5 Opensees Modeling 41

3.6 Results and Discussion 41

iv

Chapter 4 AbaqusCAE Finite Element Modelling 45

4.1 Introduction 45

4.2 Concrete 45

4.2.1. Elastic Properties 46

4.2.2. Plastic Properties 47

4.3 Fiber Reinforced Polymeric Jacket 48

4.4 Abaqus model: 49

4.4.1. Assembly 49

4.4.2. Boundary Conditions & Analysis Step 49

4.4.3. Interaction 49

4.4.4. Meshing 49

4.5 Conclusions 50

Chapter 5 Comparison Study and Analysis of Results 52

5.1 Introduction 52

5.2 Performance of the Drucker-Prager Model 54

5.3 Performance of the Modified Lam & Teng model 57

5.4 Regression Analysis 61

5.3 Field Retrofitting cases 66

Chapter 6 Conclusions 68

6.1 Summary 68

6.2 Conclusions 69

References 71

Appendix A : Numerical and Analytical modeling results (Tabular) 76

Appendix B : Numerical and Analytical modeling results (Graphical) 79

Appendix C : Graphical comparison with Confinement ratios 106

Appendix D-1 : Bridge Retrofitted Data 107

Appendix D-2 : CalTrans retrofitting guidelines 109

v

List of Figures

Figure 2-1 : Confinement by Square hoops and Circular Spirals. 4

Figure 2-2 : Confinement of concrete by spiral reinforcement. (Park & Paulay,1875) 5

Figure 2-3 : The modified Hognestad model for compressive stress-strain curve for concrete (MacGregor, 2012) 6

Figure 2-4 : Proposed stress-strain model for confined & unconfined concrete – Kent & Park (1971) model 8

Figure 2-5 : Typical Stress-Strain behavior of FRP confined concrete (Saafi & Toutanji, 1999) 9

Figure 2-6 : Failure modes of circular FRP-confined concrete cylinders (Micellli & Modarelli, 2012) 11

Figure 2-7 : Failure modes of sqauare FRP-confined concrete cylinders (Micellli & Modarelli, 2012) 11

Figure 2-8 : Tensile Coupon testing of CFRP, GFRP & Steel (Benziad et al (2010)) 12

Figure 2-9 : : Karabinis et al (2002). A2 & B2 are unconfined control specimens. 13

Figure 2-10 : Failure of concrete core ( Li et al (2006)) 14

Figure 2-11 : Parameters of Bilinear Confinement Model (Samaan & Mirmiran, 1998) 16

Figure 2-12 : Arching effect in Concrete. (Mander et al (1989)) 19

Figure 2-13 : Possible arching effect in non-uniformly confined concrete by FRP (Saadatmanesh et al (1994)) 20

Figure 2-14 : Free body diagram of FRP confinement (DeLorenzis et al (2003)) 21

Figure 2-15 : Lam & Teng’s stress-strain model for FRP-confined concrete. (Lam & Teng, 2003) 24

Figure 2-16 : Envelope curve and hysterical rule of proposed FRP confined concrete. (Lui et al, 2013) 26

Figure 2-17 : Friction model (J-F. Jiang et al (2012)) 31

Figure 2-18 : Existing models for plastic dilation rate. (J-F. Jiang et al (2012)) 32

Figure 2-19 : Typical Dilation curve (J-F. Jiang et al (2012)) 33

Figure 4-1 : Creating a Concrete cylinder 3D extrude Part in Abaqus 46

Figure 4-2 : FRP jacket created in Abaqus 48

Figure 5-1 : Typical curves for confinement ratio 5-10 53






vi

Figure 5-7 : Typical curves for confinement ratio 85 54

Figure 5-8 : Typical curves for confinement ratio 113 54

Figure 5-9 : Comparitive stress-strain curve for ID no 28 55

Figure 5-10 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = 3.9) 57

Figure 5-11 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = 135.84) 58

Figure 5-12 : Ultimate Strength comparison (Modified Lam & Teng vs Experimental) 59

Figure 5-13 : Ultimate Strain comparison (Modified Lam & Teng vs Experimental) 60

Figure 5-14 : Summary Report for Multivariable regression analysis 63

Figure 5-15 : Final Model performance report 64

Figure 5-16 : Ultimate Strain Comparison (New proposed equation vs Experimental) 65

Figure 5-17 : Histogram for retrofitted bridge columns based on confinement ratios. 67

Figure App-C-1 : Plot showing the variation of the ratio of the compressive strengths predicted to the experimental

compressive strengths with respect to the confinement ratio 106

1

Chapter 1 Introduction

1.1 Introduction

Today, many reinforced concrete structures are in a bad condition. According to the ASCE report

card 2013 for America’s infrastructure, one in nine of the bridges in the United States is structurally

deficient. (2013 Report card for America’s Infrastructure,ASCE) The report also mentions that the

average age of the bridges in the country is 42 years. Most of them need some rehabilitation and repair

work to either restore them to their full capacity or to increase their design capacity in order to meet their

growing demand.

Causes of deterioration can range from corrosive environmental conditions, damage due to

natural cause such as earthquakes & tornadoes or by human factors such as traffic accidents, use of sub-

standard quality of construction material, faulty construction practices or increase in the load demand for

the structure.

Indication of a deteriorated reinforced concrete column is the spalling action of the concrete cover

leading to exposure of the steel reinforcement in the column which leads to corrosion of the steel,

eventually leading to reduced performance of that structure element. With respect to deteriorated

reinforced concrete columns, one could conclude that the causes stated above result in deterioration

because of lack of lateral confinement. The longitudinal reinforcement in the reinforced concrete columns

provide very little lateral confinement effect, which is not adequate for most loading conditions.

As a structural designer one always tries to design the reinforced concrete structures in a manner

so that they exhibit ductile behavior. Lateral confinement in a reinforced concrete column provides the

column with the required ductility. Under seismic loading, this additional confinement could ensure

adequate strength for the column and increase its deformation capacity which improves its performance in

an event like an earthquake. (Park et al., 1982; Mander et al., 1988; Shams & Saadeghvaziri, 1997)

2

Many confinement techniques have been developed over the years; designing the columns with

steel hoops (stirrups) or by providing steel jacketing techniques. The steel jacketing technique has been

proved quite useful in the field of retrofitting the columns. However, corrosion of the steel can be of

concern. It also increases the self-weight of the structure to a great extent which is always a tradeoff. In

situations where the concrete cover is very loose and weak one cannot use the steel jacketing techniques

as it might damage the column even more due to the bolting of the jackets.

During recent decades, many researchers have been trying to replace the conventional steel

jacketing technique by usage of fiber reinforced polymer (FRP) wraps. FRP wraps used as confinement

can increase the ultimate compressive strength and the ultimate strain of the concrete. (Samaan et

al.,1998; Toutanji, 1999). A lot of research has been carried out on developing a retrofitting technique

with these FRP wraps. The main advantages these FRP wraps possess over the steel jackets are very high

strength to weight ratio & high resistivity to corrosion.

1.2 Scope of the study

The objective is achieved and restricted within the following scope of study:

1) Literature review to identify and choose the most relevant models for modeling of

concrete confined with fiber reinforced polymers in compression.

2) Modelling and finite element analysis of the confined concrete compression test in

AbaqusCAE.

3) Developing the stress-strain curve from several proposed empirical model (Modified Lam

& Teng).

4) Survey of experimental data on confined concrete with FRP in order to generate an

experimental database.

5) Comparison of the analytical results in order to define the strengths and limitations of the

empirical model chosen to study.

3

6) Validation of the model chosen and a study on its relevance for use in typical bridge

columns retrofitted with FRP jackets.

7) Proposing & validating changes to the empirical model for more accurate results.

4

Chapter 2 Literature Review

2.1 Introduction

A Literature review was conducted to determine the appropriate confinement model for concrete

in compression with FRP as confinement reinforcement. In order to understand the mechanism of

confinement by FRP the models of the typical steel confinement were reviewed and analyzed first. This

review of steel confinement models facilitated the study of FRP confined concrete models.

2.2 Mechanism for Concrete Confinement by Transverse Reinforcement

Steel spirals or hoops are quite commonly used as transverse reinforcement in concrete

compression members. Research as demonstrated that circular hoops are more effective in providing

confinement as compared to square hoops. Due to their shape circular hoops are able to provide

continuous confining pressure around the circumference of the compression members. The square hoops

have a tendency to bend the sides outwards due to the pressure of the concrete against the sides. This is

more effectively demonstrated in the figure shown below.

Figure 2-1 : Confinement by Square hoops and Circular Spirals.

5

Richard, Brandtzaeg and Brown (1928) proposed the relationship for calculating the strength of

concrete cylinders loaded axially to failure subject to confining fluid pressure.

f’cc = f’c + 4.1fl Eqn 2-1

where,

f’cc = axial compressive strength of confined specimen

f’c = uniaxial compressive strength of unconfined specimen

fl = lateral confining pressure

When the confining pressure is due to the steel circular spirals the free body diagram of half the

spiral is as shown in figure 2-2.

f’cc = f’c + 8.2Aspfl / ds.s Eqn 2-2

Figure 2-2 : Confinement of concrete by spiral reinforcement. (Park & Paulay,1875)

6

2.3 Modeling of Concrete in Compression

Many models have been proposed to capture the non-linear behavior of concrete in compression

by transverse reinforcement. For the scope of this study, the modified Hognestad model and the Kent &

Park model were studied. Both the models are quite capable of capturing the behavior of the confined and

well as unconfined concrete under compression.

2.3.1 Modified Hognestad Model:

This model was studied from MacGregor (2012). This model is capable of presenting the

behavior of concrete in compression (confined and unconfined).

Figure 2-3 : The modified Hognestad model for compressive stress-strain curve for concrete (MacGregor, 2012)

7

The modulus of elasticity for concrete Ec may be calculated as follows

Ec = w1.5

33 √(f’c ) psi Eqn 2-3

Where w = density of concrete in pounds per cubic foot, f’c is the compressive strength in psi. f’’c

is the maximum stress reached in the concrete. The extent of falling branch behavior depends on the limit

of useful concrete strain assumed. The slope of the line is affected by the amount of confinement, which

terminates at a strain of 0.0038.

2.3.2 Kent and Park model

Kent and Park (1971) proposed a stress-strain equation that can present both unconfined and

confined concrete behavior under compressive loading. In this model they generalized Hognestad model

(1951) equation to more completely describe the post-peak behavior. The ascending curve is represented

by : (Region : ԑc ≤ 0.002)

𝑓’c = 𝑓’c [2ԑ𝑐

0.002− (

ԑ𝑐

0.002)

2

]

Eqn 2-4

This curve is obtained modifying Hognestad second degree parabola by replacing 0.85 f’c by f’c and ԑco

by 0.002.

The post-peak branch is assumed to be straight line which has a slope that is primarily defined as

a function of the strength of concrete. (Region : 0.002 ≤ ԑc ≤ ԑ20c

fc = f’c [1-Z(ԑc - ԑco)] Eqn 2-5

where, 𝑍 = [0.5

ԑ50𝑢− ԑ𝑐𝑜0.002] Eqn 2-6

8

ԑ50u = the strains at 50% of the maximum concrete strength for unconfined concrete.

ԑ50u =3 + 0.002 f’c(in Psi)

f’c − 1000

ԑ50u =3 + 0.29 f’c (𝑖𝑛 𝑀𝑃𝐴)

145 f’c − 1000

Eqn 2-7

Figure 2-4 : Proposed stress-strain model for confined & unconfined concrete – Kent & Park (1971) model

9

2.4 Stress-Strain Response of FRP-Confined Concrete

The behavior of concrete confined by FRP differs with respect to its stress-strain response as

compared to normal concrete or steel-stirrup confined concrete. Basically, the stress-strain response of

FRP-confined concrete can be studied by dividing it in four parts

Figure 2-5 : Typical Stress-Strain behavior of FRP confined concrete (Saafi & Toutanji, 1999)

2.4.1 First Zone

In this zone the behavior is the same as that of unconfined concrete. The concrete takes up all the

axial load and the slope of the curve is the same as the slope of the stress-strain curve for unconfined

concrete. One can say during this phase the concrete behaves in a way that the FRP-confinement is not

present. One can also conclude by saying the bond between concrete and FRP-confinement is passive and

the FRP-confinement jacket is not yet activated.

10

The relations between axial stress and lateral strain can be derived from the conventional tri-axial

stress-strain state in case of this zone which is the elastic zone. ԑr (Park Kihoon, 2004)

𝜎𝑧 = −ԑr𝐸

𝑣[1−1

𝐸(1−𝑣)(

2𝑡𝐸𝑓𝑟𝑝

𝑑)]

Eqn 2-8

Where, σz = P/A or axial stress, ԑr = radial strain in concrete, v = Poisson’s Ratio of concrete

E = Elastic Modulus of concrete , Efrp = Elastic Modulus of FRP

d = diameter of concrete cylinder (Consistent system of units)

2.4.2 Transition Point

At the maximum level of unconfinement, this transition point occurs that indicates that the

concrete crack has taken place. This point can be termed as the first failure point of the concrete core. At

this point the FRP-confinement jacket starts developing its confinement effect.

2.4.3 Second Zone

The second region has the concrete core which has already started to fail. The FRP-confinement

jacket is activated and it confines the concrete core. The FRP-confinement jacket applies a continuously

increasing pressure on the concrete core until the jacket reaches its first point of failure. The amount of

confining pressure that would be exerted by the jacket will depend on the amount of FRO material in it.

The concrete is tri-axially stressed and the FRP-confinement jacket is uni-axially stressed.

2.4.4 Failure Mode

FRP material is a very brittle material, which means the failure of this material is accompanied by

a large release of energy. Failure usually starts at the middle of the specimen with a sudden or gradual

development of the crack towards the end. Ideally the failure point is assumed as that axial strain in the

specimen for which the lateral strain in the concrete reaches the strain at the fracture of the FRP confining

11

reinforcement. This means that the failure strength of the confined concrete is very closely related to the

failure strength of the FRP strengthening material used. (Karabinis & Rousakis, 2002).

However, experimental evidence shows that the value of the concrete hoop failure strain very

much lower than ultimate failure strain of the FRP material. The predicted reasons for these are: (CEB-

FIP)

1) The stress change in the jacket due to confinement pressure has a certain influence on the ultimate

strength.

2) Due to inadequate surface preparation the specimens are of poor quality.

3) Size effects

Figure 2-6 : Failure modes of circular FRP-confined concrete cylinders (Micellli & Modarelli, 2012)

Figure 2-7 : Failure modes of sqauare FRP-confined concrete cylinders (Micellli & Modarelli, 2012)

12

One can observe from the figures above at the failure point of the FRP-confined concrete the

concrete core has completely failed and the failure of the FRP-confinement starts from the middle.

2.4.5 Post Failure

As mentioned earlier FRP is a very elastic-brittle material. The constitutive property of FRP

tensile coupon testing can be represented in terms of the stress-strain curve as shown below.

Figure 2-8 : Tensile Coupon testing of CFRP, GFRP & Steel (Benziad et al (2010))

One can infer from the graph shown above that the failure of FRP material is sudden and

accompanied by a large amount of energy. The failure point of FRP confined concrete cylinders is

defined as the point where the confining material, i.e. FRP fails in tension. Hence, the failure of the FRP

confined concrete cylinder is also sudden and is accompanied by a large release of energy.

Owing to this the, behavior of the confined cylinders post-failure would be expected to not be

able to sustain any more loading pressure. This can be evident from the experimental results as shown in

the research by Karabinis et al (2002)

13

Figure 2-9 : : Karabinis et al (2002). A2 & B2 are unconfined control specimens.

14

For Concrete type A (Unconfined concrete strength = 38.5 MPa)

The three sets of specimens for this case (1 layer , 2 layers and 3 layers of CFRP) show that the

confined specimen in the post failure regime failure to exhibit any resistance to the loading until as low as

5 MPa. This clearly shows, that yielding of this specimen does not take place.

For Concrete type B (Unconfined concrete strength = 35.7 MPa)

Similarly in this case, the three sets of specimens for this case (1 layer , 2 layers and 3 layers of

CFRP) show that the confined specimen in the post failure regime failure to exhibit any resistance to the

loading until as low as 15 MPa. The specimen with three layers shows some yielding behavior. The

reason for this was however cited as different points of failure of multiple FRP fiber layers.

Experimental data available is tested upto the the failure point in a uniaxial strain controlled

compression test. Even though the concrete core is still intact and appear not to be crushed, the load

carrying capacity of the same is negligible as one can see from the graphs above. Further more a closer

representation of the concrete core would look as shown in the figure below.

Figure 2-10 : Failure of concrete core ( Li et al (2006))

From the graphs and the figure shown above one can assume that the concrete looses its load

carrying capacity during the loading regime

15

2.5 Proposed models

In the early days the constitutive models for FRP-confined concrete were the same as those for

the steel confined concrete. However, a number of research studies showed that this approach is not

accurate and a significant difference exists between the behaviors of both these specimens. As the steel in

the jacket of steel confined concrete is assumed to yield, one is safe to assume that the confining pressure

exerted by steel jacket on the concrete is constant. (Mander et al., 1988b) However, this is not true in the

case of FRP-confined concrete. FRP as a material does not yield and hence, the confining pressure

exerted by the FRP jackets is not constant and keeps increasing until its failure point.

Various FRP-confined concrete models have been proposed to date. All these models can be

classified under two major categories; a) design-oriented models b) analysis-orientated models. Design-

oriented models are geared toward their use in the engineering design practices whereas analysis-oriented

models can capture the detailed mechanical behavior exhibited by the specimen. According to a study

(Ozbakkaloglu et al., 2013), it was concluded that the design-oriented models have a better capability to

predict the ultimate strength and strain of the specimen. Thus this literature review study will explore

design-oriented models. A few of these models were reviewed and studied thoroughly. Each model that

was studied is presented in detail and the equations used are also shown.

16

2.5.1 Samaan and Mirmiran Model (1998)

The Richard & Abott model (1975) was used and calibarated by Samaan & Mirmiran to represent

the bilinear response of FRP-confined concrete.

𝑓𝑐 =(𝐸1−𝐸2)ԑ𝑐

(1+((𝐸1−𝐸2)ԑ𝑐

𝑓𝑜)

𝑛)

1𝑛

+ 𝐸2ԑ𝑐 Eqn 2-9

Figure 2-11 : Parameters of Bilinear Confinement Model (Samaan & Mirmiran, 1998)

The confining pressure is given by

𝑓𝑙 =2𝑓′𝑙𝑡𝑗

𝑑 Eqn 2-10

Where, f’l = hoop strength of the tube; tj = Tube Thickness & d = core diameter.

The strength of confined concrete can be linked to the confining pressure by FRP in the following way:

𝑓′𝑐𝑢 = 𝑓𝑐′ + 3.38 𝑓𝑙

0.7 (𝐾𝑠𝑖) Eqn 2-11

To evaluate the first slope (E1), this model adopts the formula proposed by Ahmad & Shah (1982)

to predict the secant elastic modulus.

𝐸1 = 47586√1000𝑓𝑐′ (𝐾𝑠𝑖) Eqn 2-12

17

The secant modulus changes at a point where the concrete reaches its unconfined strength. As

shown in Fig 2-6, the second slope (E2) is a function of the stiffness of the confining tube. Here, this slope

depends more on the properties of the confining tube rather than the properties of unconfined concrete

core.

𝐸2 = 52.411𝑓𝑐′0.2

+ 1.3456𝐸𝑗𝑡𝑗

𝐷 (𝐾𝑠𝑖) Eqn 2-13

Where, Ej = effective modulus of elasticity of the tube in the hoop direction.

The intercept in this model, fo is a function of the strength of the unconfined concrete & the

confining pressure provided by the FRP-tube. This was estimated as:

𝑓𝑜 = 0.872𝑓𝑐′ + 0.371𝑓𝑙 + 0.908 (𝐾𝑠𝑖) Eqn 2-14

The ultimate strain ԑcu is given as

ԑ𝑐𝑢 = (𝑓𝑐𝑢′ − 𝑓𝑜)/𝐸2 Eqn 2-15

This model is however not very sensitive to the curve-shape parameter n, and has a constant value

of 1.5 which was suggested by Samaan and Mirmiran (1998). This parameter is an important factor in

understanding the ductility and the change in the behavior of FRP-confined concrete.

18

2.5.1 Mander’s Model (1984)

Mander et al (1984) proposed a unified stress-strain mathematical model for predicting the

behavior of confined concrete. The basis for this model were the equations that were suggested by

Popovics (1973). This approach was based on energy balance method.

The stress equation suggested by this model was given by:

𝑓𝑐 =𝑓𝑐𝑐

′ 𝑥𝑟

𝑟−1+𝑥′ Eqn 2-16

Where: f’cc = compressive strength of confined concrete.

𝑥 =ԑ𝑐

ԑ𝑐𝑐 Eqn 2-17

Where ԑc = longitudinal compressive concrete strain

ԑ𝑐𝑐 = ԑ𝑐𝑜(1 + 5 (𝑓𝑐𝑐

′

𝑓𝑐𝑜′ − 1)) Eqn 2-18

f’co & ԑco are the unconfined concrete strength and the corresponding strain respectively.

𝑟 =𝐸𝑐

𝐸𝑐−𝐸(se c) Eqn 2-19

𝐸sec = 𝑓𝑐𝑐

′

ԑ𝑐𝑐 Eqn 2-20

19

2.5.2.1 Approach to compute the Effective lateral confining pressure

The approach to compute the effective lateral confining pressure was based on the effective area

which was confined between two steel stirrups. In the case of steel stirrups arching action takes place as

shown in the figure.

Figure 2-12 : Arching effect in Concrete. (Mander et al (1989))

The effective confined area of concrete reduces as we move towards the mid portion of from one

stirrup and is least at the midpoint. Hence the relationship that was proposed to compute the lateral

confining pressure reflected the fact that the effective lateral confining pressure exerted on the concrete

will be a fraction of the confining pressure generated in the stirrup. Hence,

𝑓𝑙′ = 𝑘𝑒𝑓𝑙 Eqn 2-21

Where, fl = lateral pressure generated in the transverse reinforcement.

f'l = lateral pressure generated in the transverse reinforcement.

Further, Mander et al (1989) defined ke as the ratio of the effective confined concrete area (Ae) to the area

of concrete present between the center lines of two stirrups (Acc)

𝐴𝑐𝑐 = 𝐴𝑐(1 − 𝜌𝑐𝑐) Eqn 2-22

Ρcc = ratio of area of longitudinal reinforcement to area of core section

Ac = area of core enclosed between center lines of two stirrups

20

The lateral confining pressure would be found by considering half body of the stirrup. The

assumption made in calculating the lateral confinement pressure is that the hoop tension is uniform and

exerts a uniform pressure on the concrete. Also, the steel is assumed to have yielded which leads to a

constant pressure exerted on the concrete.

2𝑓𝑦ℎ𝐴𝑠𝑝 = 𝑓𝑙𝑠𝑑𝑠 Eqn 2-23

fyh = yield strength of steel used as lateral reinforcement.

Ρs can be defined as the ratio of the volume of transverse confining steel to volume of confined concrete.

Therefore,

𝑓𝑙′ =

1

2𝑘𝑒𝜌𝑠𝑓𝑦ℎ Eqn 2-24

In the case of FRP confined concrete arching does not take place and hence the effective

confined area would be the same as the area between two boundaries of confinement. Hence it would be

correct to assume that ke would be 1. However this is only true for completely wrapped concrete

cylinders. One could compute ke in the same way as mentioned above for concrete column confined with

spiral FRP.

Figure 2-13 : Possible arching effect in non-uniformly confined concrete by FRP (Saadatmanesh et al (1994))

21

Also the confinement pressure exerted by FRP would vary as the strength developed in FRP

would depend on the level of hoop strain in FRP. Hence the equation for lateral confinement pressure

would have to be modified.

2.5.2.2 Mander’s model for FRP confined concrete

Saadatmanesh et al (1994) extended the Mander’s model to the case of FRP confinement by

computing the lateral confinement pressure applied by the FRP jacket.

Figure 2-14 : Free body diagram of FRP confinement (DeLorenzis et al (2003))

By the Equilibrium of forces:

𝑝 = 𝐸𝑙ԑ𝑙 = 𝐸𝑙ԑ𝑓 Eqn 2-25

El = Confinement/lateral modulus

𝐸𝑙 =2𝐸𝑓𝑛𝑡

𝐷 Eqn 2-26

Therefore,

𝑝𝑢 =2𝐸𝑓𝑛𝑡ԑ𝑓

𝐷=

2𝑓𝑓𝑢𝑛𝑡

𝐷 Eqn 2-27

As mentioned earlier ke would be 1. Hence

𝑓𝑙 =1

2𝜌𝑠𝑓𝑓𝑟𝑝 Eqn 2-28

22

Where, ffrp = stress in FRP at a particular point of time.

This depends on the hoop strain developed as FRP is an elastic brittle material and hence ffrp

would not be constant. For design purposes or to predict the ultimate confinement one could use the

ultimate stress in FRP(fus ) in the above equation.

𝑓𝑙 =1

2𝜌𝑠𝑓𝑢𝑠 Eqn 2-29

2.5.2.3 Expression for Compressive strength of Confined concrete

Mander’s model suggests a nonlinear relationship between confined concrete strength and the

confinement pressure applied based on the ultimate surface strength developed by Elwi & Murray (1979).

The failure concepts applied in development of this model of concrete under tri-axial state of stress was

based on the elastic perfectly plastic behavior of concrete in the compression regime. Also the lateral

stress conditions assumed in this constitutive model are uniform. Saadatmanesh et al (1994) used the

same model to predict the compressive strength of FRP confined concrete.

𝑓𝑐𝑐′ = 𝑓𝑐𝑜

′ (−1.254 + 2.254√1 +7.94𝑓𝑙

′

𝑓𝑐𝑜′ −

2𝑓𝑙′

𝑓𝑐𝑜′ ) Eqn 2-30

Where, f’l is used from the equation specified above. This yields us the results at the ultimate state

of FRP confined concrete. Hence, as shown by Imran & Pantazopoulou (1996) & Lan and Guo (1997),

the Mander’s model can be used to predict the ultimate condition of confined concrete. Further they

concluded that this was true because the confined concrete strength was essentially independent of the

shape of the loading path. Further they showed that this model can very accurately predict the ultimate

condition provided the hoop strain at the failure is very close to the tensile strain of failure of FRP during

the coupon testing. Research by Lam & Teng (2003) shows that, the two strains are not close to each

other. They are related to each other by an FRP efficiency factor k ԑ which was approximated to a specific

value of 0.586. One could use the Mander’s model at every increment of hoop strain in the FRP to

compute the stress-strain curve which will basically be an envelope of family of Mander’s model curves.

23

Research by Mirmiran et al. (1996) shows that the energy balance equation (Popovics (1973)),

which considers concrete ductility to be proportional to the stored energy in the confining material cannot

be applied in the case of FRP confinement. This was confirmed by Spoelstra et al. (1999) by comparing

the results obtained by the Mander’s model against the experimental data from the literature available.

24

2.5.3 Lam & Teng Model (2003)

Lam & Teng proposed a design-oriented model which describes the stress-strain relationship of

FRP-confined concrete. This model is only for uniformly confined concrete.

The relationship is given by:

𝑓𝑐 = 𝐸𝑐ԑ𝑐−

(𝐸𝑐−𝐸2)2

4𝑓𝑐𝑜′ ԑ𝑐

2 𝑓𝑜𝑟 0 ≤ ԑ𝑐 ≤ ԑ𝑡 Eqn 2-31

𝑓𝑐 = 𝑓𝑐𝑜′ + 𝐸2ԑ𝑐 𝑓𝑜𝑟 ԑ𝑡 ≤ ԑ𝑐 ≤ ԑ𝑐𝑢

Eqn 2-32

Where, fc & ԑc are the axial stress and the axial strain of confined concrete respectively, ԑt is the

axial strain at the transition point & E2 is the slope of the straight second portion.

ԑ𝑡 =2𝑓𝑐𝑜

′

𝐸𝑐−𝐸2 Eqn 2-33

𝐸2 =𝑓

𝑐𝑐−𝑓𝑐𝑜′

′

ԑ𝑐𝑢 Eqn 2-34

Figure 2-15 : Lam & Teng’s stress-strain model for FRP-confined concrete. (Lam & Teng, 2003)

25

The compressive strength of FRP-confined concrete f’cc is predicted using:

𝑓𝑐𝑐′ = 𝑓𝑐𝑜

′ + 3.3𝑓𝑙0.7 Eqn 2-35

𝑓𝑙 =2𝜎𝑗𝑡

𝑑=

2𝐸𝑓𝑟𝑝ԑ𝑗𝑡

𝑑 Eqn 2-36

ԑj = ԑh,rup ԑh,rup = kԑ ԑfrp Eqn 2-37

kԑ is the FRP efficiency factor and has a value of 0.586. Lam & Teng proposed that the ԑj should

be taken as the actual hoop rupture strain ԑh,rup measured in the FRP jacket and not the ultimate FRP

tensile strain ԑfrp as is assumed ideally. The ultimate concrete axial strain of uniformly confined concrete,

ԑcu is given by:

ԑ𝑐𝑢

ԑ𝑐𝑜= 1.75 + 12

𝑓𝑙

𝑓𝑐𝑜′ (

ԑℎ𝑟𝑢𝑝

ԑ𝑐𝑜)

0.45 Eqn 2-38

Here, the axial strain (ԑco ) at the compressive strength of unconfined concrete is taken as 0.002

.

26

2.5.4 Modified Lam & Teng (Liu et al., 2013)

The constitutive model proposed by Lam & Teng (2003) is a very sophisticated model capable of

capturing the behavior of FRP-confined concrete to a promising level. However, the model cannot

represent the process of gradual development of confinement by the FRP-tube. He et al.,(2013) made an

attempt to modify the parabola in a way which would be able to represent the transition more accurately.

This model uses the slope and intercept proposed by Samaan et al. (1998) in Lam & Teng (2003) model.

The first branch is parabolic which transitions into the second branch which is linear. The

transition occurs at smoothly at a transition strain ԑt

ԑ𝑡 =2𝑓𝑜

(𝐸𝑐−𝐸2) Eqn 2-39

𝐸2 =𝑓𝑐𝑐

′ −𝑓𝑜

ԑ𝑐𝑢 Eqn 2-40

Figure 2-16 : Envelope curve and hysterical rule of proposed FRP confined concrete. (Lui et al, 2013)

27

This model requires the definition of three parameters based on the general shape of stress-strain

curve described above. The three parameters include: the ultimate strength (f’cc), the ultimate strain (ԑcu)

and the intercept stress( fo ). These are the three critical parameters critical to the constitutive model. Lam

& Teng (2003) proposes that the intercept stress fo be taken equal to the compressive strength of the

unconfined concrete. (fo = fco*)

The condition at ultimate state of FRP confined concrete is directly related to the ultimate

transverse tension of FRP material which provides the confining pressure on the concrete. The amount of

the maximum confining pressure that will be applied by the FRP tube on the concrete will be controlled

by the ultimate strain (ԑh,rup ) in the FRP. This strain is not equal to the ԑFRP which is the ultimate tensile

strain in the coupon testing. Hence a FRP-strain reduction factor (ke ) is defined to describe the

relationship between ԑh,rup & ԑFRP.

ԑℎ,𝑟𝑢𝑝

ԑ𝑓𝑟𝑝= 𝑘𝑒 Eqn 2-41

Now the actual confining pressure (fl,a ) at ultimate is calculated by using ԑh,rup

𝑓𝑙,𝑎 = 𝑘𝑒𝑓𝑙 =2𝐸𝑓𝑟𝑝𝑡𝑓𝑟𝑝ԑℎ,𝑓𝑟𝑝

𝐷 Eqn 2-42

Where, D = Diameter, Efrp = Youngs modulus of FRP & tfrp is the thickness of FRP layer.

FRP strain reduction factor( k e ) is very important in order to accurately describe the shape of the

model. The values suggested by Lam & Teng (2003) for different types of confinement is not general and

accurate enough. Lim & Ozbakkaloglu (2013) proposed an equation for the strain reduction factor which

relates to the Youngs modulus of FRP and the unconfined concrete compressive strength.

𝑘𝑒 = 0.9 − 2.3𝑓𝑐𝑜′ 𝑥 10−3 − 0.75𝐸𝑓𝑥 10−6 (𝑀𝑃𝑎) 𝑓𝑜𝑟 105𝑀𝑃𝑎 ≤ 𝐸𝑓 ≤ 6.4 𝑥 105𝑀𝑃𝑎

Eqn 2-43

28

This expression is able to predict the strain reduction factor for FRP-confined concrete with

concrete of unconfined compressive strength upto 120MPa. This expression is also valid for GFRP,

CFRP and aramid FRP. With this, the maximum confining pressure can be accurately calculated.

Lam & Teng (2003) model was verified based on the database specified in Teng et al.,(2003)

which had concrete specimens of unconfined concrete specimens less than 43 MPa. With increase in the

unconfined concrete strength fco*, the ultimate coefficients of stress and strain are needed to be modified.

Hence the expression for the ultimate strain that has been proposed is:

ԑ𝑐𝑢 = 𝑐2ԑ𝑐𝑜 + 12 (𝑓𝑙,𝑎

𝑓𝑐𝑜∗ ) ԑ0.45ℎ,𝑟𝑢𝑝 ԑ𝑐0

0.55 Eqn 2-44

He et al (2013) used the database from Lim & Ozabakkaloglu (2013) which covered concrete

specimens of unconfined concrete strength of 6.2MPa to 169.7 MPa. With this database the normalized

coefficients were statically determined:

𝑐2 = 2 − 0.01 (𝑓𝑐𝑜∗ − 20)& 𝑐2 ≥ 1 Eqn 2-45

The minimum threshold for FRP confined concrete to display complete strain-hardening is

defined in terms of the FRP confinement stiffness K1. With the help of this, we define the FRP stiffness

threshold Klo.

𝐾1 =2𝐸𝑓𝑟𝑝𝑡𝑓𝑟𝑝

𝐷

𝐾𝑙𝑜 = 𝑓𝑐𝑜∗1.65

Eqn 2-46-1 & 2

If K1 ≥ Klo

f’cc = c1 f*co

+ 2 k1 f’l,a / kԑ Eqn 2-47

c1 = 1 + 0.0058 fl,a / (f*co ԑh,rup) Eqn 2-48

29

where k1 =1 for wrapped FRP & k1 = 0.9 for FRP tube

If K1 < Klo

f’cc = c1 f*co

+ k1 (f’l,a - flo ) Eqn 2-49

c1 = (K1 / f*co

1.6 )

0.2 Eqn 2-50

ԑlo = 24 (f*co

/ K1

1.6)

0.4 ԑc0 Eqn 2-51

flo = K1 ԑlo Eqn 2-52

where k1 =3.18 for wrapped FRP & k1 = 2.89 for FRP tube

A specific relationship is proposed between the stress intercept (fo ) and the unconfined concrete

strength (f*co

).

fo = 1.105 f

*co

Eqn 2-53

30

2.5.5 Drucker-Prager Plasticity Model

Drucker & Prager proposed this model in 1952. There are three criteria that control the

framework of this model. Every plasticity model is governed by a yielding and hardening criteria; the

flow rule; path dependence; limited tensile strength and the pressure dependence critieria. In this model,

the parameters that control the yielding and hardening criteria are the friction angle and the cohesion. The

plastic dilation controls the flow rule of the plasticity model. A limited amount of study has been carried

out regarding the plastic dilation rate for FRP-confined concrete. Mirmiran et al., (2000) & Karabinis et

al., (2008) used a constant dilation rate which is not true. Yu et al., (2010) demonstrated with his research

that the dilation rate varies with the plastic strains and the lateral stiffness. This was further verified by J-

F. Jiang et al (2012). In order to implement this model in Abaqus we need to define three parameters : 1)

Friction angle model, 2) hardening/softening function & 3) Plastic Dilation model.

2.5.5.1 Friction angle model

The friction angle φ can be related to the internal friction angle Φ defined in the Mohr-Coloumb

theory as

𝑡𝑎𝑛𝜑 =6𝑠𝑖𝑛𝛷

3−𝑠𝑖𝑛𝛷 Eqn 2-54

The internal friction angle is the angle that the tangent line which is drawn to the Mohr’s circle at

the failure state makes with the X-axis or with the normal stress axis. In case of concrete the failure state

is defined as the onset of peak strength. Mohr’s circle is used to describe the behavior of brittle materials.

While considering the plasticity model, we consider the softening behavior of concrete. Brittleness

reduces under increasing deformation due to increasing softness. This causes the internal friction angle to

reduce and the assumption of it being constant is not quite accurate.

31

Research by J-F. Jiang et al (2012) proposes an equation to determine the friction angle φ.

𝜑 = 𝜑𝑜 + 𝑘ԑ𝑝 Eqn 2-55

Where φo = 56.44o , k = 226 ԑ𝑝 = 𝑃𝑙𝑎𝑠𝑡𝑖𝑐 𝑆𝑡𝑟𝑎𝑖𝑛

Figure 2-17 : Friction model (J-F. Jiang et al (2012))

2.5.5.2 Cohesion Model

In the Drucker-Prager Plasticity Model, k is the hardening/softening which controls the

development of the surface that yields. Most of the hardening-softening functions are based on the

assumption that the internal friction angle φ is constant. Hence a different function is required which

considers the variation in the internal friction angle φ. Based on the friction model described above the

relationship between the normalized cohesion k/f’c & ԑp can be described as follows:

𝑘(ԑ𝑝𝜌)

𝑓𝐶′ = 𝑘𝑜 + 𝐸𝑝 (

ԑ𝑝

1+ 𝜂ԑ𝑝) + 𝑝1(𝜌)ԑ𝑝

2 + 𝑝2(𝜌)ԑ𝑝2 Eqn 2-56

Where, ko = 1/8 , Ep = 2700 (initial slope of the cohesion curve), η = 6587,

𝑝1 =𝜌

𝑎1+𝑎2𝜌+𝑎3√𝜌 Eqn 2-57

𝑝2 =(𝑏1𝜌+𝑏2)

𝜌+𝑏3 Eqn 2-58

Where, a1 = 0.12, a2 = 0.0044, a3 = -0.0023, b1 = -0.75, b2 = -41.06, & b3 = 2.52

32

2.5.5.3 Dilation Angle (Plastic Dilation Model)

The FRP jacket induces a passive type of confinement on the concrete. Dilation is the measure of

the volume change. In the case of FRP-confined concrete the plastic volumetric deformation/strain is

critical as it controls the amount of confining pressure that will be generated by the FRP tube. Mirmiran et

al.,(2000) pointed out that a zero plastic dilation rate , which is true for steel confined concrete can predict

reasonably the behavior of FRP confined concrete but not accurately. Karabinis & Rousakis (2002) used

an asymptotic function, where the plastic dilation angle β was assumed to decrease from -27.4o to -56.3

o.

Rousakis et al., (2008) later assumed a constant rate which depends on unconfined concrete strength.

Figure 2-18 : Existing models for plastic dilation rate. (J-F. Jiang et al (2012))

J-F. Jiang et al (2012) proposed a mathematical model for the dilation angle. Test results from

various research studies were used in this case and a model accurately satisfying these test results was

arrived at.

33

Figure 2-19 : Typical Dilation curve (J-F. Jiang et al (2012))

𝛽 = 𝛽0 + (𝑀𝑜 + 𝜆1𝛽𝑜)(ԑ𝑐

𝑝) + 𝜆2𝛽𝑢(ԑ𝑐

𝑝)

2

1 + 𝜆1ԑ𝑐𝑝

+ 𝜆2(ԑ𝑐𝑝

)2

Eqn 2-59

The coefficients λ1 , λ2 & βu are functions of βo , Mo & ρ.

βo = -37, Mo = 157000 & 𝜌 =2𝐸𝑓𝑡𝑓

𝐷𝑓𝑐′

& 𝜆1 = 11.61𝜌 + 980

𝜆2 = 5700𝜌 + 225000

𝛽𝑢 = 101.66 exp(−0.06𝜌) − 37.5

Eqn 2-60 (a) (b) (c)

34

2.6 Conclusions

As discussed in this chapter the stress-strain curve of a FRP confined concrete cylinder specimen

can be divided into four portions: The first zone, transition point, second zone and the ultimate/failure

point. The Samaan and Mirmiran, 1998 model provides the stress-strain curve and the ultimate stress and

strain value. It provides two equations for the ultimate condition of the stress-strain curve. The stress

equation is derived from experimental tests and the ultimate strain is derived from the geometric shape of

the curve.

The Manders model, 1984 is primarily derived for steel confinement, wherein the second zone is

derived from the maximum confining pressure exerted by steel confinement. The Lam & Teng model and

the Modified Lam & Teng’s model provide several equations for each point on the stress strain curve.

Accordingly, this study would be aiming at validating the modified Lam & Teng’s model against

the experimental stress-strain curves and those obtained via finite element simuation in AbaqusCAE by

using the Drucker-Prager plasticity model.

35

Chapter 3 Experimental Database & Preliminary results.

3.1 Introduction

In order to validate the reliability of the model chosen for this study it was necessary to gather an

experimental database. The selection of the experimental database was subjected to the availability of the

complete information related to the specimens, such as its geometric configuration, the material properties

of concrete and FRP and the final results including the stress-strain curve. The experimental database

chosen for this study includes 51 concrete cylinder specimens wrapped with Glass FRP and 54 wrapped

with Carbon FRP. The database comprises of concrete cylinders having unconfined compression strength

varying from 20 MPa to 107.8 MPa.

Source ID

Type

of

FRP

D

(mm)

H

(mm)

f'co

(MPa)

ԑco

(%)

Efrp

(GPa)

t

(mm) ρ ԑh,rup

f'cc

(MPa) ԑcu

Lam &

Teng

(2004)

1 Carbon 152 305 35.9 0.203 250.5 0.165 15.15 0.969 47.2 1.106

2 Carbon 152 305 35.9 0.203 250.5 0.165 15.15 0.981 53.2 1.292

3 Carbon 152 305 35.9 0.203 250.5 0.165 15.15 1.147 50.4 1.273

4 Carbon 152 305 35.9 0.203 250.5 0.33 30.30 0.949 71.6 1.85

5 Carbon 152 305 35.9 0.203 250.5 0.33 30.30 0.988 68.7 1.683

6 Carbon 152 305 35.9 0.203 250.5 0.33 30.30 1.001 69.9 1.962

7 Carbon 152 305 34.3 0.188 250.5 0.495 47.57 0.799 82.6 2.046

8 Carbon 152 305 34.3 0.188 250.5 0.495 47.57 0.884 90.4 2.413

9 Carbon 152 305 34.3 0.188 250.5 0.495 47.57 0.968 97.3 2.516

10 Glass 152 305 38.5 0.223 21.8 1.27 9.46 1.44 51.9 1.315

11 Glass 152 305 38.5 0.223 21.8 1.27 9.46 1.89 58.3 1.459

12 Glass 152 305 38.5 0.223 21.8 2.54 18.92 1.67 77.3 2.188

13 Glass 152 305 38.5 0.223 21.8 2.54 18.92 1.76 75.7 2.457

36

Source I

D

Type of

FRP

D

(m

m)

H

(mm

)

f'co

(MPa)

ԑco

(%)

Efrp

(GPa

)

t

(mm

)

ρ ԑh,rup

f'cc

(MPa

)

ԑcu

Lam et

al

(2006)

14 Carbon 152 305 41.1 0.256 250 0.165 13.21 0.81 52.6 0.9

15 Carbon 152 305 41.1 0.256 250 0.165 13.21 1.08 57 1.21

16 Carbon 152 305 41.1 0.256 250 0.165 13.21 1.07 55.4 1.11

17 Carbon 152 305 38.9 0.25 247 0.33 27.57 1.06 76.8 1.91

18 Carbon 152 305 38.9 0.25 247 0.33 27.57 1.13 79.1 2.08

19 Carbon 152 305 38.9 0.25 247 0.33 27.57 0.79 65.8 1.25

Teng et

al

(2007)

20 Glass 152 305 39.6 0.263 80.1 0.17 4.52 1.869 41.5 0.825

21 Glass 152 305 39.6 0.263 80.1 0.17 4.52 1.609 40.8 0.942

22 Glass 152 305 39.6 0.263 80.1 0.34 9.05 2.04 54.6 2.13

23 Glass 152 305 39.6 0.263 80.1 0.34 9.05 2.061 56.3 1.825

24 Glass 152 305 39.6 0.263 80.1 0.51 13.57 1.955 65.7 2.558

25 Glass 152 305 39.6 0.263 80.1 0.51 13.57 1.667 60.9 1.792

Jiang et

al

(2007)

26 Glass 152 305 33.1 0.309 80.1 0.17 5.41 2.08 42.4 1.303

27 Glass 152 305 33.1 0.309 80.1 0.17 5.41 1.758 41.6 1.268

28 Glass 152 305 45.9 0.243 80.1 0.17 3.90 1.523 48.4 0.813

29 Glass 152 305 45.9 0.243 80.1 0.17 3.90 1.915 46 1.063

30 Glass 152 305 45.9 0.243 80.1 0.34 7.81 1.639 52.8 1.203

31 Glass 152 305 45.9 0.243 80.1 0.34 7.81 1.799 55.2 1.254

32 Glass 152 305 45.9 0.243 80.1 0.51 11.71 1.594 64.6 1.554

33 Glass 152 305 45.9 0.243 80.1 0.51 11.71 1.94 65.9 1.904

34 Carbon 152 305 38 0.217 240.7 0.68 56.67 0.977 110.1 2.551

35 Carbon 152 305 38 0.217 240.7 0.68 56.67 0.965 107.4 2.613

36 Carbon 152 305 38 0.217 240.7 1.02 85.01 0.892 129 2.794

37 Carbon 152 305 38 0.217 240.7 1.02 85.01 0.927 135.7 3.082

38 Carbon 152 305 38 0.217 240.7 1.36 113.3

5 0.872 161.3 3.7

39 Carbon 152 305 38 0.217 240.7 1.36 113.3

5 0.877 158.5 3.544

40 Carbon 152 305 37.7 0.275 260 0.11 9.98 0.935 48.5 0.895

41 Carbon 152 305 37.7 0.275 260 0.11 9.98 1.092 50.3 0.914

42 Carbon 152 305 44.2 0.26 260 0.11 8.51 0.734 48.1 0.691

43 Carbon 152 305 44.2 0.26 260 0.11 8.51 0.969 51.1 0.888

44 Carbon 152 305 44.2 0.26 260 0.22 17.03 1.184 65.7 1.304

45 Carbon 152 305 44.2 0.26 260 0.22 17.03 0.938 62.9 1.025

46 Carbon 152 305 47.6 0.279 250.5 0.33 22.85 0.902 82.7 1.304

47 Carbon 152 305 47.6 0.279 250.5 0.33 22.85 1.13 85.5 1.936

48 Carbon 152 305 47.6 0.279 250.5 0.33 22.85 1.064 85.5 1.821

37

Source I

D

Type

of FRP

D

(mm

)

H

(mm

)

f'co

(MPa

)

ԑco

(%)

Efrp

(GPa

)

t

(mm) ρ ԑh,rup

f'cc

(MPa) ԑcu

Harries

&

Kharel

(2003)

49 Carbon 152 305 31.9 0.28 230 0.165 15.65 1.03 32.9 0.6

50 Carbon 152 305 31.9 0.28 230 0.33 31.31 1.19 35.8 0.8

51 Carbon 152 305 31.9 0.28 230 0.495 46.96 1.55 52.2 1.38

Shah-

way et

al

(2000)

52 Carbon 152.2 305 20 0.2 82.7 0.5 27.17 0.578 33.8 1.59

53 Carbon 152.2 305 20 0.278 82.7 1 54.34 0.578 46.4 2.21

54 Carbon 152.2 305 20 0.233 82.7 1.5 81.50 0.578 62.6 2.58

55 Carbon 152.2 305 20 0.276 82.7 2 108.7 0.578 75.7 3.56

56 Carbon 152.2 305 20 0.192 82.7 2.5 135.8 0.578 80.2 3.42

57 Carbon 152.2 305 40 0.036 82.7 0.5 13.58 0.743 59.1 0.62

58 Carbon 152.2 305 40 0.313 82.7 1 27.17 0.743 76.5 0.97

59 Carbon 152.2 305 40 0.26 82.7 1.5 40.75 0.743 98.8 1.26

60 Carbon 152.2 305 40 0.302 82.7 2 54.34 0.743 112.7 1.9

Almu-

Sallam

(2007)

61 Glass 150 300 47.7 0.2 27 1.3 9.81 0.84 56.7 1.5

62 Glass 150 300 47.7 0.2 27 3.9 29.43 0.8 100.1 2.72

63 Glass 150 300 50.8 0.2 27 1.3 9.21 1 55.5 1.21

64 Glass 150 300 50.8 0.2 27 3.9 27.64 0.8 90.8 1.88

65 Glass 150 300 60 0.2 27 1.3 7.80 0.5 62.4 0.5

66 Glass 150 300 60 0.2 27 3.9 23.40 0.7 99.6 1.67

67 Glass 150 300 80.58 0.2 27 1.3 5.81 0.24 88.9 0.36

68 Glass 150 300 80.58 0.2 27 3.9 17.42 0.86 100.9 0.63

69 Glass 150 300 90.3 0.2 27 1.3 5.18 0.26 97 0.32

70 Glass 150 300 90.3 0.2 27 3.9 15.55 0.82 110 0.93

71 Glass 150 300 107.8 0.2 27 1.3 4.34 0.3 116 0.29

72 Glass 150 300 107.8 0.2 27 3.9 13.02 0.3 125.2 0.26

38

Source ID Type

of FRP

D

(mm

)

H

(mm

)

f'co

(MPa

)

ԑco

(%)

Efrp

(GPa

)

t

(mm

)

ρ ԑh,rup

f'cc

(M

Pa)

ԑcu

Nanni

&

Brad-

ford

(1995)

73 Glass 150 300 36.3 0.2 52 0.3 5.73 46 2.29

74 Glass 150 300 36.3 0.2 52 0.3 5.73 41.2 1.89

75 Glass 150 300 36.3 0.2 52 0.6 11.46 60.5 3.09

76 Glass 150 300 36.3 0.2 52 0.6 11.46 59.2 3.41

77 Glass 150 300 36.3 0.2 52 0.6 11.46 59.8 2.74

78 Glass 150 300 36.3 0.2 52 0.6 11.46 60.2 2.89

79 Glass 150 300 36.3 0.2 52 0.6 11.46 69 3.1

80 Glass 150 300 36.3 0.2 52 0.6 11.46 55.8 2.49

81 Glass 150 300 36.3 0.2 52 0.6 11.46 56.4 2.97

82 Glass 150 300 36.3 0.2 52 1.2 22.92 84.9 3.15

83 Glass 150 300 36.3 0.2 52 1.2 22.92 84.3 4.15

84 Glass 150 300 36.3 0.2 52 1.2 22.92 73.6 4.1

85 Glass 150 300 36.3 0.2 52 2.4 45.84 106.

9 5.24

86 Glass 150 300 36.3 0.2 52 2.4 45.84 104.

6 5.45

87 Glass 150 300 36.3 0.2 52 2.4 45.84 107.

9 4.51

J.F.

Berthet

et al

(2005)

88 Carbon 160 320 22.18 0.233 230 0.165 21.39 9.57 42.8 1.633

89 Carbon 160 320 25.03 0.233 230 0.165 18.95 9.64 37.8 0.932

90 Carbon 160 320 25.03 0.233 230 0.165 18.95 9.6 45.8 1.674

91 Carbon 160 320 40.07 0.2 230 0.22 15.79 7.88 59.7 0.599

92 Carbon 160 320 40.2 0.2 230 0.22 15.73 8.33 60.7 0.693

93 Carbon 160 320 40.13 0.2 230 0.22 15.76 8.09 60.2 0.73

94 Carbon 160 320 40.18 0.2 230 0.44 31.49 9.24 91.6 1.443

95 Carbon 160 320 40.18 0.2 230 0.44 31.48 9.67 89.6 1.364

96 Carbon 160 320 40.09 0.2 230 0.44 31.55 8.85 86.6 1.166

97 Glass 160 320 40.36 0.2 74 0.22 5.04 13.69 44.8 0.526

98 Glass 160 320 40.26 0.2 74 0.22 5.05 12.46 46.3 0.467

99 Glass 160 320 40.16 0.2 74 0.22 5.07 10.75 49.8 0.496

100 Carbon 160 320 51.92 0.23 230 0.66 36.54 6.67 108 1.141

101 Carbon 160 320 52.09 0.23 230 0.66 36.43 8.71 112 1.124

102 Carbon 160 320 51.88 0.23 230 0.66 36.58 8.82 107.

9 1.121

103 Glass 160 320 20 0.233 74 0.33 15.26 16.55 42.8 1.698

104 Glass 160 320 20 0.233 74 0.33 15.26 16.43 42.3 1.687

105 Glass 160 320 20 0.233 74 0.33 15.26 16.71 43.1 1.711 Table 3-3-1 : Experimental Database (ID 01 to ID 105)

The literature review considered for this experimental database were from research studies that

made an attempt to understand the behavior of FRP-confined concrete under uniaxial compression.

39

3.2 Preliminary study

It was necessary that the procedure to be carried out for analytical and numerical modeling of the

FRP confined concrete cylinders was feasible and valid. This was tested and verified by modeling the

experimental setup of uniaxial compression test for unconfined concrete. The experimental stress-strain

curve for unconfined concrete was chosen from the CSUF Test Report# SRRS-SCCI-OP1202., March

2003. This report was very detailed about the concrete properties and the findings from the uniaxial

compression test setup. Hence the data from the CSUF Test Report# SRRS-SCCI-OP1202., March 2003

was used.

3.3 Test Database

The cylinders under consideration were 6”x12” concrete cylinders. All the cylinders were

allowed to cure for 28 days. The average compressive strength of the concrete used after 28 days of

curing was 6200 psi (42.8MPa). Following was the database used:

Specimen ID Compressive Strength, ksi (MPa) Axial Strain at Rupture

CC-1 6.14 (42.3) 0.00304

CC-2 6.00 (41.41) 0.00301

CC-3 6.37 (42.8) 0.00305

CC-4 5.28 (43.92) 0.00281

CC-5 6.3 (36.40) 0.00323

CC-6 6.2 (43.40) 0.00314

Table 3-2 : Unconfined Concrete Specimen Data (CSUF Test Report 2003)

40

Following are the experimental stress-strain curves to be used:

Fig 3.1: Experimental Stress Strain Plots (CSUF Test Report 2003)

3.4 Abaqus modeling

The concrete was modeled in Abaqus as a 3D extrude cylindrical part with dimensions as per the

the report under consideration. The concrete material was modeled as a Concrete damaged plasticity

model. The concrete compression model was chosen to be Hognestad model as described in Section 2.3.1.

The Hognestad model used in Abaqus represents the behavior of confined concrete. The Kent-Park model

used in Concrete02 material code in Opensees is derived by generalizing the Hognestad model. Hence it

was believed that using Hognestad model in Abaqus would yield the most consistent results.

CC-1

CC-2

CC-3

CC-6 CC-4

CC-5

41

The cylinder was partitioned into a concentric cylinder for the purpose of uniform meshing. The

mesh elements used were solid elements of wedge type and the interpolation was quadratic. A general

static loading condition was created in which a displacement was applied in a equal time steps. The stress-

strain data was exported from the output which was plotted with the experimental and Opensees stress

strain curve.

3.5 Opensees Modeling

The concrete cylinder was modeled in opensees using the pushover analysis code. Concrete02

model was used which follows the Kent-Park model. The analysis was performed on node to node

element which is essentially a collection of 2D elements which replicates the behavior of a 3D object as in

our case. The properties like unconfined compressive strength and ultimate strain were used from the

report under consideration.

3.6 Results and Discussion

Concrete specimen CC-1 was deemed to be an outlier pertaining to the extremely brittle failure.

For the remaining specimens the results from Opensees and Experimental curve were very much in

agreement with each other.

The initial stiffness from the Abaqus stress strain plot was much higher than that obtained from

the experimental and opensees stress strain plot. The Hognestad model used in Abaqus represents the

behavior of confined concrete. It is expected that the stiffness of confined concrete be higher. However ,

the concrete cylinders tested are unconfined. Due to this difference in the modeling and actual scenario

one could explain the difference between the stiffness obtained by Abaqus simulation and actual

experimental testing. A rigorous study for mesh sensitivity would also help us to better understand the

deviation in the stiffness in the results from Abaqus.

The failure point in the Abaqus simulation is at a higher strain that in the experimental results and

Opensees simulation. This can also be attributed to the Hognestad model exhibiting a confined concrete

42

behavior in which the confinement will cause the cylinder to rupture at a higher strain in comparison to

the unconfined concrete cylinder. The ultimate stress values were consistent in all three results. However,

the ultimate strain was lower in Abaqus simulation as compared to Opensees and Experimental results

which yielded the same ultimate strain. Also, the rupture stress is less in Abaqus simulation as compared

to Opensess and Abaqus.

The shape of the stress strain curve is however similar in all three cases except for the point

where the Hognestad model transitions from a linear behavior to a non-linear behavior. It is predicted that

choosing a Todeschini model for Abaqus might prove more accurate. This could be verified in future

studies.

Figure 3-2 : Comparison of Stress Strain Plots of Abaqus, Opensees and Experimental Data CC2

43



44



45

Chapter 4 AbaqusCAE Finite Element Modelling

4.1 Introduction

A lot of researchers have been exploring models that are accurate enough to portray the behavior

of FRP-confined concrete in order to study its behavior. Based on the theory of plasticity, some

constitutive models can capture the principle features of the highly complex nonlinear behavior of

concrete. (Pekau et al.,1992). Karabinis et al., (2002) demonstrated that the behavior of confined concrete

can be accurately demonstrated by the Drucker-Prager (DP) plasticity model. Therefore, for this study,

the Drucker-Prager model was selected to model FRP-confined concrete in Abaqus.

4.2 Concrete

The concrete cylinder is modelled as a full cylinder according to its geometric configurations as

per the information obtained from the respective published literature which is the source of the

experimental database as mentioned in Chapter 3.The cylinder is modeled as a 3D extrude part (see

Figure 5-1). The cylinder is partition into another concentric cylinder. This assembly of 2 concentric

cylinders is then partition into four equal parts. This procedure is carried out so that we can have a

uniform mesh which will help us obtaining better results at a lower computational time.

46

Figure 4-1 : Creating a Concrete cylinder 3D extrude Part in Abaqus

4.2.1. Elastic Properties

The concrete material is modeled as a isotropic material. The ACI 318-11 design equation was

chosen to input the elastic modulus of concrete and a Poisson’s ratio of 0.2 was adopted.

𝐸𝑐 = 4734√𝑓𝑐′ (in MPa) Eqn 4-1

47

4.2.2. Plastic Properties

The Extended Drucker-Prager plasticity model will be used to assign plasticity to our model. We

would be using the linear Plasticity model. The Drucker-Prager plasticity parameters based on the strain

in concrete, was implemented using the subroutine option of SDFV in Abaqus CAE. The parameters were

calculated using the equations mentioned in Section (Eqn 2-55 for friction angle, Eqn 2-56 for Cohesion

& Eqn 2-59 for the Dilation angle) and were used in each FE input file. The parameters used in Abaqus

for modelling the material Concrete are listed in Table 5-1

Material Property Notation Values used in this

study

Concrete

Elastic Modulus Ec 21171MPa to 49151MPa

Poisson's Ratio νc 0.2

Unconfined Compressive Strength f'co 20MPa to 107.8MPA

Failure Strain ԑco

Cohesion k

Dilation Angle β

Angle of Friction ϕ

Table 4-1 : Concrete DP-model parameters

48

4.3 Fiber Reinforced Polymeric Jacket

The FRP jacket is modeled as a shell in Abaqus. Abaqus allows the user to specify the shell

thickness for the FRP jacket and hence the feature of varying thickness of FRP jackets can be taken into

consideration with different specimens. The FRP sheet is modeled as elastic laminar with orthotropic

elasticity in plane stress without bending and bending stiffness. The elastic modulus is only in the

direction of the fibers which is along the circumferential direction. A Poisson’s ratio of 0 was assigned.

Modelling the FRP-tube as a shell-extrusion part and assigning it an elastic laminar behavior,

following inputs are required.

Material Property Notation

FRP

Elastic Modulus E1

Elastic Modulus E2

Poisson's Ratio νc

Shear Modulus G12

Shear Modulus G13

Shear Modulus G23

Table 4-2 : Material Properties for FRP

Figure 4-2 : FRP jacket created in Abaqus

49

4.4 Abaqus model:

4.4.1. Assembly

The 3D concrete cylinder and the FRP shell was assembled between two analytically rigid plates.

These plates were included in the analysis in order to measure the strain and stress in accordance with the

actual compression test. A reference point was created in the top plate (loading plate) in order to be able

to record the history output of displacement along the height of the cylinder. Similarly a reference point

was created in the bottom plate in order to record the reaction force.

4.4.2. Boundary Conditions & Analysis Step

Boundary conditions were applied. An encastre (fixed) condition was applied at the reference

point of bottom plate and a displacement was applied at the reference point on the top plate. A dynamic

explicit step was created as convergence performance with the Drucker-prager plasticity algorithm is

better with explicit analysis. Loading was applied by specifying the displacement as a constant rate which

is in accordance with the uniaxial compression tests carried out in the experimental program from which

we are using the experimental results.

4.4.3. Interaction

There were two types of interfaces which needed to be defined. The interaction between FRP

jacket and concrete surface was defined as a tied connection at edges and a no slip property was assigned

along the surface. As for the interaction between the plates and concrete a tie connection property was

assigned.

4.4.4. Meshing

As mentioned earlier the concrete cylinder was partitioned for the purpose of uniform meshing.

The central portion of the cylinder was assigned with tetrahedral type of elements and the outer portion

was assigned with Hex-type elements. Results were obtained according to the history output requested at

the two reference points.

50

Total number of elements present was 900 and number of nodes were 976. Followinf are the

meshing details:

Concrete Cylinder : Hexahedral Elements (C3D8R): 360

Concrete Cylinder : Tetrahedral Elements (C3D6) : 180

FRP Jacket : Quadrilateral Elements (S4R) : 360

4.5 Conclusions

During the course of modeling the uniaxial compression test of a concrete cylinder confined with

FRP in Abaqus it was learnt that there was a need for a more robust model which would very accurately

replicate the behavior of FRP confined concrete under uniaxial compression loading. The model used in

this study had the following limitations.

1) Elastic Modulus of Concrete: The elastic modulus of concrete was not available as per the

experimental testing of the concrete batch that was used in the study. Hence, the empirical

relationship which is provided by ACI 318-11 was used. This equation is found to be a good

estimate for design purposes. But with respect to this study, where there was a need to have

experimentally verified material properties in order to validate the chosen empirical model,

the use of this equation was not desirable.

2) Poisson’s Ratio: As discussed earlier, the behavior of FRP confined concrete can be

explained in terms of the Drucker-Prager plasticity model. Also, at different stages of loading

the confinement pressure exerted by the FRP would vary which would affect the Poisson’s

ratio. Providing a Poisson’s ratio of 0.2 as a constant would be an approximation. A study

needs to be carried out regarding the Poisson’s ratio at different stages of loading in order to

implement the exact material behavior in Abaqus.

51

3) Negative Dilation Rate: The equations used in the implementation of the Drucker-Prager

plasticity model in Abaqus show that the dilation rate changes from a negative value to a

regime of positive values depending on the FRP material and the grade of concrete used.

(refer fig 2-19). Abaqus, by default does not consider negative dilation rate and assumes a

dilation rate of 0 in this case. This would not represent the exact behavior of FRP confined

concrete as verified by J-F Jiang et al.,(2012).

4) Mesh Convergence Studies: A systematic mesh convergence study was needed to be carried

out. Finite element results are sensitive to the type and size of meshing that is used in the

analysis. A mesh convergence study would help in determining the optimum meshing for the

results to be accurate.

52

Chapter 5 Comparison Study and Analysis of Results

5.1 Introduction

The ultimate stress and the ultimate strain obtained by the modified Lam & Teng’s model and

from the Abaqus simulation was compared with those obtained from the experimental results separately.

The ultimate stress and strain values used in comparison study were the reported values in the literature

that was used for the experimental database. The tabulated results have been listed in Appendix A. A

comparison study was conducted between the results from the modified Lam & Teng’s model,

AbaqusCAE finite element simulation and the experimental database. From the comparison between

these three results, the accuracy of the analytical models was examined. The comparison of the model was

assessed based on the ultimate strength and ultimate strain. For design purposes it is of utmost importance

that the empirical models used predict the ultimate condition accurately. In the case of FRP confined

concrete, the failure point is accompanied by large release of energy which indicates a failure highly

brittle in nature. With respect to civil structures such a failure can be catastrophic and hence an accurate

assessment of this particular stage while design structures with FRP confined concrete is of paramount

importance.

Typical curves for various confinement ratios are shown below (from ρ = 3.9 to ρ = 135.84).

These are the representative stress-strain curves obtained for various confinement ratios. Confinement

ratio is a quantitative measure of the amount of confinement available from the FRP jacket in comparison

to the unconfined strength of concrete. It is given by the following equation.

𝜌 =2𝐸𝑓𝑡𝑓

𝐷𝑓𝑐′ Eqn 5-1

The complete set of stress-strain curves are listed in Appendix B for all the 105 specimens.

53

Figure 5-1 : Typical curves for confinement ratio 5-10






Abaqus Experimental Modified Lam & Teng






54

Figure 5-7 : Typical curves for confinement ratio 85

Figure 5-8 : Typical curves for confinement ratio 113

5.2 Performance of the Drucker-Prager Model

As mentioned earlier, the material constitutive model used for concrete in the Abaqus simulation

is the Drucker-Prager hardening model. This model essentially exhibits the hardening behavior of the

FRP confined concrete. However, in the case of FRP confined concrete the amount of hardening and its

characteristics are governed by the amount of confinement provided. The Drucker-Prager model, which

depends on parameters like friction angle and plastic dilation which are governed by the type and the

amount of confinement present as discussed in section 5.2. It was observed in this study that this model

fails to accurately represent the behavior of FRP confined concrete with lower (ρ < 5) and higher (ρ < 5)

confinement ratios.

Specimens with ID number 20, 21, 28, 29 have low confinement ratios (ρ = 3.9 to 4.52). The

experimental stress-strain curve shows a curve shape similar to an unconfined concrete cylinder

(Appendix B) . A typical curve of Specimen with ID no 28 is shown below.

Experimental Modified Lam & Teng Abaqus

Experimental Abaqus Modified Lam & Teng

55

Figure 5-9 : Comparitive stress-strain curve for ID no 28

This is expected as the amount of FRP confinement provided is not enough to impose any

significant confinement pressure on the concrete cylinder and hence the actual behavior exhibited is

similar to that of an unconfined concrete cylinder. However, the Drucker-Prager model still exhibits a

certain amount of hardening behavior which causes the Abaqus simulations not to accurately replicate the

experimental stress-strain curve. The equations governing the plastic dilation as mentioned in Section

5.2.5, show that for lower confinement ratios the plastic dilation does not change considerably throughout

the loading regime which ensures a hardening branch of the stress strain curve with a positive slope.

Specimens with ID number 36, 37, 38, 39, 54, 55 & 56 have very high confinement ratios (ρ =

85.01 to 135.84). There is a considerable deviation between the stress-strain curves obtained from the

actual experimental study and Abaqus simulations. (Figure 6-7 & 6-8) In the case of higher confinement

ratios, the complete capacity of confinement is not used.

There can be two cases of higher confinement ratios

1) High thickness of the FRP jacket: In this case the outermost fibers are not stretched to their

full capacity before the concrete core disintegrates completely. Hence, the amount of


56

hardening computed by the equations specified in Section 5.2 is much more than the actual

confinement pressure experienced by the concrete cylinder.

2) High Ef/f’c ratio : This is a case of having a low strength concrete confined by a high strength

FRP jacket. In this case the concrete core tends to significantly deteriorate before the

complete development of the FRP jacket can occur. However, the hardening criterion does

not take this into account which leads to an estimation of higher confined strength of

concrete.

From the model chosen (mentioned in Section 2.5.5.3) to represent the changing plastic dilation

angle it was evident that specimens having higher confinement ratios had a negative dilation angles (see

Figure 2-19). Abaqus does not take into consideration a negative dilation angle and assumes it to 0. This

is also believed to be a reason for the Abaqus simulation results for higher confinement ratios not

complying with the actual experimental results.

57

5.3 Performance of the Modified Lam & Teng model

The modified Lam & Teng model is a design-oriented model. The model is expected to give

accurate ultimate state conditions. It was observed that the model failed to predict the ultimate state in

cases of heavily confined concrete or specimens with high confinement ratios.

It was observed that the accuracy of the modified Lam & Teng model significantly depended on

the amount of confinement that was provided by the FRP jackets. The model performed well and could

replicate the ultimate states for confinement ratios up to ρ = 85. Specimens with ID number 36, 37, 38,

39, 54, 55 & 56 (ρ = 85.01 to 135.84) showed deviation as compared to the experiments. However, the

model tends to work considerably well in terms of specimens having low confinement ratios. The

graphical results are shown in Appendix B for all the specimens. Typical figures are shown for specimens

with lower confinement ratio and higher confinement ratios.

Figure 5-10 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = 3.9)

Abaqus Modified Lam & Teng Experimental

58

Figure 5-11 : Comparative Stress-Strain curve for Specimen ID29 (Confinement Ratio = 135.84)

As observed, Eqn 2-44 fails to predict the ultimate condition for higher confinement ratio (ρ >

85). The reason for this was believed that the equation fails to take into consideration that at higher

confinement ratios the complete FRP confinement pressure that is available is not utilized because the

concrete core disintegrates much before the complete FRP confinement is used. This leads to the Eqn 2-

44 predicting a higher ultimate strengths for highly confined specimens. There is a need for more

sophisticated equations that take into consideration the effect of higher confinement ratios.

Experimental Abaqus Modified Lam & Teng

59

Following is a plot showing the ultimate strength predicted by modified Lam & Teng’s model in

comparison to the experimental results used.

Figure 5-12 : Ultimate Strength comparison (Modified Lam & Teng vs Experimental)

The modified Lam & Teng model gives us an over prediction of strength which is not

conservative. It was observed that more than 85% of the specimens showed an accuracy up to a difference

of 15% in prediction of the ultimate strength as compared to the experimental results. Hence according to

this study the performance of modified Lam & Teng’s model in prediction of ultimate strength was

acceptable.

0

50

100

150

200

250

0 50 100 150 200 250

ρ < 5

5 < ρ < 85

ρ > 85

Experimental ԑcc (%)

Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)

Ultimate Strain Comparison (New Proposed Equation)


Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)

Ultimate Strain Comparison (New Proposed Equation) (ρ > 85)


Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)



Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)

Ultimate Strength Comparison (Modified Lam & Teng)

Modif

ied L

am &

Ten

g f

' cc(M

Pa)

Experimental f'cc

(MPa)

60

A similar plot was constructed for comparing the ultimate strain prediction by the modified Lam

& Teng’s model.

Figure 5-13 : Ultimate Strain comparison (Modified Lam & Teng vs Experimental)

From the above graph it is evident that Eqn 2-44 proposed in the modified Lam & Teng’s model

yielded non conservative values of ultimate strain. It was observed that less than 65% of the specimens

showed an accuracy upto a difference of 15% in prediction of the ultimate strain as compared to the

experimental results (Appendix A). Hence a need to modify the equation for the prediction of ultimate

strain condition was needed. Following this, a regression analysis and optimization was carried out in

order to introduce a new equation for the ultimate strain for FRP confined concrete which was in better

accordance with the experimental findings.

0

2

4

6

8

10

12

0 2 4 6 8 10 12

ρ < 5

5 < ρ < 85

ρ > 85


Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)



Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)

Ultimate Strain Comparison (New Proposed Equation) (ρ > 85)


Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)



Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)

Ultimate Strain Comparison (Modified Lam & Teng)

Modif

ied L

am &

Ten

g ԑ

cc (%

)

Experimental ԑcc

(%)

Ultimate Strain Comparison (Modified Lam & Teng vs Experimental)

61

5.4 Regression Analysis

Regression analysis is a statistical process for estimating the relationships among variables. As

seen earlier the equation provided by Modified Lam & Teng’s model (Eqn 2-44) was not able to predict

the ultimate strain condition accurately. Hence there was a need to perform a regression analysis in order

to establish a more robust relationship between the material properties of the FRP confined concrete and

the ultimate strain which agrees with the experimentally observed values for the same. Minitab17

software was used to carry out the regression analysis.

In the case of this study, it is expected that the ultimate strain depends on many variables. Hence,

this was a case of multivariable regression analysis. The same experimental database (105 data points)

used earlier was used in this case. The response for our analysis was set as the experimentally observed

values of ultimate strain. As mentioned in the literature review, Lam & Teng.,(2003) proposed the

ultimate strain equation as mentioned in Eqn 2-38

This equation was modified by Lui et al.,2013 by introducing a coefficient c2 in place of the

constant value of 1.75. Hence the ultimate strain equation was modified as mentioned in Eqn 2-44

It was determined that a further modification, with respect to this coefficient, c2 is needed in order

to improve the performance of the modified Lam & Teng’s model. Hence it was decided to carry out a

multi-regression analysis for c2.

A correlation analysis was carried out with respect to several parameters which included

unconfined concrete strength (f’c), unconfined ultimate concrete strain (ԑc), modulus of elasticity of FRP

(Efrp), thickness of the FRP jacket (tfrp), ultimate strain of FRP (ԑfrp), diameter of the specimen (D) & the

ultimate hoop strain for FRP confined concrete (ԑh,rup). A correlation analysis measures the extent to

which variables tend to change the response. The correlation analysis provides us with two values; 1)

Pearson Coefficient & 2) P-Value. Pearson Coefficient is a type of correlation coefficient that represents

the relationship between two variables that are measured on the same interval or ratio scale. The value of

62

the pearson coefficient will vary between -1 to +1. A value of 0 denotes no correlation between the

variables. One can determine the effect a particular variable on the value of response (ultimate strain, ԑcu)

based on this value. Higher the absolute value of Pearson coefficient higher is the correlation between the

two variables. The p-value is defined as the probability, under the assumption of a hypothesis, of

obtaining a result equal to or more extreme than what is actually supposed to be observed. The smaller the

p-value, the larger is the significance because it tells us that the hypothesis under consideration may not

adequately explains the observation. It is also referred to as the level of significance of a particular

variable with respect to the response variable.

Variables for the regression analysis were chosen based on these two values obtained via a

correlation analysis. The variables chosen were; unconfined concrete strength (f’c), unconfined ultimate

concrete strain (ԑc), modulus of elasticity of FRP (Efrp), thickness of the FRP jacket (tfrp), ultimate hoop

strain for FRP confined concrete (ԑh,rup). A multiple regression analysis was carried out with respect to the

5 variables stated above. This analysis considered terms upto a degree of 2 (quadratic). A diagnostic was

carried out on the regression model that was fitted by the analysis procedure. The regression model was

able to explain 88.64% of the variation in the response quantity (ultimate strain). Also the P-value was

less than 0.001 which indicates all the parameters are significant to the model.

63

Figure 5-14 : Summary Report for Multivariable regression analysis

The regression model is as follows

𝑐2 = 𝑓𝑐𝑜(0.772 − 0.001511𝐸𝑓𝑟𝑝 − 59.63)

+ ԑ𝑐𝑜(54288 − 5439055ԑ𝑐𝑜 − 46.7𝐸𝑓𝑟𝑝 − 5062𝑡𝑓𝑟𝑝 − 1877431ԑℎ𝑟𝑢𝑝)

+ ԑℎ𝑟𝑢𝑝(15609 − 233618ԑℎ𝑟𝑢𝑝) + 𝑡𝑓𝑟𝑝(21.43 − 1.94𝑡𝑓𝑟𝑝 − 10.08ԑℎ𝑟𝑢𝑝)

+ 𝐸𝑓𝑟𝑝(0.5660 − 0.00711𝐸𝑓𝑟𝑝)

Eqn 5-2

64

Figure 5-15 : Final Model performance report

Based on this equation the ultimate strain was computed and compared against the experimental

values. It was found out that more than 75% of the specimens have the predicted ultimate strain with an

error of +-15% as compared to the earlier equation which showed only less than 65% having predicted

ultimate strain with the same margin of error. One can also observe the data values on the graph below are

less scattered as compared to the ones calculated by the equation proposed in the modified Lam & Teng’s

model. (Eqn 2-44)

65

Figure 5-16 : Ultimate Strain Comparison (New proposed equation vs Experimental)

0

2

4

6

8

10

12

0 2 4 6 8 10 12

ρ < 5

5 < ρ < 85

ρ > 85


Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)



Mo

dif

ied

Lam

& T

en

g ԑ c

c (%

)


66

5.3 Field Retrofitting cases

Based on the observations of the results obtained, it was to be assessed if this model could be

used to field retrofitting design cases. As mentioned earlier the Drucker-Prager model works well with

specimens having confining ratios from ρ = 5 to ρ = 85 & the Modified Lam & Teng’s model works well

for specimens having confining ratios less than ρ = 85. Hence there was a need for assessing if the above

mentioned models could be applied to the actual retrofitted bridge columns. Appendix D -1 and D-2 show

the data of 157 bridge retrofitted columns. These data were collected from various research publications

mentioned below which conducted a study on actual bridge columns retrofitted with FRP jackets and

from CalTrans design guidelines.

Further data was collected from the design guideline for FRP confined concrete columns in the

bridge manuals as a bridge standard detailing sheets published by Caltrans. A few documents published

by Caltrans for the design guide for bridge piers were studied. The bridge manual for standard designs for

column casing using FRP composite system provides standard FRP jacketing system used for design with

varying geometry of the column. The Standard Specifications (2010) published by the Department of

Transportation of State of California mentions in section 51-1.03B that the strength of the concrete should

be more than 3250 psi (22MPa) if it has to permit vehicles weighing more than 4000lbs. This is the

average weight of a sedan. Assuming bridges on interstates which are designed for multi-axel vehicles

weighing much more than 4000lbs the grade of concrete expected to be more than 22MPa. Following is a

histogram (fig 6-17) showing the distribution of retrofitted bridge columns with respect to confinement

ratios.

67

Figure 5-17 : Histogram for retrofitted bridge columns based on confinement ratios.

One can observe that the more than 85% of the bridge columns that are retrofitted with FRP, have

confinement ratios lower than 35. (ρ < 35). It was also observed that the retrofitted bridge column data

used for this study have confinement ratios less than ρ = 80. An observation can be made that the design

specifications do not allow for a higher confinement ratios where the modified Lam & Teng’s model is

found not to perform well.

However, the above study is carried out for a pure concrete cylinder. The behavior of reinforced

confined with steel reinforcement will not be the same as before. Presence of confining steel is going to

cause to the slope of the first zone to change until a point where steel yields. Also the ultimate strength of

the specimen will be a contribution from the steel confining and FRP wrap confining. Future studies

should modify the equations for the model state above to include the effect of steel reinforcement

(longitudinal & transverse).

0

10

20

30

40

50

60

5 10 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

No

of

retr

ofi

tte

d b

rid

ge c

olu

mn

s

Confinement Ratios

Histogram for Retrofitted bridge columns based on confinement ratios

68

Chapter 6 Conclusions

6.1 Summary

We discussed a few empirical models a few researchers have proposed. Based on a literature

review we chose to carry a comparative study on the empirical model; the modified Lam & Tengs model

(Liu et al.,2013), a database of experimental results and results from finite element simulation procedure

carried out in AbaqusCAE using the Drucker-Prager model for FRP confined concrete.

The objective of the study is to validate the results obtained by these models in comparison of the

experimental database collected and Abaqus simulation. Each of the 105 specimens mentioned earlier in

the experimental database was modeled in AbaqusCAE. This procedure was described in Chapter 5. The

specimens were modeled according to their geometric & material properties. The stress-strain results so

obtained were plotted and compared with the experimental stress-strain curves and the one obtained by

implementing the mathematical equations proposed by Liu et al.,(2013). Key parameters which affect the

stress-strain behavior of concrete cylinders confined with fiber reinforced plastics were thoroughly

investigated. These key parameters included the ultimate condition of the stress-strain curve. A validation

study is performed on this model and the behavior of the same is to be compared with Abaqus and

Experimental data.

69

6.2 Conclusions

1) Confinement Ratio effect: The validation analysis for the modified Lam & Teng’s model

indicated that the performance of this empirical model depended highly on the confinement ratio.

The empirical model failed to replicate the actual behavior of highly confined concrete cylinders.

According to the observations in this study, the modified Lam & Teng’s model fails to predict

results in accordance with actual behavior for specimens having confinement ratio higher than 85.

However, as mentioned in section 6.3, in actual practice it is observed that the bridge columns

that are retrofitted with FRP jackets have a lower confinement ratio. Hence, one would not be

designing columns retrofitted with confinement ratios higher than 85.

2) Non conservative prediction: As observed from Figure 6-12 and 6-13 most of the data points lie

above the x=y line. This means that the modified Lam & Teng’s model over estimates the

ultimate condition (both the ultimate strength and ultimate strain) of the FRP confined concrete.

For design purposes it is necessary that the model predict the ultimate condition on the

conservative side. The model is over conservative.

3) Ultimate Strain Equation: The ultimate strain equation provided by modified Lam & Teng’s

model yielded inaccurate results. There was a considerable deviation (±15%) for 35% of the

specimens in comparison with the experimentally observed values. The importance of predicting

the ultimate strain accurately was discussed in the previous section. Hence a need for a new

equation which more accurately predicts the ultimate strain was needed. In this regard a

regression analysis was carried out based on the same experimental database that is used for this

study. A new modified equation was proposed which is derived from the equation proposed in the

modified Lam & Teng’s model.

70

ԑ𝑐𝑢 = 𝑐2ԑ𝑐𝑜 + 12 (𝑓𝑙,𝑎

𝑓𝑐𝑜∗ ) ԑ0.45ℎ,𝑟𝑢𝑝 ԑ𝑐0

0.55

Where,

𝑐2 = 𝑓𝑐𝑜(0.772 − 0.001511𝐸𝑓𝑟𝑝 − 59.63)

+ ԑ𝑐𝑜(54288 − 5439055ԑ𝑐𝑜 − 46.7𝐸𝑓𝑟𝑝 − 5062𝑡𝑓𝑟𝑝 − 1877431ԑℎ𝑟𝑢𝑝)

+ ԑℎ𝑟𝑢𝑝(15609 − 233618ԑℎ𝑟𝑢𝑝) + 𝑡𝑓𝑟𝑝(21.43 − 1.94𝑡𝑓𝑟𝑝 − 10.08ԑℎ𝑟𝑢𝑝)

+ 𝐸𝑓𝑟𝑝(0.5660 − 0.00711𝐸𝑓𝑟𝑝)

Eqn 6-1

Using this equation more than 75% of the specimens have predicted ultimate strain with a

deviation of less than +- 15%. This study concludes that Eqn 6-1 should be used to calculate the ultimate

strain for FRP confined concrete which has an improved performance in prediction of ultimate strain.

71

References

American Concrete Institute (ACI) Committee 318.(2011) “ Building Code Requirements for Reinforced

Concrete.” ACI 318-11, American Concrete Institute, Farmington Hills, Michigan.

Ahmad, S. H., and Shah, S. P. (1982). “Stress-strain curves of concrete confined by spiral reinforcement.”

ACI J., 79(6), 484–490

Almusallam, Tarek H. "Behavior of normal and high-strength concrete cylinders confined with E-

glass/epoxy composite laminates." Composites Part B: Engineering 38.5 (2007): 629-639.

Berthet, J. F., E. Ferrier, and P. Hamelin. "Compressive behavior of concrete externally confined by

composite jackets. Part A: experimental study."Construction and Building Materials 19.3 (2005): 223-

232.

Benzaid, R., Mesbah, H. and Chikh, N.E. (2010), “FRP-confined concrete cylinders: axial compression

experiments and strength model”, J. Reinf. Plast. Compos., 29(16), 2469-2488.

Brena, S. F., and Schlick, B. M. _2007_. “Hysteretic behavior of bridge columns with FRP-jacketed lap

splices designed for moderate ductility enhancement.” J. Compos. Constr., 11_6_, 565–574.

Chang, K. C., et al. "Seismic retrofit study of RC bridge columns." International Training Program for

Seismic Design of Building Structures (2000).

Drucker, D.C.., Prager, W., 1952. Soil Mechanics and plastic analysis or limit design. Quart. Appl. Math.

10., 157-165

De Lorenzis L., and Tepfers R.. Comparative study of models on confinement of concrete cylinders with

fiber-reinforced polymer composites. J. Compos. Constr., 2003, 7(3), 219-234.

De Lorenzis, L., F. Micelli, and A. La Tegola. "4.1 Influence of specimen size and resin type on the

behaviour of FRP-confined concrete cylinders." Advanced Polymer Composites for Structural

Applications in Construction: Proceedings of the First International Conference, Held at Southampton

University, UK, on 15-17 April 2002. Thomas Telford, 2002.

Elsanadedy, Hussein M. Seismic performance and analysis of ductile composite-jacketed reinforced

concrete bridge columns. Diss. University of California, Irvine, 2002.

Elwi, A. A., and Murray, D. W. (1979). "A 3D hypoelastic concrete constitutive relationship." /. Engrg.

Mech. Div., ASCE, 105(4), 623-641.

Fahmy, Mohamed FM, Zhishen Wu, and Gang Wu. "Seismic performance assessment of damage-

controlled FRP-retrofitted RC bridge columns using residual deformations." Journal of Composites for

construction 13.6 (2009): 498-513.

Fib, (federation international du beton). (2001), “Bulletin 14, Externally bonded FRP reinforcement for

RC structures.” Technical report, Case Postale 88, CH-1015 Lausanne July, 2001.

72

Ghosh, K. K., and Sheikh, S. A. _2007_. “Seismic upgrade with carbon fiber reinforced polymer of

columns containing lap-spliced reinforcing bars.” ACI Struct. J., 104_2_, 227–236.

Harajli, M. H. _2008_. “Seismic behavior of RC columns with bondcritical regions: Criteria for bond

strengthening using external FRP

jackets.” J. Compos. Constr., 12_1_, 69–79.

Haroun, Medhat A., and Hussein M. Elsanadedy. "Fiber-reinforced plastic jackets for ductility

enhancement of reinforced concrete bridge columns with poor lap-splice detailing." Journal of Bridge

Engineering 10.6 (2005): 749-757.

Harries, Kent A., and Gayatri Kharel. "Experimental investigation of the behavior of variably confined

concrete." Cement and Concrete research 33.6 (2003): 873-880.

Imran, I., and Pantazopoulou, S. (1996). ‘‘Experimental study of plain concrete under triaxial stress.’’

ACI Mat. J., 93(6), 589–601.

J.G. Teng, T. Yu, Y.L. Wong, and S.L. Dong (2007). Hybrid FRP-concrete–steel tubular columns:

Concept and behaviour.

Construction and Building Materials. 21(4):846–54.

Jiang, Jia-Fei, and Yu-Fei Wu. "Identification of material parameters for Drucker–Prager plasticity model

for FRP confined circular concrete columns."International Journal of Solids and Structures 49.3 (2012):

445-456.

Karabinis, A. I., and T. C. Rousakis. "Concrete confined by FRP material: a plasticity

approach." Engineering structures 24.7 (2002): 923-932.

Karabinis, Athanasios I., Theodoros C. Rousakis, and Georgia E. Manolitsi. "3D finite-element analysis

of substandard RC columns strengthened by fiber-reinforced polymer sheets." Journal of Composites for

Construction 12.5 (2008): 531-540.

Kent, D.C. and Park, R. (1971), “Flexural members with confined concrete”, J. Struct. Division, 97(7),

1969-1990

Park, Kihoon, & Maria Lopez. Modeling of Concrete Confined with FRP Wraps. Thesis. The

Pennsylvania State University, University Park, 2004

Popovics, Sandor. "A numerical approach to the complete stress-strain curve of concrete." Cement and

concrete research 3.5 (1973): 583-599.

Kawashima, K. _2000_. “Seismic design and retrofit of bridges.” Proc., 12th World Conf. on Earthquake

Engineering (WCEE) _CD-ROM_, New Zealand Society for Earthquake Engineering, Auckland, New

Zealand.

Krishan, A. L., and M. A. Krishan. "Strength of Axially Loaded Concrete-Filled Steel Tubular Columns

with Circular Cross-Section." Advances in Environmental Biology 8.7 (2014): 1991-1994.

Lam, L., Teng, J.G., 2003. Design-oriented stress-strain model for FRP-confined concrete. Construction

and Building Materials, 17(6-7):471-489. [doi:10.1016/S0950-0618 (03)00045-X]

73

Lam, L., Teng, J.G., Cheng, C.H. and Xiao, Y. (2006). “FRP-confined concrete under axial cyclic

compression”, Cement and Concrete Composites, 28(10), 949-958.

Lan, S., and Guo, Z. (1997). “Experimental investigation of multiaxial compressive strength of concrete

under different stress paths.” ACI Mater. J., 94(5), 427–434

Li, Guoqiang. "Experimental study of FRP confined concrete cylinders."Engineering structures 28.7

(2006): 1001-1008.

Liu, Hui, et al. "A modified constitutive model for FRP confined concrete in circular sections and its

implementation with OpenSees programming." Journal of Zhejiang University SCIENCE A 14.12 (2013):

856-866.

MacGregor, J.G. (1992). Reinforced Concrete: mechanics and design, Prentice-Hall, Inc., New Jersey.

Mander, J.B., Priestley, M.J.N., Park, R., 1988b. Theoretical stress-strain model for confined concrete.

Journal of Structural Engineering, 114(8):1804-1826. [doi:10.1061/ (ASCE)0733-9445(1988)114:8(1804)

Matthys, Stijn, Luc Taerwe, and Katrien Audenaert. "Tests on axially loaded concrete columns confined

by fiber reinforced polymer sheet wrapping." ACI Special Publication 188 (1999).

Micelli, Francesco, and Rossella Modarelli. "Experimental and analytical study on properties affecting the

behaviour of FRP-confined concrete." Composites Part B: Engineering 45.1 (2013): 1420-1431.

Nanni, Antonio, and Nick M. Bradford. "FRP jacketed concrete under uniaxial

compression." Construction and Building Materials 9.2 (1995): 115-124.

Ozbakkaloglu, Togay, Jian C. Lim, and Thomas Vincent. "FRP-confined concrete in circular sections:

Review and assessment of stress–strain models." Eng Struct 49 (2013): 1068-1088.

Park, R., Negel Priestly, M.J., Gill, W.D., 1982. Ductility of square-confined concrete columns. Journal

of the Structural Division, 108(4):929-950.

Pekau, O. A., Z. X. Zhang, and G. T. Liu. "Constitutive model for concrete in strain space." Journal of

engineering mechanics 118.9 (1992): 1907-1927.

Richard, R. M., and Abbott, B. J. (1975). “Versatile elastic-plastic stress strain formula.” J. Eng. Mech.

Div., 101(4), 511–515.

Rochette, Pierre, and Pierre Labossiere. "Axial testing of rectangular column models confined with

composites." Journal of Composites for Construction 4.3 (2000): 129-136.

Saadatmanesh, Hamid, Mohammad R. Ehsani, and Mu-Wen Li. "Strength and ductility of concrete

columns externally reinforced with fiber composite straps."ACI Structural journal 91.4 (1994).

Samaan, M., Mirimiran, A., Shahawy, M., 1998. Model of concrete confined by fiber composites. Journal

of Structural Engineering, 124(9):1025-1031. [doi:10.1061/(ASCE)0733-9445(1998)124:9(1025)]

Seible, Frieder, et al. "Seismic retrofit of RC columns with continuous carbon fiber jackets." Journal of

composites for construction 1.2 (1997): 52-62.

74

Shahawy, M., A. Mirmiran, and T. Beitelman. "Tests and modeling of carbon-wrapped concrete

columns." Composites Part B: Engineering 31.6 (2000): 471-480.

Shams, M., Saadeghvaziri, M.A., 1997. State of the art of concrete-filled steel tubular columns. ACI

Structural Journal, 94(5):558-571.

Shan, Bo, Yan Xiao, and Yurong Guo. "Residual performance of FRP-retrofitted RC columns after being

subjected to cyclic loading damage." Journal of Composites for construction 10.4 (2006): 304-312.

Sheikh, Shamim A., and Grace Yau. "Seismic behavior of concrete columns confined with steel and fiber-

reinforced polymers." ACI Structural Journal 99.1 (2002).

Spoelstra, M.R., and Monti, G. (1999). “FRP-confined concrete model”, Journal of Composites for

Construction, ASCE, 3(3), 143-150.

T. Jiang and J.G. Teng (2007). Analysis-oriented stress-strain models for FRP-confined concrete.

Engineering Structures.

29(11), pp. 2968-2986.

Toutanji, H., 1999. Stress-strain characteristics of concrete columns externally confined with advanced

fiber composite sheets. ACI Materials Journal, 96(3):397-404.

Triantafyllou, Garyfalia G., Theodoros C. Rousakis, and Athanasios I. Karabinis. "Axially Loaded

Reinforced Concrete Columns with a Square Section Partially Confined by Light GFRP Straps." Journal

of Composites for Construction (2014).

Wang, Lei-Ming, and Yu-Fei Wu. "Effect of corner radius on the performance of CFRP-confined square

concrete columns: Test." Engineering Structures 30.2 (2008): 493-505.

Watanabe, K., et al. "Confinement effect of FRP sheet on strength and ductility of concrete cylinders

under uniaxial compression." Non-metallic (FRP) reinforcement for concrete structures 1 (1997): 233-

240.

Wu, G., Wu, Z. S., Lu, Z. T., and Gu, D. S. _2006a_. “Seismic retrofit of large scale circular RC columns

wrapped with CFRP sheets.” Proc.,3rd Int. Conf. on FRP Composites in Civil Engineering (CICE

2006)_CD-ROM_, Dept. of Civil and Environmental Engineering, Florida International Univ., Miami,

547–550.

Wu, G., Gu, D. S., Wu, Z. S., Jiang, J. B., and Hu, X. Q. _2007_. “Comparative study on seismic

performance of circular concrete columns strengthened with BFRP and CFRP composites.” Proc., Asia-

Pacific Conf. on FRP in Structures (APFIS 2007), Vol. 1, Dept. of Civil Engineering, The Univ. of Hong

Kong, Hong Kong, China, 199–204.

Wu, Z. S., Gu, D. S., Wu, G., and Hirahata, H. _2006b_. “Seismic performance of RC columns

strengthened with dyneema fiber-reinforced polymer sheets.” Proc., 4th Int. Conf. of Earthquake

Engineering (ICEE2006), National Center for Research on Earthquake Engineering _NCREE_, Taipei,

Taiwan.

Xiao, Yan, Hui Wu, and G. R. Martin. "Prefabricated composite jacketing of RC columns for enhanced

shear strength." Journal of structural engineering 125.3 (1999): 255-264.

75

Xiao, Yan, and Rui Ma. "Seismic retrofit of RC circular columns using prefabricated composite

jacketing." Journal of structural engineering 123.10 (1997): 1357-1364.

Yu, T., et al. "Finite element modeling of confined concrete-I: Drucker–Prager type plasticity

model." Engineering Structures 32.3 (2010): 665-679.

76

Appendix A : Numerical and Analytical modeling results (Tabular)

ID Experimental Abaqus Results

Modified Lam & Teng

Ratio f'cc

(Abaqus)/f'cc (Exp)

Ratio f'cc

(MLT)/f'cc (Exp) f'cc

(MPa) ԑcu (%)

f'cc (MPa)

ԑcu (%) f'cc

(MPa) ԑcu (%)

1 47.2 1.106 58.48 0.902 55.49 0.898 1.24 1.18

2 53.2 1.292 58.48 0.902 55.49 0.898 1.1 1.04

3 50.4 1.273 58.48 0.902 55.49 0.898 1.16 1.1

4 71.6 1.85 73.24 1.616 75.08 1.610 1.02 1.05

5 68.7 1.683 73.24 1.616 75.08 1.610 1.07 1.09

6 69.9 1.962 73.24 1.616 75.08 1.610 1.05 1.07

7 82.6 2.046 104.2 2.32 92.99 2.323 1.26 1.13

8 90.4 2.413 104.2 2.32 92.99 2.323 1.26 1.03

9 97.3 2.516 104.2 2.32 92.99 2.323 1.26 0.96

10 51.9 1.315 63.14 1.440 60.17 1.435 1.22 1.16

11 58.3 1.459 63.14 1.440 60.17 1.435 1.08 1.03

12 77.3 2.188 77.88 2.640 76.60 2.648 1.01 0.99

13 75.7 2.457 77.88 2.640 76.60 2.648 1.03 1.01

14 52.6 0.9 66.55 1.002 61.19 0.997 1.27 1.16

15 57 1.21 66.55 1.002 61.19 0.997 1.17 1.07

16 55.4 1.11 66.55 1.002 61.19 0.997 1.2 1.1

17 76.8 1.91 79.16 1.851 80.08 1.837 1.03 1.04

18 79.1 2.08 79.16 1.851 80.08 1.837 1 1.01

19 65.8 1.25 79.16 1.851 80.08 1.837 1.2 1.22

20 41.5 0.825 63.37 0.848 44.01 0.845 1.53 1.06

21 40.8 0.942 63.37 0.848 44.01 0.845 1.55 1.08

22 54.6 2.13 64.12 1.428 59.17 1.427 1.17 1.08

23 56.3 1.825 64.12 1.428 59.17 1.427 1.14 1.05

24 65.7 2.558 70.89 2.012 67.71 2.009 1.08 1.03

25 60.9 1.792 70.89 2.012 67.71 2.009 1.16 1.11

26 42.4 1.303 48.85 1.090 40.26 1.092 1.15 0.95

27 41.6 1.268 48.85 1.090 40.26 1.092 1.17 0.97

28 48.4 0.813 66.18 0.723 47.43 0.710 1.37 0.98

29 46 1.063 66.18 0.723 47.43 0.710 1.44 1.03

30 52.8 1.203 71.72 1.182 62.95 1.178 1.36 1.19

77

ID Experimental Abaqus Results Modified Lam & Teng

Ratio f'cc

(Abaqus)/f'cc (Exp)

Ratio f'cc (MLT)/f'cc

(Exp) f'cc (MPa)

ԑcu (%) f'cc

(MPa) ԑcu (%) f'cc (MPa) ԑcu (%)

31 55.20 1.254 71.72 1.182 62.95 1.178 1.3 1.14

32 64.60 1.554 79.13 1.648 76.42 1.645 1.22 1.18

33 65.90 1.904 79.13 1.648 76.42 1.645 1.2 1.16

34 110.10 2.551 116.29 2.952 115.57 2.963 1.06 1.05

35 107.40 2.613 116.29 2.952 115.57 2.963 1.08 1.08

36 129.00 2.794 186.48 4.322 154.36 4.336 1.45 1.2

37 135.70 3.082 186.48 4.322 154.36 4.336 1.37 1.14

38 161.30 3.700 294.87 5.706 193.15 5.709 1.83 1.2

39 158.50 3.544 294.87 5.706 193.15 5.709 1.86 1.22

40 48.50 0.895 54.67 0.798 49.59 0.809 1.13 1.02

41 50.30 0.914 54.67 0.798 49.59 0.809 1.09 0.99

42 48.10 0.691 58.98 0.689 53.71 0.686 1.23 1.12

43 51.10 0.888 58.98 0.689 53.71 0.686 1.15 1.05

44 65.70 1.304 76.04 1.114 71.31 1.112 1.16 1.09

45 92.90 1.025 76.04 1.114 71.31 1.112 0.82 0.77

46 82.70 1.304 89.91 1.473 86.78 1.465 1.09 1.05

47 85.50 1.936 89.91 1.473 86.78 1.465 1.05 1.01

48 85.50 1.821 89.91 1.473 86.78 1.465 1.05 1.01

49 32.90 0.600 45.7579 0.752 37.75 0.750 1.39 1.14

50 35.80 0.800 58.6883 1.2152 46.646 1.2 1.63 1.3

51 52.20 1.380 71.52 1.678 54.00618 1.66 1.37 1.03

52 33.80 1.590 40.41 1.942 39.00 2.006 1.2 1.15

53 46.40 2.210 78.67 4.591 58.47 4.593 1.7 1.26

54 62.60 2.580 134.51 6.128 78.19 6.134 2.15 1.25

55 75.70 3.560 269.5714 8.9003 97.79 8.901 3.56 1.29

56 80.20 3.420 321.35 8.999 117.39 9.026 4.01 1.46

57 59.10 0.620 59.65 0.299 73.08 0.783 1.01 1.24

58 76.50 0.970 97.11 1.918 88.20 1.919 1.27 1.15

59 98.80 1.260 115.97 2.440 107.79 2.435 1.17 1.09

60 112.70 1.900 146.62 3.441 127.39 3.450 1.3 1.13

61 56.70 1.500 78.39 1.099 70.10 1.109 1.38 1.24

62 100.10 2.720 133.39 2.921 112.00 2.926 1.33 1.12

63 55.50 1.210 82.15 1.050 71.95 1.042 1.48 1.3

64 90.80 1.880 118.73 2.720 115.10 2.725 1.31 1.27

65 62.40 0.500 90.87 0.890 77.21 0.884 1.46 1.24

66 99.60 1.670 126.46 2.203 124.30 2.253 1.27 1.25

67 88.90 0.360 101.14 0.660 88.12 0.661 1.14 0.99

68 100.90 0.630 155.19 1.589 145.10 1.584 1.54 1.44

78

ID Experimental Abaqus Results

Modified Lam & Teng

Ratio f'cc (Abaqus)/f'cc

(Exp)

Ratio f'cc

(MLT)/f'cc (Exp)

f'cc (MPa) ԑcu (%) f'cc (MPa) ԑcu (%) f'cc

(MPa) ԑcu (%)

69 97 0.32 105.15 0.599 92.76 0.594 1.08 0.96

70 110 0.93 167.07 1.390 150.90 1.382 1.52 1.37

71 116 0.29 121.95 0.508 100.85 0.502 1.05 0.87

72 125.2 0.26 186.34 1.112 158.92 1.105 1.49 1.27

73 46 2.29 52.47 0.806 45.18 0.869 1.14 0.98

74 41.2 1.89 52.47 0.860 45.18 0.869 1.27 1.1

75 60.5 3.09 59.86 1.539 58.06 1.539 0.99 0.96

76 59.2 3.41 59.86 1.539 58.06 1.539 1.01 0.98

77 59.8 2.74 59.86 1.539 58.06 1.539 1 0.97

78 60.2 2.89 59.86 1.539 58.06 1.539 0.99 0.96

79 69 3.1 59.86 1.539 58.06 1.539 0.87 0.84

80 55.8 2.49 59.86 1.539 58.06 1.539 1.07 1.04

81 56.4 2.97 59.86 1.539 58.06 1.539 1.06 1.03

82 84.9 3.15 70.19 2.880 79.81 2.878 0.83 0.94

83 84.3 4.15 70.19 2.880 79.81 2.878 0.83 0.95

84 73.6 4.1 70.19 2.880 79.81 2.878 0.95 1.08

85 106.9 5.24 123.52 5.511 123.33 5.556 1.16 1.15

86 104.6 5.45 123.52 5.511 123.33 5.556 1.18 1.18

87 107.9 4.51 123.52 5.511 123.33 5.556 1.14 1.14

88 42.8 1.633 38.13 1.205 37.37 1.201 0.89 0.87

89 37.8 0.932 41.76 1.075 40.22 1.078 1.1 1.06

90 45.8 1.674 41.76 1.075 40.22 1.079 0.91 0.88

91 59.7 0.599 62.40 0.842 60.32 0.801 1.05 1.01

92 60.7 0.693 62.27 0.793 60.45 0.798 1.03 1

93 60.2 0.73 62.16 0.792 60.39 0.798 1.03 1

94 91.6 1.443 79.46 1.399 80.68 1.396 0.87 0.88

95 89.6 1.364 79.46 1.399 80.68 1.396 0.89 0.9

96 86.6 1.166 79.46 1.399 80.68 1.396 0.92 0.93

97 44.8 0.526 73.15 1.153 51.77 1.147 1.63 1.16

98 46.3 0.467 73.15 1.153 51.77 1.147 1.58 1.12

99 49.8 0.496 73.15 1.153 51.77 1.147 1.47 1.04

100 108 1.141 106.97 1.634 112.63 1.639 0.99 1.04

101 112 1.124 106.97 1.634 112.63 1.639 0.96 1.01

102 107.9 1.121 106.97 1.634 112.63 1.639 0.99 1.04

103 42.8 1.698 40.56 3.625 42.14 3.637 0.95 0.98

104 42.3 1.687 40.56 3.625 42.14 3.637 0.96 1

105 43.1 1.711 40.56 3.625 42.14 3.637 0.94 0.98

Table 0-1 : Comparitive results (ID01 to ID105)

79

Appendix B : Numerical and Analytical modeling results (Graphical)

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

***Experimental graphs were not available for specimens ID73, ID74, ID97, ID98 and ID99.

106

Appendix C : Graphical comparison with Confinement ratios

Following are the results plotted for the stress-strain behavior of concrete cylinders confined with

fiber reinforced concrete. The experimental results are not available for specimens ID73, ID74, ID97,

ID98 & ID99.

The following graphical representation was plotted in order to show the performance of both the models

with the ratio of the predicted ultimate strength to the experimental ultimate strength plotted against the

confinement ratio.

Figure App-C-1 : Plot showing the variation of the ratio of the compressive strengths predicted to

the experimental compressive strengths with respect to the confinement ratio

One can observe that the prediction of both the Abaqus results and the modified Lam & Tengs

results yields accurate predictions with respect to the ultimate strength of confined concrete for the

specimens having confinement ratios ranging from 10 to 45

107

Appendix D-1 : Bridge Retrofitted Data

Following is a table showing the data of retrofitted bridge columns.

Source Specimen D (mm) L (mm) f'c (Mpa) Type of FRP Ef (Gpa) tf (mm) ρ

Seible et al (1997) 1 610 3658 34.5 CFRP 124 6.12 72.11974

2 610 3658 34.5 CFRP 124 5.1 60.09979

Xiao and Ma (1997)

3 610 2440 44.8 GFRP 48.3 9.6 33.93443

4 610 2440 44.8 GFRP 48.3 12.8 45.2459

5 610 2440 44.8 GFRP 48.3 3.6 12.72541

Xiao et al (1999) 6 610 1830 37.1 GFRP 38 7.62 25.58968

7 610 1830 37.1 GFRP 38 5.08 17.05979

Kawashima et al (2000)

8 400 1360 30 CFRP 230 0.11 4.216667

9 400 1360 37.5 CFRP 230 0.22 6.746667

10 400 1360 30 CFRP 230 0.11 4.216667

11 400 1360 27.5 CFRP 230 0.22 9.2

Sheikh and Yua (2002)

12 356 1470 40.4 GFRP 19.7 1.25 3.424324

13 356 1470 40.4 GFRP 19.7 2.5 6.848648

14 356 1470 40.4 CFRP 72.5 1 10.08177

15 356 1470 44.8 CFRP 145 0.5 9.091593

16 356 1470 40.8 GFRP 19.7 1.25 3.390752

17 356 1470 41.6 CFRP 72.5 1 9.790946

18 356 1470 42.8 GFRP 19.7 2.5 6.464612

19 356 1470 43.9 CFRP 72.5 1 9.277981

Elsanadedy (2002)

20 610 2440 44.8 GFRP 48.2 12.7 44.79947

21 610 2440 44.8 GFRP 48.2 15.9 56.08753

22 610 2440 31 GFRP 37.9 12.7 50.90746

23 610 2440 31 GFRP 37.9 12.7 50.90746

24 610 1830 37.1 CFRP 111.6 4.3 42.40908

25 1829 3660 29.6 GFRP 23.4 9.8 8.471621

26 610 2440 39.3 CFRP 115.1 4.1 39.37012

Chang et al (2003)

27 760 3250 20 CFRP 236 0.83 25.77368

28 760 3250 20 CFRP 236 0.55 17.07895

29 760 3250 20 CFRP 236 0.55 17.07895

30 760 1750 16.7 CFRP 236 0.55 20.45383

31 760 1750 16.7 CFRP 236 0.41 15.2474

32 760 1750 16.7 CFRP 236 0.28 10.41286

108

Source Specimen D (mm) L (mm) f'c (Mpa) Type of FRP Ef (Gpa) tf (mm) ρ

Haroun and Elsanadedy

(2005b)

33 610 3657.5 36 CFRP 231.5 0.7 14.75865

34 610 3657.5 36.9 CFRP 230.1 0.7 14.31161

35 610 3657.5 32.8 GFRP 36.5 11.4 41.59336

36 610 3657.5 37.7 CFRP 226 1.7 33.41305

37 610 3657.5 39.7 GFRP 36.4 12.7 38.17814

38 610 3657.5 33.1 CFRP 63 8.3 51.79535

39 610 2440 40.8 CFRP 231.5 0.7 13.02234

40 610 2440 39.2 CFRP 230.1 0.7 13.4719

41 610 2440 34.2 GFRP 18.5 10.3 18.26766

42 610 2440 37.6 CFRP 103.8 1.2 10.86153

43 610 2440 35.7 CFRP 103.8 2.3 21.92589

Shan et al (2006) 44 375 1500 30.9 CFRP 220 0.88 33.41532

45 375 1500 38.7 GFRP 33 4 18.19121

Wu et al (2006a)

46 360 800 34.9 CFRP 249.6 0.17 6.754537

47 360 800 34.9 CFRP 249.6 0.42 16.68768

48 360 800 34.9 CFRP 249.6 0.75 29.79943

Wu et al (2006b)

49 360 1100 34.9 DFRP 59.6 0.39 3.700096

50 360 1100 34.9 DFRP 59.6 0.65 6.166826

51 360 800 34.9 DFRP 59.6 1.03 9.772047

Wu et al (2007) 52 360 800 34.9 BFRP 106 2 33.74721

Brena and Schlick (2007)

53 240 1085 23.9 CFRP 227 0.02 1.582985

54 240 1085 23.9 AFRP 120 0.03 1.25523

55 240 1085 23.9 CFRP 227 0.02 1.582985

56 240 1085 23.9 AFRP 120 0.03 1.25523

Ghosh and Sheikh (2007)

57 356 2010 24.9 CFRP 79 1 17.82411

58 356 2010 25.1 CFRP 79 1 17.68208

59 356 2010 26.5 CFRP 79 1 16.74793

Harajli (2008) 60 200 1800 39 CFRP 230 0.13 7.666667

61 200 1800 39 CFRP 230 0.26 15.33333

109

Appendix D-2 : CalTrans retrofitting guidelines

Following is a table showing the possible combinations of bridge retrofitted columns according to

CalTrans design guidelines.

GFRP CFRP

D (mm) f'c (Mpa)

Ef (Gpa)

tf (mm) ρ

D (mm) f'c (Mpa)

Ef (Gpa)

tf (mm) ρ

304.8 30 80 0.4953 8.666667 304.8 30 230 0.4953 24.91667

304.8 30 80 0.4953 8.666667 304.8 30 230 0.4953 24.91667

609.6 30 80 0.4953 4.333333 609.6 30 230 0.4953 12.45833

609.6 30 80 0.9906 8.666667 609.6 30 230 0.9906 24.91667

914.4 30 80 0.6604 3.851852 914.4 30 230 0.6604 11.07407

914.4 30 80 1.3208 7.703704 914.4 30 230 1.3208 22.14815

1219.2 30 80 0.9906 4.333333 1219.2 30 230 0.9906 12.45833

1219.2 30 80 1.8161 7.944444 1219.2 30 230 1.8161 22.84028

1524 30 80 1.1557 4.044444 1524 30 230 1.1557 11.62778

1524 30 80 2.3114 8.088889 1524 30 230 2.3114 23.25556

1828.8 30 80 1.3208 3.851852 1828.8 30 230 1.3208 11.07407

1828.8 30 80 2.6416 7.703704 1828.8 30 230 2.6416 22.14815

304.8 40 80 0.3302 4.333333 304.8 40 230 0.4953 18.6875

304.8 40 80 0.6604 8.666667 304.8 40 230 0.4953 18.6875

609.6 40 80 0.6604 4.333333 609.6 40 230 0.4953 9.34375

609.6 40 80 1.1557 7.583333 609.6 40 230 0.9906 18.6875

914.4 40 80 0.9906 4.333333 914.4 40 230 0.6604 8.305556

914.4 40 80 1.8161 7.944444 914.4 40 230 1.3208 16.61111

1219.2 40 80 1.1557 3.791667 1219.2 40 230 0.9906 9.34375

1219.2 40 80 2.3114 7.583333 1219.2 40 230 1.8161 17.13021

1524 40 80 1.4859 3.9 1524 40 230 1.1557 8.720833

1524 40 80 2.8067 7.366667 1524 40 230 2.3114 17.44167

1828.8 40 80 1.8161 3.972222 1828.8 40 230 1.3208 8.305556

1828.8 40 80 3.4671 7.583333 1828.8 40 230 2.6416 16.61111

110

GFRP CFRP

D (mm) f'c (Mpa)

Ef (Gpa)

tf (mm) ρ

D (mm) f'c (Mpa)

Ef (Gpa)

tf (mm) ρ

304.8 50 80 0.4953 5.2 304.8 50 230 0.4953 14.95

304.8 50 80 0.4953 5.2 304.8 50 230 0.4953 14.95

609.6 50 80 0.4953 2.6 609.6 50 230 0.4953 7.475

609.6 50 80 0.9906 5.2 609.6 50 230 0.9906 14.95

914.4 50 80 0.6604 2.311111 914.4 50 230 0.6604 6.644444

914.4 50 80 1.3208 4.622222 914.4 50 230 1.3208 13.28889

1219.2 50 80 0.9906 2.6 1219.2 50 230 0.9906 7.475

1219.2 50 80 1.8161 4.766667 1219.2 50 230 1.8161 13.70417

1524 50 80 1.1557 2.426667 1524 50 230 1.1557 6.976667

1524 50 80 2.3114 4.853333 1524 50 230 2.3114 13.95333

1828.8 50 80 1.3208 2.311111 1828.8 50 230 1.3208 6.644444

1828.8 50 80 2.6416 4.622222 1828.8 50 230 2.6416 13.28889

304.8 60 80 0.4953 4.333333 304.8 60 230 0.4953 12.45833

304.8 60 80 0.4953 4.333333 304.8 60 230 0.4953 12.45833

609.6 60 80 0.4953 2.166667 609.6 60 230 0.4953 6.229167

609.6 60 80 0.9906 4.333333 609.6 60 230 0.9906 12.45833

914.4 60 80 0.6604 1.925926 914.4 60 230 0.6604 5.537037

914.4 60 80 1.3208 3.851852 914.4 60 230 1.3208 11.07407

1219.2 60 80 0.9906 2.166667 1219.2 60 230 0.9906 6.229167

1219.2 60 80 1.8161 3.972222 1219.2 60 230 1.8161 11.42014

1524 60 80 1.1557 2.022222 1524 60 230 1.1557 5.813889

1524 60 80 2.3114 4.044444 1524 60 230 2.3114 11.62778

1828.8 60 80 1.3208 1.925926 1828.8 60 230 1.3208 5.537037

1828.8 60 80 2.6416 3.851852 1828.8 60 230 2.6416 11.07407

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NUMERICAL AND ANALYTICAL MODELING OF CONCRETE …